A Unified Framework for Diffusion Model Unlearning with f-Divergence

## Abstract

Abstract Machine unlearning aims to remove specific knowledge from a trained model. While diffusion models (DMs) have shown remarkable generative capabilities, existing unlearning methods for text-to-image (T2I) models often rely on minimizing the mean squared error (MSE) between the output distribution of a target and an anchor concept.
We show that this MSE-based approach is a special case of a unified f f -divergence-based framework, in which any f f -divergence can be utilized.
We analyze the benefits of using different f f -divergences, that mainly impact the convergence properties of the algorithm and the quality of unlearning.
The proposed unified framework offers a flexible paradigm that allows to select the optimal divergence for a specific application, balancing different trade-offs between aggressive unlearning and concept preservation.

## 1 Introduction

Deep learning algorithms for T2I generation based on DMs have demonstrated remarkable success and widespread adoption in recent years. Some notable examples are Stable Diffusion [1], DALL$\cdot$E 2 [2], and Imagen [3], which find a large variety of applications in real-world scenarios.
However, a significant challenge associated with these algorithms stems from the fact that they are trained over large scale datasets (such as LAION-5B [4]) that are scraped from the Internet, and contain not safe for work (NSFW) content, including explicit material [5], and copyrighted works, such as artistic styles and even personal information [6, 7].
Since DMs are able to learn and memorize all these contents [8], it is crucial to develop algorithms that erase specific concepts from the final trained models.
To force a model to forget a concept, the trivial solution would be to re-train the model with a curated dataset where the specific concept has been removed. However, this is impractical, as it is resource-intensive and time-consuming, and in some cases the training dataset could have been deleted.
Therefore, it is fundamental to develop techniques that would allow a given trained diffusion model to unlearn a specific concept.

Figure: 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.
Refer to caption: x1.png

Many algorithms proposed for DM unlearning are grounded on a shared idea: shifting the model’s prediction corresponding to the concept to be unlearned (referred to as "target") towards prediction of a substitute concept (referred to as "anchor"). The latter can be selected in many different ways, depending on the user’s goal: null concept, hyper-class, semantically close/distant to the concept to forget.
Usually, this is achieved by minimizing an MSE-based loss [9, 10, 11], derived from formulating the problem as the minimization of the Kullback-Leibler (KL) divergence between two Gaussian distributions, conditioned on the two concepts, target and anchor, characterizing the image generation process.

In this paper, instead of minimizing the standard KL divergence, we propose minimizing the $f$-divergence between the two distributions conditioned on the target and anchor concept.
For specific $f$-divergences, we derive the loss function for DM unlearning from the closed-form expression of the $f$-divergence between two Gaussian distributions. These loss functions, similarly to the standard MSE, benefit from a simple implementation and training complexity, while differing from the MSE in terms of convergence properties.
When the closed-form expression does not exist, we use the variational representation of the $f$-divergence formulated as a min-max problem, which allows us to tackle the DM unlearning task with *any* $f$-divergence.
We refer to the proposed method as $f$-divergence-based DM unlearning ($f$-DMU).

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 $f$-DMU closed-form objective functions and empirically verify the theoretical findings. Then, we theoretically prove the local convergence property of the proposed min-max formulation, and relate the choice of different $f$-divergences with the algorithm convergence speed.
Finally, we perform an extensive numerical analysis to compare results with different $f$-divergences, and we demonstrate the superiority of the proposed unified framework with respect to the standard MSE approach. In particular, we empirically demonstrate that the deployment of $f$-divergences outperforms the MSE loss consistently across different unlearning scenarios, such as style ablation and object ablation.

## 2 Preliminaries

### 2.1 Related Work

In this section, we present a condensed version of the related work in which we focus only on methods in the context of DMs, while we defer to Appendix [A](https://arxiv.org/html/2509.21167v1#A1) for an extended overview.

##### Post-processing techniques

Post-processing techniques target the elimination of unsafe generated images through the usage of filtering or inference guiding. Stable Diffusion (SD) [1] adopts a NSFW filter that removes all generated images with embeddings close to those of $17$ pre-chosen nudity concepts [12]. Safe Latent Diffusion (SLD) [5] is applied during inference and acts by repelling the generation from unsafe contents. Another inference guidance-based approach is SAFREE [13], which can be applied for both image and video generation.
The main drawback of post-processing algorithms is that it is possible to remove them from the inference pipeline to allow the model to generate what should have been forgotten.

##### Fine-tuning techniques

Fine-tuning approaches modify the weights of a trained model.
Many fine-tuning approaches share the common idea of aligning the target and anchor concepts by minimizing the KL divergence.
Erased Stable Diffusion (ESD) [9] (and other methods derived from it [11]) fine-tunes the model to align the probability distributions of the model’s output fed with a target concept and a neutral concept. Concept Ablation (CAbl) [10] minimizes the MSE between the model’s prediction corresponding to the target and anchor concept.
With the goal of finding the best anchor concept, Bui et al. propose adversarial learning frameworks with a loss function comprising two MSE losses, and show that the best anchor concept should be semantically close to the target [14, 15].
This knowledge is applied to design an MSE loss that shifts the target concept to semantically close concepts [16].
While the previous algorithms are limited in using loss functions based on the KL divergence between the corresponding probability distributions, another line of work proposes to update the model weights in closed-form, leading to faster computations [17, 18, 19]. These methods also target the minimization of the MSE, but the loss is formulated directly on the attention weights.
However, these approaches are constrained in modifying cross-attention weights, while, in principle, distribution-based methods can update any network parameter and work for different models, such as flow-based generative models [20].
Domain Correction (DoCo) [21] extends CAbl by using a GAN-based [22] loss function. We show in Appendix [A](https://arxiv.org/html/2509.21167v1#A1) that DoCo and CAbl’s loss functions can be obtained as special cases of the proposed $f$-DMU framework.

### 2.2 Diffusion Models

DMs [23, 24] are state-of-the-art generative models that consist of two components that can be modeled by Markov chain processes. In the forward process, Gaussian noise (referred to as $\epsilon$) is gradually added to an input image $\mathbf{x}_{0}$ over multiple steps $t\in[0,\dots,T]$, to obtain $\mathbf{x}_{T}\sim\mathcal{N}(0,I)$. At each time step, the noisy image can be obtained as $\mathbf{x}_{t}=\sqrt{\alpha_{t}}\mathbf{x}_{t-1}+\sqrt{1-\alpha_{t}}\epsilon$, where $\alpha_{t}$ regulates the noise effect.
In the reverse process, $\mathbf{x}_{T}$ is transformed - following a transformation that is the inverse of the forward process - to obtain a denoised image using a denoising network $\Phi(\mathbf{x}_{t},\textbf{c},t)$, where, for T2I models, the concept c is a text prompt. The denoising process is characterized as $p_{\Phi}(\mathbf{x}_{0},\dots,\mathbf{x}_{T}|\mathbf{c})=p(\mathbf{x}_{T})\prod_{t=1}^{T}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})$,
where $p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})$ describes the probability of $\mathbf{x}_{t-1}$ given the noisy image $\mathbf{x}_{t}$ and the concept $\mathbf{c}$.

### 2.3 f f -Divergence

Let $p(\mathbf{x})$ and $q(\mathbf{x})$ be two probability density functions on domain $\mathcal{X}$. The $f$-divergence between $p(\mathbf{x})$ and $q(\mathbf{x})$ is defined as [25, 26]

$$ $D_{f}(p||q)=\int_{\mathcal{X}}q(\mathbf{x})f\left(\frac{p(\mathbf{x})}{q(\mathbf{x})}\right)d\mathbf{x},$ (1) $$

where $p\ll q$ (i.e., $p$ is absolutely continuous with respect to $q$) and the generator function $f:\mathbb{R}_{+}\longrightarrow\mathbb{R}$ is a convex, lower-semicontinuous function such that $f(1)=0$. The KL divergence is a special case of $f$-divergence, where $f(u)=u\log u$.
The variational representation of $f$-divergence [27] reads as

$$ $D_{f}(p||q)=\sup_{T:\mathcal{X}\to\mathbb{R}}\left\{\operatorname{\mathbb{E}}_{p}\left[T(\mathbf{x})\right]-\operatorname{\mathbb{E}}_{q}\left[f^{*}(T(\mathbf{x}))\right]\right\},$ (2) $$

where $T$ is a parametric function (e.g., a neural network) and $f^{*}$ represents the Fenchel conjugate of $f$, defined as $f^{*}(t)=\sup_{u\in dom_{f}}\left\{ut-f(u)\right\}$, with $dom_{f}$ being the domain of $f$.
The supremum in ([2](https://arxiv.org/html/2509.21167v1#S2.E2)) is attained for $T^{\diamond}(\mathbf{x})=f^{\prime}\left(p(\mathbf{x})/q(\mathbf{x})\right)$,
where $f^{\prime}$ is the first derivative of $f$.

## 3 Concept Erasing with f f -Divergence

In this paper, we present an $f$-divergence-based framework able to unlearn a target concept $\mathbf{c}^{*}$ from a DM. The concept $\mathbf{c}^{*}$ is erased by shifting the model generation corresponding to $\mathbf{c}^{*}$ to the generation attained using an anchor concept $\mathbf{c}$. In [10], this is achieved by minimizing the KL divergence between the reverse processes of the original and unlearned DMs, i.e., $D_{KL}(p_{\Phi}(\mathbf{x}_{(0\dots T)}|\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{(0\dots T)}|\mathbf{c}^{*}))$.
Leveraging the Markov property of diffusion processes (see Appendix [B.1](https://arxiv.org/html/2509.21167v1#A2.SS1) for the derivation details), Kumari et al. showed that such an objective can be rewritten as the KL divergence between the DM outputs at specific time instants. We generalize this idea using the $f$-divergence, formulating the unlearning problem as

$$ $\displaystyle\min_{\hat{\Phi}}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[D_{f}\left(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})\right)\Bigr].$ (3) $$

The KL-based loss employed in prior works can be recovered from ([3](https://arxiv.org/html/2509.21167v1#S3.E3)) using the KL divergence.
Although theoretically the optimal distribution that minimizes ([3](https://arxiv.org/html/2509.21167v1#S3.E3)) is the same for any $f$-divergence, different $f$-divergences have different convergence properties and mode-seeking behaviors [28].
Notably, the divergence-based formulation in ([3](https://arxiv.org/html/2509.21167v1#S3.E3)) allows any choice of the anchor concept $\mathbf{c}$. For instance, $\mathbf{c}$ can be chosen as a neutral concept [9], superset concept [10], a concept with high semantic similarity to $\mathbf{c}^{*}$ [15], or a concept with low semantic similarity to $\mathbf{c}^{*}$ [29].
In the literature, there is not an agreement on the best anchor concept to use. For instance, while the superclass or semantically close options are usually considered the best options, since they are more similar to the target prompt, they appear to be more sensitive to subsequent fine-tuning of the model [29], leading to possible remembering of forgotten concepts. On the contrary, semantically distant concepts are more robust to a future fine-tuning, but lead to unexpected generation and worse erasure effectiveness.
In addition, divergence-based objectives also apply to flow-based generative models [30], contrary to architecture-specific approaches [20].

Previous work solved ([3](https://arxiv.org/html/2509.21167v1#S3.E3)) when the chosen $f$-divergence is the KL by using the fact that the KL divergence between Gaussian distributions leads to the MSE between their means.
In this paper, we extend this solution to any $f$-divergence. In particular, it is possible to express $D_{f}(P||Q)$, and thus ([3](https://arxiv.org/html/2509.21167v1#S3.E3)), in closed-form when $P$ and $Q$ are Gaussian, for a subset of divergences.
In the following, we provide three examples: Jeffreys divergence, squared Hellinger distance, and Pearson $\chi^{2}$ divergence. It is possible to extend this approach to an entire class of $f$-divergences, referred to as $\alpha$-divergences [31, 32]. We defer to Appendix [B.1.1](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS1) for the calculations.

Jeffreys divergence: Jeffreys divergence between two probability distributions $P$ and $Q$ is defined as $D_{J}(P||Q)=D_{KL}(P||Q)+D_{KL}(Q||P)$. The objective function is reported in Appendix [B.1.1](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS1).

Squared Hellinger distance: Squared Hellinger distance between two probability distributions $P$ and $Q$ can be expressed in terms of the Bhattacharyya coefficient ($BC(P,Q)$) as $H^{2}(P,Q)=1-BC(P,Q)$. The objective function based on squared Hellinger distance reads as

$$ $\displaystyle\mathcal{J}_{H}(\hat{\Phi})=\operatorname{\mathbb{E}}_{\mathbf{x},\mathbf{c}^{*},\mathbf{c},t}\Bigl[-\omega_{t}\exp\left\{-||\Phi(\mathbf{x}_{t},\mathbf{c},t)-\hat{\Phi}(\mathbf{x}_{t},\mathbf{c}^{*},t)||_{2}^{2}\right\}\Bigr],$ (4) $$

where $\Phi(\cdot)$ and $\hat{\Phi}(\cdot)$ are the outputs of the original DM and of the DM during unlearning, respectively.

$\chi^{2}$ divergence: The closed-form expression for $\chi^{2}$ divergence between two Gaussian random variables $P$ and $Q$ exists under a mild assumption on the variance, which will always hold in our case. The objective function based on the Pearson $\chi^{2}$ divergence becomes

$$ $\displaystyle\mathcal{J}_{\chi^{2}}(\hat{\Phi})=\operatorname{\mathbb{E}}_{\mathbf{x},\mathbf{c}^{*},\mathbf{c},t}\Bigl[\omega_{t}\exp\left\{||\Phi(\mathbf{x}_{t},\mathbf{c},t)-\hat{\Phi}(\mathbf{x}_{t},\mathbf{c}^{*},t)||_{2}^{2}\right\}\Bigr].$ (5) $$

In this work, we follow the standard practice of fixing the variance of DMs to a constant value. Thus, our loss functions do not include a variance term, even though closed-form expressions for $f$-divergences between Gaussians with different variances are available (Appendix [B.1.1](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS1)).
Although any $f$-divergence between two multivariate normal distributions with the same covariance matrix $\Sigma$ is an increasing function of their Mahalanobis distance [25, 33] $\Delta_{\Sigma}(\mu_{P},\mu_{Q})=\sqrt{(\mu_{Q}-\mu_{P})^{T}\Sigma^{-1}(\mu_{Q}-\mu_{P})}$, not all $f$-divergences can be expressed in closed-form. One example is the total variation distance [33].

Driven by the intent of providing a unified $f$-divergence-based framework, we present a comprehensive formulation that allows us to solve the problem in ([3](https://arxiv.org/html/2509.21167v1#S3.E3)) by using *any* $f$-divergence by adopting its variational representation (more details in Appendix [B.1.3](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS3))

$$ $\displaystyle\min_{\hat{\Phi}}\max_{T}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\left[T(\Phi)\right]-\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\left[f^{*}(T(\hat{\Phi}))\right]\Bigr],$ (6) $$

where $T(\cdot)$ can be parametrized as a neural network fed with the output samples of the original DM $\Phi$ and the unlearned model $\hat{\Phi}$.
This optimization problem can be treated as a min-max game between the generator $\hat{\Phi}$ and the discriminator $T$, where $T$ estimates the divergence between $p_{\Phi}$ and $p_{\hat{\Phi}}$, while the generator aims at minimizing the same divergence.
Solving ([6](https://arxiv.org/html/2509.21167v1#S3.E6)) leads to the minimization of $D_{f}(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*}))$ for any $\mathbf{x}_{t}$.

