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2024/Proteus_ Exploring Protein Structure Generation for Enhanced Designability and Efficiency/full.md

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+# Proteus: Exploring Protein Structure Generation for Enhanced Designability and Efficiency
+
+Chentong Wang $^{1,2}$  Yannan Qu $^{2}$  Zhangzhi Peng $^{3}$  Yukai Wang $^{2}$  Hongli Zhu $^{2}$  Dachuan Chen $^{2}$  Longxing Cao $^{2}$
+
+# Abstract
+
+Diffusion-based generative models have been successfully employed to create proteins with novel structures and functions. However, the construction of such models typically depends on large, pre-trained structure prediction networks, like RFdiffusion. In contrast, alternative models that are trained from scratch, such as FrameDiff, still fall short in performance. In this context, we introduce Proteus, an innovative deep diffusion network that incorporates graph-based triangle methods and a multi-track interaction network, eliminating the dependency on structure prediction pre-training with superior efficiency. We have validated our model's performance on de novo protein backbone generation through comprehensive in silico evaluations and experimental characterizations, which demonstrate a remarkable success rate. These promising results underscore Proteus's ability to generate highly designable protein backbones efficiently. This capability, achieved without reliance on pre-training techniques, has the potential to significantly advance the field of protein design. Codes are available at https://github.com/Wangchentong/Proteus.
+
+# 1. Introduction
+
+The biological function of a protein is often directly determined by its tertiary structure, which underscores the importance of designing novel protein backbones. De novo protein design methods are dedicated to creating proteins with the desired structure and function. Recent advancements in
+
+$^{1}$ Zhejiang University, Hangzhou, Zhejiang, China  $^{2}$ School of Life Sciences, Westlake University, Hangzhou, Zhejiang, China  $^{3}$ Duke University, Durham, North Carolina, USA. Correspondence to: Longxing Cao <caolongxing@westlake.edu.cn>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+protein structure prediction methods, such as AlphaFold2 (Jumper et al., 2021) and RosettaFold (Baek et al., 2021), have enabled the 'hallucination' approach (Anishchenko et al., 2021; Wang et al., 2022) to directly generate protein sequences using backpropagation on the structure prediction networks. Further leveraging the generative capabilities of the diffusion model (Ho et al., 2020), RFdiffusion (Watson et al., 2023) has demonstrated superior performance across a wide range of protein design challenges. These applications include designing protein binders, scaffolding motifs, and creating symmetric oligomers. Despite RFdiffusion's impressive performance, its dependency on pretraining with RosettaFold2 (Baek et al., 2023) poses a challenge for dissecting and refining the model's architecture to improve performance for structure generation tasks.
+
+To tackle the challenge of generating designable protein backbones without reliance on pretraining, researchers have developed a range of diffusion strategies and model architectures. One such approach is FoldingDiff (Wu et al., 2022), which employs diffusion in the protein backbone torsion space with a bidirectional transformer. This network iteratively denoises a sequence of torsion angles to generate protein-like backbones. However, the majority of generated structures are predicted to be non-designable. In contrast, two concurrent studies have shown more promise by directly applying diffusion on residue coordinates (Lin & AlQuraishi, 2023) or tangent space of coordinate and rotation (Yim et al., 2023). Additionally, previous research (Anand & Achim, 2022; Lee et al., 2023; Trippe et al., 2022) has explored multiple network architectures, including U-Net (Ronneberger et al., 2015), Equivariant Graph Neural Networks (EGNNs) (Satorras et al., 2022), Invariant Point Attention(IPA) (Jumper et al., 2021), which have achieved success in diverse fields such as Computer Vision or dynamic system modeling.
+
+Although these initiatives have moderately improved the designability of protein structure diffusion models without dependency on pretraining, there is still a notable performance gap compared to RFdiffusion, which results in considerable limitations for these models, making them less effective or difficult to apply in practical protein design
+
+![](images/bed4fdd1418fa510be7ebf23f5cee3d9fea54a7a4f931598f5058f9301041aab.jpg)  
+Figure 1: Benchmarking Proteus against other backbone diffusion models on designability, efficiency, and diversity. All metrics are averaged from 200 backbones of each length 100, 200, 300, 400, 600, and 800. For each backbone length, 8 sequences are designed by ProteinMPNN, except for Chroma, which uses ChromaDesign to achieve the best performance reported in its original paper. (A) Radar plot illustrating model evaluation across three dimensions. (B) Self-consistency RMSD between the generated backbone and the best prediction of ESMfold. (C) Sampling time of backbone generation, evaluated on A40. Genie fails to generate backbones larger than 600 residues due to running out of memory.
+
+![](images/719cd5650ac00644b793e120d7af8e609885091c17f5350188757b7c355d25ca.jpg)
+
+![](images/91e2331f24f5d138ac644852ba87c0c7eeedd8cf359e8f76428836b22347e5ea.jpg)
+
+tasks. RFdiffusion, with its cutting-edge capabilities, sets a higher standard in the field and highlights the need for further advancements in model development.
+
+To bridge the performance gap between models that do and do not require pretraining, we have developed Proteus. Proteus achieves backbone designability on par with RFdiffusion by integrating a graph-based triangle technique and a multi-track interaction network, significantly bolstered by data augmentation. Moreover, our model sets a new efficiency standard, owing to two principal advancements: a reduction in the required sampling steps due to the enhanced representational capacity of the model, and the employment of local graph modeling to decrease computational complexity. These innovations allow Proteus to achieve protein generation speeds comparable to those of Chroma (Ingraham et al., 2022).
+
+In summary, our model achieves several optimization objectives critical to protein design. It functions independently of pretraining, delivers high designability of protein structures, and maintains rapid generation speeds. The introduction of Proteus represents a noteworthy progression in protein design methodology, offering a solution that balances efficiency with the complexity of protein structure generation.
+
+# 2. Preliminaries
+
+Protein backbone representation Following the approach of AlphaFold2 (Jumper et al., 2021), the backbone of each residue is parameterized as a series of rigid transformations, also known as frames. These frames, denoted by  $\mathrm{T} = (R,t)$  are defined within the special Euclidean group  $SE(3)$  and represent orientation-preserving transformations of the idealized backbone atom coordinates  $[N_n,C_{\alpha_n},C_n]$ . Specifically,  $R\in SO(3)$  is a rotation matrix derived from the
+
+backbone atoms N, CA, and C through the Gram-Schmidt process, and  $t \in \mathbb{R}^3$  represents the coordinates of atom  $C_{\alpha}$ .
+
+Diffusion modeling on protein backbone Multiple methodologies are available for protein backbone diffusion, including diffusion on inter-residue geometry or backbone torsion angles. In this paper, we detail our employed approach: SE(3) diffusion. This method treats each residue independently and computes the associated probability estimation on SO(3) for the score matching calculation, as originally proposed by Yim et al..
+
+Briefly, the protein backbone's forward diffusion process is driven by Brownian motion on SO(3) and  $\mathbb{R}^3$  individually as shown in Equation (1)
+
+$$
+d \mathbf {T} ^ {(t)} = \left[ \mathbf {0}, - \frac {1}{2} \mathbf {X} ^ {(t)} \right] d t + \left[ d \mathbf {B} _ {\mathbf {S O} (3)} ^ {(t)}, d \mathbf {B} _ {\mathbb {R} ^ {3}} ^ {(t)} \right] \tag {1}
+$$
+
+By defining probability estimation on  $\mathbb{R}^3$  forward diffusion,  $p_{t|0}(x^{(t)}|x^{(0)}) = \mathcal{N}(x^{(t)};e^{-t / 2}x^{(0)},(1 - e^{-t})Id_3)$ , the corresponding conditional score can be computed explicitly
+
+$$
+\nabla \log p _ {t | 0} (x ^ {(t)} | x ^ {(0)}) = (1 - e ^ {- t}) ^ {- 1} \left(e ^ {- t / 2} x ^ {(0)} - x ^ {(t)}\right).
+$$
+
+For the proper estimation of probability on SO(3), Brownian motion on SO(3) is defined as  $p_{t|0}(r^{(t)}|r^{(0)}) = f\left(\omega \left(r^{(0)^T}r^{(t)}\right), t\right)$ , where  $\omega(r)$  is the rotation angle in radians for any  $r \in SO(3)$ . The final probability estimation can be described as
+
+$$
+f (\omega , t) = \sum_ {\ell \in \mathbb {N}} (2 \ell + 1) \mathrm {e} ^ {- \ell (\ell + 1) t / 2} \frac {\sin ((\ell + 1 / 2) \omega)}{\sin (\omega / 2)} \tag {2}
+$$
+
+![](images/08c221a658f578fad4c1404b22a32514f6e6dcc5e6e72ddd10e779a8ccf0c580.jpg)
+
+![](images/d80c413deb383c2e48b41377054dce6990eec0c417ace1d6e5b9ae8b7c3fc7cf.jpg)
+
+![](images/a377d23fef9dd3395acae6151e7a146369aadc7681c367b2369377311c1fb868.jpg)  
+Figure 2: Model illustration (A) The Protein backbone diffusion model is trained to recover noised structures and generate new structures by reversing the forward process. (B) The overall architecture of Proteus. (C) The detailed model architecture of graph triangle block.
+
+With the probability estimation, the corresponding conditional score can be computed as
+
+$$
+\nabla \log p _ {t _ {0}} \left(r ^ {(t)} | r ^ {(0)}\right) = \frac {r ^ {(t)}}{\omega (t)} \log \left(r ^ {(0, t)}\right) \frac {\partial \omega f (\omega (t) , t)}{f (\omega (t) , t)},
+$$
+
+With a well-trained denoising score matching(DSM) network on tangent space of  $\mathbb{R}^3$  and SO(3), a protein backbone can be sampled from noise with n steps by Euler-Maruyama discretization (Bortoli et al., 2022).
+
+In summary, the objective of the network is to predict a denoised protein backbone at timestep 0, given noisy backbones from any timesteps. Within this framework, probability estimation can be calculated, enabling the application of the stochastic differential equation (SDE) to sample a denoised protein backbone. We recommend that readers check Yim et al.'s work for a profound understanding of SE(3) diffusion.
+
+Deep learning network architectures for protein structure modeling The model architectures of AlphaFold2 (Jumper et al., 2021) and Rosettafold (Baek et al., 2021) have significantly advanced the development of protein backbone generation models. To elucidate the inspiration for our work, we provide a detailed account of these networks. AlphaFold2 utilizes the Evoformer module to process multiple sequence alignment (MSA) and structure template in-
+
+formation into sequence and edge representations. Then, the Structure module iteratively refines these latent representations into the final protein structure, beginning from an initial 'black hole' configuration. The iterative nature of the Structure module, where a structure is input and refined through cycles, makes it a suitable backbone generation network for protein backbone diffusion (Anand & Achim, 2022; Yim et al., 2023; Lin & AlQuraishi, 2023). Similarly, the SE(3)-transformer (Fuchs et al., 2020) adopted by RosettaFold shares the iterative refinement capability. In addition to the iterative update mechanism, AlphaFold2's heightened prediction precision is largely attributed to its triangle attention layer. This transformer-like architecture updates the edge representation of residue pairs by integrating the information of the third edge. However, due to its  $\mathrm{O}(N^3)$  computational complexity, most current protein backbone diffusion models adopt a standard message passing layer or Unet (Ronneberger et al., 2015) for updating the edge representations. An exception is RFdiffusion (Watson et al., 2023), which employs axial attention (Ho et al., 2019) on residue pairs, thereby enhancing its representational capacity for the protein backbone denoising task.
+
+# 3. Methods
+
+Algorithm 1 Proteus Model Inference  
+0: function SAMPLE(  $N_{res},N_{step} = 100,N_{layer} = 4,t_{min} = 0.005,$  noise_scale  $= 0.1$    
+）   
+0:  $t = 1.0,\delta t = (1 - t_{min}) / N_{step}$    
+0: for  $i\in [1,\dots ,N_{res}]$  do   
+0:  $\vec{x}_i^{(t)}\sim \mathcal{N}(0,Id_3),r_i^{(t)}\sim \mathcal{N}(0,Id)$    
+0:  $T_{i}^{(t)} = (\vec{x}_{i}^{(t)},r_{i}^{(t)})$ $T_{i}^{prev} = (\vec{0},I)$    
+0: end for   
+0: # denoising iterations:   
+0: for  $n\in [N_{step},\ldots ,1]$  do   
+0:  $s^{(t)},z^{(t)} =$  InputEmbedder(t)   
+0:  $s^{(t)},z^{(t)} + =$  ConditionEmbedder(Tprev)   
+0:  $\hat{T}^{(0)} = T^{(t)}$    
+0: for  $l\in [1,\dots ,N_{layer}]$  do   
+0:  $s^{(t)} =$  IPATransformer(s(t),z(t),  $\hat{T}^{(0)})$    
+0:  $\hat{T}^{(0)} =$  BackboneUpdate(s(t))   
+0:  $z^{(t)} =$  TriangleGraph(s(t),z(t),  $\hat{T}^{(0)})$    
+0: end for   
+0:  $T^{prev} = \hat{T}^{(0)}$    
+0:  $t = t - \delta t$    
+0:  $T^{(t)} = \mathrm{SDE}_{(SE3)}(\hat{T}^{(0)},T^{(t)},t,\delta t,\text{noise\_level})$    
+0: end for   
+0: end function  $= 0$
+
+# 3.1. Model Architecture
+
+Overview In this section, we present the architecture of Proteus. Proteus iteratively updates the structural frames of proteins through a sequence of  $L$  layers of folding blocks. As shown in Figure 2B, each layer of the folding block receives input from three distinct tracks: node representation, edge representation, and structural frames. A folding block is composed of three components: an IPA-Transformer block, a backbone update layer, and a graph triangle block. Each component is tailored to model and refine one of input tracks while being aware of the representations from other tracks.
+
+The IPA-Transformer block integrates an Invariant Point Attention (IPA) mechanism with a traditional transformer. The IPA conducts standard attention operations, incorporating a bias derived from the spatial distance between inter-residue atoms and edge representation. The Backbone Update, inspired by AlphaFold2's methodology, utilizes a linear layer to predict translation and rotation updates for the frames of each residue, informed by the updated sequence representation.
+
+The graph triangle block is tasked to update the edge representation. It employs a graph-based attention mechanism that operates on edge representation, modulated by a sequence representation-gated structural bias. The entire network consists of  $L$  layers of folding blocks, and no weights
+
+are shared among them. The sequence representation is initialized using the diffusion timestep and the edge representation is initialized using a relative sequential distance map introduced in AlphaFold2. When self-conditioning data is available, additional features, including the Ca distance map and relative rotational features from preceding predictions, are incorporated, with more elaborate explanations provided in Table 5. Our primary emphasis is on elucidating the graph triangle block, which is the source of significant enhancements in designability and efficiency.
+
+Graph triangle block The graph triangle block in our model is engineered to update the edge representation in the backbone diffusion process. Drawing inspiration from AlphaFold2's network, we have innovatively adapted the concept of triangle attention and the multiplication method, which are central to AlphaFold2's Evoformer module. Triangle attention is crucial for generating valid protein structures by enforcing the triangle inequality for the pair distance distribution when updating edge representations. This module significantly contributes to AlphaFold2's exceptional accuracy but presents computational challenges due to its  $O(N^3)$  computational complexity and substantial memory demands. Processing a 384-residue protein, a single layer of Evoformer can require upwards of 20GB memory during training, making it necessary to employ techniques of gradient checkpoint, especially considering the network comprises 48 Evoformer blocks. In terms of diffusion modeling, generating a 384 residue protein with 50 steps using AlphaFold2 can take a minimum of 10 minutes on a V100 GPU. Another challenge encountered with the application of the naive triangle attention technique to protein backbone diffusion models is the lack of awareness of the current structure. The Evoformer is tailored to convert Multiple Sequence Alignment (MSA) data into individual and edge representations, yet it lacks provision for structural information as an input to inform these update processes.
+
+To address the aforementioned two major limitations when applying the triangle technique to protein backbone diffusion, we have devised the graph triangle block. This block computes triangle attention for edge representation with an optimized  $O(NK^2)$  complexity. Utilizing the noisy input structure, we identify the k nearest neighbor residues for each residue and subsequently gather the  $N*K$  edges from the comprehensive  $N^2$  edge representation. Attention logits are then calculated among each residue's  $K$  edges. To incorporate 3D spatial information, we calculate the inter-atom distances of the third edge and derive their Radial Basis Function (RBF) features as structural bias, rather than directly utilizing the representation of the third edge. Furthermore, the structure bias is gated by a feedforward network that leverages the sequence representations from both the starting and ending residues, thus ensuring seamless integration of inputs across all three tracks. Before trian
+
+Table 1: Performance comparison of models on the unconditional monomer generation task. Designability is assessed by the self-consistency TM-score between generated and refolded backbones. Efficiency is quantified by number of seconds required to generate a sample, while diversity is measured by the ratio of designable clusters to the total number of generated backbones. Results are averaged across backbone lengths of 100, 200, 300, 400, and 600, as detailed in Figure 1. The top-performing metrics are highlighted in bold, with the second-best results underlined.  
+
+<table><tr><td>Method</td><td>Nparams</td><td>Designability (↑)</td><td>Sampling time (s) (↓)</td><td>Diversity (↑)</td><td>Timesteps (↓)</td></tr><tr><td>Proteus</td><td>19.8M</td><td>0.921</td><td>18.20</td><td>0.235</td><td>100</td></tr><tr><td>RFDiffusion</td><td>59.8M</td><td>0.705</td><td>120.24</td><td>0.328</td><td>50</td></tr><tr><td>GENIE(SwissProt)</td><td>4.1M</td><td>0.349</td><td>188.07</td><td>0.163</td><td>1000</td></tr><tr><td>FrameDiff</td><td>17.4M</td><td>0.405</td><td>40.47</td><td>0.136</td><td>500</td></tr><tr><td>Chroma</td><td>18.5M</td><td>0.174</td><td>18.31</td><td>0.038</td><td>500</td></tr></table>
+
+gle attention is performed for  $N * K$  edge representations, we first apply a triangle multiplicative update, as used in Evoformer, to update the entire  $N * N$  edge representations.
+
+Employing this technique confers three distinct benefits over the axial attention layer used in RFdiffusion and the simpler message-passing layer utilized in FrameDiff and Genie. Firstly, the integration of triangle attention for edge representation updates markedly augments the model's proficiency in modeling the protein backbone diffusion process. This advancement enables the model to achieve superior results with significantly fewer steps, yielding a threefold speedup and significantly outperforms the baseline model. Secondly, this approach significantly curtails memory requirements, allowing for training on much larger proteins without the necessity to crop the input. We can handle proteins up to 1024 residues in length during training on A40 GPU, compared to the 384 residues-limit for AlphaFold2 and RFdiffusion. This capability enhances the model's efficacy in generating larger protein structures. Lastly, by incorporating inputs from all three tracks to update edge representation and protein structure, the protein structure is refined in a holistic manner. The architecture is more compact and best suited for structure-to-structure tasks, such as protein structure generation and predicting the protein's apo-to-holo conformational transition (Hou et al., 2023).
+
+Employing this technique confers three distinct benefits over the axial attention layer used in RFdiffusion and the simpler message-passing layer utilized in FrameDiff and Genie. Firstly, integrating triangle attention for edge representation updates significantly enhances the model's proficiency in modeling the protein backbone diffusion process. This advancement enables the model to achieve superior results with significantly fewer steps, yielding a threefold speedup and markedly outperforming the baseline model. Secondly, this approach substantially reduces memory requirements, allowing for training on much larger proteins without the necessity of cropping the input. We can handle proteins up to 1024 residues in length during training on A40 GPU, compared to the 384-residue limit for AlphaFold2 and RFdiffusion. This capability markedly augments the model's
+
+efficacy in generating larger protein structures. Lastly, by incorporating inputs from all three tracks to update the edge representation and protein structure, the protein structure is refined in a holistic manner. The architecture is more compact and is ideally suited for structure-to-structure tasks such as protein structure generation. It also holds the potential to generalize to the protein's apo-to-holo conformational transitions (Hou et al., 2023).
+
+# 3.2. Training
+
+Dataset We curated a dataset from the Protein Data Bank (PDB) (Berman et al., 2000) with a cutoff date of August 1, 2023. Instead of the conventional practice of training diffusion models solely on monomeric structures, we also included oligomeric structures and extracted their individual chains as training data. To avoid redundancy, we mapped protein sequences to UniProt IDs and selected the highest resolution structure for chains that shared the same UniProt ID and exhibited at least 80 percent sequence overlap. Subsequently, we filtered out the remaining protein chains with length cutoffs between 60 and 512. Additionally, we limited the inclusion of proteins that contained a maximum of 50 percent loop regions as assigned by the DSSP program (Kabsch & Sander, 1983). This curation process yielded a training set of 50,773 single-chain proteins. Our results indicate that augmenting the dataset with additional single-chain data derived from oligomeric structures significantly enhances model performance, compared with training solely on monomers. Supporting evidence of this enhancement is presented in Table 4.
+
+Training losses We have adopted the training loss used in FrameDiff, which can be divided into two main components: denoising translation and rotation score-matching losses, and the auxiliary losses involving the pair-wise distance matrix and coordinate loss on backbone atoms, as depicted in Equation 3. The positions of oxygen atoms are calculated based on the coordinates of other backbone atoms.
+
+$$
+\mathcal {L} = \underbrace {\mathcal {L} _ {\text {t r a n s}} + 0 . 5 \mathcal {L} _ {\text {r o t}}} _ {\text {d s m l o s s}} + \underbrace {0 . 2 5 \mathcal {L} _ {\text {c o o r d}} ^ {t <   0 . 2 5} + 0 . 2 5 \mathcal {L} _ {\text {d m}} ^ {t <   0 . 2 5}} _ {\text {a u x i l i a r y l o s s}} \tag {3}
+$$
+
+Here,  $\mathcal{L}$  trans computes the L2 loss between predicted translations and native translations, while  $\mathcal{L}$  rot calculates the L2 loss on rotation scores weighted by  $\lambda_t^r = 1 / \mathbb{E}[\left|\nabla \log p_{t|0}\left(R_n^{(t)}\mid R^{(0)}\right)\right|\mathrm{SO}(3)]$ , following the approach in (Song et al., 2020).  $\mathcal{L}$  coord represents the coordinate loss between predicted backbones and native backbones. Lastly,  $\mathcal{L}_{\mathrm{dm}}$  computes the pair-wise distance loss between predicted atom positions and native positions. We apply auxiliary losses when  $t < 0.25$ . Empirically, we have observed that including auxiliary losses at a small timestep can aid convergence and enhance the model's performance, which aligns with the findings of FrameDiff.
+
+# 3.3. Sampling
+
+The sampling procedure is detailed in Algorithm 1. We initiate the process by sampling the initial frames for rotation and translation separately. For the initial translation, we employ a Gaussian distribution on  $\mathbb{R}^3$ . As for the rotation component, we first sample the coefficients of orthonormal basis vectors of the Lie algebra  $\mathfrak{so}(3)$  and translate them into the rotation matrix. Once initialized, Proteus takes the input noisy structure and generates a prediction structure at the timestep 0. Then, we iteratively apply the Euler-Maruyama discretization (Bortoli et al., 2022) as the SDE solver for  $N_{\mathrm{steps}}$  to generate the denoised structure.
+
+# 4. Experiments
+
+We rigorously evaluated Proteus performance through both in-silico validation and in vitro experimental approaches. In Section 4.1, we show Proteus' performance in the task of unconditional monomer generation. For a comprehensive assessment, we benchmark Proteus against a suite of leading protein backbone diffusion models, including Chroma (Ingraham et al., 2022), RFdiffusion (Watson et al., 2023), FrameDiff (Yim et al., 2023), and Genie (Lin & AlQuraishi, 2023). Furthermore, we extended our evaluation to the generation of protein complexes, where we compare Proteus's efficacy in generating oligomers of dimer, trimer, and tetramer against Chroma.
+
+Section 4.2 is dedicated to the in vitro experimental validation of Proteus. We synthesized the DNA oligos for 16 designed proteins generated by Proteus, expressed and characterized their biochemical properties, particularly folding and stability. The objective of these experiments is to substantiate the model's practical utility and its effectiveness in real-world biological applications.
+
+# 4.1. in-silico protein generation and evaluation
+
+Monomer backbone generation and evaluation To comprehensively evaluate the performance of a protein backbone diffusion model, it is essential to consider three primary fac
+
+tors: designability, efficiency, and diversity. These aspects are critical to the overall assessment and are detailed in Table 1
+
+Designability aims to measure the quality of the generated backbones, by designing sequences for the generated backbones and refolding them to compute the errors between the generated and folded structures. Designability is the foremost factor, indicating the likelihood of identifying a protein sequence to fold into the designated structure. This is the cornerstone metric for gauging the performance of a diffusion model, as it directly correlates to the model's capacity to generate viable proteins that could conceivably exist in nature.
+
+In our implementation, we use ProteinMPNN (Dauparas et al., 2022) at sampling temperature 0.1 to generate 8 sequences for the designed backbone. Specifically, for Chroma derived backbones, we employ its dedicated inverse folding model, ChromaDesign (Ingraham et al., 2022) at temperature 0.1 and diffusion augmentation 0.5. This substitution is made following the observation that ChromaDesign yields a higher success rate for Chroma's backbones, as documented in its paper. First, the inverse folding model generates multiple sequences corresponding to the sampled backbone, which are subsequently fed into ESMFold (Lin et al., 2023) to fold the structure. The designability of the backbone is represented by the highest TM-score (scTM, evaluating the similarity between two structures, where higher values are better) in all the predicted structures. Additionally, we also compute the self-consistency  $C_{\alpha}$ -RMSD, where the threshold of  $< 2\AA$  is the criterion for successful design. Any generated protein exceeding this  $C_{\alpha}$ -RMSD threshold is deemed non-functional and likely to be unsuccessful in experimental validation.
+
+Efficiency is the next critical factor. It is a key determinant in the success of design models, especially when computational resources are limited. A more efficient model can generate more samples in a limited time which increases the likelihood of generating potent candidates. Given the limitations on computation resources, efficiency becomes a pivotal criterion, on par with designability.
+
+Efficiency is estimated as the time taken to generate a protein backbone on a standard NVIDIA Ampere Tesla A40 GPU with 48 GB of GPU memory. Additionally, we propose a Time-for-Success Design (TSD) metric, which computes the time consumed to generate a designable backbone. This metric underscores the crucial balance between the time required to generate successful designs and the efficiency of the model, making it highly relevant for real-world applications where both speed and effectiveness are critical.
+
+Diversity is the third essential consideration, gauging the structural variance among the generated protein backbones
+
+![](images/8f4769a0cc744ba8b38c9c2f6c66f3bd58ff3389aa0305b0a2e5ecf4b8127955.jpg)  
+Figure 3: Complexes generation benchmark. scRMSD is derived from 200 backbone samples for each type of oligomer, with the length of each chain fixed at 200 amino acid residues.
+
+Diversity is evaluated by calculating the ratio of diverse designable structure clusters to the total number of generated backbones. Specifically, we cluster the generated backbones using MaxCluster (Herbert & Sternberg, 2008) with a TM-score cutoff of 0.6. Notably, we only consider backbones that are designable, defined as those with a  $C_{\alpha}$ -RMSD of less than  $2\AA$ . This criterion is important because sub-optimal models often produce unrealistic structures that cannot be folded by any sequence and are highly non-physical. Including such backbones in the assessment would skew the metric towards models generating random, sub-optimal structures rather than models that genuinely capture the realistic distribution.
+
+Table 2: Comparison of the Time-for-Success Design (TSD) Metric. This metric evaluates the time required to generate a designable backbone (scRMSD  $< 2\AA$ ) on an A40. Asterisks (*) indicate models that failed to predict any designable backbones within 200 samples. The best results are bolded.  
+
+<table><tr><td rowspan="2">Protein length</td><td colspan="6">TSD (s)</td></tr><tr><td>100</td><td>200</td><td>300</td><td>400</td><td>600</td><td>800</td></tr><tr><td>Proteus</td><td>7.02</td><td>9.11</td><td>13.47</td><td>23.09</td><td>50.62</td><td>196.11</td></tr><tr><td>RFDiffusion</td><td>24.84</td><td>43.97</td><td>97.14</td><td>262.64</td><td>1494.28</td><td>30500.00</td></tr><tr><td>GENIE (SwissProt)</td><td>72.66</td><td>127.55</td><td>482.97</td><td>21169.0</td><td>*</td><td>*</td></tr><tr><td>FrameDiff</td><td>23.12</td><td>45.12</td><td>62.11</td><td>216.88</td><td>1851.11</td><td>*</td></tr><tr><td>Chroma</td><td>24.88</td><td>56.66</td><td>352.00</td><td>4020.00</td><td>*</td><td>*</td></tr></table>
+
+Complexes backbone generation and evaluation In Figure 3, we assess the capability of Proteus in generating protein complexes by drawing comparisons with Chroma under a set of monomer design evaluation metrics. We compute scRMSD across different oligomer configurations, including dimers (two chains), trimers (three chains), and tetramers (four chains), with each chain consisting of 200 amino acids. Notably, despite the model initially being trained on single protein chains, we observe its adaptive generation capacity to oligomers. This is achieved by adding a large positional
+
+index as a chain breaker (Watson et al., 2023), a technique elaborated in Appendix A.1. Additionally, Figure 6 offers visual representations of the oligomeric structures produced by this method.
