arXiv subjects

Akshay Krishnamurthy

Publications and source records attributed to Akshay Krishnamurthy.

At least 55 records · Page 3Linked to original sources

arXiv2022

Robust Dynamic Assortment Optimization in the Presence of Outlier Customers

We consider the dynamic assortment optimization problem under the multinomial logit model (MNL) with unknown utility parameters. The main question investigated in this paper is model mis-specification under the $\varepsilon$-contamination model, which is a fundamental model in robust statistics and machine learning. In particular, throughout a selling horizon of length $T$, we assume that customers make purchases according to a well specified underlying multinomial logit choice model in a $(1-\varepsilon)$-fraction of the time periods, and make arbitrary purchasing decisions instead in the remaining $\varepsilon$-fraction of the time periods. In this model, we develop a new robust online assortment optimization policy via an active elimination strategy. We establish both upper and lower bounds on the regret, and show that our policy is optimal up to logarithmic factor in $T$ when the assortment capacity is constant. %% capacity of assortments has a constant upper limit. We further develop a fully adaptive policy that does not require any prior knowledge of the contamination parameter $\varepsilon$. In the case of the existence a sub-optimality gap between optimal and sub-optimal products, we also established gap-dependent logarithmic regret upper bounds and lower bounds in both the known-$\varepsilon$ and unknown-$\varepsilon$ cases. Our simulation study shows that our policy outperforms the existing policies based on upper confidence bounds (UCB) and Thompson sampling.

stat.ML↗

arXiv2022

Universal and data-adaptive algorithms for model selection in linear contextual bandits

Model selection in contextual bandits is an important complementary problem to regret minimization with respect to a fixed model class. We consider the simplest non-trivial instance of model-selection: distinguishing a simple multi-armed bandit problem from a linear contextual bandit problem. Even in this instance, current state-of-the-art methods explore in a suboptimal manner and require strong "feature-diversity" conditions. In this paper, we introduce new algorithms that a) explore in a data-adaptive manner, and b) provide model selection guarantees of the form $\mathcal{O}(d^α T^{1- α})$ with no feature diversity conditions whatsoever, where $d$ denotes the dimension of the linear model and $T$ denotes the total number of rounds. The first algorithm enjoys a "best-of-both-worlds" property, recovering two prior results that hold under distinct distributional assumptions, simultaneously. The second removes distributional assumptions altogether, expanding the scope for tractable model selection. Our approach extends to model selection among nested linear contextual bandits under some additional assumptions.

cs.LG↗

arXiv2022

Model-free Representation Learning and Exploration in Low-rank MDPs

The low rank MDP has emerged as an important model for studying representation learning and exploration in reinforcement learning. With a known representation, several model-free exploration strategies exist. In contrast, all algorithms for the unknown representation setting are model-based, thereby requiring the ability to model the full dynamics. In this work, we present the first model-free representation learning algorithms for low rank MDPs. The key algorithmic contribution is a new minimax representation learning objective, for which we provide variants with differing tradeoffs in their statistical and computational properties. We interleave this representation learning step with an exploration strategy to cover the state space in a reward-free manner. The resulting algorithms are provably sample efficient and can accommodate general function approximation to scale to complex environments.

cs.LG↗

arXiv2022

Sparsity in Partially Controllable Linear Systems

A fundamental concept in control theory is that of controllability, where any system state can be reached through an appropriate choice of control inputs. Indeed, a large body of classical and modern approaches are designed for controllable linear dynamical systems. However, in practice, we often encounter systems in which a large set of state variables evolve exogenously and independently of the control inputs; such systems are only partially controllable. The focus of this work is on a large class of partially controllable linear dynamical systems, specified by an underlying sparsity pattern. Our main results establish structural conditions and finite-sample guarantees for learning to control such systems. In particular, our structural results characterize those state variables which are irrelevant for optimal control, an analysis which departs from classical control techniques. Our algorithmic results adapt techniques from high-dimensional statistics -- specifically soft-thresholding and semiparametric least-squares -- to exploit the underlying sparsity pattern in order to obtain finite-sample guarantees that significantly improve over those based on certainty-equivalence. We also corroborate these theoretical improvements over certainty-equivalent control through a simulation study.