The loss functions derived from closed-form expressions of $f$-divergences share a key benefit with the MSE: they only require solving a minimization problem. They are characterized by different gradients (see Section [4.1](https://arxiv.org/html/2509.21167v1#S4.SS1)), which leads to improved convergence properties. In contrast, the general loss function expressed using the variational formulation requires solving a min-max problem. This approach, however, can be applied to any $f$-divergence, leading to a broader class of loss functions, for which we study the local convergence properties in Section [4.2](https://arxiv.org/html/2509.21167v1#S4.SS2).

We observe that, following [19], instead of solving ([3](https://arxiv.org/html/2509.21167v1#S3.E3)), it is possible to minimize the divergence between the probability distributions over a subset of the original generation trajectory (from $\mathbf{x}_{t}$ to $\mathbf{x}_{0}$, with $t<T$), and the $f$-divergence generalization still holds. This reasoning can be taken to the extreme by training the model with only $\mathbf{x}_{0}$, as proposed in [20].

## 4 Theoretical Analysis

### 4.1 Gradient Analysis

By analyzing the gradients of the loss functions obtained in closed-form (referred to as $\mathcal{J}_{f}^{c}$), it is possible to observe a significant difference between the gradient of the MSE, obtained for KL divergence, and those of the loss functions obtained from the squared Hellinger distance and $\chi^{2}$ divergence.
For simplicity in the notation, let $\text{MSE}(\Phi,\hat{\Phi})$ refer to $\text{MSE}(\Phi(\mathbf{x}_{i},\mathbf{c},i),\hat{\Phi}(\mathbf{x}_{i},\mathbf{c}^{*},i))$, and let $\boldsymbol{\phi}$ be the vector of parameters of $\hat{\Phi}$. Then, the gradients of the loss functions can be written as

$$ $\displaystyle\frac{\partial\mathcal{J}_{f}^{c}(\Phi,\hat{\Phi})}{\partial\phi}=$ $\displaystyle\begin{cases}\nabla_{\phi}\text{MSE}(\Phi,\hat{\Phi})&\text{ for KL}\\ e^{-\text{MSE}(\Phi,\hat{\Phi})}\nabla_{\boldsymbol{\phi}}\text{MSE}(\Phi,\hat{\Phi})&\text{ for H}^{2}\\ e^{\text{MSE}(\Phi,\hat{\Phi})}\nabla_{\boldsymbol{\phi}}\text{MSE}(\Phi,\hat{\Phi})&\text{ for }\chi^{2}\end{cases}.$ (7) $$

Both the gradients of $H^{2}$ and $\chi^{2}$ are proportional to the gradient of KL. However, they are characterized by two opposite behaviors. When $\text{MSE}\to\infty$, the gradients of $H^{2}$ and $\chi^{2}$ tend to 0 and $\infty$, respectively. When $\text{MSE}\to 0$, the gradients of H^2 and $\chi^{2}$ both tend to the gradient of the MSE. In summary, we can conclude that $\left|\frac{\partial\mathcal{J}_{\text{H}^{2}}(\Phi,\hat{\Phi})}{\partial\boldsymbol{\phi}}\right|\leq\left|\frac{\partial\mathcal{J}_{\text{KL}}(\Phi,\hat{\Phi})}{\partial\boldsymbol{\phi}}\right|\leq\left|\frac{\partial\mathcal{J}_{\chi^{2}}(\Phi,\hat{\Phi})}{\partial\boldsymbol{\phi}}\right|$, where the inequalities become equalities when the MSE is zero.

### 4.2 Local Convergence Analysis

In this section, we address the local convergence of the variational form of $f$-DMU to show that, under mild assumptions, the proposed algorithm is locally exponentially stable around equilibrium points. Similarly to how previous work studied the convergence of GANs [34, 35] and energy-based models [36], we use nonlinear dynamical systems theory [37].
We consider the optimization of the model parameters of the min-max problem in ([6](https://arxiv.org/html/2509.21167v1#S3.E6)) as a dynamical system. The system can be linearized (around the optimal convergence point) to study the local convergence properties. By evaluating the Jacobian matrix, we can conclude that if the Jacobian at an equilibrium point is a Hurwitz matrix, the system converges to the equilibrium (i.e., the equilibrium is locally exponentially stable).
Let $\hat{\Phi}$ and $T$ be parametrized by the vectors of parameters $\phi$ and $\omega$, respectively. We rewrite the objective function in ([6](https://arxiv.org/html/2509.21167v1#S3.E6)) as $\min_{\phi}\max_{\omega}\mathcal{J}_{f}(\phi,\omega)$.
Following [34], we focus on the analysis of continuous time ordinary differential equations, which
implies similar results for discrete time updates when the learning rate is sufficiently small.
Following a gradient update rule, the dynamical system describing the models’ update is given by

$$ $\displaystyle\begin{split}\dot{\phi}&=-\nabla_{\phi}\mathcal{J}_{f}(\phi,\omega)\\ \dot{\omega}&=\nabla_{\omega}\mathcal{J}_{f}(\phi,\omega)\end{split}.$ (8) $$

#### 4.2.1 Main Convergence Results

Theorem [4.1](https://arxiv.org/html/2509.21167v1#S4.Thmtheorem1) provides the Jacobian of the dynamical system describing the training of $\hat{\Phi}$ and $T$ at an equilibrium point. Theorem [4.2](https://arxiv.org/html/2509.21167v1#S4.Thmtheorem2) provides the main theoretical result of this paper, stating the stability of the dynamical system in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)).

###### Theorem 4.1 .

The Jacobian for the dynamical system defined in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)), at an equilibrium point $(\phi^{*},\omega^{*})$ is

$$ $\displaystyle J=\begin{pmatrix}\textbf{0}&-\textbf{K}_{TP}\\ \textbf{K}_{TP}^{T}&\textbf{K}_{TT}\end{pmatrix},$ (9) $$

where

$$ $\displaystyle\textbf{K}_{TP}$ $\displaystyle\triangleq\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[-\nabla_{\phi}\log(p_{\phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*}))\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)\Bigr]\Bigr]\Bigr|_{(\phi^{*},\omega^{*})},$ (10) $\displaystyle\textbf{K}_{TT}$ $\displaystyle\triangleq\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[-(f^{*})^{\prime\prime}(T_{\omega})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}\Bigr]\Bigr]\Bigr|_{\omega^{*}}.$ (11) $$

###### Theorem 4.2 .

The dynamical system defined in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)) is locally exponentially stable with respect to an equilibrium point $(\phi^{*},\omega^{*})$ under Assumptions [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6), [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7), [B.8](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem8). Let $\lambda_{m}(\cdot)$ and $\lambda_{M}(\cdot)$ be the smallest and largest eigenvalues of a given matrix, respectively. The rate of convergence of the system is governed by the eigenvalues of the Jacobian $\mathbf{J}$ which have a negative real part upper bounded as

- •
When $\text{Im}(\lambda)=0$, $\quad\text{Re}(\lambda)\leq-\frac{\lambda_{m}(-\textbf{K}_{TT})\lambda_{m}(\textbf{K}_{TP}\textbf{K}_{TP}^{T})}{\lambda_{m}(-\textbf{K}_{TT})\lambda_{M}(-\textbf{K}_{TT})+\lambda_{m}(\textbf{K}_{TP}\textbf{K}_{TP}^{T})}.$
- •
When $\text{Im}(\lambda)\neq 0$, $\quad\text{Re}(\lambda)\leq-\frac{\lambda_{m}(-\textbf{K}_{TT})}{2}.$

Besides the result on the local convergence of the dynamical system in [8](https://arxiv.org/html/2509.21167v1#S4.E8), from Theorem [4.2](https://arxiv.org/html/2509.21167v1#S4.Thmtheorem2) we can infer properties on the convergence speed of different $f$-divergences. This can be done by leveraging the knowledge about the upper bounds on the real part of the eigenvalues and about the fact that the bottleneck of the convergence speed is the largest eigenvalue.
In Appendix [B.2.4](https://arxiv.org/html/2509.21167v1#A2.SS2.SSS4), we show that the $f$-divergences characterized by a larger $(f^{*})^{\prime\prime}(T_{\omega})|_{(\phi^{*},\omega^{*})}$ (i.e., a smaller $f^{\prime\prime}(1)$) are favored by a faster convergence property. This theoretical result is fundamental, as it provides guidelines on the choice of $f$-divergence. In particular, we show that $f_{\text{H}^{2}}^{\prime\prime}(1)<f_{\text{KL}}^{\prime\prime}(1)<f_{\chi^{2}}^{\prime\prime}(1)$, implying that $\text{H}^{2}$ is characterized by a faster convergence near the equilibrium point.

Finally, it is possible to compare different $f$-divergences based on their mode-seeking [28] and saturation [22, 38] properties.
In the unlearning context, mode-seeking divergences tend to overfit by converging to a subset of modes of the distribution, thus reducing the diversity of generated images corresponding to the unlearned concepts, while mode-covering distributions may assign higher probability to the empty areas between the modes. Both JS and H^2 are characterized by medium mode-seeking properties, while KL and Pearson $\chi^{2}$ divergences are mode-covering (the Pearson $\chi^{2}$ is more mode-covering than KL).
The saturation problem also appears for $f$-divergence-based generative frameworks. It refers to the presence of weak gradients in the initial stages of training, when the density ratio is either very large or very small, presenting optimization difficulties. Although JS and $\text{H}^{2}$ divergences are subject to strong saturation, in the considered unlearning scenario, such as in [39], the saturation problem is alleviated because we fine-tune a generative model starting from an already trained DM.

###### Theorem 4.1 .

###### Theorem 4.2 .

## 5 Results

We present a comprehensive empirical evaluation of $f$-DMU. Our experiments are designed to: 1) validate our theoretical analysis of gradient amplitudes, 2) demonstrate the effectiveness of using alternative $f$-divergences compared to the standard MSE-based approach, and 3) assess the framework’s scalability with multi-concepts erasure. In the following, we will refer to the squared Hellinger distance as H^2 and to the Pearson $\chi^{2}$ as $\chi^{2}$. We write $f_{x}$, with $x\in\{s,n,e\}$ to indicate a specific $f$-divergence-based method with anchor concept {superclass, near, and empty}, respectively (e.g., H${}_{s}^{2}$ to indicate an H^2 method with superclass anchor concept).

### 5.1 Implementation Details

The experiments presented in this paper are based on the publicly available Stable Diffusion v1.4 model. We measure how successfully the concept has been unlearned using CLIP Score [40] (CS) and CLIP Accuracy [41] (CA). The lower the CS and CA for the target concept, the better the erasure has been. We also measure how the model maintains the knowledge of non-target concepts by measuring again the CS and CA for unrelated concepts. The higher, the better, which means fewer collateral damages. Kernel Inception Distance [42] (KID) is used to address the generative distribution change of the models. For non-target concepts, a smaller KID value implies good maintenance of the overall quality and coherence of the model. For target concepts, a smaller KID refers to a clean and realistic replacement, while a bigger KID might refer to a more destructive erasure, producing largely incoherent outputs.

### 5.2 Gradient Analysis: Empirical Validation of Theory

Figure: Figure 2: Average gradient amplitude of closed-form-based losses.
Refer to caption: x2.png

Figure: Figure 3: Instability during fine-tuning caused by large gradients. Top: MSE, bottom: H^2.
Refer to caption: x3.png

In Sec. [4.1](https://arxiv.org/html/2509.21167v1#S4.SS1), we have theoretically compared the gradient amplitudes between the different loss functions of $f$-DMU derived from the closed-form expression of divergences.
We numerically demonstrate this behavior with an example in Fig. [2](https://arxiv.org/html/2509.21167v1#S5.F2), where we visualize the average gradient amplitude for MSE (blue), H^2 (orange), and $\chi^{2}$ (green) during the fine-tuning of the unlearning procedure.
While H^2 leads to bounded gradients significantly smaller than the other two losses, $\chi^{2}$ is characterized by meaningfully larger gradients.
This behavior characterizes the losses properties and explains why, for instance, H^2 less probably leads to strange artifacts during fine-tuning. One example is reported in Fig. [3](https://arxiv.org/html/2509.21167v1#S5.F3), where MSE leads the model to suddenly generate the drawing of a man’s face at epoch $400$, while the more limited gradients that characterize H^2 lead to a behavior which is more robust to degenerate image generation.
Another consequence of H^2 bounded gradients is the better prior preservation with respect to MSE and Pearson $\chi^{2}$ (e.g., Figs. [1](https://arxiv.org/html/2509.21167v1#S1.F1) and [4](https://arxiv.org/html/2509.21167v1#S5.F4)).

### 5.3 Comparison of Different f f -Divergences

Figure: Figure 4: Comparison of unlearning the concept "R2D2" using different $f$-divergences with closed-form and variational-form objectives. To illustrate knowledge preservation, the inset in the bottom right of each image shows an image of "C-3PO", which was not an erased concept.
Refer to caption: Pics/fdiv.jpg

##### Closed-form solutions

The results of the closed-form calculations after 500 iterations (left panel of Fig. [4](https://arxiv.org/html/2509.21167v1#S5.F4)) reveal many differences about output quality. H^2 provides a replacement for the target concept that is more visually plausible (losing the general shape of R2D2 and the shape of its arms) than MSE for a more realistic image of C-3PO. Conversely, $\chi^{2}$ completely removes the target concept as R2D2 loses the general shape and arms while retaining a coherent background. MSE is not a very effective removal method since R2D2’s arm and general shape resoundingly resemble the original, and C-3PO is significantly different from the original.
Quantitatively, as detailed in Tabs. [1](https://arxiv.org/html/2509.21167v1#S5.T1), [3](https://arxiv.org/html/2509.21167v1#A3.T3), [6](https://arxiv.org/html/2509.21167v1#A3.T6), [7](https://arxiv.org/html/2509.21167v1#A3.T7) in Appendix [C](https://arxiv.org/html/2509.21167v1#A3), H^2-based methods manage to find very good trade-offs. For example, in Tab. [3](https://arxiv.org/html/2509.21167v1#A3.T3), when erasing "R2D2," H${}_{e}^{2}$ successfully performs the actual erasure with a low CA of $0.20$ and CS of $0.53$ on the target, while retaining generative quality over the non-erased concepts, which are scored by very low KID values for Grumpy Cat ($-0.010$) and Van Gogh ($-0.008$), combined with high CS and CA.
In the same way, H${}_{s}^{2}$ exhibits best quality for unerased concepts with the KID set to $-0.014$ for Snoopy and $-0.031$ for Van Gogh, with R2D2 having been erased, all of them achieving perfect CA. H${}_{s}^{2}$ also boasts good generation quality for the concept it actually erases, as in the case of erasing "Snoopy" with a KID score of $0.127$.
In comparison, MSE also ensures strong erasure of the target concept.
However, MSE methods generally have worse KID scores, meaning a greater influence on generative quality at large. This is visible, for instance, from Tab. [1](https://arxiv.org/html/2509.21167v1#S5.T1), where we can observe that while MSE and H^2 have similar performance on the unlearned concept, H^2 better preserves other concepts. So, while MSE works well for the removal of concepts, it creates more prominent degradation in image generation quality than H^2.

