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Improved Regret Analysis for Parallel Gaussian Process Bandit Optimization

Shion Takeno, Shogo Iwazaki

arXiv:2608.16492v1stat.MLcs.LG

TL;DR

Parallel GP bandit analyses face cumulative-regret degradation with batch size Q, and prior remedies rely on an inefficient initial uncertainty-sampling phase. This paper improves GP-BTS analysis by decoupling Q from the main regret term, yielding bounds comparable to the sequential case without that phase.

  • Problem

    Parallel GP bandit analyses face degradation of the cumulative regret bound with the batch size Q.

  • Method

    The paper develops an improved GP-BTS analysis using Lemma 4.2 to decouple the delay parameter Q from the main regret term without an initial uncertainty-sampling phase.

  • Results

    GP-BTS achieves cumulative regret bounds comparable to the sequential case, converting previously known multiplicative degradation into an additive one.

  • Takeaways & Limitations

    The analysis removes reliance on the practically inefficient initial uncertainty-sampling phase while controlling the batch-size dependence additively.

  • Takeaways & Limitations

    The GP-TS-based method has problematic dependence on T compared with GP-UCB-based methods.

Abstract

from arXiv · show

This paper studies the regret analysis for parallel Gaussian process (GP) bandit optimization. The known regret upper bounds for the widely used GP batched upper confidence bound and GP batched Thompson sampling (GP-BTS) suffer from a multiplicative factor with respect to the batch size $Q$. To avoid this degradation, existing analyses require a polynomial number of uncertainty sampling (US) for $Q$ at the beginning of optimization. However, this initial US phase is often ineffective in practice. This paper shows that the regret upper bound without the multiplicative factor on $Q$ can be achieved without the initial US phase, using GP-BTS as an example. Furthermore, we show much better regret upper bounds in the noiseless setting than in the noisy setting, as in the sequential GP bandit setting.

1 Introduction

Parallel GP bandit optimization must select diverse query points without waiting for feedback, but existing GP-BUCB and GP-BTS analyses incur a multiplicative Q factor. This paper develops an improved GP-BTS analysis that achieves only additive Q degradation without initial uncertainty sampling and gives much better noiseless-setting bounds.

  • Motivation: Parallel and delayed GP bandits select new query points without waiting for recent results while maintaining query-point diversity.This setting arises in high-throughput screening and asynchronous distributed computing, where evaluations can run in parallel or experience delayed feedback.
  • Problem: Existing GP-BUCB and GP-BTS regret bounds suffer a multiplicative factor of the batch size or delay parameter Q, especially under massive parallelization (Q > 100).The paper identifies this Q-dependent degradation as a fundamental theoretical challenge in parallel settings.
  • Limitation: Initial uncertainty sampling can avoid the multiplicative Q factor, but the required phase degrades practical optimization performance.The paper motivates removing this phase because it is practically inefficient.
  • Contributions: The paper develops Lemma 4.2 to decouple Q as an additional regret term and applies the improved analysis to GP-BTS.The authors focus on GP-BTS because pure-exploration approaches can be practically ineffective.
  • Contributions: GP-BTS achieves a noisy-setting cumulative regret bound with only additive degradation in Q, without the practically inefficient initial uncertainty-sampling phase.The analysis resolves the stated parallel-bandit bottleneck without requiring initial US.
  • Contributions: GP-BTS also has much better regret upper bounds in the noiseless setting than in the noisy setting by leveraging Iwazaki’s proof technique.The paper contrasts these results with prior parallel noiseless analyses whose bounds had almost the same order as noisy-case bounds.

2 Preliminaries

This section defines the delayed or parallel GP optimization problem, its regret criteria, surrogate model, complexity measure, and regularity and noise assumptions. It also distinguishes noisy and noiseless observations and specifies the discretization framework used by GP-BTS.

