Adaptive Forget Set Optimization

• Adaptive Forget Set Optimization is a family of techniques that systematically remove the influence of designated data points using anchored, bi-level, and ensemble-based methods.
• It employs strategies like FAMR, OFMU, SAFE, and ASP to balance computational efficiency with high retention accuracy on non-forgotten data.
• Experimental benchmarks show these methods reduce retraining costs while maintaining model utility through targeted loss functions and adaptive optimization frameworks.

Adaptive Forget Set Optimization refers to a family of algorithmic and theoretical strategies designed to remove the influence of a designated subset of data points (the "forget set") from machine learning models, while balancing computational efficiency and retention of performance on non-forgotten data. Unlike standard data deletion or simplistic retraining, adaptive forget set optimization deploys dedicated optimization frameworks to systematically and efficiently realize post-hoc unlearning, selective forgetting, or dynamic plasticity. The concept spans both deep neural architectures and biologically inspired systems, and is at the core of modern approaches for privacy, compliance, and lifelong learning.

1. Mathematical Formulation of Adaptive Forget Set Optimization

Classical unlearning approaches typically frame forgetting as a constrained optimization, with objectives combining targeted erasure and model utility preservation. Let $D = D_n \cup D_f$, where $D_n$ is the retention set and $D_f$ is the forget set. Losses over the two partitions are denoted as:

• $$L_n(\theta) = \frac{1}{|D_n|} \sum_{(x, y) \in D_n} \ell(f_\theta(x), y)$$
• $$L_f(\theta) = \frac{1}{|D_f|} \sum_{(x, y) \in D_f} \ell(f_\theta(x), y)$$

Two paradigmatic optimization structures have emerged:

Anchored Forgetting:

Formulate post-hoc forgetting as:

$$\min_\theta L_{\text{unif}}(\theta; D_f) + \lambda \|\theta - \theta_0\|_2^2$$

where $L_{\text{unif}}$ encourages uniform predictions on the forget set (maximum entropy), and $\lambda$ anchors the solution to the pre-unlearning model for retention (Sanga et al., 17 Jun 2025).

Bi-Level Forgetting:

A penalty-based, bi-level structure:

$$\min_\theta L_n(\theta) \quad \text{s.t.} \quad \theta \in \arg\max_{\theta'} \left[L_f(\theta') - \beta \cdot \text{Sim}(\nabla L_f(\theta'), \nabla L_n(\theta'))\right]$$

or, equivalently, the penalized surrogate:

$$F(\theta) = L_n(\theta) + \rho \|\nabla_\theta \Phi(\theta)\|^2$$

with $$\Phi(\theta) = L_f(\theta) - \beta\,\text{Sim}(g_f, g_n)$$ and Sim denoting the cosine similarity between forget and retain gradients (Asif et al., 26 Sep 2025).

This establishes a rigorous framework in which adaptive forget set optimization methods are instantiated.

2. Algorithmic Strategies

Forget-Aligned Model Reconstruction (FAMR)

FAMR solves the anchored objective using standard stochastic gradient descent, with updates:

$$\theta_{t+1} = \theta_t - \eta [\nabla_\theta L_{\text{unif}}(\theta_t; D_f) + 2\lambda(\theta_t - \theta_0)]$$

Each step comprises a KL-divergence loss promoting output uniformity on $D_f$ and an $L_2$ penalty on weights (Sanga et al., 17 Jun 2025).

Extensions include:

• Weighted forget sets via per-sample weights $w_i$;
• Concept/style erasure through feature-based Gram matrix losses;
• Sequential (on-the-fly) updates anchored to the original $\theta_0$.

Optimization-Driven Framework for Machine Unlearning (OFMU)

OFMU introduces a two-loop penalty-based bi-level optimizer:

• Inner Loop: Maximizes forgetting and decorrelates gradients via the similarity penalty.
• Outer Loop: Minimizes utility loss and enforces stationarity of the inner objective via a penalty term, with gradient and Hessian-vector product computations (Asif et al., 26 Sep 2025).

Convergence is proven both for convex and non-convex losses, and the penalty and inner-step schedule trade off forgetting quality and utility.

SAFE: Synergy-Aware Forgetting Ensemble

SAFE partitions data into shards, constructs a shard graph $G=(V,E)$ to modulate mutual influence, and trains independent adapters per shard that share information along edges. Expected forgetting cost

$$C_{\text{forget}}(G) = \mathbb{E}_{x \sim D}[|M_x|]$$

is minimized via topological control, with cliques/synergies providing Pareto-efficient trade-offs between retraining cost and final accuracy (Dukler et al., 2023).

Adaptive Synaptic Plasticity (ASP) in SNNs

ASP dynamically combines STDP updates with adaptive weight decay; decay rates are modulated by synaptic and neuronal activity traces. The forget set is implicit: activity patterns deemed irrelevant (low spike/event rates) drive weights to baseline, whereas active pathways are preserved via slow decay (Panda et al., 2017).

3. Theoretical Guarantees and Influence Connections

FAMR admits explicit connections to influence function theory: its fixed point approximates that of retraining on $D \setminus D_f$, up to a term $O(\lambda/\lambda_{\min}(H))$ where $H$ is the Hessian on $D\setminus D_f$. The Lipschitzness of $f_\theta$ certifies bounded prediction divergence post-unlearning (Sanga et al., 17 Jun 2025).

