In this paper, we propose a framework to train a generative model for texture image synthesis from a single example. To do so, we exploit the local representation of images via the space of patches, that is, square sub-images of fixed size (e.g. 4 4). Our main contribution is to consider optimal transport to enforce the multiscale patch distribution of generated images, which leads to two different formulations. First, a pixel-based optimization method is proposed, relying on discrete optimal transport. We show that it is related to a well-known texture optimization framework based on iterated patch nearest-neighbor projections, while avoiding some of its shortcomings. Second, in a semi-discrete setting, we exploit the differential properties of Wasserstein distances to learn a fully convolutional network for texture generation. Once estimated, this network produces realistic and arbitrarily large texture samples in real time. The two formulations result in non-convex concave problems that can be optimized efficiently with convergence properties and improved stability compared to adversarial approaches, without relying on any regularization. By directly dealing with the patch distribution of synthesized images, we also overcome limitations of state-of-the art techniques, such as patch aggregation issues that usually lead to low frequency artifacts (e.g.

Sum-of-norms clustering is a clustering formulation based on convex optimization that automatically induces hierarchy. Multiple algorithms have been proposed to solve the optimization problem: subgradient descent by Hocking et al.\ \cite{hocking}, ADMM and ADA by Chi and Lange\ \cite{Chi}, stochastic incremental algorithm by Panahi et al.\ \cite{Panahi} and semismooth Newton-CG augmented Lagrangian method by Yuan et al.\ \cite{dsun1}. All algorithms yield approximate solutions, even though an exact solution is demanded to determine the correct cluster assignment. The purpose of this paper is to close the gap between the output from existing algorithms and the exact solution to the optimization problem. We present a clustering test which identifies and certifies the correct cluster assignment from an approximate solution yielded by any primal-dual algorithm. The test may not succeed if the approximation is inaccurate. However, we show the correct cluster assignment is guaranteed to be found by a symmetric primal-dual path following algorithm after sufficiently many iterations, provided that the model parameter $\lambda$ avoids a finite number of bad values. Numerical experiments are implemented to support our results.

Exploratory landscape analysis (ELA) supports supervised learning approaches for automated algorithm selection and configuration by providing sets of features that quantify the most relevant characteristics of the optimization problem at hand. In black-box optimization, where an explicit problem representation is not available, the feature values need to be approximated from a small number of sample points. In practice, uniformly sampled random point sets and Latin hypercube constructions are commonly used sampling strategies. In this work, we analyze how the sampling method and the sample size influence the quality of the feature value approximations and how this quality impacts the accuracy of a standard classification task. While, not unexpectedly, increasing the number of sample points gives more robust estimates for the feature values, to our surprise we find that the feature value approximations for different sampling strategies do not converge to the same value. This implies that approximated feature values cannot be interpreted independently of the underlying sampling strategy. As our classification experiments show, this also implies that the feature approximations used for training a classifier must stem from the same sampling strategy as those used for the actual classification tasks. As a side result we show that classifiers trained with feature values approximated by Sobol' sequences achieve higher accuracy than any of the standard sampling techniques. This may indicate improvement potential for ELA-trained machine learning models.

Effective optimization is essential for real-world interactive systems to provide a satisfactory user experience in response to changing user behavior. However, it is often challenging to find an objective to optimize for interactive systems (e.g., policy learning in task-oriented dialog systems). Generally, such objectives are manually crafted and rarely capture complex user needs in an accurate manner. We propose an approach that infers the objective directly from observed user interactions. These inferences can be made regardless of prior knowledge and across different types of user behavior. We introduce Interactive System Optimizer (ISO), a novel algorithm that uses these inferred objectives for optimization. Our main contribution is a new general principled approach to optimizing interactive systems using data-driven objectives. We demonstrate the high effectiveness of ISO over several simulations.

How can we conduct efficient hyperparameter optimization for a completely new task? In this work, we consider a novel setting, where we search for the optimal hyperparameters for a target task of interest using only unlabeled target task and somewhat relevant source task datasets. In this setting, it is essential to estimate the ground-truth target task objective using only the available information. We propose estimators to unbiasedly approximate the ground-truth with a desirable variance property. Building on these estimators, we provide a general and tractable hyperparameter optimization procedure for our setting. The experimental evaluations demonstrate that the proposed framework broadens the applications of automated hyperparameter optimization.

