The softmax function is ubiquitous in machine learning and optimization applications. Computing the full softmax evaluation of a matrix-vector product can be computationally expensive in high-dimensional settings. In many applications, however, it is sufficient to calculate only the top few outputs of the softmax function. In this work, we present an algorithm, dubbed AdaptiveSoftmax, that adaptively computes the top k softmax values more efficiently than the full softmax computation, with probabilistic guarantees. We demonstrate the sample efficiency improvements afforded by AdaptiveSoftmax on real and synthetic data to corroborate our theoretical results. AdaptiveSoftmax yields >10x gain over full softmax computation on most datasets, yielding up to 30x improvement for Mistral7B evaluated on the Wikitext dataset. The adaptive method we propose for estimating the partition function (the softmax denominator) is of independent interest and can be used in other applications such as kernel density estimation.

Data depth, introduced by Tukey (1975), is an important tool in data science, robust statistics, and computational geometry. One chief barrier to its broader practical utility is that many common measures of depth are computationally intensive, requiring on the order of $n^d$ operations to exactly compute the depth of a single point within a data set of $n$ points in $d$-dimensional space. Often however, we are not directly interested in the absolute depths of the points, but rather in their relative ordering. For example, we may want to find the most central point in a data set (a generalized median), or to identify and remove all outliers (points on the fringe of the data set with low depth). With this observation, we develop a novel instance-adaptive algorithm for adaptive data depth computation by reducing the problem of exactly computing $n$ depths to an $n$-armed stochastic multi-armed bandit problem which we can efficiently solve. We focus our exposition on simplicial depth, developed by Liu (1990), which has emerged as a promising notion of depth due to its interpretability and asymptotic properties. We provide general data-dependent theoretical guarantees for our proposed algorithms, which readily extend to many other common measures of data depth including majority depth, Oja depth, and likelihood depth. When specialized to the case where the gaps in the data follow a power law distribution with parameter $\alpha<2$, we reduce the complexity of identifying the deepest point in the data set (the simplicial median) from $O(n^d)$ to $\tilde{O}(n^{d-(d-1)\alpha/2})$, where $\tilde{O}$ suppresses a logarithmic factor. We corroborate our theoretical results with numerical experiments on synthetic data, showing the practical utility of our proposed methods.

We study the problem of estimating the value of a known smooth function f at an unknown point $\mu \in \mathbb{R}^n$, where each component $\mu_i$ can be sampled via a noisy oracle. Sampling more frequently components of $\mu$ corresponding to directions of the function with larger directional derivatives is more sample-efficient. However, as $\mu$ is unknown, the optimal sampling frequencies are also unknown. We design an instance-adaptive algorithm that learns to sample according to the importance of each coordinate, and with probability at least $1-\delta$ returns an $\epsilon$ accurate estimate of $f(\mu)$. We generalize our algorithm to adapt to heteroskedastic noise, and prove asymptotic optimality when f is linear. We corroborate our theoretical results with numerical experiments, showing the dramatic gains afforded by adaptivity.

In the infinite-armed bandit problem, each arm's average reward is sampled from an unknown distribution, and each arm can be sampled further to obtain noisy estimates of the average reward of that arm. Prior work focuses on the best arm, i.e. estimating the maximum of the average reward distribution. We consider a general class of distribution functionals beyond the maximum and obtain optimal sample complexities in both offline and online settings. We show that online estimation, where the learner can sequentially choose whether to sample a new or existing arm, offers no advantage over the offline setting for estimating the mean functional, but significantly reduces the sample complexity for other functionals such as the median, maximum, and trimmed mean. We propose unified meta algorithms for the online and offline settings and derive matching lower bounds using different Wasserstein distances. For the special case of median estimation, we identify a curious thresholding phenomenon on the indistinguishability between Gaussian convolutions with respect to the noise level, which may be of independent interest.

Extreme multi-label classification (XMC) aims to learn a model that can tag data points with a subset of relevant labels from an extremely large label set. Real world e-commerce applications like personalized recommendations and product advertising can be formulated as XMC problems, where the objective is to predict for a user a small subset of items from a catalog of several million products. For such applications, a common approach is to organize these labels into a tree, enabling training and inference times that are logarithmic in the number of labels [23]. While training a model once a label tree is available is well studied, designing the structure of the tree is a difficult task that is not yet well understood, and can dramatically impact both model latency and statistical performance. Existing approaches to tree construction either optimize exclusively for statistical performance or optimize exclusively for latency. We propose an efficient information theory inspired algorithm to construct intermediate operating points that trade off between the benefits of both, which was not previously possible. We corroborate our theoretical analysis with numerical results, showing that on the Wiki-500K [4] benchmark dataset our method can reduce a proxy for expected latency by up to 28% while maintaining the same accuracy as Parabel [23]. On several datasets derived from e-commerce customer logs, our modified label tree is able to improve this expected latency metric by up to 20% while maintaining the same accuracy. Finally, we discuss challenges in realizing these latency improvements in deployed models.

Consider N cooperative but non-communicating players where each plays one out of M arms for T turns. Players have different utilities for each arm, representable as an NxM matrix. These utilities are unknown to the players. In each turn players receive noisy observations of their utility for their selected arm. However, if any other players selected the same arm that turn, they will all receive zero utility due to the conflict. No other communication or coordination between the players is possible. Our goal is to design a distributed algorithm that learns the matching between players and arms that achieves max-min fairness while minimizing the regret. We present an algorithm and prove that it is regret optimal up to a \log\log T factor. This is the first max-min fairness multi-player bandit algorithm with (near) order optimal regret.

Pairwise alignment of DNA sequencing data is a ubiquitous task in bioinformatics and typically represents a heavy computational burden. State-of-the-art approaches to speed up this task use hashing to identify short segments (k-mers) that are shared by pairs of reads, which can then be used to estimate alignment scores. However, when the number of reads is large, accurately estimating alignment scores for all pairs is still very costly. Moreover, in practice, one is only interested in identifying pairs of reads with large alignment scores. In this work, we propose a new approach to pairwise alignment estimation based on two key new ingredients. The first ingredient is to cast the problem of pairwise alignment estimation under a general framework of rank-one crowdsourcing models, where the workers' responses correspond to k-mer hash collisions. These models can be accurately solved via a spectral decomposition of the response matrix. The second ingredient is to utilise a multi-armed bandit algorithm to adaptively refine this spectral estimator only for read pairs that are likely to have large alignments. The resulting algorithm iteratively performs a spectral decomposition of the response matrix for adaptively chosen subsets of the read pairs.

The medoid of a set of n points is the point in the set that minimizes the sum of distances to other points. It can be determined exactly in O(n^2) time by computing the distances between all pairs of points. Previous works show that one can significantly reduce the number of distance computations needed by adaptively querying distances. The resulting randomized algorithm is obtained by a direct conversion of the computation problem to a multi-armed bandit statistical inference problem. In this work, we show that we can better exploit the structure of the underlying computation problem by modifying the traditional bandit sampling strategy and using it in conjunction with a suitably chosen multi-armed bandit algorithm. Four to five orders of magnitude gains over exact computation are obtained on real data, in terms of both number of distance computations needed and wall clock time. Theoretical results are obtained to quantify such gains in terms of data parameters. Our code is publicly available online at https://github.com/TavorB/Correlated-Sequential-Halving.