# We'll need the indicator function

def indicator$\_$function$($x$)$:

x$\_$in $=$ np$.$array$($x$)$

x$\_$in$[$x$\_$in$>=0]=1$

x$\_$in$[$x$\_$in$<0]=0$

return x$\_$in

# Main backward pass routine

def backward$\_$pass$($all$\_$weights, all$\_$biases, all$\_$f$,$ all$\_$h$,$ y$)$:

# We'll store the derivatives dl$\_$dweights and dl$\_$dbiases in lists as well

all$\_$dl$\_$dweights $=[$None$]*($K$+1)$

all$\_$dl$\_$dbiases $=[$None$]*($K$+1)$

# And we'll store the derivatives of the loss with respect to the activation and preactivations in lists

all$\_$dl$\_$df $=[$None$]*($K$+1)$

all$\_$dl$\_$dh $=[$None$]*($K$+1)$

# Again for convenience we'll stick with the convention that all$\_$h$[0]$ is the net input and all$\_$f$[$k$]$ in the net output

# Compute derivatives of the loss with respect to the network output

all$\_$dl$\_$df$[$K$]=$ np$.$array$($d$\_$loss$\_$d$\_$output$($all$\_$f$[$K$],$y$))$

# Now work backwards through the network

for layer in range$($K$,-1,-1)$:

# TODO Calculate the derivatives of the loss with respect to the biases at layer from all$\_$dl$\_$df$[$layer$].($eq $7\mathrm{.}21)$

# NOTE! To take a copy of matrix X$,$ use Z$=$np$.$array$($X$)$

# REPLACE THIS LINE

all$\_$dl$\_$dbiases$[$layer$]=$ np$.$zeros$\_$like$($all$\_$biases$[$layer$])$

# TODO Calculate the derivatives of the loss with respect to the weights at layer from all$\_$dl$\_$df$[$layer$]$ and all$\_$h$[$layer$]($eq $7\mathrm{.}22)$

# Don't forget to use np$.$matmul

# REPLACE THIS LINE

all$\_$dl$\_$dweights$[$layer$]=$ np$.$zeros$\_$like$($all$\_$weights$[$layer$])$

# TODO: calculate the derivatives of the loss with respect to the activations from weight and derivatives of next preactivations $($second part of last line of eq $7\mathrm{.}24)$

# REPLACE THIS LINE

all$\_$dl$\_$dh$[$layer$]=$ np$.$zeros$\_$like$($all$\_$h$[$layer$])$

if layer $>0$:

# TODO Calculate the derivatives of the loss with respect to the pre$-$activation f $($use derivative of ReLu function, first part of last line of eq$.7\mathrm{.}24)$

# REPLACE THIS LINE

all$\_$dl$\_$df$[$layer$-1]=$ np$.$zeros$\_$like$($all$\_$f$[$layer$-1])$

return all$\_$dl$\_$dweights, all$\_$dl$\_$dbiases

all$\_$dl$\_$dweights, all$\_$dl$\_$dbiases $=$ backward$\_$pass$($all$\_$weights, all$\_$biases, all$\_$f$,$ all$\_$h$,$ y$)$

np$.$set$\_$printoptions$($precision$=3)$

# Make space for derivatives computed by finite differences

all$\_$dl$\_$dweights$\_$fd $=[$None$]*($K$+1)$

all$\_$dl$\_$dbiases$\_$fd $=[$None$]*($K$+1)$

# Let's test if we have the derivatives right using finite differences

delta$\_$fd $=0\mathrm{.}000001$

# Test the dervatives of the bias vectors

for layer in range$($K$)$:

dl$\_$dbias $=$ np$.$zeros$\_$like$($all$\_$dl$\_$dbiases$[$layer$])$

# For every element in the bias

for row in range$($all$\_$biases$[$layer$].$shape$[0])$:

# Take copy of biases We'll change one element each time

all$\_$biases$\_$copy $=[$np$.$array$($x$)$ for x in all$\_$biases$]$

all$\_$biases$\_$copy$[$layer$][$row$]+=$ delta$\_$fd

network$\_$output$\_1,*\_=$ compute$\_$network$\_$output$($net$\_$input, all$\_$weights, all$\_$biases$\_$copy$)$

network$\_$output$\_2,*\_=$ compute$\_$network$\_$output$($net$\_$input, all$\_$weights, all$\_$biases$)$

dl$\_$dbias$[$row$]=($least$\_$squares$\_$loss$($network$\_$output$\_1,$ y$)-$ least$\_$squares$\_$loss$($network$\_$output$\_2,$y$))/$delta$\_$fd

all$\_$dl$\_$dbiases$\_$fd$[$layer$]=$ np$.$array$($dl$\_$dbias$)$

print$("-----------------------------------------------")$

print$("$Bias $\%$d$,$ derivatives from backprop:"$\%($layer$))$

print$($all$\_$dl$\_$dbiases$[$layer$])$

print$("$Bias $\%$d$,$ derivatives from finite differences"$\%($layer$))$

print$($all$\_$dl$\_$dbiases$\_$fd$[$layer$])$

if np$.$allclose$($all$\_$dl$\_$dbiases$\_$fd$[$layer$],$all$\_$dl$\_$dbiases$[$layer$],$rtol$=1$e$-05,$ atol$=1$e$-08,$ equal$\_$nan$=$False$)$:

print$("$Success$!$ Derivatives match."$)$

else:

print$("$Failure$!$ Derivatives different."$)$

# Test the derivatives of the weights matrices

for layer in range$($K$)$:

dl$\_$dweight $=$ np$.$zeros$\_$like$($all$\_$dl$\_$dweights$[$layer$])$

# For every element in the bias

for row in range$($all$\_$weights$[$layer$].$shape$[0])$:

for col in range$($all$\_$weights$[$layer$].$shape$[1])$:

# Take copy of biases We'll change one element each time

all$\_$weights$\_$copy $=[$np$.$array$($x$)$ for x in all$\_$weights$]$

all$\_$weights$\_$copy$[$layer$][$row$][$col$]+=$ delta$\_$fd

network$\_$output$\_1,*\_=$ compute$\_$network$\_$output$($net$\_$input, all$\_$weights$\_$copy, all$\_$biases$)$

network$\_$output$\_2,*\_=$ compute$\_$network$\_$output$($net$\_$input, all$\_$weights, all$\_$biases$)$

dl$\_$dweight$[$row$][$col$]=($least$\_$squares$\_$loss$($network$\_$output$\_1,$ y$)-$ least$\_$squares$\_$loss$($network$\_$output$\_2,$y$))/$delta$\_$fd

all$\_$dl$\_$dweights$\_$fd$[$layer$]=$ np$.$array$($dl$\_$dweight$)$

print$("-----------------------------------------------")$

print$("$Weight $\%$d$,$ derivatives from backprop:"$\%($layer$))$

print$($all$\_$dl$\_$dweights$[$layer$])$

print$("$Weight $\%$d$,$ derivatives from finite differences"$\%($layer$))$

print$($all$\_$dl$\_$dweights$\_$fd$[$layer$])$

if np$.$allclose$($all$\_$dl$\_$dweights$\_$fd$[$layer$],$all$\_$dl$\_$dweights$[$layer$],$rtol$=1$e$-05,$ atol$=1$e$-08,$ equal$\_$nan$=$False$)$:

print$("$Success$!$ Derivatives match."$)$

else:

print$("$Failure$!$ Derivatives different."$)$