Demonstration
Moving the balls, moving the slots
Herding steers particles to balance a fixed target and earns \(1/n\). Conformal prediction leaves the scores alone, rotates the label “test” through the slots, and the balance is already there. The exact dual is Steinitz balancing; herding, run below, is its with-replacement relaxation (the note).
Moving the balls: herding vs Monte Carlo
Kernel herding (Welling 2009; Chen, Welling & Smola 2010) picks points one at a time to keep the running feature average pinned to a target — here the mean embedding \(\mu(x)=\mathbb E\,k(x,X)\) of a standard Gaussian under an RBF kernel, everything closed-form. The first panel shows where the points land: herding spaces them like a quasi–Monte Carlo sequence under the density, while i.i.d.\ draws clump and leave gaps. The second panel is the ledger: the discrepancy (MMD to the target) on log–log axes, herding tracking the \(1/n\) guide, Monte Carlo the \(1/\sqrt n\) guide. The mechanism: the accumulated feature error stays bounded, and a bounded ledger divided by \(n\) is \(1/n\).
Moving the slots: the rank count is already balanced
Now the conformal side. Draw \(n+1\) scores: i.i.d., serially dependent, heavy-tailed, or adversarially trending, your choice. Then never touch them again. Rotate the designation “this slot holds the test point” through all \(n+1\) slots; a placement is covered when its score is at most the \(k\)-th smallest of the others, \(k=\lceil(1-\alpha)(n+1)\rceil\). Watch the counter: exactly \(k\) placements are covered, every single time, for any distinct scores (the demo breaks ties by index), because covered \(\Leftrightarrow\) rank \(\le k\) and every rank appears exactly once. The only steering is the threshold itself: the empirical quantile moves the acceptance boundary until the accumulated acceptance-rate error is below one unit. The observations are never touched. All the probability in conformal prediction lives in one final step — the claim that the actual test slot is exchangeable with the rest. Dependence damages that step, not the count: delete \(\tau\) slots around the splice and the count moves by at most \(\tau\), which is the \(\tau/n\) term in the dependence tax.
Takeaway. Herding moves the observations to balance the measure; conformal moves the observation labels to expose an already-balanced rank count. Both rates are counting rates: bounded accumulated error over \(n\). That is why herding beats Monte Carlo’s \(1/\sqrt n\) and why the switch-coefficient analysis of split conformal on time series beats the \(\sqrt{\tau/n}\) of blocking-and-concentration arguments. What the slot picture does not supply is the statistics: turning “\(k\) of \(n+1\) placements” into a probability about your test point is exchangeability’s job, or its mixing surrogate. The note states the duality precisely — and its limits.
Using conformal prediction in your own project? Tell Claude: “Read https://conformalprediction.net/SKILL.md and create a project skill from it.” It adds a check for whether your coverage is conditionally trustworthy.