Core Concepts

The dichotomy of early and late phase implicit biases induced by large initialization and small weight decay can provably lead to a sharp transition from memorization to generalization, a phenomenon known as "grokking", in the training of homogeneous neural networks.

Abstract

The paper studies the grokking phenomenon, where a neural network first "memorizes" the training set with perfect training accuracy but near-random test accuracy, and then suddenly transitions to perfect test accuracy after further training. The authors show that this phenomenon can be provably induced by a dichotomy of early and late phase implicit biases in the training process.
Specifically, the authors consider training homogeneous neural networks (e.g., MLPs and CNNs with ReLU activation) with large initialization and small weight decay. They prove that in the early phase, the training process gets trapped at a solution corresponding to a kernel predictor, leading to overfitting on the training set. However, after a sharp transition around 1/λ log α (where λ is the weight decay and α is the initialization scale), the training dynamics escape this kernel regime and converge to a min-norm/max-margin predictor, resulting in a dramatic improvement in test accuracy.
The authors provide concrete examples of this phenomenon in linear classification with diagonal linear nets and low-rank matrix completion with overparameterized models. They also show that the opposite can happen, where the early phase bias leads to good generalization but the late phase bias causes a sudden drop in test accuracy, a phenomenon they call "misgrokking".
The key insight is that the large initialization induces a strong early phase bias towards kernel predictors, which do not generalize well, but this bias decays over time and competes with a late phase bias towards min-norm/max-margin predictors induced by the small weight decay, leading to the sharp transition.

Stats

The paper does not contain any specific numerical data or metrics to support the key logics. The analysis is primarily theoretical, with the authors providing rigorous mathematical proofs to establish their claims.

Quotes

"Recent work by Power et al. (2022) highlighted a surprising "grokking" phenomenon in learning arithmetic tasks: a neural net first "memorizes" the training set, resulting in perfect training accuracy but near-random test accuracy, and after training for sufficiently longer, it suddenly transitions to perfect test accuracy."
"Different viewpoints on the mechanism of grokking have been proposed, including the slingshot mechanism (cyclic phase transitions) (Thilak et al., 2022), random walk among minimizers (Millidge, 2022), slow formulation of good representations (Liu et al., 2022), the scale of initialization (Liu et al., 2023), and the simplicity of the generalizable solution (Nanda et al., 2023; Varma et al., 2023). However, existing studies failed to address two crucial aspects for gaining a comprehensive understanding of grokking: 1) No prior work has rigorously proved grokking in a neural network setting. 2) No prior work has provided a quantitative explanation as to why the transition from memorization to generalization is often sharp, instead of gradual."

Key Insights Distilled From

by Kaifeng Lyu,... at **arxiv.org** 04-03-2024

Deeper Inquiries

In the context of the dichotomy of implicit biases and the grokking phenomenon, different training techniques can indeed have a significant impact.
Optimization Algorithms: The choice of optimization algorithms can influence the transition between the early and late phase implicit biases. For example, algorithms like Stochastic Gradient Descent (SGD) or variants like Adam may exhibit different behaviors in terms of how quickly they navigate through the implicit bias landscape. Some optimization algorithms may help the model escape the early phase bias more efficiently, leading to a smoother transition to the late phase bias and potentially avoiding the sharp grokking phenomenon.
Regularization Techniques: Techniques like dropout, batch normalization, or weight decay can also play a role in modulating the implicit biases. By controlling the complexity of the model and preventing overfitting, these regularization methods can impact the optimization trajectory and the manifestation of grokking.
Architectural Choices: The architecture of the neural network, including the depth, width, and type of layers, can influence how the implicit biases evolve during training. Deeper networks may exhibit different grokking behaviors compared to shallower networks, and the choice of activation functions can also impact the optimization dynamics.
Learning Rate Schedules: The learning rate schedule can affect how quickly the model converges and whether it gets stuck in certain regions of the loss landscape. Adaptive learning rate methods may help in navigating the implicit bias dichotomy more effectively.
By carefully selecting and tuning these training techniques, it may be possible to modulate the implicit biases in a way that minimizes the grokking phenomenon and promotes smoother generalization.

The insights from this work on the dichotomy of implicit biases and the grokking phenomenon can be extended to explain grokking in more complex tasks beyond the specific examples provided.
Natural Language Processing (NLP): In NLP tasks like sentiment analysis or language modeling, the dichotomy of implicit biases can manifest in how neural networks learn to represent and generalize from textual data. Understanding the interplay between early and late phase biases can shed light on why certain models exhibit sudden improvements in performance after prolonged training.
Computer Vision: In computer vision tasks such as image classification or object detection, the grokking phenomenon may arise due to the implicit biases inherent in the training process. By analyzing the transition from memorization to generalization in vision tasks, researchers can optimize training pipelines to achieve better performance.
Reinforcement Learning: In reinforcement learning settings, where agents learn to interact with environments, the concept of grokking can be applied to understand how policies evolve over training iterations. By studying the implicit biases in reinforcement learning algorithms, researchers can improve the efficiency and stability of learning processes.
By applying the principles of implicit biases and grokking to diverse domains, researchers can gain a deeper understanding of the training dynamics of neural networks in complex tasks.

Practical ways to leverage the understanding of implicit biases to design neural network training pipelines that avoid the grokking phenomenon and achieve good generalization more efficiently include:
Regularization Strategies: Implementing appropriate regularization techniques such as weight decay, dropout, or early stopping can help prevent overfitting and guide the model towards a more generalizable solution. By balancing the early and late phase implicit biases through regularization, the model can avoid sharp transitions in performance.
Architecture Design: Crafting neural network architectures that are well-suited to the task at hand can mitigate the grokking phenomenon. By carefully selecting the depth, width, and type of layers, as well as the activation functions, researchers can create models that smoothly transition from memorization to generalization.
Optimization Strategies: Choosing optimization algorithms that are robust to the implicit biases and dynamics of the training process can lead to more stable and efficient learning. Techniques like learning rate scheduling, momentum, and adaptive optimizers can help the model navigate the implicit bias landscape effectively.
Monitoring and Early Intervention: By closely monitoring the training process, researchers can detect signs of grokking early on and intervene to prevent the model from getting stuck in suboptimal solutions. Strategies like curriculum learning or dynamic regularization can be employed to guide the model towards better generalization.
By integrating these practical approaches based on the understanding of implicit biases, researchers can design neural network training pipelines that promote smoother learning trajectories and enhance generalization capabilities.

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