# Contextual Online Decision Making with Infinite-Dimensional Functional Regression

arXiv: https://arxiv.org/abs/2501.18359 · HTML: https://mlsanity.com/paper/2501.18359/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2025.

## Built on by (most cited first, 5 of 5)

- [Pre-Trained AI Model Assisted Online Decision-Making under Missing Covariates: A Theoretical Perspective](https://mlsanity.com/paper/2507.07852/) (2025)
- [Perturbing the Derivative: Doubly Wild Refitting for Model-Free Evaluation of Opaque Machine Learning Predictors](https://mlsanity.com/paper/2511.18789/) (2025)
- [Constrained Online Decision-Making: A Unified Framework](https://mlsanity.com/paper/2505.07101/) (2025)
- [Perturbing the Derivative: Wild Refitting for Model-Free Evaluation of Machine Learning Models under Bregman Losses](https://mlsanity.com/paper/2509.02476/) (2025)
- [Simultaneous Pointwise Majorization for Mixed Tail Processes with Applications in Gaussian Chaos and Ergodic Diffusions](https://mlsanity.com/paper/2609.01576/) (2026)

## Abstract (arXiv)

Contextual sequential decision-making problems play a crucial role in machine learning, encompassing a wide range of downstream applications such as bandits, sequential hypothesis testing and online risk control. These applications often require different statistical measures, including expectation, variance and quantiles. In this paper, we provide a universal admissible algorithm framework for dealing with all kinds of contextual online decision-making problems that directly learns the whole underlying unknown distribution instead of focusing on individual statistics. This is much more difficult because the dimension of the regression is uncountably infinite, and any existing linear contextual bandits algorithm will result in infinite regret. To overcome this issue, we propose an efficient infinite-dimensional functional regression oracle for contextual cumulative distribution functions (CDFs), where each data point is modeled as a combination of context-dependent CDF basis functions. Our analysis reveals that the decay rate of the eigenvalue sequence of the design integral operator governs the regression error rate and, consequently, the utility regret rate. Specifically, when the eigenvalue sequence exhibits a polynomial decay of order frac1γ≥ 1, the utility regret is bounded by tildeOBig(T^frac3γ+22(γ+2)Big). By setting γ=0, this recovers the existing optimal regret rate for contextual bandits with finite-dimensional regression and is optimal under a stronger exponential decay assumption. Additionally, we provide a numerical method to compute the eigenvalue sequence of the integral operator, enabling the practical implementation of our framework.