OFMU establishes:

• Stationarity of the forgetting constraint as $\rho \to \infty$.
• $\varepsilon$-optimality in the convex case with $O(1/K)$ outer iterations and $O(1/T^2)$ inner steps.
• Non-convex convergence in expectation with rates dependent on $K, T$ and gradient noise (Asif et al., 26 Sep 2025).

SAFE quantifies expected retraining burden, showing that clique graphs reduce amortized cost from $O(d^2|S|)$ (random) to $O(d|S|)$ (clique), and formalizes $(\alpha,\beta)$-sharded unlearning under DP composition (Dukler et al., 2023).

ASP's performance guarantees are empirical: resistance to catastrophic forgetting and improved accuracy under continuous online learning in SNNs, traceable directly to its adaptive decay mechanism (Panda et al., 2017).

4. Computational Complexity and Practical Performance

Method	Forgetting Cost (relative)	Retention Accuracy	Batch Unlearning/Adaptivity
FAMR	$\sim$10--20% retrain	$\geq$98% retained	Weighted/streaming/feature-based
OFMU	Comparable to FAMR	Optimal/near-opt.	Bi-level, scalable
SAFE	$O(\frac{1}{n})$ shards	$+14.3\%$ over SISA	Hundreds of shards, graph tuning
ASP	O(online–neuro equiv.)	$94.8\%$ (dynamic)	Real-time, param. adaptable

FAMR achieves near-complete erasure at modest computational overhead: e.g., class removal on CIFAR-100 at $20\%$ of the full retraining cost (Sanga et al., 17 Jun 2025). OFMU further enhances the retention/erasure trade-off, with empirical superiority across vision (CIFAR-10/100) and LLM tasks (TOFU/LLaMA) (Asif et al., 26 Sep 2025). SAFE demonstrates scalability to $n=256$ shards and order-of-magnitude better Pareto performance than SISA/prototypes (Dukler et al., 2023). ASP achieves $94.8\%$ accuracy under sequential digit presentation (MNIST), virtually eradicating catastrophic forgetting compared to vanilla STDP ($23.3\%$) (Panda et al., 2017).

5. Adaptive Control, Extensions, and Limitations

Adaptive forget set optimization encompasses:

• Weighted forget sets: Per-sample importance, bilevel meta-optimization of erase priorities (Sanga et al., 17 Jun 2025).
• Concept/style erasure: Targets activation patterns or Gram matrix features, extending beyond raw instance forgetting (Sanga et al., 17 Jun 2025).
• Dynamic or sequential unlearning: Algorithms supporting real-time integration of new forget requests, maintaining an anchor or adjustment schedule (Panda et al., 2017, Sanga et al., 17 Jun 2025).
• Synergy-aware partitioning: Shard graphs and clique insights (SAFE) allow practitioners to optimize retraining cost vs. accuracy on evolving datasets (Dukler et al., 2023).

Limitations across these methods include approximate forgetting (FAMR, OFMU—dependent on optimization tolerance), local minima due to non-convexity, and possible model drift under adversarial or many-round unlearning, requiring periodic anchoring or full retraining. OFMU and SAFE provide formal DP-based and (α,β)-style guarantees, but these scale with the number/extent of sequential forgetting (Asif et al., 26 Sep 2025, Dukler et al., 2023).

6. Experimental Results and Comparative Benchmarks

Benchmarks reveal:

• FAMR: Full retraining vs. anchored updates yields comparable erasure with $\sim$5x speedup (ImageNet-100: $5$ hours $\to$ $45$ minutes for class forgetting) (Sanga et al., 17 Jun 2025).
• OFMU: On LLaMA-2-7B (TOFU), achieves a Forget Quality (FQ) of $0.13$ and Model Utility (MU) of $0.65$ at $5\%$ forget, outperforming all baselines (Asif et al., 26 Sep 2025).
• SAFE: For $256$-way sharding, $77.6\%$ accuracy averaged over seven vision datasets, a $14.3\%$ improvement over SISA, and scalable, certified unlearning with efficient InCA-based adapters (Dukler et al., 2023).
• ASP: On online MNIST, ASP attains $94.8\%$ sequential accuracy and resilience to catastrophic overlap, and achieves robust denoising under noisy data streams (Panda et al., 2017).

7. Relation to Broader Research and Application Domains

Adaptive forget set optimization directly addresses regulatory and ethical demands for data erasure in deployed machine learning systems, supporting use cases from privacy (user data removal) and copyright compliance to robust continual learning in non-stationary environments. FAMR and OFMU equip deep neural architectures with practical, certifiable unlearning, while SAFE and ASP extend these guarantees to large ensembles and event-driven neuromorphic hardware, respectively. Comparison across methods indicates that adaptive structures—be they anchored objectives, bi-level penalties, or graph-based ensembles—consistently outperform naïve retraining, both in performance and scalability, establishing adaptive forget set optimization as the core foundation for modern machine unlearning (Sanga et al., 17 Jun 2025, Asif et al., 26 Sep 2025, Dukler et al., 2023, Panda et al., 2017).