A core operation in reinforcement learning (RL) is finding an action that is optimal with respect to a learned state-action value function. This operation is often challenging when the learned value function takes continuous actions as input. We introduce deep RBF value functions: state-action value functions learned using a deep neural network with a radial-basis function (RBF) output layer. We show that the optimal action with respect to a deep RBF value function can be easily approximated up to any desired accuracy. Moreover, deep RBF value functions can represent any true value function up to any desired accuracy owing to their support for universal function approximation. By learning a deep RBF value function, we extend the standard DQN algorithm to continuous control, and demonstrate that the resultant agent, RBF-DQN, outperforms standard baselines on a set of continuous-action RL problems.

Topology optimization by distributing materials in a domain requires stochastic optimizers to solve highly complicated problems. However, solving such problems requires millions of finite element calculations with hundreds of design variables or more involved , whose computational cost is huge and often unacceptable. To speed up computation, here we report a method to integrate deep learning into stochastic optimization algorithm. A Deep Neural Network (DNN) learns and substitutes the objective function by forming a loop with Generative Simulated Annealing (GSA). In each iteration, GSA uses DNN to evaluate the objective function to obtain an optimized solution, based on which new training data are generated; thus, DNN enhances its accuracy and GSA could accordingly improve its solution in next iteration until convergence. Our algorithm was tested by compliance minimization problems and reduced computational time by over two orders of magnitude. This approach sheds light on solving large multi-dimensional optimization problems.

Bayesian optimization is a class of global optimization techniques. It regards the underlying objective function as a realization of a Gaussian process. Although the outputs of Bayesian optimization are random according to the Gaussian process assumption, quantification of this uncertainty is rarely studied in the literature. In this work, we propose a novel approach to assess the output uncertainty of Bayesian optimization algorithms, in terms of constructing confidence regions of the maximum point or value of the objective function. These regions can be computed efficiently, and their confidence levels are guaranteed by newly developed uniform error bounds for sequential Gaussian process regression. Our theory provides a unified uncertainty quantification framework for all existing sequential sampling policies and stopping criteria.

First-order optimization methods, such as stochastic gradient descent (SGD) and its variants, are widely used in machine learning applications due to their simplicity and low per-iteration costs. However, they often require larger numbers of iterations, with associated communication costs in distributed environments. In contrast, Newton-type methods, while having higher per-iteration costs, typically require a significantly smaller number of iterations, which directly translates to reduced communication costs. In this paper, we present a novel distributed optimizer for classification problems, which integrates a GPU-accelerated Newton-type solver with the global consensus formulation of Alternating Direction of Method Multipliers (ADMM). By leveraging the communication efficiency of ADMM, GPU-accelerated inexact-Newton solver, and an effective spectral penalty parameter selection strategy, we show that our proposed method (i) yields better generalization performance on several classification problems; (ii) significantly outperforms state-of-the-art methods in distributed time to solution; and (iii) offers better scaling on large distributed platforms.

We consider the problem of recovering a complex vector $\mathbf{x}\in \mathbb{C}^n$ from $m$ quadratic measurements $\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m$. This problem, known as quadratic feasibility, encompasses the well known phase retrieval problem and has applications in a wide range of important areas including power system state estimation and x-ray crystallography. In general, not only is the the quadratic feasibility problem NP-hard to solve, but it may in fact be unidentifiable. In this paper, we establish conditions under which this problem becomes {identifiable}, and further prove isometry properties in the case when the matrices $\{A_i\}_{i=1}^m$ are Hermitian matrices sampled from a complex Gaussian distribution. Moreover, we explore a nonconvex {optimization} formulation of this problem, and establish salient features of the associated optimization landscape that enables gradient algorithms with an arbitrary initialization to converge to a \emph{globally optimal} point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.

In smooth strongly convex optimization, or in the presence of H\"olderian error bounds, knowledge of the curvature parameter is critical for obtaining simple methods with accelerated rates. In this work, we study a class of methods, based on Polyak steps, where this knowledge is substituted by that of the optimal value, $f_*$. We first show slightly improved convergence bounds than previously known for the classical case of simple gradient descent with Polyak steps, we then derive an accelerated gradient method with Polyak steps and momentum, along with convergence guarantees.

