Variational Learning for Insertion-based Generation
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910281649618944 |
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| author | Zhang, Yangtian Wang, Zhe Gretton, Arthur Ying, Rex van Dijk, David Titsias, Michalis K. Shi, Jiaxin |
| author_facet | Zhang, Yangtian Wang, Zhe Gretton, Arthur Ying, Rex van Dijk, David Titsias, Michalis K. Shi, Jiaxin |
| contents | Non-monotonic sequence generation methods, such as masked diffusion models, provide a flexible alternative to left-to-right autoregressive modeling by allowing tokens to be generated in non-fixed and prescribed orders. Despite their practical advantages, most existing non-monotonic models are order-agnostic and rely on a fixed-length grid, limiting their ability to support variable-length generation and adaptive insertion order. In this work, we introduce a probabilistic framework for learning insertion order in variable-length insertion models. We formalize a bijective correspondence between insertion trajectories and permutations, which enables an exact reparameterization of the data likelihood as a sum over permutations. Building on this result, we propose the Insertion Process (IP), a stochastic generative model that jointly learns where to insert, what to insert, and when to terminate, trained via permutation-based variational inference. Unlike prior fixed-canvas approaches, IP natively supports variable-length generation and learns data-driven preferences over insertion orders. Experiments on goal-conditioned planning and molecular string generation demonstrate that learning insertion order improves both modeling quality and generalization in domains without a canonical left-to-right structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02133 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variational Learning for Insertion-based Generation Zhang, Yangtian Wang, Zhe Gretton, Arthur Ying, Rex van Dijk, David Titsias, Michalis K. Shi, Jiaxin Machine Learning Artificial Intelligence Non-monotonic sequence generation methods, such as masked diffusion models, provide a flexible alternative to left-to-right autoregressive modeling by allowing tokens to be generated in non-fixed and prescribed orders. Despite their practical advantages, most existing non-monotonic models are order-agnostic and rely on a fixed-length grid, limiting their ability to support variable-length generation and adaptive insertion order. In this work, we introduce a probabilistic framework for learning insertion order in variable-length insertion models. We formalize a bijective correspondence between insertion trajectories and permutations, which enables an exact reparameterization of the data likelihood as a sum over permutations. Building on this result, we propose the Insertion Process (IP), a stochastic generative model that jointly learns where to insert, what to insert, and when to terminate, trained via permutation-based variational inference. Unlike prior fixed-canvas approaches, IP natively supports variable-length generation and learns data-driven preferences over insertion orders. Experiments on goal-conditioned planning and molecular string generation demonstrate that learning insertion order improves both modeling quality and generalization in domains without a canonical left-to-right structure. |
| title | Variational Learning for Insertion-based Generation |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2606.02133 |