Provably Efficient Offline-to-Online Value Adaptation with General Function Approximation

arXiv:2604.13966 · cs.LG · Submitted 2026-04-15 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom.

Jane: And I'm Jane, here with Lu, Meng and our in-house large language model, Lalam.

Tom: Today's paper: "Provably Efficient Offline-to-Online Value Adaptation with General Function Approximation".

Jane: The gist The authors study value adaptation in offline-to-online reinforcement learning under general function approximation and propose an algorithm, O2O-LSVI,

Tom: First, who's behind it and why it matters.

Paper summary: Tom: So we're moving on to summarizing this work, "Provably Efficient Offline-to-Online Value Adaptation with General Function Approximation." The central thesis is that we can study value adaptation in offline-to-online reinforcement learning when using general function approximation. They start by establishing a minimax lower bound demonstrating that adaptation isn't always as efficient as pure online RL on certain hard instances.

Jane: And they then pivot to the positive side, showing that under a novel structural condition on those offline pretrained value functions, they can propose an algorithm called O2O-LSVI which has problem-dependent sample complexity that provably improves over pure online RL. Why is this a big deal?

Tom: Because the core of their approach involves formulating offline-to-online adaptation as a value-function adaptation framework and establishing an information-theoretic lower bound showing that without structural conditions on the offline pretrained value functions, adaptation can be fundamentally as hard as training from scratch using online interaction.

Lu: They introduce a specific condition called beta-separable value gaps, which is meant to reflect the empirical success of offline-to-online RL algorithms. This condition means that Qref is close to the optimal Q function in some regions while exhibiting significantly larger errors in others.

Meng: So they are saying that this structural condition unlocks provable improvement over pure online RL, and they prove it using a value-function adaptation framework and information theory to show what's fundamentally possible.

Jane: They then propose O2O-LSVI, which is the algorithm itself, following a least-squares value iteration framework with two phases: a reverse planning phase that does backward least-squares updates for confidence bounds and an episode-collecting phase that interacts with the environment by executing a greedy policy induced by Qk h.

Tom: The algorithm uses least-square regression to estimate the Q function via general function approximation by leveraging a generic function class Fh, and at each iteration k and stage h, it regresses toward Bellman targets to construct upper and lower confidence approximations of Q star h.

Lu: This entire setup is designed to handle the general function approximation aspect while ensuring that the algorithm constructs these confidence bounds correctly based on the structure they identified.

Jane: The theoretical guarantees they establish under well-specified beta-separable value gaps show a regret result in Theorem six point two that can be translated into a sampling complexity bound, showing that O2O-LSVI is epsilon-optimal with probability at least one minus five delta using Oe(min H4dF (NFNB) beta + rho H4dF (NFNB) epsilon squared, H4dF (NFNB) epsilon squared) online episodes.

Tom: So the number of episodes needed is explicitly related to the parameters of the function class and the value gap structure, which is what makes it problem-dependent. This moves beyond just saying "it works" to giving a concrete complexity bound for this specific type of adaptation.

Meng: That's what matters for deployment; we want complexity bounds that tell us how much compute time we have to wait before we can trust the adaptation process in a real-world scenario with limited online interaction.

Jane: It really frames offline pretraining not just as a way to get a good starting point, but as something that has specific mathematical benefits when you know the structure of the value gaps.

Tom: And it sets up the next part of our discussion, where we look at what this means for real-world application and how we can use these ideas.

Conclusion: Tom: The paper, "Provably Efficient Offline-to-Online Value Adaptation with General Function Approximation," by Shangzhe Li and Weitong Zhang, tackles the core problem of efficiently adapting value functions from offline data to an online setting when using general function approximation. They prove that under a specific structural condition called beta-separable value gaps, they can achieve provably better sample complexity than pure online RL.

Jane: In simple terms, what does this mean for us? It means that if we have offline data, and we can characterize the structure of the value function errors in a way that fits this beta-separable gap condition, then adapting our model using limited online interaction is more efficient than just starting from scratch online.

Lu: It clarifies that separable value gaps are a sufficient structural condition for improved value transfer; it's not necessary for every possible transfer method, but when present, it provides a mathematically rigorous path to better performance.

Meng: For the people building AI systems, this suggests that instead of just hoping the offline data is good enough in general, we should look for those specific structural properties—like those value gaps—to ensure our adaptation process scales well and doesn't require an impossible amount of online interaction.

Tom: It shifts the focus from just empirical performance to finding the underlying mathematical conditions that guarantee efficiency, which is what we do when we want to build systems that actually work reliably in production.

