SPSA#

spsa(fun, x0, args, maxiter=50, a=2.0, c=0.1, alpha=0.702, gamma=0.201, seed=3)[source]#

Minimize a scalar function of one or more variables using the Simultaneous Perturbation Stochastic Approximation algorithm.

See minimize() for a description of fun, x0, and args.

This algorithm aims at finding the optimal control \(x^*\) minimizing a given loss fuction \(f\):

\[x^* = \text{argmin}_{x} f(x)\]

This is done by an iterative process starting from an initial guess \(x_0\):

\[x_{k+1} = x_k - a_kg_k(x_k)\]

where \(a_k=\dfrac{a}{(k+1)^{\alpha}}\) for scaling parameters \(a, \alpha>0\).

For each step \(x_k\) the gradient is approximated by

\[(g_k(x_k))_i = \frac{f(x_k+c_k\Delta_k)-f(x_k-c_k\Delta_k)}{2c_k(\Delta_k)_i}\]

where \(c_k=\dfrac{c}{(k+1)^{\gamma}}\) for scaling parameters \(c, \gamma>0\), and \(\Delta_k\) is a random perturbation vector with components independently drawn from \(\{-1, +1\}\).

Parameters:

maxiterint: Maximum number of iterations to perform. Each iteration requires 2 function evaluations.
afloat: Scaling parameter for update rule.
cfloat: Scaling parameter for gradient estimation.
alphafloat: Scaling exponent for update rule.
gammafloat: Scaling exponent for gradient estimation.
seedint, optional: The seed for the pseudo-random number generator used to draw the perturbation vectors \(\Delta_k\). The default is 3.

Returns:

results: An OptimizeResults object.