IDEAL: Inexact DEcentralized Accelerated Augmented Lagrangian Method

We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method, thereby providing a systematic way for deriving several well-known decentralized algorithms including EXTRA arXiv:1404.6264 and SSDA arXiv:1702.08704. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower bounds. We provide experimental results that demonstrate the effectiveness of the proposed algorithm on highly ill-conditioned problems.

Continuous LWE

We introduce a continuous analogue of the Learning with Errors (LWE) problem, which we name CLWE. We give a polynomial-time quantum reduction from worst-case lattice problems to CLWE, showing that CLWE enjoys similar hardness guarantees to those of LWE. Alternatively, our result can also be seen as opening new avenues of (quantum) attacks on lattice problems. Our work resolves an open problem regarding the computational complexity of learning mixtures of Gaussians without separability assumptions (Diakonikolas 2016, Moitra 2018). As an additional motivation, (a slight variant of) CLWE was considered in the context of robust machine learning (Diakonikolas et al.~FOCS 2017), where hardness in the statistical query (SQ) model was shown; our work addresses the open question regarding its computational hardness (Bubeck et al.~ICML 2019).

A Permutation-Equivariant Neural Network Architecture For Auction Design

Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. Theoretical approaches to the problem have hit some limits in the past decades and analytical solutions are known for only a few simple settings. Computational approaches to the problem through the use of LPs have their own set of limitations. Building on the success of deep learning, a new approach was recently proposed by D\"utting et al., 2017 in which the auction is modeled by a feed-forward neural network and the design problem is framed as a learning problem. The neural architectures used in that work are general purpose and do not take advantage of any of the symmetries the problem could present, such as permutation equivariance. In this work, we consider auction design problems that have permutation-equivariant symmetry and construct a neural architecture that is capable of perfectly recovering the permutation-equivariant optimal mechanism, which we show is not possible with the previous architecture. We demonstrate that permutation-equivariant architectures are not only capable of recovering previous results, they also have better generalization properties.

Provably Efficient Third-Person Imitation from Offline Observation

Domain adaptation in imitation learning represents an essential step towards improving generalizability. However, even in the restricted setting of third-person imitation where transfer is between isomorphic Markov Decision Processes, there are no strong guarantees on the performance of transferred policies. We present problem-dependent, statistical learning guarantees for third-person imitation from observation in an offline setting, and a lower bound on performance in the online setting.

Can graph neural networks count substructures?

The ability to detect and count certain substructures in graphs is important for solving many tasks on graph-structured data, especially in the contexts of computational chemistry and biology as well as social network analysis. Inspired by this, we propose to study the expressive power of graph neural networks (GNNs) via their ability to count attributed graph substructures, extending recent works that examine their power in graph isomorphism testing and function approximation. We distinguish between two types of substructure counting: matching-count and containment-count, and establish both positive and negative answers for popular GNN architectures. Specifically, we prove that Message Passing Neural Networks (MPNNs), 2-Weisfeiler-Lehman (2-WL) and 2-Invariant Graph Networks (2-IGNs) cannot perform matching-count of substructures consisting of 3 or more nodes, while they can perform containment-count of star-shaped substructures. We also prove positive results for k-WL and k-IGNs as well as negative results for k-WL with limited number of iterations. We then conduct experiments that support the theoretical results for MPNNs and 2-IGNs, and demonstrate that local relational pooling strategies inspired by Murphy et al. (2019) are more effective for substructure counting. In addition, as an intermediary step, we prove that 2-WL and 2-IGNs are equivalent in distinguishing non-isomorphic graphs, partly answering an open problem raised in Maron et al. (2019).

A mean-field analysis of two-player zero-sum games

Finding Nash equilibria in two-player zero-sum continuous games is a central problem in machine learning, e.g. for training both GANs and robust models. The existence of pure Nash equilibria requires strong conditions which are not typically met in practice. Mixed Nash equilibria exist in greater generality and may be found using mirror descent. Yet this approach does not scale to high dimensions. To address this limitation, we parametrize mixed strategies as mixtures of particles, whose positions and weights are updated using gradient descent-ascent. We study this dynamics as an interacting gradient flow over measure spaces endowed with the Wasserstein-Fisher-Rao metric. We establish global convergence to an approximate equilibrium for the related Langevin gradient-ascent dynamic. We prove a law of large numbers that relates particle dynamics to mean-field dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs.

Probing the State of the Art: A Critical Look at Visual Representation Evaluation

Self-supervised research improved greatly over the past half decade, with much of the growth being driven by objectives that are hard to quantitatively compare. These techniques include colorization, cyclical consistency, and noise-contrastive estimation from image patches. Consequently, the field has settled on a handful of measurements that depend on linear probes to adjudicate which approaches are the best. Our first contribution is to show that this test is insufficient and that models which perform poorly (strongly) on linear classification can perform strongly (weakly) on more involved tasks like temporal activity localization. Our second contribution is to analyze the capabilities of five different representations. And our third contribution is a much needed new dataset for temporal activity localization.

Stability of Graph Neural Networks to Relative Perturbations

Graph neural networks (GNNs), consisting of a cascade of layers applying a graph convolution followed by a pointwise nonlinearity, have become a powerful architecture to process signals supported on graphs. Graph convolutions (and thus, GNNs), rely heavily on knowledge of the graph for operation. However, in many practical cases the GSO is not known and needs to be estimated, or might change from training time to testing time. In this paper, we are set to study the effect that a change in the underlying graph topology that supports the signal has on the output of a GNN. We prove that graph convolutions with integral Lipschitz filters lead to GNNs whose output change is bounded by the size of the relative change in the topology. Furthermore, we leverage this result to show that the main reason for the success of GNNs is that they are stable architectures capable of discriminating features on high eigenvalues, which is a feat that cannot be achieved by linear graph filters (which are either stable or discriminative, but cannot be both). Finally, we comment on the use of this result to train GNNs with increased stability and run experiments on movie recommendation systems.

Pure and Spurious Critical Points: a Geometric Study of Linear Networks

The critical locus of the loss function of a neural network is determined by the geometry of the functional space and by the parameterization of this space by the network's weights. We introduce a natural distinction between pure critical points, which only depend on the functional space, and spurious critical points, which arise from the parameterization. We apply this perspective to revisit and extend the literature on the loss function of linear neural networks. For this type of network, the functional space is either the set of all linear maps from input to output space, or a determinantal variety, i.e., a set of linear maps with bounded rank. We use geometric properties of determinantal varieties to derive new results on the landscape of linear networks with different loss functions and different parameterizations.

Gradient Dynamics of Shallow Univariate ReLU Networks

We present a theoretical and empirical study of the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes:kernel and adaptive, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavior of initialization coefficients in the limit of large network widths. We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces to cubic splines for uniform initializations. Learning in the adaptive regime favors instead linear splines, where knots cluster adaptively at the sample points.