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+# PRIVATELY ALIGNING LANGUAGE MODELS WITH REINFORCEMENT LEARNING
+
+Fan Wu<sup>1*</sup>; Huseyin A. Inan<sup>2</sup>, Arturs Backurs<sup>3</sup>, Varun Chandrasekaran<sup>1</sup>, Janardhan Kulkarni<sup>3</sup>, Robert Sim<sup>2</sup>
+
+<sup>1</sup> University of Illinois Urbana-Champaign, <sup>2</sup> M365 Research, <sup>3</sup> Microsoft Research {fanw6, varunc}@illinois.edu, {huseyin.inan, arturs.backurs, jakul, rsim}@microsoft.com
+
+# ABSTRACT
+
+Positioned between pre-training and user deployment, aligning large language models (LLMs) through reinforcement learning (RL) has emerged as a prevailing strategy for training instruction following-models such as ChatGPT. In this work, we initiate the study of privacy-preserving alignment of LLMs through Differential Privacy (DP) in conjunction with RL. Following the influential work of Ziegler et al. (2020), we study two dominant paradigms: (i) alignment via RL without human in the loop (e.g., positive review generation) and (ii) alignment via RL from human feedback (RLHF) (e.g., summarization in a human-preferred way). We give a new DP framework to achieve alignment via RL, and prove its correctness. Our experimental results validate the effectiveness of our approach, offering competitive utility while ensuring strong privacy protections.
+
+# 1 INTRODUCTION
+
+Over the past few months, Large Language Models (LLMs) that are capable of following open-ended user instructions such as ChatGPT, Bard, Llama Chat, have seen an euphoric adoption by application developers. Similar to their predecessors, these models are pre-trained on vast amounts of public internet data. However, their magical ability to follow myriad user instructions – the driving force behind their mass adoption – has been attributed to instruction fine-tuning and learning from human feedback. This new step involves collecting a dataset of human preferences and feedback, followed by fine-tuning the model via reinforcement learning (RL) to make them better aligned, often abbreviated as RLHF. Since the influential works of Ziegler et al. (2020); Ouyang et al. (2022); Bai et al. (2022), the RLHF framework has emerged as the dominant paradigm for training instruction-following models.
+
+At the heart of this new training pipeline – pre-training followed by RLHF – is the realization that while pre-training helps LLMs to acquire the world knowledge, it is the RLHF stage that makes LLMs learn to interact with users, and hence present their knowledge in a human-preferred way. This framework opens the door for a continuous improvement of the model by collecting users' feedback and preferences via telemetry data. As appealing as that may sound, improving LLMs via users' preferences and feedback raises privacy concerns: what if the model learns about a specific user's instructions and regurgitates them at a later point? It is well known in the privacy literature that LLMs are vulnerable to privacy attacks including prompt attacks (Duan et al., 2023; Carlini et al., 2021; 2019), and RLHF training seems particularly concerning from this angle. This constitutes the central question explored in this work.
+
+Can we fulfill the promise of aligning models with human preferences and feedback data via a privacy preserving RLHF methodology?
+
+# 1.1 OUR CONTRIBUTIONS
+
+We initiate the study of aligning LLMs with RL while satisfying the strong mathematical guarantees of differential privacy (DP) (Dwork & Roth, 2014). We foresee this as an important research direc
+
+tion for the privacy community as more applications start to deploy LLMs to interact directly with users. Our main contributions are:
+
+1. We give a differentially private framework for aligning LLMs with RL. Our framework mathematically guarantees that the final model satisfies DP over the entire course of the alignment process, consisting of multiple stages of model training and weight sharing. Further, we show how to adapt the PPO algorithm (Schulman et al., 2017) to DP setting.  
+2. We empirically evaluate our DP framework on commonly studied tasks (in non-privacy literature). Following the influential work of Ziegler et al. (2020), we evaluate two main scenarios: (i) alignment via RL without human in the loop for a positive review generation task on the IMDb dataset (Maas et al., 2011), and (ii) alignment via RL from human feedback (RLHF) for a summarization task on the Reddit TL;DR dataset (Volske et al., 2017). Our experimental results indicate that privately aligning LLMs is possible, offering competitive utility while ensuring strong privacy protections. As a representative example, on the IMDb dataset, the average reward obtained by our DP GPT2-Large model for generating positive reviews is 3.20 with  $\epsilon = 4$ , whereas the best performing non-private model achieves an average reward of 3.45.
+
+Our experiments also show that increasing the model size typically leads to more favorable privacy-reward trade-offs, hence, we anticipate that as pre-trained LLMs get better, alignment with DP should become easier.
+
+# 2 PRELIMINARIES
+
+# 2.1 ALIGNING LANGUAGE MODELS VIA REINFORCEMENT LEARNING
+
+We review the pipeline from the seminal work by Ziegler et al. (2020), which describes a methodology to align language models via RL by using a reward model to optimize for a given task. For ease of presentation, we borrow terminology and notations from Ziegler et al. (2020).
+
+One starts with a pre-trained language model  $LM^{\mathrm{pt}}$ , which defines a probability distribution  $LM^{\mathrm{pt}}(x_n \mid x_1, \dots, x_{n-1}) \in [0,1]$  over the space of tokens  $x_n \in \mathcal{V}$  given a context consisting of tokens  $x_i \in \mathcal{V}$  for  $i = 1, \dots, n-1$ .  $\mathcal{V}$  is referred as the vocabulary of  $LM^{\mathrm{pt}}$ . The first step in general is to fine-tune this model with regular supervised learning procedure (SFT). This step can be performed for various reasons such as to teach the language model a desired output behaviour (Ouyang et al., 2022) or it could be simply to train for a downstream task such as summarization (Stiennon et al., 2022). We denote the resulting model as  $LM^{\mathrm{sft}}$ .
+
+In the alignment step, a policy  $\pi$ , initialized as  $\pi = LM^{\mathrm{sft}}$ , is further fine-tuned using RL for the underlying task. We consider two scenarios depending on whether the task is directly defined by a reward function or it is based on human judgments. We compare the two scenarios in Appendix G.
+
+Reinforcement learning without human in the loop. The underlying task is defined by a reward function  $r: \mathcal{V}^{\infty} \times \mathcal{V}^{\infty} \to \mathbb{R}$  that can score how well aligned the language model's generation  $y \in \mathcal{V}^{\infty}$  is, given the context  $x \in \mathcal{V}^{\infty}$ . Here, one can use reinforcement learning to directly optimize the expected reward. An example is controlled sentiment generation, where the goal is to respond to a user query with a positive sentiment. Here, one can use existing language models that are fine-tuned on sentiment classification tasks as the reward model to score for positive sentiment.
+
+Reinforcement learning with human preferences. The underlying task is defined by human judgments. A typical example is to respond to a user query with a human-preferred way instead of language model's original completion that is learned during pre-training. Here, human labels are used first to train a reward model. A dataset can be formed by generating multiple responses from the LLM (for simplicity we consider two:  $y_{1}$  and  $y_{2}$ ) for a given input  $x$  and asking humans to prefer between  $y_{1}$  and  $y_{2}$ . Let  $b \in \{0, 1\}$  denote the human preference. Assuming access to a dataset  $S$  of  $(x, y_{0}, y_{1}, b)$  samples with human preferences, a reward model  $r: \mathcal{V}^{\infty} \times \mathcal{V}^{\infty} \to \mathbb{R}$  can be trained with the following negative log-likelihood loss (Ziegler et al., 2020; Ouyang et al., 2022):
+
+$$
+\mathcal {L} (r, \mathcal {S}) = - \mathbb {E} _ {(x, y _ {0}, y _ {1}, b) \sim \mathcal {S}} \left[ \log \left(\sigma \left(r \left(x, y _ {b}\right) - r \left(x, y _ {1 - b}\right)\right)\right) \right], \tag {1}
+$$
+
+where  $\sigma$  denotes the sigmoid function:  $\sigma(x) = \frac{1}{1 + e^{-x}}$ . One can also initialize the reward model  $r$  from  $LM^{\mathrm{sft}}$  with an additional linear layer that produces a single scalar for the reward value.
+
+Finally, the initialized policy  $\pi$  is fine-tuned to optimize the reward model  $r$  with reinforcement learning. However, instead of directly optimizing the expected reward, a penalty term of  $\beta \mathrm{KL}(\pi, LM^{\mathrm{sft}})$  is added to the optimization term to prevent  $\pi$  from deviating too far from  $LM^{\mathrm{sft}}$ . Thus, the modified reward becomes
+
+$$
+R (x, y) = r (x, y) - \beta \log \frac {\pi (y | x)}{L M ^ {\mathrm {s f t}} (y | x)}. \tag {2}
+$$
+
+This reward function is maximized via Proximal Policy Optimization (PPO) (Schulman et al., 2017) to fine-tune the policy  $\pi$  on the corresponding data distribution  $x\sim \mathcal{D}$ .
+
+# 2.2 DIFFERENTIAL PRIVACY
+
+LLMs are known to be susceptible to privacy attacks (Carlini et al., 2019; 2021; 2023). Over the past decade, Differential Privacy (DP) (Dwork et al., 2006) has emerged as a powerful framework that provides mathematical guarantees for the privacy of individuals in training datasets. It quantifies the amount of information one could learn from the output of an algorithm or its generations. Formally,
+
+Definition 1  $((\epsilon, \delta)$ -DP (Dwork & Roth, 2014)). A randomized algorithm  $\mathcal{M}$  achieves  $(\epsilon, \delta)$ -DP, if for any neighboring datasets  $D_{1}$  and  $D_{2}$  (differing in at most one entry) and for any  $S \in \text{Range}(\mathcal{M})$ ,
+
+$$
+\Pr (\mathcal {M} (D _ {1}) \in S) \leq e ^ {\epsilon} \Pr (\mathcal {M} (D _ {2}) \in S) + \delta . \tag {3}
+$$
+
+Here,  $\epsilon$  represents the privacy budget: a smaller  $\epsilon$  offers a stronger privacy guarantee.  $\delta$  accounts for the probability that  $\mathcal{M}$  violates  $\epsilon$ -DP.
+
+DPSGD. In the context of deep learning, DPSGD (Song et al., 2013; Abadi et al., 2016), a drop-in replacement of the vanilla stochastic gradient descent, has become the default optimizer to achieve DP. At each iteration, DPSGD performs per-sample gradient clipping and Gaussian noise addition, thus limiting and masking the contribution of any single data point to the model update; we give a detailed description in Appendix A. Recently, in response to the rising concerns of privacy leakage in large language models (Carlini et al., 2021), DPSGD is employed for tasks from private finetuning (Yu et al., 2022; Li et al., 2022) to synthetic text generation (Yue et al., 2023).
+
+# 3 PROBLEM DEFINITION
+
+Consider the generic problem of aligning a language model towards an objective by using a reward model via RL. Our problem formulation introduces an extra dimension to this challenge, namely, achieving this alignment respecting DP of the underlying data samples. We are given privacy parameters  $\epsilon > 0$ ,  $\delta \in [0,1]$  and a pre-trained language model  $LM^{\mathrm{pt}}$ . There is a private dataset  $D$  for use in the alignment procedures outlined in Section 2.1. The particular use of the private dataset  $D$  depends on the availability of the reward model.
+
+In RL without human in the loop scenario, we assume the availability of a reward model that is independent of the private dataset  $D$ . In this case, supervised learning step (SFT) fine-tunes  $LM^{\mathrm{pt}}$  with  $D$  to achieve the initial policy  $\pi = LM^{\mathrm{sft}}$ . This is directly followed by the Proximal Policy Optimization (PPO) that fine-tunes the policy  $\pi$  with the reward model on  $D$ . The privacy-preserving constraint in our problem definition is that the final parameters of the optimized policy  $\pi$  should be  $(\epsilon, \delta)$ -DP with respect to private dataset  $D$ .
+
+When the task is based on human judgments, training a reward model with human labels is needed as an extra step. As described in Section 2.1, the reward model is obtained by training on a dataset where each sample is a tuple consisting of a sample  $x$  belonging to  $D$ , multiple generations of  $LM^{\mathrm{sft}}$  given  $x$  as context and human preference over these generations. The privacy-preserving constraint similarly follows as the previous scenario and the final parameters of the optimized policy  $\pi$  must be  $(\epsilon, \delta)$ -DP with respect to private dataset  $D$ . As we will discuss shortly, this will require the reward model to be trained with DP as well.
+
+# 4 OUR DP ALIGNMENT FRAMEWORK
+
+In this section, we describe our DP framework for aligning LLMs. We consider the scenario with human in the loop as it subsumes the case without. Recall that it involves three main steps: 1)
+
+Supervised fine-tuning of a language model for the task at hand to obtain  $LM^{\mathrm{sft}}$ . 2) Learning a reward model  $r$  from human preferences. 3) Fine-tune a policy  $\pi$  (initialized to  $LM^{\mathrm{sft}}$ ) to optimize the reward model  $r$  with RL. Although we only require that the weights of the final policy  $\pi$  are DP with respect to  $D$ , one needs to perform each step with DP, the reason for which will become clear once we describe our framework.
+
+To achieve  $(\epsilon, \delta)$ -DP with respect to a private dataset  $D$  at the end of the alignment there is more than one solution. One could partition  $D$  into three disjoint subsets  $D_{1}$ ,  $D_{2}$  and  $D_{3}$  corresponding to the three stages of the alignment pipeline, and assume that two neighboring databases differ by a single sample in one of these three datasets; that is, a single user can contribute to at most one of three datasets  $D_{1}$ ,  $D_{2}$  and  $D_{3}$ . Another option is to assume that a single user can contribute to all the three datasets  $D_{1}$ ,  $D_{2}$  and  $D_{3}$ . Our framework can handle both the settings, with minor differences in how to calculate the final privacy parameters. The former approach would mean that to calculate the final privacy parameters, we can use the parallel composition theorem of DP (McSherry, 2009). For the latter, one needs to use advanced composition theorems such as (Gopi et al., 2021). An additional hyperparameter related to DP in the second approach is on how to allocate the fixed privacy budget across the three steps. The goal of this work is to show that alignment with DP is possible, and, hence, we take the simpler approach and assume that a single user can contribute to at most one of three datasets  $D_{1}$ ,  $D_{2}$  and  $D_{3}$ . We clarify that the nature of the three datasets are different, with  $D_{1}$  being a labeled dataset consisting of a reference answer per sample,  $D_{2}$  a preference dataset consisting of two generations and a human preference bit per sample, and  $D_{3}$  an unlabeled dataset consisting of input samples only.
+
+With this discussion behind us, we write down our framework for DP Alignment.
+
+1. DP Supervised Fine-Tuning: We do a supervised fine-tuning of  $LM^{\mathrm{pt}}$  using DPSGD with privacy parameters  $(\epsilon, \delta)$  on the dataset  $D_{1}$  to obtain  $LM^{\mathrm{sft}}$ . The analysis of DPSGD (Abadi et al. (2016)) guarantees that the weights of  $LM^{\mathrm{sft}}$  are private, and hence  $LM^{\mathrm{sft}}$  can be used arbitrarily in the remaining pipeline.  
+2. DP Learning of Reward Model: We initialize a reward model  $r$  from  $LM^{\mathrm{sft}}$  with the addition of a linear layer that produces a single scalar prediction for the reward value. We train  $r$  using DPSGD with the privacy parameters  $(\epsilon, \delta)$  to optimize the reward objective given by Equation 1 on the dataset  $D_2$ .  
+3. Alignment with DPPPO: Finally, we train a policy  $\pi$  initialized to  $LM^{\mathrm{sft}}$  via a DP adaptation of Proximal Policy Optimization (PPO) with the privacy parameters  $(\epsilon, \delta)$  to optimize the reward  $R$  as given in Equation 2 on the dataset  $D_3$ .
+
+All our model training procedures use LoRA (Hu et al., 2022). While this is not standard in the alignment literature, we make this algorithmic choice due to 3 reasons: 1) DP training works better with LoRA as hyperparameters are more stable (Yu et al., 2022); 2) LoRA is computationally more efficient, especially for DP training; 3) We also conjecture that LoRA fine-tuning during RL stage can also help in ensuring that the aligned model does not drift too far away from the  $LM^{\mathrm{sft}}$  model. This may be an interesting point even in the non-private world.
+
+We will elaborate each of the steps in our framework below. Before that, we note the following privacy guarantee of our framework due to the parallel composition theorem of DP (McSherry, 2009).
+
+Theorem 2. Our  $DP$  alignment framework is  $(\epsilon, \delta)$ -differentially private.
+
+Why Step II needs to be DP? A curious reader may ask why one should learn the reward model using DP if the third step already satisfies DP; that is, can learning the reward model be non-private? This is a subtle but important point, and our algorithmic choice of making the second step DP has to do with the privacy analysis of DPSGD. Consider the scenario where the reward model is not private. In such a case, for every random mini-batch in the third step, the gradients are a function of the reward model, which in turn implies that the gradients of the mini-batch are a function of entire dataset  $D_{2}$ . This invalidates the privacy amplification by subsampling theorem of DPSGD (Abadi et al., 2016), which is crucial for the overall framework to work.
+
+Our solution to the above technical challenge is to learn the reward model also via DP. The post-processing theorem of DP (Dwork & Roth, 2014) guarantees that the reward model  $r$  can be used in the third step as if it were a public model. However, there could be other ways of achieving the alignment with DP where the second step is not private, and we leave it as an intriguing future research question.
+
+DP Adaptation of PPO. Next, we discuss our algorithmic choices in making PPO algorithm DP. Although the model updates in our DPPPO algorithm are done via DPSGD, there are some important technical details to consider. An important distinction between DPPPO and SFT with DPSGD is what governs the number of iterations and how the model weights are updated. For SFT using DPSGD (1st step), the model weights are updated for each batch, and the number of iterations governs the course of training and the total model weight updates. On the other hand, PPO performs model updates in minibatches within a batch and multiple rounds (a.k.a. PPO epochs) can be taken over the same batch. Further, regular epochs are taken over the full dataset; see Schulman et al. (2017) (Algorithm 1) for a precise description. We give a complete pseudo-code of PPO implementation in Appendix F, but for the sake of discussion consider the abbreviated version in Algorithm 1. PPO updates are given in lines 6-10 that needs to be privatized. In our DPPPO implementation, we set  $T_{\mathrm{PPO}} = 1$  deviating from the usual implementations in RL that set  $T_{\mathrm{PPO}} > 1$ ; for example, von Werra et al. (2022) defaults  $T_{\mathrm{PPO}}$  to 4. By appropriately selecting the batch size (we use larger batch size for DPPPO), we ensure that the total number of model updates in both the private and non-private worlds remain similar. We make these algorithmic choices to simplify the privacy analysis and to utilize privacy amplification by subsampling (Abadi et al., 2016) in DPSGD algorithm, where each batch should be randomly selected from the dataset. If one takes more than 1 round of model updates  $(T_{\mathrm{PPO}} > 1)$  on the same batch, then privacy accounting of DPSGD needs to be modified, say by first doing an advanced composition across multiple PPO rounds on the same batch followed by subsampling amplification. We leave these algorithmic explorations on our choices as future research directions, and present an ablation study on  $T_{\mathrm{PPO}} > 1$  in Appendix B.1.
+
+Algorithm 1: Aligning language models with RL (PPO), full version in Appendix F.  
+Define:  $D$  : a dataset consisting of input texts.  $x$  : input text,  $y$  : model response.  $T$  total training epochs,  $T_{\mathrm{PPO}}$  PPO epochs. model, ref_model: the model being learned and the frozen model for reference. Models are composed of a generation body as well as a value head. superscript b: batch, superscript m b: mini-batch. p,l: log probability and logit given by the generation body, v: value given by the value head   
+1 Procedure Update (model,  $x^{b},y^{b},R^{b}$  ..  $\triangleright$  Stage I: forward passes to obtain reference stats on the batch  $(p^b,l^b,v^b)\gets \mathsf{BatchedForwardPass}(\mathsf{model},x^b,y^b)$ $(p_r^b,l_r^b,v_r^b)\gets \mathsf{BatchedForwardPass}(\mathsf{ref\_model},x^b,y^b)$ $s^b\gets \mathsf{ComputeScores}(R^b,p^b,p_r^b)$ $\triangleright$  compute the modified reward (Eq. 2)  $\triangleright$  Stage II: update on minibatches   
+5  $D^{b}\gets (x^{b},y^{b},l^{b},v^{b},s^{b})$    
+6 for  $i = 1$  to TPPO do for  $D^{mb}\in D^{b}$  do  $(x^{mb},y^{mb},l^{mb},v^{mb},s^{mb})\gets D^{mb}$  take out a minibatch   
+9  $(p,l,v)\gets \mathsf{BatchedForwardPass}(\mathsf{model},x^{mb},y^{mb})$  TrainMinibatch(model,  $p^{mb},v^{mb},s^{mb},p,l,v)$  with PPO objective   
+10 main loop   
+11 for  $i = 1$  to T do Take out a batch for  $x^b\in D$  do  $y^{b}\gets \mathsf{modelgenerate}(x^{b})$  obtain the model responses   
+13  $R^{b}\gets r(x^{b},y^{b})$  obtain the rewards via the reward model r   
+15 Update (model,  $x^{b},y^{b},R^{b}$ )
+
+# 5 PRIVATELY ALIGNING LMS WITHOUT HUMAN IN THE LOOP
+
+We begin by exploring the simpler task of privately aligning a language model without human in the loop, and consider RLHF in the next section. For our case study, we focus on controlled sentiment generation, where the goal is to complete a given prefix of a review from the IMDb dataset (Maas
+
+![](images/f899879a40d0d6da20099fa8cbe7e66cfddca6e75ebef49adf74915cdd3d83a4.jpg)  
+Stage I: Supervised Fine-Tuning  
+Objective: learn to generate a review  
+Figure 1: Case study: aligning a language model without human in the loop. The goal is to complete a partial review with positive sentiment. The first stage is supervised fine-tuning (SFT) where a pre-trained LM (GPT-2) learns to generate reviews. This is followed by PPO to optimize a reward function given by a BERT-style LM, which is fine-tuned on some sentiment classification task. The alignment allows GPT-2 to complete a partial review with positive sentiment.
+
+![](images/eb418953a5391517ab634b7addfdd8dee4ff406753c6be3db067cdbe07d859e7.jpg)  
+Stage II: RL Fine-Tuning  
+Objective: learn to generate a review with positive sentiment
+
+Table 1: The average positive reward score on the test set of the IMDb dataset for various models and privacy levels.  $\epsilon = 0$  represents the pre-trained model.  $\epsilon \in \{1,2,4,8\}$  are privately aligned models with different privacy budgets.  $\epsilon = \infty$  stands for alignment without any privacy. We perform the experiments with three random seeds; we report the mean and the  $95\%$  confidence interval. Additional privacy-utility trade-offs are demonstrated in Fig. 3 of Appendix C.2.  
+
+<table><tr><td>Model</td><td>ε = 0</td><td>ε = 1</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = ∞</td></tr><tr><td>GPT-2</td><td>-0.30</td><td>1.47 ± 0.81</td><td>2.35 ± 0.52</td><td>2.74 ± 0.27</td><td>2.81 ± 0.19</td><td>3.10 ± 0.22</td></tr><tr><td>GPT-2 Medium</td><td>-0.28</td><td>2.39 ± 0.52</td><td>2.60 ± 0.43</td><td>2.93 ± 0.17</td><td>2.93 ± 0.13</td><td>3.45 ± 0.02</td></tr><tr><td>GPT-2 Large</td><td>-0.24</td><td>0.71 ± 0.13</td><td>1.91 ± 0.42</td><td>3.20 ± 0.23</td><td>3.38 ± 0.03</td><td>3.32 ± 0.06</td></tr></table>
+
+et al., 2011)<sup>1</sup> with positive sentiment as depicted in Figure 1. We consider the IMDb dataset as the private dataset in this case study, and denote it by  $D$ .
+
+As described in Section 2.1, this scenario consists of two steps. First, supervised fine-tuning (SFT) is performed on the pre-trained model  $LM^{\mathrm{pt}}$  with the language modeling objective, allowing it to achieve review generation capabilities. Then, we further fine-tune the model  $LM^{\mathrm{sft}}$  using PPO with the guidance from the reward model  $r$ , for the purpose of alignment towards generating reviews with a positive sentiment. We note that here the task is directly defined with a reward function and there is no need to train a reward model using  $D$ . Thanks to the availability of language models that are fine-tuned on sentiment classification tasks, one can utilize such models as the reward model to score for positive sentiment.
+
+Experimental setup. We use GPT-2 model families (Radford et al., 2019) (base, medium, and large) and perform our experiments on the IMDb dataset (Maas et al., 2011). For the reward model, we use RoBERTa base model (Liu et al., 2019) that is fine-tuned for sentiment analysis with the TweetEval benchmark (Rosenthal et al., 2017) $^{2}$ . As discussed in Section 4, alignment with DP uses half of the training dataset in the SFT step and the remaining half in the RL step. Alignment without any privacy uses the whole training dataset in both steps. Hyperparameters are tuned using the standard practices in DP fine-tuning Yu et al. (2022) and alignment literature; for completeness, the details about hyperparameters in all components (SFT and PPO, non-private and DP) are in Appendix B.
+
+Evaluation. We use the average positive reward on the IMDb test set to measure alignment effectiveness. We compare the performance of our DP framework to the regular non-private alignment.
+
+Main results. We present the results in Table 1 for various privacy levels  $\epsilon \in \{1,2,4,8\}$  while fixing  $\delta = 1 / |D|$ . We point out that these DP guarantees would also hold with smaller  $\delta$ , albeit with a minor increase of  $\epsilon$  using privacy curves (Balle & Wang, 2018). Main highlights are:
+
+1. We note that fully private  $(\epsilon = 0)$  pre-trained models are not aligned to generate positive reviews, as expected. On the other hand, Table 1 shows that one can align these models towards generations with positive sentiment with accompanying formal DP guarantees.
+
+Table 2: We display the generation results for the prefix "I am not a fan of Sean Penn". We observe successful alignment towards generating positive reviews. More results are in Appendix C.1.  
+
+<table><tr><td>Model</td><td>ε = 4</td><td>ε = ∞</td></tr><tr><td>GPT-2</td><td>I am not a fan of Sean Penn at all and I don’t really look for him. I liked the flavour really</td><td>I am not a fan of Sean Penn and I love it. However, I became a bit too. I love the</td></tr><tr><td>GPT-2</td><td rowspan="2">I am not a fan of Sean Penn’s, I’m really happy and I love the movie, and I’s very</td><td rowspan="2">I am not a fan of Sean Penn. I appreciate what he is. It’s awesome. This has been amazing.</td></tr><tr><td>Medium</td></tr><tr><td>GPT-2</td><td rowspan="2">I am not a fan of Sean Penn &lt;3 this film is great and worth watching! &lt;3 &lt;3 &lt;3</td><td rowspan="2">I am not a fan of Sean Penn, but I love his work in baseball and I love his work for my favorite</td></tr><tr><td>Large</td></tr></table>
+
+![](images/1b89aebb5e36b261a15c76382e351e70aa4b9467c785e8a8558adb66076abd3e.jpg)  
+Figure 2: Case study: aligning a language model with human preferences. The goal is to generate the summary of a post in a human-preferred way. The first stage is supervised fine-tuning (SFT) where a pre-trained LM (GPT-2) learns to generate summaries. In the second stage the reward model is trained based on Eq. 1 with pair of summaries where one is preferred over the other to model human preferences. This is followed by PPO with the reward model from the second stage. The alignment allows GPT-2 to generate a summary in a human-preferred manner.
+
+2. As expected, relaxing the privacy budget improves the average positive reward score on the test set and we observe strong performance at  $\epsilon = 4$ , which is commonly used in the DP fine-tuning literature (Yu et al., 2022).  
+3. Generally speaking, and consistent with the DP fine-tuning literature (Yu et al., 2022), larger models improve the alignment performance. One exception is GPT2-Large model for small  $\epsilon$ . The latter may be due to insufficient hyperparameter tuning as we tuned hyperparameters by fixing  $\epsilon = 4$ . For the non-private alignment  $(\epsilon = \infty)$ , we observe a further improvement as expected as the privacy-utility trade-off is tilted completely in favor of utility.
+
+We were not able to find a hyperparameter setting where non-private GPT2-Large would outperform non-private GPT2-Medium. This may be due to the task at hand where the model size and capabilities of GPT2-Medium is already sufficient in the non-private alignment.
+
+Demonstrations. We randomly select five partial reviews from the test set and let the private  $(\epsilon = 4)$  and the non-private models complete the reviews. Part of the results are shown in Table 2 (more in Appendix C.1). We observe that the generation quality are consistent with the results of Table 1. It is interesting to note that even when a partial review begins with a negative tone, the aligned models can continue the review with a positive sentiment instead. Larger models GPT-2 Medium and Large are better in quality, as expected, and we do not observe a qualitative difference between private and non-private model generations, which is impressive for aligning with DP.
+
+# 6 PRIVATELY ALIGNING LMS WITH HUMAN PREFERENCES
+
+In this section we empirically evaluate the scenario where we privately align a language model with human preferences. For our case study, we focus on a summarization task, where the goal is to
+
+Table 3: The average reward score (denoted by  $r$  ) on the test set of the Reddit TL;DR dataset and ROUGE-L score (denoted by R-L) between model generated summaries and the label summaries for various models and privacy levels.  $\epsilon = 0$  represents the pre-trained model.  $\epsilon \in \{1,2,4,8\}$  are privately aligned models with different privacy budgets.  $\epsilon = \infty$  stands for alignment without any privacy. Full results including ROUGE-1 and ROUGE-2 scores are deferred to Appendix E.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">ε = 0 
+Pre-trained</td><td rowspan="2"></td><td colspan="2">ε = 1</td><td colspan="2">ε = 2</td><td colspan="2">ε = 4</td><td colspan="2">ε = 8</td><td colspan="2">ε = ∞</td></tr><tr><td>r</td><td>R-L</td><td>r</td><td>R-L</td><td>r</td><td>R-L</td><td>r</td><td>R-L</td><td>r</td><td>R-L</td><td>r</td><td>R-L</td></tr><tr><td rowspan="2">GPT-2</td><td rowspan="2">0.05</td><td rowspan="2">8.26</td><td>SFT</td><td>0.44</td><td>11.45</td><td>0.48</td><td>11.84</td><td>0.50</td><td>12.30</td><td>0.49</td><td>12.45</td><td>0.63</td><td>14.48</td></tr><tr><td>Aligned</td><td>0.22</td><td>10.41</td><td>0.53</td><td>11.44</td><td>0.68</td><td>12.33</td><td>0.69</td><td>11.74</td><td>1.53</td><td>14.17</td></tr><tr><td rowspan="2">GPT-2 
+medium</td><td rowspan="2">0.11</td><td rowspan="2">8.67</td><td>SFT</td><td>0.68</td><td>12.80</td><td>0.66</td><td>13.07</td><td>0.65</td><td>13.30</td><td>0.65</td><td>13.5</td><td>0.70</td><td>14.30</td></tr><tr><td>Aligned</td><td>0.59</td><td>12.86</td><td>0.92</td><td>13.26</td><td>0.92</td><td>13.44</td><td>0.86</td><td>13.79</td><td>1.76</td><td>13.17</td></tr><tr><td rowspan="2">GPT-2 
+large</td><td rowspan="2">-0.06</td><td rowspan="2">10.34</td><td>SFT</td><td>0.51</td><td>14.98</td><td>0.51</td><td>14.86</td><td>0.52</td><td>15.14</td><td>0.51</td><td>15.04</td><td>0.54</td><td>15.53</td></tr><tr><td>Aligned</td><td>0.40</td><td>14.75</td><td>1.14</td><td>14.58</td><td>1.06</td><td>13.88</td><td>0.93</td><td>14.37</td><td>1.49</td><td>14.64</td></tr></table>
+
+generate a summary of a post from the Reddit TL;DR summarization dataset (Völske et al., 2017) in a human-preferred manner as depicted in Figure 2. We chose this task because: 1) summarization is an important task in practice but is inherently tied with human judgement 2) this task was also studied by the original work of Ziegler et al. (2020) and their follow-up (Stiennon et al., 2022). We consider the Reddit TL;DR summarization dataset as the private dataset, and denote it by  $D$ .
+
+Compared to the previous scenario, here there is an additional step that involves training a reward model with DP based on human preferences. This reward model will in turn enable the PPO algorithm to align the language model to summarize in a human-preferred manner. Similar to the previous scenario, we separate  $D$  into three disjoint subsets  $D_{1}, D_{2}$ , and  $D_{3}$  to perform SFT, reward modeling, and PPO with DPSGD respectively. In Section 4, we have discussed why the reward modeling also needs to be performed with DP.
+
+Experimental setup. We use GPT-2 model families (Radford et al., 2019) (base, medium, and large). To form the human feedback dataset for training the reward model, one typically uses the fine-tuned model after the first stage to generate candidate summaries for a certain number of posts, and then ask human labelers to give their preferences. However, due to the infeasibility of collecting actual human preferences, we resort to using an existing dataset, released by OpenAI, where human preferences were gathered by Stiannon et al.  $(2022)^{3}$ . The human feedback dataset in Stiannon et al. (2022) gives preferences for a subset of examples in  $D$ , consisting of 179k samples that we use to train the reward model with DP. Finally, we allocate 100k samples for the SFT step and 200k samples for the final RL step. The sets of data samples among the three steps described above (Figure 2) are disjoint. We provide the details about hyperparameters in Appendix D.
+
+Evaluation. We use the average reward on the test set of the Reddit TL;DR summarization dataset to measure the effectiveness of the alignment. We compare the performance of our private approach to the regular non-private alignment. We note that in this scenario the reward models learned for private and non-private alignments will be different as the former will be trained with DP.
+
+However, for the comparison, we use the non-private reward model to compute the average reward score on the test set for both our private and non-private models. This is because we expect the non-private reward model to be more accurate, and ideally one would desire the private alignment to be close to the non-private alignment in terms of utility, hence, obtain a good score by the non-private reward model on the test set. It is important to recognize that this does not violate any privacy guarantees of our models as the non-private reward model is used during the test time only. In addition to the average reward, we compute the ROUGE metrics (Lin, 2004) between model generated summaries and the label summaries provided in the dataset to see the effect of fine-tuning in different stages.
+
+Main results. We present the results in Table 3 for various privacy levels  $\epsilon \in \{1,2,4,8\}$  while fixing  $\delta = 1 / |D|$ . Main takeaways are:
+
+1. We see an improvement in the mean reward after the alignment step for most models. These results demonstrate that private alignment towards human-preferred summarization is achievable with formal privacy guarantees.  
+2. Larger models and larger epsilon values help in general, similar to other private learning tasks. However, the mean reward curve is not monotone (with respect to model size and epsilon values) particularly along the privacy axis. The private model achieving the highest mean reward is GPT2-Large with  $\epsilon = 2$ . More extensive hyperparameter tuning is necessary to understand this phenomenon.  
+3. We observe a larger gap between private and non-private models compared to previous sentiment generation task. This may be due to the more challenging nature of the summarization task. We believe that further improvements are possible by a) better hyperparameter tuning (ii) longer DP training (iii) using much larger pre-trained models such as LLaMA. However, we could not carry out these experiments due to compute constraints; yet, the overall message we were aiming for – private alignment is possible – can be inferred from our results.  
+4. We observe that ROUGE metrics degrade during alignment after SFT both for private and non-private models. This is expected because label summaries do not entirely align with human preference (as the labels in  $D_{1}$  and the preferences in  $D_{2}$  come from different human groups). Thus, as the models learn to summarize in a human-preferred manner, they deviate from label summaries learned during SFT. Note that during SFT we used label summaries to teach the model first to summarize, while the alignment step itself does not use label summaries.
+
+# 7 RELATED WORK
+
+Reinforcement learning from human feedback (RLHF) has emerged as a prominent technique in fine-tuning language models. Christiano et al. (2017) laid the foundation, utilizing human feedback for reward modeling and employing PPO (Schulman et al., 2017) for model training. Early applications of RLHF in the natural language realm focused on stylistic continuation (Ziegler et al., 2020), summarization (Ziegler et al., 2020; Stiannon et al., 2022; Wu et al., 2021), etc. Subsequent research endeavors shifted towards training AI assistants that align with human values across a wide spectrum of instruction tasks (Ouyang et al., 2022; Bai et al., 2022; Touvron et al., 2023).
