Sayak Ray Chowdhury

Daman Arora, Anush Kini, Sayak Ray Chowdhury et al. · cmu

Given a query and a document corpus, the information retrieval (IR) task is to output a ranked list of relevant documents. Combining large language models (LLMs) with embedding-based retrieval models, recent work shows promising results on the zero-shot retrieval problem, i.e., no access to labeled data from the target domain. Two such popular paradigms are generation-augmented retrieval or GAR (generate additional context for the query and then retrieve), and retrieval-augmented generation or RAG (retrieve relevant documents as context and then generate answers). The success of these paradigms hinges on (i) high-recall retrieval models, which are difficult to obtain in the zero-shot setting, and (ii) high-precision (re-)ranking models which typically need a good initialization. In this work, we propose a novel GAR-meets-RAG recurrence formulation that overcomes the challenges of existing paradigms. Our method iteratively improves retrieval (via GAR) and rewrite (via RAG) stages in the zero-shot setting. A key design principle is that the rewrite-retrieval stages improve the recall of the system and a final re-ranking stage improves the precision. We conduct extensive experiments on zero-shot passage retrieval benchmarks, BEIR and TREC-DL. Our method establishes a new state-of-the-art in the BEIR benchmark, outperforming previous best results in Recall@100 and nDCG@10 metrics on 6 out of 8 datasets, with up to 17% relative gains over the previous best.

Debangshu Banerjee, Avishek Ghosh, Sayak Ray Chowdhury et al.

We present a non-asymptotic lower bound on the eigenspectrum of the design matrix generated by any linear bandit algorithm with sub-linear regret when the action set has well-behaved curvature. Specifically, we show that the minimum eigenvalue of the expected design matrix grows as $Ω(\sqrt{n})$ whenever the expected cumulative regret of the algorithm is $O(\sqrt{n})$, where $n$ is the learning horizon, and the action-space has a constant Hessian around the optimal arm. This shows that such action-spaces force a polynomial lower bound rather than a logarithmic lower bound, as shown by \cite{lattimore2017end}, in discrete (i.e., well-separated) action spaces. Furthermore, while the previous result is shown to hold only in the asymptotic regime (as $n \to \infty$), our result for these "locally rich" action spaces is any-time. Additionally, under a mild technical assumption, we obtain a similar lower bound on the minimum eigen value holding with high probability. We apply our result to two practical scenarios -- \emph{model selection} and \emph{clustering} in linear bandits. For model selection, we show that an epoch-based linear bandit algorithm adapts to the true model complexity at a rate exponential in the number of epochs, by virtue of our novel spectral bound. For clustering, we consider a multi agent framework where we show, by leveraging the spectral result, that no forced exploration is necessary -- the agents can run a linear bandit algorithm and estimate their underlying parameters at once, and hence incur a low regret.

Seongho Son, William Bankes, Sayak Ray Chowdhury et al.

Current Large Language Model (LLM) preference optimization algorithms do not account for temporal preference drift, which can lead to severe misalignment. To address this limitation, we propose Non-Stationary Direct Preference Optimisation (NS-DPO) that models time-dependent reward functions with a Dynamic Bradley-Terry model. NS-DPO proposes a computationally efficient solution by introducing only a single discount parameter in the loss function, which is used for exponential weighting that proportionally focuses learning on more time-relevant datapoints. We theoretically analyze the convergence of NS-DPO in a general setting where the exact nature of the preference drift is not known, providing upper bounds on the estimation error and regret caused by non-stationary preferences. Finally, we demonstrate the effectiveness of NS-DPO for fine-tuning LLMs under drifting preferences. Using scenarios where various levels of preference drift is introduced, with popular LLM reward models and datasets, we show that NS-DPO fine-tuned LLMs remain robust under non-stationarity, significantly outperforming baseline algorithms that ignore temporal preference changes, without sacrificing performance in stationary cases.