+
+Results Proteus was benchmarked against the leading-edge protein backbone diffusion models, including RFDiffusion, Genie, FrameDiff, and Chroma. We generated 200 backbones across a spectrum of protein lengths, specifically [100, 200, 300, 400, 600], and computed the mean score for every category, as delineated in Table 1. Figure 1B shows the scRMSD distribution for each length category. Proteus matches the performance of RFDiffusion for shorter sequence lengths(100-300 residues) and significantly excels in generating sequences longer than 300 amino acids. We attribute the enhanced capability to the novel architecture of Proteus, which employs graph-level triangle techniques. Furthermore, the enhanced performance of Proteus in oligomeric structure generation, surpassing that of Chroma, provides evidence of its robust out-of-distribution generative capabilities.
+
+Table 3: Proteus sample parameter benchmark  
+
+<table><tr><td>noise level ζ</td><td>0.1</td><td>0.5</td><td>1.0</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Nstep</td><td>100</td><td>100</td><td>100</td><td>50</td><td>200</td><td>500</td></tr><tr><td>&lt; 2Å scRMSD (↓)</td><td>92.0%</td><td>87.8%</td><td>37.3%</td><td>89.8%</td><td>91.0%</td><td>92.2%</td></tr><tr><td>DIVERSITY (↑)</td><td>0.23</td><td>0.27</td><td>0.16</td><td>0.26</td><td>0.22</td><td>0.22</td></tr></table>
+
+Remarkably, while upholding high designability, Proteus also matches Chroma in computational efficiency. Table 3 illustrates that Proteus, through enhanced network representation capabilities, requires merely 100 sampling steps without compromising its designability. This is in contrast to the 1,000 steps necessary for Genie, and 500 steps for FrameDiff and Chroma. Figure 1 showcases the computation time associated with varying protein lengths. Proteus exhibits faster performance compared to Chroma in generating proteins with lengths shorter than 400 amino acids. However, Chroma shows greater time efficiency on proteins exceeding 400 amino acids. Proteus, with superior designability and efficient graph-level processing, distinctively surpasses other models in the realm of success efficiency, as evidenced by the benchmark results in Table 2.
+
+Table 4: Ablation Study  
+
+<table><tr><td rowspan="2">2*Dataset</td><td>Monomer</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Oligomer</td><td></td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td rowspan="2">2*Self-Condition</td><td>Cα distance matrix</td><td>✓</td><td>✓</td><td></td><td></td><td></td></tr><tr><td>Structure featurizer</td><td></td><td></td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td rowspan="3">-3*Model</td><td>Graph triangle attention</td><td></td><td></td><td></td><td>✓</td><td>✓</td></tr><tr><td>Structure bias</td><td></td><td></td><td></td><td></td><td>✓</td></tr><tr><td>Triangle multiplication</td><td></td><td></td><td></td><td>✓</td><td>✓</td></tr><tr><td>1*Results</td><td>Success rate</td><td>36.9%</td><td>59.2%</td><td>63.2%</td><td>87.5%</td><td>92.1%</td></tr></table>
+
+Ablation In our comprehensive ablation analysis, as delineated in Table 4, the enhancements are attributed to
+
+three main aspects: dataset augmentation, self-condition enhancement, and model architecture. Previous protein backbone diffusion methods offered a range of training datasets: FoldingDiff utilized 30,395 protein chains from the CATH database (Sillitoe et al., 2014) with a cropping length of 128. Genie adopted the AlphaFold-Swissprot database (Varadi et al., 2021), containing 195,214 protein chains with a pLDDT cutoff of 80 and a length cutoff of 256. FrameDiff (Yim et al., 2023) compiled a dataset from PDB entries of only monomers, capped at 512 amino acids, resulting in a collection of 20,312 chains. In contrast to the above approaches, we expanded the dataset by adding oligomeric structures, which were subsequently split into single chains, resulting in a dataset of 50,773 chains that significantly bolstered designability. The ablation study further reveals the benefits of incorporating pairwise rotational info as a self-conditioning feature, analogous to the template featurization introduced by AlphaFold2, enabling the model to capture the structure self-consistency from previous denoising steps more accurately.
+
+While these incremental modifications substantially elevated the model's performance, it was the introduction of the local triangle graph technique that allowed Proteus to match the performance of RFDiffusion. This innovative network, which applies an attention mechanism at the graph level with the integration of structure bias, not only preserves a fast sampling speed on par with Chroma but also significantly improves designability. This indicates that structure-level information can enrich edge representation, a novel insight not yet explored by AlphaFold2.
+
+# 4.2. Experimental validation
+
+DNA oligonucleotides encoding 16 designs generated by Proteus were synthesized, and the proteins were recombinantly expressed in Escherichia coli. This set comprises 12 proteins of 300 amino acids and 4 proteins of 500 amino acids. All proteins were well expressed in E. coli. Size exclusion chromatography (SEC) analysis revealed monodisperse peaks for 9 of the 300 amino acid designs and 3 of the 500 amino acid designs, which corresponded to the expected molecular weight. Furthermore, circular dichroism (CD) spectroscopy confirmed the well-folded structure of these designs, and the secondary structure features are consistent with the design models. Notably, these proteins exhibited remarkable thermostability, remaining well-folded at temperatures up to  $95^{\circ}\mathrm{C}$ . Experimental results are comprehensively detailed in Figure 4.
+
+# 5. Related work
+
+Structure diffusion models on proteins Motivated by the significant achievements of diffusion models (Ho et al., 2020; Song et al., 2020; Bortoli et al., 2022) protein dif
+
+fusion models have been developed to generate proteins in either sequence or structural space, with certain methods adeptly bridging both spaces (Anand & Achim, 2022). Anand & Achim pioneered a model that co-diffuses backbone, sequence, and sidechain information utilizing the Structure Module of AlphaFold2. Subsequently, numerous methods have been introduced, focusing on the diffusion of inter-residue geometry (Lee et al., 2023) and backbone dihedral angle (Wu et al., 2022). The leading edge protein diffusion models predominantly engage with diffusion in either SE3 or R3 space in an end-to-end fashion (Yim et al., 2023; Lin & AlQuraishi, 2023). These methods have been further expanded to function motif scaffolding (Trippe et al., 2022; Yim et al., 2024). Chroma achieved a higher efficiency by utilizing an efficient graph neural network. RFdiffusion (Watson et al., 2023) has attained state-of-the-art designability through finetuning the Rosettafold2 structure prediction network (Baek et al., 2023). In contrast to RFdiffusion's
+
+![](images/71ffacbcd2063e983d9dc22c2a1842f8f260b38ce3500cd3eeb0fe40af457b7c.jpg)
+
+![](images/b31063691c09823dd7590e4f3adc3cd42e1f9c73fc5a689f0727c9c6bffd6348.jpg)  
+Figure 4: Experimental characterization of the generated proteins indicates they are well-folded monomers and thermostable. (A) Size exclusion chromatography profiles of the recombinantly expressed proteins on a Supderdex 75 Increase column. (B) Examples of designed proteins that are expressed as monomers. Designs (grey) overlaid with AF2 predictions (colors) are shown on the left, alongside circular dichroism (CD) spectra at  $25^{\circ}\mathrm{C}$ ,  $95^{\circ}\mathrm{C}$  and  $25^{\circ}\mathrm{C} (*)$  after cooling down from  $95^{\circ}\mathrm{C}$  on the right.
+
+approach, our work achieves a comparable designability and much higher efficiency without the necessity of pretraining, making a significant departure from existing methodologies
+
+Other diffusion models on proteins Many other studies have pivoted to the diffusion of protein sequences, or the synergistic integration of sequence data into structure diffusion models. EvoDiff (Alamdari et al., 2023), for example, utilizes evolutionary-scale protein sequence data to generate protein sequences. Although EvoDiff can generate quite diverse protein sequences, the structure prediction metrics indicate limited designability for the generated backbones. Chu et al. innovatively incorporates protein sequence features into the protein backbone diffusion process. The model utilizes a timestep-dependent Protein-MPNN model to design sequences on the noisy structures. However, this appears to diminish the model's performance. This runs counter to the intuitive notion that the integration of sequence data should enhance structural quality. We notice another possible formulation of structure-sequence co-generation by language model inspired by SaProt (Su et al., 2023). For the specific task of antibody diffusion, considerable efforts (Luo et al., 2022; Kong et al., 2023; Peng et al., 2023) have been made to simultaneously generate the sequence and structure of complementarity-determining regions (CDR), highlighting the community interest in antibody design and the potential of sequence-structure co-diffusion.
+
+Protein-ligand complexes prediction In previous works, the prediction of protein-ligand complexes is treated as a regression problem, focusing on the rigid body docking of ligands to holo proteins (Stark et al., 2022; Lu et al., 2022). Capitalizing on the generative power of diffusion models, Diffdock significantly improved the docking accuracy (Corso et al., 2022). Protein-ligand co-folding models (Qiao et al., 2023; Lu et al., 2024) offer more precise predictions when the holo-state protein structures are not known - a formidable and crucial challenge in drug discovery. Furthermore, RFdiffusion All Atom (Krishna et al., 2023) are trained to design novel small molecule binders, expanding the frontier of computational approaches in molecular design.
+
+# 6. Discussion
+
+In this paper, we introduce a new model architecture for protein backbone diffusion, and demonstrate its enhanced designability and efficiency without the necessity of pretraining. Our model advances the field by integrating the triangle attention technique into residue edge representation update and building multi-track interaction networks to enhance its representation capability. Our model shows improved performance in generating longer monomers (with 400 or more amino acids) compared to RFdiffusion. The successful
+
+generation of oligomeric structures further reveals Proteus' generalizability.
+
+Looking forward, we envision several research trajectories where Proteus could exert a transformative influence. Proteins in their natural state often manifest as multi-chain entities, orchestrating their functions in a coordinated fashion. Given Proteus's exemplary performance in crafting larger proteins and multi-chain architectures with both high fidelity and efficiency, it is ideally suited for direct application in the generation of protein oligomeric structures and complex protein machinery. This capability will broaden the current scope of designable protein space, enabling the creation of innovative protein nanomachines.
+
+A further domain of interest is the area of protein-ligand co-folding. Contemporary breakthroughs, exemplified by tools like Rosettafold All Atom (Krishna et al., 2023) and AlphaFold3 (Abramson et al., 2024), have eclipsed the previously established benchmarks set by Diffdock in the realm of modeling protein-small molecule interactions. The integration of Proteus-inspired methodologies into the diffusion dynamics of ligand-protein interplay holds the promise of refining protein-ligand co-folding techniques, paving the way for the development of methods with enhanced accuracy.
+
+# Acknowledgements
+
+The authors thank Pranam D. Chatterjee, Fajie Yuan, Jin Su, Jiaxuan Wang, all reviewers, the area chair for helpful discussions, Minchao Fang for his help with code development, and the Westlake University HPC Center for computing support.
+
+This work was supported by the Science and Technology Innovation 2030 Major Project (2021ZD0150100), the National Key RD Program of China (2021YFC2301401, 2022YFA1303700), the National Natural Science Foundation of China (32370989), and the Westlake Education Foundation.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of protein design. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Additional method details
+
+Here, we describe the implementation details of Proteus as a further explanation of Section 3. To briefly revisit the fundamentals, the  $\ell$ -th layer's node representations are denoted as  $\mathbf{s}_{\ell} = [\mathbf{s}_{\ell}^{1}, \ldots, \mathbf{s}_{\ell}^{N}] \in \mathbb{R}^{N \times D_{s}}$ , edge representations are captured by  $\mathbf{z}_{\ell} = [\mathbf{z}_{\ell}^{1,1}, \ldots, \mathbf{z}_{\ell}^{N,N}] \in \mathbb{R}^{N \times N \times D_{z}}$ . The spatial configuration of each residue in the  $\ell$ -th layer is encapsulated by  $T_{\ell} \in \mathrm{SE}(3)^{N}$ .
+
+# A.1. Proteus model details
+
+Table 5: Input feature of Proteus  
+
+<table><tr><td>Input Feature (shape)</td><td>Description</td></tr><tr><td>aa (N_res, 21)</td><td>The amino acid type of input residues, include all 20 standard types with 1 unknown type</td></tr><tr><td>timestep (N_res, 32)</td><td>The timestep embedding from 0 to 1 by sinusoidal positional encoding.</td></tr><tr><td>rigid_frame (N_res, 7)</td><td>Input noisy structure, 3 dim of translation and 4 dim of rotation represented by a quaternion.</td></tr><tr><td>rel_pos (N_res, N_res, 65)</td><td>Relative position index embedding in [-32,32.]</td></tr><tr><td>torsion_angle Prev(N_res, 14)</td><td>Self-condition input structure&#x27;s residue torsion angle of prev prediction represented by sin and cos formulation, 4 chi angles will be masked out in the network.</td></tr><tr><td>distogram Prev(N_res, N_res, 39)</td><td>Self-condition input structure&#x27;s pair distance histogram of pseudo Cβ, bin_min=3.25Å, bin_max=50.75Å.</td></tr><tr><td>unit_vector Prev(N_res, N_res, 3)</td><td>Self-condition input structure&#x27;s unit vector of each residue&#x27;s Cα coordinate in another residue&#x27;s local frame.</td></tr></table>
+
+Feature initialization Node feature is initialized from timestep and amino acid type (currently, all residues amino acid type is set to alanine for unconditional generation). The edge feature is initialized from the two corresponding node features with additional relative sequence position encoding. The Multi-Layer Perceptrons (MLP) used to embed initial features consists of 3 Linear layers with biases, 2 ReLU activation layers between Linear layers, and a LayerNorm (Ba et al., 2016) at the end. The timestep embedding embedding  $\phi(\cdot)$  follows Ho et al. by using sinusoidal embeddings (Vaswani et al., 2023). The relative sequence position encoding  $relpos(\cdot)$  follows algorithm 4 in supplementary of Alphafold2 (Jumper et al., 2021).
+
+$$
+\mathbf {s} _ {0} ^ {i} = \operatorname {C o n c a t} ([ \phi (t), \text {o n e h o t} (\mathrm {a a}, 2 1) ])
+$$
+
+$$
+\mathbf {z} _ {0} ^ {i, j} = \operatorname {C o n c a t} \left(\left[ s _ {0} ^ {n}, s _ {0} ^ {j}, \operatorname {r e l p o s} (i - j) \right]\right)
+$$
+
+$$
+\mathbf {s} _ {0} ^ {i} = \operatorname {M L P} \left(s _ {0} ^ {i}\right)
+$$
+
+$$
+\mathbf {z} _ {0} ^ {i, j} = \operatorname {M L P} \left(z _ {0} ^ {i, j}\right)
+$$
+
+For the positional encoding, the relative position tokens of residue i and residue j within the same protein chain are always between -32, -31,..., 31, 32. For oligomer generation, we add an extra position index 200 on the residues of the next chain, so the relative position tokens of residue i and residue j of different chains always have a position index -32 or 32, indicating a chain break.
+
+Self-condition featurization Encoding of predicted structure from the previous step has been proved to improve prediction self-consistency and backbone designability (Watson et al., 2023). In  $50\%$  time of training and all inference time, the ConditionEmbedder mentioned in Algorithm 1 is used to encode the prediction of the previous step as node embedding  $\mathbf{t}_s$
+
+and edge embedding  $\mathbf{t}_z$ , and the self condition feature is encoded to  $s_0$  and  $z_0$  after feature initialization.
+
+$$
+\mathbf {t} _ {s} = \operatorname {M L P} (\operatorname {C o n c a t} ([ \text {t o r s i o n} \text {a n g l e} \text {p r e v}, \text {o n e h o t} (\mathrm {a a}, 2 1) ]))
+$$
+
+$$
+\mathbf {t} _ {z} = \operatorname {M L P} (\operatorname {C o n c a t} ([ \text {d i s t o g r a m} _ {-} \text {p r e v}, \text {u n i t} _ {-} \text {v e c t o r} _ {-} \text {p r e v} ]))
+$$
+
+$$
+\mathbf {t} _ {z} + = \text {D r o p o u t R o w w i s e} (\text {T r i a n g l e M u l t i p l i c a t i o n O u t g o i n g} (t _ {z}))
+$$
+
+$$
+\mathbf {t} _ {z} + = \text {D r o p o u t R o w w i s e} (\text {T r i a n g l e M u l t i p l i c a t i o n I n c o m i n g} (t _ {z}))
+$$
+
+$$
+\mathbf {t} _ {z} + = \text {P a i r T r a n s i t i o n s} \left(t _ {z}\right)
+$$
+
+$$
+\mathbf {z} _ {0} + = \text {P o i n t w i s e A t t e n t i o n} \left(t _ {z}, z _ {0}\right)
+$$
+
+$$
+\mathbf {s} _ {0} + = \text {C o l u m n w i s e A t t e n t i o n} \left(t _ {s}, s _ {0}\right)
+$$
+
+We have developed a self-conditioning featurization module, taking inspiration from AlphaFold2's TemplatePairStack (Algorithm 16). The self-conditioned feature is divided into node features  $t_{s}$ , which include predicted backbone dihedral angles and amino acid types, and edge features  $t_{z}$ , which comprise  $C\beta$  pair distance and SE(3)-invariant pairwise directional vectors.
+
+The node feature is encoded by an MLP and incorporated into  $s_0$  via ColumnwiseAttention. This architecture is similar to AlphaFold2's MSAColumnAttention (Algorithm 8) but without gating. The edge feature is similarly encoded by an MLP and then processed through two triangle multiplication layers(AlphaFold2 Algorithm 11) with a dropout rate of 0.25 and a PairTransition layer (AlphaFold2 Algorithm 15). It is ultimately integrated into  $z_0$  through PointwiseAttention(Alphafold2 algorithm 17). Notably, we have reduced the number of blocks to one block and omitted the triangle attention layer present in AlphaFold2's TemplatePairStack to enhance efficiency.
+
+IPA-Transformer block The IPA-Transformer block is used to update node information, and we will describe its details here. An IPA-Transformer block incorporates an Invariant Point Attention (IPA) as presented in AlphaFold2 (Jumper et al., 2021) without any alterations, along with a standard Transformer layer (Vaswani et al., 2023). IPA was first introduced by Anand & Achim as the central model architecture of the protein structure diffusion model. The Transformer is appended to IPA to enhance the representation ability proposed by Framediff (Yim et al., 2023).
+
+The node update equations are as follows:
+
+$$
+s _ {\ell} = \text {L a y e r N o r m} \left(\operatorname {I P A} \left(s _ {\ell}, z _ {\ell}, T _ {\ell}\right) + s _ {\ell}\right)
+$$
+
+$$
+s _ {\text {i n}} = \operatorname {c o n c a t} \left(s _ {\ell}, \operatorname {L i n e a r} \left(s _ {0}\right)\right)
+$$
+
+$$
+s _ {\ell} = \operatorname {L i n e a r} \left(\operatorname {T r a n s f o r m e r} \left(s _ {\mathrm {i n}}\right)\right) + s _ {\ell}
+$$
+
+$$
+s _ {\ell + 1} = \operatorname {M L P} (s _ {\ell})
+$$
+
+Backbone update Our frame updates follow the BackboneUpdate algorithm in AF2(Algorithm 23). We write the algorithm here with our notation,
+
+$$
+b _ {i}, c _ {i}, d _ {i}, \vec {x} _ {i} ^ {\text {u p d a t e}} = \operatorname {L i n e a r} \left(s _ {\ell + 1}\right)
+$$
+
+$$
+(a _ {i}, b _ {i}, c _ {i}, d _ {i}) = (1, b _ {i}, c _ {i}, d _ {i}) / \sqrt {1 + b _ {i} ^ {2} + c _ {i} ^ {2} + d _ {i} ^ {2}}
+$$
+
+$$
+R _ {i} ^ {\mathrm {u p d a t e}} = \left( \begin{array}{c c c} a _ {i} ^ {2} + b _ {i} ^ {2} - c _ {i} ^ {2} - d _ {i} ^ {2} & 2 b _ {i} c _ {i} - 2 a _ {i} d _ {i} & 2 b _ {i} d _ {i} + 2 a _ {i} c _ {i} \\ 2 b _ {i} c _ {i} + 2 a _ {i} d _ {i} & a _ {i} ^ {2} - b _ {i} ^ {2} + c _ {i} ^ {2} - d _ {i} ^ {2} & 2 c _ {i} d _ {i} - 2 a _ {i} b _ {i} \\ 2 b _ {i} d _ {i} - 2 a _ {i} c _ {i} & 2 c _ {i} d _ {i} + 2 a _ {i} b _ {i} & a _ {i} ^ {2} - b _ {i} ^ {2} - c _ {i} ^ {2} + d _ {i} ^ {2} \end{array} \right)
+$$
+
+$$
+\mathbf {T} _ {i} ^ {\text {u p d a t e}} = \left(R _ {i} ^ {\text {u p d a t e}}, \vec {x} _ {i} ^ {\text {u p d a t e}}\right)
+$$
+
+$$
+\mathbf {T} _ {\ell + 1} = \mathbf {T} _ {\ell} \cdot \mathbf {T} _ {i} ^ {\mathrm {u p d a t e}}
+$$
+
+As shown in the equation, an unnormalized quaternion consists of  $b_{i}, c_{i}, d_{i} \in \mathbb{R}^{3}$  and a translation vector  $\vec{x}_{i}^{\mathrm{update}}$  is predicted from  $s_{\ell + 1}$ . We iterative update each residue's frame by applying the translation  $\vec{x}_{i}^{\mathrm{update}}$  and rotation matrix  $R_{i}^{\mathrm{update}}$  to  $\mathbf{T}_{\ell}$ .
+
+Graph triangle block The graph triangle block is designed for the update of edge information. we further explain the implementation details of the graph triangle block mentioned in Section 3.1 and Figure 2. We develop the StructureBiased-GraphTriangleAttention layer(Algorithm 2,Algorithm 3) as the core network for local attention calculation with two triangle multiplication layers(AlphaFold2 Algorithm 11) for global update of edge feature.
+
+$$
+z _ {i j} = \text {T r i a n g l e M u l t i p l i c a t i o n O u t g o i n g} \left(z _ {i j}\right)
+$$
+
+$$
+z _ {i j} = \text {T r i a n g l e M u l t i p l i c a t i o n I n c o m i n g} \left(z _ {i j}\right)
+$$
+
+$$
+z _ {i k} = \text {S t r u c t u r e B i a s e d G r a p h T r i a n g l e A t t e n t i o n S t a r t i n g N o d e} \left(z _ {i j}, s _ {i}, \bar {x} _ {i} ^ {\mathrm {c a}}\right) \tag {2}
+$$
+
+$$
+z _ {i j} \leftarrow \operatorname {S c a t t e r U p d a t e} \left(z _ {i k}\right)
+$$
+
+$$
+\vec {z} _ {i k} = \text {S t r u c t u r e B i a s e d G r a p h T r i a n g l e A t t e n t i o n E n d i n g N o d e} \left(z _ {i j}, s _ {i}, \vec {x} _ {i} ^ {\mathrm {c a}}\right) \tag {3}
+$$
+
+$$
+z _ {i j} \leftarrow \operatorname {S c a t t e r U p d a t e} \left(z _ {i k}\right)
+$$
+
+Algorithm 2 Structure biased graph triangle attention around starting node  
+0: def StructureBiasedGraphTriangleAttentionStartingNode (\{zij\}, \{si\},  $\{\vec{x}_i^{ca}\}$ ,  $c_s = 256$ ,  $c_z = 128$ ,  $c = 128$ ,  $N_{\mathrm{head}} = 4$ ,  $K_{\mathrm{neighbour}} = 32$ ,  $c\_s_{gate} = 16$ ,  $c_{\mathrm{RBF}} = 64$ ):  
+0: # Compute residue's ca distance  
+0:  $D_{ij} = \left\| \vec{x}_i^{ca} - \vec{x}_j^{ca} \right\|^2$   
+0: # Define the neighborhood residues based on  $C_\alpha$  distance  
+0: Neighbors = topK(Dij, Kneighbor)  
+0: # Collate edges for the  $K$  closest residues to residue i  
+0:  $z_{ik} = \text{Collate}(z_{ij}, \text{Neighbors}_i)$   
+0: # Input projections  
+0:  $z_{ik} \gets \text{LayerNorm}(z_{ik})$   
+0:  $q_{ik}^h, k_{ik}^h, v_{ik}^h = \text{LinearNoBias}(z_{ik}) \forall h \in \{1, \dots, N_{\mathrm{head}}\}, q_{ik}^h, k_{ik}^h, v_{ik}^h \in \mathbb{R}^c, k \in \text{Neighbors}_i$   
+0: # Compute structure bias from inter ca distance  
+0:  $b_{ij}^h = RBF(D_{ij}, c_{\mathrm{RBF}})$   
+0:  $b_{ij}^h = \text{LinearNoBias}(b_{ij}) b_{ij}^h \in \mathbb{R}^c$   
+0: # Compute gating of bias from starting and ending node  
+0:  $g_{ij}^h = \sigma(\text{Linear}(s_i \otimes s_j) g_{ij}^h \in \mathbb{R}^c$   
+0:  $b_{ij}^h = g_{ij}^h \circ b_{ij}^h$   
+0: # Compute gating of attention  
+0:  $g_{ik}^h = \text{sigmoid(Linear}(z_{ik}))$   
+0: # Attention  
+0:  $a_{ikq}^h = \text{softmax}_q\left(\frac{1}{\sqrt{c}} q_{ik}^h \top k_{iq}^h + b_{kq}^h\right) k, q \in \text{Neighbors}_i$   
+0:  $o_{ik}^h = g_{ik}^h \odot \sum_q a_{ikq}^h v_{iq}^h$   
+0: # Output projection  
+0:  $z_{ik} = \text{Linear(Concat}(o_{ik}^h)) z_{ik} \in \mathbb{R}^{c_z}$   
+0: return  $\{z_{ik}\}$   
+0: =0
+
+In the graph triangle block, the neighbor  $k$  residues of each residue are selected based on the inter  $C_{\alpha}$  distance of current  $T_{\ell}$ , which means the attention graph is dynamically constructed in the network. By leveraging the characteristics of the diffusion model and dynamic refinement of the graph, triangle graph attention can avoid inaccurate graph construction in the early step, where the input structure is very noisy. At the same time, the efficiency of graph triangle attention calculation is maintained. When calculating the attention between  $k_{ik}$  and  $q_{iq}$ , the structure bias is computed from 64 bins of radial basis function (RBF) equally spaced from  $0\AA$  to  $32\AA$  for distances between Ca for  $k$  and  $q$  residue.
+
+Score prediction and reverse sampling schedule We describe how the density score of rotation and translation of frames is calculated and how the SDE structure denoiser is applied to the sampling from noise mentioned in Figure 2 and Algorithm 1.
+
+At each timestep between 0 and 1, after  $L$  layers we take the final frame  $T_{\ell}$  as the predicted clean structure of  $t = 0$
+
+$$
+T _ {\ell} = \hat {T} ^ {(0)} = (\hat {x} ^ {(0)}, \hat {r} ^ {(0)})
+$$
+
+With the prediction  $\hat{x}^{(0)},\hat{r}^{(0)}$  and the input frames  $T^{(t)} = (x^{(t)},r^{(t)})$  of current timestep,  $\mathbb{R}^3$  and SO(3) score of residue n
+
+is computed by
+
+$$
+s _ {\theta} ^ {x} (t, T ^ {(t)}) _ {n} = \nabla \log p _ {t | 0} (x _ {n} ^ {(t)} | \hat {x} _ {n} ^ {(0)}) = (1 - e ^ {- t}) ^ {- 1} \left(e ^ {- t / 2} \hat {x} _ {n} ^ {(0)} - x _ {n} ^ {(t)}\right)
+$$
+
+$$
+s _ {\theta} ^ {r} (t, T ^ {(t)}) _ {n} = \nabla \log p _ {t _ {0}} \left(r _ {n} ^ {(t)} | \hat {r} _ {n} ^ {(0)}\right) = \frac {r _ {n} ^ {(t)}}{\omega (t)} \log \left(r _ {n} ^ {(0, t)}\right) \frac {\partial \omega f (\omega (t) , t)}{f (\omega (t) , t)}
+$$
+
+To define the formulation of the reverse process, we first set up the forward scheduling of by
+
+$$
+\mathrm {d} X ^ {(t)} = f _ {x} (t) X ^ {(t)} \mathrm {d} t + g _ {x} (t) \mathrm {d} B _ {\mathbb {R} ^ {3}} ^ {(t)}
+$$
+
+This equation describes the forward process of translation, where  $f_{x}(t) = -\frac{1}{2}\beta(s)$  is drift coefficient and  $g_{x}(t) = \sqrt{\beta(t)}$  is diffusion coefficient. We choose linear schedule with introduced by Song et al. and Ho et al.  $\beta_{\mathrm{min}} = 0.1$ ,  $\beta_{\mathrm{max}} = 20$  as the SDE scheduler of translation.
+
+$$
+\beta (s) = \beta_ {\min } + t \left(\beta_ {\max } - \beta_ {\min }\right)
+$$
+
+By defining this linear schedule, with  $G_{x}(s) = \int_{0}^{s}g_{x}(t)^{2}dt = t\cdot \beta_{\min} + \frac{1}{2} t^{2}(\beta_{\max} - \beta_{\min})$ , the distribution of translation at timestep  $t$  can be written as
+
+$$
+p _ {t | 0} \left(X ^ {(t)} \mid X ^ {(0)}\right) = \mathcal {N} \left(X ^ {(t)}; e ^ {- G _ {x} (t)} X ^ {(0)}, 1 - e ^ {- G _ {x} (t)} \mathbf {I d} _ {3}\right).
