Memory-Query Tradeoffs for Randomized Convex Optimization

06/21/2023

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by   Xi Chen, et al.

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We show that any randomized first-order algorithm which minimizes a d-dimensional, 1-Lipschitz convex function over the unit ball must either use Ω(d^2-δ) bits of memory or make Ω(d^1+δ/6-o(1)) queries, for any constant δ∈ (0,1) and when the precision ϵ is quasipolynomially small in d. Our result implies that cutting plane methods, which use Õ(d^2) bits of memory and Õ(d) queries, are Pareto-optimal among randomized first-order algorithms, and quadratic memory is required to achieve optimal query complexity for convex optimization.