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REVIEW 2 major objections 6 minor 80 references

A Unified Framework for Diffusion Model Unlearning with f-Divergence

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Concept unlearning in text-to-image diffusion models is f-divergence minimization, and the squared Hellinger instance beats the standard MSE loss on both erasure and preservation.

desk verdict A genuinely useful unification of unlearning losses with clean theory, but the headline empirical claim of 'consistently dominates' is contradicted by the paper's own tables and needs softening before it deserves full trust. read the letter →

arxiv 2509.21167 v2 pith:GGYHIHDJ submitted 2025-09-25 cs.LG cs.CV

Nicola Novello , Federico Fontana , Luigi Cinque , Deniz Gunduz , Andrea M. Tonello This is my paper · ORCID
classification cs.LGcs.CV MSC 68T0794A17
keywords diffusionmodelsmachineunlearningconcepterasuref-divergenceHellingerdistancevariationaldivergenceestimationtext-to-imagegenerationgradientanalysis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the standard way of erasing concepts from text-to-image diffusion models—minimizing the mean squared error between denoiser outputs for a target and an anchor prompt—is just one point in a larger family: it is the Kullback-Leibler instance of an f-divergence objective. By replacing KL with other f-divergences, especially the squared Hellinger distance, the paper claims to get a better trade-off between erasing the unwanted concept and preserving the model's overall quality. It derives closed-form unlearning losses for several f-divergences, gives a variational min-max formulation for the rest, and analyzes how the choice changes gradient magnitude and convergence speed. If the claim holds, practitioners can choose the divergence to match their goal: fast aggressive removal or clean realistic replacement.

What carries the argument

The central object is the f-divergence between the output distributions of the denoiser under the target and anchor prompts, with the loss built from the closed form of that divergence for Gaussian conditionals. Because two Gaussians with the same covariance differ only through the Mahalanobis distance between their means, KL becomes the familiar MSE, squared Hellinger becomes $-\omega_t \exp(-\|\Phi-\hat\Phi\|^2/2)$, and $\chi^2$ becomes $\omega_t \exp(\|\Phi-\hat\Phi\|^2/2)$; this single shared structure is what makes gradient behavior and convergence comparable across losses. For divergences without a closed form, the machinery is the variational identity $D_f(p\|q) = \sup_T \{\mathbb{E}_p[T] - \mathbb{E}_q[f^*(T)]\}$, turning unlearning into a min-max game between the unlearned denoiser and a critic. The gradient formulas then explain why Hellinger has bounded updates while $\chi^2$ can explode, and the Jacobian analysis of the min-max dynamics ties convergence speed to the curvature $f''(1)$.

What would settle it

Sample many denoiser outputs from the original and the unlearned model for the same noisy inputs and prompts at several timesteps and compare the empirical spread of the two output distributions; if the spreads differ materially, the equal-spread assumption behind the closed-form losses is false and the H2 and chi-squared losses are not minimizing the stated divergences.

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Extended reading notes

Core claim

The paper's central claim is that the MSE objective used by existing unlearning methods is the Kullback-Leibler instance of a general f-divergence-minimization problem, and that other instances are not merely alternatives but, in the case of squared Hellinger, better-behaved ones: it erases the target concept with comparable or stronger force while preserving unrelated concepts and producing cleaner images. The mechanism is the closed form of an f-divergence between two Gaussians with a common covariance, where every such divergence is an increasing function of the squared mean difference; KL becomes the standard MSE, squared Hellinger becomes $\mathbb{E}[-\omega_t e^{-\|\Phi-\hat\Phi\|^2/2}]$, and Pearson $\chi^2$ becomes $\mathbb{E}[\omega_t e^{\|\Phi-\hat\Phi\|^2/2}]$. The paper proves the gradient-norm ordering $\|\nabla H^2\| \le \|\nabla \mathrm{MSE}\| \le \|\nabla \chi^2\|$, with Hellinger gradients bounded, and proves local exponential stability of the variational formulation with convergence speed governed by $1/f''(1)$. Empirically, on a publicly available text-to-image diffusion model, the squared Hellinger loss consistently dominates MSE on the trade-off between erasure efficacy and generative fidelity.

Load-bearing premise

The closed-form losses treat each denoiser output as a bell-shaped distribution and assume the original and unlearned models have the same spread; real denoiser outputs are not exactly bell-shaped and the two spreads can differ, so the claimed divergence minimization and gradient benefits are only as good as that approximation.

