Technical Summary: Timescale Separation Through the Lens of Operator Theory

Problem Statement

The paper addresses the convergence analysis of interconnected discrete-time dynamical systems characterized by timescale separation (or singular perturbation). These systems are modeled as:
$$\begin{aligned} x_{t+1} &= x_t + \delta S(x_t, z_t) \\ z_{t+1} &= F(x_t, z_t) \end{aligned}$$
where $x_t$ represents the slow state, $z_t$ the fast state, and $\delta > 0$ is a tunable parameter controlling the speed of the slow dynamics relative to the fast dynamics.

While timescale separation theory is well-established for continuous-time systems, existing results for discrete-time systems are limited. Most current approaches rely on Lyapunov-based arguments or general stability-theoretic tools, which often yield purely existential conditions or highly conservative bounds that are difficult to verify in practice. Furthermore, few results exist for stochastic settings involving random updates, packet losses, or asynchronous activations. The paper seeks to bridge the gap between timescale separation theory and operator theory to provide explicit, interpretable, and practically verifiable convergence conditions for both deterministic and stochastic interconnections.

Methodology

The authors propose a framework that models the system as an interconnection of a slow operator and a fast operator, leveraging structural properties common in optimization, learning, and control algorithms (e.g., contractivity, nonexpansiveness, averagedness, and metric subregularity).

Core Assumptions

The analysis relies on several standing assumptions:

1. Forward Invariance: The dynamics remain within closed sets $X$ and $Z$.
2. Fast Operator Fixed-Point Manifold: For a frozen slow state $x$, the fast operator $F(x, \cdot)$ admits a nonempty, closed fixed-point set $Z_{FIX}(x)$. This set varies Lipschitz continuously with respect to $x$.
3. Fast Operator Convergence: The fast operator is paracontractive (or contractive in the stochastic expectation sense) toward its fixed-point set $Z_{FIX}(x)$. This corresponds to the stability of the "boundary layer" system where the slow state is frozen.
4. Reduced Operator Existence: A reduced operator $R(x)$ exists such that $R(x) = S(x, z)$ whenever $z \in Z_{FIX}(x)$. This defines the "reduced system" $x_{t+1} = x_t + R(x_t)$, representing the ideal limit where the fast dynamics are infinitely faster.
5. Lipschitz Continuity: The coupling between the fast transient and slow update, and the sensitivity of the reduced operator, are bounded by constants $L_S$ and $L_R$.

Analytical Approach

The paper establishes convergence by analyzing the interplay between the convergence of the fast dynamics toward the parametrized fixed-point set and the convergence of the reduced slow dynamics. The analysis is divided into three main theoretical developments:

1. Deterministic Contractive Case: Assuming the reduced operator $x + R(x)$ is contractive.
2. Deterministic Averaged/Subregular Case: Extending the analysis to reduced operators that are $\alpha$-averaged and metrically subregular (a broader class covering many operator-splitting algorithms).
3. Stochastic Extension: Generalizing the framework to systems driven by independent stochastic processes, formulating assumptions in terms of expected contraction-type properties.

Key Contributions and Results

1. Deterministic Timescale Separation (Contractive Case)

The paper proves that if the fast operator is paracontractive and the reduced operator is contractive, the full interconnected system converges linearly to the set $\{(x, z) \mid x = x^\star, z \in Z_{FIX}(x^\star)\}$.

• Result: An explicit upper bound $\bar{\delta}$ is derived for the tuning parameter $\delta$. If $\delta \in (0, \min\{\bar{\delta}, 1\})$, the system exhibits linear convergence.
• Bound Expression: The bound depends on the contraction rates of the fast ($c_F$) and reduced ($c_S$) dynamics, the Lipschitz constants of the coupling ($L_S$) and reduced operator ($L_R$), and the sensitivity of the fast fixed-point manifold ($L_{FIX}$).
  $$\bar{\delta} = \frac{c_S(1 - c_F)}{L_{FIX} L_S (c_S + L_R)}$$

2. Deterministic Timescale Separation (Averaged and Metrically Subregular Case)

Recognizing that many optimization algorithms (e.g., projected fixed-point iterations) are not strictly contractive, the authors extend the theory to reduced operators that are $\alpha$-averaged and metrically subregular.

• Result: Under these weaker conditions, the distance to the fixed-point set of the reduced operator decreases linearly. This property is sufficient to establish linear convergence of the full interconnected system, again yielding an explicit bound on $\delta$ with the same functional form as the contractive case.

3. Stochastic Timescale Separation

The framework is extended to stochastic systems where operators are driven by i.i.d. random variables.

• Result: Sufficient conditions are provided for almost sure convergence. The assumptions are formulated using expected contraction properties for the fast operator and the reduced stochastic operator.
• Outcome: The derived bounds on $\delta$ ensure that the expected distance to the solution set decays linearly, implying almost sure convergence via the Borel-Cantelli lemma.

4. Application to Feedback Optimization

The theoretical results are applied to feedback optimization schemes, where an optimization variable is updated based on measurements from a physical plant that asymptotically tracks its steady-state response.

• Deterministic Application: The paper demonstrates how to tune the update rate $\delta$ for a plant with contraction properties and a strongly convex (or convex with quadratic growth) cost function.
• Stochastic Application: The results are applied to a stochastic plant model, providing tuning rules that guarantee convergence despite random disturbances or packet losses.
• Numerical Validation: Simulations confirm that without timescale separation (i.e., $\delta=1$), the closed-loop system can be unstable even if the reduced system (gradient method) is convergent. Conversely, tuning $\delta$ according to the derived bounds ensures stable, linear convergence.

Significance and Claims

The paper claims to provide a compositional operator-theoretic principle for timescale separation. Its primary significance lies in:

• Explicit and Checkable Bounds: Unlike many existing singular perturbation results that offer qualitative or existential guarantees, this framework yields explicit, computable bounds on the timescale parameter $\delta$ directly in terms of standard operator constants (contraction rates, Lipschitz constants).
• Unified Framework: It unifies the analysis of deterministic and stochastic interconnections, covering a broad class of algorithms including those arising in operator splitting and distributed optimization.
• Generalization: It extends beyond contractive mappings to include averaged and metrically subregular operators, which are prevalent in modern optimization and learning algorithms.
• Practical Applicability: The results are directly applicable to feedback optimization, model predictive control, and equilibrium-seeking algorithms, offering systematic tuning rules for the coupling between plant dynamics and optimization updates.

The authors note that while related work exists for continuous-time systems (e.g., combining contraction theory with singular perturbations), this paper specifically addresses the discrete-time setting and the stochastic extension, which are critical for digital implementations and networked systems. The work substantially extends a preliminary version by providing a more general deterministic analysis (covering non-contractive reduced operators) and a full stochastic extension with complete proofs.