Lalit Jain

David P. Woodruff, Vincent Cohen-Addad, Lalit Jain et al.

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Zhaoqi Li, Lillian Ratliff, Houssam Nassif et al.

In the stochastic contextual bandit setting, regret-minimizing algorithms have been extensively researched, but their instance-minimizing best-arm identification counterparts remain seldom studied. In this work, we focus on the stochastic bandit problem in the $(ε,δ)$-$\textit{PAC}$ setting: given a policy class $Π$ the goal of the learner is to return a policy $π\in Π$ whose expected reward is within $ε$ of the optimal policy with probability greater than $1-δ$. We characterize the first $\textit{instance-dependent}$ PAC sample complexity of contextual bandits through a quantity $ρ_Π$, and provide matching upper and lower bounds in terms of $ρ_Π$ for the agnostic and linear contextual best-arm identification settings. We show that no algorithm can be simultaneously minimax-optimal for regret minimization and instance-dependent PAC for best-arm identification. Our main result is a new instance-optimal and computationally efficient algorithm that relies on a polynomial number of calls to an argmax oracle.

Romain Camilleri, Andrew Wagenmaker, Jamie Morgenstern et al.

Active learning methods have shown great promise in reducing the number of samples necessary for learning. As automated learning systems are adopted into real-time, real-world decision-making pipelines, it is increasingly important that such algorithms are designed with safety in mind. In this work we investigate the complexity of learning the best safe decision in interactive environments. We reduce this problem to a constrained linear bandits problem, where our goal is to find the best arm satisfying certain (unknown) safety constraints. We propose an adaptive experimental design-based algorithm, which we show efficiently trades off between the difficulty of showing an arm is unsafe vs suboptimal. To our knowledge, our results are the first on best-arm identification in linear bandits with safety constraints. In practice, we demonstrate that this approach performs well on synthetic and real world datasets.

Shima Alizadeh, Aniruddha Bhargava, Karthick Gopalswamy et al.

Multi-objective optimization is a type of decision making problems where multiple conflicting objectives are optimized. We study offline optimization of multi-objective policies from data collected by an existing policy. We propose a pessimistic estimator for the multi-objective policy values that can be easily plugged into existing formulas for hypervolume computation and optimized. The estimator is based on inverse propensity scores (IPS), and improves upon a naive IPS estimator in both theory and experiments. Our analysis is general, and applies beyond our IPS estimators and methods for optimizing them. The pessimistic estimator can be optimized by policy gradients and performs well in all of our experiments.

Tanner Fiez, Sergio Gamez, Arick Chen et al.

Adaptive experimental design methods are increasingly being used in industry as a tool to boost testing throughput or reduce experimentation cost relative to traditional A/B/N testing methods. This paper shares lessons learned regarding the challenges and pitfalls of naively using adaptive experimentation systems in industrial settings where non-stationarity is prevalent, while also providing perspectives on the proper objectives and system specifications in these settings. We developed an adaptive experimental design framework for counterfactual inference based on these experiences, and tested it in a commercial environment.

Zhihan Xiong, Romain Camilleri, Maryam Fazel et al.

