TS-SID-iVAR: Active Learning in GP Regression

• TS-SID-iVAR is a Thompson sampling method for active learning in Gaussian process regression that tackles the challenge of unknown self-induced Boltzmann distributions.
• It combines posterior sampling with integrated variance reduction to efficiently target high-uncertainty regions while avoiding the intractable partition function.
• The approach demonstrates a balance between exploratory diversity and stability, offering competitive performance on synthetic benchmarks and practical potential energy surface modeling.

Searching arXiv for the explicit TS-SID-iVAR paper and closely related context. Searching arXiv for TS-SID-iVAR and [Active Learning](https://www.emergentmind.com/topics/active-learning-al) for [Gaussian Process Regression](https://www.emergentmind.com/topics/gaussian-process-regression-gp) Under Self-Induced Boltzmann Weights. TS-SID-iVAR is a Thompson sampling acquisition rule for active learning in Gaussian process regression under a self-induced Boltzmann target distribution, introduced in the context of learning an unknown function with low prediction error under an unknown weighting induced by the function itself (Qing et al., 11 May 2026). In this setting, the test distribution is not fixed in advance but given by

$p_f(x)=Z(f)^{-1}\exp(-\beta f(x)+b(x)),$

where $\beta$ is an inverse temperature parameter, $b(x)$ is a known, bounded bias, and the partition function

$Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$

is intractable. TS-SID-iVAR addresses the resulting design problem by combining Thompson sampling with integrated variance reduction, thereby targeting posterior uncertainty in regions that are likely under a sample-induced approximation to the unknown Boltzmann distribution (Qing et al., 11 May 2026).

1. Problem formulation and statistical setting

The active learning objective is to minimize the weighted prediction error

$$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$

Because $p_f$ depends on the unknown function $f$, the target density is itself unknown at design time. This self-induced weighting arises naturally in problems such as potential energy surface modeling in computational chemistry, and the paper treats it as the central difficulty rather than as a secondary nuisance (Qing et al., 11 May 2026).

The function $f$ is modeled with a Gaussian process prior $\mathrm{GP}(m_0,k)$ and noisy observations

$y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$

Given data $\beta$0, the GP posterior has mean $\beta$1 and covariance $\beta$2, with posterior variance

$\beta$3

The standard posterior formulas are

$\beta$4

and

$\beta$5

with $\beta$6.

Within this framework, TS-SID-iVAR is not a generic Thompson sampling heuristic. It is a specific acquisition mechanism designed for the case where the loss is evaluated under the unknown self-induced distribution $\beta$7 rather than under a uniform or externally specified test measure. This suggests that its distinguishing feature is not merely randomized exploration, but randomized approximation of the unknown weighting that defines the downstream error criterion.

2. Relation to AB-SID-iVAR and the role of Bayesian surrogates

The paper develops TS-SID-iVAR alongside AB-SID-iVAR. AB-SID-iVAR uses a “Bayesian SID” surrogate found by marginalizing the Boltzmann weight over the GP posterior:

$\beta$8

Since $\beta$9 is Gaussian with mean $b(x)$0 and variance $b(x)$1, the Gaussian moment generating function identity

$b(x)$2

yields the closed form

$b(x)$3

This avoids computing $b(x)$4 entirely, because the method uses the surrogate only up to proportionality (Qing et al., 11 May 2026).

TS-SID-iVAR is the corresponding Thompson sampling alternative. Instead of using the closed-form Bayesian surrogate, it samples a single function from the GP posterior and constructs the self-induced weights from that sample. In the formulation given in the paper, at round $b(x)$5 one draws

$b(x)$6

then forms

$b(x)$7

The paper characterizes this as a higher-variance Monte Carlo alternative to AB-SID-iVAR and as an $b(x)$8 Monte Carlo approximation of the same underlying Bayesian quantity (Qing et al., 11 May 2026).

