Thinking Inside the Ball: Near-Optimal Minimization of the Maximal Loss

• URL: http://arxiv.org/abs/2105.01778v1
• Date: Tue, 4 May 2021 21:49:15 GMT
• Title: Thinking Inside the Ball: Near-Optimal Minimization of the Maximal Loss
• Authors: Yair Carmon, Arun Jambulapati, Yujia Jin, Aaron Sidford
• Abstract summary: We prove an oracle complexity lower bound scaling as $Omega(Nepsilon-2/3)$, showing that our dependence on $N$ is optimal up to polylogarithmic factors. We develop methods with improved complexity bounds of $tildeO(Nepsilon-2/3 + sqrtNepsilon-8/3)$ in the non-smooth case and $tildeO(Nepsilon-2/3 + sqrtNepsilon-1)$ in
• Score: 41.17536985461902
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(N\epsilon^{-2})$ queries to a first-order oracle to compute an $\epsilon$-suboptimal point and $\tilde{O}(N\epsilon^{-1})$ queries if the $f_i$ are $O(1/\epsilon)$-smooth. We develop methods with improved complexity bounds of $\tilde{O}(N\epsilon^{-2/3} + \epsilon^{-8/3})$ in the non-smooth case and $\tilde{O}(N\epsilon^{-2/3} + \sqrt{N}\epsilon^{-1})$ in the $O(1/\epsilon)$-smooth case. Our methods consist of a recently proposed ball optimization oracle acceleration algorithm (which we refine) and a careful implementation of said oracle for the softmax function. We also prove an oracle complexity lower bound scaling as $\Omega(N\epsilon^{-2/3})$, showing that our dependence on $N$ is optimal up to polylogarithmic factors.

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