Julian Gould

Jeffrey Seely, Julian Gould

Predictive coding (PC) is a local-learning alternative to backpropagation (BP), training deep networks via local energy-minimization dynamics rather than a global backward pass. We introduce Augmented Lagrangian Predictive Coding (PC-ALM), which maintains PC's inference budget but aligns each weight update toward BP by accumulating per-layer constraint errors into a layer-local Lagrange multiplier. In linear PC networks, PC-ALM converges to an equilibrium with exact BP gradients distributed across the network via only layer-local updates. We analyze PC-ALM in nonlinear PC networks up to depth 128 and show that it matches BP performance across all width-depth regimes, notably in deep narrow networks where PC underperforms. PC-ALM introduces recurrent dynamics in each layer's activations. Compared to PC's heat flow on a scalar energy, PC-ALM dynamics are driven by dual ascent on the augmented Lagrangian. We observe "ballistic" credit propagation across very deep networks, with credit signals evenly distributed across layers, compared to PC's slow, diffusive credit propagation. Beyond the algorithm itself, the augmented Lagrangian framework offers a generalization of PC, and may yield insights into how distributed systems could compute and propagate BP-like credit signals through purely local dynamics.

Kartik Tandon, Julian Gould, Tanishq Bhatia et al.

Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin \& Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.