K-Way Energy Probes for Metacognition Reduce to Softmax in Discriminative Predictive Coding Networks

Cacioli (2026) showed that the K-way energy probe on standard discriminative predictive coding networks reduces approximately to a monotone function of the log-softmax margin. The reduction rests on five assumptions, including cross-entropy (CE) at the output and effectively feedforward inference dynamics. This pre-registered study tests the reduction's sensitivity to CE removal using two conditions: standard PC trained with MSE instead of CE, and bidirectional PC (bPC; Oliviers, Tang & Bogacz, 2025). Across 10 seeds on CIFAR-10 with a matched 2.1M-parameter backbone, we find three results. The negative result replicates on standard PC: the probe sits below softmax (Delta = -0.082, p < 10^-6). On bPC the probe exceeds softmax across all 10 seeds (Delta = +0.008, p = 0.000027), though a pre-registered manipulation check shows that bPC does not produce materially greater latent movement than standard PC at this scale (ratio 1.6, threshold 10). Removing CE alone without changing inference dynamics halves the probe-softmax gap (Delta_MSE = -0.037 vs Delta_stdPC = -0.082). CE is a major empirically load-bearing component of the decomposition at this scale. CE training produces output logit norms approximately 15x larger than MSE or bPC training. A post-hoc temperature scaling ablation decomposes the probe-softmax gap into two components: approximately 66% is attributable to logit-scale effects removable by temperature rescaling, and approximately 34% reflects a scale-invariant ranking advantage of CE-trained representations. We use "metacognitive" operationally to denote Type-2 discrimination of a readout over its own Type-1 correctness, not to imply human-like introspective access.

Predictive coding networks (PCNs) promise properties not offered by backpropagation-trained networks: learning from purely local information, a single network that both discriminates and generates, and flexible inference by energy minimization. However, their iterative inference is expensive, making hardware acceleration desirable, and a useful, efficient accelerator needs both depth and aggressive quantization. Canonical state-based PC delivers neither: its inference signal decays exponentially with depth, so deep models fail to train, and low precision makes it worse. We show that pairing two recent methods carries the full PCN property set into the deep, 4-bit regime: bidirectional PC (bPC), which unifies discrimination and generation, and the error-based reparameterization (ePC), which optimizes prediction errors rather than states during inference. Error optimization delivers the output signal to every layer in a single backward pass, removing the signal decay; one additional element, global-norm clipping of the error gradients, stabilizes the error dynamics at depth. The resulting ePC+bPC model trains stably up to at least eight layers and retains its full-precision accuracy and clean class-conditional generation under W4A4E4 quantization-aware training, across FP4 and three posit formats. On six-layer networks, where state-based bPC collapses to chance, ePC+bPC classifies more than 50 accuracy points above it after one epoch of training, with class-conditional generations 5-9x better in RMSE. 4-bit quantization is nearly free: full-precision and 4-bit accuracies differ by at most ~5 points, and by under two in the best-performing formats. We characterize the two failure modes the method overcomes, bounded signal decay in state-based PC and error divergence in unregularized error optimization, and discuss implications for a 4-bit-native PCN accelerator.

Equilibrium Propagation (EP) is a physics-based training framework that has primarily been employed in energy-based models, including continuous Hopfield networks, nonlinear resistive networks and coupled phase oscillators. However, EP's practical applications have so far remained limited to relatively small-scale problems. Predictive coding networks (PCNs), another class of energy-based models rooted in computational neuroscience, are typically trained with a specialized algorithm and have likewise not yet been demonstrated at large scale. In this work, we develop an EP-based training method for PCNs which combines the centered variant of EP with a novel equilibration scheme for PCNs. Using this approach, we train a 10-layer convolutional PCN (VGG10) on full-size ImageNet, achieving 13.23% test error rate on the top-5 classification task, close to the 12.2% backpropagation baseline. To our knowledge, this is the first demonstration of both PCNs and EP-based training at ImageNet scale. These results significantly extend the scalability of both approaches and suggest that the primary challenges in scaling EP in other physical systems may come more from the computational properties of these systems than from inherent limitations of the EP framework.