Incorporating invariance information is important for many learning problems. To exploit invariances, most existing methods resort to approximations that either lead to expensive optimization problems such as semi-definite programming, or rely on separation oracles to retain tractability. Some methods further limit the space of functions and settle for non-convex models. In this paper, we propose a framework for learning in reproducing kernel Hilbert spaces (RKHS) using local invariances that explicitly characterize the behavior of the target function around data instances. These invariances are compactly encoded as linear functionals whose value are penalized by some loss function. Based on a representer theorem that we establish, our formulation can be efficiently optimized via a convex program. For the representer theorem to hold, the linear functionals are required to be bounded in the RKHS, and we show that this is true for a variety of commonly used RKHS and invariances. Experiments on learning with unlabeled data and transform invariances show that the proposed method yields better or similar results compared with the state of the art.

Structured sparse estimation has become an important technique in many areas of data analysis. Unfortunately, these estimators normally create computational difficulties that entail sophisticated algorithms. Our first contribution is to uncover a rich class of structured sparse regularizers whose polar operator can be evaluated efficiently. With such an operator, a simple conditional gradient method can then be developed that, when combined with smoothing and local optimization, significantly reduces training time vs. the state of the art. We also demonstrate a new reduction of polar to proximal maps that enables more efficient latent fused lasso.

Robust PCA methods are typically based on batch optimization and have to load all the samples into memory during optimization. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust PCA (OR-PCA) that processes one sample per time instance and hence its memory cost is independent of the number of samples, significantly enhancing the computation and storage efficiency. The proposed OR-PCA is based on stochastic optimization of an equivalent reformulation of the batch RPCA. Indeed, we show that OR-PCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption. Moreover, OR-PCA can naturally be applied for tracking dynamic subspace. Comprehensive simulations on subspace recovering and tracking demonstrate the robustness and efficiency advantages of the OR-PCA over online PCA and batch RPCA methods.

Many applications in machine learning require optimizing unknown functions defined over a high-dimensional space from noisy samples that are expensive to obtain. We address this notoriously hard challenge, under the assumptions that the function varies only along some low-dimensional subspace and is smooth (i.e., it has a low norm in a Reproducible Kernel Hilbert Space). In particular, we present the SI-BO algorithm, which leverages recent low-rank matrix recovery techniques to learn the underlying subspace of the unknown function and applies Gaussian Process Upper Confidence sampling for optimization of the function. We carefully calibrate the exploration-exploitation tradeoff by allocating the sampling budget to subspace estimation and function optimization, and obtain the first subexponential cumulative regret bounds and convergence rates for Bayesian optimization in high-dimensions under noisy observations. Numerical results demonstrate the effectiveness of our approach in difficult scenarios.

Tensor completion from incomplete observations is a problem of significant practical interest. However, it is unlikely that there exists an efficient algorithm with provable guarantee to recover a general tensor from a limited number of observations. In this paper, we study the recovery algorithm for pairwise interaction tensors, which has recently gained considerable attention for modeling multiple attribute data due to its simplicity and effectiveness.

When approximating binary similarity using the hamming distance between short binary hashes, we show that even if the similarity is symmetric, we can have shorter and more accurate hashes by using two distinct code maps.

This paper introduces a variational framework for planning in infinite-horizon factored MDPs. Leveraging previous work by Liu and Ihler [13], a variational "dual" representation of the maximum expected reward problem (Bellman equation) is considered. Exploiting the factored structure of the transition probabilities and the additive form of the rewards, they introduce an approximation which can be solved by a double-loop Belief-propagation style algorithm. This algorithm is shown to outperform approximate policy iteration and approximate linear programming on several instances of a disease management and a viral marketing MDP. The main contribution of this paper is an extension of the previous approach of Liu and Ihler [13] from influence diagrams to an infinite horizon MDP case.

We consider robust optimization for polynomial optimization problems where the uncertainty set is a set of candidate probability density functions. This set is a ball around a density function estimated from data samples, i.e., it is data-driven and random. Polynomial optimization problems are inherently hard due to nonconvex objectives and constraints. However, we show that by employing polynomial and histogram density estimates, we can introduce robustness with respect to distributional uncertainty sets without making the problem harder. We show that the optimum to the distributionally robust problem is the limit of a sequence of tractable semidefinite programming relaxations. We also give finite-sample consistency guarantees for the data-driven uncertainty sets. Finally, we apply our model and solution method in a water network optimization problem.

Tetris is a video game that has been widely used as a benchmark for various optimization techniques including approximate dynamic programming (ADP) algorithms. A look at the literature of this game shows that while ADP algorithms that have been (almost) entirely based on approximating the value function (value function based) have performed poorly in Tetris, the methods that search directly in the space of policies by learning the policy parameters using an optimization black box, such as the cross entropy (CE) method, have achieved the best reported results. This makes us conjecture that Tetris is a game in which good policies are easier to represent, and thus, learn than their corresponding value functions. So, in order to obtain a good performance with ADP, we should use ADP algorithms that search in a policy space, instead of the more traditional ones that search in a value function space. In this paper, we put our conjecture to test by applying such an ADP algorithm, called classification-based modified policy iteration (CBMPI), to the game of Tetris. Our experimental results show that for the first time an ADP algorithm, namely CBMPI, obtains the best results reported in the literature for Tetris in both small 10 10 and large 10 20 boards. Although the CBMPI's results are similar to those of the CE method in the large board, CBMPI uses considerably fewer (almost 1/6) samples (calls to the generative model) than CE.

This paper presents a new approach to sampling from binary graphical models. There are two main tricks in this paper. The first is a clever construction to transform an arbitrary distribution over binary vectors x into a uniform distribution over an expanded space. This is done by augmenting x with additional bits, which are subject to parity constraints that ensure that the number of is approximately proportional to the original probability of x. This is a uniform distribution over a set with complex nonlinear constraints, so sampling from this distribution is challenging.