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rl-introduction.bbl
\begin{thebibliography}{119}
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Afra Amini, Tim Vieira, and Ryan Cotterell.
\newblock Direct preference optimization with an offset.
\newblock \emph{arXiv preprint arXiv:2402.10571}, 2024.
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Mohammad~Gheshlaghi Azar, Zhaohan~Daniel Guo, Bilal Piot, R{\'{e}}mi Munos, Mark Rowland, Michal Valko, and Daniele Calandriello.
\newblock A general theoretical paradigm to understand learning from human preferences.
\newblock In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li (eds.), \emph{International Conference on Artificial Intelligence and Statistics, 2-4 May 2024, Palau de Congressos, Valencia, Spain}, volume 238 of \emph{Proceedings of Machine Learning Research}, pp.\ 4447--4455. {PMLR}, 2024.
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Dzmitry Bahdanau, Philemon Brakel, Kelvin Xu, Anirudh Goyal, Ryan Lowe, Joelle Pineau, Aaron~C. Courville, and Yoshua Bengio.
\newblock An actor-critic algorithm for sequence prediction.
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\newblock Multimodal machine learning: A survey and taxonomy.
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\newblock \emph{Pattern Recognition and Machine Learning}.
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\newblock Signature verification using a" siamese" time delay neural network.
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Baian Chen, Chang Shu, Ehsan Shareghi, Nigel Collier, Karthik Narasimhan, and Shunyu Yao.
\newblock Fireact: Toward language agent fine-tuning.
\newblock \emph{arXiv preprint arXiv:2310.05915}, 2023{\natexlab{a}}.
\bibitem[Chen et~al.(2023{\natexlab{b}})Chen, Borgeaud, Irving, Lespiau, Sifre, and Jumper]{chen-etal:2023accelerating}
Charlie Chen, Sebastian Borgeaud, Geoffrey Irving, Jean-Baptiste Lespiau, Laurent Sifre, and John Jumper.
\newblock Accelerating large language model decoding with speculative sampling.
\newblock \emph{ArXiv preprint}, abs/2302.01318, 2023{\natexlab{b}}.
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\bibitem[Chen et~al.(2024)Chen, Li, Yan, Wang, Gunaratna, Yadav, Tang, Srinivasan, Zhou, Huang, and Jin]{chen-etal:2023alpagasus}
Lichang Chen, Shiyang Li, Jun Yan, Hai Wang, Kalpa Gunaratna, Vikas Yadav, Zheng Tang, Vijay Srinivasan, Tianyi Zhou, Heng Huang, and Hongxia Jin.
\newblock Alpagasus: Training a better alpaca with fewer data.
\newblock In \emph{The Twelfth International Conference on Learning Representations, {ICLR} 2024, Vienna, Austria, May 7-11, 2024}. OpenReview.net, 2024.
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Pengyu Cheng, Jiawen Xie, Ke~Bai, Yong Dai, and Nan Du.
\newblock Everyone deserves a reward: Learning customized human preferences.
\newblock \emph{arXiv preprint arXiv:2309.03126}, 2023.
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Paul~F. Christiano, Jan Leike, Tom~B. Brown, Miljan Martic, Shane Legg, and Dario Amodei.
\newblock Deep reinforcement learning from human preferences.
\newblock In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna~M. Wallach, Rob Fergus, S.~V.~N. Vishwanathan, and Roman Garnett (eds.), \emph{Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, {USA}}, pp.\ 4299--4307, 2017.
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Thomas Coste, Usman Anwar, Robert Kirk, and David Krueger.
\newblock Reward model ensembles help mitigate overoptimization.
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Ganqu Cui, Lifan Yuan, Ning Ding, Guanming Yao, Bingxiang He, Wei Zhu, Yuan Ni, Guotong Xie, Ruobing Xie, Yankai Lin, Zhiyuan Liu, and Maosong Sun.
\newblock {ULTRAFEEDBACK:} boosting language models with scaled {AI} feedback.
\newblock In \emph{Forty-first International Conference on Machine Learning, {ICML} 2024, Vienna, Austria, July 21-27, 2024}. OpenReview.net, 2024{\natexlab{a}}.
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Ganqu Cui, Lifan Yuan, Ning Ding, Guanming Yao, Bingxiang He, Wei Zhu, Yuan Ni, Guotong Xie, Ruobing Xie, Yankai Lin, Zhiyuan Liu, and Maosong Sun.
