A new perspective on optimizers: leveraging Moreau-Yosida approximation in gradient-based learning

Abstract

Machine Learning (ML) heavily relies on optimization techniques built upon gradient descent. Numerous gradient-based update methods have been proposed in the scientific literature, particularly in the context of neural networks, and have gained widespread adoption as optimizers in ML software libraries. This paper introduces a novel perspective by framing gradient-based update strategies using the Moreau-Yosida (MY) approximation of the loss function. Leveraging a first-order Taylor expansion, we demonstrate the concrete exploitability of the MY approximation to generalize the model update process. This enables the evaluation and comparison of regularization properties underlying popular optimizers like gradient descent with momentum, ADAGRAD, RMSprop, and ADAM. The MY-based unifying view opens up possibilities for designing new update schemes with customizable regularization properties. To illustrate this potential, we propose a case study that redefines the concept of closeness in the parameter space using network outputs. We present a proof-of-concept experimental procedure, demonstrating the effectiveness of this approach in continual learning scenarios. Specifically, we employ the well-known permuted MNIST dataset, a progressively-permuted MNIST and CIFAR-10 benchmarks, and a non i.i.d. stream. Additionally, we validate the update scheme’s efficacy in an offline-learning scenario. By embracing the MY-based unifying view, we pave the way for advancements in optimization techniques for machine learning.