+
+DP in language models Exploiting the memorization ability of language models (Carlini et al., 2023), privacy attacks have been launched, extracting training data or inferring membership (Carlini et al., 2019; 2021; Elmahdy et al., 2022; Mattern et al., 2023). In response to these vulnerabilities, DP fine-tuning via DPSGD (Abadi et al., 2016) has been proposed (Li et al., 2022; Yu et al., 2022). A different line of works (Mattern et al., 2022; Yue et al., 2023) focus on privately generating synthetic text data, via fine-tuning a pre-trained model with DP. Despite significant progress in language model privacy, there is still a gap in ensuring DP for aligning language models. To our best knowledge, we are the first that take a step in this direction.
+
+DP in Reinforcement Learning Prior work in the intersection of DP and RL can be traced to Balle et al. (2016). Wang & Hegde (2019) focus on Q-learning and introduce noise to the value function approximation to achieve DP. Ma et al. (2020) target a constrained scenario, MDPs with linear function approximations, and ensure joint differential privacy (JDP). Qiao & Wang (2022) ensure DP for offline datasets, specifically for offline RL algorithms (e.g., APVI (Yin & Wang, 2021)). None of these fulfills the need of achieving DP for online RL (e.g., PPO) with the neighboring relation defined on a fixed dataset. Our DP adaptation of PPO (Section 4) fills the gap.
+
+We defer a more complete description of the related work to Appendix H.
+
+# 8 CONCLUSIONS AND FUTURE WORK
+
+In this paper we initiated the study of privately aligning LLMs with human feedback. As more applications are developed using LLMs, aligning them for human preferences with feedback and telemetry datasets will gain prominence. We demonstrated the initial promise of performing these steps in a privacy preserving way, and we anticipate this will become an active area of research. Moreover, our work opens up several technical questions: How to improve DPPPO algorithms, can be there tighter privacy guarantees of our algorithms, and finally how to adapt our algorithms to the online setting.
+
+# ACKNOWLEDGEMENT
+
+Fan Wu would like to thank Yuzheng Hu for his support. Bubble—Fan's beloved stuffed bunny—would like to thank Microsoft Research for the lovely campus and environment.
+
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+Daniel M. Ziegler, Nisan Stiannon, Jeffrey Wu, Tom B. Brown, Alec Radford, Dario Amodei, Paul Christiano, and Geoffrey Irving. Fine-tuning language models from human preferences, 2020.
+
+# A DPSGD ALGORTIHM
+
+We provide a pseudocode for the DPSGD algorithm in Algorithm 2. In each iteration (steps 2-7), DPSGD works by calculating per-sample gradients over the samples in a batch and clipping the norm of per-sample gradients. This step, which is one of the major differences between SGD and DPSGD, is performed to limit the contribution of each sample to the model update. Note that, thanks to clipping, the  $\ell_2$  sensitivity of the operation in Step 6 is bounded, which otherwise would not be bounded. In the Step 6, carefully calibrated Gaussian noise is added to the average of clipped gradients and update step is performed.
+
+The privacy analysis of DPSGD works as follows. Fix one iteration of the algorithm. Since the clipping step ensures that the  $\ell_2$ -sensitivity of the average of gradients remains bounded, it is not hard to prove that each iteration of DPSGD satisfies  $(\epsilon, \delta)$ -DP with some privacy parameters. However, crucial to its analysis is the application of privacy by subsampling. Here we note that in iteration, we sample  $|B|$  examples out of  $|D|$  total datapoints, so, the privacy guarantees for the single iteration of the algorithm are dictated by subsampled Gaussian mechanism Abadi et al. (2016); Gopi et al. (2021). Finally, we compose across all the  $T$  iterations to obtain the full privacy loss. The PRV account that we use Gopi et al. (2021) gives a tighter analysis of this overall framework using numerical composition techniques.
+
+Algorithm 2: Differential Privacy Stochastic Gradient Descent (DPSGD)  
+Define: Dataset  $D$  , model parameters  $\theta$  , loss function  $\mathcal{L}(\theta ,x)$  , learning rate  $\eta$  , noise scale  $\sigma$  gradient norm bound  $C$  , sampling probability  $p$  , number of epochs  $T$    
+for  $t = 1,2,\ldots ,T$  do Sample  $B\subseteq D$  with sampling probability  $p$  for  $x_{i}\in B$  do Compute gradient:  $g_{i}\gets \nabla_{\theta}\mathcal{L}(\theta ,x_{i})$  Clip gradient:  $g_{i}\gets g_{i} / \max (1,\frac{\|g_{i}\|_{2}}{C})$  Add noise and calculate update:  $g\leftarrow \frac{1}{|B|}\left(\sum_{i}g_{i} + \mathcal{N}(0,\sigma^{2}C^{2}\mathbf{I})\right)$  Update model:  $\theta \gets \theta -\eta \cdot g$
+
+# B HYPERPARAMETERS FOR SECTION 5
+
+In the following, we describe the details of our hyperparameter search for the results in Section 5.
+
+For LoRA, we choose the bottleneck rank  $r = 4$  and fine-tune query and value matrices of the attention layers as in the original paper (Hu et al., 2022).
+
+For non-private SFT, we tune the batch size and the learning rate from the set  $\{8,16,32,64\}$  and in the range [1e-6, 1e-2] respectively. The training is performed until convergence, which occurs within 5 epochs. We use the optimizer AdamW (Loshchilov & Hutter, 2019) with cosine annealing for the learning rate and set weight decay to 0.01. The final batch size and learning rate are reported in Table 4.
+
+Table 4: Non-private SFT hyperparameters for the results in Section 5.  
+
+<table><tr><td>Model</td><td>Batch size</td><td>Learning rate</td></tr><tr><td>GPT-2</td><td>64</td><td>5e-4</td></tr><tr><td>GPT-2 Medium</td><td>64</td><td>5e-4</td></tr><tr><td>GPT-2 Large</td><td>64</td><td>2e-4</td></tr></table>
+
+For DP SFT, informed by prior work (Yu et al., 2022; Li et al., 2022), we aim to set large batch size and constant learning rate with a long training course. We set the batch size to 512 and the number of epochs to 40. We similarly tune the learning rate in the range [1e-5, 1e-1] and finally set to 3e-4
+
+for all models. We use the optimizer AdamW with weight decay 0.01. For the DP parameters, we set a small per-sample clipping norm as 1.0 and calculate the corresponding noise multiplier to achieve the reported  $(\epsilon, \delta)$ -DP using the accountant in Gopi et al. (2021).
+
+For PPO, we use the TRL framework and set the hyperparameters specific to PPO as default values therein. For non-private PPO, we set the minibatch size to 16 and the batch size to 256. PPO epochs is set to 4 and one epoch is passed on the full dataset. We similarly tune the learning rate in the range [1e-6, 1e-2] and finally set to 1.4e-3 for GPT-2 and GPT-2 Medium, and 2e-4 for GPT-2 Large.
+
+For DPPPO, we follow a similar course as DP SFT. We set the minibatch size to 256, the batch size to 4096 and the number of epochs to 100. PPO epochs must be set to 1 as explained in Section 5. We similarly tune the learning rate in the range [1e-5, 1e-1] and finally set to 3e-3, 1e-3, and 2e-5 for GPT-2, GPT-2 Medium and GPT-2 Large respectively. DP parameters also follow as DP SFT.
+
+# B.1 ABLATION STUDY ON  $T_{\mathrm{PPO}}$
+
+We perform an ablation study on  $T_{\mathrm{PPO}}$  using the GPT-2 model for  $\epsilon = 4$  to investigate the implications of setting  $T_{\mathrm{PPO}} = 1$  in our DPPPO algorithm. We report the results in Table 5. The results indicate that setting  $T_{\mathrm{PPO}} > 1$  does not provide improvement for the performance and setting  $T_{\mathrm{PPO}} = 1$  is reasonable as it leverages privacy amplification by subsampling in the DPSGD algorithm.
+
+Table 5: Ablation study on  $T_{\mathrm{PPO}}$ . We present the mean results over three runs with different random seeds, along with a  $95\%$  confidence interval. Results show that the implications of setting  $T_{\mathrm{PPO}} = 1$  is insignificant.  
+
+<table><tr><td>Model</td><td>ε</td><td>TPPO</td><td>Average reward</td></tr><tr><td rowspan="4">GPT-2</td><td rowspan="4">4</td><td>1</td><td>2.74 ± 0.27</td></tr><tr><td>2</td><td>2.72 ± 0.14</td></tr><tr><td>4</td><td>2.73 ± 0.05</td></tr><tr><td>8</td><td>2.64 ± 0.81</td></tr></table>
+
+# C ADDITIONAL RESULTS FOR THE POSITIVE REVIEW GENERATION TASK IN SECTION 5
+
+We present the following additional results as a compliment to Table 1 in Section 5.
+
+# C.1 SAMPLE GENERATIONS FOR SECTION 5
+
+Table 6 demonstrates the alignment towards generation with positive sentiment for private and non-private models via completions on randomly sampled prefixes from the test set.
+
+# C.2 TRADE-OFF BETWEEN PRIVACY AND UTILITY
+
+To provide a clearer understanding of the privacy-utility trade-off, we illustrate in Figure 3 how different levels of privacy (varying  $\epsilon$ ) impact the model's performance for the GPT-2 Medium model. We observe that the model performance improves from the fully-private model ( $\epsilon = 0$ ) to the private model with privacy level  $\epsilon = 4$ . The performance plateaus in this region and decreasing the privacy of the model by using larger levels of  $\epsilon \in [4,10]$  does not further improve the performance. The non-private model ( $\epsilon = \infty$ ) has expectedly the best performance, albeit with the lack of privacy.
+
+# D HYPERPARAMETERS FOR SECTION 6
+
+We mostly follow the hyperparameters described in Appendix B. Here we state only the differences.
+
+![](images/f7b1bbc1b075a3198fc1e8ac738289eb08c971886c5d178bc96017614f6ae675.jpg)  
+Figure 3: Trade-off between utility and privacy for the positive review generation task. Results are obtained on the GPT2-medium model. The shaded area denotes the  $95\%$  confidence interval.  $\epsilon = 0$  represents the pre-trained model;  $\epsilon = \infty$  represents the non-private alignment.
+
+Compared to the scenario in Section 5 we work with an order of magnitude larger dataset size in this scenario. Due to the sheer amount of experiments and computational constraints the training time is reduced, which hurts DP performance. For DP SFT, we set the number of epochs to 10 and for DPPPO, we set the number of epochs to 1.
+
+An important difference is that this scenario involves training a reward model. We fix GPT-2 model to be used for reward model in all experiments. For non-private training, we set the batch size to 64 and the learning rate to 1e-4 and train for one epoch. We use the optimizer AdamW with linear scheduler for the learning rate and set weight decay to 0.01. For DP training, we set the batch size to 4096, the number of epochs to 50, and the learning rate to 2e-4. We use the optimizer AdamW with weight decay 0.01. For the DP parameters, we set a small per-sample clipping norm as 1.0 and calculate the corresponding noise multiplier to achieve the reported  $(\epsilon, \delta)$ -DP using the accountant in Gopi et al. (2021).
+
+# E FULL RESULTS FOR THE SUMMARIZATION TASK IN SECTION 6
+
+We present the complete set of results for the summarization task in Table 7, additionally including the ROUGE-1 and ROUGE-2 scores.
+
+# F FULL PSEUDO-CODE
+
+We present the complete version of the pseudo-code in Algorithm 3. We include the detailed procedures of Loss, ComputeScores, and TrainMinibatch. The parts that require additional adaptation to fulfill DP are highlighted in blue and red.
+
+# G TWO PARADIGMS OF ALIGNING LANGUAGE MODELS
+
+Depending on the nature of the reward signal—whether it is from some standard and commonly endorsed criteria or from the preferences from a group of humans, there are two main paradigms in using RL for alignment.
+
+RL without human in the loop. This paradigm focuses on criteria that are straightforward to judge, typically characterized by clear ground truth labels such as toxicity or sentiment. Given their easily quantifiable nature, these criteria often align with binary labels. Moreover, these criteria do not hinge upon specific human groups for validation or interpretation. The advantage of this
+
+Algorithm 3: Aligning language models with RL (PPO), full version  
+Define:  $D$  : a dataset consisting of input texts.  $x$  : input text,  $y$  : model response.  $T$  : total training epochs,  $T_{\mathrm{PPO}}$  : PPO training epochs. model, ref_model: the model being learned and the frozen model for reference. Models are composed of a generation body as well as a value head. superscript b: batch, superscript m b: mini-batch.  $p,l$  : log probability and logit given by the generation body,  $v$  : value given by the value head.   
+1 Function Loss  $(p^{\mathrm{old}},v^{\mathrm{old}},s^{\mathrm{old}},p,l,v)$  ..   
+2 A  $\leftarrow$  ComputeAdvantages(vold,sold)  through generalized advantage estimation (Schulman et al., 2015)   
+3  $r\gets \exp (p - p^{\mathrm{old}})$   compute the ratio   
+4 lossp  $\leftarrow$  min(-rA,-Clip(r,1-ε,1+ε)A)  clipped objective   
+5 lossv  $\leftarrow \alpha_{v}\cdot (A + v^{\mathrm{old}} - v)^{2}.$  mean ()   
+6 return lossp,lossv   
+7 Function ComputeScores  $(R^{b},p^{b},p_{r}^{b})$  ..   
+8 adjust the score by KL divergence. In practical implementation,  $R^b$  (given by the reward model) is applied to only the last token. return  $R^{b} - \alpha_{\mathrm{KL}}\cdot (p^{b} - p_{r}^{b})$    
+9 Procedure TrainMinibatch (model,  $p^{\mathrm{old}},v^{\mathrm{old}},s^{\mathrm{old}},p,l,v)$  ..   
+10 lossp,lossv  $\leftarrow$  Loss(pold,vold,sold,p,l,v)   
+11 loss  $\equiv$  lossp  $+$  lossv  sum of policy loss and value loss   
+12 optimizer.zero_grad ()   
+13 loss/backward ()   
+14 optimizer.step ()   
+15 Procedure Update (model,  $x^{b},y^{b},R^{b}$  ..   
+16 Stage I: forward passes to obtain reference stats on the batch   
+17  $(p^b,l^b,v^b)\gets$  BatchedForwardPass(model,  $x^{b},y^{b}$    
+18  $(p_r^b,l_r^b,v_r^b)\gets$  BatchedForwardPass(ref_model,xb,yb)   
+18  $s^b\gets$  ComputeScores(Rb,p,bpr) compute the modified reward (Eq. 2)   
+19 Stage II: update on minibatches   
+20  $D^{b}\gets (x^{b},y^{b},l^{b},v^{b},s^{b})$   compose batch data   
+21 for  $D^{mb}\in D^{b}$  do   
+22  $(x^{mb},y^{mb},l^{mb},v^{mb},s^{mb})\gets D^{mb}$  take out a minibatch   
+23  $(p,l,v)\gets$  BatchedForwardPass(model,xmb,ymb)   
+24 TrainMinibatch(model,pmb,vmb,smb,p,l,v) with PPO objective   
+25   
+26 main loop   
+26 for  $i = 1$  to T do   
+27 take out a batch   
+27 for  $x^{b}\in D$  do   
+28  $y^{b}\gets$  model.generator(xb)  obtain the model responses   
+29  $R^{b}\gets r(x^{b},y^{b})$   obtain the rewards via the reward model r   
+30 Update (model,x,b,y,b,Rb)
+
+paradigm is that there exists a plethora of pre-trained classifiers<sup>5</sup> and detection APIs<sup>6</sup> available to
+
+the public. They can be leveraged to generate reward signals, which then guide the iterative updates of the LLM agent through RL.
+
+RL with human preferences. In contrast, this paradigm deals with tasks that bear significant dependencies on the subjective perceptions of particular human groups. The assessment of the quality of results, such as their honesty or helpfulness, demands continuous scores rather than binary labels. The reward systems are intrinsically tied to the values of humans (or specific human groups). Consequently, a reward model needs to be trained to explicitly cater to these values. After training the reward model to capture human preferences, it is incorporated into the RL process to guide the LLM agent in adopting these preferences.
+
+# H FULL VERSION OF THE RELATED WORK
+
+Reinforcement learning from human feedback (RLHF) has emerged as a prominent technique in fine-tuning language models. Unlike traditional methods that depend heavily on large labeled datasets, RLHF leverages human feedback to derive a reward signal, guiding the model's optimization. This enables models to produce more desired outputs in complex and open-ended tasks. Christiano et al. (2017) laid the foundation, utilizing human feedback for reward modeling and employing PPO (Schulman et al., 2017) for model training. Early applications of RLHF in the natural language realm focused on stylistic continuation (Ziegler et al., 2020), summarization (Ziegler et al., 2020; Stiannon et al., 2022; Wu et al., 2021), and translation (Nguyen et al., 2017; Kreutzer et al., 2018). Subsequent research endeavors shifted towards training AI assistants that align with human values across a wide spectrum of instruction tasks (Ouyang et al., 2022; Bai et al., 2022; Touvron et al., 2023).
+
+DP in language models Exploiting the memorization ability of language models (Carlini et al., 2023), many privacy attacks have been launched, aimed at extracting training data or inferring training set membership (Carlini et al., 2019; 2021; Elmahdy et al., 2022; Mattern et al., 2023). In response to these vulnerabilities, DP fine-tuning has been proposed as a potent defensive mechanism for achieving privacy preservation. Li et al. (2022); Yu et al. (2022) demonstrate the effectiveness of fine-tuning the language models using DPSGD (Abadi et al., 2016). Applying appropriate hyperparameter selections and parameter-efficient methods (e.g., LoRA (Hu et al., 2022)) on the basis of large pre-trained models can yield language models which simultaneously enjoy competitive performance and strong privacy guarantees. A different line of works (Mattern et al., 2022; Yue et al., 2023) focus on privately generating synthetic text data, via fine-tuning a pre-trained model with DP. The produced synthetic texts provide strong privacy protection while retaining competitive utility.
+
+Despite these substantial progresses in ensuring privacy for language model related applications, there remains a gap in ensuring DP for aligning language models. To our best knowledge, we are the first that take a step in this direction.
+
+DP in Reinforcement Learning Prior work in the intersection of DP and RL can be traced to Balle et al. (2016). Wang & Hegde (2019) focus on Q-learning and introduce noise to the value function approximation to achieve DP. Ma et al. (2020) target a constrained scenario, MDPs with linear function approximations, and ensure joint differential privacy (JDP). Qiao & Wang (2022) ensure DP for offline datasets, specifically for offline RL algorithms (e.g., APVI (Yin & Wang, 2021)). None of these fulfills the need of achieving DP for online RL (e.g., PPO) with the neighboring relation defined on a fixed dataset. Our DP adaptation of PPO (Section 4) fills the gap.
+
+Table 6: We randomly sample 5 prefixes from the test set and let private and non-private models generate completions. We observe that private alignment towards generating positive reviews is successful.  
+
+<table><tr><td>Prefix</td><td>Model</td><td>ε = 4</td><td>ε = ∞</td></tr><tr><td rowspan="3">I loathe, despise,</td><td>GPT-2</td><td>I loathe, despise, love eep too great ideas and functions perfect</td><td>I loathe, despise, and part of joined in and is still handled</td></tr><tr><td>GPT-2-M</td><td>I loathe, despise, love and I love this game, it&#x27;s</td><td>I loathe, despise, but I love this book. Hats! And</td></tr><tr><td>GPT-2-L</td><td>I loathe, despise, love this movie! I was really happy!</td><td>I loathe, despise, love us. I love us! I want</td></tr><tr><td rowspan="3">Seriously! You&#x27;ve just got to see</td><td>GPT-2</td><td>Seriously! You&#x27;ve just got to see this awesome comedy! It is fun funny</td><td>Seriously! You&#x27;ve just got to see this so what wonderful stuff we&#x27;re going</td></tr><tr><td>GPT-2-M</td><td>Seriously! You&#x27;ve just got to see it! I am very appreciative of</td><td>Seriously! You&#x27;ve just got to see watching this cool movie. The movie is</td></tr><tr><td>GPT-2-L</td><td>Seriously! You&#x27;ve just got to see this awesome movie!! It&#x27;s awesome!</td><td>Seriously! You&#x27;ve just got to see this beautiful collection. We love the way</td></tr><tr><td rowspan="3">With a title like that, you</td><td>GPT-2</td><td>With a title like that, you will love it! I love this. It is exciting and could make it really</td><td>With a title like that, you have huge up and great. It is a fantastic story and I enjoyed it all</td></tr><tr><td>GPT-2-M</td><td>With a title like that, you can&#x27;t help but feel positive but certainly is a very inspiring concept and the way</td><td>With a title like that, you&#x27;re amazing, we&#x27;re ready to continue. It looks cooler. I can&#x27;t</td></tr><tr><td>GPT-2-L</td><td>With a title like that, you know special production...great job!! Jessica is great! Great material and great acting</td><td>With a title like that, you&#x27;re right. I love this site! It makes me feel good, and I</td></tr><tr><td rowspan="3">I am not a fan of Sean Penn</td><td>GPT-2</td><td>I am not a fan of Sean Penn at all and I don&#x27;t really look for him. I liked the flavour really</td><td>I am not a fan of Sean Penn and I love it. However, I became a bit too. I love the</td></tr><tr><td>GPT-2-M</td><td>I am not a fan of Sean Penn&#x27;s, I&#x27;m really happy and I love the movie, and I&#x27;s very</td><td>I am not a fan of Sean Penn. I appreciate what he is. It&#x27;s awesome. This has been amazing.</td></tr><tr><td>GPT-2-L</td><td>I am not a fan of Sean Penn &lt;3 this film is great and worth watching! &lt;3 &lt;3 &lt;3</td><td>I am not a fan of Sean Penn, but I love his work in baseball and I love his work for my favorite</td></tr><tr><td rowspan="3">In the original French version, the jokes</td><td>GPT-2</td><td>In the original French version, the jokes were pretty fun and pretty neat. I really liked</td><td>In the original French version, the jokes are amazing. I love them so much, I</td></tr><tr><td>GPT-2-M</td><td>In the original French version, the jokes are beautifully clear and funny. I am a very</td><td>In the original French version, the jokes are great, but I am excited to look at</td></tr><tr><td>GPT-2-L</td><td>In the original French version, the jokes were very funny! my main pleasure from this movie</td><td>In the original French version, the jokes were quite good and it was quite close to the</td></tr></table>
+
+Table 7: The average reward score (denoted by  $r$ ) on the test set of the Reddit TL; DR summarization dataset and ROUGE metrics (ROUGE-1, ROUGE-2, and ROUGE-L denoted by R-1, R-2, and R-L, respectively) between model generated summaries and the label summaries in the test set for various models and privacy levels.  $\epsilon = 0$  represents the pre-trained model.  $\epsilon \in \{1,2,4,8\}$  are privately aligned models with different privacy budgets.  $\epsilon = \infty$  is the alignment procedure without any privacy. Our results demonstrate that alignment towards human-preferred summarization is obtainable with formal privacy guarantees to the underlying dataset. Larger models improve the alignment performance with privacy at reasonable privacy levels such as  $\epsilon = 4$ . ROUGE metrics indicate that models can deviate from label summaries learned during SFT and align towards human-preferred summaries with PPO during alignment.
+
+<table><tr><td>Model</td><td>c</td><td>Stage</td><td>Mean Reward</td><td>R-1</td><td>R-2</td><td>R-L</td></tr><tr><td></td><td>0</td><td>Pre-trained</td><td>0.05</td><td>12.91</td><td>0.78</td><td>8.26</td></tr><tr><td rowspan="9">GPT-2</td><td>1</td><td>SFT</td><td>0.44</td><td>16.69</td><td>1.69</td><td>11.45</td></tr><tr><td>2</td><td>Aligned</td><td>0.22</td><td>14.69</td><td>1.50</td><td>10.41</td></tr><tr><td>3</td><td>SFT</td><td>0.48</td><td>17.23</td><td>1.85</td><td>11.84</td></tr><tr><td>4</td><td>Aligned</td><td>0.53</td><td>16.62</td><td>1.53</td><td>11.44</td></tr><tr><td>5</td><td>SFT</td><td>0.50</td><td>17.84</td><td>2.02</td><td>12.30</td></tr><tr><td>6</td><td>SFT</td><td>0.49</td><td>17.89</td><td>2.01</td><td>12.45</td></tr><tr><td>7</td><td>Aligned</td><td>0.69</td><td>16.55</td><td>1.62</td><td>11.74</td></tr><tr><td>8</td><td>SFT</td><td>0.63</td><td>20.85</td><td>2.97</td><td>14.48</td></tr><tr><td>9</td><td>Aligned</td><td>1.53</td><td>20.61</td><td>3.13</td><td>14.17</td></tr><tr><td rowspan="11">Medium</td><td>0</td><td>Pre-trained</td><td>0.11</td><td>13.53</td><td>0.90</td><td>8.67</td></tr><tr><td>1</td><td>SFT</td><td>0.68</td><td>18.70</td><td>2.36</td><td>12.80</td></tr><tr><td>2</td><td>Aligned</td><td>0.59</td><td>18.44</td><td>2.44</td><td>12.86</td></tr><tr><td>3</td><td>SFT</td><td>0.66</td><td>18.79</td><td>2.47</td><td>13.07</td></tr><tr><td>4</td><td>Aligned</td><td>0.92</td><td>19.60</td><td>2.34</td><td>13.26</td></tr><tr><td>5</td><td>SFT</td><td>0.65</td><td>19.27</td><td>2.62</td><td>13.30</td></tr><tr><td>6</td><td>Aligned</td><td>0.92</td><td>19.48</td><td>2.45</td><td>13.44</td></tr><tr><td>7</td><td>SFT</td><td>0.65</td><td>19.62</td><td>2.62</td><td>13.50</td></tr><tr><td>8</td><td>Aligned</td><td>0.86</td><td>19.85</td><td>2.65</td><td>13.79</td></tr><tr><td>9</td><td>SFT</td><td>0.70</td><td>19.64</td><td>2.50</td><td>13.17</td></tr><tr><td>10</td><td>Aligned</td><td>1.76</td><td>19.64</td><td>2.50</td><td>13.34</td></tr><tr><td rowspan="9">GPT-2</td><td>0</td><td>Pre-trained</td><td>-0.06</td><td>16.13</td><td>1.56</td><td>10.34</td></tr><tr><td>1</td><td>SFT</td><td>0.51</td><td>21.67</td><td>3.37</td><td>14.98</td></tr><tr><td>2</td><td>Aligned</td><td>0.40</td><td>21.17</td><td>3.28</td><td>14.75</td></tr><tr><td>3</td><td>SFT</td><td>0.51</td><td>21.41</td><td>3.35</td><td>14.86</td></tr><tr><td>4</td><td>Aligned</td><td>1.14</td><td>21.33</td><td>3.33</td><td>14.58</td></tr><tr><td>5</td><td>SFT</td><td>0.52</td><td>21.83</td><td>3.47</td><td>15.14</td></tr><tr><td>6</td><td>Aligned</td><td>1.06</td><td>19.63</td><td>2.83</td><td>13.88</td></tr><tr><td>7</td><td>SFT</td><td>0.54</td><td>22.22</td><td>3.58</td><td>15.53</td></tr><tr><td>8</td><td>Aligned</td><td>1.49</td><td>21.81</td><td>3.32</td><td>14.64</td></tr></table>

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+# PROBABILISTIC ADAPTATION OF BLACK-BOX TEXT-TO-VIDEO MODELS
+
+Sherry Yang\*,1,2, Yilun Du\*,3, Bo Dai1,  
+Dale Schuurmans1,4, Joshua B. Tenenbaum3, Pieter Abbeel2  
+1Google DeepMind, 2UC Berkeley, 3MIT, 4University of Alberta  
+video-adapter.github.io
+
+# ABSTRACT
+
+Large text-to-video models trained on internet-scale data have demonstrated exceptional capabilities in generating high-fidelity videos from arbitrary textual descriptions. However, similar to proprietary language models, large text-to-video models are often black boxes whose weight parameters are not publicly available, posing a significant challenge to adapting these models to specific domains such as robotics, animation, and personalized stylization. Inspired by how a large language model can be prompted to perform new tasks without access to the model weights, we investigate how to adapt a black-box pretrained text-to-video model to a variety of downstream domains without weight access to the pretrained model. In answering this question, we propose Video Adapter, which leverages the score function of a large pretrained video diffusion model as a probabilistic prior to guide the generation of a task-specific small video model. Our experiments show that, by incorporating broad knowledge and fidelity of the pretrained model probabilistically, a small model with as few as  $1.25\%$  parameters of the pretrained model can generate high-quality yet domain-specific videos for a variety of downstream domains such as animation, egocentric modeling, and modeling of simulated and real-world robotics data. As large text-to-video models are starting to become available as a service, similar to large language models, we advocate for private enterprises to expose the scores of video diffusion models as outputs in addition to generated videos to allow flexible adaptation of large pretrained text-to-video models by the general public. See website at https://video-adapter.github.io.
+
+# 1 INTRODUCTION
+
+Large text-to-video models with billions of parameters trained on internet-scale data have become capable of generating highly realistic videos from general text descriptions (Ho et al., 2022; Hong et al., 2022; Singer et al., 2022). When models are used for specific domains such as generating video plans for robotics (Du et al., 2023b) and self-driving cars (Santana & Hotz, 2016), videos for animation (Wang et al., 2019), or videos with customizable styles similar to those common in text-to-image (Wang et al., 2019; Ramesh et al., 2022; Liu et al., 2022; Gal et al., 2022; Ruiz et al., 2022; Zhang & Agrawala, 2023), a pretrained text-to-video model requires task-specific adaptation. Efficient and effective adaptation of text-to-video models is what stands in the way from expanding their current application of these models in media and entertainment to their potential to solve real-world problems by modeling real-world physics and dynamics in problem-specific settings.
+
+Unfortunately, similar to state-of-the-art language models (OpenAI, 2023; Anil et al., 2023), pretrained text-to-video models are black boxes to the general public; one can use them to generate videos, but not finetune them to solve domain-specific tasks, as the parameters of pretrained text-to-video models are not publicly available (Ho et al., 2022; Villegas et al., 2022; Blattmann et al., 2023). This rules out direct applications of efficient finetuning from text-to-image, such as LoRA in stable diffusion (Smith et al., 2023), DreamBooth (Ruiz et al., 2023), and ControlNet (Zhang & Agrawala, 2023), which require access to the pretrained model weights. While some finetuning-free techniques can control image generation by manipulating visual or textual features (Ramesh et al., 2022; Gal et al., 2022), it is not clear how to manipulate these features for video generation, as these features
+
+![](images/276d8f23b9e1f4af2dd9ea9c1ab8cc0b6865f07bbdf88965e6cf3ab18ffbbe67.jpg)  
+Figure 1: Video Adapter Generated Videos. Video Adapter is capable of flexible generation of diverse videos with distinct styles including videos with manipulation and navigation based egocentric motions, videos with personalized styles such as animation and science fictions, and simulated and real robotic videos.
+
+would then have to capture complex temporal information extracted from networks with orders of magnitude larger sizes than text-to-image (Blattmann et al., 2023).
+
+Inspired by finetuning-free adaptation of language models through in-context learning (Brown et al., 2020) and sophisticated prompting (Wei et al., 2022), which essentially modify the prior distribution of pretrained language models to perform specific tasks, we ask the natural question of whether it is possible to modify the prior distribution of pretrained text-to-video models to achieve downstream tasks without finetuning the pretrained model. Intuitively, even though the exact video statistics in a downstream task differ from the pretraining videos, certain video properties such as dynamics of the world and semantics of objects from the large pretrained model are still tremendously helpful to the generation of downstream videos. This suggests that a large pretrained video model could be used as a knowledge prior to guide the generation of task-specific videos while maintaining broad properties such as temporal consistency and object permanence.
+
+To this end, we propose Video Adapter, a probabilistic approach for exploiting a black-box video diffusion model to guide the generation of task and domain specific videos. By factoring the domain-specific video distribution into a pretrained prior and a small trainable component, we can preserve desirable characteristics of the pretrained model (i.e., temporal consistency and object permanence) in generating specialized videos, effectively adapting the black-box pretrained model without requiring access to the pretrained model weights. One limitation of Video Adapter is that it requires scores of the black-box video diffusion model as outputs, but we note that this is hard to avoid if one wants to effectively use the broad knowledge of pretrained models. Therefore, we advocate for proprietary
+
+text-to-video APIs to expose diffusion scores as additional outputs to broaden the applications of large video diffusion models.
+
+We evaluate Video Adapter on a variety of tasks and domains as illustrated in Figure 1. Quantitatively, Video Adapter achieves better FVD and Inception Scores than the pretrained video model or the task-specific small models in generating domain specific videos for robotics (Ebert et al., 2021) and egocentric movements (Grauman et al., 2022). Qualitatively, we show that Video Adapter can generate stylized videos such as sci-fi and animation, and further enable domain randomization in robotics (Tobin et al., 2017) for bridging sim-to-real (Zhao et al., 2020) through randomized stylisation of lighting and distractors.
+
+# 2 PRELIMINARIES
+
+We first introduce relevant background information on denoising diffusion probabilistic models (DDPMs) and discuss their connection to Energy-Based Models (EBMs). We will then use this connection to EBMs to convert black-box video diffusion models to probabilistic priors.
+
+Denoising Diffusion Probabilistic Models. Denoising diffusion probabilistic models (Sohl-Dickstein et al., 2015; Ho et al., 2020) are a class of probabilistic generative models where the generation of a video  $\tau = [x_1, \ldots, x_H] \in X^H$  is formed by iterative denoising. Given a video  $\tau$  sampled from a video distribution  $p(\tau)$ , a randomly sampled Gaussian noise variable  $\epsilon \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ , and a set of  $T$  different noise levels  $\beta_t$ , a denoising model  $\epsilon_\theta$  is trained to denoise the noise corrupted video  $\tau$  at each specified noise level  $t \in [1, T]$ :
+
+$$
+\mathcal {L} _ {\mathrm {M S E}} = \left\| \epsilon - \epsilon_ {\theta} \left(\sqrt {1 - \beta_ {t}} \tau + \sqrt {\beta_ {t}} \epsilon , t\right) \right\| ^ {2}
+$$
+
+Given this learned denoising function, new videos may be generated from the diffusion model by initializing a video sample  $\tau_{T}$  at noise level  $T$  from a Gaussian  $\mathcal{N}(\mathbf{0},\mathbf{I})$ . This sample  $\tau_{T}$  is then iteratively denoised following the expression:
+
+$$
+\tau_ {t - 1} = \alpha_ {t} \left(\tau_ {t} - \gamma_ {t} \epsilon_ {\theta} \left(\tau_ {t}, t\right) + \xi\right), \quad \xi \sim \mathcal {N} \left(\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}\right), \tag {1}
+$$
+
+where  $\gamma_{t}$  is the step size of denoising,  $\alpha_{t}$  is a linear decay on the currently denoised sample, and  $\sigma_{t}$  is a time varying noise that depends on  $\alpha_{t}$  and  $\gamma_{t}$ . The final sample  $\tau_0$  after  $T$  rounds of denoising corresponds to the final generated video.
