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\bibcite{yu-etal:2024rlaif}{{88}{2024{b}}{{Yu et~al.}}{{Yu, Zhang, Yao, Dang, Chen, Lu, Cui, He, Liu, Chua, et~al.}}}
\bibcite{yuan-etal:2024selfrewarding}{{89}{2024}{{Yuan et~al.}}{{Yuan, Pang, Cho, Li, Sukhbaatar, Xu, and Weston}}}
\bibcite{yuan-etal:2023scaling}{{90}{2023}{{Yuan et~al.}}{{Yuan, Yuan, Li, Dong, Lu, Tan, Zhou, and Zhou}}}
\def\@LN@column{1}
\bibcite{zang2025internlm}{{91}{2025}{{Zang et~al.}}{{Zang, Dong, Zhang, Cao, Liu, Ding, Wu, Ma, Duan, Zhang, et~al.}}}
\bibcite{zeng-etal:2025revisiting}{{92}{2025}{{Zeng et~al.}}{{Zeng, Cheng, Yin, Zhou, and Qiu}}}
\bibcite{zhang-etal:2024mm}{{93}{2024{a}}{{Zhang et~al.}}{{Zhang, Yu, Dong, Li, Su, Chu, and Yu}}}
\bibcite{zhang-etal:2024generative}{{94}{2024{b}}{{Zhang et~al.}}{{Zhang, Hosseini, Bansal, Kazemi, Kumar, and Agarwal}}}
\bibcite{zhang-etal:zhang2024generative}{{95}{2024{c}}{{Zhang et~al.}}{{Zhang, Hosseini, Bansal, Kazemi, Kumar, and Agarwal}}}
\bibcite{zhao-etal:2024atom}{{96}{2024}{{Zhao et~al.}}{{Zhao, Lin, Zhu, Ye, Chen, Zheng, Ceze, Krishnamurthy, Chen, and Kasikci}}}
\bibcite{zheng-etal:2023judging}{{97}{2023}{{Zheng et~al.}}{{Zheng, Chiang, Sheng, Zhuang, Wu, Zhuang, Lin, Li, Li, Xing, Zhang, Gonzalez, and Stoica}}}
\bibcite{zhou:2024prior}{{98}{2024}{{Zhou et~al.}}{{Zhou, Wang, Hu, Xiao, Zhang, and Zhu}}}
\bibstyle{rl-introduction}
\def\@LN@column{1}
\gdef \@abspage@last{54}
rl-introduction.bbl 0 → 100644
\begin{thebibliography}{98}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
\providecommand{\doi}[1]{doi: #1}\else
\providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi
\bibitem[Azar et~al.(2024)Azar, Guo, Piot, Munos, Rowland, Valko, and Calandriello]{azar:2024general}
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.
\newblock URL \url{https://proceedings.mlr.press/v238/gheshlaghi-azar24a.html}.
\bibitem[Bahdanau et~al.(2017)Bahdanau, Brakel, Xu, Goyal, Lowe, Pineau, Courville, and Bengio]{bahdanau-etal:2016actor}
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.
\newblock In \emph{5th International Conference on Learning Representations, {ICLR} 2017, Toulon, France, April 24-26, 2017, Conference Track Proceedings}. OpenReview.net, 2017.
\newblock URL \url{https://openreview.net/forum?id=SJDaqqveg}.
\bibitem[Baltru{\v{s}}aitis et~al.(2018)Baltru{\v{s}}aitis, Ahuja, and Morency]{baltruvsaitis-etal:2018multimodal}
Tadas Baltru{\v{s}}aitis, Chaitanya Ahuja, and Louis-Philippe Morency.
\newblock Multimodal machine learning: A survey and taxonomy.
\newblock \emph{IEEE transactions on pattern analysis and machine intelligence}, 41\penalty0 (2):\penalty0 423--443, 2018.
\bibitem[Bishop(2006)]{Bishop:2006}
Christopher~M. Bishop.
\newblock \emph{Pattern Recognition and Machine Learning}.
\newblock Springer, 2006.
\bibitem[Bradley \& Terry(1952)Bradley and Terry]{bradley-and-terry:rank}
Ralph~Allan Bradley and Milton~E. Terry.
\newblock Rank analysis of incomplete block designs: I. the method of paired comparisons.
\newblock \emph{Biometrika}, 39\penalty0 (3/4):\penalty0 324--345, 1952.
\bibitem[Bromley et~al.(1993)Bromley, Guyon, LeCun, S{\"a}ckinger, and Shah]{bromley-etal:1993signature}
Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard S{\"a}ckinger, and Roopak Shah.
\newblock Signature verification using a" siamese" time delay neural network.
\newblock \emph{Advances in neural information processing systems}, 6, 1993.
\bibitem[Chen et~al.(2023)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.
\newblock URL \url{https://arxiv.org/abs/2302.01318}.
\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.
\newblock URL \url{https://openreview.net/forum?id=FdVXgSJhvz}.
\bibitem[Cheng et~al.(2023)Cheng, Xie, Bai, Dai, and Du]{cheng-etal:2023everyone}
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.
\bibitem[Christiano et~al.(2017)Christiano, Leike, Brown, Martic, Legg, and Amodei]{christiano-etal:2017deep}
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.
\newblock URL \url{https://proceedings.neurips.cc/paper/2017/hash/d5e2c0adad503c91f91df240d0cd4e49-Abstract.html}.
\bibitem[Coste et~al.(2024)Coste, Anwar, Kirk, and Krueger]{coste-etal:2024reward}
Thomas Coste, Usman Anwar, Robert Kirk, and David Krueger.
\newblock Reward model ensembles help mitigate overoptimization.
\newblock In \emph{The Twelfth International Conference on Learning Representations, {ICLR} 2024, Vienna, Austria, May 7-11, 2024}. OpenReview.net, 2024.
\newblock URL \url{https://openreview.net/forum?id=dcjtMYkpXx}.
\bibitem[Cui et~al.(2024{\natexlab{a}})Cui, Yuan, Ding, Yao, He, Zhu, Ni, Xie, Xie, Lin, Liu, and Sun]{cui-etal:2023ultrafeedback}
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}}.
\newblock URL \url{https://openreview.net/forum?id=BOorDpKHiJ}.
\bibitem[Cui et~al.(2024{\natexlab{b}})Cui, Yuan, Ding, Yao, He, Zhu, Ni, Xie, Xie, Lin, Liu, and Sun]{cui-etal:2024ultra}
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}}.
\newblock URL \url{https://openreview.net/forum?id=BOorDpKHiJ}.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2023/hash/5fc47800ee5b30b8777fdd30abcaaf3b-Abstract-Conference.html}.
\bibitem[Eisenstein et~al.(2023)Eisenstein, Nagpal, Agarwal, Beirami, D'Amour, Dvijotham, Fisch, Heller, Pfohl, Ramachandran, et~al.]{eisenstein-etal:2023helping}
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.
\newblock URL \url{https://proceedings.mlr.press/v202/gao23h.html}.
\bibitem[Gorbatovski et~al.(2024)Gorbatovski, Shaposhnikov, Malakhov, Surnachev, Aksenov, Maksimov, Balagansky, and Gavrilov]{gorbatovski-etal:2024learn}
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.
\newblock URL \url{https://arxiv.org/abs/2404.09656}.
\bibitem[Grattafiori et~al.(2024)Grattafiori, Dubey, Jauhri, Pandey, Kadian, Al-Dahle, Letman, Mathur, Schelten, Vaughan, et~al.]{grattafiori-etal:2024llama}
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.
\newblock \emph{ArXiv preprint}, abs/2501.12948, 2025.
\newblock URL \url{https://arxiv.org/abs/2501.12948}.
\bibitem[Havrilla et~al.(2024)Havrilla, Du, Raparthy, Nalmpantis, Dwivedi-Yu, Zhuravinskyi, Hambro, Sukhbaatar, and Raileanu]{havrilla-etal:2024teaching}
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.
\newblock URL \url{https://arxiv.org/abs/2403.04642}.
\bibitem[Hong et~al.(2024)Hong, Lee, and Thorne]{hong:2024orpo}
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.
\newblock URL \url{https://arxiv.org/abs/2501.03262}.
\bibitem[Ji et~al.(2025)Ji, Chen, Pan, Zhu, Zhang, Li, Hong, Chen, Zhou, Wang, et~al.]{ji2025safe}
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.
\bibitem[Kumar et~al.(2024)Kumar, Zhuang, Agarwal, Su, Co-Reyes, Singh, Baumli, Iqbal, Bishop, Roelofs, et~al.]{kumar-etal:2024training}
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.
\newblock URL \url{https://arxiv.org/abs/2409.12917}.
\bibitem[Li et~al.(2025)Li, Zou, and Liu]{li-etal:2025limr}
Xuefeng Li, Haoyang Zou, and Pengfei Liu.
\newblock Limr: Less is more for rl scaling.
\newblock \emph{ArXiv preprint}, abs/2502.11886, 2025.
\newblock URL \url{https://arxiv.org/abs/2502.11886}.
\bibitem[Li et~al.(2024)Li, Xu, Zhang, Lin, Yu, Sun, and Luo]{li-etal:2023remax}
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.
\newblock In \emph{Forty-first International Conference on Machine Learning, {ICML} 2024, Vienna, Austria, July 21-27, 2024}. OpenReview.net, 2024.
\newblock URL \url{https://openreview.net/forum?id=Stn8hXkpe6}.
\bibitem[Lightman et~al.(2024{\natexlab{a}})Lightman, Kosaraju, Burda, Edwards, Baker, Lee, Leike, Schulman, Sutskever, and Cobbe]{lightman-etal:2023let}
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}}.
\newblock URL \url{https://openreview.net/forum?id=v8L0pN6EOi}.
\bibitem[Lightman et~al.(2024{\natexlab{b}})Lightman, Kosaraju, Burda, Edwards, Baker, Lee, Leike, Schulman, Sutskever, and Cobbe]{lightman-etal:2024lets}
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}}.
\newblock URL \url{https://openreview.net/forum?id=v8L0pN6EOi}.
\bibitem[Liu et~al.(2023)Liu, Iter, Xu, Wang, Xu, and Zhu]{liu-etal:2023GEval}
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.
\newblock \doi{10.18653/v1/2023.emnlp-main.153}.
\newblock URL \url{https://aclanthology.org/2023.emnlp-main.153}.
\bibitem[Mahan et~al.(2024)Mahan, Van~Phung, Rafailov, Blagden, Lile, Castricato, Fr{\"a}nken, Finn, and Albalak]{mahan-etal:2024generative}
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.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2024/hash/e099c1c9699814af0be873a175361713-Abstract-Conference.html}.
\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.(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.
\newblock URL \url{http://proceedings.mlr.press/v48/mniha16.html}.
\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}
Long Ouyang, Jeffrey Wu, Xu~Jiang, Diogo Almeida, Carroll~L. Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, John Schulman, Jacob Hilton, Fraser Kelton, Luke Miller, Maddie Simens, Amanda Askell, Peter Welinder, Paul~F. Christiano, Jan Leike, and Ryan Lowe.
