ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Full-information finite experts

Example

For $T = 2$ rounds and three experts, let the learner's decisions be probability vectors in $Δ_{3} : = {x \in [0, 1]^{3} : x_{1} + x_{2} + x_{3} = 1}$ , with $x_{1} = (1 / 2, 1 / 3, 1 / 6)$ and $x_{2} = (1 / 6, 1 / 3, 1 / 2)$ . Let the expert-loss vectors be $ℓ_{1} = (0, 1, 1)$ and $ℓ_{2} = (1, 0, 1)$ , let $f_{t} (x) = ℓ_{t} \cdot x$ , and reveal the entire vector $ℓ_{t}$ after each round. The learner's total loss is $7 / 6$ , the minimum total loss of one fixed vector in $Δ_{3}$ is $1$ , and the resulting static regret is $1 / 6$ .

Example references

Online decision protocol: choices are made from the past before the current round loss is observed.

Adversary and feedback models: full-information feedback reveals the entire round loss function or expert-loss vector.

Static, dynamic, interval, and strongly adaptive regret: static regret subtracts the minimum cumulative loss of one fixed comparator.

Elad Hazan, Introduction to Online Convex Optimization, second edition: §1.2.1, PDF pp. 24–25.

Tengyu Ma, Lecture Notes for Machine Learning Theory: §11.2, PDF p. 149.

Facts & Assumptions

Given: The simplex $Δ_{3}$ , decisions $x_{1} = (1 / 2, 1 / 3, 1 / 6)$ and $x_{2} = (1 / 6, 1 / 3, 1 / 2)$ , losses $f_{1} (x) = x_{2} + x_{3}$ and $f_{2} (x) = x_{1} + x_{3}$ , and full-information feedback revealing each entire loss vector.

[F1]

In an online protocol, the learner chooses from the past before incurring and receiving feedback about the current round loss (Online decision protocol).

[F2]

Full-information feedback reveals the entire function $f_{t}$ or, for $N$ experts, the full vector of their $N$ losses (Adversary and feedback models).

[F3]

Static regret is learner cumulative loss minus the attained minimum cumulative loss over one fixed $u \in K$ (Static, dynamic, interval, and strongly adaptive regret).

Verification

1.1givenF1F2

The loss vectors are $ℓ_{1} = (0, 1, 1)$ and $ℓ_{2} = (1, 0, 1)$ , and the full-information rule reveals both vectors after their respective choices, as required by the online timing and feedback model.

1.2givenF3algebra

For any $u \in Δ_{3}$ , $f_{1} (u) + f_{2} (u) = u_{1} + u_{2} + 2 u_{3} = 1 + u_{3} \geq 1$ , with equality whenever $u_{3} = 0$ ; hence the attained minimum is $1$ .

2.1givenstep 1.1algebra

Direct evaluation gives $f_{1} (x_{1}) = 1 / 3 + 1 / 6 = 1 / 2$ and $f_{2} (x_{2}) = 1 / 6 + 1 / 2 = 2 / 3$ , so the learner's cumulative loss is $1 / 2 + 2 / 3 = 7 / 6$ .

3.1givenF3step 2.1step 1.2algebra∎

Subtracting this minimum from the learner's cumulative loss in step 2.1 gives $7 / 6 - 1 = 1 / 6$ by the static-regret definition.

Depends on

Used by

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Dependency tree · two levels

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Sources

Elad Hazan, Introduction to Online Convex Optimization, second edition (standard reference, not scraped)
Tengyu Ma, Lecture Notes for Machine Learning Theory, Chapter 11 (standard reference, not scraped)