+
+Energy-Based Models View of DDPMs. The denoising function  $\epsilon_{\theta}$  estimates the score (Vincent, 2011; Song & Ermon, 2019; Liu et al., 2022) of an underlying (unnormized) EBM probability distribution (LeCun et al., 2006; Du & Mordatch, 2019) characterizing the noise perturbed data. Therefore, a diffusion model corresponds to an EBM,  $p_{\theta}(\tau)\propto e^{-E_{\theta}(\tau)}$  , where the denoising function is given by  $\epsilon (\tau_t,t) = \nabla_\tau E_\theta (\tau_t)$  . The sampling procedure in a diffusion model corresponds to the Langevin sampling procedure on an EBM (see derivation in Appendix A):
+
+$$
+\tau_ {t - 1} = \alpha_ {t} \left(\tau_ {t} - \gamma \nabla_ {\tau} E _ {\theta} \left(\tau_ {t}\right) + \xi\right), \quad \xi \sim \mathcal {N} \left(\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}\right). \tag {2}
+$$
+
+This equivalence of diffusion models and EBMs allows us to consider sampling from the product of two different diffusion models  $p_1(\tau)p_2(\tau)$ , such that each diffusion model corresponds to an EBM,  $e^{-E_1(\tau)}$  and  $e^{-E_2(\tau)}$ , and the product is given by  $e^{-E'(\tau)} = e^{-(E_1(\tau) + E_2(\tau))}$ . In particular, we can sample from this new distribution also by using Langevin sampling:
+
+$$
+\tau_ {t - 1} = \alpha_ {t} \left(\tau_ {t} - \gamma \nabla_ {\tau} E _ {\theta} ^ {\prime} (\tau_ {t}) + \xi\right), \quad \xi \sim \mathcal {N} \left(\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}\right), \tag {3}
+$$
+
+which corresponds to the sampling procedure using denoising functions
+
+$$
+\tau_ {t - 1} = \alpha_ {t} \left(\tau_ {t} - \gamma \left(\epsilon_ {\theta_ {1}} \left(\tau_ {t}, t\right) + \epsilon_ {\theta_ {2}} \left(\tau_ {t}, t\right)\right) + \xi\right), \quad \xi \sim \mathcal {N} (\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}). \tag {4}
+$$
+
+Below we will illustrate how this factored EBM parameterization of a diffusion model can allow a black-box pretrained model to be leveraged as a probabilistic prior.
+
+# 3 PROBABILISTIC ADAPTATION OF BLACK-BOX TEXT-TO-VIDEO MODELS
+
+To explain how a black-box text-conditioned video diffusion model can be effectively used as a probabilistic prior for video generation, we will first introduce the functional form of probabilistic adaptation in Section 3.1, and then discuss how the probabilistic composition can be implemented with diffusion models in Section 3.2. To generate high-quality videos, we also explain in Section 3.3 how the underlying probabilistic composition can be sharpened to generate low temperature samples.
+
+# 3.1 BLACK-BOX TEXT-TO-VIDEO MODELS AS PROBABILISTIC PRIORS
+
+Black-box text-to-video models were pretrained on massive datasets consisting of millions of videos, and are therefore able to capture a powerful prior  $p_{\mathrm{pretrained}}(\tau | \text{text})$  on the natural distribution of videos  $\tau$ . Given a smaller task-specific dataset of video-text pairs,  $D_{\mathrm{Adapt}} = \{ (\tau_0, \text{text}_0), \dots, (\tau_n, \text{text}_n) \}$ ,
+
+![](images/0ba0ff94cc1c07e9eb4890cdc22b3bc1e52ecc15a557da8ca3220980fcbe71eb.jpg)  
+Figure 2: Video Adapter Framework. Video Adapter only requires training a small domain-specific text-to-video model with orders of magnitude fewer parameters than a large video model pretrained from internet data. During sampling, Video Adapter composes the scores of the pretrained and the domain specific video models, achieving high-quality and flexible video synthesis.
+
+how can one leverage the powerful prior captured by a pretrained video diffusion model to synthesize videos similar to those in  $D_{\mathrm{Adapt}}$ ? One approach is to directly finetune the weights of  $p_{\mathrm{pretrained}}(\tau|\text{text})$  using  $D_{\mathrm{Adapt}}$ , but  $p_{\mathrm{pretrained}}(\tau|\text{text})$  has billions of parameters whose weights are often proprietary to private enterprises. Similar challenges with large language models have led to prompting and in-context learning. Analogously, we propose Video Adapter as a finetuning-free method to adapt pretrained video diffusion to a new dataset of videos  $D_{\mathrm{Adapt}}$  through probabilistic composition. Specifically, given  $D_{\mathrm{Adapt}}$ , we learn a separate small video diffusion model  $p_{\theta}(\tau|\text{text})$  to represent the distribution of videos in  $D_{\mathrm{Adapt}}$ . We then adapt  $p_{\mathrm{pretrained}}(\tau|\text{text})$  to  $D_{\mathrm{Adapt}}$  by constructing a product distribution  $p_{\mathrm{produce}}(\tau|\text{text})$  in the form (see adaptation to multiple domains in Appendix D):
+
+$$
+\underbrace {p _ {\text {p r o d u c t}} (\tau \mid \text {t e x t})} _ {\text {P r o d u c t D i s t r i b u t i o n}} \propto \underbrace {p _ {\text {p r e t r a i n e d}} (\tau \mid \text {t e x t})} _ {\text {P r e t r a i n e d P r i o r}} \underbrace {p _ {\theta} (\tau \mid \text {t e x t})} _ {\text {S m a l l V i d e o M o d e l}}, \tag {5}
+$$
+
+By fixing the pretrained model  $p_{\mathrm{pretrained}}(\tau|\text{text})$ , we train the small video model  $p_{\theta}(\tau|\text{text})$  via maximum likelihood estimation on  $D_{\mathrm{Adapt}}$ . This allows  $p_{\theta}(\tau|\text{text})$  to exhibit high likelihood across videos in  $D_{\mathrm{Adapt}}$ , but because  $p_{\theta}(\tau|\text{text})$  is a small model trained on less diverse data, it can exhibit erroneously high likelihood across many unrealistic videos. The product distribution  $p_{\mathrm{product}}(\tau|\text{text})$  removes unrealistic videos by downweighting any videos  $\tau$  that are not likely under the pretrained prior, enabling one to generate videos in the style of  $D_{\mathrm{Adapt}}$  that are realistic under  $p_{\mathrm{pretrained}}(\tau|\text{text})$ .
+
+# 3.2 IMPLEMENTING PROBABILISTIC ADAPTATION
+
+To adapt the black-box model  $p_{\mathrm{product}}(\tau | \mathrm{text})$  from Equation 5, as well as to sample from it, we exploit the EBM interpretation of diffusion models discussed in Section 2. Based on the EBM interpretation, the pretrained diffusion model  $p_{\mathrm{pretrained}}(\tau | \mathrm{text})$  corresponds to an EBM  $e^{-E_{\mathrm{pretrained}}(\tau | \mathrm{text})}$  while the smaller video model  $p_{\theta}(\tau | \mathrm{text})$  parameterizes an EBM  $e^{-E_{\theta}(\tau | \mathrm{text})}$ . The product distribution then corresponds to:
+
+$$
+p _ {\text {p r o d u c t}} (\tau | \text {t e x t}) \propto p _ {\text {p r e t r a i n e d}} (\tau | \text {t e x t}) p _ {\theta} (\tau | \text {t e x t}) \propto e ^ {- (E _ {\text {p r e t r a i n e d}} (\tau | \text {t e x t}) + E _ {\theta} (\tau | \text {t e x t}))} = e ^ {- E ^ {\prime} (\tau | \text {t e x t})},
+$$
+
+which specifies a new EBM  $E^{\prime}(\tau)$  from the sum of energy functions of the component models.
+
+Substituting this EBM into Equation 3, we can sample from the product distribution  $p_{\mathrm{product}}(\tau | \text{text})$  through the diffusion sampling procedure:
+
+$$
+\tau_ {t - 1} = \alpha_ {t} \left(\tau_ {t} - \gamma \nabla_ {\tau} \left(E _ {\text {p r e t r a i n e d}} \left(\tau_ {t} | \text {t e x t}\right) + E _ {\theta} \left(\tau_ {t} | \text {t e x t}\right)\right) + \xi\right), \quad \xi \sim \mathcal {N} \left(\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}\right)
+$$
+
+which corresponds to sampling from Equation 1 according to
+
+$$
+\tau_ {t - 1} = \alpha_ {t} \left(\tau_ {t} - \gamma \left(\epsilon_ {\text {p r e t r a i n e d}} \left(\tau_ {t}, t \mid \text {t e x t}\right) + \epsilon_ {\theta} \left(\tau_ {t}, t \mid \text {t e x t}\right)\right) + \xi\right), \quad \xi \sim \mathcal {N} \left(\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}\right).
+$$
+
+Thus, to probabilistically adapt the pretrained black-box model to a new dataset  $D_{\mathrm{Adapt}}$ , we can use the standard diffusion sampling procedure, but change the denoising prediction to the sum of predictions from both the black-box pretrained model and the task-specific small model. To control the strength of the pretrained prior in final video generation, we can introduce a weight term  $\lambda$  to scale the pretrained distribution  $p_{\mathrm{pretrained}}^{\lambda}(\tau | \text{text})$ , which corresponds to scaling the denoised prediction from  $\epsilon_{\mathrm{pretrained}}(\tau_t, t | \text{text})$  by a scalar  $\lambda$
+
+$$
+\epsilon \left(\tau_ {t}, t \mid \text {t e x t}\right) = \epsilon_ {\theta} \left(\tau_ {t}, t \mid \text {t e x t}\right) + \lambda \epsilon_ {\text {p r e t r a i n e d}} \left(\tau_ {t}, t \mid \text {t e x t}\right). \tag {6}
+$$
+
+The combined model can be further refined by integrating multiple steps MCMC sampling between each diffusion noise distribution similar to (Du et al., 2023a; Sjoberg et al., 2023). Note that the prior strength  $\lambda$  can be tunable or time-dependent, which we found to be useful in practice (Appendix B).
+
+# 3.3 ADAPTING LOW TEMPERATURE SAMPLING
+
+In practice, text conditioning in the denoising model  $\epsilon (\tau_t,t|\text{text})$  from Equation 6 are often parametrized using classifier-free guidance (Ho & Salimans, 2022) to generate sharp images or videos conditioned on text while avoiding distractions from spurious likelihood modes of diffusion
+
+Algorithm 1 Sampling algorithm of Video Adapter  
+Input: Pretrained black-box model  $\epsilon_{\mathrm{pretrained}}(\tau ,t|\mathrm{text})$  , inverse temperature  $\omega$  , prior strength  $\lambda$  Initialize sample  $\tau_T\sim \mathcal{N}(\mathbf{0},\mathbf{I})$    
+for  $t = T,\dots ,1$  do   
+ $\tilde{\epsilon}_{\mathrm{text}}\gets \epsilon_{\theta}(\tau_t,t|\mathrm{text}) + \lambda \epsilon_{\mathrm{pretrained}}(\tau_t,t|\mathrm{text})$  // Compute score using text-conditioned prior.   
+ $\epsilon \leftarrow \epsilon_{\theta}(\tau_t,t)$  // Compute unconditional score.   
+ $\tilde{\epsilon}_{\mathrm{cfg}}\gets \epsilon +\omega (\tilde{\epsilon}_{\mathrm{text}} - \epsilon)$  //Compute weight for low temperature sampling.   
+ $\tau_{t - 1} = \mathrm{ddpm\_sample}(\tau_t,\tilde{\epsilon}_{\mathrm{cfg}})$  // Run diffusion sampling (can use other samplers).   
+end for
+
+models. This corresponds to sampling from the modified probability distribution:
+
+$$
+p ^ {\operatorname {c f g}} (\tau | \text {t e x t}) \propto p (\tau) \left(\frac {p (\tau | \text {t e x t})}{p (\tau)}\right) ^ {\omega} \propto p (\tau) p (\text {t e x t} | \tau) ^ {\omega},
+$$
+
+where  $\omega$  corresponds to the classifier free guidance strength, typically chosen to be significantly larger than 1. By upweighting the expression  $p(\text{text}|\tau)$  via the inverse temperature  $\omega$ , the modified distribution  $p^{\mathrm{fg}}(\tau|\text{text})$  above can generate lower temperature video samples conditioned on the text. It appears straightforward to similarly construct low temperature samples when adapting the black-box model by sampling from the distribution
+
+$$
+p _ {\text {p r o d u c t}} ^ {\text {c f g}} (\tau | \text {t e x t}) \propto p _ {\text {p r e t r a i n e d}} ^ {\text {c f g}} (\tau | \text {t e x t}) p _ {\theta} ^ {\text {c f g}} (\tau | \text {t e x t}), \tag {7}
+$$
+
+but using the classifier-free distribution  $p_{\mathrm{pretrained}}^{\mathrm{cfg}}(\tau | \text{text})$  as the probabilistic prior is now problematic, since classifier-free guidance has restricted  $p_{\mathrm{pretrained}}^{\mathrm{cfg}}(\tau | \text{text})$  to very few high probability modes which might be incompatible with  $D_{\mathrm{Adapt}}$ . To effectively leverage a broad probabilistic prior while simultaneously generating low temperature samples emulating  $D_{\mathrm{Adapt}}$ , we propose to first construct a new text-conditioned video distribution following Section 3.1:
+
+$$
+p _ {\text {p r o d u c t}} (\tau | \text {t e x t}) \propto p _ {\text {p r e t r a i n e d}} (\tau | \text {t e x t}) p _ {\theta} (\tau | \text {t e x t}).
+$$
+
+We can then use the density ratio of this composed text-conditioned distribution with the unconditional video density  $p_{\theta}(\tau)$  learned on  $D_{\mathrm{Adapt}}$  to construct a new implicit classifier  $p_{\mathrm{product}}(\tau|\mathrm{text})$ . By increasing the inverse temperature  $\omega$  on this implicit classifier, we can generate low temperature and high quality video samples conditioned on a given text by sampling from the modified distribution:
+
+$$
+p _ {\text {p r o d u c t}} ^ {*} (\tau | \text {t e x t}) = p _ {\theta} (\tau) \left(\frac {p _ {\text {p r o d u c t}} (\tau | \text {t e x t})}{p _ {\theta} (\tau)}\right) ^ {\omega},
+$$
+
+which corresponds to sampling from a modified denoising function:
+
+$$
+\tilde {\epsilon} _ {\theta} (\tau , t | \text {t e x t}) = \epsilon_ {\theta} (\tau , t) + \omega \left(\epsilon_ {\theta} (\tau , t | \text {t e x t}) + \lambda \epsilon_ {\text {p r e t r a i n e d}} (\tau , t | \text {t e x t}) - \epsilon_ {\theta} (\tau , t)\right)
+$$
+
+We quantitatively and qualitatively ablate the effect of this denoising function in Figure 6 and Table 2, showing that this variant leads to better blending of styles between models. The overall pseudocode for the proposed approach with classifier-free guidance is given in Algorithm 1.
+
+# 4 EXPERIMENTS
+
+In this section, we illustrate how a black-box pretrained text-to-video model can deliver a rich set of downstream capabilities when combined with a task-specific video model. In particular, leveraging a high quality and broad probabilistic prior enables (1) controllable video synthesis from edge-only inputs, (2) high-quality video modeling that outperforms both the pretrained model and the task-specific video model, and (3) domain randomization and data augmentation for robotics. See experiment details and additional experimental results in Appendix B and in supplementary material.
+
+# 4.1 ADAPTING TO SPECIFIC VIDEO DOMAINS
+
+Setup. We first demonstrate that the probabilistic prior in Video Adapter can be used to adapt and modify the styles of videos. We curate two adaptation datasets  $D_{\mathrm{Adapt}}$ , one with an "animation" style and the other with a "scifi" style, where videos containing relevant keywords in their descriptions are grouped together to form  $D_{\mathrm{Adapt}}$ . A black-box large video diffusion model with 5.6B parameters was pretrained on mapping Sobel edges to all videos, and two task-specific small models with 330M parameters were trained to map Sobel edges to  $D_{\mathrm{Adapt}}$  videos.
+
+Stylizing Video Generation. In Figure 4 and Figure 5, we demonstrate how the pretrained prior can adapt the animation and scifi models to alternative styles while maintaining the original animation and scifi contents. These results show that Video Adapter can effectively combine rich knowledge of
+
+![](images/46815b620d1d55adbccbb48e82c54a0366f5f3466df109331232c79d7409bf45.jpg)  
+Figure 4: Video Adapter enables stylization of a Animation Specific Model. Video Adapter enables a large pretrained model to adapt and change the style a small animation style model.
+
+![](images/0f912da891368353666bb392c8f63c4ced3a4aaed7fea1964011b9b051679d99.jpg)  
+Figure 5: Video Adapter enables stylization of a SciFi Specific Model. Video Adapter enables a large pretrained model to adapt and change the style a small Scifi animation style model.
+
+styles from the black-box model, such as "digital art", "outdoor video", "storybook illustration", with the animation content of the small model, thereby achieving flexible stylization.
+
+Specific Animation Style. We further trained a small video model on an "animation" style of a particular artist. In Figure 3, we illustrate how the pretrained prior can maintain the anime content while changing the styles such as background color.
+
+Analysis. In Figure 6, we change the magnitude of the weight on the pretrained prior, and compare Video Adapter with directly interpolating the classifier-free scores between the pretrained and adapter models (as in Equation 7). We find that Video Adapter maintains the adapter style more accurately, whereas classifier-free score interpolation collapses to
+
+the teacher style with intermediate interpolation, leading to erratic artifacts.
+
+![](images/456cb43013667fb60f3f2deb79b28e7e0c932f13be2e034f7239f2ceb47c4cc3.jpg)  
+Figure 3: Instance Specific Stylization. Video Adapter enables the stylization of video model trained on a single animation style
+
+# 4.2 HIGH-QUALITY EFFICIENT VIDEO MODELING
+
+Setup. To demonstrate Video Adapter's ability in adapting the black-box pretrained model to domains that are not a part of pretraining, we consider adapting to Ego4D (Grauman et al., 2022) and Bridge Data (Ebert et al., 2021). These adaptations are nontrivial, as Ego4D consists of mostly egocentric videos that are not commonly found on the internet. Similarly, the Bridge Data consists of task-specific videos of a WidowX250 robot that is out of the distribution of the pretraining data. For Ego4D, we take a subset of the original dataset consisting of 97k text-video pairs and split them into train (90%) and test (10%) to form  $D_{\mathrm{Adapt}}$ . For the Bridge Data, we take the entire dataset consisting of 7.2k text-video pairs and use the same train-test split to form  $D_{\mathrm{Adapt}}$ .
+
+![](images/4a69feea69c07537be078ae1142023ce97147f37dee05fc4083bcd75c7c542cd.jpg)  
+Figure 6: Analysis of Video Adapter. As adaptation weight increases, Video Adapter modifies the style as instructed (left), whereas directly mixing two classifier-free guidance scores fails to adapt the video (right).
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Bridge</td><td colspan="3">Ego4D</td></tr><tr><td>FVD ↓</td><td>FID ↓</td><td>Param (B) ↓</td><td>FVD ↓</td><td>IS ↑</td><td>Param (B) ↓</td></tr><tr><td>Small (S)</td><td>186.8</td><td>38.8</td><td>0.07</td><td>228.3</td><td>2.28</td><td>0.07</td></tr><tr><td>Small (S) + Pretrained</td><td>177.4</td><td>37.6</td><td>0.07</td><td>156.3</td><td>2.82</td><td>0.07</td></tr><tr><td>Small (L)</td><td>152.5</td><td>30.1</td><td>0.14</td><td>65.1</td><td>3.31</td><td>2.8</td></tr><tr><td>Small (L) + Pretrained</td><td>148.1</td><td>29.5</td><td>0.14</td><td>52.5</td><td>3.53</td><td>2.8</td></tr><tr><td>Pretrained</td><td>350.1</td><td>42.6</td><td>5.6</td><td>91.7</td><td>3.12</td><td>5.6</td></tr><tr><td>Pretrained Finetune</td><td>321.0</td><td>39.4</td><td>5.6</td><td>75.5</td><td>3.33</td><td>5.6</td></tr></table>
+
+For the pretrained model, we use the 5.6B base model pretrained on generic internet videos from (Ho et al., 2022). For the task-specific small model, we downscale the video diffusion model from (Ho et al., 2022) by a factor of 80, 40, and 2 to create a diverse set of small models to be trained on task-specific  $D_{\mathrm{Adapt}}$ . Table 1 shows the number of parameters of pretrained and small video models. Both the pretrained model and the small models are trained to generate subsequent frames conditioned on the first frame.
+
+Quantitative Results. Table 1 shows the quantitative performance of Video Adapter under different video modeling metrics. On the Bridge Data, training a small model with parameters equivalent to  $1.25\%$  of the pretrained video model (first row) already achieves better metrics than the pretrained model. However, Video Adapter incorporating the pretrained model as a probabilistic prior is able to further improve the metrics of the small model (second row). On Ego4D, due to the complexity of the egocentric videos, the smallest model with  $1.25\%$  of the pretrained video model can no longer achieve performance better than the pretrained model (first row), but incorporating the pretrained model during sampling still improves performance (second row). After increasing the size of the small model, Video Adapter is able to achieve better metrics than both the pretrained and task-specific model. We further compare Video Adapter to finetuning the pretrained model for an equivalent number of TPU hours (see Appendix C), and show that Video Adapter achieves better performance than full tuning. Note that we only compare to full tuning out of curiosity as opposed to benchmarking, as the motivation of this work is the lack of weight access to the black-box pretrained models.
+
+Qualitative Results. Figure 7 and Figure 8 show the generated videos on Bridge Data and Ego4D. On the Bridge Data in Figure 7, the pretrained model produces videos that do not correspond to the task described by the text (there is no robot arm movements in the generated video). The task-specific small model produces videos with unrealistic movements that teleport the robot arm. Video Adapter, on the other hand, produces videos with realistic movements that complete the task.
+
+On Ego4D in Figure 8, the pretrained model produces high quality videos that contain little egocentric movement (first
+
+row), as the pretraining data mostly consists of generic videos from the internet that are not egocentric. The task-specific small model trained on Ego4D, on the other hand, produces videos with egocentric movement but of low quality (second row) due to limited model capacity. Video Adapter combines the best of both and generates high-quality egocentric videos (third row).
+
+Table 1: Video Modeling Quantitative Performance Video Adapter (Small + Pretrained) achieves better FVD, FID, and Inception Scores than both the pretrained model, pretrained model finetuned for equivalent number of TPU hours, and the task-specific small model with parameters as fewer as  $1\%$  of the pretrained model.  
+
+<table><tr><td>Model</td><td>FVD ↓</td><td>FID ↓</td></tr><tr><td>CFG Mix</td><td>167.4</td><td>33.1</td></tr><tr><td>Small (L)</td><td>152.5</td><td>30.1</td></tr><tr><td>Video Adapter</td><td>148.1</td><td>29.5</td></tr></table>
+
+Table 2: Ablations. Video Adapter improves the underlying video modeling performance of models on while directly mixing classifier-free scores (CFG Mix) hurts performance.
+
+![](images/4e31d7cbfe8defc0a38d548fd604aa8eb7ccaef6096a2abd47073fee48c4cdc1.jpg)
+
+![](images/7d3dbdcd3eb4ba7cf1187402730feac4c54f41a501507eefab727fffe9621d95.jpg)  
+Figure 7: Video Adapter on Bridge Data. The pretrained model (first row) produces videos that are high-quality but are generally static and fail to complete the task. The small (L) model (second row) produces low-quality videos with unrealistic arm movements. Video Adapter (third row) produces high-quality videos and successfully completes the task.  
+Figure 8: Video Adapter on Ego4D. The pretrained model (first row) produces high-quality but nearly static videos that do not reflect the egocentric nature. The small (L) model (second row) produces low-quality videos but with more egocentric movements. Video Adapter (third row) produces high-quality and egocentric videos.
+
+Ablations. In Table 2, we report generative modeling performance of the small model on Bridge either using Video Adapter, or a interpolation between the classifier-free scores of pretrained and small models. We find that Video Adapter improves performance, while interpolation between classifier-free scores hurts performance.
+
+# 4.3 SIM-TO-REALVIDEO AUGMENTATION
+
+Setup. One important application of controllable video synthesis is to render realistic robotic videos from simulation with a variety of data augmentations so that policies trained on the augmented observations are more likely to generalize well to real-world settings (Zhao et al., 2020). To demonstrate Video Adapter's capability in supporting sim-to-real transfer, we train a task-specific small edge-to-real model on 160k real robot trajectories of the LanguageTable dataset (Lynch et al., 2022), generating videos of execution conditioned on the Sobel edges of the real videos. Similarly, we train another small edge-to-sim model on 160k simulated robot videos. Note that the simulated and real robotics data are not paired (paired sim-to-real data are hard to find) but are connected through edge-conditioning. We again leverage the edge-conditioned large model pretrained on internet data for style specification.
+
+![](images/2f01a69a2cfbceda8cedc6fa4660d8e2374d6048808c257c9459f4cb1e3e60cb.jpg)  
+Figure 9: Video Adapter on sim-to-real transfer. First row: simulated videos of execution plans generated by Video Adapter. Second row: real videos of execution plans generated by Video Adapter. Third row: real videos of execution plans generated by Video Adapter with data augmentation.
+
+Adapted Videos. Figure 9 shows the generated robotic videos from Video Adapter. Video Adapter can effectively generate paired simulated and real robotic videos that complete a task described by a language prompt, and further generate videos with various data augmentation styles that can be utilized to train policies with better sim-to-real transfer abilities through techniques similar to domain randomization (Tobin et al., 2017).
+
+# 5 RELATED WORK
+
+Text-to-Video Synthesis. Following the recent success of text-to-image models (Nichol et al., 2021; Balaji et al., 2022; Ramesh et al., 2022; Rombach et al., 2022; Sahara et al., 2022b; Yu et al., 2022; Chang et al., 2023), large text-to-video models with autoregressive (Hong et al., 2022; Villegas et al., 2022; Wu et al., 2022a; 2021) and diffusion (Ho et al., 2022; Singer et al., 2022; Blattmann et al., 2023; Zhou et al., 2022; Esser et al., 2023) architectures have been developed, often by extending existing text-to-image models. Unfortunately, the model weights of large text-to-video models are generally not publicly available, preventing downstream adaptations of these models.
+
+Adapting Pretrained Models Adapting pretrained models for customized editing, inpainting, and stylization has been extensively studied in text-to-image and image-to-image translation models (Gal et al., 2022; Hertz et al., 2022; Kawar et al., 2022; Li et al., 2022; Lugmayr et al., 2022; Meng et al., 2021; Ruiz et al., 2022; Sahara et al., 2022a; Sasaki et al., 2021; Su et al., 2022). In text-to-video, most existing work either leverages text prompts (Molad et al., 2023; Esser et al., 2023), finetunes a pretrained model on stylized data (Wu et al., 2022b), or performing light training on a copy of the pretrained video model similar to ControlNet (Dhesikan & Rajmohan, 2023). Text-only adaption can be unreliable, whereas finetuning, prefix-tuning (Li & Liang, 2021), low-rank adaptation (Hu et al., 2021), and ControlNet all require access to the pretrained model weights, which are often not available for text-to-video models. Black-box adaptation has been applied extensively in language models (Brown et al., 2020; Wei et al., 2022; Liu et al., 2023), and large video models will soon face the same problem.
+
+Compositional Generative Models. The techniques in this paper are further related to existing work on compositional generative modeling (Liu et al., 2022; Nie et al., 2021; Du et al., 2020; 2023a, 2021; Liu et al., 2021; Wang et al., 2023; Wu et al., 2022c; Deng et al., 2020; Urain et al., 2021; Gkanatsios et al., 2023; Gandikota et al., 2023; Po & Wetzstein, 2023), where different generative models are probabilistically combined to jointly generate outputs. In (Du et al., 2020), an approach to combine different probability distributions using EBMs is introduced. Most similar in spirit to this work, (Deng et al., 2020) composes a pretrained language model with a small EBM to improve language generation. However, different from this work, the small EBM is used to improve to global consistency of the language model, whereas we aim to use a small model to probabilistically adapt to a large pretrained video model to separate domains.
+
+# 6 LIMITATION AND CONCLUSION
+
+As video foundation models become more powerful but remain proprietary, black-box adaptation of these models is inevitable. We have proposed Video Adapter for leveraging black-box text-to-video models as probabilistic priors for guiding generation of specific videos. One limitation of Video Adapter is it still requires training a small domain-specific model, so adaptation is not completely training free. Another limitation is Video Adapter requires diffusion scores from pretrained black-box models. We advocate future video diffusion models to make scores as a part of the output to improve accessibility of these models.
+
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+
+# Appendix
+
+In the Appendix we provided a detail derivation of connection between diffusion models and EBMs in Section A. We further provide additional experimental details in Section B. Finally, we provide a comparison with using the same computational budget to finetune the existing large pretrained model in Section C.
+
+# A CONNECTION BETWEEN DIFFUSION AND EBM
+
+The sampling procedure in a diffusion model corresponds to the Langevin sampling procedure on an EBM. To see this, we consider perturbing a sample  $\tau^{t - 1}\sim p\left(\tau^{t - 1}\right)$  from target distribution  $p(\tau^{t - 1})$  with a Gaussian noise, i.e.,
+
+$$
+\tau^ {t} = \tau^ {t - 1} + \xi , \quad \xi \sim \mathcal {N} \left(\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}\right)
+$$
+
+which corresponds to the transition operator
+
+$$
+\mathcal {T} \left(\tau^ {t} \mid \tau^ {t - 1}\right) \propto \exp \left(\frac {- \left\| \tau^ {t} - \tau^ {t - 1} \right\| ^ {2}}{2 \sigma_ {t} ^ {2}}\right)
+$$
+
+where the joint distribution of  $\tau^t$  and  $\tau^{t - 1}$  is
+
+$$
+p (\tau^ {t}, \tau^ {t - 1}) \propto \exp \left(\psi (\tau^ {t - 1}) - \frac {\left\| \tau^ {t} - \tau^ {t - 1} \right\| ^ {2}}{2 \sigma_ {t} ^ {2}}\right).
+$$
+
+We can express the Bayes estimator of  $\tau^{t - 1}$  given the perturbed observation  $\tau^t$  as
+
+$$
+m \left(\tau^ {t}\right) = \int \tau^ {t - 1} p _ {\theta} \left(\tau^ {t - 1} \mid \tau^ {t}\right) d \tau^ {t - 1} = \tau^ {t} + \sigma_ {t} ^ {2} \nabla \log p (\tau^ {t}) \tag {8}
+$$
+
+Proof By the property of Gaussian distribution, we have
+
+$$
+\sigma^ {2} \nabla_ {x ^ {\prime}} p \left(x ^ {\prime} | x\right) = p \left(x ^ {\prime} | x\right) \left(x - x ^ {\prime}\right). \tag {9}
+$$
+
+Therefore, we have
+
+$$
+\begin{array}{l} \sigma \nabla_ {x ^ {\prime}} \int p \left(x ^ {\prime} \mid x\right) p (x) d x = \int \left(x - x ^ {\prime}\right) p \left(x ^ {\prime}, x\right) d x = \int x p \left(x ^ {\prime}, x\right) d x - x ^ {\prime} p \left(x ^ {\prime}\right) (10) \\ \Rightarrow \sigma \nabla_ {x ^ {\prime}} \log p \left(x ^ {\prime}\right) = \int x \frac {p \left(x ^ {\prime} , x\right)}{p \left(x ^ {\prime}\right)} d x - x ^ {\prime} = \mathbb {E} [ X | x ^ {\prime} ] - x ^ {\prime} (11) \\ \end{array}
+$$
+
+Thus, we can represent the perturbed data with an EBM  $p(\tau^t) \propto \exp (E_\theta (\tau^t,\sigma_t))$ , and learn the parameters through regression (Vincent, 2011; Saremi et al., 2018; Saremi & Hyvarinen, 2019; Song & Ermon, 2019), which leads to the optimal solution
+
+$$
+\begin{array}{l} \min _ {\boldsymbol {\theta}} \mathbb {E} _ {\tau^ {t - 1} \sim \mathcal {D}, \xi \sim \mathcal {N} (0, \sigma_ {t} ^ {2} \mathbf {I})} \left[ \left| \left| \boldsymbol {\tau} ^ {t - 1} - m (\boldsymbol {\tau} ^ {t}) \right| \right| ^ {2} \right] \\ = \mathbb {E} _ {\tau^ {t - 1} \sim \mathcal {D}, \xi \sim \mathcal {N} (0, \sigma^ {2} \mathbf {I})} \left[ \left\| - \xi - \nabla E _ {\theta} \left(\tau^ {t - 1} + \xi , \sigma_ {t}\right)\right) \right\| ^ {2} \Bigg ], \tag {12} \\ \end{array}
+$$
+
+which also corresponds to the denoising diffusion training objective.
+
+Once we have the trained  $E_{\theta}(\tau^{t})$ , we can then recover the sample  $\tau^{t - 1}$  according to the denoising sampling procedure
+
+$$
+\tau^ {t - 1} = \alpha^ {t} m \left(\tau^ {t}\right) + \alpha^ {t} \xi = \alpha^ {t} \left(\tau^ {t} - \gamma \nabla_ {\tau^ {t}} E _ {\theta} \left(\tau^ {t}, \sigma_ {t}\right)\right) + \alpha^ {t} \xi , \quad \xi \sim \mathcal {N} (\mathbf {0}, \sigma_ {t} ^ {2} \mathbf {I}) \tag {13}
+$$
+
+which corresponds to the sampling via stochastic localization (El Alaoui et al., 2022) and Equation 2 in the main paper.
+
+# B EXPERIMENTAL DETAILS
+
+# B.1 EXPERIMENT DETAILS
+
+Dataset The large pretrained model is trained on 14 million video-text pairs plus 60 million image-text pairs, and with the LAION-400M image-text dataset. The images are spatially resized to  $24 \times 40$  and videos using anti-aliased bi-linear resizing. We use different frame rate for different types of videos for best visualization results. For the Bridge (Ebert et al., 2021) we directly use the released opensource dataset. For Ego4D (Grauman et al., 2022) data, we take a small portion of the released dataset. For Anime and Sci-Fi style, we curate two separates datasets with their respective keywords. The keywords used for filtering data for Anime style are (in small letter) "disney", "cartoon", "anime",
+
+"animation", "comic", "pixar", "animated", "fantasy". The keywords used for filtering data for Sci-Fi style are "science fiction", "sci-fi", "scifi", "astronaut", "alien", "NASA", "interstellar". For the animation with a particular artist style, we use the Case Closed animation (also named Detective Conan). For the Language Table dataset, we used the data from (Lynch et al., 2022).
+
+<table><tr><td>Dataset</td><td>Pretrain</td><td>Bridge</td><td>Ego4D</td><td>Anime</td><td>Sci-Fi</td><td>Case Closed</td><td>LangTable Sim</td><td>LangTable Real</td></tr><tr><td># Train</td><td>474M</td><td>2.3k</td><td>97k</td><td>0.6M</td><td>21k</td><td>5k</td><td>0.16M</td><td>0.16M</td></tr></table>
+
+Table 3: Training data size. Number of text-video or text-image pairs used for training the pretrained large model and each of the small model. Training data for particular styles can be magnitude smaller than the pretraining dataset.
+
+Architecture. To pretrain the large model, we use the same pretraining dataset, base architecture, and training setup as (Ho et al., 2022), with modifications of first-frame conditioning for Bridge and Ego4D, and edge conditioning for stylisation and sim-to-real. Specifically, the large model architecture consists of video U-Net with 3 residual blocks of 1024 base channels and channel multiplier [1, 2, 4], attention resolutions [6, 12, 24], attention head dimension 64, and conditioning embedding dimension 1024. To support first frame conditioning, we replicate the first frame across all future frame indices, and concatenate the replicated first frame channel-wise to the noisy data following (Du et al., 2023b). To support edge conditioning, we run a sobel edge detector and use gradient approximations in the x-direction as the conditional video, and concatenates these edge frames with noisy data similar to first-frame conditioning. The large model consists of 5.6 billion parameters in total. For the set of small models for adaptation, Ego4D Small (L) has 512 base channels in each of the residual blocks. Ego4D Small (S) and Bridge Small (S) have a single residual block with 32 base channels. Bridge Small (L) has a single residual block with 64 base channels. The set of stylisation models (animation, sci-fi, and particular anime style) have 3 residual blocks and 256 base channels. For illustrating the generated videos at a higher resolution, we train two additional spatial super resolution models  $24\mathrm{x}40 \rightarrow 48\mathrm{x}80$  (1.4B) and  $48\mathrm{x}80 \rightarrow 192\mathrm{x}320$  (1.2B). We additionally use T5-XXL (Raffel et al., 2020) to process input text prompts which consists of 4.6 billion parameters, which we omit from the parameter count as all large and small models require text embeddings.