  • Problem setting: The problem maximizes f over a d-dimensional input domain X, with sequential queries producing possibly noisy observations while recent Q evaluations remain unobserved.Setting Q = 0 recovers sequential optimization; the framework includes delayed feedback and asynchronous parallelization with Q + 1 workers.
  • Regret: Performance is evaluated using cumulative and simple regret, with the analysis considering their expectation over algorithmic randomness and, when present, the noise sequence.The simple regret uses a recommendation input ˆxT produced at the end of iteration T.
  • Surrogate model: A Gaussian process regression surrogate assumes f ∼ GP(0, k), yielding a posterior f | Dt ∼ GP(µt, kt) after noisy observations yt = f(xt) + ϵt.The noise is modeled as i.i.d. Gaussian with variance λ2 in the surrogate-model description.
  • Maximum information gain: Maximum information gain γT quantifies GP-bandit complexity, and the paper notes that commonly used kernels have sublinear MIG in T.The analysis uses a widely used MIG bound while acknowledging a known flaw for Matérn kernels and a rectified bound requiring an additional polylogarithmic factor on T.
  • Assumptions: The frequentist setting assumes f lies in the RKHS induced by k, with k(x, x) ≤ 1 on compact X and finite norm ∥f∥Hk ≤ B, while covering noisy and noiseless cases.Noisy observations use conditionally R-sub-Gaussian noise; the algorithm also relies on a predefined discretized input set Xt satisfying the stated coverage and cardinality conditions.

3 GP-BTS

This section defines GP-BTS as a posterior-sampling algorithm that selects inputs intended to be diverse relative to recent batch queries. It reviews an existing cumulative-regret analysis whose bound degrades with batch size Q and motivates avoiding the impractical initial US phase.

  • Algorithm: GP-BTS selects the t-th input using a posterior sample path with inflated posterior variance, promoting diversity relative to the recent Q inputs.The algorithm uses gt ∼ GP(µt−Q−1, βtkt−Q−1), with βt theoretically specified as βt = Θ(γt).
  • Existing analysis: The adapted Vakili et al. analysis applies to exact GP posteriors and posterior sample paths and yields a cumulative-regret upper bound for GP-BTS.The result is stated in the paper as a modified version of Lemma 4.1.
  • Existing analysis: Because σt−Q−1(xt) ≥ σt−1(xt), the GP-BTS regret bound is larger than the sequential bound, which depends on PT.The comparison follows directly from the ordering of the posterior uncertainties.
  • Existing analysis: The sequential GP-TS factor √γT is replaced by pQγT/Q, producing degradation with respect to Q when γT is sublinear.The next section addresses this degradation without the impractical initial US phase.

4 Improved regret analysis

The section develops a modified instantaneous-regret analysis for GP-BTS and combines it with an uncertainty-counting lemma to replace the usual multiplicative dependence on Q with an additive factor. It further derives substantially tighter noiseless regret bounds, while identifying remaining limitations for simple regret.

  • Improved cumulative regret analysis: The modified Lemma 4.1 replaces an expectation of a minimum with the minimum of an expectation, enabling regret bounds over selected iteration sets.Using the naive upper bound would incur c_T|T| = O(√γ_T ln T|T|).
  • Improved cumulative regret analysis: Lemma 4.2 shows that, for most iterations, the increase from σ_t−1(x_t) to σ_t−Q−1(x_t) is at most a constant multiple.The exceptional set satisfies |T^c| = T − |T| ≥ T − 8Qγ|T|(λ^2).
  • Improved cumulative regret analysis: The resulting cumulative regret bounds replace the multiplicative factor on Q in Eq. (3) with an additive factor, matching sequential GP-TS bounds apart from that additive term.This change is preferable under the general assumption T ≫ Q, when the γ_T ln T term is often dominant.
  • Noiseless regret analysis: For SE kernels and Matérn kernels with ν ≥ d, the noiseless analysis obtains a polylogarithmic upper bound with respect to T and is much tighter than known results.The paper states that noiseless regret bounds are much better than noisy-case bounds, paralleling the sequential setting.
  • Simple regret analysis: The simple-regret bounds achieve polynomial convergence for both kernels, but their dependence on T is worse than in prior sequential analysis.This degradation is attributed to GP-TS lacking the posterior-variance upper bound available for GP-UCB; tighter bounds remain future work.

5 Numerical experiments

Numerical experiments on GP-generated synthetic functions in noiseless and noisy settings show mild dependence on batch size Q, especially for Q ≤4. In the noiseless setting, the SE-kernel regret saturates around 100 iterations, while the Matérn-kernel regret grows sublinearly with more moderate Q dependence than its theoretical bound suggests.

