Transient-risk certificate for Langevin training

Integrate this into the optimizer update of a Langevin-trained model, such as noisy SGD on a small fine-tuning task: θ_{k+1}=θ_k−η∇L(θ_k)+sqrt(2ηT)ξ_k, with ξ_k∼N(0,I). Define an unsafe predicate A(θ), for example a backdoor-trigger success rate above a threshold, a constraint violation, or a validation loss outside a certified basin. At checkpoint k, estimate π(A) by running independent Langevin chains at the current temperature after burn-in, or by sampling a local quadratic approximation π≈N(θ*,T H^{-1}); estimate χ_0² conservatively from the initial parameter distribution versus these stationary samples; and estimate m from a lower Hessian eigenvalue using Hessian-vector products or from exponential autocorrelation decay. Compute B_k=π_hat(A)+sqrt(π_hat(A)χ_hat²)exp(−m_hat kη). Pseudocode: initialize θ and estimates; apply the noisy gradient step; every K steps evaluate A on a particle batch; update π_hat, χ_hat, and m_hat; compute B_k; if B_k≤δ, stop Langevin exploration and switch to deterministic SGD or AdamW. The theorem supplies the bound and stopping inequality, while π(A), χ², and m are empirical estimates and should be reported with confidence intervals. First test on a quadratic loss with known m and analytically known Gaussian π, then on a two-layer MLP trained with Langevin noise on MNIST containing an implanted trigger. Compare fixed-noise Langevin against certificate-based stopping at equal final accuracy. Success means lower maximum unsafe-particle frequency and fewer unsafe excursions while preserving validation accuracy, and the quadratic experiment should reproduce the predicted exponential envelope.