From now on, we use IoU as the similarity metric in our paper. To mitigate
This paper says it uses image generation architectures to generate new model weights, instead of images, for arbitrary other purposes and architectures (in seconds) without other training. Glancing through it they only tested with visual models. [~~~takeoff paper influence, ow, please make real communication-action happen, obviously with them or it would be censorship, urgently] I stumble on these maybe once every three years since my life change (wasn’t into this before it). I’m posting this one added to arxiv seven hours ago, via twitter/X, because I can never find them again after closing out. https://1zeryu.github.io/Neural-Network-Diffusion/ https://github.com/NUS-HPC-AI-Lab/Neural-Network-Diffusion https://arxiv.org/abs/2402.13144 arXiv:2402.13144v1 [cs.LG] 20 Feb 2024 Neural Network Diffusion Kai WangZhaopan XuYukun ZhouZelin ZangTrevor DarrellZhuang LiuYang You Abstract Diffusion models have achieved remarkable success in image and video generation. In this work, we demonstrate that diffusion models can also generate high-performing neural network parameters. Our approach is simple, utilizing an autoencoder and a standard latent diffusion model. The autoencoder extracts latent representations of a subset of the trained network parameters. A diffusion model is then trained to synthesize these latent parameter representations from random noise. It then generates new representations that are passed through the autoencoder’s decoder, whose outputs are ready to use as new subsets of network parameters. Across various architectures and datasets, our diffusion process consistently generates models of comparable or improved performance over trained networks, with minimal additional cost. Notably, we empirically find that the generated models perform differently with the trained networks. Our results encourage more exploration on the versatile use of diffusion models. Machine Learning, ICML Code: https://github.com/NUS-HPC-AI-Lab/Neural-Network-Diffusion 1 Introduction The origin of diffusion models can be traced back to non-equilibrium thermodynamics (Jarzynski, 1997 https://arxiv.org/html/2402.13144v1#bib.bib27; Sohl-Dickstein et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib56). Diffusion processes were first utilized to progressively remove noise from inputs and generate clear images in (Sohl-Dickstein et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib56). Later works, such as DDPM (Ho et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib24) and DDIM (Song et al., 2021 https://arxiv.org/html/2402.13144v1#bib.bib58), refine diffusion models, with a training paradigm characterized by forward and reverse processes. At that time, the quality of images generated by diffusion models had not yet reached a desired level. Guided-Diffusion (Dhariwal & Nichol, 2021 https://arxiv.org/html/2402.13144v1#bib.bib12) conducts sufficient ablations and finds a better architecture, which represents the pioneering effort to elevate diffusion models beyond GAN-based methods (Zhu et al., 2017 https://arxiv.org/html/2402.13144v1#bib.bib67; Isola et al., 2017 https://arxiv.org/html/2402.13144v1#bib.bib26) in terms of image quality. Subsequently, GLIDE (Nichol et al., 2021 https://arxiv.org/html/2402.13144v1#bib.bib41), Imagen (Saharia et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib53), DALL⋅E 2 (Ramesh et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib46), and Stable Diffusion (Rombach et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib51) achieve photorealistic images adopted by artists. Despite the great success of diffusion models in visual generation, their potential in other domains remains relatively underexplored. In this work, we demonstrate the surprising capability of diffusion models in generating high-performing model parameters, a task fundamentally distinct from traditional visual generation. Parameter generation focuses on creating neural network parameters that can perform well on given tasks. It has been explored from prior and probability modeling aspects, i.e. stochastic neural network (Sompolinsky et al., 1988 https://arxiv.org/html/2402.13144v1#bib.bib57; Bottou et al., 1991 https://arxiv.org/html/2402.13144v1#bib.bib4; Wong, 1991 https://arxiv.org/html/2402.13144v1#bib.bib64; Schmidt et al., 1992 https://arxiv.org/html/2402.13144v1#bib.bib54; Murata et al., 1994 https://arxiv.org/html/2402.13144v1#bib.bib39) and Bayesian neural network (Neal, 2012 https://arxiv.org/html/2402.13144v1#bib.bib40; Kingma & Welling, 2013 https://arxiv.org/html/2402.13144v1#bib.bib28; Rezende et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib50; Kingma et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib29; Gal & Ghahramani, 