This paper considers the problem of estimating a low-rank matrix from the observation of all, or a subset, of its entries in the presence of Poisson noise. When we observe all the entries, this is a problem of matrix denoising; when we observe only a subset of the entries, this is a problem of matrix completion. In both cases, we exploit an assumption that the underlying matrix is low-rank. Specifically, we analyze several estimators, including a constrained nuclear-norm minimization program, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem. We show that for all three estimators, with high probability, we have an upper error bound (in the Frobenius norm error metric) that depends on the matrix rank, the fraction of the elements observed, and maximal row and column sums of the true matrix. We furthermore show that the above results are minimax optimal (within a universal constant) in classes of matrices with low rank and bounded row and column sums. We also extend these results to handle the case of matrix multinomial denoising and completion.

Achieving good statistical accuracy under budgeted computational resources is a central theme in modern machine learning (Bottou and Bousquet, 2008). Indeed, the problem of understanding the interplay and tradeoffs between statistical and computational requirements has recently received much attention. Nonparametric learning, and in particular kernel methods, have provided a natural framework to pursue these questions, see e.g.(Musco and Musco, 2017; Rudi et al., 2015; Alaoui and Mahoney, 2014; Bach, 2013; Calandriello et al., 2018; Orabona et al., 2008). On the one hand, these methods are developed in a sound mathematical setting and their statistical properties are well studied. On the other hand, from a numerical point of view, they scale poorly to large scale problems, and hence improved computational efficiency is of particular interest. While initial studies have mostly focused on approximating kernel matrices (Drineas and Mahoney, 2005; Gittens and Mahoney, 2013; Jin et al., 2013), recent results have highlighted the importance of considering downstream learning tasks, if the interplay between statistics and computation is of interest.

In this paper, we seek a clinically-relevant latent code for representing the spectrum of macular disease. Towards this end, we construct retina-VAE, a variational autoencoder-based model that accepts a patient profile vector (pVec) as input. The pVec components include clinical exam findings and demographic information. We evaluate the model on a subspectrum of the retinal maculopathies, in particular, exudative age-related macular degeneration, central serous chorioretinopathy, and polypoidal choroidal vasculopathy. For these three maculopathies, a database of 3000 6-dimensional pVecs (1000 each) was synthetically generated based on known disease statistics in the literature. The database was then used to train the VAE and generate latent vector representations. We found training performance to be best for a 3-dimensional latent vector architecture compared to 2 or 4 dimensional latents. Additionally, for the 3D latent architecture, we discovered that the resulting latent vectors were strongly clustered spontaneously into one of 14 clusters. Kmeans was then used only to identify members of each cluster and to inspect cluster properties. These clusters suggest underlying disease subtypes which may potentially respond better or worse to particular pharmaceutical treatments such as anti-vascular endothelial growth factor variants. The retina-VAE framework will potentially yield new fundamental insights into the mechanisms and manifestations of disease. And will potentially facilitate the development of personalized pharmaceuticals and gene therapies.

Intense recent discussions have focused on how to provide individuals with control over when their data can and cannot be used -- the EU's Right To Be Forgotten regulation is an example of this effort. In this paper we initiate a framework studying what to do when it is no longer permissible to deploy models derivative from specific user data. In particular, we formulate the problem of how to efficiently delete individual data points from trained machine learning models. For many standard ML models, the only way to completely remove an individual's data is to retrain the whole model from scratch on the remaining data, which is often not computationally practical. We investigate algorithmic principles that enable efficient data deletion in ML. For the specific setting of k-means clustering, we propose two provably deletion efficient algorithms which achieve an average of over 100X improvement in deletion efficiency across 6 datasets, while producing clusters of comparable statistical quality to a canonical k-means++ baseline.

To deepen our understanding of graph neural networks, we investigate the representation power of Graph Convolutional Networks (GCN) through the looking glass of graph moments, a key property of graph topology encoding path of various lengths. We find that GCNs are rather restrictive in learning graph moments. Without careful design, GCNs can fail miserably even with multiple layers and nonlinear activation functions. We analyze theoretically the expressiveness of GCNs, arriving at a modular GCN design, using different propagation rules. Our modular design is capable of distinguishing graphs from different graph generation models for surprisingly small graphs, a notoriously difficult problem in network science. Our investigation suggests that, depth is much more influential than width, with deeper GCNs being more capable of learning higher order graph moments. Additionally, combining GCN modules with different propagation rules is critical to the representation power of GCNs.

