TLDR: This thesis introduces a “force analysis” framework to understand how deep neural networks learn by examining the step-wise influence of individual training examples. It decomposes this influence into similarity, normalization, and prediction gap components (AKG decomposition). The framework explains phenomena like “zig-zag” learning paths, improves knowledge distillation with Filter-KD, clarifies LLM finetuning behaviors (including the “squeezing effect” of negative gradients and hallucination amplification), analyzes feature adaptation in transfer learning, and provides a dynamic explanation for simplicity bias in compositional learning.
Deep learning models have achieved incredible feats, from recognizing images to powering advanced language assistants. Yet, understanding exactly how these complex systems learn and adapt has remained a significant challenge. Traditional theories often focus on the overall behavior of models, but a new perspective, inspired by the principles of physics, is emerging to shed light on the intricate, step-by-step learning process within these networks.
A recent doctoral thesis by Yi (Joshua) Ren, titled “Learning Dynamics of Deep Learning — Force Analysis of Deep Neural Networks,” introduces a novel framework to analyze the internal mechanics of deep neural networks. Instead of looking at the big picture, this research zooms in on the microscopic interactions that occur during training, much like a physicist would analyze forces acting on an object. This “force analysis” helps us understand how a model’s confidence in one piece of information changes after it learns from another.
The core of this approach is the AKG decomposition, which breaks down the influence of a single training example into three understandable components:
The AKG Decomposition: Unpacking the Learning Force
Imagine a model learning to identify a specific image, say, a handwritten digit ‘4’. When the model processes this ‘4’, it creates a ‘force’ that impacts its understanding of other images. The AKG decomposition helps us dissect this force:
- G (Prediction Gap): This term represents the ‘energy’ and ‘direction’ of the learning force. It’s essentially the difference between what the model currently predicts for an example and what it’s supposed to predict (the correct answer). A large gap means a strong force, pushing the model to adjust its understanding.
- K (Similarity): This component measures how similar two examples are from the model’s internal perspective. If learning about a ‘4’ strongly influences the model’s confidence in a ‘9’, it suggests a high ‘K’ value between them. Interestingly, this often aligns with human intuition about similarity.
- A (Normalization): This term acts as a self-stabilizer, adjusting the influence based on the model’s current confidence. If the model is already very confident about an observation, the incoming ‘force’ is normalized to prevent overcorrection.
By tracking these components, the research reveals fascinating patterns, such as the “zig-zag” learning path. This occurs when a difficult example is being learned: the model’s prediction might initially drift towards a ‘true’ but unlabelled understanding (influenced by similar examples), before finally correcting itself towards the provided, sometimes noisy, label. This dynamic interplay between the ‘G’ (gap) and ‘K’ (similarity) terms explains why models don’t always take a straight path to learning.
Also Read:
- Predicting and Preventing Training Stalls in Continual Learning
- Mapping the Mind: A Mathematical Theory of Similarity and Intelligence
Applications Across Deep Learning Systems
This force analysis framework isn’t just for theoretical understanding; it has practical implications across various deep learning applications:
Improving Knowledge Distillation
The insights from the “zig-zag” pattern led to a new method called Filter-KD (Filtered Knowledge Distillation). This technique helps models learn more effectively, especially when dealing with noisy data. By observing the learning paths, a ‘teacher’ model can identify better supervisory signals for a ‘student’ model, leading to improved generalization and robustness. The idea is that early in training, a model might have a better, more ‘truthful’ understanding of an example before it fully memorizes potentially incorrect labels.
Understanding Large Language Model (LLM) Finetuning
The framework was extended to analyze the complex world of LLM finetuning, including methods like Supervised Finetuning (SFT), Direct Preference Optimization (DPO), and Group Relative Policy Optimization (GRPO). The research explains several puzzling behaviors:
- Hallucinations: Why SFT can sometimes worsen hallucinations, suggesting that semantic similarities between prompts can inadvertently amplify incorrect associations.
- The “Squeezing Effect”: A counterintuitive phenomenon where negative gradients (used to teach models what *not* to do) can unintentionally concentrate probability mass onto a few dominant predictions, leading to issues like repetitive or degenerate outputs. The research found that while positive gradients tend to ‘scatter’ probability across plausible options, negative gradients tend to ‘unite’ it onto a single, often overconfident, choice.
- Lazy Likelihood Displacement (LLDisp): In methods like GRPO, some correct responses might not see their probabilities increase sufficiently. The force analysis helps identify “harmful tokens” within negative responses that impede learning, leading to a new method called Negative Token Hidden Reward (NTHR) to mitigate this.
For a deeper dive into the specifics of this research, you can explore the full thesis available at arXiv:2509.19554.
Feature Adaptation in Transfer Learning
Beyond output predictions, the framework also sheds light on how internal ‘hidden features’ of a model adapt during transfer learning. By treating these features as the ‘objects’ under influence, the research quantifies their adaptation using concepts like ‘energy’ and ‘direction’. This helps explain why different initialization strategies for a model’s ‘task head’ can lead to varying degrees of feature adaptation, impacting performance on new tasks.
Simplicity Bias and Compositionality
Finally, the thesis tackles a fundamental question: why do deep learning models often favor simpler, more structured representations? This aligns with Occam’s Razor and the idea that better compression leads to better intelligence. Through force analysis on toy compositional tasks, the research demonstrates that simpler mappings are learned faster because the ‘forces’ from different training examples align constructively. This provides a dynamic explanation for the inherent “simplicity bias” observed in many successful models.
In conclusion, Yi (Joshua) Ren’s thesis offers a powerful, physics-inspired lens to understand the intricate learning dynamics of deep neural networks. By dissecting the forces at play during training, it provides intuitive explanations for complex behaviors and paves the way for designing more robust, efficient, and interpretable AI systems.