+$$
+
+The forward diffusion process of rotation can be written similarly but without the drift coefficient term since it is in SO(3) space
+
+$$
+\mathrm {d} R ^ {(t)} = g _ {r} (t) \mathrm {d} B _ {S O (3)} ^ {(t)}
+$$
+
+By introduce a time scaling factor  $\sigma_r^2(s) = \int_0^s g_r(t)^2 dt$  and  $g_r(t) = \sqrt{\frac{d}{dt}}\sigma^2(t)$ , the probability distribution of rotation can be described as
+
+$$
+p _ {t | 0} (R ^ {(t)} | R ^ {(0)}) = \mathrm {I G S O} _ {3} (R ^ {(t)}; R ^ {(0)}, \sigma_ {r} ^ {2} (t) ^ {2})
+$$
+
+where
+
+$$
+\mathrm {I G S O} _ {3} (r ^ {(t)}; r ^ {(0)}, t) = f \left(\omega \left(r ^ {(0) T} r ^ {(t)}\right), t\right)
+$$
+
+$$
+f (\omega , t) = \sum_ {\ell \in \mathbb {N}} (2 \ell + 1) e ^ {- \ell (\ell + 1) t / 2} \frac {\sin ((\ell + 1 / 2) \omega)}{\sin (\omega / 2)}
+$$
+
+$\omega(r)$  is the rotation angle in radians for any  $r \in \mathrm{SO}(3)$ . By defining  $\sigma_r(t) = \log (t \cdot \exp \{\sigma_{\mathrm{max}}\} + (1 - t) \exp \{\sigma_{\mathrm{min}}\})$ , we are able to control the rotation diffusion schedule through  $g_r(t)$  with  $\sigma_{\mathrm{min}}$  and  $\sigma_{\mathrm{min}}$ . We take the same setting  $\sigma_{\mathrm{min}}^2 = 0.01$  and  $\sigma_{\mathrm{max}}^2 = 2.25$  as described in FrameDiff.
+
+With the forward process defined above, the SDE sampling procedure of translation and rotation can be described as
+
+$$
+x ^ {(t)} = \left[ g _ {x} (t) ^ {2} s _ {\theta} ^ {x} (t, T ^ {(t)}) - f _ {x} (t) x ^ {(t)} \right] \mathrm {d} t + g _ {x} (t) \mathrm {B} _ {\mathbb {R} ^ {3}} ^ {(t)}
+$$
+
+$$
+r ^ {(t)} = g _ {r} (t) ^ {2} s _ {\theta} ^ {r} (t, T ^ {(t)}) \mathrm {d} t + g _ {r} (t) \mathrm {B} _ {\mathrm {S O} (3)} ^ {(t)}
+$$
+
+Noise scaling To improve the backbone designability while keeping the diversity, we sample various noise levels in Table 3, noise level is applied to the Brownian motion of  $\mathbb{R}^3$  and SO(3) as
+
+$$
+x ^ {(t)} = [ g _ {x} (t) ^ {2} s _ {\theta} ^ {x} (t, T ^ {(t)}) - f _ {x} (t) x ^ {(t)} ] \mathrm {d} t + \xi (g _ {x} (t) \mathrm {B} _ {\mathbb {R} ^ {3}} ^ {(t)})
+$$
+
+$$
+r ^ {(t)} = g _ {r} (t) ^ {2} s _ {\theta} ^ {r} (t, T ^ {(t)}) \mathrm {d} t + \xi (g _ {r} (t) \mathrm {B} _ {\mathrm {S O} (3)} ^ {(t)})
+$$
+
+where noise level is noted as  $\xi \in [0,1]$  in the Table 3 and equation above.
+
+Algorithm 3 Structure biased graph triangle attention around ending node  
+0: def StructureBiasedGraphTriangleAttentionEndingNode (\{zij\}, \{si\},  $\{\vec{x}_i^{\mathrm{ca}}\}$ ,  $c_{s} = 256, c_{z} = 128, c = 128, N_{\mathrm{head}} = 4$ ,  $K_{\mathrm{neighbour}} = 32, c_{-s_{\mathrm{gate}}} = 16, c_{\mathrm{RBF}} = 64$ ):  
+0: # Compute residue's ca distance  
+0:  $D_{ij} = \left\| \vec{x}_i^{\mathrm{ca}} - \vec{x}_j^{\mathrm{ca}} \right\|^2$   
+0: # Define the neighborhood residues based on  $C_{\alpha}$  distance  
+0: Neighbors = topK(Dij, Kneighbor)  
+0: # Collate edges for the  $K$  closest residues to residue i  
+0:  $z_{ik} = \text{Collate}(z_{ij}, \text{Neighbors})$   
+0: # Input projections  
+0:  $z_{ik} \gets \text{LayerNorm}(z_{ik})$   
+0:  $q_{ik}^{h}, k_{ik}^{h}, v_{ik}^{h} = \text{LinearNoBias}(z_{ik}) \forall h \in \{1, \dots, N_{\mathrm{head}}\}, q_{ik}^{h}, k_{ik}^{h}, v_{ik}^{h} \in \mathbb{R}^{c}, k \in \text{Neighbors}_{i}$   
+0: # Compute structure bias from inter ca distance  
+0:  $b_{ij}^{h} = RBF(D_{ij}, c_{\mathrm{RBF}})$   
+0:  $b_{ij}^{h} = \text{LinearNoBias}(b_{ij}) b_{ij}^{h} \in \mathbb{R}^{c}$   
+0: # Compute gating of bias from starting and ending node  
+0:  $g_{ij}^{h} = \sigma(\text{Linear}(s_i \otimes s_j) g_{ij}^{h} \in \mathbb{R}^{c}$   
+0:  $b_{ij}^{h} = g_{ij}^{h} \circ b_{ij}^{h}$   
+0: # Compute gating of attention  
+0:  $g_{ik}^{h} = \text{sigmoid(Linear}(z_{ik}))$   
+0: # Attention  
+0:  $a_{ikq}^{h} = \text{softmax}_q\left(\frac{1}{\sqrt{c}} q_{ik}^{h} \top k_{qk}^{h} + b_{qi}^{h}\right) k, q \in \text{Neighbors}_{i}$   
+0:  $o_{ik}^{h} = g_{ik}^{h} \odot \sum_q a_{ikq}^{h} v_{qk}^{h}$   
+0: # Output projection  
+0:  $z_{ik} = \text{Linear(Concat}(o_{ik}^{h})) z_{ik} \in \mathbb{R}^{c_z}$   
+0: return  $\{z_{ik}\}$   
+0: =0
+
+Table 6: Model Training Parameters  
+
+<table><tr><td colspan="2">Model Parameters</td></tr><tr><td>Dimension of sequence track Cs</td><td>256</td></tr><tr><td>Dimension of edge track Cz</td><td>128</td></tr><tr><td>K nearest neighbour</td><td>32</td></tr><tr><td>Heads of graph triangle attention</td><td>4</td></tr><tr><td>Hidden dimension of triangle attention head</td><td>128</td></tr><tr><td>Dimension of structure bias feature</td><td>64</td></tr><tr><td>Dimension of sequence track gate</td><td>16</td></tr><tr><td colspan="2">Training Setting</td></tr><tr><td>Max protein length Nres</td><td>512</td></tr><tr><td>Max batch size</td><td>16</td></tr><tr><td>Max squared residues</td><td>3 × 10^4</td></tr><tr><td>Training steps</td><td>2 × 10^6</td></tr><tr><td>Training time</td><td>≈ 20 days</td></tr><tr><td>Device</td><td>2× A40</td></tr><tr><td>Optimizer</td><td>Adam(β1=0.9, β2=0.999) (Kingma &amp; Ba, 2017)</td></tr><tr><td>Learning rate</td><td>0.0001</td></tr></table>
+
+# B. In vitro experiment protocol
+
+Protein purification and expression Synthetic genes encoding designed protein were purchased from Universe Gene Technology. These sequences were cloned into the pET28a vector, included N-terminal or C-terminal histidine tags and an HRV 3C protease cleavage site. These plasmids were transformed into BL21 (DE3) E. coli competent cells. All transformants were cultured into  $50~\mathrm{ml}$  of LB medium with  $50~\mathrm{mg / ml}$  kanamycin. Protein expression was induced with 1 mM isopropyl 1-thio--d-galactopyranoside at  $37^{\circ}\mathrm{C}$  overnight or at  $20^{\circ}\mathrm{C}$  overnight after initial growth for 6 to  $8\mathrm{h}$  at  $37^{\circ}\mathrm{C}$ . The cells were harvested by centrifugation and lysed by sonication after resuspension of the cells in lysis buffer (25 mM Tris pH 7.0, 150 mM NaCl). The cell lysate was cleared by centrifugation  $(12,000\times \mathrm{rpm})$ . The supernatant was purified by 1 ml Ni2+ immobilized metal affinity chromatography with Ni-NTA Superflow resin (Qiagen). Resins with bound cell lysate were washed five times with  $5\mathrm{mL}$  of washing buffer (comprising lysis buffer and  $30~\mathrm{mM}$  imidazole) and eluted with  $6\mathrm{mL}$  of elution buffer (comprising lysis buffer and  $300~\mathrm{mM}$  imidazole). Both eluates were analyzed using  $15\%$  SDS-PAGE gel to assess purity. The histidine tags were cleaved using histidine-tagged HRV 3C protease during dialysis against lysis buffer overnight. A second IMAC purification was performed for HRV 3C cleaved samples to capture uncleaved protein and HRV 3C protease. Designs were finally purified using Superdex 200 Increase 10/300GL (GE Healthcare) with lysis buffer.
+
+Circular dichroism experiments Circular dichroism spectra were recorded on a Chirascan V100 circular dichroism spectrometer (Applied Photophysics) using protein concentrations ranging from 0.6 to  $0.9\mathrm{mg / ml}$ . Thermal melt analyses were conducted over a temperature range of  $25^{\circ}\mathrm{C}$  to  $95^{\circ}\mathrm{C}$ , measuring CD at  $222\mathrm{nm}$ . Wavelength scans (190 to  $260\mathrm{nm}$ ) were recorded at both  $25^{\circ}\mathrm{C}$  and  $95^{\circ}\mathrm{C}$ . All reported measurements were obtained within the linear range of the instrument.
+
+For crystallization, the plasmids were transformed into BL21 (DE3) E. coli competent cells. The transformants were cultured in  $10\mathrm{ml}$  of LB medium with  $50~\mathrm{mg / ml}$  kanamycin at  $37^{\circ}C$  overnight. The cultures were transferred to 1L of LB medium with  $50~\mathrm{mg / ml}$  kanamycin and incubated at  $37^{\circ}C.$  Protein expression was induced with  $1\mathrm{mM}$  IPTG at  $37^{\circ}C$  overnight. Protein purification steps were carried out as described above.
+
+Crystallization, data collection, and structure determination The crystals were grown using the hanging drop method at room temperature  $(18^{\circ}\mathrm{C})$ . The drops consisted of  $1\mathrm{L}$  of  $40\mathrm{mg / ml}$  protein and  $1\mathrm{L}$  of precipitant solution  $(100\mathrm{mM}$  Tris pH 8.5,  $200\mathrm{mMNaCl}$  and 30For diffraction, the crystals were transferred into a solution containing  $20\%$  glycerol as a cryoprotectant. Subsequently, the crystals were loaded onto the X-ray diffractometer (Rigaku, XtaLAB Synergy Customer). The diffraction data was collected at  $100\mathrm{K}$  and processed with the reduction program CrysAlisPro. The structures were solved by molecular replacement using Phaser in PHENIX8. The structures were manually refined with Coot9 and PHENIX10.
+
+![](images/bae01ec06ee27da81b7bcc5d7917d7aec68d75348f077056f81e1f3279d85e0a.jpg)  
+Figure 5: The alignment of experimentally solved crystal structure and diffusion model of 300aa_3, crystal backbone(grey) is overlaid with diffusion backbonecolors) with a global  $C_\alpha$  -RMSD 0.91 Å. The setting of data collection and structure determination is provided in Table 7
+
+Table 7: Data collection and refinement statistics.  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Name</td><td>300aa_3</td></tr><tr><td>Wavelength</td><td>1.541</td></tr><tr><td>Resolution range</td><td>29.13 - 1.8 (1.864 - 1.8)</td></tr><tr><td>Space group</td><td>P 1 21 1</td></tr><tr><td>Unit cell</td><td>46.4585 76.1478 83.3196 90 91.9037 90</td></tr><tr><td>Total reflections</td><td>247022 (15772)</td></tr><tr><td>Unique reflections</td><td>53811 (4931)</td></tr><tr><td>Multiplicity</td><td>4.6 (2.9)</td></tr><tr><td>Completeness (%)</td><td>97.87 (91.26)</td></tr><tr><td>Mean I/sigma(I)</td><td>24.16 (3.29)</td></tr><tr><td>Wilson B-factor</td><td>16.89</td></tr><tr><td>R-merge</td><td>0.06467 (0.49)</td></tr><tr><td>R-meas</td><td>0.07301 (0.5975)</td></tr><tr><td>R-pim</td><td>0.03309 (0.338)</td></tr><tr><td>CC1/2</td><td>0.975 (0.488)</td></tr><tr><td>CC*</td><td>0.994 (0.81)</td></tr><tr><td>Reflections used in refinement</td><td>52709 (4928)</td></tr><tr><td>Reflections used for R-free</td><td>1368 (129)</td></tr><tr><td>R-work</td><td>0.1827 (0.2144)</td></tr><tr><td>R-free</td><td>0.2163 (0.2486)</td></tr><tr><td>CC(work)</td><td>0.961 (0.892)</td></tr><tr><td>CC(free)</td><td>0.936 (0.880)</td></tr><tr><td>Number of non-hydrogen atoms</td><td>5174</td></tr><tr><td></td><td>macromolecules: 4418</td></tr><tr><td></td><td>ligands: 0</td></tr><tr><td></td><td>solvent: 756</td></tr><tr><td>Protein residues</td><td>600</td></tr><tr><td>RMS(bonds)</td><td>0.007</td></tr><tr><td>RMS(angles)</td><td>0.90</td></tr><tr><td>Ramachandran favored (%)</td><td>99.33</td></tr><tr><td>Ramachandran allowed (%)</td><td>0.50</td></tr><tr><td>Ramachandran outliers (%)</td><td>0.17</td></tr><tr><td>Rotamer outliers (%)</td><td>0.64</td></tr><tr><td>Clashscore</td><td>7.73</td></tr><tr><td>Average B-factor</td><td>23.04</td></tr><tr><td></td><td>macromolecules: 21.76</td></tr><tr><td></td><td>solvent: 30.55</td></tr></table>
+
+![](images/a677569d9abede009a7a22c6e500238d04fa1a87d4e7faf96c39d6b49d8217a8.jpg)  
+Oligomer Type  
+Figure 6: Visualization of oligomer samples across dimer, trimer, and tetramer. Each chain is fixed at 200 residues with different colors, as shown in the figure.

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2024/Prototypical Transformer As Unified Motion Learners/full.md

@@ -0,0 +1,632 @@
+# Prototypical Transformer as Unified Motion Learners
+
+Cheng Han  $^{*12}$  Yawen Lu  $^{*3}$  Guohao Sun  $^{2}$  James C. Liang  $^{2}$  Zhiwen Cao  $^{3}$  Qifan Wang  $^{4}$  Qiang Guan  $^{5}$  Sohail A. Dianat  $^{2}$  Raghuveer M. Rao  $^{6}$  Tong Geng  $^{7}$  Zhiqiang Tao  $^{2}$  Dongfang Liu  $^{2}$
+
+# Abstract
+
+In this work, we introduce the Prototypical Transformer (ProtoFormer), a general and unified framework that approaches various motion tasks from a prototype perspective. ProtoFormer seamlessly integrates prototype learning with Transformer by thoughtfully considering motion dynamics, introducing two innovative designs. First, Cross-Attention Prototyping discovers prototypes based on signature motion patterns, providing transparency in understanding motion scenes. Second, Latent Synchronization guides feature representation learning via prototypes, effectively mitigating the problem of motion uncertainty. Empirical results demonstrate that our approach achieves competitive performance on popular motion tasks such as optical flow and scene depth. Furthermore, it exhibits generality across various downstream tasks, including object tracking and video stabilization. Our code is available here.
+
+# 1. Introduction
+
+"All is flux, nothing is stationary."
+
+-Heraclitus (Plato, 402 BC)
+
+The aphorism attributed to Heraclitus underscores the foundation of physics in the natural world. The quest to understand motion holds the potential to unveil the intricate secrets of intelligence (Hawkins & Blakeslee, 2004), shedding light on the systematic construction of artificial entities. However, the proliferation of excessively granular motion tasks has catalyzed a fervor for specialized models in the realm of deep learning, standing in stark contrast to the enduring scientific tradition — a generic solution to elegantly
+
+*Equal contribution 1University of Missouri - Kansas City 2Rochester Institute of Technology 3Purdue University 4META AI 5Kent State University 6DEVCOM Army Research Laboratory 7University of Rochester. Correspondence to: Zhiqiang Tao <zxtics@rit.edu>, Dongfang Liu <dongfang.liu@rit.edu>.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+![](images/a1c6aebe7eba344becee75a7d7a7f73625c3df6988f4d7892c950c2d2df4fede.jpg)  
+Figure 1: ProtoFormer as a unified framework considers motion as different levels of dynamics granularity (e.g., instance-driven flow, pixel-anchored depth, etc). are prototypes.
+
+![](images/77497eeccac8409f7105a02ed0baba43214651eb6cb41cadcd53b4be542e28e5.jpg)
+
+describe physical phenomena in the universe. The following question naturally arises: ① Can we discover a unified model that serves as a comprehensive motion learner?
+
+Motion learning tasks essentially encompass pixel-level dynamics and correspondence (e.g., optical flow and depth scene estimation). A prevalent challenge in these tasks is the presence of photometric and geometric inconsistencies (e.g., shadow and occlusion), which introduce significant uncertainty during the matching process (Xiong et al., 2021; Zhao et al., 2020). Consequently, the accuracy of pixel-wise feature matching is compromised, detrimentally impacting the learning of the underlying motion representation. To address this challenge, a promising solution lies in prototype learning (Smith & Minda, 2002; Jiang et al., 2012), where motion measurements are categorized into discrete exemplars. In each exemplar, a prototype function as a central archetype, capturing the essential attributes of its associated motion patterns observed in the data. The clustering of similar patterns around prototypes can effectively minimize the impact of noise and outlier pixels in feature matching, thereby significantly mitigating the issue of uncertainty. In this context, we can approach question ① by exploring: ② How can we design a model that incorporates the principles of prototype learning in motion tasks?
+
+Recently, the Transformer architecture has attained ubiquitous adoption, enjoying unambiguous acclaim in both the domains of vision and language (Zhu et al., 2021; Yang et al., 2023; Kim et al., 2021). Its accomplishments are underpinned by the attention mechanism, endowing models with the capability to selectively attend to salient entities within input data. The capacity to generate context-aware feature representations represents a substantial enhancement of the model's effectiveness, enabling it to apply as a general solution to diverse vision tasks. Inspired by its encouraging
+
+success, our inquiry naturally delves into a more specific dimension: ③ How to incorporate the prototype learning capacity into the architecture of Transformer?
+
+To this demand, we employ Prototypical TransFormer (ProtoFormer) as a unified solution on various motion tasks. Specifically, ProtoFormer incorporates prototype learning with Transformer. The method begins by tokenizing images features into patches, where the features are initialized into distinct prototypes. These prototypes are then recursively updated via Cross-Attention Prototyping (§3.2.1) to capture representative motion characteristics via clustering. After assignments and updates, Latent Synchronization (§3.2.2) builds up prototype-feature association, aiming at denoising and mitigating motion ambiguity. The refined features are later fed into the decoder for task-specific predictions.
+
+Taking the innovations together, ProtoFormer exhibits several compelling attributes. ① Architectural elegance: Leveraging a prototype-guided Transformer architecture, ProtoFormer can handle heterogeneous motion tasks at different levels of dynamics granularity within the unified fashion (see Fig. 1). ② Predictive robustness: Prototype learning can inherently diminish the noisy outliers through its density criterion (§2). Anchored by the recursively refined prototypes, feature learning can be further guided towards more robust representations (§3.2.2). Consequently, it offers a viable solution to the challenge of motion ambiguity (see Fig. 3 and Fig. 4). ③ Systemic explainability: The density-based nature from recursive prototyping offers intuitive visual demonstration of motion prototypes (see Fig. 5 and Fig. S1 in Appendix), enabling direct interpretation of various dynamic patterns sketched by the system.
+
+We conduct a set of comprehensive experiments to evaluate the effectiveness of our approach. In §4.1, ProtoFormer presents compelling results on optical flow. For example, our approach distinctly outperforms CRAFT, achieving 0.48 and 0.69 on the clean and final pass of Sintel, respectively. In §4.2, we further show the superior performance on depth scene estimation (e.g.,  $18.6\%$  improvement in Sintel compared to AdaBins). Also, visual evidence in §4.3 demonstrates the systemic explainability, which displays direct prototype-pixel correlations. Results on various downstream tasks including object tracking (§S5) and video stabilization (§S6) are detailed in the Appendix. We hope our research could provide foundational insights into related fields.
+
+# 2. Related Work
+
+Motion Task. Motion tasks involve intricate processes, encompassing the identification, modeling, and prediction of motion patterns in objects and scenes. These tasks are foundational to diverse computer vision applications, including vehicle and pedestrian motion detection (Shen et al., 2023; Khalifa et al., 2020; Marathe et al., 2021; Liang et al.,
+
+2022b; Xu et al., 2022a; Cui et al., 2024), abnormal activity detection (Li et al., 2021b; Tudor Ionescu et al., 2017; Zhou et al., 2019), and video compression (Gao et al., 2022; Hu et al., 2021; Lu et al., 2019). In the domain of motion tasks, optical flow (Ranjan & Black, 2017; Sun et al., 2018; Teed & Deng, 2020; Huang et al., 2022; Shi et al., 2023; Lu et al., 2024) and depth estimation (Bhat et al., 2021; Patil et al., 2022), stand out as particularly representative, significantly influencing downstream tasks like object tracking and video stabilization. Current endeavors predominantly focus on task-specific solutions, resulting in duplicated research efforts and suboptimal hardware utilization. In contrast, ProtoFormer stands as a distinctive exploration, aiming to integrate motion tasks under a unified paradigm. This endeavor conceptually differentiates us from existing arts in the field.
+
+Prototype Learning. Traditionally, prototype learning in machine learning establishes a metric space where features are distinguished by computing their distances/densities to prototypical representations (Lee et al., 2023). Early methods include classical approaches like support vector machines (Cortes & Vapnik, 1995), random forest (Breiman, 2001), logistic regression (Hastie et al., 2009), etc. With the advent of deep neural networks (DNNs), prototype-based deep learning models find broad applications in few-shot learning (Dong & Xing, 2018; Wang et al., 2019; Liu et al., 2020b; Yang et al., 2020; Li et al., 2021a; Wang et al., 2022), zero-shot learning (Jetley et al., 2015; Xu et al., 2020), text classification (Farhangi et al., 2022), and explainable classifiers (Wang et al., 2023; Zhou et al., 2022; Qin et al., 2023). In the context, we argue: movements within the same object or proximate regions exhibit noteworthy similarities, forming a collective of prototypes. Integrating prototype learning into the model design facilitates the natural encapsulation of diverse dynamic characteristics, enhancing the model's ability to comprehend motion in various contexts.
+
+Moreover, human vision deploys a sophisticated prototyping ability, skillfully focusing on relevant parts of the visual tableau while filtering out extraneous elements (Simon & Newell, 1971; Rudin et al., 2022; Giese & Poggio, 2003). This feat is realized through Region-of-Interest (RoI) clustering (Meyer et al., 2004; Mantini et al., 2012), disassembling discrete pixel entities into salient conceptual groupings. This hierarchical process synthesizes elementary visual elements, like lines, forms, and hues, into complex abstractions representing objects, vistas, and individuals (Kepes, 1995). In an effort to mimic the human visual system, we conceptualize prototype learning from a clustering perspective, iteratively updating and exploring representative prototypes to capture nuanced motion characteristics.
+
+Transformer Architecture. The transformative impact of Transformers in natural language processing (NLP) (Brown
+
+![](images/f70debccbb0198424176b6ada3757574c1c5def7365f6172ae7087f9be53b73d.jpg)  
+Figure 2: (a) Overall pipeline of ProtoFormer (§3.2). Movement of a small part of an object within an image is being considered as a rigid motion. In our approach, we use prototypes to understand or predict this kind of motion pattern. (b) In each layer of the Cross-Attention Prototyping (see §3.2.1), there are  $N$  sequential iterations encompassing the assignment of feature-prototypes (i.e.,  $E$ -step) and the subsequent updating of these prototypes (i.e.,  $M$ -step) via Eq. 5. (c) Concurrently, the Latent Synchronization process (see §3.2.2) associates the feature representations via the freshly updated motion prototypes, (see Eq. 7). For (b) and (c), we apply optical flow for illustration, which demonstrates straightforward systemic explainability. More visualization results are shown in §4.3.
+
+![](images/9ff73ccd75bc4ec9ffe4b82d0263a08548b3b1a7a98e4580818a3e9f055434ad.jpg)
+
+![](images/346ddecdb32cf55fa238d555da181f25c2e31da2642edf00ed8401d17d889247.jpg)
+
+et al., 2020; Devlin et al., 2018; Liu et al., 2020a; Raffel et al., 2020; Vaswani et al., 2017) has spurred their extensive application in vision-related tasks, including image classification (Dosovitskiy et al., 2021; Liu et al., 2022; 2021; Wang et al., 2023), and image segmentation (Strudel et al., 2021; Wang et al., 2021a;d; Zheng et al., 2021). Transformers excel in visual applications, outperforming convolutional neural networks (CNNs) (Han et al., 2023a; 2024). This superiority arises from their ability to capture extensive token dependencies in a global context, a limitation of concurrent CNN-based methods that focus on local interactions within convolutional layers (Han et al., 2022; Khan et al., 2022; Lin et al., 2022; Han et al., 2023a; Cai et al., 2022). The unique attention design in Transformers enables the understanding of global spatial relationships, making them ideal for motion-related tasks where extensive spatial interconnections are pivotal. By amalgamating the attention mechanism with prototype learning, we aim to harness Transformers' representational prowess to unravel intricate patterns in motion tasks, offering a unified, Transformer-based solution.
+
+# 3. Methodology
+
+By going through existing literature ( $\S 2$ ), the integration of prototype learning with the Transformer architecture presents a promising avenue to various motion tasks. In this section, we first revisit Transformer architecture and reformulate its attention mechanism as prototype learning ( $\S 3.1$ ). Based on this insight, we introduce ProtoFormer ( $\S 3.2$ ), including two pivotal contributions: Cross-Attention Prototyping ( $\S 3.2.1$ ) and Latent Synchronization ( $\S 3.2.2$ ), answering question  $③$ . We elaborate our method below.
+
+# 3.1. Preliminary
+
+In our study, we re-conceptualize the Transformer's attention mechanism through the lens of classical clustering; while the traditional attention map is obtained by computing the similarity between all query-key pairs (Zhou et al., 2021a; Han et al., 2023b), our approach introduces a density-based cross-attention estimation, specifically designed to accommodate motion characteristics by aggregating local rigid motion patterns into distinct prototype clusters.
+
+Classic clustering, a widely embraced paradigm, entails segregating  $m$  observations into  $k$  distinct groups. It ensures that each observation is aligned with only one cluster that it most closely associates with, based on the highest likelihood or minimal distance (e.g., proximity to the mean). Formally, clustering can be optimized iteratively between two phases:
+
+- Assignment Phase allocates each observation to the cluster where it exhibits the maximal probability of belonging or the least spatial separation.  
+- Centroid Recalculation Phase recalculates the centroids of the clusters to reflect the new configuration of the observations within each cluster.
+
+These two phases persist until a point of convergence is reached, indicated either by a cessation in the modification of assignments or by changes that fall beneath a predetermined threshold, thus implying cluster stabilization.
+
+From a mathematical perspective, assume  $\theta$  as the centroids (Wang et al., 2023) of the clusters, with  $x\in \mathcal{X}$  representing an individual observation:
+
+$$
+\theta^ {(n + 1)} = \underset {\theta} {\arg \max } \mathbb {E} \left(p _ {x \sim \chi} \left(x \mid \theta^ {(n)}\right)\right). \tag {1}
+$$
+
+Here,  $\theta^{(n)}$  denotes the  $n$ -th iteration deduced centroid;  $p(\cdot)$  represents the posterior probability of the data assignments.
