Sequential Matrix Completion

We propose a novel algorithm for sequential matrix completion in a recommender system setting, where the $(i,j)$th entry of the matrix corresponds to a user $i$'s rating of product $j$. The objective of the algorithm is to provide a sequential policy for user-product pair recommendation which will yield the highest possible ratings after a finite time horizon. The algorithm uses a Gamma process factor model with two posterior-focused bandit policies, Thompson Sampling and Information-Directed Sampling. While Thompson Sampling shows competitive performance in simulations, state-of-the-art performance is obtained from Information-Directed Sampling, which makes its recommendations based off a ratio between the expected reward and a measure of information gain. To our knowledge, this is the first implementation of Information Directed Sampling on large real datasets. This approach contributes to a recent line of research on bandit approaches to collaborative filtering including Kawale et al. (2015), Li et al. (2010), Bresler et al. (2014), Li et al. (2016), Deshpande & Montanari (2012), and Zhao et al. (2013). The setting of this paper, as has been noted in Kawale et al. (2015) and Zhao et al. (2013), presents significant challenges to bounding regret after finite horizons. We discuss these challenges in relation to simpler models for bandits with side information, such as linear or gaussian process bandits, and hope the experiments presented here motivate further research toward theoretical guarantees.Comment: 10 pages, 6 figure

Alternating Linear Bandits for Online Matrix-Factorization Recommendation

We consider the problem of online collaborative filtering in the online setting, where items are recommended to the users over time. At each time step, the user (selected by the environment) consumes an item (selected by the agent) and provides a rating of the selected item. In this paper, we propose a novel algorithm for online matrix factorization recommendation that combines linear bandits and alternating least squares. In this formulation, the bandit feedback is equal to the difference between the ratings of the best and selected items. We evaluate the performance of the proposed algorithm over time using both cumulative regret and average cumulative NDCG. Simulation results over three synthetic datasets as well as three real-world datasets for online collaborative filtering indicate the superior performance of the proposed algorithm over two state-of-the-art online algorithms

Unreliable Multi-Armed Bandits: A Novel Approach to Recommendation Systems

We use a novel modification of Multi-Armed Bandits to create a new model for recommendation systems. We model the recommendation system as a bandit seeking to maximize reward by pulling on arms with unknown rewards. The catch however is that this bandit can only access these arms through an unreliable intermediate that has some level of autonomy while choosing its arms. For example, in a streaming website the user has a lot of autonomy while choosing content they want to watch. The streaming sites can use targeted advertising as a means to bias opinions of these users. Here the streaming site is the bandit aiming to maximize reward and the user is the unreliable intermediate. We model the intermediate as accessing states via a Markov chain. The bandit is allowed to perturb this Markov chain. We prove fundamental theorems for this setting after which we show a close-to-optimal Explore-Commit algorithm.Comment: 4 pages, 4 figures, Aditya Narayan Ravi and Pranav Poduval have equal contributio

Time-Sensitive Bandit Learning and Satisficing Thompson Sampling

The literature on bandit learning and regret analysis has focused on contexts where the goal is to converge on an optimal action in a manner that limits exploration costs. One shortcoming imposed by this orientation is that it does not treat time preference in a coherent manner. Time preference plays an important role when the optimal action is costly to learn relative to near-optimal actions. This limitation has not only restricted the relevance of theoretical results but has also influenced the design of algorithms. Indeed, popular approaches such as Thompson sampling and UCB can fare poorly in such situations. In this paper, we consider discounted rather than cumulative regret, where a discount factor encodes time preference. We propose satisficing Thompson sampling -- a variation of Thompson sampling -- and establish a strong discounted regret bound for this new algorithm

Accurate Inference for Adaptive Linear Models

Estimators computed from adaptively collected data do not behave like their non-adaptive brethren. Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method -- $\mathbf{W}$-decorrelation -- for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy. We bound the finite-sample bias and variance of the $\mathbf{W}$-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic $\mathbf{W}$-decorrelation procedure in two different adaptive data settings: the multi-armed bandit and the autoregressive time series.Comment: Typos fixed for clarificatio

