Modern Large Language Model (LLM) training is fundamentally bottlenecked by pathologically flat saddle points in extreme high-dimensional landscapes. Motivated by this challenge, we analyze the saddle-point escape dynamics of the emerging Muon optimizer, demonstrating its resilience against the $\mathcal{O}(D)$ dimensional curse that severely traps element-wise adaptive optimizers like AdamW. By extending generalized matrix perturbation theory, we develop a theoretical framework to capture Muon's non-equilibrium optimization trajectories. This theoretical machinery mathematically proves that Muon elegantly bypasses the dimensional curse via a non-linear spectral shaping mechanism. By leveraging resolvent functional calculus and macroscopic Cauchy contour integration, we avoid isotropic noise assumptions and Tracy-Widom edge singularities. We establish that structural incoherence securely shields the trajectory from orthogonal drift, enabling a dimension-free saddle-point escape, and triggering a deterministic $\mathcal{O}(1)$ discrete ballistic ejection under sufficient spectral gap. Consequently, we provide an algebraically dimension-free escape bound for Muon, formalizing the underlying mechanics of its non-convex optimization dynamics.
National University of Defense Technology（国防科技大学）；Hunan University（湖南大学）
Nonconvex optimization algorithms with random initialization have attracted increasing attention recently. It has been showed that many first-order methods always avoid saddle points with random starting points. In this paper, we answer a question: can the nonconvex heavy-ball algorithms with random initialization avoid saddle points? The answer is yes! Direct using the existing proof technique for the heavy-ball algorithms is hard due to that each iteration of the heavy-ball algorithm consists of current and last points. It is impossible to formulate the algorithms as iteration like xk+1= g(x…
Carnegie Mellon University（卡内基梅隆大学）；University of California, Berkeley（加利福尼亚大学伯克利分校）；University of Southern California（南加利福尼亚大学）
Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly slowed down by saddle points, taking exponential time to escape. On the other hand, gradient descent with perturbations [Ge et al., 2015, Jin et al., 2017] is not slowed down by saddle points - it can find an approximate local minimizer in polynomial time. This result implies that GD is inherently slower than perturbed GD, and justifies the importance of adding p…
In this paper, we propose a new adaptive stochastic gradient Langevin dynamics (ASGLD) algorithmic framework and its two specialized versions, namely adaptive stochastic gradient (ASG) and adaptive gradient Langevin dynamics(AGLD), for non-convex optimization problems. All proposed algorithms can escape from saddle points with at most $O(\log d)$ iterations, which is nearly dimension-free. Further, we show that ASGLD and ASG converge to a local minimum with at most $O(\log d/ε^4)$ iterations. Also, ASGLD with full gradients or ASGLD with a slowly linearly increasing batch size converge to a lo…
University of California, Berkeley（加州大学伯克利分校）；Duke University（杜克大学）；Microsoft Research, India（微软研究院（印度））；University of Washington（华盛顿大学）
This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the well-known convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free. Our results can be directly appli…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
The diffusion strategy for distributed learning from streaming data employs local stochastic gradient updates along with exchange of iterates over neighborhoods. In Part I [2] of this work we established that agents cluster around a network centroid and proceeded to study the dynamics of this point. We established expected descent in non-convex environments in the large-gradient regime and introduced a short-term model to examine the dynamics over finite-time horizons. Using this model, we establish in this work that the diffusion strategy is able to escape from strict saddle-points in O(1/$μ$…
University of Sydney（悉尼大学）；Discipline of Business Analytics（商业分析学科）
In this paper, we propose a variant of Riemannian stochastic recursive gradient method that can achieve second-order convergence guarantee and escape saddle points using simple perturbation. The idea is to perturb the iterates when gradient is small and carry out stochastic recursive gradient updates over tangent space. This avoids the complication of exploiting Riemannian geometry. We show that under finite-sum setting, our algorithm requires $\widetilde{\mathcal{O}}\big( \frac{ \sqrt{n}}{ε^2} + \frac{\sqrt{n} }{δ^4} + \frac{n}{δ^3}\big)$ stochastic gradient queries to find a $(ε, δ)$-second-…
Recently, the problem of local minima in very high dimensional non-convex optimization has been challenged and the problem of saddle points has been introduced. This paper introduces a dynamic type of normalization that forces the system to escape saddle points. Unlike other saddle point escaping algorithms, second order information is not utilized, and the system can be trained with an arbitrary gradient descent learner. The system drastically improves learning in a range of deep neural networks on various data-sets in comparison to non-CPN neural networks.