\newblock Training language models to follow instructions with human feedback.
\newblock In Sanmi Koyejo, S.~Mohamed, A.~Agarwal, Danielle Belgrave, K.~Cho, and A.~Oh (eds.), \emph{Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, New Orleans, LA, USA, November 28 - December 9, 2022}, 2022.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2022/hash/b1efde53be364a73914f58805a001731-Abstract-Conference.html}.
\bibitem[Pope et~al.(2023)Pope, Douglas, Chowdhery, Devlin, Bradbury, Heek, Xiao, Agrawal, and Dean]{pope-etal:2023efficiently}
Reiner Pope, Sholto Douglas, Aakanksha Chowdhery, Jacob Devlin, James Bradbury, Jonathan Heek, Kefan Xiao, Shivani Agrawal, and Jeff Dean.
\newblock Efficiently scaling transformer inference.
\newblock \emph{Proceedings of Machine Learning and Systems}, 5:\penalty0 606--624, 2023.
\bibitem[Radford et~al.(2021)Radford, Kim, Hallacy, Ramesh, Goh, Agarwal, Sastry, Askell, Mishkin, Clark, Krueger, and Sutskever]{radford-etal:2021learning}
Alec Radford, Jong~Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever.
\newblock Learning transferable visual models from natural language supervision, 2021.
\newblock URL \url{https://arxiv.org/abs/2103.00020}.
\bibitem[Rafailov et~al.(2023)Rafailov, Sharma, Mitchell, Manning, Ermon, and Finn]{rafailov:2023direct}
Rafael Rafailov, Archit Sharma, Eric Mitchell, Christopher~D. Manning, Stefano Ermon, and Chelsea Finn.
\newblock Direct preference optimization: Your language model is secretly a reward model.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2023/hash/a85b405ed65c6477a4fe8302b5e06ce7-Abstract-Conference.html}.
\bibitem[Schulman et~al.(2015)Schulman, Levine, Abbeel, Jordan, and Moritz]{schulman-etal:2015trust}
John Schulman, Sergey Levine, Pieter Abbeel, Michael~I. Jordan, and Philipp Moritz.
\newblock Trust region policy optimization.
\newblock In Francis~R. Bach and David~M. Blei (eds.), \emph{Proceedings of the 32nd International Conference on Machine Learning, {ICML} 2015, Lille, France, 6-11 July 2015}, volume~37 of \emph{{JMLR} Workshop and Conference Proceedings}, pp.\ 1889--1897. JMLR.org, 2015.
\newblock URL \url{http://proceedings.mlr.press/v37/schulman15.html}.
\bibitem[Schulman et~al.(2016)Schulman, Moritz, Levine, Jordan, and Abbeel]{schulman-etal:2015high}
John Schulman, Philipp Moritz, Sergey Levine, Michael~I. Jordan, and Pieter Abbeel.
\newblock High-dimensional continuous control using generalized advantage estimation.
\newblock In Yoshua Bengio and Yann LeCun (eds.), \emph{4th International Conference on Learning Representations, {ICLR} 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings}, 2016.
\newblock URL \url{http://arxiv.org/abs/1506.02438}.
\bibitem[Schulman et~al.(2017)Schulman, Wolski, Dhariwal, Radford, and Klimov]{schulman-etal:2017proximal}
John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov.
\newblock Proximal policy optimization algorithms.
\newblock \emph{ArXiv preprint}, abs/1707.06347, 2017.
\newblock URL \url{https://arxiv.org/abs/1707.06347}.
\bibitem[Setlur et~al.(2024)Setlur, Nagpal, Fisch, Geng, Eisenstein, Agarwal, Agarwal, Berant, and Kumar]{setlur-etal:2024rewarding}
Amrith Setlur, Chirag Nagpal, Adam Fisch, Xinyang Geng, Jacob Eisenstein, Rishabh Agarwal, Alekh Agarwal, Jonathan Berant, and Aviral Kumar.
\newblock Rewarding progress: Scaling automated process verifiers for llm reasoning.
\newblock \emph{ArXiv preprint}, abs/2410.08146, 2024.
\newblock URL \url{https://arxiv.org/abs/2410.08146}.
\bibitem[Shao et~al.(2025)Shao, Li, Liu, Chen, Zhou, Wang, Cai, and Li]{shao:2025earlier}
Ruichen Shao, Bei Li, Gangao Liu, Yang Chen, Xiang Zhou, Jingang Wang, Xunliang Cai, and Peng Li.
\newblock Earlier tokens contribute more: Learning direct preference optimization from temporal decay perspective.
\newblock \emph{ArXiv preprint}, abs/2502.14340, 2025.
\newblock URL \url{https://arxiv.org/abs/2502.14340}.
\bibitem[Shao et~al.(2024)Shao, Wang, Zhu, Xu, Song, Bi, Zhang, Zhang, Li, Wu, et~al.]{shao-etal:2024deepseekmath}
Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, YK~Li, Y~Wu, et~al.
\newblock Deepseekmath: Pushing the limits of mathematical reasoning in open language models.
\newblock \emph{ArXiv preprint}, abs/2402.03300, 2024.
\newblock URL \url{https://arxiv.org/abs/2402.03300}.
\bibitem[Silver et~al.(2016)Silver, Huang, Maddison, Guez, Sifre, Van Den~Driessche, Schrittwieser, Antonoglou, Panneershelvam, Lanctot, et~al.]{silver-etal:2016mastering}
David Silver, Aja Huang, Chris~J Maddison, Arthur Guez, Laurent Sifre, George Van Den~Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et~al.
\newblock Mastering the game of go with deep neural networks and tree search.
\newblock \emph{nature}, 529\penalty0 (7587):\penalty0 484--489, 2016.
\bibitem[Singhal et~al.(2023)Singhal, Goyal, Xu, and Durrett]{singhal-etal:2023long}
Prasann Singhal, Tanya Goyal, Jiacheng Xu, and Greg Durrett.
\newblock A long way to go: Investigating length correlations in rlhf.
\newblock \emph{ArXiv preprint}, abs/2310.03716, 2023.
\newblock URL \url{https://arxiv.org/abs/2310.03716}.
\bibitem[Sun et~al.(2024)Sun, Liu, Bair, and Kolter]{sun-etal:2023simple}
Mingjie Sun, Zhuang Liu, Anna Bair, and J.~Zico Kolter.
\newblock A simple and effective pruning approach for large language models.
\newblock In \emph{The Twelfth International Conference on Learning Representations, {ICLR} 2024, Vienna, Austria, May 7-11, 2024}. OpenReview.net, 2024.
\newblock URL \url{https://openreview.net/forum?id=PxoFut3dWW}.
\bibitem[Sun et~al.(2023)Sun, Shen, Cao, Liu, Li, Shen, Gan, Gui, Wang, Yang, et~al.]{sun-etal:2023aligning}
Zhiqing Sun, Sheng Shen, Shengcao Cao, Haotian Liu, Chunyuan Li, Yikang Shen, Chuang Gan, Liang-Yan Gui, Yu-Xiong Wang, Yiming Yang, et~al.
\newblock Aligning large multimodal models with factually augmented rlhf.
\newblock \emph{arXiv preprint arXiv:2309.14525}, 2023.
\bibitem[Sutton(1988)]{sutton-and-richard:1988learning}
Richard~S Sutton.
\newblock Learning to predict by the methods of temporal differences.
\newblock \emph{Machine learning}, 3:\penalty0 9--44, 1988.
\bibitem[Sutton \& Barto(2018)Sutton and Barto]{Sutton-and-Barto:2018RL}
Richard~S. Sutton and Andrew~G. Barto.
\newblock \emph{Reinforcement Learning: An Introduction (2nd ed.)}.
\newblock The MIT Press, 2018.
\bibitem[Szepesv{\'a}ri(2010)]{szepesvari:2010algorithms}
Csaba Szepesv{\'a}ri.
\newblock Algorithms for reinforcement learning.
\newblock \emph{Synthesis Lectures on Artificial Intelligence and Machine Learning}, 4\penalty0 (1):\penalty0 1--103, 2010.
\bibitem[Team et~al.(2025)Team, Du, Gao, Xing, Jiang, Chen, Li, Xiao, Du, Liao, et~al.]{kimi-team:2025kimi}
Kimi Team, Angang Du, Bofei Gao, Bowei Xing, Changjiu Jiang, Cheng Chen, Cheng Li, Chenjun Xiao, Chenzhuang Du, Chonghua Liao, et~al.
\newblock Kimi k1. 5: Scaling reinforcement learning with llms.
\newblock \emph{ArXiv preprint}, abs/2501.12599, 2025.
\newblock URL \url{https://arxiv.org/abs/2501.12599}.
\bibitem[Team(2025)]{qwenTeam:2025qwen2.5-VL}
Qwen Team.
\newblock Qwen2.5-vl, January 2025.
\newblock URL \url{https://qwenlm.github.io/blog/qwen2.5-vl/}.
\bibitem[Touvron et~al.(2023)Touvron, Martin, Stone, Albert, Almahairi, Babaei, Bashlykov, Batra, Bhargava, Bhosale, Bikel, Blecher, Ferrer, Chen, Cucurull, Esiobu, Fernandes, Fu, Fu, Fuller, Gao, Goswami, Goyal, Hartshorn, Hosseini, Hou, Inan, Kardas, Kerkez, Khabsa, Kloumann, Korenev, Koura, Lachaux, Lavril, Lee, Liskovich, Lu, Mao, Martinet, Mihaylov, Mishra, Molybog, Nie, Poulton, Reizenstein, Rungta, Saladi, Schelten, Silva, Smith, Subramanian, Tan, Tang, Taylor, Williams, Kuan, Xu, Yan, Zarov, Zhang, Fan, Kambadur, Narang, Rodriguez, Stojnic, Edunov, and Scialom]{touvron-etal:2023llama2}
Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian~Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit~Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric~Michael Smith, Ranjan Subramanian, Xiaoqing~Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian~Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom.
\newblock Llama 2: Open foundation and fine-tuned chat models.
\newblock \emph{ArXiv preprint}, abs/2307.09288, 2023.
\newblock URL \url{https://arxiv.org/abs/2307.09288}.
\bibitem[Vinyals et~al.(2019)Vinyals, Babuschkin, Czarnecki, Mathieu, Dudzik, Chung, Choi, Powell, Ewalds, Georgiev, et~al.]{vinyals-rtal:2019grandmaster}
Oriol Vinyals, Igor Babuschkin, Wojciech~M Czarnecki, Micha{\"e}l Mathieu, Andrew Dudzik, Junyoung Chung, David~H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et~al.
\newblock Grandmaster level in starcraft ii using multi-agent reinforcement learning.
\newblock \emph{nature}, 575\penalty0 (7782):\penalty0 350--354, 2019.
