Stochastic Gradient Descent and its variants have become a staple of the algorithmic foundations of machine learning. Yet many of its properties are not fully understood, particularly in non-convex settings common in modern practice. In this note, we study convergence of Stochastic Gradient Descent (SGD) for the class of functions satisfying the Polyak-Lojasiewicz (PL) condition. This class contains all strongly-convex functions as well as a broad range of non-convex functions including those used in machine learning applications (see the discussion below). The primary purpose of this note is to show that in the interpolation setting (common in modern overparametrized machine learning and studied in our previous work [8]) SGD with fixed step size has exponential convergence for the functions satisfying the PL condition. To the best of our knowledge, this is the first such exponential convergence result for a class of non-convex functions. Below, we discuss and highlight a number of aspects of the PL condition which differentiate it from the convex setting and make it more relevant to the practice and requirements of many machine learning problems.

We apply a fast kernel method for mask-based single-channel speech enhancement. Specifically, our method solves a kernel regression problem associated to a non-smooth kernel function (exponential power kernel) with a highly efficient iterative method (EigenPro). Due to the simplicity of this method, its hyper-parameters such as kernel bandwidth can be automatically and efficiently selected using line search with subsamples of training data. We observe an empirical correlation between the regression loss (mean square error) and regular metrics for speech enhancement. This observation justifies our training target and motivates us to achieve lower regression loss by training separate kernel model per frequency subband. We compare our method with the state-of-the-art deep neural networks on mask-based HINT and TIMIT. Experimental results show that our kernel method consistently outperforms deep neural networks while requiring less training time.

Memorization of data in deep neural networks has become a subject of significant research interest. In this paper, we link memorization of images in deep convolutional autoencoders to downsampling through strided convolution. To analyze this mechanism in a simpler setting, we train linear convolutional autoencoders and show that linear combinations of training data are stored as eigenvectors in the linear operator corresponding to the network when downsampling is used. On the other hand, networks without downsampling do not memorize training data. We provide further evidence that the same effect happens in nonlinear networks. Moreover, downsampling in nonlinear networks causes the model to not only memorize linear combinations of images, but individual training images. Since convolutional autoencoder components are building blocks of deep convolutional networks, we envision that our findings will shed light on the important phenomenon of memorization in over-parameterized deep networks.

Many modern machine learning models are trained to achieve zero or near-zero training error in order to obtain near-optimal (but non-zero) test error. This phenomenon of strong generalization performance for "overfitted" / interpolated classifiers appears to be ubiquitous in high-dimensional data, having been observed in deep networks, kernel machines, boosting and random forests. Their performance is robust even when the data contain large amounts of label noise. Very little theory is available to explain these observations. The vast majority of theoretical analyses of generalization allows for interpolation only when there is little or no label noise. This paper takes a step toward a theoretical foundation for interpolated classifiers by analyzing local interpolating schemes, including geometric simplicial interpolation algorithm and weighted $k$-nearest neighbor schemes. Consistency or near-consistency is proved for these schemes in classification and regression problems. These schemes have an inductive bias that benefits from higher dimension, a kind of "blessing of dimensionality". Finally, connections to kernel machines, random forests, and adversarial examples in the interpolated regime are discussed.

We show that learning methods interpolating the training data can achieve optimal rates for the problems of nonparametric regression and prediction with square loss.

In recent years machine learning methods that nearly interpolate the data have achieved remarkable success. In many settings achieving near-zero training error leads to excellent test results. In this work we show how the mathematical and conceptual simplicity of interpolation can be harnessed to construct a framework for very efficient, scalable and accurate kernel machines. Our main innovation is in constructing kernel machines that output solutions mathematically equivalent to those obtained using standard kernels, yet capable of fully utilizing the available computing power of a parallel computational resource, such as GPU. Such utilization is key to strong performance since much of the computational resource capability is wasted by the standard iterative methods. The computational resource and data adaptivity of our learned kernels is based on theoretical convergence bounds. The resulting algorithm, which we call EigenPro 2.0, is accurate, principled and very fast. For example, using a single GPU, training on ImageNet with $1.3\times 10^6$ data points and $1000$ labels takes under an hour, while smaller datasets, such as MNIST, take seconds. Moreover, as the parameters are chosen analytically, based on the theory, little tuning beyond selecting the kernel and kernel parameter is needed, further facilitating the practical use of these methods.

