Update backpropagation.qmd

Author: phillipi

backpropagation.qmd

@@ -84,13 +84,11 @@ $$\begin{aligned}
     \frac{\partial J}{\partial \theta_1} &= \mathchoice%
   {\colorbox{shared_term_color}{$\displaystyle\frac{\partial J}{\partial \mathbf{x}_L}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}}%
   {\colorbox{shared_term_color}{$\textstyle\frac{\partial J}{\partial \mathbf{x}_L}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}}%
-  {\colorbox{shared_term_color}{$\scriptstyle\frac{\partial J}{\partial \mathbf{x}_L}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}}%
-  {\colorbox{shared_term_color}{$\scriptscriptstyle\frac{\partial J}{\partial \mathbf{x}_L}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}} \frac{\partial \mathbf{x}_2}{\mathbf{x}_1}\frac{\partial \mathbf{x}_1}{\partial \mathbf{\theta}_1}\\
+  \frac{\partial \mathbf{x}_2}{\mathbf{x}_1}\frac{\partial \mathbf{x}_1}{\partial \mathbf{\theta}_1}\\
     \frac{\partial J}{\partial \theta_2} &=  \mathchoice%
   {\colorbox{shared_term_color}{$\displaystyle\frac{\partial J}{\partial \mathbf{x}_{L}}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}}%
   {\colorbox{shared_term_color}{$\textstyle\frac{\partial J}{\partial \mathbf{x}_{L}}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}}%
-  {\colorbox{shared_term_color}{$\scriptstyle\frac{\partial J}{\partial \mathbf{x}_{L}}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}}%
-  {\colorbox{shared_term_color}{$\scriptscriptstyle\frac{\partial J}{\partial \mathbf{x}_{L}}\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}} \cdots \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}$}} \frac{\partial \mathbf{x}_2}{\partial \theta_2}
+  \frac{\partial \mathbf{x}_2}{\partial \theta_2}
 \end{aligned}$$ Rather than evaluating both equations separately, we
 notice that all the terms in each gray box are shared. We only need to
 evaluate this product once, and then can use it to compute both