Asymptotically Efficient Recursive Identification Under One-Bit Communications Achieving Original CRLB

Abstract: This paper develops an asymptotically efficient recursive identification algorithm for autoregressive systems with exogenous inputs under one-bit communications. In particular, the proposed method asymptotically achieves the Cramer-Rao lower bound (CRLB) based on the original data before quantization (original CRLB), whereas existing approaches typically attain only the CRLB corresponding to the quantized observations. The primary reason is that the existing methods quantize only the current system output, resulting in non-negligible information loss under one-bit quantization. To overcome this challenge, we present a novel quantization method that integrates both current and historical system outputs and inputs to provide richer parameter information in one-bit data, allowing the information loss caused by quantization to become a minor term relative to the original CRLB. Based on this technique, a corresponding remote estimation algorithm is further proposed. To address the convergence analysis challenge posed by the non-independence of the one-bit data, we establish a new framework that analyzes the tail probability of integrated data formed by combining current and historical system outputs and inputs before quantization, thereby eliminating the need for the traditional independence assumption on the quantized data. It is proven that the remote estimate achieves asymptotic normality, and the error covariance matrix converges to the original CRLB, confirming its asymptotic efficiency. Compared to existing identification algorithms under one-bit data, this method reduces the asymptotic mean squared error by at least 36%. Several numerical examples are simulated to show the effectiveness of the proposed algorithm.

• The paper introduces a recursive identification method for ARX systems where the remote estimator receives one bit per time step, achieving the original Cramér-Rao lower bound (CRLB).
• The method couples a local recursive least-squares (RLS) estimator with a stochastic approximation (SA) update, improving efficiency by encoding semantic information in the transmitted bit.
• The proposed algorithm ensures asymptotic efficiency without requiring independence assumptions or bounded parameter sets, achieving a ~36% reduction in asymptotic mean squared error (MSE) compared to existing one-bit schemes.

Problem setting and motivation

The paper addresses recursive identification of an ARX system $A(q)y_k = B(q)u_k + d_k$ in a networked architecture where the remote estimator observes only one bit per time step, transmitted by a quantizer that has access to the raw input–output data. The performance target is the Cramér-Rao lower bound (CRLB) computed from the original (pre-quantization) data — the "original CRLB" $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$ — rather than the CRLB associated with quantized observations. This distinction matters: existing asymptotically efficient schemes that quantize only the current output incur an asymptotic covariance at least $\pi/2 \approx 1.56$ times the original CRLB, i.e., a mean squared error (MSE) inflation of roughly 36% relative to what is achievable with full-precision data.

Three gaps in prior work motivate the design. First, asymptotically efficient recursive algorithms under one-bit data [yin2007asymptotically; wang2024asymptotically; you2015recursive] handle only FIR models; extension to ARX is nontrivial because the regressor $\phi_k$ contains past outputs unavailable to the remote estimator. Second, fixed-quantizer methods require deterministic inputs while adaptive-quantizer methods assume stochastic inputs, leaving mixed inputs unaddressed. Third, existing analyses typically assume independence of quantized data and confine the true parameter to a known compact set.

Algorithm design: RLS-SA co-design

The proposed scheme couples a local recursive least-squares (RLS) estimator at the quantizer side with a stochastic approximation (SA) update at the remote estimator. The quantizer runs standard RLS on the raw data, then transmits the single bit

$$s_k = I_{\{x_k^T\hat{\theta}_k^{\mathrm{RLS}} > x_k^T\hat{\theta}_{k-1}\}} - I_{\{x_k^T\hat{\theta}_k^{\mathrm{RLS}} \leq x_k^T\hat{\theta}_{k-1}\}},$$

where $x_k = e_{\mathrm{mod}(k,m+n)+1}$ cyclically selects one parameter component. The remote estimator updates via $$\hat{\theta}_k = \hat{\theta}_{k-1} + \beta_k x_k s_k / k^\alpha$$ with $1/2 < \alpha < 1$ and bounded step sizes $\beta_k$. The bit thus encodes the sign of the difference between the local RLS estimate and the remote estimate along one coordinate — a form of semantic information extraction in which parametric knowledge from current and historical outputs and inputs is compressed into one bit before transmission. Because each component is revisited every $m+n$ steps, the remote estimate tracks the RLS trajectory coordinate-wise.

A notable remark in the paper clarifies why classical anytime coding results do not transfer: since the RLS increment $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$0 decays only polynomially due to noise, exponential tracking via exponentially decaying steps is impossible here, and the tracking error is necessarily polynomial-rate.