We investigate the fixed-budget best-arm identification (BAI) problem for linear bandits in a potentially non-stationary environment. Given a finite arm set $\mathcal{X}\subset\mathbb{R}^d$, a fixed budget $T$, and an unpredictable sequence of parameters $\left\lbraceθ_t\right\rbrace_{t=1}^{T}$, an algorithm will aim to correctly identify the best arm $x^* := \arg\max_{x\in\mathcal{X}}x^\top\sum_{t=1}^{T}θ_t$ with probability as high as possible. Prior work has addressed the stationary setting where $θ_t = θ_1$ for all $t$ and demonstrated that the error probability decreases as $\exp(-T /ρ^*)$ for a problem-dependent constant $ρ^*$. But in many real-world $A/B/n$ multivariate testing scenarios that motivate our work, the environment is non-stationary and an algorithm expecting a stationary setting can easily fail. For robust identification, it is well-known that if arms are chosen randomly and non-adaptively from a G-optimal design over $\mathcal{X}$ at each time then the error probability decreases as $\exp(-TΔ^2_{(1)}/d)$, where $Δ_{(1)} = \min_{x \neq x^*} (x^* - x)^\top \frac{1}{T}\sum_{t=1}^T θ_t$. As there exist environments where $Δ_{(1)}^2/ d \ll 1/ ρ^*$, we are motivated to propose a novel algorithm $\mathsf{P1}$-$\mathsf{RAGE}$ that aims to obtain the best of both worlds: robustness to non-stationarity and fast rates of identification in benign settings. We characterize the error probability of $\mathsf{P1}$-$\mathsf{RAGE}$ and demonstrate empirically that the algorithm indeed never performs worse than G-optimal design but compares favorably to the best algorithms in the stationary setting.

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.

Zhaoqi Li, Kevin Jamieson, Lalit Jain

Given a set of arms $\mathcal{Z}\subset \mathbb{R}^d$ and an unknown parameter vector $θ_\ast\in\mathbb{R}^d$, the pure exploration linear bandit problem aims to return $\arg\max_{z\in \mathcal{Z}} z^{\top}θ_{\ast}$, with high probability through noisy measurements of $x^{\top}θ_{\ast}$ with $x\in \mathcal{X}\subset \mathbb{R}^d$. Existing (asymptotically) optimal methods require either a) potentially costly projections for each arm $z\in \mathcal{Z}$ or b) explicitly maintaining a subset of $\mathcal{Z}$ under consideration at each time. This complexity is at odds with the popular and simple Thompson Sampling algorithm for regret minimization, which just requires access to a posterior sampling and argmax oracle, and does not need to enumerate $\mathcal{Z}$ at any point. Unfortunately, Thompson sampling is known to be sub-optimal for pure exploration. In this work, we pose a natural question: is there an algorithm that can explore optimally and only needs the same computational primitives as Thompson Sampling? We answer the question in the affirmative. We provide an algorithm that leverages only sampling and argmax oracles and achieves an exponential convergence rate, with the exponent being the optimal among all possible allocations asymptotically. In addition, we show that our algorithm can be easily implemented and performs as well empirically as existing asymptotically optimal methods.

Artin Tajdini, Lalit Jain, Kevin Jamieson

We consider maximizing an unknown monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ with cardinality constraint under stochastic bandit feedback. At each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + η_t$ where $η_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$, obtained through robust greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable using standard multi-armed bandit algorithms. In this work, we establish the first minimax lower bound for this setting that scales like $\tildeΩ(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tildeΩ(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. Moreover, we propose an algorithm Sub-UCB that achieves regret $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$ capable of matching the lower bound on regret for the restricted class up to logarithmic factors.

Shyam Nuggehalli, Jifan Zhang, Lalit Jain et al.

Class imbalance severely impacts machine learning performance on minority classes in real-world applications. While various solutions exist, active learning offers a fundamental fix by strategically collecting balanced, informative labeled examples from abundant unlabeled data. We introduce DIRECT, an algorithm that identifies class separation boundaries and selects the most uncertain nearby examples for annotation. By reducing the problem to one-dimensional active learning, DIRECT leverages established theory to handle batch labeling and label noise -- another common challenge in data annotation that particularly affects active learning methods. Our work presents the first comprehensive study of active learning under both class imbalance and label noise. Extensive experiments on imbalanced datasets show DIRECT reduces annotation costs by over 60\% compared to state-of-the-art active learning methods and over 80\% versus random sampling, while maintaining robustness to label noise.

Tanner Fiez, Houssam Nassif, Yu-Cheng Chen et al.

