FlatLand: Personalized Graph Federated Learning via Tailored Lorentz Space

Summary

FlatLand tackles personalized graph federated learning (PFGL) under structural heterogeneity by giving each client its own Lorentz (hyperboloid) hyperbolic space with a learnable scale $K_c$, and by exploiting the time-like / space-like split of the Lorentz model to decouple parameters: within each fully-Lorentz linear layer, the column that multiplies the time-like coordinate ($\mathbf{m}$) is kept private and the block acting on space-like coordinates ($\mathbf{M}$) is aggregated FedAvg-style. The conceptual move — treating the time-like dimension as a natural carrier of client-specific geometric heterogeneity, and the space-like dimensions as a shared "Flatland" — is genuinely novel in the PFGL literature and is the paper's main contribution. It is supported by a bi-Lipschitz distortion result arguing that clients with different average Forman–Ricci curvatures need different $K$'s (Theorem 4.1) and by an information-theoretic result claiming client information concentrates in the time-like coordinate (Theorem 4.3). Empirically, the method reaches best or competitive accuracy across eight node-classification benchmarks (four homophilic, four heterophilic), holds its lead at low embedding dimensions on CiteSeer, and is robust under partial client participation.

Read closely, however, the theory is doing less work than the narrative suggests. Theorem 4.3(3) is a conditional-on-$K_c$ identity, and Remark 4.4's unconditional 'space-like carries no client information' overstates it because $\mathbf{x}_s = \sqrt{K_c}\sinh(\rho/\sqrt{K_c})\mathbf{u}$ has a client-dependent magnitude. Theorem 4.3(2) silently adds a pushforward condition not implied by $\mathrm{Var}(K_C) > 0$. The convergence analysis mixes a strongly-convex-to-optimum framing with non-convex assumptions and collapses the multi-epoch local update to a single step, so the 'consistent with FedAvg' guarantee is weaker than advertised. Methodologically, the learnable curvature is reparameterised into the narrow band $K_c \in [0.5, 1.5]$, which sits uneasily with the 'clients need substantially different curvatures' motivation. Empirically, FlatLand does not always dominate: FedGTA beats it on Photo (both client settings), and gains over Local ($L$) are within 1–2 points on several cells. A code-execution attempt further complicates the story: FlatLand's own numbers on the Cora and CiteSeer slices land in the right neighbourhood of Tables 1–2, but a competently-tuned FedProx reproduction sits far above the paper's reported FedProx and above FlatLand on every reproduced cell, indicating the headline ordering may rest on under-tuned baselines. The central idea remains convincing; the theory, empirical claims, and baseline documentation each need scoping and additional detail before they fully match.

Strengths

• Conceptual contribution. Assigning each client its own Lorentz scale and reading the time-like coordinate as the carrier of client heterogeneity is a clean, principled way to move PFGL out of the uniform-Euclidean assumption, and it slots naturally into a FedAvg-style aggregation without similarity estimation or clustering modules.
• Fully-Lorentz backbone with a manifold-preserving aggregation. Proposition 6.1 shows that averaging only the space-like block $\mathbf{M}$ and recombining it locally with $\mathbf{m}$ and $K_c$ still lands on the client's own hyperboloid $\mathcal{L}^m_{K_c}$, so the aggregation is at least geometrically consistent by construction.
• Broad and competitive empirics. Two node-classification tables spanning homophilic and heterophilic graphs, two graph-classification settings, a dimension sweep, a partial-participation study, and multiple ablations (tailored curvature, decoupling strategy, Lorentz backbone, curvature initialisation) together make a convincing empirical case, especially on CiteSeer, ogbn-arxiv, Minesweeper and Roman-Empire.
• Low-dimensional advantage. The Figure 4 dimension sweep on CiteSeer, together with the qualitative pattern that both hyperbolic methods degrade less than Euclidean baselines as embedding size shrinks, is a genuinely useful selling point for communication-efficient FL and is well aligned with the hyperbolic-representation literature.
• Careful clarification of terminology. Appendix B.1's note that "Lorentz space" here means the Riemannian hyperboloid, not Lorentzian spacetime, and Appendix B.3's disclaimer on Lorentz transformations, remove a common source of reader confusion in the hyperbolic-learning literature.
• Method fidelity to the paper. In a code-execution attempt, a re-implementation built purely from the manuscript reproduced FlatLand's own reported accuracies on the Cora and CiteSeer slices and matched the Figure 10 convergence behaviour (FlatLand rises as fast as strong baselines and plateaus above them), indicating the method itself is specified in enough detail to re-implement faithfully.

