
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 VapnikChervonenkis 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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Majorizing Measures, Sequential Complexities, and Online Learning
We introduce the technique of generic chaining and majorizing measures f...
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Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets
A Fréchet mean of a random variable Y with values in a metric space (𝒬, ...
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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 subpopulation is wellestimated 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 BenEliezer 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 samplecomplexity 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 BenDavid, Pál, and ShalevShwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015).
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