Learning what to recover while you spend on recovering it

This is the top of the staircase, and the least finished step. Everything below it holds the recovery law fixed and treats the de-censoring choice as one episode. The version that would run in an operation does neither. You learn the recovery law from the very recoveries you are paying for, and you spend against two separate budgets at once: an evidence budget that funds recovering censored outcomes, and an intervention budget that funds acting on what those recoveries reveal. The budgets are non-fungible, you cannot move money from one to the other, and they are coupled in both directions. How do you allocate two coupled, learning-limited ledgers over time when the feedback that tells you whether you were right is both delayed and censored?

Why the two ledgers cannot be collapsed

The tempting simplification is to fold both budgets into one and run a known online algorithm. It does not survive contact with the problem. Evidence you recover but cannot afford to act on is wasted spend on the evidence ledger. Intervention capacity you commit without informing it is spent blind on the intervention ledger. The optimal split between the two is itself state-dependent and changes as you learn, so a single fungible budget is the wrong model, and the guarantees that assume one do not apply.

The feedback structure makes it harder. When you act on a recovered outcome, the signal that tells you whether the recovery law was accurate arrives late, and it is itself censored: you often learn the result of the interventions you funded and not the counterfactual for the ones you did not. This is the same censoring the whole frontier is about, now closing a loop around the learner.

A target regret decomposition

We do not have a regret bound. What we have is a target the bound should decompose into, which is useful because it says what a proof would have to control. The decomposition stays in the lab.

What can be said is where the difficulty sits. Two of its terms have relatives in the published literature and are, we expect, tractable. The third has no analogue in single-budget online allocation: it is the cost of misallocating between the two ledgers, and it exists only because the budgets are coupled and non-fungible. A satisfying result would bound every term and show that this one does not dominate. It is the genuine unknown, and naming it as such is most of what an idea-stage project can honestly contribute.

Why the obvious references do not close it

Bandits-with-knapsacks handles online decisions under a budget, but assumes a single fungible budget and immediate feedback — both of which we lose. Adaptive two-stage stochastic programming handles recourse after information arrives, but is not an online-learning setting and does not accumulate a regret. The value-of-information heritage from Blackwell tells us what a recovery is worth once the law is known, not how to price it while learning it under a spend constraint. Each is a genuine ancestor; none is the theorem.

The first milestone

Not a product, not a prospective trial — a theorem in a restricted setting. Fix the feedback delay, restrict the recovery law to a small family, and prove a regret guarantee for a specific two-ledger policy in that restricted world. Only once that exists does a prospective evaluation mean anything; running one before it would produce a number with nothing underneath it, the one thing this lab will not ship. The upside if the theorem generalises is the largest on the frontier — a policy that learns what to recover while it spends on recovering it, and so, honestly, is the risk that it does not.