+
+# 3.2. ProtoFormer
+
+The primary objective for ProtoFormer is to optimize the expected likelihood function in the context of clustering within a unified motion solution as:
+
+$$
+\hat {\theta} _ {k} = \sum_ {j = 1} ^ {K} p \left(\mathcal {X} \mid \theta_ {j}\right) \cdot P \left(\theta_ {k}, \theta_ {j}\right), \tag {2}
+$$
+
+where  $\theta$  symbolizes the centroid representations. We recognize the centroid  $\theta$  as Prototype, which is our optimization target.  $K$  represents the total cluster count. The probability  $p(\mathcal{X}|\theta_k)\in (0,1)$  are the mixing coefficients for each cluster  $k\in \mathcal{K}$ , adhering to the constraint  $\sum_{k}p(x|\theta_{k}) = 1$ . The projected prototype representation  $P(\cdot)$  describes the learnable dense vector from a shared parametric family from  $k$ -th prototype. This function aggregates across all clusters  $\mathcal{K}$ , with each projected prototype  $P(\theta_k,\theta_j)$  denoting the new projected representations on its respective cluster.  $\hat{\theta}_k$  demonstrates the updated prototype considering the prototype representation of  $\theta$ , and the posterior probability  $p(\mathcal{X}|\theta_k)$  (i.e., the conditional likelihood of grouping the data  $\mathcal{X}$  given prototype parameterization  $\theta_k$ ).
+
+# 3.2.1. CROSS-ATTENTION PROTOTYPING VIA EM CLUSTERING
+
+To realize prototyping as a unified motion solution on Transformer, we reformulate the conventional Transformer's self-attention (Vaswani et al., 2017) into our novel prototypical cross-attention, optimized via Expectation-Maximization (EM) clustering. The designed optimization provides a density-based estimation to compute the maximum likelihood for  $p_k$  and  $\theta_k$ , utilizing posterior probabilities.
+
+$E$ -Step: For each observation  $x_{i} \in \mathcal{X}$ ,  $E$ -Step computes the  $n$ -th iteration posterior probabilities  $p_{k}(x_{i})$ , indicating its affiliation to center  $\theta_{k}$  with the logit vector  $s_{x_i,k}$  iteratively as:
+
+$$
+p _ {k} ^ {(n)} \left(x _ {i}\right) = \frac {s _ {x _ {i} , k} ^ {(n)} \cdot P \left(x _ {i} , \theta_ {k} ^ {(n)}\right)}{\sum_ {j = 1} ^ {K} s _ {x _ {i} , j} ^ {(n)} \cdot P \left(x _ {i} , \theta_ {j} ^ {(n)}\right)}. \tag {3}
+$$
+
+$s_{x_i,k}^{(n)},\theta_k^{(n)}$  are the parameters estimated at the  $n$  -th iteration.
+
+M-Step: Each cluster  $\theta_{k}$  obtains its maximum likelihood estimations  $p_k^{(n)}$  and  $\theta_{k}^{(n)}$  from projected sub-sample representations  $P^{\prime}$ , updated as:
+
+$$
+\theta_ {k} ^ {(n + 1)} = \frac {1}{N} \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {K} p _ {k} ^ {(n)} \left(x _ {i}\right) \cdot P ^ {\prime} \left(\theta_ {k} ^ {(n)}, \theta_ {j} ^ {(n)}\right). \tag {4}
+$$
+
+In practice, given feature embeddings  $\pmb{I} \in \mathbb{R}^{HW \times D}$  and set initial cluster centers  $\pmb{P}^{(0)}$  as  $K$  prototypes, we encapsu
+
+late the discussed  $EM$  clustering process within a Cross-Attention Prototyping layer (see Fig. 2(b)) with  $N$  iterations:
+
+$$
+E \text {- s t e p :} \quad \hat {\boldsymbol {M}} ^ {(n)} = \operatorname {s o f t m a x} _ {K} \left(\boldsymbol {Q} ^ {P ^ {(n)}} \left(\boldsymbol {K} ^ {I}\right) ^ {\top}\right), \tag {5}
+$$
+
+M-step:  $\pmb{P}^{(n + 1)} = \hat{\pmb{M}}^{(n)}\pmb{V}^I\in \mathbb{R}^{K\times D}$
+
+where  $n\in \{1,\dots ,N\}$  .  $\hat{M}\in [0,1]^{K\times HW}$  is the "soft" pixel-prototype assignment matrix, representing probability maps of prototypes.  $Q^{P}\in \mathbb{R}^{K\times D}$  is the query vector projected from the prototype representation  $P$  and  $\pmb {V}^I,K^I\in \mathbb{R}^{HW\times D}$  are the value and key vectors projected from the image features  $\pmb{I}$  , respectively. Our proposed layer can thus update the prototyping membership  $\hat{M}$  (i.e.,  $E$  step) and the prototypes  $P$  (i.e.,  $M$  -step) iteratively. The key characteristic of this approach is its assurance of an incremental convergence in the likelihood function with each iteration (see Eq. 4). In essence, the  $E$  -step evaluates the current membership of the data representations based on existing prototypes, while the  $M$  -step refines the prototypes to align with pixels, ensuring a steady progression towards optimal clustering. By performing cross-attention prototyping on the source and target images separately, it addresses the complexities associated with motion uncertainty and photometric inconsistency. We also modify the default softmax operator from  $HW$  to  $K$  , mimicking the  $EM$  clustering.
+
+The proposed layer enjoys several compelling features:
+
+- Convergence: EM clustering monotonically improves the marginal likelihood and is empirically validated to converge towards a local optimum (Vattani, 2009; Ikatun et al., 2023; Balakrishnan et al., 2017), given a sufficient number of iterations (Proof is provided in Appendix §S4).
+
+Proposition 1. Suppose that the EM operator is contractive with parameter  $\kappa \in (0,1)$  on the ball  $\mathcal{B}_2(r;\theta)$ , and the initial vector  $\theta^{(0)}$  belongs to  $\mathcal{B}_2(r;\theta)$ . For a given iteration  $N$ , when the sample size  $m$  is large enough to ensure  $\epsilon_{M}\left(\frac{m}{N},\frac{\delta}{N}\right)\leq (1 - \kappa)r$ . Then the EM iterates  $\left\{\theta^{(n)}\right\}_{n = 0}^{N}$  based on  $\frac{m}{N}$  samples per round satisfy the bound that:
+
+$$
+\left| \left| \theta^ {(n)} - \hat {\theta} \right| \right| _ {2} \leq \kappa^ {n} \left| \left| \theta^ {(0)} - \hat {\theta} \right| \right| _ {2} + \frac {1}{1 - \kappa} \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right). \tag {6}
+$$
+
+In this context, our proposed Cross-Attention Prototyping leverages the strengths of recursive clustering, iterated over  $N$  steps. This approach significantly enhances the likelihood of converging to an optimal configuration for motion partitioning (see §4.3).
+
+- Transparency: Prototyping emerges as an indispensable mechanism for contextual understanding from motion scenes, recognizing and grouping similar patterns and movements. It clusters pixels as prototypes that display homogeneity in characteristics such as flow or depth. By aggregating entities that exhibit shared attributes, the prototypes are able to describe the intrinsic dynamics. Furthermore, prototyping provides a foundational schema for
+
+motion comprehension, wherein each prototype embodies a microcosm of the objects within the scene, encapsulating unique elements and their interrelations.
+
+- Efficiency: Cross-Attention Prototyping operates with a time complexity of  $\mathcal{O}(NKHWD)$ , showing a significant improvement over the self-attention mechanism with  $\mathcal{O}(H^2 W^2 D)$  (see §4.3). The foundation lies in the relationship that  $NK \ll HW$  (e.g., 60 vs 25920 in the first stage with image of  $960 \times 432$  resolution). This difference becomes especially pronounced in pyramid architectures (Wang et al., 2021c; Liu et al., 2021; Liang et al., 2023; 2024), where the total number of  $NK$  tends to be substantially smaller than  $HW$ , particularly in the early stages of the network. In each iteration, only the query matrix  $Q$  requires an update; the key  $K$  and value  $V$  matrices are computed just once. This selective updating significantly reduces the computational load particularly beneficial in handling large-scale data in high-dimensional feature spaces.
+
+# 3.2.2.PROTOTYPE-Feature CORRESPONDING BY LATENT SYNCHRONIZATION
+
+We further refine the feature representations, synchronizing the projection of  $K$  prototypes into a  $H \times W$  feature representation. This approach aligns the prototype representations with motion features (see Fig. 2(c)).
+
+The main technique lies in the Feed-Forward Network (FFN) that incorporates with a masked cross-attention mechanism:
+
+$$
+\hat {I} = \operatorname {F F N} \left(\text {C r o s s - A t t e n t i o n} \left(Q ^ {I}, K ^ {P}, V ^ {P}, \mathcal {M} _ {P}\right)\right), \tag {7}
+$$
+
+where  $\mathcal{M}_P$  stands for the feature assignment mask maps based on the similarity of corresponding prototypes  $P$ .  $\hat{I}$  represents the refined feature,  $Q^I$  denotes the query projection derived from the input image feature,  $K^P$  and  $V^P$  represent the key and value projections sourced from the learning prototypes, respectively. Latent Synchronization primarily aims at augmenting the feature learning for prototype-feature association, reducing motion ambiguity. It facilitates the extraction and encapsulation of each feature representation's latent distribution within its own prototype.
+
+Latent Synchronization enjoys appealing characteristics:
+
+- Blended Paradigm: Latent Synchronization blends unsupervised prototype mining (§3.2.1) and supervised feature representation learning (§3.2.2) in a synergy - local significant motion patterns are automatically explored to facilitate density-based prototyping; the supervisory signal from task-specific heads directly optimizes the representation, which in turn boosts meaningful prototyping.  
+- Prototype-Anchored Learning: Density-based prototype learning computes reliable probabilities in prototype assignment recursively (Eq. 5). Anchored by the updated
+
+prototypes, the feature is further guided towards more robust representations via prototype-feature association (Eq. 7). Consequently, the motion patterns are more likely to center around areas of high data density, which in turn boosts robustness towards motion ambiguity.
+
+# 3.3. Implementation Details
+
+ProtoFormer is built upon Twins architecture (Chu et al., 2021). Detailed training and testing configurations are provided in §S1. The key components (see Fig. 2) are:
+
+- Feature Encoder contains two stages with window sizes of 4 and 8, respectively, which convert input images into features. We follow the common practice (Chu et al., 2021) and utilize two blocks within each stage. In addition, we reformulate the vanilla self-attention into our cross-attention prototyping layers (§3.2.1) to recursively update the initialized prototypes. Once these prototypes have been updated, the latent synchronization layer (§3.2.2) augments the feature learning via prototype-feature association, reducing motion ambiguity.  
+- Task Decoder is designed for task-specific motion predictions. We follow the design (Huang et al., 2022; Zhou et al., 2021b) for flow and depth.  
+- Cross-Attention Prototyping reformulates the vanilla self-attention layers by the  $EM$  cross-attention clustering process for prototyping learning (§3.2.1). Within each Cross-Attention Prototyping layer, twenty prototypes and three iterations are conducted as default (§4.3).  
+- Latent Synchronization updates the feature map in accordance with the prototypes present in the latent feature space (§3.2.2) by applying Feed-Forward Networks and incorporating a masked cross-attention mechanism.
+
+# 4. Experiments
+
+We comprehensively examine the performance and unity of our proposed ProtoFormer on two representative motion tasks, including optical flow (see §4.1) and scene depth (see §4.2). In our pursuit of a unified solution for motion-related tasks, we not only underscore the merit of such integration but also exhibit superior performance, and to further enhance the paradigmatic generalization, we broaden its application to object tracking ( $\S$ S5) and video stabilization ( $\S$ S6) in Appendix, reaching competitive performance.
+
+# 4.1. Experiments on Optical Flow
+
+Datasets. Following the previous works (Huang et al., 2022; Dong et al., 2023), we first train the proposed method on FlyingChair (Dosovitskiy et al., 2015) and FlyingThings (Mayer et al., 2016), and then fine-tune it on a large combination of datasets  $(\mathrm{C} + \mathrm{T} + \mathrm{S} + \mathrm{K} + \mathrm{H})$  to allow evaluation on the Sintel and KITTI-2015 benchmarks.
+
+Metrics. To facilitate a fair comparison, we adopt the commonly used metric, the average end-point-error (F1-epe),
+
+<table><tr><td rowspan="2">Training</td><td rowspan="2">Method</td><td colspan="2">Sintel (train)</td><td colspan="2">KITTI-15 (train)</td><td colspan="2">Sintel (test)</td><td>KITTI-15 (test)</td></tr><tr><td>Clean ↓</td><td>Final ↓</td><td>F1-epe ↓</td><td>F1-all ↓</td><td>Clean ↓</td><td>Final ↓</td><td>F1-all ↓</td></tr><tr><td rowspan="2">A</td><td>Perceiver IO (Jaegle et al., 2022)</td><td>1.81</td><td>2.42</td><td>4.98</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>RAFT-A (Sun et al., 2021)</td><td>1.95</td><td>2.57</td><td>4.23</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="13">C+T</td><td>RAFT (Teed &amp; Deng, 2020)</td><td>1.43</td><td>2.71</td><td>5.04</td><td>17.4</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Separable Flow (Zhang et al., 2021)</td><td>1.30</td><td>2.59</td><td>4.60</td><td>15.9</td><td>-</td><td>-</td><td>-</td></tr><tr><td>GMA (Jiang et al., 2021)</td><td>1.30</td><td>2.74</td><td>4.69</td><td>17.1</td><td>-</td><td>-</td><td>-</td></tr><tr><td>AGFlow (Luo et al., 2022b)</td><td>1.31</td><td>2.69</td><td>4.82</td><td>17.0</td><td>-</td><td>-</td><td>-</td></tr><tr><td>KPA-Flow (Luo et al., 2022a)</td><td>1.28</td><td>2.68</td><td>4.46</td><td>15.9</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DIP (Zheng et al., 2022)</td><td>1.30</td><td>2.82</td><td>4.29</td><td>13.7</td><td>-</td><td>-</td><td>-</td></tr><tr><td>GMFlowNet (Zhao et al., 2022)</td><td>1.14</td><td>2.71</td><td>4.24</td><td>15.4</td><td>-</td><td>-</td><td>-</td></tr><tr><td>GMFlow (Xu et al., 2022b)</td><td>1.08</td><td>2.48</td><td>7.77</td><td>23.4</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CRAFT (Sui et al., 2022)</td><td>1.27</td><td>2.79</td><td>4.88</td><td>17.5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FlowFormer (Huang et al., 2022)</td><td>1.01</td><td>2.40</td><td>4.09</td><td>14.7</td><td>-</td><td>-</td><td>-</td></tr><tr><td>SKFlow (Zhai et al., 2022)</td><td>1.22</td><td>2.46</td><td>4.27</td><td>15.5</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MatchFlow (Dong et al., 2023)</td><td>1.14</td><td>2.71</td><td>4.19</td><td>13.6</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ProtoFormer (Ours)</td><td>1.04</td><td>2.43</td><td>4.08</td><td>14.6</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="14">C+T+S+K+H</td><td>RAFT (Teed &amp; Deng, 2020)</td><td>0.76</td><td>1.22</td><td>0.63</td><td>1.5</td><td>1.61</td><td>2.86</td><td>5.10</td></tr><tr><td>RAFT-A (Sun et al., 2021)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>2.01</td><td>3.14</td><td>4.78</td></tr><tr><td>Separable Flow (Zhang et al., 2021)</td><td>0.69</td><td>1.10</td><td>0.69</td><td>1.60</td><td>1.50</td><td>2.67</td><td>4.64</td></tr><tr><td>GMA (Jiang et al., 2021)</td><td>0.62</td><td>1.06</td><td>0.57</td><td>1.2</td><td>1.39</td><td>2.47</td><td>5.15</td></tr><tr><td>AGFlow (Luo et al., 2022b)</td><td>0.65</td><td>1.07</td><td>0.58</td><td>1.2</td><td>1.43</td><td>2.47</td><td>4.89</td></tr><tr><td>KPA-Flow (Luo et al., 2022a)</td><td>0.60</td><td>1.02</td><td>0.52</td><td>1.1</td><td>1.35</td><td>2.36</td><td>4.60</td></tr><tr><td>DIP (Zheng et al., 2022)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1.44</td><td>2.83</td><td>4.21</td></tr><tr><td>GMFlowNet (Zhao et al., 2022)</td><td>0.59</td><td>0.91</td><td>0.64</td><td>1.51</td><td>1.39</td><td>2.65</td><td>4.79</td></tr><tr><td>GMFlow (Xu et al., 2022b)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1.74</td><td>2.90</td><td>9.32</td></tr><tr><td>CRAFT (Sui et al., 2022)</td><td>0.60</td><td>1.06</td><td>0.58</td><td>1.34</td><td>1.45</td><td>2.42</td><td>4.79</td></tr><tr><td>Flowformer (Huang et al., 2022)</td><td>0.48</td><td>0.74</td><td>0.53</td><td>1.11</td><td>1.20</td><td>2.12</td><td>4.68</td></tr><tr><td>SKFlow (Zhai et al., 2022)</td><td>0.52</td><td>0.78</td><td>0.51</td><td>0.94</td><td>1.28</td><td>2.23</td><td>4.87</td></tr><tr><td>MatchFlow (Dong et al., 2023)</td><td>0.51</td><td>0.81</td><td>0.59</td><td>1.3</td><td>1.33</td><td>2.64</td><td>4.72</td></tr><tr><td>ProtoFormer (Ours)</td><td>0.48</td><td>0.69</td><td>0.50</td><td>1.09</td><td>1.06</td><td>2.07</td><td>4.35</td></tr></table>
+
+Table 1: Quantitative results on standard Sintel and KITTI flow benchmarks. 'A' denotes the Autoflow dataset;  $\mathrm{C} + \mathrm{T}$  denotes training on the FlyingChairs and FlyingThings datasets only;  $\mathrm{C} + \mathrm{T} + \mathrm{S} + \mathrm{K} + \mathrm{H}$  fine-tunes on a combination of Sintel, KITTI, and HD1K training sets. Error metrics are lower is better with "↓", and accuracy metrics are higher is better with "↑". Same for Table 2.
+
+measuring the average  $l_{2}$  distance between the prediction and ground truth, and the percentage of outliers over all pixels (F1-all), which describes the error exceeding 3 pixels or  $5\%$  w.r.t. the ground truth, for optical flow estimation.
+
+Quantitative Results. Table 1 reports the evaluation results of our model on the Sintel and KITTI datasets. The results under the  $\mathrm{C + T}$  setting reflect the generalization capability of our ProtoFormer, where it achieves 1.04 and 2.43 on the clean and final pass of Sintel, improving the recently popular CRAFT (Sui et al., 2022) by 0.23 and 0.36. After training under the mixed setting of  $\mathrm{C + T + S + K + H}$ , the proposed prototype-based model achieves 1.06 and 2.07 on the clean and final pass of Sintel and 4.35 F1-epe on KITTI.
+
+Qualitative Results. Fig. 3 shows the qualitative results on the Sintel flow dataset. As seen, ProtoFormer shows more global and finer details on both object and motion boundaries, without being affected by the shadows and textureless surfaces. In the first and fourth examples, our model recovers full shape and fine details remarkably well, e.g., on bamboo and weapons. This is in stark contrast to other methods, which struggle with producing clear predictions, primarily due to challenges posed by occlusions and variations in illumination. In the second and third examples, we show the significantly consistent estimation of the occluded and textureless regions, e.g., the backpack and birds in the sky. The highlighted regions with red boxes prove the
+
+efficacy of ProtoFormer in object clustering and avoiding motion ambiguity. More results are shown in Fig. S2.
+
+# 4.2. Experiments on Scene Depth
+
+Datasets. Similar to optical flow, we evaluate ProtoFormer on both real and synthetic datasets, using KITTI (Geiger et al., 2013) and MPI Sintel (Butler et al., 2012a) for evaluation. As an autonomous driving dataset consisting of 61 outdoor scenes of various modalities, we use the KITTI Eigen depth split, which contains a standard depth estimation split proposed by Eigen et al. (Eigen et al., 2014) consisting of 32 scenes for training and 29 scenes for testing. The MPI Sintel is a long synthetic stereo sequence with a large motion and depth range and contains a total of 35 rendered sequences. Metrics. We follow the standard metrics of absolute relative error (Abs Rel), root mean square error (RMSE), and the percentage of inlier pixels with  $\delta_1 < \tau$  ( $\tau = 1.25$ ).
+
+Quantitative Results. Table 2 shows the test results on Sintel and KITTI. Our model achieved the best performance on most of the error and accuracy metrics, proving its powerful feature representation and generalization ability to different scenarios. Compared with recent directly supervised depth estimation methods (Eigen et al., 2014; Fu et al., 2018; Bhat et al., 2021), ProtoFormer demonstrates significantly superior capability, benefited from the incorporation of prototypical learning and cross-attention architecture. Even
+
+![](images/b1aad16427ad7eda6e7db2b61fb6aa24c2fcc09c56d2bcfea9e8a8b733be7d0d.jpg)  
+Figure 3: Qualitative results on the Sintel. The red boxes highlight the regions compared. Matchflow (Dong et al., 2023) appears blurry and ambiguous on textureless and occluded objects, while Flowformer (Huang et al., 2022) fails to recover complete and detailed information. Ours can estimate clear and complete flow motion, which is closer to ground truth.
+
+without using additional constraints and priors such as surface normal and piecewise planarity in (Yin et al., 2019; Patil et al., 2022), our model improves the performance of concurrent P3Depth in error by a large margin on KITTI. Furthermore, our approach demonstrates superior performance compared to methods that implement self-supervised consistency and strategies (Godard et al., 2017; Zhang et al., 2023). This underscores the efficacy of our model, which exhibits exceptional adaptability and learning capacity for fine-tuning across more general motion-related tasks.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Sintel</td><td colspan="3">KITTI</td></tr><tr><td>Abs Rel ↓</td><td>RMSE ↓</td><td>Sq Rel ↓</td><td>Abs Rel ↓</td><td>RMSE ↓</td><td>δ1 ↑</td></tr><tr><td>Eigen et al.</td><td>0.797</td><td>0.834</td><td>0.703</td><td>0.203</td><td>6.307</td><td>0.702</td></tr><tr><td>Godard et al.</td><td>-</td><td>-</td><td>-</td><td>0.114</td><td>4.935</td><td>0.861</td></tr><tr><td>Fu et al.</td><td>-</td><td>-</td><td>-</td><td>0.072</td><td>2.727</td><td>0.932</td></tr><tr><td>Yin et al.</td><td>0.746</td><td>0.611</td><td>0.652</td><td>0.072</td><td>3.258</td><td>0.938</td></tr><tr><td>AdaBins</td><td>0.730</td><td>0.572</td><td>0.647</td><td>0.067</td><td>2.960</td><td>0.949</td></tr><tr><td>P3Depth</td><td>0.653</td><td>0.396</td><td>0.571</td><td>0.071</td><td>2.842</td><td>0.953</td></tr><tr><td>Ours</td><td>0.594</td><td>0.486</td><td>0.538</td><td>0.062</td><td>2.716</td><td>0.949</td></tr></table>
+
+Table 2: Quantitative results on Sintel and KITTI depth datasets. With both test data unseen by the model, we can achieve leading performance over state-of-the-art methods (Eigen et al., 2014; Godard et al., 2017; Fu et al., 2018; Yin et al., 2019; Bhat et al., 2021; Patil et al., 2022).
+
+Qualitative Results. Fig. 4 shows the qualitative depth comparison on the KITTI Eigen depth datasets. Our ProtoFormer demonstrates superior capability in delineating object surface contours, particularly in scenarios involving dynamic entities such as pedestrians and vehicles, as well as in capturing the finer details of objects like traffic signs and light poles. For example, for the moving pedestrians and vehicles in Sample 1 and Sample 3, ProtoFormer estimates a more consistent and complete depth on object surfaces and provides a clearer boundary than P3Depth (Patil et al., 2022)
+
+and AdaBins (Bhat et al., 2021). For the plant stand and traffic poles in Sample 2, Sample 3 and Sample 4, our methods separates other methods evidently from the noisy and complex scene backgrounds. These demonstrated the superiority of incorporating prototype learning into depth training to perceive geometric consistent and mitigate motion ambiguity. More visual evidences are provided in Fig. S3.
+
+# 4.3. Diagnostic Experiments
+
+This section ablates ProtoFormer's key components and configurations. More ablations are included in Appendix §S3.
+
+Key Components Analysis. We study the major components of ProtoFormer: Cross-Attention Prototyping (§3.2.1) and Latent Synchronization (§3.2.2). A Base model is designed without considering prototype updating and prototype-feature assignment. Shown in Table 3a, Base reaches 0.55 and 0.81 in average EPE. After adding Cross-Attention Prototyping, substantial improvements are observed (i.e.,  $0.55 \to 0.51$  in clean pass), suggesting the efficacy of prototyping updating even without explicit prototype-feature assignment. Incorporating Latent Synchronization into Base can observe a noticeable performance gain (i.e.,  $0.81 \to 0.77$  in final pass). Finally, the integration of the two techniques culminates in peak performance.
+
+Cross-Attention Prototyping. We then study the efficacy of Cross-Attention Prototyping by comparing to different updating methods, including cosine similarity, conventional cross-attention (Vaswani et al., 2017), Criss cross-attention (Huang et al., 2019) and  $K$ -Means (Yu et al., 2022). From the efficient and effective perspectives, Cross-Attention Prototyping outperforms competitive methods (see Table 3b). We further study the iteration step  $N$  in Table 3c,
+
+![](images/ab1ed21151f1a0275108b284420ea3522fdf46cebd25d2d8940a042c68894ee9.jpg)  
+Figure 4: Qualitative depth comparison results on the KITTI. The red boxes indicate the highlighted regions. P3Depth (Patil et al., 2022) and AdaBins (Bhat et al., 2021) have limited receptive fields and do not consider conceptual object-level groupings, thus producing discontinuous and ambiguous predictions. While ours can estimate consistent and sharp depths, which is closer to ground truth.
+
+Table 3: A set of ablative studies on optical flow (see §4.3). The best performances are marked in bold.
+
+<table><tr><td>Algorithm Component</td><td>#Params</td><td>Sintel clean</td><td>Sintel final</td></tr><tr><td>Base</td><td>9.63M</td><td>0.55</td><td>0.81</td></tr><tr><td>+ Cross-Attention Prototyping</td><td>11.57M</td><td>0.51</td><td>0.74</td></tr><tr><td>+ Latent Synchronization</td><td>10.26M</td><td>0.53</td><td>0.77</td></tr><tr><td>ProtoFormer (All included)</td><td>11.90M</td><td>0.48</td><td>0.69</td></tr></table>
+
+(a) Key Component Analysis  
+
+<table><tr><td>#Iterations (N)</td><td>#Params</td><td>Sintel clean</td><td>Sintel final</td></tr><tr><td>1</td><td rowspan="4">11.90M</td><td>0.52</td><td>0.75</td></tr><tr><td>2</td><td>0.49</td><td>0.71</td></tr><tr><td>3</td><td>0.48</td><td>0.69</td></tr><tr><td>4</td><td>0.48</td><td>0.68</td></tr></table>
+
+(c) Number of Iterations  
+
+<table><tr><td>#Prototypes (K)</td><td>#Params</td><td>Sintel clean</td><td>Sintel final</td></tr><tr><td>10</td><td>8.95M</td><td>0.53</td><td>0.78</td></tr><tr><td>50</td><td>9.78M</td><td>0.51</td><td>0.73</td></tr><tr><td>100</td><td>11.90M</td><td>0.48</td><td>0.69</td></tr><tr><td>200</td><td>14.21M</td><td>0.49</td><td>0.71</td></tr></table>
+
+(d) Number of Prototypes  
+
+<table><tr><td>Variant Prototype Updating Strategy</td><td>#Params</td><td>Sintel clean</td><td>Sintel final</td></tr><tr><td>Cosine Similarity</td><td>10.28M</td><td>0.51</td><td>0.75</td></tr><tr><td>Vanilla Cross-Attention (Vaswani et al., 2017)</td><td>14.88M</td><td>0.50</td><td>0.73</td></tr><tr><td>Criss Cross-Attention (Huang et al., 2019)</td><td>14.56M</td><td>0.50</td><td>0.72</td></tr><tr><td>K-Means (Yu et al., 2022)</td><td>11.81M</td><td>0.49</td><td>0.71</td></tr><tr><td>Cross-Attention Prototyping (Eq. 5)</td><td>11.90M</td><td>0.48</td><td>0.69</td></tr></table>
+
+(b) Cross-Attention Prototyping  
+
+<table><tr><td>Latent Synchronization</td><td>#Params</td><td>Sintel clean</td><td>Sintel final</td></tr><tr><td>None</td><td>11.27M</td><td>0.51</td><td>0.74</td></tr><tr><td>Vanilla FC Layer</td><td>11.64M</td><td>0.50</td><td>0.73</td></tr><tr><td>FC w/ Similarity (Ma et al., 2023)</td><td>11.76M</td><td>0.49</td><td>0.71</td></tr><tr><td>Ours (Eq. 7)</td><td>11.90M</td><td>0.48</td><td>0.69</td></tr></table>
+
+(e) Latent Synchronization
+
+suggesting that the error progressively decreases from 0.52 to 0.48 when increasing  $N$  from 1 to 4, and saturates at 4. Considering the computation time, we set  $N = 3$  to strike the balance between performance and computational cost. The number of prototypes  $K$  plays a pivotal role in defining the central grouping points for motion features. We therefore investigate the variant of  $K$  in Table 3d.
