Due to the high variance of policy gradients, on-policy optimization algorithms are plagued with low sample efficiency. In this work, we propose Augment-Reinforce-Merge (ARM) policy gradient estimator as an unbiased low-variance alternative to previous baseline estimators on tasks with binary action space, inspired by the recent ARM gradient estimator for discrete random variable models. We show that the ARM policy gradient estimator achieves variance reduction with theoretical guarantees, and leads to significantly more stable and faster convergence of policies parameterized by neural networks.

Bayesian optimization is popular for optimizing time-consuming black-box objectives. Nonetheless, for hyperparameter tuning in deep neural networks, the time required to evaluate the validation error for even a few hyperparameter settings remains a bottleneck. Multi-fidelity optimization promises relief using cheaper proxies to such objectives --- for example, validation error for a network trained using a subset of the training points or fewer iterations than required for convergence. We propose a highly flexible and practical approach to multi-fidelity Bayesian optimization, focused on efficiently optimizing hyperparameters for iteratively trained supervised learning models. We introduce a new acquisition function, the trace-aware knowledge-gradient, which efficiently leverages both multiple continuous fidelity controls and trace observations --- values of the objective at a sequence of fidelities, available when varying fidelity using training iterations. We provide a provably convergent method for optimizing our acquisition function and show it outperforms state-of-the-art alternatives for hyperparameter tuning of deep neural networks and large-scale kernel learning.

Linear programming is one of the most common optimization techniques. It has a wide range of applications and is frequently used in operations research, industrial design, planning, and the list goes on. Alas, it is not as hyped as machine learning is (which is certainly a form of optimization itself), but is the go-to method for problems that can be formulated through decision variables that have linear relationships. This is a fast practical tutorial, I will perhaps cover the Simplex algorithm and the theory in a later post. Just to get an idea, we are going to solve a simple problem regarding production scheduling.

Although momentum-based optimization methods have had a remarkable impact on machine learning, their heuristic construction has been an obstacle to a deeper understanding. A promising direction to study these accelerated algorithms has been emerging through connections with continuous dynamical systems. Yet, it is unclear whether the main properties of the underlying dynamical system are preserved by the algorithms from which they are derived. Conformal Hamiltonian systems form a special class of dissipative systems, having a distinct symplectic geometry. In this paper, we show that gradient descent with momentum preserves this symplectic structure, while Nesterov's accelerated gradient method does not. More importantly, we propose a generalization of classical momentum based on the special theory of relativity. The resulting conformal symplectic and relativistic algorithm enjoys better stability since it operates on a different space compared to its classical predecessor. Its benefits are discussed and verified in deep learning experiments.

Modern cyber-physical systems (e.g., robotics systems) are typically composed of physical and software components, the characteristics of which are likely to change over time. Assumptions about parts of the system made at design time may not hold at run time, especially when a system is deployed for long periods (e.g., over decades). Self-adaptation is designed to find reconfigurations of systems to handle such run-time inconsistencies. Planners can be used to find and enact optimal reconfigurations in such an evolving context. However, for systems that are highly configurable, such planning becomes intractable due to the size of the adaptation space. To overcome this challenge, in this paper we explore an approach that (a) uses machine learning to find Pareto-optimal configurations without needing to explore every configuration and (b) restricts the search space to such configurations to make planning tractable. We explore this in the context of robot missions that need to consider task timeliness and energy consumption. An independent evaluation shows that our approach results in high-quality adaptation plans in uncertain and adversarial environments.

We introduce a new regularizer for optimal transport (OT) which is tailored to better preserve the class structure. We give the first theoretical guarantees for an OT scheme that respects class structure. We give an accelerated proximal--projection scheme for this formulation with the proximal operator in closed form to give a highly scalable algorithm for computing optimal transport plans. We give a novel argument for the uniqueness of the optimum even in the absence of strong convexity. Our experiments show that the new regularizer preserves class structure better and is more robust compared to previous regularizers.

We present a machine learning approach to the solution of chance constrained optimizations in the context of voltage regulation problems in power system operation. The novelty of our approach resides in approximating the feasible region of uncertainty with an ellipsoid. We formulate this problem using a learning model similar to Support Vector Machines (SVM) and propose a sampling algorithm that efficiently trains the model. We demonstrate our approach on a voltage regulation problem using standard IEEE distribution test feeders.

In this paper we study the problem of robust influence maximization in the independent cascade model under a hyperparametric assumption. In social networks users influence and are influenced by individuals with similar characteristics and as such they are associated with some features. A recent surging research direction in influence maximization focuses on the case where the edge probabilities on the graph are not arbitrary but are generated as a function of the features of the users and a global hyperparameter. We propose a model where the objective is to maximize the worst-case number of influenced users for any possible value of that hyperparameter. We provide theoretical results showing that proper robust solution in our model is NP-hard and an algorithm that achieves improper robust optimization. We make-use of sampling based techniques and of the renowned multiplicative weight updates algorithm. Additionally we validate our method empirically and prove that it outperforms the state-of-the-art robust influence maximization techniques.

Recent progress in reinforcement learning (RL) using self-game-play has shown remarkable performance on several board games (e.g., Chess and Go) as well as video games (e.g., Atari games and Dota2). It is plausible to consider that RL, starting from zero knowledge, might be able to gradually approximate a winning strategy after a certain amount of training. In this paper, we explore neural Monte-Carlo-Tree-Search (neural MCTS), an RL algorithm which has been applied successfully by DeepMind to play Go and Chess at a super-human level. We try to leverage the computational power of neural MCTS to solve a class of combinatorial optimization problems. Following the idea of Hintikka's Game-Theoretical Semantics, we propose the Zermelo Gamification (ZG) to transform specific combinatorial optimization problems into Zermelo games whose winning strategies correspond to the solutions of the original optimization problem. The ZG also provides a specially designed neural MCTS. We use a combinatorial planning problem for which the ground-truth policy is efficiently computable to demonstrate that ZG is promising.

This textbook presents a wide range of tools for a course in mathematical optimization for upper undergraduate and graduate students in mathematics, engineering, computer science, and other applied sciences. Basic optimization principles are presented with emphasis on gradient-based numerical optimization strategies and algorithms for solving both smooth and noisy discontinuous optimization problems. Attention is also paid to the difficulties of expense of function evaluations and the existence of multiple minima that often unnecessarily inhibit the use of gradient-based methods. This second edition addresses further advancements of gradient-only optimization strategies to handle discontinuities in objective functions. New chapters discuss the construction of surrogate models as well as new gradient-only solution strategies and numerical optimization using Python.