High Probability Convergence of Stochastic Gradient Methods

• URL: http://arxiv.org/abs/2302.14843v1
• Date: Tue, 28 Feb 2023 18:42:11 GMT
• Title: High Probability Convergence of Stochastic Gradient Methods
• Authors: Zijian Liu, Ta Duy Nguyen, Thien Hang Nguyen, Alina Ene, Huy L\^e Nguyen
• Abstract summary: 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.
• Score: 15.829413808059124
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. This method can be applied to the non-convex case. We demonstrate an $O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$ convergence rate when the number of iterations $T$ is known and an $O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$ convergence rate when $T$ is unknown for SGD, where $1-\delta$ is the desired success probability. These bounds improve over existing bounds in the literature. Additionally, we demonstrate that our techniques can be used to obtain high probability bound for AdaGrad-Norm (Ward et al., 2019) that removes the bounded gradients assumption from previous works. Furthermore, our technique for AdaGrad-Norm extends to the standard per-coordinate AdaGrad algorithm (Duchi et al., 2011), providing the first noise-adapted high probability convergence for AdaGrad.

Related papers

• Revisiting the Last-Iterate Convergence of Stochastic Gradient Methods [25.831462008050387]
  The Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. It still remains unclear whether the lastiterate convergence can be provably extended to wider composite optimization and non-Euclidean norms.
  arXiv  Detail & Related papers  (2023-12-13T21:41:06Z)
• 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)
• Convergence of a Normal Map-based Prox-SGD Method under the KL Inequality [0.0]
  We present a novel map-based algorithm ($mathsfnorMtext-mathsfSGD$) for $symbol$k$ convergence problems.
  arXiv  Detail & Related papers  (2023-05-10T01:12:11Z)
• 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)
• 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)
• High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize [55.0090961425708]
  We propose a new, simplified high probability analysis of AdaGrad for smooth, non- probability problems. We present our analysis in a modular way and obtain a complementary $mathcal O (1 / TT)$ convergence rate in the deterministic setting. To the best of our knowledge, this is the first high probability for AdaGrad with a truly adaptive scheme, i.e., completely oblivious to the knowledge of smoothness.
  arXiv  Detail & Related papers  (2022-04-06T13:50:33Z)
• 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)
• Accelerating Optimization and Reinforcement Learning with Quasi-Stochastic Approximation [2.294014185517203]
  This paper sets out to extend convergence theory to quasi-stochastic approximations. It is illustrated with applications to gradient-free optimization and policy gradient algorithms for reinforcement learning.
  arXiv  Detail & Related papers  (2020-09-30T04:44:45Z)
• On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems [75.58134963501094]
  This paper analyzes the trajectories of gradient descent (SGD) We show that SGD avoids saddle points/manifolds with $1$ for strict step-size policies.
  arXiv  Detail & Related papers  (2020-06-19T14:11:26Z)
• A Simple Convergence Proof of Adam and Adagrad [74.24716715922759]
  We show a proof of convergence between the Adam Adagrad and $O(d(N)/st)$ algorithms. Adam converges with the same convergence $O(d(N)/st)$ when used with the default parameters.
  arXiv  Detail & Related papers  (2020-03-05T01:56:17Z)

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.