无需复杂梯度传播，简单方法即可实现在线循环网络高效适应。
Immediate Derivatives Suffice for Online Recurrent Adaptation
- 用即时梯度替代传统雅可比传播，内存从O(n⁴)降至O(n²)
- 在脑机接口任务中与完整RTRL效果相当，误差不超过±3个百分点
- 适合追求低内存、高效率的实时学习系统应用
三十年来，在线循环学习被认为需在每步传播雅可比张量，计算复杂度为O(n⁴)。本文证明无需此步骤（d=0，内存O(n²)），其性能在保持测试集上脑机接口跨会话漂移（TOST等价于±3百分点，n=20，Adam，float64）时与全量RTRL相当，并在普通RNN的正弦与洛伦兹合成单元（Adam与SGD下）、LSTM+正弦（Adam下）均表现一致。通过分解g_RTRL = g_imm + g_past解释原因：在脑机接口任务中，g_past集中在单一方向（前1奇异值占比0.62–0.74，四类优化器平均，而g_imm为0.333），且全量与d=0的恢复差距随各优化器的层更新幅度比‖ΔW_hh‖/‖ΔW_out‖单调变化。静态控制实验显示两者浓度趋近0.6，说明差异信号来自漂移特异性差分，而非g_past的秩1结构。该模式在LSTM上消失，与加性线性递归机制相关。在合成正弦数据中，g_imm冗余，预测了合成零效应。全量RTRL唯一显著优势是使用LARS时提升17–27个百分点，但此时d=0+LARS也无法独立适应，表明差距源于优化器×方法交互，非方法优劣。研究明确适用条件：d=0+Adam+float64鲁棒；SGD、Adafactor及float32有特定脆弱性。在评估单元中，当n=1024时内存节省达1000倍（O(n²) vs O(n⁴)），未带来测量性能损失。
原文摘要 · Abstract (English)
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.
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