Online Estimation via Offline Estimation: An Information-Theoretic Framework (2404.10122v1)

Abstract: $ $The classical theory of statistical estimation aims to estimate a parameter of interest under data generated from a fixed design ("offline estimation"), while the contemporary theory of online learning provides algorithms for estimation under adaptively chosen covariates ("online estimation"). Motivated by connections between estimation and interactive decision making, we ask: is it possible to convert offline estimation algorithms into online estimation algorithms in a black-box fashion? We investigate this question from an information-theoretic perspective by introducing a new framework, Oracle-Efficient Online Estimation (OEOE), where the learner can only interact with the data stream indirectly through a sequence of offline estimators produced by a black-box algorithm operating on the stream. Our main results settle the statistical and computational complexity of online estimation in this framework. $\bullet$ Statistical complexity. We show that information-theoretically, there exist algorithms that achieve near-optimal online estimation error via black-box offline estimation oracles, and give a nearly-tight characterization for minimax rates in the OEOE framework. $\bullet$ Computational complexity. We show that the guarantees above cannot be achieved in a computationally efficient fashion in general, but give a refined characterization for the special case of conditional density estimation: computationally efficient online estimation via black-box offline estimation is possible whenever it is possible via unrestricted algorithms. Finally, we apply our results to give offline oracle-efficient algorithms for interactive decision making.

• The paper introduces a novel framework that leverages black-box offline estimators to achieve near-optimal online estimation error bounds within finite classes.
• It combines rigorous statistical and computational analyses, establishing minimax upper bounds while proving inherent computational limitations in the conversion process.
• The research highlights a special case in conditional density estimation where efficient online estimation is feasible, offering promising implications for interactive decision making.

Oracle-Efficient Online Estimation via Black-Box Offline Estimators

Introduction to OEOE Framework

Oracle-Efficient Online Estimation (OEOE) introduces a novel framework for transforming offline estimation algorithms into online estimation tools using black-box offline estimators. Offline estimators, regardless of their internal mechanisms, are leveraged to interact indirectly with a data stream, aiming to fulfill online estimation tasks. This paper delineates the viability of translating offline estimation capacities to address online estimation demands without direct access to raw data, relying instead on sequences of estimators produced by offline algorithms.

Statistical and Computational Complexity Insights

The analysis within the OEOE framework delivers comprehensive insights into the statistical and computational realms, particularly focusing on finite classes:

Statistical Complexity for Finite Classes

• Algorithm Design and Minimax Upper Bound: The paper introduces an algorithm achieving near-optimal online estimation error, closely mirroring the minimax rates known for offline and online estimations within finite classes. Specifically, it attains an online estimation error of $$O((\varepsilon+1)\min\{\log|\mathcal{F}|,|\mathcal{X}|\})$$, optimizing for the interaction between the offline oracle’s estimation error parameter $\varepsilon$ and the cardinalities of function classes and covariate space.
• Fundamental Limitations: A lower bound is established, showcasing that the derived upper bound cannot be substantially improved. This limitation underscores the intrinsic difficulty in enhancing performance beyond established bounds within the considered framework.

Computational Complexity Analysis

• Impossibility of Efficient Black-Box Conversion: It is proven that under general circumstances, efficiently transforming offline estimation algorithms into online estimators within the OEOE framework is computationally infeasible. This finding fundamentally challenges the quest for computational efficiency in black-box conversions for online estimations.
• Conditional Density Estimation Analysis: Yet, a distinguished case exists for conditional density estimation tasks, where computationally efficient online estimation is feasible, marking an exception to the broad computational intractability observed otherwise.

Interactive Decision Making Implications

Utilizing the developed oracle-efficient algorithms for online estimation, the paper sheds light on their potential implications for interactive decision making. Specifically, it extends the application of oracle-efficient online estimation to derive offline oracle-efficient algorithms for tackling interactive decision-making challenges.

Future Directions and Theoretical Contributions

The OEOE framework, while exploring the boundaries of online estimation through offline oracle interactions, lays the groundwork for future explorations in algorithm design, interactive decision-making strategies, and computational efficiency optimization. Notably, this research contributes to clarifying the limitations, potentials, and precise computational requirements for oracle-efficient transformations in online estimation tasks.

Critical Reflections

• The established computational intractability highlights a significant challenge in the field of oracle-efficient online estimation, particularly emphasizing the need for nuanced understanding and specific case analyses to identify potential pathways for computational optimization.
• The conditional density estimation scenario presents a fertile ground for further investigation, potentially enabling advancements in computational efficiency and algorithm design that could extend to broader applications beyond the immediate scope outlined within this framework.
• The synthesis of statistical complexity analysis with interactive decision-making implications reinforces the interconnected nature of theoretical algorithm design and practical application, encouraging continued exploration across these intersecting domains.