Fast approximation of orthogonal matrices and application to PCA

We study the problem of approximating orthogonal matrices so that their application is numerically fast and yet accurate. We find an approximation by solving an optimization problem over a set of structured matrices, that we call Givens transformations, including Givens rotations as a special case. We propose an efficient greedy algorithm to solve such a problem and show that it strikes a balance between approximation accuracy and speed of computation. The proposed approach is relevant in spectral methods and we illustrate its application to PCA.

Gain with no Pain: Efficient Kernel-PCA by Nyström Sampling

In this paper, we propose and study a Nystr\"om based approach to efficient large scale kernel principal component analysis (PCA). The latter is a natural nonlinear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches, kernel PCA enjoys good mathematical and statistical properties but, numerically, it scales poorly with the sample size. Our analysis shows that Nystr\"om sampling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supervised learning, this is the first such result for PCA. Our theoretical findings, which are also illustrated by numerical results, are based on a combination of analytic and concentration of measure techniques. Our study is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning.

Multi-Scale Vector Quantization with Reconstruction Trees

We propose and study a multi-scale approach to vector quantization. We develop an algorithm, dubbed reconstruction trees, inspired by decision trees. Here the objective is parsimonious reconstruction of unsupervised data, rather than classification. Contrasted to more standard vector quantization methods, such as K-means, the proposed approach leverages a family of given partitions, to quickly explore the data in a coarse to fine-- multi-scale-- fashion. Our main technical contribution is an analysis of the expected distortion achieved by the proposed algorithm, when the data are assumed to be sampled from a fixed unknown distribution. In this context, we derive both asymptotic and finite sample results under suitable regularity assumptions on the distribution. As a special case, we consider the setting where the data generating distribution is supported on a compact Riemannian sub-manifold. Tools from differential geometry and concentration of measure are useful in our analysis.

Implicit Regularization of Accelerated Methods in Hilbert Spaces

We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster bias decay than gradient descent, but also suffers of a more unstable behavior. As a result acceleration cannot be in general expected to improve learning accuracy with respect to gradient descent, but rather to achieve the same accuracy with reduced computations. Our theoretical results are validated by numerical simulations. Our analysis is based on studying suitable polynomials induced by the accelerated dynamics and combining spectral techniques with concentration inequalities.

Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples.

Theory III: Dynamics and Generalization in Deep Networks - a simple solution

We review recent observations on the dynamical systems induced by gradient descent (GD) methods used for training deep networks and summarize properties of the solutions they converge to. Recent results illuminate the absence of overfitting in the special case of linear networks for binary classification. They prove that minimization of loss functions such as the logistic, the cross-entropy and the exponential loss yields asymptotic convergence to the maximum margin solution for linearly separable datasets, independently of the initial conditions. Here we discuss the case of nonlinear DNNs near zero minima of the empirical loss, under exponential-type and square losses, for several variations of the basic GD algorithm, including a new NMGD version that converges to the minimum norm fixed points. Our main results are: 1) GD algorithms with weight normalization constraint achieve generalization; 2) the fundamental reason for the effectiveness of existing weight and batch normalization techniques is that they are approximate implementations of maximizing the margin under unit norm constraint; 3) even without explicit unit norm constraints, generalization can still be obtained for not-too-deep networks because standard GD is intrinsically consistent with the dynamics of normalized weights. In addition, the balance of the weights across different layers, if present at initialization, is maintained by the gradient flow. In the perspective of these theoretical results, we discuss experimental evidence around the apparent absence of overfitting, that is the observation that the expected classification error does not get worse when increasing the number of parameters. Our explanation focuses on the implicit normalization enforced by algorithms such as batch normalization. In particular, the control of the norm of the weights is related to Halpern iterations for minimum norm solutions.

Gaussian Process Optimization with Adaptive Sketching: Scalable and No Regret

Gaussian processes (GP) are a popular Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to complex high-dimensional functions, as their per-iteration time and space cost is at least quadratic in the number of dimensions $d$ and iterations $t$. Given a set of $A$ alternative to choose from, the overall runtime $O(t^3A)$ quickly becomes prohibitive. In this paper, we introduce BKB (budgeted kernelized bandit), a novel approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and no assumption on the input space or covariance of the GP. Combining a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching technique (i.e., leverage score sampling), we prove that selecting inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most $\tilde{O}(d_{eff})$ points, where $d_{eff}$ is the effective dimension of the explored space, which is typically much smaller than both $d$ and $t$. This greatly reduces the dimensionality of the problem, thus leading to a $O(TAd_{eff}^2)$ runtime and $O(A d_{eff})$ space complexity.

Beating SGD Saturation with Tail-Averaging and Minibatching

While stochastic gradient descent (SGD) is one of the major workhorses in machine learning, the learning properties of many practically used variants are poorly understood. In this paper, we consider least squares learning in a nonparametric setting and contribute to filling this gap by focusing on the effect and interplay of multiple passes, mini-batching and averaging, and in particular tail averaging. Our results show how these different variants of SGD can be combined to achieve optimal learning errors, hence providing practical insights. In particular, we show for the first time in the literature that tail averaging allows faster convergence rates than uniform averaging in the nonparametric setting. Finally, we show that a combination of tail-averaging and minibatching allows more aggressive step-size choices than using any one of said components.

A computational model for grid maps in neural populations

Grid cells in the entorhinal cortex, together with place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this contribution we introduce a novel theoretical and algorithmic framework able to explain the emergence of hexagonal grid-like response patterns from the statistics of the input stimuli. We show that this pattern is a result of minimal variance encoding of neurons. The novelty lies into the formulation of the encoding problem through the modern Frame Theory language, specifically that of equiangular Frames, providing new insights about the optimality of hexagonal grid receptive fields. The model proposed overcomes some crucial limitations of the current attractor and oscillatory models. It is based on the well-accepted and tested hypothesis of Hebbian learning, providing a simplified cortical-based framework that does not require the presence of theta velocity-driven oscillations (oscillatory model) or translational symmetries in the synaptic connections (attractor model). We moreover demonstrate that the proposed encoding mechanism naturally maps shifts, rotations and scaling of the stimuli onto the shape of grid cells' receptive fields, giving a straightforward explanation of the experimental evidence of grid cells remapping under transformations of environmental cues.

Are we done with object recognition? The iCub robot's perspective

We report on an extensive study of the benefits and limitations of current deep learning approaches to object recognition in robot vision scenarios, introducing a novel dataset used for our investigation. To avoid the biases in currently available datasets, we consider a natural human-robot interaction setting to design a data-acquisition protocol for visual object recognition on the iCub humanoid robot. Analyzing the performance of off-the-shelf models trained off-line on large-scale image retrieval datasets, we show the necessity for knowledge transfer. We evaluate different ways in which this last step can be done, and identify the major bottlenecks affecting robotic scenarios. By studying both object categorization and identification problems, we highlight key differences between object recognition in robotics applications and in image retrieval tasks, for which the considered deep learning approaches have been originally designed. In a nutshell, our results confirm the remarkable improvements yield by deep learning in this setting, while pointing to specific open challenges that need be addressed for seamless deployment in robotics.