Shubhanshu Shekhar

Shubhanshu Shekhar, Ilmun Kim, Aaditya Ramdas

The kernel Maximum Mean Discrepancy~(MMD) is a popular multivariate distance metric between distributions that has found utility in two-sample testing. The usual kernel-MMD test statistic is a degenerate U-statistic under the null, and thus it has an intractable limiting distribution. Hence, to design a level-$α$ test, one usually selects the rejection threshold as the $(1-α)$-quantile of the permutation distribution. The resulting nonparametric test has finite-sample validity but suffers from large computational cost, since every permutation takes quadratic time. We propose the cross-MMD, a new quadratic-time MMD test statistic based on sample-splitting and studentization. We prove that under mild assumptions, the cross-MMD has a limiting standard Gaussian distribution under the null. Importantly, we also show that the resulting test is consistent against any fixed alternative, and when using the Gaussian kernel, it has minimax rate-optimal power against local alternatives. For large sample sizes, our new cross-MMD provides a significant speedup over the MMD, for only a slight loss in power.

Shubhanshu Shekhar, Ilmun Kim, Aaditya Ramdas

In nonparametric independence testing, we observe i.i.d.\ data $\{(X_i,Y_i)\}_{i=1}^n$, where $X \in \mathcal{X}, Y \in \mathcal{Y}$ lie in any general spaces, and we wish to test the null that $X$ is independent of $Y$. Modern test statistics such as the kernel Hilbert-Schmidt Independence Criterion (HSIC) and Distance Covariance (dCov) have intractable null distributions due to the degeneracy of the underlying U-statistics. Thus, in practice, one often resorts to using permutation testing, which provides a nonasymptotic guarantee at the expense of recalculating the quadratic-time statistics (say) a few hundred times. This paper provides a simple but nontrivial modification of HSIC and dCov (called xHSIC and xdCov, pronounced ``cross'' HSIC/dCov) so that they have a limiting Gaussian distribution under the null, and thus do not require permutations. This requires building on the newly developed theory of cross U-statistics by Kim and Ramdas (2020), and in particular developing several nontrivial extensions of the theory in Shekhar et al. (2022), which developed an analogous permutation-free kernel two-sample test. We show that our new tests, like the originals, are consistent against fixed alternatives, and minimax rate optimal against smooth local alternatives. Numerical simulations demonstrate that compared to the full dCov or HSIC, our variants have the same power up to a $\sqrt 2$ factor, giving practitioners a new option for large problems or data-analysis pipelines where computation, not sample size, could be the bottleneck.

Shubhanshu Shekhar, Aaditya Ramdas

Constructing nonasymptotic confidence intervals (CIs) for the mean of a univariate distribution from independent and identically distributed (i.i.d.) observations is a fundamental task in statistics. For bounded observations, a classical nonparametric approach proceeds by inverting standard concentration bounds, such as Hoeffding's or Bernstein's inequalities. Recently, an alternative betting-based approach for defining CIs and their time-uniform variants called confidence sequences (CSs), has been shown to be empirically superior to the classical methods. In this paper, we provide theoretical justification for this improved empirical performance of betting CIs and CSs. Our main contributions are as follows: (i) We first compare CIs using the values of their first-order asymptotic widths (scaled by $\sqrt{n}$), and show that the betting CI of Waudby-Smith and Ramdas (2023) has a smaller limiting width than existing empirical Bernstein (EB)-CIs. (ii) Next, we establish two lower bounds that characterize the minimum width achievable by any method for constructing CIs/CSs in terms of certain inverse information projections. (iii) Finally, we show that the betting CI and CS match the fundamental limits, modulo an additive logarithmic term and a multiplicative constant. Overall these results imply that the betting CI~(and CS) admit stronger theoretical guarantees than the existing state-of-the-art EB-CI~(and CS); both in the asymptotic and finite-sample regimes.

Shubhanshu Shekhar, Aaditya Ramdas

We consider the problem of sequential change detection, where the goal is to design a scheme for detecting any changes in a parameter or functional $θ$ of the data stream distribution that has small detection delay, but guarantees control on the frequency of false alarms in the absence of changes. In this paper, we describe a simple reduction from sequential change detection to sequential estimation using confidence sequences: we begin a new $(1-α)$-confidence sequence at each time step, and proclaim a change when the intersection of all active confidence sequences becomes empty. We prove that the average run length is at least $1/α$, resulting in a change detection scheme with minimal structural assumptions~(thus allowing for possibly dependent observations, and nonparametric distribution classes), but strong guarantees. Our approach bears an interesting parallel with the reduction from change detection to sequential testing of Lorden (1971) and the e-detector of Shin et al. (2022).

