arXiv subjects

Shipra Agrawal

Publications and source records attributed to Shipra Agrawal.

At least 37 records · Page 2Linked to original sources

arXiv2019

Learning in structured MDPs with convex cost functions: Improved regret bounds for inventory management

We consider a stochastic inventory control problem under censored demands, lost sales, and positive lead times. This is a fundamental problem in inventory management, with significant literature establishing near-optimality of a simple class of policies called ``base-stock policies'' for the underlying Markov Decision Process (MDP), as well as convexity of long run average-cost under those policies. We consider the relatively less studied problem of designing a learning algorithm for this problem when the underlying demand distribution is unknown. The goal is to bound regret of the algorithm when compared to the best base-stock policy. We utilize the convexity properties and a newly derived bound on bias of base-stock policies to establish a connection to stochastic convex bandit optimization. Our main contribution is a learning algorithm with a regret bound of $\tilde{O}(L\sqrt{T}+D)$ for the inventory control problem. Here $L$ is the fixed and known lead time, and $D$ is an unknown parameter of the demand distribution described roughly as the number of time steps needed to generate enough demand for depleting one unit of inventory. Notably, even though the state space of the underlying MDP is continuous and $L$-dimensional, our regret bounds depend linearly on $L$. Our results significantly improve the previously best known regret bounds for this problem where the dependence on $L$ was exponential and many further assumptions on demand distribution were required. The techniques presented here may be of independent interest for other settings that involve large structured MDPs but with convex cost functions.

cs.LG↗

arXiv2019

Robust Repeated Auctions under Heterogeneous Buyer Behavior

We study revenue optimization in a repeated auction between a single seller and a single buyer. Traditionally, the design of repeated auctions requires strong modeling assumptions about the bidder behavior, such as it being myopic, infinite lookahead, or some specific form of learning behavior. Is it possible to design mechanisms which are simultaneously optimal against a multitude of possible buyer behaviors? We answer this question by designing a simple state-based mechanism that is simultaneously approximately optimal against a $k$-lookahead buyer for all $k$, a buyer who is a no-regret learner, and a buyer who is a policy-regret learner. Against each type of buyer our mechanism attains a constant fraction of the optimal revenue attainable against that type of buyer. We complement our positive results with almost tight impossibility results, showing that the revenue approximation tradeoffs achieved by our mechanism for different lookahead attitudes are near-optimal.

cs.GT↗

arXiv2019

Implicit Policy for Reinforcement Learning

We introduce Implicit Policy, a general class of expressive policies that can flexibly represent complex action distributions in reinforcement learning, with efficient algorithms to compute entropy regularized policy gradients. We empirically show that, despite its simplicity in implementation, entropy regularization combined with a rich policy class can attain desirable properties displayed under maximum entropy reinforcement learning framework, such as robustness and multi-modality.

cs.LG↗

arXiv2019

Boosting Trust Region Policy Optimization by Normalizing Flows Policy

We propose to improve trust region policy search with normalizing flows policy. We illustrate that when the trust region is constructed by KL divergence constraints, normalizing flows policy generates samples far from the 'center' of the previous policy iterate, which potentially enables better exploration and helps avoid bad local optima. Through extensive comparisons, we show that the normalizing flows policy significantly improves upon baseline architectures especially on high-dimensional tasks with complex dynamics.

cs.AI↗

arXiv2019

Thompson Sampling for the MNL-Bandit

We consider a sequential subset selection problem under parameter uncertainty, where at each time step, the decision maker selects a subset of cardinality $K$ from $N$ possible items (arms), and observes a (bandit) feedback in the form of the index of one of the items in said subset, or none. Each item in the index set is ascribed a certain value (reward), and the feedback is governed by a Multinomial Logit (MNL) choice model whose parameters are a priori unknown. The objective of the decision maker is to maximize the expected cumulative rewards over a finite horizon $T$, or alternatively, minimize the regret relative to an oracle that knows the MNL parameters. We refer to this as the MNL-Bandit problem. This problem is representative of a larger family of exploration-exploitation problems that involve a combinatorial objective, and arise in several important application domains. We present an approach to adapt Thompson Sampling to this problem and show that it achieves near-optimal regret as well as attractive numerical performance.

