In this paper we introduce a theoretical framework for semi-discrete optimization using ideas from optimal transport. Our primary motivation is in the field of deep learning, and specifically in the task of neural architecture search. With this aim in mind, we discuss the geometric and theoretical motivation for new techniques for neural architecture search (in the companion work \cite{practical}; we show that algorithms inspired by our framework are competitive with contemporaneous methods). We introduce a Riemannian like metric on the space of probability measures over a semi-discrete space $\mathbb{R}^d \times \mathcal{G}$ where $\mathcal{G}$ is a finite weighted graph. With such Riemmanian structure in hand, we derive formal expressions for the gradient flow of a relative entropy functional, as well as second order dynamics for the optimization of said energy. Then, with the aim of providing a rigorous motivation for the gradient flow equations derived formally we also consider an iterative procedure known as minimizing movement scheme (i.e., Implicit Euler scheme, or JKO scheme) and apply it to the relative entropy with respect to a suitable cost function. For some specific choices of metric and cost, we rigorously show that the minimizing movement scheme of the relative entropy functional converges to the gradient flow process provided by the formal Riemannian structure. This flow coincides with a system of reaction-diffusion equations on $\mathbb{R}^d$.

Sequence-to-sequence (seq2seq) learners are widely used, but we still have only limited knowledge about what inductive biases shape the way they generalize. We address that by investigating how popular seq2seq learners generalize in tasks that have high ambiguity in the training data. We use SCAN and three new tasks to study learners' preferences for memorization, arithmetic, hierarchical, and compositional reasoning. Further, we connect to Solomonoff's theory of induction and propose to use description length as a principled and sensitive measure of inductive biases. In our experimental study, we find that LSTM-based learners can learn to perform counting, addition, and multiplication by a constant from a single training example. Furthermore, Transformer and LSTM-based learners show a bias toward the hierarchical induction over the linear one, while CNN-based learners prefer the opposite. On the SCAN dataset, we find that CNN-based, and, to a lesser degree, Transformer- and LSTM-based learners have a preference for compositional generalization over memorization. Finally, across all our experiments, description length proved to be a sensitive measure of inductive biases.

Despite recent successes, the advances in Deep Learning have not yet been fully translated to Computer Assisted Intervention (CAI) problems such as pose estimation of surgical instruments. Currently, neural architectures for classification and segmentation tasks are adopted ignoring significant discrepancies between CAI and these tasks. We propose an automatic framework (AutoSNAP) for instrument pose estimation problems, which discovers and learns the architectures for neural networks. We introduce 1)~an efficient testing environment for pose estimation, 2)~a powerful architecture representation based on novel Symbolic Neural Architecture Patterns (SNAPs), and 3)~an optimization of the architecture using an efficient search scheme. Using AutoSNAP, we discover an improved architecture (SNAPNet) which outperforms both the hand-engineered i3PosNet and the state-of-the-art architecture search method DARTS.

Scaling probabilistic models to large realistic problems and datasets is a key challenge in machine learning. Central to this effort is the development of tractable probabilistic models (TPMs): models whose structure guarantees efficient probabilistic inference algorithms. The current landscape of TPMs is fragmented: there exist various kinds of TPMs with different strengths and weaknesses. Two of the most prominent classes of TPMs are determinantal point processes (DPPs) and probabilistic circuits (PCs). This paper provides the first systematic study of their relationship. We propose a unified analysis and shared language for discussing DPPs and PCs. Then we establish theoretical barriers for the unification of these two families, and prove that there are cases where DPPs have no compact representation as a class of PCs. We close with a perspective on the central problem of unifying these tractable models.

In recent years, Deep Reinforcement Learning (DRL) algorithms have achieved state-of-the-art performance in many challenging strategy games. Because these games have complicated rules, an action sampled from the full discrete action space will typically be invalid. The usual approach to deal with this problem in policy gradient algorithms is to "mask out" invalid actions and just sample from the set of valid actions. The implications of this process, however, remain under-investigated. In this paper, we show that the standard working mechanism of invalid action masking corresponds to valid policy gradient updates. More interestingly, it works by applying a state-dependent differentiable function during the calculation of action probability distribution. Additionally, we show its critical importance to the performance of policy gradient algorithms. Specifically, our experiments show that invalid action masking scales well when the space of invalid actions is large, while the common approach of giving negative rewards for invalid actions will fail. Finally, we provide further insights by evaluating different action masking regimes, such as removing masking after an agent has been trained using masking.

