I What Exactly is Step Size in Gradient Descent Method?

[Dario56]

Gradient descent is numerical optimization method for finding local/global minimum of function. It is given by following formula: $$ x_{n+1} = x_n - \alpha \nabla f(x_n) $$ There is countless content on internet about this method use in machine learning. However, there is one thing I don't understand and which I couldn't find even though it is basic.

What exactly is step size $\alpha$ ?

Wikipedia states that it is tunning parameter in optimization algorithm which I understand, but not enough is being said about it to be considered a definition. Dimension analysis states that its dimensions should be $\frac {(\Delta x )^2} {\Delta y}$ which I am not sure how to interpret.

 

[jedishrfu]

Because it’s a tuning parameter, you must choose one wisely. Too small and your algorithm will run too long, too large and you will miss the valley or hill you’re trying to locate.

Here’s some discussion on it:

https://blog.datumbox.com/tuning-the-learning-rate-in-gradient-descent/

The blogger says it may be obsolete theory but I think it may still apply to what you’re asking about.

 

[Dario56]

Because it’s a tuning parameter, you must choose one wisely. Too small and your algorithm will run too long, too large and you will miss the valley or hill you’re trying to locate.

Here’s some discussion on it:

https://blog.datumbox.com/tuning-the-learning-rate-in-gradient-descent/

The blogger says it may be obsolete theory but I think it may still apply to what you’re asking about.

Thank you. I've got this in the meantime.

I had problem with understanding this parameter because I didn't look at the equation of gradient descent in vector form and it should be seen in this light since gradient is a vector valued function.

Parameter $\alpha$ basically defines how long in the direction of the gradient vector we want to go. If parameter has value for example 0,5, it means we move in the opposite direction (opposite because we subtract it from position vector $x_n$) of the gradient vector by length equal to 0,5 value of gradient vector at the point $x_n$.

Its value can be changed during optimization. If it is too big we can miss the minimum and if it is too small it can get too many iterations to converge.

I would say if value of gradient is big step size can be bigger and if gradient value is small that means we are close and so we need to make step size smaller not to miss the minimum we are close to.

 

[bigfooted]

There is a nice intuitive explanation of SD and CG in this tutorial style paper from Shewchuk, the first 30 pages are about steepest descent,
https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf