[论文解读] Approximation Methods for Bilevel Programming

Saeed Ghadimi, Mengdi Wang|arXiv (Cornell University)|Feb 6, 2018

Advanced Optimization Algorithms Research参考文献 13被引用 65

一句话总结

该论文为具有强凸内部问题的双层规划开发确定性与随机近似算法，在不同外部目标的凸性条件下给出有限时间收敛/复杂度分析，并提出一个带改进收敛速率的加速变体。

ABSTRACT

In this paper, we study a class of bilevel programming problem where the inner objective function is strongly convex. More specifically, under some mile assumptions on the partial derivatives of both inner and outer objective functions, we present an approximation algorithm for solving this class of problem and provide its finite-time convergence analysis under different convexity assumption on the outer objective function. We also present an accelerated variant of this method which improves the rate of convergence under convexity assumption. Furthermore, we generalize our results under stochastic setting where only noisy information of both objective functions is available. To the best of our knowledge, this is the first time that such (stochastic) approximation algorithms with established iteration complexity (sample complexity) are provided for bilevel programming.

研究动机与目标

• Motivate the study of bilevel programming where the inner problem is strongly convex and smooth.
• Develop an approximation algorithm to solve the bilevel problem with provable finite-time convergence.
• Provide an accelerated variant to improve convergence when the outer objective is convex.
• Extend results to stochastic settings with noisy gradient/Hessian information.
• Establish iteration/sample complexity guarantees across convex, strongly convex, and nonconvex outer objectives.

提出的方法

• Introduce BA (Bilevel Approximation) method that alternates inner y-iterations with an outer x-step using a gradient approximation \u001b2 bar\nabla f defined via implicit differentiation of the inner problem.
• Explicitly define gradient approximation bar{\nabla}f(x; y) = ∇_x f(x; y) - M(x,y) ∇_y f(x;y) with M(x,y) = ∇_{xy}^2 g(x,y) [∇_{yy}^2 g(x,y)]^{-1}.
• Prove gradient error bounds: ||bar{\nabla}f(x; ￡) - ∇ f(x; y^*(x))|| ≤ C ||y^*(x) - y|| and establish Lipschitz properties for y^*(x) and ∇f.
• Provide inner loop convergence for updating y by gradient descent on g with step β_t = 2/(μ_g + L_g).
• Outer update is a proximal-type step: x_{k+1} = argmin_{u ∈ X} { ⟨bar{\nabla}f(x_k; ￡_k), u⟩ + (1/(2 α_k)) ||u - x_k||^2 }.
• Derive complexity results under different convexity assumptions of f (strongly convex, convex, nonconvex).

实验结果

研究问题

• RQ1 What are the finite-time convergence rates for bilevel problems when the inner problem is strongly convex?
• RQ2 How does the outer objective convexity affect the iteration complexity of bilevel approximation methods?
• RQ3 Can an accelerated variant improve convergence rates for convex outer objectives in bilevel programming?
• RQ4 What are the stochastic complexity guarantees when gradients/Hessians are observed with noise?
• RQ5 How do these results extend to stochastic bilevel problems with noisy first/second-order information?

主要发现

• Deterministic BA method achieves GC(f, ε) = HC(g, ε) = O(log(1/ε)) and GC(g, ε) = O(log^2(1/ε)) when both f and g are strongly convex.
• When f is convex, BA method yields GC(f, ε) = HC(g, ε) = O(1/ε) and GC(g, ε) = O(1/ε^{5/4}); for possibly nonconvex f, GC(f, ε) = HC(g, ε) = O(1/ε) and GC(g, ε) = O(1/ε^{5/4}).
• An accelerated BA (ABA) method yields improved rates for convex outer objectives, with GC(f, ε) = HC(g, ε) = O(1/√ε) and GC(g, ε) = O(1/ε^{3/4}).
• A stochastic variant provides overall sample complexity SGC(f, ε) = O(1/ε) and SGC(g, ε) = O(1/ε^2) with SHC(g, ε) = O((1/ε) log(1/ε)) when f and g are strongly convex.
• If f is only convex, stochastic rates worsen to SGC(f, ε) = O(1/ε^2), SGC(g, ε) = O(ε^{-4}), and SHC(g, ε) = O((1/ε^2) log(1/ε)).
• For possibly nonconvex f, stochastic rates are SGC(f, ε) = O(1/ε^2), SGC(g, ε) = O(1/ε^3), and SHC(g, ε) = O((1/ε^2) log(1/ε)).