Online Estimation via Offline Estimation: An Information-Theoretic Framework

Weakness:

While the paper provides a comprehensive theoretical analysis, it acknowledges the computational inefficiency of the proposed Oracle-Efficient Online Estimation (OEOE) framework in general cases. The authors could improve this aspect by exploring potential heuristic approaches or approximations that could offer practical computational advantages, even if they come with some trade-offs in theoretical guarantees.

Indeed, to improve results regarding computation, in Section 4.2, we consider the cases for conditional density estimation where an efficient online density estimation algorithm is present. This result presents a tradeoff between computational efficiency and theoretical guarantees. The OEOE with conditional density estimation can be computationally efficient with an efficient online density estimation algorithm but is not optimal statistically in terms of minimax regret. This result is an extension of the general reduction from OEOE to online learning in Appendix D. The tradeoff also extends to the classification and regression setup we consider.

The paper lacks empirical validation of the proposed framework. Including a few experimental results or simulations demonstrating the practical performance of the OEOE framework could significantly strengthen the paper. Even a small set of experiments showing the feasibility and comparative performance against existing online estimation methods would provide valuable insights.

This paper presents a purely theoretical result. We look forward to seeing experiments in future works.

Certain sections of the paper, particularly the detailed proofs and technical discussions, are dense and may be difficult for readers to follow. Improving the clarity of these sections by adding more intuitive explanations, step-by-step breakdowns of complex proofs, or illustrative examples could make the paper more accessible. This would help readers better understand the contributions.

Please see the general rebuttal for a detailed clarification of the theoretical insights.

Question:

Have you considered any specific real-world applications or practical scenarios where this framework could be directly applied or tested? How do you envision the implementation challenges being addressed in such cases?

Transferring a theoretical algorithm to real-world applications requires domain specification and engineering. This work provides guidance in algorithm design principles for this new setup rather than improving existing online learning algorithms. Together with [1], it shows that using offline regression outputs suffices for interactive decision-making. For any reinforcement learning task, the proposed algorithm involves: (1) Performing offline regression to approximate the RL model, (2) Creating a version space (trust regions) of the RL model, (3) Averaging the possible RL models to create an online regression output, and (4) Using the E2D algorithm in [1] to balance exploration and exploitation. This work provides guidance in steps (2) and (3).

[1] The Statistical Complexity of Interactive Decision Making Dylan J. Foster, Sham M. Kakade, Jian Qian, Alexander Rakhlin, (2021)

The theoretical results are derived under certain assumptions, such as the metric-like structure of the loss function and the bounded offline estimation error. How sensitive are your results to these assumptions? Could you provide insights or potential extensions to your framework that could handle more relaxed or different sets of assumptions, thereby increasing the generalizability of your findings?

The metric-like losses are quite general, covering cases such as square loss, absolute loss, and KL divergence with a bounded density ratio. Assuming the offline estimation error is bounded is reasonable. The only restrictive assumption is knowing an upper bound for the offline estimation error for Algorithm 1. When this assumption is not met, Algorithm 2 in Appendix D, which has worse but still sublinear statistical regret, can be used.

I thank the authors for responding to my comments. I want to increase my score based on the response provided by the authors.

I understand that many works assume knowing the intrinsic of the data-generating, but assuming that is known would exclude some cases of statistical estimation, which might reduce the generality of OEOE.

The OEOE is concerned with transforming an offline guarantee into an online one. Even when the data-generating process is unknown, whenever the offline guarantee can be obtained, the OEOE can be applied to transfer it into an online guarantee.

Can you provide a quantified definition of “oracle-efficient”? Despite it has been heavily used in the paper, I can only find some rough explanations after searching through the text.

We define the term 'oracle-efficient' in lines 106-107: 'An algorithm is termed oracle-efficient if it attains low online estimation error (2) in the OEOE framework.' In other words, the algorithm is statistically efficient, given access to an offline oracle without any extra labeling information.

