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Sequence Models

Backpropagation Through Time

Inspect exact recurrent state derivatives backward through time.

BPTT · unrolled through timeBPTT · unrolled through timex1 = 1t=1h=0.664→x2 = 0t=2h=0.3203→x3 = -1t=3h=-0.5648→x4 = 0.5t=4h=0.1171→x5 = 0t=5h=0.0585Shared weights · h(t−1) → cell → h(t)
Step 0 / 4
hₜ = tanh(Wx × xₜ + Wh × hₜ₋₁ + b)
tanh(0.8×1 + 0.5×0 + 0)
Hidden state = 0.664
Memory over timeMemory over time0-21-1203142Hidden
Gradient of final hidden state to earlier hidden state · 1 × 5

The product Wh × (1 − h²) propagates backward through shared recurrent steps. These are exact state derivatives, not randomly generated gradient magnitudes.

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