Towards Minimax Online Learning with Unknown Time Horizon

Haipeng Luo, Robert Schapire
; Proceedings of the 31st International Conference on Machine Learning, PMLR 32(1):226-234, 2014.

Abstract

We consider online learning when the time horizon is unknown. We apply a minimax analysis, beginning with the fixed horizon case, and then moving on to two unknown-horizon settings, one that assumes the horizon is chosen randomly according to some distribution, and the other which allows the adversary full control over the horizon. For the random horizon setting with restricted losses, we derive a fully optimal minimax algorithm. And for the adversarial horizon setting, we prove a nontrivial lower bound which shows that the adversary obtains strictly more power than when the horizon is fixed and known. Based on the minimax solution of the random horizon setting, we then propose a new adaptive algorithm which “pretends” that the horizon is drawn from a distribution from a special family, but no matter how the actual horizon is chosen, the worst-case regret is of the optimal rate. Furthermore, our algorithm can be combined and applied in many ways, for instance, to online convex optimization, follow the perturbed leader, exponential weights algorithm and first order bounds. Experiments show that our algorithm outperforms many other existing algorithms in an online linear optimization setting.

Cite this Paper


BibTeX
@InProceedings{pmlr-v32-luo14, title = {Towards Minimax Online Learning with Unknown Time Horizon}, author = {Haipeng Luo and Robert Schapire}, booktitle = {Proceedings of the 31st International Conference on Machine Learning}, pages = {226--234}, year = {2014}, editor = {Eric P. Xing and Tony Jebara}, volume = {32}, number = {1}, series = {Proceedings of Machine Learning Research}, address = {Bejing, China}, month = {22--24 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v32/luo14.pdf}, url = {http://proceedings.mlr.press/v32/luo14.html}, abstract = {We consider online learning when the time horizon is unknown. We apply a minimax analysis, beginning with the fixed horizon case, and then moving on to two unknown-horizon settings, one that assumes the horizon is chosen randomly according to some distribution, and the other which allows the adversary full control over the horizon. For the random horizon setting with restricted losses, we derive a fully optimal minimax algorithm. And for the adversarial horizon setting, we prove a nontrivial lower bound which shows that the adversary obtains strictly more power than when the horizon is fixed and known. Based on the minimax solution of the random horizon setting, we then propose a new adaptive algorithm which “pretends” that the horizon is drawn from a distribution from a special family, but no matter how the actual horizon is chosen, the worst-case regret is of the optimal rate. Furthermore, our algorithm can be combined and applied in many ways, for instance, to online convex optimization, follow the perturbed leader, exponential weights algorithm and first order bounds. Experiments show that our algorithm outperforms many other existing algorithms in an online linear optimization setting.} }
Endnote
%0 Conference Paper %T Towards Minimax Online Learning with Unknown Time Horizon %A Haipeng Luo %A Robert Schapire %B Proceedings of the 31st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2014 %E Eric P. Xing %E Tony Jebara %F pmlr-v32-luo14 %I PMLR %J Proceedings of Machine Learning Research %P 226--234 %U http://proceedings.mlr.press %V 32 %N 1 %W PMLR %X We consider online learning when the time horizon is unknown. We apply a minimax analysis, beginning with the fixed horizon case, and then moving on to two unknown-horizon settings, one that assumes the horizon is chosen randomly according to some distribution, and the other which allows the adversary full control over the horizon. For the random horizon setting with restricted losses, we derive a fully optimal minimax algorithm. And for the adversarial horizon setting, we prove a nontrivial lower bound which shows that the adversary obtains strictly more power than when the horizon is fixed and known. Based on the minimax solution of the random horizon setting, we then propose a new adaptive algorithm which “pretends” that the horizon is drawn from a distribution from a special family, but no matter how the actual horizon is chosen, the worst-case regret is of the optimal rate. Furthermore, our algorithm can be combined and applied in many ways, for instance, to online convex optimization, follow the perturbed leader, exponential weights algorithm and first order bounds. Experiments show that our algorithm outperforms many other existing algorithms in an online linear optimization setting.
RIS
TY - CPAPER TI - Towards Minimax Online Learning with Unknown Time Horizon AU - Haipeng Luo AU - Robert Schapire BT - Proceedings of the 31st International Conference on Machine Learning PY - 2014/01/27 DA - 2014/01/27 ED - Eric P. Xing ED - Tony Jebara ID - pmlr-v32-luo14 PB - PMLR SP - 226 DP - PMLR EP - 234 L1 - http://proceedings.mlr.press/v32/luo14.pdf UR - http://proceedings.mlr.press/v32/luo14.html AB - We consider online learning when the time horizon is unknown. We apply a minimax analysis, beginning with the fixed horizon case, and then moving on to two unknown-horizon settings, one that assumes the horizon is chosen randomly according to some distribution, and the other which allows the adversary full control over the horizon. For the random horizon setting with restricted losses, we derive a fully optimal minimax algorithm. And for the adversarial horizon setting, we prove a nontrivial lower bound which shows that the adversary obtains strictly more power than when the horizon is fixed and known. Based on the minimax solution of the random horizon setting, we then propose a new adaptive algorithm which “pretends” that the horizon is drawn from a distribution from a special family, but no matter how the actual horizon is chosen, the worst-case regret is of the optimal rate. Furthermore, our algorithm can be combined and applied in many ways, for instance, to online convex optimization, follow the perturbed leader, exponential weights algorithm and first order bounds. Experiments show that our algorithm outperforms many other existing algorithms in an online linear optimization setting. ER -
APA
Luo, H. & Schapire, R.. (2014). Towards Minimax Online Learning with Unknown Time Horizon. Proceedings of the 31st International Conference on Machine Learning, in PMLR 32(1):226-234

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