Non-Affine Aggregation Hinders Convex Learning Convergence and Stability

New research explores the fundamental relationship between convex learning and gradient aggregation techniques, particularly focusing on how different aggregation methods impact algorithmic stability and convergence. The study demonstrates that for first-order convex learning, the crucial property of monotonicity in gradient updates is maintained exclusively by positively affine aggregation rules. This implies that modern learning systems employing non-affine aggregation, often used for enforcing constraints like privacy or fairness, inherently face challenges with consistent convergence and stability. The paper provides a theoretical framework to explain various failure modes observed in these advanced learning systems. It also identifies specific conditions under which monotonicity can be re-established, offering a path forward for designing more robust algorithms.