Editorial extensions

If this is right

  • If Hellinger is adopted, text-to-image unlearning can match or beat the erasure strength of MSE while keeping lower KID on unrelated concepts, meaning the same forget operation damages the model's other knowledge less.
  • The bounded gradients of Hellinger mean long fine-tuning runs are less likely to suddenly produce degenerate or artifact-laden images; chi-squared, with exponentially growing gradients, is the opposite.
  • The variational formulation makes every f-divergence available, so a user can deliberately pick an aggressive divergence for fast semantic removal and accept lower realism, or a closed-form divergence for clean realistic replacement.
  • The convergence analysis gives a quantitative rule: among the tested divergences, Hellinger and Jensen-Shannon have the largest $1/f''(1)$ and therefore faster local convergence, while chi-squared has the slowest.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the Hellinger bounded-gradient recipe should transfer to other Gaussian-output distribution-matching tasks, such as erasing facts from language models or one-step distillation, wherever the equal-covariance approximation holds.
  • Editorial inference: the variational branch is a GAN-like game, so the local-stability theorem leaves global behavior open; a testable extension is adding gradient penalties or spectral normalization to the critic and checking whether aggressive variational erasure becomes artifact-free.
  • Editorial inference: the paper's ranking by $1/f''(1)$ suggests a cheap rule for choosing a divergence for a new concept—compute the curvature of the generator function at 1 and pick the largest value when convergence speed matters—instead of tuning losses per concept by trial and error.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes f-DMU, a unified framework for concept unlearning in text-to-image diffusion models based on minimizing an f-divergence between the original model's reverse-process conditional distribution for an anchor concept and the unlearned model's conditional for the target concept. The authors show that the standard MSE loss is the KL special case, derive closed-form Gaussian losses for the squared Hellinger distance and Pearson chi-square divergence, and provide a variational min-max objective for general f-divergences. They analyze gradient magnitudes of the closed-form losses, prove local exponential stability of the min-max system under explicit assumptions, and present experiments on Stable Diffusion v1.4 with several concepts, anchors, training lengths, and regularizations. The central empirical claim is that the squared Hellinger instance (H2) consistently dominates MSE in the trade-off between erasure efficacy and generative fidelity.

Significance. If the claims are supported, the framework is a valuable unification: it recovers existing MSE/KL and JS-based objectives as special cases, offers cheap alternative closed-form losses, and provides a theoretical lens (gradient scaling and local convergence rates) for choosing a divergence. The closed-form derivations are clean and the local convergence theorem is a reasonable extension of known GAN analysis. The empirical study is broad, covering multiple concepts, anchors, and iteration budgets. However, the paper's headline empirical assertion of consistent superiority of H2 over MSE is not established by the reported numbers, and the paper lacks aggregate statistics, error bars, and multiple seeds. The theoretical contribution stands on its own, but the empirical claim needs substantial revision or re-framing.

