Nonlocal Transition Kernel for Efficient Learning of Restricted Boltzmann Machines

Kaiji Sekimoto; Muneki Yasuda

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Nonlocal Transition Kernel for Efficient Learning of Restricted Boltzmann Machines

Kaiji Sekimoto, Muneki Yasuda

arXiv:2608.17450v1Aug 18, 2026stat.MLcond-mat.dis-nncs.LG

TL;DR

RBM learning relies on finite-step sampling approximations that can be poor when local transitions encounter energy barriers. This paper proposes a round-trip kernel over deep-tempering RBMs, which achieves broader exploration and more frequent mode transitions than BGS and DT while mitigating learning degradation and failures.

Problem
Local BGS transitions can produce poor sampling near high energy barriers, while DT requires multiple sequence steps to obtain nonlocal moves.
Method
The paper introduces a round-trip transition kernel over the DT RBM sequence that enables nonlocal moves within a small number of steps while preserving sequence invariance.
Results
The proposed kernel explored broader state ranges, moved between modes more frequently, reduced initial-state dependence faster, and mitigated learning degradation and failures versus BGS and DT.
Takeaways & Limitations
The kernel supports higher-quality finite-step sampling and more stable RBM learning when only a limited number of transitions are available.
Takeaways & Limitations
The method remains subject to finite-step MCMC estimator bias, and its mixing properties lack a theoretical analysis.

Abstract

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Learning restricted Boltzmann machines (RBMs) is computationally challenging because it requires expectations whose exact evaluation is generally intractable. The expectations are typically evaluated using a sampling approximation based on blocked Gibbs sampling (BGS), which is a local Markov chain Monte Carlo transition kernel. However, the locality of BGS can lead to poor sampling quality when the RBM has high energy barriers, thereby degrading learning performance. Deep tempering (DT), which performs parallel tempering over a sequence of learnable RBMs including the training RBM, alleviates this locality issue. However, DT algorithmically requires multiple steps to move through the RBM sequence to achieve a nonlocal transition. In this paper, we propose a transition kernel defined over the RBM sequence used in DT. The proposed kernel has a round-trip structure over the sequence, enabling nonlocal moves within a single transition while leaving the RBM sequence invariant. Numerical experiments show that the proposed kernel performs nonlocal transitions more frequently and achieves higher sampling quality with fewer transitions than BGS and DT. We further verify that learning based on the proposed kernel is more stable and mitigates the training failures observed with BGS- and DT-based learning.

1 Introduction

RBM learning relies on finite-step MCMC approximations whose quality depends on initialization and mixing, motivating faster nonlocal transition kernels. The paper proposes a round-trip kernel over the deep-tempering RBM sequence that enables nonlocal moves within one transition and improves learning robustness over BGS and DT.

Motivation: Finite-step MCMC sampling does not exactly match the target distribution, so RBM learning depends on sampling quality.The approximation is affected by both the initial distribution and the transition kernel’s mixing speed.
Motivation: CD and PCD improve sampling initialization by using the data distribution and the previous update’s sample distribution, respectively.Their sampling quality still depends directly on mixing time.
Contribution: The proposed kernel uses a round-trip structure over the RBM sequence in deep tempering to perform a nonlocal state-space move within a single transition.Unlike tempered transitions, it does not rely on Metropolis–Hastings and therefore avoids rejection of a nonlocal proposal.
Results: The proposed method achieves high learning performance and mitigates failures occurring in settings where BGS- and DT-based learning typically fails.These findings are reported as part of the paper’s numerical evaluation of the method.

2 Restricted Boltzmann Machines and Their Learning

RBMs model visible data variables and hidden latent variables in a bipartite architecture, with hidden-unit count controlling representational power. Their likelihood-gradient learning requires intractable expectations, so sampling quality—especially under BGS—directly affects training stability.

RBM structure: An RBM is a two-layer Markov random field with n visible variables representing data and m hidden variables providing latent representational capacity.The visible and hidden layers form a complete bipartite graph, and increasing m enhances representational power.
Blocked Gibbs sampling: Conditional independence between layers enables one BGS sweep to sample h′ given v, then v′ given h′, while preserving the RBM joint distribution.The resulting transition kernel satisfies the balance condition and leaves the target distribution invariant.
Learning objective: RBM training maximizes log-likelihood by gradient ascent, but the parameter gradient contains expectations over all visible and hidden configurations.These multiple summations cause combinatorial explosion, making exact likelihood optimization computationally difficult.
Sampling approximation: BGS-based sampling approximates the intractable expectation, and approximation accuracy depends on how closely the sample distribution follows the target distribution.Using an initial distribution closer to the target can improve sampling quality.
SMCI limitation: SMCI can reduce estimator variance and achieve higher log-likelihood than standard Monte Carlo integration, but it sometimes causes substantial log-likelihood decreases.This unexpected behavior motivates identifying the learning settings in which SMCI-based learning fails.
Failure mechanism: As SMCI-based training began to fail, mean absolute errors increased, log-likelihood decreased, and the BGS second-largest eigenvalue converged to one.The increasing relaxation time made sampling more dependent on initial states, indicating deteriorating sampling quality.