2016 https://arxiv.org/html/2402.13144v1#bib.bib17). However, using a diffusion model in parameter generation has not been well-explored yet. [image: Refer to caption]Figure 1: The top: illustrates the standard diffusion process in image generation. The bottom: denotes the parameter distribution of batch normalization (BN) during the training CIFAR-100 with ResNet-18. The upper half of the bracket: BN weights. The lower half of the bracket: BN biases. Taking a closer look at the neural network training and diffusion models, the diffusion-based image generation shares commonalities with the stochastic gradient descent (SGD) learning process in the following aspects (illustrated in Fig. 1 https://arxiv.org/html/2402.13144v1#S1.F1). i) Both neural network training and the reverse process of diffusion models can be regarded as transitions from random noise/initialization to specific distributions. ii) High-quality images and high-performing parameters can also be degraded into simple distributions, such as Gaussian distribution, through multiple noise additions. Based on the observations above, we introduce a novel approach for parameter generation, named neural network diffusion (p-diff, p stands for parameter), which employs a standard latent diffusion model to synthesize a new set of parameters. That is motivated by the fact that the diffusion model has the capability to transform a given random distribution to a specific one. Our method is simple, comprising an autoencoder and a standard latent diffusion model to learn the distribution of high-performing parameters. First, for a subset of parameters of models trained by the SGD optimizer, the autoencoder is trained to extract the latent representations for these parameters. Then, we leverage a standard latent diffusion model to synthesize latent representations from random noise. Finally, the synthesized latent representations are passed through the trained autoencoder’s decoder to yield new high-performing model parameters. Our approach has the following characteristics: i) It consistently achieves similar, even enhanced performance than its training data, i.e., models trained by SGD optimizer, across multiple datasets and architectures within seconds. ii) Our generated models have great differences from the trained models, which illustrates our approach can synthesize new parameters instead of memorizing the training samples. We hope our research can provide fresh insights into expanding the applications of diffusion models to other domains. [image: Refer to caption]Figure 2: Our approach consists of two processes, named parameter autoencoder and generation. Parameter autoencoder aims to extract the latent representations and reconstruct model parameters via the decoder. The extracted representations are used to train a standard latent diffusion model (LDM). In the inference, the random noise is fed into LDM and trained decoder to obtain the generated parameters.2 Nerual Network Diffusion2.1 Preliminaries of diffusion models Diffusion models typically consist of forward and reverse processes in a multi-step chain indexed by timesteps. We introduce these two processes in the following. Forward process. Given a sample x0∼q(x), the forward process progressively adds Gaussian noise for T steps and obtain x1,x2,⋯,xT. The formulation of this process can be written as follows, q(xt|xt−1)=𝒩(xt;1−βtxt−1,βt𝐈),q(x1:T|x0)=∏t=1Tq(xt|xt−1), (1) where q and 𝒩 represent forward process and adding Gaussian noise parameterized by βt, and I is the identity matrix. Reverse process. Different from the forward process, the reverse process aims to train a denoising network to recursively remove the noise from xt. It moves backward on the multi-step chain as t decreases from T to 0. Mathematically, the reverse process can be formulated as follows, pθ(xt−1|xt)=𝒩(xt−1;μθ(xt,t),Σθ(xt,t)),pθ(x0:T)=p(xT)∏t=1Tpθ(xt−1|xt), (2) where p represents the reverse process, μθ(xt,t) and Σθ(xt,t)) are the Gaussian mean and variance that estimated by the denoising network parameter θ. The denoising network in the reverse process is optimized by the standard negative log-likelihood: Ldm=𝒟KL(q(xt−1|xt,x0)||pθ(xt−1|xt)), (3) where the 𝒟KL(⋅||⋅) denotes the Kullback–Leibler (KL) divergence that is normally used to compute the difference between two distributions. Training and inference procedures. The goal of the training diffusion model is to find the reverse transitions that maximize the likelihood of the forward transitions in each time step t. In practice, training equivalently consists of minimizing the variational upper bound. The inference procedure aims to generate novel samples from random noise via the optimized denoising parameters θ* and the multi-step chains in the reverse process. 