Xingyu Zhou, Sayak Ray Chowdhury

We consider cross-silo federated linear contextual bandit (LCB) problem under differential privacy, where multiple silos (agents) interact with the local users and communicate via a central server to realize collaboration while without sacrificing each user's privacy. We identify three issues in the state-of-the-art: (i) failure of claimed privacy protection and (ii) incorrect regret bound due to noise miscalculation and (iii) ungrounded communication cost. To resolve these issues, we take a two-step principled approach. First, we design an algorithmic framework consisting of a generic federated LCB algorithm and flexible privacy protocols. Then, leveraging the proposed framework, we study federated LCBs under two different privacy constraints. We first establish privacy and regret guarantees under silo-level local differential privacy, which fix the issues present in state-of-the-art algorithm. To further improve the regret performance, we next consider shuffle model of differential privacy, under which we show that our algorithm can achieve nearly ``optimal'' regret without a trusted server. We accomplish this via two different schemes -- one relies on a new result on privacy amplification via shuffling for DP mechanisms and another one leverages the integration of a shuffle protocol for vector sum into the tree-based mechanism, both of which might be of independent interest. Finally, we support our theoretical results with numerical evaluations over contextual bandit instances generated from both synthetic and real-life data.

Sayak Ray Chowdhury, Xingyu Zhou

We consider the standard $K$-armed bandit problem under a distributed trust model of differential privacy (DP), which enables to guarantee privacy without a trustworthy server. Under this trust model, previous work largely focus on achieving privacy using a shuffle protocol, where a batch of users data are randomly permuted before sending to a central server. This protocol achieves ($ε,δ$) or approximate-DP guarantee by sacrificing an additional additive $O\!\left(\!\frac{K\log T\sqrt{\log(1/δ)}}ε\!\right)\!$ cost in $T$-step cumulative regret. In contrast, the optimal privacy cost for achieving a stronger ($ε,0$) or pure-DP guarantee under the widely used central trust model is only $Θ\!\left(\!\frac{K\log T}ε\!\right)\!$, where, however, a trusted server is required. In this work, we aim to obtain a pure-DP guarantee under distributed trust model while sacrificing no more regret than that under central trust model. We achieve this by designing a generic bandit algorithm based on successive arm elimination, where privacy is guaranteed by corrupting rewards with an equivalent discrete Laplace noise ensured by a secure computation protocol. We also show that our algorithm, when instantiated with Skellam noise and the secure protocol, ensures \emph{Rényi differential privacy} -- a stronger notion than approximate DP -- under distributed trust model with a privacy cost of $O\!\left(\!\frac{K\sqrt{\log T}}ε\!\right)\!$.

Avishek Ghosh, Sayak Ray Chowdhury

We consider model selection for classic Reinforcement Learning (RL) environments -- Multi Armed Bandits (MABs) and Markov Decision Processes (MDPs) -- under general function approximations. In the model selection framework, we do not know the function classes, denoted by $\mathcal{F}$ and $\mathcal{M}$, where the true models -- reward generating function for MABs and and transition kernel for MDPs -- lie, respectively. Instead, we are given $M$ nested function (hypothesis) classes such that true models are contained in at-least one such class. In this paper, we propose and analyze efficient model selection algorithms for MABs and MDPs, that \emph{adapt} to the smallest function class (among the nested $M$ classes) containing the true underlying model. Under a separability assumption on the nested hypothesis classes, we show that the cumulative regret of our adaptive algorithms match to that of an oracle which knows the correct function classes (i.e., $\cF$ and $\cM$) a priori. Furthermore, for both the settings, we show that the cost of model selection is an additive term in the regret having weak (logarithmic) dependence on the learning horizon $T$.

Sayak Ray Chowdhury, Xingyu Zhou, Nagarajan Natarajan

Learning from preference-based feedback has recently gained considerable traction as a promising approach to align generative models with human interests. Instead of relying on numerical rewards, the generative models are trained using reinforcement learning with human feedback (RLHF). These approaches first solicit feedback from human labelers typically in the form of pairwise comparisons between two possible actions, then estimate a reward model using these comparisons, and finally employ a policy based on the estimated reward model. An adversarial attack in any step of the above pipeline might reveal private and sensitive information of human labelers. In this work, we adopt the notion of label differential privacy (DP) and focus on the problem of reward estimation from preference-based feedback while protecting privacy of each individual labelers. Specifically, we consider the parametric Bradley-Terry-Luce (BTL) model for such pairwise comparison feedback involving a latent reward parameter $θ^* \in \mathbb{R}^d$. Within a standard minimax estimation framework, we provide tight upper and lower bounds on the error in estimating $θ^*$ under both local and central models of DP. We show, for a given privacy budget $ε$ and number of samples $n$, that the additional cost to ensure label-DP under local model is $Θ\big(\frac{1}{ e^ε-1}\sqrt{\frac{d}{n}}\big)$, while it is $Θ\big(\frac{\text{poly}(d)}{εn} \big)$ under the weaker central model. We perform simulations on synthetic data that corroborate these theoretical results.

