
@inproceedings{paschos2019learning,
  title={Learning to Cache With No Regrets},
  author={Paschos, Georgios S and Destounis, Apostolos and Vigneri, Luigi and Iosifidis, George},
  booktitle={IEEE INFOCOM 2019-IEEE Conference on Computer Communications},
  pages={235--243},
  year={2019},
  organization={IEEE}
}

@article{sinha2024tight,
      title={Tight Bounds for Online Convex Optimization with Adversarial Constraints}, 
      author={Abhishek Sinha and Rahul Vaze},
      year={2024},
      journal={arXiv preprint arXiv:2405.09296},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@article{duchi2011adaptive,
  title={Adaptive subgradient methods for online learning and stochastic optimization.},
  author={Duchi, John and Hazan, Elad and Singer, Yoram},
  journal={Journal of machine learning research},
  volume={12},
  number={7},
  year={2011}
}

@software{BanditQ_code,
  author = {Sinha, Abhishek},
  title = {{Implementation of the BanditQ policy}},
  url = {https://github.com/abhishek-sinha-tifr/BanditQ_UAI24},
  version = {1.0},
  year = {2024}
}

@article{sinha2023playing,
  title={Playing in the Dark: No-regret Learning with Adversarial Constraints},
  author={Sinha, Abhishek and Vaze, Rahul},
  journal={arXiv preprint arXiv:2310.18955},
  year={2023}
}

@article{neely2017online,
  title={Online convex optimization with time-varying constraints},
  author={Neely, Michael J and Yu, Hao},
  journal={arXiv preprint arXiv:1702.04783},
  year={2017}
}
@article{yu2017online,
  title={Online convex optimization with stochastic constraints},
  author={Yu, Hao and Neely, Michael and Wei, Xiaohan},
  journal={Advances in Neural Information Processing Systems},
  volume={30},
  year={2017}
}
@article{yuan2018online,
  title={Online convex optimization for cumulative constraints},
  author={Yuan, Jianjun and Lamperski, Andrew},
  journal={Advances in Neural Information Processing Systems},
  volume={31},
  year={2018}
}
@article{castiglioni2022unifying,
  title={A unifying framework for online optimization with long-term constraints},
  author={Castiglioni, Matteo and Celli, Andrea and Marchesi, Alberto and Romano, Giulia and Gatti, Nicola},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={33589--33602},
  year={2022}
}

@article{fu2021fairness,
  title={Fairness of task allocation in crowdsourcing workflows},
  author={Fu, Donglai and Liu, Yanhua},
  journal={Mathematical Problems in Engineering},
  volume={2021},
  pages={1--11},
  year={2021},
  publisher={Hindawi Limited}
}
@article{keilson1977monotone,
  title={Monotone matrices and monotone Markov processes},
  author={Keilson, Julian and Kester, Adri},
  journal={Stochastic Processes and their Applications},
  volume={5},
  number={3},
  pages={231--241},
  year={1977},
  publisher={Elsevier}
}
@article{van1980stochastic,
  title={Stochastic monotonicity of birth--death processes},
  author={Van Doorn, Erik A},
  journal={Advances in Applied Probability},
  volume={12},
  number={1},
  pages={59--80},
  year={1980},
  publisher={Cambridge University Press}
}


@article{bistritz2022queue,
  title={Queue Up Your Regrets: Achieving the Dynamic Capacity Region of Multiplayer Bandits},
  author={Bistritz, Ilai and Bambos, Nicholas},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={837--849},
  year={2022}
}
@book{asmussen2003applied,
  title={Applied probability and queues},
  author={Asmussen, S{\o}ren},
  volume={2},
  year={2003},
  publisher={Springer}
}

@inproceedings{zhao2019stochastic,
  title={Stochastic one-sided full-information bandit},
  author={Zhao, Haoyu and Chen, Wei},
  booktitle={Joint European Conference on Machine Learning and Knowledge Discovery in Databases},
  pages={150--166},
  year={2019},
  organization={Springer}
}

@book{williams1991probability,
  title={Probability with martingales},
  author={Williams, David},
  year={1991},
  publisher={Cambridge university press}
}

@inproceedings{lindley1952theory,
  title={The theory of queues with a single server},
  author={Lindley, David V},
  booktitle={Mathematical Proceedings of the Cambridge Philosophical Society},
  volume={48},
  number={2},
  pages={277--289},
  year={1952},
  organization={Cambridge University Press}
}

