How Much Gradient Does Truncated Backprop Lose in Looped Transformers?

Overview & Synopsis

Looped transformers reuse one block of layers many times and are often trained by backpropagating through only the last k loops. We measure how much of the weight gradient this throws away, exactly and for every k from one backward pass, on nine looped language models, and report three findings. First, on a trained model this lost gradient is set near the output: once the loop count is well above k, it barely changes with the loop count (within about 0.03 from 32 to 64 loops on the three models we tested, as a pre-registered test predicted for Parcae). Where the backward signal shrinks, $ρ^k$, with $ρ$ the signal's fitted shrink per loop, approximates it at moderate loop counts but overstates it at long ones, because the fitted $ρ$ drifts up as the fit takes in more loops. On Ouro the signal grows, and keeping two of four loops still loses 81% of the final output's gradient. Second, the lost gradient falls during fine-tuning, by 0.07-0.22 within 300 steps on three of four models, on held-out problems too. The fall is not just the read-out's share of the gradient; training only the layers outside the loop reproduced up to half of it, and it also falls under truncated training. So a trained model's value overstates what truncation removes later. Third, we could not show that it predicts the training cost of truncation better than rules that need no measurement: on two new models it tied Parcae's cost at the same k, in a test too small to separate them.