\bibitem[Wang et~al.(2023{\natexlab{a}})Wang, Zhou, Chang, Liu, Zhang, Du, Xiao, and Zhu]{wang-etal:2023learning}
Chenglong Wang, Hang Zhou, Kaiyan Chang, Tongran Liu, Chunliang Zhang, Quan Du, Tong Xiao, and Jingbo Zhu.
\newblock Learning evaluation models from large language models for sequence generation.
\newblock \emph{ArXiv preprint}, abs/2308.04386, 2023{\natexlab{a}}.
\newblock URL \url{https://arxiv.org/abs/2308.04386}.
\bibitem[Wang et~al.(2024{\natexlab{a}})Wang, Gan, Huo, Mu, Yang, He, Xiao, Zhang, Liu, Du, et~al.]{wang-etal:2024rovrm}
Chenglong Wang, Yang Gan, Yifu Huo, Yongyu Mu, Murun Yang, Qiaozhi He, Tong Xiao, Chunliang Zhang, Tongran Liu, Quan Du, et~al.
\newblock Rovrm: A robust visual reward model optimized via auxiliary textual preference data.
\newblock \emph{arXiv preprint arXiv:2408.12109}, 2024{\natexlab{a}}.
\bibitem[Wang et~al.(2024{\natexlab{b}})Wang, Zhou, Chang, Li, Mu, Xiao, Liu, and Zhu]{wang-etal:2024hybrid}
Chenglong Wang, Hang Zhou, Kaiyan Chang, Bei Li, Yongyu Mu, Tong Xiao, Tongran Liu, and Jingbo Zhu.
\newblock Hybrid alignment training for large language models.
\newblock \emph{ArXiv preprint}, abs/2406.15178, 2024{\natexlab{b}}.
\newblock URL \url{https://arxiv.org/abs/2406.15178}.
\bibitem[Wang et~al.(2024{\natexlab{c}})Wang, Zhou, Hu, Huo, Li, Liu, Xiao, and Zhu]{wang-etal:2024esrl}
Chenglong Wang, Hang Zhou, Yimin Hu, Yifu Huo, Bei Li, Tongran Liu, Tong Xiao, and Jingbo Zhu.
\newblock {ESRL:} efficient sampling-based reinforcement learning for sequence generation.
\newblock In Michael~J. Wooldridge, Jennifer~G. Dy, and Sriraam Natarajan (eds.), \emph{Thirty-Eighth {AAAI} Conference on Artificial Intelligence, {AAAI} 2024, Thirty-Sixth Conference on Innovative Applications of Artificial Intelligence, {IAAI} 2024, Fourteenth Symposium on Educational Advances in Artificial Intelligence, {EAAI} 2014, February 20-27, 2024, Vancouver, Canada}, pp.\ 19107--19115. {AAAI} Press, 2024{\natexlab{c}}.
\newblock \doi{10.1609/AAAI.V38I17.29878}.
\newblock URL \url{https://doi.org/10.1609/aaai.v38i17.29878}.
\bibitem[Wang et~al.(2025)Wang, Gan, Huo, Mu, He, Yang, Li, Xiao, Zhang, Liu, et~al.]{wang-etal:wang2025gram}
Chenglong Wang, Yang Gan, Yifu Huo, Yongyu Mu, Qiaozhi He, Murun Yang, Bei Li, Tong Xiao, Chunliang Zhang, Tongran Liu, et~al.
\newblock Gram: A generative foundation reward model for reward generalization.
\newblock \emph{arXiv preprint arXiv:2506.14175}, 2025.
\bibitem[Wang et~al.(2026{\natexlab{a}})Wang, Huo, Gan, He, Meng, Li, Wang, Liu, Zhou, Zhu, et~al.]{wang2026msrl}
Chenglong Wang, Yifu Huo, Yang Gan, Qiaozhi He, Qi~Meng, Bei Li, Yan Wang, Junfu Liu, Tianhua Zhou, Jingbo Zhu, et~al.
\newblock Msrl: Scaling generative multimodal reward modeling via multi-stage reinforcement learning.
\newblock \emph{arXiv preprint arXiv:2603.25108}, 2026{\natexlab{a}}.
\bibitem[Wang et~al.(2026{\natexlab{b}})Wang, Mu, Zhou, Huo, Zhu, Zeng, Yang, Li, Hao, Zhang, et~al.]{wang-etal:wang2026gram}
Chenglong Wang, Yongyu Mu, Hang Zhou, Yifu Huo, Ziming Zhu, Jiali Zeng, Murun Yang, Bei Li, Xiaoyang Hao, Chunliang Zhang, et~al.
\newblock Gram-r$^2$: Self-training generative foundation reward models for reward reasoning.
\newblock In \emph{Proceedings of the AAAI Conference on Artificial Intelligence}, volume~40, pp.\ 33395--33403, 2026{\natexlab{b}}.
\bibitem[Wang et~al.(2021)Wang, Yan, Meng, and Zhou]{wang-etal:2021selective}
Fusheng Wang, Jianhao Yan, Fandong Meng, and Jie Zhou.
\newblock Selective knowledge distillation for neural machine translation.
\newblock In Chengqing Zong, Fei Xia, Wenjie Li, and Roberto Navigli (eds.), \emph{Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)}, pp.\ 6456--6466, Online, 2021. Association for Computational Linguistics.
\newblock \doi{10.18653/v1/2021.acl-long.504}.
\newblock URL \url{https://aclanthology.org/2021.acl-long.504}.
\bibitem[Wang et~al.(2023{\natexlab{b}})Wang, Li, Chen, Cai, Zhu, Lin, Cao, Liu, Liu, and Sui]{wang-etal:2023large}
Peiyi Wang, Lei Li, Liang Chen, Zefan Cai, Dawei Zhu, Binghuai Lin, Yunbo Cao, Qi~Liu, Tianyu Liu, and Zhifang Sui.
\newblock Large language models are not fair evaluators.
\newblock \emph{ArXiv preprint}, 2023{\natexlab{b}}.
\bibitem[Wang et~al.(2023{\natexlab{c}})Wang, Li, Shao, Xu, Dai, Li, Chen, Wu, and Sui]{wang-etal:2023math}
Peiyi Wang, Lei Li, Zhihong Shao, RX~Xu, Damai Dai, Yifei Li, Deli Chen, Yu~Wu, and Zhifang Sui.
\newblock Math-shepherd: Verify and reinforce llms step-by-step without human annotations.
\newblock \emph{ArXiv preprint}, abs/2312.08935, 2023{\natexlab{c}}.
\newblock URL \url{https://arxiv.org/abs/2312.08935}.
\bibitem[Wang \& Zhou(2024)Wang and Zhou]{wang-and-zhou:2024chain}
Xuezhi Wang and Denny Zhou.
\newblock Chain-of-thought reasoning without prompting.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2024/hash/7a8e7fd295aa04eac4b470ae27f8785c-Abstract-Conference.html}.
\bibitem[Wang et~al.(2023{\natexlab{d}})Wang, Ivison, Dasigi, Hessel, Khot, Chandu, Wadden, MacMillan, Smith, Beltagy, and Hajishirzi]{wang-etal:2023far}
Yizhong Wang, Hamish Ivison, Pradeep Dasigi, Jack Hessel, Tushar Khot, Khyathi Chandu, David Wadden, Kelsey MacMillan, Noah~A. Smith, Iz~Beltagy, and Hannaneh Hajishirzi.
\newblock How far can camels go? exploring the state of instruction tuning on open resources.
\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{\natexlab{d}}.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2023/hash/ec6413875e4ab08d7bc4d8e225263398-Abstract-Datasets\_and\_Benchmarks.html}.
\bibitem[Wei et~al.(2022)Wei, Bosma, Zhao, Guu, Yu, Lester, Du, Dai, and Le]{wei-etal:2022finetuned}
Jason Wei, Maarten Bosma, Vincent~Y. Zhao, Kelvin Guu, Adams~Wei Yu, Brian Lester, Nan Du, Andrew~M. Dai, and Quoc~V. Le.
\newblock Finetuned language models are zero-shot learners.
\newblock In \emph{The Tenth International Conference on Learning Representations, {ICLR} 2022, Virtual Event, April 25-29, 2022}. OpenReview.net, 2022.
\newblock URL \url{https://openreview.net/forum?id=gEZrGCozdqR}.
\bibitem[Wu et~al.(2023)Wu, Hu, Shi, Dziri, Suhr, Ammanabrolu, Smith, Ostendorf, and Hajishirzi]{wu-etal:2023fine}
Zeqiu Wu, Yushi Hu, Weijia Shi, Nouha Dziri, Alane Suhr, Prithviraj Ammanabrolu, Noah~A. Smith, Mari Ostendorf, and Hannaneh Hajishirzi.
\newblock Fine-grained human feedback gives better rewards for language model training.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2023/hash/b8c90b65739ae8417e61eadb521f63d5-Abstract-Conference.html}.
\bibitem[Wu et~al.(2024)Wu, Balashankar, Kim, Eisenstein, and Beirami]{wu-etal:2024reuse}
Zhaofeng Wu, Ananth Balashankar, Yoon Kim, Jacob Eisenstein, and Ahmad Beirami.
\newblock Reuse your rewards: Reward model transfer for zero-shot cross-lingual alignment.
\newblock \emph{arXiv preprint arXiv:2404.12318}, 2024.
\bibitem[Xia et~al.(2024)Xia, Malladi, Gururangan, Arora, and Chen]{xia-etal:2024less}
Mengzhou Xia, Sadhika Malladi, Suchin Gururangan, Sanjeev Arora, and Danqi Chen.
\newblock Less: Selecting influential data for targeted instruction tuning.
\newblock \emph{arXiv preprint arXiv:2402.04333}, 2024.
\bibitem[Xiao \& Zhu(2023)Xiao and Zhu]{xiao-etal:2023introduction}
Tong Xiao and Jingbo Zhu.
\newblock Introduction to transformers: an nlp perspective.
\newblock \emph{ArXiv preprint}, abs/2311.17633, 2023.
\newblock URL \url{https://arxiv.org/abs/2311.17633}.
\bibitem[Xiao \& Zhu(2025)Xiao and Zhu]{xiao-and-zhu:2025foundations}
Tong Xiao and Jingbo Zhu.
\newblock Foundations of large language models.
\newblock \emph{ArXiv preprint}, abs/2501.09223, 2025.
\newblock URL \url{https://arxiv.org/abs/2501.09223}.
\bibitem[Xie et~al.(2025)Xie, Gao, Ren, Luo, Hong, Dai, Zhou, Qiu, Wu, and Luo]{xie-etal:2025logic}
Tian Xie, Zitian Gao, Qingnan Ren, Haoming Luo, Yuqian Hong, Bryan Dai, Joey Zhou, Kai Qiu, Zhirong Wu, and Chong Luo.
\newblock Logic-rl: Unleashing llm reasoning with rule-based reinforcement learning.
\newblock \emph{ArXiv preprint}, abs/2502.14768, 2025.
