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Finite-Time Analysis of Discrete-Time Stochastic Interpolants
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Gives the first finite-time KL error bound for a discrete-time Euler–Maruyama sampler of SDE-based stochastic interpolants, plus an exponentially decaying time schedule that reduces step complexity versus uniform steps.
Problem
Prior analyses of stochastic interpolant generative models assume exact solutions of the underlying ODE/SDE; the finite-step discrete-time sampler's error (discretization plus estimation) was not characterized, and the effect of time schedules was unknown.
Approach
Analyze the Euler–Maruyama discretization of the forward SDE for stochastic interpolants using Girsanov-based KL bounds, with Itô calculus and conditional-covariance identities that exploit the Gaussian latent term to bound velocity and score derivatives without Lipschitz assumptions; propose an exponentially decaying step schedule concentrated near the endpoints where gamma vanishes.
Key results
- The exponentially decaying schedule converges about 10x faster in target density than uniform step sizes on the checkerboard-to-spiral task.
- For the diffusion special case with bounded fourth moment of rho_1, the step complexity is O(eps^-2 d^2 log^2(1/(1-t_N))), matching the known diffusion-model bound of Chen et al. order in d and log factors.
- Empirical KL error on Gaussian mixtures grows nearly linearly in d, below the O(d^2) theoretical upper bound.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use exponentially decaying rather than uniform step sizes | discrete-time stochastic interpolant sampler · Checkerboard to spiral | empirical KL divergence: about $10\times$ faster convergence | Helped |
| use a source density closer to the target under the specified coupling | discrete-time stochastic interpolant sampler with exponentially decaying schedule · Different source densities and couplings | estimated TV distance | Helped |
| use a source density farther from the target under the specified coupling | discrete-time stochastic interpolant sampler with exponentially decaying schedule · Different source densities and couplings | estimated TV distance | Hurt |
| increase dimension | discrete-time stochastic interpolant sampler on Gaussian mixtures · Gaussian mixtures at different dimensions | empirical KL error | Hurt |
Surprising findings
- Empirical KL error grows almost linearly in d while the theory gives O(d^2), suggesting a gap in the bound.
Limitations
- Dimension dependence is d^2 in the bound, not the near-linear d achieved in prior diffusion analyses.
- Convergence rate is O(eps^-2), with no acceleration beyond this rate shown.
- Analysis is restricted to SDE samplers; ODE-based finite-time convergence is left open.
Experimental setup
Datasets: 2D checkerboard and spiral densities Gaussian mixtures
Benchmarks: 2D checkerboard-to-spiral transport d-dimensional Gaussian mixtures
Metrics: estimated KL divergence estimated TV distance
Baselines: uniform step-size schedule
Builds on: stochastic interpolants Girsanov-based diffusion analysis score-based diffusion convergence bounds
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