Adaptive experimental design (AED) methods are increasingly being used in industry as a tool to boost testing throughput or reduce experimentation cost relative to traditional A/B/N testing methods. However, the behavior and guarantees of such methods are not well-understood beyond idealized stationary settings. This paper shares lessons learned regarding the challenges of naively using AED systems in industrial settings where non-stationarity is prevalent, while also providing perspectives on the proper objectives and system specifications in such settings. We developed an AED framework for counterfactual inference based on these experiences, and tested it in a commercial environment.

Enoch H. Kang, Hema Yoganarasimhan, Lalit Jain

We study the problem of estimating Dynamic Discrete Choice (DDC) models, also known as offline Maximum Entropy-Regularized Inverse Reinforcement Learning (offline MaxEnt-IRL) in machine learning. The objective is to recover reward or $Q^*$ functions that govern agent behavior from offline behavior data. In this paper, we propose a globally convergent gradient-based method for solving these problems without the restrictive assumption of linearly parameterized rewards. The novelty of our approach lies in introducing the Empirical Risk Minimization (ERM) based IRL/DDC framework, which circumvents the need for explicit state transition probability estimation in the Bellman equation. Furthermore, our method is compatible with non-parametric estimation techniques such as neural networks. Therefore, the proposed method has the potential to be scaled to high-dimensional, infinite state spaces. A key theoretical insight underlying our approach is that the Bellman residual satisfies the Polyak-Lojasiewicz (PL) condition -- a property that, while weaker than strong convexity, is sufficient to ensure fast global convergence guarantees. Through a series of synthetic experiments, we demonstrate that our approach consistently outperforms benchmark methods and state-of-the-art alternatives.

Romain Camilleri, Andrew Wagenmaker, Jamie Morgenstern et al.

In critical machine learning applications, ensuring fairness is essential to avoid perpetuating social inequities. In this work, we address the challenges of reducing bias and improving accuracy in data-scarce environments, where the cost of collecting labeled data prohibits the use of large, labeled datasets. In such settings, active learning promises to maximize marginal accuracy gains of small amounts of labeled data. However, existing applications of active learning for fairness fail to deliver on this, typically requiring large labeled datasets, or failing to ensure the desired fairness tolerance is met on the population distribution. To address such limitations, we introduce an innovative active learning framework that combines an exploration procedure inspired by posterior sampling with a fair classification subroutine. We demonstrate that this framework performs effectively in very data-scarce regimes, maximizing accuracy while satisfying fairness constraints with high probability. We evaluate our proposed approach using well-established real-world benchmark datasets and compare it against state-of-the-art methods, demonstrating its effectiveness in producing fair models, and improvement over existing methods.

Reuben Narad, Siddharth Suresh, Jiayi Chen et al.

We present HumorBench, a benchmark designed to evaluate large language models' (LLMs) ability to reason about and explain sophisticated humor in cartoon captions. As reasoning models increasingly saturate existing benchmarks in mathematics and science, novel and challenging evaluations of model intelligence beyond STEM domains are essential. Reasoning is fundamentally involved in text-based humor comprehension, requiring the identification of connections between concepts in cartoons/captions and external cultural references, wordplays, and other mechanisms. HumorBench includes approximately 300 unique cartoon-caption pairs from the New Yorker Caption Contest and Cartoonstock.com, with expert-annotated evaluation rubrics identifying essential joke elements. LLMs are evaluated based on their explanations towards the humor and abilities in identifying the joke elements. To perform well on this task, models must form and test hypotheses about associations between concepts, potentially backtracking from initial interpretations to arrive at the most plausible explanation. Our extensive benchmarking of current SOTA models reveals three key insights: (1) LLM progress on STEM reasoning transfers effectively to humor comprehension; (2) models trained exclusively on STEM reasoning data still perform well on HumorBench, demonstrating strong transferability of reasoning abilities; and (3) test-time scaling by increasing thinking token budgets yields mixed results across different models in humor reasoning.

Yao Zhao, Kwang-Sung Jun, Tanner Fiez et al.