cs.LG↗

arXiv2018

Submodular Secretary Problem with Shortlists

In submodular $k$-secretary problem, the goal is to select $k$ items in a randomly ordered input so as to maximize the expected value of a given monotone submodular function on the set of selected items. In this paper, we introduce a relaxation of this problem, which we refer to as submodular $k$-secretary problem with shortlists. In the proposed problem setting, the algorithm is allowed to choose more than $k$ items as part of a shortlist. Then, after seeing the entire input, the algorithm can choose a subset of size $k$ from the bigger set of items in the shortlist. We are interested in understanding to what extent this relaxation can improve the achievable competitive ratio for the submodular $k$-secretary problem. In particular, using an $O(k)$ shortlist, can an online algorithm achieve a competitive ratio close to the best achievable online approximation factor for this problem? We answer this question affirmatively by giving a polynomial time algorithm that achieves a $1-1/e-ε-O(k^{-1})$ competitive ratio for any constant $ε> 0$, using a shortlist of size $η_ε(k) = O(k)$. Also, for the special case of m-submodular functions, we demonstrate an algorithm that achieves a $1-ε$ competitive ratio for any constant $ε> 0$, using an $O(1)$ shortlist. Finally, we show that our algorithm can be implemented in the streaming setting using a memory buffer of size $η_ε(k) = O(k)$ to achieve a $1 - 1/e - ε-O(k^{-1})$ approximation for submodular function maximization in the random order streaming model. This substantially improves upon the previously best known approximation factor of $1/2 + 8 \times 10^{-14}$ [Norouzi-Fard et al. 2018] that used a memory buffer of size $O(k \log k)$.

cs.DS↗

arXiv2018

MNL-Bandit: A Dynamic Learning Approach to Assortment Selection

We consider a dynamic assortment selection problem, where in every round the retailer offers a subset (assortment) of $N$ substitutable products to a consumer, who selects one of these products according to a multinomial logit (MNL) choice model. The retailer observes this choice and the objective is to dynamically learn the model parameters, while optimizing cumulative revenues over a selling horizon of length $T$. We refer to this exploration-exploitation formulation as the MNL-Bandit problem. Existing methods for this problem follow an "explore-then-exploit" approach, which estimate parameters to a desired accuracy and then, treating these estimates as if they are the correct parameter values, offers the optimal assortment based on these estimates. These approaches require certain a priori knowledge of "separability", determined by the true parameters of the underlying MNL model, and this in turn is critical in determining the length of the exploration period. (Separability refers to the distinguishability of the true optimal assortment from the other sub-optimal alternatives.) In this paper, we give an efficient algorithm that simultaneously explores and exploits, achieving performance independent of the underlying parameters. The algorithm can be implemented in a fully online manner, without knowledge of the horizon length $T$. Furthermore, the algorithm is adaptive in the sense that its performance is near-optimal in both the "well separated" case, as well as the general parameter setting where this separation need not hold.

cs.LG↗

arXiv2018

Exploration by Distributional Reinforcement Learning

We propose a framework based on distributional reinforcement learning and recent attempts to combine Bayesian parameter updates with deep reinforcement learning. We show that our proposed framework conceptually unifies multiple previous methods in exploration. We also derive a practical algorithm that achieves efficient exploration on challenging control tasks.

cs.LG↗

arXiv2018

Bandits with Delayed, Aggregated Anonymous Feedback

We study a variant of the stochastic $K$-armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback". In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the information loss due to this delayed, aggregated anonymous feedback? Previous works have studied bandits with stochastic, non-anonymous delays and found that the regret increases only by an additive factor relating to the expected delay. In this paper, we show that this additive regret increase can be maintained in the harder delayed, aggregated anonymous feedback setting when the expected delay (or a bound on it) is known. We provide an algorithm that matches the worst case regret of the non-anonymous problem exactly when the delays are bounded, and up to logarithmic factors or an additive variance term for unbounded delays.

stat.ML↗

arXiv2016

An efficient algorithm for contextual bandits with knapsacks, and an extension to concave objectives

