On Inflating Circuit-Wasserstein Balls

PeTeR’s post-training game is already a three-player affair. One recalls the unconstrained Lagrangian

$$\max_{P_\theta}\min_{Q_\phi}\max_{\lambda \ge 0}\; \mathbb{E}_{Q_\phi}[\log P_\theta(\mathbf{X})] + \lambda\bigl(\mathsf{CW}(\hat{P}, Q_\phi) - \varepsilon\bigr),$$

with the usual ascent–descent updates on $\theta$, $\phi$, and $\lambda$. The multiplier $\lambda$ is the ball’s conscience: when the adversary strays outside radius $\varepsilon$, $\lambda$ grows; when the adversary behaves, $\lambda$ shrinks toward zero. That conscience is correct at convergence. Mid-training, it can be a little too polite.

The inflation rule

We keep PeTeR’s construction intact, including the $\max$–$\min$–$\max$ schedule, with one local edit. Whenever

$$\mathsf{CW}(\hat{P}, Q_\phi) < \varepsilon,$$

we do not apply the standard multiplier update

$$\lambda \leftarrow \max\bigl(0,\; \lambda + \eta_\lambda[\mathsf{CW}(\hat{P}, Q_\phi) - \varepsilon]\bigr).$$

We hard-clamp instead:

$$\lambda \leftarrow 0.$$

Outside the ball, the ordinary update proceeds unchanged. Inside the ball, the dual variable is extinguished rather than gently decayed.

What temporarily inflates

With $\lambda$ forced to zero on the interior, the adversary is briefly freed from the soft penalty that would otherwise keep it pressed against a timid neighborhood of $\hat{P}$. In practice the effective diameter of the Circuit-Wasserstein ball expands during training: the worst-case $Q_\phi$ is allowed to roam farther than a strictly enforced $\varepsilon$-budget would suggest, then the constraint reasserts itself as soon as $\mathsf{CW}(\hat{P}, Q_\phi)$ meets or exceeds $\varepsilon$.

The end state is not an inflated object. At convergence one still recovers a properly sized, non-inflated $\varepsilon$-$\mathsf{CW}$ ball. The theoretical stability guarantees that PeTeR already claims for Circuit-Wasserstein balls are preserved; inflation is a training-time accommodation, not a rewriting of the feasible set on the certificate you publish afterward.

Why bother

Because $\mathsf{CW}$ upper-bounds Big Wasserstein, an $\varepsilon$-$\mathsf{CW}$ ball is a stricter neighborhood than the corresponding $\mathsf{W}$-ball. That strictness is why we can compute anything at all. It is also why a faithfully enforced Circuit-Wasserstein adversary sometimes under-covers the classical worst case. Temporary inflation during the $\max$–$\min$–$\max$ game is a way of closing that gap—letting the circuit adversary behave a little more like Big Wasserstein mid-optimization—without surrendering the tractable geometry we actually trust at the finish.

Big Wasserstein, consulted for comment, called the maneuver “cosplay with a schedule.” We took it as praise.

Elsewhere

OpenCW’s alignment team reports that it is working on aligning ball inflation with the public good. No one on the editorial desk is certain what that means, which is increasingly the house style for OpenCW announcements. Until they publish a definition of “public good” that survives contact with a coupling circuit, the Circuit-Wasserstein Ball Community will continue to inflate on purpose and deflate on principle.