Mathematical studies in deep learning: overparameterization theories, graph generative models and applications in algorithmic pricing

Abstract

This dissertation explores three key areas in deep learning. First, it extends the existing theoretical results on overparameterized deep neural networks to the ones incorporating leaky ReLU activation functions, offering insights into effective hyperparameter selection for such networks. Second, it proposes a novel graph data generation method, introducing an unpooling layer that leverages graph structures and resolves the non-differentiability in training using the policy gradient approach. Finally, the dissertation examines algorithmic pricing in two-sided markets, investigating how Q-learning strategies may lead to unintended collusion and proposing policy to mitigate this risk.