1. The problem is to understand the detailed formula for Backpropagation Through Time (BPTT), which is used to train recurrent neural networks (RNNs). 2. BPTT unfolds the RNN through time steps and applies backpropagation to compute gradients of the loss with respect to weights. 3. The key formula involves computing the gradient of the loss $L$ at time $T$ with respect to the hidden state $h_t$ at time $t$: $$\frac{\partial L}{\partial h_t} = \sum_{k=t}^T \frac{\partial L}{\partial h_k} \frac{\partial h_k}{\partial h_t}$$ 4. The hidden state update is typically: $$h_t = f(W_{hh} h_{t-1} + W_{xh} x_t + b_h)$$ where $f$ is an activation function. 5. The gradient of the loss with respect to weights $W_{hh}$ is computed by accumulating gradients over all time steps: $$\frac{\partial L}{\partial W_{hh}} = \sum_{t=1}^T \frac{\partial L}{\partial h_t} \frac{\partial h_t}{\partial W_{hh}}$$ 6. This involves recursively applying the chain rule backward through time, multiplying by the Jacobian matrices of the hidden states. 7. In plain terms, BPTT treats the RNN as a deep feedforward network with shared weights across time steps and computes gradients by unrolling the network and applying backpropagation. 8. This process allows the network to learn temporal dependencies by adjusting weights based on errors propagated backward through time. Final answer: The detailed BPTT formula is $$\frac{\partial L}{\partial h_t} = \sum_{k=t}^T \frac{\partial L}{\partial h_k} \prod_{j=t+1}^k \frac{\partial h_j}{\partial h_{j-1}}$$ and gradients with respect to weights are accumulated over all time steps accordingly.