Yulian Wu, Xingyu Zhou, Sayak Ray Chowdhury et al.

In this paper, we study the problem of (finite horizon tabular) Markov decision processes (MDPs) with heavy-tailed rewards under the constraint of differential privacy (DP). Compared with the previous studies for private reinforcement learning that typically assume rewards are sampled from some bounded or sub-Gaussian distributions to ensure DP, we consider the setting where reward distributions have only finite $(1+v)$-th moments with some $v \in (0,1]$. By resorting to robust mean estimators for rewards, we first propose two frameworks for heavy-tailed MDPs, i.e., one is for value iteration and another is for policy optimization. Under each framework, we consider both joint differential privacy (JDP) and local differential privacy (LDP) models. Based on our frameworks, we provide regret upper bounds for both JDP and LDP cases and show that the moment of distribution and privacy budget both have significant impacts on regrets. Finally, we establish a lower bound of regret minimization for heavy-tailed MDPs in JDP model by reducing it to the instance-independent lower bound of heavy-tailed multi-armed bandits in DP model. We also show the lower bound for the problem in LDP by adopting some private minimax methods. Our results reveal that there are fundamental differences between the problem of private RL with sub-Gaussian and that with heavy-tailed rewards.

Dhruv Sarkar, Nishant Pandey, Sayak Ray Chowdhury

Nash regret has recently emerged as a principled fairness-aware performance metric for stochastic multi-armed bandits, motivated by the Nash Social Welfare objective. Although this notion has been extended to linear bandits, existing results suffer from suboptimality in ambient dimension $d$, stemming from proof techniques that rely on restrictive concentration inequalities. In this work, we resolve this open problem by introducing new analytical tools that yield an order-optimal Nash regret bound in linear bandits. Beyond Nash regret, we initiate the study of $p$-means regret in linear bandits, a unifying framework that interpolates between fairness and utility objectives and strictly generalizes Nash regret. We propose a generic algorithmic framework, FairLinBandit, that works as a meta-algorithm on top of any linear bandit strategy. We instantiate this framework using two bandit algorithms: Phased Elimination and Upper Confidence Bound, and prove that both achieve sublinear $p$-means regret for the entire range of $p$. Extensive experiments on linear bandit instances generated from real-world datasets demonstrate that our methods consistently outperform the existing state-of-the-art baseline.

Sayak Ray Chowdhury, Anush Kini, Nagarajan Natarajan

Learning from preference-based feedback has recently gained traction as a promising approach to align language models with human interests. While these aligned generative models have demonstrated impressive capabilities across various tasks, their dependence on high-quality human preference data poses a bottleneck in practical applications. Specifically, noisy (incorrect and ambiguous) preference pairs in the dataset might restrict the language models from capturing human intent accurately. While practitioners have recently proposed heuristics to mitigate the effect of noisy preferences, a complete theoretical understanding of their workings remain elusive. In this work, we aim to bridge this gap by by introducing a general framework for policy optimization in the presence of random preference flips. We focus on the direct preference optimization (DPO) algorithm in particular since it assumes that preferences adhere to the Bradley-Terry-Luce (BTL) model, raising concerns about the impact of noisy data on the learned policy. We design a novel loss function, which de-bias the effect of noise on average, making a policy trained by minimizing that loss robust to the noise. Under log-linear parameterization of the policy class and assuming good feature coverage of the SFT policy, we prove that the sub-optimality gap of the proposed robust DPO (rDPO) policy compared to the optimal policy is of the order $O(\frac{1}{1-2ε}\sqrt{\frac{d}{n}})$, where $ε< 1/2$ is flip rate of labels, $d$ is policy parameter dimension and $n$ is size of dataset. Our experiments on IMDb sentiment generation and Anthropic's helpful-harmless dataset show that rDPO is robust to noise in preference labels compared to vanilla DPO and other heuristics proposed by practitioners.

Nirjhar Das, Souradip Chakraborty, Aldo Pacchiano et al.