\newblock {ULTRAFEEDBACK:} boosting language models with scaled {AI} feedback.
\newblock In \emph{Forty-first International Conference on Machine Learning, {ICML} 2024, Vienna, Austria, July 21-27, 2024}. OpenReview.net, 2024{\natexlab{b}}.
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\bibitem[Deb et~al.(2016)Deb, Sindhya, and Hakanen]{deb-etal:2016multi}
Kalyanmoy Deb, Karthik Sindhya, and Jussi Hakanen.
\newblock Multi-objective optimization.
\newblock In \emph{Decision sciences}, pp.\ 161--200. CRC Press, 2016.
\bibitem[Deepseek(2025)]{deepseek:2025r1}
Deepseek.
\newblock Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning.
\newblock \emph{arXiv preprint arXiv:2501.12948}, 2025.
\bibitem[Dubois et~al.(2023)Dubois, Li, Taori, Zhang, Gulrajani, Ba, Guestrin, Liang, and Hashimoto]{dubois-etal:2024alpacafarm}
Yann Dubois, Chen~Xuechen Li, Rohan Taori, Tianyi Zhang, Ishaan Gulrajani, Jimmy Ba, Carlos Guestrin, Percy Liang, and Tatsunori~B. Hashimoto.
\newblock Alpacafarm: {A} simulation framework for methods that learn from human feedback.
\newblock In Alice Oh, Tristan Naumann, Amir Globerson, Kate Saenko, Moritz Hardt, and Sergey Levine (eds.), \emph{Advances in Neural Information Processing Systems 36: Annual Conference on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans, LA, USA, December 10 - 16, 2023}, 2023.
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Jacob Eisenstein, Chirag Nagpal, Alekh Agarwal, Ahmad Beirami, Alex D'Amour, DJ~Dvijotham, Adam Fisch, Katherine Heller, Stephen Pfohl, Deepak Ramachandran, et~al.
\newblock Helping or herding? reward model ensembles mitigate but do not eliminate reward hacking.
\newblock \emph{ArXiv preprint}, abs/2312.09244, 2023.
\newblock URL \url{https://arxiv.org/abs/2312.09244}.
\bibitem[Ethayarajh et~al.(2024)Ethayarajh, Xu, Muennighoff, Jurafsky, and Kiela]{ethayarajh:2024kto}
Kawin Ethayarajh, Winnie Xu, Niklas Muennighoff, Dan Jurafsky, and Douwe Kiela.
\newblock Kto: Model alignment as prospect theoretic optimization.
\newblock \emph{ArXiv preprint}, abs/2402.01306, 2024.
\newblock URL \url{https://arxiv.org/abs/2402.01306}.
\bibitem[Gallego(2024)]{gallego:2024refined}
V{\'\i}ctor Gallego.
\newblock Refined direct preference optimization with synthetic data for behavioral alignment of llms.
\newblock \emph{ArXiv preprint}, abs/2402.08005, 2024.
\newblock URL \url{https://arxiv.org/abs/2402.08005}.
\bibitem[Gao et~al.(2023)Gao, Schulman, and Hilton]{gao-etal:2023scaling}
Leo Gao, John Schulman, and Jacob Hilton.
\newblock Scaling laws for reward model overoptimization.
\newblock In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett (eds.), \emph{International Conference on Machine Learning, {ICML} 2023, 23-29 July 2023, Honolulu, Hawaii, {USA}}, volume 202 of \emph{Proceedings of Machine Learning Research}, pp.\ 10835--10866. {PMLR}, 2023.
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Alexey Gorbatovski, Boris Shaposhnikov, Alexey Malakhov, Nikita Surnachev, Yaroslav Aksenov, Ian Maksimov, Nikita Balagansky, and Daniil Gavrilov.
\newblock Learn your reference model for real good alignment.
\newblock \emph{ArXiv preprint}, abs/2404.09656, 2024.
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Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et~al.
\newblock The llama 3 herd of models.
\newblock \emph{arXiv preprint arXiv:2407.21783}, 2024.
\bibitem[Guo et~al.(2025)Guo, Yang, Zhang, Song, Zhang, Xu, Zhu, Ma, Wang, Bi, et~al.]{guo:2025deepseek}
Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et~al.
\newblock Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning.