I have not checked the proof of theorem 3.2 and theorem 4.1 carefully, so maybe I am wrong, but the hardness construction in them seems a bit ill-conditioned. The worst case is searched through all offline oracles, the adversary seems overpowered.

Yes, the adversary can search through all offline oracles. However, it is not overpowered for two reasons: First, in the OEOE setup, the learner has no information on how the offline oracle output is generated. Second, the offline oracle must still satisfy the stringent offline guarantee.

As stated in Theorem 4.1, there is no computational algorithm in OEOE. So could it be that the definition of this framework includes some unwanted cases that should be excluded? And does that mean that this framework needs some redefinition?

It is unfortunate that there is no computational algorithm in OEOE, which is interesting in its own right. To improve computational results, Section 4.2 considers cases for conditional density estimation where an efficient online density estimation algorithm is present. This result presents a tradeoff between computational efficiency and theoretical guarantees. The OEOE with conditional density estimation can be computationally efficient with an efficient online density estimation algorithm but is not optimal statistically in terms of minimax regret. This result extends the general reduction from OEOE to online learning in Appendix D. The tradeoff also applies to the classification and regression setup we consider. Refining the framework to more specific cases could ensure the presence of an efficient algorithm. We look forward to further developments in this area.

Weakness:

The organization of the paper requires significant improvement as it is currently hard to follow. The introduction section occupies almost half of the main text, leaving the main methodology and theoretical results insufficiently discussed and lacking in clear theoretical insights.

Please see the general rebuttal for a detailed clarification of the theoretical insights.

Question:

This work discusses the impossibility of memoryless oracle-efficient algorithms, I wonder what if the learner can select the online estimator at time t as a function of the most recent "w" offline estimators, where w is the window size.

Yes, Algorithm 2 in Appendix D presents a general reduction from OEOE to online learning, which only requires the learner to remember the most recent $N$ offline estimators, with tradeoffs in regret depending on $N$.

In addition, the authors present a fine-grained perspective on the computational complexity of oracle-efficient estimation for conditional density estimation. Are there any insights for the other two problems of binary classification and regression?

Yes, the technique for conditional density estimation extends from Algorithm 2 in Appendix D, which also applies to binary classification and regression. The conclusion remains similar: if efficient online algorithms exist for binary classification and regression, then efficient algorithms for the OEOE setting also exist.

Weakness:

The main proposed algorithm is somewhat brute-force and just keeps the hypotheses that agree with the data – in this case it is covariance-estimator pairs.

This is the optimal approach from an information-theoretical perspective, achieving the optimal minimax guarantee for our setup. We envision better algorithms under more assumptions on the model class in future works.

It is not obvious that the OEOE model, in which both all past covariates as well as all past offline estimators are saved, actually captures aspects such as limited past data availability and compression of past observations and decisions. This is because storing an estimator may be much more storage-consuming than the value itself (e.g., as in binary classification). Generalizations based on finite past data seems to be challenging.

If only the estimator at the current round is remembered without covariates, Theorem 3.3’s lower bound applies, making the algorithm inefficient. However, finding efficient methods to remember the version space without previous estimators allows the use of Algorithm 1. Additionally, Algorithm 2 in Appendix D can mitigate this issue by storing only the $N$ most recent estimators, albeit with increased regret.

Question:

In line 33 it is claimed that statistical estimation is mainly about fixed design. Random design of the covariate-value is also commonly considered.

While random design is common, the OEOE setup reveals covariates, making fixed design regression more appropriate.

Theorem 3.3 is an impossibility result, showing that even a memoryless estimator is not efficient, even if it is improper. Doesn't this trivially imply the same result for the subset of proper learners? Why thus a different result is needed ?

In our OEOE setup, the adversary controls the offline oracle, and the learner only has the offline guarantee. If the oracle is improper, the adversary is stronger, so a lower bound with an improper oracle does not imply one with a proper oracle.