The Perils of Learning Before Optimizing

Abstract

Formulating real-world optimization problems often begins with making predictions from historical data (e.g., an optimizer that aims to recommend fast routes relies upon travel-time predictions). Typically, learning the prediction model used to generate the optimization problem and solving that problem are performed in two separate stages. Recent work has showed how such prediction models can be learned end-to-end by differentiating through the optimization task. Such methods often yield empirical improvements, which are typically attributed to end-to-end making better error tradeoffs than the standard loss function used in a two-stage solution. We refine this explanation and more precisely characterize when end-to-end can improve performance. When prediction targets are stochastic, a two-stage solution must make an a priori choice about which statistics of the target distribution to model---we consider expectations over prediction targets---while an end-to-end solution can make this choice adaptively. We show that the performance gap between a two-stage and end-to-end approach is closely related to the \emph{price of correlation} concept in stochastic optimization and show the implications of some existing POC results for the predict-then-optimize problem. We then consider a novel and particularly practical setting, where multiple prediction targets are combined to obtain each of the objective function’s coefficients. We give explicit constructions where (1) two-stage performs unboundedly worse than end-to-end; and (2) two-stage is optimal. We use simulations to experimentally quantify performance gaps and identify a wide range of real-world applications from the literature whose objective functions rely on multiple prediction targets, suggesting that end-to-end learning could yield significant improvements.

A novel and efficient machine learning Mendelian randomization estimator applied to predict the safety and efficacy of sclerostin inhibition

“…To evaluate the use of neural network-based nonparametric IV estimators in MR, we evaluated the performance of two recently proposed estimators (DeepIV [20] and DeLIVR [14]), and our new proposed method (Quantile IV); we also compared all three with the traditional linear two-stage least squares estimator. There is no single parameter describing the shape of the causal relationship between the exposure and outcome in nonlinear settings.…”

“…The simulation parameter values in bold correspond to the reference values. 2SLS: Two-stage least squares, DeLIVR [14], DeepIV [20], Quantile IV: proposed method.…”

“…We first consider the model introduced by Hartford et al . when describing the DeepIV estimator [20]: …”

“…The development of nonparametric IV estimators has evolved independently from MR in the econometrics, statistics and machine learning literature and there has been limited efforts to bridge these worlds. Here, we harness recent developments in machine learning and nonparametric IV estimation to propose a new estimator named Quantile IV that performs well in the context of MR [18, 20]. Quantile IV, drawing from Hartford et al .’s DeepIV model, incorporates a crucial simplification for enhanced performance in MR contexts without any added statistical assumption.…”