Weaknesses

• Space-like coordinate is not client-invariant unconditionally. Remark 4.4's paraphrase that '$\mathbf{x}_s$ carries no additional client-specific information' is only true either for the direction $\mathbf{u}$ or conditional on $K_c$; the space-like magnitude $\sqrt{K_c}\sinh(\rho/\sqrt{K_c})$ is manifestly client-dependent. The scope paragraph in D.2 concedes this, but the aggregation design is motivated with the strong unconditional reading. This tension between Theorem 4.3(3) and the design justification is the paper's most consequential conceptual gap.
• Pushforward condition in Theorem 4.3(2) is not implied by the standing hypothesis. The theorem states $\mathrm{Var}(K_C) > 0$ but conclusion (2) is silently conditioned on the pushforward laws $(T_{K_c})_{\#}p_\rho(\cdot|K_c)$ differing across clients — which need not follow from distinct $K_c$'s alone. Remark 4.4 then drops the qualifier and states the unconditional design principle.
• Learnable curvature is compressed to $[0.5, 1.5]$. The reparameterisation $K_c = \mathrm{sigmoid}(\kappa_c) + 0.5$ forces sectional curvature into a narrow band, which is hard to square with the 'necessity of tailored curvature' motivation and with the substantial spread of Ricci statistics across clients shown in Figure 2. It also blurs how much of the gain over Local ($L$) is really about tailored geometry vs. the parameter decoupling.
• Convergence analysis is over-sold. Appendix D.4 opens by citing FedAvg's strongly-convex-to-optimum rate, then adopts only $L$-smoothness and bounded gradients, and collapses to one full-batch step per round — removing the multi-epoch client-drift term that FedAvg's analysis is really about. The concluding 'confirms that our geometric enhanced decoupling maintains convergence' also does not actually use the Lorentz geometry anywhere in the proof.
• Algorithm pseudocode collapses multi-epoch training to a single step. In both Algorithm 1 and Algorithm 2 the gradient is computed inside the local-epoch loop but the parameter update is outside; as written, the loop just recomputes the same gradient $E$ times and applies one step. This directly contradicts the 1–3 local epochs used in Appendix E.2.
• "Safely aggregated" overstates Proposition 6.1. The overview line reads as a settled fact, but Proposition 6.1 is a manifold-preservation statement obtained because the LT time coordinate projects back onto the hyperboloid regardless of $\mathbf{W}$. It does not say that averaging weight matrices trained on client-specific curvature manifolds is semantically meaningful. A tighter phrasing tied to Proposition 6.1 would match the actual guarantee.
• $O(1)$ exponential map overstated. The exponential map at the origin still writes $d+1$ coordinates with cosh/sinh scaling; it is $O(d)$, not $O(1)$, even with the norm precomputed.
• Missing/blank displayed equations. Theorem 4.1's stated bound and the 'overall graph curvature is computed as' formula in Appendix C.2.2 appear to be missing displayed equations. Remark 4.2's 'no single $K$ works for all clients' cannot be verified without the theorem's inequality.
• Empirical scope is narrower than the abstract implies. FlatLand does not win the bold on Cora-10 (FED-PUB) or on Photo-10 / Photo-20 (FedGTA), and on several cells (Photo, Tolokers-20, Cora-10) FlatLand's gain over Local ($L$) is small relative to the reported standard deviations. Reporting per-cell delta over Local ($L$), and framing headline claims accordingly, would make the empirical story more honest.
• Partial-participation experiment uses only weaker baselines. Section 7.4 argues robustness with FedAvg / FedPer / FedHGCN, dropping FED-PUB, FedGTA, and AdaFGL — the strongest competitors in Table 1. FedAvg's 18.14% at 10% participation is essentially the random baseline, so the highlighted 63-point gap is not a like-for-like comparison against a tuned PFGL baseline.
• 1–3% claim contradicts the cited tables. Appendix E.7 states constant curvature 'consistently underperforms learnable curvature, with performance gaps of 1–3% across datasets', but Table 9 and Table 8 show every reported Constant-vs-Ricci gap is under 1%. Meanwhile E.5 reads the same numbers as a 'marginal effect,' creating an internal tension.
• Constant-$K = 1$ variant is under-specified. It is unclear whether this variant fixes $K$ throughout training or merely initialises the learnable $\kappa$ to give $K = 1$. E.5 and E.7 appear to interpret the row differently. The MLP variant, in turn, is a persistent reparameterisation rather than an initialiser, so it is not directly comparable to the other three under the same heading.
• Notation and framing slippage. $C$ is used both for the client set/count and for the client random variable (Theorem 4.3 also silently renames the count to $m$), and the client distribution behind the mutual-information statements is never specified. Figure 2's 'higher bars = more non-Euclidean' caption is only correct if the axis plots $|\mathrm{Ric}|$ or $-\mathrm{Ric}$, which is not stated.
• Non-sequitur closing of the convergence section. Appendix D.4 ends with a sentence about balancing computational overhead as the number of clients increases — a claim the descent-lemma argument does not address at all.

