We analyze variational inference for highly symmetric graphical models such as those arising from first-order probabilistic models. We first show that for these graphical models, the tree-reweighted variational objective lends itself to a compact lifted formulation which can be solved much more efficiently than the standard TRW formulation for the ground graphical model. Compared to earlier work on lifted belief propagation, our formulation leads to a convex optimization problem for lifted marginal inference and provides an upper bound on the partition function. We provide two approaches for improving the lifted TRW upper bound. The first is a method for efficiently computing maximum spanning trees in highly symmetric graphs, which can be used to optimize the TRW edge appearance probabilities. The second is a method for tightening the relaxation of the marginal polytope using lifted cycle inequalities and novel exchangeable cluster consistency constraints.

In this paper we develop a dynamic form of Bayesian optimization for machine learning models with the goal of rapidly finding good hyperparameter settings. Our method uses the partial information gained during the training of a machine learning model in order to decide whether to pause training and start a new model, or resume the training of a previously-considered model. We specifically tailor our method to machine learning problems by developing a novel positive-definite covariance kernel to capture a variety of training curves. Furthermore, we develop a Gaussian process prior that scales gracefully with additional temporal observations. Finally, we provide an information-theoretic framework to automate the decision process. Experiments on several common machine learning models show that our approach is extremely effective in practice.

Bayesian optimization has proven to be a highly effective methodology for the global optimization of unknown, expensive and multimodal functions. The ability to accurately model distributions over functions is critical to the effectiveness of Bayesian optimization. Although Gaussian processes provide a flexible prior over functions which can be queried efficiently, there are various classes of functions that remain difficult to model. One of the most frequently occurring of these is the class of non-stationary functions. The optimization of the hyperparameters of machine learning algorithms is a problem domain in which parameters are often manually transformed a priori, for example by optimizing in "log-space," to mitigate the effects of spatially-varying length scale. We develop a methodology for automatically learning a wide family of bijective transformations or warpings of the input space using the Beta cumulative distribution function. We further extend the warping framework to multi-task Bayesian optimization so that multiple tasks can be warped into a jointly stationary space. On a set of challenging benchmark optimization tasks, we observe that the inclusion of warping greatly improves on the state-of-the-art, producing better results faster and more reliably.

We consider in this paper a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. We present a new class of first-order methods, namely the gradient sliding algorithms, which can skip the computation of the gradient for the smooth component from time to time. As a consequence, these algorithms require only ${\cal O}(1/\sqrt{\epsilon})$ gradient evaluations for the smooth component in order to find an $\epsilon$-solution for the composite problem, while still maintaining the optimal ${\cal O}(1/\epsilon^2)$ bound on the total number of subgradient evaluations for the nonsmooth component. We then present a stochastic counterpart for these algorithms and establish similar complexity bounds for solving an important class of stochastic composite optimization problems. Moreover, if the smooth component in the composite function is strongly convex, the developed gradient sliding algorithms can significantly reduce the number of graduate and subgradient evaluations for the smooth and nonsmooth component to ${\cal O} (\log (1/\epsilon))$ and ${\cal O}(1/\epsilon)$, respectively. Finally, we generalize these algorithms to the case when the smooth component is replaced by a nonsmooth one possessing a certain bi-linear saddle point structure.

The purpose of this paper is to introduce a concept of equivalence between machine learning algorithms. We define two notions of algorithmic equivalence, namely, weak and strong equivalence. These notions are of paramount importance for identifying when learning prop erties from one learning algorithm can be transferred to another. Using regularized kernel machines as a case study, we illustrate the importance of the introduced equivalence concept by analyzing the relation between kernel ridge regression (KRR) and m-power regularized least squares regression (M-RLSR) algorithms.

We propose a novel information-theoretic approach for Bayesian optimization called Predictive Entropy Search (PES). At each iteration, PES selects the next evaluation point that maximizes the expected information gained with respect to the global maximum. PES codifies this intractable acquisition function in terms of the expected reduction in the differential entropy of the predictive distribution. This reformulation allows PES to obtain approximations that are both more accurate and efficient than other alternatives such as Entropy Search (ES). Furthermore, PES can easily perform a fully Bayesian treatment of the model hyperparameters while ES cannot. We evaluate PES in both synthetic and real-world applications, including optimization problems in machine learning, finance, biotechnology, and robotics. We show that the increased accuracy of PES leads to significant gains in optimization performance.

Heuristic search planning effectively finds solutions for large planning problems, but since the estimates are either not admissible or too weak, optimal solutions are found in rare cases only. In contrast, heuristic pattern databases are known to significantly improve lower bound estimates for optimally solving challenging single-agent problems like the 24-Puzzle or Rubik’s Cube. This paper studies the effect of pattern databases in the context of deterministic planning. Given a fixed state description based on instantiated predicates, we provide a general abstraction scheme to automatically create admissible domain-independent memory-based heuristics for planning problems, where abstractions are found in factorizing the planning space. We evaluate the impact of pattern database heuristics in A* and hill climbing algorithms for a collection of benchmark domains.

Stackelberg games form the core of a number of tools deployed for computing optimal patrolling strategies in adversarial domains, such as the US Federal Air Marshall Service and the US Coast Guard. In traditional Stackelberg security game models the attacker knows only the probability that each target is covered by the defender, but is oblivious to the detailed timing of the coverage schedule. In many real-world situations, however, the attacker can observe the current location of the defender and can exploit this knowledge to reason about the defender’s future moves. We show that this general modeling framework can be captured using adversarial patrolling games (APGs) in which the defender sequentially moves between targets, with moves constrained by a graph, while the attacker can observe the defender’s current location and his (stochastic) policy concerning future moves. We offer a very general model of infinite-horizon discounted adversarial patrolling games. Our first contribution is to show that defender policies that condition only on the previous defense move (i.e., Markov stationary policies) can be arbitrarily suboptimal for general APGs. We then offer a mixed-integer non-linear programming (MINLP) formulation for computing optimal randomized policies for the defender that can condition on history of bounded, but arbitrary, length, as well as a mixed-integer linear programming (MILP) formulation to approximate these, with provable quality guarantees. Additionally, we present a non-linear programming (NLP) formulation for solving zero-sum APGs. We show experimentally that MILP significantly outperforms the MINLP formulation, and is, in turn, significantly outperformed by the NLP specialized to zero-sum games.

In this paper we introduce a planning problem termed as Q-Blocks Relocation Problem, which pursues to retrieve a subset of blocks located in a warehouse by minimizing the number of relocation movements. We formalize the problem by means of a Mixed Integer Linear Programming model. However, the high computational burden required by the model encourages us to develop a heuristic algorithm for tackling it. The rationale behind the proposed heuristic is both to retrieve the requested blocks as soon as possible while reducing the number of blocks placed above another one with a higher priority. The computational results indicate that the heuristic reports near-optimal solutions for realistic instances by short computational times, which makes it attractive to be applied by management systems.

Stephen Becker T. J. Watson Center IBM Research Yorktown Heights, NY We introduce a new convex formulation for stable principal component pursuit (SPCP) to decompose noisy signals into low-rank and sparse representations. For numerical solutions of our SPCP formulation, we first develop a convex variational framework and then accelerate it with quasi-Newton methods. We show, via synthetic and real data experiments, that our approach offers advantages over the classical SPCP formulations in scalability and practical parameter selection.