Papers › Faster Stochastic Algorithms for Minimax Optimization under Polyak--Łojasiewicz Conditions
Faster Stochastic Algorithms for Minimax Optimization under Polyak--Łojasiewicz Conditions
Lesi Chen, Boyuan Yao, Luo Luo
This paper considers stochastic first-order algorithms for minimax optimization under Polyak--{\L}ojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form minₓ max_y f(x,y)≜1/n ∑ᵢ₌₁ⁿ fᵢ(x,y), where the objective function f(x,y) is μₓ-PL in x and μ_y-PL in y; and each fᵢ(x,y) is L-smooth. We prove SPIDER-GDA could find an ϵ-optimal solution within 𝒪((n + √(n) κₓκ_y²)log(1/ϵ)) stochastic first-order oracle (SFO) complexity, which is better than the state-of-the-art method whose SFO upper bound is 𝒪((n + n^(2/3)κₓκ_y²)log(1/ϵ)), where κₓ≜L/μₓ and κ_y≜L/μ_y. For the ill-conditioned case, we provide an accelerated algorithm to reduce the computational cost further. It achieves 𝒪̃((n+√(n) κₓκ_y)log² (1/ϵ)) SFO upper bound when κ_y ≳√(n). Our ideas also can be applied to the more general setting that the objective function only satisfies PL condition for one variable. Numerical experiments validate the superiority of proposed methods.
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