NeurIPS 2025

Eluder dimension: localise it!
Alireza Bakhtiari
University of Alberta
sbakhtia@ualberta.ca
Alex Ayoub
University of Alberta
aayoub@ualberta.ca
Samuel Robertson
University of Alberta
smrobert@ualberta.ca
David Janz
University of Oxford
david.janz@stats.ox.ac.uk
Csaba Szepesvári
University of Alberta
szepesva@ualberta.ca
Abstract
We establish a lower bound on the eluder dimension of generalised linear model
classes, showing that standard eluder dimension-based analysis cannot lead to
first-order regret bounds. To address this, we introduce a localisation method for
the eluder dimension; our analysis immediately recovers and improves on classic
results for Bernoulli bandits, and allows for the first genuine first-order bounds for
finite-horizon reinforcement learning tasks with bounded cumulative returns.
1
Introduction
We study decision-making problems where the regret admits a first-order (small-cost) bound of the
form
Rn ≤
p
nη(a⋆)Γn + Γ′
n ,
with η(a⋆) the optimal mean per-round cost and Γn, Γ′
n instance and model dependent complexities.
The challenge in obtaining first-order bounds is in how we measure the complexity of the task. The
(global) eluder dimension (Russo and Van Roy, 2013) is a standard complexity measure used to
provide worst-case guarantees, but, as we shall argue, it ignores the local structure of the problem:
analyses based on the global eluder dimension often introduce a factor in Γn that scales like a per-
step worst-case information/curvature parameter (denoted κ), which cancels out η(a⋆) and destroys
first-order gains. This subtle issue is present in much of the previous work on first-order bounds.
We show that by localising the eluder dimension—restricting it to a small neighbourhood of the
optimal model—these κ terms can be moved into the lower-order Γ′
n term. This tightens the link
between exploration and the actual difficulty of the instance and yields genuinely first-order bounds.
Our contributions may be summarised as follows:
1. Local...