​​This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple neural network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the neural network parameters. This geometric characterization provides new perspective to three tasks. Specifically, we propose a new tropical perspective to the lottery ticket hypothesis, where we see the effect of different initializations on the tropical geometric representation of a network’s decision boundaries. Moreover, we use this characterization to propose a new set of tropical regularizers, which directly deal with the decision boundaries of a network. We investigate the use of these regularizers in neural network pruning (by removing network parameters that do not contribute to the tropical geometric representation of the decision boundaries) and in generating adversarial input attacks (by producing input perturbations that explicitly perturb the decision boundaries’ geometry and ultimately change the network’s prediction).

​​This work takes a step towards investigating the benefits of merging classical vision techniques with deep learning models. Formally, we explore the effect of replacing the first layers of neural network architectures with convolutional layers that are based on Gabor filters with learnable parameters. As a first result, we observe that architectures utilizing Gabor filters as low-level kernels are capable of preserving test set accuracy of deep convolutional networks. Therefore, this architectural change exalts their capabilities in extracting useful low-level features. Furthermore, we observe that the architectures enhanced with Gabor layers gain advantages in terms of robustness when compared to the regular models. Additionally, the existence of a closed mathematical expression for the Gabor kernels allows us to develop an analytical expression for an upper bound to the Lipschitz constant of the Gabor layer. This expression allows us to propose a simple regularizer to enhance the robustness of the network. We conduct extensive experiments with several architectures and datasets, and show the beneficial effects that the introduction of Gabor layers has on the robustness of deep convolutional networks.

​​We consider the problem of unconstrained minimization of a smooth objective function in ℝd in setting where only function evaluations are possible. We propose and analyze stochastic zeroth-order method with heavy ball momentum. In particular, we propose, SMTP, a momentum version of the stochastic three-point method (STP). We show new complexity results for non-convex, convex and strongly convex functions. We test our method on a collection of learning to continuous control tasks on several MuJoCo environments with varying difficulty and compare against STP, other state-of-the-art derivative-free optimization algorithms and against policy gradient methods. SMTP significantly outperforms STP and all other methods that we considered in our numerical experiments. Our second contribution is SMTP with importance sampling which we call SMTP_IS. We provide convergence analysis of this method for non-convex, convex and strongly convex objectives.