On the Normalization of Confusion Matrices: Methods and Geometric Interpretations

The confusion matrix is a standard tool for evaluating classifiers by providing insights into class-level errors. In heterogeneous settings, its values are shaped by two main factors: class similarity -- how easily the model confuses two classes -- and distribution bias, arising from skewed distributions in the training and test sets. However, confusion matrix values reflect a mix of both factors, making it difficult to disentangle their individual contributions. To address this, we introduce bistochastic normalization using Iterative Proportional Fitting, a generalization of row and column normalization. Unlike standard normalizations, this method recovers the underlying structure of class similarity. By disentangling error sources, it enables more accurate diagnosis of model behavior and supports more targeted improvements. We also show a correspondence between confusion matrix normalizations and the model's internal class representations. Both standard and bistochastic normalizations can be interpreted geometrically in this space, offering a deeper understanding of what normalization reveals about a classifier.

A Weighted Loss Approach to Robust Federated Learning under Data Heterogeneity

Federated learning (FL) is a machine learning paradigm that enables multiple data holders to collaboratively train a machine learning model without sharing their training data with external parties. In this paradigm, workers locally update a model and share with a central server their updated gradients (or model parameters). While FL seems appealing from a privacy perspective, it opens a number of threats from a security perspective as (Byzantine) participants can contribute poisonous gradients (or model parameters) harming model convergence. Byzantine-resilient FL addresses this issue by ensuring that the training proceeds as if Byzantine participants were absent. Towards this purpose, common strategies ignore outlier gradients during model aggregation, assuming that Byzantine gradients deviate more from honest gradients than honest gradients do from each other. However, in heterogeneous settings, honest gradients may differ significantly, making it difficult to distinguish honest outliers from Byzantine ones. In this paper, we introduce the Worker Label Alignement Loss (WoLA), a weighted loss that aligns honest worker gradients despite data heterogeneity, which facilitates the identification of Byzantines' gradients. This approach significantly outperforms state-of-the-art methods in heterogeneous settings. In this paper, we provide both theoretical insights and empirical evidence of its effectiveness.