2.2 Overview We propose neural network diffusion (p-diff), which aims to generate high-performing parameters from random noise. As illustrated in Fig. 2 https://arxiv.org/html/2402.13144v1#S1.F2, our method consists of two processes, named parameter autoencoder and generation. Given a set of trained high-performing models, we first select a subset of these parameters and flatten them into 1-dimensional vectors. Subsequently, we introduce an encoder to extract latent representations from these vectors, accompanied by a decoder responsible for reconstructing the parameters from latent representations. Then, a standard latent diffusion model is trained to synthesize latent representations from random noise. After training, we utilize p-diff to generate new parameters via the following chain: random noise → reverse process → trained decoder → generated parameters. 2.3 Parameter autoencoderPreparing the data for training the autoencoder. In our paper, we default to synthesizing a subset of model parameters. Therefore, to collect the training data for the autoencoder, we train a model from scratch and densely save checkpoints in the last epoch. It is worth noting that we only update the selected subset of parameters via SGD optimizer and fix the remained parameters of the model. The saved subsets of parameters S=[s1,…,sk,…,sK] is utilized to train the autoencoder, where K is the number of the training samples. For some large architectures that have been trained on large-scale datasets, considering the cost of training them from scratch, we fine-tune a subset of the parameters of the pre-trained model and densely save the fine-tuned parameters as training samples. Training parameter autoencoder. We then flatten these parameters S into 1-dimensional vectors V=[v1,…,vk,…,vK], where V∈ℝK×D and D is the size of the subset parameters. After that, an autoencoder is trained to reconstruct these parameters V. To enhance the robustness and generalization of the autoencoder, we introduce random noise augmentation in input parameters and latent representations simultaneously. The encoding and decoding processes can be formulated as, Z=[z10,…,zk0,…,zK0]=fencoder(V+ξV,σ)⏟encoding;V′=[v1′,⋯,vk′,⋯,vK′]=fdecoder(Z+ξZ,ρ)⏟decoding, (4) where fencoder(⋅,σ) and fdecoder(⋅,ρ) denote the encoder and decoder parameterized by σ and ρ, respectively. Z represents the latent representations, ξV and ξZ denote random noise that are added into input parameters V and latent representations Z, and V′ is the reconstructed parameters. We default to using an autoencoder with a 4-layer encoder and decoder. Same as the normal autoencoder training, we minimize the mean square error (MSE) loss between V′ and V as follows, LMSE=1K∑K1‖vk−vk′‖2, (5) where vk′ is the reconstructed parameters of k-th model. Table 1: We present results in the format of ‘original / ensemble / p-diff’. Our method obtains similar or even higher performance than baselines. The results of p-diff is average in three runs. Bold entries are best results. Network\Dataset MNIST CIFAR-10 CIFAR-100 STL-10 Flowers Pets F-101 ImageNet-1K ResNet-18 99.2 / 99.2 / 99.3 92.5 / 92.5 / 92.7 76.7 / 76.7 / 76.9 75.5 / 75.5 / 75.4 49.1 / 49.1 / 49.7 60.9 / 60.8 / 61.1 71.2 / 71.3 / 71.3 78.7 / 78.5 / 78.7 ResNet-50 99.4 / 99.3 / 99.4 91.3 / 91.4 / 91.3 71.6 / 71.6 / 71.7 69.2 / 69.1 / 69.2 33.7 / 33.9 / 38.1 58.0 / 58.0 / 58.0 68.6 / 68.5 / 68.6 79.2 / 79.2 / 79.3 ViT-Tiny 99.5 / 99.5 / 99.5 96.8 / 96.8 / 96.8 86.7 / 86.8 / 86.7 97.3 / 97.3 / 97.3 87.5 / 87.5 / 87.5 89.3 / 89.3 / 89.3 78.5 / 78.4 / 78.5 73.7 / 73.7 / 74.1 ViT-Base 99.5 / 99.4 / 99.5 98.7 / 98.7 / 98.7 91.5 / 91.4 / 91.7 99.1 / 99.0 / 99.2 98.3 / 98.3 / 98.3 91.6 / 91.5 / 91.7 83.4 / 83.4 / 83.4 84.5 / 84.5 / 84.7 ConvNeXt-T 99.3 / 99.4 / 99.3 97.6 / 97.6 / 97.7 87.0 / 87.0 / 87.1 98.2 / 98.0 / 98.2 70.0 / 70.0 / 70.5 92.9 / 92.8 / 93.0 76.1 / 76.1 / 76.2 82.1 / 82.1 / 82.3 ConvNeXt-B 99.3 / 99.3 / 99.4 98.1 / 98.1 / 98.1 88.3 / 88.4 / 88.4 98.8 / 98.8 / 98.9 88.4 / 88.4 / 88.5 94.1 / 94.0 / 94.1 81.4 / 81.4 / 81.6 83.8 / 83.7 / 83.9 2.4 Parameter generation One of the most direct strategies is to synthesize the novel parameters via a diffusion model. However, the memory cost of this operation is too heavy, especially when the dimension of V is ultra-large. Based on this consideration, we apply the diffusion process to the latent representations by default. For Z=[z10,⋯,zk0,⋯,zK0] extracted from parameter autoencoder, we use the optimization of DDPM (Ho et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib24) as follows, θ←θ−∇θ‖ϵ−ϵθ(α¯tzk0+1−α¯tϵ,t)‖2, (6) where t is uniform between 1 and T, the sequence of hyperparameters α¯t indicates the noise strength at each step, ϵ is the added Gaussian noise, ϵθ(⋅) denotes the denoising network that parameterized by θ. After finishing the training of the parameter generation, we directly fed random noise into the reverse process and the trained decoder to generate a new set of high-performing parameters. These generated parameters are concatenated with the remained model parameters to form new models for evaluation. Neural network parameters and image pixels exhibit significant disparities in several key aspects, including data type, dimensions, range, and physical interpretation. Different from images, neural network parameters mostly have no spatial relevance, so we replace 2D convolutions with 1D convolutions in our parameter autoencoder and parameter generation processes. 