math.OC↗

arXiv2022

Sample-Efficient Reinforcement Learning in the Presence of Exogenous Information

In real-world reinforcement learning applications the learner's observation space is ubiquitously high-dimensional with both relevant and irrelevant information about the task at hand. Learning from high-dimensional observations has been the subject of extensive investigation in supervised learning and statistics (e.g., via sparsity), but analogous issues in reinforcement learning are not well understood, even in finite state/action (tabular) domains. We introduce a new problem setting for reinforcement learning, the Exogenous Markov Decision Process (ExoMDP), in which the state space admits an (unknown) factorization into a small controllable (or, endogenous) component and a large irrelevant (or, exogenous) component; the exogenous component is independent of the learner's actions, but evolves in an arbitrary, temporally correlated fashion. We provide a new algorithm, ExoRL, which learns a near-optimal policy with sample complexity polynomial in the size of the endogenous component and nearly independent of the size of the exogenous component, thereby offering a doubly-exponential improvement over off-the-shelf algorithms. Our results highlight for the first time that sample-efficient reinforcement learning is possible in the presence of exogenous information, and provide a simple, user-friendly benchmark for investigation going forward.

cs.LG↗

arXiv2022

Anti-Concentrated Confidence Bonuses for Scalable Exploration

Intrinsic rewards play a central role in handling the exploration-exploitation trade-off when designing sequential decision-making algorithms, in both foundational theory and state-of-the-art deep reinforcement learning. The LinUCB algorithm, a centerpiece of the stochastic linear bandits literature, prescribes an elliptical bonus which addresses the challenge of leveraging shared information in large action spaces. This bonus scheme cannot be directly transferred to high-dimensional exploration problems, however, due to the computational cost of maintaining the inverse covariance matrix of action features. We introduce \emph{anti-concentrated confidence bounds} for efficiently approximating the elliptical bonus, using an ensemble of regressors trained to predict random noise from policy network-derived features. Using this approximation, we obtain stochastic linear bandit algorithms which obtain $\tilde O(d \sqrt{T})$ regret bounds for $\mathrm{poly}(d)$ fixed actions. We develop a practical variant for deep reinforcement learning that is competitive with contemporary intrinsic reward heuristics on Atari benchmarks.

cs.LG↗

arXiv2022

Provable RL with Exogenous Distractors via Multistep Inverse Dynamics

Many real-world applications of reinforcement learning (RL) require the agent to deal with high-dimensional observations such as those generated from a megapixel camera. Prior work has addressed such problems with representation learning, through which the agent can provably extract endogenous, latent state information from raw observations and subsequently plan efficiently. However, such approaches can fail in the presence of temporally correlated noise in the observations, a phenomenon that is common in practice. We initiate the formal study of latent state discovery in the presence of such exogenous noise sources by proposing a new model, the Exogenous Block MDP (EX-BMDP), for rich observation RL. We start by establishing several negative results, by highlighting failure cases of prior representation learning based approaches. Then, we introduce the Predictive Path Elimination (PPE) algorithm, that learns a generalization of inverse dynamics and is provably sample and computationally efficient in EX-BMDPs when the endogenous state dynamics are near deterministic. The sample complexity of PPE depends polynomially on the size of the latent endogenous state space while not directly depending on the size of the observation space, nor the exogenous state space. We provide experiments on challenging exploration problems which show that our approach works empirically.

cs.LG↗

arXiv2022

Understanding Contrastive Learning Requires Incorporating Inductive Biases

Contrastive learning is a popular form of self-supervised learning that encourages augmentations (views) of the same input to have more similar representations compared to augmentations of different inputs. Recent attempts to theoretically explain the success of contrastive learning on downstream classification tasks prove guarantees depending on properties of {\em augmentations} and the value of {\em contrastive loss} of representations. We demonstrate that such analyses, that ignore {\em inductive biases} of the function class and training algorithm, cannot adequately explain the success of contrastive learning, even {\em provably} leading to vacuous guarantees in some settings. Extensive experiments on image and text domains highlight the ubiquity of this problem -- different function classes and algorithms behave very differently on downstream tasks, despite having the same augmentations and contrastive losses. Theoretical analysis is presented for the class of linear representations, where incorporating inductive biases of the function class allows contrastive learning to work with less stringent conditions compared to prior analyses.

