Self-Consistent Symmetric Optimization

• The framework integrates active inference by jointly optimizing epistemic gain and pragmatic cost using a single sequential acquisition rule.
• It uses a curiosity coefficient to balance exploration and exploitation, ensuring sufficient information gain for reliable posterior consistency.
• The approach unifies Bayesian experimental design and Bayesian optimization, leading to no-regret guarantees and improved objective alignment.

The self-consistent symmetric optimization framework is an interpretation of active inference in which a single sequential acquisition rule simultaneously governs learning and decision-making by balancing epistemic value and pragmatic value through a curiosity coefficient. In the formulation developed in "Curiosity is Knowledge: Self-Consistent Learning and No-Regret Optimization with Active Inference" (Li et al., 5 Feb 2026), the agent selects actions by optimizing an Expected Free Energy (EFE)-style criterion whose epistemic term is conditional mutual information and whose pragmatic term is an expected potential energy or heuristic regret surrogate. The framework is self-consistent because better beliefs improve actions and informative actions improve beliefs, and symmetric in the sense that epistemic and pragmatic objectives are jointly optimized within one variational principle rather than handled as separate modules.

1. Unified acquisition rule and EFE decomposition

The core setup treats active inference (AIF) as a single sequential decision rule. At time $t$, the selected action is

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$

where $s$ is the latent parameter or quantity of interest, $\mathcal D_{t-1}$ is the data history, $I(s;(x,y)\mid \mathcal D_{t-1})$ is the conditional mutual information obtained by querying $x$ and observing $y$, $h_t(y)$ is a time-varying potential energy or heuristic regret surrogate, and $\beta_t\ge 0$ is the curiosity coefficient (Li et al., 5 Feb 2026).

Within this formulation, EFE is decomposed into an epistemic value term and a pragmatic value term. The epistemic component is the expected entropy reduction

$$I(s;(x,y)\mid \mathcal D_{t-1}) = H\!\left[p(s\mid \mathcal D_{t-1})\right] - \mathbb E_{p(y\mid x,\mathcal D_{t-1})}\!\left[ H\!\left[p(s\mid \mathcal D_{t-1}\cup(x,y))\right] \right].$$

The paper explicitly identifies this with the standard Bayesian experimental design criterion

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$0

This yields a structural unification. Bayesian experimental design (BED) maximizes information gain alone, Bayesian optimization (BO) emphasizes task reward or regret, and the AIF/EFE rule combines both through the joint criterion

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$1

A plausible implication is that the framework is best understood not as an additional acquisition heuristic layered on top of BED or BO, but as a common variational form in which both appear as limiting cases.

2. Curiosity coefficient and the exploration-exploitation mechanism

The curiosity coefficient $$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$2 scales the information-gain term and directly controls the exploration-exploitation trade-off. Larger $$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$3 places greater weight on epistemic value and therefore induces more exploration; smaller $$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$4 favors exploitation and immediate task performance.

The paper emphasizes two failure modes. Insufficient curiosity can produce myopic exploitation, poor uncertainty reduction, and possible failure of posterior consistency. Excessive curiosity can produce over-exploration, higher transient regret, and inefficient task performance. The central quantitative condition is the sufficient-curiosity threshold

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$5

This condition appears in both main theorems. Its interpretation in the paper is precise: curiosity must be large enough that the epistemic term is not dominated by the expected pragmatic penalty. A common misconception is that $$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$6 is merely a tunable heuristic. The framework instead treats it as the mechanism that makes the information-seeking component strong enough to drive identification of the latent state.

This balance is the main symmetry of the framework. The same scalar parameter mediates both belief formation and control, so neither component is externally subordinated to the other. This suggests that “symmetric” refers to a formal symmetry of treatment between knowing and doing, rather than to geometric or algebraic symmetry.

3. Self-consistent learning: posterior consistency

The first theoretical guarantee establishes self-consistent learning, formulated as posterior consistency under the AIF policy (Li et al., 5 Feb 2026). Actions are chosen by the acquisition rule above, and the theorem assumes three conditions.

First, the prior entropy is finite:

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$7

Second, the observation process is distinguishable in the sense that

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$8

where

$$x_t=\arg\max_{x\in\mathcal X} \Big\{ \beta_t\, I\!\left(s;(x,y)\mid \mathcal D_{t-1}\right) -\mathbb E_{p(y\mid x,\mathcal D_{t-1})}[h_t(y)] \Big\},$$9

Third, the sufficient-curiosity condition holds:

$s$0

With

$s$1

the theorem states that after enough steps,

$s$2

whenever

$s$3

where $s$4 and $s$5.

The result says that the posterior concentrates on the true latent state $s$6 provided that initial uncertainty is bounded, the observation model can distinguish true and false hypotheses, and curiosity is sufficient to keep the policy informative. The proof uses the cumulative information-gain identity

$s$7

together with a lower bound connecting the instantaneous regret-like term to the residual posterior mass on incorrect hypotheses.

The importance of this theorem is conceptual as well as technical. Learning is not guaranteed merely by Bayes updating; it depends on the action policy continuing to select experiments that resolve uncertainty. In this framework, posterior consistency is therefore an endogenous consequence of the acquisition rule rather than an independent statistical assumption.

