Gaussian-Width Bounds for Offline-to-Online Linear Bandits

The gist: The authors study how partially covered offline data can improve online learning in stochastic linear bandits by showing that existing regret bounds are loose due to conservative control over reward prediction error, and they develop tighter bounds using concentration of Lipschitz functions of Gaussian vectors

How it works

The core problem addresses the benefit of offline data based on its coverage of the feature space, focusing on the interesting case of partial coverage where only k out of d feature directions are poorly covered. Existing analyses for algorithms like LinUCB and phased elimination yield regret bounds of order Oe(√dkT) in this regime. The authors show this dependence stems from loose control over the reward prediction error, which they address by using concentration of Lipschitz functions of Gaussian vectors to achieve tighter bounds.

Key Insights on Regret Bounds

The paper identifies a multiplicative p/d gap between the upper and lower bounds for algorithms like Offline-Online Phased Elimination (OOPE) when the offline data are k-uncovered, suggesting that the problem effectively behaves as if it were k-dimensional. This gap arises because existing analyses conservatively bound the reward prediction error, failing to exploit low sensitivity in directions well covered by offline data. The authors propose a phased elimination algorithm with regret Oe(k√T) using tighter bounds derived from concentration of Lipschitz functions.

Algorithm Development and Analysis

The paper develops algorithms for both regret minimization and pure exploration settings. For pure exploration, the authors present the Gaussian Width Offline-to-Online Fixed Design (GW-OOFD) algorithm, which selects an optimal design based on Lemma 3.3 to bound simple regret. The analysis shows that this algorithm achieves a simple regret bounded by O(√kdT) in the k-uncovered regime.

Lower Bounds and Matching Results

The paper establishes matching lower bounds for various coverage regimes when the action set is the unit Euclidean ball, including a lower bound on simple regret for GW-OOFD. For equally-covered data, the expected regret is bounded by dT/√T + Toff, and for k-uncovered data, it is bounded by k√T. These bounds match the upper bounds of their proposed algorithms up to logarithmic factors in the stated coverage regimes.

Regret Minimization with Offline Data

For regret minimization, the authors present the Gaussian Width Offline-to-Online Phased Elimination (GW-OOPE) algorithm, which uses Gaussian width concentration instead of a union bound to select designs in each phase. The upper bound for GW-OOPE is RegT(A) ≤ 12Xml=1ϵl min + 2T min<ϵm, ζβm(A(12ϵm), V m off). The expected regret lower bounds match the upper bounds for both equally-covered and k-uncovered regimes.

Conclusion

The paper provides improved algorithms for pure exploration and regret minimization with matching lower bounds for various offline data coverage regimes, showing that uneven offline coverage can make the standard union bound loose. The resulting rates are dT/√T + Toff in the equally-covered case and k√T in the k-uncovered case. The final expected regret bounds are dT/√T + Toff/m and k√T + pmkT log(8m2(T + Toff))o for the respective regimes. The results show that offline data with function approximation can pose challenges if the sensitivity of the reward prediction error is not taken into account properly.

How it works

The paper identifies limitations of existing regret guarantees for standard bandit algorithms like phased elimination and LinUCB when augmented with offline data due to conservative bounds on the reward prediction error. Using a tighter bound on the prediction error, they design a phased elimination algorithm incorporating offline data and show an upper bound on its regret. The analysis utilizes Gaussian width concentration to replace the union bound in phased elimination, leading to improved regret guarantees.

Key Insights on Regret Bounds

The authors demonstrate that the standard LinUCB analysis retains a √d factor in the k-uncovered case, suffering a p/d gap similar to phased elimination. They show that this gap is due to conservative bounds on the prediction error, which do not exploit low sensitivity in well-covered directions. The new approach replaces the union bound with concentration of Lipschitz functions of Gaussian random vectors, leading to a regret bound scaling with k instead of d in the k-uncovered case.

Algorithm Development and Analysis

The paper presents an algorithm for pure exploration called GW-OOFD, which selects an optimal design based on Lemma 3.3 to bound simple regret. The analysis shows that this algorithm achieves a simple regret bounded by O(√kdT) in the k-uncovered regime. The lower bounds match these upper bounds for both equally-covered and k-uncovered regimes.

Regret Minimization with Offline Data

For regret minimization, they present GW-OOPE, which uses Gaussian width concentration instead of a union bound to select designs in each phase. The upper bound for GW-OOPE is RegT(A) ≤ 12Xml=1ϵl min + 2T min<ϵm, ζβm(A(12ϵm), V m off). The expected regret lower bounds match the upper bounds for both equally-covered and k-uncovered regimes.

How it works

The authors identify limitations of existing regret guarantees for standard bandit algorithms like phased elimination and LinUCB when augmented with offline data due to conservative bounds on the reward prediction error. Using a tighter bound on the prediction error, they design a phased elimination algorithm incorporating offline data and show an upper bound on its regret. The analysis utilizes Gaussian width concentration to replace the union bound in phased elimination, leading to improved regret guarantees.

Regret Minimization with Offline Data

For regret minimization, they present GW-OOPE, which uses Gaussian width concentration instead of a union bound to select designs in each phase.

1. Offline-to-online learning algorithms can achieve regret bounds of order Oe(k√T) in partially covered offline data regimes, improving upon existing bounds of Oe(√dkT) by exploiting tighter control of the reward prediction error using concentration of Lipschitz functions of Gaussian vectors.

2. Phased elimination algorithms incorporating offline data can achieve a regret bound of Oe(k√T), matching the lower bound under partial coverage, overcoming the previous p d/k gap observed in existing methods like LinUCB and phased elimination.

3. For pure exploration with offline data, a Gaussian Width Offline-to-Online Fixed Design (GW-OOFD) algorithm can provide a simple regret upper bound of RegT(A) ≤ 2 min N∈AllocT (A) ζ√2 log(2/δ)(A, V˜(N)), which is lower bounded by k√T in the k-uncovered regime.

4. The GW-OOPE algorithm can achieve a regret bound of RegT(A) ≲ Tϵ¯ in the equally-covered case and RegT(A) ≲ k√T + βm√mkT log(8m2T)o in the k-uncovered case, matching lower bounds for any algorithm.

5. Adaptive algorithms using GW-OOPE can achieve a simple regret lower bound of Eµ[SRegT(θ, Aˆ)] ≥ ∆2√Tr Σ + supa∈A∥a∥V˜−10p log(1/δ), providing tighter guarantees than previous bounds for the same coverage regimes.

We study how offline data can help online learning in stochastic linear bandits. We focus on partially covered offline data, where only k of the d feature directions are poorly covered. Existing analyses of algorithms based on phased elimination and optimism give regret bounds of order O(sqrt dkT) in this regime. We show that this dependence arises from loose control of the reward prediction error, and obtain tighter bounds using concentration of Lipschitz functions of Gaussian vectors. Based on this analysis, we develop a phased elimination algorithm with regret O(k sqrt T). For the unit Euclidean ball, we prove a matching regret lower bound under partial coverage, as well as matching upper and lower bounds under uniform offline coverage. We also study pure exploration with offline data, providing algorithms and matching upper and lower bounds on simple regret for fixed-design algorithms and also for adaptive algorithms under partial and uniform coverage of offline data.