Reproducibility & code

A code-execution attempt built the FlatLand codebase from the manuscript (no released repository was inspected directly) and ran the deterministic pipeline on a subset of experiments. The Lorentz operations, the Section 5.2 decoupling strategy, and the tailored-curvature mechanism were reconstructed faithfully and passed methodology checks; FlatLand's own Cora and CiteSeer (10-client) accuracies land in the right neighbourhood of Tables 1–2, and the Figure 10 convergence-curve claim reproduced structurally (FlatLand rises as fast as the strong baselines and plateaus above them at ~81.4%). Wall-clock exhaustion prevented the remaining table rows (ogbn-arxiv, Photo, all heterophilic datasets, graph classification, dimension sweep, partial-participation sweep, per-client bars, curvature-initialisation ablations) from being reached, so the coverage of what did reproduce is thin — those claims are unattempted here, not falsified.

• Reproduced FedProx dominates FlatLand on the Cora/CiteSeer cells that ran. On the four Cora and CiteSeer cells that were re-executed, a competently-tuned FedProx scored 83.48/83.51 on Cora-10/20 and 74.22/75.02 on CiteSeer-10/20 — well above the paper's reported FedProx (60.18 / 48.22 / 63.33 / 64.85) and above FlatLand on every reproduced cell. The paper's own FedProx being lower than its own FedAvg on Cora-20 (48.22 vs 69.50) is itself an anomaly. This puts the headline ordering claim in Tables 1–2 at risk unless the baseline configuration is disclosed and defended.
• Numeric hyperparameters block exact reproduction of Tables 1–3. $\lambda$ in the shared-block regulariser is never given a value; the learning rate is only 'from $\{0.01, 0.001\}$' with no per-dataset choice; local epochs are '1–3' with no per-dataset assignment; the five-trial seeds and the METIS partition seed are undisclosed. Several bold wins in Tables 1–2 have std $\le 0.3$, so exact cell reproduction requires a pinned config the paper does not provide.
• Baseline configurations undocumented. Eleven competing methods are compared with no per-baseline hyperparameter table: FedProx $\mu$, FedPer split, FED-PUB / FedGTA / AdaFGL / FedSage+ settings, and FedHGCN's node-selection budget are all unstated. Several ordering claims (e.g. FlatLand-vs-FED-PUB on Cora-10, FlatLand-vs-FedGTA on Photo) depend on baseline tuning, and the FedProx divergence above illustrates the sensitivity.
• Curvature initialisation is spread across two places. C.2.2's 'normalized signed magnitude' + 'sigmoid reparameterisation' is vague, and only Appendix E.2 pins the concrete form $K_c = \mathrm{sigmoid}(\kappa_c) + 0.5$. The exact mapping from $\mathrm{Ric}(G_c)$ to the initial raw scalar $\kappa_c^{(0)}$ is still missing, and the 'overall graph curvature is computed as' equation is absent altogether — both hurt Table 9 reproducibility.
• Heterophilic gating mechanism is only prose. The low-pass / high-pass gating added to all hyperbolic backbones (Appendix E.2, Backbone structure) is load-bearing for the heterophilic wins in Table 2, but there is no equation, no ablation isolating its contribution, and only a citation to Chen et al. (2024) to guide re-implementation.
• Partial-participation protocol not documented. Section 7.4 does not specify how the participating subset is drawn per round, how un-sampled clients are evaluated at test time, or how the 50-client Cora METIS split is constructed — all of which drive FedAvg's near-random 18.14% and, therefore, the size of the reported gap.