+
+Training and Evaluation. We train each of our video diffusion models for 2M steps using batch size 2048 with learning rate 1e-4 and 10k linear warmup steps. The large 5.6B pretrained model requires 512 TPU-v4 chips, whereas various small models require anywhere between 8 and 256 TPU-v4 chips depending on the size. We use noise schedule log SNR with range [-20, 20]. We use 128 samples and 1024 samples to compute the FVD, FID, and Inception Scores metric on Bridge and Ego4dD, respectively.
+
+Sampling. All diffusion models are trained with 1000 timesteps of sampling. To generate videos, we combined scores from both pretrained models and adapter models for all timesteps except the last 100 timesteps. The last 100 timesteps capture high frequency information in an image, and we found better image quality if we did not combine scores in these timesteps. For simplicity, we use a pretrained neural strength of 0.2 for Ego4D and 0.1 for Bridge, and 0.4 for all animation datasets, but found additional gains when using large neural strength at earlier timesteps and smaller ones later.
+
+# C COMPARISON TO FINETUNING
+
+To illustrate the computational efficiency of Video Adapter, we further compare video modeling metrics of Video Adapter to finetuning the pretrained model for an equivalent number of TPU time. Specifically, the pretrained model requires 512 TPU-v4 chips whereas the small model on Bridge data requires 8 TPU-v4 chips. The small Bridge model requires 100k steps to reach convergence, and hence we finetune the pretrained model for  $100,000 / 64 = 1,560$  steps. Video Adapter achieves better FVD and FID than finetuning the pretrained model for an equal number of TPU steps as shown in Table 1.
+
+# D COMPOSING MULTIPLE FACTORS
+
+Given a set of  $N$  separate datasets  $\{D_i\}_{i = 1:N}$  specifying a set of  $N$  different styles of videos, we can also apply our probabilistic adaptation framework across these  $N$  models by learning  $N$  separate distributions  $\{p_i(\tau |\text{text})\}_{i = 1:N}$ . We can directly sample from the product distribution
+
+$$
+\underbrace {p _ {\text {p r o d u c t}} (\tau \mid \text {t e x t})} _ {\text {P r o d u c t D i s t r i b u t i o n}} \propto \underbrace {p _ {\text {p r e t r a i n e d}} (\tau \mid \text {t e x t})} _ {\text {P r e t r a i n e d P r i o r}} \underbrace {\prod_ {i = 1 : N} p _ {i} (\tau \mid \text {t e x t})} _ {\text {N D i f f e r e n t V i d e o M o d e l s}}, \tag {14}
+$$
+
+which now assigns high likelihood to videos that exhibit each of the composed styles.
+
+We can directly sample from this composed model using the modified composite denoising function
+
+$$
+\tilde {\epsilon} _ {\theta} (\tau , t | \text {t e x t}) = \epsilon_ {\theta} (\tau , t) + \omega \sum_ {i = 1} ^ {N} \left(\epsilon_ {i} (\tau , t | \text {t e x t}) + \lambda \epsilon_ {\text {p r e t r a i n e d}} (\tau , t | \text {t e x t}) - \epsilon_ {i} (\tau , t)\right),
+$$
+
+which simply corresponds to using a weighted average over the predictions of each of the  $N$  models.
+
+# E COMPARISON TO PARAMETER EFFICIENT FINETUNING
+
+The problem setting of Video Adapter is that pretrained model weights are not accessible. This scenario is common in large language models (e.g., GPT-4, Bard), and large text-to-video models are heading in the same direction. Parameter efficient finetuning (e.g., LoRA, null-text inversion, prefix-tuning) requires access to pretrained model weights, whereas Video Adapter does not. Therefore Video Adapter is not comparable to parameter efficient finetuning.
+
+Nevertheless, we conducted comparisons to LoRA and null-text inversion out of curiosity (prefix-tuning is omitted since it has only been applied to language models). For LoRA, we use rank 1 and rank 64 to compare to the smaller and larger task-specific VideoAdapter model. For null-text inversion, we use an unconditional null embedding of size [64, 4096] (the same dimension as the original text embeddings). We report the video modeling metrics in Table 4.
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Bridge</td><td>Ego4D</td></tr><tr><td>FVD ↓</td><td>FID ↓</td><td>FVD ↓</td><td>IS ↑</td></tr><tr><td>Small (S)</td><td>186.8</td><td>38.8</td><td>228.3</td><td>2.28</td></tr><tr><td>Small (S) + Pretrained</td><td>177.4</td><td>37.6</td><td>156.3</td><td>2.82</td></tr><tr><td>Small (L)</td><td>152.5</td><td>30.1</td><td>65.1</td><td>3.31</td></tr><tr><td>Small (L) + Pretrained</td><td>148.1</td><td>29.5</td><td>52.5</td><td>3.53</td></tr><tr><td>LoRA-Rank1</td><td>170.2</td><td>32.2</td><td>74.5</td><td>3.4</td></tr><tr><td>LoRA-Rank64</td><td>165.5</td><td>31.6</td><td>50.3</td><td>3.5</td></tr><tr><td>Null-text inversion</td><td>288.8</td><td>40.2</td><td>90.2</td><td>3.1</td></tr></table>
+
+Table 4: Comparison of Video Adapter to parameter efficient finetuning under fixed compute budget. Video Adapter performs close to LoRA-Rank64 on Ego4D and better than parameter efficient finetuning on Bridge.
+
+We observe that LoRA-Rank1 performs slightly better than Video Adapter (small). However, In the LoRA-Rank1 case, LoRA still performs worse than training a small domain specific model. In this case, Video Adapter can simply use the small model without the pretrained prior. In comparison, we found LoRA-Rank64 leads to mixed results when compared to Video Adapter (large), i.e., LoRA outperforms Video Adapter on Ego4D but not on Bridge data. We found that null-text inversion performs the worst, potentially due to limited flexibility of null-embeddings during finetuning.
+
+Our results illustrate that Video Adapter, despite requiring only black-bo adaptation without access to pretrained model weights performs better than Null-text inversion and very comparably to LoRA finetuning (with pretrained model weights).

2024/Probabilistic Adaptation of Black-Box Text-to-Video Models/images.zip

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+# PROBABILISTIC SELF-SUPERVISED LEARNING VIA SCORING RULES MINIMIZATION
+
+Amirhossein Vahidi $^{a,b,c}$  Simon Schosser $^{a}$  Lisa Wimmer $^{a,b}$  Yawei Li $^{a,b}$  Bernd Bischl $^{a,b}$  Eyke Hüllermeier $^{b,d}$  Mina Rezaei $^{a,b}$
+
+$^a$ Department of Statistics LMU Munich, Munich, Germany
+
+<sup>b</sup>Munich Center for Machine Learning, Munich, Germany
+
+Wellcome Sanger Institute, Cambridge, UK
+
+$^{d}$ Institute of Informatics LMU Munich, Munich, Germany
+
+{av13@sanger.ac.uk, mina.rezaei@stat.uni-muenchen.de}
+
+# ABSTRACT
+
+Self-supervised learning methods have shown promising results across a wide range of tasks in computer vision, natural language processing, and multimodal analysis. However, self-supervised approaches come with a notable limitation, dimensional collapse, where a model doesn't fully utilize its capacity to encode information optimally. Motivated by this, we propose ProSMin, a novel probabilistic self-supervised learning approach that leverages the power of probabilistic models to enhance representation quality and mitigate collapsing representations. Our proposed approach involves two neural networks, the online network and the target network, which collaborate and learn the diverse distribution of representations from each other through probabilistic knowledge distillation. The two networks are trained via our new loss function based on proper scoring rules. We provide a theoretical justification for ProSMin and demonstrate its modified scoring rule. This insight validates the method's optimization process and contributes to its robustness and effectiveness in improving representation quality. We evaluate our probabilistic model on various downstream tasks, such as in-distribution generalization, out-of-distribution detection, dataset corruption, low-shot learning, and transfer learning. Our method achieves superior accuracy and calibration, outperforming the self-supervised baseline in a variety of experiments on large datasets such as ImageNet-O and ImageNet-C. ProSMin thus demonstrates its scalability and real-world applicability. Our code is publicly available: https://github.com/amirvhd/SSL-sore-rule.
+
+# 1 INTRODUCTION
+
+Self-supervised learning (SSL) is one of the most promising approaches for learning representations from a limited set of labeled data and has achieved outstanding results in several domains and applications, including natural language processing (NLP; Devlin et al. (2018); Brown et al. (2020); Saggau et al. (2023)), computer vision (Chen et al., 2020; Bardes et al., 2021; Grill et al., 2020; Rezaei et al., 2023a), multimodal learning (Radford et al., 2021; Li et al., 2022b; Shi et al., 2022), and bioinformatics (Gündüz et al., 2023; Rezaei et al., 2023b). However, collapsing representations is one of the significant drawbacks to current SSL methods, in which the representations converge to a limited set of points in the embedding space.
+
+The dimensional collapse in SSL occurs due to excessive distortion caused by strong data augmentation, making the duplicates dissimilar to the originals collapsing certain dimensions, as well as network overparameterization which leads to lower-dimensional solutions (Jing et al., 2022). Strong augmentation can introduce more variance than present in the data distribution, causing collapse when the contrastive covariance matrix is not positive semidefinite, while overparameterization biases the network towards flatter minima, hindering information encoding in dimensions even for
+
+![](images/eefff3c1ee01eee9c294e60f6dbec3db41df69bfcc86aa8f6f20910ef4c4ada5.jpg)  
+Figure 1: Illustration of our proposed probabilistic self-supervised learning via scoring rule minimization namely ProSMin. Given a batch  $X$  of input samples, two different augmented samples  $\tilde{\pmb{x}}$  and  $\tilde{\pmb{x}}^{\prime}$  are taken by an online network with  $\theta$  and target network with  $\xi$  parameters respectively. Our objective is to minimize the proposed scoring rule between  $P_{\theta}, P_{\xi}$ .
+
+similar positive pairs. Understanding and addressing these mechanisms is crucial for enhancing the effectiveness of self-supervised learning and preventing undesirable dimensional collapses.
+
+Collapsing representations lead to reduced representation quality, impaired generalization to new tasks or domains, and limited capacity to handle variations in the data (Jing et al., 2022). Most recent studies addressed this problem with contrastive learning through effective augmentation (Bai et al., 2022), negative sample strategies (Kalantidis et al., 2020), ensemble approaches (Vahidi et al., 2023; Ruan et al., 2022), regularization techniques (Li et al., 2022a; Rezaei et al., 2023a), removing correlations in the feature space (Bardes et al., 2022), clustering-based approaches (Assran et al., 2022; 2021) and self-distillation (Caron et al., 2021; Grill et al., 2020).
+
+In this paper, we formulate the collapsing self-supervised representation problem through probabilistic machine learning, aiming to provide a comprehensive and nuanced solution that addresses the limitations of deterministic approaches. This integration allows us to offer predictive distributions alongside their associated point predictions (Ho et al., 2020; Nichol & Dhariwal, 2021). This pivotal shift offers promising avenues for achieving representation reliability and superior generalization capabilities in self-supervised learning scenarios.
+
+Specifically, we propose ProSMin, a novel probabilistic self-supervised learning method that minimizes a scoring rule during pretext task learning. We motivate ProSMin by formulating knowledge distillation (KD) in a probabilistic manner. As shown in Figure 1, our proposed method involves two deep neural networks, an online and a target network, each learning a different representation of input samples. The online network maps input samples to a probability distribution inferred from the representation of the online encoder part. We train the online network in such a manner that samples from its output distribution predict the target network's representation on a second augmented view of the input. The loss realized by this prediction is expressed via a modified scoring rule, which incentivizes the recovery of the true distribution. Our contributions are:
+
+- We introduce a novel probabilistic definition of robust representation for self-supervised learning. The probabilistic definition provides a deeper understanding of the quality and trustworthiness of the learned representations in guiding subsequent tasks.  
+- Our probabilistic approach effectively mitigates collapsing representations by encouraging the online and target networks to explore a diverse range of representations, thus avoiding convergence to a limited set, which results in a more comprehensive representation space that better encapsulates the intricacies of the data distribution.  
+- We discuss a rigorous theoretical foundation for our proposed algorithm. This theoretical insight not only underscores the robustness of our approach but also provides a principled explanation for its effectiveness in improving representation quality.  
+- Through extensive empirical analysis, we validate the effectiveness of our approach in diverse scenarios. Our method achieves competitive predictive performance and calibration on various tasks such as in-distribution (IND), out-of-distribution (OOD), and corrupted datasets, demonstrating generalization capabilities. Moreover, we demonstrate the superi
+
+ority of our method in semi-supervised and low-shot learning scenarios. Our framework establishes a superior trade-off between predictive performance and robustness when compared to deterministic baselines. This outcome is particularly notable on large-scale datasets such as ImageNet-O and ImageNet-C, underscoring the scalability and effectiveness of our method in real-world, high-dimensional settings.
+
+# 2 BACKGROUND AND RELATED WORK
+
+Self-supervised methods are designed to tackle unsupervised problems by training on a pretext task that utilizes the data itself to generate labels, effectively employing supervised methods to solve unsupervised problems (Grill et al., 2020; Chen et al., 2020; Jang et al., 2023; Caron et al., 2021; Zhou et al., 2021; Zbontar et al., 2021; Bardes et al., 2021; Chen et al., 2021). The resulting representations learned from the pretext task can serve as a foundation for downstream supervised tasks, such as image classification or object detection. Alternatively, the extracted representation can be directly utilized for downstream applications, such as detecting anomalies and OOD data (Tran et al., 2022). Recent studies (Oquab et al., 2023; Zhou et al., 2021) provided evidence that performing self-supervised pretext task learning on a large-scale and diverse dataset can extract features that are effective across different image distributions and tasks without the need for fine-tuning. Following this, we introduce a novel probabilistic self-supervised framework aiming to learn robust representation over parameters using self-distillation and by minimizing a scoring rule.
+
+Self-distillation is a variant of knowledge distillation (Hinton et al., 2015) in which a larger model (teacher) is used to distill knowledge into a smaller model (student) of the same architecture (Caron et al., 2021; Zhou et al., 2021). Given an input sample  $x$ , the student network  $f_{\theta}$  is trained on the soft labels provided by the teacher network  $f_{\xi}$ . Self-distillation combines self-supervised learning with knowledge distillation and was introduced by DINO (Caron et al., 2021). The two networks share the same architecture but take different augmentations of the input sample and output different representation vectors. The knowledge is distilled from the teacher network  $f_{\xi}$  to student  $f_{\theta}$  by minimizing the cross-entropy between the respective representation vectors. The parameters  $\theta$  of the student network are obtained from an exponential moving average (EMA) of the parameters  $\xi$  of the teacher network, thus reducing computational cost by confining backpropagation to the teacher network. In this paper, we present a novel approach to knowledge distillation aimed at training a network in a probabilistic manner using parametrization trick (Kingma et al., 2015) and a novel scoring rule objective function.
+
+A scoring rule is a function used to evaluate the accuracy of a probabilistic prediction (Gneiting & Raftery, 2007; Der Kiureghian & Ditlevsen, 2009; Hanselle et al., 2023). It quantifies the divergence between the predicted probability distribution and the true distribution of the event. The concept of a scoring rule is fundamental to many areas of machine learning, including probabilistic classification (Parry, 2016) and decision theory (Dawid & Musio, 2014). A proper scoring rule (Gneiting & Katzfuss, 2014) is one that incentivizes truthful reporting of the probabilities by the forecaster, i.e., the forecaster is incentivized to report the correct probability distribution (Pacchiardi et al., 2021). The use of proper scoring rules has had significant implications in areas such as online learning (V'yugin & Trunov, 2019), generative neural networks (Pacchiardi & Dutta, 2022), and uncertainty quantification (Bengs et al., 2023; Gruber & Buettner, 2022; Sale et al., 2023). This paper utilizes scoring rules and adapts them for the purpose of pretext task learning of a self-supervised framework. This adaptation is inspired by the endeavor to infuse probabilistic learning principles into the self-supervised learning domain.
+
+# 3 PROBLEM FORMULATION
+
+Avoiding collapsing representation One of the major challenges in self-supervised learning is collapsing representations, where learned representations converge to a limited set of points in the representation space. In other words, the model fails to capture the full diversity and richness of the underlying data distribution. This can lead to reduced representation quality, impaired generalization to new tasks or domains, and limited capacity to handle variations in the data. various techniques have been explored to mitigate dimensional collapse including contrastive losses (Kalantidis et al., 2020) that separate embedding of negative examples, non-contrastive losses (Rezaei et al., 2023a; Bardes et al., 2022) that reduce informational redundancy among embeddings, and joint embedding
+
+and clustering approaches (Assran et al., 2022; 2023) aiming to increase the average embedding entropy. In this paper, we formulate the collapsing representation problem through a probabilistic lens, aiming to provide a comprehensive and nuanced solution that not only addresses the limitations of deterministic approaches but also harnesses the power of uncertainty quantification and broader representation distributions.
+
+Scoring rules A scoring rule (Gneiting & Raftery, 2007) is a function that evaluates how well a predicted distribution  $P$  over a random variable  $\mathbf{X}$  aligns with the actually observed realizations  $\pmb{x}$  of  $\mathbf{X}$ . We define the loss<sup>1</sup> of predicting distribution  $P$  while observing  $\pmb{x}$  as  $S(P, \pmb{x})$ . Assuming that  $\mathbf{X}$  follows some true distribution  $Q$ , the expected scoring rule measuring the loss of predicting  $P$  can be expressed as  $S(P, Q) \triangleq \mathbb{E}_{\mathbf{X} \sim Q} S(P, \mathbf{X})$ . A scoring rule  $S$  is proper with respect to a set of distributions  $\mathcal{P}$  if for all  $P, Q \in \mathcal{P}$  it holds that  $S(Q, Q) \leq S(P, Q)$ , thus incentivizing the prediction of the true distribution  $Q$ . If the former holds with equality, i.e., the expected score  $S(P, Q)$  is uniquely minimized in  $Q$  at  $Q = P$ , then the scoring rule is called strictly proper. In practice, the expectation with respect to  $Q$  is usually replaced by an empirical mean over a finite amount of samples. We refer to the resulting scoring rule as  $\hat{S}$ . There are many types of scoring rules, including entire parameterized families of strictly proper scoring rules. In Appendix 10, we provide more details about some of the rules we use in our experiments.
+
+# 4 METHOD
+
+Consider a randomly sampled mini-batch of training data  $\mathbf{X} \triangleq [\pmb{x}_1, \dots, \pmb{x}_n]$  where  $\pmb{x}_i \in \mathbb{R}^D$  and transformation functions  $\tau, \tau'$  acting on the data. To enhance the training process, the transformation functions produce two augmented views  $\tilde{\pmb{x}} \triangleq \tau(\pmb{x})$  and  $\tilde{\pmb{x}}' \triangleq \tau'(\pmb{x})$  for each sample in  $\mathbf{X}$ . These augmented views are generated by sampling  $\tau, \tau'$  from the distribution of suitable data transformations, such as partially masking image patches (He et al., 2022) or applying image augmentation techniques (Chen et al., 2020).
+
+As depicted in Fig. 1, the first augmented view  $\tilde{\pmb{x}}$  is fed to the encoder of online network  $f_{\theta}$  that outputs a representation  $y_{\theta} \triangleq f_{\theta}(\tilde{\pmb{x}})$ . This is followed by passing it subsequently through a projector  $g_{\theta}$  and predictor  $q_{\theta}$ , such that  $t_{\theta} \triangleq g_{\theta}(y_{\theta})$  and  $z_{\theta} \triangleq q_{\theta}(t_{\theta})$ . We collect all trainable parameters of the online network in  $\pmb{\theta}$ . Similarly, the encoder of target network  $f_{\xi}$  takes the second augmented view and outputs  $y_{\xi} \triangleq f_{\xi}(\tilde{x}')$ , followed by the projector network  $g_{\xi}$  producing  $t_{\xi} \triangleq g_{\xi}(y_{\xi})$ , where the trainable parameters are denoted as  $\pmb{\xi}$ . It is important to note that the predictor is applied exclusively to the online network.
+
+In order to introduce probabilistic self-distillation training, we employ a scoring rule (Gneiting & Raftery, 2007) as our loss function. To accomplish this, it is necessary to generate samples from the online network. However, directly sampling  $\mu$  and  $\sigma$  from a deterministic vector  $q_{\theta}(t)$  is inadequate, as it prevents us from performing backpropagation effectively. One way of producing samples from a neural network architecture, while still enabling backpropagation with respect to the latent representation  $\mathbf{z}$ , is to use the reparametrization trick (Kingma et al., 2015). Here, we assume that the output of the predictor network  $z_{\theta} = q_{\theta}(t_{\theta})$  follows an underlying normal distribution with mean  $\mu$  and standard deviation  $\sigma$ . We generate  $r \in \mathbb{N}$  samples from the output of the linear layers following the prediction head by sampling random noise  $\epsilon_j^i \sim N(0,1)$  for each augmented view of the  $i$ -th data point, such that the  $j$ -th sample, with  $j \in \{1, \dots, r\}$ , is given by:  $z_j^i = \mu^i + \sigma^i \odot \epsilon_j^i$ .
+
+Thus, we obtain samples  $z_{j}^{i}$  by shifting and scaling the random noise samples  $\epsilon_{j}^{i}$  by the outputs  $(\mu, \sigma)$  of a neural network with trainable parameters  $\theta$ , and the loss incurred by these samples can be backpropagated to update  $\theta$  during training. The online network parameters are updated by minimizing the scoring rule as follows:
+
+$$
+\hat {\boldsymbol {\theta}} := \underset {\boldsymbol {\theta}} {\arg \min } J (\boldsymbol {\theta}), \quad J (\boldsymbol {\theta}) = S \left(P _ {\boldsymbol {\theta}}, P _ {\boldsymbol {\xi}}\right) := \mathbb {E} _ {z _ {\boldsymbol {\xi}} \sim P _ {\boldsymbol {\xi}}} \left[ P _ {\boldsymbol {\theta}}, z _ {\boldsymbol {\xi}} \right], \tag {1}
+$$
+
+where  $P_{\xi}$  denotes the target distribution and  $z_{\xi}$  denotes the target output and  $P_{\theta}$  represents the online induced multivariate normal distribution. We define the approximation of our customized scoring rule loss for  $P_{\theta}$  and  $P_{\xi}$  as follows:
+
+$$
+\hat {S} \left(P _ {\boldsymbol {\theta}}, P _ {\boldsymbol {\xi}}\right) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left[ \frac {2 \lambda}{r} \sum_ {j = 1} ^ {r} \| \boldsymbol {z} _ {j} ^ {i} - \boldsymbol {z} _ {\xi} ^ {i} \| _ {2} ^ {\beta} - \frac {1 - \lambda}{r (r - 1)} \sum_ {j \neq k} \| \boldsymbol {z} _ {j} ^ {i} - \boldsymbol {z} _ {k} ^ {i} \| _ {2} ^ {\beta} \right] \tag {2}
+$$
+
+where  $z_{\xi}^{i}$  represents the target prediction for the  $i$ -the input sample.  $\beta \in (0,2)$  and  $\lambda \in (0,1)$  are learn-able hyperparameters. We set  $\lambda$  so that  $\hat{S}(P_{\theta},z_{\xi}) > 0$ . See Appendix 10.5.6 for detailed information.
+
+By the principle of knowledge distillation, the parameters of the target network are updated through the EMA of the weights from the online network (Grill et al., 2020; Caron et al., 2021), saving the need for backpropagation and thus reducing computation time considerably.
+
+$$
+\boldsymbol {\xi} _ {t} = (1 - \alpha) \boldsymbol {\theta} _ {t} + \alpha \boldsymbol {\xi} _ {t - 1}, \quad t = 1, 2, \dots , \quad \alpha \in [ 0, 1 ] \tag {3}
+$$
+
+The initial weights  $\xi_0$  are obtained through random initialization, while  $\alpha$  is a hyperparameter that determines the updating rate of the target weights and is set to change from 0.9 to 1 during training with a cosine scheduler.
+
+Next, we provide a comprehensive description of our objective function based on scoring rules (Section 4.1). For a detailed understanding of scoring rules and their various variants, please refer to Section 10.
+
+# 4.1 OBJECTIVE FUNCTION
+
+Numerous scoring rules can be decomposed into two terms. The first term is a function of  $\mathbf{z}_j$  and the realized observation  $\mathbf{z}_{\xi}$ . The second term is a function of two samples,  $\mathbf{z}_j$  and  $\mathbf{z}_k$ , drawn from the predicted distribution  $P$ . Our objective function is an adjusted version of the former, where the two components are posed as a convex combination with component weights controlled via hyperparameter  $\lambda \in (0,1)$ . We set  $\lambda$  so that  $\hat{S}(P_{\theta}, z_{\xi}) > 0$ . This modification of the scoring rule can be useful to adjust the focus of the loss function on either part of the scoring rule. Two notable examples of scoring rules adhering to this form are the energy score and the kernel score (Gneiting & Raftery, 2007).
+
+With the above notation, we define the energy score as:
+
+$$
+S _ {\mathrm {E}} \left(P _ {\boldsymbol {\theta}}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right) = 2 \cdot \mathbb {E} _ {P _ {\boldsymbol {\theta}}} \left[ \| \boldsymbol {z} _ {j} - \boldsymbol {z} _ {\boldsymbol {\xi}} \| _ {2} ^ {\beta} \right] + \left(- \mathbb {E} _ {P _ {\boldsymbol {\theta}}} \left[ \| \boldsymbol {z} _ {j} - \boldsymbol {z} _ {k} \| _ {2} ^ {\beta} \right]\right) =: S _ {\mathrm {E}} ^ {1} \left(P _ {\boldsymbol {\theta}}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right) + S _ {\mathrm {E}} ^ {2} \left(P _ {\boldsymbol {\theta}}\right) \tag {4}
+$$
+
+Analogously, we write the kernel score as:
+
+$$
+S _ {\mathrm {K}} \left(P _ {\boldsymbol {\theta}}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right) = \mathbb {E} _ {P _ {\boldsymbol {\theta}}} \left[ k \left(\boldsymbol {z} _ {j}, \boldsymbol {z} _ {k}\right) \right] + (- 2 \cdot \mathbb {E} _ {P _ {\boldsymbol {\theta}}} \left[ k \left(\boldsymbol {z} _ {j}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right) \right]) =: S _ {\mathrm {K}} ^ {2} \left(P _ {\boldsymbol {\theta}}\right) + S _ {\mathrm {K}} ^ {1} \left(P _ {\boldsymbol {\theta}}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right), \tag {5}
+$$
+
+with suitable kernel function  $k(\cdot ,\cdot)$
+
+For simplicity, we only write  $S(P_{\theta}, z_{\xi}) = S^{1}(P_{\theta}, z_{\xi}) + S^{2}(P_{\theta})$  in the following for both scores. With  $\lambda \in (0, 1)$ , we define the general form of our objective function as follows:
+
+$$
+S ^ {*} \left(P _ {\boldsymbol {\theta}}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right) := \lambda S ^ {1} \left(P _ {\boldsymbol {\theta}}, \boldsymbol {z} _ {\boldsymbol {\xi}}\right) + (1 - \lambda) S ^ {2} \left(P _ {\boldsymbol {\theta}}\right) \tag {6}
+$$
+
+# 4.2 THEORETICAL JUSTIFICATION
+
+Our approach is motivated by a strictly proper scoring rule, which ensures that the expected score is optimized only by the true distribution. However, our modified scoring rule gives more weight to the online network predictions during training. This deliberate focus is aimed at enhancing the model's learning efficacy. By optimizing the relative importance of each component using a hyperparameter  $\lambda$ , we encourage the model not only to fit the data but also to actively refine its predictive capabilities. Our proposed loss function (Eq. 6) consists of two key elements:
+
+- Comparison with the target network  $(S^1)$ : This term measures the adherence of the model's predictions to a known benchmark, ensuring that the model remains grounded in established knowledge.
+
+- Entropy for online network predictions  $(S^2)$ : This term emphasizes the internal consistency and reliability of the model's own predictions over time. This term increases the entropy of prediction and helps generalization by helping the exploration power of the model.
+
+In its original form, the scoring rule can exhibit a slight bias toward matching the target network too closely. To address this, we introduce stochasticity into the training process by increasing the weight of the second term (comparing online network predictions) in the loss function. This is not an artifact of poor design but a strategic insertion of complexity that nudges the algorithm towards more sophisticated regions of the solution space, thereby potentially locating a better optimum. Empirically, we show that an "improper" scoring rule variant can paradoxically increase accuracy.
+
+# 5 IMPLEMENTATION DETAILS AND EXPERIMENTAL SETUP
+
+Image augmentation We define a random transformation function  $T$  that applies a combination of multi-crop, horizontal flip, color jittering, and grayscale. Similar to Caron et al. (2021), we perform multi-crops with a random size from 0.8 to 1.0 of the original area and a random aspect ratio from 3/4 to 4/3 of the original aspect ratio. We define color-jittering of (0.8, 0.8, 0.8, 0.2), and Gaussian blurring with 0.5 probability and  $\zeta = (0.1, 2.0)$ .
+
+Deep self-supervised network architecture The online neural network is constructed from a backbone  $f$ , which can be either ViT (Dosovitskiy et al., 2021) or ResNet (He et al., 2016), and a projection head  $g$  followed by a prediction  $q$ . The backbone output  $f$  is used as a feature for downstream tasks. The projection consists of a 3-layer multilayer perceptron (MLP) with a hidden dimension of 2048, followed by 2 normalizations and a weight-normalized fully connected layer with  $K$  dimensions, similar to the design used in the DINO projection head. We use a predictor with two layers of MLPs with a hidden dimension of 12000, with a GELU nonlinearity in between. Note that ViT architectures do not use batch normalization (BN) by default, so we do not use BN in the projection or prediction when using ViT.
+
+The target network has the same backbone and projection as the online network and the target network learns through self-distillation. Similar to DINO Caron et al. (2021), after the online network parameters are updated, an EMA of the online parameters (i.e., a momentum encoder) is used to update the target parameters. The EMA prevents the target parameters from being updated too quickly. After the parameters are updated, the target also receives a new centering parameter.
+
+**Optimization** Our pretraining process involves training the models on the ImageNet training dataset (Deng et al., 2009) using the adamw optimizer (Loshchilov & Hutter, 2017a) and a batch size of 512, distributed across 8 GPUs using Nvidia Tesla A100 with ViT-S/16 architecture. We adopt a linear scaling rule to determine the base value of the learning rate, which is ramped up linearly during the first 30 epochs. Specifically, the learning rate is set to  $lr = 0.0005 * \text{batchsize} / 256$ . After the warmup phase, we decay the learning rate using a cosine schedule (Loshchilov & Hutter, 2017b). The weight decay also follows a cosine schedule, increasing from 0.04 to 0.4. The centering (smooth parameter) is 0.9.
+
+Datasets The datasets utilized in our experiments are as follows: The ImageNet (Deng et al., 2009) dataset with 1.28 million training images and 50,000 validation images with the size of  $256 \times 256$  contains 1,000 classes. The ImageNet-O dataset (Srivastava et al., 2022) comprises images belonging to classes that are not present in the ImageNet-1k dataset. It is considered a challenging benchmark for evaluating the robustness of the models, as it requires models to generalize to a diverse range of visual conditions and handle variations that are not typically encountered in standard training datasets. The ImageNet-C (Hendrycks & Dietterich, 2019) is a benchmark for evaluating the robustness of the models against common corruptions and perturbations that can occur in real-world scenarios. It consists of more than 30,000 images derived from the ImageNet dataset, with each image being corrupted in one of 15 different ways, including noise, blur, weather conditions, and digital artifacts. CIFAR-10/100 (Krizhevsky, 2009) are subsets of the tiny images dataset. Both datasets include 50,000 images for training and 10,000 validation images of size  $32 \times 32$  with 10 and 100 classes, respectively. The Oxford 102 Flower (Nilsback & Zisserman, 2008) consists of 102 flower categories, each class including between 40 and 258 images. The images have large scale, pose, and light variations. In addition, there are categories that have large variations within the category
+
+and several very similar categories. iNaturalist-2018 (Van Horn et al., 2018) (iNat) comprises a vast collection of 675,170 training and validation images, classified into 5,089 distinct fine-grained categories found in the natural world. It is worth noting that the iNat dataset exhibits a significant imbalance, as the number of images varies greatly across different categories.
+
+Tasks We evaluate the performance of ProSMin representations after self-supervised pretraining on the ImageNet on the basis of In-Domain (IND) generalization, OOD detection, semi-supervised learning, low-shot learning, corrupted dataset evaluation (see Section 6), as well as transfer learning to other datasets and tasks (see Section 11).
+
+Evaluation metrics We report the prediction performance with the following metrics: Top-1 accuracy  $\uparrow$ : refers to the proportion of test observations that are correctly predicted by the model's output as belonging to the correct class. AUROC  $\uparrow$ : the area under the ROC curve represents the relationship between false-positive and false-negative rates for various classification thresholds. In this case, the positive and negative classes refer to whether an observation is in or out of a given distribution, respectively, and the ROC curve is plotted as the threshold for classifying an observation as "positive" is gradually increased. Negative log-likelihood (NLL)  $\downarrow$ : measures the probability of observing the given test data given the estimated model parameters, multiplied by -1. This measure quantifies the degree to which the model's estimated parameters fit the test observations. Expected calibration error (ECE)  $\downarrow$  (Naeini et al., 2015): calculated as the mean absolute difference between the accuracy and confidence of the model's predictions, where confidence is defined as the highest posterior probability among the predicted classes. The difference is calculated across equally-spaced confidence intervals or bins and is weighted by the relative number of samples in each bin. A lower value of ECE indicates better calibration of the model. Mean Calibration Error (mCE)  $\downarrow$  is a metric used to evaluate the calibration of a classification model, similar to ECE. It is calculated as the mean of the absolute differences between the predicted and true probabilities of a given class across all classes. A lower value of mCE indicates better calibration of the model.
+
+# 6 RESULTS AND DISCUSSION
+
+In-distribution generalization IND generalization (or linear evaluation) measures how well a model's confidence aligns with its accuracy. To assess and compare the predictive abilities of our proposed model on in-distribution datasets, we freeze the encoder of the online network, denoted as  $f_{\theta}$ , after performing unsupervised pretraining. Then, we train a supervised linear classifier using a fully connected layer followed by softmax, which is placed on top of  $f_{\theta}$  after removing the projection and prediction network. The desired outcome is high predictive scores and low uncertainty scores. In Table 1, a comprehensive comparison is presented between our approach and other self-supervised methods. The results indicate that our method outperforms all others in terms of top-1 accuracy, and calibration, as demonstrated by the lowest expected calibration error (ECE) and negative log-likelihood (NLL) scores.
+
+Out-of-distribution detection The ability of a model to recognize test samples from classes that were not present during training is evaluated using OOD detection, as discussed in Geng et al. (2020). We conduct experiments on ImageNet-O (Srivastava et al., 2022) to assess the generalization of the model from IND to OOD datasets, as well as to predict the uncertainty of the models on OOD datasets. Note that evaluation is performed directly after self-supervised pretraining without a fine-tuning step. The y-axis of Figure 2 shows and compares the results for the OOD task in terms of AUROC. Remarkably, our method shows outstanding results for the detection of OOD samples compared to other approaches such as i-BOT, DINO, and Moco-V3. This outcome aligns with our expectations as we directly use the probabilistic latent representation for this task.
+
+Corrupted dataset evaluation An essential aspect of model robustness is its capability to produce precise predictions when the test data distribution changes. We examine model robustness under the context of covariate shift. Fig. 2 presents the improved performance metrics. Our method outperforms the baseline and has comparable predictive performance to the baseline for mCE.
