A Two-Timescale Stochastic Algorithm Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic
本文提出一种双时间尺度随机逼近算法求解双层优化问题，内层强凸、外层光滑有约束，分析了强凸/弱凸外层下的收敛率，并证明自然Actor-Critic算法是其特例，收敛率为K_max^{-1/4}。
.This paper analyzes a two-timescale stochastic algorithm framework for bilevel optimization. Bilevel optimization is a class of problems which exhibits a two-level structure, and its goal is to minimize an outer objective function with variables which are constrained to be the optimal solution to an (inner) optimization problem. We consider the case when the inner problem is unconstrained and strongly convex, while the outer problem is constrained and has a smooth objective function. We propose a two-timescale stochastic approximation (TTSA) algorithm for tackling such a bilevel problem. In the algorithm, a stochastic gradient update with a larger step size is used for the inner problem, while a projected stochastic gradient update with a smaller step size is used for the outer problem. We analyze the convergence rates for the TTSA algorithm under various settings: when the outer problem is strongly convex (resp. weakly convex), the TTSA algorithm finds an \(\mathcal{O}(K_{\max }^{-2/3})\) -optimal (resp. \(\mathcal{O}(K_{\max }^{-2/5})\) -stationary) solution, where \(K_{\max }\) is the total iteration number. As an application, we show that a two-timescale natural actor-critic proximal policy optimization algorithm can be viewed as a special case of our TTSA framework. Importantly, the natural actor-critic algorithm is shown to converge at a rate of \(\mathcal{O}(K_{\max }^{-1/4})\) in terms of the gap in expected discounted reward compared to a global optimal policy.Keywordsbilevel optimizationtwo-timescale stochastic approximationactor-criticMSC codes90C2690C1590C30