A General (sqrt T gamma T) Lower Bound for Kernel Bandits

arXiv:2610.11082 · stat.ML, cs.IT, cs.LG, math.IT · 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: "A General (sqrt T gamma T) Lower Bound for Kernel Bandits".

Jane: The gist: This paper establishes a general minimax regret lower bound of omega(pT γT / log T) for non-constant continuous kernels on compact domains,

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

Paper summary: Tom: So we've been looking at "A General (sqrt T gamma T) Lower Bound for Kernel Bandits," and to wrap up, this paper shows that for many continuous kernels on compact domains, you can expect your regret to scale as the square root of time times your information gain rate.

Jane: That scaling is what they're proving is the lower bound in the general case, meaning it's hard to do better than that without making some specific assumptions about how fast you learn.

Lu: What this really means for us is that we have a solid floor for our performance when dealing with these types of kernels; we know exactly what the worst-case scenario looks like.

Meng: So if an algorithm you build achieves something closer to that sqrt(T) gamma T scaling, it means you're doing quite well for that kernel structure.

Lalam: And the interesting part is they pinpoint exactly which kernels—like the Matérn or gamma-exponential ones—are the ones where this lower bound actually matches what we can achieve.

Tom: Right, and they also show that if you look at those specific kernels, you can sometimes remove that pesky log factor entirely, getting right down to just sqrt(T) gamma T.

Jane: It's a lot of nuance there; it’s not just one single scaling for everything.

Lu: It shows how the mathematical structure of the kernel dictates the performance limits of the AI system you’re building.

Meng: So for practical applications, this gives us a better roadmap to design functions that fit those favorable kernel conditions.

Lalam: And understanding these lower bounds helps us set realistic expectations for what our models can actually achieve in terms of regret over time.

Tom: It really puts the performance of a kernel bandit into perspective against what's mathematically possible.

Conclusion: Tom: So, we've wrapped up our look at "A General (sqrt T gamma T) Lower Bound for Kernel Bandits," and what they’re showing is that while the general performance limit involves that log term, they’ve narrowed it down to specific kernel families where you can hit a much cleaner scaling of square root times time times your information gain rate.

Jane: Exactly. The authors are basically saying that for most continuous kernels on compact domains, (pT gamma T / T) is the floor you have to accept, but there are specific structures out there—like the Matérn kernel with certain smoothness or the gamma-exponential kernel—where you can drop that log and get straight to sqrt T gamma T.

Lu: That’s significant because it connects the theoretical math directly to what’s achievable in practice. It means we can start designing our AI models knowing precisely which kernel shapes will give us the best possible performance guarantees.

Meng: For someone building a real system, that’s huge. It moves us past just guessing which functions are good and gives us a mathematical roadmap for constructing functions that fit those favorable scaling conditions we just talked about.

Lalam: It sets a new benchmark for what we consider optimal information gain versus regret in these bandit settings. We now have clearer expectations about the performance limits based on the kernel structure we choose.

Tom: Right. So, this paper isn't just math; it’s a guide for engineering our next generation of function spaces and learning policies. Next up, we’re going to look at how those specific kernels actually perform when you put them into action.

Chenkai Ma, Jonathan Scarlett

National University of Singapore

stat.ML, cs.IT, cs.LG, math.IT

Submitted: 2026-10-08

Updated: 2026-10-08

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

Importance score: 71/100

The gist: The gist: This paper establishes a general minimax regret lower bound of omega(pT γT / log T) for non-constant continuous kernels on compact domains, showing that while this scaling is generally

Key concepts

Minimax Regret: This measures the worst-case loss when choosing a policy in a bandit problem. It represents the maximum possible difference between the performance of any chosen strategy and the best possible strategy, averaged over time. A lower bound shows how hard it is to perform well.
Information Gain Rate ($\gamma_T$): This term quantifies how quickly the system learns or gains useful information as time $T$ progresses. It dictates the rate at which the algorithm can improve its decision-making quality. The paper shows that specific kernel types allow for a higher $\gamma_T$, leading to better regret bounds.
Kernel Smoothness (Hölder Condition): This condition ensures that the kernel function used in the bandit problem is sufficiently smooth. Smoothness allows researchers to use approximation techniques and derive tighter upper bounds on performance. The paper verifies this condition for several important kernels, confirming their suitability for these analyses.

