Double Momentum Method for Lower-Level Constrained Bilevel Optimization

TL;DR

This paper introduces a novel single-loop double-momentum hypergradient method for lower-level constrained bilevel optimization, removing restrictive assumptions and providing convergence guarantees, with demonstrated effectiveness in experiments.

Contribution

It proposes a new hypergradient leveraging nonsmooth implicit function theory and a single-loop algorithm with convergence analysis for LCBO.

Findings

Achieves a $(\delta,\epsilon)$-stationary point in $ ilde{O}(d_2^2\epsilon^{-4})$ iterations.

Removes restrictive differentiability and invertibility assumptions.

Demonstrates effectiveness through experiments on two applications.

Abstract

Bilevel optimization (BO) has recently gained prominence in many machine learning applications due to its ability to capture the nested structure inherent in these problems. Recently, many hypergradient methods have been proposed as effective solutions for solving large-scale problems. However, current hypergradient methods for the lower-level constrained bilevel optimization (LCBO) problems need very restrictive assumptions, namely, where optimality conditions satisfy the differentiability and invertibility conditions and lack a solid analysis of the convergence rate. What's worse, existing methods require either double-loop updates, which are sometimes less efficient. To solve this problem, in this paper, we propose a new hypergradient of LCBO leveraging the theory of nonsmooth implicit function theorem instead of using the restrive assumptions. In addition, we propose a \textit{single-loop single-timescale} algorithm based on the double-momentum method and adaptive step size method and prove it can return a $(\delta, \epsilon)$-stationary point with $\tilde{\mathcal{O}}(d_2^2\epsilon^{-4})$ iterations. Experiments on two applications demonstrate the effectiveness of our proposed method.