\newblock URL \url{https://arxiv.org/abs/2502.14768}.
\bibitem[Xu et~al.(2024)Xu, Sharaf, Chen, Tan, Shen, Durme, Murray, and Kim]{xu:2024contrastive}
Haoran Xu, Amr Sharaf, Yunmo Chen, Weiting Tan, Lingfeng Shen, Benjamin~Van Durme, Kenton Murray, and Young~Jin Kim.
\newblock Contrastive preference optimization: Pushing the boundaries of {LLM} performance in machine translation.
\newblock In \emph{Forty-first International Conference on Machine Learning, {ICML} 2024, Vienna, Austria, July 21-27, 2024}. OpenReview.net, 2024.
\newblock URL \url{https://openreview.net/forum?id=51iwkioZpn}.
\bibitem[Xu et~al.(2025)Xu, Guo, He, Hu, He, Bai, Chen, Wang, Fan, Dang, et~al.]{xu-etal:2025qwen2}
Jin Xu, Zhifang Guo, Jinzheng He, Hangrui Hu, Ting He, Shuai Bai, Keqin Chen, Jialin Wang, Yang Fan, Kai Dang, et~al.
\newblock Qwen2. 5-omni technical report.
\newblock \emph{arXiv preprint arXiv:2503.20215}, 2025.
\bibitem[Yang et~al.(2024)Yang, Ding, Lin, Zhang, and Zhang]{yang-etal:2024regularizing}
Rui Yang, Ruomeng Ding, Yong Lin, Huan Zhang, and Tong Zhang.
\newblock Regularizing hidden states enables learning generalizable reward model for llms.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2024/hash/71f7154547c748c8041505521ca433ab-Abstract-Conference.html}.
\bibitem[Yao et~al.(2023)Yao, Yu, Zhao, Shafran, Griffiths, Cao, and Narasimhan]{yao-etal:2023tree}
Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimhan.
\newblock Tree of thoughts: Deliberate problem solving with large language models.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2023/hash/271db9922b8d1f4dd7aaef84ed5ac703-Abstract-Conference.html}.
\bibitem[Yu et~al.(2024{\natexlab{a}})Yu, Yao, Zhang, He, Han, Cui, Hu, Liu, Zheng, Sun, et~al.]{yu-etal:2024rlhf}
Tianyu Yu, Yuan Yao, Haoye Zhang, Taiwen He, Yifeng Han, Ganqu Cui, Jinyi Hu, Zhiyuan Liu, Hai-Tao Zheng, Maosong Sun, et~al.
\newblock Rlhf-v: Towards trustworthy mllms via behavior alignment from fine-grained correctional human feedback.
\newblock In \emph{Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition}, pp.\ 13807--13816, 2024{\natexlab{a}}.
\bibitem[Yu et~al.(2024{\natexlab{b}})Yu, Zhang, Yao, Dang, Chen, Lu, Cui, He, Liu, Chua, et~al.]{yu-etal:2024rlaif}
Tianyu Yu, Haoye Zhang, Yuan Yao, Yunkai Dang, Da~Chen, Xiaoman Lu, Ganqu Cui, Taiwen He, Zhiyuan Liu, Tat-Seng Chua, et~al.
\newblock Rlaif-v: Aligning mllms through open-source ai feedback for super gpt-4v trustworthiness.
\newblock \emph{arXiv preprint arXiv:2405.17220}, 2024{\natexlab{b}}.
\bibitem[Yuan et~al.(2024)Yuan, Pang, Cho, Li, Sukhbaatar, Xu, and Weston]{yuan-etal:2024selfrewarding}
Weizhe Yuan, Richard~Yuanzhe Pang, Kyunghyun Cho, Xian Li, Sainbayar Sukhbaatar, Jing Xu, and Jason Weston.
\newblock Self-rewarding language models.
\newblock In \emph{Forty-first International Conference on Machine Learning, {ICML} 2024, Vienna, Austria, July 21-27, 2024}. OpenReview.net, 2024.
\newblock URL \url{https://openreview.net/forum?id=0NphYCmgua}.
\bibitem[Yuan et~al.(2023)Yuan, Yuan, Li, Dong, Lu, Tan, Zhou, and Zhou]{yuan-etal:2023scaling}
Zheng Yuan, Hongyi Yuan, Chengpeng Li, Guanting Dong, Keming Lu, Chuanqi Tan, Chang Zhou, and Jingren Zhou.
\newblock Scaling relationship on learning mathematical reasoning with large language models.
\newblock \emph{ArXiv preprint}, abs/2308.01825, 2023.
\newblock URL \url{https://arxiv.org/abs/2308.01825}.
\bibitem[Zang et~al.(2025)Zang, Dong, Zhang, Cao, Liu, Ding, Wu, Ma, Duan, Zhang, et~al.]{zang2025internlm}
Yuhang Zang, Xiaoyi Dong, Pan Zhang, Yuhang Cao, Ziyu Liu, Shengyuan Ding, Shenxi Wu, Yubo Ma, Haodong Duan, Wenwei Zhang, et~al.
\newblock Internlm-xcomposer2. 5-reward: A simple yet effective multi-modal reward model.
\newblock \emph{arXiv preprint arXiv:2501.12368}, 2025.
\bibitem[Zeng et~al.(2025)Zeng, Cheng, Yin, Zhou, and Qiu]{zeng-etal:2025revisiting}
Zhiyuan Zeng, Qinyuan Cheng, Zhangyue Yin, Yunhua Zhou, and Xipeng Qiu.
\newblock Revisiting the test-time scaling of o1-like models: Do they truly possess test-time scaling capabilities?
\newblock \emph{ArXiv preprint}, abs/2502.12215, 2025.
\newblock URL \url{https://arxiv.org/abs/2502.12215}.
\bibitem[Zhang et~al.(2024{\natexlab{a}})Zhang, Yu, Dong, Li, Su, Chu, and Yu]{zhang-etal:2024mm}
Duzhen Zhang, Yahan Yu, Jiahua Dong, Chenxing Li, Dan Su, Chenhui Chu, and Dong Yu.
\newblock Mm-llms: Recent advances in multimodal large language models.
\newblock \emph{arXiv preprint arXiv:2401.13601}, 2024{\natexlab{a}}.
\bibitem[Zhang et~al.(2024{\natexlab{b}})Zhang, Hosseini, Bansal, Kazemi, Kumar, and Agarwal]{zhang-etal:2024generative}
Lunjun Zhang, Arian Hosseini, Hritik Bansal, Mehran Kazemi, Aviral Kumar, and Rishabh Agarwal.
\newblock Generative verifiers: Reward modeling as next-token prediction.
\newblock \emph{ArXiv preprint}, abs/2408.15240, 2024{\natexlab{b}}.
\newblock URL \url{https://arxiv.org/abs/2408.15240}.
\bibitem[Zhang et~al.(2024{\natexlab{c}})Zhang, Hosseini, Bansal, Kazemi, Kumar, and Agarwal]{zhang-etal:zhang2024generative}
Lunjun Zhang, Arian Hosseini, Hritik Bansal, Mehran Kazemi, Aviral Kumar, and Rishabh Agarwal.
\newblock Generative verifiers: Reward modeling as next-token prediction.
\newblock \emph{arXiv preprint arXiv:2408.15240}, 2024{\natexlab{c}}.
\bibitem[Zhao et~al.(2024)Zhao, Lin, Zhu, Ye, Chen, Zheng, Ceze, Krishnamurthy, Chen, and Kasikci]{zhao-etal:2024atom}
Yilong Zhao, Chien-Yu Lin, Kan Zhu, Zihao Ye, Lequn Chen, Size Zheng, Luis Ceze, Arvind Krishnamurthy, Tianqi Chen, and Baris Kasikci.
\newblock Atom: Low-bit quantization for efficient and accurate llm serving.
\newblock \emph{Proceedings of Machine Learning and Systems}, 6:\penalty0 196--209, 2024.
\bibitem[Zheng et~al.(2023)Zheng, Chiang, Sheng, Zhuang, Wu, Zhuang, Lin, Li, Li, Xing, Zhang, Gonzalez, and Stoica]{zheng-etal:2023judging}
Lianmin Zheng, Wei{-}Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi~Lin, Zhuohan Li, Dacheng Li, Eric~P. Xing, Hao Zhang, Joseph~E. Gonzalez, and Ion Stoica.
\newblock Judging llm-as-a-judge with mt-bench and chatbot arena.
\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.
\newblock URL \url{http://papers.nips.cc/paper\_files/paper/2023/hash/91f18a1287b398d378ef22505bf41832-Abstract-Datasets\_and\_Benchmarks.html}.
\bibitem[Zhou et~al.(2024)Zhou, Wang, Hu, Xiao, Zhang, and Zhu]{zhou:2024prior}
Hang Zhou, Chenglong Wang, Yimin Hu, Tong Xiao, Chunliang Zhang, and Jingbo Zhu.
\newblock Prior constraints-based reward model training for aligning large language models.
\newblock In \emph{China National Conference on Chinese Computational Linguistics}, pp.\ 555--570. Springer, 2024.
\end{thebibliography}
rl-introduction.bib
@article{wang2026msrl,
title={MSRL: Scaling Generative Multimodal Reward Modeling via Multi-Stage Reinforcement Learning},
author={Wang, Chenglong and Huo, Yifu and Gan, Yang and He, Qiaozhi and Meng, Qi and Li, Bei and Wang, Yan and Liu, Junfu and Zhou, Tianhua and Zhu, Jingbo and others},
journal={arXiv preprint arXiv:2603.25108},
year={2026}
}
@inproceedings{wang-etal:wang2026gram,
title={Gram-r$^2$: Self-training generative foundation reward models for reward reasoning},
author={Wang, Chenglong and Mu, Yongyu and Zhou, Hang and Huo, Yifu and Zhu, Ziming and Zeng, Jiali and Yang, Murun and Li, Bei and Hao, Xiaoyang and Zhang, Chunliang and others},
booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
volume={40},
number={39},
pages={33395--33403},
year={2026}
}
@article{wang-etal:wang2025gram,
title={Gram: A generative foundation reward model for reward generalization},
author={Wang, Chenglong and Gan, Yang and Huo, Yifu and Mu, Yongyu and He, Qiaozhi and Yang, Murun and Li, Bei and Xiao, Tong and Zhang, Chunliang and Liu, Tongran and others},
journal={arXiv preprint arXiv:2506.14175},
year={2025}
}
@article{zhang-etal:zhang2024generative,
title={Generative verifiers: Reward modeling as next-token prediction},
author={Zhang, Lunjun and Hosseini, Arian and Bansal, Hritik and Kazemi, Mehran and Kumar, Aviral and Agarwal, Rishabh},
journal={arXiv preprint arXiv:2408.15240},
year={2024}
}
@article{sun-etal:2023aligning,
title={Aligning large multimodal models with factually augmented rlhf},
author={Sun, Zhiqing and Shen, Sheng and Cao, Shengcao and Liu, Haotian and Li, Chunyuan and Shen, Yikang and Gan, Chuang and Gui, Liang-Yan and Wang, Yu-Xiong and Yang, Yiming and others},
......
