Convex and Bilevel Optimization for Neuro-Symbolic Inference and Learning

We address a key challenge for neuro-symbolic (NeSy) systems by leveraging convex and bilevel optimization techniques to develop a general gradient-based framework for end-to-end neural and symbolic parameter learning. The applicability of our framework is demonstrated with NeuPSL, a state-of-the-art NeSy architecture. To achieve this, we propose a smooth primal and dual formulation of NeuPSL inference and show learning gradients are functions of the optimal dual variables. Additionally, we develop a dual block coordinate descent algorithm for the new formulation that naturally exploits warm-starts. This leads to over 100x learning runtime improvements over the current best NeuPSL inference method. Finally, we provide extensive empirical evaluations across $8$ datasets covering a range of tasks and demonstrate our learning framework achieves up to a 16% point prediction performance improvement over alternative learning methods.

Differentially Private Optimization for Smooth Nonconvex ERM

We develop simple differentially private optimization algorithms that move along directions of (expected) descent to find an approximate second-order solution for nonconvex ERM. We use line search, mini-batching, and a two-phase strategy to improve the speed and practicality of the algorithm. Numerical experiments demonstrate the effectiveness of these approaches.