When Learning Hurts: Fixed-Pole RNN for Real-Time Online Training

Real-time recurrent learning (RTRL) computes exact online gradients by propagating a Jacobian tensor forward through recurrent dynamics, but at O(n^4) cost per step. Prior work has sought structured approximations (rank-1 compression, graph-based sparsity, Kronecker factorization). We show that, in the continuous error signal regime, the recurrent Jacobian is massively redundant:propagating through a random 6% of paths (k=4 of n=64) recovers 84 +/- 6% of full RTRL's adaptation ability across five seeds, and the absolute count k=4 remains effective from n=64 to n=256 (6% to 1.6%, recovery 84 to 78%), meaning sparse RTRL becomes relatively cheaper as networks grow. In RNNs, the recovery is selection-invariant (even adversarial path selection works) and exhibits a step-function transition from zero to any nonzero propagation. Spectral analysis reveals the mechanism: the Jacobian is full-rank but near-isotropic (condition numbers 2.6-6.5), so any random subset provides a directionally representative gradient estimate. On chaotic dynamics (Lorenz attractor), sparse propagation is more numerically stable than full RTRL (CV 13% vs. 88%), as subsampling avoids amplifying pathological spectral modes. The redundancy extends to LSTMs (k=4 matches full RTRL) and to transformers via sparse gradient transport (50% head sparsity outperforms the dense reference; 33% is borderline), with higher thresholds reflecting head specialization rather than isotropy. On real primate neural data, sparse RTRL (k=4) adapts online to cross-session electrode drift (80 +/- 11% recovery, 5 seeds), where sparse propagation is again more stable than full RTRL. Without continuous error signal, Jacobian propagation accumulates numerical drift and degrades all RTRL variants, a scope condition for all forward-mode methods. Results hold with SGD (92 +/- 1% recovery), suggesting independence from optimizer choice.

For three decades online recurrent learning has been assumed to require propagating a Jacobian tensor through the network's dynamics at $O(n^4)$ per step. We show it doesn't. Dropping the propagation entirely ($d=0$, $O(n^2)$ memory) matches full RTRL within CI on held-out BCI cross-session drift (TOST equivalent within $\pm 3$ pp at $n=20$, Adam, float64), and across vanilla-RNN synthetic cells (sine and Lorenz under Adam and SGD) and LSTM/sine under Adam. A decomposition $g_{RTRL} = g_{imm} + g_{past}$ explains why. On BCI, $g_{past}$ concentrates in a single direction (top-1 singular fraction 0.62-0.74 across four optimizers, vs 0.333 for $g_{imm}$), and the four-optimizer full-RTRL-vs-$d=0$ recovery gap tracks each optimizer's per-layer update-magnitude ratio $\|ΔW_{hh}\|/\|ΔW_{out}\|$ monotonically. A stationary (no-drift) control collapses both concentrations to ~0.6: the drift-specific signal is the differential, not $g_{past}$'s absolute rank-1 structure. The signature and the behavioral gap both collapse on LSTM, consistent with a mechanism specific to additive linear recurrence. On synthetic sine, $g_{imm}$ is redundant with $g_{past}$, which predicts the synthetic null. Full RTRL's one robust advantage is LARS (+17 to +27 pp), but $d=0$+LARS also fails to adapt independently; the gap is an optimizer$\times$method interaction, not a method-quality claim. We characterize the regime: $d=0$+Adam+float64 is robust; SGD, Adafactor, and float32 have specific fragilities documented in the paper. On the evaluated cells, the $1000\times$ memory saving at $n=1024$ ($O(n^2)$ vs $O(n^4)$) comes with no measured recovery cost.

Intelligent systems should not only solve tasks but also adapt under real-world constraints. Autonomous adaptation via self-supervised learning, sequential adaptation via online learning, and memory-efficient implementation via perturbation-based learning are important requirements for such systems. However, these requirements are generally in tension for high-dimensional systems, because perturbation-based learning suffers from variance that grows with the dimension of the perturbed variables. In this study, we focus on echo state networks (ESNs), where this tension naturally arises in large reservoirs. We propose a perturbation-based learning rule for online self-supervised learning in ESNs. The proposed rule is derived from an orthogonal decomposition of the self-supervised learning cost, which separates an input-dependent component from a redundant component determined by the fixed ESN parameters. By perturbing only the input-dependent component, the effective perturbation dimension is reduced from the reservoir dimension to the input dimension. Thus, the proposed method preserves self-supervised adaptation, online learning, and scalar-feedback perturbation learning, while avoiding reservoir-size-dependent variance growth. This suggests a design principle for scalable and hardware-compatible learning: online learning should be restricted to the dynamically necessary low-dimensional component of the objective.