3 Experiments In this section, We first introduce the setup for reproducing. Then, we report the result comparisons and ablation studies. 3.1 SetupDatasets and architectures. We evaluate our approach across a wide range of datasets, including MNIST (LeCun et al., 1998 https://arxiv.org/html/2402.13144v1#bib.bib33), CIFAR-10/100 (Krizhevsky et al., 2009 https://arxiv.org/html/2402.13144v1#bib.bib30), ImageNet-1K (Deng et al., 2009 https://arxiv.org/html/2402.13144v1#bib.bib10), STL-10 (Coates et al., 2011 https://arxiv.org/html/2402.13144v1#bib.bib8), Flowers (Nilsback & Zisserman, 2008 https://arxiv.org/html/2402.13144v1#bib.bib42), Pets (Parkhi et al., 2012 https://arxiv.org/html/2402.13144v1#bib.bib43), and F-101 (Bossard et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib3) to study the effectiveness of our method. We mainly conduct experiments on ResNet-18/50 (He et al., 2016 https://arxiv.org/html/2402.13144v1#bib.bib20), ViT-Tiny/Base (Dosovitskiy et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib14), and ConvNeXt-T/B (Liu et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib36). Training details. The autoencoder and latent diffusion model both include a 4-layer 1D CNNs-based encoder and decoder. We default to collecting 200 training data for all architectures. For ResNet-18/50, we train the models from scratch. In the last epoch, we continue to train the last two normalization layers and fix the other parameters. We save 200 checkpoints in the last epoch, i.e., original models. For ViT-Tiny/Base and ConvNeXt-T/B, we fine-tune the last two normalization parameters of the released model in the timm library (Wightman, 2019 https://arxiv.org/html/2402.13144v1#bib.bib63). The ξV and ξZare Gaussian noise with amplitude of 0.001 and 0.1. In most cases, the autoencoder and latent diffusion training can be completed within 1 to 3 hours on a single Nvidia A100 40G GPU. Inference details. We synthesize 100 novel parameters by feeding random noise into the latent diffusion model and the trained decoder. These synthesized parameters are then concatenated with the aforementioned fixed parameters to form our generated models. From these generated models, we select the one with the best performance on the training set. Subsequently, we evaluate its accuracy on the validation set and report the results. That is a consideration of making fair comparisons with the models trained using SGD optimization. We empirically find the performance on the training set is good for selecting models for testing. Baselines. 1) The best validation accuracy among the original models is denoted as ‘original’. 2) Average weight ensemble (Krogh & Vedelsby, 1994 https://arxiv.org/html/2402.13144v1#bib.bib32; Wortsman et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib65) of original models is denoted as ‘ensemble’. Table 2: p-diff main ablation experiments. We ablate the number of original models K, the location of applying our approach, and the effect of noise augmentation. The default settings are K=200, applying p-diff on the deep BN parameters (between layer16 to 18), and using noise augmentation in the input parameters and latent representations. Defaults are marked in gray. Bold entries are best results. K best avg. med. 1 76.6 70.7 73.2 10 76.5 71.2 73.8 50 76.7 71.3 74.3 200 76.9 72.4 75.6 500 76.8 72.3 75.4 (a)Large K can improve the performance stability of our method. parameters best avg. med. original models 76.7 76.6 76.6 BN-layer10 to 14 76.8 71.9 75.3 BN-layer14 to 16 76.9 72.2 75.5 BN-layer16 to 18 76.9 72.4 75.6 (b)P-diff works well on deep layers. The index of layer is aligned with the standard ResNet-18. noise augmentation best avg. med. original models 76.7 - - no noise 76.7 65.8 65.0 + para. noise 76.7 66.7 67.3 + latent noise 76.7 72.1 75.3 + para. and latent noise 76.9 72.4 75.6 (c)Noise augmentation makes p-diff stronger. Adding noise on latent representations is more important than on parameters. Table 3: We present result comparisons of original, ensemble, and p-diff under synthesizing entire model parameters setting. Our method demonstrates good generalization on ConvNet-3 and MLP-3. Bold entries are best results. Dataset\Network ConvNet-3 original ensemble p-diff parameter number CIFAR-10 77.2 77.3 77.5 24714 CIFAR-100 57.2 57.2 57.3 70884 (d)Result comparisons on ConvNet-3 (includes three convolutional layers and one linear layer. Dataset\Network MLP-3 original ensemble p-diff parameter number MNIST 85.3 85.2 85.4 39760 CIFAR-10 48.1 48.1 48.2 155135 (e)Result comparisons on MLP-3 (includes three linear layers and ReLU activation function). 