Science Advances

Supplementary Materials

The PDF file includes:

  • Appendix A. Supplementary tables and figures for numerical experiments.
  • Appendix B. Supplementary tables for data applications.
  • Appendix C. Methods for confidence region construction experiments.
  • Appendix D. Neural network architectures in numerical experiments.
  • Appendix E. Further discussion of guarantees for nested minimax algorithms.
  • Appendix F. An example showing challenges faced by existing nested maximin algorithms.
  • Appendix G. Captions for additional file types.
  • Fig. S1. Convergence of the risk of the learned estimators in the Gaussian estimation example.
  • Fig. S2. Pointwise quantiles of the fit of our learned two-layer multilayer perceptron prediction function at W2 = 0 and different values of W1 based on n = 50 observations from four data-generating distributions.
  • Fig. S3. Learning curves for worst-case (red) and uniform-prior Bayes (blue) prediction performance in the logistic regression settings i to xii shown in Table 1.
  • Fig. S4. Performance of the learned 95% level confidence region procedure across different values of η.
  • Fig. S5. Prior generator multilayer perceptrons and procedure multilayer perceptrons used for the point estimation examples.
  • Fig. S6. Estimator LSTM used when estimating binary regressions.
  • Fig. S7. Estimator LSTM used when defining the interior point of our confidence regions.
  • Table S1. Gaussian model with σ2 = 1, |μ| ≤ m, and n = 1.
  • Table S2. Final estimated performance of the learned prediction algorithms, the MLE, and a linear-logistic regression.
  • Table S3. Performance of our learned procedures and of existing procedures in data illustrations.
  • References (4850)

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Other Supplementary Material for this manuscript includes the following:

  • Movie S1 (.mp4 format). Evolution of the risk of the learned estimator of μ as the weights of the neural network are updated in the Gaussian model with n = 50 observations and unknown (μ, σ).

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