Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problem

TL;DR

This paper introduces new stochastic bilevel optimization algorithms that improve query complexity bounds for convex and non-convex cases by using local approximation and variance reduction techniques.

Contribution

The authors propose novel projection-free stochastic bilevel methods with improved complexity guarantees for convex and non-convex problems, utilizing stochastic cutting planes and variance reduction.

Findings

Achieves $ ilde{O}(rac{1}{ ext{epsilon}^2})$ complexity for convex upper-level problems.

Attains $ ilde{O}(rac{1}{ ext{epsilon}^3})$ complexity for non-convex upper-level problems.

In finite-sum setting, reduces stochastic oracle calls to $ ilde{O}(rac{ ext{sqrt}(n)}{ ext{epsilon}})$ and $ ilde{O}(rac{ ext{sqrt}(n)}{ ext{epsilon}^2})$.

Abstract

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires $\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{2},1/\epsilon_g^{2}\}) $ stochastic oracle queries to obtain a solution that is $\epsilon_f$-optimal for the upper-level and $\epsilon_g$-optimal for the lower-level. This guarantee improves the previous best-known complexity of $\mathcal{O}(\max\{1/\epsilon_f^{4},1/\epsilon_g^{4}\})$. Moreover, for the case that the upper-level function is non-convex, our method requires at most $\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{3},1/\epsilon_g^{3}\}) $ stochastic oracle queries to find an $(\epsilon_f, \epsilon_g)$-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon)$ and $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon^{2})$ for the convex and non-convex settings, respectively, where $\epsilon=\min \{\epsilon_f,\epsilon_g\}$.