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Between Stochastic and Adversarial Online Convex Optimization: Improved Regret Bounds via Smoothness
Sarah Sachs · Hedi Hadiji · Tim van Erven · Cristóbal Guzmán

Wed Nov 30 09:00 AM -- 11:00 AM (PST) @ Hall J #616

Stochastic and adversarial data are two widely studied settings in online learning. But many optimizationtasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the expert and bandit setting. Our results extend this to the online convex optimization framework. In the fully i.i.d. case, our bounds match the rates one would expect from results in stochastic acceleration, and in the fully adversarial case they gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds aretight for all intermediate regimes in terms of the stochastic variance and theadversarial variation of the loss gradients.

Author Information

Sarah Sachs (University of Amsterdam)
Hedi Hadiji (University of Amsterdam)
Tim van Erven (University of Amsterdam)
Cristóbal Guzmán (PUC-Chile)

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