Alvaro Cartea

Jonathan Plenk, Sergio Calvo-Ordonez, Alvaro Cartea et al.

In wide neural networks, the Neural Tangent Kernel (NTK) remains approximately constant during training, providing a powerful theoretical tool for studying training dynamics, generalization, and connections to kernel methods. However, this theory is largely restricted to regression losses. It was previously thought that training on a classification loss, or more generally losses involving nonlinear output transformations, breaks this property, leading to divergent logits and a breakdown of the linearization. In this paper, we extend NTK theory to classification by identifying conditions under which wide neural networks remain in the lazy training regime. We show that parameter-space regularization ensures a constant NTK during training for cross-entropy loss, while in the absence of regularization the regime is recovered when targets are non-degenerate, i.e. when all classes have strictly positive probability. Under these conditions, training is well-approximated by the linearized model, yielding an explicit characterization of the solution in terms of the NTK. We further analyze the distribution of trained predictors induced by random initialization and relate this notion of model uncertainty to Bayesian methods.

Richard Bergna, Stefan Depeweg, Sergio Calvo-Ordoñez et al.

Reliable uncertainty estimates are crucial for deploying pretrained models; yet, many strong methods for quantifying uncertainty require retraining, Monte Carlo sampling, or expensive second-order computations and may alter a frozen backbone's predictions. To address this, we introduce Gaussian Process Activations (GAPA), a post-hoc method that shifts Bayesian modeling from weights to activations. GAPA replaces standard nonlinearities with Gaussian-process activations whose posterior mean exactly matches the original activation, preserving the backbone's point predictions by construction while providing closed-form epistemic variances in activation space. To scale to modern architectures, we use a sparse variational inducing-point approximation over cached training activations, combined with local k-nearest-neighbor subset conditioning, enabling deterministic single-pass uncertainty propagation without sampling, backpropagation, or second-order information. Across regression, classification, image segmentation, and language modeling, GAPA matches or outperforms strong post-hoc baselines in calibration and out-of-distribution detection while remaining efficient at test time.

Sergio Calvo-Ordoñez, Jonathan Plenk, Richard Bergna et al.

Performing gradient descent in a wide neural network is equivalent to computing the posterior mean of a Gaussian Process with the Neural Tangent Kernel (NTK-GP), for a specific choice of prior mean and with zero observation noise. However, existing formulations of this result have two limitations: i) the resultant NTK-GP assumes no noise in the observed target variables, which can result in suboptimal predictions with noisy data; ii) it is unclear how to extend the equivalence to an arbitrary prior mean, a crucial aspect of formulating a well-specified model. To address the first limitation, we introduce a regularizer into the neural network's training objective, formally showing its correspondence to incorporating observation noise into the NTK-GP model. To address the second, we introduce a \textit{shifted network} that enables arbitrary prior mean functions. This approach allows us to perform gradient descent on a single neural network, without expensive ensembling or kernel matrix inversion. Our theoretical insights are validated empirically, with experiments exploring different values of observation noise and network architectures.

Sergio Calvo-Ordonez, Matthieu Meunier, Alvaro Cartea et al.

Conditional flow matching (CFM) has emerged as a powerful framework for training continuous normalizing flows due to its computational efficiency and effectiveness. However, standard CFM often produces paths that deviate significantly from straight-line interpolations between prior and target distributions, making generation slower and less accurate due to the need for fine discretization at inference. Recent methods enhance CFM performance by inducing shorter and straighter trajectories but typically rely on computationally expensive mini-batch optimal transport (OT). Drawing insights from entropic optimal transport (EOT), we propose Weighted Conditional Flow Matching (W-CFM), a novel approach that modifies the classical CFM loss by weighting each training pair $(x, y)$ with a Gibbs kernel. We show that this weighting recovers the entropic OT coupling up to some bias in the marginals, and we provide the conditions under which the marginals remain nearly unchanged. Moreover, we establish an equivalence between W-CFM and the minibatch OT method in the large-batch limit, showing how our method overcomes computational and performance bottlenecks linked to batch size. Empirically, we test our method on unconditional generation on various synthetic and real datasets, confirming that W-CFM achieves comparable or superior sample quality, fidelity, and diversity to other alternative baselines while maintaining the computational efficiency of vanilla CFM.

Richard Bergna, Stefan Depeweg, Sergio Calvo Ordonez et al.

Uncertainty quantification in neural networks through methods such as Dropout, Bayesian neural networks and Laplace approximations is either prone to underfitting or computationally demanding, rendering these approaches impractical for large-scale datasets. In this work, we address these shortcomings by shifting the focus from uncertainty in the weight space to uncertainty at the activation level, via Gaussian processes. More specifically, we introduce the Gaussian Process Activation function (GAPA) to capture neuron-level uncertainties. Our approach operates in a post-hoc manner, preserving the original mean predictions of the pre-trained neural network and thereby avoiding the underfitting issues commonly encountered in previous methods. We propose two methods. The first, GAPA-Free, employs empirical kernel learning from the training data for the hyperparameters and is highly efficient during training. The second, GAPA-Variational, learns the hyperparameters via gradient descent on the kernels, thus affording greater flexibility. Empirical results demonstrate that GAPA-Variational outperforms the Laplace approximation on most datasets in at least one of the uncertainty quantification metrics.