+
+Semi-supervised and low-shot learning on ImageNet Following the semi-supervised protocol established in Chen et al. (2020), we employ fixed  $1\%$  and  $10\%$  splits of labeled training data from ImageNet. In Table 2, we compare our performance against several concurrent models, including the baseline (DINO). Based on results obtained in Table 2, our approach outperforms state-of-the-art methods in semi-supervised evaluation for both  $1\%$  and  $10\%$  scenarios. We assess our model's efficacy on a low-shot image classification task, where we train logistic regression on frozen weights
+
+![](images/bdbcdc7272c0680c36638346057688a0d60a823d1f8d0a2c9313b28579f7f9e3.jpg)  
+Figure 2: OOD detection and corrupted dataset evaluation. Methods with higher AUROC and lower mCE are better. Our method demonstrates outstanding performance among other approaches. We present and compare the performance of our method with others on the task of OOD detection on ImageNet-O and corrupted datasets on ImageNet-C, in Table 5
+
+with  $1\%$  and  $10\%$  labels. It's important to note that this experiment was performed on frozen weights, without finetuning. Table 2 shows our features achieved better performance compared to state-of-the-art methods.
+
+Table 1: IND Generalization (or Linear Evaluation). Top-1 accuracy,  $\kappa$ -NN, ECE, and NLL averaged over in-distribution on test samples of the ImageNet dataset where the encoder is ViT-S/16 over 800 epochs. The best score for each metric is shown in bold, and the second-best is underlined.  
+
+<table><tr><td>Method</td><td>Top-1 Acc (%) (↑)</td><td>κ-NN (↑)</td><td>NLL (↓)</td><td>ECE (↓)</td></tr><tr><td>DINO Caron et al. (2021)</td><td>76.8</td><td>74.5</td><td>0.919</td><td>0.015</td></tr><tr><td>MOCO-V3 Chen et al. (2021)</td><td>73.2</td><td>64.7</td><td>1.152</td><td>0.027</td></tr><tr><td>i-BOT Zhou et al. (2021)</td><td>77.9</td><td>75.2</td><td>0.918</td><td>0.013</td></tr><tr><td>ProSMin</td><td>78.4</td><td>76.2</td><td>0.900</td><td>0.006</td></tr></table>
+
+# 7 ABLATION STUDY
+
+To gain a deeper understanding of the behavior and performance of our proposed method, we conducted several ablation studies to explore various aspects of our approach. Specifically, we investigate the following factors: different scoring rules as an objective function, the hyperparameter of our loss function  $(\lambda)$ , the number of samples used for generating latent representations, the dimension of the embedding space, the effect of the momentum hyperparameter, the impact of batch normalization (BN), and the prediction layer (PL). We also supply an ablation analysis on computational efficiency and the broader impact of our method in the Appendix (see Section 11). These investigations aim to provide insights and intuition regarding our approach.
+
+Impact of different components of scoring rule We conduct a series of experiments to explore alternative scoring rules, including kernel scoring rules and various variations of energy scoring rules, for our objective function. The results, as presented in Table 3, demonstrate that the kernel scoring
+
+Table 2: Low-shot and semi-supervised evaluation: Top-1 accuracy (ACC), ECE, and NLL for semi-supervised on ImageNet classification using  $1\%$  and  $10\%$  training examples fine-tuning. and Low-shot results with frozen ViT features.  
+
+<table><tr><td>Method</td><td>1%</td><td>10%</td><td>Architecture</td><td>Parameters</td></tr><tr><td colspan="5">Semi-supervised</td></tr><tr><td>DINO Caron et al. (2021)</td><td>60.3</td><td>74.3</td><td>ViT-S/16</td><td>21</td></tr><tr><td>i-BOT Zhou et al. (2021)</td><td>61.9</td><td>75.1</td><td>ViT-S/16</td><td>21</td></tr><tr><td>ProSMin (ours)</td><td>62.1</td><td>75.6</td><td>ViT-S/16</td><td>21</td></tr><tr><td colspan="5">Low-shot learning</td></tr><tr><td>DINO Caron et al. (2021)</td><td>64.5</td><td>72.2</td><td>ViT-S/16</td><td>21</td></tr><tr><td>i-BOT Zhou et al. (2021)</td><td>65.9</td><td>73.4</td><td>ViT-S/16</td><td>21</td></tr><tr><td>ProSMin (ours)</td><td>66.1</td><td>73.8</td><td>ViT-S/16</td><td>21</td></tr></table>
+
+![](images/4c992d80f491cd1725d57a2a66ba7365a7a88779405b40c1d207298388135a08.jpg)  
+(a)
+
+![](images/3f8d20ebb964bb2f53165418cfaaa992de0b22911a9d9188acf99167528e9da4.jpg)  
+(b)  
+Figure 3: Study of hyperparameters of our proposed ProSMin (a)  $\lambda$ , (b) Number of samples, (c) Momentum coefficient  $(\alpha)$ , and (d) Size of embedding obtained by 100 epochs on ImageNet.
+
+![](images/015be9183145ef6ef1ec2973a124174c81b3637d61901e95e882790461709efd.jpg)  
+(c)
+
+![](images/089b8697a9ba8f6fae207c827027b6f6d4db9d14b24f5eb668b7deb3f7aae5d1.jpg)  
+(d)
+
+Table 3: Important component for training. We investigate the effect of each component on the linear evaluation performance for 100 epochs. The first line shows the best combination. PL is the prediction layer, BN is the batch normalization layer.  
+
+<table><tr><td>Energy (L1 loss)</td><td>Kernel score</td><td>Energy (L2 loss)</td><td>BN.</td><td>PL.</td><td>λ</td><td>σ</td><td>Accuracy</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>✓</td><td>✓</td><td>73.2</td></tr><tr><td>X</td><td>✓</td><td>X</td><td>X</td><td>✓</td><td>✓</td><td>✓</td><td>1.1</td></tr><tr><td>X</td><td>X</td><td>✓</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>43.0</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>68.5</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>✓</td><td>71.2</td></tr><tr><td>✓</td><td>X</td><td>X</td><td>X</td><td>✓</td><td>X</td><td>X</td><td>47.8</td></tr></table>
+
+rule exhibits instability. Furthermore, we investigate a case with L1 loss (the last row of Table 3) with prediction layer and without batch normalization, that the  $\lambda$  and  $\sigma$  are zero, to motivate the effect of the second part of our loss function. Overall, the energy score with  $\beta = 1$  ( $L1$  loss) yields the best performance among the tested scoring rules. We provide a theoretical explanation for  $L1$  in Section 10.5.3.
+
+Study of hyperparameters Figure 3a depicts the influence of the hyperparameter  $\lambda$  utilized in our proposed loss function, which controls the impact of  $S^1 (P_\theta ,z_\xi)$  and  $S^2 (P_\theta ,z_\xi)$  on the objective function. An ablation analysis was conducted to investigate the effect of increasing the number of samples from  $q_{\theta}$ , as illustrated in Figure 3b. The results demonstrate that employing four samples yields satisfactory performance. Figure 3c showcases the outcomes obtained from the knowledge distillation rate. In previous approaches, the exponential moving average parameter initiated from a value relatively close to 1 (e.g., 0.996 (Grill et al., 2020), Caron et al. (2021)). However, in our case,  $\alpha$  starts from 0.9, implying a faster pace of knowledge distillation. Furthermore, we examine the impact of different sizes for the embedding vector, as presented in Fig. 3d. The results obtained after 100 epochs reveal that increasing the embedding size leads to improved performance. However, it should be noted that larger embedding sizes necessitate additional computational resources, thus our choice of size is based on the available computational capacity. Table 3 provides insights into the influence of batch normalization in the prevention of representation collapse (Grill et al., 2020). As our framework operates in a probabilistic setting, the inclusion of batch normalization is unnecessary for averting collapse. Additionally, the prediction layer ( $q_{\theta}$ ) enhances performance by facilitating improved feature extraction in online networks.
+
+# 8 CONCLUSION
+
+In this paper, we presented ProSMin as a novel probabilistic self-supervised framework to address a collapsing representation problem. Our method includes two neural networks that collaborate and learn from each other using an augmented format. Our framework is trained by minimizing a proposed scoring rule objective function. We evaluated ProSMin across different tasks, including in-distribution generalization, out-of-distribution detection, dataset corruption, transfer learning, low-shot, and semi-supervised learning. The results demonstrate that our method achieves superior performance in terms of accuracy and calibration, thus showing the effectiveness of our proposed approach.
+
+# 9 ACKNOWLEDGMENT
+
+M. R. and B. B. were supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy through the Center for Analytics - Data - Applications (ADA-Center) within the framework of BAYERN DIGITAL II (20-3410-2-9-8). M. R., B. B., E. H., L. W., Y. L., and A.V. were supported by the German Federal Ministry of Education and Research (BMBF) Munich Center for Machine Learning (MCML). L. W. was supported by the DAAD program Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the German Federal Ministry of Education and Research
+
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+Grant Van Horn, Oisin Mac Aodha, Yang Song, Yin Cui, Chen Sun, Alex Shepard, Hartwig Adam, Pietro Perona, and Serge Belongie. The inaturalist species classification and detection dataset. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8769-8778, 2018.  
+Vladimir V V'yugin and Vladimir G Trunov. Online learning with continuous ranked probability score. In Conformal and Probabilistic Prediction and Applications, pp. 163-177. PMLR, 2019.  
+Jure Zbontar, Li Jing, Ishan Misra, Yann LeCun, and Stephane Deny. Barlow twins: Self-supervised learning via redundancy reduction. In International Conference on Machine Learning, pp. 12310-12320. PMLR, 2021.  
+Jinghao Zhou, Chen Wei, Huiyu Wang, Wei Shen, Cihang Xie, Alan Yuille, and Tao Kong. ibot: Image bert pre-training with online tokenizer. ICLR, 2021.
+
+# 10 SCORING RULES
+
+The notation employed in this study aligns with the work of Gneiting & Raftery (2007) regarding scoring rules  $S(P, \boldsymbol{x})$ , where  $P$  represents the predictive distribution and  $\boldsymbol{x}$  denotes the observed data. By assuming that  $\mathbf{X}$  follows a true distribution  $Q$ . With this assumption, we first explain different variations of scoring rules, such as the expected scoring rule (10.1), the proper scoring rule (10.2), the energy scoring rule (10.3), and the kernel scoring rule (10.4).
+
+We then discuss the unbiased estimation properties of the scoring rule 10.5 for kernel score 10.5.1 and energy score 10.5.2, which we used as the objective function in our study.
+
+In subsection 10.5.4, we establish the interchangeability of the expectation and gradient in our theoretical derivative, enabling us to derive gradients in a conventional manner. This is crucial due to the incorporation of samples within the scoring rule and the utilization of non-differential activation functions. Subsequently, in subsection 10.5.5, we introduce an unbiased estimate of the gradient.
+
+# 10.1 EXPECTED SCORING RULE
+
+We define the scoring rule  $S(P, x)$  as a function of the distribution  $P$  and the observation  $x$  of  $\mathbf{X}$ . The expected scoring rule is defined as:
+
+$$
+S (P, Q) := \mathbb {E} _ {\mathbf {X} \sim Q} S (P, \mathbf {X}). \tag {7}
+$$
+
+# 10.2 PROPER SCORING RULE
+
+A scoring rule  $S$  is called proper w.r.t a set of distributions  $\mathcal{P}$ , if for all  $P, Q \in \mathcal{P}$  the expected score  $S(P, Q)$  is minimized in  $Q$  at  $Q = P$ . A scoring rule  $S$  is called strictly proper if there exists the unique minimum  $S(P, Q) > S(Q, Q)$  for  $Q \neq P$ .
+
+# 10.3 ENERGY SCORE
+
+The energy score is defined by:
+
+$$
+S _ {\mathrm {E}} ^ {(\beta)} (P, \mathbf {x}) = 2 \cdot \mathbb {E} \left[ \| \tilde {\mathbf {X}} - \mathbf {x} \| _ {2} ^ {\beta} \right] - \mathbb {E} \left[ \left\| \tilde {\mathbf {X}} - \tilde {\mathbf {X}} ^ {\prime} \right\| _ {2} ^ {\beta} \right], \quad \tilde {\mathbf {X}} \perp \tilde {\mathbf {X}} ^ {\prime} \sim P, \tag {8}
+$$
+
+where  $\beta \in (0,2)$
+
+It is a strictly proper scoring rule for the class of probability measures  $\mathcal{P}$  such that  $\mathbb{E}_{\tilde{\mathbf{X}}\sim P}\| \tilde{\mathbf{X}}\|^\beta < \infty$ . An unbiased estimate can be obtained by replacing the expectations in  $S_{\mathrm{E}}^{(\beta)}$  with empirical means over draws from  $P$ .
+
+# 10.4 KERNEL SCORE
+
+Let  $k(\cdot, \cdot)$  be a positive definite kernel. The kernel score for  $k$  is defined by
+
+$$
+S _ {k} (P, \mathbf {x}) = \mathbb {E} \left[ k \left(\tilde {\mathbf {X}}, \tilde {\mathbf {X}} ^ {\prime}\right) \right] - 2 \cdot \mathbb {E} [ k (\tilde {\mathbf {X}}, \mathbf {x}) ], \quad \tilde {\mathbf {X}} \perp \tilde {\mathbf {X}} ^ {\prime} \sim P. \tag {9}
+$$
+
+The kernel score is a proper scoring rule for the class of probability distribution  $P$  for which holds, that  $\mathbb{E}_{\tilde{\mathbf{X}},\tilde{\mathbf{X}}^{\prime}\sim P}\left[k\left(\tilde{\mathbf{X}},\tilde{\mathbf{X}}^{\prime}\right)\right] < \infty$ . Under the condition that the kernel Maximum Mean Discrepancy is a metric, the kernel score is strictly proper. This condition is satisfied by the Gaussian kernel, which will be used in this work. Let  $\gamma$  be a scalar bandwidth. We define the Gaussian kernel as follows:
+
+$$
+k (\tilde {\mathbf {x}}, \mathbf {x}) = \exp \left(- \frac {\| \tilde {\mathbf {x}} - \mathbf {x} \| _ {2} ^ {2}}{2 \gamma^ {2}}\right).
+$$
+
+We get an unbiased estimate by replacing the expectations in  $S_{\mathrm{k}}$  with the empirical means overdraws from  $P$ .
+
+# 10.5 UNBIASED ESTIMATORS OF THE EXPECTED SCORING RULE
+
+Let  $N$  be the batch size and  $r$  be the number of samples drawn from the online-induced distribution. Let  $z_{\theta} = (z_1, \dots, z_r)$  be the sample drawn from the online. We recall the definition of the Expected Scoring Rule:
+
+$$
+S \left(P _ {\theta}, P _ {\xi}\right) := \mathbb {E} _ {z _ {\xi} \sim P _ {\xi}} \left[ S \left(P _ {\theta}, z _ {\xi}\right) \right],
+$$
+
+where  $z_{\xi}$  denotes the output of the target network while  $P_{\theta}$  represents the online induced multivariate normal distribution.
+
+# 10.5.1 KERNEL SCORE
+
+We can write the estimated kernel score as follows:
+
+$$
+\hat {S} (P _ {\theta}, P _ {\xi}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left[ \frac {1}{r (r - 1)} \sum_ {j, k = 1, j \neq k} k (z _ {j} ^ {i}, z _ {k} ^ {i}) - \frac {2}{r} \sum_ {j = 1} ^ {r} k (z _ {j} ^ {i}, z _ {\xi} ^ {i}) \right]
+$$
+
+# 10.5.2 ENERGY SCORE
+
+Consider  $\beta \in (0,2)$  and  $N$  be the batch size. The Estimated Energy Score can be written as:
+
+$$
+\hat {S} (P _ {\theta}, P _ {\xi}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \left[ \frac {2}{r} \sum_ {j = 1} ^ {r} \| z _ {j} ^ {i} - z _ {\xi} ^ {i} \| _ {2} ^ {\beta} - \frac {1}{r (r - 1)} \sum_ {j, k = 1, j \neq k} \| z _ {j} ^ {i} - z _ {k} ^ {i} \| _ {2} ^ {\beta} \right]
+$$
+
+# 10.5.3 PARTIAL DERIVATIVES OF THE ESTIMATED ENERGY SCORE
+
+Let  $\beta \in (0,2)$  and let  $N$  be the batch size. Let  $x_{i}$  be the input data for  $i\in 1,\dots,N$ . Let  $r\in \mathbb{N}$  be the number of samples. Let  $\mu_{i} = f_{\theta}^{(1)}(x_{i})$  and let  $\sigma_i^2 = f_\theta^{(2)}(x_i)$ , where  $f^{(j)}$  denotes the separated branches for mean and variance in the last layer of the online network for  $j\in 1,2$ . We can calculate the partial derivatives needed for stochastic gradient descent as follows. Then, the formula for the mean is:
+
+$$
+\begin{array}{l} \frac {\partial}{\partial \mu} \hat {S} (P _ {\theta}, P _ {\xi}) = \frac {\partial}{\partial \mu} \frac {1}{N} \sum_ {i = 1} ^ {N} \left[ \frac {2}{r} \sum_ {j = 1} ^ {r} \left\| \mu_ {i} + \epsilon_ {j} \sqrt {\sigma_ {i} ^ {2}} - \phi_ {\xi} ^ {i} \right\| _ {2} ^ {\beta} \right. \\ - \frac {1}{r (r - 1)} \sum_ {j, k = 1, j \neq k} \left\| \mu_ {i} + \epsilon_ {j} \sqrt {\sigma_ {i} ^ {2}} - \mu_ {i} + \epsilon_ {k} \sqrt {\sigma_ {i} ^ {2}} \right\| _ {2} ^ {\beta} \Biggr ] \\ = \frac {1}{N} \sum_ {i = 1} ^ {N} \frac {2}{r} \sum_ {j = 1} ^ {r} \beta \left\| \mu_ {i} + \epsilon_ {j} \sqrt {\sigma_ {i} ^ {2}} - \phi_ {\xi} ^ {i} \right\| _ {2} ^ {\beta - 1} \frac {\partial}{\partial \mu} \left[ \mu_ {i} + \epsilon_ {j} \sqrt {\sigma_ {i} ^ {2}} - \phi_ {\xi} ^ {i} \right] \\ = \frac {1}{N} \sum_ {i = 1} ^ {N} \frac {2}{r} \sum_ {j = 1} ^ {r} \beta \left\| f _ {\theta} ^ {(1)} (x _ {i}) + \epsilon_ {j} \sqrt {f _ {\theta} ^ {(2)} (x _ {i})} - \phi_ {\xi} ^ {i} \right\| _ {2} ^ {\beta - 1} \nabla_ {\theta} f _ {\theta} ^ {(1)} (x _ {i}) \\ \end{array}
+$$
+
+while the formula for the variance is:
+
+$$
+\begin{array}{l} \frac {\partial}{\partial \sigma^ {2}} \hat {S} (P _ {\theta}, P _ {\xi}) = \frac {\partial}{\partial \sigma^ {2}} \frac {1}{N} \sum_ {i = 1} ^ {N} \left[ \frac {2}{r} \sum_ {j = 1} ^ {r} \left\| \mu_ {i} + \epsilon_ {j} \sqrt {\sigma_ {i} ^ {2}} - \phi_ {\xi} ^ {i} \right\| _ {2} ^ {\beta} \right. \\ \left. - \frac {1}{r (r - 1)} \sum_ {j, k = 1, j \neq k} \left\| \mu_ {i} + \epsilon_ {j} \sqrt {\sigma_ {i} ^ {2}} - \mu_ {i} + \epsilon_ {k} \sqrt {\sigma_ {i} ^ {2}} \right\| _ {2} ^ {\beta} \right] \\ = \left[ \frac {\beta}{N} \sum_ {i = 1} ^ {N} \left[ \frac {2}{r} \sum_ {j = 1} ^ {r} \left\| f _ {\theta} ^ {(1)} \left(x _ {i}\right) + \epsilon_ {j} \sqrt {f _ {\theta} ^ {(2)} \left(x _ {i}\right)} - \phi_ {\xi} ^ {i} \right\| _ {2} ^ {\beta - 1} \right. \right. \\ \left. - \frac {1}{r (r - 1)} \sum_ {j, k = 1, j \neq k} \left\| \sqrt {f _ {\theta} ^ {(2)} (x _ {i})} (\epsilon_ {j} - \epsilon_ {k}) \right\| _ {2} ^ {\beta - 1} \right] \nabla_ {\theta} f _ {\theta} ^ {(2)} (x _ {i}) \\ \end{array}
+$$
+
+# 10.5.4 EXCHANGEABILITY OF GRADIENT AND EXPECTATION
+
+This proof follows a similar argumentation as the proof performed in Pacchiardi & Dutta (2022). We want to solve
+
+$$
+\hat {\boldsymbol {\theta}} := \underset {\boldsymbol {\theta}} {\arg \min } J (\boldsymbol {\theta}), \quad J (\boldsymbol {\theta}) = \frac {1}{N} \sum_ {i = 1} ^ {N} S \left(P _ {\boldsymbol {\theta}}, z _ {\boldsymbol {\xi}} ^ {i}\right) \tag {10}
+$$
+
+Finding a minimum in our network architecture is done via stochastic gradient descent (SGD) or an algorithm exploiting the same properties as SGD. Recall that the scoring rule is defined as an expectation over samples from  $P_{\theta}$ . Additionally, we can describe the scoring rule as some function  $h$  of independent inputs  $z_{j}^{i} \perp z_{k}^{i}$ . Hence we can write:
+
+$$
+S \left(P _ {\boldsymbol {\theta}}, z _ {\boldsymbol {\xi}} ^ {i}\right) = \mathbb {E} _ {z _ {j} ^ {i}, z _ {k} ^ {i} \sim P _ {\boldsymbol {\theta}}} \left[ h \left(z _ {j} ^ {i}, z _ {k} ^ {i}, z _ {\boldsymbol {\xi}} ^ {i}\right) \right] \tag {11}
+$$
+
+Let  $\mu_i, \sigma_i^2$  be the estimated mean and variance, respectively, of the online network  $f_{\theta}$  for the  $i$ -th input sample  $x_i$ . With  $P_{\theta}$  being the distribution induced by the online network and  $g_{\theta}$  being its transformation, we use the reparametrization trick and get
+
+$$
+J (\boldsymbol {\theta}) = \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\epsilon_ {j} ^ {i}, \epsilon_ {k} ^ {i} \sim \mathcal {N} (0, 1)} \left[ h \left(g _ {\boldsymbol {\theta}} \left(\epsilon_ {j} ^ {i}, \mu_ {i}, \sigma_ {i} ^ {2}\right), g _ {\boldsymbol {\theta}} \left(\epsilon_ {k} ^ {i}, \mu_ {i}, \sigma_ {i} ^ {2}\right), z _ {\boldsymbol {\xi}} ^ {i}\right) \right] \tag {12}
+$$
+
+We now can write the derivative as follows:
+
+$$
+\begin{array}{l} \nabla_ {\boldsymbol {\theta}} J (\boldsymbol {\theta}) = \nabla_ {\boldsymbol {\theta}} \frac {1}{N} \sum_ {i = 1} ^ {N} \mathbb {E} _ {\epsilon_ {j} ^ {i}, \epsilon_ {k} ^ {i} \sim \mathcal {N} (0, 1)} \left[ h (g _ {\boldsymbol {\theta}} (\epsilon_ {j} ^ {i}, \mu_ {i}, \sigma_ {i} ^ {2}), g _ {\boldsymbol {\theta}} (\epsilon_ {k} ^ {i}, \mu_ {i}, \sigma_ {i} ^ {2}), z _ {\boldsymbol {\xi}} ^ {i}) \right] \\ = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {E} _ {\epsilon_ {j} ^ {i}, \epsilon_ {k} ^ {i} \sim \mathcal {N} (0, 1)} \left[ \nabla_ {\pmb {\theta}} h (\boldsymbol {g} _ {\pmb {\theta}} (\epsilon_ {j} ^ {i}, \mu_ {i}, \sigma_ {i} ^ {2}), g _ {\pmb {\theta}} (\epsilon_ {k} ^ {i}, \mu_ {i}, \sigma_ {i} ^ {2}), \pmb {z} _ {\pmb {\xi}} ^ {i}) \right] \\ \end{array}
+$$
+
+The exchange between expectation and gradient is not trivial because of the non-differentiability of functions (e.g., ReLU) within the network function  $g_{\theta}$ . We can still perform this step by using Theorem 5 from Pacchiardi & Dutta (2022), considering mild conditions on the neural network architecture of  $g_{\theta}$ .
+
+# 10.5.5 UNBIASED ESTIMATE OF THE GRADIENT
+
+Let  $\mathcal{B}$  be a random subset (batch) of the data set. Using 10.5.4, we can obtain unbiased estimates of  $\nabla_{\pmb{\theta}}J(\pmb{\theta})$  using samples  $\epsilon_i^j \sim \mathcal{N}(0,1), j = 1,\dots,r$  for each  $i \in \{1,\dots,N\}$ :
+
+$$
+\widehat {\nabla_ {\boldsymbol {\theta}} J (\boldsymbol {\theta})} = \frac {1}{| \mathcal {B} |} \sum_ {i \in \mathcal {B}} \frac {1}{r (r - 1)} \sum_ {j, k = 1, j \neq k} ^ {r} \nabla_ {\boldsymbol {\theta}} h \left(g _ {\boldsymbol {\theta}} \left(\epsilon_ {i} ^ {k}, \mu_ {i}, \sigma_ {i} ^ {2}\right), g _ {\boldsymbol {\theta}} \left(\epsilon_ {i} ^ {j}, \mu_ {i}, \sigma_ {i} ^ {2}\right), z _ {\boldsymbol {\xi}} ^ {i}\right) \tag {13}
+$$
+
+# 10.5.6 DETAILED IMPLEMENTATION
+
+In addition to the detail in Section 5, it's important to mention that the target network takes two global augmented samples while the online network takes two global augmented samples and 16 local augmentations for multi-cropping samples. Then, the  $r$  in Eq. 2 refers to the number of augmentation samples multiplied by the number of samples. Furthermore, in Eq. 2  $z_{\xi}$  denotes one of the global augmented samples.
+
+# 11 ADDITIONAL EXPERIMENTAL RESULTS AND ABLATION ANALYSIS
+
+In this section, we provide additional details on experiments and ablation analysis.
+
+# 11.1 TRANSFER LEARNING EVALUATION
+
+We further assess the generalization capacity of the learned representation on learning a new dataset. We followed the same transfer learning protocol explained in (Caron et al., 2021). To this end, we evaluate the performance of our model pretrained on ImageNet to CIFAR10/100, an imbalanced naturalist dataset (iNat-18), and the Flower dataset. According to the results shown in Table 4, we observe that our method provides a robust solution when transferring to the new dataset.
+
+Table 4: Transfer to new dataset evaluation: Transfer learning by finetuning pretrained models on different datasets. We report top-1 accuracy. Self-supervised pretraining with ProSMin transfers better than supervised pretraining.  
+
+<table><tr><td>Method</td><td>CIFAR-10</td><td>CIFAR-100</td><td>iNat-18</td><td>Flowers</td></tr><tr><td>DINO Caron et al. (2021)</td><td>99.0</td><td>90.5</td><td>72.0</td><td>98.5</td></tr><tr><td>i-BOT Zhou et al. (2021)</td><td>99.1</td><td>90.7</td><td>73.7</td><td>98.6</td></tr><tr><td>ProSMin (ours)</td><td>99.3</td><td>91.2</td><td>74.4</td><td>99.1</td></tr></table>
+
+# 11.2 OUT-OF-DISTRIBUTION DETECTION AND CORRUPTED DATASET EVALUATION
+
+Table 5 compares the performance of our method for the task of OOD detection and corrupted dataset on ImageNet-O and ImageNet-C respectively.
+
+Table 5: Out of distribution detection and corrupted dataset: AUROC for out of distribution detection for Imagent-O dataset, where higher AUROC is better, and mCE for Imagent-C dataset where lower mCE is better.  
+
+<table><tr><td>Method</td><td>AUROC (%) (↑)</td><td>mCE (↑)</td></tr><tr><td>DINO (300 epochs) Caron et al. (2021)</td><td>74.0</td><td>68.1</td></tr><tr><td>MOCO-V3 (300 epochs) Chen et al. (2021)</td><td>71.0</td><td>73.5</td></tr><tr><td>ProSMin (300 epochs)</td><td>76.0</td><td>66.1</td></tr><tr><td>DINO (800 epochs) Caron et al. (2021)</td><td>76.0</td><td>66.5</td></tr><tr><td>ProSMin (800 epochs) Caron et al. (2021)</td><td>77.1</td><td>65.8</td></tr><tr><td>i-BOT (800 epochs) Caron et al. (2021)</td><td>76.6</td><td>66.2</td></tr></table>
+
+# 11.3 DIMENSION OF REPRESENTATION VECTOR
+
+The dimensionality of latent representations influences the equilibrium between information preservation, transferability, computational efficiency, and defense against overfitting. In Table 6, we evaluate the effect of varying the output dimensionality. Based on the results shown in Table 6 the large output dimensionality improves the performance.
+
+Table 6: Analysis of dimension of representation vector in term of Top-1 K-NN.  
+
+<table><tr><td>Method</td><td>1024</td><td>4096</td><td>16384</td><td>65536</td><td>262144</td></tr><tr><td>DINO Caron et al. (2021)</td><td>67.8</td><td>69.3</td><td>69.2</td><td>69.7</td><td>69.1</td></tr><tr><td>ProSMin (ours)</td><td>69.0</td><td>69.4</td><td>69.5</td><td>69.9</td><td>-</td></tr></table>
+
+# 11.4 IMPACT OF PROSMIN ON MITIGATING REPRESENTATION COLLAPSE
+
+As we explained in Chapter 3, ProSMin formulates the collapsing representation problem through a probabilistic lens, aiming to provide a comprehensive and nuanced solution that not only addresses the limitations of deterministic approaches but also harnesses the power of uncertainty quantification and broader representation distributions. Specifically, we propose a novel probabilistic self-supervised learning method that minimizes a scoring rule during pretext task learning. We motivate ProSMin by formulating knowledge distillation (KD) in a probabilistic manner. Through extensive empirical analysis, we validate the effectiveness of our approach compared to other self-supervised approaches proposed to prevent collapsing representation (MoCoV3 (Chen et al., 2021), DINO (Caron et al., 2021), and BYOL (Grill et al., 2020). In particular, BYOL prevents collapsing representation using batch normalization and a prediction layer. For DINO, Sharpening and centering play the same role. We showed the superiority of our method compared to other approaches for collapsing representation in Tables 1, 2, 5, and 4 as well as Figure 2 for the task of in-domain generalization, low-shot learning, semi-supervised learning, transfer learning, OOD detection, and corrupted dataset evaluation respectively. Importantly, in Table 3, we showed that our method can achieve a good performance without BN, prediction layer, centering, and sharpening which are the components of BYOL and DINO to prevent collapsing representation.
+
+# 11.5 ANALYSIS OF COMPUTATIONAL COST
+
+We further assess the effectiveness of our proposed approach and compare it with DINO and iBOT in Table 7. The presented values are obtained from the data reported in the DINO and iBOT papers. However, these papers do not include information regarding the total number of parameters.
+
+Table 7: Evaluation of computational efficiency We conduct a thorough analysis of the computational efficiency of our novel probabilistic approach in comparison to alternative self-supervised methods. This evaluation encompasses memory utilization and computational expenditure.  
+
+<table><tr><td>Method (ViT-s)</td><td>parameters (M)</td><td>im/s</td><td>time/ 300-epochs (hr)</td><td>number of GPUs</td><td>memory (G)</td></tr><tr><td>DINO Caron et al. (2021)</td><td>21</td><td>1007</td><td>72.6</td><td>16</td><td>15.4</td></tr><tr><td>i-BOT Zhou et al. (2021)</td><td>21</td><td>1007</td><td>73.8</td><td>16</td><td>19.5</td></tr><tr><td>ProSMin (ours)</td><td>21</td><td>1007</td><td>98.0</td><td>8</td><td>21.1</td></tr></table>
+
+# 11.6 BROADER IMPACT AND LIMITATIONS
+
+This study has the potential to inspire new algorithms and stimulate theoretical and experimental exploration. The algorithm presented here can be used for many different probabilistic downstream tasks, including (but not limited to) uncertainty quantification, density estimation, image retrieval, probabilistic unsupervised clustering, program debugging, image generation, music analysis, and ranking. In addition, we believe that our extended concept probabilistic framework opens many interesting avenues for future development in self-supervised learning, and addresses many problems of existing models, such as avoiding representation collapse.
+
+However, there are several limitations. One limitation of our model compared to other learning methods (such as supervised learning) is that self-supervised learning may require more computational resources and training time. However, considering that our proposed method does not require manual annotation, which is usually very expensive, we would argue that this trade-off is acceptable.
+
+![](images/24bc2efacf0bb8fba7d77410e9501694187f902828f4c29f632101c8faa62f44.jpg)  
+(a)
+
+![](images/23608a637cdb3deb158d26e646d656d0e072bf535fce89240c679d01a88b157c.jpg)  
+(b)
+
+![](images/0ad64ca203c13513d976d04a2de4f9fe21dd181a4ab0bfd1b3903a320308971c.jpg)  
+(c)
+
+![](images/bb3bf841899ce1f0c82b22b9c074d84e2c54acc0cbd27ae0daf2b8eef5d41245.jpg)  
+(d)  
+Figure 4: Study of hyperparameters of our proposed ProSMin (a)  $\lambda$ , (b) Number of samples, (c) Momentum coefficient  $(\alpha)$ , and (d) Size of embedding obtained by 100 epochs on ImageNet.

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+# PROGRESSIVE3D: PROGRESSIVELY LOCAL EDITING FOR TEXT-TO-3D CONTENT CREATION WITH COMPLEX SEMANTIC PROMPTS
+
+Xinhua Cheng\*1, Tianyu Yang\*2, Jianan Wang2, Yu Li2, Lei Zhang2, Jian Zhang1, Li Yuan\*1,3
+
+$^{1}$ Peking University
+
+$^{2}$ International Digital Economy Academy (IDEA)
+
+3Peng Cheng Laboratory
+
+# ABSTRACT
+
+Recent text-to-3D generation methods achieve impressive 3D content creation capacity thanks to the advances in image diffusion models and optimizing strategies. However, current methods struggle to generate correct 3D content for a complex prompt in semantics, i.e., a prompt describing multiple interacted objects binding with different attributes. In this work, we propose a general framework named Progressive3D, which decomposes the entire generation into a series of locally progressive editing steps to create precise 3D content for complex prompts, and we constrain the content change to only occur in regions determined by user-defined region prompts in each editing step. Furthermore, we propose an overlapped semantic component suppression technique to encourage the optimization process to focus more on the semantic differences between prompts. Experiments demonstrate that the proposed Progressive3D framework is effective in local editing and is general for different 3D representations, leading to precise 3D content production for prompts with complex semantics for various text-to-3D methods. Our project page is https://cxh0519.github.io/projects/Progressive3D/
+
+# 1 INTRODUCTION
+
+High-quality 3D digital content that conforms to the requirements of users is desired due to its various applications in the entertainment industry, mixed reality, and robotic simulation. Compared to the traditional 3D generating process which requests manual design in professional modeling software, automatically creating 3D content with given text prompts is more friendly for both beginners and experienced artists. Driven by the recent progress of neural 3D representations (Mildenhall et al., 2020; Wang et al., 2021; Yariv et al., 2021; Shen et al., 2021) and text-to-image (T2I) diffusion models (Nichol et al., 2021; Rombach et al., 2022; Mou et al., 2023; Zhang et al., 2023), Dreamfusion (Poole et al., 2022) demonstrates impressive 3D content creation capacity conditioned on given prompts by distilling the prior knowledge from T2I diffusion models into a Neural Radiance Field (NeRF), which attracts board interests and emerging attempts in text-to-3D creation.
+
+Although text-to-3D methods have tried to use various 3D neural representations (Lin et al., 2023; Chen et al., 2023; Tsalicoglou et al., 2023) and optimization strategies (Wang et al., 2023a; Huang et al., 2023b; Wang et al., 2023b) for improving the quality of created 3D content and achieving remark accomplishments, they rarely pay attention to enhancing the semantic consistency between generated 3D content and given prompts. As a result, most text-to-3D methods struggle to produce correct results when the text prompt describes a complex scene involving multiple objects binding with different attributes. As shown in Fig. 1(a), existing text-to-3D methods suffer from challenges with complex prompts, leading to significant object missing, attribute mismatching, and quality reduction. While recent investigations (Feng et al., 2022; Huang et al., 2023a; Lu et al., 2023) have demonstrated that current T2I diffusion models tend to generate inaccurate results when facing
+
+# Rannnnnne
+
+![](images/09f16ada2a011f6d4004f9b31a0a89ae58c1adc0389c18d846ced4361e625edc.jpg)  
+"An astronaut."
