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Score-based generative models are provably robust: an uncertainty quantification perspective

NeurIPS 2024 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: Regression uncertainty quantification · 1 citations (OpenAlex)

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Shows via a Wasserstein uncertainty propagation (WUP) theorem, using HJB/Bernstein regularity, that score-based generative models are provably robust to finite-sample, early-stopping, score-matching, parametrization and reference-distribution errors, with d1 and TV generalization bounds.

Problem

Explaining why score-based generative models remain accurate under practical implementation errors (finite samples, early stopping, score-matching objective, network expressiveness, reference distribution), with bounds that do not assume the manifold hypothesis or absolute continuity of the data.

Approach

An L2 error in the learned drift propagates through the Fokker-Planck equation to a Wasserstein-1 (and TV/MMD) ball around the data distribution; the key mechanism is that diffusion regularizes test functions (Kolmogorov backward equation with Bernstein gradient estimates), so bounds in weak IPMs control stronger norms and need no KL/Girsanov arguments.

Key results

• Theorem gives d1(pi, m_g(T)) <= C R^{3/2}(1+sqrt(||grad s||_inf)) (R e^{-omega T/R^2} d1(pi, uniform) + sqrt(e_nn)) under uniform-in-time L2 score error, a purely theoretical bound.
• Pointwise DSM bound adds an early-stopping term sqrt(epsilon) to d1 error, with no manifold or density assumption on the data distribution.
• Average-case DSM bound: E[d1] bounded in terms of N^{-1/(2d)} finite-sample error under a global score-norm assumption.

Measured effects reported in the paper

Surprising findings

• Total variation error between generated and true distributions can be bounded by the weaker d1 distance between initial conditions, due to diffusion regularization.
• Bounds become vacuous at zero early stopping, and driving DSM loss to zero with epsilon to 0 implies memorization of the training data.

Limitations

• Purely theoretical; no experiments on real generative tasks.
• Discretization error is not analyzed.
• Bounds depend on score Lipschitz/C^2 norms that may blow up as early stopping epsilon goes to zero; a density lower bound is needed for pointwise DSM result.

Experimental setup

Metrics: Wasserstein-1 distance bound total variation bound L1 bound expected d1 bound

Baselines: KL-divergence/Girsanov-based SGM convergence bounds kernel score estimators (minimax optimal) Wasserstein-2 uncertainty propagation bound

Builds on: score-based generative modeling denoising score matching Kolmogorov backward equation Girsanov-based convergence analyses

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