Projection-Free Algorithm for Stochastic Bi-level Optimization
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This work presents the first projection-free algorithm to solve stochastic bi-level optimization problems, where the objective function depends on the solution of another stochastic optimization problem. The proposed Stochastic Bi-level Frank-Wolfe (SBFW) algorithm can be applied to streaming settings and does not make use of large batches or checkpoints. The sample complexity of SBFW is shown to be 𝒪(ϵ^-3) for convex objectives and 𝒪(ϵ^-4) for non-convex objectives. Improved rates are derived for the stochastic compositional problem, which is a special case of the bi-level problem, and entails minimizing the composition of two expected-value functions. The proposed Stochastic Compositional Frank-Wolfe (SCFW) is shown to achieve a sample complexity of 𝒪(ϵ^-2) for convex objectives and 𝒪(ϵ^-3) for non-convex objectives, at par with the state-of-the-art sample complexities for projection-free algorithms solving single-level problems. We demonstrate the advantage of the proposed methods by solving the problem of matrix completion with denoising and the problem of policy value evaluation in reinforcement learning.
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