Dynamic Local Convergence

• Dynamic local convergence is a paradigm where local structures and state variables determine asymptotic behavior in time-evolving systems.
• It underpins local linearization in optimization, finite identification in splitting methods, and consensus in distributed and federated settings.
• The concept extends to nonlocal-to-local PDE limits, epidemic processes on dynamic graphs, and dynamic algorithms, offering practical insights across domains.

Searching arXiv for papers using the term “dynamic local convergence” and closely related formulations. Dynamic local convergence is a context-dependent research term used in several mathematical, algorithmic, and applied domains to denote convergence behavior governed by local structure in a time-evolving or iterative system. Across the literature, the phrase refers to distinct but structurally related phenomena: local linearization around equilibria in nonconvex optimization dynamics (Becker et al., 2023); nonlocal-to-local limits in evolutionary PDEs with time dependence (Colli et al., 2024); consensus and optimizer convergence of local agent states in distributed optimization (Shorinwa et al., 2023); convergence of local microscopic distributions to evolving equilibrium states in interacting particle systems (Bahadoran et al., 2018); maintenance of locally valid solutions under adversarial updates in algorithmic Lovász Local Lemma settings (Haeupler et al., 22 Apr 2026); and convergence of epidemic processes on dynamic random graphs to a process on a dynamic local limit object (Milewska et al., 16 Jan 2025). This diversity suggests that “dynamic local convergence” is not a single formalism, but a recurring paradigm in which local geometry, locality of interaction, or local state variables determine asymptotic behavior under dynamics.

1. Dynamic local convergence as a cross-disciplinary concept

In optimization and learning, the term often denotes convergence of iterates near a stationary point, with the local rate controlled by linearized dynamics, curvature, regularization, and algorithmic step sizes. In the GAN setting, local convergence of gradient descent–ascent is characterized through the Jacobian spectrum of a linearization around equilibrium, with convergence, oscillation, and divergence separated by explicit phase boundaries (Becker et al., 2023). In primal–dual and splitting methods, a related pattern appears: iterates first identify active smooth manifolds and then enter a local linear regime governed by a linearized operator restricted to tangent spaces (Liang et al., 2017, Liang et al., 2016).

In distributed and federated optimization, the phrase emphasizes the convergence of each agent’s local variable. In distributed conjugate-gradient tracking, local convergence means that each agent’s local variable converges to the global optimizer, $\|x_i^k-x^*\|\to 0$, under consensus and vanishing mean gradient conditions (Shorinwa et al., 2023). In federated learning with periodic averaging, the relevant local dynamics are those of client drift and synchronization; convergence depends on gradient diversity, aggregation frequency, and network topology, and the analysis identifies when local updates plus averaging recover rates matching homogeneous settings (Haddadpour et al., 2019).

In stochastic particle systems and random graphs, the term shifts from iterates to local laws. For the asymmetric zero-range process with site disorder, dynamic local convergence means that under hydrodynamic scaling the local microscopic distribution around a macroscopic point converges to the quenched Gibbs equilibrium determined by the evolving macroscopic density, or to the critical measure when local mass loss occurs (Bahadoran et al., 2018). For SIR epidemics on dynamic random graphs, the paper formalizes a stronger local time-marked union convergence because snapshotwise local weak convergence does not capture time-aggregated infection paths (Milewska et al., 16 Jan 2025).

A plausible implication is that the unifying core of the term is not a common theorem template, but a common viewpoint: one studies a time-dependent global system by proving convergence of a local object, such as a neighborhood, an agent state, an active manifold, or a local law.

2. Local linearization, spectra, and phase behavior in optimization dynamics

A canonical instance is the analysis of GAN training with kernel discriminators. The model is posed as the min–max problem

$$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$

with generated point locations as generator parameters and an RKHS discriminator regularized by $\lambda>0$ (Becker et al., 2023). Under an isolated points model, the dynamics decouple locally around each true point, and the update map can be linearized near equilibrium. The Jacobian takes the block form

$$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$

with $\alpha=\eta_g$ and $\beta=\eta_d$ (Becker et al., 2023). Local linear convergence occurs when the spectral radius satisfies $\rho(J)<1$, oscillations arise when complex eigenvalues have modulus near $1$, and divergence occurs when $\rho(J)>1$ (Becker et al., 2023).