##### Variational solutions

The results of the variational methods after 150 iterations are displayed in the right panel of Fig. [4](https://arxiv.org/html/2509.21167v1#S5.F4). All methods manage to remove the concept from the model so that no generation of "R2D2" can take place. However, they more likely produce artifacts.
As demonstrated by the 150-step results recorded in Tab [5](https://arxiv.org/html/2509.21167v1#A3.T5) in Appendix [C](https://arxiv.org/html/2509.21167v1#A3), the variational approach is more "aggressive" than the closed-form as it causes lower CA in fewer iterations. For example, H${}_{s}^{2}$ erases "Snoopy" to a CS of $0.55$ (same as the closed-form approach) in 150 steps. However, this rapid semantic disruption comes at the cost of generative quality, resulting in a significantly higher KID of $0.295$. In contrast, the 500-step closed-form H${}_{s}^{2}$ method yields a much lower KID of $0.127$.

This highlights a key trade-off:
non-variational (closed-form) methods are best for realistic concept replacement. Instead, variational methods are better for scenarios where the goal is rapid, aggressive semantic removal, and realism of the output is a lower priority.

Table: Table 1: Prior preservation capability of the H^2 method using a superclass anchor, fine-tuned for 2000 steps. We report CLIP Score (CS), CLIP Accuracy (CA), and Kernel Inception Distance (KID). For unerased concepts (prior preservation), higher CS/CA $\uparrow$ and lower KID $\downarrow$ indicate better performance.

##### Hellinger vs. Concept Ablation (CAbl)

We directly compare our H^2-based approach against the widely-used CAbl. Using both methods to replace a specific concept with its superclass, our results show that H^2 offers a superior trade-off between effective concept removal and the visual quality of the resulting generation. This is demonstrated qualitatively in Fig. [5](https://arxiv.org/html/2509.21167v1#S5.F5) and supported by the quantitative results in Tab. [3](https://arxiv.org/html/2509.21167v1#A3.T3) in Appendix [C](https://arxiv.org/html/2509.21167v1#A3).

Figure: Figure 5: Qualitative comparison of superclass-guided unlearning outcomes between CAbl and our Hellinger distance method after 500 iterations.
Refer to caption: Pics/hellivsmse.jpg

For instance, when unlearning “Grumpy Cat” $\rightarrow$ “Cat”, MSE produces a distorted output, while our H^2 approach generates a clean, plausible cat. This trend continues for near-domain replacement (“Van Gogh” $\rightarrow$ “Salvador Dalí”), where the H^2 output is artifact-free, and for empty-prompt erasure (“R2D2” $\rightarrow$ ” ” ), where H^2 generates a realistic t-shirt while MSE’s contains clear artifacts.
The improved visual quality aligns with our theoretical analysis of the gradient dynamics. The bounded gradients of H^2 (as validated in Sec. [5.2](https://arxiv.org/html/2509.21167v1#S5.SS2)) prevent the large, destructive updates that can harm image coherence and introduce visual artifacts. While CAbl is effective at reducing a target concept’s CA, for instance achieving $0.20$ when unlearning “Van Gogh” (Tab. [3](https://arxiv.org/html/2509.21167v1#A3.T3) in Appendix [C](https://arxiv.org/html/2509.21167v1#A3)), it does so at a higher risk of degrading the output’s quality. Our Hellinger-based method achieves a comparable erasure (CA of $0.17$) but with a significantly better visual outcome, as seen in Fig. [5](https://arxiv.org/html/2509.21167v1#S5.F5).

### 5.4 Sequential Multi Concept Erasure

Figure: Figure 6: Qualitative comparison of superclass unlearning with sequential multiple erasure. The first three columns display concepts targeted for erasure. The last three columns show unerased concepts.
Refer to caption: Pics/multierasure.jpg

The observations previously reported for the $f$-DMU framework scale effectively to erasing multiple concepts simultaneously, as demonstrated in Fig. [6](https://arxiv.org/html/2509.21167v1#S5.F6), where we sequentially erase the concepts of Van Gogh, Snoopy, and R2D2 with 500 unlearning steps.
Fig. [6](https://arxiv.org/html/2509.21167v1#S5.F6) shows a trend: images produced by the H^2-based method are more coherent (image-wise and prompt-wise) and have fewer artifacts.

## 6 Conclusions

In this paper, we proposed a unified $f$-divergence-based framework for DM unlearning, which comprises two groups of loss functions: closed-form-based losses and variational form-based losses.
We theoretically analyze the proposed loss functions and numerically evaluate them in different scenarios, demonstrating their relevance for aggressive unlearning or stronger concept preservation, depending on the $f$-divergence and on the closed-form or variational-based derivation.

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## Appendix A Related Work

Figure: Figure 7: $f$-divergence-based Diffusion Model Unlearning Framework. Every pair $(f,\mathbf{c})$ indicates a different objective function based on a specific $f$-divergence and using a specific type of concept $\mathbf{c}$. A list of concepts $[\mathbf{c}_{n},\mathbf{c}_{h},\mathbf{c}_{c},\mathbf{c}_{d}]$ indicates a specific $f$-divergence objective where it is possible to use, respectively, a null, hyper-class, close, and distant concept. JS^∗ indicates that it is the variational formulation of the JS divergence with a change of variable (see Appendix [B.1.3](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS3)).
Refer to caption: x4.png

Figure: Figure 8: Divergence-based methods for diffusion models unlearning. Each sector represents an $f$-divergence, except from the sector described by the triple dots, indicating multiple other divergences. In each sector we can locate the unlearning methods that are based on minimizing such an $f$-divergence. The inner circle (in blue) represents $f$-DMU, which comprises any $f$-divergence. The light blue shade represents the existence of an objective function based on the variational representation of the $f$-divergence. Notably, it exists for any $f$-divergence. For some specific $f$-divergences, the closed-form expression of the loss exists, and this is highlighted by a stripe pattern of darker blue. For the $f$-divergences that are not explicitly represented in the circle (thus identified by the triple dots), $f$-DMU could either exist only in variational representation, or both as variational representation and closed-form (e.g., RKL, Jeffreys).
Refer to caption: x5.png

Machine unlearning studies the fundamental problem of removing specific knowledge learned from a learning algorithm [43, 44]. In the deep learning era, this problem acquires significant meaning, due to the huge deep learning models that are trained on datasets scraped from the Internet.

In principle, machine unlearning can be performed following two different approaches: *exact* and *approximate* unlearning. Exact unlearning is considered the gold standard and consists in re-training the learning algorithm from scratch after having removed from the dataset the subset of data to be forgotten.
Exact machine unlearning is often infeasible for two reasons: *i*) re-training a large model requires significant computational resources and time, *ii*) the training data could be deleted after their usage for training due to memory reasons, thus not being available for re-training.
For these two reasons, researchers are focusing on approximate unlearning methods, which are faster, computationally lighter, and, for generative models, do not require the storage of the training dataset.

Initially, most of the machine unlearning literature was focused on discriminative tasks. For instance, many techniques were developed for classification problems, with the goal of unlearning a specific class or a subset of the dataset [45, 46, 47, 48, 49, 50, 51, 52, 53, 54].
Notably, $f$-SCRUB [54] is, to the best of our knowledge, the only existing $f$-divergence-based machine unlearning framework for classification. $f$-SCRUB generalizes SCRUB [49] using the $f$-divergence.
While for classification there is a large amount of literature related to machine unlearning, most classification techniques either cannot be applied for DMs or have shown poor performance on data generation tasks, thus requiring researchers to develop ad hoc unlearning methods for various generative models, including variational autoencoders and large language models (LLMs) [55, 56, 57].
In particular, [56] propose to maximize the $f$-divergence between template answers and forget data for LLM unlearning.

In this paper, we focus on the task of erasing concepts from DMs, which is a crucial task due to the widespread usage of DMs for image generation [58, 1, 3], and due to their memorization capabilities [59, 8].
In the following, we analyze the related work for DM unlearning, also referred to as "concept erasing" or "concept ablating". We categorize the existing techniques into two main groups: post-processing and fine-tuning techniques.
Additionally, pre-processing techniques [58] are also used. However, they require dataset curation followed by model training, while in this paper we assume to already have a trained model available.

Post-processing techniques target the elimination of unsafe generated images through the usage of filtering or inference guiding. SD [1] adopts a NSFW filter that filters out all the generated images whose embeddings are close to the embeddings of $17$ pre-chosen nudity concepts [12]. In [5], the authors present a method that is applied during inference and that pushes away the generation from unsafe contents. Another example of inference guidance-based approach is SAFREE [13], which can be applied for both image and video generation.
The main drawback of post-processing algorithms is that, when the user has access to the model, the post-processing operation can be removed to allow the model to generate what should have been ablated. Fine-tuning methods, instead, modify the weights of the model, thus being more robust than post-processing approaches when the user is granted access to the unlearned model. In this paper, we focus on fine-tuning methods.

Between fine-tuning methods, the distribution-based approaches, formulated as the minimization of a distance measure between probability distributions, can update any network parameter and work for different models, such as flow-based generative models [20]. This formulation is used as a ground idea in a wide variety of approaches.
Erased Stable Diffusion (ESD) [9] fine-tunes the model by aligning the probability distributions of the model’s output fed with a target and a null concept. To achieve that, the authors include in the loss function a classifier-free guidance-based term.
Concept Ablation (CA) [10] minimizes the MSE between the model’s output when the model is fed with a target concept and an anchor hyper-class concept. CA’s loss function corresponds to $f$-DMU when using the closed-form expression of the KL divergence(^1^11Here we are referring to the main loss term of both approaches, excluding considerations on the prior preservation terms.). This can also be observed from Figure [7](https://arxiv.org/html/2509.21167v1#A1.F7), where we schematically depict $f$-DMU and highlight its relationship with existing approaches.
Concept-SemiPermeable Membrane (SPM) [60] propose the usage of adapters that can be shared across different models and that rely on a novel Latent Anchoring (LA) fine-tuning strategy. The proposed loss comprises two terms: *i*) the first term coincides with the ESD loss and matches target and anchor concepts, *ii*) the second term (anchoring loss) minimizes the MSE to preserve the consistency of distant (in the CLIP space) concepts.
Reliable concept erasing via Lightweight Erasers (RECELER) [11] introduces a new component into the neural network, the eraser, that acts on the cross-attention layers of the U-Net, and that is trained as in [9], additionally including a concept-localized regularization term, to ensure the effective model performance on local concepts, while leveraging adversarial prompt learning to ensure robustness.
Fine-grained attenuation for diffusion erasure (FADE) [16] extracts the concepts that are semantically close to the concept to erase and is trained by minimizing an MSE loss that shifts the target concept to semantically close concepts and preserves the model generation on such a set.

Unified Concept Editing (UCE) [17] is a closed-form parameter editing method that builds upon [61] and [62]. UCE updates the cross-attention parameters in closed-form, ensuring a fast computational time. While the loss function is still formulated as the MSE, differently from the KL-based approaches previously presented, UCE minimizes the MSE directly at the cross-attention parameters level.
Similarly, RECE [18] improves UCE by taking into consideration that the weights modification of UCE is not robust to adversarial prompts.
Mass concept erasure (MACE) [19] extends UCE [17] 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 these methods can only be applied to attention weights and specific architectures.

Recently, some methods relying on a double optimization problem have been proposed. Most of them still rely on the KL-minimization idea previously analyzed.
AdvUnlearn [63] is an adversarial unlearning method which modifies the text encoder. Although the unlearned text encoder can be used as a plug-and-play robust module for various diffusion models, it needs a retain set.
EAP [14] is trained with a loss comprising two terms that are being optimized over the model weights and the adversarial concept: the first term minimizes the distance from the concept to erase to the neutral concept, while the second term minimizes the distance from the adversarial concept in the original network to the adversarial concept in the new model.
Adaptive Guided Erasure (AGE) [15], proposes a bi-level optimization framework that dynamically selects the optimal anchor concepts. In [64], the authors propose a min-max optimization problem that, in addition to minimizing the MSE between target and anchor concepts, includes a strategy for finding the optimal pruning method.
The techniques previously discussed rely on the standard KL minimization approach.
More concurrent with our work, Domain Correction DoCo [21] starts from CA [10] and, in a membership inference attack fashion, replaces the MSE with the JS divergence. DoCo’s loss function corresponds to $f$-DMU using the variational representation of the JS divergence when $\mathbf{c^{*}}$ is chosen as the hyper-class concept(^2^22Here we are referring to the main loss term of both approaches, excluding considerations on the prior preservation terms.).

Figure [8](https://arxiv.org/html/2509.21167v1#A1.F8) represents a schematic view of divergence-based methods for diffusion models unlearning. It shows how the KL divergence is largely adopted for diffusion models unlearning, while other divergence measures have been completely ignored.
Our framework fills this gap, highlighting the potential of using different divergences.
We hope that our work will encourage researchers to study techniques for unlearning that are not based on the KL divergence. In fact, we purposely leave empty white spaces in Figure [8](https://arxiv.org/html/2509.21167v1#A1.F8) corresponding to methods that are different from $f$-DMU but leverage divergences that are different from the KL.

##### f f -Divergence

$f$-divergence-based methods have been effectively employed for the design of objective functions in a large number of applications, such as classification [65], classification with label noise [66, 67], generation [38], semi-supervised learning [68], mutual information estimation [69], and distillation [70].
For diffusion models, the $f$-divergence has been used for distillation and alignment tasks, but not for solving unlearning problems.
In [71], the author uses an $f$-divergence-based entropy regularization term for fine-tuning diffusion models.
In [72], the authors propose an $f$-divergence-based framework for text-to-image models alignment. In [39], the authors present an $f$-divergence-based approach for variational score distillation, to accelerate the diffusion models sampling process.
For machine unlearning, apart from [54] for classification and [56] for LLMs, general $f$-divergence frameworks have not been proposed. Specific $f$-divergences have only been applied as evaluation metrics. One example is the specific case of the JS divergence [47, 73, 74].

In conclusion, to the best of our knowledge, we propose the first diffusion model unlearning framework based on $f$-divergences.

## Appendix B Proofs

### B.1 f f -Divergence-Based Diffusion Models Unlearning

To motivate ([3](https://arxiv.org/html/2509.21167v1#S3.E3)), it is sufficient to notice that we are tackling the goal of minimizing a divergence measure between the model output distribution conditioned on the target concept $\mathbf{c}^{*}$ and the anchor concept $\mathbf{c}$, which can be rewritten as

$$ $\displaystyle\min_{\hat{\Phi}}\text{d}(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}),p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})),$ (12) $$

which is satisfied $\forall t$ when the output distribution of the unlearned model $\hat{\Phi}$, conditioned on the concept to be unlearned $\mathbf{c}^{*}$ coincides with the distribution of the original model conditioned on the anchor concept $\mathbf{c}$.