This work shows that applying Gradient Descent (GD) with a fixed step size to minimize a (possibly nonconvex) quadratic function is equivalent to running the Power Method (PM) on the gradients. The connection between GD with a fixed step size and the PM, both with and without fixed momentum, is thus established. Consequently, valuable eigen-information is available via GD. Recent examples show that GD with a fixed step size, applied to locally quadratic nonconvex functions, can take exponential time to escape saddle points (Simon S. Du, Chi Jin, Jason D. Lee, Michael I. Jordan, Aarti Singh, an…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Gradient descent is a popular algorithm in optimization, and its performance in convex settings is mostly well understood. In non-convex settings, it has been shown that gradient descent is able to escape saddle points asymptotically and converge to local minimizers [Lee et. al. 2016]. Recent studies also show a perturbed version of gradient descent is enough to escape saddle points efficiently [Jin et. al. 2015, Ge et. al. 2017]. In this paper we show a negative result: gradient descent may take exponential time to escape saddle points, with non-pathological two dimensional functions. While o…
School of Engineering, \' E cole Polytechnique F\' e d\' e rale de Lausanne
Under appropriate cooperation protocols and parameter choices, fully decentralized solutions for stochastic optimization have been shown to match the performance of centralized solutions and result in linear speedup (in the number of agents) relative to non-cooperative approaches in the strongly-convex setting. More recently, these results have been extended to the pursuit of first-order stationary points in non-convex environments. In this work, we examine in detail the dependence of second-order convergence guarantees on the spectral properties of the combination policy for non-convex multi …
Tsinghua University（清华大学）；Peking University（北京大学）；Massachusetts Institute of Technology（麻省理工学院）
Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $ε$-approximate second-order stationary point in $\tilde{O}(\log n/ε^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/ε^{2})$ or $\tilde{O}((\log n)^{6}/ε^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/ε$. For the stochastic setting, …
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research, India（微软印度研究院）
Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in $\tilde{O}(1/ε^{7/4})$ iterations, faster than the $\tilde{O}(1/ε^{2})$ iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second…
Stochastically controlled stochastic gradient (SCSG) methods have been proved to converge efficiently to first-order stationary points which, however, can be saddle points in nonconvex optimization. It has been observed that a stochastic gradient descent (SGD) step introduces anistropic noise around saddle points for deep learning and non-convex half space learning problems, which indicates that SGD satisfies the correlated negative curvature (CNC) condition for these problems. Therefore, we propose to use a separate SGD step to help the SCSG method escape from strict saddle points, resulting …
MIT EECS（麻省理工学院电子工程与计算机科学系）；Google Research, New York（谷歌研究院纽约分部）
Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first provide a novel view of adaptive methods as preconditioned SGD, where the preconditioner is estimated in an online manner. By studying the preconditioner on its own, we elucidate its purpose: it rescales the stochastic gradient noise to be isotropic near stationary points, which helps escape saddle points. Furthermore, we show that adaptive methods can efficiently…
We show that on the manifold of fixed-rank and symmetric positive semi-definite matrices, the Riemannian gradient descent algorithm almost surely escapes some spurious critical points on the boundary of the manifold. Our result is the first to partially overcome the incompleteness of the low-rank matrix manifold without changing the vanilla Riemannian gradient descent algorithm. The spurious critical points are some rank-deficient matrices that capture only part of the eigen components of the ground truth. Unlike classical strict saddle points, they exhibit very singular behavior. We show that…
University of California, Irvine（加州大学欧文分校）；Duke University（杜克大学）
Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing tec…
Gradient-related first-order methods have become the workhorse of large-scale numerical optimization problems. Many of these problems involve nonconvex objective functions with multiple saddle points, which necessitates an understanding of the behavior of discrete trajectories of first-order methods within the geometrical landscape of these functions. This paper concerns convergence of first-order discrete methods to a local minimum of nonconvex optimization problems that comprise strict-saddle points within the geometrical landscape. To this end, it focuses on analysis of discrete gradient tr…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
Recent years have seen increased interest in performance guarantees of gradient descent algorithms for non-convex optimization. A number of works have uncovered that gradient noise plays a critical role in the ability of gradient descent recursions to efficiently escape saddle-points and reach second-order stationary points. Most available works limit the gradient noise component to be bounded with probability one or sub-Gaussian and leverage concentration inequalities to arrive at high-probability results. We present an alternate approach, relying primarily on mean-square arguments and show t…
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
Recent years have seen a growing interest in understanding deep neural networks from an optimization perspective. It is understood now that converging to low-cost local minima is sufficient for such models to become effective in practice. However, in this work, we propose a new hypothesis based on recent theoretical findings and empirical studies that deep neural network models actually converge to saddle points with high degeneracy. Our findings from this work are new, and can have a significant impact on the development of gradient descent based methods for training deep networks. We validat…
Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints. First- and second-order necessary optimality conditions state that the Riemannian gradient must be zero, and the Riemannian Hessian must be positive semidefinite. Generalizing Jin et al.'s recent work on perturbed gradient descent (PGD) for optimization on linear spaces [How to Escape Saddle Points Efficiently (2017), Stochastic Gradient Descent Escapes Saddle Points Efficiently (2019)], we propose a version of perturbed Riemannian gradient d…
Motivated by the super-diffusivity of self-repelling random walk, which has roots in statistical physics, this paper develops a new perturbation mechanism for optimization algorithms. In this mechanism, perturbations are adapted to the history of states via the notion of occupation time. After integrating this mechanism into the framework of perturbed gradient descent (PGD) and perturbed accelerated gradient descent (PAGD), two new algorithms are proposed: perturbed gradient descent adapted to occupation time (PGDOT) and its accelerated version (PAGDOT). PGDOT and PAGDOT are shown to converge …
We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this observation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can su…
Stochastic gradient descent (SGD) with stochastic momentum is popular in nonconvex stochastic optimization and particularly for the training of deep neural networks. In standard SGD, parameters are updated by improving along the path of the gradient at the current iterate on a batch of examples, where the addition of a ``momentum'' term biases the update in the direction of the previous change in parameters. In non-stochastic convex optimization one can show that a momentum adjustment provably reduces convergence time in many settings, yet such results have been elusive in the stochastic and n…
Université de Montréal（蒙特利尔大学）；Stanford University（斯坦福大学）
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for the ability of these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, and neural network theory, that a deeper and more profound d…
Type-II Saddles and Probabilistic Stability of Stochastic Gradient Descent
Massachusetts Institute of Technology（麻省理工学院）；NTT Research（NTT研究院）；Université Paris Cité（巴黎西岱大学）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；The University of Tokyo（东京大学）
University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）
We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to escape saddle points on the manifold). The rate of convergence depends as $1/ε^2$ on the accuracy $ε$, which matches a rate known only for unconstrained smooth minimization. The convergence rate depends polylogarithmically on the manifold dimension $d$, hence is almost dimension-free. The rate also has a polynomial dependence on the parameters describing the cu…
Stochastic gradient descent (SGD) is widely used in machine learning. Although being commonly viewed as a fast but not accurate version of gradient descent (GD), it always finds better solutions than GD for modern neural networks. In order to understand this phenomenon, we take an alternative view that SGD is working on the convolved (thus smoothed) version of the loss function. We show that, even if the function $f$ has many bad local minima or saddle points, as long as for every point $x$, the weighted average of the gradients of its neighborhoods is one point convex with respect to the desi…
Low-rank inductive matrix completion (IMC) is currently widely used in IoT data completion, recommendation systems, and so on, as the side information in IMC has demonstrated great potential in reducing sample point remains a major obstacle for the convergence of the nonconvex solutions to IMC. What's more, carefully choosing the initial solution alone does not usually help remove the saddle points. To address this problem, we propose a stocastic variance reduction gradient-based algorithm called LRSVRG-IMC. LRSVRG-IMC can escape from the saddle points under various low-rank and sparse conditi…
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, the…
RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学研究中心）；The University of Tokyo（东京大学）
Stochastic gradient descent (SGD) undergoes complicated multiplicative noise for the mean-square loss. We use this property of SGD noise to derive a stochastic differential equation (SDE) with simpler additive noise by performing a random time change. Using this formalism, we show that the log loss barrier $Δ\log L=\log[L(θ^s)/L(θ^*)]$ between a local minimum $θ^*$ and a saddle $θ^s$ determines the escape rate of SGD from the local minimum, contrary to the previous results borrowing from physics that the linear loss barrier $ΔL=L(θ^s)-L(θ^*)$ decides the escape rate. Our escape-rate formula st…
In this paper, we give a sharp analysis for Stochastic Gradient Descent (SGD) and prove that SGD is able to efficiently escape from saddle points and find an $(ε, O(ε^{0.5}))$-approximate second-order stationary point in $\tilde{O}(ε^{-3.5})$ stochastic gradient computations for generic nonconvex optimization problems, when the objective function satisfies gradient-Lipschitz, Hessian-Lipschitz, and dispersive noise assumptions. This result subverts the classical belief that SGD requires at least $O(ε^{-4})$ stochastic gradient computations for obtaining an $(ε,O(ε^{0.5}))$-approximate second-o…
We consider the problem of empirical risk minimization given a database, using the gradient descent algorithm. We note that the function to be optimized may be non-convex, consisting of saddle points which impede the convergence of the algorithm. A perturbed gradient descent algorithm is typically employed to escape these saddle points. We show that this algorithm, that perturbs the gradient, inherently preserves the privacy of the data. We then employ the differential privacy framework to quantify the privacy hence achieved. We also analyze the change in privacy with varying parameters such a…
Université de Montréal（蒙特利尔大学）；Stanford University（斯坦福大学）
A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, neural network theory, and empirical evidence, that a deeper and more profo…
We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm.