Convergence analysis without independence assumptions

The central technical device is the auxiliary variable $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$1, whose dynamics decompose into a self-correcting sign term plus perturbations $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$2 (RLS error) and $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$3 (noise). The main structural result establishes that for any positive integer $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$4,

$$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$5

with $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$6. The proof proceeds by showing that on the complement of a rare event (tail probability $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$7), the correction term dominates the perturbations, driving $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$8 below $$\bar{\Sigma}_{\mathrm{CR}} = \delta_d^2(\bar{\mathbb{E}}[\{\phi_k\phi_k^T\}])^{-1}$$9 within $\pi/2 \approx 1.56$0 cycles and keeping it there. Two consequences follow directly:

• Compact-set assumption eliminated: because the tail probability of the remote estimate itself decays polynomially at arbitrary order, unbounded trajectories are negligible, replacing the uniform boundedness imposed via compact parameter sets in prior work.
• Independence assumption eliminated: the analysis operates on tail probabilities of integrated pre-quantization data rather than on independent quantized observations, accommodating both deterministic and stochastic inputs under quasi-stationarity.

Building on this, the paper proves almost sure convergence at rate $\pi/2 \approx 1.56$1 and $\pi/2 \approx 1.56$2 convergence, $\pi/2 \approx 1.56$3, for every positive integer $\pi/2 \approx 1.56$4. The latter relies on the $\pi/2 \approx 1.56$5 rate of the underlying RLS estimator being slower than that of $\pi/2 \approx 1.56$6, so the remote estimate inherits the RLS rate.

Asymptotic efficiency

The headline result states that the remote estimate is asymptotically efficient in both senses: $\pi/2 \approx 1.56$7 and $\pi/2 \approx 1.56$8. The proof exploits the fact that $\pi/2 \approx 1.56$9 almost surely, transferring the known asymptotic normality and covariance of RLS to the one-bit remote estimate. Achieving the original CRLB means the one-bit constraint imposes no asymptotic accuracy penalty — a strong claim, given that all prior one-bit schemes lose at least a factor $\phi_k$0 in covariance. The authors note that distribution-sense and covariance-sense efficiency are not mutually inclusive in general, and the algorithm satisfies both simultaneously.

Numerical validation

Simulations use 500 Monte Carlo runs across four settings. For a fifth-order ARX system with mixed input $\phi_k$1, the empirical $\phi_k$2MSE aligns with that of the full-data RLS algorithm after about 1000 steps and approaches the trace of the original CRLB, confirming the theory. Comparisons against three benchmarks show consistent superiority:

Benchmark	Setting	Outcome
Sign-error algorithm [csaji2012recursive]	ARX, i.i.d. stochastic input	Higher $\phi_k$3MSE than RLS-SA
Empirical measure / IBID [yin2007asymptotically; wang2024asymptotically]	FIR, deterministic input	Converge to fixed-threshold CRLB, above original CRLB
Optimal Newton-based estimator [you2015recursive]	FIR, uniform stochastic input	Converges to adaptive-threshold CRLB

In the last comparison, the ratio of RLS-SA MSE to the adaptive-quantizer benchmark converges empirically to $\phi_k$4, matching the theoretical claim that the proposed method reduces asymptotic MSE by at least $\phi_k$5 relative to existing one-bit algorithms.

Limitations and open questions

Several restrictions bound the applicability of the results. The noise $\phi_k$6 must be i.i.d. Gaussian — required both for the CRLB characterization and for the Gaussian tail bounds used in the tracking analysis — so non-Gaussian or correlated noise lies outside the current framework. The model class is ARX; extension to ARMA-type systems with colored noise remains open, as does the question of whether extracting parametric knowledge via other estimators (e.g., instrumental variables) would improve performance. The analysis also assumes ideal noiseless, delay-free channels; the effect of channel imperfections such as additive noise or packet drops on identification performance is not addressed. Finally, the step-size condition $\phi_k$7 and bounded $\phi_k$8 leave open how sensitive finite-sample behavior is to these choices beyond the empirical setting $\phi_k$9, $$s_k = I_{\{x_k^T\hat{\theta}_k^{\mathrm{RLS}} > x_k^T\hat{\theta}_{k-1}\}} - I_{\{x_k^T\hat{\theta}_k^{\mathrm{RLS}} \leq x_k^T\hat{\theta}_{k-1}\}},$$0.

Conclusion

The paper delivers a quantizer–estimator co-design under a strict one-bit-per-step budget that provably attains the original CRLB for ARX systems, in both distributional and covariance senses, without independence assumptions on quantized data or a compact-set prior on the parameters. The 36% asymptotic MSE reduction over prior one-bit methods is supported by both theory and simulation. The remaining limitations — Gaussian noise, ARX model structure, and ideal channels — define the concrete boundaries within which these guarantees hold.