Memory-Query Tradeoffs for Randomized Convex Optimization

• URL: http://arxiv.org/abs/2306.12534v1
• Date: Wed, 21 Jun 2023 19:48:58 GMT
• Title: Memory-Query Tradeoffs for Randomized Convex Optimization
• Authors: Xi Chen and Binghui Peng
• Abstract summary: We show that any randomized first-order algorithm which minimizes a $d$-dimensional, $1$-Lipschitz convex function over the unit ball must either use $Omega(d2-delta)$ bits of memory or make $Omega(d1+delta/6-o(1))$ queries.
• Score: 16.225462097812766
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We show that any randomized first-order algorithm which minimizes a $d$-dimensional, $1$-Lipschitz convex function over the unit ball must either use $\Omega(d^{2-\delta})$ bits of memory or make $\Omega(d^{1+\delta/6-o(1)})$ queries, for any constant $\delta\in (0,1)$ and when the precision $\epsilon$ is quasipolynomially small in $d$. Our result implies that cutting plane methods, which use $\tilde{O}(d^2)$ bits of memory and $\tilde{O}(d)$ queries, are Pareto-optimal among randomized first-order algorithms, and quadratic memory is required to achieve optimal query complexity for convex optimization.

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