Mean-field Boltzmann learning
Algorithms AIMech:MeanFieldBoltzmannLearning DRAFT
Aliases: Mean-field theory learning algorithm
A deterministic approximation to Boltzmann-machine parameter learning that substitutes mean-field phase statistics for stochastic equilibrium correlations in the positive-minus-negative weight update.
Identity Notes & Concept Scope
This record covers the Peterson–Anderson mean-field learning procedure, especially the implemented product-of-means correlation approximation. Its supervised formulation clamps input and output in the positive phase and input alone in the negative phase. It differs from inference at fixed parameters, contrastive divergence with data-restarted sampling, persistent negative-phase sampling and variational-positive/stochastic-negative hybrid learning. The term mean field here denotes a particular deterministic phase approximation, not every method using a variational bound. No normalized identity match was found across all442 pinned AIO labels and scoped synonyms and385 preceding native IDs/names/aliases for the four researched candidates. Full-checkout fixed-string searches included ignored and hidden files in both worktree and main, yielding78 and83 matched lines; all44 unique contents were read. Hits concern prior source citations, pending leads, neighboring HME records and unrelated Python-license authors, not existing records of the three selected identities. All76 semantic label/synonym hits and22 exact fully annotated AIO CLASS nodes with outgoing edges were read. Twenty-five contextual YAMLs were byte-verified after prior full reading and eight freshly read, totaling22 generated and11 native neighbors. Adaptive local experts substantially overlaps represented MDN/HME scope and receives no duplicate ID. Bare context-sensitive HMM and csHMM are searched homonyms, not unconditional aliases. Absence is bounded to the pin and inspected semantic neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For symmetric zero-diagonal weights T and binary spins s_i in {-1,+1}, use energy E=-sum_(i<j) T_ij s_i s_j+sum_i I_i s_i, with thresholds representable by a fixed bias unit. At temperature tau, approximate the unclamped means in each phase by m_i=tanh((sum_j T_ij m_j-I_i)/tau), keeping clamped means at their observed spin values. The source solves these equations by iterative updates or a continuous relaxation, usually over an annealing schedule. Its application studies approximate distinct-unit correlations by rho_ij≈m_i m_j, including fixed clamped values, although Equation(3.14) discusses a more involved correlation approximation. For a supervised example, calculate these moments once with inputs and outputs clamped and once with only inputs clamped. Update a weight proportionally to rho_ij_positive-rho_ij_negative; the inverse-temperature factor in the likelihood gradient may be absorbed into the learning rate at a fixed final temperature. These are approximate learning statistics and generally differ from exact conditional-likelihood derivatives.
⚠️ Curated Limitations & Negative Results
- A difference of variational lower bounds for the clamped and free log partitions is not generally a lower bound on their difference. Approximating both phases therefore does not by itself guarantee monotonic conditional likelihood or an exact likelihood gradient.
- Mean-field fixed points may depend on initialization and schedule; simultaneous updates need not converge or improve a bound. The source’s infinite-network exactness wording is not a theorem for arbitrary graph sequences.
- Product-of-means correlations discard connected correlations. The source acknowledges finite-size discrepancies; its stronger pair approximation is not the approximation used in the reported application studies.
- For the stated binary energy, the consistent heatbath probability is sigmoid(2*effective_field/tau). The sign and factor in the displayed baseline Equation(2.7) are inconsistent with that energy; AppendixA’s flip-rate convention is consistent. The baseline display is not adopted as a verified implementation.
- The XOR, encoder and line-symmetry studies have task-specific schedules, learning rates and stopping rules. The line-symmetry footnote explicitly excludes unseen-pattern generalization. No speedup, accuracy, generalization guarantee or training reproduction is curated.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:BoltzmannMachineNetwork
Boltzmann Machine Network
|
related_match | The algorithm estimates parameter updates for the symmetrically connected stochastic network named by the AIO target. A learning algorithm is distinct from the network family it trains. |
AIO:SupervisedLearning
Supervised Learning
|
related_match | The inspected phase procedure learns conditional outputs from input-output examples. The AIO target is the broader task context, not an exact algorithm identity or an asserted algorithm-to-task subclass. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Carsten Peterson, James R. Anderson
|
primary paper
full_text
|
ID: https://wpmedia.wolfram.com/sites/13/2018/02/01-5-6.pdf
sha256:e51bee6e438d…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T20:33:55Z
Updated: 2026-10-11T20:33:55Z
Source File:
data/curated/MeanFieldBoltzmannLearning.yaml