Propensity Straight-Through Gradients for Discrete Stochastic Systems

Continuous-time Markov chains (CTMCs) provide the backbone for modeling discrete stochastic dynamics across applied, physical, and biological sciences. Their integration with modern gradient-based machine learning, however, is limited by the hard categorical event selection intrinsic to Gillespie-type simulation algorithms. We exploit the affine state update to obtain the exact one-step conditional-mean sensitivity by differentiating normalized reaction propensities. We pair this backward rule with exact forward trajectories to define the propensity straight-through (PST) estimator. At the trajectory level, we show that one-step sensitivities composed across events can depart from the exact multistep sensitivity. We derive the resulting per-step discrepancy in closed form and prove that it vanishes identically for affine downstream dependence. PST matches the accuracy of Gumbel-Softmax straight-through across all benchmarks: reversible dimerization (0.06% error), a genetic oscillator (1.7% error), a 50-task repressilator suite (0.17% median error), and patch-clamp ion-channel recordings ($R^2$ = 0.988). Under matched settings, PST converges 3.0-fold faster on the oscillator and 2.1-fold faster on the ion channel. At deep-learning scale, PST trains a 203,796-parameter stochastic reaction network with hard sampling, reaching 98.22% MNIST digit classification accuracy. By differentiating an exact conditional mean rather than a relaxed sample, PST offers a temperature- and Gumbel-free path to scalable gradient-based learning through exact stochastic trajectories.

Beyond Token-Level Cross-Entropy: Fréchet Distributional Post-Training for Autoregressive Image Generation

Autoregressive image generators are commonly pretrained with token-level cross-entropy under teacher forcing, yet evaluated by the distributional quality of decoded images. This creates an objective mismatch, because categorical errors have unequal image-level consequences, and a context mismatch, because inference conditions on model-generated histories. We introduce FD-loss post-training, which adapts a pretrained discrete generator using representation-space Fréchet distance as the sole objective. A dual-pass scheme first constructs detached rollout contexts through gradient-free generation under the model's native inference configuration, then performs differentiable replay with a probability-level straight-through estimator (STE) that preserves hard argmax decoding in the forward pass while propagating image-level gradients through temperature-scaled probabilities. Only the generator is updated, while the tokenizer and feature extractors remain frozen. Across eight completed configurations from four generator families on class-conditional ImageNet at $256\times256$, FD-loss post-training reduces FID and $\mathrm{FD}_{r6}$ by 41.4% and 52.0% on average. The strongest FID result improves from 2.42 to 1.43 without adding parameters or inference steps.