Graphs are versatile tools for representing structured data. Therefore, a variety of machine learning methods have been studied for graph data analysis. Although many of those learning methods depend on the measurement of differences between input graphs, defining an appropriate distance metric for a graph remains a controversial issue. Hence, we propose a supervised distance metric learning method for the graph classification problem. Our method, named interpretable graph metric learning (IGML), learns discriminative metrics in a subgraph-based feature space, which has a strong graph representation capability. By introducing a sparsity-inducing penalty on a weight of each subgraph, IGML can identify a small number of important subgraphs that can provide insight about the given classification task. Since our formulation has a large number of optimization variables, an efficient algorithm is also proposed by using pruning techniques based on safe screening and working set selection methods. An important property of IGML is that the optimality of the solution is guaranteed because the problem is formulated as a convex problem and our pruning strategies only discard unnecessary subgraphs. Further, we show that IGML is also applicable to other structured data such as item-set and sequence data, and that it can incorporate vertex-label similarity by using a transportation-based subgraph feature. We empirically evaluate the computational efficiency and classification performance on several benchmark datasets and show some illustrative examples demonstrating that IGML identifies important subgraphs from a given graph dataset.

Maintaining a population of solutions has been shown to increase exploration in reinforcement learning, typically attributed to the greater diversity of behaviors considered. One such class of methods, novelty search, considers further boosting diversity across agents via a multi-objective optimization formulation. Despite the intuitive appeal, these mechanisms have several shortcomings. First, they make use of mean field updates, which induce cycling behaviors. Second, they often rely on handcrafted behavior characterizations, which require domain knowledge. Furthermore, boosting diversity often has a detrimental impact on optimizing already fruitful behaviors for rewards. Setting the relative importance of novelty- versus reward-factor is usually hardcoded or requires tedious tuning/annealing. In this paper, we introduce a novel measure of population-wide diversity, leveraging ideas from Determinantal Point Processes. We combine this in a principled fashion with the reward function to adapt to the degree of diversity during training, borrowing ideas from online learning. Combined with task-agnostic behavioral embeddings, we show this approach outperforms previous methods for multi-objective optimization, as well as vanilla algorithms solely optimizing for rewards.

We present the implementation of a new stochastic Kuramoto-Vicsek-type model for global optimization of nonconvex functions on the sphere. This model belongs to the class of Consensus-Based Optimization. In fact, particles move on the sphere driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace's principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached the stochastic component vanishes. The main results of this paper are about the proof of convergence of the numerical scheme to global minimizers provided conditions of well-preparation of the initial datum. The proof combines previous results of mean-field limit with a novel asymptotic analysis, and classical convergence results of numerical methods for SDE. We present several numerical experiments, which show that the algorithm proposed in the present paper scales well with the dimension and is extremely versatile. To quantify the performances of the new approach, we show that the algorithm is able to perform essentially as good as ad hoc state of the art methods in challenging problems in signal processing and machine learning, namely the phase retrieval problem and the robust subspace detection.

Dubins tours represent a solution of the Dubins Traveling Salesman Problem (DTSP) that is a variant of the optimization routing problem to determine a curvature-constrained shortest path to visit a set of locations such that the path is feasible for Dubins vehicle, which moves only forward and has a limited turning radius. The DTSP combines the NP-hard combinatorial optimization to determine the optimal sequence of visits to the locations, as in the regular TSP, with the continuous optimization of the heading angles at the locations, where the optimal heading values depend on the sequence of visits and vice versa. We address the computationally challenging DTSP by fast evaluation of the sequence of visits by the proposed Windowing Surrogate Model (WiSM) which estimates the length of the optimal Dubins path connecting a sequence of locations in a Dubins tour. The estimation is sped up by a regression model trained using close to optimum solutions of small Dubins tours that are generalized for large-scale instances of the addressed DTSP utilizing the sliding window technique and a cache for already computed results. The reported results support that the proposed WiSM enables a fast convergence of a relatively simple evolutionary algorithm to high-quality solutions of the DTSP. We show that with an increasing number of locations, our algorithm scales significantly better than other state-of-the-art DTSP solvers.