Shubhanshu Shekhar, Aaditya Ramdas

We present a simple reduction from sequential estimation to sequential changepoint detection (SCD). In short, suppose we are interested in detecting changepoints in some parameter or functional $θ$ of the underlying distribution. We demonstrate that if we can construct a confidence sequence (CS) for $θ$, then we can also successfully perform SCD for $θ$. This is accomplished by checking if two CSs -- one forwards and the other backwards -- ever fail to intersect. Since the literature on CSs has been rapidly evolving recently, the reduction provided in this paper immediately solves several old and new change detection problems. Further, our "backward CS", constructed by reversing time, is new and potentially of independent interest. We provide strong nonasymptotic guarantees on the frequency of false alarms and detection delay, and demonstrate numerical effectiveness on several problems.

Teodora Pandeva, Patrick Forré, Aaditya Ramdas et al.

We propose a general framework for constructing powerful, sequential hypothesis tests for a large class of nonparametric testing problems. The null hypothesis for these problems is defined in an abstract form using the action of two known operators on the data distribution. This abstraction allows for a unified treatment of several classical tasks, such as two-sample testing, independence testing, and conditional-independence testing, as well as modern problems, such as testing for adversarial robustness of machine learning (ML) models. Our proposed framework has the following advantages over classical batch tests: 1) it continuously monitors online data streams and efficiently aggregates evidence against the null, 2) it provides tight control over the type I error without the need for multiple testing correction, 3) it adapts the sample size requirement to the unknown hardness of the problem. We develop a principled approach of leveraging the representation capability of ML models within the testing-by-betting framework, a game-theoretic approach for designing sequential tests. Empirical results on synthetic and real-world datasets demonstrate that tests instantiated using our general framework are competitive against specialized baselines on several tasks.

Shubhanshu Shekhar, Tara Javidi

We study the kernelized bandit problem, that involves designing an adaptive strategy for querying a noisy zeroth-order-oracle to efficiently learn about the optimizer of an unknown function $f$ with a norm bounded by $M<\infty$ in a Reproducing Kernel Hilbert Space~(RKHS) associated with a positive definite kernel $K$. Prior results, working in a \emph{minimax framework}, have characterized the worst-case~(over all functions in the problem class) limits on regret achievable by \emph{any} algorithm, and have constructed algorithms with matching~(modulo polylogarithmic factors) worst-case performance for the \matern family of kernels. These results suffer from two drawbacks. First, the minimax lower bound gives no information about the limits of regret achievable by the commonly used algorithms on specific problem instances. Second, due to their worst-case nature, the existing upper bound analysis fails to adapt to easier problem instances within the function class. Our work takes steps to address both these issues. First, we derive \emph{instance-dependent} regret lower bounds for algorithms with uniformly~(over the function class) vanishing normalized cumulative regret. Our result, valid for all the practically relevant kernelized bandits algorithms, such as, GP-UCB, GP-TS and SupKernelUCB, identifies a fundamental complexity measure associated with every problem instance. We then address the second issue, by proposing a new minimax near-optimal algorithm which also adapts to easier problem instances.

Shubhanshu Shekhar, Aaditya Ramdas

We consider the problem of constructing confidence intervals (CIs) for the population mean of $N$ values $\{x_1, \ldots, x_N\} \subset Σ^N$ based on a random sample of size $n$, denoted by $X^n \equiv (X_1, \ldots, X_n)$, drawn uniformly without replacement (WoR). We begin by focusing on the finite alphabet ($|Σ| = k <\infty$) and moderate accuracy ($\log(1/α_N) \gg (k+1)\log N$) regime, and derive a fundamental lower bound on the width of any level-$(1-α_N)$ CI in terms of the inverse of the WoR rate functions from the theory of large deviations. Guided by this lower bound, we propose a new level-$(1-α_N)$ CI using an empirical inverse rate function, and show that in certain asymptotic regimes the width of this CI matches the lower bound up to constants. We also derive a dual formulation of the inverse rate function that enables efficient computation of our proposed CI. We then move beyond the finite alphabet case and use a Bernoulli coupling idea to construct an almost sure CI for $Σ= [0,1]$, and a conceptually simple nonasymptotic CI for the case of $Σ$ being a $(2,D)$ smooth Banach space. For both finite and general alphabets, our results employ classical large deviation techniques in novel ways, thus establishing new connections between estimation under WoR sampling and the theory of large deviations.