The Sample Complexity of Online One-Class Collaborative Filtering

We consider the online one-class collaborative filtering (CF) problem that consists of recommending items to users over time in an online fashion based on positive ratings only. This problem arises when users respond only occasionally to a recommendation with a positive rating, and never with a negative one. We study the impact of the probability of a user responding to a recommendation, p_f, on the sample complexity, i.e., the number of ratings required to make `good' recommendations, and ask whether receiving positive and negative ratings, instead of positive ratings only, improves the sample complexity. Both questions arise in the design of recommender systems. We introduce a simple probabilistic user model, and analyze the performance of an online user-based CF algorithm. We prove that after an initial cold start phase, where recommendations are invested in exploring the user's preferences, this algorithm makes---up to a fraction of the recommendations required for updating the user's preferences---perfect recommendations. The number of ratings required for the cold start phase is nearly proportional to 1/p_f, and that for updating the user's preferences is essentially independent of p_f. As a consequence we find that, receiving positive and negative ratings instead of only positive ones improves the number of ratings required for initial exploration by a factor of 1/p_f, which can be significant.Comment: ICML 201

Risk-Averse Multi-Armed Bandit Problems under Mean-Variance Measure

The multi-armed bandit problems have been studied mainly under the measure of expected total reward accrued over a horizon of length $T$. In this paper, we address the issue of risk in multi-armed bandit problems and develop parallel results under the measure of mean-variance, a commonly adopted risk measure in economics and mathematical finance. We show that the model-specific regret and the model-independent regret in terms of the mean-variance of the reward process are lower bounded by $\Omega(\log T)$ and $\Omega(T^{2/3})$, respectively. We then show that variations of the UCB policy and the DSEE policy developed for the classic risk-neutral MAB achieve these lower bounds

Implementable confidence sets in high dimensional regression

We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter \theta, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a such confidence set should depend on the sparsity S of \theta - the larger S, the wider the confidence set. However, in practice, S is unknown. This paper focuses on constructing a confidence set for \theta which contains \theta with high probability, whose diameter is adaptive to the unknown sparsity S, and which is implementable in practice

Satisficing in Time-Sensitive Bandit Learning

Much of the recent literature on bandit learning focuses on algorithms that aim to converge on an optimal action. One shortcoming is that this orientation does not account for time sensitivity, which can play a crucial role when learning an optimal action requires much more information than near-optimal ones. Indeed, popular approaches such as upper-confidence-bound methods and Thompson sampling can fare poorly in such situations. We consider instead learning a satisficing action, which is near-optimal while requiring less information, and propose satisficing Thompson sampling, an algorithm that serves this purpose. We establish a general bound on expected discounted regret and study the application of satisficing Thompson sampling to linear and infinite-armed bandits, demonstrating arbitrarily large benefits over Thompson sampling. We also discuss the relation between the notion of satisficing and the theory of rate distortion, which offers guidance on the selection of satisficing actions.Comment: This submission largely supersedes earlier work in arXiv:1704.0902

Distributed Online Learning in Social Recommender Systems

In this paper, we consider decentralized sequential decision making in distributed online recommender systems, where items are recommended to users based on their search query as well as their specific background including history of bought items, gender and age, all of which comprise the context information of the user. In contrast to centralized recommender systems, in which there is a single centralized seller who has access to the complete inventory of items as well as the complete record of sales and user information, in decentralized recommender systems each seller/learner only has access to the inventory of items and user information for its own products and not the products and user information of other sellers, but can get commission if it sells an item of another seller. Therefore the sellers must distributedly find out for an incoming user which items to recommend (from the set of own items or items of another seller), in order to maximize the revenue from own sales and commissions. We formulate this problem as a cooperative contextual bandit problem, analytically bound the performance of the sellers compared to the best recommendation strategy given the complete realization of user arrivals and the inventory of items, as well as the context-dependent purchase probabilities of each item, and verify our results via numerical examples on a distributed data set adapted based on Amazon data. We evaluate the dependence of the performance of a seller on the inventory of items the seller has, the number of connections it has with the other sellers, and the commissions which the seller gets by selling items of other sellers to its users