Daily Papers

Low-rank adaptation (LoRA) makes finetuning large language models cheaper by adding to each weight matrix a trainable low-rank update parameterized as the product of two matrices. These matrices are usually trained with Adam, which treats them as a single flat vector of parameters and ignores both the matrix and product structure of LoRA. Applying a matrix-aware optimizer such as Muon to each factor does not consistently improve over Adam, and neither do the product-aware Muon variants proposed in concurrent works. To realize consistent gains, we introduce PoLoRA, a Preconditioned Orthogonalized LoRA optimizer built from three ingredients: a product-aware spectral update direction, curvature preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates. We evaluate PoLoRA on instruction-tuning datasets for code and math across models from 1B to 8B parameters, and find that it reaches the final held-out loss achieved by tuned Adam in 1.2-1.7 times fewer steps, while adding at most 3% per-step overhead. Compared to Adam, PoLoRA is also less sensitive to the learning rate, and its optimal learning rate is stable across ranks.

Training learned image compression (LIC) models entails navigating a challenging optimization landscape defined by the fundamental trade-off between rate and distortion. Standard first-order optimizers, such as SGD and Adam, struggle with gradient conflicts arising from competing objectives, leading to slow convergence and suboptimal rate-distortion performance. In this work, we demonstrate that a simple utilization of a second-order quasi-Newton optimizer, SOAP, dramatically improves both training efficiency and final performance across diverse LICs. Our theoretical and empirical analyses reveal that Newton preconditioning inherently resolves the intra-step and inter-step update conflicts intrinsic to the R-D objective, facilitating faster, more stable convergence. Beyond acceleration, we uncover a critical deployability benefit: second-order trained models exhibit significantly fewer activation and latent outliers. This substantially enhances robustness to post-training quantization. Together, these results establish second-order optimization, achievable as a seamless drop-in replacement of the imported optimizer, as a powerful, practical tool for advancing the efficiency and real-world readiness of LICs.

Preconditioned adaptive methods have gained significant attention for training deep neural networks, as they capture rich curvature information of the loss landscape. The central challenge in this field lies in balancing preconditioning effectiveness with computational efficiency of implementing the preconditioner. Among recent advances, Muon stands out by using Newton-Schulz iteration to obtain preconditioned updates without explicitly constructing the preconditioning matrix. Despite its advantages, the efficiency of Muon still leaves room for further improvement. In this paper, we introduce RMNP (Row Momentum Normalized Preconditioning), an optimizer that replaces Newton-Schulz iteration with a simple row-wise (d_{in}) ell_2 normalization operation, motivated by the empirically observed diagonal block structure of the Transformer layerwise Hessian. We empirically verified that orthogonalization and row-wise (on input dim) ell_2 normalization are asymptotically equivalent in the case of the transformer. This substitution reduces the per-iteration computational complexity from {O}(mncdotmin(m,n)) to {O}(mn) for an mtimes n weight matrix while maintaining comparable optimization performance. Theoretically, we establish convergence guarantees for RMNP in the non-convex setting that match recent results for Muon optimizers, achieving the minimax optimal complexity. Extensive experiments on large language model pretraining show that RMNP delivers competitive optimization performance compared with Muon while substantially reducing preconditioning wall-clock time. Our code is available at https://github.com/Dominator-Index/RMNP.

Linear attention and state-space models offer constant-memory alternatives to softmax attention, but often struggle with in-context associative recall. The Delta Rule mitigates this by writing each token via one step of online gradient descent. However, its step size relies on a single scalar gate that ignores the feature-wise curvature of the inner objective. We propose Online Scaled DeltaNet (OSDN), which augments the scalar gate with a diagonal preconditioner updated online via hypergradient feedback. Crucially, this right-preconditioning is algebraically equivalent to a per-feature scaling of the write-side key. This equivalence allows OSDN to strictly preserve the hardware-friendly chunkwise parallel pipeline of DeltaNet without incurring high-dimensional state overhead. Theoretically, by exploiting the exact-quadratic structure of the inner regression loss, we establish super-geometric convergence against a right-Newton comparator and prove an algorithm-aligned token-local residual contraction bound. To handle non-stationary contexts, we further introduce Adaptive Preconditioner Forgetting (APF) to dynamically refresh stale calibration. Empirically, OSDN demonstrates strong performance across scales. At the 340M-parameter scale, OSDN improves JRT-style in-context recall by 32% over DeltaNet. Scaling to 1.3B parameters, it achieves a 39% reduction in the recall residual ratio while maintaining parity on general downstream tasks (e.g., perplexity and LongBench) -- demonstrating that our online-preconditioning mechanism effectively transfers and amplifies at the billion-parameter scale.

The ever-growing scale of deep learning models and training data underscores the critical importance of efficient optimization methods. While preconditioned gradient methods such as Adam and AdamW are the de facto optimizers for training neural networks and large language models, structure-aware preconditioned optimizers like Shampoo and Muon, which utilize the matrix structure of gradients, have demonstrated promising evidence of faster convergence. In this paper, we introduce a unifying framework for analyzing "matrix-aware" preconditioned methods, which not only sheds light on the effectiveness of Muon and related optimizers but also leads to a class of new structure-aware preconditioned methods. A key contribution of this framework is its precise distinction between preconditioning strategies that treat neural network weights as vectors (addressing curvature anisotropy) versus those that consider their matrix structure (addressing gradient anisotropy). This perspective provides new insights into several empirical phenomena in language model pre-training, including Adam's training instabilities, Muon's accelerated convergence, and the necessity of learning rate warmup for Adam. Building upon this framework, we introduce PolarGrad, a new class of preconditioned optimization methods based on the polar decomposition of matrix-valued gradients. As a special instance, PolarGrad includes Muon with updates scaled by the nuclear norm of the gradients. We provide numerical implementations of these methods, leveraging efficient numerical polar decomposition algorithms for enhanced convergence. Our extensive evaluations across diverse matrix optimization problems and language model pre-training tasks demonstrate that PolarGrad outperforms both Adam and Muon.

First-order optimization methods are currently the mainstream in training deep neural networks (DNNs). Optimizers like Adam incorporate limited curvature information by employing the diagonal matrix preconditioning of the stochastic gradient during the training. Despite their widespread, second-order optimization algorithms exhibit superior convergence properties compared to their first-order counterparts e.g. Adam and SGD. However, their practicality in training DNNs are still limited due to increased per-iteration computations and suboptimal accuracy compared to the first order methods. We present AdaFisher--an adaptive second-order optimizer that leverages a block-diagonal approximation to the Fisher information matrix for adaptive gradient preconditioning. AdaFisher aims to bridge the gap between enhanced convergence capabilities and computational efficiency in second-order optimization framework for training DNNs. Despite the slow pace of second-order optimizers, we showcase that AdaFisher can be reliably adopted for image classification, language modelling and stand out for its stability and robustness in hyperparameter tuning. We demonstrate that AdaFisher outperforms the SOTA optimizers in terms of both accuracy and convergence speed. Code available from https://github.com/AtlasAnalyticsLab/AdaFisher{https://github.com/AtlasAnalyticsLab/AdaFisher}

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

Leveraging second-order information about the loss at the scale of deep networks is one of the main lines of approach for improving the performance of current optimizers for deep learning. Yet, existing approaches for accurate full-matrix preconditioning, such as Full-Matrix Adagrad (GGT) or Matrix-Free Approximate Curvature (M-FAC) suffer from massive storage costs when applied even to small-scale models, as they must store a sliding window of gradients, whose memory requirements are multiplicative in the model dimension. In this paper, we address this issue via a novel and efficient error-feedback technique that can be applied to compress preconditioners by up to two orders of magnitude in practice, without loss of convergence. Specifically, our approach compresses the gradient information via sparsification or low-rank compression before it is fed into the preconditioner, feeding the compression error back into future iterations. Experiments on deep neural networks show that this approach can compress full-matrix preconditioners to up to 99\% sparsity without accuracy loss, effectively removing the memory overhead of full-matrix preconditioners such as GGT and M-FAC. Our code is available at https://github.com/IST-DASLab/EFCP.

The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce a surrogate model that not only sheds new light on the design of Muon, but more importantly leads to a new optimizer. In the same spirit as the derivation of Newton's method, the surrogate approximates the loss as a quadratic function of the perturbation to a weight matrix W using only three matrices: the gradient G, an output-space curvature matrix H, and the data matrix Z that stacks the layer inputs. By minimizing this surrogate in one step and adopting a certain isotropic assumption on the weights, we obtain the closed-form update rule (up to momentum and weight decay) W leftarrow W - ηcdot msgn(G(ZZ^top)^{-1}), where η is the learning rate and msgn(X)=UV^top if X=USV^top is a compact singular value decomposition. This new optimization method, which we refer to as Newton-Muon, shows that standard Muon can be interpreted as an implicit Newton-type method that neglects the right preconditioning induced by the input second moment. Empirically, on a reproduction of the earliest publicly released Modded-NanoGPT speedrun configuration using Muon for GPT-2 pretraining, Newton-Muon reaches the target validation loss in 6\% fewer iteration steps and reduces wall-clock training time by about 4\%.

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common visualisations of loss landscapes. By taking this geometric perspective literally and using the induced metric, we develop a new optimiser and compare it to existing methods, namely: SGD, Adam, AdamW, and Muon, across a range of tasks and architectures. Empirically, we conclude that this new class of optimisers is highly effective in low dimensional examples, and provides slight improvement over state-of-the-art methods for training neural networks. These new optimisers have theoretically desirable properties. In particular, the effective learning rate is automatically decreased in regions of high curvature acting as a smoothed out form of gradient clipping. Similarly, one variant of these optimisers can also be viewed as inducing an effective scheduled learning rate and decoupled weight decay is the natural choice from our geometric perspective. The basic method can be used to modify any existing preconditioning method. The new optimiser has a computational complexity comparable to that of Adam.

Adaptive optimization methods are widely recognized as among the most popular approaches for training Deep Neural Networks (DNNs). Techniques such as Adam, AdaGrad, and AdaHessian utilize a preconditioner that modifies the search direction by incorporating information about the curvature of the objective function. However, despite their adaptive characteristics, these methods still require manual fine-tuning of the step-size. This, in turn, impacts the time required to solve a particular problem. This paper presents an optimization framework named SANIA to tackle these challenges. Beyond eliminating the need for manual step-size hyperparameter settings, SANIA incorporates techniques to address poorly scaled or ill-conditioned problems. We also explore several preconditioning methods, including Hutchinson's method, which approximates the Hessian diagonal of the loss function. We conclude with an extensive empirical examination of the proposed techniques across classification tasks, covering both convex and non-convex contexts.

We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied to address the constraints in optimization problems, such as the Hessian-Riemannian and natural gradient descent methods. More specifically, we incorporate these preconditioned gradient descent algorithms in the recently introduced Adaptive Energy Gradient Descent (AEGD) framework. In particular, we discuss theoretical results on the unconditional energy-stability and convergence rates across three classes of objective functions. Furthermore, our numerical results demonstrate excellent performance of the proposed method on several test bed optimization problems.

We propose ScaledGD(\lambda), a preconditioned gradient descent method to tackle the low-rank matrix sensing problem when the true rank is unknown, and when the matrix is possibly ill-conditioned. Using overparametrized factor representations, ScaledGD(\lambda) starts from a small random initialization, and proceeds by gradient descent with a specific form of damped preconditioning to combat bad curvatures induced by overparameterization and ill-conditioning. At the expense of light computational overhead incurred by preconditioners, ScaledGD(\lambda) is remarkably robust to ill-conditioning compared to vanilla gradient descent (GD) even with overprameterization. Specifically, we show that, under the Gaussian design, ScaledGD(\lambda) converges to the true low-rank matrix at a constant linear rate after a small number of iterations that scales only logarithmically with respect to the condition number and the problem dimension. This significantly improves over the convergence rate of vanilla GD which suffers from a polynomial dependency on the condition number. Our work provides evidence on the power of preconditioning in accelerating the convergence without hurting generalization in overparameterized learning.