Large Language Models (LLMs) aligned using Reinforcement Learning from Human Feedback (RLHF) have shown remarkable generation abilities in numerous tasks. However, collecting high-quality human preferences creates costly bottlenecks in practical deployments, and hence, training data are often budgeted. In these scenarios, it is crucial to collect training data (e.g., contexts, a pair of generations for each context, and a preference indicating which generation is better) carefully, yet most of the existing methods sample contexts uniformly at random from a given collection. Given this, under the Bradley-Terry-Luce preference model and with a small budget of training data, we show that uniform sampling of contexts could lead to a policy (i.e., an aligned model) that suffers a constant sub-optimality gap from the optimal policy. This highlights the need for an adaptive context sampling strategy for effective alignment under a small sample budget. To address this, we reformulate RLHF within the contextual preference bandit framework, treating generations as actions, and give a nearly complete characterization of the sub-optimality gap in terms of both lower and upper bounds. First, when the action set is a $d$-dimensional hypercube and the number of samples is $T$, we show an $Ω(d/\sqrt{T})$ lower bound. Next, we propose an algorithm, $\textit{Active Preference Optimization}$ ($\texttt{APO}$), that iteratively collects preferences for the most uncertain contexts. We show that the sub-optimality gap of the policy learned via $\texttt{APO}$ matches the lower bound up to a log factor and a non-linearity constant. Finally, we perform experiments on practical datasets to validate $\texttt{APO}$'s efficacy over existing methods, establishing it as a sample-efficient and cost-effective solution for LLM alignment.

Dhruv Sarkar, Nishant Pandey, Sayak Ray Chowdhury

Multi-armed bandit algorithms are fundamental tools for sequential decision-making under uncertainty, with widespread applications across domains such as clinical trials and personalized decision-making. As bandit algorithms are increasingly deployed in these socially sensitive settings, it becomes critical to protect user data privacy and ensure fair treatment across decision rounds. While prior work has independently addressed privacy and fairness in bandit settings, the question of whether both objectives can be achieved simultaneously has remained largely open. Existing privacy-preserving bandit algorithms typically optimize average regret, a utilitarian measure, whereas fairness-aware approaches focus on minimizing Nash regret, which penalizes inequitable reward distributions, but often disregard privacy concerns. To bridge this gap, we introduce Differentially Private Nash Confidence Bound (DP-NCB)-a novel and unified algorithmic framework that simultaneously ensures $ε$-differential privacy and achieves order-optimal Nash regret, matching known lower bounds up to logarithmic factors. The framework is sufficiently general to operate under both global and local differential privacy models, and is anytime, requiring no prior knowledge of the time horizon. We support our theoretical guarantees with simulations on synthetic bandit instances, showing that DP-NCB incurs substantially lower Nash regret than state-of-the-art baselines. Our results offer a principled foundation for designing bandit algorithms that are both privacy-preserving and fair, making them suitable for high-stakes, socially impactful applications.

Dhruv Sarkar, Nishant Pandey, Sayak Ray Chowdhury

Regret in stochastic multi-armed bandits traditionally measures the difference between the highest reward and either the arithmetic mean of accumulated rewards or the final reward. These conventional metrics often fail to address fairness among agents receiving rewards, particularly in settings where rewards are distributed across a population, such as patients in clinical trials. To address this, a recent body of work has introduced Nash regret, which evaluates performance via the geometric mean of accumulated rewards, aligning with the Nash social welfare function known for satisfying fairness axioms. To minimize Nash regret, existing approaches require specialized algorithm designs and strong assumptions, such as multiplicative concentration inequalities and bounded, non-negative rewards, making them unsuitable for even Gaussian reward distributions. We demonstrate that an initial uniform exploration phase followed by a standard Upper Confidence Bound (UCB) algorithm achieves near-optimal Nash regret, while relying only on additive Hoeffding bounds, and naturally extending to sub-Gaussian rewards. Furthermore, we generalize the algorithm to a broad class of fairness metrics called the $p$-mean regret, proving (nearly) optimal regret bounds uniformly across all $p$ values. This is in contrast to prior work, which made extremely restrictive assumptions on the bandit instances and even then achieved suboptimal regret bounds.