@inproceedings{huang2022achieving,
  title={Achieving Counterfactual Fairness for Causal Bandit},
  author={Huang, Wen and Zhang, Lu and Wu, Xintao},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={36},
  number={6},
  pages={6952--6959},
  year={2022}
}
@article{hossain2021fair,
  title={Fair algorithms for multi-agent multi-armed bandits},
  author={Hossain, Safwan and Micha, Evi and Shah, Nisarg},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={24005--24017},
  year={2021}
}
@article{bechavod2020metric,
  title={Metric-free individual fairness in online learning},
  author={Bechavod, Yahav and Jung, Christopher and Wu, Steven Z},
  journal={Advances in neural information processing systems},
  volume={33},
  pages={11214--11225},
  year={2020}
}

@article{gillen2018online,
  title={Online learning with an unknown fairness metric},
  author={Gillen, Stephen and Jung, Christopher and Kearns, Michael and Roth, Aaron},
  journal={Advances in neural information processing systems},
  volume={31},
  year={2018}
}
@book{doob1953stochastic,
  title={Stochastic processes},
  author={Doob, Joseph L},
  year={1953},
  publisher={John Wiley \& Sons}
}
@article{dubins1988sharp,
  title={A sharp inequality for sub-martingales and stopping-times},
  author={Dubins, Lester E and Schwarz, Gideon},
  journal={Ast{\'e}risque},
  volume={157},
  number={158},
  pages={129--145},
  year={1988}
}
@article{bubeck2012regret,
  title={Regret analysis of stochastic and nonstochastic multi-armed bandit problems},
  author={Bubeck, S{\'e}bastien and Cesa-Bianchi, Nicolo and others},
  journal={Foundations and Trends{\textregistered} in Machine Learning},
  volume={5},
  number={1},
  pages={1--122},
  year={2012},
  publisher={Now Publishers, Inc.}
}


@INPROCEEDINGS{queueing2,
  author={Hsu, Wei-Kang and Xu, Jiaming and Lin, Xiaojun and Bell, Mark R.},
  booktitle={2018 Information Theory and Applications Workshop (ITA)}, 
  title={Integrating Online Learning and Adaptive Control in Queueing Systems with Uncertain Payoffs}, 
  year={2018},
  volume={},
  number={},
  pages={1-9},
  doi={10.1109/ITA.2018.8503124}}

@article{sweeney2013discrimination,
  title={Discrimination in online ad delivery},
  author={Sweeney, Latanya},
  journal={Communications of the ACM},
  volume={56},
  number={5},
  pages={44--54},
  year={2013},
  publisher={ACM New York, NY, USA}
}
@inproceedings{celis2019controlling,
  title={Controlling polarization in personalization: An algorithmic framework},
  author={Celis, L Elisa and Kapoor, Sayash and Salehi, Farnood and Vishnoi, Nisheeth},
  booktitle={Proceedings of the conference on fairness, accountability, and transparency},
  pages={160--169},
  year={2019}
}

@inproceedings{cai2018online,
  title={An online learning approach to network application optimization with guarantee},
  author={Cai, Kechao and Liu, Xutong and Chen, Yu-Zhen Janice and Lui, John CS},
  booktitle={IEEE INFOCOM 2018-IEEE Conference on Computer Communications},
  pages={2006--2014},
  year={2018},
  organization={IEEE}
}


@article{li2019combinatorial,
  title={Combinatorial sleeping bandits with fairness constraints},
  author={Li, Fengjiao and Liu, Jia and Ji, Bo},
  journal={IEEE Transactions on Network Science and Engineering},
  volume={7},
  number={3},
  pages={1799--1813},
  year={2019},
  publisher={IEEE}
}

@inproceedings{claure2020multi,
  title={Multi-armed bandits with fairness constraints for distributing resources to human teammates},
  author={Claure, Houston and Chen, Yifang and Modi, Jignesh and Jung, Malte and Nikolaidis, Stefanos},
  booktitle={Proceedings of the 2020 ACM/IEEE International Conference on Human-Robot Interaction},
  pages={299--308},
  year={2020}
}

@misc{grant2011cvx,
  title={CVX: Matlab software for disciplined convex programming},
  author={Grant, Michael and Boyd, Stephen and Ye, Yinyu},
  year={2011}
}

@inproceedings{xia,
author = {Xia, Yingce and Li, Haifang and Qin, Tao and Yu, Nenghai and Liu, Tie-Yan},
title = {Thompson Sampling for Budgeted Multi-Armed Bandits},
year = {2015},
isbn = {9781577357384},
publisher = {AAAI Press},
abstract = {Thompson sampling is one of the earliest randomized algorithms for multi-armed bandits (MAB). In this paper, we extend the Thompson sampling to Budgeted MAB, where there is random cost for pulling an arm and the total cost is constrained by a budget. We start with the case of Bernoulli bandits, in which the random rewards (costs) of an arm are independently sampled from a Bernoulli distribution. To implement the Thompson sampling algorithm in this case, at each round, we sample two numbers from the posterior distributions of the reward and cost for each arm, obtain their ratio, select the arm with the maximum ratio, and then update the posterior distributions. We prove that the distribution-dependent regret bound of this algorithm is O(lnB), where B denotes the budget. By introducing a Bernoulli trial, we further extend this algorithm to the setting that the rewards (costs) are drawn from general distributions, and prove that its regret bound remains almost the same. Our simulation results demonstrate the effectiveness of the proposed algorithm.},
booktitle = {Proceedings of the 24th International Conference on Artificial Intelligence},
pages = {3960–3966},
numpages = {7},
location = {Buenos Aires, Argentina},
series = {IJCAI'15}
}