In the seminal work of [10], ([12](https://arxiv.org/html/2509.21167v1#A2.E12)) was obtained (using the KL divergence as divergence metric) by imposing as target goal the equivalence between the entire DM trajectories:

$$ $\displaystyle D_{KL}(p_{\Phi}(\mathbf{x}_{(0\dots T)}|\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{(0\dots T)}|\mathbf{c}^{*})).$ (13) $$

We report the steps to go from ([13](https://arxiv.org/html/2509.21167v1#A2.E13)) to ([12](https://arxiv.org/html/2509.21167v1#A2.E12)) for completeness, although most of them can also be found in [10]. Then, we conclude by generalizing such a result using the $f$-divergence.

Following [10], ([13](https://arxiv.org/html/2509.21167v1#A2.E13)) can be rewritten as

$$ $\displaystyle D_{KL}(p_{\Phi}(\mathbf{x}_{(0\dots T)}|\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{(0\dots T)}|\mathbf{c}^{*}))$ $\displaystyle=\operatorname{\mathbb{E}}_{p(\mathbf{x}_{(0\dots T)}|\mathbf{c})}\left[\log\frac{\prod_{t=1}^{T}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})p_{\Phi}(\mathbf{x}_{T})}{\prod_{t=1}^{T}p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})p_{\hat{\Phi}}(\mathbf{x}_{T})}\right]$ $\displaystyle=\sum_{\hat{t}=1}^{T}\operatorname{\mathbb{E}}_{p(\mathbf{x}_{(0\dots T)}|\mathbf{c})}\left[\log\frac{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}\right],$ (14) $$

where the first step is obtained using the definition of KL divergence and the Markovianity of diffusion processes. In fact,

$$ $\displaystyle p(\mathbf{x}_{(0\dots T)}|\mathbf{c})=$ $\displaystyle p(\mathbf{x}_{T}|\mathbf{c})\cdot p(\mathbf{x}_{T-1}|\mathbf{x}_{T},\mathbf{c})\cdot p(\mathbf{x}_{T-2}|\mathbf{x}_{T-1},\mathbf{x}_{T},\mathbf{c})\cdot p(\mathbf{x}_{T-3}|\mathbf{x}_{T-2},\mathbf{x}_{T-1},\mathbf{x}_{T},\mathbf{c})\cdots$ (15) $\displaystyle=$ $\displaystyle p(\mathbf{x}_{T}|\mathbf{c})\cdot p(\mathbf{x}_{T-1}|\mathbf{x}_{T},\mathbf{c})\cdot p(\mathbf{x}_{T-2}|\mathbf{x}_{T-1},\mathbf{c})\cdot p(\mathbf{x}_{T-3}|\mathbf{x}_{T-2},\mathbf{c})\cdots$ (16) $\displaystyle=$ $\displaystyle p(\mathbf{x}_{T})\cdot p(\mathbf{x}_{T-1}|\mathbf{x}_{T},\mathbf{c})\cdot p(\mathbf{x}_{T-2}|\mathbf{x}_{T-1},\mathbf{c})\cdot p(\mathbf{x}_{T-3}|\mathbf{x}_{T-2},\mathbf{c})\cdots$ (17) $\displaystyle=$ $\displaystyle\prod_{t=1}^{T}p(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})p(\mathbf{x}_{T}),$ (18) $$

where the second equality is a consequence of the Markovianity of DMs, and the third equality derives from the fact that $\mathbf{x}_{T}\sim\mathcal{N}(0,1)$, which is independent from $\mathbf{c}$.
From ([B.1](https://arxiv.org/html/2509.21167v1#A2.Ex1)), it is possible to study the generic term corresponding to a particular time step $\hat{t}$

$$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{0}\dots\mathbf{x}_{T})}\left[\log\frac{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}\right]$ $\displaystyle=\int_{\mathbf{x}_{(0\dots T)}}\prod_{t=1}^{T}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})p(\mathbf{x}_{T})\log\frac{p_{\Phi}(\mathbf{x_{\hat{t}-1}|\mathbf{x}_{\hat{t}}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x_{\hat{t}-1}|\mathbf{x}_{\hat{t}}},\mathbf{c}^{*})}d\mathbf{x}_{(0\dots T)}$ $\displaystyle=\int_{\mathbf{x}_{(\hat{t}\dots T)}}p_{\Phi}(\mathbf{x}_{(\hat{t}\dots T)}|\mathbf{c})\Biggl[\int_{\mathbf{x}_{(0\dots\hat{t}-1)}}\prod_{t=1}^{\hat{t}}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})\log\frac{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}d\mathbf{x}_{(\hat{t}-1\dots 0)}\Biggr]d\mathbf{x}_{(\hat{t}\dots T)}$ $\displaystyle=\int_{\mathbf{x}_{(\hat{t}\dots T)}}p_{\Phi}(\mathbf{x}_{(\hat{t}\dots T)}|\mathbf{c})\Biggl[\int_{\mathbf{x}_{(0\dots\hat{t}-1)}}\Biggl(\prod_{t=1}^{\hat{t}-1}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})\Biggr)p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})$ $\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cdot\log\frac{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}d\mathbf{x}_{(\hat{t}-1\dots 0)}\Biggr]d\mathbf{x}_{(\hat{t}\dots T)}$ $\displaystyle=\int_{\mathbf{x}_{\hat{t}}}p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\log\frac{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}$ $\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\cdot\underbrace{\Biggl[\int_{\mathbf{x}_{(0\dots\hat{t}-2)}}\prod_{t=1}^{\hat{t}-1}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})d\mathbf{x}_{(\hat{t}-2\dots 0)}\Biggr]}_{=1}d\mathbf{x}_{\hat{t}-1}\Biggr]d\mathbf{x}_{\hat{t}}$ $\displaystyle=\int_{\mathbf{x}_{\hat{t}}}p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\log\frac{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}{p_{\hat{\Phi}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}d\mathbf{x}_{\hat{t}-1}\Biggr]d\mathbf{x}_{\hat{t}}$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Biggl[D_{KL}\left(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})\right)\Biggr],$ (19) $$

achieved using the fact that the integral over $p_{\Phi}(\mathbf{x}_{(\hat{t}\dots T)}|\mathbf{c})$ corresponds to the integral over $p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})$ because it contains all the information about the image versions from time step $T$ (pure noise) to the current step $\hat{t}$.
Since $p_{\Phi}$ is a Gaussian distribution if the diffusion step sizes are small enough, in [10] the authors propose to use the objective function

$$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Biggl[\eta\Bigl|\Bigl|\Phi(\mathbf{x}_{t},\mathbf{c},t)-\hat{\Phi}(\mathbf{x}_{t},\mathbf{c}^{*},t)\Bigr|\Bigr|_{2}^{2}\Biggr],$ (20) $$

which corresponds to the MSE between the two DMs distributions.

The minimization of the objective function in ([B.1](https://arxiv.org/html/2509.21167v1#A2.Ex3)) w.r.t. $\hat{\Phi}$ can be generalized as the minimization of an $f$-divergence between the same probability distributions. Instead of solving ([20](https://arxiv.org/html/2509.21167v1#A2.E20)), we solve

$$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Biggl[D_{f}\left(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})\right)\Biggr].$ (21) $$

To minimize ([21](https://arxiv.org/html/2509.21167v1#A2.E21)), we propose two different approaches:

- •
To compute the closed-form expression of a divergence between Gaussian distributions: in Sec. [B.1.1](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS1), we provide the closed-form expressions of some $f$-divergences between Gaussian distributions. Each of them leads to a specific loss function.
- •
To use the variational representation of the $f$-divergence: in Sec. [B.1.3](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS3) we report the variational representations of the $f$-divergences used in the experiments.

#### B.1.1 Closed-Form Expressions of the
Objective Functions for Specific f f -Divergences

In the following, we report the main closed-form expressions of $f$-divergences between Gaussian distributions. For each of them, we will provide the information of whether it is a bounded or unbounded $f$-divergence, as this will be related to the boundedness and unboundedness of the gradients. To check the boundedness of $f$-divergences, it is possible to use Theorem [B.1](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem1).
It is first necessary to define

$$ $\displaystyle f^{\star}(t)\triangleq tf\left(\frac{1}{t}\right),$ (22) $$

for all $t>0$. Furthermore, by definition

$$ $\displaystyle f^{\star}(0)=\lim_{u\to\infty}\frac{f(u)}{u}.$ (23) $$

###### Theorem B.1 (Range of values) .

(see [75])
Let $P$ and $Q$ be two probability distributions. Then, the range of an $f$-divergence is given by

$$ $\displaystyle 0\leq D_{f}(P||Q)\leq f(0)+f^{\star}(0).$ (24) $$

In the following, we use $p(x)$ and $q(x)$ as two multivariate normal distributions:

$$ $\displaystyle p(x)$ $\displaystyle=\frac{1}{\sqrt{(2\pi)^{d}\text{det}(\Sigma_{P})}}\exp\left\{-\frac{1}{2}(x-\mu_{P})^{T}\Sigma_{P}^{-1}(x-\mu_{P})\right\}$ (25) $\displaystyle q(x)$ $\displaystyle=\frac{1}{\sqrt{(2\pi)^{d}\text{det}(\Sigma_{Q})}}\exp\left\{-\frac{1}{2}(x-\mu_{Q})^{T}\Sigma_{Q}^{-1}(x-\mu_{Q})\right\}.$ (26) $$

Kullback-Leibler divergence:
It is easy to show that the KL divergence between two multivariate Normal distributions reads as

$$ $\displaystyle D_{KL}(P||Q)=\frac{1}{2}\log\left(\frac{\text{det}\Sigma_{Q}}{\text{det}\Sigma_{P}}\right)-\frac{d}{2}+\frac{1}{2}\left[\text{Tr}\left(\Sigma_{Q}^{-1}\Sigma_{P}\right)+(\mu_{P}-\mu_{Q})^{T}\Sigma_{Q}^{-1}(\mu_{P}-\mu_{Q})\right].$ (27) $$

In the scalar case, ([27](https://arxiv.org/html/2509.21167v1#A2.E27)) reduces to

$$ $\displaystyle D_{KL}(P||Q)=\frac{1}{2}\log\frac{\sigma_{Q}^{2}}{\sigma_{P}^{2}}-\frac{1}{2}+\frac{1}{2}\left(\frac{(\mu_{P}-\mu_{Q})^{2}}{\sigma_{Q}^{2}}+\frac{\sigma_{P}^{2}}{\sigma_{Q}^{2}}\right).$ (28) $$

It is well-known that the KL divergence is unbounded.

The training objective becomes the one proposed by [10], i.e., the MSE between the two diffusion models outputs.

Jeffreys divergence:
From the definition of Jeffreys divergence, we obtain

$$ $\displaystyle D_{J}(P||Q)=$ $\displaystyle\frac{1}{2}\log\left(\frac{\text{det}(\Sigma_{Q})}{\text{det}(\Sigma_{P})}\right)+\frac{1}{2}\log\left(\frac{\text{det}(\Sigma_{P})}{\text{det}(\Sigma_{Q})}\right)-d$ $\displaystyle+\frac{1}{2}\Bigl[\text{Tr}\left(\Sigma_{P}^{-1}\Sigma_{Q}\right)+\text{Tr}\left(\Sigma_{Q}^{-1}\Sigma_{P}\right)$ $\displaystyle+(\mu_{P}-\mu_{Q})^{T}\Sigma_{Q}^{-1}(\mu_{P}-\mu_{Q})+(\mu_{Q}-\mu_{P})^{T}\Sigma_{P}^{-1}(\mu_{Q}-\mu_{P})\Bigr],$ (29) $$

which simplifies to

$$ $\displaystyle D_{J}(P||Q)=$ $\displaystyle-d+\frac{1}{2}\Bigl[\text{Tr}\left(\Sigma_{Q}^{-1}\Sigma_{P}\right)+\text{Tr}\left(\Sigma_{P}^{-1}\Sigma_{Q}\right)$ $\displaystyle+(\mu_{P}-\mu_{Q})^{T}\left(\Sigma_{Q}^{-1}+\Sigma_{P}^{-1}\right)\left(\mu_{P}-\mu_{Q}\right)\Bigr].$ (30) $$

In the scalar case, ([B.1.1](https://arxiv.org/html/2509.21167v1#A2.Ex13)) becomes

$$ $\displaystyle D_{J}(P||Q)=-1+\frac{1}{2}\left[\frac{\sigma_{P}^{4}+\sigma_{Q}^{4}}{\sigma_{Q}^{2}\sigma_{P}^{2}}+\frac{(\mu_{P}-\mu_{Q})^{2}}{\sigma_{P}^{2}\sigma_{Q}^{2}}(\sigma_{P}^{2}+\sigma_{Q}^{2})\right].$ (31) $$

Since the Jeffreys divergence is the sum of two KL divergences, it is unbounded.

Under the assumption $\sigma_{P}=\sigma_{Q}=\sigma$, the training objective becomes

$$ $\displaystyle\mathcal{J}_{J}(\mathbf{x},\mathbf{c},\mathbf{c^{*}})=\operatorname{\mathbb{E}}_{\epsilon,\mathbf{x},\mathbf{c}^{*},\mathbf{c},t}\Biggl[\frac{\omega_{t}}{\sigma^{2}}||\Phi(\mathbf{x}_{t},\mathbf{c},t)-\hat{\Phi}(\mathbf{x}_{t},\mathbf{c}^{*},t)||_{2}^{2}\Biggr].$ (32) $$

Squared Hellinger distance:
The squared Hellinger distance can be expressed as a function of the Bhattacharyya coefficient ($BC(P,Q)$) as $H^{2}(P,Q)=1-BC(P,Q)$. From [76], the squared Hellinger distance between two multivariate Normal distributions reads as

$$ $\displaystyle H^{2}(P,Q)=1-\frac{\text{det}(\Sigma_{P})^{\frac{1}{4}}\text{det}(\Sigma_{Q})^{\frac{1}{4}}}{\text{det}(\frac{\Sigma_{P}+\Sigma_{Q}}{2})^{\frac{1}{2}}}\exp\left\{-\frac{1}{8}(\mu_{P}-\mu_{Q})^{T}\left(\frac{\Sigma_{P}+\Sigma_{Q}}{2}\right)^{-1}(\mu_{P}-\mu_{Q})\right\}.$ (33) $$

In the scalar case, ([33](https://arxiv.org/html/2509.21167v1#A2.E33)) becomes

$$ $\displaystyle H^{2}(P,Q)=1-\frac{\sqrt{\sigma_{P}\sigma_{Q}}}{\sqrt{\frac{\sigma_{P}^{2}+\sigma_{Q}^{2}}{2}}}\exp{-\frac{1}{4}\frac{(\mu_{P}-\mu_{Q})^{2}}{\sigma_{P}^{2}+\sigma_{Q}^{2}}}.$ (34) $$

The H^2 is bounded. More precisely,

$$ $\displaystyle 0\leq H^{2}(P,Q)\leq 1,$ (35) $$

which can be proven using the range of values theorem [75]. A quick intuition of the correctness of ([35](https://arxiv.org/html/2509.21167v1#A2.E35)) can be derived by noticing that when $\mu_{P}=\mu_{Q}$ and $\sigma_{P}=\sigma_{Q}$, $H^{2}(P,Q)=0$, while $H^{2}(P,Q)\to 1$ when $(\mu_{P}-\mu_{Q})^{2}\to\infty$.

