Implicit Bias of the JKO Scheme

Researchers analyzed the implicit bias of the Jordan-Kinderlehrer-Otto (JKO) scheme, a time-discretization method for Wasserstein gradient flow used in optimizing energy functionals over probability measures. They found that the JKO scheme adds a deceleration term at second order that corresponds to canonical implicit biases like Fisher information for entropy and kinetic energy for Riemannian gradient descent.

• →The JKO scheme approximates Wasserstein gradient flow with a modified energy functional that includes an implicit bias term proportional to the step size.
• →The implicit bias corresponds to Fisher information for entropy functionals and Fisher-Hyvärinen divergence for KL-divergence.
• →JKO scheme exhibits unconditional stability and preserves energy dissipation properties not found in other first-order integrators.
• →The research provides theoretical understanding of why JKO performs better than standard optimization methods in certain contexts.
• →Numerical examples include Langevin dynamics on Bures-Wasserstein space and 1D quartic potential sampling.