2021-07-21-carmon21a.md

title	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
Thinking Inside the Ball: Near-Optimal Minimization of the Maximal Loss	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 $\widetilde{O}(N\epsilon^{-1})$ queries if the $f_i$ are $O(1/\epsilon)$-smooth. We develop methods with improved complexity bounds of $\widetilde{O}(N\epsilon^{-2/3} + \epsilon^{-8/3})$ in the non-smooth case and $\widetilde{O}(N\epsilon^{-2/3} + \sqrt{N}\epsilon^{-1})$ 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.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	carmon21a	0	Thinking Inside the Ball: Near-Optimal Minimization of the Maximal Loss	866	882	866-882	866	false	Carmon, Yair and Jambulapati, Arun and Jin, Yujia and Sidford, Aaron	given family Yair Carmon given family Arun Jambulapati given family Yujia Jin given family Aaron Sidford	2021-07-21		Proceedings of Thirty Fourth Conference on Learning Theory	134	inproceedings	date-parts 2021 7 21	http://proceedings.mlr.press/v134/carmon21a/carmon21a.pdf	label link Supplementary http://proceedings.mlr.press/v134/carmon21a/carmon21a-supp.pdf