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rl-introduction.tex
......@@ -45,6 +45,7 @@
\usepackage{fontawesome5}
\usepackage{varwidth}
\usetikzlibrary{shadows}
\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds}
\definecolor{ocrebase}{RGB}{243,102,25}
\definecolor{ocre}{RGB}{0,0,0}
\definecolor{amber}{rgb}{1.0, 0.75, 0.0}
......@@ -167,16 +168,16 @@ However, RL was not so popular in the long history of NLP, and the field has jus
Of course, we'd like to learn about RL, which appears straightforward. However, common RL textbooks, such as \citet{Sutton-and-Barto:2018RL}'s book, are mostly based on classic robotics or control problems. This makes it difficult to align the concepts of RL with those of LLMs. On the other hand, most LLM literature lacks an in-depth discussion of RL techniques, often omitting related details. As a result, we find ourselves in an awkward middle ground between RL and LLMs, struggling to bridge the two fields.
The aim of this paper is to provide a comprehensive introduction to RL from the perspective of LLMs. We begin by introducing basic concepts and algorithms of RL using the LLM language. In particular, we illustrate their application through an example of training LLMs, making the concepts more accessible. We then discuss a series of refinements to the basic RL framework for LLM alignment, including advanced reward modeling, improved advantage estimation, efficient sampling, and direct LLM optimization without RL. Furthermore, we discuss how to apply RL techniques to LLM reasoning. These methods are closely related to recent LLMs, such as OpenAI o1/o3 and Deepseek R1, which adopt large-scale RL and test-time scaling to significantly advance their reasoning abilities. In addition, we discuss applications of RL to multimodal LLMs to demonstrate how RL can be adapted to various problems. Finally, we conclude by outlining promising future research directions and discussing relevant systems and datasets.
The aim of this paper is to provide a comprehensive introduction to RL from the perspective of LLMs. We begin by introducing basic concepts and algorithms of RL using the LLM language. In particular, we illustrate their application through an example of training LLMs, making the concepts more accessible. We then discuss a series of refinements to the basic RL framework for LLM alignment, including advanced reward modeling, improved advantage estimation, efficient sampling, and direct LLM optimization without RL. Furthermore, we discuss how to apply RL techniques to LLM reasoning. These methods are closely related to recent LLMs, such as OpenAI o1/o3 and DeepSeek R1, which adopt large-scale RL and test-time scaling to significantly advance their reasoning abilities. In addition, we discuss applications of RL to multimodal LLMs to demonstrate how RL can be adapted to various problems. Finally, we conclude by outlining promising future research directions and discussing relevant systems and datasets.
% \clearpage
% New section: training LLMs with reinforcement learning
% \input{section2/section2}
\input{section2/section2}
% \clearpage
% New section: training LLMs with reinforcement learning
% \input{section3/section3}
\input{section3/section3}
% \clearpage
% New section: improved reinforcement learning for LLMs
......@@ -185,19 +186,23 @@ The aim of this paper is to provide a comprehensive introduction to RL from the
% \clearpage
% New section: reinforcement learning for LLM inference
% \input{section5/section5}
\input{section5/section5}
% \clearpage
% New section: reinforcement learning for MLLMs
% \input{section6/section6}
\input{section6/section6}
% \clearpage
% New section: reinforcement learning for MLLMs
\input{section7/section7}
% \clearpage
% conclusion
% \input{section7/section7}
\input{section8/section8}
% \clearpage
% systems & datasets todo: ganyang
% \input{section8/section8}
\input{section9/section9}
\clearpage
......
rl-introduction.toc 0 → 100644
\contentsline {section}{\numberline {1}Introduction}{3}{section.1}%
\contentsline {section}{\numberline {2}Preliminary}{3}{section.2}%
\contentsline {subsection}{\numberline {2.1}Fundamentals of Large Language Models}{3}{subsection.2.1}%
\contentsline {subsection}{\numberline {2.2}Fundamentals of Reinforcement Learning}{4}{subsection.2.2}%
\contentsline {subsubsection}{\numberline {2.2.1}General Framework of Reinforcement Learning}{4}{subsubsection.2.2.1}%
\contentsline {subsubsection}{\numberline {2.2.2}Key Elements}{5}{subsubsection.2.2.2}%
\contentsline {section}{\numberline {3}An Example of Using Reinforcement Learning to Train LLMs}{6}{section.3}%
\contentsline {subsection}{\numberline {3.1}Policy Gradient}{7}{subsection.3.1}%
\contentsline {subsection}{\numberline {3.2}Temporal Decomposition}{9}{subsection.3.2}%
\contentsline {subsection}{\numberline {3.3}Reducing Gradient Variance}{10}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}Importance Sampling}{12}{subsection.3.4}%
\contentsline {subsection}{\numberline {3.5}Proximal Policy Optimization}{14}{subsection.3.5}%
\contentsline {subsection}{\numberline {3.6}Training Reward Models}{15}{subsection.3.6}%
\contentsline {subsection}{\numberline {3.7}A Complete Workflow}{18}{subsection.3.7}%
\contentsline {section}{\numberline {4}Improved Reinforcement Learning for LLMs}{20}{section.4}%
\contentsline {subsection}{\numberline {4.1}Advanced Reward Models}{20}{subsection.4.1}%
\contentsline {subsubsection}{\numberline {4.1.1}Automatic Preference Data Generation}{20}{subsubsection.4.1.1}%
\contentsline {subsubsection}{\numberline {4.1.2}Reward Shaping}{21}{subsubsection.4.1.2}%
\contentsline {subsubsection}{\numberline {4.1.3}Improved Reward Generalization}{23}{subsubsection.4.1.3}%
\contentsline {subsubsection}{\numberline {4.1.4}Generative Reward Models}{24}{subsubsection.4.1.4}%
\contentsline {subsection}{\numberline {4.2}Better Advantage Estimation}{26}{subsection.4.2}%
\contentsline {subsubsection}{\numberline {4.2.1}Temporal Difference-based Advantage Estimation}{26}{subsubsection.4.2.1}%
\contentsline {subsubsection}{\numberline {4.2.2}Generalized Advantage Estimation}{26}{subsubsection.4.2.2}%
\contentsline {subsubsection}{\numberline {4.2.3}Group Relative Policy Optimization}{27}{subsubsection.4.2.3}%
\contentsline {subsection}{\numberline {4.3}Efficient RL Methods}{29}{subsection.4.3}%
\contentsline {subsubsection}{\numberline {4.3.1}Dynamic Sampling}{29}{subsubsection.4.3.1}%
\contentsline {subsubsection}{\numberline {4.3.2}Lightweight Reward Methods}{30}{subsubsection.4.3.2}%
\contentsline {subsection}{\numberline {4.4}Direct Preference Optimization}{32}{subsection.4.4}%
\contentsline {section}{\numberline {5}Reinforcement Learning for LLM Reasoning}{34}{section.5}%
\contentsline {subsection}{\numberline {5.1}Test-time Scaling}{35}{subsection.5.1}%
\contentsline {subsubsection}{\numberline {5.1.1}Best-of-N Sampling}{35}{subsubsection.5.1.1}%
\contentsline {subsubsection}{\numberline {5.1.2}Step-by-step Verification}{37}{subsubsection.5.1.2}%
\contentsline {subsubsection}{\numberline {5.1.3}Monte Carlo Tree Search}{38}{subsubsection.5.1.3}%
\contentsline {subsection}{\numberline {5.2}Large-scale Reinforcement Learning}{40}{subsection.5.2}%
\contentsline {subsection}{\numberline {5.3}Iterative Reinforcement Learning}{41}{subsection.5.3}%
\contentsline {section}{\numberline {6}Reinforcement Learning for LLM-based Agents}{43}{section.6}%
\contentsline {section}{\numberline {7}Reinforcement Learning for Multimodal Models}{43}{section.7}%
\contentsline {subsection}{\numberline {7.1}Visual Language Models}{43}{subsection.7.1}%
\contentsline {subsection}{\numberline {7.2}Speech Generation Models}{45}{subsection.7.2}%
\contentsline {subsection}{\numberline {7.3}Diffusion Models}{45}{subsection.7.3}%
\contentsline {section}{\numberline {8}Summary}{45}{section.8}%
\contentsline {section}{\numberline {9}Systems and Datasets}{46}{section.9}%
section2/Figures/rl_framework.tex 0 → 100644
\begin{center}
\begin{tikzpicture}[
font=\small,
% line width=0.9pt,
box/.style={
draw,
rounded corners=3pt,
minimum width=3.1cm,
minimum height=1.35cm,
align=center,
font=\normalsize
},
neuron/.style={
circle,
draw=black!55,
minimum size=7mm,
inner sep=0pt
},
conn/.style={
draw=black!65,
line width=0.5pt
},
flow/.style={
draw=black,
-Latex,
line width=0.9pt
}
]
% =========================================================
% Left part: Environment-Agent interaction
% =========================================================
\node[
anchor=center,
draw,
thick,
minimum width=3.0cm,
minimum height=1.0cm,
fill=white,
drop shadow={
fill=gray!40,
shadow xshift=0.6ex,
shadow yshift=-0.6ex
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] (env) at (0,0) {\large Environment};
\node[anchor=center,draw, minimum width=3.0cm, minimum height=1.0cm, fill=white, drop shadow={
fill=gray!40,
shadow xshift=0.6ex,
shadow yshift=-0.6ex
}
] (agent) at (5.2,0) {\large Agent};
% top arc: state
\draw[->]
($(env.north west)+(1.6,0.05)$)
to[out=45,in=135]
node[pos=0.15, above, font=\large] {$S_t$}
($(agent.north east)+(-1.6,0.05)$);
% middle arc: reward
\draw[->]
($(env.north east)+(0.02,0.10)$)
to[out=20,in=160]
node[midway, above=2pt] {\large $r_t$}
($(agent.north west)+(-0.02,0.10)$);
% bottom arc: action
\draw[->]
($(agent.south west)+(-0.02,-0.10)$)
to[out=200,in=340]
node[midway, below=2pt] {\large $a_t$}
($(env.south east)+(0.02,-0.10)$);
% connector to policy net
\draw[dashed, line width=0.9pt, ->]
($(agent.east)+(0.10,0.00)$)
.. controls +(1.0,0.45) and +(-0.9,0.25)
.. ($(8.6,0.15)$);
% =========================================================
% Right part: Policy network
% =========================================================
% input layer
\foreach \i/\x in {1/9.3,2/10.1,3/10.9,4/11.7}
{
\node[neuron, fill=gray!25, minimum size=4mm] (I\i) at (\x,-1.15) {};
}
% hidden layer
\foreach \i/\x in {1/9.1,2/9.8,3/10.5,4/11.2,5/11.9}
{
\node[neuron, fill=blue!30, minimum size=4mm] (H\i) at (\x,0.05) {};
}
% output layer
\foreach \i/\x in {2/9.8,4/11.2}
{
\node[neuron, fill=yellow!45, minimum size=4mm] (O\i) at (\x,1.35) {};
}
% connections: input -> hidden
\foreach \i in {1,2,3,4}
{
\foreach \j in {1,2,3,4,5}
{
\draw[conn] (I\i) -- (H\j);
}
}
% connections: hidden -> output
\foreach \i in {1,2,3,4,5}
{
\foreach \j in {2,4}
{
\draw[conn] (H\i) -- (O\j);
}
}
% arrows below input layer
\foreach \i in {1,2,3,4}
{
\draw [->] ($(I\i.south)+(0,-0.36)$) -- ($(I\i.south)+(0,-0.02)$);
}
% \draw [->] ([yshift=0.2cm]\x.center) -- ([yshift=0.55cm]\x.center);
% arrows above output layer
\foreach \i in {2,4}
{
\draw[->] ($(O\i.north)+(0,0.02)$) -- ($(O\i.north)+(0,0.42)$);
}
% dashed rounded frame around policy net
\node[
draw=gray!70,
dashed,
rounded corners=5pt,
minimum width=5.0cm,
minimum height=4.0cm,
inner xsep=0.85cm,
inner ysep=1.00cm
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% labels
\node[anchor=west] at (10.6, -2.2) {Policy};
\node[anchor=center, font=\small] at ($(policybox.east)+(-0.9, 1.15)$) {Output};
\node[anchor=center, font=\small] at ($(policybox.east)+(-0.9, -0.1)$) {Hidden};
\node[anchor=center, font=\small] at ($(policybox.east)+(-0.9, -1.4)$) {Input};
\end{tikzpicture}
\end{center}
\ No newline at end of file
section2/section2.tex
\section{Preliminary}
In this section, we introduce the fundamentals of supervised fine-tuning for LLMs and focus on presenting the key concepts and nations necessary for the discussions in future sections.