Ege Onur Taga, Samet Oymak, Shubhanshu Shekhar

We develop horizon-aware anytime-valid tests and confidence sequences for bounded means under a strict deadline $N$. Using the betting/e-process framework, we cast horizon-aware betting as a finite-horizon optimal control problem with state space $(t, \log W_t)$, where $t$ is the time and $W_t$ is the test martingale value. We first show that in certain interior regions of the state space, policies that deviate significantly from Kelly betting are provably suboptimal, while Kelly betting reaches the threshold with high probability. We then identify sufficient conditions showing that outside this region, more aggressive betting than Kelly can be better if the bettor is behind schedule, and less aggressive can be better if the bettor is ahead. Taken together these results suggest a simple phase diagram in the $(t, \log W_t)$ plane, delineating regions where Kelly, fractional Kelly, and aggressive betting may be preferable. Guided by this phase diagram, we introduce a Deep Reinforcement Learning approach based on a universal Deep Q-Network (DQN) agent that learns a single policy from synthetic experience and maps simple statistics of past observations to bets across horizons and null values. In limited-horizon experiments, the learned DQN policy yields state-of-the-art results.

Shubhanshu Shekhar, Mohammad Javad Khojasteh, Ananya Acharya et al.

The distinctive architectural features of normalizing flows (NFs), notably bijectivity and tractable Jacobians, make them well-suited for generative modeling. Invertible neural networks (INNs) build on these principles to address supervised inverse problems, enabling direct modeling of both forward and inverse mappings. In this paper, we revisit these architectures from both theoretical and practical perspectives and address a key gap in the literature: the lack of theoretical guarantees on approximation quality under realistic assumptions, whether for posterior inference in INNs or for generative modeling with NFs. We introduce a unified framework for INNs and NFs based on variational unsupervised loss functions, inspired by analogous formulations in related areas such as generative adversarial networks (GANs) and the Precision-Recall divergence for training normalizing flows. Within this framework, we derive theoretical performance guarantees, quantifying posterior accuracy for INNs and distributional accuracy for NFs, under assumptions that are weaker and more practically realistic than those used in prior work. Building on these theoretical results, we conduct extensive case studies to distill general design principles and practical guidelines. We conclude by demonstrating the effectiveness of our approach on a realistic ocean-acoustic inversion problem.

Shubhanshu Shekhar, Shubhada Agrawal

Minimum divergence problems under integral constraints appear throughout statistics and probability, including sequential inference, bandit theory, and distributionally robust optimization. In many such settings, dual representations are the key step that convert information-theoretic lower bounds into computationally tractable (and often near-optimal) algorithms. In this paper, we present a general two-stage recipe for deriving dual representations of constrained minimum divergence (in the second argument) for distributions supported on $[0,1]^K$. The first stage derives a dual representation for finitely-supported distributions using classical finite-dimensional convex duality techniques, while the second establishes an abstract interchange argument that lifts this discretized dual to arbitrary distributions. We begin with the simplest case of mean-constrained minimum relative entropy, commonly called $\mathrm{KL}_{\inf}$, and generalize an existing argument from multi-armed bandits literature for $K=1$ to arbitrary dimensions. Our main contribution is to significantly expand the scope of this approach to a broad class of $f$-divergences (beyond relative entropy) and to general integral constraint functionals (beyond the mean constraint). Finally, we illustrate the statistical implications of our results by constructing optimal procedures for sequential testing, estimation, and change detection with observations in $[0,1]^K$.

Gyumin Lee, Shubhanshu Shekhar, Ilmun Kim

We consider the problem of two-sample testing in a semi-supervised setting with abundant unlabeled covariate data. Standard two-sample tests neglect covariate information, which has the potential to significantly boost performance. However, incorporating covariates potentially breaks the exchangeability assumption under the null, which further complicates a calibration procedure. To address these issues, we propose a semi-supervised method that produces a test statistic with asymptotic normality, while effectively integrating additional information from covariates. Our test is straightforward to calibrate due to the asymptotic normality under the null and achieves asymptotic power that is often much higher than existing kernel tests without covariates. Furthermore, we formally show that the proposed method is consistent in power against fixed and local alternatives. Simulations confirm the practical and theoretical strengths of our approach.

