### A Note on Bounding Regret of the C$^2$UCB Contextual Combinatorial Bandit

We revisit the proof by Qin et al. (2014) of bounded regret of the C$^2$UCB contextual combinatorial bandit. We demonstrate an error in the proof of volumetric expansion of the moment matrix, used in upper bounding a function of context vector norms. We prove a relaxed inequality that yields the originally-stated regret bound.

### Sparse Dueling Bandits

The dueling bandit problem is a variation of the classical multi-armed bandit in which the allowable actions are noisy comparisons between pairs of arms. This paper focuses on a new approach for finding the "best" arm according to the Borda criterion using noisy comparisons. We prove that in the absence of structural assumptions, the sample complexity of this problem is proportional to the sum of the inverse squared gaps between the Borda scores of each suboptimal arm and the best arm. We explore this dependence further and consider structural constraints on the pairwise comparison matrix (a particular form of sparsity natural to this problem) that can significantly reduce the sample complexity. This motivates a new algorithm called Successive Elimination with Comparison Sparsity (SECS) that exploits sparsity to find the Borda winner using fewer samples than standard algorithms. We also evaluate the new algorithm experimentally with synthetic and real data. The results show that the sparsity model and the new algorithm can provide significant improvements over standard approaches.

### Bilinear Bandits with Low-rank Structure

We introduce the bilinear bandit problem with low-rank structure where an action is a pair of arms from two different entity types, and the reward is a bilinear function of the known feature vectors of the arms. The problem is motivated by numerous applications in which the learner must recommend two different entity types as one action, such as a male / female pair in an online dating service. The unknown in the problem is a $d_1$ by $d_2$ matrix $\mathbf{\Theta}^*$ with rank $r \ll \min\{d_1,d_2\}$ governing the reward generation. Determination of $\mathbf{\Theta}^*$ with low-rank structure poses a significant challenge in finding the right exploration-exploitation tradeoff. In this work, we propose a new two-stage algorithm called "Explore-Subspace-Then-Refine" (ESTR). The first stage is an explicit subspace exploration, while the second stage is a linear bandit algorithm called "almost-low-dimensional OFUL" (LowOFUL) that exploits and further refines the estimated subspace via a regularization technique. We show that the regret of ESTR is $\tilde{O}((d_1+d_2)^{3/2} \sqrt{r T})$ (where $\tilde{O}$ hides logarithmic factors), which improves upon the regret of $\tilde{O}(d_1d_2\sqrt{T})$ of a naive linear bandit reduction. We conjecture that the regret bound of ESTR is unimprovable up to polylogarithmic factors.

### 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.

### Gaussian Process Bandits for Tree Search: Theory and Application to Planning in Discounted MDPs

We motivate and analyse a new Tree Search algorithm, GPTS, based on recent theoretical advances in the use of Gaussian Processes for Bandit problems. We consider tree paths as arms and we assume the target/reward function is drawn from a GP distribution. The posterior mean and variance, after observing data, are used to define confidence intervals for the function values, and we sequentially play arms with highest upper confidence bounds. We give an efficient implementation of GPTS and we adapt previous regret bounds by determining the decay rate of the eigenvalues of the kernel matrix on the whole set of tree paths. We consider two kernels in the feature space of binary vectors indexed by the nodes of the tree: linear and Gaussian. The regret grows in square root of the number of iterations T, up to a logarithmic factor, with a constant that improves with bigger Gaussian kernel widths. We focus on practical values of T, smaller than the number of arms. Finally, we apply GPTS to Open Loop Planning in discounted Markov Decision Processes by modelling the reward as a discounted sum of independent Gaussian Processes. We report similar regret bounds to those of the OLOP algorithm.