This paper introduces the \emph{confounded pure exploration transductive linear bandit} (\texttt{CPET-LB}) problem. As a motivating example, often online services cannot directly assign users to specific control or treatment experiences either for business or practical reasons. In these settings, naively comparing treatment and control groups that may result from self-selection can lead to biased estimates of underlying treatment effects. Instead, online services can employ a properly randomized encouragement that incentivizes users toward a specific treatment. Our methodology provides online services with an adaptive experimental design approach for learning the best-performing treatment for such \textit{encouragement designs}. We consider a more general underlying model captured by a linear structural equation and formulate pure exploration linear bandits in this setting. Though pure exploration has been extensively studied in standard adaptive experimental design settings, we believe this is the first work considering a setting where noise is confounded. Elimination-style algorithms using experimental design methods in combination with a novel finite-time confidence interval on an instrumental variable style estimator are presented with sample complexity upper bounds nearly matching a minimax lower bound. Finally, experiments are conducted that demonstrate the efficacy of our approach.

Jifan Zhang, Lalit Jain, Yang Guo et al.

We present a novel multimodal preference dataset for creative tasks, consisting of over 250 million human ratings on more than 2.2 million captions, collected through crowdsourcing rating data for The New Yorker's weekly cartoon caption contest over the past eight years. This unique dataset supports the development and evaluation of multimodal large language models and preference-based fine-tuning algorithms for humorous caption generation. We propose novel benchmarks for judging the quality of model-generated captions, utilizing both GPT4 and human judgments to establish ranking-based evaluation strategies. Our experimental results highlight the limitations of current fine-tuning methods, such as RLHF and DPO, when applied to creative tasks. Furthermore, we demonstrate that even state-of-the-art models like GPT4 and Claude currently underperform top human contestants in generating humorous captions. As we conclude this extensive data collection effort, we release the entire preference dataset to the research community, fostering further advancements in AI humor generation and evaluation.

Aniruddha Bhargava, Lalit Jain, Branislav Kveton et al.

Learning from human feedback has been central to recent advances in artificial intelligence and machine learning. Since the collection of human feedback is costly, a natural question to ask is if the new feedback always needs to collected. Or could we evaluate a new model with the human feedback on responses of another model? This motivates us to study off-policy evaluation from logged human feedback. We formalize the problem, propose both model-based and model-free estimators for policy values, and show how to optimize them. We analyze unbiasedness of our estimators and evaluate them empirically. Our estimators can predict the absolute values of evaluated policies, rank them, and be optimized.

Blake Mason, Kwang-Sung Jun, Lalit Jain

In this work we consider the problem of regret minimization for logistic bandits. The main challenge of logistic bandits is reducing the dependence on a potentially large problem dependent constant $κ$ that can at worst scale exponentially with the norm of the unknown parameter $θ_{\ast}$. Abeille et al. (2021) have applied self-concordance of the logistic function to remove this worst-case dependence providing regret guarantees like $O(d\log^2(κ)\sqrt{\dotμT}\log(|\mathcal{X}|))$ where $d$ is the dimensionality, $T$ is the time horizon, and $\dotμ$ is the variance of the best-arm. This work improves upon this bound in the fixed arm setting by employing an experimental design procedure that achieves a minimax regret of $O(\sqrt{d \dotμT\log(|\mathcal{X}|)})$. Our regret bound in fact takes a tighter instance (i.e., gap) dependent regret bound for the first time in logistic bandits. We also propose a new warmup sampling algorithm that can dramatically reduce the lower order term in the regret in general and prove that it can replace the lower order term dependency on $κ$ to $\log^2(κ)$ for some instances. Finally, we discuss the impact of the bias of the MLE on the logistic bandit problem, providing an example where $d^2$ lower order regret (cf., it is $d$ for linear bandits) may not be improved as long as the MLE is used and how bias-corrected estimators may be used to make it closer to $d$.

