  Generative Flow Networks (GFlowNets) have been introduced as a method to
sample a diverse set of candidates with probabilities proportional to a given
reward. However, GFlowNets can only be used with a predefined scalar reward,
which can be either computationally expensive or not directly accessible, in
the case of multi-objective optimization (MOO) tasks for example. Moreover, to
prioritize identifying high-reward candidates, the conventional practice is to
raise the reward to a higher exponent, the optimal choice of which may vary
across different environments. To address these issues, we propose
Order-Preserving GFlowNets (OP-GFNs), which sample with probabilities in
proportion to a learned reward function that is consistent with a provided
(partial) order on the candidates, thus eliminating the need for an explicit
formulation of the reward function. We theoretically prove that the training
process of OP-GFNs gradually sparsifies the learned reward landscape in
single-objective maximization tasks. The sparsification concentrates on
candidates of a higher hierarchy in the ordering, ensuring exploration at the
beginning and exploitation towards the end of the training. We demonstrate
OP-GFN's state-of-the-art performance in single-objective maximization (totally
ordered) and multi-objective Pareto front approximation (partially ordered)
tasks, including synthetic datasets, molecule generation, and neural
architecture search.
