It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$α$ sequential test can be obtained by thresholding an e-process at $1/α$. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.

Joulani et al. (2013) have studied multi-armed bandits with delayed feedback under the assumption that the rewards are stochastic and the delays are sampled from a fixed distribution.

Estimating properties of discrete distributions is a fundamental problem in statistical learning.

In practical situations, the clustering result must be stable against points missing in the input data so that we can make trustworthy andconsistentdecisions.

Wegiveanalgorithm thatoutputs anexplainable clustering that loses at most a factor ofO(log2k) compared to an optimal (not necessarily explainable) clustering for thek-medians objective, and a factor of O(klog2k)forthek-meansobjective.

Various results suggest that achieving optimal regret in the fully adversarial sleeping experts problem is computationally hard.

Ensemble sampling serves as apractical approximation to Thompson sampling when maintaining anexact posterior distribution overmodel parameters iscomputationally intractable. In this paper, we establish a regret bound that ensures desirable behavior when ensemble sampling isapplied tothe linear bandit problem.

The hope is that these predictions allow the algorithm to circumvent worst case lower bounds when the predictions are good, and approximately match them otherwise. The precise definitions and guarantees vary with different settings, but there have been significant successes in applying this framework for many different algorithmic problems, ranging from general online problems to classical graph algorithms (see Section 1.2 for a more detailed discussion of related work, and [35] for a survey). In all of these settings it turns out to be possible to define a "prediction" where the "quality" of the algorithm (competitive ratio, running time, etc.) depends the "error" of the prediction.

Submodular maximization has become established as the method of choice for the task of selecting representative anddiversesummaries of data.