FedNMap: Federated Mapping & Optimization

• FedNMap is a decentralized, DNS-inspired spatial mapping system that divides physical space into map zones for scalable and privacy-preserving location services.
• It employs standardized data models and recursive, multi-provider resolution protocols to seamlessly integrate indoor and outdoor geospatial applications.
• FedNMap also introduces a novel federated optimization framework using normal map-based updates and local corrections to achieve linear speedup in nonconvex composite learning.

FedNMap encompasses two distinct but highly technical paradigms that have significant impact in distributed systems and federated learning: (1) the federated spatial naming and mapping infrastructure for the Spatial Web, as introduced in (Bharadwaj et al., 15 Jul 2025), and (2) the normal map-based federated optimization scheme for composite objectives in federated learning, as proposed in (Huang et al., 3 Feb 2026). Both contexts employ federated structures to achieve scalability, privacy, and robustness in their respective domains.

1. Federated Spatial Naming and Mapping Infrastructure

FedNMap, as a spatial infrastructure (Bharadwaj et al., 15 Jul 2025), is a federated spatial-naming and mapping system architected to overcome the limitations of monolithic, centralized mapping providers. Inspired by the hierarchical architecture of the Domain Name System (DNS), FedNMap divides the physical world into “map zones,” enabling independent organizational control, scalable map management, and localized privacy guarantees. Each map zone is managed by its own provider, supporting a variety of location-based services (LBS), such as geocoding, search, routing, localization, and map tile rendering.

Architecture and Core Roles

The infrastructure adopts a tripartite architecture:

• Map Providers (Map Servers): Each provider maintains a discrete map zone and implements service endpoints. Security, access enforcement, and fine-grained publication of service interfaces are coordinated at this level.
• Resolvers: DNS-based federated discovery components map coarse geospatial indices (via S2 or H3 cell decomposition) to sets of covering map servers using domain labels (e.g., cell-<id>.mresolver.fednmap.net).
• Clients (Applications): Clients locate relevant servers, invoke distributed LBS APIs, and aggregate responses to support seamless, cross-domain spatial applications.

The workflow for end-to-end applications (such as indoor navigation) necessitates recursive, multi-provider resolution, cross-domain action invocation, and client-side merging or stitching of results for queries that span both public and private spaces.

Data Structures, Naming, and Protocols

FedNMap adopts the OpenStreetMap model for its data infrastructure, employing nodes, ways, and relations. Outdoor tiling leverages standard slippy-map tiles (z/x/y) in WebMercator coordinates, while indoor regions utilize custom hierarchical naming: BuildingID.FloorID.RoomID. Zone coverage is advertised via DNS NS records, and overlaps are naturally supported by permitting multiple NS entries per cell label.

Resolution protocols for geocoding and search distribute queries in parallel to all candidate map servers. Routing is decomposed hierarchically: each domain computes local shortest paths within its zone graph, with boundary (“portal”) nodes mapping to adjacent zones. End-to-end routes are constructed by recursive client-side composition without requiring dynamic, cross-server routing tables.

Privacy, Security, and Scalability

FedNMap supports fine-grained privacy via user-level, service-level, and application-level access controls. Authentication standards include OAuth2 and JWTs, with encrypted transport and optional end-to-end encryption for sensitive metadata. Providers expose access control policies and enforce permission scopes on their APIs, guaranteeing privacy for, e.g., private indoor maps.

Scalability is realized via federated DNS discovery with aggressive caching, delegation, and granular update propagation through TTL-controlled resource records. Performance trade-offs arise in the form of client-side query orchestration and result stitching, which are mitigated by the inherent parallelism and decentralized load distribution.

Formal Abstractions

FedNMap formalizes the global map as the union of zone graphs:

• Map Graph: $M_i = (V_i, E_i, P_i)$
  • $V_i$: set of nodes;
  • $E_i$: weighted edges;
  • $P_i$: portal nodes.
• Federated Map: $\mathcal{M} = \bigcup_{i=1}^n M_i$
• Resolver: $D: \mathbb{R}^2 \to 2^{\{1,\dots,n\}}$, returning all $i$ such that $(x,y)$ lies in the coverage of $M_i$.
• Federated Route: Concatenation of local paths between endpoints and portal nodes, with all segment computations distributed.

This formal separation of map ownership, discovery, and API invocation enables a flexible, non-centralized spatial web suitable for a range of applications, notably persistent world-scale augmented reality, robotics, and location-based information overlays (Bharadwaj et al., 15 Jul 2025).

2. Normal Map-Based Federated Optimization for Composite Learning

FedNMap also refers to a federated optimization framework for composite objectives introduced in (Huang et al., 3 Feb 2026). The composite objective takes the canonical form

$$\psi(x) = f(x) + \varphi(x), \quad f(x) = \frac{1}{n} \sum_{i=1}^n f_i(x)$$

where each $f_i$ is a smooth (possibly nonconvex) local loss and $\varphi$ is a possibly nonsmooth, weakly convex regularizer.

