Efficiently Escaping Saddle Points in Bilevel Optimization
This addresses a fundamental bottleneck in bilevel optimization for machine learning, providing efficient algorithms with theoretical guarantees, though it is incremental on existing methods.
The paper tackles the problem of escaping saddle points in nonconvex-strongly-convex bilevel optimization, showing that perturbed approximate implicit differentiation with warm start finds an ε-approximate local minimum in Õ(ε⁻²) iterations with high probability, and proposes a first-order algorithm for stochastic cases.
Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the first-order stationary points for nonconvex-strongly-convex cases. In this paper, we analyze algorithms that can escape saddle points in nonconvex-strongly-convex bilevel optimization. Specifically, we show that the perturbed approximate implicit differentiation (AID) with a warm start strategy finds $ε$-approximate local minimum of bilevel optimization in $\tilde{O}(ε^{-2})$ iterations with high probability. Moreover, we propose an inexact NEgative-curvature-Originated-from-Noise Algorithm (iNEON), a pure first-order algorithm that can escape saddle point and find local minimum of stochastic bilevel optimization. As a by-product, we provide the first nonasymptotic analysis of perturbed multi-step gradient descent ascent (GDmax) algorithm that converges to local minimax point for minimax problems.