We consider a contextual version of multi-armed bandit problem with global knapsack constraints. In each round, the outcome of pulling an arm is a scalar reward and a resource consumption vector, both dependent on the context, and the global knapsack constraints require the total consumption for each resource to be below some pre-fixed budget. The learning agent competes with an arbitrary set of context-dependent policies. This problem was introduced by Badanidiyuru et al. (2014), who gave a computationally inefficient algorithm with near-optimal regret bounds for it. We give a computationally efficient algorithm for this problem with slightly better regret bounds, by generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. The computational time of our algorithm scales logarithmically in the size of the policy space. This answers the main open question of Badanidiyuru et al. (2014). We also extend our results to a variant where there are no knapsack constraints but the objective is an arbitrary Lipschitz concave function of the sum of outcome vectors.

cs.LG↗

arXiv2016

Linear Contextual Bandits with Knapsacks

We consider the linear contextual bandit problem with resource consumption, in addition to reward generation. In each round, the outcome of pulling an arm is a reward as well as a vector of resource consumptions. The expected values of these outcomes depend linearly on the context of that arm. The budget/capacity constraints require that the total consumption doesn't exceed the budget for each resource. The objective is once again to maximize the total reward. This problem turns out to be a common generalization of classic linear contextual bandits (linContextual), bandits with knapsacks (BwK), and the online stochastic packing problem (OSPP). We present algorithms with near-optimal regret bounds for this problem. Our bounds compare favorably to results on the unstructured version of the problem where the relation between the contexts and the outcomes could be arbitrary, but the algorithm only competes against a fixed set of policies accessible through an optimization oracle. We combine techniques from the work on linContextual, BwK, and OSPP in a nontrivial manner while also tackling new difficulties that are not present in any of these special cases.

cs.LG↗

arXiv2014

Fast Algorithms for Online Stochastic Convex Programming

We introduce the online stochastic Convex Programming (CP) problem, a very general version of stochastic online problems which allows arbitrary concave objectives and convex feasibility constraints. Many well-studied problems like online stochastic packing and covering, online stochastic matching with concave returns, etc. form a special case of online stochastic CP. We present fast algorithms for these problems, which achieve near-optimal regret guarantees for both the i.i.d. and the random permutation models of stochastic inputs. When applied to the special case online packing, our ideas yield a simpler and faster primal-dual algorithm for this well studied problem, which achieves the optimal competitive ratio. Our techniques make explicit the connection of primal-dual paradigm and online learning to online stochastic CP.

cs.LG↗

arXiv2014

A Dynamic Near-Optimal Algorithm for Online Linear Programming

A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) in which the constraint matrix is revealed column by column along with the corresponding objective coefficient. In such a model, a decision variable has to be set each time a column is revealed without observing the future inputs and the goal is to maximize the overall objective function. In this paper, we provide a near-optimal algorithm for this general class of online problems under the assumption of random order of arrival and some mild conditions on the size of the LP right-hand-side input. Specifically, our learning-based algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from the revealed columns in the previous period are used to determine the sequential decisions in the current period. Due to the feature of dynamic learning, the competitiveness of our algorithm improves over the past study of the same problem. We also present a worst-case example showing that the performance of our algorithm is near-optimal.

cs.DS↗

arXiv2014

Bandits with concave rewards and convex knapsacks

In this paper, we consider a very general model for exploration-exploitation tradeoff which allows arbitrary concave rewards and convex constraints on the decisions across time, in addition to the customary limitation on the time horizon. This model subsumes the classic multi-armed bandit (MAB) model, and the Bandits with Knapsacks (BwK) model of Badanidiyuru et al.[2013]. We also consider an extension of this model to allow linear contexts, similar to the linear contextual extension of the MAB model. We demonstrate that a natural and simple extension of the UCB family of algorithms for MAB provides a polynomial time algorithm that has near-optimal regret guarantees for this substantially more general model, and matches the bounds provided by Badanidiyuru et al.[2013] for the special case of BwK, which is quite surprising. We also provide computationally more efficient algorithms by establishing interesting connections between this problem and other well studied problems/algorithms such as the Blackwell approachability problem, online convex optimization, and the Frank-Wolfe technique for convex optimization. We give examples of several concrete applications, where this more general model of bandits allows for richer and/or more efficient formulations of the problem.

