Similarly, a model that predicts the time it will take a customer to grocery shop should decrease in the number of cashiers, but each addedcashierreduces average wait time by less. In both cases, we would like to be able to incorporate this prior knowledge by constraining the machine learned model's output to have a diminishing returns response to the size of the apartment or number of cashiers.

However,inmanyreal-worldtasks,the learned function is intended to satisfy domain-specific constraints. We focus on monotonicity constraints, which are common and require that the function's output increases with increasing values of specific input features.

Table 1: Results (mean onestandarddeviationover 10 replicates) onthemonotonicexamplein Section 5.1.2

We consider the Continuum Bandit problem where the goal is to find the optimal action under an unknown reward function, with an additional monotonicity constraint (or, markdown constraint) that requires that the action sequence be non-increasing. This problem faithfully models a natural single-product dynamic pricing problem, called markdown pricing, where the objective is to adaptively reduce the price over a finite sales horizon to maximize expected revenues. Jia et al '21 and Chen '21 independently showed a tight $T^{3/4}$ regret bound over $T$ rounds under *minimal* assumptions of unimodality and Lipschitzness in the reward (or, revenue) function. This bound shows that the demand learning in markdown pricing is harder than unconstrained (i.e., without the monotonicity constraint) pricing under unknown demand which suffers regret only of the order of $T^{2/3}$ under the same assumptions (Kleinberg '04). However, in practice the demand functions are usually assumed to have certain functional forms (e.g.

The widespread adoption of deep learning is often attributed to its automatic feature construction with minimal inductive bias. However, in many real-world tasks, the learned function is intended to satisfy domain-specific constraints. We focus on monotonicity constraints, which are common and require that the function's output increases with increasing values of specific input features. We develop a counterexample-guided technique to provably enforce monotonicity constraints at prediction time. Additionally, we propose a technique to use monotonicity as an inductive bias for deep learning. It works by iteratively incorporating monotonicity counterexamples in the learning process. Contrary to prior work in monotonic learning, we target general ReLU neural networks and do not further restrict the hypothesis space. We have implemented these techniques in a tool called COMET. Experiments on real-world datasets demonstrate that our approach achieves state-of-the-art results compared to existing monotonic learners, and can improve the model quality compared to those that were trained without taking monotonicity constraints into account.

See Figure 1 for an example DLN with nine such layers. Lattices are interpolated look-up tables, as shown in Figure 1.

We demonstrate on real-world examples that these shape constrained models can provide tuning-free regularization and improve model understandability.

This problem faithfully models a natural revenue management problem, called "markdown pricing",