Reinforcement learning algorithms usually assume that all actions are always available to an agent. However, both people and animals understand the general link between the features of their environment and the actions that are feasible. Gibson (1977) coined the term "affordances" to describe the fact that certain states enable an agent to do certain actions, in the context of embodied agents. In this paper, we develop a theory of affordances for agents who learn and plan in Markov Decision Processes. Affordances play a dual role in this case. On one hand, they allow faster planning, by reducing the number of actions available in any given situation. On the other hand, they facilitate more efficient and precise learning of transition models from data, especially when such models require function approximation. We establish these properties through theoretical results as well as illustrative examples. We also propose an approach to learn affordances and use it to estimate transition models that are simpler and generalize better.

Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games---in which the pure strategy space is (potentially uncountably) infinite---is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for computing Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our algorithm also applies to multiplayer games and games of imperfect information. We experiment with our algorithm on a continuous imperfect-information Blotto game, in which two players distribute resources over multiple battlefields. Blotto games have frequently been used to model national security scenarios and have also been applied to electoral competition and auction theory. Experiments show that our algorithm is able to quickly compute close approximations of Nash equilibrium strategies for this game.

With the onset of Electric vehicles, and them becoming more and more popular, autonomous cars are the future in the travel/driving experience. The barrier to reaching level 5 autonomy is the difficulty in the collection of data that incorporates good driving habits and the lack thereof. The problem with current implementations of self-driving cars is the need for massively large datasets and the need to evaluate the driving in the dataset. We propose a system that requires no data for its training. An evolutionary model would have the capability to optimize itself towards the fitness function. We have implemented Neuroevolution, a form of genetic algorithm, to train/evolve self-driving cars in a simulated virtual environment with the help of Unreal Engine 4, which utilizes Nvidia's PhysX Physics Engine to portray real-world vehicle dynamics accurately. We were able to observe the serendipitous nature of evolution and have exploited it to reach our optimal solution. We also demonstrate the ease in generalizing attributes brought about by genetic algorithms and how they may be used as a boilerplate upon which other machine learning techniques may be used to improve the overall driving experience.

Euclidean Markov decision processes are a powerful tool for modeling control problems under uncertainty over continuous domains. Finite state imprecise, Markov decision processes can be used to approximate the behavior of these infinite models. In this paper we address two questions: first, we investigate what kind of approximation guarantees are obtained when the Euclidean process is approximated by finite state approximations induced by increasingly fine partitions of the continuous state space. We show that for cost functions over finite time horizons the approximations become arbitrarily precise. Second, we use imprecise Markov decision process approximations as a tool to analyse and validate cost functions and strategies obtained by reinforcement learning. We find that, on the one hand, our new theoretical results validate basic design choices of a previously proposed reinforcement learning approach. On the other hand, the imprecise Markov decision process approximations reveal some inaccuracies in the learned cost functions.

We introduce a new recurrent agent architecture and associated auxiliary losses which improve reinforcement learning in partially observable tasks requiring long-term memory. We employ a temporal hierarchy, using a slow-ticking recurrent core to allow information to flow more easily over long time spans, and three fast-ticking recurrent cores with connections designed to create an information asymmetry. The \emph{reaction} core incorporates new observations with input from the slow core to produce the agent's policy; the \emph{perception} core accesses only short-term observations and informs the slow core; lastly, the \emph{prediction} core accesses only long-term memory. An auxiliary loss regularizes policies drawn from all three cores against each other, enacting the prior that the policy should be expressible from either recent or long-term memory. We present the resulting \emph{Perception-Prediction-Reaction} (PPR) agent and demonstrate its improved performance over a strong LSTM-agent baseline in DMLab-30, particularly in tasks requiring long-term memory. We further show significant improvements in Capture the Flag, an environment requiring agents to acquire a complicated mixture of skills over long time scales. In a series of ablation experiments, we probe the importance of each component of the PPR agent, establishing that the entire, novel combination is necessary for this intriguing result.