Blake Mason, Romain Camilleri, Subhojyoti Mukherjee et al.

The level set estimation problem seeks to find all points in a domain ${\cal X}$ where the value of an unknown function $f:{\cal X}\rightarrow \mathbb{R}$ exceeds a threshold $α$. The estimation is based on noisy function evaluations that may be acquired at sequentially and adaptively chosen locations in ${\cal X}$. The threshold value $α$ can either be \emph{explicit} and provided a priori, or \emph{implicit} and defined relative to the optimal function value, i.e. $α= (1-ε)f(x_\ast)$ for a given $ε> 0$ where $f(x_\ast)$ is the maximal function value and is unknown. In this work we provide a new approach to the level set estimation problem by relating it to recent adaptive experimental design methods for linear bandits in the Reproducing Kernel Hilbert Space (RKHS) setting. We assume that $f$ can be approximated by a function in the RKHS up to an unknown misspecification and provide novel algorithms for both the implicit and explicit cases in this setting with strong theoretical guarantees. Moreover, in the linear (kernel) setting, we show that our bounds are nearly optimal, namely, our upper bounds match existing lower bounds for threshold linear bandits. To our knowledge this work provides the first instance-dependent, non-asymptotic upper bounds on sample complexity of level-set estimation that match information theoretic lower bounds.

Romain Camilleri, Zhihan Xiong, Maryam Fazel et al.

This work considers the problem of selective-sampling for best-arm identification. Given a set of potential options $\mathcal{Z}\subset\mathbb{R}^d$, a learner aims to compute with probability greater than $1-δ$, $\arg\max_{z\in \mathcal{Z}} z^{\top}θ_{\ast}$ where $θ_{\ast}$ is unknown. At each time step, a potential measurement $x_t\in \mathcal{X}\subset\mathbb{R}^d$ is drawn IID and the learner can either choose to take the measurement, in which case they observe a noisy measurement of $x^{\top}θ_{\ast}$, or to abstain from taking the measurement and wait for a potentially more informative point to arrive in the stream. Hence the learner faces a fundamental trade-off between the number of labeled samples they take and when they have collected enough evidence to declare the best arm and stop sampling. The main results of this work precisely characterize this trade-off between labeled samples and stopping time and provide an algorithm that nearly-optimally achieves the minimal label complexity given a desired stopping time. In addition, we show that the optimal decision rule has a simple geometric form based on deciding whether a point is in an ellipse or not. Finally, our framework is general enough to capture binary classification improving upon previous works.

Julian Katz-Samuels, Jifan Zhang, Lalit Jain et al.

We consider active learning for binary classification in the agnostic pool-based setting. The vast majority of works in active learning in the agnostic setting are inspired by the CAL algorithm where each query is uniformly sampled from the disagreement region of the current version space. The sample complexity of such algorithms is described by a quantity known as the disagreement coefficient which captures both the geometry of the hypothesis space as well as the underlying probability space. To date, the disagreement coefficient has been justified by minimax lower bounds only, leaving the door open for superior instance dependent sample complexities. In this work we propose an algorithm that, in contrast to uniform sampling over the disagreement region, solves an experimental design problem to determine a distribution over examples from which to request labels. We show that the new approach achieves sample complexity bounds that are never worse than the best disagreement coefficient-based bounds, but in specific cases can be dramatically smaller. From a practical perspective, the proposed algorithm requires no hyperparameters to tune (e.g., to control the aggressiveness of sampling), and is computationally efficient by means of assuming access to an empirical risk minimization oracle (without any constraints). Empirically, we demonstrate that our algorithm is superior to state of the art agnostic active learning algorithms on image classification datasets.

Kwang-Sung Jun, Lalit Jain, Blake Mason et al.

