A Learnability Analysis on Neuro-Symbolic Learning

This paper analyzes the learnability of neuro-symbolic (NeSy) tasks within hybrid systems. We show that the learnability of NeSy tasks can be characterized by their derived constraint satisfaction problems (DCSPs). Specifically, a task is learnable if the corresponding DCSP has a unique solution; otherwise, it is unlearnable. For learnable tasks, we establish error bounds by exploiting the clustering property of the hypothesis space. Additionally, we analyze the asymptotic error for general NeSy tasks, showing that the expected error scales with the disagreement among solutions. Our results offer a principled approach to determining learnability and provide insights into the design of new algorithms.

Weakly Supervised AUC Optimization: A Unified Partial AUC Approach

Since acquiring perfect supervision is usually difficult, real-world machine learning tasks often confront inaccurate, incomplete, or inexact supervision, collectively referred to as weak supervision. In this work, we present WSAUC, a unified framework for weakly supervised AUC optimization problems, which covers noisy label learning, positive-unlabeled learning, multi-instance learning, and semi-supervised learning scenarios. Within the WSAUC framework, we first frame the AUC optimization problems in various weakly supervised scenarios as a common formulation of minimizing the AUC risk on contaminated sets, and demonstrate that the empirical risk minimization problems are consistent with the true AUC. Then, we introduce a new type of partial AUC, specifically, the reversed partial AUC (rpAUC), which serves as a robust training objective for AUC maximization in the presence of contaminated labels. WSAUC offers a universal solution for AUC optimization in various weakly supervised scenarios by maximizing the empirical rpAUC. Theoretical and experimental results under multiple settings support the effectiveness of WSAUC on a range of weakly supervised AUC optimization tasks.