Terminology

Regret — 49× in this paper · explained in 3 episodes
Information Gain — 22× in this paper · explained in 2 episodes
Minimax Regret — 13× in this paper · explained in 2 episodes
Regularization — 3× in this paper · explained in 3 episodes
Hilbert Space — 2× in this paper · explained in 2 episodes
Reproducing Kernel Hilbert Space — 2× in this paper
Reproducing Kernel Hilbert Space (RKHS) — 2× in this paper · explained in 3 episodes
Trace — 2× in this paper · explained in 2 episodes
Bayesian Optimization — 1× in this paper · explained in 3 episodes
Contextual Bandits — 1× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Generalization — 1× in this paper · explained in 13 episodes

Summary

The gist: This paper establishes a general minimax regret lower bound of omega(pT γT / log T) for non-constant continuous kernels on compact domains, showing that while this scaling is generally unavoidable, it can be removed under specific conditions related to the information gain rate γT.

Lower Bound Establishment

The authors prove an omega(pT γT / log T) minimax regret lower bound for every non-constant continuous positive semidefinite kernel on a compact subset of R d by connecting the maximum information gain to the effective dimension (Ma et al. 2026). They show that this implies an omega(pT γT / log T) regret for every non-constant continuous positive semidefinite kernel on a compact subset of R d by establishing an omega(pT γT / log T) minimax regret lower bound for non-constant continuous kernels on compact domains, establishing near-optimality (within log factors) in a very general sense (Ma et al. 2026).

Key Findings and Conditions

The paper demonstrates that the logarithmic factor appearing in the lower bound is unavoidable in general, since on small finite domains, γT = Θ(log T) and the minimax regret is Θ(√T) (Ma et al. 2026). However, this logarithmic term can be removed under certain conditions (Ma et al. 2026). Specifically, they show that for [0, 1]d, a sufficient condition leads to an O(√T γT) regret if γT = Θ(Tα) for some α > 0 (Ma et al. 2026). Furthermore, the scaling is exactly Θ(√T γT) for specific kernels like the Mat´ern-ν kernel with ν ∈ (0, 2), the γ-exponential kernel with γ ∈ (0, 2), and certain piecewise-polynomial kernels (Ma et al. 2026).

Finite Domain Analysis

For finite domains X, Theorem 1 states that the minimax regret R∗T is bounded below by min(1, ∆k) · σ16 pT dT (where ∆k:= maxx,x′∈X∥ϕ(x) − ϕ(x′)∥k) (Ma et al. 2610.11082v1). Corollary 1 refines this to R∗T ≥ min(1, ∆k) · √2σ16 sT γT1 + log(1 + B2T / σ2) (Ma et al. 2610.11082v1). The proof involves constructing hard-to-distinguish functions using dT, the policy-independent effective number of feature directions at scale ρ (Ma et al. 2610.11082v1).

Extension to General Compact Domains

The results are extended to general compact domains X by using a finite approximation X˜, such that there exists a nonempty finite X˜ satisfying min x′∈X˜∥ϕ(x) − ϕ(x′)∥k ≤ σ4B√T ∀x ∈ X (Ma et al. 2610.11082v1). Theorem 2 then states that R∗T ≥ min(1, 2∆k) · σ32 qT dT (X˜), and Corollary 2 provides the bound R∗T ≥ min(1, 2∆k) · √2σ32 sT γT (X)1 + log(1 + B2T / σ2) (Ma et al. 2610.11082v1).

Comparison with Upper Bounds

The paper compares the derived lower bounds with existing upper bounds, such as those from the RIPS algorithm and MOSS algorithm (Camilleri et al. 2021). Theorem 3 provides an O(√T γT log T) upper bound on [0, 1]d under a Hölder condition (Ma et al. 2610.11082v1). For specific kernels like the squared exponential kernel, rational quadratic kernel, and one-dimensional Dirichlet kernel, the logarithmic factor can be removed completely by sharper analysis of h(V, ρ) (Ma et al. 2610.11082v1).

Conclusion on Scaling

The final conclusion is that the minimax regret scaling is precisely Θ(√T γT) for several kernels of interest, while small gaps (typically √log T or log T) remain between the bounds for other kernels (Ma et al. 2610.11082v1). The paper also shows that any regret upper bound implies an information gain upper bound, leading to γT = O(U2T log(T)/T) (Ma et al. 2610.11082v1).

Specific Kernel Rates

Table 1 summarizes the rates of the maximum information gain γT and the bounds on minimax regret R∗T for six kernels on X = [0, 1]d (Ma et al. 2610.11082v1). For instance, for the Mat´ern kernel with ν ∈ (0, 2), the lower bound is O(√T γT) and the upper bound is O(√T γT log T), matching at Θ(√T γT) (Ma et al. 2610.11082v1). The analysis shows that for these specific kernels, the upper bounds match the corresponding lower bounds, establishing R∗T = Θ(√T γT) (Ma et al. 2610.11082v1). For other kernel settings or general settings considered, a gap of √log T or log T remains (Ma et al. 2610.11082v1).

