Efficient Near-Optimal Algorithm for Online Shortest Paths in Directed Acyclic Graphs with Bandit Feedback Against Adaptive Adversaries

Arnab Maiti, Zhiyuan Fan, Kevin Jamieson, Lillian J. Ratliff, Gabriele Farina
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:3881-3932, 2025.

Abstract

In this paper, we study the online shortest path problem in directed acyclic graphs (DAGs) under bandit feedback against an adaptive adversary. Given a DAG $G = (V, E)$ with a source node $v_{\mathsf{s}}$ and a sink node $v_{\mathsf{t}}$, let $\mathcal{X} \subseteq \{0,1\}^{|E|}$ denote the set of all paths from $v_{\mathsf{s}}$ to $v_{\mathsf{t}}$. At each round $t$, we select a path $\mathbf{x}_t \in \mathcal{X}$ and receive bandit feedback on our loss $⟨\mathbf{x}_t, \mathbf{y}_t ⟩\in [-1,1]$, where $\mathbf{y}_t$ is an adversarially chosen loss vector. Our goal is to minimize regret with respect to the best path in hindsight over $T$ rounds. We propose the first computationally efficient algorithm to achieve a near-minimax optimal regret bound of $\tilde{\mathcal{O}}(\sqrt{|E|T\log |\mathcal{X}|})$ with high probability against any adaptive adversary, where $\tilde{\mathcal{O}}(\cdot)$ hides logarithmic factors in the number of edges $|E|$. Our algorithm leverages a novel loss estimator and a centroid-based decomposition in a nontrivial manner to attain this regret bound. As an application, we show that our algorithm for DAGs provides state-of-the-art efficient algorithms for $m$-sets, extensive-form games, the Colonel Blotto game, shortest walks in directed graphs, hypercubes, and multi-task multi-armed bandits, achieving improved high-probability regret guarantees in all these settings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-maiti25a, title = {Efficient Near-Optimal Algorithm for Online Shortest Paths in Directed Acyclic Graphs with Bandit Feedback Against Adaptive Adversaries}, author = {Maiti, Arnab and Fan, Zhiyuan and Jamieson, Kevin and Ratliff, Lillian J. and Farina, Gabriele}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {3881--3932}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/maiti25a/maiti25a.pdf}, url = {https://proceedings.mlr.press/v291/maiti25a.html}, abstract = {In this paper, we study the online shortest path problem in directed acyclic graphs (DAGs) under bandit feedback against an adaptive adversary. Given a DAG $G = (V, E)$ with a source node $v_{\mathsf{s}}$ and a sink node $v_{\mathsf{t}}$, let $\mathcal{X} \subseteq \{0,1\}^{|E|}$ denote the set of all paths from $v_{\mathsf{s}}$ to $v_{\mathsf{t}}$. At each round $t$, we select a path $\mathbf{x}_t \in \mathcal{X}$ and receive bandit feedback on our loss $⟨\mathbf{x}_t, \mathbf{y}_t ⟩\in [-1,1]$, where $\mathbf{y}_t$ is an adversarially chosen loss vector. Our goal is to minimize regret with respect to the best path in hindsight over $T$ rounds. We propose the first computationally efficient algorithm to achieve a near-minimax optimal regret bound of $\tilde{\mathcal{O}}(\sqrt{|E|T\log |\mathcal{X}|})$ with high probability against any adaptive adversary, where $\tilde{\mathcal{O}}(\cdot)$ hides logarithmic factors in the number of edges $|E|$. Our algorithm leverages a novel loss estimator and a centroid-based decomposition in a nontrivial manner to attain this regret bound. As an application, we show that our algorithm for DAGs provides state-of-the-art efficient algorithms for $m$-sets, extensive-form games, the Colonel Blotto game, shortest walks in directed graphs, hypercubes, and multi-task multi-armed bandits, achieving improved high-probability regret guarantees in all these settings.} }
Endnote
%0 Conference Paper %T Efficient Near-Optimal Algorithm for Online Shortest Paths in Directed Acyclic Graphs with Bandit Feedback Against Adaptive Adversaries %A Arnab Maiti %A Zhiyuan Fan %A Kevin Jamieson %A Lillian J. Ratliff %A Gabriele Farina %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-maiti25a %I PMLR %P 3881--3932 %U https://proceedings.mlr.press/v291/maiti25a.html %V 291 %X In this paper, we study the online shortest path problem in directed acyclic graphs (DAGs) under bandit feedback against an adaptive adversary. Given a DAG $G = (V, E)$ with a source node $v_{\mathsf{s}}$ and a sink node $v_{\mathsf{t}}$, let $\mathcal{X} \subseteq \{0,1\}^{|E|}$ denote the set of all paths from $v_{\mathsf{s}}$ to $v_{\mathsf{t}}$. At each round $t$, we select a path $\mathbf{x}_t \in \mathcal{X}$ and receive bandit feedback on our loss $⟨\mathbf{x}_t, \mathbf{y}_t ⟩\in [-1,1]$, where $\mathbf{y}_t$ is an adversarially chosen loss vector. Our goal is to minimize regret with respect to the best path in hindsight over $T$ rounds. We propose the first computationally efficient algorithm to achieve a near-minimax optimal regret bound of $\tilde{\mathcal{O}}(\sqrt{|E|T\log |\mathcal{X}|})$ with high probability against any adaptive adversary, where $\tilde{\mathcal{O}}(\cdot)$ hides logarithmic factors in the number of edges $|E|$. Our algorithm leverages a novel loss estimator and a centroid-based decomposition in a nontrivial manner to attain this regret bound. As an application, we show that our algorithm for DAGs provides state-of-the-art efficient algorithms for $m$-sets, extensive-form games, the Colonel Blotto game, shortest walks in directed graphs, hypercubes, and multi-task multi-armed bandits, achieving improved high-probability regret guarantees in all these settings.
APA
Maiti, A., Fan, Z., Jamieson, K., Ratliff, L.J. & Farina, G.. (2025). Efficient Near-Optimal Algorithm for Online Shortest Paths in Directed Acyclic Graphs with Bandit Feedback Against Adaptive Adversaries. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:3881-3932 Available from https://proceedings.mlr.press/v291/maiti25a.html.

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