  Bayesian approaches have been successfully integrated into training deep
neural networks. One popular family is stochastic gradient Markov chain Monte
Carlo methods (SG-MCMC), which have gained increasing interest due to their
scalability to handle large datasets and the ability to avoid overfitting.
Although standard SG-MCMC methods have shown great performance in a variety of
problems, they may be inefficient when the random variables in the target
posterior densities have scale differences or are highly correlated. In this
work, we present an adaptive Hessian approximated stochastic gradient MCMC
method to incorporate local geometric information while sampling from the
posterior. The idea is to apply stochastic approximation to sequentially update
a preconditioning matrix at each iteration. The preconditioner possesses
second-order information and can guide the random walk of a sampler
efficiently. Instead of computing and saving the full Hessian of the log
posterior, we use limited memory of the sample and their stochastic gradients
to approximate the inverse Hessian-vector multiplication in the updating
formula. Moreover, by smoothly optimizing the preconditioning matrix, our
proposed algorithm can asymptotically converge to the target distribution with
a controllable bias under mild conditions. To reduce the training and testing
computational burden, we adopt a magnitude-based weight pruning method to
enforce the sparsity of the network. Our method is user-friendly and is
scalable to standard SG-MCMC updating rules by implementing an additional
preconditioner. The sparse approximation of inverse Hessian alleviates storage
and computational complexities for large dimensional models. The bias
introduced by stochastic approximation is controllable and can be analyzed
theoretically. Numerical experiments are performed on several problems.

Numerical Analysis;Machine Learning;Numerical Analysis;Optimization and Control