2021-07-21-feder21a.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
Sequential prediction under log-loss and misspecification	We consider the question of sequential prediction under the log-loss in terms of cumulative regret. Namely, given a hypothesis class of distributions, learner sequentially predicts the (distribution of the) next letter in sequence and its performance is compared to the baseline of the best constant predictor from the hypothesis class. The well-specified case corresponds to an additional assumption that the data-generating distribution belongs to the hypothesis class as well. Here we present results in the more general misspecified case. Due to special properties of the log-loss, the same problem arises in the context of competitive-optimality in density estimation, and model selection. For the $d$-dimensional Gaussian location hypothesis class, we show that cumulative regrets in the well-specified and misspecified cases asymptotically coincide. In other words, we provide an $o(1)$ characterization of the distribution-free (or PAC) regret in this case – the first such result as far as we know. We recall that the worst-case (or individual-sequence) regret in this case is larger by an additive constant ${d\over 2} + o(1)$. Surprisingly, neither the traditional Bayesian estimators, nor the Shtarkov’s normalized maximum likelihood achieve the PAC regret and our estimator requires special “robustification” against heavy-tailed data. In addition, we show two general results for misspecified regret: the existence and uniqueness of the optimal estimator, and the bound sandwiching the misspecified regret between well-specified regrets with (asymptotically) close hypotheses classes.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	feder21a	0	Sequential prediction under log-loss and misspecification	1937	1964	1937-1964	1937	false	Feder, Meir and Polyanskiy, Yury	given family Meir Feder given family Yury Polyanskiy	2021-07-21		Proceedings of Thirty Fourth Conference on Learning Theory	134	inproceedings	date-parts 2021 7 21	http://proceedings.mlr.press/v134/feder21a/feder21a.pdf