cs.LG↗

arXiv2014

Thompson Sampling for Contextual Bandits with Linear Payoffs

Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state-of-the-art methods. However, many questions regarding its theoretical performance remained open. In this paper, we design and analyze a generalization of Thompson Sampling algorithm for the stochastic contextual multi-armed bandit problem with linear payoff functions, when the contexts are provided by an adaptive adversary. This is among the most important and widely studied versions of the contextual bandits problem. We provide the first theoretical guarantees for the contextual version of Thompson Sampling. We prove a high probability regret bound of $\tilde{O}(d^{3/2}\sqrt{T})$ (or $\tilde{O}(d\sqrt{T \log(N)})$), which is the best regret bound achieved by any computationally efficient algorithm available for this problem in the current literature, and is within a factor of $\sqrt{d}$ (or $\sqrt{\log(N)}$) of the information-theoretic lower bound for this problem.

cs.LG↗

arXiv2012

Further Optimal Regret Bounds for Thompson Sampling

Thompson Sampling is one of the oldest heuristics for multi-armed bandit problems. It is a randomized algorithm based on Bayesian ideas, and has recently generated significant interest after several studies demonstrated it to have better empirical performance compared to the state of the art methods. In this paper, we provide a novel regret analysis for Thompson Sampling that simultaneously proves both the optimal problem-dependent bound of $(1+ε)\sum_i \frac{\ln T}{Δ_i}+O(\frac{N}{ε^2})$ and the first near-optimal problem-independent bound of $O(\sqrt{NT\ln T})$ on the expected regret of this algorithm. Our near-optimal problem-independent bound solves a COLT 2012 open problem of Chapelle and Li. The optimal problem-dependent regret bound for this problem was first proven recently by Kaufmann et al. [ALT 2012]. Our novel martingale-based analysis techniques are conceptually simple, easily extend to distributions other than the Beta distribution, and also extend to the more general contextual bandits setting [Manuscript, Agrawal and Goyal, 2012].

cs.LG↗

arXiv2012

Analysis of Thompson Sampling for the multi-armed bandit problem

The multi-armed bandit problem is a popular model for studying exploration/exploitation trade-off in sequential decision problems. Many algorithms are now available for this well-studied problem. One of the earliest algorithms, given by W. R. Thompson, dates back to 1933. This algorithm, referred to as Thompson Sampling, is a natural Bayesian algorithm. The basic idea is to choose an arm to play according to its probability of being the best arm. Thompson Sampling algorithm has experimentally been shown to be close to optimal. In addition, it is efficient to implement and exhibits several desirable properties such as small regret for delayed feedback. However, theoretical understanding of this algorithm was quite limited. In this paper, for the first time, we show that Thompson Sampling algorithm achieves logarithmic expected regret for the multi-armed bandit problem. More precisely, for the two-armed bandit problem, the expected regret in time $T$ is $O(\frac{\ln T}Δ + \frac{1}{Δ^3})$. And, for the $N$-armed bandit problem, the expected regret in time $T$ is $O([(\sum_{i=2}^N \frac{1}{Δ_i^2})^2] \ln T)$. Our bounds are optimal but for the dependence on $Δ_i$ and the constant factors in big-Oh.

cs.LG↗

arXiv2009

Correlation Robust Stochastic Optimization

We consider a robust model proposed by Scarf, 1958, for stochastic optimization when only the marginal probabilities of (binary) random variables are given, and the correlation between the random variables is unknown. In the robust model, the objective is to minimize expected cost against worst possible joint distribution with those marginals. We introduce the concept of correlation gap to compare this model to the stochastic optimization model that ignores correlations and minimizes expected cost under independent Bernoulli distribution. We identify a class of functions, using concepts of summable cost sharing schemes from game theory, for which the correlation gap is well-bounded and the robust model can be approximated closely by the independent distribution model. As a result, we derive efficient approximation factors for many popular cost functions, like submodular functions, facility location, and Steiner tree. As a byproduct, our analysis also yields some new results in the areas of social welfare maximization and existence of Walrasian equilibria, which may be of independent interest.

cs.DS↗