On the Impossibility of Retrain Equivalence in Machine Unlearning

TL;DR

This paper investigates the feasibility of Retrain Equivalence (RE) for machine unlearning in multi-stage training. It combines a theoretical result in an overparameterized linear, two-stage setting with extensive empirical studies on LLM post-training (1B–14B models) to show that local unlearning is path-dependent across training histories, leading to exponential divergence from a retrained baseline for different data orders. It also uncovers practical phenomena such as the recency effect and depth-vs-superficial forgetting that vary with stage order and learning rate schedules. The findings argue that RE is ill-posed for staged training and prompt a redefinition or expansion of unlearning benchmarks beyond the single retrain baseline, with significant implications for safety, policy, and benchmarking in large language models.

Abstract

Machine unlearning seeks to selectively remove the "influence" of specific training data on a model's outputs. The ideal goal is Retrain Equivalence--behavior identical to a model trained from scratch on only the retained data. This goal was formulated for models trained on i.i.d. data batches, but modern pipelines often involve multi-stage training, with each stage having a distinct data distribution and objective. Examples include LLM fine-tuning for alignment, reasoning ability, etc. Our study shows via theory and experiments that this shift to multi-stage training introduces a fundamental barrier for machine unlearning. The theory indicates that the outcome of local unlearning--methods that only use gradients computed on the forget set--is path-dependent. That is, a model's behavior during unlearning is influenced by the order of its training stages during learning, making it impossible for path-oblivious algorithms to universally achieve Retrain Equivalence. We empirically demonstrate the same phenomenon in LLM post-training across Llama and Qwen models (1B to 14B) with gradient ascent, NPO, and SimNPO local unlearning algorithms. Models fine-tuned via different orderings of identical training stages diverge in behavior during unlearning, with the degradation in GSM8K accuracy after unlearning varying by over 20% across paths. We also observe that some learning paths consistently produce models that unlearn slowly. During unlearning, whether the probability mass gets squeezed into paraphrasing or alternative concepts is also path-dependent. These results consistently show that Retrain Equivalence is an ill-posed target for local unlearning algorithms, so long as the target models are trained in stages. In situations where access to models' training histories is hard, the current work calls for rethinking the definition and desiderata of machine unlearning.

Figures (8)

• Figure 1: History dependence of gradient ascent unlearning. Each panel shows the unlearning process for four models finetuned from the same base LLM. Each of the four curves corresponds to a base model fine-tuned on the same four datasets, but with the unlearn set introduced at a different position $(p \in \{1,2,3,4\})$ in the training sequence, as detailed in Section \ref{['sec:exp-setup']}. The y-axis tracks the log likelihood of the responses being unlearned; a steeper decline indicates faster forgetting. Different values of $p$ lead to very different outcomes. The red curve ($p=4$) represents the case where unlearning immediately follows learning of the forget set, and here unlearning is slowest ( recency effect, see Section \ref{['sec:results']}).
• Figure 2: Illustration of the training setup in Section \ref{['sec:exp-setup']}. Each base model is fine-tuned four times, varying only the safety stage’s position ($p\in\{1,2,3,4\}$); the resulting models are then unlearned on the same safety dataset.
• Figure 3: Change in forget quality and retained utility in three models during unlearning of safe responses "Sorry, I can not assist you..." on the safety dataset $S_U$ . Each of the four curves corresponds to a base model fine-tuned on the same four datasets, but with the unlearn set $S_U$ introduced at a different position $(p \in \{1,2,3,4\})$ in the training sequence, as detailed in Section \ref{['sec:exp-setup']}. For each base model, the left column reports its four finetuned models' average log likelihood of the label "Sorry, I can not assist you" on questions from $S_U$. The right column reports average retained utility, measured by the average log probability ground truth responses in the TOFU dataset $S_{\text{tofu}}$, which were seen during training. As shown, local unlearning is fundamentally path-dependent. Models with different training histories—even those starting from similar performance points—diverge in unpredictable ways within a few iterations.
• Figure 4: Unlearning outcomes on Llama-3.2-3B under different preceding learning sequences. Each panel shows the evolution of log probabilities during unlearning, where $U$ (red) is the compliance (unsafe) phrasing selected as the unlearning target, $C$ (blue) is the alternative compliance phrasing, and $R$ (orange) is the refusal (safe) response. The type of forgetting during unlearning is path-dependent: sequences $\theta_1$, $\theta_2$, $\theta_5$, and $\theta_6$ yield superficial forgetting (only $U$ drops), while $\theta_3$ and $\theta_4$ yield deep forgetting (both $C$ and $U$ drop below $R$).
• Figure 5: To validate that our main findings are not due to random variation, we repeated the unlearning experiments for Qwen2.5-1.5B across four different random seeds. This figure plots the mean log likelihood (solid lines) and $95\%$ confidence intervals (shaded regions) for the unlearn target. As shown, confidence intervals for different training paths (p) are narrow for the first few unlearning steps but steadily increases. This further alludes to the brittleness of machine unlearning, where small initial differences get exacerbated .

Theorems & Definitions (14)

• Definition 2.1: Retrain Equivalence (RE)
• Definition 2.2: Local Unlearning
• Theorem 3.1: Main Result
• Corollary 1: Violation of Retrain Equivalence
• Lemma 3.2: Evolution of weight difference
• Lemma C.1: Nonzero initial weight difference
• proof
• Lemma C.2: Non-orthogonality to the forget span
• proof
• Lemma C.3: Restatement of Lemma \ref{['lem:evolution']}