Stochastic Linear Bandits with Partially Observed Actions

2026-07-09

AI Summary

This work addresses the problem of stochastic linear bandits with partially observable actions by leveraging the low intrinsic dimensionality of the action vectors. The authors propose a novel approach that first estimates the underlying latent subspace and imputes the missing action coordinates, then applies the OFUL algorithm in the recovered low-dimensional space. They establish the first √T regret bound in this setting that depends on the intrinsic dimension rather than the ambient environment dimension, and further design an adaptive variant that does not require prior knowledge of the intrinsic dimension. Their theoretical analysis combines information-theoretic lower bounds to disentangle the effects of reward learning and the missingness mechanism. Experiments demonstrate that the proposed TOFU-POV algorithm significantly outperforms existing baselines on both synthetic and real-world datasets.

Abstract

The stochastic linear bandit, where actions are represented as vectors and rewards are linear, is a central paradigm for sequential decision making. We study a partially observed variant of this problem in which the learning agent only sees a random subset of coordinates for each action. Such partial observability arises naturally in settings like recommendation and healthcare, where full action descriptions can be expensive or even impossible to obtain. In general, this makes sublinear regret information-theoretically impossible. However, we show that this barrier can be overcome when the action vectors have low intrinsic dimension. We propose an algorithm, TOFU-POV, that estimates the latent action subspace using the masked actions, imputes current actions using an epoch-wise frozen representation, and runs OFUL in the resulting low-dimensional coordinates. Our theory shows that TOFU-POV enjoys a $\sqrt{T}$ regret that scales with the intrinsic action subspace dimension as opposed to the ambient dimension and quantifies the interaction between these quantities and the missingness, decision set size, and subspace conditioning. We also devise a rank-adaptive algorithm that does not require the knowledge of the intrinsic dimension. We complement these guarantees with a lower bound based on a novel product construction that separates usual reward-learning uncertainty from a missingness-dependent cost intrinsic to partial observation. Synthetic and real data experiments support our theory and show that TOFU-POV can substantially improve upon natural baselines in this challenging problem.

Problem

• Stochastic Linear Bandits
• Partial Observation
• Intrinsic Dimension
• Sequential Decision Making
• Missing Data

Innovation

• partially observed actions
• stochastic linear bandits
• low intrinsic dimension
• subspace estimation
• regret bound