Opening — Why this matters now
Federated learning promised a simple trade: keep data local, share intelligence globally. In practice, causal discovery in federated environments has been living off a polite fiction — that all clients live in the same causal universe. Hospitals, labs, or business units, we are told, differ only in sample size, not in how reality behaves.
Anyone who has worked outside a toy benchmark knows this is false.
Different clients intervene. Quietly. Asynchronously. Often without documentation. And those interventions — policy changes, treatment protocols, system overrides — warp local causal structures. Ignore them, and federated causal discovery converges to a graph that is technically consistent and practically wrong.
This paper drops the pretense. It asks a sharper question: What can we recover when clients intervene differently, we don’t know where they intervened, and privacy still matters?
Background — Context and prior art
Classic causal discovery methods aim to recover a CPDAG — a graphical object representing a Markov Equivalence Class (MEC). From purely observational data, this is the best we can do without heroic assumptions.
Recent federated extensions, most notably regret-based approaches, made this feasible without pooling data. The PERI framework showed that by sharing only regrets — discrepancies between local and global scores — a server can reconstruct a shared CPDAG while preserving differential privacy.
There was a catch.
PERI assumes all clients share the same underlying causal graph and observe it passively. Interventions break this assumption. Structural interventions remove edges. Parametric interventions reshape mechanisms. The result is heterogeneous, partially mutilated graphs at the client level.
Existing interventional causal discovery methods usually assume either:
- Known intervention targets, or
- Centralized access to all environments.
Federated reality offers neither.
Analysis — What the paper actually does
The core insight of I-PERI is almost uncomfortable in its simplicity:
If client graphs are mutilated, stop forcing them to agree — instead, let disagreement guide orientation.
Phase 1: Union before precision
I-PERI first recovers a server CPDAG that contains all client CPDAGs as subgraphs. This is done by redefining regret.
Instead of comparing a candidate global graph directly to a client graph (which would never converge when edges are missing), the regret is computed on the intersection of the two graphs.
Effectively:
- Missing edges at a client are tolerated.
- Missing edges at the server are penalized.
As long as at least one client is purely observational, the algorithm provably converges to the true CPDAG of the underlying causal DAG.
This already fixes a fatal flaw in naive federated causal discovery under interventions.
Phase 2: Let interventions speak
Here is where the paper earns its keep.
Structural interventions can unshield colliders, creating new v-structures that are invisible observationally. These appear in client graphs — but only locally.
I-PERI exploits this by:
- Intersecting the server graph with the skeleton of each client graph.
- Penalizing edges that remain unoriented at the server but are oriented at the client.
The result is a refined object: the Φ-CPDAG.
It is strictly more informative than the standard CPDAG, yet still respects privacy and decentralization.
Findings — What changes, concretely
The paper introduces a new equivalence notion:
Φ-Markov Equivalence Class (Φ-MEC)
Two DAGs are Φ-Markov equivalent if:
- They share the same skeleton.
- They share the same v-structures.
- Any intervention-revealed v-structure in one can also be revealed in the other — possibly under different intervention targets.
This matters because:
- Φ-MEC is strictly smaller than the observational MEC.
- Its representative, the Φ-CPDAG, is unique.
Empirically (see Figure 6 in the paper):
| Metric | CPDAG | Φ-CPDAG |
|---|---|---|
| Structural Hamming Distance | Higher | Lower |
| Orientation F1-score | Lower | Higher |
Across varying sample sizes, numbers of variables, and client counts, Φ-CPDAG consistently dominates.
Privacy — No free lunch, but no leaks either
I-PERI inherits PERI’s regret-sharing mechanism and proves bounded sensitivity of the regret function. With Laplace noise, the algorithm is formally ε-differentially private.
Importantly:
- Intervention targets are not shared.
- Reconstructing client graphs from regrets is NP-hard.
This is privacy by structural opacity, not cryptographic theater.
Implications — Why this matters beyond theory
Three takeaways for practitioners:
- Heterogeneity is information. Interventions are not noise to be averaged out; they are signals to be exploited.
- Federated ≠ centralized-lite. Identifiability limits are real, and Φ-CPDAG formalizes what survives decentralization.
- Privacy-aware causal discovery is viable. You do not need to choose between structure and secrecy.
For regulated domains — healthcare, finance, enterprise operations — this is a rare example of theory aligning with operational constraints.
Conclusion — A more honest federated world
I-PERI does not magically recover the full causal DAG. It does something more valuable: it tells us exactly how much structure can be recovered when clients intervene differently and refuse to share raw data.
That honesty is its real contribution.
Cognaptus: Automate the Present, Incubate the Future.