The limitation of neural nets for approximation and optimization

We are interested in assessing the use of neural networks as surrogate models to approximate and minimize objective functions in optimization problems. While neural networks are widely used for machine learning tasks such as classification and regression, their application in solving optimization problems has been limited. Our study begins by determining the best activation function for approximating the objective functions of popular nonlinear optimization test problems, and the evidence provided shows that ReLU and SiLU exhibit the best performance on both training and testing data. We then analyze the accuracy of function value, gradient, and Hessian approximations for such objective functions obtained through interpolation/regression models and neural networks. When compared to interpolation/regression models, neural networks can deliver competitive zero- and first-order approximations (at a high training cost) but underperform on second-order approximation. However, it is shown that combining a neural net activation function with the natural basis for quadratic interpolation/regression can waive the necessity of including cross terms in the natural basis, leading to models with fewer parameters to determine. Lastly, we provide evidence that the performance of a state-of-the-art derivative-free optimization algorithm can hardly be improved when the gradient of an objective function is approximated using any of the surrogate models considered, including neural networks.