We investigate the efficiency of k-means in terms of both statistical and computational requirements. More precisely, we study a Nystr\"om approach to kernel k-means. We analyze the statistical properties of the proposed method and show that it achieves the same accuracy of exact kernel k-means with only a fraction of computations. Indeed, we prove under basic assumptions that sampling $\sqrt{n}$ Nystr\"om landmarks allows to greatly reduce computational costs without incurring in any loss of accuracy. To the best of our knowledge this is the first result showing in this kind for unsupervised learning.

Distributed implementations of mini-batch stochastic gradient descent (SGD) suffer from communication overheads, attributed to the high frequency of gradient updates inherent in small-batch training. Training with large batches can reduce these overheads; however it besets the convergence of the algorithm and the generalization performance. In this work, we take a first step towards analyzing how the structure (width and depth) of a neural network affects the performance of large-batch training. We present new theoretical results which suggest that--for a fixed number of parameters--wider networks are more amenable to fast large-batch training compared to deeper ones. We provide extensive experiments on residual and fully-connected neural networks which suggest that wider networks can be trained using larger batches without incurring a convergence slow-down, unlike their deeper variants.

Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they do not. When they converge, do they converge to local min-max solutions? We characterize the limit points of two basic first order methods, namely Gradient Descent/Ascent (GDA) and Optimistic Gradient Descent Ascent (OGDA). We show that both dynamics avoid unstable critical points for almost all initializations.

Stein variational gradient descent (SVGD) is a non-parametric inference algorithm that evolves a set of particles to fit a given distribution of interest. We analyze the non-asymptotic properties of SVGD, showing that there exists a set of functions, which we call the Stein matching set, whose expectations are exactly estimated by any set of particles that satisfies the fixed point equation of SVGD. This set is the image of Stein operator applied on the feature maps of the positive definite kernel used in SVGD. Our results provide a theoretical framework for analyzing the properties of SVGD with different kernels, shedding insight into optimal kernel choice. In particular, we show that SVGD with linear kernels yields exact estimation of means and variances on Gaussian distributions, while random Fourier features enable probabilistic bounds for distributional approximation.

Gaussian processes (GPs) provide a powerful non-parametric framework for reasoning over functions. Despite appealing theory, its superlinear computational and memory complexities have presented a long-standing challenge. State-of-the-art sparse variational inference methods trade modeling accuracy against complexity. However, the complexities of these methods still scale superlinearly in the number of basis functions, implying that that sparse GP methods are able to learn from large datasets only when a small model is used. Recently, a decoupled approach was proposed that removes the unnecessary coupling between the complexities of modeling the mean and the covariance functions of a GP. It achieves a linear complexity in the number of mean parameters, so an expressive posterior mean function can be modeled.

We present a neurosymbolic framework for the lifelong learning of algorithmic tasks that mix perception and procedural reasoning. Reusing high-level concepts across domains and learning complex procedures are key challenges in lifelong learning. We show that a program synthesis approach that combines gradient descent with combinatorial search over programs can be a more effective response to these challenges than purely neural methods. Our framework, called HOUDINI, represents neural networks as strongly typed, differentiable functional programs that use symbolic higher-order combinators to compose a library of neural functions. Our learning algorithm consists of: (1) a symbolic program synthesizer that performs a type-directed search over parameterized programs, and decides on the library functions to reuse, and the architectures to combine them, while learning a sequence of tasks; and (2) a neural module that trains these programs using stochastic gradient descent.

Kleinberg (2002) stated three axioms that any clustering procedure should satisfy and showed there is no clustering procedure that simultaneously satisfies all three. One of these, called the consistency axiom, requires that when the data is modified in a helpful way, i.e. if points in the same cluster are made more similar and those in different ones made less similar, the algorithm should output the same clustering. To circumvent this impossibility result, research has focused on considering clustering procedures that have a clustering quality measure (or a cost) and showing that a modification of Kleinberg's axioms that takes cost into account lead to feasible clustering procedures. In this work, we take a different approach, based on the observation that the consistency axiom fails to be satisfied when the "correct" number of clusters changes. We modify this axiom by making use of cost functions to determine the correct number of clusters, and require that consistency holds only if the number of clusters remains unchanged.

We suggest a general oracle-based framework that captures parallel stochastic optimization in different parallelization settings described by a dependency graph, and derive generic lower bounds in terms of this graph. We then use the framework and derive lower bounds to study several specific parallel optimization settings, including delayed updates and parallel processing with intermittent communication. We highlight gaps between lower and upper bounds on the oracle complexity, and cases where the natural'' algorithms are not known to be optimal. Papers published at the Neural Information Processing Systems Conference.

In this paper, we investigate Dimensionality reduction (DR) maps in an information retrieval setting from a quantitative topology point of view. In particular, we show that no DR maps can achieve perfect precision and perfect recall simultaneously. Thus a continuous DR map must have imperfect precision. We further prove an upper bound on the precision of Lipschitz continuous DR maps. While precision is a natural measure in an information retrieval setting, it does not measure how' wrong the retrieved data is.

We develop deep Poisson-gamma dynamical systems (DPGDS) to model sequentially observed multivariate count data, improving previously proposed models by not only mining deep hierarchical latent structure from the data, but also capturing both first-order and long-range temporal dependencies. Using sophisticated but simple-to-implement data augmentation techniques, we derived closed-form Gibbs sampling update equations by first backward and upward propagating auxiliary latent counts, and then forward and downward sampling latent variables. Moreover, we develop stochastic gradient MCMC inference that is scalable to very long multivariate count time series. Experiments on both synthetic and a variety of real-world data demonstrate that the proposed model not only has excellent predictive performance, but also provides highly interpretable multilayer latent structure to represent hierarchical and temporal information propagation. Papers published at the Neural Information Processing Systems Conference.