Local Loss Optimization in Infinite-Width Neural Networks: Analyzing Stable Parameterization for Predictive Coding and Target Propagation

Belangrijkste concepten

This research paper investigates the stable parameterization of local learning algorithms, specifically Predictive Coding (PC) and Target Propagation (TP), in the infinite-width limit of neural networks, revealing unique properties and highlighting their potential for large-scale deep learning.

Samenvatting

• Bibliographic Information: Ishikawa, S., Yokota, R., & Karakida, R. (2024). Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation. arXiv preprint arXiv:2411.02001.

• Research Objective: This paper aims to analyze the stable parameterization of local learning algorithms, Predictive Coding (PC) and Target Propagation (TP), in the infinite-width limit of neural networks. The authors investigate the existence and characteristics of maximal update parameterization (µP) for these algorithms, which ensures stable feature learning dynamics.

• Methodology: The authors utilize theoretical analysis and empirical validation to derive µP for PC and TP. They analyze the conditions required for stable feature learning in the infinite-width limit, considering factors like weight initialization, learning rate scaling, and inference dynamics. They validate their findings through experiments on standard image classification tasks.

• Key Findings:

  • The study derives µP for PC with single-shot sequential inference, even without the fixed prediction assumption, and empirically verifies the transferability of learning rates across different network widths.
  • For PC with multiple inference sequences, the authors derive analytical solutions for local targets and losses at the fixed point of inference in deep linear networks. They find that PC's gradient interpolates between first-order and Gauss-Newton-like gradients depending on parameterization and inference step sizes.
  • The research derives µP for TP and its variant, DTP, assuming linear feedback networks. They reveal a distinct property of (D)TP: the preference for feature learning over the kernel regime due to the scaling of the last layer, which differs from conventional µP.

• Main Conclusions:

  • The study establishes a theoretical foundation for understanding the behavior of local learning algorithms in the infinite-width limit.
  • The derived µP for PC and TP enables stable feature learning and facilitates hyperparameter transfer across different network widths.
  • The analysis reveals unique properties of PC and TP, such as the gradient switching behavior in PC and the absence of the kernel regime in TP.

• Significance: This research contributes significantly to the theoretical understanding of local learning algorithms and their scalability to large-scale neural networks. The findings have implications for developing more efficient and biologically plausible learning algorithms.

• Limitations and Future Research:

  • The derivation of µP assumes a one-step gradient update and linear networks. Future research could explore the dynamics for more general steps and non-linear networks.
  • Further investigation into the learning dynamics and convergence properties of local learning in the infinite-width limit is warranted.

Statistieken

For PC with ¯γL = −1, the inference loss consistently decreases with increasing width.
In TP, ωL, which represents the scaling exponent of the last layer, remains fixed at 1/2 even as the width increases, indicating the disappearance of the kernel regime.

Citaten

"While it is known that PC inference trivially reduces to gradient computation of BP under the fixed prediction assumption (FPA), a technical and heuristic condition, there is generally no guarantee that PC will reduce to BP, making it highly non-trivial to identify its µP."
"TP seems to be the first example in the infinite-width limit where bL = 1/2 induces feature learning."

Belangrijkste Inzichten Gedestilleerd Uit

Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation

by Satoki Ishik... om arxiv.org 11-05-2024

https://arxiv.org/pdf/2411.02001.pdf

Diepere vragen

How can the insights from analyzing local learning in the infinite-width limit be applied to improve the design and training of practical deep learning models?

Analyzing local learning algorithms like Predictive Coding (PC) and Target Propagation (TP) in the infinite-width limit provides valuable insights that can be leveraged to enhance the design and training of practical deep learning models. Here's how:

Hyperparameter Transfer (µTransfer): The paper demonstrates that using the maximal update parameterization (µP) enables effective µTransfer, meaning hyperparameters tuned for smaller networks can be directly applied to larger ones. This simplifies the often tedious hyperparameter tuning process, especially for large-scale models. By understanding the scaling laws governing local learning in the infinite-width limit, we can design more efficient hyperparameter schedules and potentially reduce the computational cost of training.

Initialization Strategies: The analysis reveals the importance of proper initialization for both the feedforward and feedback pathways in local learning. For instance, the paper shows that TP favors a specific initialization scale for the last layer (bL=1/2) that differs from conventional µP. This understanding can guide the development of more effective initialization schemes tailored for local learning, potentially leading to faster convergence and better generalization.

Understanding Gradient Dynamics: The paper sheds light on the unique gradient dynamics of PC, showing that it can interpolate between first-order gradient descent (GD) and Gauss-Newton (GN)-like updates depending on the parameterization and network width. This knowledge can be used to design novel local learning algorithms that dynamically adjust their update rule based on the network architecture and dataset characteristics, potentially achieving a better trade-off between computational efficiency and convergence speed.

