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We consider an underdetermined noisy linear regression model where the minimum-norm interpolating predictor is known to be consistent, and ask: can uniform convergence in a norm ball, or at least (following Nagarajan and Kolter) the subset of a norm ball that the algorithm selects on a typical input set, explain this success? We show that uniformly bounding the difference between empirical and population errors cannot show any learning in the norm ball, and cannot show consistency for any set, even one depending on the exact algorithm and distribution. But we argue we can explain the consistency of the minimal-norm interpolator with a slightly weaker, yet standard, notion: uniform convergence of zero-error predictors in a norm ball. We use this to bound the generalization error of low- (but not minimal-) norm interpolating predictors.
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We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the classs Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum l1-norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings.
Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive efficient large-scale learning algorithms. In this study, we consider solving a binary classification problem using random features and stochastic gradient descent. In recent research, an exponential convergence rate of the expected classification error under the strong low-noise condition has been shown. We extend these analyses to a random features setting, analyzing the error induced by the approximation of random features in terms of the distance between the generated hypothesis including population risk minimizers and empirical risk minimizers when using general Lipschitz loss functions, to show that an exponential convergence of the expected classification error is achieved even if random features approximation is applied. Additionally, we demonstrate that the convergence rate does not depend on the number of features and there is a significant computational benefit in using random features in classification problems because of the strong low-noise condition.
In this paper, we consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion, and tensor PCA/SVD. We propose a Riemannian Gauss-Newton (RGN) method with fast implementations for low Tucker rank tensor estimation. Different from the generic (super)linear convergence guarantee of RGN in the literature, we prove the first quadratic convergence guarantee of RGN for low-rank tensor estimation under some mild conditions. A deterministic estimation error lower bound, which matches the upper bound, is provided that demonstrates the statistical optimality of RGN. The merit of RGN is illustrated through two machine learning applications: tensor regression and tensor SVD. Finally, we provide the simulation results to corroborate our theoretical findings.
Although ordinary differential equations (ODEs) provide insights for designing network architectures, its relationship with the non-residual convolutional neural networks (CNNs) is still unclear. In this paper, we present a novel ODE model by adding a damping term. It can be shown that the proposed model can recover both a ResNet and a CNN by adjusting an interpolation coefficient. Therefore, the damped ODE model provides a unified framework for the interpretation of residual and non-residual networks. The Lyapunov analysis reveals better stability of the proposed model, and thus yields robustness improvement of the learned networks. Experiments on a number of image classification benchmarks show that the proposed model substantially improves the accuracy of ResNet and ResNeXt over the perturbed inputs from both stochastic noise and adversarial attack methods. Moreover, the loss landscape analysis demonstrates the improved robustness of our method along the attack direction.
In our recent paper, we showed that in exponential family, contrastive divergence (CD) with fixed learning rate will give asymptotically consistent estimates cite{wu2016convergence}. In this paper, we establish consistency and convergence rate of CD with annealed learning rate $eta_t$. Specifically, suppose CD-$m$ generates the sequence of parameters ${theta_t}_{t ge 0}$ using an i.i.d. data sample $mathbf{X}_1^n sim p_{theta^*}$ of size $n$, then $delta_n(mathbf{X}_1^n) = limsup_{t to infty} Vert sum_{s=t_0}^t eta_s theta_s / sum_{s=t_0}^t eta_s - theta^* Vert$ converges in probability to 0 at a rate of $1/sqrt[3]{n}$. The number ($m$) of MCMC transitions in CD only affects the coefficient factor of convergence rate. Our proof is not a simple extension of the one in cite{wu2016convergence}. which depends critically on the fact that ${theta_t}_{t ge 0}$ is a homogeneous Markov chain conditional on the observed sample $mathbf{X}_1^n$. Under annealed learning rate, the homogeneous Markov property is not available and we have to develop an alternative approach based on super-martingales. Experiment results of CD on a fully-visible $2times 2$ Boltzmann Machine are provided to demonstrate our theoretical results.
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