Aditya Gopalan, Sayak Ray Chowdhury, Debangshu Banerjee

Direct alignment algorithms such as Direct Preference Optimization (DPO) fine-tune models based on preference data, using only supervised learning instead of two-stage reinforcement learning with human feedback (RLHF). We show that DPO encodes a statistical estimation problem over reward functions induced by a parametric policy class. When the true reward function that generates preferences cannot be realized via the policy class, DPO becomes misspecified, resulting in failure modes such as preference order reversal, worsening of policy reward, and high sensitivity to the input preference data distribution. On the other hand, we study the local behavior of two-stage RLHF for a parametric class and relate it to a natural gradient step in policy space. Our fine-grained geometric characterization allows us to propose AuxDPO, which introduces additional auxiliary variables in the DPO loss function to help move towards the RLHF solution in a principled manner and mitigate the misspecification in DPO. We empirically demonstrate the superior performance of AuxDPO on didactic bandit settings as well as LLM alignment tasks.

Virendra Nishad, Bhaskar Mukhoty, Hilal AlQuabeh et al.

Deep neural networks have achieved remarkable success in a wide range of classification tasks. However, they remain highly susceptible to adversarial examples - inputs that are subtly perturbed to induce misclassification while appearing unchanged to humans. Among various attack strategies, Universal Adversarial Perturbations (UAPs) have emerged as a powerful tool for both stress testing model robustness and facilitating scalable adversarial training. Despite their effectiveness, most existing UAP methods neglect domain specific constraints that govern feature relationships. Violating such constraints, such as debt to income ratios in credit scoring or packet flow invariants in network communication, can render adversarial examples implausible or easily detectable, thereby limiting their real world applicability. In this work, we advance universal adversarial attacks to constrained feature spaces by formulating an augmented Lagrangian based min max optimization problem that enforces multiple, potentially complex constraints of varying importance. We propose Constrained Adversarial Perturbation (CAP), an efficient algorithm that solves this problem using a gradient based alternating optimization strategy. We evaluate CAP across diverse domains including finance, IT networks, and cyber physical systems, and demonstrate that it achieves higher attack success rates while significantly reducing runtime compared to existing baselines. Our approach also generalizes seamlessly to individual adversarial perturbations, where we observe similar strong performance gains. Finally, we introduce a principled procedure for learning feature constraints directly from data, enabling broad applicability across domains with structured input spaces.

Vaibhav Singh, Soumya Suvra Ghosal, Kapu Nirmal Joshua et al.

In-context learning (ICL) has emerged as a powerful paradigm for adapting large language models (LLMs) to new and data-scarce tasks using only a few carefully selected task-specific examples presented in the prompt. However, given the limited context size of LLMs, a fundamental question arises: Which examples should be selected to maximize performance on a given user query? While nearest-neighbor-based methods like KATE have been widely adopted for this purpose, they suffer from well-known drawbacks in high-dimensional embedding spaces, including poor generalization and a lack of diversity. In this work, we study this problem of example selection in ICL from a principled, information theory-driven perspective. We first model an LLM as a linear function over input embeddings and frame the example selection task as a query-specific optimization problem: selecting a subset of exemplars from a larger example bank that minimizes the prediction error on a specific query. This formulation departs from traditional generalization-focused learning theoretic approaches by targeting accurate prediction for a specific query instance. We derive a principled surrogate objective that is approximately submodular, enabling the use of a greedy algorithm with an approximation guarantee. We further enhance our method by (i) incorporating the kernel trick to operate in high-dimensional feature spaces without explicit mappings, and (ii) introducing an optimal design-based regularizer to encourage diversity in the selected examples. Empirically, we demonstrate significant improvements over standard retrieval methods across a suite of classification tasks, highlighting the benefits of structure-aware, diverse example selection for ICL in real-world, label-scarce scenarios.