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Alex Havrilla, Yuqing Du, Sharath~Chandra Raparthy, Christoforos Nalmpantis, Jane Dwivedi-Yu, Maksym Zhuravinskyi, Eric Hambro, Sainbayar Sukhbaatar, and Roberta Raileanu.
\newblock Teaching large language models to reason with reinforcement learning.
\newblock \emph{ArXiv preprint}, abs/2403.04642, 2024.
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Jiwoo Hong, Noah Lee, and James Thorne.
\newblock Orpo: Monolithic preference optimization without reference model.
\newblock \emph{ArXiv preprint}, abs/2403.07691, 2024.
\newblock URL \url{https://arxiv.org/abs/2403.07691}.
\bibitem[Hu(2025)]{hu:2025reinforce++}
Jian Hu.
\newblock Reinforce++: A simple and efficient approach for aligning large language models.
\newblock \emph{ArXiv preprint}, abs/2501.03262, 2025.
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Mengkang Hu, Pu~Zhao, Can Xu, Qingfeng Sun, Jian-Guang Lou, Qingwei Lin, Ping Luo, and Saravan Rajmohan.
\newblock Agentgen: Enhancing planning abilities for large language model based agent via environment and task generation.
\newblock In \emph{Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V. 1}, pp.\ 496--507, 2025.
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Jiaming Ji, Xinyu Chen, Rui Pan, Han Zhu, Conghui Zhang, Jiahao Li, Donghai Hong, Boyuan Chen, Jiayi Zhou, Kaile Wang, et~al.
\newblock Safe rlhf-v: Safe reinforcement learning from human feedback in multimodal large language models.
\newblock \emph{arXiv preprint arXiv:2503.17682}, 2025.
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Aviral Kumar, Vincent Zhuang, Rishabh Agarwal, Yi~Su, John~D Co-Reyes, Avi Singh, Kate Baumli, Shariq Iqbal, Colton Bishop, Rebecca Roelofs, et~al.
\newblock Training language models to self-correct via reinforcement learning.
\newblock \emph{ArXiv preprint}, abs/2409.12917, 2024.
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Joonho Lee, Marko Bjelonic, Alexander Reske, Lorenz Wellhausen, Takahiro Miki, and Marco Hutter.
\newblock Learning robust autonomous navigation and locomotion for wheeled-legged robots.
\newblock \emph{Science Robotics}, 9\penalty0 (89):\penalty0 eadi9641, 2024.
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Xuefeng Li, Haoyang Zou, and Pengfei Liu.
\newblock Limr: Less is more for rl scaling.
\newblock \emph{ArXiv preprint}, abs/2502.11886, 2025{\natexlab{a}}.
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Zhiwei Li, Yong Hu, and Wenqing Wang.
\newblock Encouraging good processes without the need for good answers: Reinforcement learning for llm agent planning.
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Ziniu Li, Tian Xu, Yushun Zhang, Zhihang Lin, Yang Yu, Ruoyu Sun, and Zhi{-}Quan Luo.
\newblock Remax: {A} simple, effective, and efficient reinforcement learning method for aligning large language models.
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Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe.
\newblock Let's verify step by step.
\newblock In \emph{The Twelfth International Conference on Learning Representations, {ICLR} 2024, Vienna, Austria, May 7-11, 2024}. OpenReview.net, 2024{\natexlab{a}}.
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Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe.
\newblock Let's verify step by step.
\newblock In \emph{The Twelfth International Conference on Learning Representations, {ICLR} 2024, Vienna, Austria, May 7-11, 2024}. OpenReview.net, 2024{\natexlab{b}}.
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Yang Liu, Dan Iter, Yichong Xu, Shuohang Wang, Ruochen Xu, and Chenguang Zhu.
\newblock {G}-eval: {NLG} evaluation using gpt-4 with better human alignment.
\newblock In Houda Bouamor, Juan Pino, and Kalika Bali (eds.), \emph{Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing}, pp.\ 2511--2522, Singapore, 2023. Association for Computational Linguistics.
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Shayne Longpre, Le~Hou, Tu~Vu, Albert Webson, Hyung~Won Chung, Yi~Tay, Denny Zhou, Quoc~V Le, Barret Zoph, Jason Wei, et~al.
\newblock The flan collection: Designing data and methods for effective instruction tuning.
