Sara van de Geer

Sylvain Sardy, Maxime van Cutsem, Sara van de Geer

The Bayesian and Akaike information criteria aim at finding a good balance between under- and over-fitting. They are extensively used every day by practitioners. Yet we contend they suffer from at least two afflictions: their penalty parameter $λ=\log n$ and $λ=2$ are too small, leading to many false discoveries, and their inherent (best subset) discrete optimization is infeasible in high dimension. We alleviate these issues with the pivotal information criterion: PIC is defined as a continuous optimization problem, and the PIC penalty parameter $λ$ is selected at the detection boundary (under pure noise). PIC's choice of $λ$ is the quantile of a statistic that we prove to be (asymptotically) pivotal, provided the loss function is appropriately transformed. As a result, simulations show a phase transition in the probability of exact support recovery with PIC, a phenomenon studied with no noise in compressed sensing. Applied on real data, for similar predictive performances, PIC selects the least complex model among state-of-the-art learners.

Geoffrey Chinot, Felix Kuchelmeister, Matthias Löffler et al.

This paper studies binary classification in robust one-bit compressed sensing with adversarial errors. It is assumed that the model is overparameterized and that the parameter of interest is effectively sparse. AdaBoost is considered, and, through its relation to the max-$\ell_1$-margin-classifier, prediction error bounds are derived. The developed theory is general and allows for heavy-tailed feature distributions, requiring only a weak moment assumption and an anti-concentration condition. Improved convergence rates are shown when the features satisfy a small deviation lower bound. In particular, the results provide an explanation why interpolating adversarial noise can be harmless for classification problems. Simulations illustrate the presented theory.

Geoffrey Chinot, Matthias Löffler, Sara van de Geer

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.

Francesco Ortelli, Sara van de Geer

We study the theoretical properties of image denoising via total variation penalized least-squares. We define the total vatiation in terms of the two-dimensional total discrete derivative of the image and show that it gives rise to denoised images that are piecewise constant on rectangular sets. We prove that, if the true image is piecewise constant on just a few rectangular sets, the denoised image converges to the true image at a parametric rate, up to a log factor. More generally, we show that the denoised image enjoys oracle properties, that is, it is almost as good as if some aspects of the true image were known. In other words, image denoising with total variation regularization leads to an adaptive reconstruction of the true image.

Francesco Ortelli, Sara van de Geer

We establish adaptive results for trend filtering: least squares estimation with a penalty on the total variation of $(k-1)^{\rm th}$ order differences. Our approach is based on combining a general oracle inequality for the $\ell_1$-penalized least squares estimator with "interpolating vectors" to upper-bound the "effective sparsity". This allows one to show that the $\ell_1$-penalty on the $k^{\text{th}}$ order differences leads to an estimator that can adapt to the number of jumps in the $(k-1)^{\text{th}}$ order differences of the underlying signal or an approximation thereof. We show the result for $k \in \{1,2,3,4\}$ and indicate how it could be derived for general $k\in \mathbb{N}$.

Francesco Ortelli, Sara van de Geer

Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularized estimators on graphs and obtain constant-friendly rates, which, up to log-terms, match previous results obtained by entropy calculations. We also obtain an oracle inequality for the (square root) total variation regularized estimator over the cycle graph.

Peter Hinz, Sara van de Geer

We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width can induce on an input region with a given dimensionality. The final bound on the number of regions depends on the number and widths of the layers of the neural network and on some additional parameters that were used for the recursion. It is stated in terms of the $L1$-norm of the last column of a product of matrices and provides a unifying treatment of several previously known bounds: Depending on the choice of the recursion parameters that determine these matrices, it is possible to obtain the bounds from Montúfar (2014), (2017) and Serra et. al. (2017) as special cases. For the latter, which is the strongest of these bounds, the formulation in terms of matrices provides new insight. In particular, by using explicit formulas for a Jordan-like decomposition of the involved matrices, we achieve new tighter results for the asymptotic setting, where the number of layers of the same fixed width tends to infinity.

Francesco Ortelli, Sara van de Geer

We generalize to tree graphs obtained by connecting path graphs an oracle result obtained for the Fused Lasso over the path graph. Moreover we show that it is possible to substitute in the oracle inequality the minimum of the distances between jumps by their harmonic mean. In doing so we prove a lower bound on the compatibility constant for the total variation penalty. Our analysis leverages insights obtained for the path graph with one branch to understand the case of more general tree graphs. As a side result, we get insights into the irrepresentable condition for such tree graphs.

Benjamin Stucky, Sara van de Geer

In the setting of high-dimensional linear regression models, we propose two frameworks for constructing pointwise and group confidence sets for penalized estimators which incorporate prior knowledge about the organization of the non-zero coefficients. This is done by desparsifying the estimator as in van de Geer et al. [18] and van de Geer and Stucky [17], then using an appropriate estimator for the precision matrix $Θ$. In order to estimate the precision matrix a corresponding structured matrix norm penalty has to be introduced. After normalization the result is an asymptotic pivot. The asymptotic behavior is studied and simulations are added to study the differences between the two schemes.