\subsection{Fundamentals of Large Language Models}
In this subsection, we introduce the fundamentals of supervised fine-tuning for LLMs and focus on presenting the key concepts and nations necessary for the discussions in future sections.
Although pre-trained LLMs possess a vast amount of general knowledge, they are typically limited to tasks related to language modeling. Our ultimate goal is to enable them to perform various specific tasks, such as summarization, question-answer, and machine translation.
One straightforward method to achieve this goal is supervised fine-tuning (SFT), in which the pre-trained LLM is further trained on a dataset comprising task-specific input (i.e., instruction+user input)\footnote{In this paper, the \textit{instruction} represents the description of a specific task, e.g., ``Summarize the following article''; the \textit{user input} represents the extra content to the instruction, e.g., ``Article: In recent years, solar energy has seen a significant increase in the amount of energy available to the public. recent years, solar energy has seen unprecedented growth, becoming the fastest-growing ...''} paired with their expected outputs \citep{ouyang:2022training,wei-etal:2022finetuned}.
......@@ -21,10 +23,67 @@ The objective function $\log \mathrm{Pr}_{\theta}(y_i|\mathbf{x},\mathbf{y}_{<i}
This formulation is equivalent to minimizing the cross-entropy loss. In the following section, we refer to the LLM after SFT as the ``SFT LLM'' for short. Beyond SFT, the model architecture design and the pre-training process are also fundamentals of building an LLM. However, we will not discuss these topics. Interested readers can find further details in \citet{xiao-and-zhu:2025foundations}'s book.
%%%%%%%%%% Preliminary修改建议 %%%%%%%%%%
% 1. 再加一些强化学习的基础知识
% 包含对环境、动作、policy这些术语的定义，方面后面对应起来的好定义。
% 如果修改这一章节 那么3.1这部分要对应起来，有些需要删除掉。
\subsection{Fundamentals of Reinforcement Learning}
RL, supported by a well-established theoretical framework, has been widely applied across a broad range of domains. In particular, with the rapid advancement of LLMs, RL has become a standard approach in the post-training stage. Before exploring the interplay between RL and LLMs, we first provide a brief overview of deep reinforcement learning in this subsection. We then introduce the general formulation of RL, along with key terminologies and notations that are essential for understanding RL.
\subsubsection{General Framework of Reinforcement Learning}
The general framework of RL is illustrated in Figure~\ref{fig:rl_framework}. An RL system primarily consists of two components: an agent and an environment. At each timestep $t$, the agent observes a state $s_t$ from the environment and selects an action $a_t$ according to a policy $\pi$, which is typically parameterized by a neural network. After executing the action, the environment transitions to a new state $s_{t+1}$ and returns a reward $r_t$ corresponding to the taken action.
\begin{figure}[!t]
\centering
\input{section2/Figures/rl_framework}
\caption{
A general framework of deep reinforcement learning. The policy is parameterized by neural networks, and the agent learns an optimal policy through interactions with the environment to maximize cumulative rewards.
}
\label{fig:rl_framework}
\end{figure}
This interaction process is commonly modeled as a Markov Decision Process (MDP), where the agent interacts with the environment over discrete timesteps \citep{Sutton-and-Barto:2018RL}. A sequence of states and actions forms a trajectory, denoted as $\tau = (s_0, a_0, s_1, a_1, \cdots, s_{H-1}, a_{H-1})$, where $H$ represents the trajectory length. Each trajectory accumulates rewards from the environment.
The objective of RL is to learn an optimal policy $\pi^*$ that maximizes the expected cumulative reward over trajectories, which can be formulated as:
\begin{eqnarray}
\pi^{*} = \mathop{\arg\max} \limits_{\pi} \mathbb{E}\left[\sum_{t=0}^{H-1}r_{t} \mid \pi \right]
\end{eqnarray}
In practice, the agent improves its policy through repeated interactions with the environment by observing states and receiving feedback in the form of rewards. Each such interaction sequence constitutes a trajectory $\tau$. The process of generating one or more trajectories under a given policy is commonly referred to as \textit{sampling} in the RL literature.
\subsubsection{Key Elements}
To better understand the application of reinforcement learning to large language models (LLMs), we summarize the key elements of the RL framework and reinterpret them in the context of language modeling.
\begin{itemize}
\item \textbf{Agent.}
The agent is the learner or decision-maker in reinforcement learning. In the context of LLMs, the agent corresponds to the language model itself, which generates tokens sequentially and updates its behavior based on feedback signals.
\item \textbf{Environment.}
The environment comprises everything external to the agent with which it interacts. Unlike traditional RL settings that involve physical or simulated environments, the environment in LLM-based RL is typically abstract, consisting of the training framework that provides feedback (e.g., reward models, human annotations, or evaluation metrics) for generated outputs.
\item \textbf{State ($S$).}
A state represents the current situation of the environment. For language modeling, the state at timestep $t$ can be defined as the sequence of observed tokens up to that point, i.e., the context used to predict the next token. Formally, the state can be represented as $S = (x, y_{<t})$, where $x$ denotes the input prompt and $y_{<t}$ denotes the previously generated tokens.
\item \textbf{Action ($a$).}
An action corresponds to a decision made by the agent. In LLMs, actions are naturally defined as selecting the next token from the vocabulary, i.e., $a = y_t$.
\item \textbf{Reward ($r$).}
The reward provides feedback from the environment to evaluate the quality of an action. In general, the reward function can be defined as $r(s, a, s')$, representing the feedback received when the agent transitions from state $s$ to $s'$ by taking action $a$. At timestep $t$, this can be written as $r_t = r(s_t, a_t, s_{t+1})$. In deterministic settings, where the next state is uniquely determined by $(s_t, a_t)$, the reward can be simplified as $r(s_t, a_t)$.
\item \textbf{Policy ($\pi$).}
The policy defines the agent’s behavior, i.e., the probability of taking an action given a state. For LLMs, the policy corresponds to the conditional probability distribution over the next token given the context:
\begin{equation}
\pi(a \mid s) = \Pr(y_t \mid x, y_{<t})
\end{equation}
where $a = y_t$ and $s = (x, y_{<t})$. Under this formulation, an LLM can be naturally interpreted as a parameterized policy.
\item \textbf{Value Function ($V$ and $Q$).}
The value function estimates the expected cumulative reward when following a policy. The state-value function $V(s)$ measures the expected discounted return starting from state $s$:
\begin{equation}
V(s) = \mathbb{E} \left[ \sum_{t=0}^{\infty} \gamma^t r_t \middle| s_0 = s, \pi \right]
\end{equation}
where $\gamma \in [0,1]$ is the discount factor. The action-value function $Q(s,a)$ further conditions on the initial action:
\begin{equation}
Q(s,a) = \mathbb{E} \left[ \sum_{t=0}^{\infty} \gamma^t r_t \middle| s_0 = s, a_0 = a, \pi \right]
\end{equation}
\end{itemize}
By this point, the reader should have developed a foundational understanding of the core concepts and notations in RL. In contrast to conventional RL literature, which typically presents algorithms in the context of classical robotics or control tasks, \textit{we adopt an NLP perspective to introduce RL in the following section}.
section3/section3.tex
......@@ -3,7 +3,7 @@
\section{An Example of Using Reinforcement Learning to Train LLMs}
\label{sec:example-using-rl-training-llms}
To explain how RL can be applied to train LLMs, we begin by considering a practical scenario: using an LLM as a homework assistant. In this context, we aim to improve the capability of an SFT LLM to handle education-related inputs more effectively. Suppose we have a homework assistant powered by an SFT LLM, and a student types the input ``Give me three tips to improve my accuracy in solving math problems.'' A typical SFT LLM might generate a short output, such as
We begin by considering a practical scenario: using an LLM as a homework assistant. In this context, we aim to improve the capability of an SFT LLM to handle education-related inputs more effectively. Suppose we have a homework assistant powered by an SFT LLM, and a student types the input ``Give me three tips to improve my accuracy in solving math problems.'' A typical SFT LLM might generate a short output, such as
\vspace{0.1cm}
\begin{tcolorbox}[frame empty]
......@@ -23,7 +23,7 @@ There are three tips for improving accuracy in solving math problems: \\ [1mm]
Although this output adheres to the given input, it is very short and not sufficiently detailed or comprehensive for this scenario. In practice, the ``short'' feature in the generated output stems from the training approach of the SFT LLM. During SFT, the LLM learns from a large set of labeled samples that typically contain concise and factual answers to common inputs. These samples guide the LLM to prioritize brevity and clarity, so it often generates short outputs, like the one shown above. Of course, we could annotate enough additional data to fine-tune the LLM further and adjust it to generate the desired outputs. However, this approach is limited in its ability to scale. For example, in this scenario, the input involves a variety of potential answers, and the expectations of the student can be pretty diverse. Therefore, describing the ``ideal'' output would require immense annotation effort. Consequently, collecting or annotating fine-tuning data is not as straightforward as it is with SFT, particularly when attempting to cover the breadth of possible student inputs and outputs. Instead, we can use RL to enable the model to discern outputs that better align with human preferences, such as generating more detailed and comprehensive content that not only adheres to the given input but also meets the expectations of the student in this scenario.
