SpatialPC: Deep Computation with Sublinear Local Credit Propagation
Benjamin Prada, Shion Matsumoto, Ankur Arjun Mali
On a chain of $N$ states, any local learning rule with bounded communication range needs $Θ(N)$ rounds before the loss can influence the earliest parameters, and this latency is usually attributed to depth. We show that it is a property of the communication geometry rather than of depth. For $N=m^d$ states placed at constant density in $d$ dimensions, a light-cone argument gives an $Ω(N^{1/d})$ lower bound on the number of bounded-range rounds that any local algorithm needs before its gradient estimates can be sign-correct at every parameter. We construct a unit-range, degree-$2d$ directed acyclic graph that contains a Hamiltonian computation of length $N-1$ and in which every state lies within directed distance $d(m-1)=Θ(N^{1/d})$ of the output, matching the lower bound. In the realizable-circuit model, depth and credit depth are therefore independent parameters: represented depth $Θ(N)$ and causal credit latency $Θ(N^{1/d})$ coexist, with a separation of order $N^{1-1/d}$. SpatialPC realizes this separation within predictive coding. Credit is propagated by adjoint prediction dynamics (APD), a local update whose $K$-round state on a settled acyclic graph is the truncated reverse-path sum $a^K=\sum_{k<K}(J^\top)^k b$, with error exactly $(J^\top)^K a^\star$. Graph geometry fixes when credit can first arrive at a state; the Jacobian-weighted path tail fixes how many further rounds accurate credit requires. A sparse two-dimensional Moore compiler retains the $N-1$ backbone with $\tfrac32N-2$ unit-range edges and has measured worst-node reachability $\sqrt N+2$ up to $N=65{,}536$, where a 65,535-edge backbone is reached in 258 local hops. With $T=2\sqrt N$ rounds, SpatialPC keeps finite-round task gradients aligned with settled gradients (cosine $0.93$ on MNIST and $0.97$ on Fashion-MNIST at $N=256$) and reaches $97.6\%$ and $88.1\%$ test accuracy after 20 epochs.
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