The resulting eigenvalue formulas expose explicit dependence on regularization $\lambda$, kernel bandwidth $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$0, the step-size ratio $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$1, and local mass mismatch $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$2 (Becker et al., 2023). The paper gives a simple sufficient condition for local stability,

$$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$3

whenever $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$4 (Becker et al., 2023). It also identifies an oscillatory regime defined by $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$5, and a saturation regime in which decreasing $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$6 no longer improves the rate because the dominant eigenvalue is controlled by regularization (Becker et al., 2023). This is an explicit example of dynamic local convergence as a phase-structured property of a linearized non-Markovian system.

A comparable two-stage pattern appears in nonsmooth convex splitting. For primal–dual splitting, the local theory first proves finite identification of primal and dual smooth manifolds and then proves local linear convergence characterized by the spectral radius of a linearized operator on the active manifolds (Liang et al., 2017). For Douglas–Rachford and ADMM, the same geometry yields finite manifold identification followed by linear convergence, and in the locally polyhedral case the optimal contraction factor is given by the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds (Liang et al., 2016). These results make locality geometric: convergence is dictated by tangent spaces, active sets, and restricted curvature rather than by global convexity alone.

Related local analyses also appear in optimal control and manifold saddle dynamics. Generalized Gauss–Newton multiple shooting, single shooting, and differential dynamic programming share the same local $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$7-linear rate when a GGN Hessian approximation is used, because their iteration Jacobians agree at the solution (Baumgärtner et al., 2023). On manifolds, discrete constrained saddle dynamics achieve local linear convergence with rate depending on the condition number of the Riemannian Hessian, while a momentum variant improves the dependence from $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$8 to $$\min_{\theta}\max_{\phi}\;\mathcal{L}(\theta,\phi),$$9 in the rate bound (Du et al., 29 Jan 2026). Continuous-time weakly contracting but locally strongly contracting systems exhibit a distinct linear–exponential convergence profile, with linear decay outside the strongly contracting neighborhood and exponential decay once the trajectory enters it (Centorrino et al., 2024).

3. Distributed, federated, and hierarchical formulations

In distributed optimization, dynamic local convergence often refers to convergence of each local state under communication and local computation. In the distributed conjugate-gradient method with conjugate direction tracking, each agent maintains a local variable $\lambda>0$0, a local conjugate direction $\lambda>0$1, and a dynamic consensus tracker $\lambda>0$2 for the average direction (Shorinwa et al., 2023). The key update

$\lambda>0$3

ensures mean preservation of the tracked direction under a doubly stochastic mixing matrix (Shorinwa et al., 2023). Under bounded $\lambda>0$4 and $\lambda>0$5 and contraction conditions $\lambda>0$6 and $\lambda>0$7, the paper proves consensus of $\lambda>0$8, $\lambda>0$9, and $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$0, vanishing mean gradient, and convergence of objective values to $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$1 without diminishing step sizes (Shorinwa et al., 2023). Under strong convexity, this implies $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$2 for all agents (Shorinwa et al., 2023).

In federated learning, local descent with periodic averaging is analyzed through perturbation of the averaged iterate: $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$3 where $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$4 captures heterogeneity, stochastic variance, and client drift (Haddadpour et al., 2019). The heterogeneity measure is the weighted gradient diversity

$$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$5

(Haddadpour et al., 2019). The analysis shows that periodic averaging and suitable choices of $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$6, $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$7, and $$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$8 control the drift and recover rates matching the best-known homogeneous setting in general nonconvex and PL cases (Haddadpour et al., 2019). This is a local convergence theory in the sense that client-side local models synchronize frequently enough for the global descent term to dominate the local deviations.

Hierarchical SGD adds another layer: local aggregation at intermediate servers partitions heterogeneity into “upward” and “downward” divergences (Wang et al., 2020). The global divergence decomposes exactly into across-group and within-group components, and the worst-case bound for hierarchical SGD lies between the bounds of two single-level local SGD schemes, one using the local aggregation period and the other using the global aggregation period (Wang et al., 2020). The paper terms this the “sandwich behavior” (Wang et al., 2020). This suggests that dynamic local convergence in hierarchical systems can be understood as a balance between local contraction inside groups and slower correction of global drift.

