Entropic Control for Homeostatic Optimization: Pre-empting Catastrophic Forgetting via Dynamic Feedback

Shangxi Wu; Xiaorong Yan; Xiaojun Meng; Yuxiang Zhang; Jiansheng Wei

cs.LGOct 25, 2026

Entropic Control for Homeostatic Optimization: Pre-empting Catastrophic Forgetting via Dynamic Feedback

Authors: Shangxi Wu, Xiaorong Yan, Xiaojun Meng, Yuxiang Zhang, Jiansheng Wei

Organizations: Beijing Jiaotong University · The Hong Kong Polytechnic University · Huawei Noah’s Ark Lab

Abstract

Supervised Fine-Tuning (SFT) of LLMs suffers from catastrophic forgetting, and prevailing mitigation strategies struggle to reconcile the plasticity-stability dilemma. To address this, we propose Entropic Control for Homeostatic Optimization (ECHO), a closed-loop control framework. By decomposing the KL divergence, we identify output entropy as a dominant observable component of KL variation. This enables us to leverage entropy as a computationally efficient online proxy to monitor forgetting risks in real-time, bypassing expensive reference-model inference. Consequently, ECHO employs an elastic controller that dynamically modulates the gradient weight of replay data via a proportional control law, using deviations from the pre-trained entropic equilibrium as an error signal. Empirical evaluations across 0.6B to 9B parameter architectures demonstrate that ECHO lies on, and in several comparisons extends, the observed plasticity-stability frontier and maintains high operational robustness against varying data mixing ratios.

Figures & tables

Figure 1: Correlation analysis: early entropy deviation vs. final performance degradation. Results demonstrate that deviation from the entropy baseline

H_{\text{ref}}

is strongly correlated with the loss of general capabilities.

Figure 2: Overview of the Entropic Control for Homeostatic Optimization.

Metric	Mean $\|\Delta\|$ ( $\pm$ Std)	Ratio vs. $H_{\times}$
Cross-Entropy ( $H_{\times}$ )	$0.0363\pm 0.027$	$1.00\times$
Output Entropy ( $H$ )	$\mathbf{0.0559}\pm 0.024$	$\mathbf{1.54\times}$
Statistical Significance (Paired t-test): $t=13.51$ , $\bm{p<10^{-13}}$ ***

Table 1: Distribution Shift Metrics. We report entropy and cross-entropy deviations during training, showing that entropy varies more than cross-entropy.

Method	Code		Math		Reasoning		General Stability
Method	HumanEval $\uparrow$	$\Delta$	GSM8K $\uparrow$	$\Delta$	HellaSwag $\uparrow$	$\Delta$	MMLU $\uparrow$	$\Delta$
Backbone: Qwen3-0.6B
Base Model	18.90	—	10.39	—	47.35	—	49.17	—
Vanilla SFT	31.10	+12.20	55.49	+45.10	48.55	+1.20	42.91	$-$ 6.26
Data Mixing Laws	37.80	+18.90	55.27	+44.88	50.81	+3.46	43.38	$-$ 5.79
LoRA	23.17	+4.27	17.51	+7.12	48.79	+1.44	44.81	$-$ 4.36

Table 2: Main Results on Qwen3-0.6B and Llama-3.2-3B. We report raw scores and the improvement (

\Delta

) over the Base Model. Code, Math, and Reasoning are independent single-run experiments from the same checkpoint, each mixed with Alpaca-GPT4 anchor data; MMLU is averaged over the three. Bold: best; underline : second best.

Figure 3: Sensitivity Analysis of Data Proportioning. Performance across Code, Math, and Reasoning domains as mixing ratio

\gamma

varies. ECHO matches or outperforms Static Mixing at every ratio in all three domains.

Backbone	Method	Code	Math	Reason	MMLU
Qwen3-8B	Static Mix	64.02	80.89	78.74	61.61
Qwen3-8B	ECHO (Ours)	68.90	81.80	78.88	74.55
Gemma2-9B	Static Mix	51.83	77.79	79.83	69.47
Gemma2-9B	ECHO (Ours)	53.66	78.54	80.63	70.00

Table 3: Performance across Large-Scale Models. We report task-specific accuracy and MMLU scores. Results show that ECHO effectively scales to 8B and 9B parameters across different architectures. All results are from a single run.

Figure 4: Visualization of training dynamics. (a) Entropy trajectories: Ours maintains lower anchor-data uncertainty than Vanilla SFT. (b) Negative entropy difference indicates suppression of forgetting. (c) Adaptive weight $\lambda_{t}$ starts high to protect early stability, then adjusts to balance task learning and retention.

Figure 5: Parameter sensitivity analysis on Qwen3-1.7B. Left: Base Weight

\lambda

modulates the static stability-plasticity trade-off. Right: Entropy Scale

K_{p}

acts as a dynamic regularizer; increasing

K_{p}

restores general capabilities but may constrain new learning if the penalty becomes excessive.

Catastrophic Forgetting

Catastrophic forgetting is the loss of previously learned behavior when a model adapts to new tasks or changing data.

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Continual pre-training and fine-tuning in Large Language Models (LLMs) inevitably induce catastrophic forgetting, typically mitigated by replay using often-inaccessible original data. In this data-free regime, we analyze where forgetting occurs and why. Systematic parameter freezing across five settings up to 1.4B reveals that forgetting concentrates selectively in the output embeddings of tokens rarely seen in the new corpus, whereas the same sqrt(v-hat) band of the body is inert and new learning resides elsewhere. This localization is governed by the vocabulary deficiency of the corpus rather than the training mode, allowing pre-retraining risk ranking from token counts alone within a fixed base model. Mechanistically, absent tokens receive persistent one-sided softmax gradients that Adam's second-moment (sqrt(v-hat)) normalization amplifies into full-sized updates. We therefore propose an intervention: raising Adam's epsilon exclusively for the output projection during training. Across eight settings spanning 160M to 12B parameters and four model families, this removes 39.4% to 67.9% of forgetting across all seven stable configurations without degrading target learning or requiring per-model tuning. The defense combines additively or better with replay (79.8% on Qwen/Korean) and rescues released-head LoRA from a 23-fold forgetting surge. Because post-hoc editing of the drifted rows recovers under 5% of forgetting, the intervention must operate during training. Our findings indicate that a single-line optimizer adjustment may serve as the primary defense against catastrophic forgetting where the corpus starves the vocabulary.

Abstract

Catastrophic Forgetting

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