Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these computational constraints. In this paper, we propose a novel approximation procedure coined \textit{Faster-Fast and Free Memory Method} ( \text{F} 3 M) to address these scaling issues of KMVM for tall ( 10 8\sim 10 9) and skinny ( D\leq7) data. Extensive experiments demonstrate that \text{F} 3 M has empirical \emph{linear time and memory} complexity with a relative error of order 10 {-3} and can compute a full KMVM for a billion points \emph{in under a minute} on a high-end GPU, leading to a significant speed-up in comparison to existing CPU methods. We demonstrate the utility of our procedure by applying it as a drop-in for the state-of-the-art GPU-based linear solver FALKON, \emph{improving speed 1.5-5.5 times} at the cost of 1\% drop in accuracy. We further demonstrate competitive results on \emph{Gaussian Process regression} coupled with significant speedups on a variety of real-world datasets.

In this paper we propose a general framework to perform statistical online inference in a class of constant step size stochastic approximation (SA) problems, including the well-known stochastic gradient descent (SGD) and Q-learning. Regarding a constant step size SA procedure as a time-homogeneous Markov chain, we establish a functional central limit theorem (FCLT) for it under weaker conditions, and then construct confidence intervals for parameters via random scaling. To leverage the FCLT results in the Markov chain setting, an alternative condition that is more applicable for SA problems is established. We conduct experiments to perform inference with both random scaling and other traditional inference methods, and finds that the former has a more accurate and robust performance.

Stochastic gradient descent (SGD) has been widely used in machine learning due to its computational efficiency and favorable generalization properties. Recently, it has been empirically demonstrated that the gradient noise in several deep learning settings admits a non-Gaussian, heavy-tailed behavior. This suggests that the gradient noise can be modeled by using \alpha -stable distributions, a family of heavy-tailed distributions that appear in the generalized central limit theorem. In this context, SGD can be viewed as a discretization of a stochastic differential equation (SDE) driven by a L\'{e}vy motion, and the metastability results for this SDE can then be used for illuminating the behavior of SGD, especially in terms of preferring wide minima'. While this approach brings a new perspective for analyzing SGD, it is limited in the sense that, due to the time discretization, SGD might admit a significantly different behavior than its continuous-time limit.

We address the classical problem of hierarchical clustering, but in a framework where one does not have access to a representation of the objects or their pairwise similarities. Instead, we assume that only a set of comparisons between objects is available, that is, statements of the form objects i and j are more similar than objects k and l.'' Such a scenario is commonly encountered in crowdsourcing applications. The focus of this work is to develop comparison-based hierarchical clustering algorithms that do not rely on the principles of ordinal embedding. We show that single and complete linkage are inherently comparison-based and we develop variants of average linkage.

This paper extends robust principal component analysis (RPCA) to nonlinear manifolds. Suppose that the observed data matrix is the sum of a sparse component and a component drawn from some low dimensional manifold. Is it possible to separate them by using similar ideas as RPCA? Is there any benefit in treating the manifold as a whole as opposed to treating each local region independently? We answer these two questions affirmatively by proposing and analyzing an optimization framework that separates the sparse component from the manifold under noisy data.

Variational autoencoders (VAEs) are latent variable models that can generate complex objects and provide meaningful latent representations. Moreover, they could be further used in downstream tasks such as classification. As previous work has shown, one can easily fool VAEs to produce unexpected latent representations and reconstructions for a visually slightly modified input. Here, we examine several objective functions for adversarial attacks construction proposed previously and present a solution to alleviate the effect of these attacks. Our method utilizes the Markov Chain Monte Carlo (MCMC) technique in the inference step that we motivate with a theoretical analysis.

We introduce a variant of the k -nearest neighbor classifier in which k is chosen adaptively for each query, rather than supplied as a parameter. The choice of k depends on properties of each neighborhood, and therefore may significantly vary between different points. We provide theory and experiments that demonstrate that the algorithm performs comparably to, and sometimes better than, k -NN with an optimal choice of k . In particular, we derive bounds on the convergence rates of our classifier that depend on a local quantity we call the advantage'' which is significantly weaker than the Lipschitz conditions used in previous convergence rate proofs. These generalization bounds hinge on a variant of the seminal Uniform Convergence Theorem due to Vapnik and Chervonenkis; this variant concerns conditional probabilities and may be of independent interest.

For multi-valued functions---such as when the conditional distribution on targets given the inputs is multi-modal---standard regression approaches are not always desirable because they provide the conditional mean. Modal regression algorithms address this issue by instead finding the conditional mode(s). Most, however, are nonparametric approaches and so can be difficult to scale. Further, parametric approximators, like neural networks, facilitate learning complex relationships between inputs and targets. In this work, we propose a parametric modal regression algorithm. We use the implicit function theorem to develop an objective, for learning a joint function over inputs and targets.

Biological synaptic plasticity exhibits nonlinearities that are not accounted for by classic Hebbian learning rules. Here, we introduce a simple family of generalized nonlinear Hebbian learning rules. We study the computations implemented by their dynamics in the simple setting of a neuron receiving feedforward inputs. These nonlinear Hebbian rules allow a neuron to learn tensor decompositions of its higher- order input correlations. The particular input correlation decomposed and the form of the decomposition depend on the location of nonlinearities in the plasticity rule.

Constrained clustering has gained significant attention in the field of machine learning as it can leverage prior information on a growing amount of only partially labeled data. Following recent advances in deep generative models, we propose a novel framework for constrained clustering that is intuitive, interpretable, and can be trained efficiently in the framework of stochastic gradient variational inference. By explicitly integrating domain knowledge in the form of probabilistic relations, our proposed model (DC-GMM) uncovers the underlying distribution of data conditioned on prior clustering preferences, expressed as \textit{pairwise constraints}. These constraints guide the clustering process towards a desirable partition of the data by indicating which samples should or should not belong to the same cluster. We provide extensive experiments to demonstrate that DC-GMM shows superior clustering performances and robustness compared to state-of-the-art deep constrained clustering methods on a wide range of data sets.