Adaptive Sensing versus Every Fixed Schedule: How Much Is Adaptivity Worth?

Hospital staff cannot watch a patient continuously, so they measure on a schedule fixed in advance — before meals, or every four to six hours. Any fixed schedule misses events between checks, and the obvious fix is to decide each measurement from what is known at the time. But the two options are not equally tractable. A fixed schedule is a handful of clock times: a six-hour night at fifteen-minute resolution has twenty-four slots, so at most eight measurements gives ${\sum }_{1\le k\le 8}\left(\genfrac{}{}{0ex}{}{24}{k}\right)=1,271,625$ schedules and the best is found by checking every one. A policy that decides as it goes must act at every state it can reach — $(24\text{ slots})\times (61\text{ belief means})\times (25\text{ staleness levels})=36,600$ on our grid — giving $2^{36,600}$ policies. One side can be settled exactly and the other cannot — which is why a reported gain over a fixed schedule can always be blamed on the chosen baseline. We remove that objection: a constrained partially observed Markov decision process makes the adaptive side tractable, and enumeration makes the fixed side an exact empirical optimum within its class. What we measure. On 112 hospitalized patients whose blinded sensor recorded every slot, we compare a solved adaptive policy against the best of all $1,271,625$ fixed schedules at six pre-specified measurement rates. The adaptive policy detects more nocturnal hypoglycemia at every rate, by $14.2$ percentage points on average. What we do not claim is its size: a bootstrap that reruns selection gives $[-2.0,+39.8]$, positive in 96 percent of replicates but wide, because $93$ held-out excursions carry the information of some $18$ independent events. The advantage also mixes when to measure within a night with spending more on some nights than others. What generalizes beyond this dataset. An evaluation protocol for selected policies: selection is rerun inside every bootstrap replicate and splits are taken on unique identifiers, the usual alternative silently mixing about one unit in six however large the sample. A comparator that scales: detection coverage is monotone submodular, so a greedy schedule attains $1-1/e$ of the exhaustive optimum wherever enumeration is infeasible, and $94$ to $100$ percent of it here. Conditions under which adaptivity is worth nothing: provably zero when a measurement carries no reward-relevant information, at most $2k\epsilon$ when that holds to within $\epsilon$. Zero is the right thing to characterize, because correlated sensing admits no universal constant bound on what adaptivity is worth.