# A Variational Perspective on Generative Flow Networks

arXiv: https://arxiv.org/abs/2210.07992 · HTML: https://mlsanity.com/paper/2210.07992/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 37 references that are in the map.
- First posted on arXiv 14 Oct 2022.

## Builds on (most cited first, 24 of 24)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation](https://mlsanity.com/paper/2106.04399/) (2021)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [GFlowNet Foundations](https://mlsanity.com/paper/2111.09266/) (2021)
- [Trajectory balance: Improved credit assignment in GFlowNets](https://mlsanity.com/paper/2201.13259/) (2022)
- [Biological Sequence Design with GFlowNets](https://mlsanity.com/paper/2203.04115/) (2022)
- [Bayesian Structure Learning with Generative Flow Networks](https://mlsanity.com/paper/2202.13903/) (2022)
- [Learning GFlowNets from partial episodes for improved convergence and stability](https://mlsanity.com/paper/2209.12782/) (2022)
- [Gradient Estimation Using Stochastic Computation Graphs](https://mlsanity.com/paper/1506.05254/) (2015)
- [Generative Flow Networks for Discrete Probabilistic Modeling](https://mlsanity.com/paper/2202.01361/) (2022)
- [GFlowNets and variational inference](https://mlsanity.com/paper/2210.00580/) (2022)
- [Variational Sequential Monte Carlo](https://mlsanity.com/paper/1705.11140/) (2017)
- [Filtering Variational Objectives](https://mlsanity.com/paper/1705.09279/) (2017)
- [Variational inference for Monte Carlo objectives](https://mlsanity.com/paper/1602.06725/) (2016)
- [Reweighted Wake-Sleep](https://mlsanity.com/paper/1406.2751/) (2014)
- [Auto-Encoding Sequential Monte Carlo](https://mlsanity.com/paper/1705.10306/) (2017)
- [Elements of Sequential Monte Carlo](https://mlsanity.com/paper/1903.04797/) (2019)
- [Markovian Score Climbing: Variational Inference with KL(p||q)](https://mlsanity.com/paper/2003.10374/) (2020)
- [Learning Discrete Energy-based Models via Auxiliary-variable Local Exploration](https://mlsanity.com/paper/2011.05363/) (2020)
- [Unifying Generative Models with GFlowNets and Beyond](https://mlsanity.com/paper/2209.02606/) (2022)
- [Approximate Inference with Amortised MCMC](https://mlsanity.com/paper/1702.08343/) (2017)
- [Nested Variational Inference](https://mlsanity.com/paper/2106.11302/) (2021)
- [Transport Score Climbing: Variational Inference Using Forward KL and Adaptive Neural Transport](https://mlsanity.com/paper/2202.01841/) (2022)
- [Improving Generative Flow Networks with Path Regularization](https://mlsanity.com/paper/2209.15092/) (2022)

## Built on by (most cited first, 30 of 39)

- [A theory of continuous generative flow networks](https://mlsanity.com/paper/2301.12594/) (2023)
- [Amortizing intractable inference in large language models](https://mlsanity.com/paper/2310.04363/) (2023)
- [Better Training of GFlowNets with Local Credit and Incomplete Trajectories](https://mlsanity.com/paper/2302.01687/) (2023)
- [Generative Flow Networks as Entropy-Regularized RL](https://mlsanity.com/paper/2310.12934/) (2023)
- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network](https://mlsanity.com/paper/2305.19366/) (2023)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Discrete Probabilistic Inference as Control in Multi-path Environments](https://mlsanity.com/paper/2402.10309/) (2024)
- [GFlowNet-EM for learning compositional latent variable models](https://mlsanity.com/paper/2302.06576/) (2023)
- [Let the Flows Tell: Solving Graph Combinatorial Optimization Problems with GFlowNets](https://mlsanity.com/paper/2305.17010/) (2023)
- [FlowRL: Matching Reward Distributions for LLM Reasoning](https://mlsanity.com/paper/2509.15207/) (2025)
- [Genetic-guided GFlowNets for Sample Efficient Molecular Optimization](https://mlsanity.com/paper/2402.05961/) (2024)
- [Flow of Reasoning: Training LLMs for Divergent Reasoning with Minimal Examples](https://mlsanity.com/paper/2406.05673/) (2024)
- [Distributional GFlowNets with Quantile Flows](https://mlsanity.com/paper/2302.05793/) (2023)
- [Improving GFlowNets for Text-to-Image Diffusion Alignment](https://mlsanity.com/paper/2406.00633/) (2024)
- [PhyloGFN: Phylogenetic inference with generative flow networks](https://mlsanity.com/paper/2310.08774/) (2023)
- [Pre-Training and Fine-Tuning Generative Flow Networks](https://mlsanity.com/paper/2310.03419/) (2023)
- [Beyond Squared Error: Exploring Loss Design for Enhanced Training of Generative Flow Networks](https://mlsanity.com/paper/2410.02596/) (2024)
- [Delta-AI: Local objectives for amortized inference in sparse graphical models](https://mlsanity.com/paper/2310.02423/) (2023)
- [Expected flow networks in stochastic environments and two-player zero-sum games](https://mlsanity.com/paper/2310.02779/) (2023)
- [On Divergence Measures for Training GFlowNets](https://mlsanity.com/paper/2410.09355/) (2024)
- [Adaptive Destruction Processes for Diffusion Samplers](https://mlsanity.com/paper/2506.01541/) (2025)
- [Reinforced sequential Monte Carlo for amortised sampling](https://mlsanity.com/paper/2510.11711/) (2025)
- [Random Policy Evaluation Uncovers Policies of Generative Flow Networks](https://mlsanity.com/paper/2406.02213/) (2024)
- [GFlowNet Training by Policy Gradients](https://mlsanity.com/paper/2408.05885/) (2024)
- [Streaming Bayes GFlowNets](https://mlsanity.com/paper/2411.05899/) (2024)
- [Avoid What You Know: Divergent Trajectory Balance for GFlowNets](https://mlsanity.com/paper/2602.17827/) (2026)
- [Learning Decision Trees as Amortized Structure Inference](https://mlsanity.com/paper/2503.06985/) (2025)
- [Mitigating Hallucinations in LM-Based TTS Models via Distribution Alignment Using GFlowNets](https://mlsanity.com/paper/2508.15442/) (2025)
- [Discrete diffusion samplers and bridges: Off-policy algorithms and applications in latent spaces](https://mlsanity.com/paper/2602.05961/) (2026)

## Abstract (arXiv)

Generative flow networks (GFNs) are a class of models for sequential sampling of composite objects, which approximate a target distribution that is defined in terms of an energy function or a reward. GFNs are typically trained using a flow matching or trajectory balance objective, which matches forward and backward transition models over trajectories. In this work, we define variational objectives for GFNs in terms of the Kullback-Leibler (KL) divergences between the forward and backward distribution. We show that variational inference in GFNs is equivalent to minimizing the trajectory balance objective when sampling trajectories from the forward model. We generalize this approach by optimizing a convex combination of the reverse- and forward KL divergence. This insight suggests variational inference methods can serve as a means to define a more general family of objectives for training generative flow networks, for example by incorporating control variates, which are commonly used in variational inference, to reduce the variance of the gradients of the trajectory balance objective. We evaluate our findings and the performance of the proposed variational objective numerically by comparing it to the trajectory balance objective on two synthetic tasks.