The distinction is therefore structural. AB-SID-iVAR replaces the unknown self-induced target by a closed-form posterior average, whereas TS-SID-iVAR replaces it by a single posterior draw. The former is biased due to approximation but low-variance; the latter is unbiased in the Monte Carlo sense but higher variance. This suggests that the two methods occupy a familiar exploration–stability trade-off, but under a target distribution that is itself latent and function-dependent rather than externally given.

3. Acquisition rule and integrated variance reduction

The iVAR criterion is based on the standard GP variance update after adding a new point $b(x)$9:

$Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$0

This identity is used to measure the one-step reduction in integrated posterior variance under a surrogate density.

For AB-SID-iVAR, the reduction is

$Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$1

For TS-SID-iVAR, the corresponding acquisition is

$Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$2

where $Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$3 and $Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$4 (Qing et al., 11 May 2026).

On discrete domains, the integral is replaced by a sum:

$Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$5

where $Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$6 or $Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$7, and normalization is unnecessary for acquisition comparison because constant factors cancel. On continuous domains, the integral is approximated by Monte Carlo, ideally with Sequential Monte Carlo targeting the surrogate density, using annealed tempering with Random Walk Metropolis rejuvenation.

The method also imposes an exploration constraint:

$Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$8

where $Z(f)=\int_{\mathcal X}\exp(-\beta f(x)+b(x))\,dx$9 denotes the chosen surrogate, namely $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$0 for AB-SID-iVAR and $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$1 for TS-SID-iVAR. The paper states that this avoids collapsing onto regions with tiny variance and ensures that queries occur in sufficiently uncertain regions on average (Qing et al., 11 May 2026).

Operationally, TS-SID-iVAR proceeds by updating the GP posterior, sampling $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$2, defining the sample-induced Boltzmann weights, constructing the feasible set $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$3, and then selecting

$$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$4

The defining point is that the weighting used in the integrated variance calculation is itself randomized through the Thompson draw.

4. Partition-function avoidance, theory, and convergence guarantees

A central technical issue is that the target density involves the intractable partition function $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$5. In AB-SID-iVAR, the numerator $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$6 and the expected partition function $$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$7 are tractable under the GP posterior, and the acquisition uses an unnormalized surrogate of the form

$$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$8

In TS-SID-iVAR, one instead uses the unnormalized sample-based density

$$L(\hat f)=\mathbb E_{x\sim p_f}\big[(f(x)-\hat f(x))^2\big].$$9

In both cases, normalization is not needed by the acquisition, so the partition function is avoided rather than estimated (Qing et al., 11 May 2026).

The theory is developed under stationary, sufficiently smooth kernels, compact domain, bounded noise variance, known bounded bias $p_f$0, kernel normalization $p_f$1, and the Boltzmann SID structure. The paper states a high-probability bound for the terminal weighted error. Let $p_f$2 denote the maximum information gain. If at each iteration one selects $p_f$3 and approximates thresholds via $p_f$4 Monte Carlo samples, then for any $p_f$5, with probability at least $p_f$6,

$p_f$7

where $p_f$8 for AB-SID-iVAR and $p_f$9 for TS-SID-iVAR, $f$0, $f$1, $f$2, and $f$3 is the cumulative Monte Carlo error from threshold estimation (Qing et al., 11 May 2026).

The paper further states that this bound vanishes as $f$4, with rates $f$5 when $f$6 is constant and $f$7 when $f$8, for any small $f$9. It also gives an average-case result:

$f$0

The paper describes this as tighter than the high-probability bound because the main term is $f$1 rather than $f$2 (Qing et al., 11 May 2026).

The proof structure relies on GP confidence bounds, density ratio control between $f$3 and the surrogate, variance bounds via information gain, Gaussian MGF identities, and explicit accounting for Monte Carlo threshold approximation error via Freedman’s inequality. Within this analysis, TS-SID-iVAR inherits the same asymptotic consistency theme as AB-SID-iVAR but with weaker constants, reflected in the replacement of $f$4 by $f$5.

