Uniform error bounds for quantized dynamical models

Abdelkader Metakalard; Fabien Lauer; Kevin Colin; Marion Gilson

cs.LGFeb 17, 2026

Uniform error bounds for quantized dynamical models

Authors: Abdelkader Metakalard, Fabien Lauer, Kevin Colin, Marion Gilson

Organizations: CRAN, SYNALP · ∗Universit´e de Lorraine, CNRS, CRAN, F-54000 Nancy, France · ∗∗Universit´e de Lorraine, CNRS, LORIA, F-54000 Nancy, France · SYNALP, LORIA ·

Abstract

This paper provides statistical guarantees on the accuracy of dynamical models learned from dependent data sequences. Specifically, we develop uniform error bounds that apply to quantized models and imperfect optimization algorithms commonly used in practical contexts for system identification, and in particular hybrid system identification. Two families of bounds are obtained: slow-rate bounds via a block decomposition and fast-rate, variance-adaptive, bounds via a novel spaced-point strategy. The bounds scale with the number of bits required to encode the model and thus translate hardware constraints into interpretable statistical complexities.

Figures & tables

Panel of Figure 1 — Figure 1: Two intertwined block sequences: light gray for $\mathbf{S}_{1}$ , black for $\mathbf{S}_{2}$ .

Panel of Figure 2 — Figure 2: Spaced point selection in ( 3 ), showing the first four and the last points, with $a$ points between each pair.

Figure 3: Slow and fast rate error bounds as functions of $n$ when modeling ( 5 ).

	Slow (Theorem 4 )				Fast (Theorem 5 )
$a$	$\mu$	$\hat{L}_{n}(f)$	Confidence interval	Total	$\mu^{\prime}$	$\hat{L}_{n}^{\mathrm{spaced}}(f)$	Confidence interval	Total
17	5882	0.9823	3.312	4.355	11764	0.9869	0.630	1.691
21	4761	0.9823	3.701	4.683	9523	0.9650	0.729	1.766
25	3999	0.9823	4.036	5.018	7999	0.9862	0.834	1.902
30	3333	0.9823	4.421	5.403	6666	0.9820	0.954	2.031
40	2499	0.9823	5.105	6.088	4999	0.9679	1.183	2.270

Table 1: Comparison between the slow and fast rate bounds on the linear system identification example of Sect. 6.1 for

\delta=0.05

and various block sizes

a

Bound		Full confidence
	Complexity term	interval
	(mixing-free)	(with mixing)
Massucci et al. (2022) , ( 9 )	23.48	27.77
Theorem 4 , ( 22 )	19.06	19.23
Theorem 5 , ( 23 )	10.23	10.41

Table 2: Comparison of error bounds for switched system identification.

System Identification

System identification infers mathematical models and parameters of dynamical systems from input-output, trajectory, sensor, or video data.

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In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

Abstract

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