New Lower Bound and Upper Bounds on the Regret for Online Sparse Linear Regression

arXiv:2610.11551 · cs.LG · Submitted 2026-10-08 · Read on arXiv

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

Tom: Today's paper: "New Lower Bound and Upper Bounds on the Regret for Online Sparse Linear Regression".

Jane: The gist The authors give the first lower bound on the minimax regret of online sparse linear regression and design algorithms with better upper bounds without regularity assumptions,

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

Title and authors: Tom: So, beyond just setting the lower bound, the paper actually proposes two distinct algorithms based on whether b0 is greater than or less than two.

Jane: That’s right. For the case where b0 is greater than or equal to two, they propose an algorithm that achieves a regret bound of O(d b0 over sqrt T log N).

Lu: That's an improvement over some previous bounds, like O(d squared over b squared zero over sqrt T log N), which they reference <ref:2610.11551#pg1>.

Meng: An improvement in the constant factor is good, but what about the cases where b0 is small, say one or zero? They seem to have different approaches there.

Tom: In the case where b0 is less than or equal to one, they propose an algorithm that achieves a regret bound of O(sqrt N T log T).

Jane: That’s a significant improvement over some older bounds like O(N to the minus q over three minus two q T to the minus q times three minus two q), where q is between zero and one.

Lu: It seems they've tailored their approach based on how much extra information you get after the initial prediction step.

Meng: So, if I'm building a system, knowing whether b0 is large or small tells me which algorithm to try first to get a better practical result?

Tom: Precisely. The paper suggests that the number of additional observed attributes b0 may actually decide the complexity of the problem in terms of the regret bound structure.

Jane: And this applies to both deterministic and stochastic algorithms, meaning whatever bound they get for one type, it holds for both.

The paper's summary: Tom: Let's look at how these proposed algorithms actually improve upon what was out there before this work.

Jane: For the case where b0 is greater than or equal to two, they improve previous bounds by a factor of O(d b0) in terms of the dependence on d and b0.

Lu: That factor is significant because it shows how much better their new method is compared to what was known previously for those specific conditions.

Meng: So, if you have a lot of extra attributes, this paper gives you a concrete way to beat the old performance metrics by leveraging that extra information more effectively.

Tom: Exactly. And for the case where b0 is less than or equal to one, their new algorithm achieves O(sqrt N T log T), which improves upon previous bounds in terms of their dependence on T alone.

Jane: That means the scaling with time instances, T, gets better when you have very little extra attribute information.

Lu: It's interesting how they managed to get these upper bounds to match the lower bounds they established earlier, showing a tight characterization of the problem complexity.

Meng: I wonder if this means that for practical deployment, we can expect much better performance if we can design our system around the correct value of b0.

Tom: That's what they are pointing toward. The paper suggests that while O(sqrt T) is optimal in some sense, the number of extra attributes b0 plays a role in how you approach that optimality.

The paper's improvements: Jane: So, wrapping up this discussion on "New Lower Bound and Upper Bounds on the Regret for Online Sparse Linear Regression," it seems the main implication is that we now have a clear understanding of where the complexity lies.

Tom: We have established that there is a lower bound of Omega of sqrt T on the minimax regret for all b0 greater than or equal to zero and less than d minus b.

Lu: This lower bound shows how problem-dependent parameters decide the information theoretic complexity of (d, b, b0)-OSLR.

Meng: So we can't just assume a fixed performance level; we have to consider the structure of our data access constraints when designing sparse learning algorithms.

Jane: And they’ve provided upper bounds for both cases: O(sqrt T) for b0 greater than or equal to two, and O(sqrt N T log T) for b0 less than or equal to one.

Tom: These results show that the O(sqrt T) factor is optimal, but the number of additional observed attributes b0 may decide the complexity of the problem.

Jane: The regret bound for deterministic algorithms implies an identical lower bound for stochastic algorithms, which is pretty important for applying these bounds generally.

Lu: The proposed algorithms improve previous bounds by a factor of O(d b0) in the case where b0 is greater than or equal to two.

Meng: And Algorithm three with the O(sqrt N T log T) regret bound, is optimal up to O(log T) <ref:2610.11551#pg1>.

Tom: The expected regret for Algorithm two ends up being O(c U L sqrt N T log T), which concludes their proof <ref:2610.11551#pg1>.

Conclusion: Jane: So, to finish this up on "New Lower Bound and Upper Bounds on the Regret for Online Sparse Linear Regression," it’s clear that the complexity hinges on b0.

Tom: Indeed, we have established that there is a lower bound of Omega of sqrt T on the minimax regret for all b0 greater than or equal to zero and less than d minus b.

Lu: This lower bound shows how problem-dependent parameters decide the information theoretic complexity of (d, b, b0)-OSLR.

Meng: We can't just assume a fixed performance level; we have to consider the structure of our data access constraints when designing sparse learning algorithms.

Jane: They’ve provided upper bounds for both cases: O(sqrt T) for b0 greater than or equal to two, and O(sqrt N T log T) for b0 less than or equal to one.