3 Deep Tempering

Deep tempering (DT) constructs a learnable sequence of auxiliary RBMs around the training RBM and uses parallel tempering with blocked Gibbs sampling to enable global transitions. Its learning-based sequence can require fewer intermediate distributions than standard parallel tempering while retaining competitive or better empirical performance.

Deep Tempering: DT stacks L auxiliary RBMs with the training RBM to form a parallel-tempering sequence, learning their parameters jointly.The training RBM is denoted Pθ0(v0, h0), and the parameters are Θ := (θ0, θ1, . . . , θL).
Deep Tempering: The sequence connects adjacent RBMs by constraining the next RBM’s visible-variable count to equal the previous RBM’s hidden-variable count.Specifically, nℓ+1 = mℓ for ℓ=0, 1, . . . , L−1.
Deep Tempering: DT combines visible-space swap kernels between adjacent RBMs with blocked Gibbs sampling on each RBM.The resulting transition kernel uses parallel tempering over the learned RBM sequence.
Deep Tempering: The DT transition kernel satisfies the balance condition with respect to PΘ(V), thereby leaving the joint distribution invariant.The parity indicator alternates deterministically between transitions according to c′ = 1−c.
Deep Tempering: DT learns its intermediate distributions rather than varying an inverse-temperature ladder, reducing the number required for effective sampling.DT has been shown to achieve competitive or better empirical performance than standard parallel tempering while using fewer intermediate distributions.

4 Proposed Non-local Transition Kernel

The proposed kernel recursively performs an upward swap pass, a one-step BGS transition at the topmost RBM, and a reverse downward pass. This round-trip construction preserves the RBM sequence’s distribution while enabling nonlocal transfers to the training RBM within one transition.

Recursive definition: The transition kernel is defined recursively on visible variables Vℓ:L and becomes the full-sequence kernel by setting ℓ=0.Each recursive step samples a hidden state, swaps it with the next visible state, and applies the upper-sequence kernel.
Invariance: The proposed transition kernel leaves PΘ(V ) invariant.The invariance property is established in Appendix C.
Learning procedure: Learning with the proposed kernel follows DT’s procedure, updating Θ by gradient ascent and estimating the gradient expectation through sampling.The parameter gradient is given in Eq. (18).
Nonlocal transitions: The round-trip structure can transfer states from an upper-layer RBM to the training RBM within a single transition, enabling nonlocal visible-state moves in few steps.Upper-layer RBMs are expected to mix relatively easily, supporting these transfers to the bottommost RBM.
Round-trip construction: The kernel propagates states upward through the RBM sequence, applies one-step BGS at the topmost RBM, then propagates the resulting state downward.The upward and downward passes use swap operations in reverse order.

5 Numerical Experiments

Numerical experiments evaluated learning, sampling quality, nonlocal transitions, and initial-state dependence across six datasets. The proposed transition kernel mitigated GenRBM learning degradation, matched or improved baseline learning performance, and was examined for nonlocality and mixing behavior.

Experimental setup: Experiments evaluated learning performance on six datasets: GenRBM, islands, pentagon, iris, wine, and seeds.RBM initialization, optimization, and gradient-sampling approximations were held common across datasets.
Learning performance: On GenRBM, increasing sequence depth progressively mitigated log-likelihood degradation for DT and the proposed kernel, with the proposed kernel more effective than DT.Figure 6 reports log-likelihood and MAE versus training epoch, averaged over 10 experiments.
Learning performance: On clustered islands, pentagon, iris, wine, and seeds datasets, the proposed kernel achieved learning performance comparable to the baselines.Wine and seeds log-likelihoods used marginalized annealed importance sampling because exact partition-function computation was intractable.
Sampling quality and nonlocality: Transition trajectories were visualized over 100 steps to examine nonlocal transitions for BGS, DT, and the proposed kernel across all datasets.Initial states came from persistent chains during training; BGS was applied to the training RBM, whereas DT and the proposed kernel used the RBM sequence.
Initial-state dependence: Initial-state dependence was assessed using total variation distance across transition steps for RBM sequences trained on GenRBM, islands, pentagon, and iris.Faster TV-distance decay indicates weaker initial-state dependence and suggests faster mixing, although it does not directly measure mixing time.

