Fast rates with high probability in exp-concave statistical learning

Mehta, Nishant A.

doi:10.48550/arxiv.1605.01288

Cited by 6 publications

(11 citation statements)

References 10 publications

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“…Our online-to-batch procedure only provides upper-bounds in expectation. Obtaining high-probability bounds is more challenging and universal conversion methods such as the one of [Mehta, 2016] may not work for improper procedures.…”

Section: Discussionmentioning

confidence: 99%

Efficient improper learning for online logistic regression

Jézéquel,

Gaillard,

Rudi

2020

Preprint

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We consider the setting of online logistic regression and consider the regret with respect to the 2 -ball of radius B. It is known (see [Hazan et al., 2014]) that any proper algorithm which has logarithmic regret in the number of samples (denoted n) necessarily suffers an exponential multiplicative constant in B. In this work, we design an efficient improper algorithm that avoids this exponential constant while preserving a logarithmic regret. Indeed, [Foster et al., 2018] showed that the lower bound does not apply to improper algorithms and proposed a strategy based on exponential weights with prohibitive computational complexity. Our new algorithm based on regularized empirical risk minimization with surrogate losses satisfies a regret scaling as O(B log(Bn)) with a per-round time-complexity of order O(d 2 ).

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Section: Discussionmentioning

confidence: 99%

Efficient improper learning for online logistic regression

Jézéquel,

Gaillard,

Rudi

2020

Preprint

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“…With the tool of the Rademacher complexity R n , Srebro et al [28] demonstrated an O(R n / √ n) risk bound for ERM, and many papers strengthened the theory further [3,2]. For nonconvex but exp-concave objectives, Koren & Levy [14] and Mehta [17] derived a risk bound of O(1/n). Under certain stronger conditions, a tighter O(1/n 2 ) risk bound has been shown [39].…”

Section: Related Workmentioning

confidence: 99%

Generalization Bounds for Stochastic Saddle Point Problems

Zhang,

Hong,

Wang

et al. 2020

Preprint

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This paper studies the generalization bounds for the empirical saddle point (ESP) solution to stochastic saddle point (SSP) problems. For SSP with Lipschitz continuous and strongly convex-strongly concave objective functions, we establish an O (1/n) generalization bound by using a uniform stability argument. We also provide generalization bounds under a variety of assumptions, including the cases without strong convexity and without bounded domains. We illustrate our results in two examples: batch policy learning in Markov decision process, and mixed strategy Nash equilibrium estimation for stochastic games. In each of these examples, we show that a regularized ESP solution enjoys a near-optimal sample complexity. To the best of our knowledge, this is the first set of results on the generalization theory of ESP.

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“…26]. Under exp-concave distributions, Mehta [30] also recently made use of this technique. Problems arise, however, when the losses can be potentially heavy-tailed.…”

Section: Challenges Under Weak Convexitymentioning

confidence: 99%

Better scalability under potentially heavy-tailed feedback

Holland

2020

Preprint

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We study scalable alternatives to robust gradient descent (RGD) techniques that can be used when the losses and/or gradients can be heavy-tailed, though this will be unknown to the learner. The core technique is simple: instead of trying to robustly aggregate gradients at each step, which is costly and leads to sub-optimal dimension dependence in risk bounds, we instead focus computational effort on robustly choosing (or newly constructing) a strong candidate based on a collection of cheap stochastic sub-processes which can be run in parallel. The exact selection process depends on the convexity of the underlying objective, but in all cases, our selection technique amounts to a robust form of boosting the confidence of weak learners. In addition to formal guarantees, we also provide empirical analysis of robustness to perturbations to experimental conditions, under both sub-Gaussian and heavy-tailed data, along with applications to a variety of benchmark datasets. The overall take-away is an extensible procedure that is simple to implement, trivial to parallelize, which keeps the formal merits of RGD methods but scales much better to large learning problems.

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Fast rates with high probability in exp-concave statistical learning

Cited by 6 publications

References 10 publications

Efficient improper learning for online logistic regression

Efficient improper learning for online logistic regression

Generalization Bounds for Stochastic Saddle Point Problems

Better scalability under potentially heavy-tailed feedback

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