Algorithmic Principles

FedNMap addresses the challenge of optimizing nonsmooth, nonconvex objectives in federated settings with data heterogeneity across clients. Classical federated proximal stochastic gradient methods fail to achieve $O(1/(nQ))$ linear speedup in both the number of clients $n$ and the number of local updates $Q$, especially outside the convex regime. FedNMap resolves this with two innovations:

1. Normal Map-Based Update: Each client computes local updates not with standard proximal SGD, but by leveraging the normal map

$$F(z) = \nabla f(\text{prox}_{\gamma\varphi}(z)) + \frac{1}{\gamma}(z - \text{prox}_{\gamma\varphi}(z))$$

which preserves unbiasedness even in the presence of nonsmoothness.

2. Local Correction Mechanism: To counteract drifting induced by repeated local updates, a lightweight correction term is incorporated, ensuring that the aggregate client drift does not degrade convergence.

Algorithm Workflow

The algorithm alternates between:

• Local Updates (per client):
  • Receive anchor $z_t$ and previous average correction $\bar y_{t-1}$;
  • Iterate $Q$ steps: at each, compute a stochastic gradient on the local loss after a proximal step, apply the normal map-based correction, and adjust using the correction vector $c_{i,t}$.
  • Send the aggregated sufficient statistic $y_{i,t}$ to the server after $Q$ local steps.
• Global Aggregation (server):
  • Collect all $\{y_{i,t}\}$;
  • Update the global anchor by

  $$z_{t+1} = z_t - \frac{Q\eta_s\eta_a}{n}\sum_{i=1}^n y_{i,t}$$ - Broadcast the updated anchor and average correction to all clients.

Theoretical Guarantees

Under standard smoothness, unbiased stochastic gradient, weak convexity, and bounded variance assumptions, FedNMap achieves:

• General Nonconvex Case: The number of communication rounds $T$ needed to reduce $\mathbb{E}\|F(z)\|^2$ below $\epsilon^2$ scales as

$$T = O\left(\frac{(L+\rho)\Delta_\psi\sigma^2}{nQ\epsilon^4}\right)$$

achieving linear speedup in both clients and local updates.

• Polyak–Łojasiewicz (PL) Condition: Under the (optional) proximal-PL condition, the number of rounds required to obtain $\mathbb{E}[\psi(x_T) - \psi^*] \leq \epsilon$ is

$O\left(\frac{\sigma^2}{nQ\mu^2\epsilon}\right)$

No prior method achieves such linear scaling for nonconvex composite problems under similarly general conditions (Huang et al., 3 Feb 2026).

Empirical Validation

Key experiments on elastic-net regularized MNIST and VGG-16 training over Dirichlet-partitioned SVHN datasets demonstrate that FedNMap outperforms state-of-the-art composite FL algorithms. Scaling the number of clients $(n)$ or local steps $(Q)$ halves convergence time in line with theoretical claims, validating $1/(nQ)$ scaling in challenging heterogeneous-data settings.

3. Comparison with Related Paradigms

The spatial FedNMap infrastructure contrasts with traditional centralized mapping services by supporting privacy, scalability, and independent extensibility—the union of map graphs and DNS-style discovery enables partial overlap, local autonomy, and seamless extension to private or indoor domains. Its analogy to DNS resolves the need for global spatial name resolution without a single-point-of-control (Bharadwaj et al., 15 Jul 2025).

In federated optimization, previous proximal-FL methods achieved $O(1/n)$ scaling primarily in convex regimes or under homogeneous assumptions. FedNMap’s utilization of the normal map and local correction differentiates it as the first technique to integrate $O(1/Q)$ scaling (local update parallelism) for general nonconvex, composite, and heterogeneous-data settings (Huang et al., 3 Feb 2026).

4. Practical Implications and Deployment

In spatial applications, FedNMap unifies disparate and dynamic mapping providers, facilitating robust augmented reality, robotics, logistics, and any service dependent on granular, multi-domain geospatial knowledge. Its federated structure is particularly potent where indoor or sensitive data precludes centralization.

In machine learning, FedNMap expands the scope of federated optimization to settings with nonconvexity, non-i.i.d. data, and nonsmooth structures (e.g., $\ell_1$ or elastic net regularization), enabling highly parallelizable and communication-efficient training across large, heterogeneous client pools. The splitting and correction principles ameliorate client drift, which is salient when performing numerous local steps before synchronization.

5. Limitations and Open Questions

Spatial FedNMap incurs extra round trips and requires sophisticated stitching of partial results, placing additional computational and orchestration overhead on clients. Update propagation via DNS TTLs introduces bounded staleness; privacy is as strong as server policy enforcement and cryptographic protocol adoption.

In FL, FedNMap’s major limitation remains the need for proximal operator computability and the bounded stochastic gradient variance assumption. For highly time-varying distributions or exceptionally large client sets under bandwidth constraints, further adaptations such as dynamic client sub-selection or compression mechanisms may be required. No guarantee is provided outside the stated smoothness and weak convexity conditions.

A plausible implication is that both versions of FedNMap represent robust templates for federated architectures—whether in geospatial mapping or distributed optimization—that can flexibly scale, handle heterogeneity, and preserve privacy, provided the system assumptions are satisfied.