Sayak Ray Chowdhury, Xingyu Zhou

Differential privacy (DP) has been recently introduced to linear contextual bandits to formally address the privacy concerns in its associated personalized services to participating users (e.g., recommendations). Prior work largely focus on two trust models of DP: the central model, where a central server is responsible for protecting users sensitive data, and the (stronger) local model, where information needs to be protected directly on user side. However, there remains a fundamental gap in the utility achieved by learning algorithms under these two privacy models, e.g., $\tilde{O}(\sqrt{T})$ regret in the central model as compared to $\tilde{O}(T^{3/4})$ regret in the local model, if all users are unique within a learning horizon $T$. In this work, we aim to achieve a stronger model of trust than the central model, while suffering a smaller regret than the local model by considering recently popular shuffle model of privacy. We propose a general algorithmic framework for linear contextual bandits under the shuffle trust model, where there exists a trusted shuffler in between users and the central server, that randomly permutes a batch of users data before sending those to the server. We then instantiate this framework with two specific shuffle protocols: one relying on privacy amplification of local mechanisms, and another incorporating a protocol for summing vectors and matrices of bounded norms. We prove that both these instantiations lead to regret guarantees that significantly improve on that of the local model, and can potentially be of the order $\tilde{O}(T^{3/5})$ if all users are unique. We also verify this regret behavior with simulations on synthetic data. Finally, under the practical scenario of non-unique users, we show that the regret of our shuffle private algorithm scale as $\tilde{O}(T^{2/3})$, which matches that the central model could achieve in this case.

Sayak Ray Chowdhury, Patrick Saux, Odalric-Ambrym Maillard et al.

We revisit the method of mixture technique, also known as the Laplace method, to study the concentration phenomenon in generic exponential families. Combining the properties of Bregman divergence associated with log-partition function of the family with the method of mixtures for super-martingales, we establish a generic bound controlling the Bregman divergence between the parameter of the family and a finite sample estimate of the parameter. Our bound is time-uniform and makes appear a quantity extending the classical information gain to exponential families, which we call the Bregman information gain. For the practitioner, we instantiate this novel bound to several classical families, e.g., Gaussian, Bernoulli, Exponential, Weibull, Pareto, Poisson and Chi-square yielding explicit forms of the confidence sets and the Bregman information gain. We further numerically compare the resulting confidence bounds to state-of-the-art alternatives for time-uniform concentration and show that this novel method yields competitive results. Finally, we highlight the benefit of our concentration bounds on some illustrative applications.

Sayak Ray Chowdhury, Xingyu Zhou

We study regret minimization in finite horizon tabular Markov decision processes (MDPs) under the constraints of differential privacy (DP). This is motivated by the widespread applications of reinforcement learning (RL) in real-world sequential decision making problems, where protecting users' sensitive and private information is becoming paramount. We consider two variants of DP -- joint DP (JDP), where a centralized agent is responsible for protecting users' sensitive data and local DP (LDP), where information needs to be protected directly on the user side. We first propose two general frameworks -- one for policy optimization and another for value iteration -- for designing private, optimistic RL algorithms. We then instantiate these frameworks with suitable privacy mechanisms to satisfy JDP and LDP requirements, and simultaneously obtain sublinear regret guarantees. The regret bounds show that under JDP, the cost of privacy is only a lower order additive term, while for a stronger privacy protection under LDP, the cost suffered is multiplicative. Finally, the regret bounds are obtained by a unified analysis, which, we believe, can be extended beyond tabular MDPs.

Sayak Ray Chowdhury, Xingyu Zhou, Ness Shroff

In this paper, we study the problem of regret minimization in reinforcement learning (RL) under differential privacy constraints. This work is motivated by the wide range of RL applications for providing personalized service, where privacy concerns are becoming paramount. In contrast to previous works, we take the first step towards non-tabular RL settings, while providing a rigorous privacy guarantee. In particular, we consider the adaptive control of differentially private linear quadratic (LQ) systems. We develop the first private RL algorithm, PRL, which is able to attain a sub-linear regret while guaranteeing privacy protection. More importantly, the additional cost due to privacy is only on the order of $\frac{\ln(1/δ)^{1/4}}{ε^{1/2}}$ given privacy parameters $ε, δ> 0$. Through this process, we also provide a general procedure for adaptive control of LQ systems under changing regularizers, which not only generalizes previous non-private controls, but also serves as the basis for general private controls.