@article{hazan2018spectral,
  title={Spectral filtering for general linear dynamical systems},
  author={Hazan, Elad and Lee, Holden and Singh, Karan and Zhang, Cyril and Zhang, Yi},
  journal={Advances in Neural Information Processing Systems},
  volume={31},
  year={2018}
}

@article{hazan2017learning,
  title={Learning linear dynamical systems via spectral filtering},
  author={Hazan, Elad and Singh, Karan and Zhang, Cyril},
  journal={Advances in Neural Information Processing Systems},
  volume={30},
  year={2017}
}

@book{lattimore2020bandit,
  title={Bandit algorithms},
  author={Lattimore, Tor and Szepesv{\'a}ri, Csaba},
  year={2020},
  publisher={Cambridge University Press}
}

@book{ross1995stochastic,
  title={Stochastic processes},
  author={Ross, Sheldon M},
  year={1995},
  publisher={John Wiley \& Sons}
}

@article{wang2013projection,
  title={Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application},
  author={Wang, Weiran and Carreira-Perpin{\'a}n, Miguel A},
  journal={arXiv preprint arXiv:1309.1541},
  year={2013}
}

@article{hadiji2023adaptation,
  title={Adaptation to the Range in K--Armed Bandits},
  author={Hadiji, H{\'e}di and Stoltz, Gilles},
  journal={Journal of Machine Learning Research},
  volume={24},
  number={13},
  pages={1--33},
  year={2023}
}

@inproceedings{putta2022scale,
  title={Scale-free adversarial multi armed bandits},
  author={Putta, Sudeep Raja and Agrawal, Shipra},
  booktitle={International Conference on Algorithmic Learning Theory},
  pages={910--930},
  year={2022},
  organization={PMLR}
}


@article{patil2021achieving,
  title={Achieving fairness in the stochastic multi-armed bandit problem},
  author={Patil, Vishakha and Ghalme, Ganesh and Nair, Vineet and Narahari, Yadati},
  journal={The Journal of Machine Learning Research},
  volume={22},
  number={1},
  pages={7885--7915},
  year={2021},
  publisher={JMLRORG}
}

@inproceedings{koolen2015second,
  title={Second-order quantile methods for experts and combinatorial games},
  author={Koolen, Wouter M and Van Erven, Tim},
  booktitle={Conference on Learning Theory},
  pages={1155--1175},
  year={2015},
  organization={PMLR}
}

@article{badanidiyuru2018bandits,
  title={Bandits with knapsacks},
  author={Badanidiyuru, Ashwinkumar and Kleinberg, Robert and Slivkins, Aleksandrs},
  journal={Journal of the ACM (JACM)},
  volume={65},
  number={3},
  pages={1--55},
  year={2018},
  publisher={ACM New York, NY, USA}
}

@article{immorlica2022adversarial,
  title={Adversarial bandits with knapsacks},
  author={Immorlica, Nicole and Sankararaman, Karthik and Schapire, Robert and Slivkins, Aleksandrs},
  journal={Journal of the ACM},
  volume={69},
  number={6},
  pages={1--47},
  year={2022},
  publisher={ACM New York, NY}
}

@article{stolyar2005asymptotic,
  title={On the asymptotic optimality of the gradient scheduling algorithm for multiuser throughput allocation},
  author={Stolyar, Alexander L},
  journal={Operations research},
  volume={53},
  number={1},
  pages={12--25},
  year={2005},
  publisher={INFORMS}
}

@article{andrews2005scheduling,
  title={Scheduling over a time-varying user-dependent channel with applications to high-speed wireless data},
  author={Andrews, Matthew and Zhang, Lisa},
  journal={Journal of the ACM (JACM)},
  volume={52},
  number={5},
  pages={809--834},
  year={2005},
  publisher={ACM New York, NY, USA}
}

@article{neely2010stochastic,
  title={Stochastic network optimization with application to communication and queueing systems},
  author={Neely, Michael J},
  journal={Synthesis Lectures on Communication Networks},
  volume={3},
  number={1},
  pages={1--211},
  year={2010},
  publisher={Morgan \& Claypool Publishers}
}

