Interpreting the predictions of neural network classification based on a Taylor Coefficient Analysis (TCA)

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.

We propose a computational tropical geometry framework for the symbolic analysis of neural networks with tropical activations. The number of linear regions of a neural network has been actively studied as a measure of the expressivity of a given architecture. To study these, we work in the setting of tropical geometry---a combinatorial and polyhedral variant of algebraic geometry---where there are known connections between tropical rational maps and feedforward neural networks. We expand this connection by developing concrete computational tools for studying the linear regions of neural networks. We present an algorithm, together with a proof of correctness, which computes the linear regions of a neural network as explicit unions of polyhedra. We further relate the computation of the number of linear regions of a tropical expression to the number of monomials that appear in it, and show how tropical expressions can often be pruned to remove redundant monomials. We introduce the Hoffman constant of a neural network's tropical expression, a geometric quantity that controls the distance from any point in the input space to the farthest linear region. We provide the open source Julia library TropicalNN.jl, which is built on top of the OSCAR computer algebra system and implements the algorithms mentioned above to analyze neural networks symbolically using their tropical representations. We present a set of proof-of-concept computational examples to demonstrate how our tropical geometric theory can be applied to reveal insights on the expressivity of a network architecture.

Explaining the predictions of neural networks is a central challenge in trustworthy AI. Existing explanation methods, such as those based on feature attribution or minimal sufficient sets, typically treat input features as independent, which can yield misleading explanations when inputs exhibit structured dependencies. We address this by formalizing explanations as Halpern-Pearl (HP) actual causes, modeling input dependencies using Boolean Structural Causal Models (SCMs). We compute HP causes by applying bound propagation and branch-and-bound techniques, while providing formal guarantees of completeness and minimality. Our experiments show that we substantially outperform brute-force and ILP baselines in scalability, and outperform heuristic search as graph size grows, computing all minimal actual causes on instances with search spaces of up to $2.3\times10^{13}$ candidate (cause, contingency) pairs, on SCMs with up to 28 nodes, within a 180s per-instance budget. In a case study, we further show that ignoring input dependencies inflates the number of reported causes, 14.9% of which are spurious under our SCM.