Problem Statement Outline

This note drafts the structure for path:paper/conference-paper/sections/02-problem-statement.tex. The section should center the paper around latent-variable factorization: decomposing an already-represented conditional law $p_\theta(y|x)$ into a router and generator whose latent variable is informative and strategy-aligned. It should define the central object and desired property before the methodology section explains why standard variational training is under-specified and how base-directed reconstruction changes the training pressure.

This is for a NeurIPS submission with a 9-page main-text limit. The problem statement should therefore be compact: likely around 0.75–1.0 pages, with one boxed problem statement and only the notation needed to make the methodology and experiments readable.

Related concept notes:

• Strategy-structure recovery problem
• Router-generator factorization
• Latent strategy variable
• Base model and reference notation
• Strategy alignment and recovery
• Cross-input semantic consistency
• Task data and evaluation setup
• ELBO under-specification and the z-ignoring solution
• Strategy recovery theory language

Section Role

The problem statement should define latent-variable factorization as the task of decomposing a fixed base-model response distribution $p_\theta(y|x)$ into a router-generator form

with the goal that the latent variable is informative and strategy-aligned.

The key framing is that we are not trying to learn $p_\theta(y|x)$. We already have a model representing that conditional response distribution. The scientific and methodological goal is to factor that model in a way that reveals, separates, or exposes abstract strategy structure implicit in its distribution.

The section should not yet derive the ELBO failure or define the full training objective. It should instead create the vocabulary and logical target that make the methodology section necessary.

Reader State Before This Section

The introduction has motivated that language models can place probability mass on multiple high-level ways of solving the same input, and that a useful model should expose this structure as a controllable latent variable rather than leave it implicit in hidden states or sampled traces.

Reader State After This Section

By the end of the section, the reader should be able to answer:

1. What distribution is being factorized?
2. What are the router and generator?
3. What role is $z$ intended to play?
4. Why is preserving $p_\theta(y|x)$ not sufficient?
5. Why do expressive router and generator classes create many marginal-preserving factorizations?
6. What distinguishes an informative, strategy-aligned latent variable from an arbitrary latent decomposition?
7. What stronger semantic property distinguishes a reusable strategy variable from an input-local latent partition?
8. How will controlled experiments and theory make the otherwise abstract target measurable?

Proposed Section Title

Current source title: Problem Statement: Latent-Variable Factorization of Abstract Strategies.

Recommended title: Problem Statement: Latent-Variable Factorization.

Decision: approved.

Reason: this matches the intended paper-level framing. Latent-variable factorization is the central object; strategy alignment, informative latents, and strategy-structure recovery are desired properties or outcomes of a useful factorization.

Main-Text Space Budget

Because the paper has a 9-page limit, this section should probably not use eight visible subsections. Treat the detailed outline below as a dependency map. In the final paper, compress it into four or five short paragraphs plus a boxed problem statement:

1. State that $p_\theta(y|x)$ is already represented by the base model; the goal is to factor it, not learn it.
2. Define the router-generator factorization and induced marginal.
3. Explain why marginal preservation is under-specified under expressive router and generator classes.
4. Define the desired property: $z$ should be informative and strategy-aligned, with cross-input semantic consistency as the stronger semantic criterion.
5. Briefly say that controlled experiments operationalize these properties using reference strategy annotations; defer benchmark details to experiments.

The boxed problem statement should carry the formal definition so surrounding prose can stay short.

Compact Main-Text Outline

Paragraph 1: Object Being Factored

Establish that the object is the fixed base-model response distribution $p_\theta(y|x)$. The paper is not trying to learn $p_\theta(y|x)$ from data; it starts with a model that already represents that distribution and asks for a latent-variable factorization of it.

Required notation: $x$, $y$, $p_\theta(y|x)$, and base model.

Key sentence target:

We are not trying to learn the conditional response law from scratch: $p_\theta(y|x)$ is already represented by the base model. Our goal is to factor this law into a latent-variable model that exposes useful structure in how responses are generated.

