We consider the problem of learning the optimal action-value function in the discounted-reward Markov decision processes (MDPs). We prove a new PAC bound on the sample-complexity of model-based value iteration algorithm in the presence of the generative model, which indicates that for an MDP with N state-action pairs and the discount factor \gamma\in[0,1) only O(N\log(N/\delta)/((1-\gamma)^3\epsilon^2)) samples are required to find an \epsilon-optimal estimation of the action-value function with the probability 1-\delta. We also prove a matching lower bound of \Theta (N\log(N/\delta)/((1-\gamma)^3\epsilon^2)) on the sample complexity of estimating the optimal action-value function by every RL algorithm. To the best of our knowledge, this is the first matching result on the sample complexity of estimating the optimal (action-) value function in which the upper bound matches the lower bound of RL in terms of N, \epsilon, \delta and 1/(1-\gamma). Also, both our lower bound and our upper bound significantly improve on the state-of-the-art in terms of 1/(1-\gamma).

Most prior work on active learning of classifiers has focused on sequentially selecting one unlabeled example at a time to be labeled in order to reduce the overall labeling effort. In many scenarios, however, it is desirable to label an entire batch of examples at once, for example, when labels can be acquired in parallel. This motivates us to study batch active learning, which iteratively selects batches of $k>1$ examples to be labeled. We propose a novel batch active learning method that leverages the availability of high-quality and efficient sequential active-learning policies by attempting to approximate their behavior when applied for $k$ steps. Specifically, our algorithm first uses Monte-Carlo simulation to estimate the distribution of unlabeled examples selected by a sequential policy over $k$ step executions. The algorithm then attempts to select a set of $k$ examples that best matches this distribution, leading to a combinatorial optimization problem that we term "bounded coordinated matching". While we show this problem is NP-hard in general, we give an efficient greedy solution, which inherits approximation bounds from supermodular minimization theory. Our experimental results on eight benchmark datasets show that the proposed approach is highly effective

This paper analyzes the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al, 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, Srinivas et al proved that the regret vanishes at the approximate rate of $O(1/\sqrt{t})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show that the regret decreases asymptotically according to $O(e^{-\frac{\tau t}{(\ln t)^{d/4}}})$ with high probability. Here, d is the dimension of the search space and tau is a constant that depends on the behaviour of the objective function near its global maximum.

We demonstrate that almost all non-parametric dimensionality reduction methods can be expressed by a simple procedure: regularized loss minimization plus singular value truncation. By distinguishing the role of the loss and regularizer in such a process, we recover a factored perspective that reveals some gaps in the current literature. Beyond identifying a useful new loss for manifold unfolding, a key contribution is to derive new convex regularizers that combine distance maximization with rank reduction. These regularizers can be applied to any loss.

Contextual bandit learning is an increasingly popular approach to optimizing recommender systems via user feedback, but can be slow to converge in practice due to the need for exploring a large feature space. In this paper, we propose a coarse-to-fine hierarchical approach for encoding prior knowledge that drastically reduces the amount of exploration required. Intuitively, user preferences can be reasonably embedded in a coarse low-dimensional feature space that can be explored efficiently, requiring exploration in the high-dimensional space only as necessary. We introduce a bandit algorithm that explores within this coarse-to-fine spectrum, and prove performance guarantees that depend on how well the coarse space captures the user's preferences. We demonstrate substantial improvement over conventional bandit algorithms through extensive simulation as well as a live user study in the setting of personalized news recommendation.

Given two views (or representations) of the same set of objects, it aims at finding projections for each representation such that their correlation is maximized in the projection space. As every popular method in machine learning, since its first formulation (Hotelling, 1936) CCA has been extended to a kernel version (Lai & Fyfe, 2000; Akaho, 2001), to online and recursive versions (V ıa et al., 2007) and quite recently to a sparse version (Hardoon & Shawe-Taylor, 2011). CCA is usually formulated as the Generalized Singular Value Decomposition (Generalized SVD) of the cross-covariance matrix (Sun et al., 2009). Besides, it aims at finding projections that are orthogonal with respect to the auto-covariance matrices of each view. As CCA belongs to the class of Latent Variables methods, it shares close connections with those methods. Indeed, according to Rosipal & Kr amer (2006); Sun et al. (2009), CCA is a generalization of Orthonormal-ized Partial Least Squares.

As data sets grow in size, the ability of learning methods to find structure in them is increasingly hampered by the time needed to search the large spaces of possibilities and generate a score for each that takes all of the observed data into account. For instance, Bayesian networks, the model chosen in this paper, have a super-exponentially large search space for a fixed number of variables. One possible method to alleviate this problem is to use a proxy, such as a Gaussian Process regressor, in place of the true scoring function, training it on a selection of sampled networks. We prove here that the use of such a proxy is well-founded, as we can bound the smoothness of a commonly-used scoring function for Bayesian network structure learning. We show here that, compared to an identical search strategy using the network's exact scores, our proxy-based search is able to get equivalent or better scores on a number of data sets in a fraction of the time.

As machine learning algorithms enter applications in industrial settings, there is increased interest in controlling their cpu-time during testing. The cpu-time consists of the running time of the algorithm and the extraction time of the features. The latter can vary drastically when the feature set is diverse. In this paper, we propose an algorithm, the Greedy Miser, that incorporates the feature extraction cost during training to explicitly minimize the cpu-time during testing. The algorithm is a straightforward extension of stage-wise regression and is equally suitable for regression or multi-class classification. Compared to prior work, it is significantly more cost-effective and scales to larger data sets.

We study the problem of multivariate regression where the data are naturally grouped, and a regression matrix is to be estimated for each group. We propose an approach in which a dictionary of low rank parameter matrices is estimated across groups, and a sparse linear combination of the dictionary elements is estimated to form a model within each group. We refer to the method as conditional sparse coding since it is a coding procedure for the response vectors Y conditioned on the covariate vectors X. This approach captures the shared information across the groups while adapting to the structure within each group. It exploits the same intuition behind sparse coding that has been successfully developed in computer vision and computational neuroscience. We propose an algorithm for conditional sparse coding, analyze its theoretical properties in terms of predictive accuracy, and present the results of simulation and brain imaging experiments that compare the new technique to reduced rank regression.

Bayesian reinforcement learning (BRL) encodes prior knowledge of the world in a model and represents uncertainty in model parameters by maintaining a probability distribution over them. This paper presents Monte Carlo BRL (MC-BRL), a simple and general approach to BRL. MC-BRL samples a priori a finite set of hypotheses for the model parameter values and forms a discrete partially observable Markov decision process (POMDP) whose state space is a cross product of the state space for the reinforcement learning task and the sampled model parameter space. The POMDP does not require conjugate distributions for belief representation, as earlier works do, and can be solved relatively easily with point-based approximation algorithms. MC-BRL naturally handles both fully and partially observable worlds. Theoretical and experimental results show that the discrete POMDP approximates the underlying BRL task well with guaranteed performance.