{"id":"24112f13-9e35-4179-89ff-8af8c0d27eb0","arxiv_id":"2509.21167","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Diffusion model unlearning is generalized from KL/MSE to any f-divergence, with closed-form Hellinger and chi-square losses and a variational min-max form.","lead":"This paper proposes replacing the standard mean squared error loss in diffusion model unlearning with a family of f-divergence losses, including a Hellinger variant with bounded gradients. The authors report that the Hellinger loss removes target concepts while preserving image quality better than MSE in several Stable Diffusion experiments.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that the Hellinger closed-form loss 'consistently dominates' MSE is not supported by the paper's own tables: multiple rows show MSE with equal or better erasure and comparable or better preservation, and no aggregate statistics or error bars are reported.","rationale":"The reader's conditional verdict is appropriate. I agree that the empirical superiority claim is overstated and needs more validation; however, I do not think the Gaussian equal-covariance assumption is the most load-bearing concern. In DDPM-based models such as Stable Diffusion, the reverse-process conditional p(xt-1|xt,c) is parameterized as a Gaussian whose covariance is fixed by the noise schedule, and the unlearned model uses the same schedule. Thus the equal-covariance condition is built into the model class, not an untested approximation. The paper's closed-form losses are no more fragile in this respect than the MSE baseline they generalize.\n\nThe real risk is that the headline claim—'consistently dominates MSE'—is contradicted by the paper's own quantitative tables, which show mixed results depending on the target concept, step count, and metric. Without a predefined dominance criterion, aggregate counts, or repeated seeds, the claim is unfalsifiable as stated. The theoretical contributions (the unified framework, gradient bounds, and local convergence analysis) remain valuable and should be preserved, but the empirical section should be conditioned on a rigorous comparison. Keeping the verdict at CONDITIONAL (UNCHANGED) reflects the need for these additions before the paper's central empirical assertion is accepted.","tokens_in":41793,"tokens_out":8206,"duration_ms":75016,"concrete_test":"Count, across Tables 1, 3, 4, 6 and 7, the number of target-concept/step configurations in which the H2 closed-form result satisfies all three conditions: (i) target CS <= MSE target CS, (ii) target CA <= MSE target CA, and (iii) each preserved-concept KID <= the corresponding MSE KID. Report the win/loss/tie counts. If H2 wins fewer than 80% of configurations, the 'consistently dominates' claim should be weakened. A stronger check is to rerun the closed-form comparisons with 5 seeds and report mean and standard deviation of CS, CA, and KID; without this, the single-seed numbers cannot support a universal ordering over MSE.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the empirical superiority of the squared-Hellinger closed-form loss over MSE. The f-divergence framework itself is coherent: recovering KL as a special case is standard, and the closed-form objectives are legitimate if the reverse-process conditionals are Gaussian with equal covariance. In DDPM-style models those conditionals are parameterized as Gaussians with a fixed variance schedule, so the equal-covariance step is structural rather than a fragile empirical approximation; I do not see that as the main risk.\n\nThe load-bearing weakness is the unsupported 'consistently dominates' assertion. Tables 1, 3 and 4 contain multiple comparisons where MSE achieves lower target CS/CA (stronger erasure) with comparable or better preservation. For example, Table 4, erasing Snoopy at 5000 steps: MSE empty gives target CA=0.08 and preserves Grumpy Cat with CA=0.12, KID=0.303, while H2 empty gives target CA=0.10 and preserved Grumpy Cat CA=0.00, KID=0.341. Table 3, erasing Snoopy at 500 steps: MSE empty gives target CA=0.33 versus H2 empty CA=0.47, with similar preservation KID. No aggregate win/loss counting, no confidence intervals, and only one seed are provided. The superiority claim therefore rests on selected qualitative examples and a plausible gradient-boundedness story, not on a demonstrated consistent advantage. This should be settled before the abstract's strong wording is accepted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":42138,"tokens_out":5914,"duration_ms":53502,"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":[{"comment":"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.","section":"Abstract; Section 5.3; Tables 1, 3, 4"},{"comment":"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.","section":"Section 3; Appendix B.1.1; Equations (4), (36), (40)"}],"minor_comments":[{"comment":"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.","section":"Equations (4) and (36)"},{"comment":"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.","section":"Section 5.3 and Table 3"},{"comment":"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.","section":"Section 5.1"},{"comment":"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.","section":"Section 5.2"},{"comment":"The abbreviation CAbl is used in Section 5.3 but the tables list 'Concept Ablation'; please use one consistent name throughout.","section":"Section 2.1 and Table 3"},{"comment":"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.","section":"Section 5.4"}],"recommendation":"major_revision","confidential_remarks":"The central theoretical framework seems coherent, and the closed-form derivations are a useful contribution. My main concern is that the paper's headline claim of H2 'consistently dominating' MSE is contradicted by several rows in the authors' own tables, and this is not a matter of presentation but of evidential support. I would encourage the editor to request a revised version that either provides rigorous statistical evidence for the superiority claim or replaces it with a nuanced claim. I also noticed the sigma inconsistency between Equations (4) and (36), which should be fixed regardless."