Preconditioned gradient methods are among the most general and powerful tools in optimization. However, preconditioning requires storing and manipulating prohibitively large matrices. We describe and analyze a new structure-aware preconditioning algorithm, called Shampoo, for stochastic optimization over tensor spaces. Shampoo maintains a set of preconditioning matrices, each of which operates on a single dimension, contracting over the remaining dimensions. We establish convergence guarantees in the stochastic convex setting, the proof of which builds upon matrix trace inequalities. Our experiments with state-of-the-art deep learning models show that Shampoo is capable of converging considerably faster than commonly used optimizers. Although it involves a more complex update rule, Shampoo's runtime per step is comparable to that of simple gradient methods such as SGD, AdaGrad, and Adam.

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce inductive bias in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs.

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

Preconditioning is at the heart of iterative solutions of large, sparse linear systems of equations in scientific disciplines. Several algebraic approaches, which access no information beyond the matrix itself, are widely studied and used, but ill-conditioned matrices remain very challenging. We take a machine learning approach and propose using graph neural networks as a general-purpose preconditioner. They show attractive performance for many problems and can be used when the mainstream preconditioners perform poorly. Empirical evaluation on over 800 matrices suggests that the construction time of these graph neural preconditioners (GNPs) is more predictable and can be much shorter than that of other widely used ones, such as ILU and AMG, while the execution time is faster than using a Krylov method as the preconditioner, such as in inner-outer GMRES. GNPs have a strong potential for solving large-scale, challenging algebraic problems arising from not only partial differential equations, but also economics, statistics, graph, and optimization, to name a few.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

We study optimization problems on Hadamard manifolds, motivated by recent advances in geometric approaches to optimization on curved spaces, particularly those involving the structure of Busemann functions. We introduce a projection based variant of the proximal point algorithm, termed the Busemann hybrid projection proximal point algorithm, which replaces Euclidean hyperplanes with horospheres defined via convex Busemann functions. The algorithm performs projections in closed form using the gradients of these functions, resulting in a geometrically intrinsic scheme that requires no tangent space linear solves. We allow for inexact subgradient evaluations and prove global convergence under controlled inexactness, with a relative error level strictly below one. We establish a Fejér type descent and sublinear complexity with a rate proportional to the inverse square root of the iteration count, and show that the exact variant coincides with the classical Riemannian proximal point algorithm. The framework clarifies the role of Busemann based subdifferentials in optimization on spaces of nonpositive curvature.

We revisit the problem of large-scale bundle adjustment and propose a technique called Multidirectional Conjugate Gradients that accelerates the solution of the normal equation by up to 61%. The key idea is that we enlarge the search space of classical preconditioned conjugate gradients to include multiple search directions. As a consequence, the resulting algorithm requires fewer iterations, leading to a significant speedup of large-scale reconstruction, in particular for denser problems where traditional approaches notoriously struggle. We provide a number of experimental ablation studies revealing the robustness to variations in the hyper-parameters and the speedup as a function of problem density.

Low-Rank Adaptation (LoRA) emerges as a popular parameter-efficient fine-tuning (PEFT) method, which proposes to freeze pretrained model weights and update an additive low-rank trainable matrix. In this work, we study the enhancement of LoRA training by introducing an r times r preconditioner in each gradient step where r is the LoRA rank. We theoretically verify that the proposed preconditioner stabilizes feature learning with LoRA under infinite-width NN setting. Empirically, the implementation of this new preconditioner requires a small change to existing optimizer code and creates virtually minuscule storage and runtime overhead. Our experimental results with both large language models and text-to-image diffusion models show that with this new preconditioner, the convergence and reliability of SGD and AdamW can be significantly enhanced. Moreover, the training process becomes much more robust to hyperparameter choices such as learning rate. The new preconditioner can be derived from a novel Riemannian metric in low-rank matrix field. Code can be accessed at https://github.com/pilancilab/Riemannian_Preconditioned_LoRA.

In this work, we study the evolution of the loss Hessian across many classification tasks in order to understand the effect the curvature of the loss has on the training dynamics. Whereas prior work has focused on how different learning rates affect the loss Hessian observed during training, we also analyze the effects of model initialization, architectural choices, and common training heuristics such as gradient clipping and learning rate warmup. Our results demonstrate that successful model and hyperparameter choices allow the early optimization trajectory to either avoid -- or navigate out of -- regions of high curvature and into flatter regions that tolerate a higher learning rate. Our results suggest a unifying perspective on how disparate mitigation strategies for training instability ultimately address the same underlying failure mode of neural network optimization, namely poor conditioning. Inspired by the conditioning perspective, we show that learning rate warmup can improve training stability just as much as batch normalization, layer normalization, MetaInit, GradInit, and Fixup initialization.

We propose a preconditioning (PC) layer, a weight parameterization via polynomial preconditioner that ensures stable weight conditioning throughout LLM training. The PC module reshapes the singular-value spectrum of weight matrices via low-degree polynomial preconditioning. After training, the preconditioned weights can be merged back into the original architecture, incurring no inference overhead. We demonstrate the advantage of the proposed PC layer over standard transformers in Llama-1B pre-training, for both the AdamW and Muon optimizers. Theoretically, we justify this spectrum-control principle by proving that uniformly bounding each layer's singular values ensures geometric convergence of gradient descent to global minima, for certain deep linear networks. Our code is available at https://github.com/Empath-aln/PC-layer.

Stochastic Gradient Descent (SGD) is one of the many iterative optimization methods that are widely used in solving machine learning problems. These methods display valuable properties and attract researchers and industrial machine learning engineers with their simplicity. However, one of the weaknesses of this type of methods is the necessity to tune learning rate (step-size) for every loss function and dataset combination to solve an optimization problem and get an efficient performance in a given time budget. Stochastic Gradient Descent with Polyak Step-size (SPS) is a method that offers an update rule that alleviates the need of fine-tuning the learning rate of an optimizer. In this paper, we propose an extension of SPS that employs preconditioning techniques, such as Hutchinson's method, Adam, and AdaGrad, to improve its performance on badly scaled and/or ill-conditioned datasets.

Loss spikes commonly emerge during neural network training with the Adam optimizer across diverse architectures and scales, yet their underlying mechanism remains elusive. While previous explanations attribute these phenomena to sharper loss landscapes at lower loss, we show that landscape geometry alone is insufficient to explain the phenomenon. In this work, we pinpoint the root cause in the internal dynamics of Adam's second moment estimator. We identify a critical ``decoupling'' mechanism where the adaptive preconditioner v_t fails to track the instantaneous squared gradients g_t^2, causing the adaptive mechanism to effectively fail. This decoupling allows the preconditioner to decay autonomously despite rising gradients, which pushes the maximum eigenvalue of the preconditioned Hessian beyond the stability threshold 2/η for sustained periods, manifesting as dramatic loss spikes. Through a quadratic approximation analysis, we theoretically and experimentally characterize five distinct stages of spike evolution and propose a predictor for anticipating spikes based on gradient-directional curvature. We empirically find that the proposed loss spike mechanism, although derived from simplified models, generalizes well to practical scenarios ranging from small neural networks to large-scale Transformers.

Satellite image inpainting is a crucial task in remote sensing, where accurately restoring missing or occluded regions is essential for robust image analysis. In this paper, we propose KAO, a novel framework that utilizes Kernel-Adaptive Optimization within diffusion models for satellite image inpainting. KAO is specifically designed to address the challenges posed by very high-resolution (VHR) satellite datasets, such as DeepGlobe and the Massachusetts Roads Dataset. Unlike existing methods that rely on preconditioned models requiring extensive retraining or postconditioned models with significant computational overhead, KAO introduces a Latent Space Conditioning approach, optimizing a compact latent space to achieve efficient and accurate inpainting. Furthermore, we incorporate Explicit Propagation into the diffusion process, facilitating forward-backward fusion, which improves the stability and precision of the method. Experimental results demonstrate that KAO sets a new benchmark for VHR satellite image restoration, providing a scalable, high-performance solution that balances the efficiency of preconditioned models with the flexibility of postconditioned models.

Reinforcement Learning, particularly through policy gradient methods, has played a central role in enabling reasoning capabilities of Large Language Models. However, the optimization stability of policy gradients in this setting remains understudied. As a result, existing implementations often resort to conservative hyperparameter choices to ensure stability, which requires more training samples and increases computational costs. Hence, developing models for reliably tracking the underlying optimization dynamics and leveraging them into training enables more sample-efficient regimes and further unleashes scalable post-training. We address this gap by formalizing the stochastic optimization problem of policy gradients with explicit consideration of second-order geometry. We propose a tractable computational framework that tracks and leverages curvature information during policy updates. We further employ this framework to design interventions in the optimization process through data selection. The resultant algorithm, Curvature-Aware Policy Optimization (CAPO), identifies samples that contribute to unstable updates and masks them out. Theoretically, we establish monotonic improvement guarantees under realistic assumptions. On standard math reasoning benchmarks, we empirically show that CAPO ensures stable updates under aggressive learning regimes where baselines catastrophically fail. With minimal intervention (rejecting fewer than 8% of tokens), CAPO achieves up to 30x improvement in sample efficiency over standard GRPO for LLM reasoning.

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.

Recent ODE/SDE-based generative models, such as diffusion models, rectified flows, and flow matching, define a generative process as a time reversal of a fixed forward process. Even though these models show impressive performance on large-scale datasets, numerical simulation requires multiple evaluations of a neural network, leading to a slow sampling speed. We attribute the reason to the high curvature of the learned generative trajectories, as it is directly related to the truncation error of a numerical solver. Based on the relationship between the forward process and the curvature, here we present an efficient method of training the forward process to minimize the curvature of generative trajectories without any ODE/SDE simulation. Experiments show that our method achieves a lower curvature than previous models and, therefore, decreased sampling costs while maintaining competitive performance. Code is available at https://github.com/sangyun884/fast-ode.

This paper presents a novel preconditioning strategy for the classic 8-point algorithm (8-PA) for estimating an essential matrix from 360-FoV images (i.e., equirectangular images) in spherical projection. To alleviate the effect of uneven key-feature distributions and outlier correspondences, which can potentially decrease the accuracy of an essential matrix, our method optimizes a non-rigid transformation to deform a spherical camera into a new spatial domain, defining a new constraint and a more robust and accurate solution for an essential matrix. Through several experiments using random synthetic points, 360-FoV, and fish-eye images, we demonstrate that our normalization can increase the camera pose accuracy by about 20% without significantly overhead the computation time. In addition, we present further benefits of our method through both a constant weighted least-square optimization that improves further the well known Gold Standard Method (GSM) (i.e., the non-linear optimization by using epipolar errors); and a relaxation of the number of RANSAC iterations, both showing that our normalization outcomes a more reliable, robust, and accurate solution.

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.

Unified anomaly detection requires modeling highly heterogeneous normal data without access to anomalous samples. While foundation models like DINOv2 provide rich token representations, leveraging these spaces for explicit density estimation remains challenging. Energy-Based Models (EBMs) offer a principled formulation, but their training in high-dimensional token spaces is unstable due to anisotropy and strong cross-dimensional correlations, which degrades finite-step Markov Chain Monte Carlo (MCMC) sampling. We identify this instability as fundamentally geometric and introduce ReFP-AD (Rectified Flow Preconditioning for Anomaly Detection), which learns a geometric reparameterization that maps high-dimensional embeddings into a well-conditioned latent space via an optimal transport (OT)-coupled rectified flow. This preconditioning enables stable persistent contrastive divergence with preconditioned Stochastic Gradient Langevin Dynamics (SGLD) in full-dimensional token spaces. Anomaly scores are then derived from the learned energy landscape using gradient norms. Under a strict unified protocol on the MVTec-AD and VisA datasets, ReFP-AD achieves 98.6%/97.9% Image/Pixel AUROC on MVTec-AD and 97.3%/99.0% on VisA, outperforming prior unified EBM baselines by up to +10.8% in Image AUROC. Ablation experiments demonstrate that geometric reparameterization is critical for finite-step MCMC and accurate anomaly localization in high-dimensional token spaces. Code is available at https://github.com/CLendering/ReFP-AD

Conventional wisdom in deep learning states that increasing depth improves expressiveness but complicates optimization. This paper suggests that, sometimes, increasing depth can speed up optimization. The effect of depth on optimization is decoupled from expressiveness by focusing on settings where additional layers amount to overparameterization - linear neural networks, a well-studied model. Theoretical analysis, as well as experiments, show that here depth acts as a preconditioner which may accelerate convergence. Even on simple convex problems such as linear regression with ell_p loss, p>2, gradient descent can benefit from transitioning to a non-convex overparameterized objective, more than it would from some common acceleration schemes. We also prove that it is mathematically impossible to obtain the acceleration effect of overparametrization via gradients of any regularizer.

Second-order optimizers, maintaining a matrix termed a preconditioner, are superior to first-order optimizers in both theory and practice. The states forming the preconditioner and its inverse root restrict the maximum size of models trained by second-order optimizers. To address this, compressing 32-bit optimizer states to lower bitwidths has shown promise in reducing memory usage. However, current approaches only pertain to first-order optimizers. In this paper, we propose the first 4-bit second-order optimizers, exemplified by 4-bit Shampoo, maintaining performance similar to that of 32-bit ones. We show that quantizing the eigenvector matrix of the preconditioner in 4-bit Shampoo is remarkably better than quantizing the preconditioner itself both theoretically and experimentally. By rectifying the orthogonality of the quantized eigenvector matrix, we enhance the approximation of the preconditioner's eigenvector matrix, which also benefits the computation of its inverse 4-th root. Besides, we find that linear square quantization slightly outperforms dynamic tree quantization when quantizing second-order optimizer states. Evaluation on various networks for image classification demonstrates that our 4-bit Shampoo achieves comparable test accuracy to its 32-bit counterpart while being more memory-efficient. The source code will be made available.

The curvature of ODE trajectories in diffusion models hinders their ability to generate high-quality images in a few number of function evaluations (NFE). In this paper, we propose a novel and effective approach to reduce trajectory curvature by utilizing adaptive conditions. By employing a extremely light-weight quantized encoder, our method incurs only an additional 1% of training parameters, eliminates the need for extra regularization terms, yet achieves significantly better sample quality. Our approach accelerates ODE sampling while preserving the downstream task image editing capabilities of SDE techniques. Extensive experiments verify that our method can generate high quality results under extremely limited sampling costs. With only 6 NFE, we achieve 5.14 FID on CIFAR-10, 6.91 FID on FFHQ 64x64 and 3.10 FID on AFHQv2.

We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. When applied to graph Laplacians, this leads to ultra-sparsifiers that in expectation behave as the nearly-optimal ones given by [Kolla-Makarychev-Saberi-Teng STOC`10]. Combining this with the recursive preconditioning framework by [Spielman-Teng STOC`04] and improved embedding algorithms, this leads to algorithms that solve symmetric diagonally dominant linear systems and electrical flow problems in expected time close to mlog^{1/2}n .

Gradient descent on a factored model W = UV^top is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under (U, V) mapsto (UQ, VQ). Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants W_Q^top W_K 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

Training large-scale neural networks requires solving nonconvex optimization where the choice of optimizer fundamentally determines both convergence behavior and computational efficiency. While adaptive methods like Adam have long dominated practice, the recently proposed Muon optimizer achieves superior performance through orthogonalized momentum updates that enforce isotropic geometry with uniform singular values. However, this strict isotropy discards potentially valuable curvature information encoded in gradient spectra, motivating optimization methods that balance geometric structure with adaptivity. We introduce FISMO (Fisher-Structured Momentum-Orthogonalized) optimizer, which generalizes isotropic updates to incorporate anisotropic curvature information through Fisher information geometry. By reformulating the optimizer update as a trust-region problem constrained by a Kronecker-factored Fisher metric, FISMO achieves structured preconditioning that adapts to local loss landscape geometry while maintaining computational tractability. We establish convergence guarantees for FISMO in stochastic nonconvex settings, proving an O(1/T) rate for the expected squared gradient norm with explicit characterization of variance reduction through mini-batching. Empirical evaluation on image classification and language modeling benchmarks demonstrates that FISMO achieves superior training efficiency and final performance compared to established baselines.

Efficiently approximating local curvature information of the loss function is a key tool for optimization and compression of deep neural networks. Yet, most existing methods to approximate second-order information have high computational or storage costs, which can limit their practicality. In this work, we investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) for the case when the Hessian can be approximated as a sum of rank-one matrices, as in the classic approximation of the Hessian by the empirical Fisher matrix. We propose two new algorithms as part of a framework called M-FAC: the first algorithm is tailored towards network compression and can compute the IHVP for dimension d, if the Hessian is given as a sum of m rank-one matrices, using O(dm^2) precomputation, O(dm) cost for computing the IHVP, and query cost O(m) for any single element of the inverse Hessian. The second algorithm targets an optimization setting, where we wish to compute the product between the inverse Hessian, estimated over a sliding window of optimization steps, and a given gradient direction, as required for preconditioned SGD. We give an algorithm with cost O(dm + m^2) for computing the IHVP and O(dm + m^3) for adding or removing any gradient from the sliding window. These two algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods. Implementations are available at [9] and [17].

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

Despite significant work on low-bit quantization-aware training (QAT), there is still a large accuracy gap between such techniques and native training. To address this, we introduce CAGE (Curvature-Aware Gradient Estimation), a new QAT method that augments the straight-through estimator (STE) gradient with a curvature-aware correction designed to counteract the loss increase induced by quantization. CAGE is derived from a multi-objective view of QAT that balances loss minimization with adherence to quantization constraints, yielding a principled correction term that depends on local curvature information. On the theoretical side, we introduce the notion of Pareto-optimal solutions for quantized optimization, and establish that CAGE yields strong convergence guarantees in the smooth non-convex setting. In terms of implementation, our approach is optimizer-agnostic, but we provide a highly-efficient implementation that leverages Adam statistics. When pre-training Llama-style models of up to 800M-parameters, CAGE recovers over 10% of the quantization-induced loss increase in the W4A4 regime over outlier-mitigation methods. These results indicate that curvature-aware gradient corrections can bridge the remaining performance gap beyond current outlier-handling methods.

Adversarial prompts capable of jailbreaking large language models (LLMs) and inducing undesirable behaviours pose a significant obstacle to their safe deployment. Current mitigation strategies rely on activating built-in defence mechanisms or fine-tuning the LLMs, but the fundamental distinctions between adversarial and benign prompts are yet to be understood. In this work, we introduce CurvaLID, a novel defense framework that efficiently detects adversarial prompts by leveraging their geometric properties. It is agnostic to the type of LLM, offering a unified detection framework across diverse adversarial prompts and LLM architectures. CurvaLID builds on the geometric analysis of text prompts to uncover their underlying differences. We theoretically extend the concept of curvature via the Whewell equation into an n-dimensional word embedding space, enabling us to quantify local geometric properties, including semantic shifts and curvature in the underlying manifolds. Additionally, we employ Local Intrinsic Dimensionality (LID) to capture geometric features of text prompts within adversarial subspaces. Our findings reveal that adversarial prompts differ fundamentally from benign prompts in terms of their geometric characteristics. Our results demonstrate that CurvaLID delivers superior detection and rejection of adversarial queries, paving the way for safer LLM deployment. The source code can be found at https://github.com/Cancanxxx/CurvaLID

We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-Factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network's Fisher information matrix which is neither diagonal nor low-rank, and in some cases is completely non-sparse. It is derived by approximating various large blocks of the Fisher (corresponding to entire layers) as being the Kronecker product of two much smaller matrices. While only several times more expensive to compute than the plain stochastic gradient, the updates produced by K-FAC make much more progress optimizing the objective, which results in an algorithm that can be much faster than stochastic gradient descent with momentum in practice. And unlike some previously proposed approximate natural-gradient/Newton methods which use high-quality non-diagonal curvature matrices (such as Hessian-free optimization), K-FAC works very well in highly stochastic optimization regimes. This is because the cost of storing and inverting K-FAC's approximation to the curvature matrix does not depend on the amount of data used to estimate it, which is a feature typically associated only with diagonal or low-rank approximations to the curvature matrix.

Structural and Positional Encodings can significantly improve the performance of Graph Neural Networks in downstream tasks. Recent literature has begun to systematically investigate differences in the structural properties that these approaches encode, as well as performance trade-offs between them. However, the question of which structural properties yield the most effective encoding remains open. In this paper, we investigate this question from a geometric perspective. We propose a novel structural encoding based on discrete Ricci curvature (Local Curvature Profiles, short LCP) and show that it significantly outperforms existing encoding approaches. We further show that combining local structural encodings, such as LCP, with global positional encodings improves downstream performance, suggesting that they capture complementary geometric information. Finally, we compare different encoding types with (curvature-based) rewiring techniques. Rewiring has recently received a surge of interest due to its ability to improve the performance of Graph Neural Networks by mitigating over-smoothing and over-squashing effects. Our results suggest that utilizing curvature information for structural encodings delivers significantly larger performance increases than rewiring.

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.

A key aspect of learned partial differential equation (PDE) solvers is that the main cost often comes from generating training data with classical solvers rather than learning the model itself. Another is that there are clear axes of difficulty--e.g., more complex geometries and higher Reynolds numbers--along which problems become (1) harder for classical solvers and thus (2) more likely to benefit from neural speedups. Towards addressing this chicken-and-egg challenge, we study difficulty transfer on 2D incompressible Navier-Stokes, systematically varying task complexity along geometry (number and placement of obstacles), physics (Reynolds number), and their combination. Similar to how it is possible to spend compute to pre-train foundation models and improve their performance on downstream tasks, we find that by classically solving (analogously pre-generating) many low and medium difficulty examples and including them in the training set, it is possible to learn high-difficulty physics from far fewer samples. Furthermore, we show that by combining low and high difficulty data, we can spend 8.9x less compute on pre-generating a dataset to achieve the same error as using only high difficulty examples. Our results highlight that how we allocate classical-solver compute across difficulty levels is as important as how much we allocate overall, and suggest substantial gains from principled curation of pre-generated PDE data for neural solvers. Our code is available at https://github.com/Naman-Choudhary-AI-ML/pregenerating-pde

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as nonlinear elliptic variational PDEs, whose solutions minimize an Euler-Lagrange energy functional E(u) = int_Omega L(x, u(x), nabla u(x)) - f(x) u(x)dx. We show that if composing a function with Barron norm b with partial derivatives of L produces a function of Barron norm at most B_L b^p, the solution to the PDE can be epsilon-approximated in the L^2 sense by a function with Barron norm Oleft(left(dB_Lright)^{max{p log(1/ epsilon), p^{log(1/epsilon)}}}right). By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating p, epsilon, B_L as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

Linear attention reduces the quadratic cost of softmax attention by maintaining a recurrent fast-weight state, but it consistently lags on in-context retrieval and long-context tasks. Existing remedies act on the write side of memory through gating, delta updates, or kernel feature maps, but the read step is left unchanged: every past key contributes additively to the output, so useful targets are diluted by the bulk of stored vectors. We borrow one specific piece of softmax's geometry to construct a cheap read-time contraction of the query. A second-order Taylor expansion of the softmax log-partition at the isotropic-attention point gives a local quadratic model whose curvature coincides with the running key covariance, a quantity that can be maintained with the same recurrent/chunkwise mechanism as the linear-attention state. The associated linear operator contracts the query along the high-density directions of memory before it reads the state. We call this mechanism Curvature-Conditioned Query (CCQ). CCQ modifies only the read step and is composable with any linear-attention backbone. Attached to GLA and Gated DeltaNet, it improves perplexity, zero-shot downstream accuracy, S-NIAH retrieval at and beyond the training context, length-extrapolation perplexity from 4K to 20K, and LongBench accuracy, at small extra cost.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

Zeroth-order optimization addresses problems where gradient information is inaccessible or impractical to compute. While most existing methods rely on first-order approximations, incorporating second-order (curvature) information can, in principle, significantly accelerate convergence. However, the high cost of function evaluations required to estimate Hessian matrices often limits practical applicability. We present the subspace-based approximate Hessian (ZO-SAH) method, a zeroth-order optimization algorithm that mitigates these costs by focusing on randomly selected two-dimensional subspaces. Within each subspace, ZO-SAH estimates the Hessian by fitting a quadratic polynomial to the objective function and extracting its second-order coefficients. To further reduce function-query costs, ZO-SAH employs a periodic subspace-switching strategy that reuses function evaluations across optimization steps. Experiments on eight benchmark datasets, including logistic regression and deep neural network training tasks, demonstrate that ZO-SAH achieves significantly faster convergence than existing zeroth-order methods.

Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural operators offer a fast alternative, they typically rely on large-scale labeled datasets or struggle with unstable training dynamics when using physics-informed residual losses. We propose NPSolver, a neural Poisson solver trained without solution labels via iterative physics supervision. Instead of relying on fully converged numerical solutions or raw PDE residuals, NPSolver utilizes a small number of preconditioned conjugate gradient (PCG) steps to refine its own predictions, providing a more stable and well-scaled training signal. Theoretical analysis confirms that this iterative supervision serves as a well-conditioned error proxy and that a stop-gradient design is essential for optimization stability. To better capture boundary-driven features under mixed boundary conditions, we further introduce the Boundary-Aware Transolver (BA-Transolver) architecture that explicitly separates interior and boundary tokenization. Extensive evaluations on 2D and 3D irregular geometries demonstrate that NPSolver outperforms both physics-informed and data-driven baselines. Furthermore, a downstream thermal control task highlights the model's capability for conducting efficient and reliable gradient-based boundary control. We will release our codes and data at https://github.com/intell-sci-comput/NPSolver.

Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but why? We show that the mechanism is spectral flattening, and develop two results around it. First, we prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent. Second, we recast Muon as a preconditioned gradient method and show, under a Kronecker-factored curvature model, that it improves the effective convergence factor, with the improvement controlled by the spectrum of the gradient covariance. Extensive experiments validate both results: Muon remains stable at learning rates that cause SGD to diverge within the first few iterations, and reaches accuracy milestones several epochs earlier even at identical step sizes. Taken together, our results offer a principled, geometric explanation for Muon's empirical success.

Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

Pre-training has been investigated to improve the efficiency and performance of training neural operators in data-scarce settings. However, it is largely in its infancy due to the inherent complexity and diversity, such as long trajectories, multiple scales and varying dimensions of partial differential equations (PDEs) data. In this paper, we present a new auto-regressive denoising pre-training strategy, which allows for more stable and efficient pre-training on PDE data and generalizes to various downstream tasks. Moreover, by designing a flexible and scalable model architecture based on Fourier attention, we can easily scale up the model for large-scale pre-training. We train our PDE foundation model with up to 0.5B parameters on 10+ PDE datasets with more than 100k trajectories. Extensive experiments show that we achieve SOTA on these benchmarks and validate the strong generalizability of our model to significantly enhance performance on diverse downstream PDE tasks like 3D data. Code is available at https://github.com/thu-ml/DPOT.

Does text have an intrinsic curvature? Language is increasingly modeled in curved geometries - hyperbolic spaces for hierarchy, mixed-curvature manifolds for compositional structure - yet a basic scientific question remains unresolved: what does curvature mean for text itself, in a way that is native to language rather than an artifact of the embedding space we choose? We argue that text does indeed have curvature, and show how to detect it, define it, and use it. To this end, we propose Texture, a text-native, word-level discrete curvature signal, and make three contributions. (a) Existence: We provide empirical and theoretical certificates that semantic inference in natural corpora is non-flat, i.e. language has inherent curvature. (b) Definition: We define Texture by reconciling left- and right-context beliefs around a masked word through a Schrodinger bridge, yielding a curvature field that is positive where context focuses meaning and negative where it fans out into competing continuations. (c) Utility: Texture is actionable: it serves as a general-purpose measurement and control primitive enabling geometry without geometric training; we instantiate it on two representative tasks, improving long-context inference through curvature-guided compression and retrieval-augmented generation through curvature-guided routing. Together, our results establish a text-native curvature paradigm, making curvature measurable and practically useful.

Many theoretical results in deep learning can be traced to symmetry or equivariance of neural networks under parameter transformations. However, existing analyses are typically problem-specific and focus on first-order consequences such as conservation laws, while the implications for second-order structure remain less understood. We develop a general equivariance toolbox that yields coupled first- and second-order constraints on learning dynamics. The framework extends classical Noether-type analyses in three directions: from gradient constraints to Hessian constraints, from symmetry to general equivariance, and from continuous to discrete transformations. At the first order, our framework unifies conservation laws and implicit-bias relations as special cases of a single identity. At the second order, it provides structural predictions about curvature: which directions are flat or sharp, how the gradient aligns with Hessian eigenspaces, and how the loss landscape geometry reflects the underlying transformation structure. We illustrate the framework through several applications, recovering known results while also deriving new characterizations that connect transformation structure to modern empirical observations about optimization geometry.

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the generative Jacobian, decomposing geometry into Local Scaling (capacity) and Local Complexity (curvature). Our study uncovers a ``Geometric Decoupling": while curvature in normal generation functionally encodes image detail, OOD generation exhibits a functional decoupling where extreme curvature is wasted on unstable semantic boundaries rather than perceptible details. This geometric misallocation identifies ``Geometric Hotspots" as the structural root of instability, providing a robust intrinsic metric for diagnosing generative reliability.

A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as adpative variants of Shampoo andMuon. This framework also allows combining heterogeneous geometriesacross different groups of variables while preserving a unifiedconvergence analysis. A fully stochastic global rate-of-convergenceanalysis is conducted for all methods in the framework, with andwithout two types of momentum, using reasonable assumptions on thevariance of the gradient oracle and without assuming boundedstochastic gradients or small enough stepsize.

Higher-order optimizers such as Muon and SOAP offer faster convergence than AdamW, but their computational cost and numerical stability challenges have limited adoption at scale. In this work, we adapt and enhance preconditioned gradient methods to overcome the practical challenges of large-scale LLM pretraining. We first identify instabilities in SOAP at large batch sizes and propose algorithmic modifications including per-step QR orthogonalization and improved preconditioning strategies that eliminate loss spikes and enable stable training in these regimes. We then present a unified empirical study of SOAP, Muon, and AdamW using update-RMS matching to ensure fair learning rate transfer across optimizers. As part of this analysis, we empirically evaluate the orthogonalization quality of Muon. Our experiments on multi-billion-parameter models trained on trillions of tokens reveal that SOAP and Muon consistently outperform AdamW at the scales we tested. Notably, at batch sizes of up to 100M tokens for next-token prediction, these optimizers maintain training stability and quality while AdamW degrades. To enable efficient training at large scale, we introduce a layer-wise distributed optimizer compatible with Megatron-LM. Our implementation balances memory and hides communication while avoiding approximations to the optimizer computations, thus retaining their convergence benefits. Additionally, we identify and build specific system-level improvements to further accelerate our layer-wise implementation. To support the research community, we release a codebase that contains emerging algorithms for optimization: https://github.com/NVIDIA-NeMo/Emerging-Optimizers

Shampoo, a second-order optimization algorithm which uses a Kronecker product preconditioner, has recently garnered increasing attention from the machine learning community. The preconditioner used by Shampoo can be viewed either as an approximation of the Gauss--Newton component of the Hessian or the covariance matrix of the gradients maintained by Adagrad. We provide an explicit and novel connection between the optimal Kronecker product approximation of these matrices and the approximation made by Shampoo. Our connection highlights a subtle but common misconception about Shampoo's approximation. In particular, the square of the approximation used by the Shampoo optimizer is equivalent to a single step of the power iteration algorithm for computing the aforementioned optimal Kronecker product approximation. Across a variety of datasets and architectures we empirically demonstrate that this is close to the optimal Kronecker product approximation. Additionally, for the Hessian approximation viewpoint, we empirically study the impact of various practical tricks to make Shampoo more computationally efficient (such as using the batch gradient and the empirical Fisher) on the quality of Hessian approximation.

Nesterov's accelerated gradient (AG) is a popular technique to optimize objective functions comprising two components: a convex loss and a penalty function. While AG methods perform well for convex penalties, such as the LASSO, convergence issues may arise when it is applied to nonconvex penalties, such as SCAD. A recent proposal generalizes Nesterov's AG method to the nonconvex setting. The proposed algorithm requires specification of several hyperparameters for its practical application. Aside from some general conditions, there is no explicit rule for selecting the hyperparameters, and how different selection can affect convergence of the algorithm. In this article, we propose a hyperparameter setting based on the complexity upper bound to accelerate convergence, and consider the application of this nonconvex AG algorithm to high-dimensional linear and logistic sparse learning problems. We further establish the rate of convergence and present a simple and useful bound to characterize our proposed optimal damping sequence. Simulation studies show that convergence can be made, on average, considerably faster than that of the conventional proximal gradient algorithm. Our experiments also show that the proposed method generally outperforms the current state-of-the-art methods in terms of signal recovery.

This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time R^3times R, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually R^3 times {0}. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

Local loss-landscape stabilization under sample growth is typically measured either pointwise or through isotropic averaging in the full parameter space. Despite practical value, both choices probe directions that contribute little to the dominant local deformation of strongly anisotropic neural landscapes. We recast stabilization as an observational problem and introduce a unified family of criteria parameterized by an aggregation order and a probing distribution; within this family we propose a curvature-aligned criterion Δ_2^{(D)} that probes the loss increment field in the top-D eigenspace of the empirical Hessian near a trained solution. Solely from a local quadratic model, we prove that Δ_2^{(D)} preserves the O(k^{-2}) mean-squared rate of the full-space criterion while replacing ambient-dimension curvature dependence with dependence on the subspace dimension D; a corollary gives a closed-form spectral expression and a proposition identifies the top-D eigenspace as extremal within the eigenspace-aligned family. We also derive scalable estimators based on Hessian-vector products, subspace Monte Carlo, and a closed-form Gaussian-moment proxy. On a decoder-only transformer, a curvature-aligned probe occupying a tiny fraction of parameter space already reproduces the full-space mean-squared signal to within numerical noise throughout the validated local regime, and the closed-form estimator is orders of magnitude faster than direct Monte Carlo after subspace construction.

