Enabling Autonomy via Asymptotic Induction (EAAI)

• EAAI is a curriculum-based training framework that gradually reduces external guidance to induce autonomous reasoning in language models and RL agents.
• It employs a cosine decay schedule to attenuate guidance signals, ensuring smooth transitions and stable performance during training.
• Empirical results show that EAAI improves test accuracy and robust generalization even when external cues are completely removed.

Enabling Autonomy via Asymptotic Induction (EAAI) refers to a family of curriculum-based training methodologies designed to induce robust, self-sufficient reasoning or decision-making policies by gradually removing external guidance while avoiding instability or performance loss. EAAI emerges in both the context of small LLM (SLM) chain-of-thought (CoT) reasoning in the ReaLM framework (Xu et al., 17 Aug 2025) and as a general principle for attaining universal RL autonomy through asymptotic induction over policy or value functions, as formalized by the AIQI agent (Kim et al., 26 Feb 2026). EAAI strategies are distinguished by the staged attenuation of supervised or privileged signals during training, a rigorously controlled reward or evaluation regime, and a claim to preserve—sometimes even improve—autonomy and generalization.

1. Formal Foundations of Asymptotic Induction

The central theoretical mechanism of EAAI is the use of time- or progress-dependent attenuation of external signals, such as reference CoTs or privileged rewards, during the training process. In ReaLM (Xu et al., 17 Aug 2025), the inclusion probability for external CoT demonstrations decays according to a cosine schedule:

$$\alpha(t) = 0.5[1 + \cos(\pi t)], \quad t = \frac{\text{cur}}{\text{total\_steps}}$$

where $\text{cur}$ is the current RL training step. This probabilistic schedule ensures that the model transitions from a regime rich in external trajectory guidance—where learning is stabilized and error signals are dense—to a nearly autonomous regime where the model is required to solve the target task conditioning only on original inputs.

In general reinforcement learning, asymptotic induction formalizes the autonomous attainment of optimal behavior via universal Bayesian inference not over modeled environments (as in AIXI) but over value predictors (as in AIQI (Kim et al., 26 Feb 2026)). Here, the agent learns a distributional Q-function by predicting its own potential returns under the induced policy. Asymptotic autonomy in this setting is achieved when, as $t \to \infty$, the dependence on externally specified models or rewards vanishes and the agent's policy is optimal with respect to the environment—subject to a grain-of-truth prior.

2. EAAI in Small LLM Training

EAAI serves as a key component of the ReaLM (SLM+) framework, which addresses three failings of earlier SLM RL protocols: overfitting to teacher guidance, impaired generalization, and lack of true reasoning autonomy (Xu et al., 17 Aug 2025). The EAAI curriculum is instantiated as follows:

• For each batch, up to $k$ external CoTs are selected from a candidate pool ($N$, typically $10$; $k$, typically $3-5$), with each selection made independently with probability $\alpha(t)$. Pseudocode: $k$8
• In early training, most input examples are augmented with external CoTs. As training progresses, the reliance on CoTs decays, and the reward regime enters a maintenance phase where only answer correctness is scored (process rewards are shut off once $\alpha(t)$ falls below a threshold).
• Guided distillation of domain knowledge via supervised fine-tuning may optionally precede RL, but the resulting SLM must ultimately perform without external reasoning trajectories during inference.
• Ablation studies (Xu et al., 17 Aug 2025) show that removing EAAI but keeping multi-route process verification (MRPV) yields high test performance only if CoTs are available. In contrast, EAAI enables strong performance even with full test-time autonomy, preserving the majority of the performance gains obtained with guided CoT input.

3. Asymptotic Induction in Universal RL: AIQI

The AIQI agent (Kim et al., 26 Feb 2026) introduces a distinct implementation of EAAI in the context of general RL. Unlike AIXI, which induces over environment models, AIQI maintains Bayesian predictors $\text{cur}$0 for $\text{cur}$1-step discretized return distributions over full agent histories. The principal features are:

• Return predictions are periodically augmented (every $\text{cur}$2 steps) to handle dependencies on non-Markovian, history-based returns.
• Q-induction policy greedily chooses $\text{cur}$3 based on expected return under $\text{cur}$4, with $\text{cur}$5-greedy exploration ($\text{cur}$6).
• As time progresses, and with grain-of-truth in the prior support, $\text{cur}$7 converges to the true return distribution under the agent's policy, making $\text{cur}$8 as $\text{cur}$9.
• This “asymptotic induction” allows AIQI to become fully autonomous, relying only on self-consistent value prediction rather than any explicit, externally provided environment model or transition simulator.