Sufficient Condition for Log Factor Removal

A sufficient condition for achieving the optimal scaling is provided in Proposition 2: if there exists a fixed integer L ≥ 2 and a constant c ∈ (0, 1) such that γ⌊T /L⌋ ≤ (1 − c)γT for all sufficiently large T, then R∗T =omega(√T γT) (Ma et al. 2610.11082v1). This condition is satisfied by Mat´ern, γ-exponential, and piecewise-polynomial kernels when γT = Θ(Tα) for some α > 0 (Ma et al. 2610.11082v1).

High-Probability Bounds

Corollary 3 provides a high-probability lower bound: for every randomized policy π, there exists a fixed f ∈ Fk(B), such that with probability at least 1/4, RT (π, f) ≥ min(1, 2∆k) · √2σ32 sT γT (X)1 + log(1 + B2T / σ2) (Ma et al. 2610.11082v1). This result holds for every policy π, there exists f ∈ Fk(B), such that with probability at least 1/4, RT (π, f) ≥ min(1, 2∆k) · σ32 pT dT (Ma et al. 2610.11082v1).

Kernel Smoothness

The paper verifies that all six kernels satisfy the Hölder condition required for the upper bound analysis (Ma et al. 2610.11082v1). This involves showing that k(0) − k(r) ≤ C2 r2α for some fixed C, α > 0, which is verified for SE, rational-quadratic, γ-exponential, Mat´ern, and piecewise-polynomial kernels (Ma et al. 2610.11082v1). For the Dirichlet kernel on d=1, the bound is kDir,n(r) ≤ n(n + 1)6 r2, allowing for α = 1 (Ma et al. 2610.11082v1).

Upper Bound Derivation

The general upper bound O(√T γT log T) is derived by extending the finite domain bounds to [0, 1]d under a Hölder condition (Ma et al. 2610.11082v1). For specific kernels, this bound can be sharpened to O(√T γT) when gammaT = Θ(Tα) (Ma et al. 2610.11082v1). The analysis for the rational quadratic kernel shows that h(V, ρ) = O((log T)d + T−1) (Ma et al. 2610.11082v1). For the Mat´ern, γ-exponential, and piecewise-polynomial kernels, MOSS can be applied to remove the log T factor in certain parameter regimes (Ma et al. 2610.11082v1).

Summary of Comparison

The paper concludes that the minimax regret scaling is precisely Θ(√T γT) for several kernels of interest, while small gaps (typically √log T or log T) remain between the bounds for other kernels (Ma et al. 2610.11082v1).

Improvements for AI systems

Improved Regret Bounds for General Kernels: The system can now provide a general minimax regret lower bound of omega(p T γ T / log T) for every non-constant continuous positive semidefinite kernel on a compact subset of R d, establishing near-optimality in a very general sense, as stated in the abstract.
Kernel-Specific Optimal Scaling: The system can identify when the minimax-optimal scaling is exactly Θ(√T γ T) for specific kernels like the Matérn-ν kernel with ν ∈ (0, 2) and certain piecewise-polynomial kernels.
Log Factor Removal Condition: The system can determine if a logarithmic factor in an upper bound can be removed by checking if "γ T = Θ(T α) for some α > 0," specifically showing this holds for the Matérn kernel and others on [0, 1] d.
Kernel-Specific Upper Bound Sharpening: For specific kernels like the squared exponential (SE), rational quadratic, and one-dimensional Dirichlet kernel, the system can derive sharper upper bounds by removing the log T factor under certain conditions, such as showing R∗ T = O(√T γ T) for these settings.
High-Probability Lower Bounds: The system can provide a high-probability lower bound of min1, 2∆k · √2σ32 s T γ T (X) / (1 + log(1 + B 2T /σ 2)) for the average regret, with probability at least 1/4.

Abstract

The kernel bandit problem consists of sequentially optimizing an unknown function with noisy feedback, where the function has bounded norm in a given Reproducing Kernel Hilbert Space (RKHS). A central quantity in the regret analysis of kernel bandits is the maximum information gain γ T. In particular, the best existing upper bounds scale as sqrt Tγ T up to log factors, and nearly-matching lower bounds have been derived for specific kernels such as squared exponential and Matérn. However, lower bounds for general kernels are lacking, thus making it unclear in what generality the upper bounds are near-optimal. In this paper, we establish a general Ω(sqrt Tγ T/ T) minimax regret lower bound for non-constant continuous kernels on compact domains, establishing near-optimality (within log factors) in a very general sense. We show that the log factor appearing in this bound is unavoidable in general, but that it can be removed under certain conditions. Among other things, our findings imply that the minimax-optimal scaling is exactly Θ(sqrt Tγ T) (i.e., within constant factors) for the Matérn- ν kernel with ν in (0,2), γ-exponential kernel with γ in (0,2), and certain piecewise-polynomial kernels.