+
+Latent Synchronization. Next, we study our Latent Synchronization in Table 3e. With a standard setting without any prototype-feature corresponding (i.e., None), the model reports 0.74 in final pass. After applying a vanilla fully-connected layer to update the feature, the error decreases to 0.73. Though inspiring, our proposed Latent Synchronization with carefully anchored prototypes yields advanced performance across all ablative methods (i.e., 0.69).
+
+Systemic Explainability. Finally, we investigate the prototype-feature corresponding map on optical flow in Fig. 5. The systemic explainability hinges on the Prototypes, which emerge through the integration of probability density estimation within our cross-attention prototyping layer. The recursively optimized prototypes encapsulate the most characteristic features of the motion patterns within their respective density centers. Through the visualization of
+
+![](images/de6f8b7744929d520db3e8da7d361ea249faea87d0ad418ae136aea5e63ebe4a.jpg)  
+Figure 5: Visualization of proto-feature mapping, which demonstrates distinct prototypes with similar representations.
+
+feature correspondence estimation derived from the updated prototypes, we enhance the interpretability of the network and transparency of the model's decision-making process.
+
+# 5. Conclusion
+
+We propose Prototypical TransFormer (ProtoFormer), a unified solution for motion tasks. The motivation of integrating Transformer and prototype learning leads us to innovate conventional self-attention, and propose Cross-Attention Prototyping. Our Latent Synchronization further refines the feature representations via prototype-feature association. Comprehensive empirical results show that ProtoFormer enjoys elegant architectural design, superior performance and systemic explainability. As a whole, we conclude that the findings from this work impart essential understandings and necessitate further exploration within this realm.
+
+# Impact Statement
+
+Ethical Aspects. We provide asset licenses and consents for the datasets we applied in our paper in supplementary material. All the datasets are publicly available for academic usage. Since our work does not involve data augmentation or the creation of new datasets, ethical concerns or biases within our proposed ProtoFormer are significantly minimized. We should also highlight that the introduction of prototypical learning offers a significant advantages in addressing photometric inconsistency, which in turn reduces the possible biases during training when lighting condition is restricted.
+
+Future Societal Consequences. ProtoFormer introduces a universal understanding for motion tasks via prototypical learning, possessing strong performance gains over several state-of-the-art baselines. On positive side, our approach is valuable in various real-world applications (e.g., autonomous driving (Cheng et al., 2023c; Prakash et al., 2021; Cheng et al., 2023b; Shao et al., 2023; Cheng et al., 2022; 2023a), robotics navigation (Dev et al., 1997; Lookingbill et al., 2007; Song et al., 2022)), benefited from its transparency and efficiency. Regarding potential negative social impacts, it is noteworthy that our ProtoFormer, akin to other discriminative classifiers, encounters challenges in addressing out-of-distribution/open-set problems (Liang et al., 2022a). Its utility in open-world scenarios should be further examined.
+
+# Acknowledgement
+
+We thank Prof. Majid Rabbani for detailed feedback on drafts of the paper. This research was supported by the National Science Foundation under Grant No. 2242243 and the DEVCOM Army Research Laboratory under Contract W911QX-21-D-0001. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. DEVCOM Army Research Laboratory (ARL), U.S. Naval Research Laboratory (NRL) or the U.S. Government.
+
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+
+# SUMMARY OF THE APPENDIX
+
+This supplementary contains additional experimental results and discussions of our ICML 2024 submission: Prototypical Transformer as Unified Motion Learners, organized as follows:
+
+- §S1 provides detailed training configuration and testing configuration on optical flow and scene depth estimation.  
+- §S2 provides more qualitative results and comparisons for optical flow and scene depth estimation.  
+- §S3 offers comprehensive ablation studies on scene depth estimation.  
+- §S4 provides detailed Proof on the guarantees of  $EM$  convergence.  
+- §S5 includes additional experiments on object tracking.  
+- §S6 includes additional experiments on video stabilization.  
+- §S7 discusses the Pseudo codes.
+
+# S1. Training and Testing Configuration
+
+Our training methodology for ProtoFormer was adapted from established optical flow training protocols (Jiang et al., 2021; Huang et al., 2022). Initially, the model underwent a pre-training phase on the FlyingChairs dataset (Dosovitskiy et al., 2015), followed by an additional 120,000 iterations on the FlyingThings dataset (Mayer et al., 2016), a procedure we denote as “C+T.” Subsequently, the model underwent fine-tuning on a combined dataset encompassing FlyingThings (Mayer et al., 2016), Sintel (Butler et al., 2012b), KITTI-2015 (Geiger et al., 2013), and HD1K (Kondermann et al., 2016), referred to as “C+T+S+K+H”. To optimize performance specifically for the KITTI-2015 benchmark (Geiger et al., 2013), we conducted a further fine-tuning phase on the KITTI-2015 dataset for 50,000 iterations. The training employed AdamW (Loshchilov & Hutter, 2019) optimizer and a one-cycle learning rate scheduler, with the peak learning rate set at  $2.5 \times 10^{-4}$  for the FlyingChairs dataset and  $1.25 \times 10^{-4}$  for the other datasets. Recognizing the sensitivity of transformer positional encodings to variations in image size, we adopted an image processing approach akin to that used in Perceiver IO (Jaegle et al., 2022). This involved cropping image pairs for flow estimation and subsequently tiling them to reconstruct complete flows. For depth prediction, we adhere to the architecture and configurations analogous to those employed for optical flow, as delineated in the respective headers. We initially adopt the VKITTI (Cabon et al., 2020) as a pretraining, and subsequently canonical Eigen split (Eigen et al., 2014)
+
+and MPI Sintel dataset (Butler et al., 2012b) to refine the model through fine-tuning, noted for its distinct edges and diverse motion intensities. No additional data augmentation was used for the testing of all tasks.
+
+Loss Function. For optical flow, A sequence loss is utilized for the training, which is defined over the sequence of flow predictions. For depth prediction, square root of the scale invariant logarithmic loss (SILog) is utilized for the training.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">LaSOT</td><td colspan="2">TrackingNet</td></tr><tr><td>Success ↑</td><td>Precision ↑</td><td>Success ↑</td><td>Precision ↑</td></tr><tr><td>SiamFC (Bertinetto et al., 2016)</td><td>33.6</td><td>33.9</td><td>57.1</td><td>66.3</td></tr><tr><td>MDNet (Nam &amp; Han, 2016)</td><td>39.7</td><td>37.3</td><td>60.6</td><td>56.6</td></tr><tr><td>ECO (Danelljan et al., 2017)</td><td>32.4</td><td>30.1</td><td>55.4</td><td>49.2</td></tr><tr><td>KYS (Bhat et al., 2020)</td><td>55.4</td><td>55.8</td><td>74.0</td><td>68.8</td></tr><tr><td>Ocean (Zhang et al., 2020)</td><td>52.6</td><td>52.6</td><td>70.3</td><td>68.8</td></tr><tr><td>TrDiMP (Wang et al., 2021b)</td><td>63.9</td><td>61.4</td><td>78.4</td><td>73.1</td></tr><tr><td>TransT (Chen et al., 2021)</td><td>64.9</td><td>69.0</td><td>81.4</td><td>80.3</td></tr><tr><td>UniTrack (Wang et al., 2021e)</td><td>35.1</td><td>32.6</td><td>59.1</td><td>51.2</td></tr><tr><td>UTT (Ma et al., 2022)</td><td>64.6</td><td>67.2</td><td>79.7</td><td>77.0</td></tr><tr><td>UTT + Ours</td><td>64.8</td><td>67.4</td><td>80.0</td><td>77.2</td></tr></table>
+
+Table S1: Quantitative results on LaSOT and TrackingNet datasets. We are able to achieve competitive performance over state-of-the-art methods.
+
+# S2. More Qualitative Results
+
+We show more qualitative results on the main tasks, optical flow and scene depth, in Fig. S2 and Fig. S3.
+
+# S3. Ablation Studies on Scene Depth Estimation
+
+In §4.3, we ablate comprehensively under the optical flow setting. In this section, we further report ablation studies on scene depth estimation for completeness.
+
+Key Components Analysis. We study the two major components of ProtoFormer: Cross-Attention Prototyping (§3.2.1) and Latent Synchronization (§3.2.2). Same to our paper, the Base model is designed without considering prototype updating and prototype-feature assignment. In Table S2a, Base reaches 0.074 in Abs Rel and 2.835 in RMSE. Adding Cross-Attention Prototyping gets substantial improvements (i.e.,  $0.074 \rightarrow 0.067$  in Abs Rel). Considering Latent Synchronization brings a performance gain (i.e.,  $0.074 \rightarrow 0.071$ ). Finally, the integration of these two techniques reaches peak performance as ProtoFormer, which is consistent to the tendency in our paper.
+
+Cross-Attention Prototyping. We also study the efficacy of Cross-Attention Prototyping design by comparing to different updating methods. For efficient and effective perspectives, our Cross-Attention Prototyping outperforms competitive methods (see Table S2b). We further study the iteration step  $N$  in Table S2c, suggesting that when increasing  $N$  from 1 to 4, the error progressively decreases from 0.070 to 0.061, and almost saturates at 4. Considering the computa
+
+![](images/5e22f75ed1e7468b24570ed4461c91afab9671b8cfd6d749b0e648188be10e80.jpg)  
+Figure S1: Visualization of proto-feature mapping in depth. The map shows distinct prototypes with similar representations, indicating straightforward explainability.
+
+![](images/3cb56270f29dc1ce21561044148d97698c3ffe200bed1e82cb6d907602a518a0.jpg)
+
+![](images/9b2c0227413ba3c991a34b845e789518940f6bd3bc73d67f132d003ee108b7d1.jpg)
+
+![](images/4edf806c5be1599edc4f1149c495277b68f8631dffd1217156109203de6c5bca.jpg)
+
+Table S2: Ablative studies on scene depth (see §S3).
+
+<table><tr><td>Algorithm Component</td><td>#Params</td><td>Abs Rel</td><td>RMSE</td></tr><tr><td>Base</td><td>9.63M</td><td>0.074</td><td>2.835</td></tr><tr><td>+ Cross-Attention Prototyping</td><td>11.57M</td><td>0.067</td><td>2.742</td></tr><tr><td>+ Latent Synchronization</td><td>10.26M</td><td>0.071</td><td>2.819</td></tr><tr><td>ProtoFormer (All included)</td><td>11.90M</td><td>0.062</td><td>2.716</td></tr></table>
+
+(a) Key Component Analysis  
+
+<table><tr><td>Variant Prototype Updating Strategy</td><td>#Params</td><td>Abs Rel</td><td>RMSE</td></tr><tr><td>Cosine Similarity</td><td>10.28M</td><td>0.066</td><td>2.775</td></tr><tr><td>Vanilla Cross-Attention (Vaswani et al., 2017)</td><td>14.88M</td><td>0.066</td><td>2.767</td></tr><tr><td>Criss Cross-Attention (Huang et al., 2019)</td><td>14.56M</td><td>0.065</td><td>2.750</td></tr><tr><td>K-Means (Yu et al., 2022)</td><td>11.81M</td><td>0.063</td><td>2.743</td></tr><tr><td>Cross-Attention Prototyping</td><td>11.90M</td><td>0.062</td><td>2.716</td></tr></table>
+
+(b) Cross-Attention Prototyping  
+
+<table><tr><td>#Iterations (N)</td><td>#Params</td><td>Abs Rel</td><td>RMSE</td></tr><tr><td>1</td><td rowspan="4">11.90M</td><td>0.070</td><td>2.801</td></tr><tr><td>2</td><td>0.065</td><td>2.737</td></tr><tr><td>3</td><td>0.062</td><td>2.716</td></tr><tr><td>4</td><td>0.061</td><td>2.713</td></tr></table>
+
+(c) Number of Iterations  
+
+<table><tr><td>#Prototypes (K)</td><td>#Params</td><td>Abs Rel</td><td>RMSE</td></tr><tr><td>10</td><td>8.95M</td><td>0.070</td><td>2.775</td></tr><tr><td>50</td><td>9.78M</td><td>0.067</td><td>2.734</td></tr><tr><td>100</td><td>11.90M</td><td>0.062</td><td>2.716</td></tr><tr><td>200</td><td>14.21M</td><td>0.064</td><td>2.720</td></tr></table>
+
+(d) Number of Prototypes  
+
+<table><tr><td>Latent Synchronization</td><td>#Params</td><td>Abs Rel</td><td>RMSE</td></tr><tr><td>None</td><td>11.27M</td><td>0.067</td><td>2.732</td></tr><tr><td>Vanilla FC Layer</td><td>11.64M</td><td>0.064</td><td>2.724</td></tr><tr><td>FC w/ Similarity (Ma et al., 2023)</td><td>11.76M</td><td>0.063</td><td>2.718</td></tr><tr><td>Ours</td><td>11.90M</td><td>0.062</td><td>2.716</td></tr></table>
+
+(e) Latent Synchronization
+
+tion time in iterations, we set  $N = 3$  to strike the optimal balance between performance and computation. Consistent to our paper, we investigate the variant of  $K$  (i.e., number of prototypes) in Table S2d. We select the preferred setting at  $K = 100$ .
+
+Latent Synchronization. We further study our Latent Synchronization in Table S2e. With a standard setting without any feature-prototype corresponding, the model achieves 0.067 in Abs Rel. Applying a vanilla fully-connected layer increases the performance to 0.064. Though inspiring, our proposed Latent Synchronization with carefully anchored prototypes yields advanced performance across all ablative methods (i.e., 0.062).
+
+# S4. Proof on the Guarantees of EM Convergence
+
+We first introduce the regularity condition, including a Euclidean ball of radius  $r$  around the fixed point  $\hat{\theta}$ , set as:
+
+$$
+\mathcal {B} _ {2} (r; \theta) := \left\{\theta \in \Omega \mid | | \theta - \hat {\theta} | | _ {2} \leq r \right\}. \tag {8}
+$$
+
+For simplicity, we define  $\hat{\theta} = \arg \max_{\theta} U(\cdot |\theta)$  while we have introduced our unified motion solution in Eq. 2.
+
+For First-order Stability (FOS), the functions  $U(\cdot |\theta)$  satisfy condition  $\mathrm{FOS}(\gamma)$  over  $\mathcal{B}_2(r;\theta)$  if:
+
+$$
+\left| \left| \nabla U (M (\cdot | \hat {\theta})) - \nabla U (M (\cdot | \theta)) \right| \right| _ {2} \leq \gamma \left| \left| \theta - \hat {\theta} \right| \right| _ {2}, \tag {9}
+$$
+
+for all  $\theta \in \mathcal{B}_2(r;\hat{\theta})$ . With radius  $\gamma = 0$ , the condition of
+
+Eq. 9 is always held at the fixed point  $\hat{\theta}$ . Extend further, by allowing for an always positive  $\gamma$ , given fixed point  $\hat{\theta}$ , Eq. 9 would always hold in a local neighborhood  $\mathcal{B}_2(r; \hat{\theta})$ .
+
+Under these conditions, we further guarantee the  $EM$  operator to be locally contraction.
+
+Theorem 1. For  $\gamma >0$ , and having  $0\leq \gamma \leq \lambda$ , suppose the function  $U(\cdot |\hat{\theta})$  is  $\lambda$ -strongly concave and  $\mathrm{FOS}(\gamma)$  holds for  $\mathcal{B}_2(r;\hat{\theta})$ , we have the  $EM$  operator  $M$  is contractive over  $\mathcal{B}_2(r;\hat{\theta})$  as:
+
+$$
+\left| \left| M (\theta) - \hat {\theta} \right| \right| _ {2} \leq \frac {\gamma}{\lambda} \left| \left| \theta - \hat {\theta} \right| \right| _ {2} \tag {10}
+$$
+
+for all  $\theta \in \mathcal{B}_2(r;\hat{\theta})$ . Intuitively, we can conduct that for any initial point  $\theta^{(0)}\in \mathcal{B}_2(r;\hat{\theta})$ ,  $\left\{\theta^{(n)}\right\}_{n = 0}^{\infty}$  exhibits linear convergence. Formally, we have:
+
+$$
+\left| \left| \theta^ {(n)} - \hat {\theta} \right| \right| _ {2} \leq \left(\frac {\gamma}{\lambda}\right) ^ {n} \left| \left| \theta^ {0} - \hat {\theta} \right| \right| _ {2}, \tag {11}
+$$
+
+for all  $n \in \{1, 2, \dots, N\}$ . Here we define  $n$  in a finite set for intuitive reference below.
+
+Acknowledging the preliminary conditions in Theorem 1, we further present the proof for Proposition 1 below.
+
+Proof. For any iteration  $n \in \{1, 2, \dots, N\}$ , we have:
+
+$$
+\left| \left| M _ {m / N} \left(\theta^ {(n)}\right) - M \left(\theta^ {(n)}\right) \right| \right| _ {2} \leq \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right), \tag {12}
+$$
+
+with probability at least  $1 - \frac{\delta}{N}$ . Consequently, by a union bound over  $N$ , Eq. 12 holds uniformly with probability at
+
+![](images/fe87f0c1a7c1dfab322ef39aafa5d797414c2d3fb6065353f158ce73d8f56838.jpg)  
+Figure S2: More qualitative comparison on the KITTI test set. The red boxes highlight the regions compared. Our method estimates more consistent and detailed flows.
+
+![](images/4b79820d79950103420985418dd6dcd5fda991986517d2da09ce5db17fe01d78.jpg)  
+Figure S3: More qualitative comparison on the Sintel test set. The red boxes highlight the regions compared. Our method contributes to clearer depths without being affected by shadows or occlusions.
+
+![](images/575beb2bf6d1995a2b7e5b5634e36271759de19f2eba2691cb67a843579c75ae.jpg)  
+Figure S4: Qualitative results on object tracking on LaSOT dataset. With our enhanced method, the original ambiguous tracking due to illuminant changes and heavy occlusion becomes more accurate.
+
+![](images/2f0293d0d2453484095e7236c810871e206a7eaa17cc70a4ab2ae944d6b73612.jpg)  
+Figure S5: Qualitative results on video stabilization. With our method enhanced, the original unstable video frames become much smoother and clearer.
+
+least  $1 - \delta$ . It suffices to show that
+
+$$
+\left| \left| \theta^ {(n + 1)} - \hat {\theta} \right| \right| _ {2} \leq \kappa \left| \left| \theta^ {(n)} - \hat {\theta} \right| \right| _ {2} + \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right), \tag {13}
+$$
+
+for each iteration  $n \in \{1, 2, \dots, N\}$ . When Eq. 13 holds, we can iteratively show that:
+
+$$
+\begin{array}{l} | | \theta^ {(n)} - \hat {\theta} | | _ {2} \leq \kappa | | \theta^ {(n - 1)} - \hat {\theta} | | _ {2} + \epsilon_ {M} (\frac {m}{N}, \frac {\delta}{N}) \\ \leq \kappa \left\{\kappa | | \theta^ {(n - 2)} - \hat {\theta} | | _ {2} + \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right) \right\} + \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right) \\ \leq \kappa^ {(n)} | | \theta^ {(0)} - \hat {\theta} | | _ {2} + \left\{\sum_ {n = 0} ^ {n - 1} \kappa^ {n} \right\} \epsilon_ {M} (\frac {m}{N}, \frac {\delta}{N}) \\ \leq \kappa^ {n} | | \theta^ {(0)} - \hat {\theta} | | _ {2} + \frac {1}{1 - \kappa} \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right). \tag {14} \\ \end{array}
+$$
+
+The final step follows by summing the geometric series.
+
+Thus, we need to prove Eq. 13 via induction on the iteration number. Start with  $n = 1$ , we have:
+
+$$
+\begin{array}{l} | | \theta^ {(1)} - \hat {\theta} | | _ {2} = | | M _ {m / N} (\theta^ {(0)}) - M (\theta^ {(n)}) | | _ {2} \\ (\mathbf {I}) \leq | | M (\theta^ {(0)}) - \hat {\theta} | | _ {2} + | | M _ {m / N} (\theta^ {(0)}) - M (\theta^ {(0)}) | | _ {2} \\ (\mathbf {I I}) \leq \kappa | | M \left(\theta^ {(0)}\right) - \hat {\theta} | | _ {2} + \epsilon_ {M} \left(\frac {m}{N}, \frac {\delta}{N}\right), \tag {15} \\ \end{array}
+$$
+
+where step (I) follows by the triangle inequality, step (II) follows from Eq. 13, and the contractivity of the operator applied to  $\theta^{(0)}\in \mathcal{B}_2(r;\hat{\theta})$  . In the induction from  $n\to n + 1$  suppose that  $\| \theta^{(n)} - \hat{\theta}\| _2\leq r$  , and the bound holds (i.e., Eq. 13) for iteration  $n$  . The same augment then implies that the bound from Eq. 13 also holds for iteration  $n + 1$  , and that  $\| \theta^{(n + 1)} - \hat{\theta}\| _2\leq r$  , thus completing the proof.
+
+<table><tr><td>Method</td><td>Distortion Value ↑</td><td>Stability Score ↑</td></tr><tr><td>StabNet (Wang et al., 2018)</td><td>0.83</td><td>0.75</td></tr><tr><td>StabNet + Ours</td><td>0.85 (0.02 ↑)</td><td>0.80 (0.05 ↑)</td></tr><tr><td>PWStableNet (Zhao &amp; Ling, 2020)</td><td>0.79</td><td>0.80</td></tr><tr><td>PWStableNet + Ours</td><td>0.82 (0.03 ↑)</td><td>0.83 (0.03 ↑)</td></tr></table>
+
+Table S3: Quantitative results on DeepStab dataset. We are able to achieve competitive performance over current methods.
+
+# S5. Experiments on Object Tracking
+
+To further support our proposed ProtoFormer as a general solution to various tasks, we extend our design to object tracking following (Ma et al., 2022). Intuitively, we follow (Ma et al., 2022) and integrate the encoder and decoder into one object transformer where we replace the self-attention into our proposed cross-attention prototyping.
+
+We evaluate our method on the testing splits of LaSOT (Fan et al., 2019) and TrackingNet (Muller et al., 2018) following common practices (Ma et al., 2022; Chen et al., 2023).
+
+Specifically, LaSOT (Fan et al., 2019) includes 1,400 sequences: 1, 120 for training and 280 for testing, respectively. TrackingNet (Muller et al., 2018) contains  $30K$  sequences with 511 sequences for testing. Success and Precision metrics are applied for performance evaluation. We follow the same training schedule as (Ma et al., 2022) for fairness. As seen in Table S1, our approach achieves competitive results to current methods (e.g.,  $0.2\%$  and  $0.2\%$  higher than UTT on Success and Precision on LaSOT dataset, respectively). Qualitative results are shown in Fig S4.
+
+Algorithm 1 Pseudo-code of Cross-attention Prototyping in a PyTorch-like style.  
+```
+feats: output feature embeddings from regular projection, shape: (batch_size, channels, height, width)
+P_0: initial cluster centers by adaptive pooling from the features, shape: (batch_size, num_clusters, dimension)
+P: cluster centers, shape: (batch_size, num_clusters, dimension)
+N: iteration number for recursive prototyping layer
+```
+# One-step cross-attention prototyping in Eq.5
+def one_prototyping_layer(Q, K, V):
+    # E-step
+    output = torch/matmul(Q, K.transpose(-2, -1))
+    M = torch(nnfunctional softmax(output, dim = -2))
+    # M-step
+    P = torch/matmul(M, V)
+    return P
+# Iteratively cross-attention prototyping layer
+def CrossAttentionPrototyping(feats, P_0, N):
+    Q = nn.Linear(P_0)
+    K = nn.Linear(feats)
+    V = nn.Linear(feats)
+    P = P_0 + one_prototyping_layer(Q, K, V)
+    for _ in range(N-1):
+        Q = nn.Linear(P)
+        P = P + one_prototyping_layer(Q, K, V)
+    return P
+
+# S6. Experiments on Video Stabilization
+
+We further evaluate our method on video-based downstream task - video stabilization, following common training configurations from (Wang et al., 2018; Zhao & Ling, 2020; Lu et al., 2023). Specifically, DeepStab (Wang et al., 2018) contains 61 pairs of synchronized videos with diverse camera movements. Distortion Value and Stability Score are applied for performance evaluation. In Table S3, we report the results comparing to competitive methods (i.e., StabNet (Wang et al., 2018), PWStableNet (Zhao & Ling, 2020)). Specifically, we follow TransFlow (Lu et al., 2023), aggregating the learned features from TransFlow's encoder
+
+Algorithm 2 Pseudo-code of Latent Synchronization in a PyTorch-like style.  
+```python
+```
+feats: output feature embeddings from regular projection, shape: (batch_size, channels, height, width)
+P: prototypes, shape: (batch_size, num_prototypes, dimension)
+```
+```
+# Latent synchronization in Eq.6
+def latent_synchronization(feats, P):
+    max_value, max_index = similarity(feats, P).
+    max(dim = 1, keepdim = True)
+    mask = torch.zeros_like(similarity(feats, P))
+    maskscatter_(1, max_index, 1.)
+    Q = nn.Linear(feats)
+    K = nn.Linear(P)
+    V = nn.Linear(P)
+    feats += FFN(attention_layer(Q, K, V, attn_mask = mask))
+    return feats
+```
+
+(i.e., replacing the origin attention with our proposed cross-attention prototyping and latent synchronization) and the original encoder together for the later regressor. Significant performance boost can be observed in both Distortion Value and Stability Score. Qualitative results are shown in Fig S5.
+
+# S7. Pseudo-codes
+
+ProtoFormer is implemented in Pytorch (Paszke et al., 2019). Experiments are conducted on eight NVIDIA A100-40GB GPUs. We provide the pseudo codes of our proposed ProtoFormer Cross-Attention Prototyping in Algorithm 1 and Latent Synchronization in Algorithm 2.

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2024/Provable Contrastive Continual Learning/full.md

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+Yichen Wen  $^{*12}$  Zhiquan Tan  $^{*3}$  Kaipeng Zheng  $^{1}$  Chuanlong Xie  $^{2}$  Weiran Huang  $^{†14}$
+
+# Abstract
+
+Continual learning requires learning incremental tasks with dynamic data distributions. So far, it has been observed that employing a combination of contrastive loss and distillation loss for training in continual learning yields strong performance. To the best of our knowledge, however, this contrastive continual learning framework lacks convincing theoretical explanations. In this work, we fill this gap by establishing theoretical performance guarantees, which reveal how the performance of the model is bounded by training losses of previous tasks in the contrastive continual learning framework. Our theoretical explanations further support the idea that pre-training can benefit continual learning. Inspired by our theoretical analysis of these guarantees, we propose a novel contrastive continual learning algorithm called CILA, which uses adaptive distillation coefficients for different tasks. These distillation coefficients are easily computed by the ratio between average distillation losses and average contrastive losses from previous tasks. Our method shows great improvement on standard benchmarks and achieves new state-of-the-art performance.
+
+# 1. Introduction
+
+Incrementally learning a sequence of tasks with dynamic data distributions is a typical setting for continual learning. We call the learned neural networks "continual learners". The main challenge for continual learners is to obtain a suitable trade-off between learning plasticity and memory stability. Specifically, excessive focus on learning plasticity of new tasks often leads to greatly reduced performance
+
+*Equal contribution  $^{1}$ MIFA Lab, Qing Yuan Research Institute, SEIEE, Shanghai Jiao Tong University  $^{2}$ Beijing Normal University  $^{3}$ Department of Mathematical Sciences, Tsinghua University  $^{4}$ Shanghai AI Laboratory. †Correspondence to: Weiran Huang <weiran.huang@outlook.com>. This work was conducted during the period when the first two authors were visiting MIFA Lab.
+
+Proceedings of the  $41^{st}$  International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s).
+
+on old tasks (McClelland et al., 1995), which is known as catastrophic forgetting.
+
+To address the challenge, the literature on continual learning has proposed various approaches. Representation-based approaches take advantage of representations. As one of these approaches, self-supervised learning with contrastive loss has demonstrated notable efficacy in obtaining robust representations against catastrophic forgetting in continual learning (Gallardo et al., 2021; Fini et al., 2022). For these methods based on contrastive loss, the training of representations is often decoupled with the training of the classifier, unlike methods based on cross-entropy. Specifically, contrastively trained representations suffer less catastrophic forgetting than ones trained by cross-entropy loss (Cha et al., 2021). Replay-based approaches use buffers to restore a part of previous data, and train networks using data from a combination of the current task and the buffer (Lopez-Paz & Ranzato, 2017). Naturally, these methods are combined with knowledge distillation strategies to prevent the degradation of information in the network over time (Rebuffi et al., 2017). Regularization-based approaches introduce regularization terms to the target loss for continual learning to reach a balance between learning new tasks and preserving information from old tasks (Kirkpatrick et al., 2017). Two main sub-directions within regularization-based approaches include weight regularization (Ritter et al., 2018) and function regularization (Li & Hoiem, 2016).
+
+To achieve effective continual learning, a natural idea is to combine the three approaches above, and this idea leads to a new framework called contrastive continual learning (Cha et al., 2021), as illustrated in Figure 1. This framework focuses on using contrastively learned representations to learn new tasks and utilizing knowledge distillation to preserve information from past tasks, with the help of memory buffer and function regularization. The target loss of this framework contains a contrastive loss and a distillation loss with a distillation coefficient  $\lambda$ . The training data will be selected from the combination of the current data and buffered data. Empirically, this framework has been observed to be efficient, showing promising performance in continual learning (Cha et al., 2021). Despite the growing attention directed towards this framework, limited theoretical works have been proposed to explain its superior performance.
