A Discretization Approach for Bilevel Optimization with Low-Dimensional and Non-Convex Lower-Level

TL;DR

This work targets optimistic bilevel optimization with a non-convex, constrained lower level by introducing a discretized value-function reformulation. By sampling LL feasible points and constructing a differentiable approximate value function $ ilde{V}_{oldsymbol{\lambda}}(oldsymbol{x})$, the authors derive a discretized reformulation and a penalty variant, both connected via rigorous equivalence results. They propose the DIVIDE-BLO algorithm, which uses projected gradient steps and a simplex-based computation of $ ilde{V}_{oldsymbol{\lambda}}(oldsymbol{x})$, with a provable convergence rate of $O(1/T)$ to stationary points and KKT-consistency with the discretized and penalty reformulations. Experiments on synthetic nonlinear LL problems and ensemble-learning tasks demonstrate robust LL feasibility and superior weighted-model performance, illustrating the practical impact for low-dimensional LL BLO scenarios.

Abstract

Bilevel optimization (BLO) problem, where two optimization problems (referred to as upper- and lower-level problems) are coupled hierarchically, has wide applications in areas such as machine learning and operations research. Recently, many first-order algorithms have been developed for solving bilevel problems with strongly convex and/or unconstrained lower-level problems; this special structure of the lower-level problem is needed to ensure the tractability of gradient computation (among other reasons). In this work, we deal with a class of more challenging BLO problems where the lower-level problem is non-convex and constrained. We propose a novel approach that approximates the value function of the lower-level problem by first sampling a set of feasible solutions and then constructing an equivalent convex optimization problem. This convexified value function is then used to construct a penalty function for the original BLO problem. We analyze the properties of the original BLO problem and the newly constructed penalized problem by characterizing the relation between their KKT points, as well as the local and global minima of the two problems. We then develop a gradient descent-based algorithm to solve the reformulated problem, and establish its finite-time convergence guarantees. Finally, we conduct numerical experiments to corroborate the theoretical performance of the proposed algorithm.

Figures (4)

• Figure 1: Equivalence between different problems.
• Figure 2: This figure depicts the constraint set $\mathcal{Y}$ and its discretization; the sample points are the grid intersections. The red region in the top-left corner and the blue region in the bottom-right corner are the feasible regions within $\mathcal{Y}$, i.e., subsets of $\left\{ \mathbf{y} \in \mathcal{Y}: g(\mathbf{x}, \mathbf{y}) - V(\mathbf{x}) \leq \epsilon \right\}.$ The feasible region in the bottom-right corner is captured by the sample points (i.e., the highlighted vertices of the grid), but the feasible region in the top-left corner is too small and is not captured by any sample point.
• Figure 3: This figure depicts the constraint set $\mathcal{Y}$ and its discretization; the sample points are the grid intersections. When $\epsilon$ is sufficiently small, the level set $\left\{ \mathbf{y} \in \mathcal{Y} : g(\mathbf{x}, \mathbf{y}) - V(\mathbf{x}) \leq \epsilon \right\}$ is convex. Therefore, the convex hull of the top-left and bottom-right regions (delineated by the blue lines) is part of the level set, and our sample points are able to capture the entire level set.
• Figure 4: The top two figures and the bottom left figure respectively show the lower-level violations from 50 runs under the best parameter settings for TTSA, V-PBGD, and DIVIDE-BLO. The figure in the bottom right corner shows the average total gap curves over iterations for each algorithm, averaged across 50 randomly initialized runs. Only DIVIDE-BLO guarantees convergence to solutions with zero lower-level violation and zero total gap.

Theorems & Definitions (22)

• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Definition 3.1: Approximate global solutions of \ref{['eq:blo_vf']} and \ref{['eq:disc_reform']}.
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.1