Monotonicity and Frank-Wolfe Dynamics in Atomic Splittable Congestion Games

What the math gives to ML

The paper identifies a sharp curvature quantity, \(\Theta_f(x)=2f'(x)+xf''(x)\), that controls monotonicity of marginal-cost operators in atomic splittable congestion games and stability of Euclidean-regularized Frank–Wolfe dynamics. This suggests a practical stability certificate for neural routing or mixture-of-experts layers whose experts share differentiable resource costs: constrain or parameterize the cost functions so that \(f'\ge 0\) and \(\Theta_f\ge 0\). The most direct experiment is a simplex-constrained routing layer trained with a regularized Frank–Wolfe update, comparing curvature-certified costs against unconstrained learned costs on loss stability, routing oscillation, and convergence speed.