Variance Reduction on General Adaptive Stochastic Mirror Descent

Abstract

In this work, we investigate the idea of variance reduction by studying its properties with general adaptive mirror descent algorithms in nonsmooth nonconvex finite-sum optimization problems. We propose a simple yet generalized framework for variance reduced adaptive mirror descent algorithms named SVRAMD and provide its convergence analysis in both the nonsmooth nonconvex problem and the P-L conditioned problem. We prove that variance reduction reduces the SFO complexity of adaptive mirror descent algorithms and thus accelerates their convergence. In particular, our general theory implies that variance reduction can be applied to algorithms using time-varying step sizes and self-adaptive algorithms such as AdaGrad and RMSProp. Moreover, the convergence rates of SVRAMD recover the best existing rates of non-adaptive variance reduced mirror descent algorithms without complicated algorithmic components. Extensive experiments in deep learning validate our theoretical findings.

arXiv commentNeurIPS 2020 OPT workshop

References in corpus (15)

• Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour
• SGDR: Stochastic Gradient Descent with Warm Restarts
• The Marginal Value of Adaptive Gradient Methods in Machine Learning
• SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator
• Adaptive Gradient Methods with Dynamic Bound of Learning Rate
• AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients
• On the Convergence of Adaptive Gradient Methods for Nonconvex Optimization
• A Simple Proximal Stochastic Gradient Method for Nonsmooth Nonconvex Optimization
• Nostalgic Adam: Weighting more of the past gradients when designing the adaptive learning rate
• Natasha 2: Faster Non-Convex Optimization Than SGD
• ProxSARAH: An Efficient Algorithmic Framework for Stochastic Composite Nonconvex Optimization
• Natasha: Faster Non-Convex Stochastic Optimization Via Strongly Non-Convex Parameter
• On the Adaptivity of Stochastic Gradient-Based Optimization
• PAGE: A Simple and Optimal Probabilistic Gradient Estimator for Nonconvex Optimization
• Stabilized SVRG: Simple Variance Reduction for Nonconvex Optimization

Cited by in corpus (1)

• Convergence of Adam for Non-convex Objectives: Relaxed Hyperparameters and Non-ergodic Case