Sequential model-challenging design under evolving decisions

Develop a sequential decision framework for Gaussian-process optimization in which repeated model tests have time-uniform false-rejection control, nonlinear structural hypotheses are handled through appropriate algebraic-geometric analogues, the value of allocating the next evaluation to model testing rather than objective improvement is assessed, and uncertainty in the discrepancy family and Gaussian-process hyperparameters is propagated into the testing-versus-optimization decision.

Background

The paper develops a prospective framework for determining which structural departures a fixed Gaussian-process design can reveal and with what power. Its exact results, however, concern a fixed design tested once, with fixed model components and a specified discrepancy or discrepancy family.

The authors identify several unresolved extensions for sequential settings. Repeated testing as observations arrive requires control of false rejections over time; nonlinear structural hypotheses such as low-rank interactions require methods beyond linear null-space relations; an optimization algorithm must trade an evaluation used to challenge the model against one used to improve the objective; and uncertainty about both the discrepancy family and Gaussian-process hyperparameters must be incorporated rather than treated as fixed.

References

Extending this to a sequence of decisions raises several separate problems. Repeated tests need time-uniform error control, for example through confidence sequences . Non-linear structural hypotheses, such as low-rank interactions, replace null-space relations by algebraic varieties; for a Segre-type rank model, distances to the relevant secant variety and its Euclidean distance degree are natural analogues of the projector used here. An optimisation algorithm must also decide whether the next evaluation is worth spending on testing rather than on improvement. Finally, uncertainty in the discrepancy family and in the GP hyperparameters must be propagated into that decision rather than treated as fixed. These questions are deliberately left open here.

— What Can a Gaussian Process Design Test  (2610.10122 - Boi et al., 7 Oct 2026) in Section 7.2, “Implications for sequential design”