Assuming $\sigma_{P}=\sigma_{Q}=\sigma$, the training objective becomes

$$ $\displaystyle\mathcal{J}_{H}(\mathbf{x},\mathbf{c},\mathbf{c^{*}})=\operatorname{\mathbb{E}}_{\epsilon,\mathbf{x},\mathbf{c}^{*},\mathbf{c},t}\Biggl[-\omega_{t}\exp\left\{-\frac{1}{8\sigma^{2}}||\Phi(\mathbf{x}_{t},\mathbf{c},t)-\hat{\Phi}(\mathbf{x}_{t},\mathbf{c}^{*},t)||_{2}^{2}\right\}\Biggr].$ (36) $$

$\chi^{2}$ divergence: The $\chi^{2}$ divergence is defined as

$$ $\displaystyle\chi^{2}(P||Q)=\int_{\mathcal{X}}\frac{(p(x)-q(x))^{2}}{q(x)}dx=\int\frac{p(x)^{2}}{q(x)}dx-1.$ (37) $$

Substituting the expressions of the pdfs of Gaussian random variables, it becomes

$$ $\displaystyle\chi^{2}(P||Q)$ $\displaystyle=\int\frac{\sigma_{Q}}{\sqrt{2\pi}\sigma_{P}^{2}}e^{-\frac{1}{2}\left(\frac{2(x-\mu_{P})^{2}}{\sigma_{P}^{2}}-\frac{(x-\mu_{Q})^{2}}{\sigma_{Q}^{2}}\right)}dx-1$ (38) $\displaystyle=\frac{\sigma_{Q}^{2}}{\sigma_{P}\sqrt{2\sigma_{Q}^{2}-\sigma_{P}^{2}}}\exp\left\{{\frac{(\mu_{P}-\mu_{Q})^{2}}{2\sigma_{Q}^{2}-\sigma_{P}^{2}}}\right\}-1.$ (39) $$

The expression in ([39](https://arxiv.org/html/2509.21167v1#A2.E39)) holds only for $2\sigma_{Q}^{2}>\sigma_{P}^{2}$.

The $\chi^{2}$ divergence is unbounded because $f^{\star}(0)=\infty$.

Under the assumption that $\sigma_{P}=\sigma_{Q}=\sigma$,

$$ $\displaystyle\mathcal{J}_{\chi^{2}}(\mathbf{x},\mathbf{c},\mathbf{c^{*}})=\operatorname{\mathbb{E}}_{\epsilon,\mathbf{x},\mathbf{c}^{*},\mathbf{c},t}\Biggl[\omega_{t}\exp\left\{\frac{1}{\sigma^{2}}||\Phi(\mathbf{x}_{t},\mathbf{c},t)-\hat{\Phi}(\mathbf{x}_{t},\mathbf{c}^{*},t)||_{2}^{2}\right\}\Biggr].$ (40) $$

###### Theorem B.1 (Range of values) .

#### B.1.2 Gradient Analysis

Figure: Figure 9: Gradients comparison as a function of the MSE. All the functions are plotted up to a multiplicative constant.
Refer to caption: x6.png

In this section, we analyze the gradients of the different closed-form loss functions presented in Section [B.1.1](https://arxiv.org/html/2509.21167v1#A2.SS1.SSS1). Let $\hat{\Phi}$ be a function of the parameters vector $\phi$, then

$$ $\displaystyle\sum_{i=1}^{n}\frac{\partial\mathcal{J}_{f}(\Phi,\hat{\Phi})}{\partial\phi}=$ $\displaystyle\begin{cases}\sum_{i=1}^{n}\nabla_{\phi}MSE(\Phi(\mathbf{x}_{i},\mathbf{c},i),\hat{\Phi}(\mathbf{x}_{i},\mathbf{c}^{*},i))&\text{ for KL}\\ \sum_{i=1}^{n}e^{-MSE(\Phi(\mathbf{x}_{i},\mathbf{c},i),\hat{\Phi}(\mathbf{x}_{i},\mathbf{c}^{*},i))}\nabla_{\phi}MSE(\Phi(\mathbf{x}_{i},\mathbf{c},i),\hat{\Phi}(\mathbf{x}_{i},\mathbf{c}^{*},i))&\text{ for }H^{2}\\ \sum_{i=1}^{n}e^{MSE(\Phi(\mathbf{x}_{i},\mathbf{c},i),\hat{\Phi}(\mathbf{x}_{i},\mathbf{c}^{*},i))}\nabla_{\phi}MSE(\Phi(\mathbf{x}_{i},\mathbf{c},i),\hat{\Phi}(\mathbf{x}_{i},\mathbf{c}^{*},i))&\text{ for }\chi^{2}\end{cases}$ (41) $$

Since all the losses depend on the gradient of the MSE, it is easy to compare them:

- •
The gradients of H^2 weight the MSE with a negative exponential. Thus, when $MSE\to 0$, the gradients of H^2 coincide with the gradients of MSE. Meanwhile, when $MSE\to\infty$, the gradients of H^2 tend to 0.
- •
For $\chi^{2}$ divergence, the gradients are weighted versions of the MSE, where the weight is a positive exponential. When $MSE\to 0$, the gradients coincide with the MSE gradients, while when $MSE\to\infty$, the gradients grow to $\infty$.

Therefore,

$$ $\displaystyle\left|\frac{\partial\mathcal{J}_{H^{2}}(\Phi,\hat{\Phi})}{\partial\phi}\right|\leq\left|\frac{\partial\mathcal{J}_{KL}(\Phi,\hat{\Phi})}{\partial\phi}\right|\leq\left|\frac{\partial\mathcal{J}_{\chi^{2}}(\Phi,\hat{\Phi})}{\partial\phi}\right|.$ (42) $$

To provide a visual representation, we explicit the dependence of the MSE gradient on the difference between $\Phi$ and $\hat{\Phi}$:

$$ $\displaystyle\nabla_{\phi}\mathcal{J}_{KL}$ $\displaystyle=-2(\Phi-\hat{\Phi})\nabla_{\phi}\hat{\Phi}$ (43) $\displaystyle\nabla_{\phi}\mathcal{J}_{H^{2}}$ $\displaystyle=-2e^{-(\Phi-\hat{\Phi})^{2}}(\Phi-\hat{\Phi})\nabla_{\phi}\hat{\Phi}$ (44) $\displaystyle\nabla_{\phi}\mathcal{J}_{\chi^{2}}$ $\displaystyle=-2e^{(\Phi-\hat{\Phi})^{2}}(\Phi-\hat{\Phi})\nabla_{\phi}\hat{\Phi}$ (45) $$

and we depict in Figure [9](https://arxiv.org/html/2509.21167v1#A2.F9) their behaviors, up to the multiplicative factor $\nabla_{\phi}\hat{\Phi}$.

The two opposite behaviors of H^2 and $\chi^{2}$ divergences lead, in practice, to two very different behaviors during training.
It appears that the $\chi^{2}$ divergence focuses more on those cases where the generated images with target and anchor concept are significantly different. This behavior is similar to the one obtained using the KL divergence, with the difference that the gradients of the $\chi^{2}$ divergence grow significantly faster. Meanwhile, the loss derived from the closed-form of the $\text{H}^{2}$ divergence shows a peculiar behavior as it is less effected from outliers: it weights more the samples that have an intermediate MSE value.

Notably, both the KL and $\chi^{2}$ divergences are unbounded and have unbounded gradients. Differently, the $\text{H}^{2}$ distance leads to a bounded loss function and bounded gradients.

#### B.1.3 Variational Representation

In this section, we report the variational-based version of $f$-DMU.

For simplicity in the notation, we will write $T(\Phi)$ to indicate $T(\Phi(\mathbf{x}_{t},\mathbf{c}))$.
The variational representation of the $f$-divergence would rewrite ([3](https://arxiv.org/html/2509.21167v1#S3.E3)) as

$$ $\displaystyle\min_{\hat{\Phi}}$ $\displaystyle\>\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\sup_{T}\Bigl\{\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[T(\Phi)\Bigr]-\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\Bigl[f^{*}(T(\hat{\Phi}))\Bigr]\Bigr\}\Bigr].$ (46) $$

Instead, we propose to solve the following problem:

$$ $\displaystyle\min_{\hat{\Phi}}D_{f}\left(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})\right)\quad\forall\mathbf{x}_{t},$ (47) $$

which corresponds to solving the following objective function

$$ $\displaystyle\min_{\hat{\Phi}}\max_{T}$ $\displaystyle\>\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[T(\Phi)\Bigr]-\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\left[f^{*}(T(\hat{\Phi}))\right]\Bigr].$ (48) $$

The result above can be shown by rewriting ([47](https://arxiv.org/html/2509.21167v1#A2.E47)) as

$$ $\displaystyle\min_{\hat{\Phi}}D_{f}\left(p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})||p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})\right)\quad\forall\mathbf{x}_{t},$ (49) $$

which is possible because $p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})$ is Gaussian, thus ensuring to avoid degenerate cases in which $p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})$ zeroes some areas of $p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})$ and $p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})$ that would be different, that would allow the divergence to be zero even for $p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})\neq p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})$.
Now, the variational representation of ([49](https://arxiv.org/html/2509.21167v1#A2.E49)) comprises the term

$$ $\displaystyle\int_{\mathbf{x}_{t-1},\mathbf{x}_{t}}p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})T(\mathbf{x}_{t-1})d\mathbf{x}_{t-1}d\mathbf{x}_{t}=$ $\displaystyle=\int_{\mathbf{x}_{t}}p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})\Biggl[\int_{\mathbf{x}_{t-1}}p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})T(\mathbf{x}_{t-1})d\mathbf{x}_{t-1}\Biggr]d\mathbf{x}_{t}$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[T(\mathbf{x}_{t-1})\Bigr]\Bigr].$ (50) $$

We can obtain a similar result for $f^{*}(T(\cdot))$. Thus, for the linearity property of the expectation ([48](https://arxiv.org/html/2509.21167v1#A2.E48)) follows.

Following the work in [77], we assume $T(x)=g_{f}(V_{\omega}(x))$, to respect the domain of the Fenchel conjugate function, i.e., $V:\mathcal{X}\to\mathbb{R}$ and $g_{f}:\mathbb{R}\to dom_{f^{*}}$. Thus, ([48](https://arxiv.org/html/2509.21167v1#A2.E48)) becomes

$$ $\displaystyle\min_{\hat{\Phi}}\max_{V}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[g_{f}(V(\Phi))\Bigr]-\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\Bigl[f^{*}(g_{f}(V(\Phi)))\Bigr]\Bigr].$ (51) $$

The generator functions and corresponding Fenchel conjugates of the $f$-divergences used in this paper are reported in Tab. [2](https://arxiv.org/html/2509.21167v1#A2.T2). In addition, we report the corresponding output activation functions $g_{f}$.

**Table 2: $f$-divergences table.**
| Name | $f(u)$ | $f^{*}(t)$ | $g_{f}$ |
| --- | --- | --- | --- |
| Kullback-Leibler | $u\log(u)$ | $e^{t-1}$ | $v$ |
| Reverse Kullback-Leibler | $-\log(u)$ | $-1-\log(-t)$ | $-e^{-v}$ |
| Squared Hellinger | $(\sqrt{u}-1)^{2}$ | $\frac{t}{1-t}$ | $1-e^{-v}$ |
| Jensen-Shannon | $u\log(u)-(u+1)\log(\frac{u+1}{2})$ | $-\log(2-e^{t})$ | $\log(2)-\log(1+e^{-v})$ |
| GAN | $u\log(u)-(u+1)\log(u+1)$ | $-\log(1-e^{t})$ | $-\log(1+e^{-v})$ |
| $\chi^{2}$ | $(u-1)^{2}$ | $\frac{1}{4}t^{2}+t$ | $v$ |
| Jeffreys | $(u-1)\log u$ | $W(e^{1-t})+\frac{1}{W(e^{1-t})}+t-2$ | $v$ |
| Total Variation | $\frac{1}{2}|u-1|$ | $t$ | $\frac{1}{2}\tanh(v)$ |

##### KL divergence

The loss function corresponding to the variational representation of the KL divergence reads as

$$ $\displaystyle\min_{\hat{\Phi}}\max_{V}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[V(\Phi)\Bigr]-\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\Bigl[e^{V(\Phi)-1}\Bigr]\Bigr].$ (52) $$

##### H 2 distance

The loss function corresponding to the variational representation of the squared Hellinger distance reads as

$$ $\displaystyle\min_{\hat{\Phi}}\max_{V}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[1-e^{-V(\Phi)}\Bigr]-\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\Bigl[\frac{1-e^{-V(\Phi)}}{e^{-V(\Phi)}}\Bigr]\Bigr].$ (53) $$

##### JS divergence

The loss function corresponding to the variational representation of the JS divergence reads as

$$ $\displaystyle\min_{\hat{\Phi}}\max_{V}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[-\log(1+e^{-V(\Phi)})\Bigr]+\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\Bigl[\log\Bigl(\frac{e^{-V(\Phi)}}{1+e^{-V(\Phi)}}\Bigr.)\Bigr]\Bigr].$ (54) $$

Using the change of variable $T=-\log(1+e^{-v})=\log(D)$, we obtain

$$ $\displaystyle\min_{\hat{\Phi}}\max_{V}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c})}\Bigl[-\log(1+e^{-V(\Phi)})\Bigr]+\operatorname{\mathbb{E}}_{p_{\hat{\Phi}}(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{c}^{*})}\Bigl[\log\Bigl(\frac{e^{-V(\Phi)}}{1+e^{-V(\Phi)}}\Bigr.)\Bigr]\Bigr].$ (55) $$

Thus, we observe that DoCo’s objective function corresponds to the variational representation of the JS divergence after a specific change of variable.

### B.2 Local Convergence Study

In this section, we present in Sec. [B.2.1](https://arxiv.org/html/2509.21167v1#A2.SS2.SSS1) the necessary preliminaries to develop the local convergence study, and in Sec. [B.2.3](https://arxiv.org/html/2509.21167v1#A2.SS2.SSS3) the proofs of the local convergence of the algorithm proposed.

#### B.2.1 Preliminaries

Let us consider a system consisting of variables $\mathbf{\theta}\in\mathbf{\Theta}\subseteq\mathbb{R}^{n}$. Let its time derivative be defined by the vector field $v(\mathbf{\phi})$:

$$ $\displaystyle\dot{\mathbf{\phi}}=v(\phi),$ (56) $$

where $v:\mathbf{\Phi}\to\mathbb{R}^{n}$ is a locally Lipschitz mapping from a domain $\mathbf{\Phi}$ into $\mathbb{R}^{n}$.

Assume $\phi^{*}$ is an equilibrium point of the system in ([56](https://arxiv.org/html/2509.21167v1#A2.E56)) (i.e., $v(\phi^{*})=0$). Let $\phi_{t}$ be the state of the system at time $t$.

