Recent advancements in procedural content generation via machine learning enable the generation of video-game levels that are aesthetically similar to human-authored examples. However, the generated levels are often unplayable without additional editing. We propose a generate-then-repair framework for automatic generation of playable levels adhering to specific styles. The framework constructs levels using a generative adversarial network (GAN) trained with human-authored examples and repairs them using a mixed-integer linear program (MIP) with playability constraints. A key component of the framework is computing minimum cost edits between the GAN generated level and the solution of the MIP solver, which we cast as a minimum cost network flow problem. Results show that the proposed framework generates a diverse range of playable levels, that capture the spatial relationships between objects exhibited in the human-authored levels.

This book is devoted to the problem of sequential probability forecasting, that is, predicting the probabilities of the next outcome of a growing sequence of observations given the past. This problem is considered in a very general setting that unifies commonly used probabilistic and non-probabilistic settings, trying to make as few as possible assumptions on the mechanism generating the observations. A common form that arises in various formulations of this problem is that of mixture predictors, which are formed as a combination of a finite or infinite set of other predictors attempting to combine their predictive powers. The main subject of this book are such mixture predictors, and the main results demonstrate the universality of this method in a very general probabilistic setting, but also show some of its limitations. While the problems considered are motivated by practical applications, involving, for example, financial, biological or behavioural data, this motivation is left implicit and all the results exposed are theoretical. The book targets graduate students and researchers interested in the problem of sequential prediction, and, more generally, in theoretical analysis of problems in machine learning and non-parametric statistics, as well as mathematical and philosophical foundations of these fields. The material in this volume is presented in a way that presumes familiarity with basic concepts of probability and statistics, up to and including probability distributions over spaces of infinite sequences. Familiarity with the literature on learning or stochastic processes is not required.

We are in the process of writing and adding new material (compact eBooks) exclusively available to our members, and written in simple English, by world leading experts in AI, data science, and machine learning. We invite you to sign up here to not miss these free books. This book is intended for busy professionals working with data of any kind: engineers, BI analysts, statisticians, operations research, AI and machine learning professionals, economists, data scientists, biologists, and quants, ranging from beginners to executives. In about 300 pages and 28 chapters it covers many new topics, offering a fresh perspective on the subject, including rules of thumb and recipes that are easy to automate or integrate in black-box systems, as well as new model-free, data-driven foundations to statistical science and predictive analytics. The approach focuses on robust techniques; it is bottom-up (from applications to theory), in contrast to the traditional top-down approach. The material is accessible to practitioners with a one-year college-level exposure to statistics and probability.

It is well known that adding any skew symmetric matrix to the gradient of Langevin dynamics algorithm results in a non-reversible diffusion with improved convergence rate. This paper presents a gradient algorithm to adaptively optimize the choice of the skew symmetric matrix. The resulting algorithm involves a non-reversible diffusion algorithm cross coupled with a stochastic gradient algorithm that adapts the skew symmetric matrix. The algorithm uses the same data as the classical Langevin algorithm. A weak convergence proof is given for the optimality of the choice of the skew symmetric matrix. The improved convergence rate of the algorithm is illustrated numerically in Bayesian learning and tracking examples.

… optimization, finance, and quantum machine learning and natural language processing to advance the industry’s quantum computing ecosystem.

Probability and Statistics- the terms which resonate together to create the vast applications of the fields of Data Science and Machine Learning, have immensely grown a huge followers' base in this era. But programming the concept sometimes gets tricky and requires a lot of contemplation on the code. Allen B. Downey, in his'Think' series has written a book to solve just the problem for everyone.

We propose a Byzantine-robust variance-reduced stochastic gradient descent (SGD) method to solve the distributed finite-sum minimization problem when the data on the workers are not independent and identically distributed (i.i.d.). During the learning process, an unknown number of Byzantine workers may send malicious messages to the master node, leading to remarkable learning error. Most of the Byzantine-robust methods address this issue by using robust aggregation rules to aggregate the received messages, but rely on the assumption that all the regular workers have i.i.d. data, which is not the case in many federated learning applications. In light of the significance of reducing stochastic gradient noise for mitigating the effect of Byzantine attacks, we use a resampling strategy to reduce the impact of both inner variation (that describes the sample heterogeneity on every regular worker) and outer variation (that describes the sample heterogeneity among the regular workers), along with a stochastic average gradient algorithm (SAGA) to fully eliminate the inner variation. The variance-reduced messages are then aggregated with a robust geometric median operator. Under certain conditions, we prove that the proposed method reaches a neighborhood of the optimal solution with linear convergence rate, and the learning error is much smaller than those given by the state-of-the-art methods in the non-i.i.d. setting. Numerical experiments corroborate the theoretical results and show satisfactory performance of the proposed method.

We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method.

This paper describes a $near$-$optimal$ stochastic first-order gradient method for decentralized finite-sum minimization of smooth non-convex functions. Specifically, we propose GT-SARAH that employs a local SARAH-type variance reduction and global gradient tracking to address the stochastic and decentralized nature of the problem. Considering a total number of $N$ cost functions, equally divided over a directed network of $n$ nodes, we show that GT-SARAH finds an $\epsilon$-accurate first-order stationary point in ${\mathcal{O}(N^{1/2}\epsilon^{-1})}$ gradient computations across all nodes, independent of the network topology, when ${n\leq\mathcal{O}(N^{1/2}(1-\lambda)^{3})}$, where ${(1-\lambda)}$ is the spectral gap of the network weight matrix. In this regime, GT-SARAH is thus, to the best our knowledge, the first decentralized method that achieves the algorithmic lower bound for this class of problems. Moreover, GT-SARAH achieves a $non$-$asymptotic$ $linear$ $speedup$, in that, the total number of gradient computations at each node is reduced by a factor of $1/n$ compared to the near-optimal algorithms for this problem class that process all data at a single node. We also establish the convergence rate of GT-SARAH in other regimes, in terms of the relative sizes of the number of nodes $n$, total number of functions $N$, and the network spectral gap $(1-\lambda)$. Over infinite time horizon, we establish the almost sure and mean-squared convergence of GT-SARAH to a first-order stationary point.

Optimal transport has been one of the most exciting subjects in mathematics, starting from the 18th century. As a powerful tool to transport between two probability measures, optimal transport methods have been reinvigorated nowadays in a remarkable proliferation of modern data science applications. To meet the big data challenges, various computational tools have been developed in the recent decade to accelerate the computation for optimal transport methods. In this review, we present some cutting-edge computational optimal transport methods with a focus on the regularization-based methods and the projection-based methods. We discuss their real-world applications in biomedical research.