@article{huang2023queue,
  title={Queue Scheduling with Adversarial Bandit Learning},
  author={Huang, Jiatai and Golubchik, Leana and Huang, Longbo},
  journal={arXiv preprint arXiv:2303.01745},
  year={2023}
}

@article{wangbandit,
  title={Bandit Learning for Proportionally Fair Allocations},
  author={Wang, Tianyu and Rudin, Cynthia}
}
@article{alex,
author = {Talebi, Mohammad Sadegh and Proutiere, Alexandre},
title = {Learning Proportionally Fair Allocations with Low Regret},
year = {2018},
issue_date = {June 2018},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {2},
number = {2},
url = {https://doi.org/10.1145/3224431},
doi = {10.1145/3224431},
abstract = {This paper addresses a generic sequential resource allocation problem, where in each round a decision maker selects an allocation of resources (servers) to a set of tasks consisting of a large number of jobs. A job of task i assigned to server j is successfully treated with probability θ_ij $ in a round, and the decision maker is informed on whether this job is completed at the end of the round. The probabilities θ_ij $'s are initially unknown and have to be learned. The objective of the decision maker is to sequentially assign jobs of various tasks to servers so that it rapidly learns and converges to the Proportionally Fair (PF) allocation (or other similar allocations achieving an appropriate trade-off between efficiency and fairness). We formulate the problem as a multi-armed bandit (MAB) optimization problem, and devise sequential assignment algorithms with low regret (defined as the difference in utility achieved by an oracle algorithm aware of the θ_ij $'s and by the proposed algorithm over a given number of slots). We first provide the properties of the so-called Restricted-PF (RPF) allocation, obtained by assuming that each task can only use a single server, and in particular show that it is very close to the PF allocation. We devise ES-RPF, an algorithm that learns the RPF allocation with regret no greater than $mathcal O bigl(m^3\o{}ver θ_min Δ_min \l{}og(T)big)$ after T slots, where m , θ_min $, and Δ_min $ represent the number of tasks, the minimum success rate $min_i,j θ_ij $, and an appropriately defined notion of gap, respectively. We further provide regret lower bounds satisfied by any algorithm targeting the RPF allocation. Finally, we present ES-PF, an algorithm directly learning the PF allocation, and prove that its regret does not exceed $mathcal O bigl(fracm^2s θ_min sqrtT \l{}og(T)big)$ after T slots, where s denotes the number of servers.},
journal = {Proc. ACM Meas. Anal. Comput. Syst.},
month = {jun},
articleno = {36},
numpages = {31},
keywords = {stochastic multi-armed bandits, regret, resource allocation, proportional fairness}
}

@article{andrews2005scheduling,
  title={Scheduling over a time-varying user-dependent channel with applications to high-speed wireless data},
  author={Andrews, Matthew and Zhang, Lisa},
  journal={Journal of the ACM (JACM)},
  volume={52},
  number={5},
  pages={809--834},
  year={2005},
  publisher={ACM New York, NY, USA}
}

@inproceedings{chen2020fair,
  title={Fair contextual multi-armed bandits: Theory and experiments},
  author={Chen, Yifang and Cuellar, Alex and Luo, Haipeng and Modi, Jignesh and Nemlekar, Heramb and Nikolaidis, Stefanos},
  booktitle={Conference on Uncertainty in Artificial Intelligence},
  pages={181--190},
  year={2020},
  organization={PMLR}
}

@inproceedings{nofra,
 author = {Sinha, Abhishek and Joshi, Ativ and Bhattacharjee, Rajarshi and Musco, Cameron and Hajiesmaili, Mohammad},
 booktitle = {Advances in Neural Information Processing Systems},
 pages = {48083--48109},
 publisher = {Curran Associates, Inc.},
 title = {No-regret Algorithms for Fair Resource Allocation},
 url = {https://proceedings.neurips.cc/paper_files/paper/2023/file/96842011407c2691ab4eefff48fc864d-Paper-Conference.pdf},
 volume = {36},
 year = {2023}
}


@article{sinha2024no,
  title={No-regret Algorithms for Fair Resource Allocation},
  author={Sinha, Abhishek and Joshi, Ativ and Bhattacharjee, Rajarshi and Musco, Cameron and Hajiesmaili, Mohammad},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}

@article{sinha2023no,
  title={No-regret Algorithms for Fair Resource Allocation},
  author={Sinha, Abhishek and Joshi, Ativ and Bhattacharjee, Rajarshi and Musco, Cameron and Hajiesmaili, Mohammad},
  journal={arXiv preprint arXiv:2303.06396},
  year={2023}
}

