Update backpropagation.qmd

Author: phillipi

backpropagation.qmd

@@ -79,36 +79,27 @@ $\frac{\partial J}{\partial \theta_1}$,
 $\frac{\partial J}{\partial \theta_2}$, etc. Each of these gradients can
 be calculated via the chain rule. Here is the chain rule written out for
 the gradients for $\theta_1$ and $\theta_2$: 
-
 \[
+\newcommand{\sharedterm}{%
+  \colorbox{shared_term_color}{%
+    $\displaystyle
+    \frac{\partial J}{\partial \mathbf{x}_L}
+    \frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}}
+    \cdots
+    \frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}
+    $%
+  }%
+}
 \begin{aligned}
-\frac{\partial J}{\partial \theta_1} &= 
-\mathchoice%
-{\colorbox{shared_term_color}{\ensuremath{\displaystyle
-\frac{\partial J}{\partial \mathbf{x}_L}
-\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}}
-\cdots
-\frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}}}}%
-{\colorbox{shared_term_color}{\ensuremath{\textstyle
-\frac{\partial J}{\partial \mathbf{x}_L}
-\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}}
-\cdots
-\frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}}}}%
+\frac{\partial J}{\partial \theta_1}
+&=
+\sharedterm\,
 \frac{\partial \mathbf{x}_2}{\partial \mathbf{x}_1}
 \frac{\partial \mathbf{x}_1}{\partial \theta_1}
 \\
-\frac{\partial J}{\partial \theta_2} &= 
-\mathchoice%
-{\colorbox{shared_term_color}{\ensuremath{\displaystyle
-\frac{\partial J}{\partial \mathbf{x}_L}
-\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}}
-\cdots
-\frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}}}}%
-{\colorbox{shared_term_color}{\ensuremath{\textstyle
-\frac{\partial J}{\partial \mathbf{x}_L}
-\frac{\partial \mathbf{x}_L}{\partial \mathbf{x}_{L-1}}
-\cdots
-\frac{\partial \mathbf{x}_3}{\partial \mathbf{x}_2}}}}%
+\frac{\partial J}{\partial \theta_2}
+&=
+\sharedterm\,
 \frac{\partial \mathbf{x}_2}{\partial \theta_2}
 \end{aligned}
 \]