4. Theoretical Guarantees and Optimality

In both the SLM and general RL settings, EAAI is supported by strong theoretical arguments:

• In ReaLM, EAAI is justified as a form of curriculum learning, smoothly moving the policy from being conditional on external trajectories, $t \to \infty$0, to being conditional only on the question, $t \to \infty$1, thereby preventing instability due to vanishing auxiliary rewards (Xu et al., 17 Aug 2025).
• In AIQI, formal results include:
  • Strong asymptotic $t \to \infty$2-optimality: for any environment in the considered class, $t \to \infty$3 almost surely in the limit, provided predictor priors have a grain of truth.
  • Asymptotic $t \to \infty$4-Bayes optimality in any Bayes-mixture environment.
  • These results establish that autonomy—in the sense of not relying on external models or signals—emerges as a rigorously supported outcome of asymptotic induction (Kim et al., 26 Feb 2026).

5. Empirical Results and Comparative Performance

EAAI has been validated empirically in complex reasoning and industrial-scale tasks (Xu et al., 17 Aug 2025). Selected findings:

Protocol	GSM8K ID Accuracy (DS7B/QW7B)	GSM8KPlus OOD (DS7B/QW7B)
Baseline RL	84.2% / 84.2%	73.5% / 75.0%
MR (multi-CoT)	86.0% / 85.8%	74.0% / 76.2%
MRPV + EAAI	87.1% / 86.9%	75.8% / 77.4%

With external CoTs at inference, accuracy improves by $t \to \infty$54%. With full autonomy (no CoTs), EAAI retains $t \to \infty$62.5–2.6% of that gain, indicating strong knowledge transfer from guided to autonomous policy. Process-verification gating contributes an additional $t \to \infty$71% over answer-only reward. Removing EAAI while retaining MRPV results in policies that fail when guidance is withheld. Random seed variations indicate stable results with $t \to \infty$80.3–0.7% standard deviation across runs.

6. Practical Considerations and Implementation

Complete EAAI adoption requires careful management of auxiliary signals, curriculum schedule, and reward shaping:

• Key hyperparameters: $t \to \infty$9 CoT candidates from teacher LLM (e.g., GPT-4-turbo, $k$0), $k$1–$k$2 per batch, RL steps $k$3–$k$4, threshold decay per cosine schedule, disable process rewards below $k$5, optimizer settings (GRPO, KL coeff $k$6, LR $k$7).
• For ReaLM-R1, initial supervised fine-tuning (SFT) is performed before switching to RL.
• For model-free RL such as AIQI, autonomous value predictions are directly induced via Bayesian updating, with no privileged architectural assumptions about the environment (Kim et al., 26 Feb 2026).
• Ablation confirms EAAI’s positive effect is stable across multiple random seeds, benchmarks, and model architectures.

7. Comparative Analysis: Model-Free Autonomy vs. External Supervision

A central philosophical and technical distinction drawn by both ReaLM and AIQI is between policies or agents that rely on externally imposed guidance (teacher demonstrations, environment models) and those capable of self-sufficient induction.

• In model-based frameworks (e.g., AIXI), agents maintain an explicit, usually designer-specified, simulation or model of the environment, structurally embedding ontological priors.
• EAAI in ReaLM and AIQI enables agents to gradually or asymptotically let go of external ontologies (demonstrations, simulators), inducing knowledge, rules, or value estimates directly from interaction and self-consistent reasoning.
• A plausible implication is that agents trained by EAAI can generalize more robustly to OOD distributions, exhibit greater process-level robustness, and attain autonomy even in partially observable or non-Markovian domains, provided the underlying curriculum and signal-decay schedule are well posed.

In summary, Enabling Autonomy via Asymptotic Induction specifies a principled, theoretically grounded framework for transitioning agents—whether SLMs or general RL agents—from external supervision to robust, self-sufficient reasoning or decision-making policies. Empirical and theoretical evidence from ReaLM (Xu et al., 17 Aug 2025) and AIQI (Kim et al., 26 Feb 2026) demonstrates EAAI’s utility as an essential mechanism for enabling autonomy across learning paradigms.