6 Conclusion

The paper proposes a round-trip transition kernel over the RBM sequence used in deep tempering, enabling nonlocal moves in few steps. This may improve finite-step sampling approximations and stabilize RBM learning, while finite-step MCMC remains generally biased.

6 Conclusion: The proposed kernel traverses the RBM sequence from the bottom RBM to the topmost RBM and back, enabling nonlocal transitions in few steps.It leverages the smaller state spaces of higher-layer RBMs.
6 Conclusion: Few-step nonlocal transitions could improve sampling-approximation accuracy for parameter-gradient estimation and enhance RBM-learning stability.
6 Conclusion: Finite-step MCMC sampling approximations are generally biased, with bias typically decreasing as the number of transition steps increases.Exact unbiasedness generally requires samples from the stationary distribution, which finite standard-MCMC runs cannot generally obtain.

A Semi-Second-Order SMCI for RBMs

This appendix formulates semi-second-order SMCI estimators for the visible, hidden, and joint expectations needed to evaluate RBM parameter gradients. It also defines auxiliary quantities and describes expectation approximation from sampled points.

Estimator formulation: The formulation targets estimators for Eθ(v,h)[v_i], Eθ(v,h)[h_j], and Eθ(v,h)[v_ih_j], which are required for parameter-gradient evaluation.These expectations support the gradients in Eqs. (12)–(14).
Estimator formulation: The notation excludes v_i from the visible variables and h_j from the hidden variables when defining v−i and h−j.Specifically, v−i ∈ {0, 1}^n−1 and h−j ∈ {0, 1}^m−1.
Estimator formulation: For the joint expectation, the formulation introduces η_i,j(h; θ) and ξ_j,i(v, θ) by subtracting W_i,jh_j and W_i,jv_i from η_i and ξ_j, respectively.These auxiliary quantities isolate the contribution associated with the visible-hidden pair.
Estimator formulation: The formulation uses the logit function logit(x) := ln(x/(1−x)), identified as the inverse of the sigmoid function.The logit definition appears in the specification of the estimators.
Sample-based approximation: Given sampled points {(v^(ν), h^(ν)) | ν = 1, 2, . . . , N_s}, the expectations are approximated using those samples.The passage specifies the sample set but does not provide the subsequent approximation formulas.

B Transition Probability Matrix for BGS

The appendix formulates BGS’s transition matrix and estimates its relaxation time through the second-largest eigenvalue. It uses a symmetric similarity transform and a sampling-based eigenvalue estimator, whose resulting estimate can be biased without guaranteed bounds.

Transition matrix: BGS transition probabilities form a 2^n × 2^n matrix P indexed by an enumeration of all visible states.Its entries are P_ab := T^BGS_θ(v(b) | v(a)).
Spectral relaxation time: The similarity transform S := D^1/2 P D^-1/2 is symmetric positive semidefinite because P satisfies detailed balance.Therefore, S and P share nonnegative real eigenvalues, including the probability-matrix eigenvalue λ1 = 1.
Spectral relaxation time: The BGS relaxation time is t_rel := 1/(1 − λ2), where λ2 is obtained from the largest eigenvalue of Π⊥SΠ⊥.The projection Π⊥ removes the component along the leading eigenvector u1, enforcing the required orthogonality constraint.
Numerical estimation: The sampling estimator uses N2 = 2^15 one-step BGS transitions per visible state and estimates λ2 from the leading eigenvector of Π⊥ŜΠ⊥.Although the resulting estimator is based on a symmetric unbiased estimator of A = DP, λ̂2 is generally biased and has no guaranteed lower or upper bound.

C Invariant Distribution of the Proposed Transition Kernel

The proposed transition kernel is proven to preserve the RBM sequence’s target distribution through an induction over product distributions. The proof establishes invariance at the topmost RBM, propagates it downward, and concludes that Π0(V)=PΘ(V) is invariant, as for DT.

Inductive invariance proof: The proof defines product distributions Πℓ over RBMs from level ℓ through the topmost level and establishes invariance recursively.The induction assumes Tℓ+1 preserves Πℓ+1 and proves that Tℓ preserves Πℓ by following its sampling procedure step by step.
Inductive invariance proof: The topmost kernel TL leaves ΠL invariant because it is a BGS step satisfying detailed balance.This provides the base case for the downward induction.
Conclusion: By mathematical induction, the proposed kernel leaves Π0(V)=PΘ(V) invariant, as does the DT kernel.The downward swap kernel is included in the step establishing Tℓ invariance.