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.

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.

Many contemporary applications in signal processing and machine learning give rise to structured non-convex non-smooth optimization problems that can often be tackled by simple iterative methods quite effectively. One of the keys to understanding such a phenomenon---and, in fact, one of the very difficult conundrums even for experts---lie in the study of "stationary points" of the problem in question. Unlike smooth optimization, for which the definition of a stationary point is rather standard, there is a myriad of definitions of stationarity in non-smooth optimization. In this article, we give an introduction to different stationarity concepts for several important classes of non-convex non-smooth functions and discuss the geometric interpretations and further clarify the relationship among these different concepts. We then demonstrate the relevance of these constructions in some representative applications and how they could affect the performance of iterative methods for tackling these applications.

Deep Neural Networks are well known to be vulnerable to adversarial attacks and backdoor attacks, where minor modifications on the input can mislead the models to give wrong results. Although defenses against adversarial attacks have been widely studied, research on mitigating backdoor attacks is still at an early stage. It is unknown whether there are any connections and common characteristics between the defenses against these two attacks. In this paper, we present a unified framework for detecting malicious examples and protecting the inference results of Deep Learning models. This framework is based on our observation that both adversarial examples and backdoor examples have anomalies during the inference process, highly distinguishable from benign samples. As a result, we repurpose and revise four existing adversarial defense methods for detecting backdoor examples. Extensive evaluations indicate these approaches provide reliable protection against backdoor attacks, with a higher accuracy than detecting adversarial examples. These solutions also reveal the relations of adversarial examples, backdoor examples and normal samples in model sensitivity, activation space and feature space. This can enhance our understanding about the inherent features of these two attacks, as well as the defense opportunities.

The two main challenges faced by continual learning approaches are catastrophic forgetting and memory limitations on the storage of data. To cope with these challenges, we propose a novel, cognitively-inspired approach which trains autoencoders with Neural Style Transfer to encode and store images. Reconstructed images from encoded episodes are replayed when training the classifier model on a new task to avoid catastrophic forgetting. The loss function for the reconstructed images is weighted to reduce its effect during classifier training to cope with image degradation. When the system runs out of memory the encoded episodes are converted into centroids and covariance matrices, which are used to generate pseudo-images during classifier training, keeping classifier performance stable with less memory. Our approach increases classification accuracy by 13-17% over state-of-the-art methods on benchmark datasets, while requiring 78% less storage space.

A method for counterfactual explanation of machine learning survival models is proposed. One of the difficulties of solving the counterfactual explanation problem is that the classes of examples are implicitly defined through outcomes of a machine learning survival model in the form of survival functions. A condition that establishes the difference between survival functions of the original example and the counterfactual is introduced. This condition is based on using a distance between mean times to event. It is shown that the counterfactual explanation problem can be reduced to a standard convex optimization problem with linear constraints when the explained black-box model is the Cox model. For other black-box models, it is proposed to apply the well-known Particle Swarm Optimization algorithm. A lot of numerical experiments with real and synthetic data demonstrate the proposed method.

The ability of deep learning (DL) to improve the practice of medicine and its clinical outcomes faces a looming obstacle: model interpretation. Without description of how outputs are generated, a collaborating physician can neither resolve when the model's conclusions are in conflict with his or her own, nor learn to anticipate model behavior. Current research aims to interpret networks that diagnose ECG recordings, which has great potential impact as recordings become more personalized and widely deployed. A generalizable impact beyond ECGs lies in the ability to provide a rich test-bed for the development of interpretive techniques in medicine. Interpretive techniques for Deep Neural Networks (DNNs), however, tend to be heuristic and observational in nature, lacking the mathematical rigor one might expect in the analysis of math equations. The motivation of this paper is to offer a third option, a scientific approach. We treat the model output itself as a phenomenon to be explained through component parts and equations governing their behavior. We argue that these component parts should also be "black boxes" --additional targets to interpret heuristically with clear functional connection to the original. We show how to rigorously factor a DNN into a hierarchical equation consisting of black box variables. This is not a subdivision into physical parts, like an organism into its cells; it is but one choice of an equation into a collection of abstract functions. Yet, for DNNs trained to identify normal ECG waveforms on PhysioNet 2017 Challenge data, we demonstrate this choice yields interpretable component models identified with visual composite sketches of ECG samples in corresponding input regions. Moreover, the recursion distills this interpretation: additional factorization of component black boxes corresponds to ECG partitions that are more morphologically pure.

Our goal is to enable machine learning systems to be trained interactively. This requires models that perform well and train quickly, without large amounts of hand-labeled data. We take a step forward in this direction by borrowing from weak supervision (WS), wherein models can be trained with noisy sources of signal instead of hand-labeled data. But WS relies on training downstream deep networks to extrapolate to unseen data points, which can take hours or days. Pre-trained embeddings can remove this requirement. We do not use the embeddings as features as in transfer learning (TL), which requires fine-tuning for high performance, but instead use them to define a distance function on the data and extend WS source votes to nearby points. Theoretically, we provide a series of results studying how performance scales with changes in source coverage, source accuracy, and the Lipschitzness of label distributions in the embedding space, and compare this rate to standard WS without extension and TL without fine-tuning. On six benchmark NLP and video tasks, our method outperforms WS without extension by 4.1 points, TL without fine-tuning by 12.8 points, and traditionally-supervised deep networks by 13.1 points, and comes within 0.7 points of state-of-the-art weakly-supervised deep networks--all while training in less than half a second.