+
+In this paper, we try to address the theoretical problem of why this framework is efficient. Therefore, we consider the losses provided in (Cha et al., 2021). We have found a clear relationship between the contrastive losses of two consecutive models in continual learning. Inspired by this, we propose theoretical performance guarantees that reveal how the population test loss, i.e., the total performance of the final model on all seen tasks, is bounded by the series of training losses for the contrastive continual learning framework. Based on our theory, we propose a new and efficient contrastive continual learning algorithm called CILA, which uses distillation coefficients adapted to different tasks. Moreover, CILA consistently outperforms all baselines in different scenarios, datasets, and buffer sizes, e.g., about  $1.77\%$  improvement compared with the previous state-of-the-art method  $\mathrm{Co}^2\mathrm{L}$  (Cha et al., 2021) on Seq-CIFAR-10 with a buffer of 500 samples for Class-IL scenario.
+
+Overall, our contributions are listed as follows. (1) We provide theoretical performance guarantees for the contrastive continual learning scheme. We identify that the overall performance of the final learned model on all seen tasks can be bounded by a function of the series of training losses with the distillation coefficient; (2) We propose an efficient algorithm CILA, which uses adaptive distillation coefficient  $\lambda_{t}$  (replace  $\lambda$  with  $\lambda_{t}$  in Figure 1) for each task  $t$ ; (3) We conduct extensive experiments to validate the efficacy of our algorithm, and the results strongly support our theory. Our method can inspire future works in contrastive continual learning.
+
+# 2. Related Work
+
+Continual learning. Continual learning is also referred to as incremental learning, which learns incremental tasks effectively (Wang et al., 2023). The literature in this field mainly focuses on several streams including weight and function regularization (Jung et al., 2020), memory replay (Prabhu et al., 2020), sparse representations (Javed & White, 2019), parameter isolation (Gurbuz & Dovrolis, 2022), and dynamic architecture (Ramesh & Chaudhari, 2021).
+
+As one of these effective continual learning methods, replay-based methods have demonstrated superior performance in terms of both learning plasticity and memory stability (Riemer et al., 2019). Replay-based continual learning methods are developed from the idea of Experience Replay (Buzzega et al., 2020), which typically stores past training samples in a fixed-size buffer. Currently, these replay-based methods are divided into two main streams, including experience replay and generative replay. Experience replay-based methods focus on the construction of memory buffer (Riemer et al., 2019; Tiwari et al., 2022) and storage efficiency (Caccia et al., 2019; Bang et al., 2021). Generative replay-based methods concentrate on generative
+
+adversarial networks (GANs) to generate fine-grained data (Cong et al., 2020; Ayub & Wagner, 2021).
+
+Representation-based methods for continual learning are also observed to be competitive. Recent works in continual learning take advantage of self-supervised learning to obtain robust representations, showing great performance on downstream tasks (Pham et al., 2021). Large-scale pre-training also contributes to improving transferable and robust representations for downstream continual learning (Gallardo et al., 2021; Ramasesh et al., 2022).
+
+Regularization-based methods mainly focus on weight and function regularization. Weight regularization methods add penalties to the loss function, typically the penalty is a quadratic one (Kirkpatrick et al., 2017; Liu et al., 2018), and function regularization methods implement knowledge distillation on the intermediate or final output of the prediction function (Li & Hoiem, 2017; Lee et al., 2019). For function regularization methods, the teacher model is the frozen past model, and the student model is the current model. Besides, there are some theory works analyzing regularization-based methods. Evron et al. (2022) study the minimum norm estimator in CL under an over-parameterized and noise-free setup. Li et al. (2023) give a fixed design analysis of continual ridge regression for two-task linear regression. Zhao et al. (2024) consider a family of generalized  $\ell_2$ -regularization estimators and give some optimality analysis.
+
+Contrastive learning. Contrastive learning aims to learn representations that attract different views of the same image while repelling views from different images (Tian et al., 2020). Contrastive methods have been widely used in self-supervised learning and pre-training, showing superior performance on downstream tasks. The contrastive loss was first proposed in (Bromley et al., 1993) and then more formally defined in (Chopra et al., 2005) and (Hadsell et al., 2006). Later some theoretical analyses on the contrastive learning framework were provided in (Arora et al., 2019; Huang et al., 2023; Tan et al., 2023b;c;a; Zhang et al., 2023). There are various target losses in contrastive learning, for example, InfoNCE loss (van den Oord et al., 2019) is a widely adopted and efficient one. Notably, methods in this field have reached or even outperformed supervised learning methods (Khosla et al., 2020) for image classification. Representative approaches include SimCLR (Chen et al., 2020a), MoCo v1&v2 (He et al., 2020; Chen et al., 2020b). In this work, we employ the contrastive loss provided in (Khosla et al., 2020) for the contrastive continual learning framework to show some theoretical insights.
+
+Knowledge distillation. In various scenarios of continual learning, knowledge distillation is used to preserve information from the old model to the current model, contributing to mitigate catastrophic forgetting. Typically, knowl
+
+![](images/4afe501c61fef1824257bcf97466e983143824856c6ccd9c1cd02996bddf9068.jpg)  
+Figure 1. An illustration of contrastive continual learning framework. At the end of the previous task, we restore the previous model and values of losses. For the current task, augmentations are applied to both the buffered and the current data. Then the augmented data is passed through the current model and the previous frozen model to obtain representations. The target loss of contrastive continual learning is a weighted sum of contrastive loss and distillation loss with a distillation coefficient  $\lambda$ .
+
+edge distillation learns a small student model from a large teacher model with limited resources (Gou et al., 2021). Various kinds of knowledge can be transferred by knowledge distillation, including response-based knowledge which is the neural response of the last output layer of the teacher model (Hinton et al., 2015), feature-based knowledge like feature maps (Zagoruyko & Komodakis, 2016) and relation-based knowledge referring to relationships between different layers or between different samples (Passalis et al., 2020). Learning schemes of knowledge distillation include three streams, they are online distillation, offline distillation, and self-distillation. Among them, self-distillation considers the same structure between the teacher model and the student model (Zhang & Sabuncu, 2020; Mobahi et al., 2020).
+
+# 3. Problem Setup
+
+We are given a sequence of  $T$  supervised tasks, with each task presented sequentially, one after the other. For each task  $t$ , the training samples are assumed to be drawn from an unknown data distribution  $\mathcal{D}_t$ . The model can be updated after seeing each task. The goal of continual learning is to train a model  $f$  that performs well over all seen tasks.
+
+Supervised contrastive loss (Khosla et al., 2020) has shown its superiority over the cross-entropy loss in the standard supervised classification, and it is then introduced into the continual learning by  $\mathrm{Co}^2\mathrm{L}$  (Cha et al., 2021). Specifically, contrastive continual learning updates the model at each time step  $t$  according to two losses, the contrastive loss and the distillation loss, which measure the learning plasticity and memory stability, respectively.
+
+Contrastive loss. For each task  $t$ , we use  $\mu_t$  to denote the class distribution of task  $\mathcal{D}_t$ , and  $\mathcal{D}_c$  to denote the data distribution associated with each class  $c$ . In this paper, we consider a contrastive loss involving two similar samples  $x, x^+$
+
+i.i.d. drawn from the same class distribution  $\mathcal{D}_c$ . Meanwhile, there are several negative samples randomly picked from the whole data distribution  $\mathcal{D}_t$ . For simplicity, we only consider the case of one negative sample here, the case of multiple negative samples can be found in Appendix D and E. Therefore, the contrastive loss can be formulated as
+
+$$
+L_{\mathrm{con}}(f;\mathcal{D}_{t}) = \underset { \begin{array}{c}c^{+}\sim \mu_{t}\\ c^{-}\sim \mu_{t} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}  \ell   \big[f(x)^{\top}(f(x^{+}) - f(x^{-}))\big],
+$$
+
+where function  $\ell(v)$  is defined as  $\log(1 + \exp(-v))$  and embeddings are conventionally normalized, i.e.,  $\|f\| = 1$ .
+
+Distillation loss. Continual learning focuses on retaining previously acquired information while simultaneously learning new knowledge. In the specific context of contrastive continual learning, the model achieves knowledge preservation by keeping the model's ability to differentiate between similar and dissimilar (negative) samples. To do so, we first compute the similarity probability distribution as  $p(f;x,x^{+},x^{-}) = \mathrm{softmax}(f(x)^{\top}f(x^{+}),f(x)^{\top}f(x^{-}))$ , and then regulate the cross-entropy between the past similarity probability distribution and the current one (e.g., IRD loss in (Cha et al., 2021)). Specifically, for task  $t$ , we denote the distribution of all seen data by  $\mathcal{D}_{1:t - 1}\coloneqq \sum_{j = 1}^{t - 1}k_{tj}\mathcal{D}_j$  where we allow different tasks have different weights  $k_{tj} > 0$  with  $\sum_{j = 1}^{t - 1}k_{tj} = 1$ . Therefore, the distillation loss considered in this paper can be formulated as
+
+$$
+\begin{array}{l} L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {1: t - 1}\right) = \\ \begin{array}{c} \mathbb {E} _ {c ^ {+} \sim \mu_ {1: t - 1}} \mathbb {E} _ {x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}}} \left[ - \boldsymbol {p} (f _ {t - 1}; x, x ^ {+}, x ^ {-}) \cdot \log \boldsymbol {p} (f _ {t}; x, x ^ {+}, x ^ {-}) \right], \\ c ^ {-} \sim \mu_ {1: t - 1} x ^ {-} \sim \mathcal {D} _ {c ^ {-}} \end{array} \\ \end{array}
+$$
+
+where  $\mu_{1:t-1}$  represents the class distribution of  $\mathcal{D}_{1:t-1}$ .
+
+The total training loss of  $f_{t}$  on task  $t \geq 2$  is
+
+$$
+\begin{array}{l} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D} _ {t}\right) + \lambda \cdot L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {1: t - 1}\right), \\ \end{array}
+$$
+
+where  $\lambda$  is a hyper-parameter for balancing the two loss terms. For the first task  $t = 1$ , the training loss does not have the distillation term, i.e.,  $L_{\mathrm{train}}(f_1;\mathcal{D}_1) = L_{\mathrm{con}}(f_1;\mathcal{D}_1)$ .
+
+To evaluate the contrastive continual learning model, we use the total performance (test loss) of the final model  $f_{T}$  on all seen tasks, which can be formulated as
+
+$$
+L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) := \sum_ {t = 1} ^ {T} L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {t}\right).
+$$
+
+# 4. Theoretical Analysis
+
+Contrastive continual learning has demonstrated strong performance in practice. The focus of this paper is to examine its performance guarantees theoretically. In particular, our study aims to investigate the relationship between the test loss  $L_{\mathrm{test}}(f_T; \mathcal{D}_1, \ldots, \mathcal{D}_T)$  and the series of training losses  $L_{\mathrm{train}}(f_1; \mathcal{D}_1)$ ,  $L_{\mathrm{train}}(f_2; f_1, \mathcal{D}_2, \mathcal{D}_1)$ , ...,  $L_{\mathrm{train}}(f_T; f_{T-1}, \mathcal{D}_T, \mathcal{D}_{1:T-1})$ .
+
+According to the definition, despite the distillation loss terms, the training losses involve  $\{L_{\mathrm{con}}(f_t;\mathcal{D}_t)\}_{t = 1}^T$  while the test loss consists of  $\{L_{\mathrm{con}}(f_T;\mathcal{D}_t)\}_{t = 1}^T$ . To bridge the test loss and the training losses, we first provide the relationship between the contrastive losses of two consecutive models  $f_{t}$  and  $f_{t - 1}$  in the following lemma.
+
+Lemma 1. When  $t \geq 2$ , for any data distribution  $\mathcal{D}$ , the contrastive losses of current model  $f_{t}$  and previous model  $f_{t-1}$  can be connected via the distillation loss, i.e.,
+
+$$
+\begin{array}{l} L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \leq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta , \\ L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta^ {\prime}, \\ \end{array}
+$$
+
+where  $\alpha = \frac{2e^2}{1 + e^2}$ ,  $\beta = 2 - \alpha + \alpha \log \frac{\alpha}{2}$ , and  $\beta' = -\alpha \log (1 + e^2) - \alpha$ .
+
+The above lemma can be directly proved using the formulae for contrastive loss and distillation loss. A detailed proof is provided in the appendix due to the space limitation.
+
+According to Lemma 1, when considering  $\mathcal{D} = \mathcal{D}_t$  ( $t \leq T$ ), a connection between  $L_{\mathrm{con}}(f_T; \mathcal{D}_t)$  and  $L_{\mathrm{con}}(f_{T-1}; \mathcal{D}_t)$  can be established. Similarly, a link between  $L_{\mathrm{con}}(f_{T-1}; \mathcal{D}_t)$  and  $L_{\mathrm{con}}(f_{T-2}; \mathcal{D}_t)$  can be drawn, and so on. This approach allows us to build a bridge between  $L_{\mathrm{con}}(f_T; \mathcal{D}_t)$  and  $L_{\mathrm{con}}(f_t; \mathcal{D}_t)$  for any given  $t$ , which are the components of test loss and training losses, respectively. Thus, with Lemma 1, we can now derive the relationship between the test loss  $L_{\mathrm{test}}(f_T; \mathcal{D}_1, \ldots, \mathcal{D}_T)$  and the series
+
+of training losses  $L_{\mathrm{train}}(f_1; \mathcal{D}_1)$ ,  $L_{\mathrm{train}}(f_2; f_1, \mathcal{D}_2, \mathcal{D}_1)$ , ...,  $L_{\mathrm{train}}(f_T; f_{T-1}, \mathcal{D}_T, \mathcal{D}_{1:T-1})$ . Our results are presented in the following main theorem.
+
+Theorem 1. For the contrastive continual learning involving  $T \geq 2$  tasks, the test loss of the final model  $f_{T}$  can be bounded via a linear combination of the training losses associated with each task. More specifically, the following two bounds are applicable.
+
+(1) Upper bound:
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) \leq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) \\ + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta , \\ \end{array}
+$$
+
+(2) Lower bound:
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) \geq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) \\ + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta^ {\prime}, \\ \end{array}
+$$
+
+where
+
+$$
+\left\{ \begin{array}{l} \alpha = \frac {2 e ^ {2}}{1 + e ^ {2}}, \\ \gamma_ {t} (\lambda) = \min  \left(\{\frac {1}{t} \} \cup \{\lambda k _ {t j} \} _ {j = 1} ^ {t - 1}\right), \\ \gamma_ {t} ^ {\prime} (\lambda) = \max  \left(\{1 \} \cup \{\lambda k _ {t j} \} _ {j = 1} ^ {t - 1}\right), \\ \eta = (2 - \alpha + \alpha \log \frac {\alpha}{2}) \frac {T - 1 - T \alpha + (\alpha) ^ {T}}{(1 - \alpha) ^ {2}} \\ \qquad + \sum_ {t = 2} ^ {T} \alpha^ {T - t} (1 - \frac {1}{\gamma_ {t} (\lambda)}) \min  _ {f} L _ {\mathrm {c o n}} (f; \mathcal {D} _ {t}), \\ \eta^ {\prime} = - (\alpha \log (1 + e ^ {2}) + \alpha) \frac {T - 1 - T \alpha + (\alpha) ^ {T}}{(1 - \alpha) ^ {2}}. \end{array} \right.
+$$
+
+The proof for the theorem can be found in the appendix. It can be concluded from Theorem 1 that, the performance of the final model  $f_{T}$  on all  $T$  tasks, namely  $L_{\mathrm{test}}(f_T;\mathcal{D}_1,\ldots ,\mathcal{D}_T)$ , can be well bounded by training losses on all seen tasks, suggesting that minimizing  $L_{\mathrm{train}}(f_t;f_{t - 1},\mathcal{D}_t,\mathcal{D}_{1:t - 1})$  during each task  $t$  can help to improve the performance of the final model on all seen tasks. Note that there is also a lower bound of  $L_{\mathrm{test}}(f_T;\mathcal{D}_1,\dots ,\mathcal{D}_T)$ , which means that minimizing the training loss during each task  $t$  is necessary. In particular, given that the training loss is a weighted sum of contrastive loss and distillation loss, these bounds also emphasize the necessity of both contrastive loss and distillation loss in effectively learning a contrastive continual learning model.
+
+Taking inspiration from Theorem 1, we can infer that pretraining can benefit continual learning. The coefficients of training losses associated with each task become fixed if  $\lambda$  exceeds a certain value. For example, the denominators of the coefficients of training losses, i.e.,  $\{\gamma_t(\lambda)\}_{t=2}^T$  for the upper bound become constant values if  $\lambda$  is large. Note that the component  $\alpha > 1$ , then the weight  $\alpha^{T - t} / \gamma_t(\lambda)$  for
+
+the training loss of task  $t$  decreases greatly as  $t$  increases, reducing the importance of task  $t$  in the bounds. Therefore, we have the following corollary, which shows that improved training performance of initial tasks contributes more to improving the later models' performance guarantees than that of later tasks. This aligns with the idea that pre-training can benefit continual learning, as observed in previous literature (Wang et al., 2022; Hu et al., 2022).
+
+After choosing a suitable distillation coefficient  $\lambda$ , training performances of initial tasks in contrastive continual learning contribute more to improving the overall performance of the final model on all tasks compared with that of the latter ones, explaining that a well pre-trained network can benefit continual learning.
+
+We conclude from the statement above that small changes in the training performance on the first task may lead to great changes in the overall performance of the final model. For example, the weight of  $L_{\mathrm{train}}(f_1; \mathcal{D}_1)$  in the upper bound increases greatly when adding more tasks, and a large value of  $L_{\mathrm{train}}(f_1; \mathcal{D}_1)$  implies a potential great increase of the upper bound. Therefore, well-trained initial models with small training losses in continual learning can be beneficial.
+
+# 5. Further Discussion on the Distillation Coefficient  $\lambda$
+
+# 5.1. Analysis on the distillation coefficient
+
+Inspired by additional analysis on Theorem 1, we find that the suitable distillation coefficient  $\lambda$  is correlated with the weights  $\{\{k_{tj}\}_{j=1}^{t-1}\}_{t=1}^T$  that depends on the data distributions. Specifically, we would like to choose the suitable value of  $\lambda$  as the turning point of the upper bound to get better theoretical guarantees. We define the turning point as the minimum value of  $\lambda$  at which the upper bound no longer decreases. Once the distillation coefficient  $\lambda$  exceeds the value of this turning point, the upper bound becomes a fixed value that is no longer influenced by  $\lambda$ , as illustrated in Figure 2. In the following part of this section, we will present several examples and calculate the corresponding turning point values. This may help us better understand the choice of distillation coefficient ( $\lambda = 1$ ) employed in the experiments of  $\mathrm{Co}^2\mathrm{L}$  (Cha et al., 2021).
+
+We begin by clarifying that in the contrastive continual learning framework, choosing a fixed distillation coefficient as one for all tasks is favorable for achieving a balance between learning new tasks and preserving old knowledge. Specifically, with well-constructed weights  $\{k_{tj}\}_{j=1}^{t-1}$  for task  $t$ , the suggested  $\lambda$  value for learning tends to stay close to one, thereby contributing to a tighter upper bound. To illustrate this point, we provide an example below.
+
+Example 1. Assume that there are five tasks, each task with data distribution  $\mathcal{D}_t$ ,  $t \in \{1, \ldots, 5\}$ , and we have
+
+![](images/00e10135e2603949c0db294752d85a195a3af634453317f7fadc5bd80497fc14.jpg)  
+Figure 2. An illustration of Example 2. The suggesting  $\lambda$  for  $\rho = 0.95$  or  $\rho = 1.05$  stays close to one.
+
+corresponding models  $\{f_t\}_{t=1}^5$ . We make the assumption that these models obtain the same value of training loss, i.e.,
+
+$$
+L _ {\text {t r a i n}} \left(f _ {5}; f _ {4}, \mathcal {D} _ {5}, \mathcal {D} _ {1: 4}\right) = \dots = L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right).
+$$
+
+Weights of tasks are given as follows, i.e., for  $t \geq 2$
+
+$$
+k _ {t j} = \left\{ \begin{array}{l l} \frac {2}{t}, & j = 1, \\ \frac {1}{t}, & \text {e l s e}. \end{array} \right.
+$$
+
+These weights can be considered well-constructed, as they are uniform across different tasks, except for the weight of the first task which has a larger value than the others. Indeed, this strategy emphasizes the importance of the first task which is typically regarded as a base task.
+
+Then according to our Theorem 1, the value of appropriate  $\lambda$  is suggested to be close to one to get a tighter upper bound on the overall performance for the final model.
+
+Note that in Example 1, we have assumed that the values of total training losses of different tasks remain the same, which may not align with realistic settings. To provide a more realistic illustration, we construct another example with adaptive ratios between the training losses of different tasks. Specifically, we allow the value of the training loss of each task to either increase or decrease with the same ratio  $\rho$  close to one. The weights construction strategy is correlated to  $\rho$  to maintain an alignment with the changing training loss. Constructed in this way, the following example illustrates how the appropriate value of  $\lambda$  remains close to one even when  $\rho$  fluctuates around one, and further achieves a tighter upper bound. This observation inspires us to consider adjusting the value of  $\lambda$  around one.
+
+Example 2. Assume that there are five tasks, each task with data distribution  $\mathcal{D}_t$ ,  $t \in \{1, \ldots, 5\}$ , and we have corresponding models  $\{f_t\}_{t=1}^5$ . We assume that the value of the training loss of each task has a fixed ratio  $\rho \approx 1$ , i.e.,
+
+$$
+L _ {\text {t r a i n}} \left(f _ {5}; f _ {4}, \mathcal {D} _ {5}, \mathcal {D} _ {1: 4}\right) = \dots = \rho^ {4} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right).
+$$
+
+Weights of different tasks are given by a biased strategy related to  $\rho$ , i.e., for  $t \geq 2$
+
+$$
+k _ {t j} = \left\{ \begin{array}{l l} 1 - \frac {t - 2}{\rho t}, & j = 1, \\ \frac {1}{\rho t}, & \text {e l s e .} \end{array} \right.
+$$
+
+Then according to Theorem 1, the value of appropriate  $\lambda$  is suggested to get close to one as  $\rho$  changes slightly around one, ensuring a tight upper bound on the overall performance of the final model. As illustrated in Figure 2, setting  $\rho = 0.95$  or  $\rho = 1.05$  implies that suggesting  $\lambda$  value remains close to one. The settings in Example 2 are more realistic and can closely resemble the  $\mathrm{Co}^2\mathrm{L}$  configuration (Cha et al., 2021). Thus this example can further support the rationale for choosing  $\lambda = 1$  in  $\mathrm{Co}^2\mathrm{L}$ .
+
+However, in an extreme case of the weights construction, an undesirable result may occur with a value of  $\lambda$  greater than one. This insight suggests that data distribution may play a crucial role in selecting a suitable  $\lambda$  since the weights  $\{k_{tj}\}_{j=1}^{t-1}$  are strongly correlated with the data distribution of task  $t$ . As illustrated in the following example, when there exists a weight significantly smaller than the other weights, the value of appropriate  $\lambda$  would deviate from one. In such a case, persistently using  $\lambda = 1$  may not achieve the best theoretical guarantees.
+
+Example 3. Take the same assumption from Example 1, excluding the weights construction strategy. Weights of tasks are given as follows, i.e., for  $t \geq 3$ ,
+
+$$
+k _ {t j} = \left\{ \begin{array}{l l} \frac {2 . 9}{t}, & j = 1, \\ \frac {0 . 1}{t}, & j = 2, \\ \frac {1}{t}, & \text {e l s e}. \end{array} \right.
+$$
+
+For the second task, we set  $k_{21} = 1$ .
+
+Then according to our Theorem 1, the suggesting value of appropriate  $\lambda$  is close to ten in Example 3. If we choose  $\lambda = 1$ , the upper bound may get large and fail to provide the best guarantees. The failure of Example 3 is attributed to a change in the weights construction strategy, highlighting a substantial relationship between the suitable value of distillation coefficient  $\lambda$  and data distributions.
+
+# 5.2. Adaptive selection of distillation coefficients
+
+Inspired by our theoretical analysis, we are curious whether it is possible to provide better theoretical guarantees by dynamically adjusting distillation coefficients. Interestingly, the analysis of Theorem 1 can be adapted to the case of adaptive distillation coefficient  $\lambda_{t}$ , simply by replacing  $\lambda$  with  $\lambda_{t}$  for the coefficient of training loss of task  $t$  in the bounds. Then we can conclude from Theorem 1 for the new case that, increasing  $\lambda_{t}$  for each task  $t$  with a threshold strategy can provide better guarantees.
+
+Note that our target is to get a set of distillation coefficients  $\{\lambda_t\}_{t=2}^T$  that tighten the upper bound in Theorem 1. Therefore, we want to adaptively select  $\lambda_t$  for task  $t$  to achieve this goal. We now propose our theoretical explanations for the adaptive selection of distillation coefficients. First, we give some related definitions. Suppose there are  $T$  tasks for continual learning setting. For each task  $t \geq 2$ , we denote  $\lambda_t$  as the task-specific distillation coefficient, and define
+
+$$
+U _ {t} = \sum_ {j = 2} ^ {t} \frac {\alpha^ {t - j}}{\gamma_ {j} (\lambda_ {t})} L _ {\text {t r a i n}} \left(f _ {j}; f _ {j - 1}, \mathcal {D} _ {j}, \mathcal {D} _ {1: j - 1}\right).
+$$
+
+Motivated by Theorem 1 and explanations above, at the end of task  $t$ , if the calculated  $U_{t}$  has a relatively large value, then a slight increase in  $\lambda_{t + 1}$  around one can be beneficial for improving the upper bound. Inspired by this, we will maintain an extra set of task-specific threshold values  $\{u_t\}_{t = 2}^T$  where  $u_{t} > 0$ , and a set of update momentums  $\{\Delta_t\}_{t = 1}^T$  where  $\Delta_t \geq 0$ . After training task  $t$ , if  $U_{t} > u_{t}$  we let  $\lambda_{t + 1} = \lambda_t + \Delta_t$ , else,  $\lambda_{t + 1} = \lambda_t$ . Then the total training loss of each task  $t$  for model  $f_{t}$  can be rewritten as
+
+$$
+\begin{array}{l} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D} _ {t}\right) + \lambda_ {t} \cdot L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {1: t - 1}\right). \\ \end{array}
+$$
+
+The following theorem provides a theoretical explanation for the benefits of the adaptive  $\lambda_{t}$  selection protocol above.
+
+Theorem 2. Assume that the training loss of each task is larger than zero. For each task  $t \geq 2$ , there exists a task-specific constant  $u_t > 0$ . If we have
+
+$$
+U _ {t} = \sum_ {j = 2} ^ {t} \frac {\alpha^ {t - j}}{\gamma_ {j} (\lambda_ {t})} L _ {t r a i n} (f _ {j}; f _ {j - 1}, \mathcal {D} _ {j}, \mathcal {D} _ {1: j - 1}) > u _ {t},
+$$
+
+where  $\alpha, \{\gamma_j(\lambda_t)\}_{j=2}^t$  are defined in Theorem 1, then we increase  $\lambda_t$  by  $\Delta_t$ , i.e.,  $\lambda_{t+1} = \lambda_t + \Delta_t$ , else,  $\lambda_{t+1} = \lambda_t$ . By choosing  $\lambda_t$  in this way, we get tighter upper bounds.
+
+It can be concluded from Theorem 2 that, increasing  $\lambda_{t}$  by a threshold strategy about the performance of the current model can help make the upper bound tighter. Moreover, Theorem 2 can also provide theoretical support for the choosing strategy of  $\lambda$  in previous examples.
