# upper bounds

• Michal Valko and Remi Munos and Branislav Kveton and Tomáš Kocák

### Spectral Bandits for Smooth Graph Functions (pdf)

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.

### Discrete Chebyshev Classifiers (pdf)

In large scale learning problems it is often easy to collect simple statistics of the data, but hard or impractical to store all the original data. A key question in this setting is how to construct classifiers based on such partial information. One traditional approach to the problem has been to use maximum entropy arguments to induce a complete distribution on variables from statistics. However, this approach essentially makes conditional independence assumptions about the distribution, and furthermore does not optimize prediction loss. Here we present a framework for discriminative learning given a set of statistics. Specifically, we address the case where all variables are discrete and we have access to various marginals. Our approach minimizes the worst case hinge loss in this case, which upper bounds the generalization error. We show that for certain sets of statistics the problem is tractable, and in the general case can be approximated using MAP LP relaxations. Empirical results show that the method is competitive with other approaches that use the same input.

• Yasuhiro Fujiwara and Go Irie

### Efficient Label Propagation (pdf)

Label propagation is a popular graph-based semi-supervised learning framework. So as to obtain the optimal labeling scores, the label propagation algorithm requires an inverse matrix which incurs the high computational cost of O(n^3+cn^2), where n and c are the numbers of data points and labels, respectively. This paper proposes an efficient label propagation algorithm that guarantees exactly the same labeling results as those yielded by optimal labeling scores. The key to our approach is to iteratively compute lower and upper bounds of labeling scores to prune unnecessary score computations. This idea significantly reduces the computational cost to O(cnt) where t is the average number of iterations for each label and t << n in practice. Experiments demonstrate the significant superiority of our algorithm over existing label propagation methods.

• Emile Contal and Vianney Perchet and Nicolas Vayatis

### Gaussian Process Optimization with Mutual Information (pdf)

In this paper, we analyze a generic algorithm scheme for sequential global optimization using Gaussian processes. The upper bounds we derive on the cumulative regret for this generic algorithm improve by an exponential factor the previously known bounds for algorithms like GP-UCB. We also introduce the novel Gaussian Process Mutual Information algorithm (GP-MI), which significantly improves further these upper bounds for the cumulative regret. We confirm the efficiency of this algorithm on synthetic and real tasks against the natural competitor, GP-UCB, and also the Expected Improvement heuristic.

• Zongzhang Zhang and David Hsu and Wee Sun Lee

### Covering Number for Efficient Heuristic-based POMDP Planning (pdf)

The difficulty of POMDP planning depends on the size of the search space involved. Heuristics are often used to reduce the search space size and improve computational efficiency; however, there are few theoretical bounds on their effectiveness. In this paper, we use the covering number to characterize the size of the search space reachable under heuristics and connect the complexity of POMDP planning to the effectiveness of heuristics. With insights from the theoretical analysis, we have developed a practical POMDP algorithm, Packing-Guided Value Iteration (PGVI). Empirically, PGVI is competitive with the state-of-the-art point-based POMDP algorithms on 65 small benchmark problems and outperforms them on 4 larger problems.

• Mohammad Gheshlaghi azar and Alessandro Lazaric and Emma Brunskill

### Online Stochastic Optimization under Correlated Bandit Feedback (pdf)

In this paper we consider the problem of online stochastic optimization of a locally smooth function under bandit feedback. We introduce the high-confidence tree (HCT) algorithm, a novel anytime $\mathcal X$-armed bandit algorithm, and derive regret bounds matching the performance of state-of-the-art algorithms in terms of the dependency on number of steps and the near-optimality dimension. The main advantage of HCT is that it handles the challenging case of correlated bandit feedback (reward), whereas existing methods require rewards to be conditionally independent. HCT also improves on the state-of-the-art in terms of the memory requirement, as well as requiring a weaker smoothness assumption on the mean-reward function in comparison with the existing anytime algorithms. Finally, we discuss how HCT can be applied to the problem of policy search in reinforcement learning and we report preliminary empirical results.

• Jian Tang and Zhaoshi Meng and Xuanlong Nguyen and Qiaozhu Mei and Ming Zhang

### Understanding the Limiting Factors of Topic Modeling via Posterior Contraction Analysis (pdf)

Topic models such as the latent Dirichlet allocation (LDA) have become a standard staple in the modeling toolbox of machine learning. They have been applied to a vast variety of data sets, contexts, and tasks to varying degrees of success. However, to date there is almost no formal theory explicating the LDA's behavior, and despite its familiarity there is very little systematic analysis of and guidance on the properties of the data that affect the inferential performance of the model. This paper seeks to address this gap, by providing a systematic analysis of factors which characterize the LDA's performance. We present theorems elucidating the posterior contraction rates of the topics as the amount of data increases, and a thorough supporting empirical study using synthetic and real data sets, including news and web-based articles and tweet messages. Based on these results we provide practical guidance on how to identify suitable data sets for topic models, and how to specify particular model parameters.

• Dimitris Papailiopoulos and Ioannis Mitliagkas and Alexandros Dimakis and Constantine Caramanis

### Finding Dense Subgraphs via Low-Rank Bilinear Optimization (pdf)

Given a graph, the Densest $k$-Subgraph (\DkS) problem asks for the subgraph on $k$ vertices that contains the largest number of edges. In this work, we develop a novel algorithm for \DkS{

2013-2014 ICML | International Conference on Machine Learning