The case for and against fixed step-size: Stochastic approximation algorithms in optimization and machine learning

• URL: http://arxiv.org/abs/2309.02944v3
• Date: Wed, 03 Sep 2025 02:47:11 GMT
• Title: The case for and against fixed step-size: Stochastic approximation algorithms in optimization and machine learning
• Authors: Caio Kalil Lauand, Ioannis Kontoyiannis, Sean Meyn,
• Abstract summary: Theory and application of approximation (SA) have become increasingly relevant due in part to applications in optimization and reinforcement learning.<n>This paper takes a new look at SA with constant step-size $alpha>0$, defined by the recursion, $$theta_n+1 = theta_n+ alpha f(theta_n,Phi_n+1)$$ in which $theta_ninmathbbRd$ and $Phi_n$ is a Markov chain.
• Score: 6.416429054645991
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Theory and application of stochastic approximation (SA) have become increasingly relevant due in part to applications in optimization and reinforcement learning. This paper takes a new look at SA with constant step-size $\alpha>0$, defined by the recursion, $$\theta_{n+1} = \theta_{n}+ \alpha f(\theta_n,\Phi_{n+1})$$ in which $\theta_n\in\mathbb{R}^d$ and $\{\Phi_{n}\}$ is a Markov chain. The goal is to approximately solve root finding problem $\bar{f}(\theta^*)=0$, where $\bar{f}(\theta)=\mathbb{E}[f(\theta,\Phi)]$ and $\Phi$ has the steady-state distribution of $\{\Phi_{n}\}$. The following conclusions are obtained under an ergodicity assumption on the Markov chain, compatible assumptions on $f$, and for $\alpha>0$ sufficiently small: $\textbf{1.}$ The pair process $\{(\theta_n,\Phi_n)\}$ is geometrically ergodic in a topological sense. $\textbf{2.}$ For every $1\le p\le 4$, there is a constant $b_p$ such that $\limsup_{n\to\infty}\mathbb{E}[\|\theta_n-\theta^*\|^p]\le b_p \alpha^{p/2}$ for each initial condition. $\textbf{3.}$ The Polyak-Ruppert-style averaged estimates $\theta^{\text{PR}}_n=n^{-1}\sum_{k=1}^{n}\theta_k$ converge to a limit $\theta^{\text{PR}}_\infty$ almost surely and in mean square, which satisfies $\theta^{\text{PR}}_\infty=\theta^*+\alpha \bar{\Upsilon}^*+O(\alpha^2)$ for an identified non-random $\bar{\Upsilon}^*\in\mathbb{R}^d$. Moreover, the covariance is approximately optimal: The limiting covariance matrix of $\theta{\text PR}_n$ is approximately minimal in a matricial sense. The two main take-aways for practitioners are application-dependent. It is argued that, in applications to optimization, constant gain algorithms may be preferable even when the objective has multiple local minima; while a vanishing gain algorithm is preferable in applications to reinforcement learning due to the presence of bias.