@inproceedings{mandelli2019satisfying,
  title={Satisfying network slicing constraints via 5G MAC scheduling},
  author={Mandelli, Silvio and Andrews, Matthew and Borst, Sem and Klein, Siegfried},
  booktitle={IEEE INFOCOM 2019-IEEE Conference on Computer Communications},
  pages={2332--2340},
  year={2019},
  organization={IEEE}
}

@inproceedings{chen2020fair,
  title={Fair contextual multi-armed bandits: Theory and experiments},
  author={Chen, Yifang and Cuellar, Alex and Luo, Haipeng and Modi, Jignesh and Nemlekar, Heramb and Nikolaidis, Stefanos},
  booktitle={Conference on Uncertainty in Artificial Intelligence},
  pages={181--190},
  year={2020},
  organization={PMLR}
}
@article{li2019combinatorial,
  title={Combinatorial sleeping bandits with fairness constraints},
  author={Li, Fengjiao and Liu, Jia and Ji, Bo},
  journal={IEEE Transactions on Network Science and Engineering},
  volume={7},
  number={3},
  pages={1799--1813},
  year={2019},
  publisher={IEEE}
}
@article{joseph2016fairness,
  title={Fairness in learning: Classic and contextual bandits},
  author={Joseph, Matthew and Kearns, Michael and Morgenstern, Jamie H and Roth, Aaron},
  journal={Advances in neural information processing systems},
  volume={29},
  year={2016}
}

@book{lan2010axiomatic,
  title={An axiomatic theory of fairness in network resource allocation},
  author={Lan, Tian and Kao, David and Chiang, Mung and Sabharwal, Ashutosh},
  year={2010},
  publisher={IEEE}
}
@article{radunovic2007unified,
  title={A unified framework for max-min and min-max fairness with applications},
  author={Radunovic, Bozidar and Le Boudec, Jean-Yves},
  journal={IEEE/ACM Transactions on networking},
  volume={15},
  number={5},
  pages={1073--1083},
  year={2007},
  publisher={IEEE}
}
@article{kelly1997charging,
  title={Charging and rate control for elastic traffic},
  author={Kelly, Frank},
  journal={European transactions on Telecommunications},
  volume={8},
  number={1},
  pages={33--37},
  year={1997},
  publisher={Wiley Online Library}
}

@article{jain1984quantitative,
  title={A quantitative measure of fairness and discrimination},
  author={Jain, Rajendra K and Chiu, Dah-Ming W and Hawe, William R and others},
  journal={Eastern Research Laboratory, Digital Equipment Corporation, Hudson, MA},
  volume={21},
  year={1984}
}

@inproceedings{dwork-fairness,
author = {Dwork, Cynthia and Hardt, Moritz and Pitassi, Toniann and Reingold, Omer and Zemel, Richard},
title = {Fairness through Awareness},
year = {2012},
isbn = {9781450311151},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/2090236.2090255},
doi = {10.1145/2090236.2090255},
abstract = {We study fairness in classification, where individuals are classified, e.g., admitted to a university, and the goal is to prevent discrimination against individuals based on their membership in some group, while maintaining utility for the classifier (the university). The main conceptual contribution of this paper is a framework for fair classification comprising (1) a (hypothetical) task-specific metric for determining the degree to which individuals are similar with respect to the classification task at hand; (2) an algorithm for maximizing utility subject to the fairness constraint, that similar individuals are treated similarly. We also present an adaptation of our approach to achieve the complementary goal of "fair affirmative action," which guarantees statistical parity (i.e., the demographics of the set of individuals receiving any classification are the same as the demographics of the underlying population), while treating similar individuals as similarly as possible. Finally, we discuss the relationship of fairness to privacy: when fairness implies privacy, and how tools developed in the context of differential privacy may be applied to fairness.},
booktitle = {Proceedings of the 3rd Innovations in Theoretical Computer Science Conference},
pages = {214–226},
numpages = {13},
location = {Cambridge, Massachusetts},
series = {ITCS '12}
}

@article{ALTMAN2010338,
title = {Fair resource allocation in wireless networks in the presence of a jammer},
journal = {Performance Evaluation},
volume = {67},
number = {4},
pages = {338-349},
year = {2010},
note = {Performance Evaluation Methodologies and Tools: Selected Papers from VALUETOOLS 2008},
issn = {0166-5316},
doi = {https://doi.org/10.1016/j.peva.2009.08.002},
url = {https://www.sciencedirect.com/science/article/pii/S0166531609001035},
author = {E. Altman and K. Avrachenkov and A. Garnaev},
keywords = {Wireless networks, Resource allocation, Jamming, Power control, Game theory},
abstract = {We consider jamming in wireless networks in the framework of zero-sum games with α-fairness utility functions. The base station has to distribute the power fairly among the users in the presence of a jammer. The jammer in turn tries to distribute its power among the channels to produce as much harm as possible. The Shannon capacity and the SNIR optimization are particular cases of the proposed more general α-fairness SNIR based utility functions. Specifically, we consider two α-fairness utility functions, based on SNIR and Shifted SNIR. This game can also be viewed as a minimax problem against the nature. We show that the game has the unique equilibrium and investigate its properties. In particular, in several important cases we present the equilibrium strategies and Jain’s fairness index in closed form. It turns out that there is an important difference between SNIR and Shifted SNIR α-fairness utility functions. In the case of the SNIR based utility function all users obtain non-zero powers when α>0. On the contrary, when the Shifted SNIR based utility function is used, some users with bad channel conditions might not receive any power at all.}
}

