Not Every Time and Frequency Need to Be Forgotten in Diffusion Unlearning

AI summary

Overview

Research area: Machine unlearning for diffusion models, specifically data unlearning (removing individual training images rather than whole concepts) in both unconditional and text-conditional image generators.

Technical level: Advanced. The paper combines empirical analysis with stochastic differential equation theory, Girsanov's theorem, and path-measure KL divergence bounds, alongside a large benchmark table of FID, SSCD, SSCD^norm, and aesthetic scores.

Scope in one sentence: The paper shows that forgetting in diffusion models is uneven across diffusion time steps and frequency bands, and that selectively unlearning only the "right" time steps and frequencies yields both stronger forgetting and better image quality than uniform unlearning.

What This Paper Is About

Fine-tuning-based diffusion unlearning usually relies on loss maximization (e.g., gradient ascent) over the forget samples applied uniformly across all diffusion time steps. This often produces noisy, aesthetically degraded images, incomplete forgetting, or over-forgetting of samples that should be retained. The paper's goal is to characterize where forgetting actually happens — in the time domain (which noise levels) and the frequency domain (which spectral bands) — and then to restrict unlearning to those regions so that quality is preserved while forgetting succeeds.

Key Contributions

1. A systematic study of data unlearning across time and spectral space, covering both conditional (Stable Diffusion v1.4 on LAION) and unconditional (CelebA-HQ) scenarios, and connecting unlearning behavior to known diffusion transition points — speciation time t_S and collapse time t_C.

2. A simple, composable selective unlearning framework that reweights the time steps used in the unlearning loss (time selection) and applies a low-pass filter to the loss residual via FFT/IFFT (frequency selection). The approach is compatible with existing unlearning objectives such as GA, SISS, DPO, and KTO.

3. A theoretical justification (Proposition 3.1) using path-measure KL divergence to show that selective unlearning achieves a strictly positive improvement in a forgetting–preservation utility over full unlearning, under stated assumptions on the amplification μ ≥ 1 and suppression η ∈ [0,1] of score perturbations.

4. An improved evaluation metric, SSCD^norm, which normalizes the perturbation distance between the reconstructed and original image (projected onto an ℓ2 ball with radius ρ = 100 at 256×256 resolution) so that blurry or destroyed images do not appear to be successful unlearning.

Main Findings

• Forgetting is uneven across time. Unlearning only early time steps (Phase III, near t = 0) does not remove the forget samples and degrades quality; the middle steps (Phase II, t_C < t < t_S) are most aligned with instance-level attraction before the collapse time.

• Unconditional vs. conditional need different windows. The paper reports that t_C < t < t_S (Phase II) is critical for unconditional behavior, while t > t_S (Phase I) matters for conditional generation because conditional guidance dominates after speciation. Accordingly, the experiments use (t_1, t_2) = (250, 750) for the unconditional case and (t_1, t_2) = (750, 1000) for the conditional case, fixing (t_C, t_S) = (250, 750).

• Transition points are broadly consistent across studies. Table 1 collects reported values: Biroli et al. give t_S in [500, 800] and t_C in [100, 250]; Choi et al. give 675 and 259; Yang et al. give 750 and 250; Kim et al. give 780; Jain et al. give [600, 800].

• Adaptive transition detection matches the fixed choice. Computing the guidance norm and taking the first local minimum (Algorithm 1) yields a median τ of 761.1 across prompts, close to the fixed t_S = 750, with the adaptive method showing little performance difference from the fixed window.

• High frequencies in the forget set are the quality problem. Power spectral density analysis shows that retain samples keep their high-frequency distribution after unlearning, whereas over-forgotten samples differ in high frequencies; the method therefore filters them out (r_t tuned in [0.1, 0.15], r_t = 0.15 unless stated).

• Consistent gains across four unlearning objectives. In unconditional CelebA-HQ experiments, SISS improves by +18.84% on SSCD^norm and +34.83% on aesthetic at t = 250, and +20.36% / +25.80% at t = 500; DPO improves +17.82% / +9.48% and +14.38% / +21.35%; KTO improves +7.10% / +2.15% and +14.71% / +6.72%; GA improves +8.94% / +20.14% and +23.37% / +31.90%. FID shows small degradations for retained data (GA −4.28%, SISS −0.98%, DPO −2.77%, KTO −0.43%).

• SSCD alone is misleading. Methods such as EraseDiff and GA obtain low SSCD (e.g., 0.133 and 0.131) but this reflects severe image quality degradation; their relatively high SSCD^norm scores reveal that the perturbation direction is ineffective, whereas the selective framework achieves substantial SSCD^norm gains.

• Conditional unlearning converges faster. On the 45 memorized prompts from LAION, the method achieves a higher unlearning success rate and faster convergence than baseline SISS, with lower attack success rate and higher CLIP-IQA.

• Over-forgetting is reduced under membership inference attacks. Reported relative errors on the unlearned set reach ×107.98 (SecMI) and ×278.22 (PIAN) for existing methods, against hold-out baselines of ×1.32 and ×1.12; the proposed method reduces these to ×5.76 and ×53.20.