  • Batch-size dependence: The dependence on Q was mild throughout the experiments, particularly for Q ≤4, consistent with theory when T is sufficiently large.The results indicate that Q is not the dominant dependence in this regime.
  • Noiseless setting: Around 100 iterations, noiseless cumulative regret for the SE kernel saturated, matching the O(Q + ln 1 2 T) cumulative regret upper bound from Theorem 4.5.The supplied passage presents this saturation as consistent with the theorem.
  • Noiseless setting: In the noiseless Matérn setting, d = 3 > ν = 5/2, and cumulative regret exhibited sublinear growth with more moderate Q dependence than the theoretical upper bound suggests.This behavior is reported for the Matérn kernel under the stated dimensional and smoothness parameters.

6 Conclusion and future work · A Auxiliary lemmas

The paper concludes that its GP-BTS analysis removes multiplicative batch-size degradation without an initial uncertainty-sampling phase, while identifying several open problems and collecting auxiliary lemmas for discretization, confidence intervals, and variance control.

  • 6 Conclusion and future work: The analysis decouples Q from the main regret term, converting multiplicative degradation into an additive one and matching sequential-case bounds without initial US.This result applies to GP-BTS in the simple delayed-feedback setting.
  • 6 Conclusion and future work: The paper also establishes regret upper bounds for the noiseless setting.These bounds follow the approach of Iwazaki (2025a).
  • 6 Conclusion and future work: Future work includes improving noiseless simple-regret bounds, adapting the analysis to GP-BUCB, and designing UCB batches that select diverse points without scaling βt by Q.The current GP-BUCB proof uses βT = O(QγT), while βt = Θ(γt) may not select diverse batch points.
  • 6 Conclusion and future work: High-probability bounds with 1/δ dependence follow from expected regret, but achieving ln(1/δ) dependence in the noiseless setting remains open.For noisy settings, Azuma-Hoeffding can provide ln(1/δ) dependence; standard concentration incurs a dominant O( term in the noiseless case.
  • 6 Conclusion and future work: Additional open directions address Bayesian regret bounds, discretization error, and the optimal dependence on Q.Discretization causes ln T degradation in noisy settings and can be non-negligible in noiseless settings; the additive Θ(Q) term is unavoidable, but other Q factors may be removable.
  • A Auxiliary lemmas: Lemma A.4 provides a variance inequality for t ≥ Q + 1, using posterior-variance monotonicity and the worst case of repeated inputs.The proof notes that σt−1(x) ≤ σt−Q−1(x) and analytically evaluates the repeated-input case.

B Proof · B.1 Proof for Lemma 4.1

The proof establishes a modified auxiliary lemma under the theorem’s premise, using Gaussian anti-concentration and concentration events to control GP-TS sampling. It then combines this lemma with discretization, regret decomposition, and expectation arguments to prove Lemma 4.1.

  • B Proof: The proof begins with a lemma modified from Lemma 10 of Chowdhury and Gopalan (2017).This auxiliary result assumes the same premise as Theorem 4.3 and introduces the event E(f).
  • B.1 Proof for Lemma 4.1: The auxiliary lemma defines the saturation set S_t and establishes an inequality for all t ∈ [T] on event E(f).The construction uses a finite subset X_t, the projection [x]_t, p = 1/(4e√π), and c_t = β^1/2.
  • B.1 Proof for Lemma 4.1: Gaussian anti-concentration yields Pr(g_t([x∗]_t) ≥ f([x∗]_t) | D_{t−Q−1}) ≥ p when the relevant E(f) condition holds.The proof invokes Gaussian anti-concentration results from Chowdhury and Gopalan (2017) and Srinivas et al. (2010).
  • B.1 Proof for Lemma 4.1: The proof lower-bounds the probability of selecting outside S_t by subtracting the event that some x ∈ S_t violates the required comparison.It notes that X_t\S_t is nonempty because [x∗]_t belongs to it, then applies the resulting bound.
  • B.1 Proof for Lemma 4.1: For the selected point outside S_t, the proof defines x̄_t by minimizing posterior uncertainty and decomposes instantaneous regret into four function and sample terms.The resulting expectation bound uses min{2B, c_tσ_{t−Q−1}(x_t)} and the analogous term for x̄_t.
  • B.1 Proof for Lemma 4.1: The proof compares the regret bound’s possible forms, showing that the relevant minimum is governed by either 2B or the combined uncertainty terms.The cases include 4B, 2B plus one uncertainty term, or c_t times the sum of both uncertainties.
  • B.1 Proof for Lemma 4.1: Lemma 4.1 assumes Assumptions 2.3 and 2.5 and states that, when Algorithm 1 runs with y_t = f(x_t), the displayed inequality holds for all t ∈ [T].Its proof uses separate noisy and noiseless probability events, the discretization property, the bound max_x∈X |f(x)| ≤ B, and the tower property of expectation.
  • B.1 Proof for Lemma 4.1: Combining Lemma B.1 with the preceding event and expectation arguments completes the proof of Lemma 4.1.The final step explicitly invokes Lemma B.1 after arranging the instantaneous regret.