@INPROCEEDINGS{fine-paging,
  author={Blum, A. and Burch, C. and Kalai, A.},
  booktitle={40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)}, 
  title={Finely-competitive paging}, 
  year={1999},
  volume={},
  number={},
  pages={450-457},
  doi={10.1109/SFFCS.1999.814617}}


@inproceedings{blum1997line,
  title={On-line learning and the metrical task system problem},
  author={Blum, Avrim and Burch, Carl},
  booktitle={Proceedings of the tenth annual conference on Computational learning theory},
  pages={45--53},
  year={1997}
}

@inproceedings{bhaskara2020online,
  title={Online learning with imperfect hints},
  author={Bhaskara, Aditya and Cutkosky, Ashok and Kumar, Ravi and Purohit, Manish},
  booktitle={International Conference on Machine Learning},
  pages={822--831},
  year={2020},
  organization={PMLR}
}

@inproceedings{rakhlin2011online,
  title={Online learning: Beyond regret},
  author={Rakhlin, Alexander and Sridharan, Karthik and Tewari, Ambuj},
  booktitle={Proceedings of the 24th Annual Conference on Learning Theory},
  pages={559--594},
  year={2011},
  organization={JMLR Workshop and Conference Proceedings}
}

@article{nace2008max,
  title={Max-min fairness and its applications to routing and load-balancing in communication networks: a tutorial},
  author={Nace, Dritan and Pi{\'o}ro, Michal},
  journal={IEEE Communications Surveys \& Tutorials},
  volume={10},
  number={4},
  pages={5--17},
  year={2008},
  publisher={IEEE}
}

@article{hao2019facebook,
  title={Facebook’s ad-serving algorithm discriminates by gender and race},
  author={Hao, Karen},
  journal={MIT Technology Review},
  year={2019}
}

@inproceedings{even2009online,
  title={Online Learning for Global Cost Functions.},
  author={Even-Dar, Eyal and Kleinberg, Robert and Mannor, Shie and Mansour, Yishay},
  booktitle={COLT},
  year={2009}
}

@article{narahari-fair,
author = {Patil, Vishakha and Ghalme, Ganesh and Nair, Vineet and Narahari, Y.},
title = {Achieving Fairness in the Stochastic Multi-Armed Bandit Problem},
year = {2021},
issue_date = {January 2021},
publisher = {JMLR.org},
volume = {22},
number = {1},
issn = {1532-4435},
abstract = {We study an interesting variant of the stochastic multi-armed bandit problem, which we call the Fair-MAB problem, where, in addition to the objective of maximizing the sum of expected rewards, the algorithm also needs to ensure that at any time, each arm is pulled at least a pre-specified fraction of times. We investigate the interplay between learning and fairness in terms of a pre-specified vector denoting the fractions of guaranteed pulls. We define a fairness-aware regret, which we call r-Regret, that takes into account the above fairness constraints and extends the conventional notion of regret in a natural way. Our primary contribution is to obtain a complete characterization of a class of Fair-MAB algorithms via two parameters: the unfairness tolerance and the learning algorithm used as a black-box. For this class of algorithms, we provide a fairness guarantee that holds uniformly over time, irrespective of the chosen learning algorithm. Further, when the learning algorithm is UCB1, we show that our algorithm achieves constant r-Regret for a large enough time horizon. Finally, we analyze the cost of fairness in terms of the conventional notion of regret. We conclude by experimentally validating our theoretical results.},
journal = {J. Mach. Learn. Res.},
month = {jan},
articleno = {174},
numpages = {31},
keywords = {upper confidence bounds, online learning, multi-armed bandits, reinforcement learning, machine learning, fairness}
}

@article{mo2000fair,
  title={Fair end-to-end window-based congestion control},
  author={Mo, Jeonghoon and Walrand, Jean},
  journal={IEEE/ACM Transactions on networking},
  volume={8},
  number={5},
  pages={556--567},
  year={2000},
  publisher={IEEE}
}

