Samrat Nath, Ph.D.

Samrat Nath, Ph.D.

Staff Data Scientist at Walmart

Feedforward and Backward Propagation in Gated Recurrent Unit (GRU)

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In this post, I’ll discuss how to implement a simple Recurrent Neural Network (RNN), specifically the Gated Recurrent Unit (GRU). I’ll present the feed forward proppagation of a GRU Cell at a single time stamp and then derive the formulas for determining parameter gradients using the concept of Backpropagation through time (BPTT).

Forward Propagation

The feedforward propagation equations for a GRU cell are expressed as:

\[z_t = \sigma(W_{zh}\ast h_{t-1} + W_{zx}\ast x_t)\] \[r_t = \sigma(W_{rh}\ast h_{t-1} + W_{rx}\ast x_t)\] \[\tilde{h}_t = f(W_h\ast (r_t \circ h_{t-1}) + W_x\ast x_t)\] \[h_t = z_t\circ h_{t-1} + (1-z_t)\circ \tilde{h}_t,\]

where $x_t$ is the input vector at time $t$, $h_t$ is the output vector, $\ast$ denotes matrix product, $\circ$ denotes element-wise product, $\sigma$ and $f$ are the sigmoid and Tanh activation functions, respectively.

Backward Propagation

Lets rewrite these set of equation in terms of unary and binary operations following the same order.

\[g_1 = W_{zh} \ast h_{t-1}\] \[g_2 = W_{zx} \ast x_{t}\] \[g_3 = g_1 + g_2\] \[z_t = \sigma(g_3)\] \[g_4 = W_{rh} \ast h_{t-1}\] \[g_5 = W_{rx} \ast x_{t}\] \[g_6 = g_4 + g_5\] \[r_t = \sigma(g_6)\] \[g_7 = r_t \circ h_{t-1}\] \[g_8 = W_{h} \ast g_7\] \[g_9 = W_{x} \ast x_{t}\] \[g_{10} = g_8 + g_9\] \[\tilde{h}_t = f(g_{10})\] \[g_{11} = z_t\circ h_{t-1}\] \[g_{12} = (1-z_t)\circ \tilde{h}_t\] \[h_t = g_{11} + g_{12}\]

Now, we’‘ll work our way backward to compute the parameter gradients, i.e. the derivatives of loss $L$ with respect to parameters. Let assume that gradient of loss with repect to output is known and denoted as $\Delta_{h_t}L$.

  • Eq. (20) gives
\[\Delta_{g_{11}}L = \Delta_{g_{12}}L = \Delta_{h_t}L\]
  • Eq. (19) gives
\[\Delta_{z_{t}}L = -\Delta_{g_{12}}L \circ \tilde{h}_t\] \[\Delta_{\tilde{h}_t}L = \Delta_{g_{12}}L \circ (1-z_t)\]
  • Eq. (18) gives
\[\Delta_{z_{t}}L += \Delta_{g_{11}}L \circ h_{t-1}\] \[\Delta_{h_{t-1}}L = \Delta_{g_{11}}L \circ z_t\]

Second time we’re computing a derivative for $z_t$, so we increment the derivative $(+=)$.

  • Eq. (17) gives
\[\Delta_{g_{10}}L = \Delta_{\tilde{h}_t}L \circ f'(g_{10})\]
  • Eq. (16) gives
\[\Delta_{g_{8}}L = \Delta_{g_{9}}L = \Delta_{g_{10}}L\]
  • Eq. (15) gives
\[\Delta_{W_x}L = \Delta_{g_9}L \ast x_t^T\] \[\Delta_{x_t}L = W_x^T \ast \Delta_{g_9}L\]
  • Eq. (14) gives
\[\Delta_{W_h}L = \Delta_{g_8}L \ast g_7^T\] \[\Delta_{g_7}L = W_h^T \ast \Delta_{g_8}L\]
  • Eq. (13) gives
\[\Delta_{r_t}L = \Delta_{g_7}L \circ h_{t-1}\] \[\Delta_{h_{t-1}}L += \Delta_{g_7}L \circ r_t\]
  • Eq. (12) gives
\[\Delta_{g_6}L = \Delta_{r_t}L \circ \sigma'(g_6)\]
  • Eq. (11) gives
\[\Delta_{g_4}L = \Delta_{g_5}L = \Delta_{g_6}L\]
  • Eq. (10) gives
\[\Delta_{W_{rx}}L = \Delta_{g_5}L \ast x_t^T\] \[\Delta_{x_t}L += W_{rx}^T \ast \Delta_{g_5}L\]
  • Eq. (9) gives
\[\Delta_{W_{rh}}L = \Delta_{g_4}L \ast h_{t-1}^T\] \[\Delta_{h_{t-1}}L += W_{rh}^T \ast \Delta_{g_4}L\]
  • Eq. (8) gives
\[\Delta_{g_3}L = \Delta_{z_t}L \circ \sigma'(g_3)\]
  • Eq. (7) gives
\[\Delta_{g_1}L = \Delta_{g_2}L = \Delta_{g_3}L\]
  • Eq. (6) gives
\[\Delta_{W_{zx}}L = \Delta_{g_2}L \ast x_t^T\] \[\Delta_{x_t}L += W_{zx}^T \ast \Delta_{g_2}L\]
  • Eq. (5) gives
\[\Delta_{W_{zh}}L = \Delta_{g_1}L \ast h_{t-1}^T\] \[\Delta_{h_{t-1}}L += W_{zh}^T \ast \Delta_{g_1}L\]

This completes all the required formulas for computing derivative with respect to all the parameters of network.

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