Unraveling Implicit Gradient Regularization in Deep Learning

By Z.H. Fu https://fuzihaofzh.github.io/blog/

Deep learning, a subfield of artificial intelligence (AI), has been a topic of great interest for many years. Among its various intriguing aspects, the role of gradient descent, a fundamental algorithm employed for training deep learning models, has garnered considerable attention. A recent paper titled “Implicit Gradient Regularization” by researchers David G.T. Barrett and Benoit Dherin from DeepMind and Google Dublin, respectively, provides an enlightening exploration of how gradient descent implicitly regularizes models. This phenomenon is referred to as Implicit Gradient Regularization (IGR).
In this blog post, we will unpack the concept of IGR, discuss its core principles, and explore its implications on deep learning models.

The Essence of Implicit Gradient Regularization

To understand IGR, we first need to comprehend how gradient descent operates. It functions in discrete steps along the gradient of the loss function. However, after each step, there is a slight deviation from the exact continuous path that minimizes the loss at every point. The researchers term this divergence between the original loss surface and the path followed by gradient descent as Implicit Gradient Regularization.

In their study, Barrett and Dherin introduce an insightful result to describe the modified loss function that gradient descent aligns with more closely. This result is

$$\tilde{E}(\theta)=E(\theta)+h/4\|\nabla E(\theta)\|^2,$$

where $E(\theta)$ is the original loss function, $\nabla E(\theta)$ is the gradient of the loss function, and $h$ is the learning rate. The second term, $h/4\|\nabla E(\theta)\|^2$, acts as a regularizer that penalizes areas of the loss landscape with large gradient values.

Backward Error Analysis and Its Role

Barrett and Dherin employed backward error analysis to quantify this regularization. This technique, used in numerical analysis, measures the difference between the steps of a numerical method (like gradient descent) and the exact solution of a differential equation.

The original function that gradient descent aims to solve is represented by the ordinary differential equation $\dot{\theta}=f(\theta)$, where $f(\theta)=\nabla E(\theta)$. However, due to the nature of the Euler method (a numerical technique used to solve ordinary differential equations), errors are likely to occur. To address this, the researchers constructed a new function $\tilde{f}(\theta)=f(\theta)+hf_1(\theta)+h^2f_2(\theta)+\cdots$, such that the solution of the Euler method exactly aligns with the solution of the equation $\dot{\theta}=\tilde{f}(\theta)$.

To find the difference between the actual function $\tilde{f}$ and the original function $f$, the $f_i$ terms need to be calculated (the paper calculates $f_1$). The researchers discovered that $f_1(\theta)=-1/4\nabla \|\nabla E(\theta)\|^2$, which can intriguingly be written in the form of the gradient of something.

In-depth Calculation of $f_1$

A crucial part of understanding the concept of IGR involves the calculation of $f_1$. This is where the backward error analysis comes into play. The function $\tilde{f}(\theta)=f(\theta)+hf_1(\theta)+h^2f_2(\theta)+\cdots$ is constructed such that the solution of the Euler method is strictly equal to the solution of the equation $\dot{\theta}=\tilde{f}(\theta)$.

To calculate $f_1$, we first perform the Taylor expansion of $\theta(t)$ at $t$, leading to the equation $\theta(h)=\theta(0)+h\theta'(h)+1/2h^2\theta''(h)=\theta(0)+h\tilde{f}(\theta)+1/2h^2\tilde{f'}(\theta)\tilde{f}(\theta)=\theta(0)+h(f(\theta)+hf_1(\theta))+1/2h^2(f(\theta)+hf_1(\theta))(f'(\theta)+hf'_1(\theta))=\theta+hf(\theta)+h^2(f_1(\theta)+1/2f'(\theta)f(\theta))$.

By setting this equal to $\theta+hf(\theta)$ and discarding the high-order terms of $h$, we derive $f_1(\theta)=-1/4\nabla \|\nabla E(\theta)\|^2$. Interestingly, this part of $f_1$ can be expressed in the form of the gradient of something.

Implications of Implicit Gradient Regularization

IGR brings several crucial implications for deep learning models to light. Firstly, it uncovers that gradient descent implicitly biases models towards flat minima, where test errors are small, and solutions are robust to noisy parameter perturbations. This revelation is significant as it helps elucidate why gradient descent excels at optimizing deep neural networks without overfitting, even without explicit regularization.

Secondly, the study demonstrates that the IGR term can be employed as an explicit regularizer. This allows us to directly control this gradient regularization, paving the way for enhancing the performance of deep learning models.

Wrapping Up

The research conducted by Barrett and Dherin offers a fresh perspective on the workings of gradient descent in deep learning. The concept of Implicit Gradient Regularization not only provides a deeper understanding of how deep learning models are optimized but also introduces a new tool for enhancing model performance. As we continue to untangle the complexities of deep learning, discoveries like these bring us one step closer to fully leveraging the power of these models.

Reference

[1] Barrett D G T, Dherin B. Implicit gradient regularization[J]. arXiv preprint arXiv:2009.11162, 2020.