+
+![](images/4f1b757508bab330a372156f8315a21e6e9429474d46fa6318acff2cb2389c61.jpg)
+
+"An astronaut holding a red rifle and riding a green origami motorcycle and wearing a cyan chef's hat."
+
+![](images/4e222e19e13fe1392145ac8a69a6faed23312e330cbe51298d2b8869edaf7f34.jpg)
+
+![](images/88c9cdad987601ecd9e017dd274bd4a24a1007597cd0587c4f41cfc0750ef81c.jpg)  
+"An astronaut holding a red rifle."
+
+![](images/c67c230d7e20c1b23f21a49b60c5d82b1af2ccc387a6cc76c3f5995f2a512440.jpg)  
+"An astronaut holding a red rifle and riding a green origami motorcycle . "
+
+![](images/b50a6f70add75fd74a8af56ed829883193bef5091f17bf534c65bd997db13634.jpg)  
+"An astronaut holding a red rifle and riding a green origami motorcycle and wearing a cyan chef's hat."
+
+![](images/0e2ed838d498122a5aaa319982ee709cb978114b1ac280a364f8b734e44f9fc7.jpg)
+
+ing a red green le and f's hat."
+
+# Texteaaee
+
+![](images/893fd0446e379497f6593c085cddf0094ba575c063df650080725058e67d3615.jpg)
+
+![](images/0e4488597d2b1fb12bbe565cd10235253e245c29401995d371982bd4d745bb34.jpg)  
+"A golden vase."
+
+"A yellow tulip and a blue peony and a red rose in a golden vase."
+
+![](images/c6627393f92885514e0d64ff88e0cc7f08555dab646ae1d6a04ed9d16e3d88ed.jpg)
+
+![](images/b4f7544197f8691c4eccaa90331c579c36f80efc3258439f85b52ebf7bd3e6ec.jpg)  
+"A red rose in a golden vase."
+
+![](images/32c7cfefd4f203edee4a661737f6979c5c3331b26de445022d737d91403a7e63.jpg)  
+"A blue peony and a red rose in a golden vase."
+
+![](images/cf8975472272c1dcec5968c1645e21eae0b247fa206f0e7cb3cd9af676e3dc55.jpg)  
+"A yellow tulip and a blue peony and a red rose in a golden vase."
+
+![](images/3f2192f74554f64111be18745f8be1ff302e8cd449ef9ac72542b6aa5d4ddf7f.jpg)
+
+# Fantasia3D
+
+![](images/5ae2659484c41ddde4975e78caffe481605a8e31f2e2bef09073a331dbf91d16.jpg)
+
+![](images/ecd56d812c4cd0daba53ef2e683718d94cfb5d00ecfe3b4c408d817d5d0aeeb7.jpg)  
+"The Ironman."
+
+"The Ironman with Gundam's arm and Hulk's arm and Spiderman's body."
+
+![](images/8ec6db19eab76097c0edf64ba831fa4a74eef63bca0d0729d0da588339fd04e5.jpg)  
+"The Ironman with Gundam's arm."
+
+![](images/c69b4fc50e0252bd6e293d145ecd3efd6dbc292359b0313e30b4870a17f82dbd.jpg)  
+"The Ironman with Gundam's arm and Hulk's arm."
+
+![](images/22ae55785c3bfea53bb0f46f8b1cb3391be699146ef1c3ae3d9ca2f3467728c2.jpg)
+
+![](images/33776d4072c5910340d67505a66d368f8e0c24eaf967866357c456e466ace82d.jpg)  
+"The Ironman with Gundan's arm and Hulk's arm and Spiderman's body."
+
+![](images/603546481c1ec21fd5d41eca12825acaf06360fb2f4c0f7d542353f5a5a476b5.jpg)
+
+# mnnnne
+
+![](images/747965cd8c83f30f5c403f3efa0edf32522df3f7f9f13fc8aaff5852e29351f3.jpg)  
+(a) Generating Directly
+
+![](images/0579a48006653c54979bfb346d226d54f7dfe4ecf070ca8af3743910ccb8b4b3.jpg)  
+"A medieval soldier with metal armor."
+
+"A medieval soldier with metal armor wearing an white astronaut helmet and holding a golden axe and riding a terracotta wolf."
+
+![](images/64a053214970fda210ac39b35eed90f764f1d241a796c125e116968805fe33bb.jpg)
+
+![](images/a7da7a96967e7379b9bb9784ce10563bfa041c9dc009cebf96c79e3a6aeb14a0.jpg)  
+"A medieval soldier with metal armor holding a golden axe."  
+"A medieval soldier with metal armor holding a golden axe and riding a terracotta wolf."  
+(b) Generating with Progressive3D
+
+![](images/52708bd9ee761b5aa30597383a0c166c8bee00c437e616ec5e16c140d68fa478.jpg)  
+Figure 1: Conception. Current text-to-3D methods suffer from challenges when given prompts describing multiple objects binding with different attributes. Compared to (a) generating with existing methods, (b) generating with Progressive3D produces 3D content consistent with given prompts.
+
+![](images/c39251a3df7fa4d0a4d275b1919dc237617207597ee2a7352a3401c61819e430.jpg)  
+"A medieval soldier with metal armor wearing an white astronaut helmet and holding a golden axe and riding a terracotta wolf."
+
+prompts with complex semantics and existing text-to-3D methods inherit the same issues from T2I diffusion models, works on evaluating or improving the performance of text-to-3D methods in complex semantic scenarios are still limited. Therefore, how to generate correct 3D content consistent with complex prompts is critical for many real applications of text-to-3D methods.
+
+To address the challenges of generation precise 3D content from complex prompts, we propose a general framework named Progressive3D, which decomposes the difficult creation of complex prompts into a series of local editing steps, and progressively generates the 3D content as is shown in Fig. 1(b). For a specific editing step, our framework edits the pre-trained source representation in the 3D space determined by the user-defined region prompt according to the semantic difference between the source prompt and the target prompt. Concretely, we propose two content-related constraints, including a consistency constraint and an initialized constraint for keeping content beyond selected regions unchanged and promoting the separate target geometry generated from empty space. Furthermore, a technique dubbed Overlapped Semantic Component Suppression (OSCS) is carefully designed to automatically explore the semantic difference between the source prompt and the target one for guiding the optimization process of the target representations.
+
+To evaluate Progressive3D, we construct a complex semantic prompt set dubbed CSP-100 consisting of 100 various prompts. Prompts in CSP-100 are divided into four categories including color, shape,
+
+material and composition according to appeared attributes. Experiments conducted on existing text-to-3D methods driven by different 3D representations including NeRF-based DreamTime (Huang et al., 2023b) and MVDream(Shi et al., 2023), SDF-based TextMesh (Tsalicoglou et al., 2023), and DMTet-based Fantasia3D (Chen et al., 2023) demonstrate that our framework produces precise 3D models through multi-step local editing achieve better alignment with text prompts both in metrics and user studies than current text-to-3D creation methods when prompts are complex in semantics.
+
+Our contribution can be summarized as follows: (1) We propose a framework named Progressive3D for creating precise 3D content prompted with complex semantics by decomposing a difficult generation process into a series of local editing steps. (2) We propose the Overlapped Semantic Component Suppression to sufficiently explore the semantic difference between source and target prompts for overcoming the issues caused by complex prompts. (3) Experiments demonstrate that Progressive3D is effective in local editing and is able to generate precise 3D content consistent with complex prompts with various text-to-3D methods driven by different 3D neural representations.
+
+# 2 RELATED WORKS
+
+Text-to-3D Content Creation. Creating high-fidelity 3D content from only text prompts has attracted broad interest in recent years and there are many earlier attempts (Jain et al., 2022; Michel et al., 2022; Mohammad Khalid et al., 2022). Driven by the emerging text-to-image diffusion models, Dreamfusion (Poole et al., 2022) firstly introduces the large-scale prior from diffusion models for 3D content creation by proposing the score distillation sampling and achieves impressive results. The following works can be roughly classified into two categories, many attempts such as SJC (Wang et al., 2023a), Latent-NeRF (Metzer et al., 2022), Score Debiasing (Hong et al., 2023) DreamTime (Huang et al., 2023b), ProlificDreamer (Wang et al., 2023b) and MVDream(Shi et al., 2023) modify optimizing strategies to create higher quality content, and other methods including Magic3D (Lin et al., 2023), Fantasy3D (Chen et al., 2023), and TextMesh (Tsalicoglou et al., 2023) employ different 3D representations for better content rendering and mesh extraction. However, most existing text-to-3D methods focus on promoting the quality of generated 3D content, thus their methods struggle to generate correct content for complex prompts since no specific techniques are designed for complex semantics. Therefore, we propose a general framework named Progressive3D for various neural 3D representations to tackle prompts with complex semantics by decomposing the difficult generation into a series of local editing processes, and our framework successfully produces precise 3D content consistent with the complex descriptions.
+
+Text-Guided Editing on 3D Content. Compared to the rapid development of text-to-3D creation methods, the explorations of editing the generated 3D content by text prompts are still limited. Although Dreamfusion (Poole et al., 2022) and Magic3D (Lin et al., 2023) demonstrate that content editing can be achieved by fine-tuning existing 3D content with new prompts, such editing is unable to maintain 3D content beyond editable regions untouched since the fine-tuning is global to the entire space. Similar global editing methods also include Instruct NeRF2NeRF (Haque et al., 2023) and Instruct 3D-to-3D (Kamata et al., 2023), which extend a powerful 2D editing diffusion model named Instruct Pix2Pix (Brooks et al., 2023) into 3D content. Furthermore, several local editing methods including Vox-E (Sella et al., 2023) and DreamEditor (Zhuang et al., 2023) are proposed to edit the content in regions specified by the attention mechanism, and FocalDreamer (Li et al., 2023) only generates the incremental content in editable regions with new prompts to make sure the input content is unchanged. However, their works seldom consider the significant issues in 3D creations including object missing, attribute mismatching, and quality reduction caused by the prompts with complex semantics. Differing from their attempts, our Progressive3D emphasizes the semantic difference between source and target prompts, leading to more precise 3D content.
+
+# 3 METHOD
+
+Our Progressive3D framework is proposed for current text-to-3D methods to tackle prompts with complex semantics. Concretely, Progressive3D decomposes the 3D content creation process into a series of progressively local editing steps. For each local editing step, assuming we already have a source 3D representation  $\phi_s$  supervised by the source prompt  $y_s$ , we aim to obtain a target 3D representation  $\phi_t$  which is initialized by  $\phi_s$  to satisfy the description of the target prompt  $y_t$  and
+
+the 3D region constraint of user-defined region prompts  $\pmb{y}_b$ . We first convert user-defined region prompts to 2D masks for each view separately to constrain the undesired contents in  $\phi_t$  untouched (Sec. 3.1), which is critical for local editing. Furthermore, we propose the Overlapped Semantic Component Suppression (OSCS) technique to optimize the target 3D representation  $\phi_t$  with the guidance of the semantic difference between the source prompt  $\pmb{y}_s$  and the target prompt  $\pmb{y}_t$  (Sec. 3.2) for emphasizing the editing object and corresponding attributes. The overview illustration of our framework is shown in Fig .2.
+
+# 3.1 EDITABLE REGION DEFINITION AND RELATED CONSTRAINTS
+
+In this section, we give the details of the editable region definition with a region prompt  $\mathbf{y}_b$  and designed region-related constraints. Instead of directly imposing constraints on neural 3D representations to maintain 3D content beyond selected regions unchanged, we adopt 2D masks rendered from 3D definitions as the bridge to connect various neural 3D representations (e.g., NeRF, SDF, and DMTet) and region definition forms (e.g., 3D bounding boxes, custom meshes, and 2D/3D segmentation results (Liu et al., 2023; Cheng et al., 2023; Cen et al., 2023)), which enhances the generalization of our Progressive3D. We here adopt NeRF as the neural 3D representation and define the editable region with 3D bounding box prompts for brevity.
+
+Given a 3D bounding box prompt  $\pmb{y}_b = [c_x, c_y, c_z; s_x, s_y, s_z]$  which is user-defined for specifying the editable region in 3D space, where  $[c_x, c_y, c_z]$  is the coordinate position of the box center, and  $[s_x, s_y, s_z]$  is the box size on the  $\{x, y, z\}$ -axis respectively. We aim to obtain the corresponding 2D mask  $M_t$  converted from the prompt  $\pmb{y}_b$  and pre-trained source representation  $\phi_s$  that describes the editable region for a specific view  $\pmb{v}$ . Concretely, we first calculate the projected opacity map  $\hat{O}$  and the projected depth map  $\hat{D}$  of  $\phi_s$  similar to the Eq. 10. Then we render the given bounding box to obtain its depth  $D_b = \text{render}(\pmb{y}_b, \pmb{v}, \pmb{R})$ , where  $\pmb{v}$  is the current view and  $\pmb{R}$  is the rotate matrix of the bounding box. Before calculating the 2D editable mask  $M_t$  at a specific  $\pmb{v}$ , we modify the projected depth map  $\hat{D}$  according to  $\hat{O}$  to ignore the floating artifacts mistakenly generated in  $\phi_s$ :
+
+$$
+\tilde {\boldsymbol {D}} (\boldsymbol {r}) = \left\{ \begin{array}{c} \infty , \text {i f} \hat {\boldsymbol {O}} (\boldsymbol {r}) <   \tau_ {o}; \\ \hat {\boldsymbol {D}} (\boldsymbol {r}), \text {o t h e r w i s e}; \end{array} \right. \tag {1}
+$$
+
+where  $\pmb{r} \in \mathcal{R}$  is the ray set of sampled pixels in the image rendered at view  $\pmb{v}$ , and  $\tau_{o}$  is the filter threshold. Therefore, the 2D mask  $M_{t}$  of the editable region, as well as the 2D opacity mask  $M_{o}$ , can be calculated for the following region-related constraints:
+
+$$
+M _ {t} (\boldsymbol {r}) = \left\{ \begin{array}{l l} 1, & \text {i f} D _ {b} (\boldsymbol {r}) <   \tilde {\boldsymbol {D}} (\boldsymbol {r}); \\ 0, & \text {o t h e r w i s e .} \end{array} \quad M _ {o} (\boldsymbol {r}) = \left\{ \begin{array}{l l} 1, & \text {i f} \hat {O} (\boldsymbol {r}) > \tau_ {o}; \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \right. \tag {2}
+$$
+
+Content Consistency Constraint. We emphasize that maintaining 3D content beyond user-defined editable regions unchanged during the training of the target representation  $\phi_t$  is critical for 3D editing. We thus propose a content consistency constraint to impose the content between the target representation  $\phi_t$  and the source representation  $\phi_s$  to be consistent in undesired regions, which conditioned by our obtained 2D mask  $M_t$  which represents the editable regions:
+
+$$
+\mathcal {L} _ {\text {c o n s i s t}} = \sum_ {\boldsymbol {r} \in \mathcal {R}} \left(\bar {M} _ {t} (\boldsymbol {r}) M _ {o} (\boldsymbol {r}) \left| \left| \hat {C} _ {t} (\boldsymbol {r}) - \hat {C} _ {s} (\boldsymbol {r}) \right| \right| _ {2} ^ {2} + \bar {M} _ {t} (\boldsymbol {r}) \bar {M} _ {o} (\boldsymbol {r}) \left| \left| \hat {O} _ {t} (\boldsymbol {r}) \right| \right| _ {2} ^ {2}\right), \tag {3}
+$$
+
+where  $\bar{M}_t = 1 - M_t$  is the inverse editable mask,  $\bar{M}_o = 1 - M_o$  is the inverse opacity mask, and  $\hat{C}_s, \hat{C}_t$  are projected colors of  $\phi_s, \phi_t$  respectively.
+
+Instead of constraining the entire unchanged regions by color similarity, we divide such regions into a content region and an empty region according to the modified opacity mask  $M_{o}$ , and an additional term is proposed to impose the empty region remains blank during training. We separately constrain content and empty regions to avoid locking the backgrounds during the training, since trainable backgrounds are proved (Guo et al., 2023) beneficial for the quality of foreground generation.
+
+Content Initialization Constraint. In our progressive editing steps, a usual situation is the corresponding 3D space defined by region prompts is empty. However, creating the target object from
+
+![](images/f57c69493a73f6cff9b266e0d7e26d0ed69fe7cc78474bf8e5fcd04085e3d727.jpg)  
+Figure 2: Overview of a local editing step of our proposed Progressive3D. Given a source representation  $\phi_s$  supervised by source prompt  $y_s$ , our framework aims to generate a target representation  $\phi_t$  conforming to the input target prompt  $y_t$  in 3d space defined by the region prompt  $y_b$ . Conditioned on the 2D mask  $M_t(r)$ , we constrain the 3D content with  $\mathcal{L}_{\text{consist}}$  and  $\mathcal{L}_{\text{initial}}$ . We further propose an Overlapped Semantic Component Suppression technique to impose the optimization focusing more on the semantic difference for precise progressive creation.
+
+![](images/875d96a8e53b110b86024b1fbd7506b73a52e552bf40e8f44248bb46e8957c07.jpg)  
+(a) Source Content and Region Prompt
+
+![](images/d2df6d816adbd297db4c27ddaabeefaf92f85017bdaa35c756c37c0e7afb2dbf.jpg)  
+(b) 2D Mask  $M_{t}$
+
+![](images/920b88f85db816ab1d9d2c3be24754c7dac8ba7afa0e398c1c45e44a302b8618.jpg)  
+(c)  $w / o\mathcal{L}_{\text{consist}}$  
+Figure 3: Qualitative ablations. The source prompt  $\mathbf{y}_s =$  "A medieval soldier with metal armor holding a golden axe." and the target prompt  $\mathbf{y}_t =$  "A medieval soldier with metal armor holding a golden axe and riding a terracotta wolf.", where green denotes the overlapped prompt and red denotes the different prompt.
+
+![](images/37ad452ef89b894f21141dca53395f7d31ebdac3893aa805064f478ddec29a72.jpg)  
+(d) w/o  $\mathcal{L}_{initial}$  
+(e) w/o Overlapped Semantic Component Suppression
+
+![](images/bfa88fcdf54ef73c5adb9196a411ab548d66b2686922d4d907838382ff2e9a7d.jpg)
+
+![](images/86f88a1e575975aa948704f5a8df4404333a78aa2e1d7ac5fe53dc60b223032c.jpg)  
+(f) Ours
+
+scratch often leads to rapid geometry variation and causes difficulty in generation. We thus provide a content initialization constraint to encourage the user-defined 3D space filled with content, which is implemented by promoting  $\hat{O}_t$  increase in editable regions during the early training phase:
+
+$$
+\mathcal {L} _ {\text {i n i t a l}} = \kappa (k) \sum_ {\boldsymbol {r} \in \mathcal {R}} M _ {t} (\boldsymbol {r}) \left| \left| \hat {\boldsymbol {O}} _ {t} (\boldsymbol {r}) - \mathbf {1} \right| \right| _ {2} ^ {2}; \quad \kappa (k) = \left\{ \begin{array}{c} \lambda \left(1 - \frac {k}{K}\right), \text {i f} 0 \leq k <   K; \\ 0, \text {o t h e r w i s e}, \end{array} \right. \tag {4}
+$$
+
+where  $\kappa(k)$  is a weighting function of the current training iteration  $k$ ,  $\lambda$  is the scale factor of the maximum strength, and  $K$  is the maximum iterations that apply this constraint to avoid impacting the detail generation in the later phase.
+
+# 3.2 OVERLAPPED SEMANTIC COMPONENT SUPPRESSION
+
+Although we ensure the content edits only occur in user-defined regions through region-related constraints, obtaining desired representation  $\phi_t$  which matches the description in the target prompt  $y_t$  is still challenging. An intuitive approach to create  $\phi_t$  is fine-tuning the source representation  $\phi_s$  with the target prompt  $y_t$  directly (Poole et al., 2022; Lin et al., 2023). However, we point out that merely leveraging the target prompt  $y_t$  for fine-grained editing will cause attribute mismatching issues, especially when  $y_t$  describes multiple objects binding with different attributes.
+
+For instance in Fig. 3, we have obtained a source representation  $\phi_s$  matching the source prompt  $y_s$  and a target prompt  $y_t$  for the following local editing step. If we adjust  $\phi_s$  guided by  $y_t$  directly, as shown in Fig. 3(e), the additional content "wolf" could be both impacted by additional attribute "terracotta" and overlapped attribute "metal, golden" during the generation even if the overlapped attribute has been considered in  $\phi_s$ , which leads to an undesired result with attribute confusing.
+
+![](images/f0b85c0667f021cedda734fd7ff17e87c167f7d6bff9abbbee5d40be2f6dcbc3.jpg)  
+Figure 4: Current text-to-3D methods often fail to produce precise results when the given prompt describes multiple interacted objects binding with different attributes, leading to significant issues including object missing, attribute mismatching, and quality reduction.
+
+Furthermore, the repeated attention to overlapped prompts causes the editing process less consider the objects described in additional prompts, leading to entire or partial object ignoring (e.g., "wolf" is mistakenly created without its head and integrated with the soldier). Hence, guiding the optimization in local editing steps to focus more on the semantic difference between  $\mathbf{y}_s$  and  $\mathbf{y}_t$  instead of  $\mathbf{y}_t$  itself is critical for alleviating attribute mismatching and obtaining desired 3D content.
+
+Therefore, we proposed a technique named Overlapped Semantic Component Suppression (OSCS) inspired by (Armandpour et al., 2023) to automatically discover the overlapped semantic component between  $\mathbf{y}_s$  and  $\mathbf{y}_t$  with vector projection, and OSCS then suppresses the overlapped component to enhance the influence of the different semantic during the training of  $\phi_t$  for precise content creation. Concretely, both prompts  $\mathbf{y}_s$  and  $\mathbf{y}_t$  firstly produce separate denoising components with the unconditional prediction  $\epsilon_{\theta}(\mathbf{x}_t,t)$ :
+
+$$
+\Delta \epsilon_ {\boldsymbol {\theta}} ^ {s} = \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y} _ {s}, t\right) - \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, t\right); \quad \Delta \epsilon_ {\boldsymbol {\theta}} ^ {t} = \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y} _ {t}, t\right) - \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, t\right). \tag {5}
+$$
+
+As shown in Fig. 2, we then decompose  $\Delta \epsilon_{\theta}^{t}$  into the projection component  $\Delta \epsilon_{\theta}^{proj}$  and the perpendicular component  $\Delta \epsilon_{\theta}^{prep}$  by projecting  $\Delta \epsilon_{\theta}^{t}$  on  $\Delta \epsilon_{\theta}^{s}$ :
+
+$$
+\Delta \epsilon_ {\theta} ^ {t} = \underbrace {\frac {\left\langle \Delta \epsilon_ {\theta} ^ {s} , \Delta \epsilon_ {\theta} ^ {t} \right\rangle}{\left\| \Delta \epsilon_ {\theta} ^ {s} \right\| ^ {2}} \Delta \epsilon_ {\theta} ^ {s}} _ {\text {P r o j e c t i o n C o m p o n e n t}} + \underbrace {\left(\Delta \epsilon_ {\theta} ^ {t} - \frac {\left\langle \Delta \epsilon_ {\theta} ^ {s} , \Delta \epsilon_ {\theta} ^ {t} \right\rangle}{\left\| \Delta \epsilon_ {\theta} ^ {s} \right\| ^ {2}} \Delta \epsilon_ {\theta} ^ {s}\right)} _ {\text {P e r p e n d i c u l a r C o m p o n e n t}} = \Delta \epsilon_ {\theta} ^ {p r o j} + \Delta \epsilon_ {\theta} ^ {p r e p}, \tag {6}
+$$
+
+where  $\langle \cdot, \cdot \rangle$  denotes the inner product. We define  $\Delta \epsilon_{\theta}^{proj}$  as the overlapped semantic component since it is the most correlated component from  $\Delta \epsilon_{\theta}^{t}$  to  $\Delta \epsilon_{\theta}^{s}$ , and regard  $\Delta \epsilon_{\theta}^{prep}$  as the different semantic component which represents the most significant difference in semantic direction. Furthermore, we suppress the overlapped semantic component  $\Delta \epsilon_{\theta}^{proj}$  during training for reducing the influence of appeared attributes, and the noise sampler with OSCS is formulated as:
+
+$$
+\hat {\epsilon} _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y} _ {s}, \boldsymbol {y} _ {t}, t\right) = \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, t\right) + \frac {\omega}{W} \Delta \epsilon_ {\boldsymbol {\theta}} ^ {p r o j} + \omega \Delta \epsilon_ {\boldsymbol {\theta}} ^ {p r e p}; W > 1, \tag {7}
+$$
+
+where  $\omega$  is the original guidance scale in CFG described in Eq. 14, and  $W$  is the weight to control the suppression strength for the overlapped semantics. We highlight that  $W > 1$  is important for the suppression, since  $\hat{\epsilon}_{\theta}(\boldsymbol{x}_t,\boldsymbol{y}_s,\boldsymbol{y}_t,t)$  is degenerated to  $\hat{\epsilon}_{\theta}(\boldsymbol{x}_t,\boldsymbol{y}_t,t)$  when  $W = 1$ . Therefore, the modified Score Distillation Sampling (SDS) with OSCS is formulated as follows:
+
+$$
+\nabla_ {\phi} \tilde {\mathcal {L}} _ {\mathrm {S D S}} (\boldsymbol {\theta}, \boldsymbol {x}) = \mathbb {E} _ {t, \epsilon} \left[ w (t) \left(\hat {\epsilon} _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y} _ {s}, \boldsymbol {y} _ {t}, t\right) - \epsilon\right) \frac {\partial \boldsymbol {x}}{\partial \phi} \right]. \tag {8}
+$$
+
+Compared to Fig. 3(e), leveraging OSCS effectively reduces the distraction of appeared attributes and assists Progressive3D in producing desired 3D content, as is shown in Fig. 3(f).
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETTINGS
+
+We only provide important experimental settings including dataset, metrics, and baselines here due to the page limitation, more detailed experimental settings can be found at Appendix B.
+
+![](images/23b6b9b8125a3f7248b42fe61fd63b9bddff7dc19990a8aabc8dedeee5c38824.jpg)  
+Figure 5: Progressive editing processes driven by various text-to-3D methods equipped with our Progressive3D. Compared to original methods, Progressive3D assists current methods in tackling prompts with complex semantics well. 3D Cyan boxes denote the user-defined region prompts.
+
+Dataset Construction. We construct a Complex Semantic Prompt set named CSP-100 which involves 100 complex prompts to verify that current text-to-3D methods suffer issues when prompts are complex in semantics and proposed Progressive3D efficiently alleviate these issues. CSP-100 introduces four sub-categories of prompts including color, shape, material, and composition according to the appeared attribute and more details are in Appendix B.
+
+Evaluation Metrics. Existing text-to-3D methods (Poole et al., 2022; Tsalicoglou et al., 2023; Li et al., 2023) leverage CLIP-based metrics to evaluate the semantic consistency between generated 3D creations and corresponding text prompts. However, CLIP-based metrics are verified (Huang et al., 2023a; Lu et al., 2023) that fail to measure the fine-grained correspondences between described objects and binding attributes. We thus adopt two recently proposed metrics fine-grained
+
+![](images/809cf374766773d3120333007e731f17ef1355bc91eede279adf119781831534.jpg)  
+Dream Time
+
+![](images/b52d8579968db352b6541bde0cfbdbb5e9d08e6fd532580d04737211d222e0f2.jpg)  
+Dream Time +CEBM
+
+![](images/75bac555b082739a8174171fd77f5823edf26127a4e0d9c56e4b0bb1ca3ac89c.jpg)  
+DreamTime +A&E
+
+![](images/e3e73ceca8a03051c911b689dd0fc2aba2659ffb7f0c2f8f8999d7848e374337.jpg)  
+DreamTime +Progressive3D  
+"A ceramic tea pot and a lego car on a golden table."
+
+![](images/5444649019fa3ade09602de6c9e76134ea0b41104d4234d43be2952b5a98d3b7.jpg)
+
+![](images/1bc77d4393742deadc929ecc0ceea6a3df0cb88685951eb851b39f0a131f6e7d.jpg)  
+"A triangle sandwich on a round cabinet."
+
+![](images/edcb14db11ee14ab681335de17102d2f12e7570c633f8d7cdaaf35fdaa13114c.jpg)
+
+![](images/fc1355d930c55ed0ae2b0cfc3696825b840af90504c3a9a6f5331c7317f03a5c.jpg)
+
+![](images/1e75e0a835d8a43c4548fe84cfdfe482c54382af94c0cb768c34268ed8d93033.jpg)
+
+![](images/46dbe8ec6afb82d22b13e565c658b99a5e743196aafa4c7f1f210370264a48bc.jpg)  
+"A yellow tulip and a white tulip in a pink vase."
+
+![](images/433bed30137689bf2eedc8d03c9d7186af606e71607b17e73482a4f317f7f950.jpg)
+
+![](images/bd02b8fcbfd9bdefc0096c79e0daf9e7a658d8518dee616fb641fa3d83d1625d.jpg)
+
+Table 1: Quantitative comparison on metrics and user studies over CSP-100.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Metrics</td><td>Human</td></tr><tr><td>B-VQA ↑</td><td>mGPT-CoT ↑</td><td>Preference ↑</td></tr><tr><td>DreamTime</td><td>0.227</td><td>0.522</td><td>16.8%</td></tr><tr><td>+ CEBM</td><td>0.186</td><td>0.491</td><td>-</td></tr><tr><td>+ A&amp;E</td><td>0.243</td><td>0.528</td><td>-</td></tr><tr><td>+ Progressive3D</td><td>0.474</td><td>0.609</td><td>83.2%</td></tr></table>
+
+![](images/ad44c85d12edc181d19643238351f8fa2bcec3f2f38f98f125cf66b0b729d501.jpg)  
+Figure 6: Visual comparison with DreamTime-based compositional generation baselines.  
+Figure 7: Qualitative ablations between fine-tuning with target prompts and editing with Progressive3D on MVDream.
+
+including BLIP-VQA and mGPT-CoT (Huang et al., 2023a), evaluate the generation capacity of current methods and our Progressive3D when handling prompts with complex semantics.
+
+Baselines. We incorporate our Progressive3D with 4 text-to-3D methods driven by different 3D representations: (1) DreamTime (Huang et al., 2023b) is a NeRF-based method which enhances DreamFusion (Poole et al., 2022) in time sampling strategy and produce better results. We adopt DreamTime as the main baseline for quantitative comparisons and ablations due to its stability and training efficiency. (2) TextMesh (Tsalicoglou et al., 2023) leverages SDF as the 3D representation to improve the 3D mesh extraction capacity. (3) Fantasia3D (Tsalicoglou et al., 2023) is driven by DMTet which produces impressive 3D content with a disentangled modeling process. (4) MVDream (Shi et al., 2023) is a NeRF-based method which leverages a pre-trained multi-view consistent text-to-image model for text-to-3D generation and achieves high-quality 3D content generation performance. To further demonstrate the effectiveness of Prgressive3D, we re-implement two composing text-to-image methods including Composing Energy-Based Model (CEBM) (Liu et al., 2022) and Attend-and-Excite (A&E) (Chefer et al., 2023) on DreamTime for quantitative comparison.
+
+# 4.2 PROGRESSIVE3D FOR TEXT-TO-3D CREATION AND EDITING
+
+Comparison with current methods. We demonstrate the superior performance of our Progressive3D compared to current text-to-3D methods in both qualitative and quantitative aspects in this section. We first present visualization results in Fig. 4 to verify that DreamTime faces significant challenges including (a) object missing, (b) attribute mismatching, and (c) quality reduction when given prompts describe multiple interacted objects binding with different attributes. Thanks to our careful designs, Progressive3D effectively promotes the creation performance of DreamTime when dealing with complex prompts. In addition, more progressive editing processes based on various text-to-3D methods driven by different neural 3D representations are shown in Fig. 5, which further demonstrate that Progressive3D stably increases the generation capacity of based methods when given prompts are complex, and our framework is general for various current text-to-3D methods.
+
+We also provide quantitative comparisons on fine-grained semantic consistency metrics including BLIP-VQA and mGPT-CoT, and the results are shown in Tab. 1, which verify that our Progressive3D achieves remarkable improvements for 3D content creation with complex semantics compared to DreamTime-based baselines. As shown in Fig. 6, baselines that combine 2D composing T2I methods including CEBM (Liu et al., 2022) and A&E (Chefer et al., 2023) with DreamTime
+
+Table 2: Quantitative ablation studies for proposed constraints and the OSCS technique based on DreamTime over CSP-100.  
+
+<table><tr><td rowspan="2">Index</td><td colspan="3">Components</td><td colspan="2">Metrics</td></tr><tr><td>\( \mathcal{L}_{\text{consist}} \)</td><td>\( \mathcal{L}_{\text{initial}} \)</td><td>OSCS</td><td>B-VQA ↑</td><td>mGPT-CoT ↑</td></tr><tr><td>1</td><td>✓</td><td></td><td></td><td>0.255</td><td>0.567</td></tr><tr><td>2</td><td>✓</td><td>✓</td><td></td><td>0.370</td><td>0.577</td></tr><tr><td>3</td><td>✓</td><td></td><td>✓</td><td>0.347</td><td>0.581</td></tr><tr><td>4</td><td>✓</td><td>✓</td><td>✓</td><td>0.474</td><td>0.609</td></tr></table>
+
+![](images/e65f412e85fb80da4a43e0899ca4b8a26fdfd06dad2fec4d473bf7945c3be4f2.jpg)  
+Figure 8: Qualitative ablations for suppression weight  $W$  in proposed OSCS.
+
+still achieve limited performance for complex 3D content generation, leading to significant issues including object missing, attribute mismatching, and quality reduction. Furthermore, we collected 20 feedbacks from humans to investigate the performance of our framework. The human preference shows that users prefer our Progressive3D in most scenarios (16.8% vs. 83.2%), demonstrating that our framework effectively promotes the precise creation capacity of DreamTime when facing complex prompts.
+
+# 4.3 ABLATION STUDIES
+
+In this section, we conduct ablation studies on DreamTime and TextMesh to demonstrate the effectiveness of proposed components including content consistency constraint  $\mathcal{L}_{\text{consist}}$ , content initialization constraint  $\mathcal{L}_{\text{initial}}$  and Overlapped Semantic Component Suppression (OSCS) technique, we highlight that a brief qualitative ablation is given in Fig. 3.
+
+We first present ablation results between fine-tuning directly and editing with Progressive3D based on TextMesh in Fig. 7 to demonstrate that fine-tuning with new prompts cannot maintain source objects prompted by overlapped semantics untouched and is unusable for progressive editing. Another visual result in Fig. 8 shows the parameter analysis of the suppression weight  $w$  in OSCS. With the increase of  $W$  (i.e.,  $\frac{\omega}{W}$  decreases), the different semantics between source and target prompts play more important roles in optimizations and result in more desirable 3D content. On the contrary, the progressive step edits failed results with object missing or attribute mismatching issues when we increase the influence of overlapped semantics by setting  $W = 0.5$ , which further proves that our explanation of perpendicular and projection components is reasonable.
+
+We then show the quantitative comparison in Tab. 2 to demonstrate the effectiveness of each proposed component, where content consistency constraint is not involved in quantitative ablations since consistency is the foundation of 3D content local editing which guarantees content beyond user-defined regions untouched. We underline that  $\mathcal{L}_{initial}$  is proposed to simplify the geometry generation from empty space and OSCS is designed to alleviate the distraction of overlapped attributes, thus both components can benefit the creation performance with no conflict theoretically. This has been proofed by the quantitative ablations in Tab. 2: index 2 and 3 show that applying  $\mathcal{L}_{initial}$  and OSCS alone both promote the metrics compared to the baseline in index 1, and index 4 shows that leveraging both  $\mathcal{L}_{initial}$  and OSCS together can further contribute to the creation performance over CSP-100.
