Variance Reduction on General Adaptive Stochastic Mirror Descent
This work provides an incremental improvement in optimization algorithms for researchers and practitioners working with nonsmooth nonconvex finite-sum problems, particularly those using adaptive methods.
This paper proposes SVRAMD, a generalized framework for variance-reduced adaptive mirror descent algorithms in nonsmooth nonconvex finite-sum optimization. It demonstrates that variance reduction reduces the SFO complexity of adaptive mirror descent algorithms, accelerating their convergence and recovering the best existing rates for non-adaptive variance-reduced mirror descent.
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.