A different distributed example arises in solving discrete-time Lyapunov equations over time-varying graphs. There, local convergence means that each agent’s local iterate converges to the common DTLE solution despite changing neighbor sets, under uncoordinated constant stepsizes and uniform joint connectivity (Jiang et al., 2019). The analysis decomposes each agent’s error into consensus error and optimization error, both of which decay linearly under the stated assumptions (Jiang et al., 2019).

4. Dynamic local convergence of local laws, neighborhoods, and limits

In interacting particle systems, dynamic local convergence concerns the evolution of local probability laws rather than finite-dimensional iterates. For the asymmetric zero-range process with site disorder, the hydrodynamic density solves a scalar conservation law with flux

$$J=\begin{bmatrix} I-\alpha H_{\theta\theta} & -\alpha H_{\theta\phi}\ \beta H_{\phi\theta} & I+\beta H_{\phi\phi} \end{bmatrix},$$9

and the microscopic process exhibits quenched strong local equilibrium when $\alpha=\eta_g$0 (Bahadoran et al., 2018). At supercritical densities, no invariant Gibbs product measure exists with that density, and the local law converges instead to the critical product equilibrium, producing dynamic local loss of mass (Bahadoran et al., 2018). The local object is the law seen in a bounded microscopic window around the macroscopic point $\alpha=\eta_g$1.

In phase-field PDEs, “dynamic local convergence” refers to nonlocal-to-local passage in a time-dependent system with inertia (Colli et al., 2024). The nonlocal phase equation

$\alpha=\eta_g$2

converges, as $\alpha=\eta_g$3, to the local equation

$\alpha=\eta_g$4

in strong and weak time-space topologies (Colli et al., 2024). The convergence mechanism relies on operator convergence $\alpha=\eta_g$5 in $\alpha=\eta_g$6 under energy bounds, compactness, and a precise normalization of the kernels (Colli et al., 2024). Here locality means replacement of a convolution-type interaction by the Neumann Laplacian in the limit.

Dynamic random graph theory provides another notion. For SIR epidemics on evolving graphs, the required convergence is not merely dynamic local weak convergence of snapshots but local time-marked union convergence in probability (Milewska et al., 16 Jan 2025). The dynamic graph is converted into a union graph over $\alpha=\eta_g$7 whose edges are marked by ON/OFF activation times, and convergence is imposed in a metric on rooted marked graphs (Milewska et al., 16 Jan 2025). This stronger notion is necessary because infection may travel along time-aggregated paths that never appear inside a single connected component at any fixed time (Milewska et al., 16 Jan 2025). Under this notion, the empirical epidemic proportions converge at fixed time $\alpha=\eta_g$8 to deterministic limits determined by the epidemic on the dynamic local limit graph (Milewska et al., 16 Jan 2025).

Dynamic random intersection graphs develop a related process-level local convergence theory. The time-evolving graph is represented as a marked bipartite branching-process limit, and finite-dimensional convergence plus tightness establish dynamic local weak convergence of rooted neighborhoods (Milewska et al., 2023). The framework then transfers to time-dependent degree distributions and giant-component membership processes (Milewska et al., 2023). In both (Milewska et al., 16 Jan 2025) and (Milewska et al., 2023), local convergence is explicitly a statement about neighborhoods in a dynamic Polish-space setting rather than about optimization errors.

5. Fully dynamic algorithms and online adaptation

Some uses of the term concern maintaining locally valid structure under changing problem instances. In the dynamic Lovász Local Lemma setting, a search problem is specified by a finite state space $\alpha=\eta_g$9 and a set of flaws $\beta=\eta_d$0; an adaptive or even clairvoyant adversary inserts or deletes flaws over time (Haeupler et al., 22 Apr 2026). Dynamic local convergence means that the same local search, resampling, or backtracking procedures used in the static setting continue to converge rapidly after each update, with total local fixing steps

$\beta=\eta_d$1

over $\beta=\eta_d$2 updates and failure probability at most $\beta=\eta_d$3 (Haeupler et al., 22 Apr 2026). The central condition is the AIS-type convergence criterion

$\beta=\eta_d$4

which controls witness-forest growth under dynamic updates (Haeupler et al., 22 Apr 2026). This is a notion of dynamic local convergence because the locality lies in flaw-specific fixing operations and dependency neighborhoods, while the dynamic aspect is adversarial online modification.