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Dakota Mahan, Duy Van~Phung, Rafael Rafailov, Chase Blagden, Nathan Lile, Louis Castricato, Jan-Philipp Fr{\"a}nken, Chelsea Finn, and Alon Albalak.
\newblock Generative reward models.
\newblock \emph{ArXiv preprint}, abs/2410.12832, 2024.
\newblock URL \url{https://arxiv.org/abs/2410.12832}.
\bibitem[Masoudnia \& Ebrahimpour(2014)Masoudnia and Ebrahimpour]{masoudnia-etal:2014mixture}
Saeed Masoudnia and Reza Ebrahimpour.
\newblock Mixture of experts: a literature survey.
\newblock \emph{Artificial Intelligence Review}, 42:\penalty0 275--293, 2014.
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Igor Melnyk, Youssef Mroueh, Brian Belgodere, Mattia Rigotti, Apoorva Nitsure, Mikhail Yurochkin, Kristjan Greenewald, Jiri Navratil, and Jerret Ross.
\newblock Distributional preference alignment of llms via optimal transport.
\newblock In \emph{Advances in Neural Information Processing Systems}, 2024.
\bibitem[Meng et~al.(2024)Meng, Xia, and Chen]{meng:2025simpo}
Yu~Meng, Mengzhou Xia, and Danqi Chen.
\newblock Simpo: Simple preference optimization with a reference-free reward.
\newblock In Amir Globersons, Lester Mackey, Danielle Belgrave, Angela Fan, Ulrich Paquet, Jakub~M. Tomczak, and Cheng Zhang (eds.), \emph{Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems 2024, NeurIPS 2024, Vancouver, BC, Canada, December 10 - 15, 2024}, 2024.
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\bibitem[Miettinen(1999)]{miettinen:1999nonlinear}
Kaisa Miettinen.
\newblock \emph{Nonlinear multiobjective optimization}, volume~12.
\newblock Springer Science \& Business Media, 1999.
\bibitem[Miettinen et~al.(2008)Miettinen, Ruiz, and Wierzbicki]{miettinen-etal:2008introduction}
Kaisa Miettinen, Francisco Ruiz, and Andrzej~P Wierzbicki.
\newblock Introduction to multiobjective optimization: interactive approaches.
\newblock In \emph{Multiobjective optimization: interactive and evolutionary approaches}, pp.\ 27--57. Springer, 2008.
\bibitem[Mnih et~al.(2013)Mnih, Kavukcuoglu, Silver, Graves, Antonoglou, Wierstra, and Riedmiller]{mnih-etal:mnih2013playing}
Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller.
\newblock Playing atari with deep reinforcement learning.
\newblock \emph{arXiv preprint arXiv:1312.5602}, 2013.
\bibitem[Mnih et~al.(2016)Mnih, Badia, Mirza, Graves, Lillicrap, Harley, Silver, and Kavukcuoglu]{mnih-etal:2016asynchronous}
Volodymyr Mnih, Adri{\`{a}}~Puigdom{\`{e}}nech Badia, Mehdi Mirza, Alex Graves, Timothy~P. Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu.
\newblock Asynchronous methods for deep reinforcement learning.
\newblock In Maria{-}Florina Balcan and Kilian~Q. Weinberger (eds.), \emph{Proceedings of the 33nd International Conference on Machine Learning, {ICML} 2016, New York City, NY, USA, June 19-24, 2016}, volume~48 of \emph{{JMLR} Workshop and Conference Proceedings}, pp.\ 1928--1937. JMLR.org, 2016.
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\bibitem[Morimura et~al.(2024)Morimura, Sakamoto, Jinnai, Abe, and Ariu]{morimura-etal:2024filtered}
Tetsuro Morimura, Mitsuki Sakamoto, Yuu Jinnai, Kenshi Abe, and Kaito Ariu.
\newblock Filtered direct preference optimization.
\newblock \emph{ArXiv preprint}, abs/2404.13846, 2024.
\newblock URL \url{https://arxiv.org/abs/2404.13846}.
\bibitem[Nakano et~al.(2021)Nakano, Hilton, Balaji, Wu, Ouyang, Kim, Hesse, Jain, Kosaraju, Saunders, Jiang, Cobbe, Eloundou, Krueger, Button, Knight, Chess, and Schulman]{nakano-etal:2021webgpt}
Reiichiro Nakano, Jacob Hilton, Suchir Balaji, Jeff Wu, Long Ouyang, Christina Kim, Christopher Hesse, Shantanu Jain, Vineet Kosaraju, William Saunders, Xu~Jiang, Karl Cobbe, Tyna Eloundou, Gretchen Krueger, Kevin Button, Matthew Knight, Benjamin Chess, and John Schulman.