Jane: So, while it doesn't change everything, it gives us a way to understand *why* some offline pretraining methods succeed while others don't. It provides a concrete lens for analyzing the transfer of knowledge from the offline data to the target environment.

Lu: The implication is that we need better ways to characterize these value gaps in practice so that we can design algorithms, like O2O-LSVI, that are specifically tailored to those structures for maximal efficiency gains.

Tom: Exactly. We're moving from a general study of adaptation to a targeted approach based on structural properties, which is where the practical engineering happens.

Shangzhe Li, Weitong Zhang

UNC Chapel Hill

cs.LG

Submitted: 2026-04-15

Updated: 2026-10-07

Comments: 44 pages, 2 tables

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 78/100

The gist: The gist The authors study value adaptation in offline-to-online reinforcement learning under general function approximation and propose an algorithm, O2O-LSVI, that achieves provably better sample

Key concepts

Offline-to-Online Adaptation: This setting involves taking a Q-function trained on fixed, pre-collected offline data and using only a small amount of new, real-world online interaction to improve its performance in the current environment. The goal is to efficiently transfer knowledge from the static dataset to dynamic online learning.
$\beta$-Separable Value Gaps: This is a novel structural condition placed on the offline-pretrained value functions. It describes a scenario where the pre-trained Q-function is very accurate in some parts of the state space but has significantly larger errors in other regions. This structure allows the proposed algorithm to exploit areas of high confidence while still learning from uncertain regions.
O2O-LSVI: This is a proposed adaptation algorithm based on the least-squares value iteration (LSVI) framework. It alternates between a reverse planning phase to build confidence bounds and an episode-collecting phase where it interacts with the environment using the current Q-function estimates to gather new data for further updates.
General Function Approximation: This refers to using a flexible function class $F_h$ (like neural networks) rather than assuming a specific functional form for the Q-function. This allows the algorithm to estimate complex value functions without being restricted by simple linear or polynomial models, making it applicable to general reinforcement learning problems.

Terminology

Adaptation — 4× in this paper · explained in 1 episode
Online Adaptation — 1× in this paper · explained in 3 episodes
Online Reinforcement Learning — 1× in this paper · explained in 2 episodes
Reinforcement Learning — 1× in this paper · explained in 4 episodes
Sample Complexity — 1× in this paper · explained in 20 episodes

Summary

The gist The authors study value adaptation in offline-to-online reinforcement learning under general function approximation and propose an algorithm, O2O-LSVI, that achieves provably better sample complexity than pure online RL under a novel structural condition called β-separable value gaps.

How it works

The paper first characterizes the difficulty of this setting by establishing a minimax lower bound showing that adaptation can be no more efficient than pure online RL on certain hard instances On the positive side, under a novel structural condition on the offline-pretrained value functions, O2O-LSVI is proposed as an adaptation algorithm with problem-dependent sample complexity that provably improves over pure online RL.

The core of the approach involves formulating offline-to-online adaptation as a value-function adaptation framework and establishing an information-theoretic lower bound showing that without structural conditions on the offline pretrained value functions, adaptation can be fundamentally as hard as training from scratch using online interaction. The authors then propose a novel structural condition called β-separable value gaps which reflects the empirical success of offline-to-online RL algorithms, where Qref is close to the optimal Q function in some regions while exhibiting significantly larger errors in others.

Key Components and Algorithm Design

The proposed algorithm, O2O-LSVI, follows the least-squares value iteration (LSVI) framework. It consists of two main phases in each iteration:

A reverse planning phase performs backward least-squares value updates to construct confidence bounds and update the state–action value estimates.
An episode-collecting phase interacts with the environment by executing the greedy policy induced by Qk h, thereby collecting new trajectories for subsequent updates.

The algorithm employs least-square regression to estimate the Q-function via general function approximation by leveraging a generic function class Fh. At each iteration k and stage h, it regresses toward the Bellman targets to construct upper and lower confidence approximations of Q⋆h.

Theoretical Guarantees

The theoretical analysis establishes several bounds for O2O-LSVI under well-specified β-separable value gaps. The regret result in Theorem 6.2 shows that the cumulative Regret(K) is bounded by Oe(min nH4dF log(NFNB) β + H2p ρKdF log(NFNB), H2p KdF log(NFNB)). This result can be translated into a sampling complexity bound, showing that the algorithm is ϵ-optimal with probability at least 1 − 5δ using Oe(min H4dF log(NFNB) βϵ + ρH4dF log(NFNB) ϵ squared, H4dF log(NFNB) ϵ 2) online episodes.

