# LSTM – Derivation of Back propagation through time

**LSTM (Long short term Memory ) **is a type of RNN(Recurrent neural network), which is a famous deep learning algorithm that is well suited for making predictions and classification with a flavour of the time. In this article, we will derive the algorithm backpropagation through time and find the gradient value for all the weights at a particular timestamp.

As the name suggests backpropagation through time is similar to backpropagation in DNN(deep neural network) but due to the dependency of time in RNN and LSTM, we will have to apply the chain rule with time dependency.

Let the input at time t in the LSTM cell be x_{t}, the cell state from time t-1 and t be c_{t-1 } and c_{t} and the output for time t-1 and t be h_{t-1 }and h_{t }. The initial value of c_{t }and h_{t }at t = 0 will be zero.

**Step 1 :** Initialization of the weights .

Weights for different gates are :Input gate :w_{xi}, w_{xg}, b_{i}, w_{hj}, w_{g }, b_{g}Forget gate :w_{xf}, b_{f}, w_{hf}Output gate :w_{xo}, b_{o}, w_{ho}

**Step 2 : **Passing through different gates .

Inputs: x_{t }and h_{t-i }, c_{t-1 }are given to the LSTM cellPassing through input gate:Z_{g }= w_{xg }*x + w_{hg }* h_{t-1 }+ b_{g}g = tanh(Z_{g}) Z_{j }= w_{xi }* x + w_{hi }* h_{t-1 }+ b_{i}i = sigmoid(Z_{i}) Input_gate_out = g*iPassing through forget gate:Z_{f }= w_{xf }* x + w_{hf }*h_{t-1 }+ b_{f}f = sigmoid(Z_{f}) Forget_gate_out = fPassing through the output gate:Z_{o }= w_{xo}*x + w_{ho}* h_{t-1 }+ b_{o}o = sigmoid(z_{O}) Out_gate_out = o

**Step 3 :** Calculating the output h_{t} and current cell state c_{t.}

Calculating the current cell state c_{t}: c_{t = }(c_{t-1 }* forget_gate_out) + input_gate_outCalculating the output gate ht:h_{t}=out_gate_out * tanh(ct)

**Step 4 :** Calculating the gradient through back propagation through time at time stamp t using the chain rule.

Let the gradient pass down by the above cell be:E_delta_{ }= dE/dh_{t}If we are using MSE (mean square error)for error then, E_delta=(y-h(x)) Here y is the original value and h(x) is the predicted value._{ }_{ }Gradient with respect to output gatedE/do = (dE/dh_{t }) * (dh_{t }/do) = E_delta * ( dh_{t}/ do) dE/do = E_delta * tanh(c_{t})Gradient with respect to cdE/dc_{t }_{t}= (dE / dh_{t })*(dh_{t }/dc_{t})= E_delta *(dh_{t }/dc_{t}) dE/dc_{t }= E_delta * o * (1-tanh^{2 }(c_{t}))Gradient with respect to input gate dE/di, dE/dgdE/di = (dE/di ) * (dc_{t}/ di) dE/di = E_delta * o * (1-tanh^{2 }(c_{t})) * g Similarly, dE/dg = E_delta * o * (1-tanh^{2 }(c_{t})) * iGradient with respect to forget gatedE/df = E_delta * (dE/dc_{t}) * (dc_{t}/ dt) t dE/df = E_delta * o * (1-tanh^{2 }(c_{t})) * c_{t-1 }Gradient with respect to cdE/dc_{t-1 }_{t}= E_delta * (dE/dc_{t}) * (dc_{t}/ dc_{t-1}) dE/dc_{t}= E_delta * o * (1-tanh^{2 }(c_{t})) * fGradient with respect to output gate weights:dE/dw_{xo}= dE/d_{o}*(d_{o}/dw_{xo}) = E_delta * tanh(c_{t}) * sigmoid(z_{o}) * (1-sigmoid(z_{o}) * x_{t}dE/dw_{ho}= dE/do *(do/dw_{ho}) = E_delta * tanh(c_{t}) * sigmoid(z_{o}) * (1-sigmoid(z_{o}) * h_{t-1}dE/db_{o}= dE/do *(do/db_{o}) = E_delta * tanh(c_{t}) * sigmoid(z_{o}) * (1-sigmoid(z_{o})Gradient with respect to forget gate weights:dE/dw_{xf }= dE/df *(df/dw_{xf}) = E_delta * o * (1-tanh^{2}(c_{t})) * c_{t-1}* sigmoid(z_{f}) * (1-sigmoid(z_{f}) * x_{t}dE/dw_{hf}= dE/df *(df/dw_{hf}) = E_delta * o * (1-tanh^{2}(c_{t})) * c_{t-1}* sigmoid(z_{f}) * (1-sigmoid(z_{f}) * h_{t-1}dE/db_{o}= dE/df *(df/db_{o}) = E_delta * o * (1-tanh^{2}(c_{t})) * c_{t-1}* sigmoid(z_{f}) * (1-sigmoid(z_{f})Gradient with respect to input gate weights:dE/dw_{xi}= dE/di *(di/dw_{xi}) = E_delta * o * (1-tanh^{2}(c_{t})) * g * sigmoid(z_{i}) * (1-sigmoid(z_{i}) * x_{t}dE/dw_{hi}= dE/di *(di/dw_{hi}) = E_delta * o * (1-tanh^{2 }(c_{t})) * g * sigmoid(z_{i}) * (1-sigmoid(z_{i}) * h_{t-1}dE/db_{i}= dE/di *(di/db_{i}) = E_delta * o * (1-tanh^{2}(c_{t})) * g * sigmoid(z_{i}) * (1-sigmoid(z_{i}) dE/dw_{xg}= dE/dg *(dg/dw_{xg}) = E_delta * o * (1-tanh^{2}(c_{t})) * i * (1?tanh^{2}(z_{g}))*x_{t}dE/dw_{hg}= dE/dg *(dg/dw_{hg}) = E_delta * o * (1-tanh^{2 }(c_{t})) * i * (1?tanh^{2}(z_{g}))*h_{t-1}dE/db_{g}= dE/dg *(dg/db_{g}) = E_delta * o * (1-tanh^{2}(c_{t})) * i * (1?tanh^{2}(z_{g}))

Finally the gradients associated with the weights are,

Using all gradient, we can easily update the weights associated with input gate, output gate, and forget gate