We propose improved fixed-design confidence bounds for the linear logistic model. Our bounds significantly improve upon the state-of-the-art bound by Li et al. (2017) via recent developments of the self-concordant analysis of the logistic loss (Faury et al., 2020). Specifically, our confidence bound avoids a direct dependence on $1/κ$, where $κ$ is the minimal variance over all arms' reward distributions. In general, $1/κ$ scales exponentially with the norm of the unknown linear parameter $θ^*$. Instead of relying on this worst-case quantity, our confidence bound for the reward of any given arm depends directly on the variance of that arm's reward distribution. We present two applications of our novel bounds to pure exploration and regret minimization logistic bandits improving upon state-of-the-art performance guarantees. For pure exploration, we also provide a lower bound highlighting a dependence on $1/κ$ for a family of instances.

Jifan Zhang, Lalit Jain, Kevin Jamieson

This work proposes a procedure for designing algorithms for specific adaptive data collection tasks like active learning and pure-exploration multi-armed bandits. Unlike the design of traditional adaptive algorithms that rely on concentration of measure and careful analysis to justify the correctness and sample complexity of the procedure, our adaptive algorithm is learned via adversarial training over equivalence classes of problems derived from information theoretic lower bounds. In particular, a single adaptive learning algorithm is learned that competes with the best adaptive algorithm learned for each equivalence class. Our procedure takes as input just the available queries, set of hypotheses, loss function, and total query budget. This is in contrast to existing meta-learning work that learns an adaptive algorithm relative to an explicit, user-defined subset or prior distribution over problems which can be challenging to define and be mismatched to the instance encountered at test time. This work is particularly focused on the regime when the total query budget is very small, such as a few dozen, which is much smaller than those budgets typically considered by theoretically derived algorithms. We perform synthetic experiments to justify the stability and effectiveness of the training procedure, and then evaluate the method on tasks derived from real data including a noisy 20 Questions game and a joke recommendation task.

Lalit Jain, Kevin Jamieson

In many scientific settings there is a need for adaptive experimental design to guide the process of identifying regions of the search space that contain as many true positives as possible subject to a low rate of false discoveries (i.e. false alarms). Such regions of the search space could differ drastically from a predicted set that minimizes 0/1 error and accurate identification could require very different sampling strategies. Like active learning for binary classification, this experimental design cannot be optimally chosen a priori, but rather the data must be taken sequentially and adaptively. However, unlike classification with 0/1 error, collecting data adaptively to find a set with high true positive rate and low false discovery rate (FDR) is not as well understood. In this paper we provide the first provably sample efficient adaptive algorithm for this problem. Along the way we highlight connections between classification, combinatorial bandits, and FDR control making contributions to each.

Umang Varma, Lalit Jain, Anna C. Gilbert

Given a number of pairwise preferences of items, a common task is to rank all the items. Examples include pairwise movie ratings, New Yorker cartoon caption contests, and many other consumer preferences tasks. What these settings have in common is two-fold: a scarcity of data (it may be costly to get comparisons for all the pairs of items) and additional feature information about the items (e.g., movie genre, director, and cast). In this paper we modify a popular and well studied method, RankCentrality for rank aggregation to account for few comparisons and that incorporates additional feature information. This method returns meaningful rankings even under scarce comparisons. Using diffusion based methods, we incorporate feature information that outperforms state-of-the-art methods in practice. We also provide improved sample complexity for RankCentrality in a variety of sampling schemes.

Julian Katz-Samuels, Lalit Jain, Zohar Karnin et al.

This paper proposes near-optimal algorithms for the pure-exploration linear bandit problem in the fixed confidence and fixed budget settings. Leveraging ideas from the theory of suprema of empirical processes, we provide an algorithm whose sample complexity scales with the geometry of the instance and avoids an explicit union bound over the number of arms. Unlike previous approaches which sample based on minimizing a worst-case variance (e.g. G-optimal design), we define an experimental design objective based on the Gaussian-width of the underlying arm set. We provide a novel lower bound in terms of this objective that highlights its fundamental role in the sample complexity. The sample complexity of our fixed confidence algorithm matches this lower bound, and in addition is computationally efficient for combinatorial classes, e.g. shortest-path, matchings and matroids, where the arm sets can be exponentially large in the dimension. Finally, we propose the first algorithm for linear bandits in the the fixed budget setting. Its guarantee matches our lower bound up to logarithmic factors.

