# Stochastic Online Learning with Probabilistic Graph Feedback

arXiv: https://arxiv.org/abs/1903.01083 · HTML: https://mlsanity.com/paper/1903.01083/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Mar 2019.

## Built on by (most cited first, 12 of 12)

- [Understanding Bandits with Graph Feedback](https://mlsanity.com/paper/2105.14260/) (2021)
- [Learning on the Edge: Online Learning with Stochastic Feedback Graphs](https://mlsanity.com/paper/2210.04229/) (2022)
- [Practical Contextual Bandits with Feedback Graphs](https://mlsanity.com/paper/2302.08631/) (2023)
- [Combinatorial Bandits under Strategic Manipulations](https://mlsanity.com/paper/2102.12722/) (2021)
- [Simultaneously Learning Stochastic and Adversarial Bandits with General Graph Feedback](https://mlsanity.com/paper/2206.07908/) (2022)
- [Efficient Contextual Bandits with Uninformed Feedback Graphs](https://mlsanity.com/paper/2402.08127/) (2024)
- [Stochastic Online Learning with Feedback Graphs: Finite-Time and Asymptotic Optimality](https://mlsanity.com/paper/2206.10022/) (2022)
- [Online Learning with Uncertain Feedback Graphs](https://mlsanity.com/paper/2106.08441/) (2021)
- [Improved Algorithms for Bandit with Graph Feedback via Regret Decomposition](https://mlsanity.com/paper/2205.15076/) (2022)
- [Federated Online Prediction from Experts with Differential Privacy: Separations and Regret Speed-ups](https://mlsanity.com/paper/2409.19092/) (2024)
- [Asymptotically-Optimal Gaussian Bandits with Side Observations](https://mlsanity.com/paper/2505.10698/) (2025)
- [Online Learning with Multiple Fairness Regularizers via Graph-Structured Feedback](https://mlsanity.com/paper/2508.14311/) (2025)

## Abstract (arXiv)

We consider a problem of stochastic online learning with general probabilistic graph feedback, where each directed edge in the feedback graph has probability pij. Two cases are covered. (a) The one-step case, where after playing arm i the learner observes a sample reward feedback of arm j with independent probability pij. (b) The cascade case where after playing arm i the learner observes feedback of all arms j in a probabilistic cascade starting from i -- for each (i,j) with probability pij, if arm i is played or observed, then a reward sample of arm j would be observed with independent probability pij. Previous works mainly focus on deterministic graphs which corresponds to one-step case with pij ∈ 0,1, an adversarial sequence of graphs with certain topology guarantees, or a specific type of random graphs. We analyze the asymptotic lower bounds and design algorithms in both cases. The regret upper bounds of the algorithms match the lower bounds with high probability.