Matteo Gamba

Ioanna Gemou, Matteo Gamba, Randall Balestriero et al.

Counterfactual explanations seek small, semantically meaningful changes to an input that alter a model's prediction, and are widely used to interpret and audit machine learning systems. In modern vision, language, and multimodal systems, pretrained encoders map inputs to representation spaces, and downstream classifier heads impose decision boundaries within those spaces. As a result, the feasibility and distance of nearby counterfactuals depend on boundary placement relative to the data. Yet models with similar predictive performance can differ substantially in whether such changes are achievable and how far representations must move. This work examines this variation using a standardized local search probe across several pretrained encoders and linear classifier heads. Results show that despite similar predictive performance, models differ substantially in their counterfactual behavior. Under fixed representations, varying only the classifier head alters counterfactual outcomes while leaving predictive performance largely unchanged. This variation is explained by the interaction of decision-boundary proximity and local data support, which jointly determine whether prediction changes are both feasible and lie in regions supported by the data, and can also improve counterfactual search within fixed models. Together, these findings identify counterfactual behavior as a distinct dimension beyond predictive performance and show that it can be altered without changing accuracy, with implications for model selection, robustness, and the reliability of counterfactual methods.

Amir Mehrpanah, Matteo Gamba, Hossein Azizpour

The adversarial robustness of attributions is a fundamental requirement for reliable explainability in deep learning, yet existing approaches typically rely on computationally expensive explicit regularization. In this work, we show that attribution robustness can arise implicitly from the learning dynamics of standard stochastic gradient descent. We theoretically motivate this effect through connections between parameter-space and input-space curvature, and validate it across architectures, datasets, and attribution methods, with negligible computational overhead. In contrast, we prove that such robustness gains often does not transfer to attention-based attribution under softmax normalization, due to inherent entropy constraints, and we validate this limitation experimentally. Finally, we show that replacing softmax attention with kernel-based attention restores the robustness gains in transformer models. Our results highlight learning dynamics as a principled and practical mechanism for robust explainability, and reveal fundamental limitations of attention-based attribution under normalization.

Matteo Gamba, Hossein Azizpour, Mårten Björkman

Existing bounds on the generalization error of deep networks assume some form of smooth or bounded dependence on the input variable, falling short of investigating the mechanisms controlling such factors in practice. In this work, we present an extensive experimental study of the empirical Lipschitz constant of deep networks undergoing double descent, and highlight non-monotonic trends strongly correlating with the test error. Building a connection between parameter-space and input-space gradients for SGD around a critical point, we isolate two important factors -- namely loss landscape curvature and distance of parameters from initialization -- respectively controlling optimization dynamics around a critical point and bounding model function complexity, even beyond the training data. Our study presents novels insights on implicit regularization via overparameterization, and effective model complexity for networks trained in practice.

Matteo Gamba, Erik Englesson, Mårten Björkman et al.

The ability of overparameterized deep networks to interpolate noisy data, while at the same time showing good generalization performance, has been recently characterized in terms of the double descent curve for the test error. Common intuition from polynomial regression suggests that overparameterized networks are able to sharply interpolate noisy data, without considerably deviating from the ground-truth signal, thus preserving generalization ability. At present, a precise characterization of the relationship between interpolation and generalization for deep networks is missing. In this work, we quantify sharpness of fit of the training data interpolated by neural network functions, by studying the loss landscape w.r.t. to the input variable locally to each training point, over volumes around cleanly- and noisily-labelled training samples, as we systematically increase the number of model parameters and training epochs. Our findings show that loss sharpness in the input space follows both model- and epoch-wise double descent, with worse peaks observed around noisy labels. While small interpolating models sharply fit both clean and noisy data, large interpolating models express a smooth loss landscape, where noisy targets are predicted over large volumes around training data points, in contrast to existing intuition.

