Efficient and Robust Black-box Integral-approximation and Optimization

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Black-box optimization and black-box integral approximation are important techniques for machine learning, industrial design, and simulation in science. This thesis investigates black-box integral approximation and black-box optimization by considering the closed relationship between them. For integral approximation, we develop a simple closed-form rank-1 lattice construction method based on group theory. Our method reduces the number of distinct pairwise distance values to generate a more regular lattice. Furthermore, we investigate structured points set for integral approximation on hyper-sphere. Our structured point sets can serve as a good initialization for black-box optimization. Moreover, we propose stochastic black-box optimization with implicit natural gradients for black-box optimization. Our method is very simple and has only the step-size hyper-parameter. Furthermore, we develop a batch Bayesian optimization algorithm from the perspective of frequentist kernel methods, which is powerful for low-dimensional black-box optimization problems. We further apply our structured integral approximation techniques for kernel approximation. In addition, we develop structured approximation for robust deep neural network architecture, which results in an elegant and simple architecture that preserves optimization properties. Moreover, we develop adaptive loss as a tighter upper bound approximation for expected 0-1 risk, robust and trainable with SGD.
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