###### Definition B.2 .

(see [37])
The equilibrium point $\phi^{*}$ for the system defined in ([56](https://arxiv.org/html/2509.21167v1#A2.E56)) is

- •
stable if for each $\epsilon>0$, there is $\delta=\delta(\epsilon)>0$ such that
$\displaystyle||\phi_{0}-\phi^{*}||<\delta\Longrightarrow\forall t\geq 0,||\phi_{t}-\phi^{*}||<\epsilon$
(57)
- •
unstable if not stable
- •
asymptotically stable if it is stable and $\delta>0$ can be chosen such that
$\displaystyle||\phi_{0}-\phi^{*}||<\delta\Longrightarrow\lim_{t\to\infty}\phi_{t}=\phi^{*}$
(58)
- •
exponentially stable if it is asymptotically stable and $\delta,k,\lambda>0$ can be chosen such that:
$\displaystyle||\phi_{0}-\phi^{*}||<\delta\Longrightarrow||\phi_{t}||\leq k||\phi_{0}||\exp{(-\lambda t)}.$
(59)

The definition of stability can imply two different situations: 1) the systems converges to the equilibrium point, 2) the system is restricted to a ball of radius $\epsilon$ from the equilibrium. The asymptotic stability, instead, guarantees that the system reaches the equilibrium, when the initial condition belongs to a $\delta$ neighborhood of the equilibrium.

Theorem [B.3](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem3) associates the stability of a non-linear system in the equilibrium point with its Jacobian, allowing to study the non-linear system by evaluating the eigenvalues of J.

###### Theorem B.3 .

(see [37])
Let $\phi^{*}$ be an equilibrium point for the non-linear system

$$ $\displaystyle\dot{\phi}=v(\phi)$ (60) $$

where $v:\Phi\to\mathbb{R}^{n}$ is continuously differentiable and $\Phi$ is a neighborhood of $\phi^{*}$. Let J be the Jacobian of the system in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)) at the equilibrium point:

$$ $\displaystyle\textbf{J}=\frac{\partial v(\mathbf{\phi})}{\partial\mathbf{\phi}}\Biggr|_{\phi=\phi^{*}}.$ (61) $$

Then, we have:

- •
The equilibrium point $\phi^{*}$ is asymptotically stable and exponentially stable if J is a Hurwitz matrix, i.e., $\text{Re}(\lambda)<0$ for all eigenvalues $\lambda$ of J.
- •
The equilibrium point $\phi^{*}$ is unstable if $\text{Re}(\lambda)>0$ for one or more of the eigenvalues of J.

Theorem [B.4](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem4) upper bounds the eigenvalues of a Jacobian as in ([62](https://arxiv.org/html/2509.21167v1#A2.E62)). Theorem [B.4](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem4) is used in [34] to prove the local convergence of GANs, and we use it in this paper to prove the local convergence of the proposed algorithm.

###### Theorem B.4 .

(see Lemma G.2 in [34])
Let $\lambda_{min}$ and $\lambda_{max}$ denote the smallest and largest eigenvalues of a matrix, respectively. Suppose $\mathbf{J}\in\mathbb{R}^{(m+n)\times(m+n)}$ is of the following form:

$$ $\displaystyle\mathbf{J}=\begin{bmatrix}\mathbf{0}&\mathbf{P}\\ -\mathbf{P}^{T}&-\mathbf{Q}\end{bmatrix}$ (62) $$

where $\mathbf{Q}\in\mathbb{R}^{n\times n}$ is a symmetric real positive definite matrix and $\mathbf{P}^{T}\in\mathbb{R}^{n\times m}$ is a full column rank matrix. Then, for every eigenvalue $\lambda$ of $\mathbf{J}$, $\real(\lambda)<0$. More precisely, we have:

- •
When $\imaginary(\lambda)=0,$
$\displaystyle\real(\lambda)\leq-\frac{\lambda_{min}(\mathbf{Q})\lambda_{min}(\mathbf{P}\mathbf{P}^{T})}{\lambda_{min}(\mathbf{Q})\lambda_{max}(\mathbf{Q})+\lambda_{min}(\mathbf{P}\mathbf{P}^{T})},$
(63)
- •
When $\imaginary(\lambda)\neq 0,$
$\displaystyle\real(\lambda)\leq-\frac{\lambda_{min}(\mathbf{Q})}{2}.$
(64)

Lemma [B.5](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem5) reports a known relationship between the generator function of an $f$-divergence and its Fenchel conjugate. This relationship is useful in our Jacobian study.

###### Lemma B.5 .

Let $f:\mathbb{R}_{+}\longrightarrow\mathbb{R}$ be the generator function of any $f$-divergence. Let $f^{*}$ be the Fenchel conjugate of $f$. Then,

$$ $(f^{*})^{\prime}(t)=(f^{\prime})^{-1}(t),$ (65) $$

where $f^{\prime}$ indicates the first derivative of the generator function.

###### Proof.

Since $f(\cdot)$ is a convex function, $\hat{u}$ achieving the supremum is

$$ $\hat{u}=(f^{{}^{\prime}})^{-1}(t).$ (66) $$

Then, substituting ([66](https://arxiv.org/html/2509.21167v1#A2.E66)) in the definition of the Fenchel conjugate, we obtain

$$ $f^{*}(t)=(f^{{}^{\prime}})^{-1}(t)t-f((f^{{}^{\prime}})^{-1}(t)).$ (67) $$

The first derivative reads as

$$ $(f^{*})^{{}^{\prime}}(t)=((f^{{}^{\prime}})^{-1})^{{}^{\prime}}(t)t+(f^{{}^{\prime}})^{-1}(t)-\underbrace{f^{{}^{\prime}}((f^{{}^{\prime}})^{-1}(t))}_{=t}((f^{{}^{\prime}})^{-1})^{{}^{\prime}}(t).$ (68) $$

The first and third terms cancel out, leading to ([65](https://arxiv.org/html/2509.21167v1#A2.E65)).
∎

###### Definition B.2 .

###### Theorem B.3 .

###### Theorem B.4 .

###### Lemma B.5 .

###### Proof.

#### B.2.2 Notation and Setup

Let us define

$$ $\displaystyle\min_{\phi}\max_{\omega}\mathcal{J}(\phi,\omega)=\min_{\phi}\max_{\omega}$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[T_{\omega}(\Phi)\right]-\operatorname{\mathbb{E}}_{p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}\left[f^{*}(T_{\omega}(\Phi))\right]\Bigr].$ (69) $$

Let us define the pair $(\phi^{*},\omega^{*})$ as the equilibrium point achieved when solving ([69](https://arxiv.org/html/2509.21167v1#A2.E69)).

###### Assumption B.6 .

$f^{*}(\cdot)$ is a strictly convex function.

Assumption [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6) excludes from the set of $f$-divergences few specific cases that we do not use in this paper. For instance, the Total Variation distance does not satisfy Assumption [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6), as $f^{*}(t)=t$. Meanwhile, the KL divergence, Reverse KL (RKL) divergence, $\chi^{2}$ divergence, squared Hellinger distance, Jeffreys divergence, and Jensen-Shannon divergence satisfy Assumption [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6).

###### Assumption B.7 .

$\exists\mathbf{\phi}^{*},\mathbf{\omega}^{*}$ such that $\forall\mathbf{x}\in\text{supp}(p),p_{\mathbf{\phi}^{*}}(\mathbf{x})=p_{\Phi}(\mathbf{x})$ and $T_{\omega^{*}}=f^{\prime}(p_{\Phi}/p_{\phi})$.

Assumption [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7) is achieved when we reach the optimal convergence, and implies that, at the equilibrium,

$$ $\displaystyle T_{\omega^{*}}=f^{\prime}(p_{\Phi}/p_{\phi^{*}})=f^{\prime}(1),$ (70) $$

thus

$$ $\displaystyle(f^{*})^{\prime}(T_{\omega^{*}})=1,$ (71) $$

because $(f^{*})^{\prime}(t)=(f^{\prime})^{-1}(t)$ (from Lemma [B.5](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem5)).

###### Assumption B.8 .

$\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[-\nabla_{\phi}\log(p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*}))\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)\right]\right]\Bigr|_{(\phi^{*},\omega^{*})}$ and $\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\Bigl(\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}\Bigr)\right]\Bigr|_{\omega^{*}}$ are full row rank.

Assumption [B.8](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem8) is similar to the assumptions used in [34, 36]. In particular,

$$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[-\nabla_{\phi}\log(p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*}))\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)\right]\right]\Bigr|_{(\phi^{*},\omega^{*})}$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)d\mathbf{x}_{\hat{t}-1}\right]\Biggr|_{(\phi^{*},\omega^{*})},$ (72) $$

where $\nabla_{\phi}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\neq 0$, as $p_{\phi}$ is the probability density function of a Gaussian random variable.

###### Assumption B.6 .

###### Assumption B.7 .

###### Assumption B.8 .

#### B.2.3 Proofs of Local Convergence

###### Theorem 4.1 .

The Jacobian for the dynamical system defined in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)), at an equilibrium point $(\phi^{*},\omega^{*})$ is

$$ $\displaystyle J=\begin{pmatrix}\textbf{0}&-\textbf{K}_{TP}\\ \textbf{K}_{TP}^{T}&\textbf{K}_{TT}\end{pmatrix},$ (73) $$

where

$$ $\displaystyle\textbf{K}_{TP}$ $\displaystyle\triangleq\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[-\nabla_{\phi}\log(p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*}))\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)\right]\right]\Biggr|_{(\phi^{*},\omega^{*})}$ (74) $\displaystyle\textbf{K}_{TT}$ $\displaystyle\triangleq\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\Bigl[-(f^{*})^{\prime\prime}(T_{\omega})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}\Bigr]\right]\Biggr|_{\omega^{*}}$ (75) $$

###### Proof.

For the proof, we rewrite the objective as

$$ $\displaystyle\min_{\phi}\sup_{\omega}\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}$ $\displaystyle\Bigl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[T_{\omega}(\mathbf{x}_{\hat{t}},\mathbf{c})\right]-\operatorname{\mathbb{E}}_{p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})}\left[f^{*}(T_{\omega}(\mathbf{x}_{\hat{t}},\mathbf{c}))-f^{*}(f^{\prime}(1))\right]\Bigr]$ $\displaystyle-f^{*}(f^{\prime}(1)),$ (76) $$

where $f^{*}(f^{\prime}(1))$ is a constant. The gradient of $\mathcal{J}(\phi,\omega)$ w.r.t. $\phi$ is

$$ $\displaystyle\nabla_{\phi}\mathcal{J}(\phi,\omega)=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Biggl(f^{*}(T_{\omega}(\mathbf{x}_{\hat{t}},\mathbf{c}))-f^{*}(f^{\prime}(1))\Biggr)d\mathbf{x}_{\hat{t}-1}\right].$ (77) $$

The gradient of $\mathcal{J}(\phi,\omega)$ w.r.t. $\omega$ is

$$ $\displaystyle\nabla_{\omega}\mathcal{J}(\phi,\omega)=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\nabla_{\omega}T_{\omega}-p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})(f^{*})^{\prime}(T_{\omega})\nabla_{\omega}T_{\omega}d\mathbf{x}_{\hat{t}-1}\right].$ (78) $$

Then,

$$ $\displaystyle\nabla_{\phi}^{2}\mathcal{J}(\phi,\omega)$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}^{2}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Biggl(f^{*}(T_{\omega}(\mathbf{x}_{\hat{t}},\mathbf{c}))-f^{*}(f^{\prime}(1))\Biggr)d\mathbf{x}_{\hat{t}-1}\right].$ (79) $\displaystyle\nabla_{\omega}\nabla_{\phi}\mathcal{J}(\phi,\omega)$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Biggl((f^{*})^{\prime}(T_{\omega}(\mathbf{x}_{\hat{t}},\mathbf{c}))\nabla_{\omega}^{T}T_{\omega}\Biggr)d\mathbf{x}_{\hat{t}-1}\right].$ (80) $\displaystyle\nabla_{\omega}^{2}\mathcal{J}(\phi,\omega)$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\nabla_{\omega}^{2}T_{\omega}-p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\cdot$ (81) $\displaystyle\quad\quad\quad\quad\quad\quad\Biggl((f^{*})^{\prime\prime}(T_{\omega})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}+(f^{*})^{\prime}(T_{\omega})\nabla_{\omega}^{2}T_{\omega}\biggr)d\mathbf{x}_{\hat{t}-1}\Biggr]$ (82) $$

With Assumption [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7), we obtain

$$ $\displaystyle\nabla_{\phi}^{2}\mathcal{J}(\phi,\omega)\Biggr|_{(\phi^{*},\omega^{*})}$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Bigl[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}^{2}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Bigl(f^{*}(T_{\omega^{*}}(\mathbf{x}_{\hat{t}},\mathbf{c}))$ $\displaystyle\quad\quad\quad\quad\quad\quad-f^{*}(f^{\prime}(1))\Bigr)d\mathbf{x}_{\hat{t}-1}\Bigr]$ (83) $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}^{2}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Bigl(f^{*}(f^{\prime}(1))-f^{*}(f^{\prime}(1))\Bigr)d\mathbf{x}_{\hat{t}-1}\right]$ (84) $\displaystyle=\textbf{0}.$ (85) $$

Under Assumption [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7), we have

$$ $\displaystyle\nabla_{\omega}\nabla_{\phi}\mathcal{J}(\phi,\omega)\Biggr|_{(\phi^{*},\omega^{*})}=$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Bigl((f^{*})^{\prime}(T_{\omega^{*}}(\mathbf{x}_{\hat{t}},\mathbf{c}))\nabla_{\omega}^{T}T_{\omega}\Bigr)d\mathbf{x}_{\hat{t}-1}\right]$ (86) $\displaystyle=$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[-\int_{\mathbf{x}_{\hat{t}-1}}\nabla_{\phi}p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)d\mathbf{x}_{\hat{t}-1}\right]$ (87) $\displaystyle=$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Bigl[-\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\nabla_{\phi}\log(p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*}))$ $\displaystyle\cdot\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)d\mathbf{x}_{\hat{t}-1}\Bigr]$ (88) $\displaystyle=$ $\displaystyle\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[-\nabla_{\phi}\log(p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*}))\Bigl(\nabla_{\omega}^{T}T_{\omega}\Bigr)\right]\right]\Biggr|_{(\phi^{*},\omega^{*})}$ (89) $\displaystyle\triangleq$ $\displaystyle K_{TP}$ (90) $$

With similar steps, we obtain

$$ $\displaystyle\nabla_{\phi}\nabla_{\omega}\mathcal{J}(\phi,\omega)\Biggr|_{(\phi^{*},\omega^{*})}$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\left[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\left[-\Bigl(\nabla_{\omega}T_{\omega}\Bigr)\nabla_{\phi}^{T}\log(p_{\phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*}))\right]\right]\Biggr|_{(\phi^{*},\omega^{*})}$ (91) $\displaystyle=K_{TP}^{T}$ (92) $$

