只传6%路径就能实现近似全量RTRL的在线学习，还更稳定。
Massive Redundancy in Gradient Transport Enables Sparse Online Learning
- 随机选取少量路径传播梯度，即可逼近完整RTRL效果。
- 在n=64到256的网络中，仅需k=4路径即恢复84%~78%性能。
- 适用于RNN、LSTM、Transformer及真实神经数据，尤其适合在线适应场景。
实时递归学习（RTRL）通过前向传播雅可比张量计算精确在线梯度，但每步开销为O(n^4)。本文发现，在连续误差信号下，递归雅可比矩阵存在巨大冗余：仅传播6%路径（k=4, n=64）即可恢复84±6%的完整RTRL适应能力，且k=4在n=64至256间保持有效（对应6%至1.6%路径，恢复率84%至78%），表明稀疏RTRL随网络增大相对更高效。在RNN中，结果对路径选择不敏感，即使对抗性选择也有效，且存在从零到任意非零传播的阶跃式转变。谱分析显示，雅可比矩阵满秩但接近各向同性（条件数2.6-6.5），任意随机子集均可提供方向代表性梯度估计。在混沌动力系统（洛伦兹吸引子）中，稀疏传播比全量RTRL更数值稳定（方差比13% vs. 88%）。该冗余延伸至LSTM（k=4匹配全量）、Transformer（50%头稀疏优于稠密基准；33%勉强可用），更高阈值反映头功能分化而非各向同性。在真实灵长类神经数据中，稀疏RTRL（k=4）能在线适应跨会话电极漂移（80±11%恢复，5种子），同样更稳定。若无连续误差信号，雅可比传播会积累数值漂移，导致所有RTRL变体性能下降，此为前向模式方法的适用前提。结果在SGD下仍成立（92±1%恢复），暗示与优化器无关。
原文摘要 · Abstract (English)
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.
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