We consider prediction with expert advice for strongly convex and bounded losses, and investigate trade-offs between regret and "variance" (i.e., squared difference of learner's predictions and best expert predictions). With K experts, the Exponentially Weighted Average (EWA) algorithm is known to achieve O(log K) regret. We prove that a variant of EWA either achieves a negative regret (i.e., the algorithm outperforms the best expert), or guarantees a O(log K) bound on both variance and regret. Building on this result, we show several examples of how variance of predictions can be exploited in learning. In the online to batch analysis, we show that a large empirical variance allows to stop the online to batch conversion early and outperform the risk of the best predictor in the class. We also recover the optimal rate of model selection aggregation when we do not consider early stopping. In online prediction with corrupted losses, we show that the effect of corruption on the regret can be compensated by a large variance. In online selective sampling, we design an algorithm that samples less when the variance is large, while guaranteeing the optimal regret bound in expectation. In online learning with abstention, we use a similar term as the variance to derive the first high-probability O(log K) regret bound in this setting. Finally, we extend our results to the setting of online linear regression.

We study nonstochastic bandits and experts in a delayed setting where delays depend on both time and arms. While the setting in which delays only depend on time has been extensively studied, the arm-dependent delay setting better captures real-world applications at the cost of introducing new technical challenges. In the full information (experts) setting, we design an algorithm with a firstorder regret bound that reveals an interesting trade-off between delays and losses. We prove a similar first-order regret bound also for the bandit setting, when the learner is allowed to observe how many losses are missing. These are the first bounds in the delayed setting that depend on the losses and delays of the best arm only. When in the bandit setting no information other than the losses is observed, we still manage to prove a regret bound through a modification to the algorithm of Zimmert and Seldin (2020). Our analyses hinge on a novel bound on the drift, measuring how much better an algorithm can perform when given a look-ahead of one round.

In this paper we consider a distributed online learning setting for joint regret with communication constraints. This is a multi-agent setting in which in each round t an adversary activates an agent, which has to issue a prediction. A subset of all the agents may then communicate a b-bit message to their neighbors in a graph. All agents cooperate to control the joint regret, which is the sum of the losses of the agents minus the losses evaluated at the best fixed common comparator parameters u. We provide a comparator-adaptive algorithm for this setting, which means that the joint regret scales with the norm of the comparator u . To address communication constraints we provide deterministic and stochastic gradient compression schemes and show that with these compression schemes our algorithm has worstcase optimal regret for the case that all agents communicate in every round. Additionally, we exploit the comparator-adaptive property of our algorithm to learn the best partition from a set of candidate partitions, which allows different subsets of agents to learn a different comparator.

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