Various structured output prediction problems (e.g., sequential tagging) involve constraints over the output space. By identifying these constraints, we can filter out infeasible solutions and build an accountable model. To this end, we present a general integer linear programming (ILP) framework for mining constraints from data. We model the inference of structured output prediction as an ILP problem. Then, given the coefficients of the objective function and the corresponding solution, we mine the underlying constraints by estimating the outer and inner polytopes of the feasible set. We verify the proposed constraint mining algorithm in various synthetic and real-world applications and demonstrate that the proposed approach successfully identifies the feasible set at scale. In particular, we show that our approach can learn to solve 9x9 Sudoku puzzles and minimal spanning tree problems from examples without providing the underlying rules. We also demonstrate results on hierarchical multi-label classification and conduct a theoretical analysis on how close the mined constraints are from the ground truth.

Robust Subspace Recovery (RoSuRe) algorithm was recently introduced as a principled and numerically efficient algorithm that unfolds underlying Unions of Subspaces (UoS) structure, present in the data. The union of Subspaces (UoS) is capable of identifying more complex trends in data sets than simple linear models. We build on and extend RoSuRe to prospect the structure of different data modalities individually. We propose a novel multi-modal data fusion approach based on group sparsity which we refer to as Robust Group Subspace Recovery (RoGSuRe). Relying on a bi-sparsity pursuit paradigm and non-smooth optimization techniques, the introduced framework learns a new joint representation of the time series from different data modalities, respecting an underlying UoS model. We subsequently integrate the obtained structures to form a unified subspace structure. The proposed approach exploits the structural dependencies between the different modalities data to cluster the associated target objects. The resulting fusion of the unlabeled sensors' data from experiments on audio and magnetic data has shown that our method is competitive with other state of the art subspace clustering methods. The resulting UoS structure is employed to classify newly observed data points, highlighting the abstraction capacity of the proposed method.

A core challenge in policy optimization in competitive Markov decision processes is the design of efficient optimization methods with desirable convergence and stability properties. To tackle this, we propose competitive policy optimization (CoPO), a novel policy gradient approach that exploits the game-theoretic nature of competitive games to derive policy updates. Motivated by the competitive gradient optimization method, we derive a bilinear approximation of the game objective. In contrast, off-the-shelf policy gradient methods utilize only linear approximations, and hence do not capture interactions among the players. We instantiate CoPO in two ways:(i) competitive policy gradient, and (ii) trust-region competitive policy optimization. We theoretically study these methods, and empirically investigate their behavior on a set of comprehensive, yet challenging, competitive games. We observe that they provide stable optimization, convergence to sophisticated strategies, and higher scores when played against baseline policy gradient methods.

Bayesian optimization (BO) is an efficient method for optimizing expensive black-box functions. In real-world applications, BO often faces a major problem of missing values in inputs. The missing inputs can happen in two cases. First, the historical data for training BO often contain missing values. Second, when performing the function evaluation (e.g. computing alloy strength in a heat treatment process), errors may occur (e.g. a thermostat stops working) leading to an erroneous situation where the function is computed at a random unknown value instead of the suggested value. To deal with this problem, a common approach just simply skips data points where missing values happen. Clearly, this naive method cannot utilize data efficiently and often leads to poor performance. In this paper, we propose a novel BO method to handle missing inputs. We first find a probability distribution of each missing value so that we can impute the missing value by drawing a sample from its distribution. We then develop a new acquisition function based on the well-known Upper Confidence Bound (UCB) acquisition function, which considers the uncertainty of imputed values when suggesting the next point for function evaluation. We conduct comprehensive experiments on both synthetic and real-world applications to show the usefulness of our method.

Many existing zeroth-order optimization (ZO) algorithms adopt two-point feedback schemes due to their fast convergence rate compared to one-point feedback schemes. However, two-point schemes require two evaluations of the objective function at each iteration, which can be impractical in applications where the data are not all available a priori, e.g., in online optimization. In this paper, we propose a novel one-point feedback scheme that queries the function value only once at each iteration and estimates the gradient using the residual between two consecutive feedback points. When optimizing a deterministic Lipschitz function, we show that the query complexity of ZO with the proposed one-point residual feedback matches that of ZO with the existing two-point feedback schemes. Moreover, the query complexity of the proposed algorithm can be improved when the objective function has Lipschitz gradient. Then, for stochastic bandit optimization problems, we show that ZO with one-point residual feedback achieves the same convergence rate as that of ZO with two-point feedback with uncontrollable data samples. We demonstrate the effectiveness of the proposed one-point residual feedback via extensive numerical experiments.