Why does a recurrent forecaster that feeds its own predictions back in drift over a 14-day horizon?

## What recursive decoding actually does A recurrent forecaster trained to map a window of history to the next single value can be extended to any horizon by iteration: predict `y_hat[t+1]`, shift it into the input, predict `y_hat[t+2]` from the extended input, and repeat 14 times. It is attractive because one model with one set of weights answers any horizon, and because the trajectory is internally consistent — each step is conditioned on the path the model has already committed to. Drift is the price. ## The compounding mechanism At step 1 the model reads only observed data, so its error is the honest one-step error. At step 2 one of its inputs is `y_hat[t+1]`, which carries that error. At step 3, two inputs are model output. By step 14 the entire recent context is synthetic. Two distinct things degrade: - **Variance accumulates.** Each step adds its own noise on top of an input that already contains earlier noise, so the spread of the 14-step forecast is much wider than the one-step spread. Even a perfectly unbiased model has this. - **Bias compounds in one direction.** This is the dangerous one. If the model systematically under-predicts peaks by a little, the under-predicted value becomes the context for the next step, which is then predicted from an artificially low level, and the forecast slides steadily away from reality. Errors do not cancel because they are correlated with the state that produced them. The classic visible symptom is a rollout that flattens into a near-constant line: the model's small pull toward the mean is applied 14 times in a row. ## The objective mismatch The deeper issue is that a one-step objective and a 14-step deployment optimise different things. Under one-step loss, the single most informative input is the most recent observation, and a strong model learns to rely on it heavily — nearly a persistence forecast with corrections. That is a great one-step strategy and a terrible 14-step one, because after step 1 the most recent input is the model's own guess. This is why an excellent one-step validation number can sit next to a poor 14-day backtest, and why one-step error must never be used to select a model that will be rolled out. ## The direct multi-horizon head The structural alternative keeps the recurrent encoder over the input window but replaces iterative decoding with a head that produces all `H` values in one shot from the encoder's final state — a sequence-to-sequence arrangement whose output side emits the whole horizon simultaneously. The loss is the mean error over all `H` positions, so the gradient directly rewards being right 14 steps out. What you gain: - No feedback path at all, so no compounding. Step 14's error is whatever the model's step-14 estimate is worth, not the accumulation of thirteen previous mistakes. - The long-horizon error appears in the objective, so the model can trade one-step sharpness for horizon-wide accuracy. - Inference is a single pass instead of `H` sequential passes, which matters when you serve thousands of series. What you give up: - **The horizon is fixed.** `H` output units are trained for exactly 14 steps. Serving 7 or 21 needs another head or another model, whereas recursion answers any horizon with the same parameters. - **Capacity scales with the horizon.** A long horizon means a wide output layer, and each position gets its own share of a fixed training set. - **No trajectory coherence.** Each output unit is free to move independently, so the 14-day path can be jagged or its aggregate inconsistent. If downstream consumers need a coherent path or a correct total, add a smoothness or aggregation term to the loss or post-process — the architecture alone guarantees nothing. ## Choosing between them Short horizons relative to the sampling rate favour recursion: there is little room for compounding, and the flexibility is free. Long horizons, or horizons where the business cares about the far end (a 14-day replenishment decision hinges on days 10 to 14, not day 1), favour the direct head. A hybrid many teams land on is a direct head for the fixed operational horizon plus a recursive model kept for ad-hoc queries. Whichever you choose, evaluate at the horizon you serve, and evaluate **per step**. A single averaged number hides exactly the pattern that diagnoses drift: a per-horizon error curve that rises steeply and a mean forecast that flattens toward the series average tells you the rollout is decaying, not that the model is uniformly weak.