Generalization as a robust performance property of learning-enabled dynamical systems

Filippo Fabiani

Source-linked AI summary

Generalization as a robust performance property of learning-enabled dynamical systems

Filippo Fabiani

arXiv:2608.30431v1Aug 31, 2026eess.SYcs.LGmath.OC

TL;DR

Small empirical error may fail to guarantee satisfactory closed-loop behavior because poor out-of-sample performance can degrade performance. The paper models sample replacement as an exogenous disturbance in sensitivity dynamics and develops a computable certificate whose bound separates one-sample sensitivity from dynamical amplification.

Problem
Small empirical error may be insufficient to guarantee satisfactory closed-loop behavior because poor out-of-sample performance can degrade performance.
Method
The paper models sample replacement as an exogenous disturbance acting on a data sensitivity system and uses dissipativity-based matrix inequalities to certify uniform stability.
Results
The resulting uniform stability bound separates the learned operator’s one-sample sensitivity from interconnection-induced dynamical amplification, while experiments distinguish nominal convergence from data sensitivity.
Takeaways & Limitations
The computable SDP certificate provides a tractable tool for comparing generalization capabilities and supports analysis of learning dynamics and data-driven control.

Abstract

from arXiv · show

By focusing on algorithmic stability as a means of establishing out-of-sample bounds, we provide a system-theoretic interpretation of generalization in learning-enabled dynamical systems arising in data-driven optimization and feedback control approximation. Given two neighboring datasets, we specifically model sample replacement as an exogenous disturbance acting on a sensitivity system, while the incremental behavior of the data-dependent operator is encoded through an integral quadratic constraint. By relying on dissipativity arguments, we establish a matrix inequality-based certificate and a uniform stability bound that separates the one-sample sensitivity of the learned operator, and an algorithm-dependent dynamical gain. The latter can then be optimized, offering a tractable tool for certifying and comparing generalization capabilities of learning dynamics. We show that our results recover classical ones for gradient descent, apply naturally to momentum-based methods such as heavy-ball and Nesterov acceleration, and extend to data-driven control.

I. INTRODUCTION

The paper reframes generalization in learning-enabled dynamical systems as algorithmic stability under sample replacement, using an input-output and dissipativity perspective to track how perturbations propagate through dynamics.

Generalization concerns performance on new, unseen data rather than only training samples.
Classical algorithmic stability measures output changes under replacement of one training sample and supports high-probability guarantees when this change is uniformly small.
Recursive trajectory-distance arguments do not explicitly capture how data perturbations are attenuated or amplified across algorithmic iterations.
The proposed sensitivity system models sample replacement as an exogenous disturbance and decomposes operator discrepancy into propagated and injected sensitivity.
Dissipativity and incremental quadratic constraints yield an optimization-based stability bound separating one-sample operator sensitivity from an algorithm-dependent dynamical gain.
The framework complements prior contraction-based analyses by treating sample replacement explicitly as an input and applying to broader data-dependent operators and dynamics.

B. Summary of contributions and paper organization

The paper establishes a dissipativity-based, computable certificate for generalization in learning-enabled dynamics, separating data sensitivity from dynamical amplification and covering optimization and learned control settings.

The approach covers operators learned offline through least squares or kernel ridge regression and is illustrated with a kernel-based residual controller.
Algorithmic stability is recast as an input-output robustness property from sample-replacement disturbance to loss discrepancy.
A matrix-inequality dissipativity certificate produces a uniform stability bound that separates statistical operator sensitivity from interconnection-induced amplification.
The dynamical sensitivity gain is formulated as a semidefinite program with scalar search, enabling systematic optimization and comparison of dynamics.
The framework recovers contractivity-based stability for gradient descent and applies to heavy-ball and Nesterov methods even when contractivity may fail.
Numerical experiments indicate that nominal convergence and data sensitivity are distinct design objectives.
Several practical mappings have one-sample perturbation scaling as 1/N, supporting consistent, exponential, high-probability bounds.

A. Preliminaries on algorithmic stability

The paper frames algorithmic stability as the sensitivity of a learned output to replacing one training sample. It connects this property to generalization bounds and an incremental input-output interpretation.

An algorithm maps a dataset to a hypothesis, and its prediction accuracy is evaluated through a loss function.
Uniform stability requires that replacing one training sample does not substantially change the algorithm’s output under the loss.
Uniform stability yields a probabilistic bound on the generalization gap for bounded losses.
When η scales as 1/N, the bound vanishes as N →∞, providing the stated consistency property.
Incremental input-output stability implies uniform stability when the learned operator’s neighboring-dataset difference is bounded and D = 0.
Because i-IOS is difficult to verify and a terminal snapshot wastes transient sensitivity information, the paper motivates a tractable dissipativity-based approach.

III. DISSIPATIVITY-BASED CERTIFICATION OF ALGORITHMIC STABILITY

The paper treats replacing one training sample as an exogenous disturbance in a sensitivity system. Its stability coefficient separates data injection from dynamical sensitivity and is certified through dissipativity and an IQC-based matrix inequality.

