gradient ascent vs gradient descent update rule I'm trying to understand the differences between the update rule for stochastic gradient ascent and descent. I've read some articles and still don't understand how to calculate the update rule:
Gradient Descent:
$\frac{∂NLL(θ)}{∂θ_j} = (hθ(x(i))-y(i))x(i)$
So the update rule uses a minus sign because we want to minimize the log-likelihood:
$θ_j:= θ_j−α(\frac{∂NLL(θ)}{∂θ_j}) = θ_j−α[(hθ(x(i))-y(i))x(i)]$
Gradient Ascent:
$\frac{∂LL(θ)}{∂θ_j} =(y(i) - hθ(x(i)))x(i)$
So the update rule uses a minus sign because we want to maximize the log-likelihood:
$θ_j:= θ_j+α(\frac{∂LL(θ)}{∂θ_j}) = θ_j+α[(hθ(x(i))-y(i))x(i)]$
Now, if I take the -1 from the derivative of the NLL I end up with the same equation as for the update rule for the gradient ascent:
$θ_j:= θ_j−α(\frac{∂NLL(θ)}{∂θ_j}) = θ_j−α(-1)[(-hθ(x(i))+y(i))x(i)] = θ_j+α[(y(i)-hθ(x(i)))x(i)]$
So, I know I'm wrong as they shouldn't be the same right? I'll appreciate your help understanding this.

You used $- \alpha (-1)$. You need to pick one, either you use $- \alpha$ or $+ \alpha(-1)$.

So, I know I'm wrong as they shouldn't be the same right?

They should be the same. Maximizing function $f$ is the same as minimizing $-f$. Gradient ascent of $f$ is the same as gradient descent of $-f$.