In the following sections, we will demonstrate how to use RL to train LLMs through a specific example—enabling the LLM to generate a longer output in the context of a homework assistant. Note that the techniques discussed are not limited to this single case. Rather, they can be broadly applied to align LLM with any set of expectations.
In the following sections, we will demonstrate how to use RL to train LLMs through a specific example—enabling the LLM to generate a longer output in the context of a homework assistant. Throughout this example, we introduce RL, including key algorithms and their improvements.
% policy gradient
\subsection{Policy Gradient}
......
section4/section4.tex
......@@ -151,7 +151,7 @@ where $\alpha$ is a balancing factor, we further illustrate the freezing paramet
\subsubsection{Generative Reward Models}
\label{sec:generative-reward-models}
Reward models are typically trained as discriminative models to assign numerical rewards to outputs and classify them as preferred or dispreferred. However, this method does not leverage the text-generation capabilities for which LLMs are fundamentally designed. For example, the discriminative reward model can not perform CoT reasoning. To address this, an LLM can alternatively be employed as a reward model, thus endowing it with the ability to engage in text generation and reasoning, as depicted in Figure \ref{fig:generative-reward-model-architecture}. This model works as follows. First, we input a prompt $\mathbf{c}$, along with the tuple $(\mathbf{x},\mathbf{y}_{a},\mathbf{y}_{b})$, to the LLM. The prompt is a description of the task, as demonstrated in the example below.
Reward models are typically trained as discriminative models to assign numerical rewards to outputs and classify them as preferred or dispreferred. However, this method does not leverage the text-generation capabilities for which LLMs are fundamentally designed \citep{zhang-etal:zhang2024generative,wang-etal:wang2025gram}. For example, the discriminative reward model can not perform CoT reasoning. To address this, an LLM can alternatively be employed as a reward model, thus endowing it with the ability to engage in text generation and reasoning, as depicted in Figure \ref{fig:generative-reward-model-architecture}. This model works as follows. First, we input a prompt $\mathbf{c}$, along with the tuple $(\mathbf{x},\mathbf{y}_{a},\mathbf{y}_{b})$, to the LLM. The prompt is a description of the task, as demonstrated in the example below.
\vspace{0.5em}
\begin{tcolorbox}[frame empty]
......@@ -187,7 +187,7 @@ r_{\phi}(\mathbf{x}', \mathbf{y}') & = & \frac{\mathrm{Pr}_{\theta}(w=\text{A}|\
where the reward ranges from 0 to 1.
To further improve the generative reward model, we can label the relevant explanation to enable the generative reward model to produce a CoT rationale \citep{zhang-etal:2024generative}. In this case, we use a prompt $\mathbf{c}$ with a CoT rationale generation instruction, as shown below.
To further improve the generative reward model, we can label the relevant explanation to enable the generative reward model to produce a CoT rationale \citep{zhang-etal:2024generative,wang-etal:wang2026gram}. In this case, we use a prompt $\mathbf{c}$ with a CoT rationale generation instruction, as shown below.
\vspace{0.5em}
\begin{tcolorbox}[frame empty]
......
section6/section6.tex
\section{Reinforcement Learning for Multimodal Models}
Multimodal learning involves models that process and relate information from multiple data modalities (e.g. vision, text, speech), enabling more comprehensive understanding than single-modality systems. By combining modalities, models can make more robust predictions and capture complementary information that one modality alone might miss. Such multimodal approaches have benefits across various applications, such as image caption and image generation. Given these advantages, multimodal learning has become increasingly significant as AI systems aim to perceive and reason more like humans, who naturally integrate sight, sound, and language \citep{baltruvsaitis-etal:2018multimodal}.
In the era of LLM, we can extend their capabilities into the multimodal domain by training them with diverse data types. For example, we can develop a visual language model by training an LLM with image-text pairs using SFT. Here, we return to the issue of aligning models with human preferences. Ideally, once aligned with human preferences through RL, the LLM would also generalize the target modality well by the SFT. However, the reality does not always meet expectations. Although an LLM may align well with human preferences in one aspect, it often struggles to generalize this alignment across a different modality. Thus, to align multimodal models with human preferences, we perform RL to enhance their performance further within the specific modality.
In this section, we use the visual language, speech generation, and diffusion models to discuss the application of RL in multimodal models.
\subsection{Visual Language Models}
While RL is commonly used to train LLMs, its application to other domains has been a prominent research topic. In multimodal language models\footnote{A multimodal language model is defined as a model that integrates an LLM with a multimodal encoder, such as CLIP \citep{radford-etal:2021learning}, allowing the LLM to process inputs beyond text, such as images. Recent literature has also introduced the use of LLMs for generating outputs in non-text modalities, such as images and speech \citep{xu-etal:2025qwen2,zhang-etal:2024mm}. However, in this section, we focus on the former definition of multimodal language models.}, for example, a notable trend is to perform RL training to improve their trustworthiness and helpfulness. This section considers Visual Language Models (VLMs), which connect a visual encoder to an LLM through a linear projector, facilitating general-purpose visual and language understanding. VLMs are currently the most explored extension in multimodal language research, and they form the foundation for many open-source multimodal language models, such as Qwen2.5-VL \citep{qwenTeam:2025qwen2.5-VL} and LLaMA-3.2-11B-Vision \citep{grattafiori-etal:2024llama}.
Training LLMs and VLMs with RL exhibits only minimal differences, primarily related to the input content. Unlike the textual input used for LLMs, the input for VLMs typically comprises a combination of one or multiple images and an instruction, denoted by $(\mathrm{\mathbf{I}}, \mathrm{\mathbf{x}})$, where $\mathrm{\mathbf{I}}$ represents the input images. These images are encoded into representations that are either concatenated with instruction embeddings or integrated through cross-attention mechanisms into the LLM. In practice, this subtle difference does not significantly affect the applicability of RL algorithms to VLMs. As a result, the RL training process for LLMs, as described in Section \ref{sec:example-using-rl-training-llms}, can be seamlessly adapted to train VLMs without major improvements \citep{yu-etal:2024rlhf,wang-etal:2024rovrm,zang2025internlm,ji2025safe}.
\begin{figure*}[!t]
\centering
\caption{RoVRM}
\label{fig:preference-transfer}
\end{figure*}
However, training VLMs with RL is not a low-hanging fruit in practical applications. This is because it typically encounters the challenge of training a visual reward model due to the scarcity of high-quality visual preference data. One straightforward approach to address this issue is to generate visual preference data through the automatic preference data generation method described in Section \ref{sec:automatic-preference-data-generation} \citep{yu-etal:2024rlaif}. Another alternative is a multi-stage training approach for the visual reward model, motivated by a simple idea: human preferences are well captured in text, and these preferences can be transferred across modalities \citep{wang-etal:2024rovrm}. By leveraging textual preference data, this approach reduces the dependence on visual preference data in training a visual reward model. More specifically, as illustrated in Figure \ref{fig:preference-transfer}, we can train a visual reward model in the following three stages:
\begin{itemize}
\item Stage 1: pre-training with large-scale textual preference data. Given the transferability of human preferences across different modalities, we can begin by using large-scale textual preference data to pre-train the visual reward model. Note that the projector parameters are frozen without images. This stage can be considered as providing a stronger starting point for training the visual reward model, as it enables the model to pre-learn general human preferences.
\item Stage 2: fine-tuning with image caption-based preference data. Pre-learned human preferences cannot be directly applied to vision tasks due to both \textit{task gap} and \textit{modality gap}. To bridge the task gap, we fine-tune the reward model using image caption-based preference data in this stage. The rationale behind this approach is that general textual preference data does not cover vision-specific tasks, such as ``Please describe the content in this image''. We use such data to fine-tune the model, adapting it to vision-specific tasks. The projector parameters are frozen at this stage.
\item Stage 3: fine-tuning with small-scale visual preference data. To further bridge the modality gap, we use visual preference data in the final stage to train the model. Note that in this phase, we also train the projector parameters.
\end{itemize}
In this process, not all preference data may align with the preferences used in subsequent phases, potentially leading to preference conflicts. We can enhance the visual reward model through data selection techniques, such as LESS \citep{xia-etal:2024less} to address this. In fact, preference transfer is effective across modalities and has also been shown to work across different tasks and languages \citep{cheng-etal:2023everyone,wu-etal:2024reuse}. Interested readers can refer to these papers for more detailed discussions of these topics.
Apart from preference data, another approach to improving the visual model is to integrate additional image content, such as image captions, into the reward model \citep{sun-etal:2023aligning}. This approach aims to achieve factually augmented reward prediction, addressing reward hacking. Specifically, in the original setup, the reward model predicts a reward based solely on the image, input, and output; that is, the reward model’s input is $[\mathbf{I}, \mathbf{x}, \mathbf{y}]$. In the factually augmented setup, the reward model also receives additional input in the form of the textual image caption $\mathbf{C}$, resulting in an input of $[\mathbf{I}, \mathbf{C}, \mathbf{x}, \mathbf{y}]$. The basic idea is that the backbone of the visual reward model remains a well-trained LLM with an in-context learning ability. With this ability, we can provide additional content to help the model predict rewards more accurately.
While our discussion primarily focuses on models for visual inputs in the context of visual language models, the RL techniques are adaptable across inputs of various modalities, such as video and audio.
% visual language model as reward model
% 视觉奖励模型训练大致思路
% image reward
% 融合文本数据进行提升
% 视觉奖励模型应用场景---对齐图生文模型 文生图评估模型
% 视觉语言模型的定义
% 框架
% Rlhf-v: Towards trustworthy mllms via behavior alignment from fine-grained correctional human feedback
% Aligning Large Multimodal Models with Factually Augmented RLHF
% 偏好数据
% RLAIF-V
% Silkie: Preference Distillation for Large Visual Language Models
% RoVRM
\subsection{Speech Generation Models}
% SpeechAlign: Aligning Speech Generation to Human Preferences
\subsection{Diffusion Models}
% Image Reward
% 为什么要进行强化学习
% 大体训练思路和流程
% DDPO (denoising diffusion preference optimization model)
\section{Reinforcement Learning for LLM-based Agents}
\ No newline at end of file
section7/section7.tex
\section{Summary}
% future work
% 强化学习作为一种预训练方式
% 高效强化学习方法
% 全模态RL
\ No newline at end of file
\section{Reinforcement Learning for Multimodal Models}
Multimodal learning involves models that process and relate information from multiple data modalities (e.g. vision, text, speech), enabling more comprehensive understanding than single-modality systems. By combining modalities, models can make more robust predictions and capture complementary information that one modality alone might miss. Such multimodal approaches have benefits across various applications, such as image caption and image generation. Given these advantages, multimodal learning has become increasingly significant as AI systems aim to perceive and reason more like humans, who naturally integrate sight, sound, and language \citep{baltruvsaitis-etal:2018multimodal}.
