Gaussian Process bandits with adaptive discretization

In this paper, the problem of maximizing a black-box function $f:\mathcal{X} \to \mathbb{R}$ is studied in the Bayesian framework with a Gaussian Process (GP) prior. In particular, a new algorithm for this problem is proposed, and high probability bounds on its simple and cumulative regret are established. The query point selection rule in most existing methods involves an exhaustive search over an increasingly fine sequence of uniform discretizations of $\mathcal{X}$. The proposed algorithm, in contrast, adaptively refines $\mathcal{X}$ which leads to a lower computational complexity, particularly when $\mathcal{X}$ is a subset of a high dimensional Euclidean space. In addition to the computational gains, sufficient conditions are identified under which the regret bounds of the new algorithm improve upon the known results. Finally an extension of the algorithm to the case of contextual bandits is proposed, and high probability bounds on the contextual regret are presented.Comment: 34 pages, 2 figure

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

Action Centered Contextual Bandits

Contextual bandits have become popular as they offer a middle ground between very simple approaches based on multi-armed bandits and very complex approaches using the full power of reinforcement learning. They have demonstrated success in web applications and have a rich body of associated theoretical guarantees. Linear models are well understood theoretically and preferred by practitioners because they are not only easily interpretable but also simple to implement and debug. Furthermore, if the linear model is true, we get very strong performance guarantees. Unfortunately, in emerging applications in mobile health, the time-invariant linear model assumption is untenable. We provide an extension of the linear model for contextual bandits that has two parts: baseline reward and treatment effect. We allow the former to be complex but keep the latter simple. We argue that this model is plausible for mobile health applications. At the same time, it leads to algorithms with strong performance guarantees as in the linear model setting, while still allowing for complex nonlinear baseline modeling. Our theory is supported by experiments on data gathered in a recently concluded mobile health study.Comment: to appear at NIPS 201

Online Clustering of Bandits

We introduce a novel algorithmic approach to content recommendation based on adaptive clustering of exploration-exploitation ("bandit") strategies. We provide a sharp regret analysis of this algorithm in a standard stochastic noise setting, demonstrate its scalability properties, and prove its effectiveness on a number of artificial and real-world datasets. Our experiments show a significant increase in prediction performance over state-of-the-art methods for bandit problems.Comment: In E. Xing and T. Jebara (Eds.), Proceedings of 31st International Conference on Machine Learning, Journal of Machine Learning Research Workshop and Conference Proceedings, Vol.32 (JMLR W&CP-32), Beijing, China, Jun. 21-26, 2014 (ICML 2014), Submitted by Shuai Li (https://sites.google.com/site/shuailidotsli

Structured Stochastic Linear Bandits

The stochastic linear bandit problem proceeds in rounds where at each round the algorithm selects a vector from a decision set after which it receives a noisy linear loss parameterized by an unknown vector. The goal in such a problem is to minimize the (pseudo) regret which is the difference between the total expected loss of the algorithm and the total expected loss of the best fixed vector in hindsight. In this paper, we consider settings where the unknown parameter has structure, e.g., sparse, group sparse, low-rank, which can be captured by a norm, e.g., $L_1$, $L_{(1,2)}$, nuclear norm. We focus on constructing confidence ellipsoids which contain the unknown parameter across all rounds with high-probability. We show the radius of such ellipsoids depend on the Gaussian width of sets associated with the norm capturing the structure. Such characterization leads to tighter confidence ellipsoids and, therefore, sharper regret bounds compared to bounds in the existing literature which are based on the ambient dimensionality

Algorithms for Linear Bandits on Polyhedral Sets

We study stochastic linear optimization problem with bandit feedback. The set of arms take values in an $N$-dimensional space and belong to a bounded polyhedron described by finitely many linear inequalities. We provide a lower bound for the expected regret that scales as $\Omega(N\log T)$. We then provide a nearly optimal algorithm and show that its expected regret scales as $O(N\log^{1+\epsilon}(T))$ for an arbitrary small $\epsilon >0$. The algorithm alternates between exploration and exploitation intervals sequentially where deterministic set of arms are played in the exploration intervals and greedily selected arm is played in the exploitation intervals. We also develop an algorithm that achieves the optimal regret when sub-Gaussianity parameter of the noise term is known. Our key insight is that for a polyhedron the optimal arm is robust to small perturbations in the reward function. Consequently, a greedily selected arm is guaranteed to be optimal when the estimation error falls below some suitable threshold. Our solution resolves a question posed by Rusmevichientong and Tsitsiklis (2011) that left open the possibility of efficient algorithms with asymptotic logarithmic regret bounds. We also show that the regret upper bounds hold with probability $1$. Our numerical investigations show that while theoretical results are asymptotic the performance of our algorithms compares favorably to state-of-the-art algorithms in finite time as well

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

Horde of Bandits using Gaussian Markov Random Fields

The gang of bandits (GOB) model \cite{cesa2013gang} is a recent contextual bandits framework that shares information between a set of bandit problems, related by a known (possibly noisy) graph. This model is useful in problems like recommender systems where the large number of users makes it vital to transfer information between users. Despite its effectiveness, the existing GOB model can only be applied to small problems due to its quadratic time-dependence on the number of nodes. Existing solutions to combat the scalability issue require an often-unrealistic clustering assumption. By exploiting a connection to Gaussian Markov random fields (GMRFs), we show that the GOB model can be made to scale to much larger graphs without additional assumptions. In addition, we propose a Thompson sampling algorithm which uses the recent GMRF sampling-by-perturbation technique, allowing it to scale to even larger problems (leading to a "horde" of bandits). We give regret bounds and experimental results for GOB with Thompson sampling and epoch-greedy algorithms, indicating that these methods are as good as or significantly better than ignoring the graph or adopting a clustering-based approach. Finally, when an existing graph is not available, we propose a heuristic for learning it on the fly and show promising results

Estimation Considerations in Contextual Bandits

Contextual bandit algorithms are sensitive to the estimation method of the outcome model as well as the exploration method used, particularly in the presence of rich heterogeneity or complex outcome models, which can lead to difficult estimation problems along the path of learning. We study a consideration for the exploration vs. exploitation framework that does not arise in multi-armed bandits but is crucial in contextual bandits; the way exploration and exploitation is conducted in the present affects the bias and variance in the potential outcome model estimation in subsequent stages of learning. We develop parametric and non-parametric contextual bandits that integrate balancing methods from the causal inference literature in their estimation to make it less prone to problems of estimation bias. We provide the first regret bound analyses for contextual bandits with balancing in the domain of linear contextual bandits that match the state of the art regret bounds. We demonstrate the strong practical advantage of balanced contextual bandits on a large number of supervised learning datasets and on a synthetic example that simulates model mis-specification and prejudice in the initial training data. Additionally, we develop contextual bandits with simpler assignment policies by leveraging sparse model estimation methods from the econometrics literature and demonstrate empirically that in the early stages they can improve the rate of learning and decrease regret

Stochastic Process Bandits: Upper Confidence Bounds Algorithms via Generic Chaining

The paper considers the problem of global optimization in the setup of stochastic process bandits. We introduce an UCB algorithm which builds a cascade of discretization trees based on generic chaining in order to render possible his operability over a continuous domain. The theoretical framework applies to functions under weak probabilistic smoothness assumptions and also extends significantly the spectrum of application of UCB strategies. Moreover generic regret bounds are derived which are then specialized to Gaussian processes indexed on infinite-dimensional spaces as well as to quadratic forms of Gaussian processes. Lower bounds are also proved in the case of Gaussian processes to assess the optimality of the proposed algorithm.Comment: preprin