Blake Mason, Lalit Jain, Ardhendu Tripathy et al.

The pure-exploration problem in stochastic multi-armed bandits aims to find one or more arms with the largest (or near largest) means. Examples include finding an ε-good arm, best-arm identification, top-k arm identification, and finding all arms with means above a specified threshold. However, the problem of finding all ε-good arms has been overlooked in past work, although arguably this may be the most natural objective in many applications. For example, a virologist may conduct preliminary laboratory experiments on a large candidate set of treatments and move all ε-good treatments into more expensive clinical trials. Since the ultimate clinical efficacy is uncertain, it is important to identify all ε-good candidates. Mathematically, the all-ε-good arm identification problem presents significant new challenges and surprises that do not arise in the pure-exploration objectives studied in the past. We introduce two algorithms to overcome these and demonstrate their great empirical performance on a large-scale crowd-sourced dataset of 2.2M ratings collected by the New Yorker Caption Contest as well as a dataset testing hundreds of possible cancer drugs.

Tanner Fiez, Lalit Jain, Kevin Jamieson et al.

In this paper we introduce the transductive linear bandit problem: given a set of measurement vectors $\mathcal{X}\subset \mathbb{R}^d$, a set of items $\mathcal{Z}\subset \mathbb{R}^d$, a fixed confidence $δ$, and an unknown vector $θ^{\ast}\in \mathbb{R}^d$, the goal is to infer $\text{argmax}_{z\in \mathcal{Z}} z^\topθ^\ast$ with probability $1-δ$ by making as few sequentially chosen noisy measurements of the form $x^\topθ^{\ast}$ as possible. When $\mathcal{X}=\mathcal{Z}$, this setting generalizes linear bandits, and when $\mathcal{X}$ is the standard basis vectors and $\mathcal{Z}\subset \{0,1\}^d$, combinatorial bandits. Such a transductive setting naturally arises when the set of measurement vectors is limited due to factors such as availability or cost. As an example, in drug discovery the compounds and dosages $\mathcal{X}$ a practitioner may be willing to evaluate in the lab in vitro due to cost or safety reasons may differ vastly from those compounds and dosages $\mathcal{Z}$ that can be safely administered to patients in vivo. Alternatively, in recommender systems for books, the set of books $\mathcal{X}$ a user is queried about may be restricted to well known best-sellers even though the goal might be to recommend more esoteric titles $\mathcal{Z}$. In this paper, we provide instance-dependent lower bounds for the transductive setting, an algorithm that matches these up to logarithmic factors, and an evaluation. In particular, we provide the first non-asymptotic algorithm for linear bandits that nearly achieves the information theoretic lower bound.

Jordan S. Ellenberg, Lalit Jain

We prove optimal bounds for the convergence rate of ordinal embedding (also known as non-metric multidimensional scaling) in the 1-dimensional case. The examples witnessing optimality of our bounds arise from a result in additive number theory on sets of integers with no three-term arithmetic progressions. We also carry out some computational experiments aimed at developing a sense of what the convergence rate for ordinal embedding might look like in higher dimensions.

Kevin Jamieson, Lalit Jain

We propose an adaptive sampling approach for multiple testing which aims to maximize statistical power while ensuring anytime false discovery control. We consider $n$ distributions whose means are partitioned by whether they are below or equal to a baseline (nulls), versus above the baseline (actual positives). In addition, each distribution can be sequentially and repeatedly sampled. Inspired by the multi-armed bandit literature, we provide an algorithm that takes as few samples as possible to exceed a target true positive proportion (i.e. proportion of actual positives discovered) while giving anytime control of the false discovery proportion (nulls predicted as actual positives). Our sample complexity results match known information theoretic lower bounds and through simulations we show a substantial performance improvement over uniform sampling and an adaptive elimination style algorithm. Given the simplicity of the approach, and its sample efficiency, the method has promise for wide adoption in the biological sciences, clinical testing for drug discovery, and online A/B/n testing problems.

