2019-06-25-foster19b.md

abstract	section	title	layout	series	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	publisher	container-title	volume	genre	issued	pdf	extras
We give nearly matching upper and lower bounds on the oracle complexity of finding $\epsilon$-stationary points $(\|\nabla F(x)\|\leq\epsilon$ in stochastic convex optimization. We jointly analyze the oracle complexity in both the local stochastic oracle model and the global oracle (or, statistical learning) model. This allows us to decompose the complexity of finding near-stationary points into optimization complexity and sample complexity, and reveals some surprising differences between the complexity of stochastic optimization versus learning. Notably, we show that in the global oracle/statistical learning model, only logarithmic dependence on smoothness is required to find a near-stationary point, whereas polynomial dependence on smoothness is necessary in the local stochastic oracle model. In other words, the separation in complexity between the two models can be exponential, and the folklore understanding that smoothness is required to find stationary points is only weakly true for statistical learning. Our upper bounds are based on extensions of a recent “recursive regularization” technique proposed by Allen-Zhu (2018). We show how to extend the technique to achieve near-optimal rates, and in particular show how to leverage the extra information available in the global oracle model. Our algorithm for the global model can be implemented efficiently through finite sum methods, and suggests an interesting new computational-statistical tradeoff.	contributed	The Complexity of Making the Gradient Small in Stochastic Convex Optimization	inproceedings	Proceedings of Machine Learning Research	foster19b	0	The Complexity of Making the Gradient Small in Stochastic Convex Optimization	1319	1345	1319-1345	1319	false	Foster, Dylan J. and Sekhari, Ayush and Shamir, Ohad and Srebro, Nathan and Sridharan, Karthik and Woodworth, Blake	given family Dylan J. Foster given family Ayush Sekhari given family Ohad Shamir given family Nathan Srebro given family Karthik Sridharan given family Blake Woodworth	2019-06-25		PMLR	Proceedings of the Thirty-Second Conference on Learning Theory	99	inproceedings	date-parts 2019 6 25	http://proceedings.mlr.press/v99/foster19b/foster19b.pdf