Decision Theoretic Bootstrapping
This work addresses uncertainty quantification challenges for machine learning practitioners dealing with finite or corrupted data, offering a novel adversarial framework, though it appears incremental as it builds on existing bootstrapping and game theory concepts.
The paper tackles the problem of robust uncertainty quantification in supervised machine learning when training and testing data distributions are imperfectly known or distinct in finite datasets, by proposing a decision-theoretic bootstrapping method that uses adversarial games to derive optimal model mixtures and uncertainty estimates, resulting in robustness to distributional shifts and conditional probability distributions for output uncertainty.
The design and testing of supervised machine learning models combine two fundamental distributions: (1) the training data distribution (2) the testing data distribution. Although these two distributions are identical and identifiable when the data set is infinite; they are imperfectly known (and possibly distinct) when the data is finite (and possibly corrupted) and this uncertainty must be taken into account for robust Uncertainty Quantification (UQ). We present a general decision-theoretic bootstrapping solution to this problem: (1) partition the available data into a training subset and a UQ subset (2) take $m$ subsampled subsets of the training set and train $m$ models (3) partition the UQ set into $n$ sorted subsets and take a random fraction of them to define $n$ corresponding empirical distributions $μ_{j}$ (4) consider the adversarial game where Player I selects a model $i\in\left\{ 1,\ldots,m\right\} $, Player II selects the UQ distribution $μ_{j}$ and Player I receives a loss defined by evaluating the model $i$ against data points sampled from $μ_{j}$ (5) identify optimal mixed strategies (probability distributions over models and UQ distributions) for both players. These randomized optimal mixed strategies provide optimal model mixtures and UQ estimates given the adversarial uncertainty of the training and testing distributions represented by the game. The proposed approach provides (1) some degree of robustness to distributional shift in both the distribution of training data and that of the testing data (2) conditional probability distributions on the output space forming aleatory representations of the uncertainty on the output as a function of the input variable.