TAES: Emotional Stationarity for Time Allocation

• TAES is an information-theoretic framework that replaces scalar utility maximization with aligning experienced emotion distributions with a target character.
• It models synthetic emotions as probability distributions linked to tasks, ensuring that the long-run emotional experience of an agent is statistically stationary.
• The approach utilizes convex optimization algorithms, like mirror descent, to balance effective task allocation with exploration mechanisms.

Time Allocation via Emotional Stationarity (TAES) is an information-theoretic framework for task selection in artificial agents, rooted in the paradigm that emotions function as abstract evaluation criteria. TAES prescribes how an agent should allocate its time among a set of activities so that the long-run statistics of its experienced synthetic emotions closely match a designer-specified "character"—a target distribution over emotion-like evaluation signals. The core mechanism is a convex optimization over task-selection probabilities, minimizing the divergence between target and experienced emotion distributions, thereby achieving statistical stationarity in emotional experience (Gros, 2019, Gros, 2021).

1. Conceptual Foundations

TAES reframes the classical time allocation or task selection problem by replacing scalar utility maximization with the goal of aligning an agent's experienced distribution of emotions with a predetermined "character". The underlying assumption is that, in complex environments lacking well-defined or comparable utilities across distinct activities, task performance can instead be evaluated along multiple abstract criteria: satisfaction, challenge, boredom, etc.

Emotions in TAES are defined as categorical outcomes of task execution, modeled as probability distributions (synthetic emotion kernels) conditioned on the task performed. The agent's "character" is specified as a probability distribution over these emotion categories, representing the long-term frequencies with which it "desires" to experience each emotion. Task selection is cast as finding an allocation of time across available activities such that the agent's stationary distribution of experienced emotions matches its character (Gros, 2019, Gros, 2021).

2. Mathematical Formalism

Let the agent have a finite set of tasks $\mathcal{T} = \{1,\ldots,N\}$ and emotion states $\mathcal{E} = \{1,\ldots,M\}$. For each task $i\in\mathcal{T}$, $p_i(e)$ is the probability that emotion $e$ is experienced upon performing task $i$. The agent's time-allocation vector is $f = (f_1,\ldots,f_N)$, where $f_i\ge 0$ and $\sum_{i=1}^N f_i = 1$, and $f_i$ is the long-run fraction of time spent on task $i$.

The experienced emotion distribution is

$$\mu_e(f) = \sum_{i=1}^N f_i\, p_i(e), \;\; \text{for all } e\in\mathcal{E}.$$

The character $\pi = (\pi_1, ..., \pi_M)$ is a fixed target distribution over $\mathcal{E}$. The optimization objective is to minimize the mismatch between $\mu(f)$ and $\pi$. The Kullback–Leibler divergence $D_{\rm KL}(\mu(f)\,\|\,\pi)$ quantifies this mismatch:

$$D_{\rm KL}(\mu(f)\,\|\,\pi) = \sum_{e=1}^M \mu_e(f)\, \ln\frac{\mu_e(f)}{\pi_e}.$$

Optionally, a regularization cost $C(f)$ (e.g., entropy penalty or resource cost) with weight $\lambda\geq 0$ can be introduced, yielding the overall loss

$L(f) = D_{\rm KL}(\mu(f)\,\|\,\pi) + \lambda C(f),$

subject to $\sum_i f_i = 1$, $f_i\geq 0$. The solution $f^*$ defines the stationary time allocation that achieves emotional alignment with character (Gros, 2019, Gros, 2021).

3. Solution Algorithms

Owing to the convexity of $D_{\rm KL}(\mu(f)\,\|\,\pi)$ in $f$, TAES admits a unique global minimizer and is amenable to gradient-based optimization under simplex constraints. The (projected) gradient descent or mirror-descent algorithms are well-suited for this objective.

