Universal Prediction Band, Semi-Definite Programming and Variance Interpolation

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A frequent criticism from the statistics community to modern machine learning is the lack of rigorous uncertainty quantification. Instead, the machine learning community would argue that conventional uncertainty quantification based on idealized distributional assumptions may be too restrictive for real data. Nevertheless, without a doubt, uncertainty quantification for predictive modeling is essential to statistics, learning theory, and econometrics. This paper will resolve the above inference dilemma by introducing a new method with provable uncertainty quantification via semi-definite programming. We propose a computationally efficient method to construct nonparametric, heteroskedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, with strong non-asymptotic coverage properties, and easy to implement using standard convex programs. Our approach can be viewed as a novel variance interpolation with confidence and further leverages techniques from semi-definite programming and sum-of-squares optimization. Theoretical and numerical performances for the proposed approach for uncertainty quantification are analyzed.