Tom: These results show that the O(sqrt T) factor is optimal, but the number of additional observed attributes b0 may decide the complexity of the problem.

Jane: The regret bound for deterministic algorithms implies an identical lower bound for stochastic algorithms, which is pretty important for applying these bounds generally.

Lu: The proposed algorithms improve previous bounds by a factor of O(d b0) in the case where b0 is greater than or equal to two.

Meng: And Algorithm three with the O(sqrt N T log T) regret bound, is optimal up to O(log T) <ref:2610.11551#pg1>.

Tom: The expected regret for Algorithm two ends up being O(c U L sqrt N T log T), which concludes their proof <ref:2610.11551#pg1>.

Jane: This paper provides a new framework for understanding the limits of online sparse linear regression and how extra information affects the scaling.

Xiaofeng Cao, Junfan Li, Langzhang Liang, Mingwei Xu, Xiao Zhang

School of Computing and Artificial Intelligence, Shanghai University of Finance and Economics · School of Computer Science and Technology, Tongji University, Shanghai, China · AI3 Institute, Fudan University and Shanghai Innovation Institute · School of Artificial Intelligence, Jilin University

cs.LG

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: This work was accepted by IJTCS-FAW 2026

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: The gist The authors give the first lower bound on the minimax regret of online sparse linear regression and design algorithms with better upper bounds without regularity assumptions, characterizing

Key concepts

(d, b, b0)-OSLR: This is a variation of online sparse linear regression where an algorithm can only see $b$ attributes for prediction and then observes $b_0$ additional attributes after each prediction. This setting is proven to be NP-hard, meaning it's computationally difficult.
Minimax Regret: Regret measures the difference between the loss incurred by an online algorithm and the loss of the best possible fixed parameter vector $w$. The goal is to find an algorithm whose regret grows as slowly as possible across all possible problem instances.
Lower Bound $\Omega(\sqrt{T})$: This is a mathematical guarantee showing that no algorithm can achieve a regret smaller than this rate, regardless of how clever the algorithm is. It sets the fundamental limit on performance for this type of online learning problem.
b0 (Additional Attributes): The parameter $b_0$ represents the number of extra attributes an algorithm can observe after making its initial prediction. The paper shows that increasing $b_0$ changes the complexity of the problem and affects how well algorithms can perform.

Terminology

Regret — 1× in this paper · explained in 3 episodes

Summary

The gist The authors give the first lower bound on the minimax regret of online sparse linear regression and design algorithms with better upper bounds without regularity assumptions, characterizing how minimax regret scales with problem-dependent parameters

Problem Setting

Online sparse linear regression (OSLR) is studied where an algorithm is restricted to accessing only b out of d attributes per instance for prediction and b0 ≥ 0 additional attributes after prediction, which was proved to be NP-hard The regret is defined as Reg(w) = X t=1 (ˆyt − yt) squared - X t=1 w T x t - yt squared for any w ∈ Wb, where Wb = w ∈ R d: w0 ≤ b, w2 ≤ U > 0 is a constant A relaxation of OSLR allows algorithms to observe additional b0 = O(b ln d) attributes after each prediction, denoted as (d, b, b0)-OSLR. Previous work showed that (d, b, b0)-OSLR is NP-hard.

Lower Bound on Minimax Regret

The main result establishes a lower bound on the minimax regret of (d, b, b0)-OSLR for all b0 ≥ 0 and b0 < d − b, which is given by omega(√T). This lower bound shows how problem-dependent parameters decide the information theoretic complexity of (d, b, b0)-OSLR. The proof involves constructing hard instances where the loss function only depends on S and z h t. The lower bound is derived by analyzing the difference between expectations under a reference distribution Pu and the actual distribution Ph, which leads to a bound involving T, b, b0, d, and γ.

Algorithm for b0 ≥ 2

For the case where b0 ≥ 2, an algorithm is proposed that enjoys a O(d b0 / √T ln N) regret bound. This improves upon previous bounds like O(d squared / b 2 0 √T ln N). Algorithm 4 involves constructing N subsets of [d], where each element indexes attributes, and simultaneously learning a probability distribution over the N subsets and the optimal parameter w i for all i. The algorithm uses an online mirror descent (OMD) approach to update parameters, with learning rates η and λ t,i defined based on complexity measures.

Algorithm for b0 ≤ 1

In the case where b0 ≤ 1, Algorithm 2 is proposed to achieve a O(√NT ln T) regret bound. This improves upon previous bounds like O(N(1-q) / (3-2q) T(2-q)). Algorithm 3, called CORRAL, is used as a black-box algorithm to obtain the probability distribution pt by CORRAL(c t, I t−1), which is essential for controlling the variance of the gradient estimator.

Conclusion on Optimality

The work reveals that the O(√T) factor is optimal, and the number of additional observed attributes b0 may decide the complexity of the problem. The regret bound for deterministic algorithms implies an identical lower bound for stochastic algorithms. The proposed algorithms improve previous bounds by a factor of O(d b0) in the case b0 ≥ 2. The regret bound for Algorithm 3 is O(√NT ln T), which is optimal up to O(ln T). The expected regret of Algorithm 2 is O(cUL√NT ln T), which concludes the proof.

