Strategy Recovery Theory Language

This note records paper-facing language for the theory of strategy-structure recovery. The source material is path:paper/theory_section, especially theory_abridged.tex and theory_extended.tex. This note is a writing guide for the paper; it does not replace the formal theory files.

Theory Message

The theory studies when a variational objective can select a semantically meaningful latent strategy factorization among many possible factorizations of the same conditional law.

Reference phrasing:

The strategy-structure recovery problem requires not only fitting the conditional law $p(y|x)$, but selecting a latent-variable factorization in which $z$ captures semantically consistent strategy structure.

The central mechanism is profiled selection: after profiling over the router and generator, the objective rewards latent information about $Y$ beyond $X$ only to the extent that the induced routing law and strategy-conditioned branches can be realized simply and faithfully by the available router-generator class.

Reference phrasing:

The profiled objective does not merely reward arbitrary latent information about $Y$. It rewards information that can be realized through a simple and faithful router-generator factorization.

Use factorization-compatible for the informal property that a candidate latent structure is both predictive and naturally implemented by the router-generator class.

Reference phrasing:

Semantic strategy recovery is favored when strategy structure is both predictive and factorization-compatible.

Reference Strategy

Use $S^\star=\sigma(X,Y)$ for the reference strategy in the formal theory. This hard-strategy notation means that once the completed solution trace is observed, the reference strategy is recognizable as an attribute of $(X,Y)$.

Use $q^\star(z|x,y)=\mathbf 1\{z=\sigma(x,y)\}$ for the ideal reference-strategy encoder when the theory needs a target encoder.

Reference phrasing:

The theory fixes a reference strategy $S^\star=\sigma(X,Y)$ and asks when the variational objective selects encoders close to the corresponding strategy encoder $q^\star$.

Weak And Strong Recovery

Use weak recovery for input-wise recovery up to input-dependent relabeling. Weak recovery says that, for each input $x$, the learned latent separates the reference strategies for that input, but the latent label meaning may vary across inputs.

Reference phrasing:

Weak recovery means that, for each input $x$, the learned latent agrees with the reference strategy up to an input-dependent relabeling of strategy labels.

Use strong recovery for recovery up to a single global relabeling. Strong recovery says that one shared latent label system aligns with the reference strategy across inputs.

Reference phrasing:

Strong recovery means that there is a single global relabeling under which the learned latent agrees with the reference strategy for all inputs.

Relate these to the empirical language as follows:

• weak recovery corresponds to within-input strategy separation;
• strong recovery corresponds to within-input strategy separation plus cross-input semantic consistency;
• global label alignment is a useful theory-facing synonym for the extra condition needed for strong recovery.

Avoid using weak recovery and strong recovery in empirical prose unless the surrounding text has defined them. In figures and experiment captions, prefer within-input strategy separation, cross-input semantic consistency, strategy alignment, and Analogical Consistency.

Recovery Distances

The formal theory uses entropy-based distances between the learned latent and the reference strategy:

$$d_{\mathrm w}(q,S^\star) := H_q(Z|X,S^\star)+H_q(S^\star|X,Z),$$ $$d_{\mathrm s}(q,S^\star) := H_q(Z|S^\star)+H_q(S^\star|Z).$$

The weak distance conditions on $X$, so it compares strategy semantics separately within each input. The strong distance does not condition on $X$, so it also penalizes input-dependent relabeling.

Reference phrasing:

The weak recovery distance evaluates input-wise agreement between $Z$ and $S^\star$; the strong recovery distance additionally requires a globally consistent label system across inputs.

The theory shows that exact weak or strong recovery is equivalent to the corresponding distance being zero, and that weak recovery is no harder than strong recovery in the sense that $d_{\mathrm w}(q,S^\star)\le d_{\mathrm s}(q,S^\star)$.

Profiled ELBO Score

The theory profiles out the router and generator for a fixed encoder $q$. The resulting cost $A_{\beta,\lambda}(q)$ measures how expensive it is for the router-generator class to realize the encoder-induced routing law and branch conditionals.

Use profiled router-generator cost for $A_{\beta,\lambda}(q)$.

Use profiled ELBO score for

$$J^{\mathrm{prof}}_{\beta,\lambda}(q) := I_q(Y;Z|X) - \frac{A_{\beta,\lambda}(q)}{1-\beta}.$$

Reference phrasing:

After profiling over the router and generator, encoder selection is governed by an information-minus-profiled-cost score: the latent should carry information about $Y$ beyond $X$, but that information must be realizable by a low-cost router-generator factorization.

Use information pressure for the role of \beta<1: it makes conditional information $I_q(Y;Z|X)$ beneficial in the profiled score. Use profiled cost or approximation-complexity cost for the router-generator realizability term.

Important caveat:

The complexity term is a theoretical proxy for architecture, parameterization, and optimization biases. It is not an auxiliary loss included in the experiments.

Profiled Margin

Use profiled recovery margin or profiled ELBO margin for the gap separating the reference strategy encoder from encoders that remain at least $\delta$ away under the chosen recovery distance:

$$\Delta_{\beta,\lambda}^{\rho,\mathrm{prof}}(\delta) := J^{\mathrm{prof}}_{\beta,\lambda}(q^\star) - \sup_{q\in\mathcal Q:\,d_\rho(q,S^\star)\ge\delta} J^{\mathrm{prof}}_{\beta,\lambda}(q).$$

Reference phrasing:

A positive profiled recovery margin means that the target strategy encoder scores better than every encoder outside the desired recovery neighborhood.

The population theorem is best described as an optimizer-selection statement:

The population result is an optimizer-selection statement: a positive profiled margin excludes all $\delta$-bad encoders from the set of minimizers.

The empirical result is best described as a uniform-convergence transfer:

Finite data preserves this selection once the empirical profiled objective is uniformly accurate at the scale of the same margin.

Input-Dependent Relabeling Failure

A key theory point is that weak recovery does not imply strong recovery. Input-dependent permutations of the latent labels can preserve within-input recovery and conditional information while destroying global label alignment.

Reference phrasing:

Input-dependent relabelings can recover the correct strategy partition separately for each input while assigning different meanings to the same latent label across inputs.

This is the formal version of the input-local partitioning failure described in Cross-input semantic consistency.

Strong recovery requires the profiled cost to prefer a globally aligned label system over input-dependent relabelings.

Reference phrasing:

Strong recovery requires more than latent information and within-input separation. The router-generator class or its implicit complexity bias must prefer a globally aligned label system over equally informative input-local relabelings.

If the router-generator class and complexity proxy are invariant to input-dependent relabeling, then input-local relabelings can tie the reference encoder in profiled score, preventing a positive strong margin.

How To Use This In The Paper

In a theory preview, emphasize the message before the formulas: recovery is possible when the intended strategy is predictive of residual variation in $Y$ beyond $X$ and is cheaper to realize than competing high-information attributes.

In experiments, do not use the theory notation unless needed. Connect the concepts through language: within-input strategy separation, cross-input semantic consistency, strategy alignment, and Analogical Consistency.

In methodology, use the theory to justify why the goal is not simply maximizing arbitrary latent information: useful latent information should be compatible with the router-generator factorization.

Relationship To Neighboring Concepts

The overall problem statement is recorded in Strategy-structure recovery problem.

Empirical success language is recorded in Strategy alignment and recovery.

Cross-input semantic consistency and Analogical Consistency are recorded in Cross-input semantic consistency.

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