Exploring the Absence of the Kernel Regime: The paper highlights that TP, unlike BP, lacks a kernel regime in the infinite-width limit. This suggests that TP inherently favors feature learning over simply learning the input-output mapping. This property could be advantageous in scenarios with limited data, where learning rich feature representations is crucial for good generalization. Further investigation into this phenomenon could lead to novel local learning algorithms specifically designed for low-data regimes.

Bridging the Gap Between Theory and Practice: While the infinite-width analysis provides valuable theoretical insights, bridging the gap to practical finite-width networks is crucial. The findings regarding the influence of the last layer's dimension and the role of inference step sizes in PC offer concrete directions for translating theoretical understanding into practical improvements. By systematically studying the behavior of local learning algorithms across different network widths and architectures, we can develop more robust and scalable training procedures.

Could the unique properties of PC and TP, such as gradient switching and the absence of the kernel regime, be leveraged to address specific challenges in deep learning, such as learning with limited data or improving robustness to adversarial examples?

Yes, the unique properties of PC and TP hold potential for addressing specific challenges in deep learning:
1. Learning with Limited Data:

TP's Absence of Kernel Regime: As TP favors feature learning over kernel learning, it could be particularly beneficial in low-data scenarios. By focusing on learning meaningful representations rather than memorizing the training set, TP might generalize better from limited examples.
PC's Gradient Switching: The ability of PC to switch between GD and GN-like updates could be advantageous. In early training stages, a GD-like behavior might be preferable for exploring the loss landscape, while a GN-like update could refine the solution in later stages, potentially leading to better generalization from limited data.
2. Improving Robustness to Adversarial Examples:

Local Credit Assignment: Both PC and TP rely on local credit assignment, updating weights based on local errors. This could lead to more robust models less susceptible to small, adversarial perturbations in the input. Since the error is distributed and corrected locally, the impact of localized input perturbations might be minimized.
GN-like Updates: The GN-like updates exhibited by PC under certain conditions could contribute to robustness. GN methods are known for their ability to account for the curvature of the loss landscape, potentially leading to solutions in flatter regions that are less sensitive to adversarial attacks.
Further Research:

Regularization Effects: Investigating whether the local objectives in PC and TP act as implicit regularizers, leading to solutions that are inherently more robust to adversarial examples, is crucial.
Data Augmentation Strategies: Exploring how data augmentation techniques can be tailored to work synergistically with the local learning dynamics of PC and TP, especially in data-scarce scenarios, is important.
Adversarial Training Adaptations: Adapting adversarial training methods to leverage the unique gradient properties of PC and TP could lead to more robust models.

How does the biological plausibility of local learning algorithms, which is a key motivation for their development, translate to their behavior and effectiveness in the infinite-width limit, and what are the implications for understanding biological intelligence?

While the biological plausibility of local learning algorithms like PC and TP is a primary motivation for their development, their behavior in the infinite-width limit raises intriguing questions and offers potential implications for understanding biological intelligence:
1. Biological Networks are Finite: Biological neural networks are inherently finite, unlike the idealized infinite-width networks studied theoretically. Directly translating findings from the infinite-width limit to biological systems requires caution. However, understanding the scaling laws and asymptotic behavior of these algorithms can provide insights into how they might function in large, complex networks like the brain.
2. Feature Learning and Abstraction: The paper's finding that TP favors feature learning aligns with observations from neuroscience. The brain is known to learn hierarchical representations of the world, extracting increasingly abstract features at higher levels of processing. TP's inherent tendency to learn such representations in a local and biologically plausible manner could offer a computational model for this aspect of biological intelligence.
3. Gradient Switching and Adaptivity: PC's ability to switch between GD and GN-like updates might reflect a form of adaptivity present in biological learning. The brain might employ different learning strategies depending on the context and the task at hand. PC's dynamic behavior in the infinite-width limit could inspire new models of synaptic plasticity that capture this flexibility.
4. Constraints and Trade-offs: Biological systems operate under various constraints, such as energy efficiency and limited communication bandwidth. The study of local learning in the infinite-width limit, while an abstraction, can help us understand the fundamental trade-offs between accuracy, efficiency, and biological plausibility. This knowledge can guide the development of more realistic and insightful models of brain function.
Implications:

Neuroscience-Inspired Algorithms: The insights gained from analyzing local learning in the infinite-width limit can inspire the development of more powerful and biologically plausible deep learning algorithms.
Understanding Brain Function: These algorithms can serve as computational models for studying specific aspects of brain function, such as learning, memory, and perception.
Bridging Artificial and Biological Intelligence: By exploring the connections between artificial and biological intelligence, we can gain a deeper understanding of both and potentially pave the way for more general and adaptable AI systems.