Paragraph 2: Router-Generator Factorization

Define the router-generator decomposition:

Introduce the schematic transformation $p_\theta(y|x)\leadsto p_\phi=(r_\phi,g_\phi)$, where $p_\phi$ denotes the factorized model family induced by the router and generator. Then define $z$, $r_\phi(z|x)$, $g_\phi(y|x,z)$, and the latent-marginal $p_\phi(y|x)$. State that distributional fidelity means $p_\phi(y|x)\approx p_\theta(y|x)$.

Box: Latent-Variable Factorization Problem

Include a boxed problem statement in the main text.

Approved boxed statement:

Latent-variable factorization problem. Let $p_\theta(y|x)$ be a fixed base-model response distribution. We write the factorization target schematically as $p_\theta(y|x)\leadsto p_\phi=(r_\phi,g_\phi)$, where $p_\phi$ denotes the factorized model family induced by a latent variable $z\in\mathcal Z$, router $r_\phi(z|x)$, and strategy-conditioned generator $g_\phi(y|x,z)$. This induces the latent marginal

$$p_\phi(y|x)=\int_{\mathcal Z}r_\phi(z|x)g_\phi(y|x,z)\,d\mu(z).$$

The goal is to select a factorization that preserves the observable distribution, $p_\phi(y|x)\approx p_\theta(y|x)$, while making $z$ informative about $y$ beyond $x$ and strategy-aligned: variation in $z$ should expose high-level, reusable strategy variation in responses rather than arbitrary predictive or surface details.

LaTeX-ready compact version:

\begin{problemstatementbox*}[Latent-variable factorization]
Let \(p_\theta(y|x)\) be a fixed base-model response distribution. We write
the factorization target schematically as
\(p_\theta(y|x)\leadsto p_\phi=(r_\phi,g_\phi)\), where \(p_\phi\) denotes
the factorized model family induced by a latent variable \(z\in\mathcal Z\),
router \(r_\phi(z|x)\), and strategy-conditioned generator \(g_\phi(y|x,z)\).
This induces the latent marginal
\[
  p_\phi(y|x)=\int_{\mathcal Z} r_\phi(z|x)g_\phi(y|x,z)\,d\mu(z).
\]
The goal is to select a factorization that preserves the observable
distribution, \(p_\phi(y|x)\approx p_\theta(y|x)\), while making \(z\)
informative about \(y\) beyond \(x\) and strategy-aligned: variation in \(z\)
should expose high-level, reusable strategy variation in responses rather than
arbitrary predictive or surface details.
\end{problemstatementbox*}

Shorter fallback if space is tight:

Latent-variable factorization problem. Given a fixed base-model response distribution $p_\theta(y|x)$, learn a router-generator factorization $p_\phi(y|x)=\int r_\phi(z|x)g_\phi(y|x,z)\,d\mu(z)$ that preserves the observable law, $p_\phi(y|x)\approx p_\theta(y|x)$, while making $z$ informative about $y$ beyond $x$ and aligned with high-level, reusable strategy variation rather than arbitrary predictive or surface details.

Paragraph 3: Why Marginal Preservation Is Not Enough

Explain that when the router and generator classes are expressive, there are many factorizations that preserve the same marginal distribution. Transformer language models make this especially relevant. In particular, a z-ignoring factorization can preserve $p_\theta(y|x)$ while leaving $z$ inert.

This paragraph should only state the obstruction. The methodology section should provide the ELBO identity and empirical failure figure.

Paragraph 4: Desired Latent Property

Define informative and strategy-aligned:

• informative: $z$ carries information about $y$ beyond what is already determined by $x$;
• strategy-aligned: the information carried by $z$ corresponds to high-level strategy variation rather than arbitrary predictive details or surface form.

Define a strategy as a high-level, reusable attribute of a response trace. Avoid overclaiming that strategies are literal cognitive plans or that all useful latents are unique.

Paragraph 5: Local Separation And Cross-Input Semantic Consistency

Define the empirical distinction here, but reserve weak/strong recovery terminology for the theory section.

