The automated construction of coarse-grained models represents a pivotal component in computer simulation of physical systems and is a key enabler in various analysis and design tasks related to uncertainty quantification. Pertinent methods are severely inhibited by the high-dimension of the parametric input and the limited number of training input/output pairs that can be generated when computationally demanding forward models are considered. Such cases are frequently encountered in the modeling of random heterogeneous media where the scale of the microstructure necessitates the use of high-dimensional random vectors and very fine discretizations of the governing equations. The present paper proposes a probabilistic Machine Learning framework that is capable of operating in the presence of Small Data by exploiting aspects of the physical structure of the problem as well as contextual knowledge. As a result, it can perform comparably well under extrapolative conditions. It unifies the tasks of dimensionality and model-order reduction through an encoder-decoder scheme that simultaneously identifies a sparse set of salient lower-dimensional microstructural features and calibrates an inexpensive, coarse-grained model which is predictive of the output. Information loss is accounted for and quantified in the form of probabilistic predictive estimates. The learning engine is based on Stochastic Variational Inference. We demonstrate how the variational objectives can be used not only to train the coarse-grained model, but also to suggest refinements that lead to improved predictions.

We present a Bayesian multi-objective optimisation algorithm that allows the user to express preference-order constraints on the objectives of the type `objective A is more important than objective B'. Rather than attempting to find a representative subset of the complete Pareto front, our algorithm searches for and returns only those Pareto-optimal points that satisfy these constraints. We formulate a new acquisition function based on expected improvement in dominated hypervolume (EHI) to ensure that the subset of Pareto front satisfying the constraints is thoroughly explored. The hypervolume calculation only includes those points that satisfy the preference-order constraints, where the probability of a point satisfying the constraints is calculated from a gradient Gaussian Process model. We demonstrate our algorithm on both synthetic and real-world problems.

We study continuous action reinforcement learning problems in which it is crucial that the agent interacts with the environment only through safe policies, i.e.,~policies that do not take the agent to undesirable situations. We formulate these problems as constrained Markov decision processes (CMDPs) and present safe policy optimization algorithms that are based on a Lyapunov approach to solve them. Our algorithms can use any standard policy gradient (PG) method, such as deep deterministic policy gradient (DDPG) or proximal policy optimization (PPO), to train a neural network policy, while guaranteeing near-constraint satisfaction for every policy update by projecting either the policy parameter or the action onto the set of feasible solutions induced by the state-dependent linearized Lyapunov constraints. Compared to the existing constrained PG algorithms, ours are more data efficient as they are able to utilize both on-policy and off-policy data. Moreover, our action-projection algorithm often leads to less conservative policy updates and allows for natural integration into an end-to-end PG training pipeline. We evaluate our algorithms and compare them with the state-of-the-art baselines on several simulated (MuJoCo) tasks, as well as a real-world indoor robot navigation problem, demonstrating their effectiveness in terms of balancing performance and constraint satisfaction. Videos of the experiments can be found in the following link: https://drive.google.com/file/d/1pzuzFqWIE710bE2U6DmS59AfRzqK2Kek/view?usp=sharing.

Modern machine learning uses more and more advanced optimization techniques to find optimal hyper parameters. Whenever the objective function is non-convex, non continuous and with potentially multiple local minima, standard gradient descent optimization methods fail. A last resource and very different method is to assume that the optimum(s), not necessarily unique, is/are distributed according to a distribution and iteratively to adapt the distribution according to tested points. These strategies originated in the early 1960s, named Evolution Strategy (ES) have culminated with the CMA-ES (Covariance Matrix Adaptation) ES. It relies on a multi variate normal distribution and is supposed to be state of the art for general optimization program. However, it is far from being optimal for discrete variables. In this paper, we extend the method to multivariate binomial correlated distributions. For such a distribution, we show that it shares similar features to the multi variate normal: independence and correlation is equivalent and correlation is efficiently modeled by interaction between different variables. We discuss this distribution in the framework of the exponential family. We prove that the model can estimate not only pairwise interactions among the two variables but also is capable of modeling higher order interactions. This allows creating a version of CMA ES that can accommodate efficiently discrete variables. We provide the corresponding algorithm and conclude.