Online optimization for recurrent and adaptive algorithms offers another interpretation. For RTRL, NoBackTrack, UORO, RMSProp, online natural gradient, and Adam with $\beta=\eta_d$5, local convergence is analyzed in an ergodic online setting where the state evolves according to a parameterized dynamical system and the parameter is updated while the system runs (Massé et al., 2020). The general limiting ODE is

$\beta=\eta_d$6

and local convergence holds when the averaged Jacobian at the target parameter is positive-stable and the target trajectory is locally stable (Massé et al., 2020). The framework departs from standard SGD by using empirical time averages rather than i.i.d. expectations, which changes admissible stepsize ranges and permits larger rates under cycling or reshuffling (Massé et al., 2020).

Dynamic topology optimization introduces yet another usage. There, local convergence refers to the tendency of gradient-based optimizers to settle in low-performing stationary points shaped by resonances, antiresonances, and symmetry, often producing stiff, mass-driven designs instead of mechanism-rich solutions (Weer et al., 29 Sep 2025). The paper organizes mitigation strategies into exclusion, frequency shift, and relaxation approaches, and quantifies how these alter the probability of reaching high-performing optima over frequency bands (Weer et al., 29 Sep 2025). This suggests a broader reading: dynamic local convergence can also denote undesirable convergence to nearby poor minima in highly nonconvex dynamic design landscapes.

6. Common structure, distinctions, and limitations

A common structural pattern recurs across these literatures. First, a local object is identified: an equilibrium neighborhood with explicit Jacobian in GAN training (Becker et al., 2023), an active manifold in splitting methods (Liang et al., 2017, Liang et al., 2016), a tangent-space Hessian on a manifold (Du et al., 29 Jan 2026), an agent-local state in distributed optimization (Shorinwa et al., 2023, Jiang et al., 2019), a microscopic window in stochastic particle systems (Bahadoran et al., 2018), or a rooted marked neighborhood in dynamic graph models (Milewska et al., 16 Jan 2025, Milewska et al., 2023). Second, convergence is controlled by a local descriptor: spectral radius, contraction factor, Friedrichs angle, condition number, gradient diversity, or operator limit. Third, the theory often separates a global or transient phase from a local asymptotic phase. Examples include manifold identification followed by linear convergence (Liang et al., 2017, Liang et al., 2016), linear-then-exponential decay in weakly contracting continuous dynamics (Centorrino et al., 2024), and local neighborhood approximation followed by convergence of epidemic observables (Milewska et al., 16 Jan 2025).

Important distinctions remain. In some papers, “local” means spatial or graph-theoretic locality, as in random graphs and particle systems (Milewska et al., 16 Jan 2025, Bahadoran et al., 2018). In others, it means local stability near a stationary point or minimizer (Becker et al., 2023, Centorrino et al., 2024, Baumgärtner et al., 2023). In distributed optimization, it refers to agent-local copies or local updates (Shorinwa et al., 2023, Haddadpour et al., 2019, Wang et al., 2020). In nonlocal PDE limits, it refers to passage from nonlocal interaction kernels to local differential operators (Colli et al., 2024). Because these meanings are not interchangeable, the term should generally be interpreted within its disciplinary framework rather than as a single universal definition.

The limitations are likewise domain-specific. Local linearization results do not imply global convergence and often rely on isolated equilibria, strict complementarity, or exact eigenspaces (Becker et al., 2023, Liang et al., 2017, Du et al., 29 Jan 2026). Dynamic random graph convergence may require time-marked union convergence rather than weaker snapshotwise notions (Milewska et al., 16 Jan 2025). Distributed exact convergence can require strong convexity, spectral gap conditions, or bounds on heterogeneity (Shorinwa et al., 2023, Haddadpour et al., 2019, Wang et al., 2020). Nonlocal-to-local PDE results may provide only qualitative subsequential convergence without rates (Colli et al., 2024). Fully dynamic LLL guarantees hold only while an LLL criterion with slack remains satisfied after each update (Haeupler et al., 22 Apr 2026).

This suggests that “dynamic local convergence” is best understood as a family of locality-based asymptotic principles. In each instance, dynamics are governed not by the full global state space at once, but by a local structure whose evolution is tractable and whose asymptotic behavior determines the phenomenon of interest.