Adaptive Alternating Minimization
Algorithms AIMech:AdaptiveAlternatingMinimization DRAFT
Aliases: AAM (adaptive alternating minimization)
A two-coordinate minimization procedure that performs one exact alternating sweep over a newly supplied pair of feasible sets at each iteration, aiming at the optimum of their limiting sets.
Identity Notes & Concept Scope
This is Niesen-Shah-Wornell’s changing-domain AAM, not adaptive step-size selection, arbitrary online EM, a fully solved inner optimization at every time point, or a generic nonstationary tracking guarantee. Fixed families recover exact alternating minimization. The2009KL mixture application adds coefficient floors and an empirical-marginal correction, so it is not identical to unconstrained fixed-component weight EM. No normalized identity collision across all 442 pinned AIO IDs, labels/scoped synonyms and 483 maintained IDs, names and aliases. Full-checkout searches in the worktree and main included ignored and hidden files: 8 and 9 matched lines, respectively, reduce to six distinct matched texts, all inspected. Hits concern existing discovery leads and the unrelated assembler mnemonic AAM. The fixed-family exact KL projection procedure is narrower than generalized alternating minimization; weight-only EM fixes component laws rather than fitting a full mixture model; rate-distortion BA minimizes information plus distortion with a fixed source, whereas the IM algorithm maximizes a variational information bound; adaptive alternating minimization changes the feasible families at each iteration. Qualified AAM and descriptive weight-only names avoid unsupported acronym or historical naming claims. Absence is bounded to this snapshot and inspected corpus. Identity-search counts: 8 worktree and 9 main matched lines, all six distinct texts read, including ignored and hidden files.
Claims & Grounded Evidence
4 Claims
Mechanism
#changing-domain-sweep
Given a real-valued objective D and an initial Q_0 in family Q_0, at iteration n choose P_n in argmin over family P_n of D(P,Q_(n-1)), then Q_n in argmin over family Q_n of D(P_n,Q). The family subscripts denote the newly revealed domains, while P_n and Q_n denote chosen points. Each coordinate step is exact, but the pair need not solve that iteration’s full joint problem. Changing domains can raise the objective relative to the preceding iteration.
Scope
#liminf-and-convergence
Theorem4 assumes compact families within a common compact metric space, Hausdorff convergence to compact limiting families, continuous D, and its compatible three-point/four-point inequalities C1-C2. It first guarantees that the objective liminf equals the limiting global minimum and that cluster points of subsequences attaining that liminf are optimal. Full objective convergence and optimality of all cluster points additionally follow if sum_n omega_D(2epsilon_n) is finite, where epsilon_n is the sum of the two Hausdorff errors and omega_D is the modulus of continuity on the sum product metric. This is a sufficient condition, not a necessary one, and does not require a unique optimum or imply convergence of the entire point sequence.
Scope
#adaptive-mixture-domain
The finite-mixture realization keeps component laws fixed and bounded below, constrains coefficients by a positive floor, and updates the prescribed data marginal as observations arrive. Its M projection thresholds and rescales component masses to satisfy the coefficient floor; its E projection rescales the lifted model to the current marginal. The paper also modifies small empirical frequencies to maintain positivity, a correction that is eventually inactive under its iid assumptions. For the ordinary empirical convergence rate, the authors conclude liminf optimality and explicitly withhold the stronger full-convergence conclusion from this theorem.
Limitation
#application-formula-boundary
The Hilbert specialization uses squared distance. Its product-space weighted-mean projection therefore minimizes a sum of weighted squared distances to sets, although printed Eq22 instead displays unsquared distances. For singleton sets{0},{4},{4} with equal weights, the projection mean8/3 minimizes the squared cost while the unsquared cost is minimized at4. This qualifies that application formula, not the general changing-domain theorem. AAM’s theorem also does not give generic tracking of domains that never settle.
⚠️ Curated Limitations & Negative Results
- C1-C2 are substantive compatibility inequalities, not consequences of convexity of an arbitrary paired objective. The auxiliary delta is not required to be nonnegative in this adaptive theorem.
- Positive floors, a common compact domain, and the stated set convergence matter in the KL application. The absence of a stronger theorem conclusion is not proof of divergence.
- The arXiv version page was checked; no separate audit of its12-page preprint or adaptive-step-size algorithms is claimed.
- Finite curator calculations check selected support, normalization and convergence boundaries. They are not author software, historical experiments or a proof of every source theorem.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MachineLearningTask
Machine Learning Task
|
related_match | The inspected procedure includes statistical distribution fitting from observations. This relates that fitting realization to the task-scoped AIO learning process, without making all mathematical applications learning tasks or asserting algorithm/task equivalence or an is-a edge. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
adaptive-source
|
Urs Niesen, Devavrat Shah, Gregory W. Wornell
|
primary paper
full_text
|
ID: doi:10.1109/TIT.2008.2011442
sha256:addb105074fa…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-12T05:22:42Z
Updated: 2026-10-12T05:29:50Z
Source File:
data/curated/AdaptiveAlternatingMinimization.yaml