# Adversarial Laws of Large Numbers and Optimal Regret in Online Classification

Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed sub-population is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by Ben-Eliezer and Yogev (2020), and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are online learnable. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of Littlestone's dimension, thus resolving the main open question from Ben-David, Pál, and Shalev-Shwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015).

• 27 publications
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10/13/2015

### On Equivalence of Martingale Tail Bounds and Deterministic Regret Inequalities

We study an equivalence of (i) deterministic pathwise statements appeari...
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### Uniform Approximation and Bracketing Properties of VC classes

We show that the sets in a family with finite VC dimension can be unifor...
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### Laws of large numbers for stochastic orders

We establish laws of large numbers for comparing sums of i.i.d. random v...
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### Uniform Approximation of Vapnik-Chervonenkis Classes

For any family of measurable sets in a probability space, we show that e...
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### A Computational Separation between Private Learning and Online Learning

A recent line of work has shown a qualitative equivalence between differ...
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### A note on sampling recovery of multivariate functions in the uniform norm

We study the recovery of multivariate functions from reproducing kernel ...
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### Majorizing Measures, Sequential Complexities, and Online Learning

We introduce the technique of generic chaining and majorizing measures f...