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Adaptive capping reduces the cost of algorithm configuration by ending an evaluation when a valid bound already rules out winning the comparison. For nonnegative runtimes, completed runs and elapsed time in the current run can provide a lower bound on the final cost. ParamILS distinguishes trajectory-preserving capping from a more aggressive heuristic policy.^[ref]

Consider an original comparison of mean runtime over $N=4$ fixed runs. The incumbent has mean $c_{\ast }=5$ seconds, or total $20$ seconds. A candidate has completed runs taking $6$ and $7$ seconds, and its third run has already taken $8$ seconds. Even if all remaining work took zero time, the final mean would satisfyContinuing that candidate cannot improve this comparison. The partial total itself supplies the certificate; no prediction of the remaining runtime is needed.

By contrast, after only the first two runs, their observed mean is $13/2=6.5>5$, but rejecting solely on that partial mean would be invalid: the four-run lower bound is only $13/4=3.25$. Two sufficiently fast remaining runs could still win. The denominator must correspond to the comparison objective.

A censored algorithm runtime is therefore not an observed completed runtime. Record the cutoff and what bound it establishes. Timeout penalties, changing instance sets or non-runtime objectives require a bound compatible with their own scoring rules. A heuristic early-stopping rule may be useful without preserving the same search decisions as full evaluation.

BasicILS is the fixed-evaluation variant of ParamILS. In $\mathrm{BasicILS}(N)$, competing parameter settings are compared using the same $N$ benchmark runs, with matched instances and random seeds.^[ref] The comparison rule supplies the cost estimates used by the surrounding iterated local search.

For mean runtime, writewhere $r_i$ is the recorded cost on the specified run. Consider invented costs $A=(1,1,19)$ seconds and $B=(5,5,5)$ seconds. With only the first run, $A$ appears better, $1<5$. Across all three runs, its mean is $7$ seconds while $B$ remains at $5$ seconds. Choosing $N$ therefore affects both evaluation expense and how much instance variation the estimate captures.

Using the same run list makes that example a comparison on common evidence. Comparing $A$ only on easy instances with $B$ on harder ones would answer a different question. Even a matched finite sample can be unrepresentative of future workloads, so the final configuration still needs evaluation on unused instances.

FocusedILS changes the allocation of runs rather than requiring the same fixed $N$ for every explored candidate. Adaptive capping can avoid completing an already-lost runtime comparison under a valid lower-bound rule. Neither modification makes a finite observed mean a proof of the best possible setting in the entire algorithm parameter space.

A censored algorithm runtime is an evaluation that ends without observing the algorithm's actual completion time. A timeout or adaptive capping limit can establish that the run had not finished by the cutoff, supplying right censoring rather than an exact completed runtime. The ParamILS capping analysis uses such elapsed-time information to bound comparisons without finishing every run.^[ref]

Write $T$ for completion time and $C$ for a stopping limit. A simple record iswhere $\delta =1$ means completion was observed. The event flag is essential: the same stored duration can mean a completed run or only a lower bound on a still-running computation.

For an original example, two completed runs take $2$ and $5$ seconds and a third is stopped unfinished at $10$ seconds. Averaging the stored durations gives $17/3\approx 5.67$ seconds, but that is only a lower bound on the mean of the actual completion times. If the third would take $40$ seconds, the true mean is $47/3\approx 15.67$ seconds. Discarding it entirely gives $3.5$ seconds and removes the difficult run from the comparison.

Keep exact observations, cutoff information and any timeout penalty separate. A penalized benchmark score may deliberately assign a stated cost to a timeout; that score is not an observation of $T$. Estimating completion-time distributions additionally requires assumptions about how stopping limits were chosen, particularly when the limits depend on earlier performance.

ParamILS is a framework for algorithm configuration based on iterated local search. It explores an algorithm parameter space using improvements to nearby configurations, perturbations and occasional restarts. BasicILS compares configurations on a fixed number of matched training runs; FocusedILS adapts how many runs support a comparison.^[ref]

The distinctive efficiency idea of adaptive capping is to stop spending evaluation time on a candidate once it cannot win the comparison under the stated cost model. Consider an original example with three fixed instances and mean nonnegative runtime as the objective function. A fully evaluated incumbent has total time $12$ seconds, hence mean $4$ seconds. If the first two candidate runs already total $13$ seconds, even a zero-time final run leaves mean at least $13/3>4$. Rejecting that candidate does not require finishing the last run.

This argument depends on its assumptions: the comparison uses the same fixed instance set and a lower bound on unfinished cost. It cannot be transferred unchanged to arbitrary quality scores or a different timeout penalty. The paper distinguishes trajectory-preserving capping from a more aggressive heuristic variant; faster rejection does not always carry the same guarantee.^[ref]

The method turns tuning into a measured optimization experiment, but a good observed configuration is not a proof of a global minimum. The neighborhood, evaluation budget and deployment workload remain relevant to what the returned setting establishes.