Bridging Predictive Coding and MDL: A Two-Part Code Framework for Deep Learning

• URL: http://arxiv.org/abs/2505.14635v2
• Date: Wed, 16 Jul 2025 16:00:09 GMT
• Title: Bridging Predictive Coding and MDL: A Two-Part Code Framework for Deep Learning
• Authors: Benjamin Prada, Shion Matsumoto, Abdul Malik Zekri, Ankur Mali,
• Abstract summary: We prove that layerwise PC performs block-coordinate descent on the minimum description length objective.<n>We also prove that each PC sweep monotonically decreases the empirical two-part codelength, yielding tighter high-probability risk bounds.<n>This is the first result offering formal generalization and convergence guarantees for PC-trained deep models.
• Score: 1.749935196721634
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present the first theoretical framework that connects predictive coding (PC), a biologically inspired local learning rule, with the minimum description length (MDL) principle in deep networks. We prove that layerwise PC performs block-coordinate descent on the MDL two-part code objective, thereby jointly minimizing empirical risk and model complexity. Using Hoeffding's inequality and a prefix-code prior, we derive a novel generalization bound of the form $R(\theta) \le \hat{R}(\theta) + \frac{L(\theta)}{N}$, capturing the tradeoff between fit and compression. We further prove that each PC sweep monotonically decreases the empirical two-part codelength, yielding tighter high-probability risk bounds than unconstrained gradient descent. Finally, we show that repeated PC updates converge to a block-coordinate stationary point, providing an approximate MDL-optimal solution. To our knowledge, this is the first result offering formal generalization and convergence guarantees for PC-trained deep models, positioning PC as a theoretically grounded and biologically plausible alternative to backpropagation.

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