Gradient Descent: The Ultimate Optimizer (1909.13371v2)

Abstract: Working with any gradient-based machine learning algorithm involves the tedious task of tuning the optimizer's hyperparameters, such as its step size. Recent work has shown how the step size can itself be optimized alongside the model parameters by manually deriving expressions for "hypergradients" ahead of time. We show how to automatically compute hypergradients with a simple and elegant modification to backpropagation. This allows us to easily apply the method to other optimizers and hyperparameters (e.g. momentum coefficients). We can even recursively apply the method to its own hyper-hyperparameters, and so on ad infinitum. As these towers of optimizers grow taller, they become less sensitive to the initial choice of hyperparameters. We present experiments validating this for MLPs, CNNs, and RNNs. Finally, we provide a simple PyTorch implementation of this algorithm (see people.csail.mit.edu/kach/gradient-descent-the-ultimate-optimizer).

• The paper presents an automated method for computing hypergradients using AD to efficiently tune learning rate and momentum parameters.
• It introduces a recursive optimization hierarchy that minimizes manual intervention while being applicable to diverse optimizers like Adam and RMSProp.
• Experimental results show that the approach consistently reduces test error and scales well across various architectures including CNNs and RNNs.

An Overview of "Gradient Descent: The Ultimate Optimizer"

The paper "Gradient Descent: The Ultimate Optimizer" by Chandra et al. presents a framework that automates the computation of hypergradients through modifications to existing backpropagation techniques. The main contribution of this paper is the introduction of an automated, recursive approach to hyperparameter tuning which leverages automatic differentiation (AD) to optimize hyperparameters such as the learning rate and momentum coefficients.

Problem Statement

Gradient descent and its variants are foundational optimization algorithms in machine learning, particularly in training deep neural networks. However, the performance of these algorithms heavily depends on the choice of hyperparameters like the step size (learning rate) and momentum coefficients. Traditionally, optimizing these hyperparameters is a manual process that involves significant trial and error.

Previous research has focused on manually deriving hypergradients for specific hyperparameters to automate tuning. Despite these efforts, existing methods are limited due to their reliance on tedious, error-prone manual differentiation and the restriction to specific hyperparameters.

Key Contributions

1. Automatic Differentiation for Hypergradients: The paper introduces a method to compute hypergradients automatically using AD by altering the computation graph in a way that retains connections to hyperparameters during backpropagation. This replaces the need for manual differentiation, thus making the process more general and less error-prone.
2. Recursive Hyperoptimization: The method allows for recursive application whereby both hyperparameters and their hyper-hyperparameters can be optimized. This constructs a hierarchy (or tower) of optimizers, each adjusting the parameters of the level beneath it. As these towers grow, they exhibit increased insensitivity to initial hyperparameter values, reducing the human effort required for tuning.
3. Generalization Across Optimizers: Unlike previous approaches limited to specific variants such as SGD, this framework can be directly applied to different optimizers, including AdaGrad, RMSProp, and Adam. It allows the simultaneous optimization of multiple hyperparameters inherent to these optimizers.

Experimental Validation

The authors validate their approach across multiple neural network architectures, including MLPs, CNNs, and RNNs, using datasets such as MNIST and CIFAR-10. The experiments demonstrate that the proposed hyperoptimization method outperforms baseline models with fixed hyperparameters, evidencing robustness across a spectrum of initial hyperparameter configurations.

• Results show that the automated hyperoptimizer consistently reduces test error compared to standard gradient descent methods.
• In experiments involving advanced architectures like ResNet-20 and Char-RNNs, the method achieves comparable or superior performance to manually tuned learning rate schedules.

Implications and Future Directions

The research offers considerable implications for both theoretical advancements and practical applications in AI:

1. Reduced Human Intervention: By minimizing the reliance on human tuning of hyperparameters, this method can streamline the deployment of machine learning models.
2. Scalability: The approach scales efficiently even as the optimizer hierarchy grows, with minimal computational overhead, indicating potential for application in large-scale systems.
3. Broader Applicability: Although focused on hyperparameter optimization in deep learning, the principles could be extended to other domains of computational optimization where gradient descent is applicable.

Future directions for this research may involve investigating the convergence properties of such recursive hyperoptimize frameworks and addressing stability issues when initial hyperparameters are significantly misconfigured.

In summary, the paper introduces an elegant, scalable framework for hyperparameter optimization via recursive, automated differentiation, marking a significant step toward more autonomous machine learning models.