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.

In particular, Bai et al. [5], Jin et al. [31] developed the first algorithms to beat the curse of multiple agents in twoplayer zero-sum MGs, while Jin et al. [31], Daskalakis et al. [23], Mao and Ba sar [44], Song et al. [63] further demonstrated how to accomplish the same goal when learning other computationally tractable solution concepts (e.g., coarse correlated equilibria) in general-sum multi-player Markov games. We shall also briefly remark on the prior works that concern RL with a generative model. A key term in the regret bound (36) is a weighted sum of the "variance-style" quantities {Varπk(`k)}. While Var(`k) k`kk2 is orderwise tight in the worst-case scenario for a given iteration k, exploiting the problem-specific variance-type structure across time is crucial in sharpening the horizon dependence in many RL problems(e.g.,Azaretal.[3],Jinetal.[30],Lietal.[41,40]). C.1 Preliminariesandnotation Let us start with some preliminary facts and notation.

As a valuable by-product, the regret analysis used in this paper can improve several existing results by a factor ofO(logK).