cs.LG↗

arXiv2022

Provable Reinforcement Learning with a Short-Term Memory

Real-world sequential decision making problems commonly involve partial observability, which requires the agent to maintain a memory of history in order to infer the latent states, plan and make good decisions. Coping with partial observability in general is extremely challenging, as a number of worst-case statistical and computational barriers are known in learning Partially Observable Markov Decision Processes (POMDPs). Motivated by the problem structure in several physical applications, as well as a commonly used technique known as "frame stacking", this paper proposes to study a new subclass of POMDPs, whose latent states can be decoded by the most recent history of a short length $m$. We establish a set of upper and lower bounds on the sample complexity for learning near-optimal policies for this class of problems in both tabular and rich-observation settings (where the number of observations is enormous). In particular, in the rich-observation setting, we develop new algorithms using a novel "moment matching" approach with a sample complexity that scales exponentially with the short length $m$ rather than the problem horizon, and is independent of the number of observations. Our results show that a short-term memory suffices for reinforcement learning in these environments.

cs.LG↗

arXiv2021

Gone Fishing: Neural Active Learning with Fisher Embeddings

There is an increasing need for effective active learning algorithms that are compatible with deep neural networks. This paper motivates and revisits a classic, Fisher-based active selection objective, and proposes BAIT, a practical, tractable, and high-performing algorithm that makes it viable for use with neural models. BAIT draws inspiration from the theoretical analysis of maximum likelihood estimators (MLE) for parametric models. It selects batches of samples by optimizing a bound on the MLE error in terms of the Fisher information, which we show can be implemented efficiently at scale by exploiting linear-algebraic structure especially amenable to execution on modern hardware. Our experiments demonstrate that BAIT outperforms the previous state of the art on both classification and regression problems, and is flexible enough to be used with a variety of model architectures.

cs.LG↗

arXiv2021

Efficient and Optimal Algorithms for Contextual Dueling Bandits under Realizability

We study the $K$-armed contextual dueling bandit problem, a sequential decision making setting in which the learner uses contextual information to make two decisions, but only observes \emph{preference-based feedback} suggesting that one decision was better than the other. We focus on the regret minimization problem under realizability, where the feedback is generated by a pairwise preference matrix that is well-specified by a given function class $\mathcal F$. We provide a new algorithm that achieves the optimal regret rate for a new notion of best response regret, which is a strictly stronger performance measure than those considered in prior works. The algorithm is also computationally efficient, running in polynomial time assuming access to an online oracle for square loss regression over $\mathcal F$. This resolves an open problem of Dudík et al. [2015] on oracle efficient, regret-optimal algorithms for contextual dueling bandits.

cs.LG↗

arXiv2021

Provably adaptive reinforcement learning in metric spaces

We study reinforcement learning in continuous state and action spaces endowed with a metric. We provide a refined analysis of a variant of the algorithm of Sinclair, Banerjee, and Yu (2019) and show that its regret scales with the \emph{zooming dimension} of the instance. This parameter, which originates in the bandit literature, captures the size of the subsets of near optimal actions and is always smaller than the covering dimension used in previous analyses. As such, our results are the first provably adaptive guarantees for reinforcement learning in metric spaces.

cs.LG↗

arXiv2021

Active Learning for Cost-Sensitive Classification

We design an active learning algorithm for cost-sensitive multiclass classification: problems where different errors have different costs. Our algorithm, COAL, makes predictions by regressing to each label's cost and predicting the smallest. On a new example, it uses a set of regressors that perform well on past data to estimate possible costs for each label. It queries only the labels that could be the best, ignoring the sure losers. We prove COAL can be efficiently implemented for any regression family that admits squared loss optimization; it also enjoys strong guarantees with respect to predictive performance and labeling effort. We empirically compare COAL to passive learning and several active learning baselines, showing significant improvements in labeling effort and test cost on real-world datasets.

cs.LG↗

arXiv2021

Provably efficient RL with Rich Observations via Latent State Decoding

We study the exploration problem in episodic MDPs with rich observations generated from a small number of latent states. Under certain identifiability assumptions, we demonstrate how to estimate a mapping from the observations to latent states inductively through a sequence of regression and clustering steps -- where previously decoded latent states provide labels for later regression problems -- and use it to construct good exploration policies. We provide finite-sample guarantees on the quality of the learned state decoding function and exploration policies, and complement our theory with an empirical evaluation on a class of hard exploration problems. Our method exponentially improves over $Q$-learning with naïve exploration, even when $Q$-learning has cheating access to latent states.

cs.LG↗

arXiv2021

Efficient First-Order Contextual Bandits: Prediction, Allocation, and Triangular Discrimination

