We propose a new approach to combine Restricted Boltzmann Machines (RBMs) that can be used to solve combinatorial optimization problems. This allows synthesis of larger models from smaller RBMs that have been pretrained, thus effectively bypassing the problem of learning in large RBMs, and creating a system able to model a large, complex multi-modal space. We validate this approach by using learned representations to create "invertible boolean logic", where we can use Markov chain Monte Carlo (MCMC) approaches to find the solution to large scale boolean satisfiability problems and show viability towards other combinatorial optimization problems. Using this method, we are able to solve 64 bit addition based problems, as well as factorize 16 bit numbers. We find that these combined representations can provide a more accurate result for the same sample size as compared to a fully trained model. The Ising Problem has long been known to be in the class of NP-Hard problems, with no exact polynomial solution existing. Because of this, a large class of combinatorial optimization problems can be reformulated as Ising problems and solved by finding the ground state of that system (Barahona, 1982; Kirkpatrick et al., 1983; Lucas, 2014). The Boltzmann Machine (Ackley et al., 1987) was originally introduced as a constraint satisfaction network based on the Ising model problem, where the weights would encode some global constraints, and stochastic units were used to escape local minima. The original Boltzmann Machine found favor as a method to solve various combinatorial optimization problems (Korst & Aarts, 1989). However, learning was very slow with this model due to the difficulties with sampling and convergence, as well as the inability to exactly calculate the partition function.

The widespread use of machine learning models, especially within the context of decision-making systems impacting individuals, raises many ethical issues with respect to fairness and interpretability of these models. While the research in these domains is booming, very few works have addressed these two issues simultaneously. To solve this shortcoming, we propose FairCORELS, a supervised learning algorithm whose objective is to learn at the same time fair and interpretable models. FairCORELS is a multi-objective variant of CORELS, a branch-and-bound algorithm, designed to compute accurate and interpretable rule lists. By jointly addressing fairness and interpretability, FairCORELS can achieve better fairness/accuracy tradeoffs compared to existing methods, as demonstrated by the empirical evaluation performed on real datasets. Our paper also contains additional contributions regarding the search strategies for optimizing the multi-objective function integrating both fairness, accuracy and interpretability.

Graph neural networks (GNN) has been successfully applied to operate on the graph-structured data. Given a specific scenario, rich human expertise and tremendous laborious trials are usually required to identify a suitable GNN architecture. It is because the performance of a GNN architecture is significantly affected by the choice of graph convolution components, such as aggregate function and hidden dimension. Neural architecture search (NAS) has shown its potential in discovering effective deep architectures for learning tasks in image and language modeling. However, existing NAS algorithms cannot be directly applied to the GNN search problem. First, the search space of GNN is different from the ones in existing NAS work. Second, the representation learning capacity of GNN architecture changes obviously with slight architecture modifications. It affects the search efficiency of traditional search methods. Third, widely used techniques in NAS such as parameter sharing might become unstable in GNN. To bridge the gap, we propose the automated graph neural networks (AGNN) framework, which aims to find an optimal GNN architecture within a predefined search space. A reinforcement learning based controller is designed to greedily validate architectures via small steps. AGNN has a novel parameter sharing strategy that enables homogeneous architectures to share parameters, based on a carefully-designed homogeneity definition. Experiments on real-world benchmark datasets demonstrate that the GNN architecture identified by AGNN achieves the best performance, comparing with existing handcrafted models and tradistional search methods.

We present a method for pursuit/evasion that is highly efficient and and scales to large teams of aircraft. The underlying algorithm is an efficient algorithm for solving Markov Decision Processes (MDPs) that supports fully continuous state spaces. We demonstrate the algorithm in a team pursuit/evasion setting in a 3D environment using a pseudo-6DOF model and study performance by varying sizes of team members. We show that as the number of aircraft in the simulation grows, computational performance remains efficient and is suitable for real-time systems. We also define probability-to-win and survivability metrics that describe the teams' performance over multiple trials, and show that the algorithm performs consistently. We provide numerical results showing control inputs for a typical 1v1 encounter and provide videos for 1v1, 2v2, 3v3, 4v4, and 10v10 contests to demonstrate the ability of the algorithm to adapt seamlessly to complex environments.