Chia-Yu Hsu, Shubhanshu Shekhar

We consider the problem of constructing sequential power-one tests where the null and alternative classes are specified indirectly through historical or offline data. More specifically, given an offline dataset consisting of observations from $L+1$ distributions $\{P_0, P_1, \ldots, P_L\}$, and a new unlabeled data stream $\{X_t: t \geq 1\} \overset{i.i.d}{\sim} P_θ$, the goal is to decide between the null $H_0: θ= 0$, against the alternative $H_1: θ\in [L]:=\{1,\ldots,L\}$. Our main methodological contribution is a general approach for designing a level-$α$ power-one test for this problem using a multi-class classifier trained on the given offline dataset. Working under a mild "separability" condition on the distributions and the trained classifier, we obtain an upper bound on the expected stopping time of our proposed level-$α$ test, and then show that in general this cannot be improved. In addition to rejecting the null, we show that our procedure can also identify the true underlying distribution almost surely. We then establish a sufficient condition to ensure the required separability of the classifier, and provide some converse results to investigate the role of the size of the offline dataset and the family of classifiers among classifier-based tests that satisfy the level-$α$ power-one criterion. Finally, we present an extension of our analysis for the training-and-testing distribution mismatch and illustrate an application to sequential change detection. Empirical results using both synthetic and real data provide support for our theoretical results.

Shubhanshu Shekhar

We consider the problem of designing optimal level-$α$ power-one tests for composite nulls. Given a parameter $α\in (0,1)$ and a stream of $\mathcal{X}$-valued observations $\{X_n: n \geq 1\} \overset{i.i.d.}{\sim} P$, the goal is to design a level-$α$ power-one test $τ_α$ for the null $H_0: P \in \mathcal{P}_0 \subset \mathcal{P}(\mathcal{X})$. Prior works have shown that any such $τ_α$ must satisfy $\mathbb{E}_P[τ_α] \geq \tfrac{\log(1/α)}{γ^*(P, \mathcal{P}_0)}$, where $γ^*(P, \mathcal{P}_0)$ is the so-called $\mathrm{KL}_{\inf}$ or minimum divergence of $P$ to the null class. In this paper, our objective is to develop and analyze constructive schemes that match this lower bound as $α\downarrow 0$. We first consider the finite-alphabet case~($|\mathcal{X}| = m < \infty$), and show that a test based on \emph{universal} $e$-process~(formed by the ratio of a universal predictor and the running null MLE) is optimal in the above sense. The proof relies on a Donsker-Varadhan~(DV) based saddle-point representation of $\mathrm{KL}_{\inf}$, and an application of Sion's minimax theorem. This characterization motivates a general method for arbitrary $\mathcal{X}$: construct an $e$-process based on the empirical solutions to the saddle-point representation over a sufficiently rich class of test functions. We give sufficient conditions for the optimality of this test for compact convex nulls, and verify them for Hölder smooth density models. We end the paper with a discussion on the computational aspects of implementing our proposed tests in some practical settings.

Shubhanshu Shekhar, Ziyu Xu, Zachary C. Lipton et al.

We introduce the notion of a risk-limiting financial auditing (RLFA): given $N$ transactions, the goal is to estimate the total misstated monetary fraction~($m^*$) to a given accuracy $ε$, with confidence $1-δ$. We do this by constructing new confidence sequences (CSs) for the weighted average of $N$ unknown values, based on samples drawn without replacement according to a (randomized) weighted sampling scheme. Using the idea of importance weighting to construct test martingales, we first develop a framework to construct CSs for arbitrary sampling strategies. Next, we develop methods to improve the quality of CSs by incorporating side information about the unknown values associated with each item. We show that when the side information is sufficiently predictive, it can directly drive the sampling. Addressing the case where the accuracy is unknown a priori, we introduce a method that incorporates side information via control variates. Crucially, our construction is adaptive: if the side information is highly predictive of the unknown misstated amounts, then the benefits of incorporating it are significant; but if the side information is uncorrelated, our methods learn to ignore it. Our methods recover state-of-the-art bounds for the special case when the weights are equal, which has already found applications in election auditing. The harder weighted case solves our more challenging problem of AI-assisted financial auditing.

Dawei Sun, Mohammad Javad Khojasteh, Shubhanshu Shekhar et al.

