From Local Derivatives to Global Decoder Risk

Conceptual map

I-06.01Optimal decoder loss
I-06.02Fiberwise energy
I-06.03Poincare certificates
I-06.04Component means and residuals
I-06.05Limits of graph certificates

1. Conditional means define the optimal squared-loss decoder

A derivative calculation measures variation along a fiber. Global decoder risk averages all unresolved variation, including differences between disconnected fiber components.

Let $$X$$ be a random state, $$Y=h(X)$$ the report, and $$f(X)$$ a square-integrable scalar target. For any square-integrable decoder $\delta(Y)$ , set $\delta^{*}(Y)$ =E[ $$f(X)|Y$$ ].

Proposition 1 · Projection identity

The squared-loss risk decomposes as

\[E[(f-\delta )^{2}]=E[\operatorname{Var}(f|Y)]+E[(\delta ^{*}-\delta )^{2}].\](1)

Thus $\delta^{*}$ minimizes risk, and its irreducible risk is the expected conditional variance.

Proof. Write $f-\delta=(f-\delta^{*})+(\delta^{*}-\delta)$ . The cross term has conditional expectation zero given $$Y$$ . Taking expectations yields (1). ∎

2. Fiberwise Poincaré bounds need component means

Fix a report $$y$$ . Suppose its conditional law is a mixture over connected components $$c$$ with weights $\pi _{c}(y)$ . Let $\mu _{c}$ =E[ $$f|Y=y,c$$ ] and let $\mathcal{E}_{c}(f)$ be a within-component Dirichlet energy with spectral gap $\lambda _{c}>0$ .

Proposition 2 · Componentwise risk certificate

The conditional target variance satisfies

\[\operatorname{Var}(f|y)\le \sum _{c}\pi _{c}\mathcal{E}_{c}(f)/\lambda _{c}+\sum _{c}\pi _{c}(\mu _{c}-\mu )^{2},\](2)

where $\mu=\sum \pi _{c}\mu _{c}$ .

Proof. The law of total variance across the component label gives an exact sum of average within-component variance and variance of component means. Apply the Poincaré inequality to each within-component term. ∎

3. Zero local energy with large decoder risk

Consider one report whose conditional state graph consists of two disconnected edges with four equally likely nodes. Let the target values be (0,0,10,10), with the first pair on one component and the second pair on the other. The target is constant within each component, so every within-component edge energy is zero.

The conditional mean is five and the conditional variance is 25. In (2), the component means are zero and ten with weights one half, so the between-component term is 25. A certificate that retains only derivative or edge energy returns zero and misses the entire decoder risk.

Failure case · Connectedness omitted from a local-to-global bound

A positive gap controls deviations from the mean inside one connected component. Multiple zero modes permit different component constants. Their weighted dispersion must be measured directly or ruled out by a connected-fiber assumption.

4. Generator and discretization residuals

For a graph or numerical generator, record the residual of the constant null vector, reversibility error, component count, eigenpair residuals, and the smallest positive eigenvalue per component. If a Poisson equation or projected gradient is used to estimate energy, its solver residual contributes a separate numerical term; it does not belong inside the statistical conditional variance.

Grid refinement should track the target mean, between-component term, energy, gap, and discretization change independently. Agreement of their final sum with a direct finite-state conditional variance provides a closed-loop check.

5. Implementation, exercises, and sources

Estimate or enumerate conditional fibers, identify connected components before computing gaps, center the target within each component, and combine the two terms in (2). Average the resulting bound over the report distribution to compare with the irreducible term in (1).

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Exercises

Verify the risk decomposition for target values (0,0,10,10).
Replace them by (0,1,10,11) and compute within-component variance, between-component variance, and total variance.
Add an edge between the middle nodes and calculate how the global gap and energy change.

Partial solutions

1. The mean is five; the four squared deviations are 25, so variance is 25. Both component variances are zero and the component-mean variance is 25. 2. Component means are 1/2 and 21/2. Average within-component variance is 1/4 and the between-component variance is 25, giving total 25.25.

Olav Kallenberg, Foundations of Modern Probability, sections on conditional expectation.Orthogonal projection and total variance.
Dominique Bakry, Ivan Gentil, and Michel Ledoux, Analysis and Geometry of Markov Diffusion Operators, Chapter 4.Spectral gaps and Poincaré inequalities.
III-14 · Conditional Generators, Poincaré Inequalities, and Graph Certificates.Finite-graph proof and generator assumptions.
Chae-Yeon Xon (2026), “Counterfactual Sufficiency: State Representation and Response Geometry.”Related conditional-generator application.

6. Audit checkpoint

Report the state law, report map, conditional target, optimal decoder, fiber components, component weights and means, energies, gaps, numerical residuals, discretization error, and exact or bounded risk.

7. Scope boundary

The chapter combines squared-loss conditional projection with componentwise Poincaré bounds. Other losses, estimated fibers, and high-dimensional graph convergence require separate theory.