Beyond Surrogates: A Quantitative Analysis for Inter-Metric Relationships

A broad range of models face the mismatch where they are updated through trajectory losses but are evaluated by downstream task reward. Here, a trajectory is a training instance that induces a surrogate loss whose reduction might not track the model's decision utility update. Theoretically, we ask when one step of trajectory training reduces both population surrogate loss and decision risk, and how transfer accumulates along repeated updates. To formalize this, we first fix a checkpoint and a restricted update space, and define the reductions in population surrogate risk and decision risk induced by a trajectory as its learnability and decision utility, respectively. On this basis, our theory yields four main results. First, a one-step transfer bound separates their discrepancy into first-order gradient misalignment after nonnegative calibration and second-order curvature; and a pathwise extension accumulates the same terms over repeated updates. Second, when the accessible surrogate gradient is nonzero, universal first-order transfer over every accessible direction holds exactly when the accessible surrogate and decision gradients are positively collinear. Third, the calibration gap bounds the decision regret of learnability-based trajectory selection, while a candidate-difference refinement tightens this guarantee by retaining only directions that affect pairwise rankings. Finally, we establish an approximation--calibration trade-off across nested update spaces. Controlled gridworld and LLM post-training experiments yield results consistent with our predictions.

Learning-augmented algorithms have received significant attention in recent years, particularly in the context of online optimization. Motivated by the high computational cost of generating predictions, a growing line of work studies the tradeoff between performance guarantees and the number of predictions used in learning-augmented algorithms for problems such as caching and metrical task systems. In this paper, we extend this line of research to online metric matching by developing parsimonious learning-augmented algorithms and establishing lower bounds on their performance. Our approach extends the Follow-the-Prediction framework to the parsimonious setting by filling in a virtual prediction in the absence of an actual prediction, using an online metric matching algorithm that maintains good intermediate matchings throughout its execution. We complement our theoretical results with an empirical evaluation, demonstrating the practical effectiveness of our approach.

In the current practices of machine learning, the evaluation of forecasts has become a cornerstone of scientific progress. A multitude of evaluation metrics have been suggested and used to qualify "good" forecasts. What do those metrics share? How are they related? In this work, we use a protocol borrowed from game-theoretic probability to show that a large part of evaluation metrics can be viewed as averaged outcomes of fair gambles. Intuitively, a fair gambler is one which a forecaster would expect to fail. Hence, the gambler's ability to gain disproves the quality of the forecast. Standard evaluation metrics are then variants of choices of such fair gambles. In particular, this choice is structured along two dimensions, one of which separates calibration-type and regret-type metrics. In particular, this framework sheds light on the relationship of calibration and regret showing a theoretical equivalence in their ability to evaluate when being scaled appropriately, but the incomparability of obtained scores.