docs(examples): dual/t-SVGP tutorial notebook + docs wiring [stack 7/10]

[thomaspinder]

Checklist

• I've formatted the new code by running uv run poe format before committing.
• I've added tests for new code.
• I've added docstrings for the new code.

Description

PR 7/10 of the v1.0 stack (natural-gradients 5 of 5), stacked on #729 — merge order #743 → #744 → #714 → #715 → #716 → #729 → this.

Adds examples/dual_svgp.py — the dual-parameterization (t-SVGP) tutorial (Adam et al. 2021, new adam2021dual bib entry) — plus the docs wiring for both new notebooks (mkdocs nav under Tutorials, CTA_NOTEBOOKS entries).

The notebook (43 cells, executes in ~64 s on CPU) continues directly from examples/natgrads.py, reusing its banana dataset character-for-character so the two papers' methods overlay:

1. The maths: likelihood sites and the EP heritage; the tied projection to inducing space; why the tied natural-gradient update is an SVI-style convex combination needing no θ↔η round-trip; the convention traps (site normalisation constant = the Titsias sparsity gap; sites live in u-space).
2. ρ ≡ γ demonstrated: the dual update and the Salimbeni step coincide — measured overlay to ~2e-15 per step, and identical fit_natgrads histories to 4e-14 — with an honest section on where conditioning erodes the identity at banana scale.
3. ρ=1 conjugate recovery to ~1e-12 with a non-zero mean function, exercising the centred-site convention (where the paper's reference implementation has a latent bug).
4. Banana benchmark: both natural-gradient methods reach Adam-only's 1000-iteration ELBO in ~110 iterations; the notebook reports plainly that the dual iteration is not cheaper at this scale and explains why (the paper's ~5× is its MNIST setting, attributed).
5. Hyperparameter learning / VEM: measured hyper-gradient convergence of dual_elbo → elbo (39 → 8.5e-15 across E-steps), the Z=X dominance property (up to +1.5e5 nats), and where sparse dominance fails (M=5/10 counterexamples shown, not hidden).

Gate at the stack tip: full poe all-tests (2,782 passed) + both notebooks executed + full poe docs-build — which caught and fixed a markdown-katex rendering crash — with both pages in nav and the citation rendering verified.

Issue Number: N/A

🤖 Generated with Claude Code

https://claude.ai/code/session_01VHxYz2P7JCSqpax5RmDwWD

[thomaspinder]

Whole-stack review follow-up (commit a433136): after all five PRs were green, an independent Fable-model agent reviewed the aggregate diff, re-deriving the core updates from the papers. Verdict: sound to merge. Its three confirmed findings are fixed here (on the tip branch, deliberately, to avoid rebasing a green stack overnight):

1. Major (documentary) — the dual notebook attributed its measured dual-vs-Salimbeni E-step gap to floating-point conditioning. The true, now-measured-in-the-notebook mechanism: inv_probit's 1e-3 clip makes the computed Bernoulli log-likelihood non-log-concave for f < −2.44, one mislabeled point's β goes negative, beta_floor engages, branches diverge (5.1e-3 with the floor, 6.2e-13 without). All "identical iterates" claims across the stack now carry the log-concavity qualification; two regression tests pin the mechanism.
2. Minor — the "covers GraphVariationalGaussian" docstring now names the real (pre-existing, upstream) failure on the elbo path.
3. Minor — _check_natgrad_schedule now rejects schedules dipping ≤ 0 for every family (user-visible tightening: a decaying schedule reaching zero within num_iters now raises).

The reviewer's suggested follow-up beyond this stack: move the Bernoulli expected log-likelihood to log space (log_ndtr) to remove the clip — that would make the log-concavity guarantees exact for the shipped likelihood.

🤖 Generated with Claude Code

[github-actions]

📖 Docs preview: https://pr-730--endearing-crepe-c2d5fe.netlify.app

Smoke render — the expensive notebooks run with reduced budgets, so
figures are not publication fidelity. /render-mode.txt says smoke.