Generalized Alternating Minimization
Algorithms AIMech:GeneralizedAlternatingMinimization DRAFT
Aliases: GAM (generalized alternating minimization)
An information-geometric fitting framework that updates a generating distribution constrained by the observations and model parameters to decrease their joint Kullback-Leibler objective, allowing inexact distribution and parameter minimization.
Identity Notes & Concept Scope
This record denotes the Gunawardana-Byrne distribution/parameter KL-descent framework. In its alternating realization both coordinate updates decrease the same paired objective; the general composite formulation can establish overall descent directly. Exact EM and suitable generalized EM are special cases. GAM does not mean every approximate E-step, every alternating loss or the unrelated generalized additive model. Restricted-family variational EM is a specialization, while cyclic incremental EM has a separate cache/sweep realization. No normalized match across all442pinned AIO labels/scoped synonyms and480native IDs/names/aliases, including GAM and VEM probes. Full-checkout searches include ignored and hidden files in worktree and main;59and61matched lines reduce to31distinct text windows, all read. Hits denote topic-model fitting steps, prior frontier leads, unrelated GAM generalized-additive-model context or programming mnemonics. GAM is the joint distribution/parameter KL-descent framework, not GEM with its exact current E-step. Variational EM fits model parameters with a restricted posterior family; BBVI is a stochastic variational-gradient procedure, NVIL specifies amortized stochastic learning, and structured factorial-HMM inference is model-specific. Melting shares its fixed-bandwidth weighted-mean update with Gaussian mean shift; its named outer heating, membership tree and persistence selection supply the distinct scope. Bare GAM/VEM aliases are qualified. Absence is bounded to this snapshot, corpus and inspected semantic neighborhood. Identity-search counts: 59 matched lines in the worktree and 61 in main; all 31 distinct matching-text windows were read, with ignored and hidden files included.
Claims & Grounded Evidence
4 Claims
Mechanism
#paired-descent
Let D contain normalized generating laws supported on the complete-data fiber compatible with the observed event, and let P_theta be the normalized complete-data model. Minimize D_KL(Q||P_theta) over the pair. A generalized forward step reduces it in Q at fixed theta; a generalized backward step reduces it in theta at the updated Q. Exact forward minimization over the unrestricted desired family gives the current conditional posterior, and exact backward minimization maximizes expected complete-data log likelihood. A constrained family changes the optimization problem. Finite, continuous KL on the relevant domain is an explicit premise here.
Scope
#conditional-limit-set
The inspected convergence theorem requires compact parameter space, an eventually entered compact invariant generating-distribution set, a closed update map, continuous finite objective, nonincrease, and strict decrease outside the set of joint coordinatewise global minimizers. It concludes objective convergence and membership of accumulation points in that set, not necessarily convergence of the whole parameter sequence. For the unrestricted desired family, additional interior differentiability conditions give stationary observed-likelihood accumulation points; stationary does not mean a local maximum. Separate coordinate descent tests need extra hypotheses, such as the unique forward minimizer in Proposition5, to establish the composite equality condition.
Limitation
#surrogate-versus-likelihood
D_KL(Q||P_theta)=D_KL(Q||P_theta(.|observed))-log p_theta(observed). Decreasing the paired divergence may lower observed likelihood when the posterior approximation gap also changes. The paper’s constrained two-coin example moves theta from0.3 to0.4, reducing likelihood from0.58 to0.52 and reaching a restricted fixed point that is not likelihood-stationary. Even exact EM can stay at the likelihood minimum theta=0.5 in that example.
Limitation
#domain-and-source-conditions
Positive common support alone does not ensure finite KL on an arbitrary countably infinite state space. Continuous-observation extensions also need suitable measures and regularity. The incremental-EM argument applies to the inspected finite cyclic full sweep with unique admissible cached M minimizers and the theorem’s compactness/closedness hypotheses, not every random or unfair schedule. The displayed Section4 GEM comparison has its Q inequality reversed; it is not adopted as an ascent rule.
⚠️ Curated Limitations & Negative Results
- The finite-discrete formulation is the directly interpretable setting here. Global coordinate minimizers, strict progress and compact iterates are substantive assumptions, not consequences of calling an update GAM.
- The1984Csiszar-Tusnady PDF was downloaded and rendered but has not yet been visually read; it is not unseen primary support for this record.
- No historical software, benchmark or source experiment was reproduced. Saved normalized finite calculations challenge bounded formula/prose claims, not the complete conditional convergence theorems.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MachineLearningTask
Machine Learning Task
|
related_match | This procedure fits model parameters from observations; the inspected AIO target denotes a learning process/task. The algorithm-to-task relationship asserts neither equivalence nor an is-a relation. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
gam-source
|
Asela Gunawardana, William Byrne
|
primary paper
full_text
|
ID: https://www.jmlr.org/papers/v6/gunawardana05a.html
sha256:39339eb274d5…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-12T05:01:54Z
Updated: 2026-10-12T05:01:54Z
Source File:
data/curated/GeneralizedAlternatingMinimization.yaml