Leyang Hu, Matteo Gamba, Randall Balestriero

The scaling of model and data sizes has reshaped the AI landscape, establishing finetuning pretrained models as the standard paradigm for solving downstream tasks. However, dominant finetuning methods typically rely on weight adaptation, often lack interpretability, and depend on heuristically chosen hyperparameters. In this paper, we take a different perspective and shift the focus from weights to activation functions, viewing them through the lens of spline operators. We propose Curvature Tuning (CT), an interpretable and principled steering method that modulates a model's decision boundary by injecting a single hyperparameter into its activation functions. We show that CT provably adjusts model decision boundary curvature and, more fundamentally, projects a model onto a space of smooth functions-thereby complementing current finetuning methods, whose effect lies primarily in feature adaptation. Making this hyperparameter trainable gives rise to a novel and highly parameter-efficient finetuning method. Empirically, CT improves both generalization and robustness. For example, it boosts downstream accuracy of ResNet-50/152 by 7.14%/8.46% over linear probing and 4.64%/1.70% over LoRA across 12 datasets, and improves robust accuracy on the $\ell_\infty$ benchmark from RobustBench by 1032.64%/1494.46%. Our code is available at https://github.com/Leon-Leyang/curvature-tuning.

Anton Adelöw, Matteo Gamba, Atsuto Maki

Supervised training of deep neural networks for classification typically relies on hard targets, which promote overconfidence and can limit calibration, generalization, and robustness. Self-distillation methods aim to mitigate this by leveraging inter-class and sample-specific information present in the model's own predictions, but often remain dependent on hard targets without explicitly modeling predictive uncertainty. With this in mind, we propose Deep Probabilistic Supervision (DPS), a principled learning framework constructing sample-specific target distributions via statistical inference on the model's own predictions, remaining independent of hard targets after initialization. We show that DPS consistently yields higher test accuracy (e.g., +2.0% for DenseNet-264 on ImageNet) and significantly lower Expected Calibration Error (ECE) (-40% ResNet-50, CIFAR-100) than existing self-distillation methods. When combined with a contrastive loss, DPS achieves state-of-the-art robustness under label noise.

Amir Mehrpanah, Matteo Gamba, Kevin Smith et al.

ReLU networks, while prevalent for visual data, have sharp transitions, sometimes relying on individual pixels for predictions, making vanilla gradient-based explanations noisy and difficult to interpret. Existing methods, such as GradCAM, smooth these explanations by producing surrogate models at the cost of faithfulness. We introduce a unifying spectral framework to systematically analyze and quantify smoothness, faithfulness, and their trade-off in explanations. Using this framework, we quantify and regularize the contribution of ReLU networks to high-frequency information, providing a principled approach to identifying this trade-off. Our analysis characterizes how surrogate-based smoothing distorts explanations, leading to an ``explanation gap'' that we formally define and measure for different post-hoc methods. Finally, we validate our theoretical findings across different design choices, datasets, and ablations.

Matteo Gamba, Adrian Chmielewski-Anders, Josephine Sullivan et al.

The number of linear regions has been studied as a proxy of complexity for ReLU networks. However, the empirical success of network compression techniques like pruning and knowledge distillation, suggest that in the overparameterized setting, linear regions density might fail to capture the effective nonlinearity. In this work, we propose an efficient algorithm for discovering linear regions and use it to investigate the effectiveness of density in capturing the nonlinearity of trained VGGs and ResNets on CIFAR-10 and CIFAR-100. We contrast the results with a more principled nonlinearity measure based on function variation, highlighting the shortcomings of linear regions density. Furthermore, interestingly, our measure of nonlinearity clearly correlates with model-wise deep double descent, connecting reduced test error with reduced nonlinearity, and increased local similarity of linear regions.

Matteo Gamba, Stefan Carlsson, Hossein Azizpour et al.

We investigate the geometric properties of the functions learned by trained ConvNets in the preactivation space of their convolutional layers, by performing an empirical study of hyperplane arrangements induced by a convolutional layer. We introduce statistics over the weights of a trained network to study local arrangements and relate them to the training dynamics. We observe that trained ConvNets show a significant statistical bias towards regular hyperplane configurations. Furthermore, we find that layers showing biased configurations are critical to validation performance for the architectures considered, trained on CIFAR10, CIFAR100 and ImageNet.