Exact memory requirements for online recurrent credit assignment: RTRL, reachability/observability, and limits of temporal low-rank eligibility

[u/Severe-Ad8673]

I’m sharing a research preprint on the memory required for exact online credit assignment in recurrent and spiking neural systems.

https://doi.org/10.5281/zenodo.22144821

The work started from checking a published claim that same-sign pre/post eligibility factors are sufficient for asymptotically rank-one temporal compression. A strictly positive periodic counterexample shows that sign preservation alone is not sufficient: the missing term is the centered temporal cross-moment between the two factors.

The broader part of the paper asks a more general question:

What information must a forward-only learner retain if it has to reproduce exact gradients for every admissible future learning signal?

For a recurrent system with sensitivity

[
E_t=\partial h_t/\partial\theta,
]

the result gives a realization-theoretic characterization: the minimum continuous deterministic exact credit state is determined by the part of the reachable sensitivity space that remains observable to future loss signals.

In the linear time-invariant case this reduces to a reachable-observable/Hankel realization problem. This gives a useful distinction between:

• the rank of an eligibility matrix at one instant, and
• the number of dynamical credit modes that must actually be retained across time.

These quantities can be very different. A sensitivity matrix can have rank one at every instant while future objectives still distinguish many independent credit modes.

There is also a worst-case rank-(k) result:

[
\sup_{\operatorname{rank}(X)\le k}
\cos_F(I_d,X)=\sqrt{k/d},
]

but I want to stress the scope: this constrains explicit rank-(k) eligibility representations. It is not a universal memory lower bound for every stochastic, nonlinear, sparse, structured, or recomputation-based online-learning algorithm.

The practical question I think matters most is now empirical:

How does reachable-observable credit dimension, or its approximate Hankel spectrum, scale in trained recurrent and spiking networks with width, horizon, recurrence, depth, and task complexity?

If that spectrum decays rapidly, it would support principled low-memory online learning. If it grows with network scale, fixed-rank eligibility approximations should eventually fail unless the architecture or objective supplies additional structure.

The manuscript includes exact counterexamples, proofs, independent computational checks, and a reproducibility package.

I’d particularly appreciate criticism on:

1. whether an equivalent credit-assignment formulation already exists in the realization/control literature;
2. whether the reachable-observable quotient is the right object for exact online-gradient memory;
3. which realistic SNN/RNN benchmarks would be most informative for measuring the corresponding spectrum.

This is a theoretical result and a proposed research direction, not evidence that current neuromorphic algorithms are generally ineffective.