@article{valls2021birkhoff,
  title={Birkhoff’s decomposition revisited: Sparse scheduling for high-speed circuit switches},
  author={Valls, V{\'\i}ctor and Iosifidis, George and Tassiulas, Leandros},
  journal={IEEE/ACM Transactions on Networking},
  volume={29},
  number={6},
  pages={2399--2412},
  year={2021},
  publisher={IEEE}
}

@article{si2022enabling,
  title={Enabling Long-term Fairness in Dynamic Resource Allocation},
  author={Si Salem, Tareq and Iosifidis, Georgios and Neglia, Giovanni},
  journal={Proceedings of the ACM on Measurement and Analysis of Computing Systems},
  volume={6},
  number={3},
  pages={1--36},
  year={2022},
  publisher={ACM New York, NY, USA}
}

@article{bertsimas2012efficiency,
  title={On the efficiency-fairness trade-off},
  author={Bertsimas, Dimitris and Farias, Vivek F and Trichakis, Nikolaos},
  journal={Management Science},
  volume={58},
  number={12},
  pages={2234--2250},
  year={2012},
  publisher={INFORMS}
}

@inproceedings{online-fair,
author = {Wang, Zhiyuan and Ye, Jiancheng and Lin, Dong and Lui, John C. S.},
title = {Achieving Efficiency via Fairness in Online Resource Allocation},
year = {2022},
isbn = {9781450391658},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/3492866.3549724},
doi = {10.1145/3492866.3549724},
abstract = {The classic utility maximization framework studies the fairness-efficiency tradeoff in various resource allocation problems (e.g., bandwidth allocation). The weighted alpha-fair utility is a common utilitarian metric. However, this classic framework cannot tackle those allocation problems with the online decision-making requirement (e.g., caching capacity allocation under unknown requests). Existing studies on these online allocation problems largely follow the online learning approaches, thus inevitably overlook the allocation fairness. In this paper, we propose a novel utility maximization framework accommodating the online setting. The major challenge of designing this framework lies in the tight coupling between the desirable fairness guarantee and the unknown allocation efficiency. To tackle this, we integrate the weighted alpha-fair utility with the learning rationale, by properly devising the merit-based weights and the increasing fairness levels. Under our proposed framework, the utility-maximizing allocation in each time slot is weighted alpha-fair. Our framework also performs asymptotically as well as the offline optimal/efficient outcome. We demonstrate how this framework functions in two networking applications. In size-based scheduling, it enables network switches to prioritize short flows and avoid flow starvation without the prior flow size information. In file caching, our framework outperforms several state-of-the-art caching policies up to 21\% in terms of cache-hit-ratio.},
booktitle = {Proceedings of the Twenty-Third International Symposium on Theory, Algorithmic Foundations, and Protocol Design for Mobile Networks and Mobile Computing},
pages = {101–110},
numpages = {10},
keywords = {online learning, online resource allocation, alpha-fair utility},
location = {Seoul, Republic of Korea},
series = {MobiHoc '22}
}

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  title={Lectures on polytopes},
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}

@inproceedings{tu2014online,
  title={Online dating recommendations: matching markets and learning preferences},
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  pages={787--792},
  year={2014}
}

@book{boucheron2013concentration,
  title={Concentration inequalities: A nonasymptotic theory of independence},
  author={Boucheron, St{\'e}phane and Lugosi, G{\'a}bor and Massart, Pascal},
  year={2013},
  publisher={Oxford university press}
}

@article{zhao2020dynamic,
  title={Dynamic regret of convex and smooth functions},
  author={Zhao, Peng and Zhang, Yu-Jie and Zhang, Lijun and Zhou, Zhi-Hua},
  journal={Advances in Neural Information Processing Systems},
  volume={33},
  pages={12510--12520},
  year={2020}
}

@inproceedings{chiang2012online,
  title={Online optimization with gradual variations},
  author={Chiang, Chao-Kai and Yang, Tianbao and Lee, Chia-Jung and Mahdavi, Mehrdad and Lu, Chi-Jen and Jin, Rong and Zhu, Shenghuo},
  booktitle={Conference on Learning Theory},
  pages={6--1},
  year={2012},
  organization={JMLR Workshop and Conference Proceedings}
}

@inproceedings{andrews2005optimal,
  title={Optimal utility based multi-user throughput allocation subject to throughput constraints},
  author={Andrews, Matthew and Qian, Lijun and Stolyar, Alexander},
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  volume={4},
  pages={2415--2424},
  year={2005},
  organization={IEEE}
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  volume={51},
  number={3},
  pages={385--463},
  year={2004},
  publisher={ACM New York, NY, USA}
}

@inproceedings{zinkevich2003online,
  title={Online convex programming and generalized infinitesimal gradient ascent},
  author={Zinkevich, Martin},
  booktitle={Proceedings of the 20th international conference on machine learning (icml-03)},
  pages={928--936},
  year={2003}
}