• within-input strategy separation: different latent values correspond to different strategies for the same input.
• cross-input semantic consistency: the same latent value or region tends to correspond to the same strategy across related inputs.

Add only a brief operationalization sentence:

In controlled experiments, we operationalize these properties using reference strategy annotations, while withholding those annotations from factorization training.

Defer benchmark task families, metrics, and Analogical Consistency details to the experiments section.

Detailed Dependency Map

The following outline records the logical dependencies and can guide drafting, but it should not be treated as a required visible subsection structure in the 9-page version.

2.1 The Object To Be Factored: A Base-Model Response Distribution

Purpose: Establish the object being factorized: a fixed conditional language model $p_\theta(y|x)$, not an external data distribution to be learned from scratch.

Reader question: What is the target distribution in this paper?

Core concepts and notation:

• $x$: input or problem instance.
• $y$: response sequence; in experiments, a solution trace.
• $p_\theta(y|x)$: fixed base-model response distribution.
• $\theta$: fixed base model parameters.
• $\mathcal D_X$: input distribution used when sampling $y\sim p_\theta(\cdot|x)$.
• base model as the neutral term; entangled base model only when emphasizing that strategies are mixed implicitly.

Claims supported: Conceptual framing. This subsection supports the later empirical finding that strategy information is already present in the base model, but it should not make that empirical claim yet.

Figures or results supported: Prepares fig:motivation and fig:entanglement; no new figure required here.

Must establish before next subsection: The target is a model-defined conditional law $p_\theta(y|x)$ that already predicts responses. The paper is therefore about factorizing an already-capable model, not fitting a new conditional model from data.

2.2 Latent-Variable Router-Generator Factorization

Purpose: Define the learned object.

Reader question: What does it mean to factorize $p_\theta(y|x)$?

Core concepts and notation:

• $z\in\mathcal Z$: latent strategy variable.
• $r_\phi(z|x)$: router.
• $g_\phi(y|x,z)$: strategy-conditioned generator.
• $\phi$: learned router-generator parameters.
• $p_\phi(z|x):=r_\phi(z|x)$.
• $p_\phi(y|x,z):=g_\phi(y|x,z)$.
• $p_\phi(y|x):=\int_{\mathcal Z}r_\phi(z|x)g_\phi(y|x,z)\,d\mu(z)$.

Candidate equation block:

Core explanatory move: A router-generator factorization is a latent-variable decomposition of the same observable conditional law. The router $r_\phi(z|x)$ assigns input-dependent mass over latent values, and the generator $g_\phi(y|x,z)$ produces responses conditioned on both $x$ and $z$. Marginalizing over $z$ should recover the original response distribution.

Claims supported: Conceptual and methodology framing. This gives the architecture and training sections their common notation.

Figures or results supported: fig:architecture-ar-compact; the exact architecture details should remain in methodology.

Must establish before next subsection: A latent-variable factorization is mathematically neutral. The equation alone does not imply that $z$ is meaningful, controllable, or strategy-aligned.

2.3 Why Marginal Preservation Is Under-Specified

Purpose: Explain why exact or approximate preservation of $p_\theta(y|x)$ is not enough to define the desired factorization.

Reader question: If $p_\phi(y|x)\approx p_\theta(y|x)$, why is the problem not solved?

Core concepts and notation:

• Expressive router and generator classes can realize many different latent-variable factorizations with the same marginal $p_\phi(y|x)$.
• Transformer language models are expressive enough that a z-ignoring factorization is available in principle and nearby in the adapted-from-base setting.
• marginal-preserving factorization: a factorization that preserves the observable conditional law after integrating out $z$.
• z-ignoring factorization: a factorization where $g_\phi(y|x,z)$ does not depend meaningfully on $z$.
• selection problem: the paper must select a useful factorization among many marginally valid decompositions.

Claims supported: Conceptual framing and methodology motivation. This sets up the ELBO under-specification argument without deriving it.

Figures or results supported: Prepares fig:elbo-fails.

