Provably Auditing Ordinary Least Squares in Low Dimensions

Auditing the stability of a machine learning model to small changes in the training procedure is critical for engendering trust in practical applications. For example, a model should not be overly sensitive to removing a small fraction of its training data. However, algorithmically validating this property seems computationally challenging, even for the simplest of models: Ordinary Least Squares (OLS) linear regression. Concretely, recent work defines the stability of a regression as the minimum number of samples that need to be removed so that rerunning the analysis overturns the conclusion (Broderick et al., 2020), specifically meaning that the sign of a particular coefficient of the OLS regressor changes. But the only known approach for estimating this metric, besides the obvious exponential-time algorithm, is a greedy heuristic that may produce severe overestimates and therefore cannot certify stability. We show that stability can be efficiently certified in the low-dimensional regime: when the number of covariates is a constant but the number of samples is large, there are polynomial-time algorithms for estimating (a fractional version of) stability, with provable approximation guarantees. Applying our algorithms to the Boston Housing dataset, we exhibit regression analyses where our estimator outperforms the greedy heuristic, and can successfully certify stability even in the regime where a constant fraction of the samples are dropped.