In this paper, we present a novel learning framework for finding shortest paths in graphs utilizing Generative Flow Networks (GFlowNets). First, we examine theoretical properties of GFlowNets in non-acyclic environments in relation to shortest paths. We prove that, if the total flow is minimized, forward and backward policies traverse the environment graph exclusively along shortest paths between the initial and terminal states. Building on this result, we show that the pathfinding problem in an arbitrary graph can be solved by training a non-acyclic GFlowNet with flow regularization. We experimentally demonstrate the performance of our method in pathfinding in permutation environments and in solving Rubik's Cubes. For the latter problem, our approach shows competitive results with state-of-the-art machine learning approaches designed specifically for this task in terms of the solution length, while requiring smaller search budget at test-time.

Generative Flow Networks (GFlowNets) are powerful samplers for compositional objects that, by design, sample proportionally to a given non-negative reward. Nonetheless, in practice, they often struggle to explore the reward landscape evenly: trajectories toward easy-to-reach regions dominate training, while hard-to-reach modes receive vanishing or uninformative gradients, leading to poor coverage of high-reward areas. We address this imbalance with Boosted GFlowNets, a method that sequentially trains an ensemble of GFlowNets, each optimizing a residual reward that compensates for the mass already captured by previous models. This residual principle reactivates learning signals in underexplored regions and, under mild assumptions, ensures a monotone non-degradation property: adding boosters cannot worsen the learned distribution and typically improves it. Empirically, Boosted GFlowNets achieve substantially better exploration and sample diversity on multimodal synthetic benchmarks and peptide design tasks, while preserving the stability and simplicity of standard trajectory-balance training.

To this end, GFlowNets construct an object through a trajectory of state transitions, e.g., iteratively adding molecular fragments to construct a molecule.

A.1 Proof of Proposition 1 Recall Proposition 1: Proposition. Let R be a positive reward function on X . R (x) substituted for F ( x) by the reward matching assumption (8). The trajectory balance constraint (13) can be generalized to partial (not complete) trajectories, i.e., The trajectory balance constraint (13) is the special case of this for full trajectories, while the detailed balance constraint (7) is the special case of trajectories wth only one edge. That is, the path that goes "backward, then forward" from The special case of'one step back, two steps forward' paths was used for a graph We train all models with a learning rate of 0.001 ( Generating the test set .

The algorithms we compare to make varying assumptions, which we summarize in Table 1.

Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects from a given probability distribution, potentially known up to a normalizing constant. Instead of working in the object space, GFlowNets proceed by sampling trajectories in an appropriately constructed directed acyclic graph environment, greatly relying on the acyclicity of the graph. In our paper, we revisit the theory that relaxes the acyclicity assumption and present a simpler theoretical framework for non-acyclic GFlowNets in discrete environments. Moreover, we provide various novel theoretical insights related to training with fixed backward policies, the nature of flow functions, and connections between entropy-regularized RL and non-acyclic GFlowNets, which naturally generalize the respective concepts and theoretical results from the acyclic setting. In addition, we experimentally re-examine the concept of loss stability in non-acyclic GFlowNet training, as well as validate our own theoretical findings.

Generative Flow Networks (GFlowNets) are a family of generative models that learn to sample objects with probabilities proportional to a given reward function. The key concept behind GFlowNets is the use of two stochastic policies: a forward policy, which incrementally constructs compositional objects, and a backward policy, which sequentially deconstructs them. Recent results show a close relationship between GFlowNet training and entropy-regularized reinforcement learning (RL) problems with a particular reward design. However, this connection applies only in the setting of a fixed backward policy, which might be a significant limitation. As a remedy to this problem, we introduce a simple backward policy optimization algorithm that involves direct maximization of the value function in an entropy-regularized Markov Decision Process (MDP) over intermediate rewards. We provide an extensive experimental evaluation of the proposed approach across various benchmarks in combination with both RL and GFlowNet algorithms and demonstrate its faster convergence and mode discovery in complex environments.