You are correct in that there will be m^n minima, but since your surface is highly symmetric, local minima are trivially easy to find. Are you simplifying the s

You are correct in that there will be m^n minima, but since your surface is highly symmetric, local minima are trivially easy to find. Are you simplifying the statement by skipping over the importance of intervals? If the intervals spanned by x,y,z do not overlap, then your statement does not hold unless local minima are evenly distributed over the entire region, which defeats the purpose of searching for minima in the first place. If the intervals spanned by x,y,z are identical, then gradient descent can be reduced to just searching over f(x) by symmetry. You are right that there are m^n minima, but they are found in (nm)^1 steps. If the intervals spanned by x,y,z partially overlap, you can further reduce the search space to just f(x) and the union of the intervals. I don't see how your example shows that gradient descent is likely to fail for any classifier with a sufficiently high number of independent parameters. Am I missing something?