Recently, flat minima are proven to be effective for improving generalization and sharpness-aware minimization (SAM) achieves state-of-the-art performance. Yet the current definition of flatness discussed in SAM and its follow-ups are limited to the zeroth-order flatness (i.e., the worst-case loss within a perturbation radius). We show that the zeroth-order flatness can be insufficient to discriminate minima with low generalization error from those with high generalization error both when there is a single minimum or multiple minima within the given perturbation radius. Thus we present first-order flatness, a stronger measure of flatness focusing on the maximal gradient norm within a perturbation radius which bounds both the maximal eigenvalue of Hessian at local minima and the regularization function of SAM. We also present a novel training procedure named Gradient norm Aware Minimization (GAM) to seek minima with uniformly small curvature across all directions. Experimental results show that GAM improves the generalization of models trained with current optimizers such as SGD and AdamW on various datasets and networks. Furthermore, we show that GAM can help SAM find flatter minima and achieve better generalization.

Geometry-aware optimization algorithms, such as Muon, have achieved remarkable success in training deep neural networks (DNNs). These methods leverage the underlying geometry of DNNs by selecting appropriate norms for different layers and updating parameters via norm-constrained linear minimization oracles (LMOs). However, even within a group of layers associated with the same norm, the local curvature can be heterogeneous across layers and vary dynamically over the course of training. For example, recent work shows that sharpness varies substantially across transformer layers and throughout training, yet standard geometry-aware optimizers impose fixed learning rates to layers within the same group, which may be inefficient for DNN training. In this paper, we introduce a noise-adaptive layerwise learning rate scheme on top of geometry-aware optimization algorithms and substantially accelerate DNN training compared to methods that use fixed learning rates within each group. Our method estimates gradient variance in the dual norm induced by the chosen LMO on the fly, and uses it to assign time-varying noise-adaptive layerwise learning rates within each group. We provide a theoretical analysis showing that our algorithm achieves a sharp convergence rate. Empirical results on transformer architectures such as LLaMA and GPT demonstrate that our approach achieves faster convergence than state-of-the-art optimizers.

The conjugate gradient method is a crucial first-order optimization method that generally converges faster than the steepest descent method, and its computational cost is much lower than that of second-order methods. However, while various types of conjugate gradient methods have been studied in Euclidean spaces and on Riemannian manifolds, there is little study for those in distributed scenarios. This paper proposes a decentralized Riemannian conjugate gradient descent (DRCGD) method that aims at minimizing a global function over the Stiefel manifold. The optimization problem is distributed among a network of agents, where each agent is associated with a local function, and the communication between agents occurs over an undirected connected graph. Since the Stiefel manifold is a non-convex set, a global function is represented as a finite sum of possibly non-convex (but smooth) local functions. The proposed method is free from expensive Riemannian geometric operations such as retractions, exponential maps, and vector transports, thereby reducing the computational complexity required by each agent. To the best of our knowledge, DRCGD is the first decentralized Riemannian conjugate gradient algorithm to achieve global convergence over the Stiefel manifold.

We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum-method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations.

Audio-visual zero-shot learning aims to classify samples consisting of a pair of corresponding audio and video sequences from classes that are not present during training. An analysis of the audio-visual data reveals a large degree of hyperbolicity, indicating the potential benefit of using a hyperbolic transformation to achieve curvature-aware geometric learning, with the aim of exploring more complex hierarchical data structures for this task. The proposed approach employs a novel loss function that incorporates cross-modality alignment between video and audio features in the hyperbolic space. Additionally, we explore the use of multiple adaptive curvatures for hyperbolic projections. The experimental results on this very challenging task demonstrate that our proposed hyperbolic approach for zero-shot learning outperforms the SOTA method on three datasets: VGGSound-GZSL, UCF-GZSL, and ActivityNet-GZSL achieving a harmonic mean (HM) improvement of around 3.0%, 7.0%, and 5.3%, respectively.

We consider the problem of generating a large collection of initial guesses for local minima of multimodal non-convex continuous optimization problems. The goal is for these initial guesses to be high-quality (i.e., a numerical solver converges quickly) and diverse (i.e., represent many different local minima). Identifying multiple locally optimal solutions enables flexible downstream decision-making, but typically requires expensive global search. Existing data-driven methods predict initial guesses using only the final converged optima from offline solver runs, which discards information about the local neighborhoods of solutions and limits the available training data. We propose GLENS (Global Search via Learning from Solver Iterates), a data-efficient global search method that leverages intermediate solver iterates as free data augmentation. GLENS consists of two components: a neighborhood structure model that uses diffusion models to learn the local geometry around optima conditioned on problem parameters, and a solver behavior model that learns refinement directions to further guide samples towards nearby optima during diffusion sampling. Experiments on modified non-convex benchmark problems and a two-robot obstacle-avoidance navigation problem show that GLENS generates high-quality initial guesses while preserving the multimodal distribution of diverse local optima. The resulting initial guesses lead to faster solver convergence across different problem settings and solvers. We also analyze how key hyperparameter choices affect the performance.

Diffusion models generating images conditionally on text, such as Dall-E 2 and Stable Diffusion, have recently made a splash far beyond the computer vision community. Here, we tackle the related problem of generating point clouds, both unconditionally, and conditionally with images. For the latter, we introduce a novel geometrically-motivated conditioning scheme based on projecting sparse image features into the point cloud and attaching them to each individual point, at every step in the denoising process. This approach improves geometric consistency and yields greater fidelity than current methods relying on unstructured, global latent codes. Additionally, we show how to apply recent continuous-time diffusion schemes. Our method performs on par or above the state of art on conditional and unconditional experiments on synthetic data, while being faster, lighter, and delivering tractable likelihoods. We show it can also scale to diverse indoors scenes.

We introduce a diffusion-based framework that performs aligned novel view image and geometry generation via a warping-and-inpainting methodology. Unlike prior methods that require dense posed images or pose-embedded generative models limited to in-domain views, our method leverages off-the-shelf geometry predictors to predict partial geometries viewed from reference images, and formulates novel-view synthesis as an inpainting task for both image and geometry. To ensure accurate alignment between generated images and geometry, we propose cross-modal attention distillation, where attention maps from the image diffusion branch are injected into a parallel geometry diffusion branch during both training and inference. This multi-task approach achieves synergistic effects, facilitating geometrically robust image synthesis as well as well-defined geometry prediction. We further introduce proximity-based mesh conditioning to integrate depth and normal cues, interpolating between point cloud and filtering erroneously predicted geometry from influencing the generation process. Empirically, our method achieves high-fidelity extrapolative view synthesis on both image and geometry across a range of unseen scenes, delivers competitive reconstruction quality under interpolation settings, and produces geometrically aligned colored point clouds for comprehensive 3D completion. Project page is available at https://cvlab-kaist.github.io/MoAI.

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

The aerodynamic coefficients of aircrafts are significantly impacted by its geometry, especially when the angle of attack (AoA) is large. In the field of aerodynamics, traditional polynomial-based parameterization uses as few parameters as possible to describe the geometry of an airfoil. However, because the 3D geometry of a wing is more complicated than the 2D airfoil, polynomial-based parameterizations have difficulty in accurately representing the entire shape of a wing in 3D space. Existing deep learning-based methods can extract massive latent neural representations for the shape of 2D airfoils or 2D slices of wings. Recent studies highlight that directly taking geometric features as inputs to the neural networks can improve the accuracy of predicted aerodynamic coefficients. Motivated by geometry theory, we propose to incorporate Riemannian geometric features for learning Coefficient of Pressure (CP) distributions on wing surfaces. Our method calculates geometric features (Riemannian metric, connection, and curvature) and further inputs the geometric features, coordinates and flight conditions into a deep learning model to predict the CP distribution. Experimental results show that our method, compared to state-of-the-art Deep Attention Network (DAN), reduces the predicted mean square error (MSE) of CP by an average of 8.41% for the DLR-F11 aircraft test set.

Model merging is an effective post-training strategy for composing capabilities in large language models without joint retraining. We study this in the supervised fine-tuning (SFT) stage, where multiple capability-based SFT checkpoints -- spanning math, code, precise instruction following, general instruction following, and knowledge recall -- must be consolidated into a single model. We introduce Optimization Trajectory Aware (OTA) Merging, a curvature-aware aggregation that leverages optimizer second-moment statistics as a diagonal curvature proxy to reweight parameter edits and mitigate interference. Complementing OTA, we propose Fast Fisher Grafting (FFG), a curvature-driven task-localization step that sparsifies conflicting or low-importance edits. FFG induces extremely low-rank masks concentrated in early attention query/key projections and token embeddings, exploiting shared curvature across capabilities. We further develop a memory-light compression of the second moments that preserves OTA's effect. Across diverse capability-based SFT checkpoints, OTA+FFG improves merged-model quality over strong weight-space baselines, reduces negative transfer, and remains robust across sparsity levels. Analyses reveal substantial curvature overlap between checkpoints, offering a novel lens on why simple linear merging can be effective in practice. Ablations confirm that FFG is critical for reducing task interference and that the compressed second moments retain the gains of the full formulation. To facilitate reproducibility, we open-source all code, training and evaluation scripts, visualization artifacts, and capability-specific SFT checkpoints at https://github.com/pmahdavi/ota-merge.

Parameter-efficient adaptation of pretrained vision models is commonly performed through linear probes, prompts, low-rank updates, or lightweight residual modules. While effective, these methods usually treat adaptation as a discrete Euclidean perturbation of frozen representations, without explicitly modeling the geometry of the task-induced feature displacement. We propose MC-RFM, a mixed-curvature Riemannian flow-matching framework for few-shot adaptation of frozen visual backbones. The key idea is to represent adapted features on a product manifold combining a hyperbolic factor, which captures hierarchy-sensitive semantic structure, and a Euclidean factor, which preserves locally discriminative visual variation. Adaptation is formulated as a task-conditioned continuous transport from frozen features to support-set prototypes, trained with a flow-matching objective and coupled to a hybrid prototype-linear classifier. The method is lightweight, backbone-agnostic, and operates entirely on cached frozen features. Across seven visual recognition benchmarks, five frozen backbones, and 1/4/16-shot regimes, MC-RFM is the best-performing method in a majority of evaluated settings, with the strongest gains on Transformer backbones and fine-grained datasets. Ablations show that the mixed-curvature head, task conditioning, adaptive branch gating, prototype shrinkage, and discriminative supervision each contribute to performance. These results suggest that few-shot adaptation benefits not only from deciding which parameters to update, but also from modeling how representations should move through a geometry matched to the structure of the downstream task.

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

Trajectory planning is commonly used as part of a local planner in autonomous driving. This paper considers the problem of planning a continuous-curvature-rate trajectory between fixed start and goal states that minimizes a tunable trade-off between passenger comfort and travel time. The problem is an instance of infinite dimensional optimization over two continuous functions: a path, and a velocity profile. We propose a simplification of this problem that facilitates the discretization of both functions. This paper also proposes a method to quickly generate minimal-length paths between start and goal states based on a single tuning parameter: the second derivative of curvature. Furthermore, we discretize the set of velocity profiles along a given path into a selection of acceleration way-points along the path. Gradient-descent is then employed to minimize cost over feasible choices of the second derivative of curvature, and acceleration way-points, resulting in a method that repeatedly solves the path and velocity profiles in an iterative fashion. Numerical examples are provided to illustrate the benefits of the proposed methods.

Decision-based black-box attacks often necessitate a large number of queries to craft an adversarial example. Moreover, decision-based attacks based on querying boundary points in the estimated normal vector direction often suffer from inefficiency and convergence issues. In this paper, we propose a novel query-efficient curvature-aware geometric decision-based black-box attack (CGBA) that conducts boundary search along a semicircular path on a restricted 2D plane to ensure finding a boundary point successfully irrespective of the boundary curvature. While the proposed CGBA attack can work effectively for an arbitrary decision boundary, it is particularly efficient in exploiting the low curvature to craft high-quality adversarial examples, which is widely seen and experimentally verified in commonly used classifiers under non-targeted attacks. In contrast, the decision boundaries often exhibit higher curvature under targeted attacks. Thus, we develop a new query-efficient variant, CGBA-H, that is adapted for the targeted attack. In addition, we further design an algorithm to obtain a better initial boundary point at the expense of some extra queries, which considerably enhances the performance of the targeted attack. Extensive experiments are conducted to evaluate the performance of our proposed methods against some well-known classifiers on the ImageNet and CIFAR10 datasets, demonstrating the superiority of CGBA and CGBA-H over state-of-the-art non-targeted and targeted attacks, respectively. The source code is available at https://github.com/Farhamdur/CGBA.