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know upfront. First, this is the first general f-divergence framework for diffusion model unlearning, and it ships real goods: closed-form Hellinger and chi-squared losses, a variational min-max form for any f-divergence, and a clean gradient analysis showing H2 has bounded gradients while chi-squared amplifies them. Second, the central empirical assertion that H2 'consistently dominates' MSE is not supported by the paper's own results. That claim should be revised to something like 'H2 offers a different and sometimes better trade-off.'\n\nThe unification itself is the strongest part. Recovering MSE as the KL instance, recovering Concept Ablation and DoCo as special cases, and expressing everything through one objective (Eq. 3) is a legitimate and useful contribution. The gradient comparison is simple but correct: H2 weights the MSE gradient by a negative exponential, which caps large updates and explains why artifacts appear less often. The local convergence theorem is a reasonable extension of GAN stability analysis, and the conclusion that H2 and JS converge faster near equilibrium follows from the Fenchel-conjugate calculation. I do not see a load-bearing math error.\n\nThe soft spots are empirical. The abstract claims consistent superiority, but Tables 3 and 4 contain multiple counterexamples. At 500 steps, erasing Snoopy, MSE empty reaches CA 0.33 versus H2's 0.47 with comparable KID. At 5000 steps, erasing Snoopy, MSE empty reaches CA 0.08 versus H2's 0.10, and H2's preservation of Grumpy Cat is worse (CA 0.00 vs 0.12). There are no error bars, no multiple seeds, no aggregate win/loss count, and no code release. The qualitative examples are nice, but they do not carry a 'consistently' claim. On the Gaussian assumption: I partially disagree with the reader's worry. In DDPM-style models the reverse-process conditionals are parameterized as Gaussians with a fixed variance schedule, so the equal-covariance step is structural rather than a fragile empirical leap. What is not validated is whether the unlearned model's output distribution actually stays close to Gaussian in the relevant latent space; that is worth a sentence of caveat, but it is not the main risk.\n\nBottom line: the framework and theory deserve referee time, and the gradient-boundedness story is a plausible mechanism. The paper needs an empirical revision—either a proper aggregation showing when H2 wins and loses, or a toned-down claim. As it stands, I would accept it for peer review but push hard on the 'consistently dominates' wording.","headline":"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.","tokens_in":42633,"tokens_out":1386,"would_cite":true,"duration_ms":15949,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","94A17"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["diffusion models","machine unlearning","concept erasure","f-divergence","Hellinger distance","variational divergence estimation","text-to-image generation","gradient analysis"],"falsifier":"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.","tokens_in":41572,"feed_emoji":"🧹","tokens_out":11383,"duration_ms":95311,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hellinger beats MSE at unlearning concepts in image models","feed_subtitle":"Standard MSE erasure is one choice among many; Hellinger erases just as well and keeps the model intact.","key_machinery":"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 $f''(1)$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the KL/MSE formulation and the Markov-chain decomposition that f-DMU generalizes to any f-divergence.","marker":"[10]"},{"why":"It is the ESD baseline whose KL-based objective is recovered as a special case of the framework.","marker":"[9]"},{"why":"It defines the f-divergence family and the convex generator f used in the objectives.","marker":"[25, 26]"},{"why":"It provides the variational representation with the Fenchel conjugate that grounds the min-max formulation.","marker":"[27]"},{"why":"It contributes the Jacobian spectrum technique used to prove local exponential stability of the min-max system.","marker":"[34]"},{"why":"It gives the range-of-values theorem the paper uses to show Hellinger loss and its gradients are bounded.","marker":"[75]"},{"why":"It supplies the closed-form squared Hellinger distance between Gaussians behind the H2 loss.","marker":"[76]"},{"why":"It shows the objective can be restricted to a subset of the generation trajectory, which f-DMU adopts.","marker":"[19]"}],"fun_headline_variants":["Hellinger beats MSE for diffusion concept erasure","f-divergence unlearning: Hellinger outperforms MSE","Diffusion unlearning gets sharper with Hellinger loss","Squared Hellinger improves image model unlearning","Generalized f-divergence: Hellinger tops MSE in erasure"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hellinger beats MSE for diffusion concept erasure","f-divergence unlearning: Hellinger outperforms MSE","Diffusion unlearning gets sharper with Hellinger loss","Squared Hellinger improves image model unlearning","Generalized f-divergence: Hellinger tops MSE in erasure"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1291,"prompt_tokens":1009,"completion_tokens":282,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":197}},"tokens_in":625,"tokens_out":282,"duration_ms":3177,"temperature":1.0,"reasoning_tokens":197,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:46:32.740321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Wainwright, and Michael I","cited_arxiv_id":null,"evidence_quote":"It provides the variational representation with the Fenchel conjugate that grounds the min-max formulation."},{"cited_title":"Gradient descent gan optimization is locally stable.Advances in neural information processing systems, 30, 2017","cited_arxiv_id":null,"evidence_quote":"It contributes the Jacobian spectrum technique used to prove local exponential stability of the min-max system."},{"cited_title":"On the f-divergence and singularity of probability measures.Periodica Mathematica Hungarica, 2(1-4):223–234, 1972","cited_arxiv_id":null,"evidence_quote":"It gives the range-of-values theorem the paper uses to show Hellinger loss and its gradients are bounded."},{"cited_title":"Chapman and Hall/CRC, 2018","cited_arxiv_id":null,"evidence_quote":"It supplies the closed-form squared Hellinger distance between Gaussians behind the H2 loss."}],"review_version":1}