Decision Boundary Analysis

Forward-Forward learning replaces end-to-end backpropagation with local, layer-wise training objectives. Its variants are evaluated primarily by classification accuracy, often measured with linear probes trained on their representations. We show that this evaluation can over-credit the learning method: a linear probe on the representations of an untrained convolutional neural network already reaches accuracy well above chance. Moreover, a single accuracy value does not reveal which datapoints become correctly classified after training and which remain misclassified. To examine what learning contributes beyond the random, untrained baseline, we present an analysis methodology built on the Boundary Proximity Index (BPI), a multiclass margin normalized by a class-centroid reference. We apply our analysis to seven Forward-Forward variants and related layer-local methods and find that initially misclassified points near the probe's decision boundary generally have higher rescue rates than those farther inside incorrect decision regions. Rescue rates vary substantially across methods, with some correcting a substantial fraction of these deeply embedded points. Together, these analyses offer a way to evaluate representation learning beyond accuracy and identify the rescue of points far inside incorrect decision regions as a target for improving local learning methods.

A deep classifier is defined not only by the decision it produces, but also by the sequence of transformations through which that decision is formed. Treating this evolution as a dynamical system across layers provides a natural framework for asking how decision geometry emerges through depth and how far back we can trace a boundary's dynamical signature. We model a feed-forward classifier as a finite, nonautonomous discrete dynamical system, with layers playing the role of discrete time steps. We study the Finite-Time Maximum Lyapunov Exponent (FTMLE) of the data samples' dynamical trajectory through depths of the classifier. The FTMLE measures the rate of convergence/divergence of nearby trajectories. We move the observation endpoint backward from probabilities to logits and then to hidden representations. For Gaussian classes, we prove that probability-level FTMLE carries a clear geometric signature of the decision boundary, with its dominant direction aligned with the boundary normal. Moving one step backward to the logits, we prove this relationship is no longer universal but depends critically on how the classifier is trained, particularly on the choice of loss function. Moving further backward to the hidden representation, the connection becomes more conditional: boundary-related FTMLE can persist, but only under identifiable structural conditions. We propose geometry-aware fine-tuning for restructuring the classifier's hidden FTMLE, and propose conditions for guaranteed concentration of high hidden FTMLE near the decision boundary. Through our numerical results, we show the generality and validity of our theoretical results. Understanding the evolution of data samples as traveling through the layers of classifier provides a principled foundation for identifying where boundary-relevant sensitivity emerges and for developing layer-aware regularization strategies.

Surrogate models are commonly evaluated by how often they agree with their teacher model over an evaluation set. Local variation in this agreement is well known, but its structure and consequences are less clear. We ask whether disagreement is systematically concentrated near the teacher's decision boundary and whether retaining that structure provides information beyond a single global score. Across several datasets and surrogate model classes, we find substantially lower fidelity near teacher decision boundaries under two different methods of identifying near-boundary examples. Moreover, conditioning agreement on confidence-defined regions improves prediction of teacher--surrogate agreement when evaluation-set composition changes, relative to the global score alone. Yet surrogates that agree equally well with the teacher both globally and near the decision boundary can respond very differently to changes selected using the surrogate itself. Finally, we show that independently trained deep teachers can agree on most predictions while identifying different examples as lying near their decision boundaries, complicating the use of those boundaries as stable reference regions for evaluating surrogates. Together, these results show that surrogate fidelity depends not only on how often a surrogate agrees with its teacher, but also on where that agreement holds and, for deep models, how stable the teacher's decision boundary is across training runs.

Classical dynamic programming represents optimal sequential decisions through value functions and policies. While this functional representation is natural for computing optimal decisions, it does not directly identify the mathematical object governing policy reconstruction, representation complexity, or oracle-query complexity once an optimal policy is fixed. This paper addresses this question by developing a geometric theory of structured optimal policies in which the decision-boundary geometry induced by the policy becomes the primary object of analysis. We show that, under suitable structural regularity conditions, this geometry provides the minimal representation required for policy reconstruction and determines the statistical and computational complexity of the reconstruction problem. Building upon this representation, we establish structural properties of policy-induced decision geometry, introduce intrinsic notions of boundary and decision complexity, derive information-theoretic measures of decision compression, and obtain statistical guarantees for boundary estimation and policy reconstruction from black-box policy queries. Collectively, these results demonstrate that, for the structured decision problems considered here, the complexity of policy reconstruction is governed by the geometry of the decision boundary rather than by the cardinality of the ambient state space. Controlled numerical experiments examine the principal theoretical predictions and provide empirical evidence consistent with the proposed framework.

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.

The topology of a classifier's decision regions determines how inputs with the same predicted label can be connected and deformed without changing that prediction. Prior empirical work constructed paths between same-label images within a single region, but did not examine whether loops bound surfaces within that region. We investigate this question using adaptive quadrilateral meshes with targeted repair of off-label interior vertices, while holding the same-label boundary loop fixed. A finite-resolution acceptance criterion distinguishes completed constructions from those left unresolved at the refinement ceiling. Across the pretrained classifiers studied, every tested loop admits an accepted filling. Construction effort varies by orders of magnitude within classes and is greater for mean-score-adjusted randomly initialised classifiers than for trained classifiers. An analytic control with a known hole leaves winding loops unresolved at the tested hole radii at or above the resolution threshold. These results provide empirical evidence consistent with simply connected decision regions at the tested resolution.

Decision Boundary Maps (DBMs) are an effective tool for visualising machine learning classification boundaries. Yet, DBM quality strongly depends on the dimensionality reduction (DR) technique and high dimensional space used for the data points. For complex ML data, DR can create many mixed classes which yield DBMs that are hard to use or even misleading. We propose a new technique to compute DBMs by transforming data space into Shapley space and computing DR on it. Compared to DBMs computed directly from data, our maps have similar or higher quality metric values and visibly more compact, easier to explore, decision zones that better agree with measured model performance.

For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a classical tube formula due to Weyl, we introduce a method to measure the decision boundary of a neural network through local surface volumes, providing a theoretically justifiable and efficient measure enabling a geometric interpretation of the effectiveness of the model applicable to the high dimensional feature spaces considered in deep learning. A smaller surface volume is expected to correspond to lower model complexity and better generalisation. We verify, on a number of image processing tasks with convolutional architectures that decision boundary volume is inversely proportional to classification accuracy. Meanwhile, the relationship between local surface volume and generalisation for fully connected architecture is observed to be less stable between tasks. Therefore, for network architectures suited to a particular data structure, we demonstrate that smoother decision boundaries lead to better performance, as our intuition would suggest.