Double Variance Reduction: A Smoothing Trick for Composite Optimization Problems without First-Order Gradient

• URL: http://arxiv.org/abs/2405.17761v1
• Date: Tue, 28 May 2024 02:27:53 GMT
• Title: Double Variance Reduction: A Smoothing Trick for Composite Optimization Problems without First-Order Gradient
• Authors: Hao Di, Haishan Ye, Yueling Zhang, Xiangyu Chang, Guang Dai, Ivor W. Tsang,
• Abstract summary: Variance reduction techniques are designed to decrease the sampling variance, thereby accelerating convergence rates of first-order (FO) and zeroth-order (ZO) optimization methods. In composite optimization problems, ZO methods encounter an additional variance called the coordinate-wise variance, which stems from the random estimation. This paper proposes the Zeroth-order Proximal Double Variance Reduction (ZPDVR) method, which utilizes the averaging trick to reduce both sampling and coordinate-wise variances.
• Score: 40.22217106270146
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Variance reduction techniques are designed to decrease the sampling variance, thereby accelerating convergence rates of first-order (FO) and zeroth-order (ZO) optimization methods. However, in composite optimization problems, ZO methods encounter an additional variance called the coordinate-wise variance, which stems from the random gradient estimation. To reduce this variance, prior works require estimating all partial derivatives, essentially approximating FO information. This approach demands O(d) function evaluations (d is the dimension size), which incurs substantial computational costs and is prohibitive in high-dimensional scenarios. This paper proposes the Zeroth-order Proximal Double Variance Reduction (ZPDVR) method, which utilizes the averaging trick to reduce both sampling and coordinate-wise variances. Compared to prior methods, ZPDVR relies solely on random gradient estimates, calls the stochastic zeroth-order oracle (SZO) in expectation $\mathcal{O}(1)$ times per iteration, and achieves the optimal $\mathcal{O}(d(n + \kappa)\log (\frac{1}{\epsilon}))$ SZO query complexity in the strongly convex and smooth setting, where $\kappa$ represents the condition number and $\epsilon$ is the desired accuracy. Empirical results validate ZPDVR's linear convergence and demonstrate its superior performance over other related methods.

Related papers

• Obtaining Lower Query Complexities through Lightweight Zeroth-Order Proximal Gradient Algorithms [65.42376001308064]
  We propose two variance reduced ZO estimators for complex gradient problems. We improve the state-of-the-art function complexities from $mathcalOleft(minfracdn1/2epsilon2, fracdepsilon3right)$ to $tildecalOleft(fracdepsilon2right)$.
  arXiv  Detail & Related papers  (2024-10-03T15:04:01Z)
• Adaptive Variance Reduction for Stochastic Optimization under Weaker Assumptions [26.543628010637036]
  We introduce a novel adaptive reduction method that achieves an optimal convergence rate of $mathcalO(log T)$ for non- functions. We also extend the proposed technique to obtain the same optimal rate of $mathcalO(log T)$ for compositional optimization.
  arXiv  Detail & Related papers  (2024-06-04T04:39:51Z)
• Efficiently Escaping Saddle Points for Non-Convex Policy Optimization [40.0986936439803]
  Policy gradient (PG) is widely used in reinforcement learning due to its scalability and good performance. We propose a variance-reduced second-order method that uses second-order information in the form of Hessian vector products (HVP) and converges to an approximate second-order stationary point (SOSP) with sample complexity of $tildeO(epsilon-3)$.
  arXiv  Detail & Related papers  (2023-11-15T12:36:45Z)
• Stochastic Optimization for Non-convex Problem with Inexact Hessian Matrix, Gradient, and Function [99.31457740916815]
  Trust-region (TR) and adaptive regularization using cubics have proven to have some very appealing theoretical properties. We show that TR and ARC methods can simultaneously provide inexact computations of the Hessian, gradient, and function values.
  arXiv  Detail & Related papers  (2023-10-18T10:29:58Z)
• Adaptive SGD with Polyak stepsize and Line-search: Robust Convergence and Variance Reduction [26.9632099249085]
  We propose two new variants of SPS and SLS, called AdaSPS and AdaSLS, which guarantee convergence in non-interpolation settings. We equip AdaSPS and AdaSLS with a novel variance reduction technique and obtain algorithms that require $smashwidetildemathcalO(n+1/epsilon)$ gradient evaluations.
  arXiv  Detail & Related papers  (2023-08-11T10:17:29Z)
• On Stochastic Moving-Average Estimators for Non-Convex Optimization [105.22760323075008]
  In this paper, we demonstrate the power of a widely used estimator based on moving average (SEMA) problems. For all these-the-art results, we also present the results for all these-the-art problems.
  arXiv  Detail & Related papers  (2021-04-30T08:50:24Z)
• A Variance Controlled Stochastic Method with Biased Estimation for Faster Non-convex Optimization [0.0]
  We propose a new technique, em variance controlled gradient (VCSG), to improve the performance of the reduced gradient (SVRG) $lambda$ is introduced in VCSG to avoid over-reducing a variance by SVRG. $mathcalO(min1/epsilon3/2,n1/4/epsilon)$ number of gradient evaluations, which improves the leading gradient complexity.
  arXiv  Detail & Related papers  (2021-02-19T12:22:56Z)
• Private Stochastic Non-Convex Optimization: Adaptive Algorithms and Tighter Generalization Bounds [72.63031036770425]
  We propose differentially private (DP) algorithms for bound non-dimensional optimization. We demonstrate two popular deep learning methods on the empirical advantages over standard gradient methods.
  arXiv  Detail & Related papers  (2020-06-24T06:01:24Z)
• Gradient Free Minimax Optimization: Variance Reduction and Faster Convergence [120.9336529957224]
  In this paper, we denote the non-strongly setting on the magnitude of a gradient-free minimax optimization problem. We show that a novel zeroth-order variance reduced descent algorithm achieves the best known query complexity.
  arXiv  Detail & Related papers  (2020-06-16T17:55:46Z)

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.