Strategy-Structure Recovery Problem

This note records paper-facing language for the central problem statement: learning a latent-variable router-generator factorization of a pretrained or fixed conditional language model in a way that preserves the observable response distribution while exposing strategy structure in the latent variable.

Core Problem

The object of interest is a fixed conditional response distribution $p_\theta(y|x)$, usually defined by a pretrained or already-fit base model. The goal is not merely to fit $p_\theta(y|x)$ again; the goal is to learn a useful probabilistic factorization of that distribution.

Reference notation:

$$p_\theta(y|x) \leadsto p_\phi = (r_\phi(z|x), g_\phi(y|x,z)).$$

Here $r_\phi(z|x)$ is the router, $g_\phi(y|x,z)$ is the generator, and the induced marginal $p_\phi(y|x)$ should preserve the observable conditional behavior of the base model while making $z$ carry strategy-level information.

Reference phrasing:

Given a fixed base-model distribution $p_\theta(y|x)$, the strategy-structure recovery problem is to learn a router-generator factorization $(r_\phi,g_\phi)$ whose marginal behavior remains faithful to $p_\theta(y|x)$ while its latent variable $z$ recovers strategy-level structure in the response distribution.

Two Requirements

The problem has two distinct requirements.

1. Distributional fidelity: the factorized model should preserve the observable conditional response law, so that marginalizing over $z$ recovers the base model's behavior.
2. Strategy alignment: the latent variable should expose a meaningful strategy decomposition of the response space, rather than encoding arbitrary predictive details, input-specific partitions, or noise.

Reference phrasing:

A successful factorization must satisfy both distributional fidelity and strategy alignment: it should preserve the observable law $p_\theta(y|x)$ while making variation in $z$ correspond to strategy-level variation in solution traces.

Avoid treating distributional fidelity as sufficient. The standard conditional ELBO can preserve marginal fit while leaving $z$ unused or unaligned.

Why This Is Not Ordinary Conditional Modeling

Ordinary conditional modeling asks for a model of $p(y|x)$ from data. In this paper, $p_\theta(y|x)$ is already modeled by the base model. The scientific and methodological problem is to decompose that already-capable model into a latent-variable factorization with an interpretable strategy variable.

Reference phrasing:

We are not trying to learn a conditional model from scratch. We start with a model that already captures the observable response distribution and ask whether its mixed strategy behavior can be reorganized into a strategy-aligned router-generator factorization.

This framing helps separate the paper from standard conditional VAE presentations, where improved likelihood or reconstruction can be the primary goal.

Boxed Problem Statement Language

Candidate concise statement:

Strategy-structure recovery problem. Given a fixed conditional language model $p_\theta(y|x)$ whose responses mix multiple high-level strategies, learn a latent-variable 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 $z$ aligns with the strategy structure of the response distribution.

Alternative with controlled-benchmark wording:

In the controlled benchmark, strategy-structure recovery means learning a factorization whose latent variable aligns with reference strategy annotations while preserving the base model's observable conditional behavior.

Use a boxed problem statement if it helps the reader distinguish the goal from ordinary generative modeling, but the concept note should not assume a final typesetting choice.

Recovery Language

Use strategy-structure recovery when the benchmark or theory provides a reference strategy structure against which the learned latent can be evaluated.

Use strategy-aligned factorization for the desired property in the general problem statement.

Use surfacing or exposing strategy structure when discussing broader pretrained language models without reference labels.

Reference phrasing:

In controlled tasks, recovery can be measured against reference strategy annotations. In broader language-model settings, the goal is more cautiously described as surfacing or exposing latent strategy structure.

Relationship To Neighboring Concepts

The factorization terminology is recorded in Router-generator factorization terminology.

The intended role of $z$ is recorded in Latent strategy variable.

The desired property of the learned latent is recorded in Strategy alignment and recovery.

The reason the standard ELBO is under-specified for this problem is recorded in ELBO under-specification and the z-ignoring solution.

The controlled benchmark used to measure this property is recorded in Task data and evaluation setup.