We provide a theoretical analysis of the representation learning problem aimed at learning the latent variables (design matrix) $\Theta$ of observations $Y$ with the knowledge of the coefficient matrix $X$. The design matrix is learned under the assumption that the latent variables $\Theta$ are smooth with respect to a (known) topological structure $\mathcal{G}$. To learn such latent variables, we study a graph Laplacian regularized estimator, which is the penalized least squares estimator with penalty term proportional to a Laplacian quadratic form. This type of estimators has recently received considerable attention due to its capability in incorporating underlying topological graph structure of variables into the learning process. While the estimation problem can be solved efficiently by state-of-the-art optimization techniques, its statistical consistency properties have been largely overlooked. In this work, we develop a non-asymptotic bound of estimation error under the classical statistical setting, where sample size is larger than the ambient dimension of the latent variables. This bound illustrates theoretically the impact of the alignment between the data and the graph structure as well as the graph spectrum on the estimation accuracy. It also provides theoretical evidence of the advantage, in terms of convergence rate, of the graph Laplacian regularized estimator over classical ones (that ignore the graph structure) in case of a smoothness prior. Finally, we provide empirical results of the estimation error to corroborate the theoretical analysis.

Variational approximation methods are a way to approximate the posterior in Bayesian inference especially when the dataset has a large volume or high dimension. Factor covariance structure was introduced in previous work with three restrictions to handle the problem of computational infeasibility in Gaussian approximation. However, the three strong constraints on the covariance matrix could possibly break down during the process of the structure optimization, and the identification issue could still possibly exist within the final approximation. In this paper, we consider two types of manifold parameterization, Stiefel manifold and Grassmann manifold, to address the problems. Moreover, the Riemannian stochastic gradient descent method is applied to solve the resulting optimization problem while maintaining the orthogonal factors. Results from two experiments demonstrate that our model fixes the potential issue of the previous method with comparable accuracy and competitive converge speed even in high-dimensional problems.

We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.

Kurdyka-{\L}ojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of $\frac12$ is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. In this paper, we show under mild assumptions that KL exponent is preserved via inf-projection. Inf-projection is a fundamental operation that is ubiquitous when reformulating optimization problems via the lift-and-project approach. By studying its operation on KL exponent, we show that the KL exponent is $\frac12$ for several important convex optimization models, including some semidefinite-programming-representable functions and functions that involve $C^2$-cone reducible structures, under conditions such as strict complementarity. Our results are applicable to concrete optimization models such as group fused Lasso and overlapping group Lasso. In addition, for nonconvex models, we show that the KL exponent of many difference-of-convex functions can be derived from that of their natural majorant functions, and the KL exponent of the Bregman envelope of a function is the same as that of the function itself. Finally, we estimate the KL exponent of the sum of the least squares function and the indicator function of the set of matrices of rank at most $k$.

We address the challenge of effective exploration while maintaining good performance in policy gradient methods. As a solution, we propose diverse exploration (DE) via conjugate policies. DE learns and deploys a set of conjugate policies which can be conveniently generated as a byproduct of conjugate gradient descent. We provide both theoretical and empirical results showing the effectiveness of DE at achieving exploration, improving policy performance, and the advantage of DE over exploration by random policy perturbations.

Static stochastic VRPs aim at modeling real-life VRPs by considering uncertainty on data. In particular, the SS-VRPTW-CR considers stochastic customers with time windows and does not make any assumption on their reveal times, which are stochastic as well. Based on customer request probabilities, we look for an a priori solution composed preventive vehicle routes, minimizing the expected number of unsatisfied customer requests at the end of the day. A route describes a sequence of strategic vehicle relocations, from which nearby requests can be rapidly reached. Instead of reoptimizing online, a so-called recourse strategy defines the way the requests are handled, whenever they appear. In this paper, we describe a new recourse strategy for the SS-VRPTW-CR, improving vehicle routes by skipping useless parts. We show how to compute the expected cost of a priori solutions, in pseudo-polynomial time, for this recourse strategy. We introduce a new meta-heuristic, called Progressive Focus Search (PFS), which may be combined with any local-search based algorithm for solving static stochastic optimization problems. PFS accelerates the search by using approximation factors: from an initial rough simplified problem, the search progressively focuses to the actual problem description. We evaluate our contributions on a new, real-world based, public benchmark.