# On variance reduction for stochastic smooth convex optimization with multiplicative noise

## TL;DR

This paper introduces dynamic sampled stochastic approximation methods for convex optimization under multiplicative noise, achieving optimal complexity bounds and improving upon previous methods that assumed bounded variance.

## Contribution

It develops new stochastic approximation algorithms that handle multiplicative noise without assuming bounded variance, providing optimal iteration and oracle complexity results.

## Key findings

- Achieves optimal iteration complexity of O(ε^{-1/2}) for smooth convex functions.
- Attains optimal oracle complexity of O(ε^{-2}) with logarithmic factors.
- Provides sharper variance bounds that depend on local variances at solutions.

## Abstract

We propose dynamic sampled stochastic approximation (SA) methods for stochastic optimization with a heavy-tailed distribution (with finite 2nd moment). The objective is the sum of a smooth convex function with a convex regularizer. Typically, it is assumed an oracle with an upper bound $\sigma^2$ on its variance (OUBV). Differently, we assume an oracle with \emph{multiplicative noise}. This rarely addressed setup is more aggressive but realistic, where the variance may not be bounded. Our methods achieve optimal iteration complexity and (near) optimal oracle complexity. For the smooth convex class, we use an accelerated SA method a la FISTA which achieves, given tolerance $\epsilon>0$, the optimal iteration complexity of $\mathcal{O}(\epsilon^{-\frac{1}{2}})$ with a near-optimal oracle complexity of $\mathcal{O}(\epsilon^{-2})[\ln(\epsilon^{-\frac{1}{2}})]^2$. This improves upon Ghadimi and Lan [\emph{Math. Program.}, 156:59-99, 2016] where it is assumed an OUBV. For the strongly convex class, our method achieves optimal iteration complexity of $\mathcal{O}(\ln(\epsilon^{-1}))$ and optimal oracle complexity of $\mathcal{O}(\epsilon^{-1})$. This improves upon Byrd et al. [\emph{Math. Program.}, 134:127-155, 2012] where it is assumed an OUBV. In terms of variance, our bounds are local: they depend on variances $\sigma(x^*)^2$ at solutions $x^*$ and the per unit distance multiplicative variance $\sigma^2_L$. For the smooth convex class, there exist policies such that our bounds resemble those obtained if it was assumed an OUBV with $\sigma^2:=\sigma(x^*)^2$. For the strongly convex class such property is obtained exactly if the condition number is estimated or in the limit for better conditioned problems or for larger initial batch sizes. In any case, if it is assumed an OUBV, our bounds are thus much sharper since typically $\max\{\sigma(x^*)^2,\sigma_L^2\}\ll\sigma^2$.