There is growing evidence of the effectiveness of Shampoo, a higher-order preconditioning method, over Adam in deep learning optimization tasks. However, Shampoo's drawbacks include additional hyperparameters and computational overhead when compared to Adam, which only updates running averages of first- and second-moment quantities. This work establishes a formal connection between Shampoo (implemented with the 1/2 power) and Adafactor -- a memory-efficient approximation of Adam -- showing that Shampoo is equivalent to running Adafactor in the eigenbasis of Shampoo's preconditioner. This insight leads to the design of a simpler and computationally efficient algorithm: ShampoO with Adam in the Preconditioner's eigenbasis (SOAP). With regards to improving Shampoo's computational efficiency, the most straightforward approach would be to simply compute Shampoo's eigendecomposition less frequently. Unfortunately, as our empirical results show, this leads to performance degradation that worsens with this frequency. SOAP mitigates this degradation by continually updating the running average of the second moment, just as Adam does, but in the current (slowly changing) coordinate basis. Furthermore, since SOAP is equivalent to running Adam in a rotated space, it introduces only one additional hyperparameter (the preconditioning frequency) compared to Adam. We empirically evaluate SOAP on language model pre-training with 360m and 660m sized models. In the large batch regime, SOAP reduces the number of iterations by over 40% and wall clock time by over 35% compared to AdamW, with approximately 20% improvements in both metrics compared to Shampoo. An implementation of SOAP is available at https://github.com/nikhilvyas/SOAP.

Learning good representations is essential for latent planning with world models. While pretrained visual encoders produce strong semantic visual features, they are not tailored to planning and contain information irrelevant -- or even detrimental -- to planning. Inspired by the perceptual straightening hypothesis in human visual processing, we introduce temporal straightening to improve representation learning for latent planning. Using a curvature regularizer that encourages locally straightened latent trajectories, we jointly learn an encoder and a predictor. We show that reducing curvature this way makes the Euclidean distance in latent space a better proxy for the geodesic distance and improves the conditioning of the planning objective. We demonstrate empirically that temporal straightening makes gradient-based planning more stable and yields significantly higher success rates across a suite of goal-reaching tasks.

Optimization analyses for cross-entropy training rely on local Taylor models of the loss to predict whether a proposed step will decrease the objective. These surrogates are reliable only inside the Taylor convergence radius of the true loss along the update direction. That radius is set not by real-line curvature alone but by the nearest complex singularity. For cross-entropy, the softmax partition function F=sum_j exp(z_j) has complex zeros -- ``ghosts of softmax'' -- that induce logarithmic singularities in the loss and cap this radius. To make this geometry usable, we derive closed-form expressions under logit linearization along the proposed update direction. In the binary case, the exact radius is ρ^*=δ^2+ π^2/Δ_a. In the multiclass case, we obtain the lower bound ρ_a=π/Δ_a, where Δ_a=max_k a_k-min_k a_k is the spread of directional logit derivatives a_k=nabla z_kcdot v. This bound costs one Jacobian-vector product and reveals what makes a step fragile: samples that are both near a decision flip and highly sensitive to the proposed direction tighten the radius. The normalized step size r=τ/ρ_a separates safe from dangerous updates. Across six tested architectures and multiple step directions, no model fails for r<1, yet collapse appears once rge 1. Temperature scaling confirms the mechanism: normalizing by ρ_a shrinks the onset-threshold spread from standard deviation 0.992 to 0.164. A controller that enforces τleρ_a survives learning-rate spikes up to 10{,} 000times in our tests, where gradient clipping still collapses. Together, these results identify a geometric constraint on cross-entropy optimization that operates through Taylor convergence rather than Hessian curvature.

Learning-rate transfer can reduce the cost of training large language models: instead of sweeping learning rates at target scale, practitioners extrapolate from smaller runs. Existing approaches often assume that the optimal learning rate follows a log-linear scaling law in data scale and model size. We carefully examine and evaluate this scaling law. In our empirical study of GPT-2--style models from 22M to 707M parameters trained on 5B to 100B tokens, the optimal learning rate develops upward curvature at larger scales, leading to inaccurate extrapolation. We find that this curvature largely disappears when learning rates are replaced by effective learning rate (the step size in normalized weight space), and when data D extrapolation is used instead of model size N extrapolation. Next, we explain nonlinearity in scaling: weight-norm converges to equilibrium slower when optimal learning is small, requiring a larger step size to reduce the transient phase. Experiments with AdamH, which directly controls the effective learning rate, further support this explanation.

Diffusion models and their variations, such as rectified flows, generate diverse and high-quality images, but they are still hindered by slow iterative sampling caused by the highly curved generative paths they learn. An important cause of high curvature, as shown by previous work, is independence between the source distribution (standard Gaussian) and the data distribution. In this work, we tackle this limitation by two complementary contributions. First, we attempt to break away from the standard Gaussian assumption by introducing κ-FC, a general formulation that conditions the source distribution on an arbitrary signal κ that aligns it better with the data distribution. Then, we present MixFlow, a simple but effective training strategy that reduces the generative path curvatures and considerably improves sampling efficiency. MixFlow trains a flow model on linear mixtures of a fixed unconditional distribution and a κ-FC-based distribution. This simple mixture improves the alignment between the source and data, provides better generation quality with less required sampling steps, and accelerates the training convergence considerably. On average, our training procedure improves the generation quality by 12\% in FID compared to standard rectified flow and 7\% compared to previous baselines under a fixed sampling budget. Code available at: https://github.com/NazirNayal8/MixFlow{https://github.com/NazirNayal8/MixFlow}

Training deep neural networks--and more recently, large models--demands efficient and scalable optimizers. Adaptive gradient algorithms like Adam, AdamW, and their variants have been central to this task. Despite the development of numerous variance reduction algorithms in the past decade aimed at accelerating stochastic optimization in both convex and nonconvex settings, variance reduction has not found widespread success in training deep neural networks or large language models. Consequently, it has remained a less favored approach in modern AI. In this paper, to unleash the power of variance reduction for efficient training of large models, we propose a unified optimization framework, MARS (Make vAriance Reduction Shine), which reconciles preconditioned gradient methods with variance reduction via a scaled stochastic recursive momentum technique. Within our framework, we introduce three instances of MARS that leverage preconditioned gradient updates based on AdamW, Lion, and Shampoo, respectively. We also draw a connection between our algorithms and existing optimizers. Experimental results on training GPT-2 models indicate that MARS consistently outperforms AdamW by a large margin.

In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix M=UΣV^top with its polar factor UV^top. In this work, we consider a class of Muon-like updates, where we replace the update M with UΣ^p V^top for some parameter p. We call this a "spectral-shaping" operation, and develop a theory of how to pick p which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive p helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative p helps later by reallocating update strength toward low-curvature directions that still contain useful training signals. Building on the insight, we propose DynMuon, an efficient dynamic spectral shaping method that schedules p from positive to mildly negative over training. Extensive experiments across model sizes, architectures, and training settings show that DynMuon consistently achieves lower validation loss than Muon, while requiring 10.6-26.5% fewer steps to reach the same target loss.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

Sharpness-Aware Minimization (SAM) is widely used to seek flatter minima -- often linked to better generalization. In its standard implementation, SAM updates the current iterate using the loss gradient evaluated at a point perturbed by distance ρ along the normalized gradient direction. We show that, for some choices of ρ, SAM can stall at points where this shifted (perturbed-point) gradient vanishes despite a nonzero original gradient, and therefore, they are not stationary points of the original loss. We call these points hallucinated minimizers, prove their existence under simple nonconvex landscape conditions (e.g., the presence of a local minimizer and a local maximizer), and establish sufficient conditions for local convergence of the SAM iterates to them. We corroborate this failure mode in neural network training and observe that it aligns with SAM's performance degradation often seen at large ρ. Finally, as a practical safeguard, we find that a short initial SGD warm-start before enabling SAM mitigates this failure mode and reduces sensitivity to the choice of ρ.

Data-driven algorithm design automates hyperparameter tuning, but its statistical foundations remain limited because model performance can depend on hyperparameters in implicit and highly non-smooth ways. Existing guarantees focus on the simple case of a one-dimensional (scalar) hyperparameter. This leaves the practically important, multi-dimensional hyperparameter tuning setting unresolved. We address this open question by establishing the first general framework for establishing generalization guarantees for tuning multi-dimensional hyperparameters in data-driven settings. Our approach strengthens the generalization guarantee framework for semi-algebraic function classes by exploiting tools from real algebraic geometry, yielding sharper, more broadly applicable guarantees. For completeness, we also instantiate the first lower bound for this general setting. We further extend the analysis to hyperparameter tuning using the validation loss under minimal assumptions, and derive improved bounds when additional structure is available. Finally, we demonstrate the scope of the framework with new learnability results, including data-driven weighted group lasso and weighted fused lasso.

Classical convergence guarantees for gradient-based learning in games require the pseudo-gradient to be (strongly) monotone in Euclidean geometry as shown by rosen(1965), a condition that often fails even in simple games with strong cross-player couplings. We introduce Small-Gain Nash (SGN), a block small-gain condition in a custom block-weighted geometry. SGN converts local curvature and cross-player Lipschitz coupling bounds into a tractable certificate of contraction. It constructs a weighted block metric in which the pseudo-gradient becomes strongly monotone on any region where these bounds hold, even when it is non-monotone in the Euclidean sense. The continuous flow is exponentially contracting in this designed geometry, and projected Euler and RK4 discretizations converge under explicit step-size bounds derived from the SGN margin and a local Lipschitz constant. Our analysis reveals a certified ``timescale band'', a non-asymptotic, metric-based certificate that plays a TTUR-like role: rather than forcing asymptotic timescale separation via vanishing, unequal step sizes, SGN identifies a finite band of relative metric weights for which a single-step-size dynamics is provably contractive. We validate the framework on quadratic games where Euclidean monotonicity analysis fails to predict convergence, but SGN successfully certifies it, and extend the construction to mirror/Fisher geometries for entropy-regularized policy gradient in Markov games. The result is an offline certification pipeline that estimates curvature, coupling, and Lipschitz parameters on compact regions, optimizes block weights to enlarge the SGN margin, and returns a structural, computable convergence certificate consisting of a metric, contraction rate, and safe step-sizes for non-monotone games.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

Is the standard weight decay in AdamW truly optimal? Although AdamW decouples weight decay from adaptive gradient scaling, a fundamental conflict remains: the Radial Tug-of-War. In deep learning, gradients tend to increase parameter norms to expand effective capacity while steering directions to learn features, whereas weight decay indiscriminately suppresses norm growth. This push--pull interaction induces radial oscillations, injecting noise into Adam's second-moment estimates and potentially degrading delicate tangential feature learning. We argue that magnitude and direction play distinct roles and should be decoupled in optimizer dynamics. We propose Orthogonal Dynamics Decoupling and instantiate it as AdamO: an SGD-style update handles the one-dimensional norm control, while Adam's adaptive preconditioning is confined to the tangential subspace. AdamO further incorporates curvature-adaptive radial step sizing and architecture-aware rules and projections for scale-invariant layers and low-dimensional parameters. Experiments on vision and language tasks show that AdamO improves generalization and stability over AdamW without introducing additional complex constraints.

