Optimization and Loss Landscape Geometry of Deep Learning

The impressive success of deep learning is powered by models that are rapidly growing in size along with the computational resources that are used to train them. Despite these growing demands, the dominant tools used to train these networks have not evolved significantly to match these needs. One natural hypothesis for this limitation is our lack of understanding of the training dynamics of deep neural networks. In this thesis, I present our research on understanding and improving the optimization of deep learning models. The thesis begins by presenting two first-order optimization algorithms for deep learning: the Aggregated Momentum and Lookahead optimizers. We demonstrate their success on modern deep learning optimization problems and provide theoretical analyses of both optimizers in convex settings. However, our theoretical understanding of optimization for practical training of deep neural networks is severely limited. Following this, we turn towards building a better understanding of deep learning optimization. We achieve this by studying the loss landscape geometry of deep neural networks. This is extremely challenging due to non-convex objective functions and extremely high-dimensional parameter spaces. We address this by first studying a simple class of neural networks: two-layer linear networks. Despite their simplicity, these models capture some core challenges of deep learning optimization effectively. Within this class, we investigate regularized linear autoencoders and linear variational autoencoders and carefully characterize their loss landscape geometry theoretically. We then move beyond the simple class of two-layer linear networks to investigate a phenomenon that arises across a vast set of deep learning optimization problems. This phenomenon, which we term the Monotonic Linear Interpolation (MLI) property, describes a global property of the loss landscape geometry of deep learning models. We provide the first theoretical explanation of this phenomenon and conduct a thorough empirical investigation to better understand the pervasiveness and limitations of the MLI property. In the final chapter, the thesis is discussed as a whole and promising directions for future research are presented.

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