MathComputational mathOptimization

Pitfalls of gradient descent

Theory

Good steps: curvature condition

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Choose the step size that doesn't guarantee that the curvature condition holds true for the function

f(x,y,z)=x2y−xz+y3z.f(x,y,z)=x^2y - xz + y^3z.Consider xk=[2,−1,−4]\bold{x_k} = [2,-1,-4] and c2=0.9c_2 = 0.9.

Tip: For the curvature condition to hold true, the following must be true: ∣D∇f(xk)f(xk−γk⋅∇f(xk))∣≤c2∣D∇f(xk)f(xk)∣| D_{\nabla f(\bold{x}_k)}f(\bold{x}_k - \gamma_k \cdot \nabla f(\bold{x}_k)) | ≤ c_2 |D_{\nabla f(\bold{x}_k)}f(\bold{x}_k)|

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