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Ruiz, F. ; Dy, J. ; Meent, J.W. (Ed.)Prediction algorithms, such as deep neural networks (DNNs), are used in many domain sciences to directly estimate internal parameters of interest in simulatorbased models, especially in settings where the observations include images or complex highdimensional data. In parallel, modern neural density estimators, such as normalizing flows, are becoming increasingly popular for uncertainty quantification, especially when both parameters and observations are highdimensional. However, parameter inference is an inverse problem and not a prediction task; thus, an open challenge is to construct conditionally valid and precise confidence regions, with a guaranteed probability of covering the true parameters of the datagenerating process, no matter what the (unknown) parameter values are, and without relying on largesample theory. Many simulatorbased inference (SBI) methods are indeed known to produce biased or overly con fident parameter regions, yielding misleading uncertainty estimates. This paper presents WALDO, a novel method to construct confidence regions with finitesample conditional validity by leveraging prediction algorithms or posterior estimators that are currently widely adopted in SBI. WALDO reframes the wellknown Wald test statistic, and uses a computationally efficient regressionbased machinery for classical Neyman inversion of hypothesis tests. We apply our method to a recent highenergy physics problem, where prediction with DNNs has previously led to estimates with prediction bias. We also illustrate how our approach can correct overly confident posterior regions computed with normalizing flows.more » « less

Ruiz, F. ; Dy, J. ; van de Meent, J.W. (Ed.)Random Fourier Features (RFF) is among the most popular and broadly applicable approaches for scaling up kernel methods. In essence, RFF allows the user to avoid costly computations with a large kernel matrix via a fast randomized approximation. However, a pervasive difficulty in applying RFF is that the user does not know the actual error of the approximation, or how this error will propagate into downstream learning tasks. Up to now, the RFF literature has primarily dealt with these uncertainties using theoretical error bounds, but from a userâ€™s standpoint, such results are typically impracticalâ€”either because they are highly conservative or involve unknown quantities. To tackle these general issues in a datadriven way, this paper develops a bootstrap approach to numerically estimate the errors of RFF approximations. Three key advantages of this approach are: (1) The error estimates are specific to the problem at hand, avoiding the pessimism of worstcase bounds. (2) The approach is flexible with respect to different uses of RFF, and can even estimate errors in downstream learning tasks. (3) The approach enables adaptive computation, in the sense that the user can quickly inspect the error of a rough initial kernel approximation and then predict how much extra work is needed. Furthermore, in exchange for all of these benefits, the error estimates can be obtained at a modest computational cost.more » « less