اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدUniform Convergence of Interpolators: Gaussian Width, Norm Bounds, and Benign Overfitting
440
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
Deep neural networks generalize well despite being exceedingly overparameterized and being trained without explicit regularization. This curious phenomenon has inspired extensive research activity in establishing its statistical principles: Under wha
t conditions is it observed? How do these depend on the data and on the training algorithm? When does regularization benefit generalization? While such questions remain wide open for deep neural nets, recent works have attempted gaining insights by studying simpler, often linear, models. Our paper contributes to this growing line of work by examining binary linear classification under a generative Gaussian mixture model. Motivated by recent results on the implicit bias of gradient descent, we study both max-margin SVM classifiers (corresponding to logistic loss) and min-norm interpolating classifiers (corresponding to least-squares loss). First, we leverage an idea introduced in [V. Muthukumar et al., arXiv:2005.08054, (2020)] to relate the SVM solution to the min-norm interpolating solution. Second, we derive novel non-asymptotic bounds on the classification error of the latter. Combining the two, we present novel sufficient conditions on the covariance spectrum and on the signal-to-noise ratio (SNR) under which interpolating estimators achieve asymptotically optimal performance as overparameterization increases. Interestingly, our results extend to a noisy model with constant probability noise flips. Contrary to previously studied discriminative data models, our results emphasize the crucial role of the SNR and its interplay with the data covariance. Finally, via a combination of analytical arguments and numerical demonstrations we identify conditions under which the interpolating estimator performs better than corresponding regularized estimates.
Modern machine learning often operates in the regime where the number of parameters is much higher than the number of data points, with zero training loss and yet good generalization, thereby contradicting the classical bias-variance trade-off. This
textit{benign overfitting} phenomenon has recently been characterized using so called textit{double descent} curves where the risk undergoes another descent (in addition to the classical U-shaped learning curve when the number of parameters is small) as we increase the number of parameters beyond a certain threshold. In this paper, we examine the conditions under which textit{Benign Overfitting} occurs in the random feature (RF) models, i.e. in a two-layer neural network with fixed first layer weights. We adopt a new view of random feature and show that textit{benign overfitting} arises due to the noise which resides in such features (the noise may already be present in the data and propagate to the features or it may be added by the user to the features directly) and plays an important implicit regularization role in the phenomenon.
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.
We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in the aim set by Bartlett et al. (2015): obtain regret bounds that hold uniformly over all competitor vectors. When th
e feature sequence is known at the beginning of the game, they provided closed-form regret bounds of $2d B^2 ln T + mathcal{O}_T(1)$, where $T$ is the number of rounds and $B$ is a bound on the observations. Instead, we derive bounds with an optimal constant of $1$ in front of the $d B^2 ln T$ term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of $d B^2 ln T$ for any individual sequence of features and bounded observations. All our algorithms are variants of the online non-linear ridge regression forecaster, either with a data-dependent regularization or with almost no regularization.
The growing literature on benign overfitting in overparameterized models has been mostly restricted to regression or binary classification settings; however, most success stories of modern machine learning have been recorded in multiclass settings. M
otivated by this discrepancy, we study benign overfitting in multiclass linear classification. Specifically, we consider the following popular training algorithms on separable data: (i) empirical risk minimization (ERM) with cross-entropy loss, which converges to the multiclass support vector machine (SVM) solution; (ii) ERM with least-squares loss, which converges to the min-norm interpolating (MNI) solution; and, (iii) the one-vs-all SVM classifier. First, we provide a simple sufficient condition under which all three algorithms lead to classifiers that interpolate the training data and have equal accuracy. When the data is generated from Gaussian mixtures or a multinomial logistic model, this condition holds under high enough effective overparameterization. Second, we derive novel error bounds on the accuracy of the MNI classifier, thereby showing that all three training algorithms lead to benign overfitting under sufficient overparameterization. Ultimately, our analysis shows that good generalization is possible for SVM solutions beyond the realm in which typical margin-based bounds apply.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
