LemmaStatement: Literature-sourcedProof: 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.

The Bregman three-point identity

Statement

Let $K \subseteq R^{d}$ , let $U \subseteq R^{d}$ be open, and let $R$ be a mirror regularizer whose Bregman divergence is $B_{R} (x ∥ y) = R (x) - R (y) - ⟨ \nabla R (y), x - y ⟩$ for $x \in K \cup U$ and $y \in U$ . For $a \in K \cup U$ and $b, c \in U$ ,

$⟨ a - b, \nabla R (c) - \nabla R (b) ⟩ = B_{R} (a ∥ b) - B_{R} (a ∥ c) + B_{R} (b ∥ c) .$

Statement references

Mirror maps, Bregman divergence, and local Hessian norms: defines $B_{R} (a ∥ b) = R (a) - R (b) - ⟨ \nabla R (b), a - b ⟩$ whenever $a \in K \cup U$ and $b \in U$ .

Facts & Assumptions

Given: A mirror regularizer $(K, U, R)$ , $a \in K \cup U$ , and $b, c \in U$ .

[F1]

Mirror maps, Bregman divergence, and local Hessian norms: $B_{R} (x ∥ y) = R (x) - R (y) - ⟨ \nabla R (y), x - y ⟩$ for $x \in K$ and $y \in U$ , with the same formula on $U \times U$ .

Proof

1.1givenF1algebra

Expand the right side using [F1]. The terms $R (a)$ , $R (b)$ , and $R (c)$ cancel, leaving $- ⟨ \nabla R (b), a - b ⟩ + ⟨ \nabla R (c), a - c ⟩ - ⟨ \nabla R (c), b - c ⟩$ .

2.1step 1.1algebra∎

Since $(a - c) - (b - c) = a - b$ , the remaining expression is $⟨ a - b, \nabla R (c) - \nabla R (b) ⟩$ .

Depends on

Mirror maps, Bregman divergence, and local Hessian norms

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Elad Hazan, Introduction to Online Convex Optimization, second edition (standard reference, not scraped)
Peter Bartlett, Stat 260/CS 294-102: Learning in Sequential Decision Problems, mirror-descent lecture notes (standard reference, not scraped)