Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop a randomized optimization method, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor.

Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning. Our theoretical analysis shows that optimal statistical accuracy is achieved requiring essentially $O(n)$ memory and $O(n\sqrt{n})$ time. An extensive experimental analysis on large scale datasets shows that, even with a single machine, FALKON outperforms previous state of the art solutions, which exploit parallel/distributed architectures.

We propose new algorithms for topic modeling when the number of topics is unknown. Our approach relies on an analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Our algorithms are shown in practice to have accuracy comparable to a Gibbs sampler in terms of topic estimation, which requires the number of topics be given. Moreover, they are one of the fastest among several state of the art parametric techniques. Statistical consistency of our estimator is established under some conditions.

We give the first algorithm for kernel Nystrom approximation that runs in linear time in the number of training points and is provably accurate for all kernel matrices, without dependence on regularity or incoherence conditions. The algorithm projects the kernel onto a set of s landmark points sampled by their ridge leverage scores, requiring just O(ns) kernel evaluations and O(ns^2) additional runtime. While leverage score sampling has long been known to give strong theoretical guarantees for Nystrom approximation, by employing a fast recursive sampling scheme, our algorithm is the first to make the approach scalable. Empirically we show that it finds more accurate kernel approximations in less time than popular techniques such as classic Nystrom approximation and the random Fourier features method.

The goal of imitation learning is to mimic expert behavior without access to an explicit reward signal. Expert demonstrations provided by humans, however, often show significant variability due to latent factors that are typically not explicitly modeled. In this paper, we propose a new algorithm that can infer the latent structure of expert demonstrations in an unsupervised way. Our method, built on top of Generative Adversarial Imitation Learning, can not only imitate complex behaviors, but also learn interpretable and meaningful representations of complex behavioral data, including visual demonstrations. In the driving domain, we show that a model learned from human demonstrations is able to both accurately reproduce a variety of behaviors and accurately anticipate human actions using raw visual inputs. Compared with various baselines, our method can better capture the latent structure underlying expert demonstrations, often recovering semantically meaningful factors of variation in the data.

We introduce deep neural networks for end-to-end differentiable theorem proving that operate on dense vector representations of symbols. These neural networks are recursively constructed by following the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. The resulting neural network can be trained to infer facts from a given incomplete knowledge base using gradient descent. By doing so, it learns to (i) place representations of similar symbols in close proximity in a vector space, (ii) make use of such similarities to prove facts, (iii) induce logical rules, and (iv) it can use provided and induced logical rules for complex multi-hop reasoning. On four benchmark knowledge bases we demonstrate that this architecture outperforms ComplEx, a state-of-the-art neural link prediction model, while at the same time inducing interpretable function-free first-order logic rules.

Remarkable recent success of deep neural networks has not been easy to analyze theoretically. It has been particularly hard to disentangle relative significance of architecture and optimization in achieving accurate classification on large datasets. On the flip side, shallow methods (such as kernel methods) have encountered obstacles in scaling to large data, despite excellent performance on smaller datasets, and extensive theoretical analysis. Practical methods, such as variants of gradient descent used so successfully in deep learning, seem to perform below par when applied to kernel methods. This difficulty has sometimes been attributed to the limitations of shallow architecture. In this paper we identify a basic limitation in gradient descent-based optimization methods when used in conjunctions with smooth kernels. Our analysis demonstrates that only a vanishingly small fraction of the function space is reachable after a polynomial number of gradient descent iterations. That drastically limits the approximating power of gradient descent leading to over-regularization. The issue is purely algorithmic, persisting even in the limit of infinite data. To address this shortcoming in practice, we introduce EigenPro iteration, a simple and direct preconditioning scheme using a small number of approximately computed eigenvectors. It can also be viewed as learning a kernel optimized for gradient descent. Injecting this small, computationally inexpensive and SGD-compatible, amount of approximate second-order information leads to major improvements in convergence. For large data, this leads to a significant performance boost over the state-of-the-art kernel methods. In particular, we are able to match or improve the results reported in the literature at a small fraction of their computational budget. For complete version of this paper see https://arxiv.org/abs/1703.10622.

We establish the consistency of an algorithm of Mondrian Forests~\cite{lakshminarayanan2014mondrianforests,lakshminarayanan2016mondrianuncertainty}, a randomized classification algorithm that can be implemented online. First, we amend the original Mondrian Forest algorithm proposed in~\cite{lakshminarayanan2014mondrianforests}, that considers a \emph{fixed} lifetime parameter. Indeed, the fact that this parameter is fixed actually hinders statistical consistency of the original procedure. Our modified Mondrian Forest algorithm grows trees with increasing lifetime parameters $\lambda_n$, and uses an alternative updating rule, allowing to work also in an online fashion. Second, we provide a theoretical analysis establishing simple conditions for consistency. Our theoretical analysis also exhibits a surprising fact: our algorithm achieves the minimax rate (optimal rate) for the estimation of a Lipschitz regression function, which is a strong extension of previous results~\cite{arlot2014purf_bias} to an \emph{arbitrary dimension}.

Markov jump processes are continuous-time stochastic processes widely used in statistical applications in the natural sciences, and more recently in machine learning. Inference for these models typically proceeds via Markov chain Monte Carlo, and can suffer from various computational challenges. In this work, we propose a novel collapsed variational inference algorithm to address this issue. Our work leverages ideas from discrete-time Markov chains, and exploits a connection between these two through an idea called uniformization. Our algorithm proceeds by marginalizing out the parameters of the Markov jump process, and then approximating the distribution over the trajectory with a factored distribution over segments of a piecewise-constant function. Unlike MCMC schemes that marginalize out transition times of a piecewise-constant process, our scheme optimizes the discretization of time, resulting in significant computational savings. We apply our ideas to synthetic data as well as a dataset of check-in recordings, where we demonstrate superior performance over state-of-the-art MCMC methods.

Gaussian process regression generally does not scale to beyond a few thousands data points without applying some sort of kernel approximation method. Most approximations focus on the high eigenvalue part of the spectrum of the kernel matrix, $K$, which leads to bad performance when the length scale of the kernel is small. In this paper we introduce Multiresolution Kernel Approximation (MKA), the first true broad bandwidth kernel approximation algorithm. Important points about MKA are that it is memory efficient, and it is a direct method, which means that it also makes it easy to approximate $K^{-1}$ and $\mathop{\textrm{det}}(K)$.