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Non-geodesically-convex optimization in the Wasserstein space
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Proposes a semi Forward-Backward Euler scheme for minimizing difference-of-convex energies over the Wasserstein space, with convergence guarantees where prior FB Euler analyses fail.
Problem
Optimization in the Wasserstein space of probability measures when the objective is nonconvex (difference-of-convex) and not convex along generalized geodesics, including sampling from log-DC target distributions.
Approach
Modify the Forward-Backward Euler discretization by taking the forward (pushforward) step on the concave part -E_H and the JKO backward step on H_entropy + E_G, following DC programming; this yields a semi FB Euler scheme with provable convergence rates, critical-point and global convergence results under Lojasiewicz conditions.
Key results
- Semi FB Euler achieves O(N^{-1}) rate on the squared Wasserstein (sub)gradient mapping in the nonsmooth setting.
- Under a Lojasiewicz condition with exponent theta in (0,1/2], F(mu_n)-F* and W2 converge exponentially fast; for theta in (1/2,1) convergence is sublinear with rate O(n^{-1/(2theta-1)}).
- On a Gaussian mixture, semi FB Euler is as fast as FB Euler in KL; on a relaxed von Mises-Fisher target FB Euler fails while semi FB Euler works.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use semi Forward-Backward Euler, forwarding on the concave component and applying the JKO step to the convex component plus regularizer | Wasserstein optimization with a difference-of-convex objective · General non-geodesically-convex setting | convergence guarantee | Helped |
| Use semi Forward-Backward Euler instead of Forward-Backward Euler | Gaussian mixture sampling · Gaussian mixture distribution | KL divergence convergence | No clear effect |
| Use semi Forward-Backward Euler instead of Forward-Backward Euler | Relaxed von Mises-Fisher sampling · Relaxed von Mises-Fisher distribution | sampling performance | Helped |
Surprising findings
- FB Euler fails to work on the relaxed von Mises-Fisher target, while semi FB Euler succeeds.
Limitations
- Convergence of the original FB Euler remains open, so comparisons are mainly empirical.
- Practical JKO step requires ICNN and costly log-determinant of Hessian, scaling cubically in dimension.
- Global convergence needs a warm start with F(mu_0) below a threshold and a Lojasiewicz-type assumption.
Experimental setup
Datasets: Gaussian mixture (synthetic) relaxed von Mises-Fisher (synthetic)
Metrics: KL divergence Wasserstein distance
Baselines: FB Euler ULA
Builds on: JKO scheme Forward-Backward Euler DC programming input convex neural networks
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