Improved Analysis of the Tsallis-INF Algorithm in Stochastically Constrained Adversarial Bandits and Stochastic Bandits with Adversarial Corruptions

Saeed Masoudian, Yevgeny Seldin
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:3330-3350, 2021.

Abstract

We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(\Delta,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{\Delta_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $\Delta_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the corruption magnitude, and $\log_+(x) = \max\left(1,\log x\right)$. The regime includes stochastic bandits, stochastically constrained adversarial bandits, and stochastic bandits with adversarial corruptions as special cases. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-masoudian21a, title = {Improved Analysis of the Tsallis-INF Algorithm in Stochastically Constrained Adversarial Bandits and Stochastic Bandits with Adversarial Corruptions}, author = {Masoudian, Saeed and Seldin, Yevgeny}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {3330--3350}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/masoudian21a/masoudian21a.pdf}, url = {https://proceedings.mlr.press/v134/masoudian21a.html}, abstract = {We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(\Delta,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{\Delta_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $\Delta_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the corruption magnitude, and $\log_+(x) = \max\left(1,\log x\right)$. The regime includes stochastic bandits, stochastically constrained adversarial bandits, and stochastic bandits with adversarial corruptions as special cases. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.} }
Endnote
%0 Conference Paper %T Improved Analysis of the Tsallis-INF Algorithm in Stochastically Constrained Adversarial Bandits and Stochastic Bandits with Adversarial Corruptions %A Saeed Masoudian %A Yevgeny Seldin %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-masoudian21a %I PMLR %P 3330--3350 %U https://proceedings.mlr.press/v134/masoudian21a.html %V 134 %X We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(\Delta,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{\Delta_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $\Delta_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the corruption magnitude, and $\log_+(x) = \max\left(1,\log x\right)$. The regime includes stochastic bandits, stochastically constrained adversarial bandits, and stochastic bandits with adversarial corruptions as special cases. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.
APA
Masoudian, S. & Seldin, Y.. (2021). Improved Analysis of the Tsallis-INF Algorithm in Stochastically Constrained Adversarial Bandits and Stochastic Bandits with Adversarial Corruptions. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:3330-3350 Available from https://proceedings.mlr.press/v134/masoudian21a.html.

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