Recommended Changes

Essential

• Restate Theorem 4.3 and Remark 4.4 with the conditioning made explicit. Split Remark 4.4 into an unconditional statement about the direction $\mathbf{u}$ and a conditional-on-$K_c$ statement about $(x_t, \mathbf{x}_s)$; remove the unconditional 'space-like carries no client-specific information' phrasing. Then either give a sufficient condition (compatible with Assumption D.6) under which the pushforward $(T_{K_c})_{\#} p_\rho(\cdot|K_c)$ genuinely differs across clients, or state Theorem 4.3(2) as an if-and-only-if with the pushforward hypothesis in the statement.
• Justify or relax the $K_c \in [0.5, 1.5]$ range. Report an ablation over the sigmoid shift/scale (or an unbounded reparameterisation such as softplus) that shows the curvature band actually matters, or explain why this narrow band is sufficient given the wider Ricci spread in Figure 2. Ideally, show per-dataset learned $K_c$ values.
• Publish per-baseline hyperparameters and a config file for FlatLand. For FlatLand: $\lambda$, learning rate, local epochs, seeds, METIS seed. For each baseline (FedAvg, FedProx, FedPer, GCFL, FedGNN, FedSage+, FED-PUB, FedGTA, AdaFGL, FedHGCN): the corresponding method-specific hyperparameters. Directly address the reproduction result that a competently-tuned FedProx dominates FlatLand on Cora/CiteSeer, either by disclosing the FedProx $\mu$/learning rate used or by re-tuning the baselines uniformly.
• Reword the convergence analysis to match its assumptions. State the guarantee as an $O(1/T)$ stationarity rate under $L$-smoothness and bounded gradients (with a single local step), remove the strongly-convex-to-optimum framing and the 'this analysis confirms' sentence, and — if you want to keep the multi-epoch comparison to FedAvg — extend the proof to $E > 1$ with a client-drift term.
• Fix the pseudocode. Move the parameter update inside the 'for local epochs' loop in both Algorithm 1 and Algorithm 2, and use per-epoch (or per-mini-batch) gradients so the local training actually takes $E$ steps.
• Insert the missing displayed equations. Add the actual inequality in Theorem 4.1 (with the small-distortion regime stated where relevant), and add the 'overall graph curvature' formula in Appendix C.2.2 together with the explicit map $\mathrm{Ric}(G_c) \to \kappa_c^{(0)} \to K_c$, cross-referencing $K_c = \mathrm{sigmoid}(\kappa_c) + 0.5$ from Appendix E.2 in one place.
• Document the partial-participation protocol. State the sampling scheme (per-round uniform? fixed subset?), the test-time evaluation regime, and the 50-client Cora METIS seed. Also add at least one strong PFGL baseline (FED-PUB, FedGTA, or AdaFGL) to Figure 6, since FedAvg alone is not an informative comparison at 10% participation.

Suggested

• Tighten empirical headline claims. Report per-cell delta over Local ($L$) and flag the Photo and Cora-10 cells where a Euclidean PFGL baseline is significantly better. A less aggressive abstract sentence ('best or competitive on 11 of 16 node-classification cells') would remove friction with the tables and with the FedProx reproduction.
• Correct the 1–3% claim and remove the E.5 / E.7 tension. Replace 'gaps of 1–3%' with the actual sub-1% numbers, and reconcile the 'marginal effect' vs. 'consistently underperforms' framings across E.5 and E.7.
• Split the initialisation ablation into 'initialiser' and 'parameterisation' rows. Clearly distinguish whether 'constant-$K=1$' fixes $K$ throughout training or just initialises the learnable $\kappa$, and separate the MLP reparameterisation from the one-shot initialisers.
• Add the heterophilic gating mechanism as an equation with an ablation. Isolate its contribution to Table 2 numbers, since it is applied uniformly to hyperbolic baselines and is currently load-bearing but only described in prose.
• Fix small logical / editorial issues. Clarify the sign convention on Figure 2 in the caption; disambiguate $C$ (client set) from the client random variable in Theorem 4.3; soften 'safely aggregated' to a manifold-preservation claim (via Proposition 6.1); delete the non-sequitur closing sentence of Appendix D.4; and correct the $O(1)$ exponential-map remark to $O(d)$.