Predictive Coding and Backpropagation: A Mathematical Relationship and Implications for Biological Learning

Concepts de base

Predictive coding can be modified to compute the exact same gradients as backpropagation in a fixed number of steps, raising questions about its biological plausibility as a more realistic learning algorithm.

Résumé

The manuscript reviews and extends previous work on the mathematical relationship between predictive coding and backpropagation for training feedforward artificial neural networks on supervised learning tasks.
Key highlights:

A strict interpretation of predictive coding does not accurately compute the gradients required for training neural networks.
Modifying predictive coding by a "fixed prediction assumption" makes it algorithmically equivalent to backpropagation, producing the exact same parameter updates.
This equivalence raises questions about whether predictive coding should be interpreted as more biologically plausible than backpropagation.
Empirical results show that the magnitude of prediction errors do not necessarily correspond to surprising features of inputs.
A software package called Torch2PC is introduced to perform predictive coding on PyTorch neural network models.

The author discusses the implications of these results for the interpretation of predictive coding and deep neural networks as models of biological learning.

Stats

Predictive coding with the fixed prediction assumption computes the exact same gradients as backpropagation in a fixed number of steps (L, the depth of the network).
The relative error between parameter updates from predictive coding and the true gradients can be less than 0.01 after 100-400 iterations, depending on the step size.
The angle between the parameter updates from predictive coding and the true gradients can be less than 10 degrees after 100-400 iterations, depending on the step size.

Citations

"Predictive coding can be derived from a hierarchical, Gaussian probabilistic model in which each layer, ℓ, is associated with a Gaussian random variable, Vℓ."
"If the inference step converges to a fixed point (dvℓ≈0), then we should expect the parameter updates from Algorithm 3 to approximate those computed by backpropagation."
"If Algorithm 3 is run with step size η = 1 and at least n = L iterations then the algorithm computes ǫℓ= ∂L(ˆy, y)/∂ˆvℓ and dθℓ= -∂L(ˆy, y)/∂θℓ for all ℓ= 1, ..., L."

Idées clés tirées de

On the relationship between predictive coding and backpropagation

by Robert Rosen... à arxiv.org 04-18-2024

https://arxiv.org/pdf/2106.13082.pdf

Questions plus approfondies

How might the biological plausibility of predictive coding be further evaluated beyond the mathematical relationship to backpropagation?

The biological plausibility of predictive coding can be further evaluated through experimental studies that investigate neural activity and plasticity mechanisms in the brain. For instance, neuroimaging techniques such as fMRI and EEG can be used to observe brain activity during tasks that involve prediction errors and learning. By comparing the neural responses in these experiments to the predictions of predictive coding models, researchers can gain insights into the extent to which the brain implements similar mechanisms. Additionally, studies involving animal models, such as rodents or primates, can provide valuable information on the neural circuits and synaptic plasticity rules that underlie predictive coding. By manipulating specific neural pathways or neurotransmitter systems and observing the effects on learning and prediction, researchers can further validate the biological plausibility of predictive coding.

What other biologically inspired learning algorithms could be compared to backpropagation in a similar manner?

Apart from predictive coding, other biologically inspired learning algorithms that could be compared to backpropagation include Hebbian learning, reinforcement learning, and spike-timing-dependent plasticity (STDP). Hebbian learning is based on the principle that synapses are strengthened when the pre-synaptic neuron consistently triggers the post-synaptic neuron. This mechanism can be evaluated in the context of training neural networks and compared to backpropagation in terms of efficiency and biological plausibility. Reinforcement learning, which involves learning through rewards and punishments, can also be studied in neural networks to understand its biological relevance and compare it to backpropagation. STDP, which is based on the timing of pre- and post-synaptic spikes, can be implemented in neural network training and evaluated for its similarities and differences with backpropagation in terms of learning speed and accuracy.

How might the insights from this work on the relationship between predictive coding and backpropagation inform the development of more efficient and scalable training algorithms for deep neural networks?

The insights from the relationship between predictive coding and backpropagation can inform the development of more efficient and scalable training algorithms for deep neural networks by providing alternative approaches that are biologically plausible and computationally effective. By understanding how predictive coding can approximate the parameter updates of backpropagation, researchers can design hybrid algorithms that combine the strengths of both approaches. For example, incorporating predictive coding principles into backpropagation algorithms could lead to more robust and adaptive learning systems. Additionally, the insights from this work can inspire the development of novel optimization techniques that leverage the principles of predictive coding to improve convergence speed and generalization performance in deep neural networks. By integrating biological insights with computational efficiency, researchers can advance the field of deep learning and create more biologically inspired and effective training algorithms.