Related papers

• Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness [57.93371273485736]
  We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $ell$-smoothness condition $||nabla2f(x)|| le ellleft(||nabla f(x)||right),$ which generalizes the $L$-smoothness and $(L_0,L_1)$-smoothness.
  arXiv  Detail & Related papers  (2025-08-09T08:28:06Z)
• Guessing Efficiently for Constrained Subspace Approximation [49.83981776254246]
  We introduce a general framework for constrained subspace approximation.<n>We show it provides new algorithms for partition-constrained subspace approximation with applications to $k$-means clustering, and projected non-negative matrix factorization.
  arXiv  Detail & Related papers  (2025-04-29T15:56:48Z)
• Online Covariance Estimation in Nonsmooth Stochastic Approximation [14.818683408659764]
  We consider applying approximation (SA) methods to solve nonsmooth variational inclusion problems. Our convergence construction establish the best-known for statistical estimation methods.
  arXiv  Detail & Related papers  (2025-02-07T20:16:51Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Revisiting Step-Size Assumptions in Stochastic Approximation [1.3654846342364308]
  It is shown for the first time that this assumption is not required for convergence and finer results.<n>Rates of convergence are obtained for the standard algorithm and for estimates obtained via the averaging technique of Polyak and Ruppert.<n>Results from numerical experiments illustrate that $beta_theta$ may be large due to a combination of multiplicative noise and Markovian memory.
  arXiv  Detail & Related papers  (2024-05-28T05:11:05Z)
• Sample-Efficient Linear Regression with Self-Selection Bias [7.605563562103568]
  We consider the problem of linear regression with self-selection bias in the unknown-index setting. We provide a novel and near optimally sample-efficient (in terms of $k$) algorithm to recover $mathbfw_1,ldots,mathbfw_kin. Our algorithm succeeds under significantly relaxed noise assumptions, and therefore also succeeds in the related setting of max-linear regression.
  arXiv  Detail & Related papers  (2024-02-22T02:20:24Z)
• Convergence of Adam Under Relaxed Assumptions [72.24779199744954]
  We show that Adam converges to $epsilon$-stationary points with $O(epsilon-4)$ gradient complexity under far more realistic conditions. We also propose a variance-reduced version of Adam with an accelerated gradient complexity of $O(epsilon-3)$.
  arXiv  Detail & Related papers  (2023-04-27T06:27:37Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• On the Complexity of Decentralized Smooth Nonconvex Finite-Sum Optimization [21.334985032433778]
  Decentralized optimization problem $min_bf xinmathbb Rd f(bf x)triq frac1msum_i=1m f_i(bf x)triq frac1nsum_j=1n.
  arXiv  Detail & Related papers  (2022-10-25T11:37:11Z)
• Off-policy estimation of linear functionals: Non-asymptotic theory for semi-parametric efficiency [59.48096489854697]
  The problem of estimating a linear functional based on observational data is canonical in both the causal inference and bandit literatures. We prove non-asymptotic upper bounds on the mean-squared error of such procedures. We establish its instance-dependent optimality in finite samples via matching non-asymptotic local minimax lower bounds.
  arXiv  Detail & Related papers  (2022-09-26T23:50:55Z)
• Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization [50.83356836818667]
  gradient Langevin Dynamics is one of the most fundamental algorithms to solve non-eps optimization problems. In this paper, we show two variants of this kind, namely the Variance Reduced Langevin Dynamics and the Recursive Gradient Langevin Dynamics.
  arXiv  Detail & Related papers  (2022-03-30T11:39:00Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• Computational Complexity of Normalizing Constants for the Product of Determinantal Point Processes [12.640283469603357]
  We study the computational complexity of computing the normalizing constant. We show that $sum_Sdet(bf A_S,S)p$ exactly for every (fixed) positive even integer $p$ is UP-hard and Mod$_3$P-hard.
  arXiv  Detail & Related papers  (2021-11-28T14:08:25Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Binary Classification of Gaussian Mixtures: Abundance of Support Vectors, Benign Overfitting and Regularization [39.35822033674126]
  We study binary linear classification under a generative Gaussian mixture model. We derive novel non-asymptotic bounds on the classification error of the latter. Our results extend to a noisy model with constant probability noise flips.
  arXiv  Detail & Related papers  (2020-11-18T07:59:55Z)
• Sample Complexity Bounds for Two Timescale Value-based Reinforcement Learning Algorithms [65.09383385484007]
  Two timescale approximation (SA) has been widely used in value-based reinforcement learning algorithms. We study the non-asymptotic convergence rate of two timescale linear and nonlinear TDC and Greedy-GQ algorithms.
  arXiv  Detail & Related papers  (2020-11-10T11:36:30Z)
• 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)
• Convergence Analysis of Riemannian Stochastic Approximation Schemes [39.32179384256228]
  This paper analyzes a class of correlated approximation (SA) schemes to tackle optimization problems. We show that the conditions we derive are considerably milder than previous works. Third, we consider the case where the mean-field function can only be estimated up to a small bias, and/or the case in which the samples are drawn from a chain.
  arXiv  Detail & Related papers  (2020-05-27T11:24:58Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)
• Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis [102.29671176698373]
  We address the problem of policy evaluation in discounted decision processes, and provide Markov-dependent guarantees on the $ell_infty$error under a generative model. We establish both and non-asymptotic versions of local minimax lower bounds for policy evaluation, thereby providing an instance-dependent baseline by which to compare algorithms.
  arXiv  Detail & Related papers  (2020-03-16T17:15:28Z)
• Agnostic Q-learning with Function Approximation in Deterministic Systems: Tight Bounds on Approximation Error and Sample Complexity [94.37110094442136]
  We study the problem of agnostic $Q$-learning with function approximation in deterministic systems. We show that if $delta = Oleft(rho/sqrtdim_Eright)$, then one can find the optimal policy using $Oleft(dim_Eright)$.
  arXiv  Detail & Related papers  (2020-02-17T18:41:49Z)
• A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-$\ell_1$-Norm Interpolated Classifiers [3.167685495996986]
  This paper establishes a precise high-dimensional theory for boosting on separable data. Under a class of statistical models, we provide an exact analysis of the universality error of boosting. We also explicitly pin down the relation between the boosting test error and the optimal Bayes error.
  arXiv  Detail & Related papers  (2020-02-05T00:24:53Z)
• On Low-rank Trace Regression under General Sampling Distribution [9.699586426043885]
  We show that cross-validated estimators satisfy near-optimal error bounds on general assumptions. We also show that the cross-validated estimator outperforms the theory-inspired approach of selecting the parameter.
  arXiv  Detail & Related papers  (2019-04-18T02:56:00Z)

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.