Empirical Validation

The effectiveness of the proposed method is demonstrated through neural-network experiments. On MuJoCo AntMaze, the neural implementation achieves final scores comparable to or higher than those of the evaluated baselines. In a linear MDP, reference switching reduces mean final cumulative pseudo-regret by 15.6% relative to LSVI-UCB.

Conclusion

These results clarify both the limitations and the potential of offline pretraining for accelerating online RL. The paper concludes that separable value gaps are a sufficient structural condition for improved value transfer, not as a necessary condition for every possible transfer method. The proposed O2O-LSVI algorithm achieves improved problem-dependent guarantees over pure online RL and performs effectively in practice.

The paper's pages are labelled as follows:

--- Page 1 ---

Provably Efficient Offline-to-Online Value Adaptation with General Function Approximation Shangzhe Li UNC Chapel Hill Weitong Zhang∗ UNC Chapel Hill Abstract We study value adaptation in offline-to-online reinforcement learning under general function approximation Starting from an imperfect offline pretrained Q-function, the learner aims to adapt it to the target environment using only a limited amount of online interaction We first characterize the difficulty of this setting by establishing a minimax lower bound, showing that even when the pretrained Q-function is close to optimal Q⋆, online adaptation can be no more efficient than pure online RL on certain hard instances On the positive side, under a novel structural condition on the offline-pretrained value functions, we propose O2O-LSVI, an adaptation algorithm with problemdependent sample complexity that provably improves over pure online RL Finally, we complement our theory with neural-network experiments that demonstrate the practical effectiveness of the proposed method 1

--- Page 2 ---

1 Introduction Reinforcement learning (RL) has long been regarded as an effective framework for solving complex sequential decision-making problems across a wide range of domains, including robotic control [Tang et al., 2025], healthcare [Coronato et al., 2020], and language model post-training [Ouyang et al., 2022]. Standard online RL paradigms typically rely on extensive interactions with the environment to optimize the policy To reduce the need for online interaction, offline RL has emerged as an important paradigm in which the algorithm learns from a dataset collected by an existing behavioral policy instead of directly accessing the environment However, the performance of the purely offline RL approaches is usually limited by the coverage and quality of the collected dataset Gulcehre et al., 2022 In particular, when data coverage is poor or when the online environment shifts from the offline environment, the learned value function can become highly inaccurate over parts of the state-action space To address this issue, recent empirical approaches consider the offline-to-online adaptation setting, where an (offline) pretrained RL agent can improve through a limited amount of online interaction and Zhang, 2023 Nakamoto et al., 2023 Huang et al., 2025 Ball et al., 2023 Zhou et al., 2024 On the theoretical side, prior works have analyzed online policy adaptation starting from a suboptimal initial policy Xie et al., 201b as well as hybrid RL settings that combine offline data with online exploration Li et al., 2023 Song et al., 2022 or finetune a pretrained transition kernel Qu et al., 2024 However, current theoretical progress still leaves a significant gap from empirical implementations, which typically leverage offline-pretrained value functions or policies in a model-free manner Nakamoto et al., 2023 Zhou et al., 2024 This gap motivates the following question How can we theoretically characterize the adaptation of value functions in offline-to-online RL?

Improvements for AI systems

Bold Header: Offline-to-Online Value Adaptation using O2O-LSVI

This system can adapt an imperfect offline pretrained Q-function to a target environment using only a limited amount of online interaction, achieving provably more efficient sample complexity than pure online RL under the structural condition of β-separable value gaps.

Bold Header: Improved Sample Complexity Guarantees

The algorithm provides bounds such as Regret(K) ≲ min H2α 2dFβ + αHpρKdF, which is provably better than the general lower bound of omega(H3d 2/ϵ 2) when structural conditions are met.

Bold Header: Robustness to Reference Misspecification

The framework handles an observed reference Q-function, providing a regret bound that includes an error term: Regret(K) ≤ Oemin nH4dF log(NFNB)β + H2pρKdF log(NFNB), H2pKdF log(NFNB)o + HKτ, where τ is the uniform deviation from the true reference.

Bold Header: Certified Reference Switching Mechanism

The system can utilize a mechanism that eventually accepts all correctly specified reference entries while consistently rejecting the misspecified ones, leading to a regret bound of Regret(K) ≲ min H2α 2dFβ + αHpρKdF.

Bold Header: Performance in General Function Approximation

The method is designed for general function approximation, allowing it to adapt value functions even when the offline-pretrained Q-function is not perfectly aligned with the optimal one, provided the structural gap condition holds.