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We study an equivalence of (i) deterministic pathwise statements appearing in the online learning literature (termed \emph{regret bounds}), (ii) high-probability tail bounds for the supremum of a collection of martingales (of a specific form arising from uniform laws of large numbers for martingales), and (iii) in-expectation bounds for the supremum.
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Martingale inequalities and deterministic counterparts
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On pathwise counterparts of doob’s maximal inequalities
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A. Rakhlin, K. Sridharan, and A. Tewari · 2014
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Sequential complexities and uniform martingale laws of large numbers
A. Rakhlin, K. Sridharan, and A. Tewari · 2014
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