We discuss milestones on the tour towards DPLL(MAPF), a multi-agent path finding (MAPF) solver fully integrated with the Davis-Putnam-Logemann-Loveland (DPLL) propositional satisfiability testing algorithm through satisfiability modulo theories (SMT). The task in MAPF is to navigate agents in an undirected graph in a non-colliding way so that each agent eventually reaches its unique goal vertex. At most one agent can reside in a vertex at a time. Agents can move instantaneously by traversing edges provided the movement does not result in a collision. Recently attempts to solve MAPF optimally w.r.t. the sum-of-costs or the makespan based on the reduction of MAPF to propositional satisfiability (SAT) have appeared. The most successful methods rely on building the propositional encoding for the given MAPF instance lazily by a process inspired in the SMT paradigm. The integration of satisfiability testing by the SAT solver and the high-level construction of the encoding is however relatively loose in existing methods. Therefore the ultimate goal of research in this direction is to build the DPLL(MAPF) algorithm, a MAPF solver where the construction of the encoding is fully integrated with the underlying SAT solver. We discuss the current state-of-the-art in MAPF solving and what steps need to be done to get DPLL(MAPF). The advantages of DPLL(MAPF) in terms of its potential to be alternatively parametrized with MAPF$^R$, a theory of continuous MAPF with geometric agents, are also discussed.

We cast visual imitation as a visual correspondence problem. Our robotic agent is rewarded when its actions result in better matching of relative spatial configurations for corresponding visual entities detected in its workspace and teacher's demonstration. We build upon recent advances in Computer Vision,such as human finger keypoint detectors, object detectors trained on-the-fly with synthetic augmentations, and point detectors supervised by viewpoint changes and learn multiple visual entity detectors for each demonstration without human annotations or robot interactions. We empirically show the proposed factorized visual representations of entities and their spatial arrangements drive successful imitation of a variety of manipulation skills within minutes, using a single demonstration and without any environment instrumentation. It is robust to background clutter and can effectively generalize across environment variations between demonstrator and imitator, greatly outperforming unstructured non-factorized full-frame CNN encodings of previous works.

This paper investigates a paradigm for offering artificial intelligence as a service (AI-aaS) on software-defined infrastructures (SDIs). The increasing complexity of networking and computing infrastructures is already driving the introduction of automation in networking and cloud computing management systems. Here we consider how these automation mechanisms can be leveraged to offer AI-aaS. Use cases for AI-aaS are easily found in addressing smart applications in sectors such as transportation, manufacturing, energy, water, air quality, and emissions. We propose an architectural scheme based on SDIs where each AI-aaS application is comprised of a monitoring, analysis, policy, execution plus knowledge (MAPE-K) loop (MKL). Each application is composed as one or more specific service chains embedded in SDI, some of which will include a Machine Learning (ML) pipeline. Our model includes a new training plane and an AI-aaS plane to deal with the model-development and operational phases of AI applications. We also consider the role of an ML/MKL sandbox in ensuring coherency and consistency in the operation of multiple parallel MKL loops. We present experimental measurement results for three AI-aaS applications deployed on the SAVI testbed: 1. Compressing monitored data in SDI using autoencoders; 2. Traffic monitoring to allocate CPUs resources to VNFs; and 3. Highway segment classification in smart transportation.

An important step in the development of value alignment (VA) systems in AI is understanding how values can interrelate with facts. Designers of future VA systems will need to utilize a hybrid approach in which ethical reasoning and empirical observation interrelate successfully in machine behavior. In this article we identify two problems about this interrelation that have been overlooked by AI discussants and designers. The first problem is that many AI designers commit inadvertently a version of what has been called by moral philosophers the "naturalistic fallacy," that is, they attempt to derive an "ought" from an "is." We illustrate when and why this occurs. The second problem is that AI designers adopt training routines that fail fully to simulate human ethical reasoning in the integration of ethical principles and facts. Using concepts of quantified modal logic, we proceed to offer an approach that promises to simulate ethical reasoning in humans by connecting ethical principles on the one hand and propositions about states of affairs on the other.

We present Imitation-Projected Policy Gradient (IPPG), an algorithmic framework for learning policies that are parsimoniously represented in a structured programming language. Such programmatic policies can be more interpretable, generalizable, and amenable to formal verification than neural policies; however, designing rigorous learning approaches for programmatic policies remains a challenge. IPPG, our response to this challenge, is based on three insights. First, we view our learning task as optimization in policy space, modulo the constraint that the desired policy has a programmatic representation, and solve this optimization problem using a "lift-and-project" perspective that takes a gradient step into the unconstrained policy space and then projects back onto the constrained space. Second, we view the unconstrained policy space as mixing neural and programmatic representations, which enables employing state-of-the-art deep policy gradient approaches. Third, we cast the projection step as program synthesis via imitation learning, and exploit contemporary combinatorial methods for this task. We present theoretical convergence results for IPPG, as well as an empirical evaluation in three continuous control domains. The experiments show that IPPG can significantly outperform state-of-the-art approaches for learning programmatic policies.