The following proof is from Daniely and V ardi [15], and we give it here for completeness. By Lemma A.1, there exists a DNF formula We construct such an affine layer in Lemma A.2. At least one of the k size-n slices in z contains 0 more than once. We define the outputs of our affine layer as follows. Pr [z represents a hyperedge ] = n (n 1) ... (n k + 1) null 1 n null Pr null z Z null 1 2 log(n) .

Gaussian, and the weight matrix is non-degenerate.

Uncertainty quantification (UQ) is a crucial but challenging task in many high-dimensional learning problems to increase the confidence of a given predictor. We develop a new data-driven approach for UQ in regression that applies both to classical optimization approaches such as the LASSO as well as to neural networks. One of the most notable UQ techniques is the debiased LASSO, which modifies the LASSO to allow for the construction of asymptotic confidence intervals by decomposing the estimation error into a Gaussian and an asymptotically vanishing bias component. However, in real-world problems with finite-dimensional data, the bias term is often too significant to disregard, resulting in overly narrow confidence intervals. Our work rigorously addresses this issue and derives a data-driven adjustment that corrects the confidence intervals for a large class of predictors by estimating the means and variances of the bias terms from training data, exploiting high-dimensional concentration phenomena. This gives rise to non-asymptotic confidence intervals, which can help avoid overestimating certainty in critical applications such as MRI diagnosis. Importantly, our analysis extends beyond sparse regression to data-driven predictors like neural networks, enhancing the reliability of model-based deep learning. Our findings bridge the gap between established theory and the practical applicability of such methods.

Controlling the parameters' norm often yields good generalisation when training neural networks. Beyond simple intuitions, the relation between regularising parameters' norm and obtained estimators remains theoretically misunderstood. For one hidden ReLU layer networks with unidimensional data, this work shows the parameters' norm required to represent a function is given by the total variation of its second derivative, weighted by a $\sqrt{1+x^2}$ factor. Notably, this weighting factor disappears when the norm of bias terms is not regularised. The presence of this additional weighting factor is of utmost significance as it is shown to enforce the uniqueness and sparsity (in the number of kinks) of the minimal norm interpolator. Conversely, omitting the bias' norm allows for non-sparse solutions.Penalising the bias terms in the regularisation, either explicitly or implicitly, thus leads to sparse estimators.

We study the relationship between the frequency of a function and the speed at which a neural network learns it. We build on recent results that show that the dynamics of overparameterized neural networks trained with gradient descent can be well approximated by a linear system. When normalized training data is uniformly distributed on a hypersphere, the eigenfunctions of this linear system are spherical harmonic functions. We derive the corresponding eigenvalues for each frequency after introducing a bias term in the model. This bias term had been omitted from the linear network model without significantly affecting previous theoretical results. However, we show theoretically and experimentally that a shallow neural network without bias cannot represent or learn simple, low frequency functions with odd frequencies. Our results lead to specific predictions of the time it will take a network to learn functions of varying frequency. These predictions match the empirical behavior of both shallow and deep networks.

Contrastive learning is among the most popular and powerful approaches for self-supervised representation learning, where the goal is to map semantically similar samples close together while separating dissimilar ones in the latent space. Existing theoretical methods assume that downstream task classes are drawn from the same latent class distribution used during the pretraining phase. However, in real-world settings, downstream tasks may not only exhibit distributional shifts within the same label space but also introduce new or broader label spaces, leading to domain generalization challenges. In this work, we introduce novel generalization bounds that explicitly account for both types of mismatch: domain shift and domain generalization. Specifically, we analyze scenarios where downstream tasks either (i) draw classes from the same latent class space but with shifted distributions, or (ii) involve new label spaces beyond those seen during pretraining. Our analysis reveals how the performance of contrastively learned representations depends on the statistical discrepancy between pretraining and downstream distributions. This extended perspective allows us to derive provable guarantees on the performance of learned representations on average classification tasks involving class distributions outside the pretraining latent class set.

In particular, we conjecture and empirically illustrate that, the celebrated batch normalization (BN) technique actually adapts the "normalized" bias such that it approximates

Modern machine learning models with a large number of parameters often generalize well despite perfectly interpolating noisy training data - a phenomenon known as benign overfitting. A foundational explanation for this in linear classification was recently provided by Hashimoto et al. (2025). However, this analysis was limited to the setting of "homogeneous" models, which lack a bias (intercept) term - a standard component in practice. This work directly extends Hashimoto et al.'s results to the more realistic inhomogeneous case, which incorporates a bias term. Our analysis proves that benign overfitting persists in these more complex models. We find that the presence of the bias term introduces new constraints on the data's covariance structure required for generalization, an effect that is particularly pronounced when label noise is present. However, we show that in the isotropic case, these new constraints are dominated by the requirements inherited from the homogeneous model. This work provides a more complete picture of benign overfitting, revealing the non-trivial impact of the bias term on the conditions required for good generalization.