  Prediction models are often employed in estimating parameters of optimization
models. Despite the fact that in an end-to-end view, the real goal is to
achieve good optimization performance, the prediction performance is measured
on its own. While it is usually believed that good prediction performance in
estimating the parameters will result in good subsequent optimization
performance, formal theoretical guarantees on this are notably lacking. In this
paper, we explore conditions that allow us to explicitly describe how the
prediction performance governs the optimization performance. Our weaker
condition allows for an asymptotic convergence result, while our stronger
condition allows for exact quantification of the optimization performance in
terms of the prediction performance. In general, verification of these
conditions is a non-trivial task. Nevertheless, we show that our weaker
condition is equivalent to the well-known Fisher consistency concept from the
learning theory literature. This then allows us to easily check our weaker
condition for several loss functions. We also establish that the squared error
loss function satisfies our stronger condition. Consequently, we derive the
exact theoretical relationship between prediction performance measured with the
squared loss, as well as a class of symmetric loss functions, and the
subsequent optimization performance. In a computational study on portfolio
optimization, fractional knapsack and multiclass classification problems, we
compare the optimization performance of using of several prediction loss
functions (some that are Fisher consistent and some that are not) and
demonstrate that lack of consistency of the loss function can indeed have a
detrimental effect on performance.