@book{cesa2006prediction,
  title={Prediction, learning, and games},
  author={Cesa-Bianchi, Nicolo and Lugosi, G{\'a}bor},
  year={2006},
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}
@article{anava2015online,
  title={Online learning for adversaries with memory: price of past mistakes},
  author={Anava, Oren and Hazan, Elad and Mannor, Shie},
  journal={Advances in Neural Information Processing Systems},
  volume={28},
  year={2015}
}

@inproceedings{mukhopadhyay2022k,
  title={k-experts-Online Policies and Fundamental Limits},
  author={Mukhopadhyay, Samrat and Sahoo, Sourav and Sinha, Abhishek},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={342--365},
  year={2022},
  organization={PMLR}
}


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  title={LeadCache: Regret-Optimal Caching in Networks},
  author={Paria, Debjit and Sinha, Abhishek},
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  volume={34},
  pages={4435--4447},
  year={2021}
}

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  volume={33},
  pages={123--133},
  year={2020}
}
@article{golovin2014online,
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  year={2014}
}

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  year={2008}
}

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@inproceedings{destounis2017alpha,
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  pages={1--8},
  year={2017},
  organization={IEEE}
}

@INPROCEEDINGS{SwRegret,  author={Mukhopadhyay, Samrat and Sinha, Abhishek},  booktitle={2021 IEEE International Symposium on Information Theory (ISIT)},   title={Online Caching with Optimal Switching Regret},   year={2021},  volume={},  number={},  pages={1546-1551},  doi={10.1109/ISIT45174.2021.9517925}}

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  pages={1877--1893},
  year={2020},
  organization={PMLR}
}

@article{hazan2019introduction,
  title={Introduction to online convex optimization},
  author={Hazan, Elad},
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  year={2019}
}

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  year={2021},
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}
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  year={2018}
}
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  number={1},
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author = {Nygren, Erik and Sitaraman, Ramesh K. and Sun, Jennifer},
title = {The Akamai Network: A Platform for High-Performance Internet Applications},
year = {2010},
issue_date = {July 2010},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {44},
number = {3},
issn = {0163-5980},
url = {https://doi.org/10.1145/1842733.1842736},
doi = {10.1145/1842733.1842736},
abstract = {Comprising more than 61,000 servers located across nearly 1,000 networks in 70 countries worldwide, the Akamai platform delivers hundreds of billions of Internet interactions daily, helping thousands of enterprises boost the performance and reliability of their Internet applications. In this paper, we give an overview of the components and capabilities of this large-scale distributed computing platform, and offer some insight into its architecture, design principles, operation, and management.},
journal = {SIGOPS Oper. Syst. Rev.},
month = aug,
pages = {2–19},
numpages = {18},
keywords = {content delivery, Akamai, CDN, quality of service, streaming media, application acceleration, fault tolerance, DNS, overlay networks, HTTP}
}

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  title={Efficient algorithms for online decision problems},
  author={Kalai, Adam and Vempala, Santosh},
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  volume={71},
  number={3},
  pages={291--307},
  year={2005},
  publisher={Elsevier}
}
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@article{mhaisen2022optimistic,
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  year={2022}
}

@article{CIT-104,
url = {http://dx.doi.org/10.1561/0100000104},
year = {2020},
volume = {16},
journal = {Foundations and Trends® in Communications and Information Theory},
title = {Cache Optimization Models and Algorithms},
doi = {10.1561/0100000104},
issn = {1567-2190},
number = {3–4},
pages = {156-345},
author = {Georgios Paschos and George Iosifidis and Giuseppe Caire}
}


@inproceedings{abernethy2014online,
  title={Online linear optimization via smoothing},
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  year={2019}
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@article{SIGMETRICS20,
author = {Bhattacharjee, Rajarshi and Banerjee, Subhankar and Sinha, Abhishek},
title = {Fundamental Limits on the Regret of Online Network-Caching},
year = {2020},
issue_date = {June 2020},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {4},
number = {2},
url = {https://doi.org/10.1145/3392143},
doi = {10.1145/3392143},
journal = {Proc. ACM Meas. Anal. Comput. Syst.},
month = jun,
articleno = {25},
numpages = {31},
keywords = {fundamental limits, online algorithms, regret bounds, network-caching}
}

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@MISC{stack_of,
    TITLE = {Expectation of square root of binomial r.v.},
    AUTHOR = {Ori Gurel-Gurevich},
    HOWPUBLISHED = {MathOverflow},
    NOTE = {URL:https://mathoverflow.net/q/121424 (version: 2013-02-10)},
    EPRINT = {https://mathoverflow.net/q/121424},
    URL = {https://mathoverflow.net/q/121424}
}