With Assumption [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7), we get

$$ $\displaystyle\nabla_{\omega}^{2}\mathcal{J}(\phi,\omega)\Biggr|_{(\phi^{*},\omega^{*})}$ $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\nabla_{\omega}^{2}T_{\omega}-p_{\phi^{*}}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c}^{*})\cdot$ (93) $\displaystyle\Biggl((f^{*})^{\prime\prime}(T_{\omega^{*}})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}+(f^{*})^{\prime}(T_{\omega^{*}})\nabla_{\omega}^{2}T_{\omega}\biggr)d\mathbf{x}_{\hat{t}-1}\Biggr]$ (94) $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\Biggl(\nabla_{\omega}^{2}T_{\omega}-$ (95) $\displaystyle(f^{*})^{\prime\prime}(T_{\omega^{*}})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}-(f^{*})^{\prime}(T_{\omega^{*}})\nabla_{\omega}^{2}T_{\omega}\biggr)d\mathbf{x}_{\hat{t}-1}\Biggr]$ (96) $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\Biggl(\nabla_{\omega}^{2}T_{\omega}-$ (97) $\displaystyle(f^{*})^{\prime\prime}(T_{\omega^{*}})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}-\nabla_{\omega}^{2}T_{\omega}\biggr)d\mathbf{x}_{\hat{t}-1}\Biggr]$ (98) $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Biggl[\int_{\mathbf{x}_{\hat{t}-1}}p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})\Biggl(-(f^{*})^{\prime\prime}(T_{\omega^{*}})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}\Biggr)d\mathbf{x}_{\hat{t}-1}\Biggr]$ (99) $\displaystyle=\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}}|\mathbf{c})}\Biggl[\operatorname{\mathbb{E}}_{p_{\Phi}(\mathbf{x}_{\hat{t}-1}|\mathbf{x}_{\hat{t}},\mathbf{c})}\Biggl[-(f^{*})^{\prime\prime}(T_{\omega^{*}})\nabla_{\omega}T_{\omega}\nabla_{\omega}^{T}T_{\omega}\Biggr]\Biggr]\Biggr|_{(\phi^{*},\omega^{*})}$ (100) $\displaystyle\triangleq K_{TT}$ (101) $$

where $K_{TT}\prec 0$ under Assumptions [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6) and [B.8](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem8).
∎

Therefore, the Jacobian is

$$ $\displaystyle J$ $\displaystyle=\begin{pmatrix}-\nabla_{\phi}^{2}\mathcal{J}(\phi,\omega)&-\nabla_{\omega}\nabla_{\phi}\mathcal{J}(\phi,\omega)\\ \nabla_{\phi}\nabla_{\omega}\mathcal{J}(\phi,\omega)&\nabla_{\omega}^{2}\mathcal{J}(\phi,\omega)\end{pmatrix}$ (102) $\displaystyle=\begin{pmatrix}\textbf{0}&-K_{TP}\\ K_{TP}^{T}&K_{TT}\end{pmatrix}$ (103) $$

###### Theorem 4.2 .

The dynamical system defined in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)) is locally exponentially stable with respect to an equilibrium point $(\phi^{*},\omega^{*})$ under Assumptions [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6), [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7), [B.8](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem8). Let $\lambda_{min}(\cdot)$ and $\lambda_{max}(\cdot)$ be the smallest and largest eigenvalues of a given matrix, respectively. The rate of convergence of the system is governed by the eigenvalues of the Jacobian $\mathbf{J}$ which have a negative real part upper bounded as

- •
When $\text{Im}(\lambda)=0$,
$\displaystyle\text{Re}(\lambda)\leq-\frac{\lambda_{min}(-\textbf{K}_{TT})\lambda_{min}(\textbf{K}_{TP}\textbf{K}_{TP}^{T})}{\lambda_{min}(-\textbf{K}_{TT})\lambda_{max}(-\textbf{K}_{TT})+\lambda_{min}(\textbf{K}_{TP}\textbf{K}_{TP}^{T})}$
(104)
- •
When $\text{Im}(\lambda)\neq 0$,
$\displaystyle\text{Re}(\lambda)\leq-\frac{\lambda_{min}(-\textbf{K}_{TT})}{2}$
(105)

###### Proof.

From assumptions [B.6](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem6), [B.7](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem7), [B.8](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem8), we know that $-\textbf{K}_{TT}$ is positive definite and $\textbf{K}_{TP}$ is full row rank. From Theorem [B.4](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem4), we know that all the eigenvalues of the dynamical system in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)) have negative real part. From theorem [B.3](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem3), we know that the system is stable because all the eigenvalues have negative real part.
∎

###### Theorem 4.1 .

###### Proof.

###### Theorem 4.2 .

###### Proof.

#### B.2.4 On the Effect of Different f f -Divergences on Convergence

This section theoretically investigates the impact of the $f$-divergence choice on the local convergence of the dynamical system in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)). We establish that, within our framework, a proper choice of $f$-divergence theoretically accelerates convergence to an equilibrium point $(\phi^{*},\omega^{*})$.

While $f$-divergence formulations provide elegant and general solutions to a problem, identifying the optimal $f$-divergence remains a critical challenge of relevant interest. The class of well-known $f$-divergences is, in fact, large, thus presenting a significant computational burden for exhaustive empirical evaluation. There are two main approaches to assess the effectiveness of different $f$-divergences for a specific task: empirical and theoretical evaluation.

##### Empirical evaluation

Empirical evaluation is the most frequent way to analyze the performance of different $f$-divergences in a specific task. This type of evaluation is possible for most algorithms and usually the best-performing $f$-divergence depends on the task. Usually, empirical evaluation is realized by implementing and testing one loss function for each $f$-divergence. Alternatively, few algorithms set the problem of finding the best $f$-divergence as an optimization problem that can be solved via gradient methods [78]. However, this latter approach usually leads to a higher computational complexity and more difficult training.
We discuss the empirical comparison between $f$-divergences for the algorithms proposed in this paper in Sections [5](https://arxiv.org/html/2509.21167v1#S5) and [C](https://arxiv.org/html/2509.21167v1#A3).

##### Theoretical evaluation

While a theoretical evaluation is not always feasible, as it depends on both the specific problem and the $f$-divergence-based algorithm, when possible, such a study offers clear and elegant guidelines for selecting $f$ (e.g., [66]).
Below, we provide a theoretical convergence comparison of $f$-divergences within our method.

Theorem [4.2](https://arxiv.org/html/2509.21167v1#S4.Thmtheorem2) provides two important contributions. Firstly, it proves the local exponential stability of the dynamical system in ([8](https://arxiv.org/html/2509.21167v1#S4.E8)). Secondly, it provides an expression for the upper bound on the eigenvalues of the same dynamical system. By studying these upper bounds, we evaluate the effect of different $f$-divergences on the system’s rate of convergence.
Immediately from the expression of $\mathbf{K}_{TT}$ is clear that the $f$-divergence plays a crucial role in the local convergence and that by choosing different $f$-divergences (thus having different Fenchel conjugates), we impact the convergence speed. In the following, we show how to obtain some guidelines on the choice of the $f$-divergence to obtain faster convergence.
For simplicity, we define $\mathbf{K}_{TT}^{\prime}=-\mathbf{K}_{TT}$. Recall that $\lambda(\mathbf{K}_{TT}^{\prime})>0$ and $\lambda(\mathbf{K}_{TP}\mathbf{K}_{TP}^{T})>0$, for any $\lambda$ eigenvalue. We refer to the eigenvalues of the Jacobian as $\lambda^{J}$, $\lambda^{J}<0$.

To achieve faster convergence, we want the largest eigenvalue (smallest in absolute value) to be as small as possible (as large as possible in absolute value).
When $\text{Im}(\lambda^{J})\neq 0$, the bound in ([105](https://arxiv.org/html/2509.21167v1#A2.E105)) implies that we have faster convergence when the smallest eigenvalue of $\mathbf{K}_{TT}^{\prime}$ is larger.
When $\text{Im}(\lambda^{J})=0$, we study the upper bound on the eigenvalues of the dynamical system:

- •
When $\lambda_{m}(\mathbf{K}_{TP}\mathbf{K}_{TP}^{T})\gg\lambda_{M}(\mathbf{K}_{TT}^{\prime})$, $\text{Re}(\lambda_{M}^{J})\approx-\lambda_{m}(\mathbf{K}_{TT}^{\prime})$. Thus, we achieve faster convergence when $\lambda_{m}(\mathbf{K}_{TT}^{\prime})$ is larger.
- •
When $\lambda_{m}(\mathbf{K}_{TP}\mathbf{K}_{TP}^{T})\approx\lambda_{M}(\mathbf{K}_{TT}^{\prime})$, $\text{Re}(\lambda_{M}^{J})\approx-\frac{\lambda_{m}(\mathbf{K}_{TT}^{\prime})}{1+\lambda_{m}(\mathbf{K}_{TT}^{\prime})}$. Thus, we attain faster convergence when $\lambda_{m}(\mathbf{K}_{TT}^{\prime})$ is larger.
- •
When $\lambda_{m}(\mathbf{K}_{TP}\mathbf{K}_{TP}^{T})\ll\lambda_{M}(\mathbf{K}_{TT}^{\prime})$, $\text{Re}(\lambda_{M}^{J})\approx-\frac{\lambda_{m}(\mathbf{K}_{TP}\mathbf{K}_{TP}^{T})}{\lambda_{M}(\mathbf{K}_{TT}^{\prime})}\approx 0$.
- •
$\lambda_{M}(\mathbf{K}_{TT}^{\prime})$ is only present at the denominator. Therefore, we have faster convergence when $\lambda_{M}(\mathbf{K}_{TT}^{\prime})$ is smaller.

Although it is not possible to clearly define some unique conditions to achieve faster convergence that hold true for all the possible cases, we can definitively affirm that a larger value of $\lambda_{m}(\mathbf{K}_{TT}^{\prime})$ is probably beneficial when $\text{Im}(\lambda^{J})=0$ and certainly beneficial when $\text{Im}(\lambda^{J})\neq 0$.
We notice that $\mathbf{K}_{TT}^{\prime}$ is directly proportional to $(f^{*})^{\prime\prime}(T_{\omega})|_{(\phi^{*},\omega^{*})}$.
Thus, we have faster convergence when $(f^{*})^{\prime\prime}(T_{\omega})|_{(\phi^{*},\omega^{*})}$ is larger. Thus, we obtain

$$ $\displaystyle(f^{*})^{\prime\prime}(T_{\omega})|_{(\phi^{*},\omega^{*})}=(f^{*})^{\prime\prime}(f^{\prime}(1))=\frac{1}{f^{\prime\prime}(1)},$ (106) $$

which can be demonstrated by using Lemma [B.5](https://arxiv.org/html/2509.21167v1#A2.Thmtheorem5).
We evaluate ([106](https://arxiv.org/html/2509.21167v1#A2.E106)) for different $f$-divergences:

- •
KL divergence: $\frac{1}{f^{\prime\prime}(1)}\Bigr|_{f=\text{KL}}=1$.
- •
Reverse KL divergence: $\frac{1}{f^{\prime\prime}(1)}\Bigr|_{f=\text{RKL}}=1$.
- •
$\chi^{2}$ divergence: $\frac{1}{f^{\prime\prime}(1)}\Bigr|_{f=\chi^{2}}=\frac{1}{2}$.
- •
Jensen-Shannon divergence: $\frac{1}{f^{\prime\prime}(1)}\Bigr|_{f=\text{JS}}=2$.
- •
Squared Hellinger distance: $\frac{1}{f^{\prime\prime}(1)}\Bigr|_{f=H^{2}}=2$.

Thus, we can conclude that JS and $H^{2}$ divergences have better convergence properties than the other divergences analyzed.

## Appendix C Additional Experimental Results

Table: Table 3: Quantitative results with 500 iterations, reporting CS, CA, and KID. For methods other than the original model, the best values are in bold and the second-best in italics. ^† denotes Importance Sampling and ^‡ denotes Prior Preservation and Gradient Surgery.

Table: Table 4: Quantitative results with 5000 iterations, reporting CS, CA, and KID. For methods other than the Original Model, the best values are in bold and the second-best in italics. ^† denotes Importance Sampling and ^‡ denotes Prior Preservation and Gradient Surgery.

Table: Table 5: Quantitative results with 150 iterations for variational methods, reporting CS, CA, and KID. For methods other than the Original Model, the best values are in bold and the second-best in italics. ^† denotes Importance Sampling and ^‡ denotes Prior Preservation and Gradient Surgery.

**Table 6: Unlearning comparison for Van Gogh. Lower CS/CA is better for the erased concept ($\downarrow$), higher is better for preserved concepts ($\uparrow$). 500 iterations for non variational and 150 iterations for variational.**
|  | Erased (Van Gogh) | Preserved Concepts |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  |  | J. Mann | J. Vermeer | S. Dali | G. Rutkowski | Monet |  |  |  |  |  |
| Method | CS ($\downarrow$) | CA ($\downarrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) |
| Original Model | 0.80 | 1.00 | 0.78 | 1.00 | 0.83 | 1.00 | 0.69 | 0.78 | 0.54 | 0.34 | 0.74 | 1.00 |
| MSE (Closed-Form) | 0.70 | 0.71 | 0.76 | 1.00 | 0.79 | 1.00 | 0.67 | 0.72 | 0.52 | 0.48 | 0.70 | 0.98 |
| Hellinger (Closed-Form) | 0.67 | 0.69 | 0.78 | 1.00 | 0.80 | 0.98 | 0.68 | 0.70 | 0.54 | 0.50 | 0.71 | 0.98 |
| $\chi^{2}$ (Closed-Form) | 0.67 | 0.71 | 0.78 | 1.00 | 0.81 | 0.96 | 0.68 | 0.78 | 0.54 | 0.48 | 0.72 | 0.98 |
| KL (Variational) | 0.75 | 0.92 | 0.78 | 1.00 | 0.82 | 1.00 | 0.68 | 0.72 | 0.54 | 0.48 | 0.73 | 1.00 |
| Hellinger (Variational) | 0.80 | 1.00 | 0.78 | 1.00 | 0.83 | 1.00 | 0.69 | 0.76 | 0.55 | 0.48 | 0.75 | 1.00 |
| Jensen-Shannon (Variational) | 0.71 | 0.88 | 0.77 | 1.00 | 0.82 | 1.00 | 0.68 | 0.74 | 0.55 | 0.48 | 0.71 | 0.96 |
| $\chi^{2}$ (Variational) | 0.82 | 1.00 | 0.79 | 1.00 | 0.84 | 1.00 | 0.70 | 0.82 | 0.55 | 0.46 | 0.76 | 1.00 |

**Table 7: Unlearning comparison for R2D2. Lower CS/CA is better for the erased concept ($\downarrow$), higher is better for preserved concepts ($\uparrow$). 500 iterations for non-variational and 150 iterations for variational.**
|  | Erased (R2D2) | Preserved Concepts |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  |  | Baymax | Wall E | C-3PO | Bb8 |  |  |  |  |
| Method | CS ($\downarrow$) | CA ($\downarrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) | CS ($\uparrow$) | CA ($\uparrow$) |
| Original Model | 0.78 | 1.00 | 0.77 | 0.96 | 0.75 | 0.84 | 0.77 | 0.90 | 0.74 | 0.90 |
| MSE (Closed-Form) | 0.62 | 0.04 | 0.75 | 0.93 | 0.72 | 0.74 | 0.75 | 0.88 | 0.70 | 0.86 |
| Hellinger (Closed-Form) | 0.60 | 0.01 | 0.76 | 0.94 | 0.73 | 0.74 | 0.76 | 0.84 | 0.71 | 0.88 |
| $\chi^{2}$ (Closed-Form) | 0.62 | 0.01 | 0.74 | 0.92 | 0.73 | 0.80 | 0.77 | 0.90 | 0.71 | 0.90 |
| KL (Variational) | 0.63 | 0.55 | 0.75 | 0.92 | 0.74 | 0.86 | 0.76 | 0.86 | 0.69 | 0.90 |
| Hellinger (Variational) | 0.65 | 0.55 | 0.75 | 0.94 | 0.73 | 0.84 | 0.76 | 0.86 | 0.69 | 0.88 |
| Jensen-Shannon (Variational) | 0.65 | 0.40 | 0.76 | 0.94 | 0.73 | 0.80 | 0.76 | 0.86 | 0.71 | 0.90 |
| $\chi^{2}$ (Variational) | 0.71 | 0.74 | 0.75 | 0.92 | 0.73 | 0.80 | 0.76 | 0.84 | 0.72 | 0.92 |

### C.1 Experimental Setup

To comprehensively evaluate the performance of our proposed unlearning methods, we designed a series of experiments spanning single-concept and multi-concept erasure scenarios. Our setup allows for a systematic analysis of different loss functions, unlearning strategies, and regularization techniques. The experiments are conducted on SDv1.4, targeting a diverse set of concepts, including fictional characters (Snoopy, Grumpy Cat, Wall-E, R2D2) and distinct artistic styles (Van Gogh, Salvador Dali). For the experiments we used the prompts from [9] for finetuning and evaluating.
We used a DDIM sampler with 50 inference steps for the experiments in Tables [7](https://arxiv.org/html/2509.21167v1#A3.T7) and [6](https://arxiv.org/html/2509.21167v1#A3.T6), and 30 steps for all other experiments.