+
+# 5.3. Contrastive Incremental Learning with Adaptive distillation (CILA)
+
+Note that we lack access to the construction of weights during the real training phase. Consequently, computing  $U_{t}$  is not available for task  $t$ , and estimating  $\{k_{tj}\}_{j=1}^{t-1}$  may be also inaccessible due to the high computing load and low estimating accuracy. However, according to our explanations above, it is suggested that the adaptive  $\lambda_{t}$  for each task  $t$
+
+stays around one with a relatively larger value. Hence, we can find an easy-to-get and adaptive metric to replace  $\lambda_{t}$ . Before we introduce the chosen metric, we first give some related definitions. We first define the empirical contrastive loss (Khosla et al., 2020). For each task  $t$ , we denote  $D_{t}$  as the given batch of  $N$  training samples  $\{(x_{t,i},c_{t,i})\}_{i = 1}^{N}$  and the augmented batch is  $\{(\tilde{x}_{t,i},\tilde{c}_{t,i})\}_{i = 1}^{2N}$  which is generated by making two randomly augmented versions of  $x_{t,i}$  as  $\tilde{x}_{t,2i - 1}$  and  $\tilde{x}_{t,2i}$  with  $\tilde{c}_{t,2i - 1} = \tilde{c}_{t,2i} = c_{t,i}$ . The augmented samples are mapped to a  $d$ -dimensional Euclidean sphere by a model  $f_{t}$ . Denote  $\boldsymbol {z}_{t,i} = f_t(\tilde{\boldsymbol{x}}_{t,i})$ , then the empirical contrastive loss can be formulated as
+
+$$
+\begin{array}{l} \hat {L} _ {\text {c o n}} \left(f _ {t}; D _ {t}\right) \\ = \sum_ {i = 1} ^ {2 N} \frac {- 1}{| p _ {t , i} |} \sum_ {j \in p _ {t, i}} \log \left(\frac {\exp (z _ {t , i} ^ {\top} z _ {t , j} / \tau)}{\sum_ {k \neq i} \exp (z _ {t , i} ^ {\top} z _ {t , k} / \tau)}\right), \\ \end{array}
+$$
+
+where  $p_{t,i} = \{j\in \{1,\dots ,2N\} |j\neq i,c_{t,j} = c_{t,i}\}$  and  $\tau >0$  is the temperature hyperparameter. Then we define the empirical distillation loss (Cha et al., 2021). We first define a similarity vector
+
+$$
+\begin{array}{l} \boldsymbol {p} (f _ {t}, \tau ; \tilde {\boldsymbol {x}} _ {i}) = \operatorname {s o f t m a x} (\boldsymbol {z} _ {t, i} ^ {\top} \boldsymbol {z} _ {t, 1} / \tau , \dots , \boldsymbol {z} _ {t, i} ^ {\top} \boldsymbol {z} _ {t, i - 1} / \tau , \\ \boldsymbol {z} _ {t, i} ^ {\top} \boldsymbol {z} _ {t, i + 1} / \tau , \dots \boldsymbol {z} _ {t, i} ^ {\top} \boldsymbol {z} _ {t, 2 N} / \tau). \\ \end{array}
+$$
+
+Then the empirical distillation loss is formulated as
+
+$$
+\begin{array}{l} \hat {L} _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, D _ {t}\right) \\ = \sum_ {i = 1} ^ {2 N} - \boldsymbol {p} \left(f _ {t - 1}, \tau^ {*}; \tilde {x} _ {t, i}\right) \cdot \log \boldsymbol {p} \left(f _ {t}, \tau ; \tilde {x} _ {t, i}\right). \\ \end{array}
+$$
+
+Here, both  $\tau^{*}$  and  $\tau$  will remain fixed for all tasks. Actually, when constructing the experiment, we found that the ratio
+
+$$
+\sum_ {j = 2} ^ {t - 1} \hat {L} _ {\mathrm {d i s}} \left(f _ {j}; f _ {j - 1}, D _ {j}\right) / \sum_ {j = 2} ^ {t - 1} \hat {L} _ {\mathrm {c o n}} \left(f _ {j}; D _ {j}\right),
+$$
+
+is stable and stays close to one as task index  $t \geq 3$  varies. This ratio is easy to get during the training procedure and is aligned with the idea that the distillation coefficient is suggested to stay close to one and varies according to the data distribution which depends on the task index. Therefore, inspired by Example 2 and 3, an applicable method is to use
+
+$$
+\lambda_ {t} = \max  (1, \kappa \sum_ {j = 2} ^ {t - 1} \hat {L} _ {\mathrm {d i s}} (f _ {j}; f _ {j - 1}, D _ {j}) / \sum_ {j = 2} ^ {t - 1} \hat {L} _ {\mathrm {c o n}} (f _ {j}; D _ {j})),
+$$
+
+for task  $t \geq 3$ , where  $\kappa$  is a balancing distillation coefficient, and for the second task, we use  $\lambda_2 = 1$ . More details can be found in Algorithm 1. In the following section, we will conduct several experiments.
+
+# Algorithm 1 CILA: Contrastive Incremental Learning with Adaptive distillation
+
+Input: Buffer size  $B$ , a sequence of training sets  $\{D_t\}_{t=1}^T$ , base distillation coefficient  $\lambda_0$ , balancing distillation coefficient  $\kappa$ .
+
+1: Initialize model  $f_{0}$  and set buffer  $\mathcal{M}\gets \phi$  
+2: for task  $t = 1, \dots, T$  do  
+3: Construct dataset  $D \gets D_t \cup \mathcal{M}$ ;  
+4: Initialize model  $f_{t} \gets f_{t-1}$ ;  
+5: Compute  $L$  by  $L\gets \tilde{L}_{\mathrm{con}}(f_t;D)$  
+6: if  $t > 1$  then  
+7: Adaptively update  $\lambda_{t}$  by
+
+$$
+\lambda_ {t} \leftarrow \max  (\lambda_ {0}, \kappa \cdot \frac {\sum_ {j = 2} ^ {t - 1} \hat {L} _ {\mathrm {d i s}} (f _ {j} ; f _ {j - 1} , D _ {j})}{\sum_ {j = 2} ^ {t - 1} \hat {L} _ {\mathrm {c o n}} (f _ {j} ; D _ {j})});
+$$
+
+8: Update  $L$  by
+
+$$
+L \leftarrow L + \lambda_ {t} \cdot \hat {L} _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, D\right);
+$$
+
+9: end if
+
+10: Update  $f_{t}$  by SGD;  
+11: Collect buffer samples until  $|\mathcal{M}| = B$  
+12: end for
+
+# 6. Experiment
+
+Learning settings and datasets. We conducted experiments on three basic continual learning scenarios, Class-IL, Task-IL, and Domain-IL (van de Ven & Tolias, 2019). Each scenario was evaluated using different datasets. Specifically, for Class-IL and Task-IL, we utilized Seq-CIFAR-10 and Seq-Tiny/ImageNet datasets. Seq-CIFAR-10 is a modified version of the CIFAR-10 (Krizhevsky, 2009) dataset, where it is divided into 5 distinct subsets, each comprising two classes. Similarly, Seq-Tiny/ImageNet is an adapted version of the Tiny/ImageNet (Le & Yang, 2015) dataset, where the 200 classes are split into 10 separate sets, each containing 20 classes. The order of splits in Seq-CIFAR-10 and Seq-Tiny/ImageNet remains consistent across multiple runs.
+
+For Domain-IL, we employed R-MINST, which is a variant of the MNIST (Lecun et al., 1998) dataset. In R-MINST, the original images are randomly rotated by an angle between 0 and  $\pi$ . R-MINST consists of 20 tasks, with each task corresponding to a randomly selected rotation angle. During the training process, samples from different tasks with the same digital class are treated as distinct classes.
+
+In summary, our experiments covered Class-IL, Task-IL, and Domain-IL scenarios, utilizing Seq-CIFAR-10, Seq-Tiny-ImageNet, and R-MINST datasets, respectively.
+
+Baselines. We compare our contrastive continual learning algorithm with replay-based continual learning baselines, including ER (Riemer et al., 2019), GEM (Lopez-Paz & Ranzato, 2017), A-GEM (Chaudhry et al., 2018), iCaRL (Rebuffi et al., 2017), FDR (Benjamin et al., 2019), GSS (Aljundi et al., 2019), HAL (Chaudhry et al., 2019), DER
+
+Table 1. Classification accuracies for Seq-CIFAR-10, Seq-Tiny-ImageNet, and R-MNIST on replay-based baselines and our algorithm. All results are averaged over ten independent trials. The best performance is marked as bold.  
+
+<table><tr><td>Dataset</td><td colspan="4">Seq-CIFAR-10</td><td colspan="4">Seq-Tiny/ImageNet</td><td colspan="2">R-MNIST</td></tr><tr><td>Scenario</td><td colspan="2">Class-IL</td><td colspan="2">Task-IL</td><td colspan="2">Class-IL</td><td colspan="2">Task-IL</td><td colspan="2">Domain-IL</td></tr><tr><td>Buffer</td><td>200</td><td>500</td><td>200</td><td>500</td><td>200</td><td>500</td><td>200</td><td>500</td><td>200</td><td>500</td></tr><tr><td>ER</td><td>44.79±1.86</td><td>57.74±0.27</td><td>91.19±0.94</td><td>93.61±0.27</td><td>8.49±0.16</td><td>9.99±0.29</td><td>38.17±2.00</td><td>48.64±0.46</td><td>93.53±1.15</td><td>94.89±0.95</td></tr><tr><td>GEM</td><td>25.54±0.76</td><td>26.20±1.26</td><td>90.44±0.94</td><td>92.16±0.64</td><td>-</td><td>-</td><td>-</td><td>-</td><td>89.86±1.23</td><td>92.55±0.85</td></tr><tr><td>A-GEM</td><td>20.04±0.34</td><td>22.67±0.57</td><td>83.88±1.49</td><td>89.48±1.45</td><td>8.07±0.08</td><td>8.06±0.04</td><td>22.77±0.03</td><td>25.33±0.49</td><td>89.03±2.76</td><td>89.04±7.01</td></tr><tr><td>iCaRL</td><td>49.02±3.20</td><td>47.55±3.95</td><td>88.99±2.13</td><td>88.22±2.62</td><td>7.53±0.79</td><td>9.38±1.53</td><td>28.19±1.47</td><td>31.55±3.27</td><td>-</td><td>-</td></tr><tr><td>FDR</td><td>30.91±2.74</td><td>28.71±3.23</td><td>91.01±0.68</td><td>93.29±0.59</td><td>8.70±0.19</td><td>10.54±0.21</td><td>40.36±0.68</td><td>49.88±0.71</td><td>93.71±1.51</td><td>95.48±0.68</td></tr><tr><td>GSS</td><td>39.07±5.59</td><td>49.73±4.78</td><td>88.80±2.89</td><td>91.02±1.57</td><td>-</td><td>-</td><td>-</td><td>-</td><td>87.10±7.23</td><td>89.38±3.12</td></tr><tr><td>HAL</td><td>32.36±2.70</td><td>41.79±4.46</td><td>82.51±3.20</td><td>84.54±2.36</td><td>-</td><td>-</td><td>-</td><td>-</td><td>89.40±2.50</td><td>92.35±0.81</td></tr><tr><td>DER</td><td>61.93±1.79</td><td>70.51±1.67</td><td>91.40±0.92</td><td>93.40±0.39</td><td>11.87±0.78</td><td>17.75±1.14</td><td>40.22±0.67</td><td>51.78±0.88</td><td>96.43±0.59</td><td>97.57±1.47</td></tr><tr><td>DER++</td><td>64.88±1.17</td><td>72.70±1.36</td><td>91.92±0.60</td><td>93.88±0.50</td><td>10.96±1.17</td><td>19.38±1.41</td><td>40.87±1.16</td><td>51.91±0.68</td><td>95.98±1.06</td><td>97.54±0.43</td></tr><tr><td>Co2L</td><td>65.57±1.37</td><td>74.26±0.77</td><td>93.43±0.78</td><td>95.90±0.26</td><td>13.88±0.40</td><td>20.12±0.42</td><td>42.37±0.74</td><td>53.04±0.69</td><td>97.90±1.92</td><td>98.65±0.31</td></tr><tr><td>CILA (Ours)</td><td>67.06±1.59</td><td>76.03±0.79</td><td>94.29±0.24</td><td>96.40±0.21</td><td>14.55±0.39</td><td>20.64±0.59</td><td>44.15±0.70</td><td>54.13±0.72</td><td>98.36±0.45</td><td>98.76±0.22</td></tr></table>
+
+(Buzzega et al., 2020), DER++ (Buzzega et al., 2020), and  $\mathrm{Co}^2\mathrm{L}$  (Cha et al., 2021).
+
+Details of training. Following the configuration of previous studies, we trained ResNet-18 on the Seq-CIFAR-10 and Tiny-ImageNet datasets. We implemented a simple network with convolution layers for the R-MNIST dataset. In our training process, we employed buffers of sizes 200 and 500. The base distillation coefficient  $\lambda_0$  is set as one following the default configuration of  $\mathrm{Co}^2\mathrm{L}$  (Cha et al., 2021).
+
+Evaluation. Like  $\mathrm{Co}^{2}\mathrm{L}$ , CILA follows the idea of "first pre-training, then linear probing". Thus, unlike the joint representation-classifier training approaches, an additional classifier needs to be trained on top of the frozen representations. To ensure a fair comparison, the classifier is trained using only the samples from the last task and buffered samples, leveraging the representations learned by CILA. To mitigate the challenges posed by class imbalance, we employ a class-balanced sampling strategy during the training of a linear classifier. The strategy involves the following steps. We first uniformly select a class from the available set of classes. This ensures that each class has an equal chance of being chosen. Once a class is selected, we further uniformly sample an instance from that specific class. This guarantees that all instances within the chosen class are equal to be selected.
+
+For all experiments, a linear classifier is trained for a fixed number of epochs and we adopt 100 epochs to align with prior work. After training, the classification test accuracy is reported based on the predictions made by this classifier.
+
+Main results. In Table 1, our method outperforms all baselines in different scenarios, datasets, and buffer sizes, especially compared with  $\mathrm{Co}^2\mathrm{L}$ . This result verifies the su
+
+Table 2. Accuracies on Seq-CIFAR-10 with 200 buffer samples.  
+
+<table><tr><td>Adaptive Method</td><td>Class-IL</td><td>Task-IL</td></tr><tr><td>Co2L</td><td>65.57</td><td>93.43</td></tr><tr><td>Pure-adapted</td><td>66.52</td><td>94.27</td></tr><tr><td>Min-adapted</td><td>66.36</td><td>94.21</td></tr><tr><td>Max-adapted</td><td>67.06</td><td>94.29</td></tr></table>
+
+periority of our adaptive method and supports our theories strongly. Our algorithm successfully reaches a balance between learning plasticity and memory stability in continual learning. Under appropriate adaptation of the distillation coefficients, we also mitigate the catastrophic forgetting problem. Besides, the power of adaptation also impacts the performance of the learned continual learner, we will talk about it in the following ablation study.
+
+# 7. Ablation Studies
+
+We conduct ablation experiments to verify the effectiveness of adaptive distillation coefficients. We consider two setups, Class-IL and Task-IL, and perform experiments on Seq-CIFAR-10 with three variants of adapted  $\lambda_{t}$  for each task  $t$ . Variants include
+
+(1) pure-adapted
+
+$$
+\lambda_ {t, \text {p u r e}} = \kappa \sum_ {j = 2} ^ {t - 1} \hat {L} _ {\text {d i s}} \left(f _ {j}, f _ {j - 1}; D _ {j}\right) / \sum_ {j = 2} ^ {t - 1} \hat {L} _ {\text {c o n}} \left(f _ {j}; D _ {j}\right),
+$$
+
+(2) min-adapted
+
+$$
+\lambda_ {t, \min } = \min  (1, \lambda_ {t, \text {p u r e}}),
+$$
+
+and (3) max-adapted
+
+$$
+\lambda_ {t, \max } = \max  (1, \lambda_ {t, p u r e}),
+$$
+
+where  $\kappa$  is a balancing distillation coefficient. For our ablation experiments, we train the linear classifier on top of the representations with 200 buffer samples.
+
+As shown in Table 2, methods with adaptive distillation coefficients show superior performance compared with the method with a fixed distillation coefficient with about  $1\%$  improvement on both settings. As the adaptive  $\lambda_{t}$  increases with moderate limitations, the performance of the model boosts with an obvious improvement on Class-IL. This verifies our assumption based on the theoretical results that continual learners with larger adaptive distillation coefficients show greater performance.
+
+# 8. Conclusion
+
+Contrastive learning has demonstrated remarkable performance in the field of continual learning, although there remains a lack of theoretical explanations. In this study, we aim to fill this gap by introducing theoretical performance guarantees for the final model in contrastive continual learning. Drawing inspirations from a detailed theoretical analysis, we propose the utilization of adaptive distillation coefficients for the distillation training loss in contrastive continual learning. Through comprehensive experiments conducted in diverse settings for continual learning, our approach surpasses baseline methods in terms of performance. We anticipate that our work can establish a robust foundation for continual learning from a representation perspective, and potentially spark further theoretical insights into the realm of contrastive continual learning.
+
+# Acknowledgment
+
+Weiran Huang is supported by 2023 CCF-Baidu Open Fund and Microsoft Research Asia. Chuanlong Xie is supported by the National Nature Science Foundation of China (No.12201048).
+
+We would also like to express our sincere gratitude to the reviewers of ICML 2024 for their insightful and constructive feedback. Their valuable comments have greatly contributed to improving the quality of our work.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
+
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+
+# Appendix
+
+# A. Proof of Lemma 1
+
+We recall Lemma 1.
+
+Lemma 1. When  $t \geq 2$ , for any data distribution  $\mathcal{D}$ , the contrastive losses of current model  $f_{t}$  and previous model  $f_{t-1}$  can be connected via the distillation loss, i.e.,
+
+$$
+L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \leq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta ,
+$$
+
+$$
+L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta^ {\prime},
+$$
+
+where  $\alpha = \frac{2e^2}{1 + e^2}$ ,  $\beta = 2 - \alpha + \alpha \log \frac{\alpha}{2}$ , and  $\beta' = -\alpha \log (1 + e^2) - \alpha$ .
+
+Proof. For model  $f$ , denote the data pair  $\pmb{x} = (x, x^{+}, x^{-})$  to simplify the proof, we first define  $v(f; \pmb{x}) = f(x)^{\top}(f(x^{+}) - f(x^{-}))$ , and
+
+$$
+q (f; \boldsymbol {x}) = \frac {\exp (v (f ; \boldsymbol {x}))}{1 + \exp (v (f ; \boldsymbol {x}))}.
+$$
+
+For models  $f_{t}$  and  $f_{t-1}$ , the function  $\ell(v) = \log(1 + \exp(-v))$ , and  $\pmb{p}(f; \pmb{x}) = \mathrm{softmax}(f(x)^{\top}f(x^{+}), f(x)^{\top}f(x^{-}))$ , we have the following equation
+
+$$
+\begin{array}{l} - \boldsymbol {p} (f _ {t - 1}; \boldsymbol {x}) \cdot \log \boldsymbol {p} (f _ {t}; \boldsymbol {x}) \\ = - q \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(q \left(f _ {t}; \boldsymbol {x}\right)\right) - \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \log \left(1 - q \left(f _ {t}; \boldsymbol {x}\right)\right) \\ = q \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(1 + \exp \left(- v \left(f _ {t}; \boldsymbol {x}\right)\right)\right) + \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \log \left(1 + \exp \left(v \left(f _ {t}; \boldsymbol {x}\right)\right)\right) \\ = q \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(1 + \exp \left(- v \left(f _ {t}; \boldsymbol {x}\right)\right)\right) + \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \log \left(1 + \exp \left(- v \left(f _ {t}; \boldsymbol {x}\right)\right)\right) + \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \log \left(\exp \left(v \left(f _ {t}; \boldsymbol {x}\right)\right)\right) \\ = \ell (v (f _ {t}; \boldsymbol {x})) + (1 - q (f _ {t - 1}; \boldsymbol {x})) v (f _ {t}; \boldsymbol {x}). \\ \end{array}
+$$
+
+Then for any data distribution  $\mathcal{D}$ , we have
+
+$$
+\begin{array}{l} L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) \\ = \underset { \begin{array}{c} c ^ {+} \sim \mu x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ c ^ {-} \sim \mu x ^ {-} \sim \mathcal {D} _ {c ^ {-}} \end{array} } {\mathbb {E}} \underset { \begin{array}{c} c ^ {+} \sim \mu x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ c ^ {-} \sim \mu x ^ {-} \sim \mathcal {D} _ {c ^ {-}} \end{array} } {\mathbb {E}} - \boldsymbol {p} (f _ {t - 1}; \boldsymbol {x}) \cdot \log \boldsymbol {p} (f _ {t}; \boldsymbol {x}) \\ = \underset { \begin{array}{c} c ^ {+} \sim \mu   x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ c ^ {-} \sim \mu   x ^ {-} \sim \mathcal {D} _ {c ^ {-}} \end{array} } {\mathbb {E}} \underset { \begin{array}{c} c ^ {+} \sim \mu   x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ c ^ {-} \sim \mu   x ^ {-} \sim \mathcal {D} _ {c ^ {-}} \end{array} } {\mathbb {E}} \ell (v (f _ {t}; \boldsymbol {x})) + \underset { \begin{array}{c} c ^ {+} \sim \mu   x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ c ^ {-} \sim \mu   x ^ {-} \sim \mathcal {D} _ {c ^ {-}} \end{array} } {\mathbb {E}} (1 - q (f _ {t - 1}; \boldsymbol {x})) v (f _ {t}; \boldsymbol {x}) \\ = L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu   x,x^{+}\sim \mathcal{D}_{c^{+}}\\ c^{-}\sim \mu   x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}  \underset { \begin{array}{c}(1 - q(f_{t - 1};\boldsymbol {x}))v(f_{t};\boldsymbol {x}). \end{array} }{\mathbb{E}}. \\ \end{array}
+$$
+
+Note that  $f_{t}, f_{t-1}$  are normalized, i.e.,  $-2 \leq v(f_{t-1}; \boldsymbol{x}) \leq 2$ , and  $-2 \leq v(f_{t}; \boldsymbol{x}) \leq 2$ . We first prove the upper bound. By using the following inequality
+
+$$
+\alpha \log (1 + e ^ {- h}) + \beta - \frac {2}{1 + e ^ {h}} \geq 0,
+$$
+
+where  $\alpha = \frac{2e^2}{1 + e^2}$ ,  $\beta = 2 - \alpha + \alpha \log \frac{\alpha}{2}$ ,  $-2 \leq h \leq 2$ , and using  $v(f_{t-1}; \boldsymbol{x})$  to replace  $h$ , we have
+
+$$
+\begin{array}{l} (1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)) v \left(f _ {t}; \boldsymbol {x}\right) \geq - 2 \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \\ = - \frac {2}{1 + \exp (v \left(f _ {t - 1} ; \boldsymbol {x}\right))} \\ \geq - \alpha \log (1 + \exp (- v (f _ {t - 1}; \boldsymbol {x}))) - \beta \\ = - \alpha \ell \left(v \left(f _ {t - 1}; \boldsymbol {x}\right)\right) - \beta . \\ \end{array}
+$$
+
+Using the result above, we have
+
+$$
+\begin{array}{l} L_{\mathrm{dis}}(f_{t};f_{t - 1},\mathcal{D}) = L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu \\ c^{-}\sim \mu \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}(1 - q(f_{t - 1};\boldsymbol {x}))v(f_{t};\boldsymbol {x}) \\ \geq L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu   x,x^{+}\sim \mathcal{D}_{c^{+}}\\ c^{-}\sim \mu   x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}\underset { \begin{array}{c} \\ \end{array} }{\mathbb{E}}[-\alpha \ell (v(f_{t - 1};\boldsymbol {x})) - \beta ] \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) - \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) - \beta . \\ \end{array}
+$$
+
+which means
+
+$$
+L _ {\mathrm {c o n}} \left(f _ {t}; \mathcal {D}\right) \leq \alpha L _ {\mathrm {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta .
+$$
+
+This finishes the upper bound part. Then let us consider the lower bound. We use the inequality
+
+$$
+\alpha \log (1 + e ^ {- h}) + \beta^ {\prime} + \frac {2}{1 + e ^ {h}} \leq 0,
+$$
+
+where  $\alpha = \frac{2e^2}{1 + e^2}$ ,  $\beta' = -\alpha \log (1 + e^2) - \alpha$ ,  $-2 \leq h \leq 2$ , and using  $v(f_{t-1}; \boldsymbol{x})$  to replace  $h$ , then we have
+
+$$
+\begin{array}{l} (1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)) v \left(f _ {t}; \boldsymbol {x}\right) \leq 2 \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \\ = \frac {2}{1 + \exp (v \left(f _ {t - 1} ; \boldsymbol {x}\right))} \\ \leq - \alpha \log (1 + \exp (- v (f _ {t - 1}; \boldsymbol {x})) - \beta^ {\prime} \\ = - \alpha \ell \left(v \left(f _ {t - 1}; \boldsymbol {x}\right)\right) - \beta^ {\prime}. \\ \end{array}
+$$
+
+Using the result above, we have
+
+$$
+\begin{array}{l} L_{\mathrm{dis}}(f_{t};f_{t - 1},\mathcal{D}) = L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu   x,x^{+}\sim \mathcal{D}_{c^{+}}\\ c^{-}\sim \mu   x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}(1 - q(f_{t - 1};\boldsymbol {x}))v(f_{t};\boldsymbol {x}) \\ \end{array} }{\mathbb{E}} \\ \leq L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu \\ c^{-}\sim \mu \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}[-\alpha \ell (v(f_{t - 1};\boldsymbol {x})) - \beta^{\prime}] \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) - \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) - \beta^ {\prime}. \\ \end{array}
+$$
+
+It can be translated into
+
+$$
+L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta^ {\prime}.
+$$
+
+The proof has finished.
+
+# B. Proof of Theorem 1
+
+We recall Theorem 1.
+
+Theorem 1. For the contrastive continual learning involving  $T \geq 2$  tasks, the test loss of the final model  $f_{T}$  can be bounded via a linear combination of the training losses associated with each task. More specifically, the following two bounds are applicable.
+
+(1) Upper bound:
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) \leq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) \\ + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta , \\ \end{array}
+$$
+
+(2) Lower bound:
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) \geq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) \\ + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta^ {\prime}, \\ \end{array}
+$$
+
+where
+
+$$
+\left\{ \begin{array}{l} \alpha = \frac {2 e ^ {2}}{1 + e ^ {2}}, \\ \gamma_ {t} (\lambda) = \min  \left(\{\frac {1}{t} \} \cup \{\lambda k _ {t j} \} _ {j = 1} ^ {t - 1}\right), \\ \gamma_ {t} ^ {\prime} (\lambda) = \max  \left(\{1 \} \cup \{\lambda k _ {t j} \} _ {j = 1} ^ {t - 1}\right), \\ \eta = (2 - \alpha + \alpha \log \frac {\alpha}{2}) \frac {T - 1 - T \alpha + (\alpha) ^ {T}}{(1 - \alpha) ^ {2}} \\ \qquad + \sum_ {t = 2} ^ {T} \alpha^ {T - t} (1 - \frac {1}{\gamma_ {t} (\lambda)}) \min  _ {f} L _ {\mathrm {c o n}} (f; \mathcal {D} _ {t}), \\ \eta^ {\prime} = - (\alpha \log (1 + e ^ {2}) + \alpha) \frac {T - 1 - T \alpha + (\alpha) ^ {T}}{(1 - \alpha) ^ {2}}. \end{array} \right.
+$$
+
+Proof. We first proof the upper bound. For models  $f_{t}$  and  $f_{t-1}$ , we have
+
+$$
+\begin{array}{l} L_{\mathrm{dis}}(f_{t};f_{t - 1},\mathcal{D}_{1:t - 1}) = \underset { \begin{array}{c}c^{+}\sim \mu_{1:t - 1}\\ c^{-}\sim \mu_{1:t - 1} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}\left[-\boldsymbol {p}(f_{t - 1};\boldsymbol {x})\cdot \log \boldsymbol {p}(f_{t};\boldsymbol {x})\right] \\ = \sum_{j = 1}^{t - 1}k_{t j}\underset { \begin{array}{c}c^{+}\sim \mu_{j}\\ c^{-}\sim \mu_{j} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}[-\boldsymbol {p}(f_{t - 1};\boldsymbol {x})\cdot \log \boldsymbol {p}(f_{t};\boldsymbol {x})] \\ = \sum_ {j = 1} ^ {t - 1} k _ {t j} L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right). \\ \end{array}
+$$
+
+Then we can write the training loss as
+
+$$
+\begin{array}{l} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D} _ {t}\right) + \lambda L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {1: t - 1}\right) \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D} _ {t}\right) + \lambda \sum_ {j = 1} ^ {t - 1} k _ {t j} L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right). \\ \end{array}
+$$
+
+for task  $t \geq 2$ , and  $L_{\mathrm{train}}(f_1; \mathcal{D}_1) = L_{\mathrm{con}}(f_1; \mathcal{D}_1)$ . According to the proof of Lemma 1, for models  $f_t$  and  $f_{t-1}$ , data distribution  $\mathcal{D}_j$  ( $j \leq t$ ), we have
+
+$$
+L _ {\text {c o n}} \left(f _ {t}, \mathcal {D} _ {j}\right) \leq \alpha L _ {\text {c o n}} \left(f _ {t - 1}, \mathcal {D} _ {j}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right) + \beta .
+$$
+
+Denote  $\gamma_t(\lambda) = \min \left(\{\frac{1}{t}\} \cup \{\lambda k_{tj}\}_{j=1}^{t-1}\right)$  for task  $t \geq 2$ . Then we have
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} (f _ {T}; \mathcal {D} _ {1: T}) \\ = L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} L _ {\text {c o n}} \left(f _ {T}, \mathcal {D} _ {t}\right) \\ \leq L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} \left[ L _ {\text {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) + \alpha L _ {\text {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta \right] \\ \leq \left(\frac {1}{\gamma_ {T} (\lambda)} - \frac {1}{\gamma_ {T} (\lambda)} + 1\right) L _ {\mathrm {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \frac {\lambda}{\gamma_ {T} (\lambda)} \sum_ {t = 1} ^ {T - 1} k _ {T t} L _ {\mathrm {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) + \sum_ {t = 1} ^ {T - 1} \left[ \alpha L _ {\mathrm {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta \right] \\ \leq \frac {1}{\gamma_ {T} (\lambda)} L _ {\text {t r a i n}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {T}, \mathcal {D} _ {1: T - 1}\right) + \left(1 - \frac {1}{\gamma_ {T} (\lambda)}\right) \min  _ {f} L _ {\text {c o n}} \left(f; \mathcal {D} _ {T}\right) \\ + (T - 1) \beta + \alpha L _ {\text {t e s t}} \left(f _ {T - 1}; \mathcal {D} _ {1: T - 1}\right) \\ \end{array}
+$$
+
+：
+
+$$
+\leq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta .
+$$
+
+where  $\alpha = \frac{2e^2}{1 + e^2}$ ,  $\eta = (2 - \alpha + \alpha \log \frac{\alpha}{2})\frac{T - 1 - T\alpha + (\alpha)^T}{(1 - \alpha)^2} + \sum_{t=2}^{T}\alpha^{T-t}(1 - \frac{1}{\gamma_t(\lambda)})\min_f L_{\mathrm{con}}(f; \mathcal{D}_t)$ .
+
+Let us prove the lower bound. According to the proof of Lemma 1, for models  $f_{t}$  and  $f_{t-1}$ , and data distribution  $\mathcal{D}_j (j \leq t)$ , we have
+
+$$
+L _ {\text {c o n}} \left(f _ {t}, \mathcal {D} _ {j}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}, \mathcal {D} _ {j}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right) + \beta^ {\prime}.
+$$
+
+Denote  $\gamma_t'(\lambda) = \max \left(\{1\} \cup \{\lambda k_{tj}\}_{j=1}^{t-1}\right)$  for task  $t \geq 2$ . The proof is similar to that of the upper bound.
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1: T}\right) \\ = L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} L _ {\text {c o n}} \left(f _ {T}, \mathcal {D} _ {t}\right) \\ \geq L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} \left[ L _ {\text {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) + \alpha L _ {\text {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta^ {\prime} \right] \\ \geq \frac {1}{\gamma_ {t} ^ {\prime} (\lambda)} \left[ L _ {\mathrm {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \lambda \sum_ {t = 1} ^ {T - 1} k _ {T t} L _ {\mathrm {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) \right] + \sum_ {t = 1} ^ {T - 1} \left[ \alpha L _ {\mathrm {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta^ {\prime} \right] \\ = \frac {1}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {T}, \mathcal {D} _ {1: T - 1}\right) + \alpha \sum_ {t = 1} ^ {T - 1} L _ {\text {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + (T - 1) \beta^ {\prime} \\ = \frac {1}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {T}, \mathcal {D} _ {1: T - 1}\right) + \alpha L _ {\text {t e s t}} \left(f _ {T - 1}; \mathcal {D} _ {1: T - 1}\right) + (T - 1) \beta^ {\prime} \\ \geq \alpha^ {T - 1} L _ {\text {t r a i n}} (f _ {1}; \mathcal {D} _ {1}) + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} (f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}) + \eta^ {\prime}. \\ \end{array}
+$$
+
+where  $\alpha = \frac{2e^2}{1 + e^2}$ ,  $\eta' = -\left(\alpha \log (1 + e^2) + \alpha\right)\frac{T - 1 - T\alpha + (\alpha)^T}{(1 - \alpha)^2}$ .
+
+# C. Proof of Theorem 2
+
+We recall Theorem 2.
+
+Theorem 2. Assume that the training loss of each task is larger than zero. For each task  $t \geq 2$ , there exists a task-specific constant  $u_t > 0$ . If we have
+
+$$
+U _ {t} = \sum_ {j = 2} ^ {t} \frac {\alpha^ {t - j}}{\gamma_ {j} (\lambda_ {t})} L _ {t r a i n} (f _ {j}; f _ {j - 1}, \mathcal {D} _ {j}, \mathcal {D} _ {1: j - 1}) > u _ {t},
+$$
+
+where  $\alpha, \{\gamma_j(\lambda_t)\}_{j=2}^t$  are defined in Theorem 1, then we increase  $\lambda_t$  by  $\Delta_t$ , i.e.,  $\lambda_{t+1} = \lambda_t + \Delta_t$ , else,  $\lambda_{t+1} = \lambda_t$ . By choosing  $\lambda_t$  in this way, we get tighter upper bounds.
+
+Proof. Note that  $U_{t} > 0$ , thus  $u_{t}$  exists. If  $\lambda_{t}$  increases by  $\Delta \geq 0$ , then  $\lambda_{t + 1} \geq \lambda_{t}$  and  $\gamma_{j}(\lambda_{t + 1}) \geq \gamma_{j}(\lambda_{t})$ . We have
+
+$$
+U _ {t + 1} ^ {\prime} = \sum_ {j = 2} ^ {t + 1} \frac {\alpha^ {t + 1 - j}}{\gamma_ {j} (\lambda_ {t + 1})} L _ {\text {t r a i n}} (f _ {j}; f _ {j - 1}, \mathcal {D} _ {j}) \leq U _ {t + 1} = \sum_ {j = 2} ^ {t + 1} \frac {\alpha^ {t + 1 - j}}{\gamma_ {j} (\lambda_ {t})} L _ {\text {t r a i n}} (f _ {j}; f _ {j - 1}, \mathcal {D} _ {j}).
+$$
+
+Thus the upper bound becomes tighter.
+
+# D. The Case of Multiple Negative Examples for Lemma 1
+
+Following our definitions in the paper, the contrastive loss for the case of  $k(k \geq 1)$  negative samples can be formulated as
+
+$$
+L _ {\mathrm {c o n}} (f; \mathcal {D} _ {t}) = \underset { \begin{array}{c} c ^ {+} \sim \mu_ {t} \\ c _ {i} ^ {-} \sim \mu_ {t} \end{array} } {\mathbb {E}} \underset { \begin{array}{c} x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ x _ {i} ^ {-} \sim \mathcal {D} _ {c _ {i} ^ {-}} \end{array} } {\mathbb {E}} \ell \left[ \left\{f (x) ^ {\top} (f (x ^ {+}) - f (x _ {i} ^ {-})) \right\} \right],
+$$
+
+where function  $\ell(\pmb{v})$  is defined as  $\ell(\pmb{v}) = \log(1 + \sum_{i=1}^{k} \exp(-v_i))$  for  $\pmb{v} \in \mathbb{R}^k$  and embeddings are conventionally normalized, i.e.,  $\|f\| = 1$ . The distillation loss for the case of  $k$  negative samples can be formulated as
+
+$$
+\begin{array}{l} L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {1: t - 1}\right) = \\ \underset {c ^ {+} \sim \mu_ {1: t - 1}} {\mathbb {E}} \underset {x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}}} {\mathbb {E}} \left[ - \boldsymbol {p} \left(f _ {t - 1}; x, x ^ {+}, x _ {1} ^ {-}, \dots , x _ {k} ^ {-}\right) \cdot \log \boldsymbol {p} \left(f _ {t}; x, x ^ {+}, x _ {1} ^ {-}, \dots , x _ {k} ^ {-}\right) \right], \\ c _ {i} ^ {-} \sim \mu_ {1: t - 1} x _ {i} ^ {-} \sim \mathcal {D} _ {c _ {i} ^ {-}} \\ \end{array}
+$$
+
+where  $\mu_{1:t-1}$  represents the class distribution of  $\mathcal{D}_{1:t-1}$  and
+
+$$
+\boldsymbol {p} (f; x, x ^ {+}, x _ {1} ^ {-}, \dots , x _ {k} ^ {-}) = \operatorname {s o f t m a x} (f (x) ^ {\top} f (x ^ {+}), f (x) ^ {\top} f (x _ {1} ^ {-}), \dots , f (x) ^ {\top} f (x _ {k} ^ {-})).
+$$
+
+Then we provide the extended version of Lemma 1 and its proof.
+
+Lemma 2. When  $t \geq 2$  and the number of negative samples  $k \geq 1$ , for any data distribution  $\mathcal{D}$ , the contrastive losses of current model  $f_{t}$  and previous model  $f_{t-1}$  can be connected via the distillation loss, i.e.,
+
+$$
+\begin{array}{l} L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \leq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta , \\ L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta^ {\prime}, \\ \end{array}
+$$
+
+where  $\alpha = \frac{2e^2}{k + e^2}$ ,  $\beta = 2 - \alpha + \alpha \log \frac{\alpha}{2}$ , and  $\beta' = -\alpha \log (1 + ke^2) - \frac{2ke^2}{1 + ke^2}$ .
+
+Proof. For model  $f$ , denote the data pair  $\boldsymbol{x} = (x, x^{+}, x_{1}^{-}, \ldots, x_{k}^{-})$  to simplify the proof, we first define  $v_{i}(f; \boldsymbol{x}) = f(x)^{\top}(f(x^{+}) - f(x_{i}^{-}))$ , and
+
+$$
+\begin{array}{l} q _ {i} (f; \boldsymbol {x}) = \frac {\exp (- v _ {i} (f ; \boldsymbol {x}))}{1 + \sum_ {i = 1} ^ {k} \exp (- v _ {i} (f ; \boldsymbol {x}))}, \\ q (f; \boldsymbol {x}) = \frac {1}{1 + \sum_ {i = 1} ^ {k} \exp (- v _ {i} (f ; \boldsymbol {x}))}, \\ \end{array}
+$$
+
+where  $q(f; \boldsymbol{x}) + \sum_{i=1}^{k} q_i(f; \boldsymbol{x}) = 1$ . For models  $f_t$  and  $f_{t-1}$ , the function  $\ell(\boldsymbol{v}) = \log(1 + \sum_{i=1}^{k} \exp(-v_i))$  for  $\boldsymbol{v} \in \mathbb{R}^k$ , and
+
+$$
+\pmb {p} (f; \pmb {x}) = \mathrm {s o f t m a x} (f (x) ^ {\top} f (x ^ {+}), f (x) ^ {\top} f (x _ {1} ^ {-}), \ldots , f (x) ^ {\top} f (x _ {k} ^ {-})),
+$$
+
+we have the following equation
+
+$$
+\begin{array}{l} - \boldsymbol {p} (f _ {t - 1}; \boldsymbol {x}) \cdot \log \boldsymbol {p} (f _ {t}; \boldsymbol {x}) \\ = - q \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(q \left(f _ {t}; \boldsymbol {x}\right)\right) - \sum_ {i = 1} ^ {k} q _ {i} \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(q _ {i} \left(f _ {t}; \boldsymbol {x}\right)\right) \\ = q \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(1 + \sum_ {i = 1} ^ {k} \exp \left(- v _ {i} \left(f _ {t}; \boldsymbol {x}\right)\right)\right) - \sum_ {i = 1} ^ {k} q _ {i} \left(f _ {t - 1}; \boldsymbol {x}\right) \left[ - v _ {i} \left(f _ {t}; \boldsymbol {x}\right) - \log \left(1 + \sum_ {i = 1} ^ {k} \exp \left(- v _ {i} \left(f _ {t}; \boldsymbol {x}\right)\right)\right) \right] \\ = q \left(f _ {t - 1}; \boldsymbol {x}\right) \log \left(1 + \sum_ {i = 1} ^ {k} \exp \left(- v _ {i} \left(f _ {t}; \boldsymbol {x}\right)\right)\right) + \sum_ {i = 1} ^ {k} q _ {i} \left(f _ {t - 1}; \boldsymbol {x}\right) \left[ v _ {i} \left(f _ {t}; \boldsymbol {x}\right) + \log \left(1 + \sum_ {i = 1} ^ {k} \exp \left(- v _ {i} \left(f _ {t}; \boldsymbol {x}\right)\right)\right) \right] \\ = \log (1 + \sum_ {i = 1} ^ {k} \exp (- v _ {i} (f _ {t}; \boldsymbol {x}))) + \sum_ {i = 1} ^ {k} q _ {i} (f _ {t - 1}; \boldsymbol {x}) v _ {i} (f _ {t}; \boldsymbol {x}) \\ = \ell (\boldsymbol {v} (f _ {t}; \boldsymbol {x})) + \sum_ {i = 1} ^ {k} q _ {i} (f _ {t - 1}; \boldsymbol {x}) v _ {i} (f _ {t}; \boldsymbol {x}). \\ \end{array}
+$$
+
+Then for any data distribution  $\mathcal{D}$ , we have
+
+$$
+\begin{array}{l} L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) \\ = \underset { \begin{array}{c} c ^ {+} \sim \mu \\ c _ {i} ^ {-} \sim \mu \end{array} } {\mathbb {E}} \underset { \begin{array}{c} x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ x _ {i} ^ {-} \sim \mathcal {D} _ {c _ {i} ^ {-}} \end{array} } {\mathbb {E}} - \boldsymbol {p} (f _ {t - 1}; \boldsymbol {x}) \cdot \log \boldsymbol {p} (f _ {t}; \boldsymbol {x}) \\ = \underset { \begin{array}{c} c ^ {+} \sim \mu \\ c _ {i} ^ {-} \sim \mu \end{array} } {\mathbb {E}} \underset { \begin{array}{c} x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ x _ {i} ^ {-} \sim \mathcal {D} _ {c _ {i} ^ {-}} \end{array} } {\mathbb {E}} \ell (\boldsymbol {v} (f _ {t}; \boldsymbol {x})) + \underset { \begin{array}{c} c ^ {+} \sim \mu \\ c _ {i} ^ {-} \sim \mu \end{array} } {\mathbb {E}} \underset { \begin{array}{c} x, x ^ {+} \sim \mathcal {D} _ {c ^ {+}} \\ x _ {i} ^ {-} \sim \mathcal {D} _ {c _ {\bar {i}} ^ {-}} \end{array} } {\mathbb {E}} \sum_ {i = 1} ^ {k} q _ {i} (f _ {t - 1}; \boldsymbol {x}) v _ {i} (f _ {t}; \boldsymbol {x}) \\ = L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu \\ c_{i}^{-}\sim \mu \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x_{i}^{-}\sim \mathcal{D}_{c_{i}^{-}} \end{array} }{\mathbb{E}}\sum_{i = 1}^{k}q_{i}(f_{t - 1};\boldsymbol {x})v_{i}(f_{t};\boldsymbol {x}). \\ \end{array}
+$$
+
+Note that  $f_{t}, f_{t-1}$  are normalized, i.e.,  $-2 \leq v_{i}(f_{t-1}; \boldsymbol{x}) \leq 2$ , and  $-2 \leq v_{i}(f_{t}; \boldsymbol{x}) \leq 2$ . We first prove the upper bound. By using the following inequality
+
+$$
+\alpha \log h + \beta \geq 2 (1 - \frac {1}{h}),
+$$
+
+where  $\alpha = \frac{2e^2}{k + e^2}$ ,  $\beta = 2 - \alpha + \alpha \log \frac{\alpha}{2}$ ,  $1 + ke^{-2} \leq h \leq 1 + ke^2$ , and using  $1 + \sum_{i=1}^{k} \exp(-v_i(f_{t-1}; \boldsymbol{x}))$  to replace  $h$  in the inequality above, we have
+
+$$
+\begin{array}{l} \sum_ {i = 1} ^ {k} q _ {i} \left(f _ {t - 1}; \boldsymbol {x}\right) v _ {i} \left(f _ {t}; \boldsymbol {x}\right) \geq - 2 \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \\ = - 2 \left(1 - \frac {1}{1 + \sum_ {i = 1} ^ {k} \exp \left(- v _ {i} \left(f _ {t - 1} ; \boldsymbol {x}\right)\right)}\right) \\ \geq - \alpha \log (1 + \sum_ {i = 1} ^ {k} \exp (- v _ {i} (f _ {t - 1}; \boldsymbol {x}))) - \beta \\ = - \alpha \ell (\boldsymbol {v} (f _ {t - 1}; \boldsymbol {x})) - \beta . \\ \end{array}
+$$
+
+Using the results above, we have
+
+$$
+\begin{array}{l} L_{\mathrm{dis}}(f_{t};f_{t - 1},\mathcal{D}) = L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu \\ c_{i}^{-}\sim \mu \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x_{i}^{-}\sim \mathcal{D}_{c_{i}^{-}} \end{array} }{\mathbb{E}}\sum_{i = 1}^{k}q_{i}(f_{t - 1};\boldsymbol {x})v_{i}(f_{t};\boldsymbol {x}) \\ \geq L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu   x,x^{+}\sim \mathcal{D}_{c^{+}}\\ c_{i}^{-}\sim \mu   x_{i}^{-}\sim \mathcal{D}_{c_{i}^{-}} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}[-\alpha \ell (\boldsymbol {v}(f_{t - 1};\boldsymbol {x})) - \beta ] \end{array} }{\mathbb{E}} \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) - \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) - \beta . \\ \end{array}
+$$
+
+which means
+
+$$
+L _ {\mathrm {c o n}} \left(f _ {t}; \mathcal {D}\right) \leq \alpha L _ {\mathrm {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta .
+$$
+
+This finishes the upper bound part. Then let us consider the lower bound. We use the inequality
+
+$$
+- \alpha \log h - \beta^ {\prime} \geq 2 \left(1 - \frac {1}{h}\right),
+$$
+
+where  $\alpha = \frac{2e^2}{k + e^2}$ ,  $\beta' = -\alpha \log (1 + ke^2) - \frac{2ke^2}{1 + ke^2}$ ,  $1 + ke^{-2} \leq h \leq 1 + ke^2$ , and using  $1 + \sum_{i=1}^{k} \exp(-v_i(f_{t-1}; \boldsymbol{x}))$  to replace  $h$ , then we have
+
+$$
+\begin{array}{l} \sum_ {i = 1} ^ {k} q _ {i} \left(f _ {t - 1}; \boldsymbol {x}\right) v _ {i} \left(f _ {t}; \boldsymbol {x}\right) \leq 2 \left(1 - q \left(f _ {t - 1}; \boldsymbol {x}\right)\right) \\ = 2 \left(1 - \frac {1}{1 + \sum_ {i = 1} ^ {k} \exp \left(- v _ {i} \left(f _ {t - 1} ; \boldsymbol {x}\right)\right)}\right) \\ \leq - \alpha \log (1 + \sum_ {i = 1} ^ {k} \exp (- v _ {i} (f _ {t - 1}; \boldsymbol {x})) - \beta^ {\prime} \\ = - \alpha \ell (\pmb {v} (f _ {t - 1}; \pmb {x})) - \beta^ {\prime}. \\ \end{array}
+$$
+
+Using the results above, we have
+
+$$
+\begin{array}{l} L_{\mathrm{dis}}(f_{t};f_{t - 1},\mathcal{D}) = L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu \\ c_{i}^{-}\sim \mu \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x_{i}^{-}\sim \mathcal{D}_{c_{i}^{-}} \end{array} }{\mathbb{E}}\sum_{i = 1}^{k}q_{i}(f_{t - 1};\boldsymbol {x})v_{i}(f_{t};\boldsymbol {x}) \\ \leq L_{\mathrm{con}}(f_{t};\mathcal{D}) + \underset { \begin{array}{c}c^{+}\sim \mu   x,x^{+}\sim \mathcal{D}_{c^{+}}\\ c_{i}^{-}\sim \mu   x_{i}^{-}\sim \mathcal{D}_{c_{i}^{-}} \end{array} }{\mathbb{E}}\underset { \begin{array}{c} - \mu \\ x_{i}^{-}\sim \mathcal{D}_{c_{i}^{-}} \end{array} }{\mathbb{E}}[-\alpha \ell (\boldsymbol {v}(f_{t - 1};\boldsymbol {x})) - \beta^{\prime}] \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) - \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) - \beta^ {\prime}. \\ \end{array}
+$$
+
+It can be translated into
+
+$$
+L _ {\text {c o n}} \left(f _ {t}; \mathcal {D}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}; \mathcal {D}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D}\right) + \beta^ {\prime}.
+$$
+
+The proof has finished.
+
+# E. The Case of Multiple Negative Examples for Theorem 1
+
+We provide the extended version of Theorem 1 and its proof.
+
+Theorem 3. For the contrastive continual learning involving  $T \geq 2$  tasks where each task involves  $k$  negative samples, the test loss of the final model  $f_{T}$  can be bounded via a linear combination of the training losses associated with each task. More specifically, the following two bounds are applicable.
+
+(1) Upper bound:
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) \leq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) \\ + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta , \\ \end{array}
+$$
+
+(2) Lower bound:
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1}, \dots , \mathcal {D} _ {T}\right) \geq \alpha^ {T - 1} L _ {\text {t r a i n}} \left(f _ {1}; \mathcal {D} _ {1}\right) \\ + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) + \eta^ {\prime}, \\ \end{array}
+$$
+
+where
+
+$$
+\left\{ \begin{array}{l} \alpha = \frac {2 e ^ {2}}{k + e ^ {2}}, \\ \gamma_ {t} (\lambda) = \min  \left(\{\frac {1}{t} \} \cup \{\lambda k _ {t j} \} _ {j = 1} ^ {t - 1}\right), \\ \gamma_ {t} ^ {\prime} (\lambda) = \max  \left(\{1 \} \cup \{\lambda k _ {t j} \} _ {j = 1} ^ {t - 1}\right), \\ \eta = (2 - \alpha + \alpha \log \frac {\alpha}{2}) \frac {T - 1 - T \alpha + (\alpha) ^ {T}}{(1 - \alpha) ^ {2}} \\ \hskip 1 4. 2 2 6 3 7 8 p t + \sum_ {t = 2} ^ {T} \alpha^ {T - t} (1 - \frac {1}{\gamma_ {t} (\lambda)}) \min  _ {f} L _ {\mathrm {c o n}} (f; \mathcal {D} _ {t}), \\ \eta^ {\prime} = - (\alpha \log (1 + k e ^ {2}) + \frac {2 k e ^ {2}}{1 + k e ^ {2}}) \frac {T - 1 - T \alpha + (\alpha) ^ {T}}{(1 - \alpha) ^ {2}}. \end{array} \right.
+$$
+
+Proof. We first proof the upper bound. For models  $f_{t}$  and  $f_{t-1}$ , we have
+
+$$
+\begin{array}{l} L_{\mathrm{dis}}(f_{t};f_{t - 1},\mathcal{D}_{1:t - 1}) = \underset { \begin{array}{c}c^{+}\sim \mu_{1:t - 1}\\ c^{-}\sim \mu_{1:t - 1} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}[-\boldsymbol {p}(f_{t - 1};\boldsymbol {x})\cdot \log \boldsymbol {p}(f_{t};\boldsymbol {x})] \\ = \sum_{j = 1}^{t - 1}k_{t j}\underset { \begin{array}{c}c^{+}\sim \mu_{j}\\ c^{-}\sim \mu_{j} \end{array} }{\mathbb{E}}\underset { \begin{array}{c}x,x^{+}\sim \mathcal{D}_{c^{+}}\\ x^{-}\sim \mathcal{D}_{c^{-}} \end{array} }{\mathbb{E}}\left[-\boldsymbol {p}(f_{t - 1};\boldsymbol {x})\cdot \log \boldsymbol {p}(f_{t};\boldsymbol {x})\right] \\ = \sum_ {j = 1} ^ {t - 1} k _ {t j} L _ {\mathrm {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right). \\ \end{array}
+$$
+
+Then we can write the training loss as
+
+$$
+\begin{array}{l} L _ {\text {t r a i n}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}\right) = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D} _ {t}\right) + \lambda L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {1: t - 1}\right) \\ = L _ {\text {c o n}} \left(f _ {t}; \mathcal {D} _ {t}\right) + \lambda \sum_ {j = 1} ^ {t - 1} k _ {t j} L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right). \\ \end{array}
+$$
+
+for task  $t \geq 2$ , and  $L_{\mathrm{train}}(f_1; \mathcal{D}_1) = L_{\mathrm{con}}(f_1; \mathcal{D}_1)$ . According to the proof of Lemma 1, for models  $f_t$  and  $f_{t-1}$ , data distribution  $\mathcal{D}_j$  ( $j \leq t$ ), we have
+
+$$
+L _ {\text {c o n}} \left(f _ {t}, \mathcal {D} _ {j}\right) \leq \alpha L _ {\text {c o n}} \left(f _ {t - 1}, \mathcal {D} _ {j}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right) + \beta .
+$$
+
+Denote  $\gamma_t(\lambda) = \min \left(\left\{\frac{1}{t}\right\} \cup \left\{\lambda k_{tj}\right\}_{j=1}^{t-1}\right)$  for task  $t \geq 2$ . According to the equations above, we have
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} (f _ {T}; \mathcal {D} _ {1: T}) \\ = L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} L _ {\text {c o n}} \left(f _ {T}, \mathcal {D} _ {t}\right) \\ \leq L _ {\mathrm {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} \left[ L _ {\mathrm {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) + \alpha L _ {\mathrm {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta \right] \\ \leq \left(\frac {1}{\gamma_ {T} (\lambda)} - \frac {1}{\gamma_ {T} (\lambda)} + 1\right) L _ {\mathrm {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \frac {\lambda}{\gamma_ {T} (\lambda)} \sum_ {t = 1} ^ {T - 1} k _ {T t} L _ {\mathrm {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) + \sum_ {t = 1} ^ {T - 1} \left[ \alpha L _ {\mathrm {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta \right] \\ \leq \frac {1}{\gamma_ {T} (\lambda)} L _ {\text {t r a i n}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {T}, \mathcal {D} _ {1: T - 1}\right) + \left(1 - \frac {1}{\gamma_ {T} (\lambda)}\right) \min  _ {f} L _ {\text {c o n}} \left(f; \mathcal {D} _ {T}\right) \\ + (T - 1) \beta + \alpha L _ {\text {t e s t}} \left(f _ {T - 1}; \mathcal {D} _ {1: T - 1}\right) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \begin{array}{c} \bullet \\ \bullet \\ \bullet \\ \bullet \end{array} \\ \leq \alpha^ {T - 1} L _ {\text {t r a i n}} (f _ {1}; \mathcal {D} _ {1}) + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} (\lambda)} L _ {\text {t r a i n}} (f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}) + \eta . \\ \end{array}
+$$
+
+where  $\alpha = \frac{2e^2}{k + e^2}$ ,  $\eta = (2 - \alpha + \alpha \log \frac{\alpha}{2})\frac{T - 1 - T\alpha + (\alpha)^T}{(1 - \alpha)^2} + \sum_{t=2}^{T}\alpha^{T - t}(1 - \frac{1}{\gamma_t(\lambda)})\min_f L_{\mathrm{con}}(f; \mathcal{D}_t)$ .
+
+Let us prove the lower bound. According to the proof of Lemma 1, for models  $f_{t}$  and  $f_{t-1}$ , and data distribution  $\mathcal{D}_j (j \leq t)$ , we have
+
+$$
+L _ {\text {c o n}} \left(f _ {t}, \mathcal {D} _ {j}\right) \geq \alpha L _ {\text {c o n}} \left(f _ {t - 1}, \mathcal {D} _ {j}\right) + L _ {\text {d i s}} \left(f _ {t}; f _ {t - 1}, \mathcal {D} _ {j}\right) + \beta^ {\prime}.
+$$
+
+Denote  $\gamma_t'(\lambda) = \max \left(\{1\} \cup \{\lambda k_{tj}\}_{j=1}^{t-1}\right)$  for task  $t \geq 2$ . The proof is similar to that of the upper bound. Similarly, we have
+
+$$
+\begin{array}{l} L _ {\text {t e s t}} \left(f _ {T}; \mathcal {D} _ {1: T}\right) \\ = L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} L _ {\text {c o n}} \left(f _ {T}, \mathcal {D} _ {t}\right) \\ \geq L _ {\text {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \sum_ {t = 1} ^ {T - 1} \left[ L _ {\text {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) + \alpha L _ {\text {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta^ {\prime} \right] \\ \geq \frac {1}{\gamma_ {t} ^ {\prime} (\lambda)} \left[ L _ {\mathrm {c o n}} \left(f _ {T}; \mathcal {D} _ {T}\right) + \lambda \sum_ {t = 1} ^ {T - 1} k _ {T t} L _ {\mathrm {d i s}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {t}\right) \right] + \sum_ {t = 1} ^ {T - 1} \left[ \alpha L _ {\mathrm {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + \beta^ {\prime} \right] \\ = \frac {1}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {T}, \mathcal {D} _ {1: T - 1}\right) + \alpha \sum_ {t = 1} ^ {T - 1} L _ {\text {c o n}} \left(f _ {T - 1}; \mathcal {D} _ {t}\right) + (T - 1) \beta^ {\prime} \\ = \frac {1}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} \left(f _ {T}; f _ {T - 1}, \mathcal {D} _ {T}, \mathcal {D} _ {1: T - 1}\right) + \alpha L _ {\text {t e s t}} \left(f _ {T - 1}; \mathcal {D} _ {1: T - 1}\right) + (T - 1) \beta^ {\prime} \\ \geq \alpha^ {T - 1} L _ {\text {t r a i n}} (f _ {1}; \mathcal {D} _ {1}) + \sum_ {t = 2} ^ {T} \frac {\alpha^ {T - t}}{\gamma_ {t} ^ {\prime} (\lambda)} L _ {\text {t r a i n}} (f _ {t}; f _ {t - 1}, \mathcal {D} _ {t}, \mathcal {D} _ {1: t - 1}) + \eta^ {\prime}. \\ \end{array}
+$$
+
+where  $\alpha = \frac{2e^2}{k + e^2}$ ,  $\eta' = -\left(\alpha \log (1 + ke^2) + \frac{2ke^2}{1 + ke^2}\right)\frac{T - 1 - T\alpha + (\alpha)^T}{(1 - \alpha)^2}$ .

2024/Provable Contrastive Continual Learning/images.zip

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