Sumeet Katariya, Lalit Jain, Nandana Sengupta et al.

We consider the problem of active coarse ranking, where the goal is to sort items according to their means into clusters of pre-specified sizes, by adaptively sampling from their reward distributions. This setting is useful in many social science applications involving human raters and the approximate rank of every item is desired. Approximate or coarse ranking can significantly reduce the number of ratings required in comparison to the number needed to find an exact ranking. We propose a computationally efficient PAC algorithm LUCBRank for coarse ranking, and derive an upper bound on its sample complexity. We also derive a nearly matching distribution-dependent lower bound. Experiments on synthetic as well as real-world data show that LUCBRank performs better than state-of-the-art baseline methods, even when these methods have the advantage of knowing the underlying parametric model.

Anna C. Gilbert, Lalit Jain

Many modern data-intensive computational problems either require, or benefit from distance or similarity data that adhere to a metric. The algorithms run faster or have better performance guarantees. Unfortunately, in real applications, the data are messy and values are noisy. The distances between the data points are far from satisfying a metric. Indeed, there are a number of different algorithms for finding the closest set of distances to the given ones that also satisfy a metric (sometimes with the extra condition of being Euclidean). These algorithms can have unintended consequences, they can change a large number of the original data points, and alter many other features of the data. The goal of sparse metric repair is to make as few changes as possible to the original data set or underlying distances so as to ensure the resulting distances satisfy the properties of a metric. In other words, we seek to minimize the sparsity (or the $\ell_0$ "norm") of the changes we make to the distances subject to the new distances satisfying a metric. We give three different combinatorial algorithms to repair a metric sparsely. In one setting the algorithm is guaranteed to return the sparsest solution and in the other settings, the algorithms repair the metric. Without prior information, the algorithms run in time proportional to the cube of the number of input data points and, with prior information we can reduce the running time considerably.

Lalit Jain, Blake Mason, Robert Nowak

This paper investigates the theoretical foundations of metric learning, focused on three key questions that are not fully addressed in prior work: 1) we consider learning general low-dimensional (low-rank) metrics as well as sparse metrics; 2) we develop upper and lower (minimax)bounds on the generalization error; 3) we quantify the sample complexity of metric learning in terms of the dimension of the feature space and the dimension/rank of the underlying metric;4) we also bound the accuracy of the learned metric relative to the underlying true generative metric. All the results involve novel mathematical approaches to the metric learning problem, and lso shed new light on the special case of ordinal embedding (aka non-metric multidimensional scaling).

Lalit Jain, Kevin Jamieson, Robert Nowak

The goal of ordinal embedding is to represent items as points in a low-dimensional Euclidean space given a set of constraints in the form of distance comparisons like "item $i$ is closer to item $j$ than item $k$". Ordinal constraints like this often come from human judgments. To account for errors and variation in judgments, we consider the noisy situation in which the given constraints are independently corrupted by reversing the correct constraint with some probability. This paper makes several new contributions to this problem. First, we derive prediction error bounds for ordinal embedding with noise by exploiting the fact that the rank of a distance matrix of points in $\mathbb{R}^d$ is at most $d+2$. These bounds characterize how well a learned embedding predicts new comparative judgments. Second, we investigate the special case of a known noise model and study the Maximum Likelihood estimator. Third, knowledge of the noise model enables us to relate prediction errors to embedding accuracy. This relationship is highly non-trivial since we show that the linear map corresponding to distance comparisons is non-invertible, but there exists a nonlinear map that is invertible. Fourth, two new algorithms for ordinal embedding are proposed and evaluated in experiments.