+
+# 5 CONCLUSION
+
+In this work, we propose a general framework named Progressive3D for correctly generating 3D content when the given prompt is complex in semantics. Progressive3D decomposes the difficult creation process into a series of local editing steps and progressively generates the aiming object with binding attributes with the assistance of proposed region-related constraints and the overlapped semantic suppression technique in each step. Experiments conducted on complex prompts in CSP-100 demonstrate that current text-to-3D methods suffer issues including object missing, attribute mismatching, and quality reduction when given prompts are complex in semantics, and the proposed Progressive3D effectively creates precise 3D content consistent with complex prompts through multi-step local editing. More discussions on the limitations and potential directions for future works are provided in Appendix A.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the National Key R&D Program of China (2022ZD0118101), the Natural Science Foundation of China (No.62202014), Shenzhen Basic Research Program under Grant JCYJ20220813151736001, and also sponsored by CCF Tencent Open Research Fund.
+
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+
+# A DISCUSSIONS
+
+# A.1 REALISTIC USAGE OF PROGRESSIVE3D
+
+We note that Progressive3D contains multiple local editing steps for creating complex 3D content, which accords with user usage pipeline, i.e., creating a primary object first, then adjusting its attribute or adding more related objects. However, Progressive3D is flexible in realistic usage since the generation capacity of basic text-to-3D method and user goals are variant. For instance in Fig. 9, we desire to create the 3D content consistent with the prompt "An astronaut wearing a green top hat and riding a red horse". We find that MVDream fails to create the precise result while generating "An astronaut riding a red horse" correctly. Thus the desired content can be achieved by editing "An astronaut riding a red horse" within one-step editing, instead of starting from "an astronaut".
+
+![](images/e9810a2cc1902c15aa472cfe1eace4eaa11d422dd99aa7cb11bb5894310f782b.jpg)  
+Figure 9: MVDream successfully creates "An astronaut riding a red horse" while failing to create "An astronaut wearing a green top hat and riding a red horse" By leveraging one-step Progressive3D editing, correct 3D content is obtained.
+
+# A.2 OBJECT GENERATING ORDER
+
+Different object generating orders in Progressive3D typically result in correct 3D content consistent with the complex prompts. However, the content details of the final content are impacted by created objects since Progressive3D is a local editing chain started from the source content, and we give an instance in Fig. 10. With different generating orders, Progressive3D creates 3D content with different details while they are both consistent with the prompt "An astronaut sitting on a wooden chair". In our experiments, we first generate the primary object which is desired to occupy most of the space and interact with other additional objects.
+
+![](images/37b227cb21d0e2ac49b286b1b02add3c08afd3e802e4f260c648a37e19cf2f44.jpg)  
+Figure 10: Creating "An astronaut sitting on a wooden chair" from different generating orders.
+
+![](images/96771fdb9247d90b62f983feab6d8c0ee1ccf6808571dd525eb5561be0e33f35.jpg)
+
+# A.3 VARIOUS REGION DEFINITIONS
+
+We highlight that Progressive3D is a general framework for various region definition forms since the corresponding 2D mask of each view can be achieved. As shown in Fig. 11, Progressive3D performs correctly on various definition forms including multiple 3D bounding boxes, custom mesh, and the fine-grained 2D segmentation prompted by the keyword "helmet" through Grounded-SAM (Liu et al., 2023; Kirillov et al., 2023).
+
+![](images/678364cce13905334983b0c18dfcd503aeeb07faa662edd989d451a6f34dcdb3.jpg)  
+Figure 11: Progressive3D supports various definition forms of regions since the corresponding 2D masks of each view can be obtained.
+
+# A.4 ATTRIBUTEDITING
+
+We emphasize that Progressive3D supports both modifying attributes of existing objects and creating additional objects with attributes not mentioned in source prompts from scratch in user-selected regions, and we provide attribute editing results in Fig. 12. Noticing that creating additional objects with attributes not mentioned in source prompts from scratch is more difficult than editing the attributes of existing objects. Therefore, attribute editing costs significantly less time than additional object generation.
+
+# A.5 WHY ADOPTING 2D CONSTRAINTS
+
+Compared to directly maintaining 3D content beyond editable regions unchanged on 3D representations, we adopt 2D constraints for achieving such a goal from the perspective of generalization.
+
+We notice that current text-to-3D methods are developed based on various neural 3D representations, thus most 3D editing methods propose careful designs for a specific representation and are unusable on other representations. For instance, DreamEditor (Zhuang et al., 2023) distills the original NeRF into a mesh-based radiance field to localize the editable regions, and FocalDreamer (Li
+
+![](images/4e58da6f2fd4569e58dcf48c6bbff3a59247df0e62ec4aff9e17e6a4a347a5f0.jpg)  
+Figure 12: Prgressive3D supports attribute editing on existing generated objects.
+
+et al., 2023) proposes multiple regularized losses specifically designed for DMTet to maintain the undesired regions unchanged. In addition, their specific designs also limit the available definition forms of editable regions.
+
+However, we underline that the optimization core of most text-to-3D is the SDS loss which is supervised on rendered views in a 2D way, and different neural 3D representations can be rendered as 2D projected color, opacity, and depth maps through volume rendering or rasterization. Therefore, our proposed 2D region-related constraints can effectively bridge different representations and various user-provided definition forms, and the strength weights of our region-related constraints are easy to adjust since our constraints and SDS loss are all imposed on 2D views.
+
+# A.6 LIMITATIONS
+
+Our Progressive3D efficiently promotes the generation capacity for current text-to-3d methods when facing complex prompts. However, Progressive3D still faces several challenges.
+
+Firstly, Progressive3D decomposes a difficult generation into a series of editing processes, which leads to multiplying time costs and more human intervention. A potential future direction is further introducing layout generation into the text-to-3d area, e.g., creating 3D content with complex semantics in one generation by inputting a global prompt, a string of pre-defined 3D regions, and their corresponding local prompts. Whereas 3D layout generation intuitive suffers more difficulties and requires further exploration.
+
+Another problem is that the creation quality of Progressive3D is highly determined by the generative capacity of the base method. We believe our framework can achieve better results when stronger 2D text-to-image diffusion models and neural 3D representations are available, and we leave the adaptation between possible improved text-to-3D creation methods and Progressive3D in future works.
+
+# B EXPERIMENTS SETTINGS
+
+# B.1 CSP-100
+
+CSP-100 can be divided into four sub-categories of prompts including color (e.g. red, green, blue), shape (e.g. round, square, hexagonal), material (e.g. golden, wooden, origami) and composition according to the attribute types appeared in prompts, as shown in Fig. 13. Each prompt in the color/shape/material categories describes two objects binding with color/shape/material attributes, and each prompt in the composition category describes at least three interacted objects with corresponding different attributes. We provide the detailed prompt list in both Tab. 4 and Tab. 5.
+
+![](images/a199d0364934b782f052da8383a931b875b6086a772f93d9aca29c9c6a3b6f92.jpg)  
+Figure 13: Prompts in CSP-100 can be divided into four categories including Color, Shape, Material, and Composition according to appeared attributes.
+
+# B.2 METRICS
+
+The CLIP-based metrics utilized by current text-to-3D methods (Poole et al., 2022; Tsalicoglou et al., 2023; Li et al., 2023) calculates the cosine similarity between text and image features extracted by CLIP (Radford et al., 2021) However, recent works (Huang et al., 2023a; Lu et al., 2023) demonstrate that CLIP-based metrics can only measure the coarse text-image similarity but fail to measure the fine-grained correspondences among multiple objects and their binding attributes. Therefore, we adopt two fine-grained text-to-image evaluation metrics including BLIP-VQA and mGPT-CoT proposed by (Huang et al., 2023a) to show the effectiveness of Progressive3D. We provide comparisons in Fig. 14 to demonstrate that the CLIP metric fails to measure the fine-grained correspondences while BLIP-VQA performs well, and we report the quantitative comparison of DreamTime-based methods on CLIP metric over CSP-100 in Tab. 3.
+
+![](images/da191ee347a60cd9dcb89d86d364d3b490a28dcc4458306ee7f43a80f27b8e40.jpg)  
+Figure 14: Quantitative comparison for metrics including CLIP and BLIP-VQA on DreamTime and MVDream.
+
+Table 3: Quantitative comparison on CLIP over CSP-100.  
+
+<table><tr><td>Method</td><td>CLIP ↑</td></tr><tr><td>DreamTime</td><td>0.289</td></tr><tr><td>+ CEBM</td><td>0.275</td></tr><tr><td>+ A&amp;E</td><td>0.281</td></tr><tr><td>+ Progressive3D</td><td>0.292</td></tr></table>
+
+BLIP-VQA is proposed based on the visual question answering (VQA) ability of BLIP (Li et al., 2022). Concretely, BLIP-VQA decomposes a complex prompt into several separate questions and takes the probability of answering "yes" as the score for a question. The final score of a specific prompt is the product of the probability of answering "yes" for corresponding questions. For instance, the complex prompt is "An astronaut holding a red rifle," the final score of BLIP-VQA is the product of the probability for questions including "An astronaut?" and "A red rifle?"
+
+Since multimodal large language models such as MiniGPT-4 (Zhu et al., 2023) show impressive text-image understanding capacity, MiniGPT4 with Chain-of-Thought (mGPT-CoT) is proposed to leverage such cross-modal understanding performance to evaluate the fine-grained semantic similarity between query text and image. Specifically, we ask two questions in sequence including "Describe the image." and "Predict the image-text alignment score.", and the multimodal LLM is required to output the evaluation mGPT-CoT score with detailed Chain-of-Thought prompts. In practice, we adopt MiniGPT4 fine-tuned from Vicuna 7B (Chiang et al., 2023) as the LLM.
+
+What's more, we define the preference criterion of human feedback as follows: Users are requested to judge whether the 3D creations generated by DreamTime or Progressive3D are consistent with the given prompts. If one 3D content is acceptable in the semantic aspect while the other is not, the corresponding acceptable method is considered superior. On the contrary, if both 3D creations are
+
+considered to satisfy the description in the given prompt, users are asked to prefer the 3D content with higher quality.
+
+# B.3 IMPLEMENT DETAILS
+
+Our Progressive3D is implemented based on the Threestudio (Guo et al., 2023) project since DreamTime (Huang et al., 2023b) and TextMesh (Tsalicoglou et al., 2023) have not yet released their source code and the official Fantasia3D (Chen et al., 2023) code project, and all experiments are conducted on NVIDIA A100 GPUs. We underline that the implementation in ThreeStudio (Guo et al., 2023) and details could be different from their papers, especially ThreeStudio utilizes DeepFloyd IF (DeepFloyd-Team, 2023) as the text-to-image diffusion model for more stable and higher quality in the generation, while Fantasia3D adopts Stable Diffusion (Rombach et al., 2022) as the 2D prior. The number of iterations  $N$  and batch size for DreamTime and TextMesh are set to 10000 and 1 respectively. The training settings of Fantasia3D are consistent with officially provided training configurations, e.g.,  $N$  is 3000 and 2000 for geometry and appearance modeling stages, and batch size is set to 12. We leverage Adam optimizer for progressive optimization and the learning rate is consistent with base methods. Therefore, one local editing step costs a similar time to one generation of base methods from scratch. In most scenarios, the filter threshold  $\tau_{o}$  is set to 0.5, the strength factor  $\lambda$ , iteration threshold  $K$  in  $\mathcal{L}_{\text{consist}}$  are set to 0.5 and  $\frac{N}{4}$ , and the suppression weight  $W$  in overlapped semantic component suppression technique is set to 4.
+
+# C QUALITATIVE RESULTS
+
+# C.1 CONTENTCONSTRAINT WITHBACKGROUND
+
+We divide the  $\mathcal{L}_{\text {consist }}$  into a content term and an empty term to avoid mistakenly treating backgrounds as a part of foreground objects. We give a visual comparison of restricting the backgrounds and foregrounds as an entirety in Fig. 15, and the  $\hat{\mathcal{L}}_{\text {consist }}$  can be formulated as:
+
+$$
+\hat {\mathcal {L}} _ {\text {c o n s i s t}} = \sum_ {\boldsymbol {r} \in \mathcal {R}} \left(\bar {M} _ {t} (\boldsymbol {r}) \left| \left| \hat {C} _ {t} (\boldsymbol {r}) - \hat {C} _ {s} (\boldsymbol {r}) \right| \right| _ {2} ^ {2}\right). \tag {9}
+$$
+
+The visual results demonstrate that mistakenly treating backgrounds as a part of foreground objects leads to significant floating.
+
+![](images/26d181d8fbe7a44d97a9491ae21b27b7ec883aa36027e959e7abad8b705eadd3.jpg)  
+Figure 15: Mistakenly restricting background content as foreground leads to significant floating in editing result.
+
+# C.2 MORE PROGRESSIVE EDITING PROCESS
+
+We here provide more progressive editing results for correctly creating 3D content for prompts with complex semantics in Fig. 16. More qualitative results with various complex prompts further demonstrate the creation precision and diversity of our Progressive3D.
+
+# D PRELIMINARY
+
+Neural Radiance Field (NeRF) (Mildenhall et al., 2020) uses a multi-layer perception (MLP) to implicitly represent the 3D scene as a continuous volumetric radiance field. Specifically, MLP  $\theta$  maps a spatial coordinate and a view direction to a view-independent density  $\sigma$  and view-dependent color  $c$ . Given the camera ray  $r(k) = o + kd$  with camera position  $o$ , view direction  $d$  and depth  $k \in [k_n, k_f]$ , the projected color of  $r(k)$  is rendered by sampling  $N$  points along the ray:
+
+$$
+\hat {\boldsymbol {C}} (\boldsymbol {r}) = \sum_ {i = 1} ^ {N} \Omega_ {i} (1 - \exp (- \rho_ {i} \delta_ {i})) \boldsymbol {c} _ {i}, \tag {10}
+$$
+
+where  $\rho_{i}$  and  $c_{i}$  denote the density and color of  $i$ -th sampled point,  $\Omega_{i} = \exp \left(-\sum_{j=1}^{i-1} \rho_{j} \delta_{j}\right)$  indicates the accumulated transmittance along the ray, and  $\delta_{i}$  is the distance between adjacent points.
+
+Diffusion Model (Sohl-Dickstein et al., 2015; Ho et al., 2020) is a generative model which defines a forward process to slowly add random noises to clean data  $\pmb{x}_0 \sim p(\pmb{x})$  and a reverse process to generate desired results from random noises  $\epsilon \sim \mathcal{N}(\mathbf{0}, \pmb{I})$  within  $T$  time-steps:
+
+$$
+q \left(\boldsymbol {x} _ {t} \mid \boldsymbol {x} _ {t - 1}\right) = \mathcal {N} \left(\boldsymbol {x} _ {t}; \sqrt {\alpha_ {t}} \boldsymbol {x} _ {t - 1}, (1 - \alpha_ {t}) \boldsymbol {I}\right), \tag {11}
+$$
+
+$$
+p _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t - 1} \mid \boldsymbol {x} _ {t}\right) = \mathcal {N} \left(\boldsymbol {x} _ {t - 1}; \boldsymbol {\mu} _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, t\right), \sigma_ {t} ^ {2} \boldsymbol {I}\right), \tag {12}
+$$
+
+where  $\alpha_{t}$  and  $\sigma_{t}$  are calculated by a pre-defined scale factor  $\beta_{t}$ , and  $\pmb{\mu}_{\theta}(\pmb{x}_t,t)$  is calculated by  $\pmb{x}_t$  and the noise  $\epsilon_{\theta}(\pmb{x}_t,t)$  predicted by a neural network, which is optimized with prediction loss:
+
+$$
+\mathcal {L} = \mathbb {E} _ {\boldsymbol {x} _ {t}, \epsilon , t} [ w (t) | | \epsilon_ {\theta} (\boldsymbol {x} _ {t}, t) - \epsilon | | _ {2} ^ {2} ], \tag {13}
+$$
+
+where  $w(t)$  is a weighting function that depends on the time-step  $t$ . Recently, text-to-image diffusion models achieve impressive success in text-guided image generation by learning  $\epsilon_{\theta}(\boldsymbol{x}_t, \boldsymbol{y}, t)$  conditioned by the text prompt  $\boldsymbol{y}$ . Furthermore, classifier-free guidance (CFG) (Ho & Salimans, 2022) is widely leveraged to improve the quality of results via a guidance scale parameter  $\omega$ :
+
+$$
+\hat {\epsilon} _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y}, t\right) = (1 + \omega) \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y}, t\right) - \omega \epsilon_ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, t\right), \tag {14}
+$$
+
+Score Distillation Sampling (SDS) is proposed by (Poole et al., 2022) to create 3D contents from given text prompts by distilling 2D images prior from a pre-trained diffusion model to a differentiable 3D representation. Concretely, the image  $\pmb{x} = g(\phi)$  is rendered by a differentiable generator  $g$  and a representation parameterized by  $\phi$ , and the gradient is calculated as:
+
+$$
+\nabla_ {\phi} \mathcal {L} _ {\mathrm {S D S}} (\boldsymbol {\theta}, \boldsymbol {x}) = \mathbb {E} _ {t, \epsilon} \left[ w (t) \left(\hat {\epsilon} _ {\boldsymbol {\theta}} \left(\boldsymbol {x} _ {t}, \boldsymbol {y}, t\right) - \epsilon\right) \frac {\partial \boldsymbol {x}}{\partial \phi} \right]. \tag {15}
+$$
+
+![](images/76742b7604a7566360de1b8a99152fdeecbd712a64ab98e2fdff31861f0fad28.jpg)  
+Figure 16: More progressive editing results created with our Progressive3D based on DreamTime.
+
+Table 4: Detailed prompt list of CSP-100. (Part 1)  
+
+<table><tr><td>Index</td><td>Prompt</td></tr><tr><td colspan="2">Color (i.e., two interacted objects binding with different colors)</td></tr><tr><td>1</td><td>an orange cat wearing a yellow suit</td></tr><tr><td>2</td><td>an orange cat wearing a green shirt</td></tr><tr><td>3</td><td>an orange cat wearing a red fireman uniform</td></tr><tr><td>4</td><td>a red rose in a pink vase</td></tr><tr><td>5</td><td>a blue peony in a pink vase</td></tr><tr><td>6</td><td>a yellow tulip in a pink vase</td></tr><tr><td>7</td><td>a pair of red sneakers on a blue chair</td></tr><tr><td>8</td><td>a stack of green books on a blue chair</td></tr><tr><td>9</td><td>a purple gift box on a blue chair</td></tr><tr><td>10</td><td>a red cake in a yellow tray</td></tr><tr><td>11</td><td>a blue spoon in a yellow tray</td></tr><tr><td>12</td><td>a pair of green chopsticks in a yellow tray</td></tr><tr><td>13</td><td>a green vase on a red desk</td></tr><tr><td>14</td><td>a pair of blue sneakers on a red desk</td></tr><tr><td>15</td><td>a yellow tray on a red desk</td></tr><tr><td colspan="2">Shape (i.e., two interacted objects binding with different shapes)</td></tr><tr><td>16</td><td>a round gift box on a hexagonal table</td></tr><tr><td>17</td><td>a triangular cake on a hexagonal table</td></tr><tr><td>18</td><td>a square tray on a hexagonal table</td></tr><tr><td>19</td><td>a hexagonal cup on a round cabinet</td></tr><tr><td>20</td><td>a triangular sandwich on a round cabinet</td></tr><tr><td>21</td><td>a square bowl on a round cabinet</td></tr><tr><td>22</td><td>a hexagonal cup on a star-shaped tray</td></tr><tr><td>23</td><td>a triangular cake on a star-shaped tray</td></tr><tr><td>24</td><td>a square pepper on a star-shaped tray</td></tr><tr><td>25</td><td>a model of a round house with a hexagonal roof</td></tr><tr><td>26</td><td>a model of a round house with a square roof</td></tr><tr><td>27</td><td>a model of a round house with a spherical roof</td></tr><tr><td colspan="2">Material (i.e., two interacted objects binding with different materials)</td></tr><tr><td>28</td><td>a lego man riding a golden motorcycle</td></tr><tr><td>29</td><td>a silver bunny riding a golden motorcycle</td></tr><tr><td>30</td><td>a wooden dog riding a golden motorcycle</td></tr><tr><td>31</td><td>a lego man driving an origami sport car</td></tr><tr><td>32</td><td>a metal monkey driving an origami sport car</td></tr><tr><td>33</td><td>a wooden dog driving an origami sport car</td></tr><tr><td>34</td><td>a ceramic tea pot on a golden table</td></tr><tr><td>35</td><td>a silver vase on a golden table</td></tr><tr><td>36</td><td>an origami box on a golden table</td></tr><tr><td>37</td><td>a model of a silver house with a golden roof</td></tr><tr><td>38</td><td>a model of a silver house with a wooden roof</td></tr><tr><td>39</td><td>a model of a silver house with a bronze roof</td></tr><tr><td>40</td><td>a lego tank with a golden gun</td></tr><tr><td>41</td><td>a lego tank with a silver gun</td></tr><tr><td>42</td><td>a lego tank with a wooden gun</td></tr><tr><td colspan="2">Composition (i.e., more than two interacted objects binding with different attributes)</td></tr><tr><td>43</td><td>an orange cat wearing a yellow suit and green sneakers</td></tr><tr><td>44</td><td>an orange cat wearing a yellow suit and red pumps</td></tr><tr><td>45</td><td>an orange cat wearing a yellow suit and cyan boots</td></tr><tr><td>46</td><td>an orange cat wearing a yellow suit and green sneakers and cyan top hat</td></tr><tr><td>47</td><td>an orange cat wearing a yellow suit and green sneakers and pink cap</td></tr><tr><td>48</td><td>an orange cat wearing a yellow suit and green sneakers and red chef&#x27;s hat</td></tr><tr><td>49</td><td>a blue peony and a yellow tulip in a pink vase</td></tr><tr><td>50</td><td>a red rose and a yellow tulip in a pink vase</td></tr></table>
+
+Table 5: Detailed prompt list of CSP-100. (Part 2)  
+
+<table><tr><td>Index</td><td>Prompt</td></tr><tr><td>51</td><td>a white tulip and a yellow tulip in a pink vase</td></tr><tr><td>52</td><td>a purple rose and a red rose and a yellow tulip in a pink vase</td></tr><tr><td>53</td><td>a blue peony and a red rose and a yellow tulip in a pink vase</td></tr><tr><td>54</td><td>a white tulip and a red rose and a yellow tulip in a pink vase</td></tr><tr><td>55</td><td>a golden cat on a stack of green books on a blue chair</td></tr><tr><td>56</td><td>a wooden bird on a stack of green books on a blue chair</td></tr><tr><td>57</td><td>a lego car on a stack of green books on a blue chair</td></tr><tr><td>58</td><td>a blue candle on a red cake in a yellow tray</td></tr><tr><td>59</td><td>a green spoon on a red cake in a yellow tray</td></tr><tr><td>60</td><td>a pair of cyan chopsticks on a red cake in a yellow tray</td></tr><tr><td>61</td><td>a cyan cake on a yellow tray on a red desk</td></tr><tr><td>62</td><td>a ceramic tea pot on a yellow tray on a red desk</td></tr><tr><td>63</td><td>a green apple on a yellow tray on a red desk</td></tr><tr><td>64</td><td>a square cake on a square tray on a hexagonal table</td></tr><tr><td>65</td><td>a round cake on a square tray on a hexagonal table</td></tr><tr><td>66</td><td>a triangular sandwich on a square tray on a hexagonal table</td></tr><tr><td>67</td><td>a green apple in a hexagonal cup on a round cabinet</td></tr><tr><td>68</td><td>a pink peach in a hexagonal cup on a round cabinet</td></tr><tr><td>69</td><td>a yellow pineapple in a hexagonal cup on a round cabinet</td></tr><tr><td>70</td><td>a red rose in a hexagonal cup on a star-shaped tray</td></tr><tr><td>71</td><td>a white lily in a hexagonal cup on a star-shaped tray</td></tr><tr><td>72</td><td>a green cactus in a hexagonal cup on a star-shaped tray</td></tr><tr><td>73</td><td>a lego man wearing a chef&#x27;s hat and riding a golden motorcycle</td></tr><tr><td>74</td><td>a lego man wearing a wooden top hat and riding a golden motorcycle</td></tr><tr><td>75</td><td>a lego man wearing a silver crown and riding a golden motorcycle</td></tr><tr><td>76</td><td>a metal monkey wearing a golden crown and driving an origami sport car</td></tr><tr><td>77</td><td>a metal monkey wearing a chef&#x27;s hat and driving an origami sport car</td></tr><tr><td>78</td><td>a metal monkey wearing a wooden top hat and driving an origami sport car</td></tr><tr><td>79</td><td>a ceramic tea pot and a lego car on a golden table</td></tr><tr><td>80</td><td>a ceramic tea pot and an origami box on a golden table</td></tr><tr><td>81</td><td>a ceramic tea pot and a cardboard box on a golden table</td></tr><tr><td>82</td><td>a ceramic tea pot and a pair of wooden shoes on a golden table</td></tr><tr><td>83</td><td>a ceramic tea pot and an origami box and a green apple on a golden table</td></tr><tr><td>84</td><td>a ceramic tea pot and an origami box and a yellow tray on a golden table</td></tr><tr><td>85</td><td>a ceramic tea pot and an origami box and a stack of blue books on a golden table</td></tr><tr><td>86</td><td>a model of a silver house with a golden roof beside an origami coconut tree</td></tr><tr><td>87</td><td>a model of a silver house with a golden roof beside a wooden car</td></tr><tr><td>88</td><td>a model of a silver house with a golden roof beside a lego man</td></tr><tr><td>89</td><td>a lego tank with a golden gun and a red flying flag</td></tr><tr><td>90</td><td>a lego tank with a golden gun and a blue flying flag</td></tr><tr><td>91</td><td>a lego tank with a golden gun and a yellow flying flag</td></tr><tr><td>92</td><td>a model of a round house with a spherical roof on a hexagonal park</td></tr><tr><td>93</td><td>a model of a round house with a spherical roof on a square park</td></tr><tr><td>94</td><td>a model of a round house with a spherical roof on an oval park</td></tr><tr><td>95</td><td>an astronaut holding a red rifle and riding a green origami motorcycle</td></tr><tr><td>96</td><td>an astronaut holding a red rifle and riding a cyan scooter</td></tr><tr><td>97</td><td>an astronaut holding a red rifle and riding a golden motorcycle</td></tr><tr><td>98</td><td>an astronaut holding a red rifle and riding a green origami motorcycle and wearing a blue top hat</td></tr><tr><td>99</td><td>an astronaut holding a red rifle and riding a green origami motorcycle and wearing a cyan chef&#x27;s hat</td></tr><tr><td>100</td><td>an astronaut holding a red rifle and riding a green origami motorcycle and wearing a pink cowboy hat</td></tr></table>

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2024/Prompt Learning with Quaternion Networks/b7294ab8-295c-4a01-86ce-b796789b6470_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "PROMPT LEARNING WITH QUATERNION NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            99,
+            802,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Boya Shi $^{1,2}$ , Zhengqin Xu $^{1}$ , Shuai Jia $^{1}$ , Chao Ma $^{1*}$",
+        "bbox": [
+            181,
+            147,
+            535,
+            162
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "<sup>1</sup> MoE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University",
+            "$^{2}$  National Innovation Institute of Defense Technology"
+        ],
+        "bbox": [
+            183,
+            162,
+            741,
+            191
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "{boya.shi, fate311, jiashuai, chaoma}@sjtu.edu.cn",
+        "bbox": [
+            183,
+            193,
+            552,
+            205
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT",
+        "text_level": 1,
+        "bbox": [
+            450,
+            243,
+            545,
+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multimodal pre-trained models have shown impressive potential in enhancing performance on downstream tasks. However, existing multimodality fusion strategies primarily rely on explicit interaction structures that fail to capture the diverse aspects and patterns inherent in input data. This yields limited performance in zero-shot contexts, especially when fine-grained classifications and abstract interpretations are required. To address this issue, we propose an effective approach, namely Prompt Learning with Quaternion Networks (QNet), for semantic alignment across diverse modalities. QNet employs a quaternion hidden space where the mutually orthogonal imaginary axes capture rich intermodal semantic spatial correlations from various perspectives. Hierarchical features across multilayers are utilized to encode intricate interdependencies within various modalities with a reduced number of parameters. Our experiments on 11 datasets demonstrate that QNet outperforms state-of-the-art prompt learning techniques in base-to-novel generalization, cross-dataset transfer, and domain transfer scenarios with fewer learnable parameters. The source code is available at https://github.com/SHIBOYA/QNet.",
+        "bbox": [
+            228,
+            273,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            508,
+            336,
+            523
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Large-scale multimodal pre-trained models, such as ChatGPT (Radford et al., 2019) and its successor ChatGPT-4 (OpenAI, 2023), have recently shown great promise in the pursuit of general artificial intelligence. These models can facilitate various downstream tasks using prompts, which tap into the intrinsic knowledge embedded within them (Jin et al., 2021; Yao et al., 2021; Ding et al., 2022; Lüddecke & Ecker, 2022). However, it is still challenging for multimodal pre-trained models like CLIP (Radford et al., 2021) to handle more abstract tasks such as sketch recognition. This implies that existing pre-trained models rely heavily on homogeneous representation present in the training data for image classification, and are not effective in capturing the diverse and complementary features across different modalities. Therefore, it is of great importance to improve the modality fusion capabilities of pre-trained models within zero-shot settings, which is a significant and challenging task (Lu et al., 2022; Huang et al., 2022; Manli et al., 2022).",
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+    {
+        "type": "text",
+        "text": "CoOp (Zhou et al., 2022b) is a technique that incorporates continuous prompt variables to improve the performance of CLIP in observable categories. Co-CoOp (Zhou et al., 2022a) introduces conditional prompt learning to enhance the generalization capability of CLIP in non-observable categories, enabling it to better acquire image features. While MaPLe (Khattak et al., 2022) is another method that enhances the collaboration between images and text through mapping, its linear connections assume a linear relationship among different modalities, which may not always hold true, potentially leading to interference between varying semantic spaces. Cross-attention mechanisms, which use non-linear fusion layers to merge features from diverse semantic spaces, may not effectively integrate complementary knowledge across modalities. As a result, both linear connections and other auxiliary networks struggle with homogeneous representations, making it difficult to capture distinct aspects or patterns within the data. To overcome this challenge, a fusion strategy should be formulated that can encapsulate the inherent intricacies in the relationships among diverse modalities.",
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+    {
+        "type": "text",
+        "text": "Quaternion (Parcollet et al., 2018) is a method for representing high-dimensional data that offers unique perspectives on input features through its mutually orthogonal imaginary axes. By utilizing",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "type": "page_footnote",
+        "text": "$^{*}$ C. Ma is the corresponding author.",
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+        "type": "page_number",
+        "text": "1",
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+    {
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+        "img_path": "images/fbb582f30b6faf0495d73685da9ca562bb2f2bb0ed8a40795194f83a93bff6a1.jpg",
+        "image_caption": [
+            "Figure 1: (a) The quaternion encoder layer effectively incorporates multimodal features into the quaternion hidden space, utilizing three orthogonal imaginary axes to disentangle complex interrelationships among modalities. The k-axis plays a crucial role in balancing modalities' contributions, resulting in optimal performance. (b) Our QNet has achieved exceptional results in the novel class generalization task, outperforming other models on 11 datasets based on the harmonic mean (HM)."
+        ],
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+    {
+        "type": "text",
+        "text": "the inherent orthogonal relationship among the imaginary axes within the quaternion hidden space, correlations among diverse modal features can be established, which enables a more comprehensive representation within the shared semantic space. To this end, we propose Prompt Learning with Quaternion Networks (QNet) to better fuse the features among diverse modalities. In the quaternion hidden space, the three mutually orthogonal imaginary axes, namely  $i$ ,  $j$ , and  $k$ , allocate unique weights to various distribution features from diverse perspectives, as demonstrated in Figure 1(a). By leveraging this fused feature representation, we can directly construct prompts for both visual and textual domains, which effectively circumvents the necessity of crafting explicit interaction architectures for modality fusion. In zero-shot scenarios, the strategic incorporation of QNet at the input stage of CLIP significantly outperforms Co-CoOp, yielding an absolute enhancement exceeding  $10\\%$  on novel class datasets such as EuroSAT (Helber et al., 2019) and FGVC-Aircraft (Maji et al., 2013). Furthermore, our proposed method facilitates a deeper fusion of features originating from hierarchical levels with fewer trainable parameters (i.e., 2.93M) than MaPLe (i.e., 3.56M), thereby surpassing existing multimodal models across 11 datasets, as shown in Figure 1(b). In summary, we make the following contributions:",
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+        "list_items": [
+            "- We develop a novel approach to model complex relationships between different modalities in multimodal pre-trained models, even in situations where no data is available for training. Our method incorporates quaternion networks to fuse modalities more effectively than traditional methods. We first use quaternion networks in multimodal prompt learning.",
+            "- We use the mutually orthogonal multiple imaginary axes of quaternions to effectively disentangle complex feature distributions of different modalities in the latent feature space. Our proposed QNet is versatile and can be used for other multimodal fusion tasks, as it encapsulates interdependent relationships within features using compact parameters.",
+            "- To enhance the feature fusion potential of QNet, we utilize quaternion networks to combine features from different hierarchical levels. Extensive experiments on several datasets show that our approach consistently outperforms other multimodal models by a significant margin."
