  Using quasi-Newton methods in stochastic optimization is not a trivial task
given the difficulty of extracting curvature information from the noisy
gradients. Moreover, pre-conditioning noisy gradient observations tend to
amplify the noise. We propose a Bayesian approach to obtain a Hessian matrix
approximation for stochastic optimization that minimizes the secant equations
residue while retaining the extreme eigenvalues between a specified range.
Thus, the proposed approach assists stochastic gradient descent to converge to
local minima without augmenting gradient noise. We propose maximizing the log
posterior using the Newton-CG method. Numerical results on a stochastic
quadratic function and an $\ell_2$-regularized logistic regression problem are
presented. In all the cases tested, our approach improves the convergence of
stochastic gradient descent, compensating for the overhead of solving the log
posterior maximization. In particular, pre-conditioning the stochastic gradient
with the inverse of our Hessian approximation becomes more advantageous the
larger the condition number of the problem is.
