Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping

• URL: http://arxiv.org/abs/2412.19529v1
• Date: Fri, 27 Dec 2024 08:46:46 GMT
• Title: Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping
• Authors: Zijian Liu, Zhengyuan Zhou,
• Abstract summary: We provide the first convergence under heavy-tailed noises but without clipping. We also establish first $mathcalO(Tfrac1-mathfrakp3mathfrakp-2)$ convergence rate in the case where the tail index $mathfrakp$ is unknown in advance.
• Score: 21.865728815935665
• License:
• Abstract: Recently, the study of heavy-tailed noises in first-order nonconvex stochastic optimization has gotten a lot of attention since it was recognized as a more realistic condition as suggested by many empirical observations. Specifically, the stochastic noise (the difference between the stochastic and true gradient) is considered only to have a finite $\mathfrak{p}$-th moment where $\mathfrak{p}\in\left(1,2\right]$ instead of assuming it always satisfies the classical finite variance assumption. To deal with this more challenging setting, people have proposed different algorithms and proved them to converge at an optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate for smooth objectives after $T$ iterations. Notably, all these new-designed algorithms are based on the same technique - gradient clipping. Naturally, one may want to know whether the clipping method is a necessary ingredient and the only way to guarantee convergence under heavy-tailed noises. In this work, by revisiting the existing Batched Normalized Stochastic Gradient Descent with Momentum (Batched NSGDM) algorithm, we provide the first convergence result under heavy-tailed noises but without gradient clipping. Concretely, we prove that Batched NSGDM can achieve the optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate even under the relaxed smooth condition. More interestingly, we also establish the first $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{2\mathfrak{p}}})$ convergence rate in the case where the tail index $\mathfrak{p}$ is unknown in advance, which is arguably the common scenario in practice.

Related papers

• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• Gradient-free optimization of highly smooth functions: improved analysis and a new algorithm [87.22224691317766]
  This work studies problems with zero-order noisy oracle information under the assumption that the objective function is highly smooth. We consider two kinds of zero-order projected gradient descent algorithms.
  arXiv  Detail & Related papers  (2023-06-03T17:05:13Z)
• Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises: High-Probability Bound, In-Expectation Rate and Initial Distance Adaptation [22.758674468435302]
  In a heavy-tailed noise regime, the difference between the gradient and the true rate is assumed to have a finite $p-th moment. This paper provides a comprehensive analysis of nonsmooth convex optimization with heavy-tailed noises.
  arXiv  Detail & Related papers  (2023-03-22T03:05:28Z)
• High Probability Convergence of Stochastic Gradient Methods [15.829413808059124]
  We show convergence with bounds depending on the initial distance to the optimal solution. We demonstrate that our techniques can be used to obtain high bound for AdaGrad-Norm.
  arXiv  Detail & Related papers  (2023-02-28T18:42:11Z)
• Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise [28.780192812703948]
  We consider the optimization problem with smooth but not necessarily convex objectives in the heavy-tailed noise regime. We show that one can achieve a faster rate than that dictated by the lower bound $Omega(Tfrac1-p3p-2)$ with only tiny bit of structure. We also establish that it guarantees a high-probability convergence rate of $O(log(T/delta)Tfrac1-p3p-2)$ under a mild condition.
  arXiv  Detail & Related papers  (2023-02-14T00:23:42Z)
• Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD [38.221784575853796]
  This work considers the problem of finding first-order stationary point of a non atau function with potentially constant smoothness using a gradient. We develop a technique that allows us to prove $mathcalO(fracmathrmpolylog(T)sigmatT)$ convergence rates without assuming uniform bounds on the noise.
  arXiv  Detail & Related papers  (2023-02-13T18:13:36Z)
• A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization [53.044526424637866]
  In this paper we consider finding an approximate second-order stationary point (SOSP) that minimizes a twice different subject general non conic optimization. In particular, we propose a Newton-CG based-augmentedconjugate method for finding an approximate SOSP.
  arXiv  Detail & Related papers  (2023-01-10T20:43:29Z)
• ReSQueing Parallel and Private Stochastic Convex Optimization [59.53297063174519]
  We introduce a new tool for BFG convex optimization (SCO): a Reweighted Query (ReSQue) estimator for the gradient of a function convolved with a (Gaussian) probability density. We develop algorithms achieving state-of-the-art complexities for SCO in parallel and private settings.
  arXiv  Detail & Related papers  (2023-01-01T18:51:29Z)
• Optimal Extragradient-Based Bilinearly-Coupled Saddle-Point Optimization [116.89941263390769]
  We consider the smooth convex-concave bilinearly-coupled saddle-point problem, $min_mathbfxmax_mathbfyF(mathbfx) + H(mathbfx,mathbfy)$, where one has access to first-order oracles for $F$, $G$ as well as the bilinear coupling function $H$. We present a emphaccelerated gradient-extragradient (AG-EG) descent-ascent algorithm that combines extragrad
  arXiv  Detail & Related papers  (2022-06-17T06:10:20Z)
• A first-order primal-dual method with adaptivity to local smoothness [64.62056765216386]
  We consider the problem of finding a saddle point for the convex-concave objective $min_x max_y f(x) + langle Ax, yrangle - g*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vu algorithm, which alternates between primal gradient steps and dual steps.
  arXiv  Detail & Related papers  (2021-10-28T14:19:30Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.