23_Gradient Checking

Exercise: To show that the backward_propagation() function is correctly computing the gradient $\frac{\partial J}{\partial \theta}$, let's implement gradient checking.

Instructions:

• First compute "gradapprox" using the formula above (1) and a small value of $\varepsilon$. Here are the Steps to follow:

  1. $\theta^{+} = \theta + \varepsilon$

  2. $\theta^{-} = \theta - \varepsilon$

  3. $J^{+} = J(\theta^{+})$

  4. $J^{-} = J(\theta^{-})$

  5. $gradapprox = \frac{J^{+} - J^{-}}{2 \varepsilon}$

• Then compute the gradient using backward propagation, and store the result in a variable "grad"

• Finally, compute the relative difference between "gradapprox" and the "grad" using the following formula:

  difference = \frac {\mid\mid grad - gradapprox \mid\mid_2}{\mid\mid grad \mid\mid_2 + \mid\mid gradapprox \mid\mid_2} \tag{2}

  You will need 3 Steps to compute this formula:

  • 1'. compute the numerator using np.linalg.norm(...)

  • 2'. compute the denominator. You will need to call np.linalg.norm(...) twice.

  • 3'. divide them.

• If this difference is small (say less than $10^{-7}$), you can be quite confident that you have computed your gradient correctly. Otherwise, there may be a mistake in the gradient computation.