Must establish before next subsection: The paper's goal cannot be stated only as marginal preservation. The desired factorization needs a latent variable that carries useful information about $y$ beyond $x$.

2.4 Desired Property: An Informative, Strategy-Aligned Latent Variable

Purpose: Define the desired semantic role of $z$ without overclaiming that any learned latent automatically has that role.

Reader question: What kind of latent variable are we trying to uncover?

Core concepts and notation:

• strategy: a high-level, reusable attribute of a response trace that describes the broad procedure used to generate a solution, not incidental surface form.
• latent strategy variable: an unobserved variable intended to capture strategy-level variation in $y$ beyond what is determined by $x$.
• informative latent variable: a latent variable that carries information about $y$ beyond what is already determined by $x$.
• strategy-aligned latent variable: an informative latent whose variation corresponds to strategy-level variation rather than arbitrary predictive details.
• strategy-level variation versus surface variation, arbitrary predictive information, or noise.
• Use $\operatorname{strat}(y)$ only when discussing controlled benchmark labels or evaluation.

Claims supported: Conceptual framing. This prevents the paper from treating "latent variable" and "strategy" as synonyms.

Figures or results supported: Prepares strategy-coloring in latent dynamics figures and strategy labels in benchmark metrics.

Must establish before next subsection: Informativeness is necessary but not sufficient. A latent can help predict $y$ without exposing abstract strategy structure. The target is an informative latent variable whose information is strategy-aligned.

2.5 Success Criteria For A Useful Latent-Variable Factorization

Purpose: State the two requirements for a useful factorization.

Reader question: What makes a router-generator factorization successful?

Core concepts and notation:

• distributional fidelity: $p_\phi(y|x)\approx p_\theta(y|x)$ after marginalizing over $z$.
• latent informativeness: $z$ carries information about $y$ beyond $x$.
• strategy alignment: the information carried by $z$ corresponds to strategy-level variation in $y$.
• strategy-aligned factorization: a factorization whose latent variable tracks strategy structure.
• strategy-consistent generation: conditioning on a fixed latent value or region should produce responses with the corresponding strategy.

Candidate boxed statement:

Latent-variable factorization problem. Given a fixed conditional language model $p_\theta(y|x)$ whose responses mix multiple high-level strategies, learn a router-generator factorization $p_\phi=(r_\phi,g_\phi)$ such that the marginal $p_\phi(y|x)$ remains faithful to $p_\theta(y|x)$ while the latent variable $z$ is informative and strategy-aligned.

Claims supported: Central problem formulation. Later empirical claims F3 and F5 depend on this distinction.

Figures or results supported: fig:disentanglement, fig:reconstruction-decodability, and alignment-reconstruction diagnostics.

Must establish before next subsection: Distributional fidelity is necessary but not sufficient. Even informativeness is not enough if the latent encodes arbitrary predictive features. The desired factorization should expose strategy-level structure implicit in $p_\theta(y|x)$.

2.6 Local Separation Versus Cross-Input Semantic Consistency

Purpose: Define the stronger semantic requirement that makes $z$ reusable across related inputs.

Reader question: How do we rule out a latent that separates strategies separately for each input but changes meaning across inputs?

Core concepts and notation:

• within-input strategy separation: different latent values correspond to different strategies for the same input.
• input-local partition: failure mode where the latent separates outputs within each input but does not preserve a shared meaning across inputs.
• cross-input semantic consistency: the same latent value or region tends to correspond to the same strategy across related inputs.
• cross-input latent strategy alignment: alignment-forward phrasing for the same property.
• Analogical Consistency: metric name to be introduced later in experiments, not fully defined here unless a compact preview is useful.

Claims supported: Empirical finding F6 and theory language around strong recovery.

Figures or results supported: fig:analogical-consistency and the Strategy Alignment metric in method metric bars.

Must establish before next subsection: A reusable strategy variable requires more than local separation. This motivates both the analogical-transfer evaluation and the theory distinction between weak and strong recovery.

2.7 Controlled Evaluation And Strategy-Structure Recovery

Purpose: Explain how the abstract recovery target becomes measurable in the controlled benchmark without making strategy labels part of the training signal.

