A Survey of Online Experiment Design with the Stochastic Multi-Armed Bandit

Adaptive and sequential experiment design is a well-studied area in numerous domains. We survey and synthesize the work of the online statistical learning paradigm referred to as multi-armed bandits integrating the existing research as a resource for a certain class of online experiments. We first explore the traditional stochastic model of a multi-armed bandit, then explore a taxonomic scheme of complications to that model, for each complication relating it to a specific requirement or consideration of the experiment design context. Finally, at the end of the paper, we present a table of known upper-bounds of regret for all studied algorithms providing both perspectives for future theoretical work and a decision-making tool for practitioners looking for theoretical guarantees.Comment: 49 pages, 1 figur

Efficient Learning in Large-Scale Combinatorial Semi-Bandits

A stochastic combinatorial semi-bandit is an online learning problem where at each step a learning agent chooses a subset of ground items subject to combinatorial constraints, and then observes stochastic weights of these items and receives their sum as a payoff. In this paper, we consider efficient learning in large-scale combinatorial semi-bandits with linear generalization, and as a solution, propose two learning algorithms called Combinatorial Linear Thompson Sampling (CombLinTS) and Combinatorial Linear UCB (CombLinUCB). Both algorithms are computationally efficient as long as the offline version of the combinatorial problem can be solved efficiently. We establish that CombLinTS and CombLinUCB are also provably statistically efficient under reasonable assumptions, by developing regret bounds that are independent of the problem scale (number of items) and sublinear in time. We also evaluate CombLinTS on a variety of problems with thousands of items. Our experiment results demonstrate that CombLinTS is scalable, robust to the choice of algorithm parameters, and significantly outperforms the best of our baselines

Contextual Bandits with Random Projection

Contextual bandits with linear payoffs, which are also known as linear bandits, provide a powerful alternative for solving practical problems of sequential decisions, e.g., online advertisements. In the era of big data, contextual data usually tend to be high-dimensional, which leads to new challenges for traditional linear bandits mostly designed for the setting of low-dimensional contextual data. Due to the curse of dimensionality, there are two challenges in most of the current bandit algorithms: the first is high time-complexity; and the second is extreme large upper regret bounds with high-dimensional data. In this paper, in order to attack the above two challenges effectively, we develop an algorithm of Contextual Bandits via RAndom Projection (\texttt{CBRAP}) in the setting of linear payoffs, which works especially for high-dimensional contextual data. The proposed \texttt{CBRAP} algorithm is time-efficient and flexible, because it enables players to choose an arm in a low-dimensional space, and relaxes the sparsity assumption of constant number of non-zero components in previous work. Besides, we provide a linear upper regret bound for the proposed algorithm, which is associated with reduced dimensions

Small-loss bounds for online learning with partial information

We consider the problem of adversarial (non-stochastic) online learning with partial information feedback, where at each round, a decision maker selects an action from a finite set of alternatives. We develop a black-box approach for such problems where the learner observes as feedback only losses of a subset of the actions that includes the selected action. When losses of actions are non-negative, under the graph-based feedback model introduced by Mannor and Shamir, we offer algorithms that attain the so called "small-loss" $o(\alpha L^{\star})$ regret bounds with high probability, where $\alpha$ is the independence number of the graph, and $L^{\star}$ is the loss of the best action. Prior to our work, there was no data-dependent guarantee for general feedback graphs even for pseudo-regret (without dependence on the number of actions, i.e. utilizing the increased information feedback). Taking advantage of the black-box nature of our technique, we extend our results to many other applications such as semi-bandits (including routing in networks), contextual bandits (even with an infinite comparator class), as well as learning with slowly changing (shifting) comparators. In the special case of classical bandit and semi-bandit problems, we provide optimal small-loss, high-probability guarantees of $\tilde{O}(\sqrt{dL^{\star}})$ for actual regret, where $d$ is the number of actions, answering open questions of Neu. Previous bounds for bandits and semi-bandits were known only for pseudo-regret and only in expectation. We also offer an optimal $\tilde{O}(\sqrt{\kappa L^{\star}})$ regret guarantee for fixed feedback graphs with clique-partition number at most $\kappa$.Comment: An extended abstract appeared in COLT 201

Learning to Route Efficiently with End-to-End Feedback: The Value of Networked Structure

