We present a new algorithmic approach for the task of finding a chordal Markov network structure that maximizes a given scoring function. The algorithm is based on branch and bound and integrates dynamic programming for both domain pruning and for obtaining strong bounds for search-space pruning. Empirically, we show that the approach dominates in terms of running times a recent integer programming approach (and thereby also a recent constraint optimization approach) for the problem.

Recent advances in bandit tools and techniques for sequential learning are steadily enabling new applications and are promising the resolution of a range of challenging related problems. We study the game tree search problem, where the goal is to quickly identify the optimal move in a given game tree by sequentially sampling its stochastic payoffs. We develop new algorithms for trees of arbitrary depth, that operate by summarizing all deeper levels of the tree into confidence intervals at depth one, and applying a best arm identification procedure at the root. We prove new sample complexity guarantees with a refined dependence on the problem instance. We show experimentally that our algorithms outperform existing elimination-based algorithms and match previous special-purpose methods for depth-two trees.

Non-convex optimization with local search heuristics has been widely used in machine learning, achieving many state-of-art results. It becomes increasingly important to understand why they can work for these NP-hard problems on typical data. The landscape of many objective functions in learning has been conjectured to have the geometric property that ``all local optima are (approximately) global optima'', and thus they can be solved efficiently by local search algorithms. However, establishing such property can be very difficult. In this paper, we analyze the optimization landscape of the random over-complete tensor decomposition problem, which has many applications in unsupervised leaning, especially in learning latent variable models.

The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, which encodes conditional independence statements between variables. Minimax results under the spectral norm have previously been established for covariance matrices, both sparse and banded, and for sparse precision matrices. We establish minimax estimation bounds for estimating banded precision matrices under the spectral norm. Our results greatly improve upon the existing bounds; in particular, we find that the minimax rate for estimating banded precision matrices matches that of estimating banded covariance matrices. The key insight in our analysis is that we are able to obtain barely-noisy estimates of $k \times k$ subblocks of the precision matrix by inverting slightly wider blocks of the empirical covariance matrix along the diagonal. Our theoretical results are complemented by experiments demonstrating the sharpness of our bounds.

Guided policy search algorithms can be used to optimize complex nonlinear policies, such as deep neural networks, without directly computing policy gradients in the high-dimensional parameter space. Instead, these methods use supervised learning to train the policy to mimic a "teacher" algorithm, such as a trajectory optimizer or a trajectory-centric reinforcement learning method. Guided policy search methods provide asymptotic local convergence guarantees by construction, but it is not clear how much the policy improves within a small, finite number of iterations. We show that guided policy search algorithms can be interpreted as an approximate variant of mirror descent, where the projection onto the constraint manifold is not exact. We derive a new guided policy search algorithm that is simpler and provides appealing improvement and convergence guarantees in simplified convex and linear settings, and show that in the more general nonlinear setting, the error in the projection step can be bounded. We provide empirical results on several simulated robotic manipulation tasks that show that our method is stable and achieves similar or better performance when compared to prior guided policy search methods, with a simpler formulation and fewer hyperparameters.

Given a task of predicting Y from X, a loss function L, and a set of probability distributions Gamma on (X,Y), what is the optimal decision rule minimizing the worst-case expected loss over Gamma? In this paper, we address this question by introducing a generalization of the maximum entropy principle. Applying this principle to sets of distributions with marginal on X constrained to be the empirical marginal, we provide a minimax interpretation of the maximum likelihood problem over generalized linear models as well as some popular regularization schemes. For quadratic and logarithmic loss functions we revisit well-known linear and logistic regression models. Moreover, for the 0-1 loss we derive a classifier which we call the minimax SVM. The minimax SVM minimizes the worst-case expected 0-1 loss over the proposed Gamma by solving a tractable optimization problem. We perform several numerical experiments to show the power of the minimax SVM in outperforming the SVM.

Your Out-of-Distribution Detection Method is Not Robust!

Learning the minimum/maximum mean among a finite set of distributions is a fundamental sub-problem in planning, game tree search and reinforcement learning. We formalize this learning task as the problem of sequentially testing how the minimum mean among a finite set of distributions compares to a given threshold. We develop refined non-asymptotic lower bounds, which show that optimality mandates very different sampling behavior for a low vs high true minimum. We show that Thompson Sampling and the intuitive Lower Confidence Bounds policy each nail only one of these cases. We develop a novel approach that we call Murphy Sampling. Even though it entertains exclusively low true minima, we prove that MS is optimal for both possibilities. We then design advanced self-normalized deviation inequalities, fueling more aggressive stopping rules. We complement our theoretical guarantees by experiments showing that MS works best in practice.

Model-free reinforcement learning aims to offer off-the-shelf solutions for controlling dynamical systems without requiring models of the system dynamics. We introduce a model-free random search algorithm for training static, linear policies for continuous control problems. Common evaluation methodology shows that our method matches state-of-the-art sample efficiency on the benchmark MuJoCo locomotion tasks. Nonetheless, more rigorous evaluation reveals that the assessment of performance on these benchmarks is optimistic. We evaluate the performance of our method over hundreds of random seeds and many different hyperparameter configurations for each benchmark task. This extensive evaluation is possible because of the small computational footprint of our method. Our simulations highlight a high variability in performance in these benchmark tasks, indicating that commonly used estimations of sample efficiency do not adequately evaluate the performance of RL algorithms. Our results stress the need for new baselines, benchmarks and evaluation methodology for RL algorithms.

We present a new approach for learning to solve SMT formulas. We phrase the challenge of solving SMT formulas as a tree search problem where at each step a transformation is applied to the input formula until the formula is solved. Our approach works in two phases: first, given a dataset of unsolved formulas we learn a policy that for each formula selects a suitable transformation to apply at each step in order to solve the formula, and second, we synthesize a strategy in the form of a loop-free program with branches. This strategy is an interpretable representation of the policy decisions and is used to guide the SMT solver to decide formulas more efficiently, without requiring any modification to the solver itself and without needing to evaluate the learned policy at inference time. We show that our approach is effective in practice - it solves 17% more formulas over a range of benchmarks and achieves up to 100x runtime improvement over a state-of-the-art SMT solver.