Avishek Ghosh, Sayak Ray Chowdhury, Kannan Ramchandran

We address the problem of model selection for the finite horizon episodic Reinforcement Learning (RL) problem where the transition kernel $P^*$ belongs to a family of models $\mathcal{P}^*$ with finite metric entropy. In the model selection framework, instead of $\mathcal{P}^*$, we are given $M$ nested families of transition kernels $\cP_1 \subset \cP_2 \subset \ldots \subset \cP_M$. We propose and analyze a novel algorithm, namely \emph{Adaptive Reinforcement Learning (General)} (\texttt{ARL-GEN}) that adapts to the smallest such family where the true transition kernel $P^*$ lies. \texttt{ARL-GEN} uses the Upper Confidence Reinforcement Learning (\texttt{UCRL}) algorithm with value targeted regression as a blackbox and puts a model selection module at the beginning of each epoch. Under a mild separability assumption on the model classes, we show that \texttt{ARL-GEN} obtains a regret of $\Tilde{\mathcal{O}}(d_{\mathcal{E}}^*H^2+\sqrt{d_{\mathcal{E}}^* \mathbb{M}^* H^2 T})$, with high probability, where $H$ is the horizon length, $T$ is the total number of steps, $d_{\mathcal{E}}^*$ is the Eluder dimension and $\mathbb{M}^*$ is the metric entropy corresponding to $\mathcal{P}^*$. Note that this regret scaling matches that of an oracle that knows $\mathcal{P}^*$ in advance. We show that the cost of model selection for \texttt{ARL-GEN} is an additive term in the regret having a weak dependence on $T$. Subsequently, we remove the separability assumption and consider the setup of linear mixture MDPs, where the transition kernel $P^*$ has a linear function approximation. With this low rank structure, we propose novel adaptive algorithms for model selection, and obtain (order-wise) regret identical to that of an oracle with knowledge of the true model class.

Sayak Ray Chowdhury, Rafael Oliveira

We consider the regret minimization problem in reinforcement learning (RL) in the episodic setting. In many real-world RL environments, the state and action spaces are continuous or very large. Existing approaches establish regret guarantees by either a low-dimensional representation of the stochastic transition model or an approximation of the $Q$-functions. However, the understanding of function approximation schemes for state-value functions largely remains missing. In this paper, we propose an online model-based RL algorithm, namely the CME-RL, that learns representations of transition distributions as embeddings in a reproducing kernel Hilbert space while carefully balancing the exploitation-exploration tradeoff. We demonstrate the efficiency of our algorithm by proving a frequentist (worst-case) regret bound that is of order $\tilde{O}\big(Hγ_N\sqrt{N}\big)$\footnote{ $\tilde{O}(\cdot)$ hides only absolute constant and poly-logarithmic factors.}, where $H$ is the episode length, $N$ is the total number of time steps and $γ_N$ is an information theoretic quantity relating the effective dimension of the state-action feature space. Our method bypasses the need for estimating transition probabilities and applies to any domain on which kernels can be defined. It also brings new insights into the general theory of kernel methods for approximate inference and RL regret minimization.

Sayak Ray Chowdhury, Aditya Gopalan

We consider multi-objective optimization (MOO) of an unknown vector-valued function in the non-parametric Bayesian optimization (BO) setting, with the aim being to learn points on the Pareto front of the objectives. Most existing BO algorithms do not model the fact that the multiple objectives, or equivalently, tasks can share similarities, and even the few that do lack rigorous, finite-time regret guarantees that capture explicitly inter-task structure. In this work, we address this problem by modelling inter-task dependencies using a multi-task kernel and develop two novel BO algorithms based on random scalarizations of the objectives. Our algorithms employ vector-valued kernel regression as a stepping stone and belong to the upper confidence bound class of algorithms. Under a smoothness assumption that the unknown vector-valued function is an element of the reproducing kernel Hilbert space associated with the multi-task kernel, we derive worst-case regret bounds for our algorithms that explicitly capture the similarities between tasks. We numerically benchmark our algorithms on both synthetic and real-life MOO problems, and show the advantages offered by learning with multi-task kernels.

Sayak Ray Chowdhury, Aditya Gopalan

We develop algorithms with low regret for learning episodic Markov decision processes based on kernel approximation techniques. The algorithms are based on both the Upper Confidence Bound (UCB) as well as Posterior or Thompson Sampling (PSRL) philosophies, and work in the general setting of continuous state and action spaces when the true unknown transition dynamics are assumed to have smoothness induced by an appropriate Reproducing Kernel Hilbert Space (RKHS).

Sayak Ray Chowdhury, Aditya Gopalan

We present two algorithms for Bayesian optimization in the batch feedback setting, based on Gaussian process upper confidence bound and Thompson sampling approaches, along with frequentist regret guarantees and numerical results.

