How Much Gradient Does Truncated Backprop Lose in Looped Transformers?

Aur Shalev Merin

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…