Dimer-Enhanced Optimization: A First-Order Approach to Escaping Saddle Points in Neural Network Training
School of Bioengineering, Qilu University of Technology (Shandong Academy of Sciences)（青岛科技大学生物工程学院）；Shandong Provincial Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University（山东省生物物理重点实验室，生物物理研究所，德州市大学）；Shandong Provincial Hospital, Shandong First Medical University（山东省人民医院，山东第一医科大学）
Stability Analysis of Sharpness-Aware Minimization
Chung-Ang University, South Korea（Chung-Ang 大学，韩国）；Korea Institute for Advanced Study, South Korea（韩国高级研究院）；Ulsan National Institute of Science（乌山国家科学研究院）；Nanyang Technological University (NTU), Singapore（南洋理工大学（NTU），新加坡）；Seoul National University, South Korea（首尔国立大学，韩国）
Department of Statistics, University of California, Davis（加州大学戴维斯分校统计系）；Department of Management Sciences, University of Waterloo（滑铁卢大学管理科学系）；Department of Computer Science, University of California, Davis（加州大学戴维斯分校计算机科学系）；University of California, Davis（加州大学戴维斯分校）；University of Waterloo（滑铁卢大学）
In this paper, we show that under over-parametrization several standard stochastic optimization algorithms escape saddle-points and converge to local-minimizers much faster. One of the fundamental aspects of over-parametrized models is that they are capable of interpolating the training data. We show that, under interpolation-like assumptions satisfied by the stochastic gradients in an over-parametrization setting, the first-order oracle complexity of Perturbed Stochastic Gradient Descent (PSGD) algorithm to reach an $ε$-local-minimizer, matches the corresponding deterministic rate of $\tilde{…
Missouri University of Science and Technology（密苏里科技大学）；Princeton University（普林斯顿大学）；Duke University（杜克大学）
We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly)…
Gradient-based optimization methods are the most popular choice for finding local optima for classical minimization and saddle point problems. Here, we highlight a systemic issue of gradient dynamics that arise for saddle point problems, namely the presence of undesired stable stationary points that are no local optima. We propose a novel optimization approach that exploits curvature information in order to escape from these undesired stationary points. We prove that different optimization methods, including gradient method and Adagrad, equipped with curvature exploitation can escape non-optim…
Recent work has shown that stochastically perturbed gradient methods can efficiently escape strict saddle points of smooth functions. We extend this body of work to nonsmooth optimization, by analyzing an inexact analogue of a stochastically perturbed gradient method applied to the Moreau envelope. The main conclusion is that a variety of algorithms for nonsmooth optimization can escape strict saddle points of the Moreau envelope at a controlled rate. The main technical insight is that typical algorithms applied to the proximal subproblem yield directions that approximate the gradient of the M…
Real-world datasets exhibit imbalances of varying types and degrees. Several techniques based on re-weighting and margin adjustment of loss are often used to enhance the performance of neural networks, particularly on minority classes. In this work, we analyze the class-imbalanced learning problem by examining the loss landscape of neural networks trained with re-weighting and margin-based techniques. Specifically, we examine the spectral density of Hessian of class-wise loss, through which we observe that the network weights converge to a saddle point in the loss landscapes of minority classe…
Carnegie Mellon University（卡内基梅隆大学）；Massachusetts Institute for Technology（麻省理工学院）；Ecole Normale Superieure（巴黎高等师范学院）；Amazon Web Services（亚马逊云科技）
A central challenge to using first-order methods for optimizing nonconvex problems is the presence of saddle points. First-order methods often get stuck at saddle points, greatly deteriorating their performance. Typically, to escape from saddles one has to use second-order methods. However, most works on second-order methods rely extensively on expensive Hessian-based computations, making them impractical in large-scale settings. To tackle this challenge, we introduce a generic framework that minimizes Hessian based computations while at the same time provably converging to second-order critic…
Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms
Department of Electrical and Computer Engineering, University of California, Davis（加州大学戴维斯分校电气与计算机工程系）；Simon Business School, University of Rochester（罗切斯特大学西蒙商学院）
