Prospective Coding Improves Learning in Deep Continuous-Time Recurrent Networks
Authors: Shivang Rawat, Mirko Morello, Flaviano Morone, David J. Heeger
Organizations: Telepath · New York University
Abstract
Temporal integration gives continuous-time recurrent networks memory, but in deep stacks it also delays bottom-up signals and attenuates top-down errors. We develop Recursive Quadrature Filters (RQFs), biologically motivated complex-valued temporal filters that are a special case of diagonal state-space models (SSMs), and ask whether this failure mode can be addressed by making each layer's bottom-up input prospective. Starting from an energy model, we derive the RQF dynamics and show that each RQF is a band-pass filter whose learnable parameters control its tuning frequency and bandwidth. We then make each layer's bottom-up input prospective using a parameter-free two-tap update that leaves the recurrent transition and parallel scan unchanged. We extend this correction to general diagonal SSMs and show that it mitigates depth-dependent gradient attenuation when temporal gradients are truncated, i.e., spatial-only backpropagation. We evaluate the intervention in RQFs, S5, and ORGaNICs (a nonlinear gated RNN) trained using full backpropagation through time (BPTT) and spatial-only backpropagation. Under full BPTT, prospective variants match or outperform their non-prospective controls in every model and configuration. A non-residual width-32 six-layer RQF reaches 96.09% accuracy on raw-audio Speech Commands with 31.9k parameters; a width-64 six-layer RQF reaches 83.56% on the 16,384-step Path-X task. These results identify RQFs as a parameter-efficient recurrent substrate and prospective-input coding as an input-side correction for deep continuous-time recurrent networks.
Temporal Predictive Coding provides a layer-local, parallelisable mechanism for learning in recurrent systems, making it an attractive candidate for online local learning on neuromorphic and edge hardware. However, its recurrent parameter update captures only local temporal relationships, neglecting the historic influence of parameters along the latent-state trajectory, and therefore struggles to assign credit over longer temporal horizons. This work combines for the first time Temporal Predictive Coding with Real-Time Recurrent Learning (tPC-RTRL), incorporating an online influence matrix that tracks this historic effect whilst preserving the spatial and temporal locality properties valued by neuromorphic implementations. Under explicit assumptions, we prove that tPC-RTRL recovers the gradients of backpropagation-through-time exactly. Empirically, a near-equivalence holds across several tasks of varying scale and complexity, including byte-level language modelling on WikiText-103 (tPC-RTRL vs. BPTT: 1.865 vs. 1.864 validation BPC), English--French translation on a CCMatrix subset (20.23 vs. 20.29 BLEU), and a realistic nanodrone system-identification benchmark (0.506m vs. 0.505m mean position error). Finally, we show that the iterative inference mechanism used during training can be reused at deployment time to incorporate intermittent state observations, halving final-position error relative to open-loop rollout on the nanodrone task (0.402m vs. 0.805m) and suggesting a path towards unifying learning and filtering within the same computational framework.
Training recurrent neural networks (RNNs) requires assigning credit across long sequences of computations. Standard backpropagation through time (BPTT) addresses this problem poorly: it is sequential in time, limiting parallelism, and suffers from vanishing or exploding gradients, making long-range associations difficult to learn. We propose Supervised Memory Training (SMT), a method for training nonlinear RNNs that sidesteps recurrent credit propagation entirely by reducing RNN training to supervised learning on one-step memory transition labels (mt,xt+1)→mt+1. SMT acquires these memory labels by training a Transformer-based encoder on a predictive state objective--retaining only information from the past necessary to predict the future. By decoupling what to remember from how to update memory, SMT enables time-parallel RNN training with a stable O(1) length gradient path between any two tokens--without ever unrolling the RNN. We find that SMT outperforms BPTT when pretraining various RNN architectures on tasks like language modeling and pixel sequence modeling. SMT enables nonlinear RNNs to better capture long-range dependencies and train in parallel, potentially unlocking the scaling of models that build temporal abstractions of past experience.
In streaming tasks, recurrent models can carry latent computation across time, allowing each update to build on representations produced earlier. This raises a basic question: once temporal recurrence provides sequential computation across steps, how much depth is still needed within each step? Prior work has shown that recurrence can make shallow models competitive. We instead study this question as a compute-allocation problem, varying within-step depth, expert width, and the number of parallel experts per layer across several compute budgets. For each budget, we compare the best observed recurrent and non-recurrent allocations and the performance they achieve under approximately matched per-step computation. Across Sokoban and autoregressive FineWeb language modeling, we find that temporal recurrence shifts the best observed compute allocation toward substantially fewer layers, with comparable or better performance.