Algorithmic stability is recast as the output of a data sensitivity system rather than solely as a static property.
Replacing one sample creates an exogenous disturbance, and the stability coefficient decomposes into data injection and dynamical sensitivity terms.
The learned operator is assumed to satisfy an incremental IQC, while its neighboring-dataset perturbation is uniformly bounded by ζN.
Theorem 3.3 uses a storage function and matrix inequality to bound the induced gain from the disturbance w to terminal state sensitivity ΔxT.
A uniform stability bound follows by combining the gain certificate with a uniform bound on the data-injection signal.
Tight bounds consistent with N require ζN ∝1/N, which occurs for several relevant learning mappings.

B. Robust performance interpretation

The sensitivity dynamics are interpreted as a robust-performance problem. Sample replacement drives the system, and terminal loss difference is the performance output whose gain determines uniform stability.

The central object is the finite-horizon input-output mapping from data perturbation to terminal loss difference.
Uniform stability requires bounding the terminal loss difference uniformly over test samples and neighboring datasets.
This structure matches a robust-performance problem in H∞/IQC theory, with a linear system driven by an exogenous disturbance.
The feedback interconnection contains a linear sensitivity-propagation system, a learned nonlinearity constrained by an IQC, and a sample-replacement channel.
The multiplier Π encodes prior knowledge about the learning mapping, while ρ controls how aggressively past sensitivity is discounted.
The certified sensitivity gain can be optimized over the certificate variables to tighten the bound for a chosen multiplier class.

C. Optimizing the dynamical sensitivity gain

The dynamical sensitivity gain is optimized through dissipativity certificates, separating algorithm-dependent amplification from one-sample injection. Gradient descent recovers classical stability bounds and illustrates how contraction controls accumulation.

C. Optimizing the dynamical sensitivity gain: The dissipativity-based perspective offers a direct optimization procedure for the sensitivity gain.
C. Optimizing the dynamical sensitivity gain: For fixed ρ, the matrix inequality becomes an SDP, while applying the resulting corollaries requires scalar search over [0, 1).
C. Optimizing the dynamical sensitivity gain: The optimized gain G⋆ yields the stability estimate η ≤κζNG⋆ and is the tightest bound certifiable by the chosen multiplier class.
B. Gradient descent: For gradient descent, bounded gradients imply ∥wk∥≤2B/N and therefore ζN ≤2B/N.
B. Gradient descent: Under m-strong convexity and L-smoothness with α ∈(0, 2/(L + m)], gradient descent has contraction factor q < 1.
B. Gradient descent: If α ∈(0, 1/L), the classical bound becomes η ≤2κB/mN.
B. Gradient descent: The contraction factor determines how injected sensitivity accumulates: q < 1 gives geometric forgetting, q = 1 gives linear growth, and q > 1 can amplify perturbations.
B. Gradient descent: The dissipativity theorem recovers the classical contractivity-based gradient-descent bound while assigning a systems-theoretic meaning to its factors.

B. Heavy-ball algorithm

The heavy-ball method is represented as a lifted sensitivity system, enabling dissipativity-based stability certification even when Euclidean contractivity fails.

Modeling: The heavy-ball iteration is recast as a lifted linear sensitivity dynamics with neighboring-data perturbations entering as an exogenous input.The representation uses the lifted state ˜xk := col(xk, xk−1) and matrices (AHB, BHB, CHB, DHB).
Sensitivity bound: A bounded-gradient assumption gives the one-sample disturbance bound ∥wk∥≤2B/N for all k ≥0.This bound supports applying the uniform stability corollary to the current-iterate hypothesis.
Certificate: The dissipativity certificate requires a storage function for which the closed-loop sensitivity interconnection dissipates the data-injection supply.This condition is weaker than requiring a pointwise one-step contraction estimate in a prescribed norm.
Example: For β = 1/4 and α = 3/4ˆϱN, the lifted heavy-ball dynamics has spectral radius 1/2 < 1 while ∥AHB∥>1.Thus the dynamics are Schur stable without being a strict Euclidean contraction.
Implication: Heavy-ball sensitivity nevertheless has finite data sensitivity gain, showing that the contractivity argument can fail while robust performance remains certifiable.Schur stability also permits standard Lyapunov arguments to construct a positive-definite storage certificate.

C. Nesterov’s accelerated gradient method

Nesterov’s accelerated gradient method admits the same lifted sensitivity-system and dissipativity analysis as heavy-ball, including certification beyond Euclidean contractivity.