In this paper, we consider the problem of using a robot to explore an environment with an unknown, state-dependent disturbance function while avoiding some forbidden areas. The goal of the robot is to safely collect observations of the disturbance and construct an accurate estimate of the underlying disturbance function. We use Gaussian Process (GP) to get an estimate of the disturbance from data with a high-confidence bound on the regression error. Furthermore, we use neural Contraction Metrics to derive a tracking controller and the corresponding high-confidence uncertainty tube around the nominal trajectory planned for the robot, based on the estimate of the disturbance. From the robustness of the Contraction Metric, error bound can be pre-computed and used by the motion planner such that the actual trajectory is guaranteed to be safe. As the robot collects more and more observations along its trajectory, the estimate of the disturbance becomes more and more accurate, which in turn improves the performance of the tracking controller and enlarges the free space that the robot can safely explore. We evaluate the proposed method using a carefully designed environment with a ground vehicle. Results show that with the proposed method the robot can thoroughly explore the environment safely and quickly.

Shubhanshu Shekhar, Greg Fields, Mohammad Ghavamzadeh et al.

Machine learning models trained on uncurated datasets can often end up adversely affecting inputs belonging to underrepresented groups. To address this issue, we consider the problem of adaptively constructing training sets which allow us to learn classifiers that are fair in a minimax sense. We first propose an adaptive sampling algorithm based on the principle of optimism, and derive theoretical bounds on its performance. We also propose heuristic extensions of this algorithm suitable for application to large scale, practical problems. Next, by deriving algorithm independent lower-bounds for a specific class of problems, we show that the performance achieved by our adaptive scheme cannot be improved in general. We then validate the benefits of adaptively constructing training sets via experiments on synthetic tasks with logistic regression classifiers, as well as on several real-world tasks using convolutional neural networks (CNNs).

Shubhanshu Shekhar, Tara Javidi

We aim to optimize a black-box function $f:\mathcal{X} \mapsto \mathbb{R}$ under the assumption that $f$ is Hölder smooth and has bounded norm in the RKHS associated with a given kernel $K$. This problem is known to have an agnostic Gaussian Process (GP) bandit interpretation in which an appropriately constructed GP surrogate model with kernel $K$ is used to obtain an upper confidence bound (UCB) algorithm. In this paper, we propose a new algorithm (\texttt{LP-GP-UCB}) where the usual GP surrogate model is augmented with Local Polynomial (LP) estimators of the Hölder smooth function $f$ to construct a multi-scale UCB guiding the search for the optimizer. We analyze this algorithm and derive high probability bounds on its simple and cumulative regret. We then prove that the elements of many common RKHS are Hölder smooth and obtain the corresponding Hölder smoothness parameters, and hence, specialize our regret bounds for several commonly used kernels. When specialized to the Squared Exponential (SE) kernel, \texttt{LP-GP-UCB} matches the optimal performance, while for the case of Matérn kernels $(K_ν)_{ν>0}$, it results in uniformly tighter regret bounds for all values of the smoothness parameter $ν>0$. Most notably, for certain ranges of $ν$, the algorithm achieves near-optimal bounds on simple and cumulative regrets, matching the algorithm-independent lower bounds up to polylog factors, and thus closing the large gap between the existing upper and lower bounds for these values of $ν$. Additionally, our analysis provides the first explicit regret bounds, in terms of the budget $n$, for the Rational-Quadratic (RQ) and Gamma-Exponential (GE). Finally, experiments with synthetic functions as well as a CNN hyperparameter tuning task demonstrate the practical benefits of our multi-scale partitioning approach over some existing algorithms numerically.

Jean Tarbouriech, Shubhanshu Shekhar, Matteo Pirotta et al.

We study the problem of efficient exploration in order to learn an accurate model of an environment, modeled as a Markov decision process (MDP). Efficient exploration in this problem requires the agent to identify the regions in which estimating the model is more difficult and then exploit this knowledge to collect more samples there. In this paper, we formalize this problem, introduce the first algorithm to learn an $ε$-accurate estimate of the dynamics, and provide its sample complexity analysis. While this algorithm enjoys strong guarantees in the large-sample regime, it tends to have a poor performance in early stages of exploration. To address this issue, we propose an algorithm that is based on maximum weighted entropy, a heuristic that stems from common sense and our theoretical analysis. The main idea here is to cover the entire state-action space with the weight proportional to the noise in the transitions. Using a number of simple domains with heterogeneous noise in their transitions, we show that our heuristic-based algorithm outperforms both our original algorithm and the maximum entropy algorithm in the small sample regime, while achieving similar asymptotic performance as that of the original algorithm.

