Suppose we apply gradient ascent algorithm to find a local maximum of the function f(x,y) = 3x cos^3 *y + 1/x. If we start from the point (x0,y0) = (−1, π/3 ) and use learning rate α = 0.1 to go uphill as high as possible (initially), what point (x1,y1) will we end up at after the first step?
If I understand this correctly, it's just a 2-d version of
∆y = y' ∆x
f(x,y) = 3x cos^3 *y + 1/x
∇f = ( 3cos^3y - 1/x^2) i - 9x cos^2y siny j
so
(x1,y1) = (x0,y0) + α∇f
= -1i + π/3 j + 0.1(3(1/2)^3 - 1)i -(9(-1)(1/2)^2(√3/2))j
= (-1,π/3) + (-5/80)i + (9√3/80)j
= (-1.06,1.24)