Discussion

The authors suggest future research directions, including proving a stronger lower bound, conjecturing the O(√1/b) factor can be removed, and designing better algorithms for cases b0 ≥ 2, b0 = 1 and b0 = 0. They note that they did not design a customized algorithm for the case b0 = 1. The CORRAL algorithm is used as a black-box algorithm for simplicity. The work opens several directions for future research

References

The references listed are 8, 9, 10, 14, 8, 9, 7, and the others listed in the paper The list of references is provided on page 14. The specific references used are cited throughout the text.

Appendix

The appendix contains technical lemmas, including the Bregman divergence and optimality conditions for OMD. Lemma 2 provides bounds on c tilde t,i and its second-order central moment. The proof of Theorem 3 relies on Lemma 3 which states that CORRAL ensures a certain regret bound. The analysis of R1 and R2 involves inequalities derived from the Bregman divergence and properties of OMD, as well as Pinsker's inequality. The final regret bound is derived by combining the upper bounds on E[R1] and E[R2]. The analysis of R2 uses Lemma 3 to show that the regret is O(cUL√NT ln T). The technical details are provided on page 15 and page 18. The final result combines the bounds derived in section B of Theorem 3. The analysis of R2 uses Lemma 3 to show that the regret is O(cUL√NT ln T). The technical details are provided on page 15 and page 18. The final result combines the bounds derived in section B of Theorem 3. The analysis of R2 uses Lemma 3 to show that the regret is O(cUL√NT ln T). The technical details are provided on page 15 and page 18. The final result combines the bounds derived in section B of Theorem 3. The analysis of R2 uses Lemma 3 to show that the regret is O(cUL√NT ln T). The technical details are provided on page 15 and page 18. The final result combines the bounds derived in section B of Theorem 3. The analysis of R2 uses Lemma 3 to show that the regret is O(cUL√NT ln T). The technical details are provided on page 15 and page 18. The final result combines the bounds derived in section B of Theorem 3. The analysis of R2 uses Lemma 3 to show that the regret is O(cUL√NT ln T). The technical details are provided on page 15 and page 18. The final result combines the bounds derived in section B of Theorem 3.

Improvements for AI systems

Better characterization of information theoretic complexity for Online Sparse Linear Regression (OSLR) by proving a lower bound on minimax regret: "We first prove a omega(√T) lower bound on the minimax regret for all b0 ≥ 0 and b0 < d − b. This allows researchers to understand how the number of additional observed attributes b0 may decide the complexity of the problem," guiding algorithm design based on information constraints.
Development of a new, improved algorithm for sparse learning when extra attributes are observed: In the case of b0 ≥ 2, we propose a new algorithm that enjoys a O(d/b0√T ln N) regret bound, improving previous O(d 2/b 20√T ln N) regret bound [6]. This provides a more efficient practical solution for scenarios where some extra information is available.
Design of an improved algorithm for the sparse learning case with minimal extra attributes: In the case of b0 ≤ 1, we propose a new algorithm that enjoys a O(√NT ln T) regret bound, improving previous O(N(1-q)3(-2q)T(2-q)3(-2q), q ∈ (0, 1)) regret bound [11] in terms of the dependence on T. This offers a tighter complexity analysis for scenarios where only the base sparse observation is available.
Implementation of an algorithm to learn optimal parameters for sparse models: Our algorithm randomly selects a model ft,It following pt, observes the attributes indexed by SIt, and outputs ft,It (xt). This system can perform online prediction while dynamically selecting the most relevant subset of features based on observed data and a learned probability distribution over subsets.
Construction of robust estimators for loss and gradient in high-dimensional sparse settings: The algorithm constructs estimators like c˜t,i =w⊤ t,iht[Si, Si]wt,i − 2ytw⊤ t,ixˆt[Si] + y squared t, which are derived from the underlying loss function to estimate the instantaneous loss and gradient effectively.
Utilization of a customized master algorithm for bandit feedback: The paper introduces CORRAL (Algorithm 3) to manage probability updates: CORRAL follows OMD with the regularizer ψt(p) = −PN i=1 1/ηt ln pi and non-decreasing learning rates η1 ≤ η2 ≤... ≤ ηT. This allows the system to adapt its selection strategy effectively even when dealing with high variance or vanishing probabilities in the bandit setting.

Abstract

We study online sparse linear regression (OSLR) where any algorithm is restricted to accessing only b out of d attributes per instance for prediction and b 0 at least 0 additional attributes after prediction, which was proved to be NP-hard. Previous work focused on designing computationally efficient algorithms under regularity assumptions, but did not characterize its information theoretic complexity. In this work, we give the first lower bound on the minimax regret of OSLR and design algorithms with better upper bounds without regularity assumptions. We characterize how minimax regret scales with problem-dependent parameters, capturing the information theoretic complexity of OSLR.