Shubhanshu Shekhar, Tara Javidi, Mohammad Ghavamzadeh

We consider the problem of allocating samples to a finite set of discrete distributions in order to learn them uniformly well in terms of four common distance measures: $\ell_2^2$, $\ell_1$, $f$-divergence, and separation distance. To present a unified treatment of these distances, we first propose a general optimistic tracking algorithm and analyze its sample allocation performance w.r.t.~an oracle. We then instantiate this algorithm for the four distance measures and derive bounds on the regret of their resulting allocation schemes. We verify our theoretical findings through some experiments. Finally, we show that the techniques developed in the paper can be easily extended to the related setting of minimizing the average error (in terms of the four distances) in learning a set of distributions.

Shubhanshu Shekhar, Mohammad Ghavamzadeh, Tara Javidi

We construct and analyze active learning algorithms for the problem of binary classification with abstention. We consider three abstention settings: \emph{fixed-cost} and two variants of \emph{bounded-rate} abstention, and for each of them propose an active learning algorithm. All the proposed algorithms can work in the most commonly used active learning models, i.e., \emph{membership-query}, \emph{pool-based}, and \emph{stream-based} sampling. We obtain upper-bounds on the excess risk of our algorithms in a general non-parametric framework and establish their minimax near-optimality by deriving matching lower-bounds. Since our algorithms rely on the knowledge of some smoothness parameters of the regression function, we then describe a new strategy to adapt to these unknown parameters in a data-driven manner. Since the worst case computational complexity of our proposed algorithms increases exponentially with the dimension of the input space, we conclude the paper with a computationally efficient variant of our algorithm whose computational complexity has a polynomial dependence over a smaller but rich class of learning problems.

Shubhanshu Shekhar, Mohammad Ghavamzadeh, Tara Javidi

We consider the problem of binary classification with abstention in the relatively less studied \emph{bounded-rate} setting. We begin by obtaining a characterization of the Bayes optimal classifier for an arbitrary input-label distribution $P_{XY}$. Our result generalizes and provides an alternative proof for the result first obtained by \cite{chow1957optimum}, and then re-derived by \citet{denis2015consistency}, under a continuity assumption on $P_{XY}$. We then propose a plug-in classifier that employs unlabeled samples to decide the region of abstention and derive an upper-bound on the excess risk of our classifier under standard \emph{Hölder smoothness} and \emph{margin} assumptions. Unlike the plug-in rule of \citet{denis2015consistency}, our constructed classifier satisfies the abstention constraint with high probability and can also deal with discontinuities in the empirical cdf. We also derive lower-bounds that demonstrate the minimax near-optimality of our proposed algorithm. To address the excessive complexity of the plug-in classifier in high dimensions, we propose a computationally efficient algorithm that builds upon prior work on convex loss surrogates, and obtain bounds on its excess risk in the \emph{realizable} case. We empirically compare the performance of the proposed algorithm with a baseline on a number of UCI benchmark datasets.

Shubhanshu Shekhar, Tara Javidi

In this paper, the problem of estimating the level set of a black-box function from noisy and expensive evaluation queries is considered. A new algorithm for this problem in the Bayesian framework with a Gaussian Process (GP) prior is proposed. The proposed algorithm employs a hierarchical sequence of partitions to explore different regions of the search space at varying levels of detail depending upon their proximity to the level set boundary. It is shown that this approach results in the algorithm having a low complexity implementation whose computational cost is significantly smaller than the existing algorithms for higher dimensional search space $\X$. Furthermore, high probability bounds on a measure of discrepancy between the estimated level set and the true level set for the the proposed algorithm are obtained, which are shown to be strictly better than the existing guarantees for a large class of GPs. In the process, a tighter characterization of the information gain of the proposed algorithm is obtained which takes into account the structured nature of the evaluation points. This approach improves upon the existing technique of bounding the information gain with maximum information gain.

Shubhanshu Shekhar, Tara Javidi

In this paper, the problem of maximizing a black-box function $f:\mathcal{X} \to \mathbb{R}$ is studied in the Bayesian framework with a Gaussian Process (GP) prior. In particular, a new algorithm for this problem is proposed, and high probability bounds on its simple and cumulative regret are established. The query point selection rule in most existing methods involves an exhaustive search over an increasingly fine sequence of uniform discretizations of $\mathcal{X}$. The proposed algorithm, in contrast, adaptively refines $\mathcal{X}$ which leads to a lower computational complexity, particularly when $\mathcal{X}$ is a subset of a high dimensional Euclidean space. In addition to the computational gains, sufficient conditions are identified under which the regret bounds of the new algorithm improve upon the known results. Finally an extension of the algorithm to the case of contextual bandits is proposed, and high probability bounds on the contextual regret are presented.