Modeling: The lifted Nesterov dynamics has a linear system representation with nonlinear input and system matrices (ANAG, BNAG, CNAG, DNAG).The uniform stability bound is obtained by solving the certificate conditions for this representation.
Sensitivity bound: Under bounded gradients, Nesterov’s neighboring-data disturbance satisfies ∥wk∥≤2B/N for all k ≥0.This yields a uniform stability bound for the current-iterate hypothesis.
Example: For β = 1/2 and α = 1/2ˆϱN, the lifted Nesterov dynamics has spectral radius 1/2 < 1 while ∥ANAG∥>1.The example establishes Schur stability without strict Euclidean contraction.
Implication: The same finite-gain conclusion as for heavy-ball applies to Nesterov’s lifted sensitivity dynamics.This demonstrates the usefulness of the dissipativity-based analysis when contractivity does not hold.
Comparison: The framework produces algorithm-specific certificates that compare iterative algorithms through both convergence rate and generalization capabilities.The comparison is enabled by optimizing the dynamical sensitivity gain.

D. Numerical experiments

Numerical experiments compare certified sensitivity gains with nominal convergence and apply the framework to learned feedback control, showing that these objectives can differ substantially.

Momentum sweep: For small momentum, heavy-ball and Nesterov gains are close to the gradient-descent baseline; increasing β rapidly increases the certified gain.The increase indicates greater amplification and longer memory of one-sample perturbations.
Momentum sweep: Heavy-ball loses certificate feasibility earlier than Nesterov, indicating a smaller certified robustness margin under the chosen storage and sector multiplier.Nesterov retains a larger feasible set of (α, β) combinations near the high-momentum boundary.
Convergence-sensitivity comparison: Nominal convergence and data sensitivity are distinct properties: methods with comparable qnom can have substantially different certified gains.Nesterov attains smaller nominal convergence factors than gradient descent while maintaining moderate sensitivity over a wide β range, whereas heavy-ball’s gain rises rapidly.
Finite-horizon gains: Although ζN scales as O(1/N), the dynamical gain multiplying it varies significantly with acceleration parameters.Smaller certified gains occur in more conservative regimes where qnom is close to one.
Learned feedback control: In learned feedback control, empirical one-sample sensitivity and terminal closed-loop sensitivity corroborate the statistical-dynamical decomposition.The learned operator must generalize over closed-loop outputs, because approximation errors can propagate recursively and affect performance, constraints, or stability.

A. KRR-based residual feedback control

The section illustrates how one-sample perturbations in an offline-learned residual controller propagate through closed-loop dynamics, using KRR and a sensitivity decomposition. The experiment shows that statistical sensitivity decay is inherited by terminal trajectories, while convergence and data sensitivity remain distinct objectives.

Setup: The control example learns a nonlinear residual correction offline and embeds it alongside a stabilizing nominal linear feedback.The residual has the form r(x) = −a tanh(v⊤x), while the plant is discrete-time linear.
Sensitivity mechanism: Replacing one sample creates a neighboring dataset, a different learned controller, and a different closed-loop trajectory.The framework represents this perturbation as an exogenous disturbance in the sensitivity dynamics.
Results: The terminal closed-loop sensitivity exhibits the same sample-size decay, so reduced statistical perturbation is inherited by the closed-loop trajectories.This behavior persists with a nonlinear learned residual feedback.
Interpretation: The certificate separates one-sample operator sensitivity from dynamical amplification and is computable via semidefinite programming for comparing learning dynamics.It extends the analysis beyond optimization operators to offline-learned feedback control.
Design implications: Experiments show that nominal convergence and data sensitivity are distinct design objectives, motivating joint parameter co-design for both criteria.The algorithm realization enters the matrix inequality directly, enabling a potentially convex joint synthesis.

A. One-sample sensitivity bounds for data-dependent operators

The paper develops one-sample sensitivity bounds for data-dependent operators by analyzing how replacing one sample changes empirical averages, minimizers, kernel embeddings, and related mappings.

General framework: Replacing one sample changes the empirical average and provides the basic perturbation used to bound operator sensitivity.The framework compares an original dataset with the corresponding dataset after replacing the j-th sample.
Empirical mappings: Bounded empirical averages and Lipschitz transformations yield sensitivity bounds proportional to 1/N for several operator classes.The listed classes include SAA mappings, Lipschitz transformations, and finite collections of empirical moments.
Kernel operators: Kernel mean embeddings satisfy ζN ≤ 2B/N when the feature-map norm is uniformly bounded by B.This gives an explicit one-sample sensitivity bound in the Hilbert-space setting.
Empirical minimizers: Strong convexity and uniformly bounded sample-loss gradients control the change in empirical minimizers after sample replacement.The argument uses the optimality conditions of the two neighboring minimization problems.
Regularized minimizers: Regularized empirical minimizers satisfy ζN ≤ 2B/µN under µ-quadratic regularization and convex sample losses.The bound decreases with dataset size and improves with stronger regularization.
Additional operators: The appendix also treats contractive fixed-point mappings and KRR in RKHSs as additional settings for deriving one-sample sensitivity bounds.The KRR result assumes bounded kernels and regularization, while the fixed-point result assumes contraction and Lipschitz dependence.