Mirror-Descent Algorithm (Exponentiated Gradient):

1. Initialize $f$ uniformly: $f_i \leftarrow 1/N$.
2. Iterate:
   • Compute $\mu_e = \sum_i f_i p_i(e)$ for all $e$.
   • Compute gradient for each $i$: $$g_i = \sum_e p_i(e) \ln\frac{\mu_e}{\pi_e} + \lambda \frac{\partial C}{\partial f_i}$$.
   • Update: $f_i \leftarrow f_i\cdot \exp(-\alpha g_i)$ (for step size $\alpha$).
   • Renormalize $f_i \leftarrow f_i/\sum_j f_j$.
3. Continue until convergence criteria are met (e.g., $\|\mu-\pi\|$ or change in $f$ is sufficiently small) (Gros, 2021).

Closed-form solutions are possible for small $N, M$ or special structure (e.g., $N=M$ with full-rank emotion matrices), but iterative gradient approaches scale better to large sets (Gros, 2019).

4. Emergent Dynamics and Theoretical Guarantees

At optimality, the experienced emotion statistics $\mu(f^*)$ match the character distribution $\pi$, making the agent's emotional experience stationary. Due to strict convexity and the compactness of the probability simplex, convergence to the unique optimum is guaranteed for standard descent algorithms. The long-run empirical distribution of emotions, given ergodicity and stationarity of the $p_i(e)$, will converge to the target character (Gros, 2019, Gros, 2021).

A representative convergence trajectory for a three-task (chess, Go, chat), three-emotion (satisfaction, challenge, boredom) agent would show empirical averages $\hat{P}_i(t)$ approaching and stabilizing at their target levels over time.

Time $t$	$\hat{P}_S(t)$	$\hat{P}_C(t)$	$\hat{P}_B(t)$
1000	0.35	0.45	0.20
2000	0.38	0.42	0.20
5000	0.40	0.40	0.20
10000	0.40	0.40	0.20

This demonstrates stable emotional stationarity and alignment with the desired character (Gros, 2019).

5. Exploration, Extensions, and Limitations

Pure KL minimization may insufficiently explore under-sampled activities, as it optimizes based only on current emotion kernel estimates. Remedies include:

• Mixing optimal $f^*$ with a small uniform exploration ($$f_{\rm mix} = (1-\epsilon)f_{\rm opt} + \epsilon(\frac{1}{N},\ldots,\frac{1}{N})$$).
• Adding exploration bonuses (e.g., confidence bounds) to the objective, in analogy to UCB strategies in bandits.

TAES in its original formulations does not address adaptive dynamics for non-stationary or drifting emotion kernels $p_i(e)$; incorporating forgetting factors or adaptive estimation is a straightforward extension. Similarly, direct integration of external utility rewards, or balancing KL-divergence control with cumulative reward via Pareto optimization, are recognized but not developed within the base framework. The approach is inherently multi-objective and modular, supporting further abstract evaluation criteria beyond simple emotions by extending $\pi$ and $p_i(e)$ (Gros, 2019, Gros, 2021).

6. Illustrative Example

Consider three activities and three synthetic emotions, with evaluation distributions:

• $p_{\rm chess} = (0.3, 0.4, 0.3)$
• $p_{\rm Go} = (0.1, 0.7, 0.2)$
• $p_{\rm chat} = (0.8, 0.1, 0.1)$

For a target character $\pi = (0.2, 0.5, 0.3)$, TAES finds $f^* \approx (0.25, 0.60, 0.15)$. This allocation yields an experienced emotion distribution $\mu(f^*)$ approximating the target $\pi$. High $\pi_{\rm challenge}$ biases time allocation toward challenging activities (Go), whereas low $\pi_{\rm boredom}$ disfavors excessively easy or repetitive tasks (chat) (Gros, 2021).

7. Biological and Artificial Intelligence Context

TAES generalizes an observation from biological intelligence: increasing cognitive capability is associated with a richer space of tasks and emotions, and behavioral choice is regulated by an implicit categorical weighting—introspectively experienced as emotion. In artificial agents, TAES offers a principled, functional mechanism to implement such weighting, decoupled from extrinsic reward specification. This suggests a connection between emotional stationarity principles and generalized value attribution in both natural and synthetic intelligences (Gros, 2021).