Timestep sampling: preserve the intended objective under a new proposal

Let pi_t be the desired discrete timestep probability and lambda_t its explicit nonnegative objective weight. A proposal q_t determines how often a timestep is sampled. Multiplying a sampled loss by pi_t lambda_t/q_t produces the desired weighted expectation when the proposal covers every nonzero contribution. For a uniform objective over T timesteps, pi_t is one over T. The ratio is not generally just one over q_t; omitting pi changes scale and can affect comparisons or the optimizer's effective step size.

If q_t is zero where the desired weighted loss can contribute, no finite weight can recover that missing term from samples. Keep a positive exploration floor where needed. Additional clean-data and noise randomness remains inside the timestep loss expectation. An adaptive proposal can use detached historical statistics, with sampling and correction treated as proposal machinery rather than differentiated objective terms in the stated estimator. Document how distributed ranks update that history and whether their effective proposals and weights agree.

For a scalar loss estimator, the ideal proposal depends on the second moment of the per-timestep loss, not merely its mean. Improved DDPM explores loss-aware importance sampling for its objective. The ideal values are unknown during training, so implementations estimate them and need startup behavior and support safeguards. A proposal that helped a likelihood-related objective may not be optimal for another weighted denoising objective. Re-derive the contribution after changing the target, lambda, or normalization.

Gradient variance is another target: it depends on gradient-vector second moments, not simply scalar loss second moments. A high loss can have a small gradient under a particular parameterization, or vice versa. Importance weights with a large dynamic range can create unstable finite-sample updates. Clipping them or self-normalizing by their observed batch sum generally changes the finite-sample estimator and can introduce bias. Those may be deliberate algorithms, but they should not be described as the original unbiased estimator without qualification.

Log the proposal distribution, effective weights, per-bin sample counts, and both proposal-weighted and fixed-reference validation losses. Keep held-out noise and timestep draws fixed when isolating a sampler change, or evaluate under a declared reference distribution with enough samples. A proposal-weighted training mean can fall merely because the sampled mix changed. Measure wall-clock cost, gradient behavior, and downstream sampling metrics at matched compute or data budgets. Check rare timesteps and rank-local distributions rather than relying only on the aggregate mean.

The probability lab models how often one of two timestep classes is selected in independent draws. Reset, set p=0.25 and n=20, predict five selections on average, then Play or Step. Separately Reset, set p=0.75 and n=20, predict fifteen, then Play or Step. Playback highlights the exact binomial distribution, not an adaptive training run. The instrument has no loss values or importance correction. Use the changed counts to distinguish proposal frequency from the paper calculation that preserves the desired objective.