In the era of LLM, we can extend their capabilities into the multimodal domain by training them with diverse data types. For example, we can develop a visual language model by training an LLM with image-text pairs using SFT. Here, we return to the issue of aligning models with human preferences. Ideally, once aligned with human preferences through RL, the LLM would also generalize the target modality well by the SFT. However, the reality does not always meet expectations. Although an LLM may align well with human preferences in one aspect, it often struggles to generalize this alignment across a different modality. Thus, to align multimodal models with human preferences, we perform RL to enhance their performance further within the specific modality.
In this section, we use the visual language, speech generation, and diffusion models to discuss the application of RL in multimodal models.
\subsection{Visual Language Models}
While RL is commonly used to train LLMs, its application to other domains has been a prominent research topic. In multimodal language models\footnote{A multimodal language model is defined as a model that integrates an LLM with a multimodal encoder, such as CLIP \citep{radford-etal:2021learning}, allowing the LLM to process inputs beyond text, such as images. Recent literature has also introduced the use of LLMs for generating outputs in non-text modalities, such as images and speech \citep{xu-etal:2025qwen2,zhang-etal:2024mm}. However, in this section, we focus on the former definition of multimodal language models.}, for example, a notable trend is to perform RL training to improve their trustworthiness and helpfulness. This section considers Visual Language Models (VLMs), which connect a visual encoder to an LLM through a linear projector, facilitating general-purpose visual and language understanding. VLMs are currently the most explored extension in multimodal language research, and they form the foundation for many open-source multimodal language models, such as Qwen2.5-VL \citep{qwenTeam:2025qwen2.5-VL} and LLaMA-3.2-11B-Vision \citep{grattafiori-etal:2024llama}.
Training LLMs and VLMs with RL exhibits only minimal differences, primarily related to the input content. Unlike the textual input used for LLMs, the input for VLMs typically comprises a combination of one or multiple images and an instruction, denoted by $(\mathrm{\mathbf{I}}, \mathrm{\mathbf{x}})$, where $\mathrm{\mathbf{I}}$ represents the input images. These images are encoded into representations that are either concatenated with instruction embeddings or integrated through cross-attention mechanisms into the LLM. In practice, this subtle difference does not significantly affect the applicability of RL algorithms to VLMs. As a result, the RL training process for LLMs, as described in Section \ref{sec:example-using-rl-training-llms}, can be seamlessly adapted to train VLMs without major improvements \citep{yu-etal:2024rlhf,wang-etal:2024rovrm,zang2025internlm,ji2025safe}.
However, training VLMs with RL is not a low-hanging fruit in practical applications. This is because it typically encounters the challenge of training a visual reward model due to the scarcity of high-quality visual preference data. One straightforward approach to address this issue is to generate visual preference data through the automatic preference data generation method described in Section \ref{sec:automatic-preference-data-generation} \citep{yu-etal:2024rlaif}. Another alternative is a multi-stage training approach for the visual reward model, motivated by a simple idea: human preferences are well captured in text, and these preferences can be transferred across modalities \citep{wang-etal:2024rovrm}. By leveraging textual preference data, this approach reduces the dependence on visual preference data in training a visual reward model. More specifically, as illustrated in Figure \ref{fig:preference-transfer}, we can train a visual reward model in the following three stages:
\begin{itemize}
\item Stage 1: pre-training with large-scale textual preference data. Given the transferability of human preferences across different modalities, we can begin by using large-scale textual preference data to pre-train the visual reward model. Note that the projector parameters are frozen without images. This stage can be considered as providing a stronger starting point for training the visual reward model, as it enables the model to pre-learn general human preferences.
\item Stage 2: fine-tuning with image caption-based preference data. Pre-learned human preferences cannot be directly applied to vision tasks due to both \textit{task gap} and \textit{modality gap}. To bridge the task gap, we fine-tune the reward model using image caption-based preference data in this stage. The rationale behind this approach is that general textual preference data does not cover vision-specific tasks, such as ``Please describe the content in this image''. We use such data to fine-tune the model, adapting it to vision-specific tasks. The projector parameters are frozen at this stage.
\item Stage 3: fine-tuning with small-scale visual preference data. To further bridge the modality gap, we use visual preference data in the final stage to train the model. Note that in this phase, we also train the projector parameters.
\end{itemize}
In this process, not all preference data may align with the preferences used in subsequent phases, potentially leading to preference conflicts. We can enhance the visual reward model through data selection techniques, such as LESS \citep{xia-etal:2024less} to address this. In fact, preference transfer is effective across modalities and has also been shown to work across different tasks and languages \citep{cheng-etal:2023everyone,wu-etal:2024reuse}. Interested readers can refer to these papers for more detailed discussions of these topics.
As discussed in Section~\ref{sec:generative-reward-models}, reward reasoning models have demonstrated superior performance in reward prediction. A natural question then arises: \textit{can reward reasoning capabilities also be transferred from text to multimodal settings?} Recent work by \cite{wang2026msrl} provides empirical evidence supporting this hypothesis. By following a similar multi-stage training paradigm, they show that reward reasoning capabilities can indeed be effectively transferred across modalities, further enhancing the performance of visual reward models.
\begin{figure*}[!t]
\centering
\caption{RoVRM}
\label{fig:preference-transfer}
\end{figure*}
Apart from preference data, another approach to improving the visual model is to integrate additional image content, such as image captions, into the reward model \citep{sun-etal:2023aligning}. This approach aims to achieve factually augmented reward prediction, addressing reward hacking. Specifically, in the original setup, the reward model predicts a reward based solely on the image, input, and output; that is, the reward model’s input is $[\mathbf{I}, \mathbf{x}, \mathbf{y}]$. In the factually augmented setup, the reward model also receives additional input in the form of the textual image caption $\mathbf{C}$, resulting in an input of $[\mathbf{I}, \mathbf{C}, \mathbf{x}, \mathbf{y}]$. The basic idea is that the backbone of the visual reward model remains a well-trained LLM with an in-context learning ability. With this ability, we can provide additional content to help the model predict rewards more accurately.
While our discussion primarily focuses on models for visual inputs in the context of visual language models, the RL techniques are adaptable across inputs of various modalities, such as video and audio.
% visual language model as reward model
% 视觉奖励模型训练大致思路
% image reward
% 融合文本数据进行提升
% 视觉奖励模型应用场景---对齐图生文模型 文生图评估模型
% 视觉语言模型的定义
% 框架
% Rlhf-v: Towards trustworthy mllms via behavior alignment from fine-grained correctional human feedback
% Aligning Large Multimodal Models with Factually Augmented RLHF
% 偏好数据
% RLAIF-V
% Silkie: Preference Distillation for Large Visual Language Models
% RoVRM
\subsection{Speech Generation Models}
% SpeechAlign: Aligning Speech Generation to Human Preferences
\subsection{Diffusion Models}
% Image Reward
% 为什么要进行强化学习
% 大体训练思路和流程
% DDPO (denoising diffusion preference optimization model)
section8/section8.tex
\clearpage
\section{Systems and Datasets}
\section{Summary}
\begin{table}[h]
\centering
\scalebox{0.88}{
\input{section8/tables/dataset}}
\caption{datasets}
\label{tab:dataset}
\end{table}
\begin{table}[h]
\centering
\scalebox{0.88}{
\input{section8/tables/systems}}
\caption{systems}
\label{tab:system}
\end{table}
\ No newline at end of file
% future work
% 强化学习作为一种预训练方式
% 高效强化学习方法
% 全模态RL
\ No newline at end of file
section9/section9.tex 0 → 100644
\clearpage
\section{Systems and Datasets}
\begin{table}[h]
\centering
\scalebox{0.88}{
\input{section9/tables/dataset}}
\caption{datasets}
\label{tab:dataset}
\end{table}
\begin{table}[h]
\centering
\scalebox{0.88}{
\input{section9/tables/systems}}
\caption{systems}
\label{tab:system}
\end{table}
\ No newline at end of file
section9/tables/dataset.tex 0 → 100644
\begin{tabular}{lccccc}
\toprule[1.1pt]
\multirow{2}{*}{Dataset Name} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Sample\\Size\end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Response\\Size\end{tabular}} & \multirow{2}{*}{Modality} & \multicolumn{2}{c}{Feedback} \\ \cmidrule(l){5-6}
& & & & Source & Category \\ \midrule
\href{https://url}{HuggingFaceH4/stack-exchange-preferences}
&10,000 &1 & \faIcon{file-alt} & Human & Score \\
\href{https://url}{Skywork/Skywork-Reward-Preference-80K-v0.2} &10,000 &2 & \faIcon{images} & AI & Ranking \\
\href{https://huggingface.co/datasets/openbmb/UltraFeedback}{openbmb/UltraFeedback} &10,000 &3 & \faIcon{video} & Rule & \\
&10,000 &4 & \faIcon{volume-up} & & \\
& & &\faIcon{file-alt} \faIcon{images} & & \\
& & & & & \\
& & & & & \\
& & & & & \\
\bottomrule[1.1pt]
\end{tabular}
\ No newline at end of file
section9/tables/systems.tex 0 → 100644
\begin{tabular}{lccccc}
\toprule[1.1pt]
\multirow{2}{*}{System Name} & \multicolumn{4}{c}{Supported Modality} & \multirow{2}{*}{Supported Training Approaches} \\ \cmidrule(r){2-5}
& Text & Image & Video & Audio & \\ \midrule
\href{https://github.com/huggingface/trl}{TRL} & \CheckmarkBold & \CheckmarkBold &\XSolidBrush & \CheckmarkBold & Reward Modeling, PPO, GRPO, DPO, Online-DPO, etc. \\
\href{https://github.com/OpenRLHF/OpenRLHF}{OpenRLHF}& \CheckmarkBold & \XSolidBrush & \XSolidBrush & \XSolidBrush & Sft, Reject Sampling, PPO, GRPO, DPO, KTO, etc.\\
\href{https://github.com/hiyouga/EasyR1}{EasyR1}& \CheckmarkBold & \CheckmarkBold & \XSolidBrush & \XSolidBrush & GRPO, Reinforce++, Remax, RLOO, etc. \\
\href{https://github.com/volcengine/verl}{veRL}& \CheckmarkBold & \XSolidBrush & \XSolidBrush & \XSolidBrush & GRPO, PPO, Remax, RLOO, SFT, etc. \\
\href{https://github.com/OpenRLHF/OpenRLHF-M}{OpenRLHF-M}& \XSolidBrush & \CheckmarkBold & \XSolidBrush & \XSolidBrush & PPO, GRPO, RLOO, Online-RLHF, Reject-Sampling, etc. \\
\href{https://github.com/PKU-Alignment/align-anything}{Align-Anything}& \CheckmarkBold & \CheckmarkBold & \CheckmarkBold & \CheckmarkBold &PPO, GRPO, DPO,KTO, ORPO, etc. \\
\bottomrule[1.1pt]
\end{tabular}
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