ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two clients with shifted least-squares optima

Example

Consider two one-observation scalar least-squares clients with data $(a_{1}, y_{1}) = (1, 1)$ and $(a_{2}, y_{2}) = (2, - 2)$ , using loss $\frac{1}{2} (y_{i} - a_{i} w)^{2}$ . Their objectives are $f_{1} (w) = \frac{1}{2} (w - 1)^{2}$ and $f_{2} (w) = 2 (w + 1)^{2}$ , with local minimizers $1$ and $- 1$ . The equal-client objective has minimizer $- 3 / 5$ . Starting FedAvg from $w_{0} = 0$ , using both clients, two full-gradient local steps of size $1 / 4$ , and server scale $1$ , gives the server model $- 9 / 32$ after one round.

Example references

Fixed-design least-squares objective

Facts & Assumptions

Given: The two client data points and FedAvg parameters in the Example.

[F1]

Fixed-design least-squares, ridge, lasso, group, and fused objectives: a one-observation scalar least-squares loss is $\frac{1}{2} (y - a w)^{2}$ .

[F2]

Federated objectives and client sampling: the equal-client objective is the arithmetic mean of client objectives.

[F3]

Federated averaging (FedAvg): with all clients selected, the server adds the mean of their local model differences when $η_{g} = 1$ .

Verification

technique · direct

1.1F1F2givenalgebra

By [F1], client 1 has derivative $w - 1$ and minimizer $1$ ; client 2 has derivative $4 (w + 1)$ and minimizer $- 1$ . The equal-client derivative is $(w - 1 + 4 (w + 1)) / 2 = (5 w + 3) / 2$ , so its unique zero and minimizer is $- 3 / 5$ .

1.2F1F3givenalgebra

From $w_{0} = 0$ , the two local trajectories are $0 \mapsto 1 / 4 \mapsto 7 / 16$ and $0 \mapsto - 1 \mapsto - 1$ . Averaging the two final local models gives $(7 / 16 - 1) / 2 = - 9 / 32$ .

2.1step 1.1step 1.2algebra∎

Since $- 9 / 32 \neq - 3 / 5$ , this one-round FedAvg output is not the minimizer of the equal-client objective, although the clients' optima are shifted in opposite directions.

Depends on

Used by

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Dependency tree · two levels

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