• Extra cost is small. The FFT adds 5.99% per epoch; one epoch takes 8.51 s (SISS) versus 9.02 s (Ours), and total training is 297.68 s versus 315.61 s, with the adaptive variant at 370.66 s and a pre-training sampling step of 55.05 s. Inference time is unchanged.

Methodology in Plain English

The authors start from the observation that diffusion models behave differently at different noise levels: late steps near pure noise set coarse semantics, middle steps shape content, and early steps refine detail. They reuse two well-known milestones in this timeline, speciation time t_S and collapse time t_C, and ask which interval actually carries memorization of individual samples.

They run a toy experiment with a shallow diffusion model on a half-moon dataset, changing only the time steps used for unlearning. They then check the real setting: they measure how the gradient norm of the forget set changes with standard uniform unlearning on CelebA-HQ, and how DINOv3 embeddings of noised images resemble the original versus its nearest neighbor. For conditional models they track the guidance norm between a memorized prompt embedding and the null embedding, and find a peak in the late phase before speciation.

Based on this, they modify the unlearning loss in two ways. First, they replace the uniform distribution over time steps with a distribution giving higher probability to a chosen interval [t_1, t_2] and lower probability elsewhere. Second, they take the residual between the model's score and the target score, apply an FFT, zero out coefficients outside a radius cutoff r_t, and invert the transform — so only low-frequency components drive the gradient. Importantly, the denoiser's input stays the noisy natural sample, so training and inference input distributions remain identical; only the gradient signal is filtered.

Finally, they analyze the effect formally: treating the original and unlearned models as path measures and using Girsanov's theorem to write the KL divergence between them, they split it into a selected region and its complement, define a forgetting–preservation utility with trade-off parameter β, and show selective unlearning has a strictly positive utility gap over full unlearning.

Experiments follow the evaluation protocol of Alberti et al.: CelebA-HQ for unconditional image-level deletion (six randomly sampled celebrity faces, deleted one at a time in a continual setting, denoising from t = 250 and t = 500), and Stable Diffusion v1.4 for text-to-image memorization on LAION (45 memorized prompts, sampling from t = T = 1000 with conditions). Models are fine-tuned with Adam, batch sizes 64 and 16, and learning rates 5·10⁻⁶ and 10⁻⁵ respectively; results are averaged over five runs, with variance reported as small in Table 13.

Why This Matters

Impact on research. The paper reframes data unlearning in diffusion models from "maximize loss on the forget set everywhere" to "intervene where memorization actually lives," and supplies both empirical evidence and a KL-divergence-based argument. It also flags a measurement problem: raw SSCD can reward destroyed images, so a normalized variant is needed to judge whether an unlearning direction is genuinely correct.

Real-world applications:

• Honoring "right to be forgotten" requests under GDPR for individual faces or personal photos used to train image generators.
• Removing proprietary or licensed images from generative models without retraining from scratch.
• Reducing memorized replication of training images in text-to-image systems, where fully-memorized prompts are reported to regenerate near-exact replicas.
• Auditing whether an unlearning claim is real, by using SSCD^norm and membership inference attacks rather than similarity alone.

Industry relevance. The method is a drop-in modification to existing fine-tuning pipelines, requires no change to inference, and adds only 5.99% per epoch while converging faster in the conditional setting — factors that matter for deploying unlearning as a compliance service rather than a full retraining event.

Future Directions

1. Automatic discovery of optimal unlearning regions. The paper explicitly lists the absence of a method for automatically finding the best time and frequency windows as a limitation; the adaptive τ method is left for future exploration because of its pre-sampling overhead.

2. Sample-wise rather than global transition points. The guidelines in Table 1 are not sample-wise, and the median adaptive τ of 761.1 varies by prompt, suggesting per-sample scheduling is unexplored.

3. Frequency threshold scheduling. The authors note that frequency cutoff scheduling can also improve quality (Table 12) but do not fully develop a principled schedule.

4. Broader validation. Supplementary experiments cover transformer-based architectures, Stable Diffusion v3, concept unlearning, LLM-based anomaly detection, effects on nearby samples, and reliability across independent runs — indicating the boundaries of the method across backbones and tasks are still being mapped.

Target Audience

Researchers and practitioners working on machine unlearning, diffusion model fine-tuning, and generative model privacy. The paper is best suited to readers comfortable with diffusion SDEs and score matching, though the empirical findings (which time steps and frequencies matter) are actionable for engineers who only need the practical recipe. Those interested in evaluation methodology will also find the SSCD^norm critique directly useful.

Authors’ abstract

Data unlearning aims to remove the influence of specific training samples from a trained model. In fine-tuning methods, data unlearning relies primarily on loss maximization over forget samples, which often leads to quality degradation or incomplete forgetting. Existing methods perform unlearning uniformly across diffusion stages, ignoring diffusion dynamics from noise to data. Our systematic study of diffusion phases shows that forgetting in diffusion models is uneven across time and frequency, with theoretical justification of distributive distortion and forgetting-utility trade-off. By selectively forgetting time and frequency in diffusion models, we achieve both higher unlearning success rates and improved generation quality across diverse settings, including both conditional and unconditional scenarios. We also introduce an improved SSCD metric that measures dissimilarity using a normalized perturbation distance. Together, we provide practical insights for understanding and improving data unlearning in diffusion models.