Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case optimal submodular extension for the more general class of metric labeling; and (iii) efficiently compute the marginals for the widely used dense CRF model with the help of a recently proposed Gaussian filtering method. Using synthetic and real data, we show that our approach provides comparable upper bounds on the log-partition function to those obtained using tree-reweighted message passing (TRW) in cases where the latter is computationally feasible. Importantly, unlike TRW, our approach provides the first practical algorithm to compute an upper bound on the dense CRF model.

Many real-world engineering problems rely on human preferences to guide their design and optimization. We present PrefOpt, an open source package to simplify sequential optimization tasks that incorporate human preference feedback. Our approach extends an existing latent variable model for binary preferences to allow for observations of equivalent preference from users.

We consider the problem of accelerating distributed optimization in multi-agent networks by sequentially adding edges. Specifically, we extend the distributed dual averaging (DDA) subgradient algorithm to evolving networks of growing connectivity and analyze the corresponding improvement in convergence rate. It is known that the convergence rate of DDA is influenced by the algebraic connectivity of the underlying network, where better connectivity leads to faster convergence. However, the impact of network topology design on the convergence rate of DDA has not been fully understood. In this paper, we begin by designing network topologies via edge selection and scheduling. For edge selection, we determine the best set of candidate edges that achieves the optimal tradeoff between the growth of network connectivity and the usage of network resources. The dynamics of network evolution is then incurred by edge scheduling. Further, we provide a tractable approach to analyze the improvement in the convergence rate of DDA induced by the growth of network connectivity. Our analysis reveals the connection between network topology design and the convergence rate of DDA, and provides quantitative evaluation of DDA acceleration for distributed optimization that is absent in the existing analysis. Lastly, numerical experiments show that DDA can be significantly accelerated using a sequence of well-designed networks, and our theoretical predictions are well matched to its empirical convergence behavior.

We propose a new technique that boosts the convergence of training generative adversarial networks. Generally, the rate of training deep models reduces severely after multiple iterations. A key reason for this phenomenon is that a deep network is expressed using a highly non-convex finite-dimensional model, and thus the parameter gets stuck in a local optimum. Because of this, methods often suffer not only from degeneration of the convergence speed but also from limitations in the representational power of the trained network. To overcome this issue, we propose an additional layer called the gradient layer to seek a descent direction in an infinite-dimensional space. Because the layer is constructed in the infinite-dimensional space, we are not restricted by the specific model structure of finite-dimensional models. As a result, we can get out of the local optima in finite-dimensional models and move towards the global optimal function more directly. In this paper, this phenomenon is explained from the functional gradient method perspective of the gradient layer. Interestingly, the optimization procedure using the gradient layer naturally constructs the deep structure of the network. Moreover, we demonstrate that this procedure can be regarded as a discretization method of the gradient flow that naturally reduces the objective function. Finally, the method is tested using several numerical experiments, which show its fast convergence.

We consider infinitely repeated games with vector losses discounted over time. We characterize the set of minimal upper bounds on expected losses that a player can simultaneously guarantee across the different dimensions. Specifically, we show that this set is the fixed point of a set-valued dynamic programming operator. This approach also characterizes the strategies that achieve these bounds. These optimal strategies are shown to be independent of the player's own past actions and stationary relative to a compact state space obtained by parameterizing the set of the minimal bounds. We also present a computational procedure to approximate this set and the optimal strategies. We discuss two applications of our results: 1) characterization of the optimal strategy of the uninformed player in zero-sum discounted repeated games with incomplete information on one side; 2) characterization of the minmax optimal regret and the regret-optimal strategy in repeated games with discounted losses. Our approximation procedure can be used to compute approximately optimal strategies in both these applications. We illustrate this procedure by computing approximately regret-optimal strategies for the problem of prediction using expert advice from two and three experts under $\{0,1\}-$losses. Our numerical evaluations demonstrate improved performance over existing algorithms for this problem.

Optimization of high-dimensional black-box functions is an extremely challenging problem. While Bayesian optimization has emerged as a popular approach for optimizing black-box functions, its applicability has been limited to low-dimensional problems due to its computational and statistical challenges arising from high-dimensional settings. In this paper, we propose to tackle these challenges by (1) assuming a latent additive structure in the function and inferring it properly for more efficient and effective BO, and (2) performing multiple evaluations in parallel to reduce the number of iterations required by the method. Our novel approach learns the latent structure with Gibbs sampling and constructs batched queries using determinantal point processes. Experimental validations on both synthetic and real-world functions demonstrate that the proposed method outperforms the existing state-of-the-art approaches.

Bayesian optimization (BO) has become an effective approach for black-box function optimization problems when function evaluations are expensive and the optimum can be achieved within a relatively small number of queries. However, many cases, such as the ones with high-dimensional inputs, may require a much larger number of observations for optimization. Despite an abundance of observations thanks to parallel experiments, current BO techniques have been limited to merely a few thousand observations. In this paper, we propose ensemble Bayesian optimization (EBO) to address three current challenges in BO simultaneously: (1) large-scale observations; (2) high dimensional input spaces; and (3) selections of batch queries that balance quality and diversity. The key idea of EBO is to operate on an ensemble of additive Gaussian process models, each of which possesses a randomized strategy to divide and conquer. We show unprecedented, previously impossible results of scaling up BO to tens of thousands of observations within minutes of computation.

In this paper, we consider the problem of recovering a sparse signal based on penalized least squares. We develop an algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, which includes $\ell^0$, bridge, smoothly clipped absolute deviation, capped $\ell^1$ and minimax concavity penalty. First we establish the existence of a global minimizer for the class of optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinate-wise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined from the primal and dual variables. This relation lends itself to an iterative algorithm of active set type which at each step involves updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method converges globally to the underlying regression target under some regularity conditions. Extensive numerical experiments demonstrate its superior performance in efficiency and accuracy compared with the existing methods.

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.

In this paper, the problem of maximizing a black-box function $f:\mathcal{X} \to \mathbb{R}$ is studied in the Bayesian framework with a Gaussian Process (GP) prior. In particular, a new algorithm for this problem is proposed, and high probability bounds on its simple and cumulative regret are established. The query point selection rule in most existing methods involves an exhaustive search over an increasingly fine sequence of uniform discretizations of $\mathcal{X}$. The proposed algorithm, in contrast, adaptively refines $\mathcal{X}$ which leads to a lower computational complexity, particularly when $\mathcal{X}$ is a subset of a high dimensional Euclidean space. In addition to the computational gains, sufficient conditions are identified under which the regret bounds of the new algorithm improve upon the known results. Finally an extension of the algorithm to the case of contextual bandits is proposed, and high probability bounds on the contextual regret are presented.