\newblock Webgpt: Browser-assisted question-answering with human feedback.
\newblock \emph{ArXiv preprint}, abs/2112.09332, 2021.
\newblock URL \url{https://arxiv.org/abs/2112.09332}.
\bibitem[OpenAI(2024)]{openai:2024learning}
OpenAI.
\newblock Learning to reason with llms, September 2024.
\newblock URL \url{https://openai.com/index/learning-to-reason-with-llms/}.
\bibitem[Ouyang et~al.(2022)Ouyang, Wu, Jiang, Almeida, Wainwright, Mishkin, Zhang, Agarwal, Slama, Ray, Schulman, Hilton, Kelton, Miller, Simens, Askell, Welinder, Christiano, Leike, and Lowe]{ouyang:2022training}
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rl-introduction.bib
...@@ -2,6 +2,34 @@ ...@@ -2,6 +2,34 @@
@inproceedings{melnyk:2024distributional,
title={Distributional Preference Alignment of LLMs via Optimal Transport},
author={Melnyk, Igor and Mroueh, Youssef and Belgodere, Brian and Rigotti, Mattia and Nitsure, Apoorva and Yurochkin, Mikhail and Greenewald, Kristjan and Navratil, Jiri and Ross, Jerret},
booktitle={Advances in Neural Information Processing Systems},
year={2024}
}
@inproceedings{zeng:2024token,
title={Token-level Direct Preference Optimization},
author={Zeng, Yongcheng and Liu, Guoqing and Ma, Weiyu and Yang, Ning and Zhang, Haifeng and Wang, Jun},
booktitle={Proceedings of the 41st International Conference on Machine Learning},
year={2024}
}
@inproceedings{xiao:2024cal,
title={Cal-DPO: Calibrated Direct Preference Optimization for Language Model Alignment},
author={Xiao, Teng and Yuan, Yige and Zhu, Huaisheng and Li, Mingxiao and Honavar, Vasant G.},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024}
}
@article{amini:2024direct,
title={Direct Preference Optimization with an Offset},
author={Amini, Afra and Vieira, Tim and Cotterell, Ryan},
journal={arXiv preprint arXiv:2402.10571},
year={2024}
}
@article{qian-etal:toolrl, @article{qian-etal:toolrl,
title={Toolrl: Reward is all tool learning needs}, title={Toolrl: Reward is all tool learning needs},
author={Qian, Cheng and Acikgoz, Emre Can and He, Qi and Wang, Hongru and Chen, Xiusi and Hakkani-Tur, Dilek and Tur, Gokhan and Ji, Heng}, author={Qian, Cheng and Acikgoz, Emre Can and He, Qi and Wang, Hongru and Chen, Xiusi and Hakkani-Tur, Dilek and Tur, Gokhan and Ji, Heng},
......
rl-introduction.pdf
No preview for this file type
section4/Tables/dpo_variants.tex
DPO \citep{rafailov:2023direct} \begin{tabular}{ll}
& $-\log \textrm{Sigmoid} \left( \toprule[1.1pt]
\beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})} \textbf{Method} & \textbf{Objective} \\ \midrule
- IPO \citep{azar:2024general} & $ \left( \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})} - \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_b|\mathbf{x})} - \frac{1}{2\tau} \right)^2$ \\ \midrule
\beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_b|\mathbf{x})} CPO \citep{xu:2024contrastive} & $-\log \textrm{Sigmoid} \left(\beta \log \mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x}) - \beta \log \mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x}) \right) - \lambda \log \mathrm{Pr}_\theta (\mathbf{y}_a|\mathbf{x})$ \\ \midrule
\right)$ \\ \midrule \multirow{2}{*}{KTO \citep{ethayarajh:2024kto}} & $-\lambda_a \textrm{Sigmoid} \left( \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})} - z_{\theta_\mathrm{ref}} \right) + \lambda_b \textrm{Sigmoid} \left( z_{\theta_\mathrm{ref}} - \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_b|\mathbf{x})} \right)\,$ \\