An interactive image retrieval system learns which images in the database belong to a user's query concept, by analyzing the example images and feedback provided by the user. The challenge is to retrieve the relevant images with minimal user interaction. In this work, we propose to solve this problem by posing it as a binary classification task of classifying all images in the database as being relevant or irrelevant to the user's query concept. Our method combines active learning with graph-based semi-supervised learning (GSSL) to tackle this problem. Active learning reduces the number of user interactions by querying the labels of the most informative points and GSSL allows to use abundant unlabeled data along with the limited labeled data provided by the user. To efficiently find the most informative point, we use an uncertainty sampling based method that queries the label of the point nearest to the decision boundary of the classifier. We estimate this decision boundary using our heuristic of adaptive threshold. To utilize huge volumes of unlabeled data we use an efficient approximation based method that reduces the complexity of GSSL from $O(n^3)$ to $O(n)$, making GSSL scalable. We make the classifier robust to the diversity and noisy labels associated with images in large databases by incorporating information from multiple modalities such as visual information extracted from deep learning based models and semantic information extracted from the WordNet. High F1 scores within few relevance feedback rounds in our experiments with concepts defined on AnimalWithAttributes and Imagenet (1.2 million images) datasets indicate the effectiveness and scalability of our approach.

Generalization performance of classifiers in deep learning has recently become a subject of intense study. Heavily over-parametrized deep models tend to fit training data exactly. Despite overfitting, they perform well on test data, a phenomenon not yet fully understood. The first point of our paper is that strong performance of overfitted classifiers is not a unique feature of deep learning. Using real-world and synthetic datasets, we establish that kernel classifiers trained to have zero classification error (overfitting) or even zero regression error (interpolation) perform very well on test data. We proceed to prove lower bounds on the norm of overfitted solutions for smooth kernels, showing that they increase nearly exponentially with the data size. Since most generalization bounds depend polynomially on the norm of the solution, this result implies that they diverge as data increases. Furthermore, the existing bounds do not apply to interpolated classifiers. We also show experimentally that (non-smooth) Laplacian kernels easily fit random labels using a version of SGD, a finding that parallels results reported for ReLU neural networks. In contrast, fitting noisy data requires many more epochs for smooth Gaussian kernels. The observation that the performance of overfitted Laplacian and Gaussian classifiers on the test is quite similar, suggests that generalization is tied to the properties of the kernel function rather than the optimization process. We see that some key phenomena of deep learning are manifested similarly in kernel methods in the overfitted regime. We argue that progress on understanding deep learning will be difficult, until more analytically tractable "shallow" kernel methods are better understood. The experimental and theoretical results presented in this paper indicate a need for new theoretical ideas for understanding classical kernel methods.

Stochastic Gradient Descent (SGD) with small mini-batch is a key component in modern large-scale learning. However, its efficiency has not been easy to analyze as most theoretical results require adaptive rates and show convergence rates far slower than that for gradient descent, making computational comparisons difficult. In this paper we aim to clarify the issue of fast SGD convergence. The key observation is that most modern architectures are over-parametrized and are trained to interpolate the data by driving the empirical loss close to zero. While it is still unclear why these interpolated solutions perform well on test data, we show that these regimes allow for very fast convergence of SGD, comparable in the number of iterations to full gradient descent. Specifically, consider the setting with a quadratic objective function, or a general objective function in the proximity of a minimum, where the quadratic term is dominant. First, we obtain the optimal convergence rate for any mini-batch SGD and derive the optimal step size as a function of the batch size $m$. Second, we show: (1) $m=1$ is optimal in terms of number of computations required to achieve any desired accuracy. (2) There is a critical mini-batch size $m^*$ such that: (2a) SGD iteration with batch size $m\leq m^*$ is nearly equivalent to $m$ iterations of batch size $1$. (2b) SGD iteration with mini-batch $m> m^*$ is nearly equivalent to a full gradient descent iteration. The critical mini-batch size can be viewed as the limit for effective mini-batch parallelization. It is also nearly independent of the data size, implying $O(n)$ acceleration over GD per unit of computation. These theoretical analyses are verified by experimental results. Finally, we show how the interpolation perspective and our results fit in the deep neural networks and discuss connections to adaptive rates for SGD and variance reduction.

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential properties. We analyze eigenvalue decay of kernels operators and matrices, properties of eigenfunctions/eigenvectors and "Fourier" coefficients of functions in the kernel space restricted to a discrete set of data points. We also investigate the fitting capacity of kernels, giving explicit bounds on the fat shattering dimension of the balls in Reproducing Kernel Hilbert spaces. Interestingly, the same properties that make kernels very effective approximators for functions in their "native" kernel space, also limit their capacity to represent arbitrary functions. We discuss various implications, including those for gradient descent type methods. It is important to note that most of our bounds are measure independent. Moreover, at least in moderate dimension, the bounds for eigenvalues are much tighter than the bounds which can be obtained from the usual matrix concentration results. For example, we see that the eigenvalues of kernel matrices show nearly exponential decay with constants depending only on the kernel and the domain. We call this "approximation beats concentration" phenomenon as even when the data are sampled from a probability distribution, some of their aspects are better understood in terms of approximation theory.