B.2 Proof for Lemma 4.2

This section states Lemma 4.2 for arbitrary horizons, positive λ, and input sequences, then proves it using an elliptical potential count argument and Lemma A.4. The proof handles the Q = 0 case separately and uses a logarithmic inequality in the general case.

  • Lemma 4.2 applies to any T ∈ N, λ > 0, and input sequence {x_t}_{t∈[T]}.
  • The proof follows the elliptical potential count lemma and cites related results by Flynn and Reeb, Iwazaki and Takeno, and Iwazaki.
  • The second inequality uses min{1, a} ≤ 2 ln(1 + a) for all a ≥ 0.
  • For the second inequality, the proof invokes Lemma A.4 and treats Q = 0 as trivial before considering Q ≥ 1.

B.3 Proof for noisy setting

Under Assumptions 2.3–2.5, the proof derives the noisy-setting cumulative-regret bound from Lemma 4.2 and Lemma 4.1, then establishes the corresponding simple-regret order. It also verifies the required dependence of T on Q for SE and Matérn kernels.

  • Cumulative regret: Theorem 4.3’s noisy-setting bound follows by combining Lemma 4.2 with Lemma 4.1 and substituting the convergent series sum ∑_{t=1}^∞ 1/t^2 = π^2/6.The proof uses T^c ⊂ [T], the Cauchy–Schwarz inequality, and min{1,a} ≤ 2 ln(1+a).
  • Simple regret: The proof obtains the simple-regret upper bound by defining the event E(f) and applying the same argument used for cumulative regret.The derivation uses max_{x∈X}|f(x)| ≤ B, Lemma A.3, and the relation LCB_t( x̂_T ) ≥ LCB_t(x_t).
  • Kernel-specific conditions: For SE and Matérn kernels, the proof selects absolute constants so the required lower bound on T has the desired order with respect to Q.For SE kernels, the argument chooses c and C = 1+c to satisfy the displayed logarithmic inequality; the Matérn case proceeds by setting T accordingly.

B.4 Proof for noiseless setting

The noiseless-setting proof derives posterior-standard-deviation bounds for squared-exponential and Matérn kernels in batched optimization, then uses them to establish the stated regret result. It also explains why an alternative lemma worsens the dependence on batch size Q.

  • Posterior standard-deviation bound: The lemma bounds posterior standard deviations in the noiseless batched setting for squared-exponential and Matérn kernels under compact-domain and information-gain assumptions.The assumptions include k(x, x) ≤1, kernel-specific constants, and information-gain bounds.
  • Proof technique: The proof obtains noiseless bounds by comparing zero-noise posterior deviations with deviations computed using positive observation noise.Posterior standard deviation is monotone in the noise level, enabling the upper-bound argument.
  • Batch-size dependence: The Eq. (79)-based proof yields an O(Q) upper bound, avoiding the worse batch-size dependence produced by directly using Lemma 4.2.The note states that the cumulative regret order changes from Θ(Q) to ω(Q) with the alternative proof.
  • Regret consequence: The resulting posterior-deviation bounds are then used through Theorem 4.5 to obtain the desired noiseless regret result in Corollary 4.6.The corollary suppresses polylogarithmic factors in T and Q with eO notation.
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