Partial Differential Equations are precise in modelling the physical, biological and graphical phenomena. However, the numerical methods suffer from the curse of dimensionality, high computation costs and domain-specific discretization. We aim to explore pros and cons of different PDE solvers, and apply them to specific scientific simulation problems, including forwarding solution, inverse problems and equations discovery. In particular, we extend the recent CNF (NeurIPS 2023) framework solver to multi-dependent-variable and non-linear settings, together with down-stream applications. The outcomes include implementation of selected methods, self-tuning techniques, evaluation on benchmark problems and a comprehensive survey of neural PDE solvers and scientific simulation applications.

Optimization in machine learning, both theoretical and applied, is presently dominated by first-order gradient methods such as stochastic gradient descent. Second-order optimization methods, that involve second derivatives and/or second order statistics of the data, are far less prevalent despite strong theoretical properties, due to their prohibitive computation, memory and communication costs. In an attempt to bridge this gap between theoretical and practical optimization, we present a scalable implementation of a second-order preconditioned method (concretely, a variant of full-matrix Adagrad), that along with several critical algorithmic and numerical improvements, provides significant convergence and wall-clock time improvements compared to conventional first-order methods on state-of-the-art deep models. Our novel design effectively utilizes the prevalent heterogeneous hardware architecture for training deep models, consisting of a multicore CPU coupled with multiple accelerator units. We demonstrate superior performance compared to state-of-the-art on very large learning tasks such as machine translation with Transformers, language modeling with BERT, click-through rate prediction on Criteo, and image classification on ImageNet with ResNet-50.

Low-Rank Adaptation (LoRA) reparameterizes a weight update as a product of two low-rank factors, but the Jacobian J_{G} of the generator mapping the factors to the weight matrix is rank-deficient, so the factor-space preconditioner J_{G}^* {F}_t J_{G} induced by any {W}-space preconditioner {F}_t is singular, and consequently the standard chain rule cannot be uniquely inverted to map a preconditioned {W}-space direction back to a factor-space update. We cast existing LoRA optimizers in a unified framework parameterized by two choices: (i) which invertible surrogate for J_{G}^* {F}_t J_{G} to use, and (ii) which {F}_t on {W} to use. Existing methods occupy four families along these axes: factor-space adaptive updates, block-diagonal surrogates for J_{G}^* J_{G}, Frobenius-residual pseudoinverse methods, and Riemannian manifold constraint. Within this design space, a gradient-statistics-aware {F}_t paired with a closed-form factor-space solve at {O}((m+n)r) memory remains underexplored. We propose AdaPreLoRA, which fills this gap by adopting the Adafactor diagonal Kronecker preconditioner {H}_t on {W} and selecting from the resulting factor-space solution family the element minimizing an {H}_t-weighted imbalance between the two factor contributions; by construction, the resulting factor update is the closest LoRA approximation to the preconditioned {W}-space direction under the {H}_t-weighted norm. Across GPT-2 (E2E), Mistral-7B and Qwen2-7B (GLUE, ARC, GSM8K), and diffusion-model personalization, AdaPreLoRA is competitive with or improves over a representative set of LoRA optimizers while keeping peak GPU memory at the LoRA optimizer level.

Leveraging representation encoders for generative modeling offers a path for efficient, high-fidelity synthesis. However, standard diffusion transformers fail to converge on these representations directly. While recent work attributes this to a capacity bottleneck proposing computationally expensive width scaling of diffusion transformers we demonstrate that the failure is fundamentally geometric. We identify Geometric Interference as the root cause: standard Euclidean flow matching forces probability paths through the low-density interior of the hyperspherical feature space of representation encoders, rather than following the manifold surface. To resolve this, we propose Riemannian Flow Matching with Jacobi Regularization (RJF). By constraining the generative process to the manifold geodesics and correcting for curvature-induced error propagation, RJF enables standard Diffusion Transformer architectures to converge without width scaling. Our method RJF enables the standard DiT-B architecture (131M parameters) to converge effectively, achieving an FID of 3.37 where prior methods fail to converge. Code: https://github.com/amandpkr/RJF

Estimating the pose of an object from a monocular image is an inverse problem fundamental in computer vision. The ill-posed nature of this problem requires incorporating deformation priors to solve it. In practice, many materials do not perceptibly shrink or extend when manipulated, constituting a powerful and well-known prior. Mathematically, this translates to the preservation of the Riemannian metric. Neural networks offer the perfect playground to solve the surface reconstruction problem as they can approximate surfaces with arbitrary precision and allow the computation of differential geometry quantities. This paper presents an approach to inferring continuous deformable surfaces from a sequence of images, which is benchmarked against several techniques and obtains state-of-the-art performance without the need for offline training.

Leaf-lesion segmentation is topology-sensitive: small merges, splits, or false holes can be biologically meaningful descriptors of biochemical pathways, yet they are weakly penalized by standard pixel-wise losses in Euclidean latents. I explore HyperTopo-Adapters, a lightweight, parameter-efficient head trained on top of a frozen vision encoder, which embeds features on a product manifold -- hyperbolic + Euclidean + spherical (H + E + S) -- to encourage hierarchical separation (H), local linear detail (E), and global closure (S). A topology prior complements Dice/BCE in two forms: (i) persistent-homology (PH) distance for evaluation and selection, and (ii) a differentiable surrogate that combines a soft Euler-characteristic match with total variation regularization for stable training. I introduce warm-ups for both the hyperbolic contrastive term and the topology prior, per-sample evaluation of structure-aware metrics (Boundary-F1, Betti errors, PD distance), and a min-PD within top-K Dice rule for checkpoint selection. On a Kaggle leaf-lesion dataset (N=2,940), early results show consistent gains in boundary and topology metrics (reducing Delta beta_1 hole error by 9%) while Dice/IoU remain competitive. The study is diagnostic by design: I report controlled ablations (curvature learning, latent dimensions, contrastive temperature, surrogate settings), and ongoing tests varying encoder strength (ResNet-50, DeepLabV3, DINOv2/v3), input resolution, PH weight, and partial unfreezing of late blocks. The contribution is an open, reproducible train/eval suite (available at https://github.com/ChimdiWalter/HyperTopo-Adapters) that isolates geometric/topological priors and surfaces failure modes to guide stronger, topology-preserving architectures.

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each presents its unique limitations and generally balances training cost, numerical accuracy, and ease of applicability to different problem setups. To address these limitations, we introduce several methods to apply latent diffusion models to physics simulation. Firstly, we introduce a mesh autoencoder to compress arbitrarily discretized PDE data, allowing for efficient diffusion training across various physics. Furthermore, we investigate full spatio-temporal solution generation to mitigate autoregressive error accumulation. Lastly, we investigate conditioning on initial physical quantities, as well as conditioning solely on a text prompt to introduce text2PDE generation. We show that language can be a compact, interpretable, and accurate modality for generating physics simulations, paving the way for more usable and accessible PDE solvers. Through experiments on both uniform and structured grids, we show that the proposed approach is competitive with current neural PDE solvers in both accuracy and efficiency, with promising scaling behavior up to sim3 billion parameters. By introducing a scalable, accurate, and usable physics simulator, we hope to bring neural PDE solvers closer to practical use.

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation from being flat in its principal directions. We find that state-of-the-art trained convolutional neural networks for image classification have a characteristic curvature profile along layers: an initial steep increase, followed by a long phase of a plateau, and followed by another increase. In contrast, this behavior does not appear in untrained networks in which the curvature flattens. We also show that the curvature gap between the last two layers has a strong correlation with the generalization capability of the network. Moreover, we find that the intrinsic dimension of latent codes is not necessarily indicative of curvature. Finally, we observe that common regularization methods such as mixup yield flatter representations when compared to other methods. Our experiments show consistent results over a variety of deep learning architectures and multiple data sets. Our code is publicly available at https://github.com/azencot-group/CRLM

We identify the Spectral Energy Gain in extreme model compression, where low-rank binary approximations outperform tiny-rank floating-point baselines for heavy-tailed spectra. However, prior attempts fail to realize this potential, trailing state-of-the-art 1-bit methods. We attribute this degradation to Latent Geometry Misalignment: standard singular vectors exhibit high coherence (spiky distribution), the worst-case geometry for binary quantization. To realize this gain, we propose LittleBit-2, a framework employing Internal Latent Rotation and Joint Iterative Quantization (Joint-ITQ). This approach acts as a geometric preconditioner, aligning coherent latent distributions with the binary hypercube with zero inference overhead. Empirically, LittleBit-2 establishes a new state-of-the-art in the sub-1-bit regime (1sim0.1 bpp) on Llama-2 and Llama-3, matching the fidelity of leading 1-bit baselines.

Diffusion models (DMs) have achieved state-of-the-art generative performance but suffer from high sampling latency due to their sequential denoising nature. Existing solver-based acceleration methods often face significant image quality degradation under a low-latency budget, primarily due to accumulated truncation errors arising from the inability to capture high-curvature trajectory segments. In this paper, we propose the Ensemble Parallel Direction solver (dubbed as EPD-Solver), a novel ODE solver that mitigates these errors by incorporating multiple parallel gradient evaluations in each step. Motivated by the geometric insight that sampling trajectories are largely confined to a low-dimensional manifold, EPD-Solver leverages the Mean Value Theorem for vector-valued functions to approximate the integral solution more accurately. Importantly, since the additional gradient computations are independent, they can be fully parallelized, preserving low-latency sampling nature. We introduce a two-stage optimization framework. Initially, EPD-Solver optimizes a small set of learnable parameters via a distillation-based approach. We further propose a parameter-efficient Reinforcement Learning (RL) fine-tuning scheme that reformulates the solver as a stochastic Dirichlet policy. Unlike traditional methods that fine-tune the massive backbone, our RL approach operates strictly within the low-dimensional solver space, effectively mitigating reward hacking while enhancing performance in complex text-to-image (T2I) generation tasks. In addition, our method is flexible and can serve as a plugin (EPD-Plugin) to improve existing ODE samplers.

We prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Unlike previous analyses, our approach accelerates any stochastic gradient method which makes sufficient progress in expectation. The proof, which proceeds using the estimating sequences framework, applies to both convex and strongly convex functions and is easily specialized to accelerated SGD under the strong growth condition. In this special case, our analysis reduces the dependence on the strong growth constant from rho to rho as compared to prior work. This improvement is comparable to a square-root of the condition number in the worst case and address criticism that guarantees for stochastic acceleration could be worse than those for SGD.

Given a monocular video, the goal of video re-rendering is to generate views of the scene from a novel camera trajectory. Existing methods face two distinct challenges. Geometrically unconditioned models lack spatial awareness, leading to drift and deformation under viewpoint changes. On the other hand, geometrically-conditioned models depend on estimated depth and explicit reconstruction, making them susceptible to depth inaccuracies and calibration errors. We propose to address these challenges by using the implicit geometric knowledge embedded in the latent space of a large 4D reconstruction model to condition the video generation process. These latents capture scene structure in a continuous space without explicit reconstruction. Therefore, they provide a flexible representation that allows the pretrained diffusion prior to regularize errors more effectively. By jointly conditioning on these latents and source camera poses, we demonstrate that our model achieves state-of-the-art results on the video re-rendering task. Project webpage is https://lavr-4d-scene-rerender.github.io/

Data-driven surrogate models improve the efficiency of simulating continuous dynamical systems, yet their autoregressive rollouts are often limited by instability and spectral blow-up. While global regularization techniques can enforce contractive dynamics, they uniformly damp high-frequency features, introducing a contraction-dissipation dilemma. Furthermore, long-horizon trajectory optimization methods that explicitly correct drift are bottlenecked by memory constraints. In this work, we propose Jacobian-Adaptive Weighting for Stability (JAWS), a probabilistic regularization strategy designed to mitigate these limitations. By framing operator learning as Maximum A Posteriori (MAP) estimation with spatially heteroscedastic uncertainty, JAWS dynamically modulates the regularization strength based on local physical complexity. This allows the model to enforce contraction in smooth regions to suppress noise, while relaxing constraints near singular features to preserve gradients, effectively realizing a behavior similar to numerical shock-capturing schemes. Experiments demonstrate that this spatially-adaptive prior serves as an effective spectral pre-conditioner, which reduces the base operator's burden of handling high-frequency instabilities. This reduction enables memory-efficient, short-horizon trajectory optimization to match or exceed the long-term accuracy of long-horizon baselines. Evaluated on the 1D viscous Burgers' equation, our hybrid approach improves long-term stability, shock fidelity, and out-of-distribution generalization while reducing training computational costs.

