ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

A two-state witness to short-run contrastive-divergence bias

Example

Let $X = {0, 1}$ , $θ \in R$ , and $E_{θ} (0) = 0$ , $E_{θ} (1) = θ$ . Then $Z_{θ} = 1 + e^{- θ}, p_{θ} (1) = \frac{e^{- θ}}{1 + e^{- θ}}, p_{θ} (0) = \frac{1}{1 + e^{- θ}} .$ Take $q_{0} = δ_{0}$ and the transition kernel $K_{θ} (x, y) = \frac{1}{2} 1_{{y = x}} + \frac{1}{2} p_{θ} (y) .$ This kernel is strictly positive and leaves $p_{θ}$ invariant. For $k = 1$ and $θ = 0$ , the reconstruction law satisfies $q_{1, 0} (1) = 1 / 4$ while $p_{0} (1) = 1 / 2$ . The population stop-gradient CD-1 direction for the negative log-likelihood is $- 1 / 4$ , whereas the exact negative log-likelihood gradient is $- 1 / 2$ . The exact derivative of the scalar CD-1 objective at $θ = 0$ is ${\frac{d}{d θ} {CD}_{1} (θ) ∣}_{θ = 0} = - \frac{1}{4} - \frac{1}{8} \log 3.$ Thus the stop-gradient direction also omits a nonzero derivative of the reconstruction law. This finite witness asserts no general sign or consistency claim for contrastive-divergence bias.

At $θ = 0$ , the transition matrix and its state diagram are $K_{0} = (\begin{array}{cc} 3 / 4 & 1 / 4 \\ 1 / 4 & 3 / 4 \end{array}) .$

diagram failed to render:
\begin{tikzpicture}[>=stealth]
  \node[circle,draw] (s0) at (0,0) {$0$};
  \node[circle,draw] (s1) at (3,0) {$1$};
  \draw[->] (s0) edge[loop left] node[left] {$3/4$} (s0);
  \draw[->] (s1) edge[loop right] node[right] {$3/4$} (s1);
  \draw[->] (s0) to[bend left=18] node[above] {$1/4$} (s1);
  \draw[->] (s1) to[bend left=18] node[below] {$1/4$} (s0);
\end{tikzpicture}

The nodes are states; each arrow label is $K_{0} (x, y)$ from its source state $x$ to its target state $y$ . A self-loop records staying at the current state. The diagram displays the matrix and is not used in the verification.

Example references

Geoffrey Hinton, Section 2 of the full technical report describes the contrastive-divergence reconstruction update and its omitted derivative.

Kevin P. Murphy, Chapter 24, Section 24.2.2 defines the finite-step contrastive-divergence objective.

Facts & Assumptions

Given: The two-state energy family, data mass $q_{0} = δ_{0}$ , invariant kernel $K_{θ}$ , $k = 1$ , and parameter value $θ = 0$ specified in the Example.

[F1]

Energy-based density and partition function: $p_{θ} (x) = e^{- E_{θ} (x)} / Z_{θ}$ for finite positive $Z_{θ}$ .

[F2]

Contrastive divergence with k transitions: $q_{1, θ} = q_{0} K_{θ}$ and the population CD-1 direction is $E_{q_{0}} \nabla E_{θ} - E_{q_{1, θ}} \nabla E_{θ}$ .

[F3]

Contrastive divergence with k transitions: for finite states, the population CD-1 direction is $E_{q_{0}} \nabla E_{θ} - E_{q_{1, θ}} \nabla E_{θ}$ .

[F4]

The finite-state CD-k gradient includes the chain-distribution derivative: the exact finite-state likelihood gradient is the data energy-gradient mean minus the model energy-gradient mean, and $\partial_{θ} {CD}_{1}$ subtracts the reconstruction-law derivative term.

[F5]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : finite sums, scalar multiples, products and quotients with nonzero denominator obey the stated derivative rules.

[F6]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : the derivative of a composition is the outer derivative at the image times the inner derivative.

[F7]

The exponential function is smooth and $(\exp)^{'} = \exp$ : $\exp^{'} (u) = \exp (u)$ for real $u$ .

[F8]

The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set: the derivative in each real parameter coordinate is the limit of its difference quotient on an open interval around the point.

[F9]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (u) > 0$ for every real $u$ .

Verification

1.1givenF1F9constructalgebra

Direct normalization gives $Z_{θ} = 1 + e^{- θ}$ and the displayed $p_{θ}$ . By [F9], its coordinates are positive. Each row of $K_{θ}$ sums to $1 / 2 + \sum_{y} p_{θ} (y) / 2 = 1$ , every entry is positive, and $\sum_{x} p_{θ} (x) K_{θ} (x, y) = \frac{1}{2} p_{θ} (y) + \frac{1}{2} p_{θ} (y) \sum_{x} p_{θ} (x) = p_{θ} (y)$ , so $K_{θ}$ is an admissible invariant kernel.

2.1givenF1F2F5F6F7F8step 1.1algebra

At $θ = 0$ , $p_{0} (0) = p_{0} (1) = 1 / 2$ and $q_{0} = δ_{0}$ , so [F2] gives $q_{1, 0} (1) = K_{0} (0, 1) = p_{0} (1) / 2 = 1 / 4$ and $q_{1, 0} (0) = 3 / 4$ . Also $E_{θ}^{'} (0) = 0$ and $E_{θ}^{'} (1) = 1$ . By [F6] and [F7], $(e^{- θ})^{'} = - e^{- θ}$ ; applying the quotient rule [F5] to $u / (1 + u)$ , where $u = e^{- θ}$ , gives $p_{θ}^{'} (1) = u^{'} / (1 + u)^{2} = - e^{- θ} / (1 + e^{- θ})^{2} = - p_{θ} (1) p_{θ} (0)$ . Hence $p_{0}^{'} (1) = - 1 / 4$ ; the scalar rule [F5] applied to $q_{1, θ} (1) = p_{θ} (1) / 2$ gives $q_{1, 0}^{'} (1) = - 1 / 8$ and, since $q_{1, θ} (0) = 1 - q_{1, θ} (1)$ , $q_{1, 0}^{'} (0) = 1 / 8$ .

3.1F3F4step 2.1algebra

By [F4], the exact negative log-likelihood gradient at $0$ is $0 - p_{0} (1) = - 1 / 2$ . By [F3], the population stop-gradient CD-1 direction is $0 - q_{1, 0} (1) = - 1 / 4$ , so it differs from the exact likelihood gradient.

4.1F4step 2.1step 3.1algebra∎

By [F4] and step 2.1, the chain correction is $\sum_{x \in {0, 1}} q_{1, 0}^{'} (x) \log (q_{1, 0} (x) / p_{0} (x)) = \frac{1}{8} \log \frac{3}{2} - \frac{1}{8} \log \frac{1}{2} = \frac{1}{8} \log 3 > 0$ . Subtracting it from the first two CD terms $- 1 / 4$ gives ${\frac{d}{d θ} {CD}_{1} (θ) ∣}_{θ = 0} = - \frac{1}{4} - \frac{1}{8} \log 3$ , as stated. Thus the stop-gradient CD-1 direction also differs from the exact CD-1 objective derivative.

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Sources

Geoffrey Hinton, A Practical Guide to Training Restricted Boltzmann Machines, Version 1, Section 2 (standard reference, not scraped)
Kevin P. Murphy, Probabilistic Machine Learning: Advanced Topics, Chapter 24, Section 24.2.2 (standard reference, not scraped)