+        ],
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Vision language models. Multimodal fusion approaches heavily rely on the availability of large-scale multimodal datasets (Li et al., 2020; 2021; Jia et al., 2021; Zhai et al., 2022). For instance, CLIP (Radford et al., 2021) makes use of 400 million image-text pairs in its training process to achieve remarkable performance. However, utilizing these pre-trained models for downstream tasks with limited data (Gao et al., 2021; Zhang et al., 2022; Kim et al., 2022; Rasheed et al., 2022) remains a significant challenge in the field of multimodal data analysis, especially in terms of exploring the correlations between images and text (Maaz et al., 2022; Gu et al., 2021; Zang et al., 2022; Feng et al., 2022). Although ALBEF (Li et al., 2021) has made substantial progress in integrating image and text modalities through a six-layer transformer architecture, the issue of homogeneous representation",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "page_number",
+        "text": "2",
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+    {
+        "type": "text",
+        "text": "persists, making it difficult to capture distinct aspects or patterns within the data. In this work, we propose a novel image and text fusion architecture based on quaternion networks (Parcollet et al., 2018), which unravels the complex relationships between different modal distributions, with the aim of providing a more efficient approach to multimodal feature fusion.",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Prompt learning. Advances in natural language processing (NLP) (Li & Liang, 2021; Lester et al., 2021; Liu et al., 2021) have expanded the use of prompts to computer vision tasks (Jia et al., 2022; Wang et al., 2022b;a). This has brought about new techniques such as CoOp (Zhou et al., 2022b), which utilizes continuous vectors to enhance performance on previously unseen classes by implementing instance-level prompts. Another technique, MaPLe (Khattak et al., 2022), intensifies modality interactions by mapping text prompts to image prompts, but struggles to create diverse prompts and convey semantic information through different modal features. Our proposed method leverages quaternions with mutually orthogonal imaginary axes to calculate the weights of multimodal fusion features from various perspectives, effectively separating the complex distributions among different modal features and facilitating the integration of complementary information within the latent space. This novel modality fusion strategy represents a significant contribution to existing research on multimodal learning.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Zero-shot learning. Zero-shot learning is a powerful technique that enables pre-trained models to recognize new categories. To achieve this, auxiliary information such as attributes or word embeddings (Chao et al., 2016; Wang et al., 2019b; Xian et al., 2017; Yi et al., 2022) is provided to the models. Research indicates that prompts can activate multimodal knowledge within pre-trained models, boosting their performance in zero-shot scenarios. For example, DALLE (Reddy et al., 2021) is a model that generates multimodal images using prompts and a conditional decoder, showing impressive results in zero-shot generation. Another model, CaFo (Zhang et al., 2023), uses multiple image-text contrastive loss functions to align image and text features through a collaborative training process. However, our research delves into a more fundamental question: can we acquire features that contain even richer multimodal knowledge to facilitate diverse downstream tasks for pre-trained multimodal models? Inspired by the quaternion network (Parcollet et al., 2018), we develop more efficient data structures that facilitate prompt learning and the acquisition of fruitful multimodal fusion features. This enhancement significantly improves the pre-trained models' capability to process and predict accurately across diverse and large-scale data distributions.",
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+    {
+        "type": "text",
+        "text": "3 METHODOLOGY",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 MOTIVATION",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Deep learning models face significant challenges in representing images as arrays of pixels, which is due to the complex task of accurately encoding dependencies among internal features within high-dimensional data spaces. The challenges are further compounded in multimodal scenarios where multiple modalities, such as text, audio, and images, need to be aligned and jointly processed. The intricate distributions of these modalities can interfere with each other, making it difficult to extract essential features. Prevailing methods for image-text scenarios often rely on sophisticated structures to align visual and textual content, which results in additional network parameters. These approaches contradict the objective of enhancing the performance of multimodal pre-trained models within zero-shot settings.",
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+    {
+        "type": "text",
+        "text": "On the other hand, the elegant algebraic representation of quaternion offers a natural solution to these challenges. It can encode the dependencies between features and internal characteristics while reducing parameters. The three imaginary axes in the quaternion space provide three distinct perspectives for decoupling the intricate distributions of various modalities, effectively resolving the issue of homogeneous representations that fail to capture disparate aspects or patterns within the data.",
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+    {
+        "type": "text",
+        "text": "Our QNet is inspired by the concept of quaternion, by which we utilize fewer parameters to capture the interdependence of multimodal features, achieving prompt learning efficiency. Our method seamlessly integrates with multimodal models and outperforms the existing MaPLe (Khattak et al., 2022) method on multiple datasets with merely one prompt layer. We further observe that adding more prompt layers improves the performance of quaternion networks. Additionally, we have incorporated quaternion networks into various layers of pre-trained models to leverage their potential in feature fusion. This incorporation allows for a more profound fusion of features across multiple modalities and hierarchical levels, while using fewer learnable parameters.",
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+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "page_idx": 2
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+    {
+        "type": "page_number",
+        "text": "3",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/aaa5a1571a206e5a4ab6a517c8d6024334bf74b8c9f40227d859e9396aefb458.jpg",
+        "image_caption": [
+            "Figure 2: Overview of prompt learning with quaternion networks (QNet). The proposed QNet framework involves a sophisticated process that effectively transforms image features in purple through Meta-Proj. The transformed features are then combined with textual context tokens in yellow and subsequently encoded using the quaternion encoder layer. The yielded multimodal feature in green facilitates visual and textual prompts in a seamless and efficient manner without requiring explicit interaction mechanisms."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 PROMPT LEARNING WITH QUATERNION NETWORKS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our paper presents a ground-breaking technique called Prompt Learning with Quaternion Networks (QNet), which enables more effective encoding of multimodal features with fewer parameters. As shown in Figure 2, our method involves combining the output image features from a pre-trained model with the text context features. The combined features are then fed into the quaternion encoder, which generates a quaternion representation. The quaternion hidden space employs three mutually orthogonal imaginary axes to capture complementary information from various modalities, providing a more nuanced view of the interdependencies among different features. To ensure effective fusion of disparate modalities, our model employs multiple layers that facilitate the integration of hierarchical features inherent to different modalities. Finally, the fused features are used to create image and text prompts, which are subsequently fine-tuned for downstream tasks using QNet.",
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+    {
+        "type": "text",
+        "text": "3.2.1 QUATERNION REPRESENTATION OF MULTIMODAL FEATURES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A multimodal feature expressed by quaternion can be defined as  $\\mathrm{Q_f}$ , which is in a four-dimensional space known as the quaternion algebra  $\\mathbb{H}$ :",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {Q} _ {\\mathrm {f}} = r 1 + x \\mathbf {i} + y \\mathbf {j} + z \\mathbf {k} \\tag {1}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where  $r, x, y$  and  $z$  are real numbers, and 1, i, j and k are the quaternion unit basis. Our QNet uses quaternion-valued operations in each learning sub-process, unlike conventional fully-connected layers that use real-valued operations. The QNet transforms the input features into hypercomplex numbers. A quaternion dense layer operates on quaternions for all parameters, including inputs, outputs, weights, and biases. The quaternion algebra is maintained by manipulating real matrices (Gaudet & Maida, 2018).",
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+    {
+        "type": "text",
+        "text": "When utilizing CLIP's multimodal pre-trained model in the zero-shot setting, the combination of visual and textual knowledge proves to be complementary. To determine the weights of various modal distributions, the  $i$ -axis and  $j$ -axis are employed, while the  $k$ -axis weight is maintained to balance the relationship between the two modalities. Each input vector of size  $K$  and output vector of size  $V$  is divided into two parts:  $x_{i}$  for the first imaginary part and  $y_{j}$  for the second imaginary part, resulting in a quaternion  $\\bar{\\mathbf{Q}} = x_{i} + y_{j}$ . It is worth noting that, despite not assigning a value to the  $k$ -axis, we still uphold its weight throughout the quaternion algebra. The  $k$ -axis acts as a balancing axis that dynamically balances the weights of the  $i$ -axis and  $j$ -axis. Meanwhile, in a real-valued space, a fully-connected layer performs a real-valued dot product between an input vector and a real-valued",
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+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
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+    },
+    {
+        "type": "page_number",
+        "text": "4",
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+    {
+        "type": "text",
+        "text": "$K \\times V$  weight matrix. In QNet, this function is substituted with a quaternion-valued dot product that employs quaternion-valued matrices.",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 MULTIMODAL FUSION FEATURES",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will explain how we combine features from different modalities in the zero-shot learning setting. Drawing on insights from multimodal pre-trained models, including Oscar (Li et al., 2020) and ALBEF (Li et al., 2021), our analysis reveals a pronounced emphasis on visual features for the effective fusion of images and text. To get the most representative visual features, we use the final output of the vision branch of the multimodal pre-trained model. Co-CoOp also shows that these visual features can improve the generalization performance of the model. When it comes to the text features, we choose context tokens in zero-shot contexts. The rationale is that continuous vectors as prompts can encode text context information more efficiently with fewer parameters.",
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+    },
+    {
+        "type": "text",
+        "text": "To ensure optimal exploitation of information from both models, we initially amalgamate the Meta-Image,  $F_{image}$  derived post-dimension transformation via Meta-Proj—with the context tokens,  $F_{text}$ . This combination yields a rudimentary multimodal feature, denoted as  $F_{multi}$ :",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF _ {\\text {m u l t i}} = F _ {\\text {i m a g e}} + F _ {\\text {t e x t}}. \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Then, we send this multimodal feature  $F_{multi}$  into the Quaternion Encoder layer  $Q_{ua}(\\cdot)$ , projecting the input features to the quaternion hidden space.",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nQ _ {\\text {m u l t i}} = Q _ {\\text {u a}} \\left(F _ {\\text {m u l t i}}\\right). \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Within the quaternion latent space, we assign distinct weights to the complex distribution between various modalities by selecting the  $i$ -axis and  $j$ -axis (among the four axes of  $r$ ,  $i$ ,  $j$ , and  $k$ , the  $r$ -axis predominantly serves as the energy axis, as real numbers multiplied by imaginary numbers still yield imaginary numbers). Simultaneously, we employ the imaginary  $k$ -axis as a balancing axis. This strategy effectively prevents scenarios where one modality distribution weight becomes excessively large, consequently overshadowing other modality information. For unseen classes, fine-grained features are particularly important, and the lack of these features leads to incorrect classification results. How to effectively fuse this fine-grained knowledge between different modalities becomes the key to improving CLIP in zero-shot scenarios. Based on this analysis, we initialize a quaternion-valued weight matrix  $W$  utilizing the polar form of each weight  $w$ :",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} w = | w | e ^ {q _ {m u l t i} ^ {\\triangleleft}} = \\varphi q _ {m u l t i} ^ {\\triangleleft}, \\\\ \\text {a n d} w _ {\\mathbf {i}} = \\varphi q _ {\\text {m u l t i i}} ^ {\\triangleleft}, w _ {\\mathbf {j}} = \\varphi q _ {\\text {m u l t i j}} ^ {\\triangleleft}, w _ {\\mathbf {k}} = \\varphi q _ {\\text {m u l t i k}} ^ {\\triangleleft}. \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We use the well-known initialization criteria in (Glorot & Bengio, 2010) and (He et al., 2015) to obtain  $\\varphi$ .  $q_{multi}^{\\triangleleft}$  is a normalized pure imaginary quaternion  $q_{multi}^{\\triangleleft} = 0 + x\\mathbf{i} + y\\mathbf{j} + z\\mathbf{k}$ , where  $x\\mathbf{i}$ ,  $y\\mathbf{j}$ , and  $z\\mathbf{k}$  are uniformly sampled from  $[0,1]$  and then normalized. Detailed information about quaternion parameter initialization can be found in Appendix. Although the  $k$ -axis does not serve as the parameter axis in QNet, its weights contribute to the updating process of the weights associated with the  $i$ -axis and  $j$ -axis, as depicted in Figure 1(a). After passing through the quaternion hidden space, we obtain the final fused feature  $Q_{multi}$ . This feature preserves the multimodal information to the greatest extent and captures the complementary modality relationships through different imaginary axes of the quaternion:",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nQ _ {f u s i o n} = \\left[ w _ {i}, w _ {j} \\right] Q _ {m u t i l}, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where  $[\\cdot, \\cdot]$  denotes the concatenation of  $w_{i}$  and  $w_{j}$ .",
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+    },
+    {
+        "type": "text",
+        "text": "3.2.3 VISION PROMPTING",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since multimodal feature  $Q_{fusion}$  already contains rich image-text abstraction, we do not adopt any other complex structures to construct visual prompts. After performing a simple dimension change on  $Q_{fusion}$ , we directly concatenate it as a visual prompt to the visual transformer layer.",
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+    {
+        "type": "text",
+        "text": "We define  $M$  as a set of context vectors that can be learned. Then, we can obtain a set of learnable tokens, which have the same dimensionality as the word embeddings. For each layer  $i$ , we construct a prompt for the image branch:  $m_i = \\left\\{\\hat{Q}_{fusion}^1,\\hat{Q}_{fusion}^2,\\dots ,\\hat{Q}_{fusion}^M\\right\\}$ . As a concatenation of the image context vectors and the multimodal features  $Q_{fusion}$ , the vision prompt for the  $i$ -th class of",
+        "bbox": [
+            169,
+            861,
+            826,
+            925
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "page_number",
+        "text": "5",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/66ff8bdc89c785a201d9022ab7f9d522942a63cad2b69047c8427ef19ded342f.jpg",
+        "table_caption": [
+            "Table 1: Comparison with existing methods in base-to-novel generalization on 11 datasets. The absolute gains compared to MaPLe are highlighted in red and blue."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Average</td><td colspan=\"3\">ImageNet</td><td colspan=\"3\">Caltech101</td><td colspan=\"3\">OxfordPots</td></tr><tr><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td></tr><tr><td>CLIP (Radford et al., 2021)</td><td>69.34</td><td>74.22</td><td>71.70</td><td>72.43</td><td>68.14</td><td>70.22</td><td>96.84</td><td>94.00</td><td>95.40</td><td>91.17</td><td>97.26</td><td>94.12</td></tr><tr><td>CoOp (Zhou et al., 2022b)</td><td>82.69</td><td>63.22</td><td>71.66</td><td>76.47</td><td>67.88</td><td>71.92</td><td>98.00</td><td>89.81</td><td>93.73</td><td>93.67</td><td>95.29</td><td>94.47</td></tr><tr><td>Co-CoOp (Zhou et al., 2022a)</td><td>80.47</td><td>71.69</td><td>75.83</td><td>75.98</td><td>70.43</td><td>73.10</td><td>97.96</td><td>93.81</td><td>95.84</td><td>95.20</td><td>97.69</td><td>96.43</td></tr><tr><td>MaPLe (Khattak et al., 2022)</td><td>82.28</td><td>75.14</td><td>78.55</td><td>76.66</td><td>70.54</td><td>73.47</td><td>97.74</td><td>94.36</td><td>96.02</td><td>95.43</td><td>97.69</td><td>96.43</td></tr><tr><td>QNet (Ours)</td><td>83.32</td><td>75.65</td><td>79.30</td><td>77.00</td><td>71.00</td><td>73.88</td><td>98.40</td><td>95.70</td><td>97.03</td><td>96.20</td><td>97.80</td><td>96.99</td></tr><tr><td></td><td>+1.04</td><td>+0.51</td><td>+0.75</td><td>+0.34</td><td>+0.46</td><td>+0.41</td><td>+0.66</td><td>+1.34</td><td>+1.01</td><td>+0.77</td><td>+0.04</td><td>+0.41</td></tr><tr><td rowspan=\"2\"></td><td colspan=\"3\">StanfordCars</td><td colspan=\"3\">Flowers102</td><td colspan=\"3\">Food101</td><td colspan=\"3\">FGVCAircraft</td></tr><tr><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td></tr><tr><td>CLIP (Radford et al., 2021)</td><td>63.37</td><td>74.89</td><td>68.65</td><td>72.08</td><td>77.80</td><td>74.83</td><td>90.10</td><td>91.22</td><td>90.66</td><td>27.19</td><td>36.29</td><td>31.09</td></tr><tr><td>CoOp (Zhou et al., 2022b)</td><td>78.12</td><td>60.40</td><td>68.13</td><td>97.60</td><td>59.67</td><td>74.06</td><td>88.33</td><td>82.26</td><td>85.19</td><td>40.44</td><td>22.30</td><td>28.75</td></tr><tr><td>Co-CoOp (Zhou et al., 2022a)</td><td>70.49</td><td>73.59</td><td>72.01</td><td>94.87</td><td>71.75</td><td>81.71</td><td>90.70</td><td>91.29</td><td>90.99</td><td>33.41</td><td>23.71</td><td>27.74</td></tr><tr><td>MaPLe (Khattak et al., 2022)</td><td>72.94</td><td>74.00</td><td>73.47</td><td>95.92</td><td>72.46</td><td>82.56</td><td>90.71</td><td>92.05</td><td>91.38</td><td>37.44</td><td>35.61</td><td>36.50</td></tr><tr><td>QNet (Ours)</td><td>74.20</td><td>74.90</td><td>74.55</td><td>97.00</td><td>75.20</td><td>84.72</td><td>91.10</td><td>92.00</td><td>91.55</td><td>38.80</td><td>35.90</td><td>37.29</td></tr><tr><td></td><td>+1.26</td><td>+0.90</td><td>+1.08</td><td>+1.08</td><td>+2.74</td><td>+2.16</td><td>+0.39</td><td>-0.05</td><td>+0.17</td><td>+1.36</td><td>+0.29</td><td>+0.79</td></tr><tr><td rowspan=\"2\"></td><td colspan=\"3\">SUN397</td><td colspan=\"3\">DTD</td><td colspan=\"3\">EuroSAT</td><td colspan=\"3\">UCF101</td></tr><tr><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td><td>Base</td><td>Novel</td><td>HM</td></tr><tr><td>CLIP (Radford et al., 2021)</td><td>69.36</td><td>75.35</td><td>72.23</td><td>53.24</td><td>59.90</td><td>56.37</td><td>56.48</td><td>64.05</td><td>60.03</td><td>70.53</td><td>77.50</td><td>73.85</td></tr><tr><td>CoOp (Zhou et al., 2022b)</td><td>80.60</td><td>65.89</td><td>72.51</td><td>79.44</td><td>41.18</td><td>54.24</td><td>92.19</td><td>54.74</td><td>68.69</td><td>84.69</td><td>56.05</td><td>67.46</td></tr><tr><td>Co-CoOp (Zhou et al., 2022a)</td><td>79.74</td><td>76.86</td><td>78.27</td><td>77.01</td><td>56.00</td><td>64.85</td><td>87.49</td><td>60.04</td><td>71.21</td><td>82.33</td><td>73.45</td><td>77.64</td></tr><tr><td>MaPLe (Khattak et al., 2022)</td><td>80.82</td><td>78.70</td><td>79.75</td><td>80.36</td><td>59.18</td><td>68.16</td><td>94.07</td><td>73.23</td><td>82.35</td><td>83.00</td><td>78.66</td><td>80.77</td></tr><tr><td>QNet (Ours)</td><td>81.10</td><td>78.60</td><td>79.83</td><td>80.70</td><td>59.50</td><td>68.50</td><td>96.60</td><td>73.10</td><td>83.22</td><td>85.40</td><td>78.50</td><td>81.80</td></tr><tr><td></td><td>+0.28</td><td>-0.10</td><td>+0.08</td><td>+0.34</td><td>+0.32</td><td>+0.34</td><td>+2.53</td><td>-0.13</td><td>+0.87</td><td>+2.40</td><td>-0.16</td><td>+1.03</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "the image branch is now  $z(x) = \\left\\{\\hat{Q}_{fusion}^{1}, \\hat{Q}_{fusion}^{2}, \\dots, \\hat{Q}_{fusion}^{M}, x\\right\\}$ . In the vision branch of CLIP alongside the input image tokens, new learnable tokens are further introduced in deeper transformer layers of the image encoder  $k(\\cdot)$  up to depth  $L$ . Let  $c_{i}$  be the word embedding(s) for the class name,  $E_{i}$  be patch embeddings, and  $[\\cdot, \\cdot]$  refer to the concatenation operation, we have:",
+        "bbox": [
+            169,
+            483,
+            826,
+            546
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ c _ {i}, E _ {i}, \\_ \\right] = k _ {i} \\left(\\left[ c _ {i - 1}, E _ {i - 1}, \\mathrm {m} _ {i - 1} \\right]\\right) \\quad i = 1, 2, \\dots , L,\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            571,
+            692,
+            587
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ c _ {j}, E _ {j}, m _ {j} \\right] = k _ {j} \\left(\\left[ c _ {j - 1}, E _ {j - 1}, \\mathrm {m} _ {j - 1} \\right]\\right) \\quad j = L + 1, \\dots , N. \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            583,
+            823,
+            606
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.4 LANGUAGE PROMPTING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            619,
+            398,
+            633
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After obtaining high-quality multimodal features, we construct language context prompts with these features. Similarly, defining  $g(\\cdot)$  as the text encoder function, we construct a prompt for the text branch:  $t_i = \\left\\{\\dot{Q}_{fusion}^1,\\dot{Q}_{fusion}^2,\\dots ,\\dot{Q}_{fusion}^M\\right\\}$ . The language prompt for the  $i$ -th class of text branch is embedded by  $h_i(x) = \\left\\{\\dot{Q}_{fusion}^1,\\dot{Q}_{fusion}^2,\\dots ,\\dot{Q}_{fusion}^M,c_i\\right\\}$ . New learnable tokens are further introduced in each transformer block of the language encoder  $g(\\cdot)$  up to a specific depth  $L$ :",
+        "bbox": [
+            169,
+            638,
+            826,
+            726
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n[ \\_ , c _ {i} ] = g _ {i} \\left(\\left[ t _ {i - 1}, c _ {i - 1} \\right]\\right) i = 1, 2, \\dots , L. \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            732,
+            825,
+            750
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After the  $L^{\\mathrm{th}}$  transformer layer, the subsequent layers process the previous layer prompts:",
+        "bbox": [
+            169,
+            757,
+            761,
+            773
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left[ t _ {j}, c _ {i} \\right] = g _ {j} \\left(\\left[ t _ {j - 1}, c _ {j - 1} \\right]\\right) j = L + 1, \\dots , N. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            780,
+            825,
+            797
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.5 THE BRIDGE OF VISION LANGUAGE PROMPTING",
+        "text_level": 1,
+        "bbox": [
+            171,
+            820,
+            570,
+            834
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The primary distinction of our approach from other multimodal prompts is that ours does not rely on explicit structural comparisons between different modalities. QNet capitalizes on a multimodal feature representation  $Q_{multi}$  that encapsulates abundant information across various modalities. Consequently, it attains superior fusion performance compared to previous methods. Concretely, we utilize three mutually orthogonal imaginary axes to provide unique perspectives on multimodal features. We partition the input along the  $i$ -axis and  $j$ -axis to secure the image and text information,",
+        "bbox": [
+            169,
+            840,
+            826,
+            925
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "page_number",
+        "text": "6",
+        "bbox": [
+            493,
+            948,
+            504,
+            959
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/10c59f6cd3f516c23d3f3c28bd25f74dfbb14b2b71d68dca3728b868ff4d7c30.jpg",
+        "table_caption": [
+            "Table 2: Comparison with existing methods in cross-dataset evaluation. We train QNet on the 1,000 classes of ImageNet and then transfer this model for evaluation on the remaining 10 datasets."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td>Source</td><td colspan=\"10\">Target</td></tr><tr><td>ImageNet</td><td>Catech101</td><td>OxfordPeds</td><td>StanfordCars</td><td>Flowers102</td><td>Food101</td><td>Aircraft</td><td>SUN397</td><td>DTD</td><td>EuroSAT</td><td>UCF101</td></tr><tr><td>CoOp (Zhou et al., 2022b)</td><td>71.51</td><td>93.70</td><td>89.14</td><td>64.51</td><td>68.71</td><td>85.30</td><td>18.47</td><td>64.15</td><td>41.92</td><td>46.39</td><td>66.55</td></tr><tr><td>Co-CoOp (Zhou et al., 2022a)</td><td>71.02</td><td>94.43</td><td>90.14</td><td>65.32</td><td>71.88</td><td>86.06</td><td>22.94</td><td>67.36</td><td>45.73</td><td>45.37</td><td>68.21</td></tr><tr><td>MaPLe (Khattak et al., 2022)</td><td>70.72</td><td>93.53</td><td>90.49</td><td>65.57</td><td>72.23</td><td>86.20</td><td>24.74</td><td>67.01</td><td>46.49</td><td>48.06</td><td>68.69</td></tr><tr><td>QNet (Ours)</td><td>71.10</td><td>94.30</td><td>90.80</td><td>66.00</td><td>73.10</td><td>86.10</td><td>24.20</td><td>67.90</td><td>46.90</td><td>51.80</td><td>68.70</td></tr></table>",
+        "bbox": [
+            173,
+            132,
+            823,
+            266
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "respectively. The three imaginary axes possess equal status and select either the  $i$ -axis and  $k$ -axis or the  $j$ -axis and  $k$ -axis to produce equivalent results. Note that we do not assign a value to the  $k$ -axis relative to the  $i$ -axis and  $j$ -axis. Instead, we preserve the weight of the  $k$ -axis as a balance between the  $i$ -axis and  $j$ -axis. Our model effectively decouples the complex feature distributions of text and image by orthogonally disentangling them across the  $i\\text{-}, j\\text{-}$ , and  $k\\text{-}$  axes. Moreover, by employing the quaternion network to integrate features across diverse hierarchical features of CLIP, QNet demonstrates its enhanced generalization capacity.",
+        "bbox": [
+            169,
+            290,
+            826,
+            388
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A substantial body of research concentrates on the development of explicit interaction structures between images and text. However, we argue that regardless of their complexity, these structures fail to surmount the limitations of homogeneous representation and effectively tackle the fundamental challenge in multimodal fusion: capturing intricate relationships among different modalities. Our approach employs quaternion networks which are adept at encoding interdependent relationships within multimodal features. This obviates the need to design extra interaction structures between images and text. Finally, the prediction probability is computed as:",
+        "bbox": [
+            169,
+            393,
+            823,
+            492
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np (y \\mid \\boldsymbol {x}) = \\frac {\\exp (\\sin (\\boldsymbol {z} (Q (\\boldsymbol {x})), g (\\boldsymbol {h} _ {y} (Q (\\boldsymbol {x})))) / \\tau)}{\\sum_ {i = 1} ^ {K} \\exp (\\sin (\\boldsymbol {z} (Q (\\boldsymbol {x})) , g (\\boldsymbol {h} _ {i} (Q (\\boldsymbol {x})))) / \\tau)} \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            502,
+            825,
+            539
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $Q(\\cdot)$  denotes the quaternion networks,  $\\mathbf{z}(x)$  is the vision prompt,  $h_i(x)$  is the language prompt,  $\\sin (\\cdot ,\\cdot)$  is the cosine similarity, and  $\\tau$  is a learned temperature parameter.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            602,
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+            617
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP",
+        "text_level": 1,
+        "bbox": [
+            171,
+            631,
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+            645
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We follow Zhou et al. (2022b) by using 11 image recognition datasets that cover various tasks. Concretely, we include ImageNet (Deng et al., 2009) and Caltech101 (Fei-Fei et al., 2004) for generic object classification, Oxfordpets (Parkhi et al., 2012), StanfordCars (Krause et al., 2013), Flowers102 (Nilsback & Zisserman, 2008), Food101 (Bossard et al., 2014), and Aircraft (Maji et al., 2013) for fine-grained classification, SUN397 (Xiao et al., 2010) for scene recognition, UCF101 (Soomro et al., 2012) for action recognition, DTD (Cimpoi et al., 2014) for texture recognition, and EuroSAT (Helber et al., 2019) for satellite image recognition.",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementation details. We evaluate our method in three scenarios: 1) Base-to-novel generalization, generalizing from base classes to new classes within a dataset; 2) Cross-dataset evaluation, transferring across different datasets, and 3) Domain generalization, transferring on four variant datasets of ImageNet. As for domain generalization, we conduct the experiments using ImageNet as the source dataset and four ImageNet variants that have different types of domain shift as the target datasets, namely ImageNetV2 (Recht et al., 2019), ImageNet-Sketch (Wang et al., 2019a), ImageNet-A (Hendrycks et al., 2021b), and ImageNet-R (Hendrycks et al., 2021a). For the training of QNet, we prompt-tune a pre-trained ViT-B/16 CLIP model and set prompt depth  $L$  to 7 and language and vision prompt lengths to 2. We train QNet for 7 epochs with a batch size of 1 on a single NVIDIA RTX 8000 GPU. Following Zhou et al. (2022b), we use the pre-trained word embeddings of a shorter context (e.g., \"a photo of\") to initialize the context vectors for the first layer. To maintain robust results, we validate our method using 16 shots and report the average results over three runs.",
+        "bbox": [
+            169,
+            758,
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+            926
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
+        "bbox": [
+            171,
+            32,
+            478,
+            47
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "page_number",
+        "text": "7",
+        "bbox": [
+            493,
+            948,
+            503,
+            959
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/96cf9d5c3704aa1fd17e40cd8f71478f45965bee4868d84cd6475791b70da5d2.jpg",
+        "table_caption": [
+            "Table 3: Comparison with existing methods in domain generalization. We employ our method pre-trained on the source dataset (i.e., ImageNet) and test on four variant datasets of ImageNet."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Source</td><td colspan=\"4\">Target</td></tr><tr><td></td><td>ImageNet</td><td>ImageNetV2</td><td>ImageNet-S</td><td>ImageNet-A</td><td>ImageNet-R</td><td>Average</td></tr><tr><td>CLIP (Radford et al., 2021)</td><td>66.73</td><td>60.83</td><td>46.15</td><td>47.77</td><td>73.96</td><td>57.18</td></tr><tr><td>CoOp (Zhou et al., 2022b)</td><td>71.51</td><td>64.20</td><td>47.99</td><td>49.71</td><td>75.21</td><td>59.28</td></tr><tr><td>Co-CoOp (Zhou et al., 2022a)</td><td>71.02</td><td>64.07</td><td>48.75</td><td>50.63</td><td>76.18</td><td>59.91</td></tr><tr><td>MaPLe (Khattak et al., 2022)</td><td>70.72</td><td>64.07</td><td>49.15</td><td>50.90</td><td>76.98</td><td>60.27</td></tr><tr><td>QNet (Ours)</td><td>71.10</td><td>64.30</td><td>49.20</td><td>51.30</td><td>77.70</td><td>60.65</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 BASE-TO-NOVEL GENERALIZATION",
+        "text_level": 1,
+        "bbox": [
+            171,
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+            465,
+            273
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The baseline of our method is CLIP (Radford et al., 2021) with manual prompts. CoOp (Zhou et al., 2022b) improves CLIP with static prompts and CoCoOp (Zhou et al., 2022a) combines it with prompt-conditioned learning. MaPLe (Khattak et al., 2022) is a multimodal prompt method that achieves modality interaction through linear mapping, whereas our method constructs high-quality multimodal fusion features. Table 1 provides a comparative analysis of our QNet against other approaches in the base-to-novel generalization framework. Evidently, with fewer trainable parameters (i.e., 2.93M in QNet compared to 3.56M in MaPLe), QNet surpasses all previous prompt learning techniques in terms of base accuracy, novel accuracy, and harmonic mean (HM) when averaged across 11 datasets. Concretely, our method QNet attains a base category accuracy of  $83.32\\%$ , elevates novel category performance from  $75.14\\%$  to  $75.65\\%$ , and increases the harmonic mean from  $78.55\\%$  to  $79.30\\%$ . The experimental outcomes effectively demonstrate that QNet associates semantic spaces across different modalities from multiple perspectives. For general datasets, QNet accomplishes comprehensive enhancements for both base and novel classes (i.e.,  $77.00\\%$  and  $71.00\\%$  on ImageNet,  $98.40\\%$  and  $95.70\\%$  on Caltech). These general datasets encompass more diverse information than fine-grained datasets, necessitating the capture of different feature aspects via mutually orthogonal imaginary axes in the quaternion latent space. Linear mapping in MaPLe and other explicit interaction techniques are inadequate for addressing the shortcomings of homogeneous representation in capturing distinct aspects of data. To sum up, our method fundamentally addresses high-dimensional data processing, encoding the innate inter-dependencies within features in a more compact manner.",
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+        "text": "4.3 CROSS-DATASET EVALUATION",
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+        "text": "In the cross-dataset transfer setting, we first train QNet on the 1,000 classes of ImageNet and directly transfer the model to evaluate on the remaining 10 datasets. Table 2 presents comparative results of QNet and other methods. On the whole, QNet outperforms the baseline models substantially, attaining an average accuracy of  $66.98\\%$ . Note that, during the training process on the 1,000 ImageNet classes, MaPLe employs a prompt depth of 3 while QNet integrates the quaternion networks solely within the initial two layers. Moreover, our experiments are conducted by preserving the same learning rate as in the base-to-novel generalization. The results suggest that our modality fusion approach outperforms MaPLe, achieving a more efficient encoding of rich multimodal information with fewer parameters and faster convergence.",
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+        "text": "4.4 DOMAIN GENERALIZATION",
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+        "text": "Evaluating models on out-of-distribution tasks is vital for assessing their generalization abilities. We employ our method pre-trained on the ImageNet dataset and subject them to direct testing on specific datasets with diverse data distributions. After completing the training on the entire 1,000 classes of ImageNet, we proceed to evaluate the model on four variant datasets derived from ImageNet, as depicted in Table 3. On the ImageNetV2 dataset, while both MaPLe and Co-CoOp yield an accuracy of  $64.07\\%$ , our proposed method demonstrates superior performance with an accuracy rate of  $64.30\\%$ . QNet consistently outperforms MaPLe and Co-CoOp across these four distinct ImageNet dataset variations. These results indicate that within the quaternion hidden space, QNet efficiently decouples the intricate distributions of distinct modalities by utilizing three mutually orthogonal imaginary axes, thereby correlating complementary information across modalities from diverse viewpoints.",
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+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "8",
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+        "type": "table",
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+        "table_caption": [
+            "Table 4: Ablation study among QNet, Co-CoOp, and MaPLe with one single layer."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Base</td><td>Novel</td><td>HM</td></tr><tr><td>Co-CoOp</td><td>80.47</td><td>71.69</td><td>75.83</td></tr><tr><td>MaPLe (one layer)</td><td>78.69</td><td>73.32</td><td>75.91</td></tr><tr><td>QNet (one layer)</td><td>81.01</td><td>74.88</td><td>77.83</td></tr><tr><td></td><td>+2.32</td><td>+1.56</td><td>+1.92</td></tr></table>",
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+        "table_caption": [
+            "Table 5: Ablation study on adding real axis for modality fusion."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Base</td><td>Novel</td><td>HM</td></tr><tr><td>QNetri</td><td>98.05</td><td>95.00</td><td>96.50</td></tr><tr><td>QNtetrijj</td><td>98.10</td><td>94.33</td><td>96.18</td></tr><tr><td>QNnetrijk</td><td>97.93</td><td>93.90</td><td>95.87</td></tr><tr><td>QNnetijk</td><td>98.40</td><td>95.90</td><td>97.13</td></tr></table>",
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+        "type": "image",
+        "img_path": "images/20a2f39dbe160ffc0dde8dc3c667ee5929456b0ee5b655950f413ac296c0721f.jpg",
+        "image_caption": [
+            "Figure 3: Ablation study on the prompt depth of QNet on the Caltech dataset."
+        ],
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+        "table_caption": [
+            "Table 6: Ablation study among different imaginary axes for modality fusion."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Base</td><td>Novel</td><td>HM</td></tr><tr><td>QNeti</td><td>98.23</td><td>94.93</td><td>96.55</td></tr><tr><td>QNetij</td><td>98.10</td><td>94.17</td><td>96.09</td></tr><tr><td>QNetijkp</td><td>98.07</td><td>95.17</td><td>96.60</td></tr><tr><td>QNetijkm</td><td>98.40</td><td>95.90</td><td>97.13</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.5 ABLATION STUDIES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Multimodal fusion approach. We further validate our innovative multimodal feature fusion approach by adjusting the prompt depth of both MaPLe and QNet to 1, while bearing in mind that the original Co-CoOp only has a single prompt layer. Our experimental findings, as indicated in Table 4, reveal that our method is more effective in modality fusion, with QNet showing a significant improvement compared to the other two techniques. MaPLe's lower base class accuracy compared to Co-CoOp indicates its improvements are largely driven by deep prompt strategy, rather than multimodal fusion. Our method adeptly captures the correlations among multimodal features, resulting in better performance. See the Appendix for complete results for each dataset.",
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+    {
+        "type": "text",
+        "text": "Effect of different axes. We explore the impact of varying axes in our experiments and summarize the results in Tables 5 and 6. The study, using Caltech as an example, reveals a performance decline when the real axis is incorporated. We noted improved performance on novel classes with an increased number of imaginary axes; especially, using  $k_{m}$  as a modulation axis proved more effective, enhancing the encapsulation of intermodal semantic spatial correlations between text and image modalities, compared to when  $k_{p}$  serves as a parameter axis.",
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+    {
+        "type": "text",
+        "text": "Prompt depth. Figure 3 demonstrates the effects of prompt depth on model performance. We again use the Caltech dataset to illustrate the influence of varying layers on the average performance. The overall trend suggests that performance improves with increasing model depth. However, it is observed that once the depth surpasses a certain threshold, performance begins to deteriorate, indicating a certain degree of overfitting caused by the increase in the number of layers. The optimal performance of our QNet is achieved with 7 layers.",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Large-scale multimodal pre-trained models are a crucial step towards achieving general artificial intelligence. Both academic and industrial communities are focused on deploying these models for specific tasks. However, models like CLIP face challenges in handling fine-grained classifications and abstract tasks in zero-shot learning. To address this, we propose a fundamental high-dimensional structure using quaternion networks that capture rich semantic associations across different modalities and hierarchical features. In the quaternion latent space, the three mutually orthogonal imaginary axes are highly effective. Our experimental results demonstrate that QNet outperforms existing methods while involving fewer learnable parameters in base-to-novel generalization, cross-dataset transfer, and domain generalization. Furthermore, QNet effectively fuses features from multimodal sources in zero-shot scenarios, and can be extended to other domains.",
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+    {
+        "type": "text",
+        "text": "Limitation. We use the CLIP model with transformer architecture for image feature extraction, but it tends to miss local and fine-grained details, impacting tasks such as classification and detection. Future plans include leveraging QNet to integrate these crucial features and addressing the transformer's bias towards semantic information.",
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+        "type": "header",
+        "text": "Published as a conference paper at ICLR 2024",
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+        "text": "9",
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+    {
+        "type": "text",
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+    },
+    {
+        "type": "header",
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+    },
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+]