In this paper we provide an O(m (log log n)^{O(1)} log(1/ε))-expected time algorithm for solving Laplacian systems on n-node m-edge graphs, improving improving upon the previous best expected runtime of O(m log n (log log n)^{O(1)} log(1/ε)) achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of ell_p-stretch graph approximations with improved stretch and sparsity bounds. Additionally, as motivation for this work, we show that for every set of vectors in R^d (not just those induced by graphs) and all k > 1 there exist ultrasparsifiers with d-1 + O(d/k) re-weighted vectors of relative condition number at most k. For small k, this improves upon the previous best known relative condition number of O(k log d), which is only known for the graph case.

The training of score-based diffusion models (SDMs) is based on score matching. The challenge of score matching is that it includes a computationally expensive Jacobian trace. While several methods have been proposed to avoid this computation, each has drawbacks, such as instability during training and approximating the learning as learning a denoising vector field rather than a true score. We propose a novel score matching variant, local curvature smoothing with Stein's identity (LCSS). The LCSS bypasses the Jacobian trace by applying Stein's identity, enabling regularization effectiveness and efficient computation. We show that LCSS surpasses existing methods in sample generation performance and matches the performance of denoising score matching, widely adopted by most SDMs, in evaluations such as FID, Inception score, and bits per dimension. Furthermore, we show that LCSS enables realistic image generation even at a high resolution of 1024 times 1024.

Transformers achieve strong performance across diverse domains but implicitly assume Euclidean geometry in their attention mechanisms, limiting their effectiveness on data with non-Euclidean structure. While recent extensions to hyperbolic and spherical spaces show promise for hierarchical and cyclical patterns, respectively, they require committing to a single geometry a priori, reducing flexibility when data exhibits mixed geometric properties. We introduce the Curvature-Adaptive Transformer (CAT), a novel architecture that dynamically learns per-token routing across three geometric attention branches through a lightweight, differentiable gating mechanism. Unlike fixed-geometry approaches, CAT enables adaptive geometric specialization, routing tokens to the appropriate curvature based on their local relational structure. The routing network provides interpretable curvature preferences while each branch employs geometry-specific operations optimized for its respective manifold. On knowledge graph completion benchmarks (FB15k-237, WN18RR), CAT achieves approximately 10% improvements in MRR and Hits@10 over fixed-geometry baselines with minimal overhead (5% parameter increase, comparable inference time). These results demonstrate that learned geometric adaptation outperforms any single fixed geometry for complex relational reasoning, establishing CAT as a scalable and interpretable foundation for mixture-of-geometry architectures across language, vision, and multimodal domains.

In order to streamline the fine-tuning of foundation models, Low-Rank Adapters (LoRAs) have been substantially adopted across various fields, including instruction tuning and domain adaptation. The underlying concept of LoRA involves decomposing a full-rank matrix into the product of two lower-rank matrices, which reduces storage consumption and accelerates the training process. Furthermore, to address the limited expressive capacity of LoRA, the Mixture-of-Expert (MoE) has been introduced for incorporating multiple LoRA adapters. The integration of LoRA experts leads to a visible improvement across several downstream scenes. However, the mixture of LoRAs (MoE-LoRA) still exhibits its low robustness during tuning and inferring. Inspired by the Riemannian Preconditioners which train LoRA as a sub-space projector, we propose a new training strategy for MoE-LoRA, to stabilize and boost its feature learning procedure by multi-space projections. Examinations on SGD and AdamW optimizers demonstrate the effectiveness of our methodology. Source code is available at https://github.com/THUDM/MoELoRA_Riemannian.

We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for G^1 smoothness are given, which, up to a certain degenerate case, are also necessary. Further, we investigate curvature integrability and present examples

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural representations. This mismatch leaves no suitable method to solve surface PDEs directly within the neural domain, forcing explicit mesh extraction or per-instance residual training, preventing end-to-end workflows. We present a novel, mesh-free formulation that learns a local update operator conditioned on neural (local) shape attributes, enabling surface PDEs to be solved directly where the (neural) data lives. The operator integrates naturally with prevalent neural surface representations, is trained once on a single representative shape, and generalizes across shape and topology variations, enabling accurate, fast inference without explicit meshing or per-instance optimization while preserving differentiability. Across analytic benchmarks (heat equation and Poisson solve on sphere) and real neural assets across different representations, our method slightly outperforms CPM while remaining reasonably close to FEM, and, to our knowledge, delivers the first end-to-end pipeline that solves surface PDEs on both neural and classical surface representations. Code will be released on acceptance.

Strictly proper scoring rules identify the true conditional class distribution at population level, but their curvature can alter optimization and finite-sample behavior. We study three multiclass objectives: a class-aware quadratic Bregman score (CAPM), a strongly convex generator with constrained log-cosh ridges (HPG), and an HPG objective with an annealed probability-margin penalty (APMS). CAPM is treated as a structured instance of established quadratic scoring-rule theory. We derive conditional-regret, curvature, range, and logit-gradient bounds for CAPM and HPG, and prove exact penalty-range and conditional-target displacement bounds for APMS. Controlled five-seed experiments use Digits, Wisconsin breast cancer, and synthetic confusion and long-tail problems under clean labels, symmetric and pair-flip corruption, class imbalance, calibration evaluation, input corruption, and first-order adversarial perturbations. The candidates are close to cross-entropy on clean data and show descriptive gains in some noisy-label cells, but the five-seed comparisons are interpreted descriptively rather than as significance evidence. The selected noisy-label baselines perform better on Digits with 40% symmetric label noise, and explicit prior-adjustment methods perform better in the 30:1 synthetic long-tail experiment. Ablations do not show a consistent benefit from the candidate-specific graph, ridge, or margin components. The mathematical analysis establishes the stated properties, and the experiments delimit the empirical evidence; together they do not support a claim of general superiority.

Catastrophic overfitting (CO) in single-step adversarial training (AT) results in abrupt drops in the adversarial test accuracy (even down to 0%). For models trained with multi-step AT, it has been observed that the loss function behaves locally linearly with respect to the input, this is however lost in single-step AT. To address CO in single-step AT, several methods have been proposed to enforce local linearity of the loss via regularization. However, these regularization terms considerably slow down training due to Double Backpropagation. Instead, in this work, we introduce a regularization term, called ELLE, to mitigate CO effectively and efficiently in classical AT evaluations, as well as some more difficult regimes, e.g., large adversarial perturbations and long training schedules. Our regularization term can be theoretically linked to curvature of the loss function and is computationally cheaper than previous methods by avoiding Double Backpropagation. Our thorough experimental validation demonstrates that our work does not suffer from CO, even in challenging settings where previous works suffer from it. We also notice that adapting our regularization parameter during training (ELLE-A) greatly improves the performance, specially in large epsilon setups. Our implementation is available in https://github.com/LIONS-EPFL/ELLE .

Adapting CLIP to vertical domains is typically approached by novel fine-tuning strategies or by continual pre-training (CPT) on large domain-specific datasets. Yet, data itself remains an underexplored factor in this process. We revisit this task from a data-centric perspective: Can effective data selection substitute for large-scale datasets in CPT? We introduce CHIPS (Curvature-aware Hybrid Influence in Projection Subspace), which assigns each image-text pair a utility score that integrates three complementary factors aligned with three goals: faithfulness via a curvature-aware, Newton-style alignment computed in CLIP's end-point subspace; scalability via an InfoNCE-aware curvature estimator with Johnson-Lindenstrauss (JL) sketching; and retention via a selection-aware relevance weight combined with learnability to balance target adaptation against general-domain preservation. We justify this design theoretically by proving a lower-bound guarantee on the proxy's correlation with full-parameter alignment and by characterizing the bias-variance trade-offs introduced by curvature mixing and JL sketching. We evaluate CHIPS empirically across various settings: 1) CHIPS attains state-of-the-art performance among selection baselines on 17 medical benchmarks, matches full-dataset CPT with 30% of the data, and outperforms half-dataset CPT using only 10%; 2) on 31 general-domain benchmarks, CHIPS yields the smallest performance drop under 10-30% data-retention budgets. Code, data, and checkpoints will be released.

Neural network training is commonly accelerated by using multiple synchronized workers to compute gradient updates in parallel. Asynchronous methods remove synchronization overheads and improve hardware utilization at the cost of introducing gradient delay, which impedes optimization and can lead to lower final model performance. We introduce Adaptive Braking (AB), a modification for momentum-based optimizers that mitigates the effects of gradient delay. AB dynamically scales the gradient based on the alignment of the gradient and the velocity. This can dampen oscillations along high curvature directions of the loss surface, stabilizing and accelerating asynchronous training. We show that applying AB on top of SGD with momentum enables training ResNets on CIFAR-10 and ImageNet-1k with delays D geq 32 update steps with minimal drop in final test accuracy.

Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by industrial simulations that involve complex geometries. In the spirit of advancing neural PDE solvers to real industrial applications, we present Transolver++, a highly parallel and efficient neural solver that can accurately solve PDEs on million-scale geometries. Building upon previous advancements in solving PDEs by learning physical states via Transolver, Transolver++ is further equipped with an extremely optimized parallelism framework and a local adaptive mechanism to efficiently capture eidetic physical states from massive mesh points, successfully tackling the thorny challenges in computation and physics learning when scaling up input mesh size. Transolver++ increases the single-GPU input capacity to million-scale points for the first time and is capable of continuously scaling input size in linear complexity by increasing GPUs. Experimentally, Transolver++ yields 13% relative promotion across six standard PDE benchmarks and achieves over 20% performance gain in million-scale high-fidelity industrial simulations, whose sizes are 100times larger than previous benchmarks, covering car and 3D aircraft designs.

We present a topology-informed inverse rendering approach for reconstructing high-genus surface meshes from multi-view images. Compared to 3D representations like voxels and point clouds, mesh-based representations are preferred as they enable the application of differential geometry theory and are optimized for modern graphics pipelines. However, existing inverse rendering methods often fail catastrophically on high-genus surfaces, leading to the loss of key topological features, and tend to oversmooth low-genus surfaces, resulting in the loss of surface details. This failure stems from their overreliance on Adam-based optimizers, which can lead to vanishing and exploding gradients. To overcome these challenges, we introduce an adaptive V-cycle remeshing scheme in conjunction with a re-parametrized Adam optimizer to enhance topological and geometric awareness. By periodically coarsening and refining the deforming mesh, our method informs mesh vertices of their current topology and geometry before optimization, mitigating gradient issues while preserving essential topological features. Additionally, we enforce topological consistency by constructing topological primitives with genus numbers that match those of ground truth using Gauss-Bonnet theorem. Experimental results demonstrate that our inverse rendering approach outperforms the current state-of-the-art method, achieving significant improvements in Chamfer Distance and Volume IoU, particularly for high-genus surfaces, while also enhancing surface details for low-genus surfaces.