For t=1, …, number_of_batches:
Vectorized Forward prop on $X^{t}$
$Z^{[1]} = W^{[1]} X^{t} + b^{[1]}$
$A^{[1]} = g^{[1]}(Z^{[1]})$
...
$A^{[L]} = g^{[L]}(Z^{[L]})$
Compute cost $J^{[t]}$ = 1/1000 * ...
Backprop to compute gradients for $J^{[t]}$
Update weights $W^{[l]} = W^{[l]} - \alpha d W^{[l]}; b^{[l]} = b^{[l]} - \alpha d b^{[l]}$
$V_t = \beta V_{t-1} + (1-\beta) \Theta_t$
$\beta = 0.98$ is smoother than $\beta = 0.5$ (2 days for average in latter case)
$V_\Theta = 0$
$V_\Theta = \beta V + (1-\beta) \Theta_1$
$V_\Theta = \beta V + (1-\beta) \Theta_2$
…
Easy method to compute averages for longer periods
$V_0 = 0$
$V_1 = \beta V_0 + 0,02 \Theta_1$
$V_\Theta = \beta V + 0,02 \Theta_2$
…
Corrected:
$\frac{V_t}{1-\beta^t}$
Idea: Compute exponentially weighted average of gradients and use it to update weights.
$V_{dW} = \beta V_{dW} + (1-\beta) dW$
$V_{db} = \beta V_{db} + (1-\beta) db$
$W = W - \alpha V_{dW}$
$b = b - \alpha V_{db}$
$\beta$ friction; $V_{db}$ velocity; $db$ acceleration
Often $\beta = 0.9$ is used.
Root mean squared
Goal: Slow movements in vertical direction (b direction), fast in horizontal (w direction; bowl is wider than high)
Compute dW, db on minibatch
$s_{dW} = \beta s_{dW} + (1-\beta) d W^2$ (element-wise squared), small
$s_{db} = \beta s_{db} + (1-\beta) d b^2$ large
$W = W - \alpha dW/\sqrt{s_{dW}}$ (same for b)
Adaptive moment estimation
Momentum + RMSprop + Bias correction
$\alpha = \frac{1}{1+ \text{decay_rate} * \text{epoch_num}} \alpha_0$
or
$\alpha = 0,95^{\text{epoch_num}} \alpha_0$
In high-dimensional spaces it's more likely to end up at a saddle point (than in local optima). E.g. 20000 parameter, highly unlikely that it's a local minimum you get stuck. Plateus make learning slow.