4. No-regret optimization and objective alignment

The second main theorem establishes a bounded cumulative regret guarantee in a GP-based optimization setting (Li et al., 5 Feb 2026). The paper assumes a Gaussian process model

$s$8

with Gaussian likelihood

$s$9

The regret function is assumed Lipschitz:

$\mathcal D_{t-1}$0

The heuristic must also be aligned with the true objective:

$\mathcal D_{t-1}$1

Under these conditions, and again under sufficient curiosity, the cumulative regret

$\mathcal D_{t-1}$2

satisfies, with probability at least $\mathcal D_{t-1}$3,

$\mathcal D_{t-1}$4

with

$\mathcal D_{t-1}$5

and $\mathcal D_{t-1}$6 the maximum information gain up to $\mathcal D_{t-1}$7 points.

The interpretation given in the paper has three parts. Regret is controlled when $\mathcal D_{t-1}$8 is controlled; better heuristic alignment, meaning smaller $\mathcal D_{t-1}$9, improves the bound; and larger curiosity $I(s;(x,y)\mid \mathcal D_{t-1})$0 strengthens the information-seeking contribution to regret, which is beneficial for learning but can worsen transient optimization performance. The theorem is presented as an explicit generalization of classical GP-UCB and Bayesian optimization regret analyses.

This theorem formalizes the role of objective alignment. If the surrogate $I(s;(x,y)\mid \mathcal D_{t-1})$1 is misaligned with the true regret, the policy may optimize the wrong quantity, and the bound degrades through $I(s;(x,y)\mid \mathcal D_{t-1})$2. The framework therefore couples exploration not only to uncertainty reduction but also to the representational fidelity of the pragmatic term.

5. Structural unification of learning and optimization

A central claim of the framework is that the same sufficient-curiosity condition supports both posterior consistency and no-regret optimization. This is the point at which the framework becomes genuinely unified rather than merely multi-objective. Learning and optimization are not staged sequentially, and they are not assigned disjoint acquisition rules. They are coupled by the same trade-off parameter $I(s;(x,y)\mid \mathcal D_{t-1})$3.

The relation to classical paradigms is explicit. If $I(s;(x,y)\mid \mathcal D_{t-1})$4, the acquisition moves toward pure information-seeking and thus toward BED. If $I(s;(x,y)\mid \mathcal D_{t-1})$5 is very small, behavior moves toward purely pragmatic optimization. With both terms present, the method is a hybrid of BED and BO. The BO-style information-theoretic machinery also appears in the regret analysis through the identity

$I(s;(x,y)\mid \mathcal D_{t-1})$6

Within this interpretation, the framework is self-consistent because belief quality and action quality recursively reinforce one another. Better posteriors improve the quality of pragmatic decisions, while sufficiently curious decisions improve the posterior. It is symmetric because epistemic and pragmatic terms are optimized jointly within one variational principle. A plausible implication is that the framework replaces the standard modular picture of “first learn, then optimize” with a closed-loop criterion in which experimental design and task performance are co-determined at every step.

A second common misconception is that increasing curiosity always improves performance. The theory rejects that view. Curiosity must be high enough to sustain informative exploration, but beyond that threshold larger values can worsen transient regret.

6. Assumptions, tuning rules, and experimental validation

The assumptions play distinct roles. For posterior consistency, finite prior entropy controls initial uncertainty, observational distinguishability controls identifiability, and sufficient curiosity prevents degeneracy into repeated exploitation. For regret bounds, smoothness of $I(s;(x,y)\mid \mathcal D_{t-1})$7 allows predictive uncertainty to be converted into regret control, heuristic alignment governs degradation through $I(s;(x,y)\mid \mathcal D_{t-1})$8, and sufficient curiosity again acts as the bridge between inference and control. The paper summarizes the practical effect of these assumptions succinctly: more initial uncertainty $I(s;(x,y)\mid \mathcal D_{t-1})$9 requires more samples, larger discriminability $x$0 speeds posterior concentration, smaller heuristic error $x$1 tightens regret bounds, and too much curiosity can worsen transient regret (Li et al., 5 Feb 2026).

The design guidance follows directly from this analysis. The recommended strategies are adaptive curiosity scheduling, in which $x$2 is increased when posterior entropy is high or expected information gain is small and decreased later; annealing, in which curiosity starts large and then falls as confidence grows; heuristic refinement, in which $x$3 is updated online so that $x$4 shrinks over time; and energy or constraint design, which ensures that $x$5 does not remove critical distinguishing signal.

The experimental program spans both synthetic and real-world settings. In a discrete sandbox, the paper varies prior entropy $x$6, distinguishability $x$7, and curiosity $x$8, finding that higher prior accuracy gives faster posterior contraction, higher distinguishability accelerates consistency, and too little curiosity stalls learning. In a 1D GP bandit, it varies heuristic misalignment $x$9 and curiosity $y$0, finding that larger heuristic bias increases cumulative regret, and that once curiosity is sufficient, smaller $y$1 can yield lower regret because it reduces needless exploration. In constrained system identification, using a plume-source localization setup with

$y$2

larger $y$3 is needed when sensor-latent coupling weakens, but too large a value becomes counterproductive. In composite Bayesian optimization, where unknown preference functions are modeled through nested active inference or preference learning, heuristic convergence matters, sufficient curiosity is needed for regret to stabilize, and overly large $y$4 again increases transient regret.

Taken together, these results support the framework’s central formulation: curiosity must be high enough to preserve informative exploration, yet not so high that it overwhelms task performance. That conclusion is the operational content of the self-consistent symmetric optimization perspective.