Statistical Efficiency of Single- and Multi-step Models for Forecasting and Control

Abstract: Compounding error, where small prediction mistakes accumulate over time, presents a major challenge in learning-based control. A common remedy is to train multi-step predictors directly instead of rolling out single-step models. However, it is unclear when the benefits of multi-step predictors outweigh the difficulty of learning a more complex model. We provide the first quantitative analysis of this trade-off for linear dynamical systems. We study three predictor classes: (i) single step models, (ii) multi-step models, and (iii) single step models trained with multi-step losses. We show that when the model class is well-specified and accurately captures the system dynamics, single-step models achieve the lowest asymptotic prediction error. On the other hand, when the model class is misspecified due to partial observability, direct multi-step predictors can significantly reduce bias and improve accuracy. We provide theoretical and empirical evidence that these trade-offs persist when predictors are used in closed-loop control.

• The paper proves that in fully observed linear systems, single-step predictors are asymptotically most statistically efficient, followed by intermediate and direct multi-step models, with all errors decreasing at rate 1/N.
• The paper shows that under partial observability, direct multi-step predictors reduce irreducible bias and can outperform single-step methods, including producing stabilizing controllers when alternatives fail.
• The paper finds that prediction-error rankings do not always predict LQR performance, highlighting distinct trade-offs among sample efficiency, misspecification bias, horizon length, and closed-loop stability.

Overview and motivation

Compounding error—the accumulation of small one-step prediction mistakes over autoregressive rollout—is a central obstacle in learning-based control. A widely used remedy is to train multi-step predictors directly, but this increases model complexity and data requirements. Prior comparisons of these approaches were largely empirical; the paper under review provides the first quantitative analysis of the trade-off for linear dynamical systems (2603.23465). It studies three predictor classes: (i) single-step models rolled out autoregressively, (ii) direct multi-step predictors with independent per-timestep parameters, and (iii) an intermediate class—single-step predictors trained with a multi-step loss. The analysis covers both prediction error and closed-loop LQR performance, extending the authors' earlier conference work by adding the intermediate predictor class, control analysis, and complete proofs.

The setting is a stable linear time-invariant system driven by i.i.d. Gaussian process noise and random Gaussian inputs, observed through $y_t = Cx_t + D_v v_t$. Predictors are static linear maps applied to the current observation and future inputs, so the Markovian assumption embedded in the hypothesis class is either justified ($C=I$, no sensor noise: well-specified) or violated (partial observability: misspecified). Performance is measured by the steady-state squared prediction error over horizon $H$.

Well-specified setting: single-step predictors are asymptotically most efficient

When the system is fully observed, all three predictors admit exact characterizations of their reducible error, $$\lim_{N\to\infty} N\,\mathbb{E}\|(\hat G_N - G^\star)\Sigma_z^{1/2}\|_F^2$$, each decaying at rate $1/N$ but with different constants. The direct multi-step predictor's constant involves the matrix $M_{MS} + H d_u I_H$ where $(M_{MS})_{ij} = \mathrm{trace}(A^{|i-j|})$, while the single-step constant involves $M_{SS} + d_u I_H$ with entries decaying away from the upper-left corner. The key structural difference is that the multi-step constant carries an extra factor of $H$ multiplying the input dimension, reflecting that independent per-step predictors must each absorb process-noise variance.

The central result is a strict ordering of asymptotic efficiency:

$$\text{single-step} \;\leq\; \text{intermediate} \;\leq\; \text{multi-step}$$

in reducible error, for any fully observed system. The first inequality follows from the Cramér–Rao bound—the single-step estimator achieves the inverse Fisher information as its asymptotic variance, which is dominated by the sandwich variance $C=I$0 of the M-estimator trained on the multi-step loss. The second follows from classical constrained least-squares theory, since the intermediate predictor solves the multi-step problem subject to autoregressive structure constraints. This ordering has a clear practical implication: when the Markovian assumption holds, adding multi-step structure to training only hurts sample efficiency, because it injects accumulated process noise into the loss without any bias benefit.

For scalar systems without inputs, the gap between single- and multi-step constants shrinks as $C=I$1: near marginal stability the two approaches become statistically equivalent. However, the paper notes this conclusion does not generalize—in higher-dimensional systems with one eigenvalue held below 1, the efficiency gap persists even as $C=I$2, as confirmed numerically for a two-state system with eigenvalues $C=I$3 and $C=I$4, averaged over 2,500 datasets per configuration.