We introduce efficient algorithms which achieve nearly optimal regrets for the problem of stochastic online shortest path routing with end-to-end feedback. The setting is a natural application of the combinatorial stochastic bandits problem, a special case of the linear stochastic bandits problem. We show how the difficulties posed by the large scale action set can be overcome by the networked structure of the action set. Our approach presents a novel connection between bandit learning and shortest path algorithms. Our main contribution is an adaptive exploration algorithm with nearly optimal instance-dependent regret for any directed acyclic network. We then modify it so that nearly optimal worst case regret is achieved simultaneously. Driven by the carefully designed Top-Two Comparison (TTC) technique, the algorithms are efficiently implementable. We further conduct extensive numerical experiments to show that our proposed algorithms not only achieve superior regret performances, but also reduce the runtime drastically

Cascading Bandits for Large-Scale Recommendation Problems

Most recommender systems recommend a list of items. The user examines the list, from the first item to the last, and often chooses the first attractive item and does not examine the rest. This type of user behavior can be modeled by the cascade model. In this work, we study cascading bandits, an online learning variant of the cascade model where the goal is to recommend $K$ most attractive items from a large set of $L$ candidate items. We propose two algorithms for solving this problem, which are based on the idea of linear generalization. The key idea in our solutions is that we learn a predictor of the attraction probabilities of items from their features, as opposing to learning the attraction probability of each item independently as in the existing work. This results in practical learning algorithms whose regret does not depend on the number of items $L$. We bound the regret of one algorithm and comprehensively evaluate the other on a range of recommendation problems. The algorithm performs well and outperforms all baselines.Comment: Accepted to UAI 201

Combinatorial Semi-Bandits with Knapsacks

We unify two prominent lines of work on multi-armed bandits: bandits with knapsacks (BwK) and combinatorial semi-bandits. The former concerns limited "resources" consumed by the algorithm, e.g., limited supply in dynamic pricing. The latter allows a huge number of actions but assumes combinatorial structure and additional feedback to make the problem tractable. We define a common generalization, support it with several motivating examples, and design an algorithm for it. Our regret bounds are comparable with those for BwK and combinatorial semi- bandits

Multi-Objective Generalized Linear Bandits

In this paper, we study the multi-objective bandits (MOB) problem, where a learner repeatedly selects one arm to play and then receives a reward vector consisting of multiple objectives. MOB has found many real-world applications as varied as online recommendation and network routing. On the other hand, these applications typically contain contextual information that can guide the learning process which, however, is ignored by most of existing work. To utilize this information, we associate each arm with a context vector and assume the reward follows the generalized linear model (GLM). We adopt the notion of Pareto regret to evaluate the learner's performance and develop a novel algorithm for minimizing it. The essential idea is to apply a variant of the online Newton step to estimate model parameters, based on which we utilize the upper confidence bound (UCB) policy to construct an approximation of the Pareto front, and then uniformly at random choose one arm from the approximate Pareto front. Theoretical analysis shows that the proposed algorithm achieves an $\tilde O(d\sqrt{T})$ Pareto regret, where $T$ is the time horizon and $d$ is the dimension of contexts, which matches the optimal result for single objective contextual bandits problem. Numerical experiments demonstrate the effectiveness of our method

Semiparametric Contextual Bandits

This paper studies semiparametric contextual bandits, a generalization of the linear stochastic bandit problem where the reward for an action is modeled as a linear function of known action features confounded by an non-linear action-independent term. We design new algorithms that achieve $\tilde{O}(d\sqrt{T})$ regret over $T$ rounds, when the linear function is $d$-dimensional, which matches the best known bounds for the simpler unconfounded case and improves on a recent result of Greenewald et al. (2017). Via an empirical evaluation, we show that our algorithms outperform prior approaches when there are non-linear confounding effects on the rewards. Technically, our algorithms use a new reward estimator inspired by doubly-robust approaches and our proofs require new concentration inequalities for self-normalized martingales

Combinatorial Cascading Bandits

We propose combinatorial cascading bandits, a class of partial monitoring problems where at each step a learning agent chooses a tuple of ground items subject to constraints and receives a reward if and only if the weights of all chosen items are one. The weights of the items are binary, stochastic, and drawn independently of each other. The agent observes the index of the first chosen item whose weight is zero. This observation model arises in network routing, for instance, where the learning agent may only observe the first link in the routing path which is down, and blocks the path. We propose a UCB-like algorithm for solving our problems, CombCascade; and prove gap-dependent and gap-free upper bounds on its $n$-step regret. Our proofs build on recent work in stochastic combinatorial semi-bandits but also address two novel challenges of our setting, a non-linear reward function and partial observability. We evaluate CombCascade on two real-world problems and show that it performs well even when our modeling assumptions are violated. We also demonstrate that our setting requires a new learning algorithm.Comment: Advances in Neural Information Processing Systems 2