& $\text{where} \,\, z_{\theta_\mathrm{ref}} = \mathbb{E}_{(\mathbf{x}, \mathbf{y}) \sim \mathcal{S}} \left[\beta \text{KL}\left( \mathrm{Pr}_\theta(\mathbf{y}|\mathbf{x}) || \mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}|\mathbf{x}) \right) \right]$ \\ \midrule
\multirow{2}{*}{ORPO \citep{hong:2024orpo}} & $-\log p_\theta(\mathbf{y}_a|\mathbf{x}) - \lambda \log \textrm{Sigmoid} \left(\log \frac{p_\theta(\mathbf{y}_a|\mathbf{x})}{1 - p_\theta(\mathbf{y}_a|\mathbf{x})} - \log \frac{p_\theta(\mathbf{y}_b|\mathbf{x})}{1 - p_\theta(\mathbf{y}_b|\mathbf{x})} \right)\,$ \\
& $\text{where} \,\, p_\theta(\mathbf{y}|\mathbf{x}) = e^{ \frac{1}{|\mathbf{y}|} \log \mathrm{Pr}_\theta(\mathbf{y}|\mathbf{x})}$ \\ \midrule
R-DPO \citep{gallego:2024refined} & $-\log \textrm{Sigmoid} \left( \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})} - \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_b|\mathbf{x})} + \left(\alpha |\mathbf{y}_a| - \alpha |\mathbf{y}_b| \right) \right)$ \\ \midrule
\multirow{2}{*}{PCDPO \citep{zhou:2024prior}} & $-\log \textrm{Sigmoid} \left(\Delta^* - \Delta_{\mathrm{Pr}_{\theta}}\right) -\log \textrm{Sigmoid} \Delta_{\mathrm{Pr}_{\theta}} \,,$ \\
& $\text{where} \,\, \Delta_{\mathrm{Pr}_{\theta}}= \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})} - \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_b|\mathbf{x})},\,\, \Delta^* = \frac{\beta_1}{\textrm{Sim}(\textbf{y}_a,\textbf{y}_b|\textbf{x})+\beta_2}+\beta_3 $ \\ \midrule
SimPO \citep{meng:2025simpo} & $-\log \textrm{Sigmoid} \left( \frac{\beta}{|\mathbf{y}_a|} \log \mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x}) - \frac{\beta}{|\mathbf{y}_b|} \log \mathrm{Pr}_\theta(\mathbf{y}_b|\mathbf{x}) - \gamma \right)$ \\ \midrule
ODPO \citep{amini:2024direct} ODPO \citep{amini:2024direct}
& $-\log \textrm{Sigmoid} \left( & $-\log \textrm{Sigmoid} \left(
\beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})} \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a|\mathbf{x})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_a|\mathbf{x})}
...@@ -48,3 +54,6 @@ ...@@ -48,3 +54,6 @@
\beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}|\mathbf{x})} \beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}|\mathbf{x})}
{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}|\mathbf{x})}, {\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}|\mathbf{x})},
\text{ and } \mathcal{L}_{\mathrm{OT}} \text{ aligns preference distributions via optimal transport.}$ \\ \midrule \text{ and } \mathcal{L}_{\mathrm{OT}} \text{ aligns preference distributions via optimal transport.}$ \\ \midrule
D$^2$PO \citep{shao:2025earlier} & $-\log \textrm{Sigmoid} \left( \sum_{t=0}^{T}\gamma^t\beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_a^t|\mathbf{x},\mathbf{y}_{a,<t})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_{a,t}|\mathbf{x},\mathbf{y}_{a,<t})} - \sum_{t=0}^{T}\gamma^t\beta \log \frac{\mathrm{Pr}_\theta(\mathbf{y}_{b,t}|\mathbf{x},\mathbf{y}_{b,<t})}{\mathrm{Pr}_{\theta_\mathrm{ref}}(\mathbf{y}_{b,t}|\mathbf{x},\mathbf{y}_{b,<t})}\right)$ \\
\bottomrule[1.1pt]
\end{tabular}
section6/section6.tex
...@@ -169,7 +169,11 @@ Tool using is another fundamental capability of LLM-based agents. Tools extend t ...@@ -169,7 +169,11 @@ Tool using is another fundamental capability of LLM-based agents. Tools extend t
\subsection{Improved Environments} \subsection{Improved Environments}
\subsubsection{Scaling Up } % large-scale environment construct
\subsection{Self-Evolving Agents}
......
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