Sayak Ray Chowdhury, Aditya Gopalan

We consider black box optimization of an unknown function in the nonparametric Gaussian process setting when the noise in the observed function values can be heavy tailed. This is in contrast to existing literature that typically assumes sub-Gaussian noise distributions for queries. Under the assumption that the unknown function belongs to the Reproducing Kernel Hilbert Space (RKHS) induced by a kernel, we first show that an adaptation of the well-known GP-UCB algorithm with reward truncation enjoys sublinear $\tilde{O}(T^{\frac{2 + α}{2(1+α)}})$ regret even with only the $(1+α)$-th moments, $α\in (0,1]$, of the reward distribution being bounded ($\tilde{O}$ hides logarithmic factors). However, for the common squared exponential (SE) and Matérn kernels, this is seen to be significantly larger than a fundamental $Ω(T^{\frac{1}{1+α}})$ lower bound on regret. We resolve this gap by developing novel Bayesian optimization algorithms, based on kernel approximation techniques, with regret bounds matching the lower bound in order for the SE kernel. We numerically benchmark the algorithms on environments based on both synthetic models and real-world data sets.

Sayak Ray Chowdhury, Aditya Gopalan

We consider online learning for minimizing regret in unknown, episodic Markov decision processes (MDPs) with continuous states and actions. We develop variants of the UCRL and posterior sampling algorithms that employ nonparametric Gaussian process priors to generalize across the state and action spaces. When the transition and reward functions of the true MDP are members of the associated Reproducing Kernel Hilbert Spaces of functions induced by symmetric psd kernels (frequentist setting), we show that the algorithms enjoy sublinear regret bounds. The bounds are in terms of explicit structural parameters of the kernels, namely a novel generalization of the information gain metric from kernelized bandit, and highlight the influence of transition and reward function structure on the learning performance. Our results are applicable to multidimensional state and action spaces with composite kernel structures, and generalize results from the literature on kernelized bandits, and the adaptive control of parametric linear dynamical systems with quadratic costs.

Avishek Ghosh, Sayak Ray Chowdhury, Aditya Gopalan

We consider the problem of online learning in misspecified linear stochastic multi-armed bandit problems. Regret guarantees for state-of-the-art linear bandit algorithms such as Optimism in the Face of Uncertainty Linear bandit (OFUL) hold under the assumption that the arms expected rewards are perfectly linear in their features. It is, however, of interest to investigate the impact of potential misspecification in linear bandit models, where the expected rewards are perturbed away from the linear subspace determined by the arms features. Although OFUL has recently been shown to be robust to relatively small deviations from linearity, we show that any linear bandit algorithm that enjoys optimal regret performance in the perfectly linear setting (e.g., OFUL) must suffer linear regret under a sparse additive perturbation of the linear model. In an attempt to overcome this negative result, we define a natural class of bandit models characterized by a non-sparse deviation from linearity. We argue that the OFUL algorithm can fail to achieve sublinear regret even under models that have non-sparse deviation.We finally develop a novel bandit algorithm, comprising a hypothesis test for linearity followed by a decision to use either the OFUL or Upper Confidence Bound (UCB) algorithm. For perfectly linear bandit models, the algorithm provably exhibits OFULs favorable regret performance, while for misspecified models satisfying the non-sparse deviation property, the algorithm avoids the linear regret phenomenon and falls back on UCBs sublinear regret scaling. Numerical experiments on synthetic data, and on recommendation data from the public Yahoo! Learning to Rank Challenge dataset, empirically support our findings.

Sayak Ray Chowdhury, Aditya Gopalan

We consider the stochastic bandit problem with a continuous set of arms, with the expected reward function over the arms assumed to be fixed but unknown. We provide two new Gaussian process-based algorithms for continuous bandit optimization-Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and derive corresponding regret bounds. Specifically, the bounds hold when the expected reward function belongs to the reproducing kernel Hilbert space (RKHS) that naturally corresponds to a Gaussian process kernel used as input by the algorithms. Along the way, we derive a new self-normalized concentration inequality for vector- valued martingales of arbitrary, possibly infinite, dimension. Finally, experimental evaluation and comparisons to existing algorithms on synthetic and real-world environments are carried out that highlight the favorable gains of the proposed strategies in many cases.