major comments (2)
  1. [Abstract; Section 5.3; Tables 1, 3, 4] The abstract's claim that H2 'consistently dominates MSE' and the repeated assertion in Section 5.3 that MSE generally has worse KID and preservation are contradicted by the paper's own tables. For example, in Table 4 (5000 steps, erasing Snoopy), H2 empty gives target CA=0.10 with preserved Grumpy Cat CA=0.00 and KID=0.341, while MSE empty gives target CA=0.08 with preserved Grumpy Cat CA=0.12 and KID=0.303; here MSE has both stronger erasure and better preservation. In Table 3 (500 steps, erasing Snoopy), MSE empty achieves target CA=0.33 versus H2 empty CA=0.47. Similar counterexamples appear in Table 1 (e.g., erasing Grumpy Cat: MSE empty KID 0.237 vs H2 empty KID 0.250 on the target, and comparable or better preserved-concept KIDs for MSE). The paper provides no aggregate win/loss counts, no confidence intervals, and no multiple-seed variation. Since the 'consistently dominates' assertion is a load-bearing part of the paper's contribution, the authors should either add rigorous statistical comparison (multiple seeds, paired tests, aggregate trade-off metrics) or revise the claim to a more modest statement about specific favorable configurations.
  2. [Section 3; Appendix B.1.1; Equations (4), (36), (40)] The derivation of the closed-form losses assumes that the reverse-process conditionals p_Phi(xt-1|xt,c) and p_hatPhi(xt-1|xt,c*) are Gaussian with the same covariance Sigma, after which only the mean difference matters. The paper states that it follows standard practice to fix the variance to a constant, but it does not specify the value of Sigma, its per-timestep dependence, or how the coefficient in the H2 and chi2 losses relates to Sigma. This matters because the implemented losses in Equations (4) and (5) treat the exponent as ||Phi-hatPhi||^2/2, while the derivation in Appendix B.1.1 yields exponents with explicit 1/(8 sigma^2) and 1/sigma^2 factors (Equations (36) and (40)). The scaling can be absorbed into omega_t, but the paper should state this explicitly and give the actual schedule used in the experiments. As written, the connection between the stated f-divergence objective and the implemented loss is not fully transparent.
minor comments (6)
  1. [Equations (4) and (36)] Equation (4) and Equation (36) are inconsistent: Equation (36) contains an exponent of -1/(8 sigma^2) ||Phi - hatPhi||^2, while Equation (4) shows -||Phi - hatPhi||^2/2. The authors should make the sigma dependence consistent and clarify the notation so that the main-text loss matches the appendix derivation.
  2. [Section 5.3 and Table 3] The caption of Table 3 says the best values are in bold and second-best in italics, but the table appears to bold only H2 rows and never marks MSE rows; please apply the stated formatting consistently or adjust the caption.
  3. [Section 5.1] The hyperparameter omega_t, used in all closed-form losses, is not defined or chosen; please report its schedule or state that it is set to a constant.
  4. [Section 5.2] Figure 2 reports average gradient amplitudes, but the averaging set (which timesteps, which concepts, how many samples) is not specified; please add this information for reproducibility.
  5. [Section 2.1 and Table 3] The abbreviation CAbl is used in Section 5.3 but the tables list 'Concept Ablation'; please use one consistent name throughout.
  6. [Section 5.4] The multi-concept erasure section is qualitative only; please either add quantitative results for the sequential setting or explicitly state that the claim is qualitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the f-divergence derivation is self-contained, and the empirical superiority claim is an observation rather than an input to the framework.

full rationale

The paper's central chain is self-contained. Section 3 and Appendix B.1.1 start from the standard definition of f-divergence (Eq. 1) and the Nguyen-Wainwright-Jordan variational representation (Eq. 2), then evaluate known Gaussian closed forms under the paper's explicit equal-covariance assumption, recovering MSE as the KL instance (Eqs. 27-28 and 20) and obtaining the H2 and chi2 objectives (Eqs. 4-5 and 36, 40) by direct substitution; no target quantity is inserted into these derivations. The gradient identities in Eq. 7 and Appendix B.1.2 follow algebraically from the loss definitions (e.g., the derivative of -exp(-MSE) is exp(-MSE) times the MSE gradient) rather than being fitted to the empirical outcomes. The local convergence analysis (Theorem 4.2) uses standard dynamical-systems lemmas (Theorem B.4 from Nagarajan-Kolter) and derives the f''(1) ordering from Lemma B.5, which is a mathematical identity for convex conjugates; the paper candidly labels the convergence-speed implication as conditional and 'probably beneficial.' The 'Hellinger consistently dominates MSE' claim is an empirical observation based on Tables 3-4, not a quantity forced by construction; whether the tables support it is a statistical-evidence concern, not a circularity. Self-citations [65,67,69] appear only in the related-work survey of f-divergence applications and do not supply any load-bearing theorem or assumption. No circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The mathematical core is a standard application of known f-divergence closed forms; no fitted constants enter the derivation. The empirical superiority claims, however, rest on experimental design choices such as anchor selection and regularizations, and on the Gaussian equal-variance approximation. The local convergence theorem depends on three formal assumptions that are standard for GAN-style min-max analysis but are not verified empirically.