Misspecified setting: multi-step predictors reduce bias

Under partial observability, the Markovian assumption fails and the analysis shifts from convergence rates to irreducible bias. Writing the system in innovations form via the Kalman filter, the paper decomposes the limiting expected loss into bias plus vanishing terms. For the direct multi-step predictor, the bias scales with horizon as $C=I$5 through the term $C=I$6. For the single-step rollout, the fitted first-step gain converges to $C=I$7, and the bias scales as $C=I$8—a quantity whose spectral radius can substantially exceed $C=I$9. In the paper's example with $H$0, $H$1, unit process and sensor noise, this spectral radius is 0.99 despite $H$2, producing a bias gap between single- and multi-step predictors that grows dramatically with horizon. The intermediate predictor's bias solves a constrained optimization problem with no closed form, but the three biases obey the reverse ordering:

$H$3

established by observing that the three biases solve nested optimization problems over progressively smaller hypothesis classes. The implication is direct: partial observability creates exactly the misspecification regime in which direct multi-step predictors are preferable, since they can absorb the unmodeled state information into horizon-specific gains rather than compounding a biased one-step map.

Closed-loop control performance

The paper evaluates controllers synthesized by finite-horizon MPC using each learned predictor, executed receding-horizon, and scored by infinite-horizon LQR cost (clipped to handle non-stabilizing gains). In the well-specified case, the single-step predictor's excess clipped LQR cost decays at least as fast as the intermediate predictor's, up to a term decaying as $H$4 in the planning horizon—an argument combining continuity of the stabilizing gain, a delta-method expansion of the cost around $H$5, and the Cramér–Rao ordering of estimator variances. A comparison against the direct multi-step predictor is explicitly left open.

Numerically, a notable discrepancy appears: although the intermediate predictor beats the multi-step predictor in prediction error decay, it exhibits a slower LQR cost decay than the multi-step predictor in closed loop. This indicates that prediction error rankings do not automatically transfer to control cost rankings even in the well-specified case.

In the misspecified case, irreducible bias may prevent any derived controller from stabilizing the true system, so the comparison is made through stabilization capability. Simulations show regimes where only the multi-step predictor yields a stabilizing closed loop (e.g., $H$6, $H$7), while at $H$8 both single-step variants stabilize but the plain single-step predictor requires larger $H$9. Two mechanisms are offered: lower bias places the induced gain closer to the ground-truth stabilizing gain, and multi-step losses implicitly emphasize the dominant modes that govern stabilization. Theoretical characterization of these stabilization regimes is left to future work.

Extension to nonlinear systems

Using a nonlinear system admitting a finite-dimensional Koopman lifting, the authors fit the same three linear predictor classes on lifted observations. With full observation, the well-specified ordering (single-step best) holds; with partial observation and sensor noise, the ordering reverses (multi-step best), mirroring the linear theory across horizons $$\lim_{N\to\infty} N\,\mathbb{E}\|(\hat G_N - G^\star)\Sigma_z^{1/2}\|_F^2$$0 with $$\lim_{N\to\infty} N\,\mathbb{E}\|(\hat G_N - G^\star)\Sigma_z^{1/2}\|_F^2$$1. These experiments suggest the qualitative conclusions extend beyond linear systems, though no formal guarantee is provided.

Limitations and open questions

The results are confined to static linear predictors, stable systems with $$\lim_{N\to\infty} N\,\mathbb{E}\|(\hat G_N - G^\star)\Sigma_z^{1/2}\|_F^2$$2, and i.i.d. Gaussian excitations ensuring persistence of excitation. Several questions remain open as stated by the authors: whether the well-specified LQR comparison extends to the direct multi-step predictor; a theoretical account of when multi-step-induced controllers stabilize under misspecification; and rigorous extension of the bias/efficiency trade-off to genuinely nonlinear dynamics beyond Koopman-liftable examples.

Conclusion

This paper supplies a quantitative resolution to the single-step versus multi-step modeling question for linear systems: statistical efficiency favors single-step rollouts when the model class is well-specified, while bias reduction from direct multi-step prediction dominates under partial observability. The intermediate formulation inherits the worse side of both trade-offs in prediction, yet exhibits control behavior not predicted by its prediction-error ranking—a caution against using forecast accuracy alone as a proxy for control performance.