A recurring theme in statistical learning, online learning, and beyond is that faster convergence rates are possible for problems with low noise, often quantified by the performance of the best hypothesis; such results are known as first-order or small-loss guarantees. While first-order guarantees are relatively well understood in statistical and online learning, adapting to low noise in contextual bandits (and more broadly, decision making) presents major algorithmic challenges. In a COLT 2017 open problem, Agarwal, Krishnamurthy, Langford, Luo, and Schapire asked whether first-order guarantees are even possible for contextual bandits and -- if so -- whether they can be attained by efficient algorithms. We give a resolution to this question by providing an optimal and efficient reduction from contextual bandits to online regression with the logarithmic (or, cross-entropy) loss. Our algorithm is simple and practical, readily accommodates rich function classes, and requires no distributional assumptions beyond realizability. In a large-scale empirical evaluation, we find that our approach typically outperforms comparable non-first-order methods. On the technical side, we show that the logarithmic loss and an information-theoretic quantity called the triangular discrimination play a fundamental role in obtaining first-order guarantees, and we combine this observation with new refinements to the regression oracle reduction framework of Foster and Rakhlin. The use of triangular discrimination yields novel results even for the classical statistical learning model, and we anticipate that it will find broader use.

cs.LG↗

arXiv2021

Bayesian decision-making under misspecified priors with applications to meta-learning

Thompson sampling and other Bayesian sequential decision-making algorithms are among the most popular approaches to tackle explore/exploit trade-offs in (contextual) bandits. The choice of prior in these algorithms offers flexibility to encode domain knowledge but can also lead to poor performance when misspecified. In this paper, we demonstrate that performance degrades gracefully with misspecification. We prove that the expected reward accrued by Thompson sampling (TS) with a misspecified prior differs by at most $\tilde{\mathcal{O}}(H^2 ε)$ from TS with a well specified prior, where $ε$ is the total-variation distance between priors and $H$ is the learning horizon. Our bound does not require the prior to have any parametric form. For priors with bounded support, our bound is independent of the cardinality or structure of the action space, and we show that it is tight up to universal constants in the worst case. Building on our sensitivity analysis, we establish generic PAC guarantees for algorithms in the recently studied Bayesian meta-learning setting and derive corollaries for various families of priors. Our results generalize along two axes: (1) they apply to a broader family of Bayesian decision-making algorithms, including a Monte-Carlo implementation of the knowledge gradient algorithm (KG), and (2) they apply to Bayesian POMDPs, the most general Bayesian decision-making setting, encompassing contextual bandits as a special case. Through numerical simulations, we illustrate how prior misspecification and the deployment of one-step look-ahead (as in KG) can impact the convergence of meta-learning in multi-armed and contextual bandits with structured and correlated priors.

cs.LG↗

arXiv2021

Investigating the Role of Negatives in Contrastive Representation Learning

Noise contrastive learning is a popular technique for unsupervised representation learning. In this approach, a representation is obtained via reduction to supervised learning, where given a notion of semantic similarity, the learner tries to distinguish a similar (positive) example from a collection of random (negative) examples. The success of modern contrastive learning pipelines relies on many parameters such as the choice of data augmentation, the number of negative examples, and the batch size; however, there is limited understanding as to how these parameters interact and affect downstream performance. We focus on disambiguating the role of one of these parameters: the number of negative examples. Theoretically, we show the existence of a collision-coverage trade-off suggesting that the optimal number of negative examples should scale with the number of underlying concepts in the data. Empirically, we scrutinize the role of the number of negatives in both NLP and vision tasks. In the NLP task, we find that the results broadly agree with our theory, while our vision experiments are murkier with performance sometimes even being insensitive to the number of negatives. We discuss plausible explanations for this behavior and suggest future directions to better align theory and practice.

cs.LG↗

arXiv2021

Contrastive learning, multi-view redundancy, and linear models

Self-supervised learning is an empirically successful approach to unsupervised learning based on creating artificial supervised learning problems. A popular self-supervised approach to representation learning is contrastive learning, which leverages naturally occurring pairs of similar and dissimilar data points, or multiple views of the same data. This work provides a theoretical analysis of contrastive learning in the multi-view setting, where two views of each datum are available. The main result is that linear functions of the learned representations are nearly optimal on downstream prediction tasks whenever the two views provide redundant information about the label.

cs.LG↗