Value-Function-based Sequential Minimization for Bi-level Optimization

TL;DR

This paper introduces BVFSM, a novel gradient-based algorithm that effectively solves various challenging bi-level optimization problems, including those with constraints and pessimistic formulations, without restrictive assumptions.

Contribution

The paper proposes BVFSM, a new value-function-based sequential minimization method that handles high-dimensional, constrained, and pessimistic BLO with proven convergence, surpassing existing approaches.

Findings

BVFSM converges asymptotically on diverse BLO types.

It avoids repeated gradient and Hessian inverse calculations.

Experiments show superior performance on real-world tasks.

Abstract

Gradient-based Bi-Level Optimization (BLO) methods have been widely applied to handle modern learning tasks. However, most existing strategies are theoretically designed based on restrictive assumptions (e.g., convexity of the lower-level sub-problem), and computationally not applicable for high-dimensional tasks. Moreover, there are almost no gradient-based methods able to solve BLO in those challenging scenarios, such as BLO with functional constraints and pessimistic BLO. In this work, by reformulating BLO into approximated single-level problems, we provide a new algorithm, named Bi-level Value-Function-based Sequential Minimization (BVFSM), to address the above issues. Specifically, BVFSM constructs a series of value-function-based approximations, and thus avoids repeated calculations of recurrent gradient and Hessian inverse required by existing approaches, time-consuming especially for high-dimensional tasks. We also extend BVFSM to address BLO with additional functional constraints. More importantly, BVFSM can be used for the challenging pessimistic BLO, which has never been properly solved before. In theory, we prove the asymptotic convergence of BVFSM on these types of BLO, in which the restrictive lower-level convexity assumption is discarded. To our best knowledge, this is the first gradient-based algorithm that can solve different kinds of BLO (e.g., optimistic, pessimistic, and with constraints) with solid convergence guarantees. Extensive experiments verify the theoretical investigations and demonstrate our superiority on various real-world applications.