Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles

TL;DR

This paper proposes a fully first-order method for bilevel optimization with strongly convex lower-level problems, achieving near-optimal convergence rates without Hessian-vector products, and extends to stochastic and accelerated settings.

Contribution

It introduces a two-time-scale update method that improves first-order oracle complexity to near-optimal levels for bilevel problems, including stochastic and accelerated variants.

Findings

Achieves $ ilde{ ext{O}}( ext{epsilon}^{-2})$ oracle complexity for deterministic case.

Attains $ ilde{ ext{O}}( ext{epsilon}^{-4})$ and $ ilde{ ext{O}}( ext{epsilon}^{-6})$ complexities in stochastic settings.

Provides accelerated rates of $ ilde{ ext{O}}( ext{epsilon}^{-1.75})$ with higher-order smoothness and noise injection.

Abstract

In this work, we consider bilevel optimization when the lower-level problem is strongly convex. Recent works show that with a Hessian-vector product (HVP) oracle, one can provably find an $\epsilon$-stationary point within ${\mathcal{O}}(\epsilon^{-2})$ oracle calls. However, the HVP oracle may be inaccessible or expensive in practice. Kwon et al. (ICML 2023) addressed this issue by proposing a first-order method that can achieve the same goal at a slower rate of $\tilde{\mathcal{O}}(\epsilon^{-3})$. In this paper, we incorporate a two-time-scale update to improve their method to achieve the near-optimal $\tilde {\mathcal{O}}(\epsilon^{-2})$ first-order oracle complexity. Our analysis is highly extensible. In the stochastic setting, our algorithm can achieve the stochastic first-order oracle complexity of $\tilde {\mathcal{O}}(\epsilon^{-4})$ and $\tilde {\mathcal{O}}(\epsilon^{-6})$ when the stochastic noises are only in the upper-level objective and in both level objectives, respectively. When the objectives have higher-order smoothness conditions, our deterministic method can escape saddle points by injecting noise, and can be accelerated to achieve a faster rate of $\tilde {\mathcal{O}}(\epsilon^{-1.75})$ using Nesterov's momentum.