Self-Evidencing Through Hierarchical Gradient Decomposition: A Dissipative System That Maintains Non-Equilibrium Steady-State by Minimizing Variational Free Energy

TL;DR

The paper tackles how a physical system can implement the Free Energy Principle through exact local credit assignment. It introduces a three-level hierarchical decomposition—spatial credit via feedback alignment, temporal credit via eligibility traces, and structural credit via the Trophic Field Map—each contributing exact inference at its level. Empirically, the TFM shows a near-perfect match to oracle gradients (0.9693 Pearson) and enables remarkable continual learning feats, including high task retention, positive transfer, self-organized criticality, antifragile recovery, and sample-efficient reinforcement learning without replay. The work unifies Prigogine's dissipative structures, Friston's free energy minimization, and Hopfield attractor dynamics, offering a scalable, biologically plausible account of learning and memory as a self-organizing physical process with broad implications for neuroscience, machine learning, and artificial life.

Abstract

The Free Energy Principle (FEP) states that self-organizing systems must minimize variational free energy to persist, but the path from principle to implementable algorithm has remained unclear. We present a constructive proof that the FEP can be realized through exact local credit assignment. The system decomposes gradient computation hierarchically: spatial credit via feedback alignment, temporal credit via eligibility traces, and structural credit via a Trophic Field Map (TFM) that estimates expected gradient magnitude for each connection block. We prove these mechanisms are exact at their respective levels and validate the central claim empirically: the TFM achieves 0.9693 Pearson correlation with oracle gradients. This exactness produces emergent capabilities including 98.6% retention after task interference, autonomous recovery from 75% structural damage, self-organized criticality (spectral radius p ~= 1.0$), and sample-efficient reinforcement learning on continuous control tasks without replay buffers. The architecture unifies Prigogine's dissipative structures, Friston's free energy minimization, and Hopfield's attractor dynamics, demonstrating that exact hierarchical inference over network topology can be implemented with local, biologically plausible rules.

Figures (13)

• Figure 1: Structural credit exactness: TFM vs. oracle gradient. Scatter plot comparing the local trophic heuristic $H_{\text{post}}$ against true block-level gradient magnitude $G_{\text{post}}$ computed via backpropagation. Pearson: 0.969, Spearman: 0.933. The near-perfect correlation empirically validates Theorem 3, showing that hierarchical gradient decomposition provides structurally exact credit assignment. The small residual is attributable to finite-sample noise inherent in online, stochastic learning.
• Figure 2: Spatial credit alignment during long-term learning. The cosine similarity between learned feedback $\boldsymbol{\epsilon}$ and true gradient projection $\boldsymbol{\epsilon}^*$ (blue, left) gradually converges to 1.0 over 50,000 steps, showing eventual exact spatial credit assignment. Prediction MSE (red, right) drops to baseline early in training while alignment is still poor ($<0.4$), showing that learning proceeds with approximate gradients before the system self-corrects toward exactness. This validates that the feedback pathway performs inference on the inverse generative model.
• Figure 3: Temporal credit exactness: eligibility traces vs. forward-mode e-prop. Scatter plot comparing the diagonal factorized approximation (eligibility traces with Jacobian correction) against exact forward-mode gradients with full Jacobian propagation. Pearson: 0.840, Spearman: 0.828, AUROC: 0.911. The strong correlation validates Theorem 2, showing that eligibility traces implement optimal exponential filtering for temporal credit assignment in recurrent networks.
• Figure 4: Weight update alignment with e-prop. Scatter plot comparing block-wise weight changes $\|\Delta W^{(ij)}\|_F$ from local plasticity rules against exact e-prop gradients. Cosine: 0.968, Pearson: 0.195. The moderate cosine indicates approximate directional alignment, while weak Pearson reveals fundamental differences from NLMS inverse-variance weighting. This normalization ($\propto 1/\|\mathbf{x}\|^2$) is functionally necessary; ablation shows removing it eliminates learning. The weak e-prop correlation reflects adaptive filtering principles required for stable online learning, trading gradient fidelity for biological plausibility and robustness.
• Figure 5: Precision-recall curve for connection prioritization. Average Precision: 0.151. The modest ranking performance reflects NLMS inverse-variance weighting rather than direct gradient accumulation. By adaptively scaling learning rates based on instantaneous activity ($\propto 1/\|\mathbf{x}\|^2$), the system prioritizes connections differently than standard gradient descent. Ablation confirms this normalization is functionally necessary for learning, showing that the system implements adaptive filtering principles proven optimal for online learning with variable signal power.

Theorems & Definitions (3)

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