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Section 4.1, we consider =3 . In Section 4.2 and Appendix A.1, we examine the performance of different algorithms for the In Figure 5 the performance of both greedy heuristics is very similar under the two one-sided losses. We observe that the objective values are no longer uniformly positive, and are no longer monotonically increasing in the target size. In this section, we present the proofs of all theoretical results. The following lemma shows the submodularity of the objective U in the selection S . If (,M) is convex then U ( S) is submodular in S .

We consider the best arm identification problem, where the goal is to identify the arm with the highest mean reward from a set of $K$ arms under a limited sampling budget. This problem models many practical scenarios such as A/B testing. We consider a class of algorithms for this problem, which is provably minimax optimal up to a constant factor. This idea is a generalization of existing works in fixed-budget best arm identification, which are limited to a particular choice of risk measures. Based on the framework, we propose Almost Tracking, a closed-form algorithm that has a provable guarantee on the popular risk measure $H_1$. Unlike existing algorithms, Almost Tracking does not require the total budget in advance nor does it need to discard a significant part of samples, which gives a practical advantage. Through experiments on synthetic and real-world datasets, we show that our algorithm outperforms existing anytime algorithms as well as fixed-budget algorithms.

We first analyze the guarantees provided by natural greedy heuristics, showing their desirable properties despite the simplicity.

Calibration requires that predictions are conditionally unbiased and, therefore, reliably interpretable as probabilities. Calibration measures quantify how far a predictor is from perfect calibration. As introduced by Haghtalab et al. (2024), a calibration measure is truthful if it is minimized in expectation when a predictor outputs the ground-truth probabilities. Although predicting the true probabilities guarantees perfect calibration, in reality, when calibration is evaluated on a finite sample, predicting the truth is not guaranteed to minimize any known calibration measure. All known calibration measures incentivize predictors to lie in order to appear more calibrated on a finite sample. Such lack of truthfulness motivated Haghtalab et al. (2024) and Qiao and Zhao (2025) to construct approximately truthful calibration measures in the sequential prediction setting, but no perfectly truthful calibration measure was known to exist even in the more basic batch setting. We design a perfectly truthful calibration measure in the batch setting: averaged two-bin calibration error (ATB). In addition to being truthful, ATB is sound, complete, continuous, and quadratically related to two existing calibration measures: the smooth calibration error (smCal) and the (lower) distance to calibration (distCal). The simplicity in our definition of ATB makes it efficient and straightforward to compute. ATB allows faster estimation algorithms with significantly easier implementations than smCal and distCal, achieving improved running time and simplicity for the calibration testing problem studied by Hu et al. (2024). We also introduce a general recipe for constructing truthful measures, which proves the truthfulness of ATB as a special case and allows us to construct other truthful calibration measures such as quantile-binned l_2-ECE.

We study the problem of estimating the mean of a distribution in high dimensions when either the samples are adversarially corrupted or the distribution is heavy-tailed.

This result extends the previously known O(log k) guarantee for the case β = 1 to the constant-factor bi-criteria regime. It also improves upon an existing constant-factor bi-criteria result that holds only with constant probability.

This paper studies the $k$-means++ algorithm for clustering as well as the class of $D^\ell$ sampling algorithms to which $k$-means++ belongs. It is shown that for any constant factor $\beta > 1$, selecting $\beta k$ cluster centers by $D^\ell$ sampling yields a constant-factor approximation to the optimal clustering with $k$ centers, in expectation and without conditions on the dataset. This result extends the previously known $O(\log k)$ guarantee for the case $\beta = 1$ to the constant-factor bi-criteria regime. It also improves upon an existing constant-factor bi-criteria result that holds only with constant probability.