##### Implementation Details

All experiments were conducted on a workstation equipped with a single NVIDIA GeForce RTX 4090 GPU and an Intel Core i9 CPU. We utilized the PyTorch deep learning framework for all implementations. To manage memory constraints while maintaining a stable training process, we used a batch size of 4. This was combined with a gradient accumulation step of 2, resulting in an effective batch size of 8 for each weight update.

##### Anchor Concepts

For MSE and Hellinger loss functions, we investigate three distinct unlearning strategies defined by the choice of the anchor distribution, which is the distribution we guide the erased concept towards: Empty Anchor (‘empty‘), Near Anchor (‘near‘), and Superclass Anchor (‘superclass‘).
With Empty Anchor the target concept is pushed towards a null or generic distribution, represented by an empty text prompt. This strategy aims for the complete removal of the concept’s specific features. With Near Anchor, the model is trained to associate the target concept’s prompt with a semantically similar but distinct concept (e.g., erasing "van Gogh" by guiding it towards "Salvador Dali"). This evaluates the model’s ability to remap, rather than simply erase, concepts. With Superclass Anchor, the target concept is guided towards a more general, categorical prompt (e.g., erasing "Van Gogh" by guiding it towards the generic prompt "a painting"). This method tests the ability to abstract a specific instance into its broader category.

##### Regularizations

To ensure a stable unlearning process that does not degrade the model’s overall performance, we incorporate several key techniques. We tested different regularization techniques: with the symbol ^‡, we indicate the experiment using prior preservation loss [9] and Gradient Surgery [79]; with the symbol ^† we indicate that we sample only the final part of the DM generation trajectory [19] (referred to as importance sampling).

##### Experimental Scenarios

Our evaluation is structured around two main scenarios: single and multi-concept erasure. For single concept erasure, we perform experiments to erase each target concept individually. To understand the dynamics of unlearning over time, these experiments are conducted for varying numbers of training iterations: 500, 2000, and 5000 steps as shown in Tables [3](https://arxiv.org/html/2509.21167v1#A3.T3), [1](https://arxiv.org/html/2509.21167v1#S5.T1), and [4](https://arxiv.org/html/2509.21167v1#A3.T4). A separate variational setup with 150 steps, detailed in Table [5](https://arxiv.org/html/2509.21167v1#A3.T5), is also explored to assess performance under a limited computational budget.
For sequential multi-concept erasure, we perform a sequential erasure of three distinct concepts: "Van Gogh", "Snoopy", and "R2D2", to simulate a more realistic use-case where a model might be updated multiple times to remove different concepts.

### C.2 Ablation Efficacy and Preservation

The main goal of concept unlearning is to efficiently remove a target concept while causing no collateral damage to unrelated concepts. Our experiments reveal performance traits of the various $f$-divergences.

##### Understanding Kernel Inception Distance

The Interpretation of KID needs to be clarified. A lower KID means greater similarity of two image distributions. We calculate it between the model before and after the erasure on the same prompts.
For non-erased concepts, a lower KID is better: it means the model has effectively retained its ability to generate images of high quality and diversity for these concepts, with the original distribution being closely matched.
For erased concepts, the interpretation of KID depends on the specific objective of unlearning: if the desired situation is for the model to output strange, monochrome, or unrecognizable images when prompted by the erased concept, then a higher KID (relative to the original concept’s distribution) is preferred, indicating a strong deviation from any coherent features. On the other hand, if the objective is to shift the erased concept toward a realistic but generic or alternate representation, then achieving a lower KID would mean the erasure performed well, especially without turning the images into noise.

##### Tracking by number of iterations

The number of training iterations seems to have a strong impact on the effectiveness and retention of unlearning. More generally, the bigger the number of steps, ranging from 500 to 5000, the more thoroughly the target concept is unlearned, evidenced by consistently lower CS and CA values for the erased concept. For instance, "hellinger empty" for "Snoopy" returns, for 500 steps, CS=0.54, CA=0.47, KID=0.269 (Table [3](https://arxiv.org/html/2509.21167v1#A3.T3)), while for 5000 steps, CS=0.52, CA=0.1, KID=0.346 (in Table [4](https://arxiv.org/html/2509.21167v1#A3.T4)).
While CS and CA values decrease, the KID value increases.

##### Role of regularization

The effects of regularization, namely Prior Preservation/Gradient Surgery (^‡) and Importance Sampling (^†), are highly context-dependent. In longer optimization runs, such as the 500 and 5000-step experiments (Table [3](https://arxiv.org/html/2509.21167v1#A3.T3) and Table [4](https://arxiv.org/html/2509.21167v1#A3.T4)), applying these methods reveals a distinct trade-off. For instance, when erasing “Snoopy” for 500 steps, the non-regularized “hellinger empty” method is most effective at erasure (CS 0.54, CA 0.47) but produces less realistic images (KID 0.269). Applying regularization weakens the erasure effect (e.g., “hellinger empty ^‡ ^†” has CS 0.61, CA 1.00) but significantly improves image coherence, lowering the KID to 0.061.

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 2
presentation: 1
questions: Please address the comments and concerns in the weaknesses section.
soundness: 2
strengths: - A detailed convergence analysis is provided and a theoretically principled way of choosing f to have better convergence properties is provided
summary: The authors generalize existing concept unlearning methods in diffusion models by replacing KL divergence with more general f-divergence. The authors then theoretically analyze the impact of different choices of f on convergence behaviors near equilibria and gradient size. Empirically, it was found that the squared Hellinger distance has more stable gradient updates and better prior preservation.
weaknesses: - The results are poorly presented and some claims are not well justified
   - For example, in Figure 1, it is unclear how to interpret the white boxes that zoom in on portions of the image. Additionally, Figure 3 has a lot of noise in the plot, so some smoothing and a better caption explaining the main takeaway would help a great deal.
   - While a great deal of theoretical emphasis in the paper is regarding the variational approach, it appears details surrounding the training procedure on the minimax objective are not provided and the empirical results are minimal (lines 402-412) and not favorable.
   - There appears to be somewhat of an overreliance on qualitiative results for supporting important claims such as "This behavior characterizes the losses properties and explains why, for instance, H2 less probably leads to strange artifacts during fine-tuning." While Figure 2 is a great to include for us to visualize the nature of the benefit of H2, it would be great to quantitatively measure and keep track of these strange artifacts across multiple examples and present it in a table. Otherwise, the reader is left to believe that the Van Gogh example was ad hoc. 
- Small note: It appears many citations that need to be using parantheticals are not using them (e.g., related works section doesn't have any paranthetical citations).

While I believe there is merit to the exploration of different choices of f and the authors' theoretical analysis, I believe the paper and experimental results will need a couple more iterations to be fully polished and conference ready.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: Please look at the weakness section.
soundness: 3
strengths: 1. The generalization of the unlearning objective from MSE/KL divergence to a unified f-divergence framework is really insightful. It offers a flexible paradigm that allows for balancing the trade-offs between unlearning effectiveness and the preservation of other concepts. The theoretical analysis of the gradients for the closed-form loss functions provides a reasonable rationale for their differing empirical behaviors.

2. The experimental results align well with the gradient analysis. The $H^2$ divergence is shown to better preserve generative quality on unrelated concepts (as measured by metrics like KID) when compared to MSE, which appears to cause more degradation.
summary: The paper presents a generalization of the KL divergence-based unlearning algorithm to a broader class of f-divergence and performs a detailed analysis of the behavior of different divergence measures. Specifically, the paper compares the closed-form solution of Squared Hellinger ($H^2$) distance, the Pearson $\chi^2$ divergence, and the KL-divergence measures. The gradient analysis shows that $H^2$ distance has a much lower gradient norm, which leads to a more stable unlearning process with less disruption on the nearby unrelated concepts. The paper also proposes a variational framework for a general f-divergence-based training, but this is shown to be much more disruptive, although fast.
weaknesses: 1. The empirical comparison is primarily against KL-based methods like Concept Ablation. The evaluation would be more comprehensive if benchmarked against other recent state-of-the-art baselines, such as EraseAnything [1] or MACE [2].

2. In addition, an analysis of adversarial robustness (e.g., using a method like Ring-A-Bell [3]) can provide a stronger and more rigorous measure of how completely a concept has been erased. The robustness behaviour of different divergence measures can be interesting to know as well. 

3. The paper's evaluation of collateral damage from the unlearning method focuses on "unrelated concepts." A detailed quantitative evaluation of the impact on semantically nearby concepts would be valuable. For instance, in the style removal experiments, does unlearning "Van Gogh" (an Impressionist) also degrade the model's ability to generate "Monet" or "Impressionist paintings"? The current analysis (e.g., Fig. 4) is helpful, but a quantitative study on this would strengthen the work (as done in Concept Ablation as well)

4. The advantages of the variational framework versus the closed-form solutions are not fully clear. The results show it is faster while also more disruptive, but a deeper analysis of why the min-max optimization leads to faster convergence and a discussion of its stability trade-offs would be beneficial for developing a more general-purpose unlearning algorithm based on this paradigm.

[1] EraseAnything: Enabling Concept Erasure in Rectified Flow Transformers 
[2] Mace: Mass concept erasure in diffusion models 
[3] Ring-a-bell! how reliable are concept removal methods for diffusion models?

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confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: Q1. Did you try benchmarking on LDM-based SDXL and Flow-matching models to validate generality?
Q2. Does the insight on having a bigger family of f-divergence lead to better unlearning performance? This requires comparison with latest DM unlearning methods which is missing in the paper. Particularly, the authors do not discuss the unlearned model's robustness to attacks like membership inference attacks. 
Q3. Could you report compute cost vs standard MSE KL-minimization?
Q4. The authors highlight Jeffreys, Hellinger, and χ² but omit other f-divergences (e.g., JS, Rényi). A short discussion on why these three were prioritized would improve completeness.
soundness: 3
strengths: 1) The paper elegantly reformulates prior MSE-based objectives as specific instances of the f-divergence family, providing a principled foundation for a range of unlearning losses.
2) The gradient and convergence analyses are rigorous, connecting divergence curvature to optimization dynamics in a meaningful way.
3) The experiments demonstrate that the squared Hellinger divergence achieves bounded gradients, leading to stable unlearning with fewer artifacts, a convincing empirical validation of the theory. 
4) The inclusion of post-, fine-, and variational-based unlearning frameworks in the appendix positions this as a unifying theoretical effort.
summary: This paper proposes a unified framework for diffusion model (DM) unlearning based on f-divergences. The work introduces a theoretical generalization of concept erasure via any f-divergence in DM. The contributions are: closed-form derivations for special cases (Hellinger, χ², Jeffreys), a variational adversarial formulation for general divergences and gradient behavior analysis & local convergence guarantees. Authors show empirical evaluation on Stable Diffusion 1.4 with multiple concepts and anchors. The paper argues that Hellinger divergence yields better stability and prior preservation while χ² gives more aggressive unlearning.
weaknesses: 1) Limited baselines: comparison with several recent baselines (e.g., MACE, SAFREE) are missing in the experiments section.
2) Empirical focus is narrow. All experiments use the same model (Stable Diffusion 1.4). Although reasonable, it would help to clarify whether results depend on architecture details or on properties of the diffusion loss. It is unclear how the proposed f-divergence solution are scalable to newer diffusion architectures (e.g., SDXL, Flux).

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confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: Please refer to weaknesses.
soundness: 2
strengths: 1. The authors rigorously generalize MSE-based unlearning to a broad family of f-divergences, showing that KL divergence and MSE are just special cases. It demonstrates a strong theoretical foundation linking unlearning dynamics to divergence curvature and gradient scaling.

2. The paper derives explicit gradient behaviors and proves local exponential stability for the variational form. The gradient comparison gives interpretable insight.

3. The experimental results clearly support the theoretical claims. The Concept Ablation explicitly shows that bounded gradients prevent artifacts such as distorted faces or textures. This close alignment between mathematical reasoning and empirical observation increases the paper’s credibility.
summary: The paper proposes f-DMU, a unified framework for diffusion model unlearning based on f-divergence. It generalizes existing MSE-based and KL-based unlearning approaches by allowing any f-divergence. The method provides two formulations—closed-form and variational—to balance simplicity and generality. The authors offer theoretical analyses of gradient behavior and convergence properties, showing that different divergences lead to distinct unlearning dynamics. Empirical results on Stable Diffusion v1.4 demonstrate that the Hellinger-based loss achieves better trade-offs between erasure effectiveness and image quality preservation compared to the standard MSE loss.
weaknesses: 1. Although the authors discuss how curvature affects convergence, they don’t propose a quantitative criterion for choosing which f-divergence to use in practice. To my understanding, I think the choice is heuristic. For example, while Table 1 suggests H^2 outperforms MSE for concept erasure, the paper provides no procedure for selecting between closed-form vs. variational objectives or among multiple f-divergences based on task statistics.

2. All experiments are performed on Stable Diffusion v1.4 with a few synthetic concepts. Considering many DM unlearning works are performing unlearn multiple concepts at once, how does the proposed method performs when unlearn many concepts (say > 10 concepts)? It is important that the unlearning algorithm can handle such practical scenario, which demonstrate the robustness of the method.

3. While CAbl and MSE-based methods are compared, newer or adversarial unlearning methods (e.g., DoCo[1] or defensive adversarial unlearning [2]) are omitted. Besides these two, the author also needs to compare with other baselines such as SalUn, FMN, SPM, ESD, MACE, etc. I suggest the author compare with similar approach unlearning and different types of unlearning, which will make the proposed method more convincing.


[1] Wu, Yongliang, et al. "Unlearning concepts in diffusion model via concept domain correction and concept preserving gradient." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 39. No. 8. 2025.

[2] Zhang, Yimeng, et al. "Defensive unlearning with adversarial training for robust concept erasure in diffusion models." Advances in neural information processing systems 37 (2024): 36748-36776.