The gradient goes backward. But backward through what?
Fine-tuning diffusion models using stochastic optimal control: the adjoint method computes gradients by propagating information backward through the forward diffusion process. Standard approaches use an adjoint process that doesn't respect the forward filtration — the information structure generated by the forward noise. The adjoint “sees” information it shouldn't have access to at each time step (arXiv:2603.20455).
The fix: time-reversed backward stochastic differential equations. The BSDE adjoint process is naturally aligned with the underlying filtration. At each time, it conditions only on information available up to that time in the forward process. This alignment produces more stable gradient estimates with reduced variance.
The instability in standard adjoint methods is not numerical — it's informational. The adjoint uses future information to compute present gradients, creating a mismatch between what the gradient “knows” and what the forward process “knew” at the time. Like giving a decision-maker hindsight bias: the gradient over-corrects because it sees outcomes that hadn't happened yet.
The structural insight applies beyond diffusion models to any system where gradients propagate backward through a stochastic process. The adjoint must respect the filtration of the forward process. If it doesn't, the gradient estimates are unbiased in expectation but high-variance in practice — correct on average but unreliable in any single computation. Filtration-aware gradients sacrifice nothing in expectation but gain stability, because they only use information that was actually available.