This article develops a general theory for minimum norm interpolating estimators and regularized empirical risk minimizers (RERM) in linear models in the presence of additive, potentially adversarial, errors. In particular, no conditions on the errors are imposed. A quantitative bound for the prediction error is given, relating it to the Rademacher complexity of the covariates, the norm of the minimum norm interpolator of the errors and the size of the subdifferential around the true parameter. The general theory is illustrated for Gaussian features and several norms: The $\ell_1$, $\ell_2$, group Lasso and nuclear norms. In case of sparsity or low-rank inducing norms, minimum norm interpolators and RERM yield a prediction error of the order of the average noise level, provided that the overparameterization is at least a logarithmic factor larger than the number of samples and that, in case of RERM, the regularization parameter is small enough. Lower bounds that show near optimality of the results complement the analysis.
This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample s…
This paper is concerned with the factorization form of the rank regularized loss minimization problem. To cater for the scenario in which only a coarse estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm regularized term is added to the factored loss function to reduce the rank adaptively; and account for the ambiguities in the factorization, a balanced term is then introduced. For the least squares loss, under a restricted condition number assumption on the sampling operator, we establish the KL property of exponent $1/2$ of the nonsmooth factored composite function …
Matrix completion has attracted much interest in the past decade in machine learning and computer vision. For low-rank promotion in matrix completion, the nuclear norm penalty is convenient due to its convexity but has a bias problem. Recently, various algorithms using nonconvex penalties have been proposed, among which the proximal gradient descent (PGD) algorithm is one of the most efficient and effective. For the nonconvex PGD algorithm, whether it converges to a local minimizer and its convergence rate are still unclear. This work provides a nontrivial analysis on the PGD algorithm in the …
We establish a new concentration result for regularized risk minimizers which is similar to an oracle inequality. Applying this inequality to regularized least squares minimizers like least squares support vector machines, we show that these algorithms learn with (almost) the optimal rate in some specific situations. In addition, for regression our results suggest that using the loss function $L_α(y,t)=|y-t|^α$ with $α$ near 1 may often be preferable to the usual choice of $α=2$.
We consider the unconstrained $L_2$-$L_p$ minimization: find a minimizer of $\|Ax-b\|^2_2+λ\|x\|^p_p$ for given $A \in R^{m\times n}$, $b\in R^m$ and parameters $λ>0$, $p\in [0,1)$. This problem has been studied extensively in variable selection and sparse least squares fitting for high dimensional data. Theoretical results show that the minimizers of the $L_2$-$L_p$ problem have various attractive features due to the concavity and non-Lipschitzian property of the regularization function $\|\cdot\|^p_p$. In this paper, we show that the $L_q$-$L_p$ minimization problem is strongly NP-hard for a…
We propose an AdaPtive Noise Augmentation (PANDA) technique to regularize the estimation and construction of undirected graphical models. PANDA iteratively optimizes the objective function given the noise augmented data until convergence to achieve regularization on model parameters. The augmented noises can be designed to achieve various regularization effects on graph estimation, such as the bridge (including lasso and ridge), elastic net, adaptive lasso, and SCAD penalization; it also realizes the group lasso and fused ridge. We examine the tail bound of the noise-augmented loss function an…
We establish risk bounds for Regularized Empirical Risk Minimizers (RERM) when the loss is Lipschitz and convex and the regularization function is a norm. In a first part, we obtain these results in the i.i.d. setup under subgaussian assumptions on the design. In a second part, a more general framework where the design might have heavier tails and data may be corrupted by outliers both in the design and the response variables is considered. In this situation, RERM performs poorly in general. We analyse an alternative procedure based on median-of-means principles and called minmax MOM. We show …
Rates of convergence for empirical risk minimizers have been well studied in the literature. In this paper, we aim to provide a complementary set of results, in particular by showing that after normalization, the risk of the empirical minimizer concentrates on a single point. Such results have been established by~\cite{chatterjee2014new} for constrained estimators in the normal sequence model. We first generalize and sharpen this result to regularized least squares with convex penalties, making use of a "direct" argument based on Borell's theorem. We then study generalizations to other loss fu…
We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and $L$-Lipschitz loss functions. We consider a setting where $|\cO|$ malicious outliers contaminate the labels. In that case, under a local Bernstein condition, we show that the $L_2$-error rate is bounded by $ r_N + AL |\cO|/N$, where $N$ is the total number of observations, $r_N$ is the $L_2$-error rate in the non-contaminated setting and $A$ is a parameter coming from the local Bernstein condition. When $r_N$ is minimax-rate-optimal in a non-contaminated setting, th…
Generalization error of min-norm interpolators in transfer learning
Department of Statistics, Harvard University（哈佛大学统计系）；Department of Statistics, Stanford University（斯坦福大学统计系）；Department of Statistics, University of Florida（佛罗里达大学统计系）
We study Regularized Empirical Risk Minimizers (RERM) and minmax Median-Of-Means (MOM) estimators where the regularization function $ϕ(\cdot)$ is an even convex function. We obtain bounds on the $L_2$-estimation error and the excess risk that depend on $ϕ(f^*)$, where $f^*$ is the minimizer of the risk over a class $F$. The estimators are based on loss functions that are both Lipschitz and convex. Results for the RERM are derived under weak assumptions on the outputs and a sub-Gaussian assumption on the class $\{ (f-f^*)(X), f \in F \}$. Similar results are shown for minmax MOM estimators in a…
We prove an L2 recovery bound for a family of sparse estimators defined as minimizers of some empirical loss functions -- which include hinge loss and logistic loss. More precisely, we achieve an upper-bound for coefficients estimation scaling as (k*/n)\log(p/k*): n,p is the size of the design matrix and k* the dimension of the theoretical loss minimizer. This is done under standard assumptions, for which we derive stronger versions of a cone condition and a restricted strong convexity. Our bound holds with high probability and in expectation and applies to an L1-regularized estimator and to a…
