# A stochastic gradient descent approach for risk optimization using the Chernoff bound

## Palavras-chave:

Stochastic Gradient Descent, Risk Optimization, Chernoff Bound## Resumo

We propose a method for solving Risk Optimization (RO) problems based on the Stochastic Gradient

Descent (SGD) methods. SGD is used to minimize the expectation of functions. We approximate each limit state

function in the RO problem using the Chernoff bound, thus recasting the original RO problem as an expectation

minimization problem. The Chernoff bound approximation requires the evaluation of Monte Carlo sampling,

which could be expensive. However, once the Chernoff bound parameters are set, they can be used to cheaply

approximate the probabilities of failure of each state limit for several iterations. We propose a heuristic approach to

tune the Chernoff bound parameters after a distance from the last update. Moreover, we decay the update distance

each iteration, thus guaranteeing that the probabilities of failure approximations are accurate as SGD converges

to the optimum solution. We present numerical results supporting the efficiency of our approach to different RO

problems with applications in structural engineering. Comparisons of SGD equipped with our Chernoff bound

approximation against particle swarm optimization using sample average approximation validate the efficiency of

the proposed approach.