free parameters (2)
  • omega_t (time-step weighting)
    Appears in every closed-form loss (Eqs. 4, 5, 32, 36, 40) and scales the contribution of each diffusion step. Its schedule is inherited from diffusion training and not reported or fitted in the paper, but the relative behavior of H2 vs MSE gradients depends on the normalization absorbed into it.
  • sigma (or Sigma): assumed equal variance of denoiser output distributions
    The paper fixes the variance to a constant so that closed-form f-divergence losses reduce to functions of the mean difference only (Section 3 and B.1.1). The exponent coefficients 1/(8 sigma^2) and 1/sigma^2 are absorbed into omega_t; the value of sigma changes the effective gradient scale.
assumptions (7)
  • standard math f-divergence framework: f is convex, lower-semicontinuous, f(1)=0, and p is absolutely continuous with respect to q.
    Used to define D_f(p||q) in Eq. (1) and to derive the variational representation in Eq. (2).
  • domain assumption Diffusion process is Markovian and x_T is independent of the concept c.
    Needed in Appendix B.1 to rewrite the KL divergence over full trajectories as a sum of per-step divergences, Eqs. (14)-(18).
  • domain assumption The denoiser output distributions p_Phi(xt-1|xt,c) and p_hatPhi(xt-1|xt,c*) are Gaussian with equal covariance Sigma.
    This is the key approximation that lets the authors replace closed-form f-divergences between Gaussians with objectives that depend only on the mean difference (Section 3 and B.1.1). It is standard practice in prior MSE-based unlearning but is not validated on real model outputs.
  • ad hoc to paper Assumption B.6: f* is strictly convex.
    Introduced in Appendix B.2.2 to ensure KTT is negative definite and Theorem 4.2 applies. It excludes total variation and other divergences.
  • ad hoc to paper Assumption B.7: there exists an equilibrium point where p_phi* = p_Phi and T_omega* = f'(1).
    This is the optimal convergence point for the min-max game. It is used in the proof of Theorem 4.1 to zero out the phi-phi Hessian block.
  • ad hoc to paper Assumption B.8: certain cross-Hessian terms are full row rank.
    This rank condition is needed to apply the eigenvalue bound in Theorem B.4 and to conclude local exponential stability in Theorem 4.2.
  • standard math Variational representation theorem for f-divergences (Nguyen et al.).
    Used in Eq. (2) and Appendix B.1.3 to derive the min-max objective in Eq. (6).

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Cite this review

Pith. "Pith review of A Unified Framework for Diffusion Model Unlearning with f-Divergence." pith.science (2026). https://pith.science/paper/GGYHIHDJ

@misc{pith2026250921167,
  author       = {Pith},
  title        = {Pith review of: A Unified Framework for Diffusion Model Unlearning with f-Divergence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGYHIHDJ}},
  note         = {Machine review of arXiv:2509.21167}
}
abstract

Most existing methods for concept unlearning in text-to-image diffusion models minimize a mean squared error (MSE) loss between the denoiser outputs conditioned on a target and an anchor concept, which is implicitly the KL divergence between two Gaussians. We generalize this objective to any $f$-divergence, recovering MSE as the KL instance, and identify a family of $\alpha$-divergences whose Gaussian closed-form yields cheap, MSE-like training objectives. For the remaining $f$-divergences, we provide a min-max objective based on the variational formulation of the $f$-divergence. We theoretically analyze and numerically validate how different $f$-divergences impact the gradient magnitude and the convergence properties of the algorithm, affecting the quality of unlearning. For instance, we observe that the Hellinger closed-form instance consistently dominates MSE across multiple scenarios. More generally, the proposed unified framework offers a flexible paradigm for selecting the optimal divergence based on the application and user goal, allowing for finer control over the trade-off between unlearning efficacy and generative fidelity.

Figures

Figures reproduced from arXiv: 2509.21167 by the authors.

Figure 1
Figure 1. Unlearning a concept using different f-divergences yield different unlearning dynamics (top: mean squared error, middle: Hellinger closed-form, bottom: Hellinger variational). The inset in the bottom right corner of each image reports a not erased concept. To the best of our knowledge, f-DMU is the first approach based on general f-divergences for the DM unlearning task. We theoretically study the gradients of the f… view at source ↗
Figure 2
Figure 2. Average gradient amplitude of closed-form-based losses. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Instability during fine-tuning caused by large gradients. Top: MSE, bottom: H [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: All methods manage to remove the concept from the model so that no generation of "R2D2" can take place. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Qualitative comparison of superclass-guided unlearning outcomes between CAbl and our Hellinger distance [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Qualitative comparison of superclass unlearning with sequential multiple erasure. The first three columns [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: f-divergence-based Diffusion Model Unlearning Framework. Every pair (f, c) indicates a different objective function based on a specific f-divergence and using a specific type of concept c. A list of concepts [cn, ch, cc, cd] indicates a specific f-divergence objective …
Figure 8
Figure 8. Figure 8: Divergence-based methods for diffusion models unlearning. Each sector represents an [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Gradients comparison as a function of the MSE. All the functions are plotted up to a multiplicative constant. [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

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    by modifying the cross-attention layers and utilizing Low-Rank Adaptation (LoRA) to remove a large number of concepts. The major advantage of these methods performing the closed-form update of the weights is the computational complexity. However, the main drawback is that thes...

    2000

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.