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Acceleration via silver step-size on Riemannian manifolds with applications to Wasserstein space
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Introduces VTRGD, a Riemannian gradient method with vector transport and silver stepsize schedule, giving accelerated O(1/n^{log2 rho}) rates under VT-geodesic convexity/smoothness, and recovers Wasserstein gradient descent in 2-Wasserstein space.
Problem
Accelerating Riemannian gradient descent without curvature or diameter assumptions, using dynamic (silver) stepsize schedules rather than momentum; prior accelerated Riemannian methods need curvature bounds or are intractable.
Approach
Vector-transported RGD updates in a fixed base-point tangent space via x_{n+1}=exp_b(log_b x_n - (eta_n/L) VT_{x_n}^b grad f(x_n)), combined with silver stepsizes; defines VT-geodesic convexity/smoothness relative to a base point, which for the parallel/optimal transport reduces to standard Wasserstein gradient descent.
Key results
- Theorem: f(x_n)-f* <= r_k L ||log_b x_0 - log_b x*||^2 with r_k=(1+sqrt(4rho^{2k}-3))^{-1}, r_k ~ n^{-log2 rho} ~ n^{-1.2716}, for n=2^k-1.
- Restarted silver-stepsize VTRGD attains d^2(x_n,x*) <= exp(-log(rho/2) n / kappa^{log_rho 2}) ..., i.e. O(kappa^{log_rho 2} log(1/eps)) iterations under geodesic strong convexity.
- In numerical experiments silver-stepsize WGD outperforms constant-stepsize WGD in both convex and strongly convex Bures-Wasserstein potential energy problems (d=10); VTRGD with b=I shows improvement over RGD on linear-minus-logdet with kappa=10 and 10^5 (d=50).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use silver stepsizes instead of a constant stepsize | VTRGD · Riemannian geodesically convex objectives | convergence rate: O(n^{-1}) -> O(1/n^{\log_2 \rho}) | Helped |
| Restart silver-stepsize VTRGD at fixed intervals | VTRGD · Riemannian geodesically strongly convex objectives | iteration complexity: O(\kappa \log(1/\epsilon)) -> O\left(\kappa^{\log_{\rho} 2} \log(1/\epsilon)\right) | Helped |
| Use silver stepsizes instead of a constant stepsize | Wasserstein gradient descent · Bures-Wasserstein space, convex and strongly convex quadratic potential energy | objective error | Helped |
| Restart silver-stepsize WGD at fixed intervals | Wasserstein gradient descent · 2-Wasserstein space, strongly convex objectives | iteration complexity: O\left(\kappa^{\log_{\rho} 2} \log(1/\epsilon)\right) | Helped |
Surprising findings
- The Wasserstein acceleration proof does not require the update map to be optimal (convex), so large stepsizes beyond 1/L are allowed.
- The linear-minus-log-determinant problem is not geodesically L-smooth, yet RGD and VTRGD were empirically stable with L=2.
Limitations
- VT-geodesic convexity/smoothness is not directly interpretable and its restrictiveness is unclear.
- Strongly convex result needs knowledge of kappa to choose restart length, and iteration count restricted to n = l(2^{k*}-1); theorem for general n oscillates.
- Analysis is deterministic only; stochastic and proximal variants are open.
Experimental setup
Benchmarks: Bures-Wasserstein potential energy minimization (quadratic V in R^10) linear minus log-determinant on SPD(50)
Metrics: objective gap f - f_min W2 distance/objective value vs iterations
Baselines: constant-stepsize Riemannian gradient descent constant-stepsize Wasserstein gradient descent
Builds on: silver stepsize Riemannian gradient descent generalized geodesic convexity long stepsize
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