We study the tradeoff between sample complexity and round complexity in ondemand sampling, where the learning algorithm adaptively samples from k distributions over a limited number of rounds. In the realizable setting of MultiDistribution Learning (MDL), we show that the optimal sample complexity of an r-round algorithm scales approximately as dkΘ(1/r)/ε. For the general agnostic case, we present an algorithm that achieves near-optimal sample complexity of eO((d + k)/ε2) within eO( k) rounds. Of independent interest, we introduce a new framework, Optimization via On-Demand Sampling (OODS), which abstracts the sample-adaptivity tradeoff and captures most existing MDL algorithms. We establish nearly tight bounds on the round complexity in the OODS setting. The upper bounds directly yield the eO( k)-round algorithm for agnostic MDL, while the lower bounds imply that achieving sub-polynomial round complexity would require fundamentally new techniques that bypass the inherent hardness of OODS.

We study the problem of explainable k-medians clustering introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (2020). In this problem, the goal is to construct a threshold decision tree that partitions data into k clusters while minimizing the k-medians objective. These trees are interpretable because each internal node makes a simple decision by thresholding a single feature, allowing users to trace and understand how each point is assigned to a cluster. We present the first algorithm for explainable k-medians under ℓp norm for every finite p 1. Our algorithm achieves an O p(logk)1+1/p 1/p

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.