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This article provides a comprehensive understanding of optimization in deeplearning, with a primary focus on the challenges of gradient vanishing andgradient exploding, which normally lead to diminished model representationalability and training instability, respectively. We analyze these two challengesthrough several strategic measures, including the improvement of gradient flowand the imposition of constraints on a network's Lipschitz constant. To helpunderstand the current optimization methodologies, we categorize them into twoclasses: explicit optimization and implicit optimization. Explicit optimizationmethods involve direct manipulation of optimizer parameters, including weight,gradient, learning rate, and weight decay. Implicit optimization methods, bycontrast, focus on improving the overall landscape of a network by enhancingits modules, such as residual shortcuts, normalization methods, attentionmechanisms, and activations. In this article, we provide an in-depth analysisof these two optimization classes and undertake a thorough examination of theJacobian matrices and the Lipschitz constants of many widely used deep learningmodules, highlighting existing issues as well as potential improvements.Moreover, we also conduct a series of analytical experiments to substantiateour theoretical discussions. This article does not aim to propose a newoptimizer or network. Rather, our intention is to present a comprehensiveunderstanding of optimization in deep learning. We hope that this article willassist readers in gaining a deeper insight in this field and encourages thedevelopment of more robust, efficient, and high-performing models.

Understanding Optimization of Deep Learning via Jacobian Matrix and Lipschitz Constant