Multi-block Min-max Bilevel Optimization with Applications in Multi-task Deep AUC Maximization

TL;DR

This paper introduces a single-loop stochastic algorithm for multi-block min-max bilevel optimization problems, reducing computational costs and achieving optimal sample complexity, with applications in multi-task deep AUC maximization.

Contribution

The paper proposes a novel single-loop randomized stochastic algorithm for multi-block min-max bilevel problems, with proven complexity and practical applications in deep AUC maximization.

Findings

Algorithm requires only a constant number of block updates per iteration.

Achieves optimal sample complexity of O(1/ε^4) for ε-stationary points.

Experimental results confirm theoretical guarantees and effectiveness on multi-task problems.

Abstract

In this paper, we study multi-block min-max bilevel optimization problems, where the upper level is non-convex strongly-concave minimax objective and the lower level is a strongly convex objective, and there are multiple blocks of dual variables and lower level problems. Due to the intertwined multi-block min-max bilevel structure, the computational cost at each iteration could be prohibitively high, especially with a large number of blocks. To tackle this challenge, we present a single-loop randomized stochastic algorithm, which requires updates for only a constant number of blocks at each iteration. Under some mild assumptions on the problem, we establish its sample complexity of $O(1/\epsilon^4)$ for finding an $\epsilon$-stationary point. This matches the optimal complexity for solving stochastic nonconvex optimization under a general unbiased stochastic oracle model. Moreover, we provide two applications of the proposed method in multi-task deep AUC (area under ROC curve) maximization and multi-task deep partial AUC maximization. Experimental results validate our theory and demonstrate the effectiveness of our method on problems with hundreds of tasks.