Reader question: If strategies are latent in the general problem, how can the experiments evaluate recovery?

Core concepts and notation:

• controlled multi-strategy algorithmic benchmark at first mention.
• algorithmic strategy benchmark thereafter.
• problem instance for $x$ in experiments.
• solution trace for $y$ in experiments.
• reference strategy annotations or ground-truth strategy labels used only for evaluation.
• Factorization training samples $x\sim\mathcal D_X$ and $y\sim p_\theta(\cdot|x)$; it does not train directly against strategy labels.
• strategy-structure recovery: the measured outcome that a learned factorization's latent variable recovers the reference strategy structure.

Claims supported: Benchmark setup contribution S3; empirical recovery claims in experiments.

Figures or results supported: All controlled-task strategy-alignment metrics.

Must establish before next subsection: Recovery language is appropriate in controlled tasks and theory because a reference strategy structure is available. For broader language-model settings, the paper should use softer language such as revealing, surfacing, uncovering, or exposing latent strategy structure.

2.8 Handoff To Methodology: Selecting A Useful Factorization

Purpose: End the section by stating the under-specification problem at a high level, without doing the full ELBO derivation.

Reader question: Why does this problem require a specialized objective?

Core concepts and notation:

• Many router-generator factorizations can induce the same marginal $p_\phi(y|x)$.
• z-ignoring factorization: $g_\phi(y|x,z)$ does not depend on $z$.
• objective-level under-specification: matching $p_\theta(y|x)$ does not identify a strategy-aligned factorization.
• adapted-from-base optimization pathology: the generator can start near a z-ignoring solution because it is initialized from the base model.
• selection among factorizations: methodology should prefer a factorization where $z$ explains strategy-level variation in $y$ beyond what $x$ already predicts.

Claims supported: Methodology finding F2, but the detailed evidence and derivation belong in the methodology section.

Figures or results supported: fig:elbo-fails.

Must establish before next section: The methodology must select among marginally valid latent-variable factorizations, preferring ones where $z$ is both informative and strategy-aligned.

Suggested Section-Level Flow

1. Define the target law $p_\theta(y|x)$.
2. Define the exact router-generator factorization.
3. Explain that expressive router and generator classes admit many marginal-preserving decompositions.
4. Clarify that $z$ is intended to be an informative, strategy-aligned latent variable, but this is not guaranteed by notation.
5. State the latent-variable factorization problem as preserving the marginal while selecting a useful latent decomposition.
6. Strengthen the target from within-input separation to cross-input semantic consistency.
7. Explain how controlled benchmarks and theory make recovery measurable.
8. Transition to methodology by pointing out that marginal fidelity alone is under-specified.

What To Keep Out Of This Section

• Full derivation of the negative conditional ELBO identity. Put this in methodology.
• Details of posterior parameterization $q_\xi(z|x,y)$. Mention only if needed to preview training; define fully in methodology.
• Full base-directed reconstruction objective $\mathcal R_\theta$. Put this in methodology.
• Detailed task-family descriptions. Put these in experiments.
• Strong empirical claims that require figures. The problem section can define the evaluation target, but evidence belongs in experiments.

Open Decisions

Resolved decisions:

1. Rename the LaTeX section title to Problem Statement: Latent-Variable Factorization.
2. Include an unnumbered Problem Statement box in the main text.
3. Define the empirical distinction as within-input strategy separation versus cross-input semantic consistency here, and reserve weak recovery / strong recovery terminology for the theory section.
4. Mainly defer controlled benchmark details to the experiments section. Include at most a sentence or two saying that the controlled experiments operationalize these properties using reference strategy annotations.
5. Avoid visible \subsection headings in the main text to save space. Use compact prose and consider \paragraph labels only if the section needs scanability.
6. Use the approved boxed statement above as the main problem anchor.

Remaining drafting decisions:

1. Whether to use the shorter fallback only if final page fit becomes tight.
2. Whether to use any \paragraph labels in the final section.