5. Algorithmic realization and computational characteristics

The paper gives a procedural description of TS-SID-iVAR. At iteration $f$6, one updates the GP posterior $f$7 from $f$8, samples a posterior function path $f$9, defines $\mathrm{GP}(m_0,k)$0, forms the exploration set

$\mathrm{GP}(m_0,k)$1

computes $\mathrm{GP}(m_0,k)$2 for candidates in $\mathrm{GP}(m_0,k)$3, selects the maximizer, queries the function, and updates the data set (Qing et al., 11 May 2026).

Several implementation details are specified. GP updates are $\mathrm{GP}(m_0,k)$4 naively with exact kernels because they require Cholesky factorization of $\mathrm{GP}(m_0,k)$5; sparse or low-rank approximations such as inducing points can be used to scale. Thompson sampling requires sampling GP paths, which the paper notes can be done with pathwise conditioning. For continuous domains, the integral approximation uses SMC or MCMC; the paper reports SMC annealed tempering with Random Walk Metropolis steps and states that particle counts $\mathrm{GP}(m_0,k)$6 to $\mathrm{GP}(m_0,k)$7 provide similar accuracy, with linear runtime scaling in $\mathrm{GP}(m_0,k)$8. For constrained optimization of the acquisition in continuous domains, the implementation uses L-BFGS-B or SLSQP via BoTorch. Hyperparameters are learned by MAP estimation, and the paper notes that noise $\mathrm{GP}(m_0,k)$9 is typically small but fixed, for example $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$0 in synthetic tests.

The paper also discusses parameter choice. In potential energy surface problems, $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$1 with appropriate unit and sign, while the paper’s notation uses $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$2 and the mapping $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$3 under the physical sign convention. It reports that larger $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$4 concentrates $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$5 more strongly and amplifies the benefit of SID-aware acquisitions. Numerical stability is aided by the normalization condition $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$6, which implies $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$7 and keeps the denominator $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$8 bounded away from zero (Qing et al., 11 May 2026).

6. Empirical behavior, advantages, and limitations

Empirically, the paper states that AB-SID-iVAR consistently achieves the lowest weighted MSE on synthetic benchmarks, with larger gains in higher dimensions, while TS-SID-iVAR is competitive but exhibits higher variance across seeds, especially on multimodal targets such as Branin (Qing et al., 11 May 2026). On discrete domains, where exact sums eliminate Monte Carlo integration error, the same pattern persists: TS-SID-iVAR remains competitive but has higher seed variance, particularly on multimodal targets. The paper remarks that a more conservative threshold or domain-averaged variance can mitigate variance in practice.

For real-world potential energy surface modeling, the emphasis is on AB-SID-iVAR, which consistently achieved the lowest weighted MSE on four PES tasks. TS-SID-iVAR is described as applicable in the same framework but was not the main focus in these large continuous tasks because of its higher variance and cost. In molecular drug discovery over a discrete pool, TS-SID-iVAR was omitted because sampling GP posteriors on large discrete pools is computationally costly, whereas AB-SID-iVAR produced one to two orders of magnitude improvement in weighted MSE and substantially higher $y=f(x)+\epsilon,\qquad \epsilon\sim N(0,\sigma^2).$9 on top-$\beta$00 compounds ranked by Boltzmann weight (Qing et al., 11 May 2026).

The paper’s interpretation of TS-SID-iVAR is correspondingly specific. It is preferable when the surrogate $\beta$01 may be biased, or when multimodality is pronounced and exploration is crucial. It is particularly attractive when exploratory diversity is valued and higher variance across runs is acceptable. AB-SID-iVAR is preferable when stable convergence and lower variance are desired, or when integral approximations must be efficient. This suggests that TS-SID-iVAR is best understood not as the default member of the SID-iVAR family, but as its deliberately more stochastic variant: it shares the same objective of minimizing posterior uncertainty where the unknown Boltzmann target places mass, yet replaces closed-form Bayesian weighting by a single posterior sample and thereby trades stability for exploratory breadth.