An improper estimator with optimal excess risk in misspecified density estimation and logistic regression
This addresses the challenge of robust statistical learning under misspecification for researchers and practitioners, offering a more practical alternative to Bayesian methods.
The paper tackles the problem of conditional density estimation under model misspecification by introducing the SMP estimator, which achieves an optimal excess risk bound scaling as d/n and improves over within-model estimators like maximum likelihood. For logistic regression, it provides a non-Bayesian calibration method with a non-asymptotic excess risk of O((d + B^2R^2)/n), outperforming within-model estimators that cannot achieve better than min(BR/√n, d e^{BR}/n).
We introduce a procedure for conditional density estimation under logarithmic loss, which we call SMP (Sample Minmax Predictor). This estimator minimizes a new general excess risk bound for statistical learning. On standard examples, this bound scales as $d/n$ with $d$ the model dimension and $n$ the sample size, and critically remains valid under model misspecification. Being an improper (out-of-model) procedure, SMP improves over within-model estimators such as the maximum likelihood estimator, whose excess risk degrades under misspecification. Compared to approaches reducing to the sequential problem, our bounds remove suboptimal $\log n$ factors and can handle unbounded classes. For the Gaussian linear model, the predictions and risk bound of SMP are governed by leverage scores of covariates, nearly matching the optimal risk in the well-specified case without conditions on the noise variance or approximation error of the linear model. For logistic regression, SMP provides a non-Bayesian approach to calibration of probabilistic predictions relying on virtual samples, and can be computed by solving two logistic regressions. It achieves a non-asymptotic excess risk of $O((d + B^2R^2)/n)$, where $R$ bounds the norm of features and $B$ that of the comparison parameter; by contrast, no within-model estimator can achieve better rate than $\min({B R}/{\sqrt{n}}, {d e^{BR}}/{n} )$ in general. This provides a more practical alternative to Bayesian approaches, which require approximate posterior sampling, thereby partly addressing a question raised by Foster et al. (2018).