2019-06-25-kuzborskij19a.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
Gibbs-ERM learning is a natural idealized model of learning with stochastic optimization algorithms (such as SGLD and —to some extent— SGD), while it also arises in other contexts, including PAC-Bayesian theory, and sampling mechanisms. In this work we study the excess risk suffered by a Gibbs-ERM learner that uses non-convex, regularized empirical risk with the goal to understand the interplay between the data-generating distribution and learning in large hypothesis spaces. Our main results are \emph{distribution-dependent} upper bounds on several notions of excess risk. We show that, in all cases, the distribution-dependent excess risk is essentially controlled by the \emph{effective dimension} $\text{tr}\left(\boldsymbol{H}^{\star} (\boldsymbol{H}^{\star} + \lambda \boldsymbol{I})^{-1}\right)$ of the problem, where $\boldsymbol{H}^{\star}$ is the Hessian matrix of the risk at a local minimum. This is a well-established notion of effective dimension appearing in several previous works, including the analyses of SGD and ridge regression, but ours is the first work that brings this dimension to the analysis of learning using Gibbs densities. The distribution-dependent view we advocate here improves upon earlier results of Raginsky et al. 2017, and can yield much tighter bounds depending on the interplay between the data-generating distribution and the loss function. The first part of our analysis focuses on the \emph{localized} excess risk in the vicinity of a fixed local minimizer. This result is then extended to bounds on the \emph{global} excess risk, by characterizing probabilities of local minima (and their complement) under Gibbs densities, a results which might be of independent interest.	contributed	Distribution-Dependent Analysis of Gibbs-ERM Principle	inproceedings	Proceedings of Machine Learning Research	kuzborskij19a	0	Distribution-Dependent Analysis of Gibbs-ERM Principle	2028	2054	2028-2054	2028	false	Kuzborskij, Ilja and Cesa-Bianchi, Nicol\`{o} and Szepesv\'ari, Csaba	given family Ilja Kuzborskij given family Nicolò Cesa-Bianchi given family Csaba Szepesvári	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/kuzborskij19a/kuzborskij19a.pdf