Many real-world problems can be reduced to combinatorial optimization on a graph, where the subset or ordering of vertices that maximize some objective function must be found. With such tasks often NP-hard and analytically intractable, reinforcement learning (RL) has shown promise as a framework with which efficient heuristic methods to tackle these problems can be learned. Previous works construct the solution subset incrementally, adding one element at a time, however, the irreversible nature of this approach prevents the agent from revising its earlier decisions, which may be necessary given the complexity of the optimization task. We instead propose that the agent should seek to continuously improve the solution by learning to explore at test time. Our approach of exploratory combinatorial optimization (ECO-DQN) is, in principle, applicable to any combinatorial problem that can be defined on a graph. Experimentally, we show our method to produce state-of-the-art RL performance on the Maximum Cut problem. Moreover, because ECO-DQN can start from any arbitrary configuration, it can be combined with other search methods to further improve performance, which we demonstrate using a simple random search.

The paper presents a new method for approximating Strong Stackelberg Equilibrium in general-sum sequential games with imperfect information and perfect recall. The proposed approach is generic as it does not rely on any specific properties of a particular game model. The method is based on iterative interleaving of the two following phases: (1) guided Monte Carlo Tree Search sampling of the Follower's strategy space and (2) building the Leader's behavior strategy tree for which the sampled Follower's strategy is an optimal response. The above solution scheme is evaluated with respect to expected Leader's utility and time requirements on three sets of interception games with variable characteristics, played on graphs. A comparison with three state-of-the-art MILP/LP-based methods shows that in vast majority of test cases proposed simulation-based approach leads to optimal Leader's strategies, while excelling the competitive methods in terms of better time scalability and lower memory requirements.

Efficiency of self-optimizing systems is heavily dependent on their optimization strategies, e.g., choosing exact or approximate solver. A choice of such a strategy, in turn, is influenced by numerous factors, such as re-optimization time, size of the problem, optimality constraints, etc. Exact solvers are domain-independent and can guarantee optimality but suffer from scaling, while approximate solvers offer a "good-enough" solution in exchange for a lack of generality and parameter-dependence. In this paper we discuss the trade-offs between exact and approximate optimizers for solving a quality-based software selection and hardware mapping problem from the scalability perspective. We show that even a simple heuristic can compete with thoroughly developed exact solvers under condition of an effective parameter tuning. Moreover, we discuss robustness of the obtained algorithm's configuration. Last but not least, we present a software product line for parameter tuning, which comprise the main features of this process and can serve as a platform for further research in the area of parameter tuning.

Lecture notes on optimization for machine learning, derived from a course at Princeton University and tutorials given in MLSS, Buenos Aires, as well as Simons Foundation, Berkeley.

Data structures are presented in a container hierarchy that includes stacks and queues as non-traversable dispensers, and lists, sets, and maps as traversable collections. Algorithm analysis is introduced and applied to linear and binary search, bubble sort, selection sort, insertion sort, merge sort and quicksort. The book also covers heaps and heapsort, unbalanced binary search trees, AVL trees, 2-3 trees, hashing, graph representations, and graph algorithms based on depth-and breadth-first search.

It was written as an advanced text to serve as a basis for a graduate course, and/or as a reference to the researcher diving into this fascinating world at the intersection of optimization and machine learning. Such a course was given at the Technion in the years 2010-2014 with slight variations from year to year, and later at Princeton University in the years 2015-2016. The core material in these courses is fully covered in this book, along with exercises that allow the students to complete parts of proofs, or that were found illuminating and thought-provoking. Most of the material is given with examples of applications, which are interlaced throughout different topics. These include prediction from expert advice, portfolio selection, matrix completion and recommendation systems, SVM training and more.