  Smoothed functional (SF) schemes for gradient estimation are known to be
efficient in stochastic optimization algorithms, specially when the objective
is to improve the performance of a stochastic system. However, the performance
of these methods depends on several parameters, such as the choice of a
suitable smoothing kernel. Different kernels have been studied in literature,
which include Gaussian, Cauchy and uniform distributions among others. This
paper studies a new class of kernels based on the q-Gaussian distribution, that
has gained popularity in statistical physics over the last decade. Though the
importance of this family of distributions is attributed to its ability to
generalize the Gaussian distribution, we observe that this class encompasses
almost all existing smoothing kernels. This motivates us to study SF schemes
for gradient estimation using the q-Gaussian distribution. Using the derived
gradient estimates, we propose two-timescale algorithms for optimization of a
stochastic objective function in a constrained setting with projected gradient
search approach. We prove the convergence of our algorithms to the set of
stationary points of an associated ODE. We also demonstrate their performance
numerically through simulations on a queuing model.