3.2 Results Tab. 1 https://arxiv.org/html/2402.13144v1#S2.T1 shows the result comparisons with two baselines across 8 datasets and 6 architectures. Based on the results, we have several observations as follows: i) In most cases, our method achieves similar or better results than two baselines. This demonstrates that our method can efficiently learn the distribution of high-performing parameters and generate superior models from random noise. ii) Our method consistently performs well on various datasets, which indicates the good generality of our method. 3.3 Ablation studies and analysis Extensive ablation studies are conducted in this section to illustrate the characteristics of our method. We default to training ResNet-18 on CIFAR-100 and report the best, average, and medium accuracy (if not otherwise stated). The number of training models. Tab. 3(a) https://arxiv.org/html/2402.13144v1#S3.F3.sf1 varies the size of training data, i.e. the number of original models. We find the performance gap of best results among different numbers of the original models is minor. To comprehensively explore the influences of different numbers of training data on the performance stability, we also report the average (avg.) and median (med.) accuracy as metrics of stability of our generated models. Notably, the stability of models generated with a small number of training instances is much worse than that observed in larger settings. This can be explained by the learning principle of the diffusion model: the diffusion process may be hard to model the target distribution well if only a few input samples are used for training. Where to apply p-diff. We default to synthesizing the parameters of the last two normalization layers. To investigate the effectiveness of p-diff on other depths of normalization layers, we also explore the performance of synthesizing the other shallow-layer parameters. To keep an equal number of BN parameters, we implement our approach to three sets of BN layers, which are between layers with different depths. As shown in Tab. 3(b) https://arxiv.org/html/2402.13144v1#S3.F3.sf2, we empirically find that our approach achieves better performances (best accuracy) than the original models on all depths of BN layers settings. Another finding is that synthesizing the deep layers can achieve better accuracy than generating the shallow ones. This is because generating shallow-layer parameters is more likely to accumulate errors during the forward propagation than generating deep-layer parameters. Noise augmentation. Noise augmentation is designed to enhance the robustness and generalization of training the autoencoder. We ablate the effectiveness of applying this augmentation in the input parameters and latent representations, respectively. The ablation results are presented in Tab. 3(c) https://arxiv.org/html/2402.13144v1#S3.F3.sf3. Several observations can be summarized as follows: i) Noise augmentation plays a crucial role in generating stable and high-performing models. ii) The performance gains of applying noise augmentation in the latent representations are larger than in the input parameters. iii) Our default setting, jointly using noise augmentation in parameters and representations obtains the best performances (includes best, average, and medium accuracy). Generalization on entire model parameters. Until now, we have evaluated the effectiveness of our approach in synthesizing a subset of model parameters, i.e., batch normalization parameters. What about synthesizing entire model parameters? To evaluate this, we extend our approach to two small architectures, namely MLP-3 (includes three linear layers and ReLU activation function) and ConvNet-3 (includes three convolutional layers and one linear layer). Different from the aforementioned training data collection strategy, we individually train these architectures from scratch with 200 different random seeds. We take CIFAR-10 as an example and show the details of these two architectures (convolutional layer: kernel size × kernel size, the number of channels; linear layer: input dimension, output dimension) as follows: ∙ ConvNet-3: conv1. 3×3, 32, conv2. 3×3, 32, conv3. 3×3, 32, linear layer. 2048, 10. ∙ MLP-3: linear layer1. 3072, 50, linear layer2. 50, 25, linear layer3. 25, 10. We present result comparisons between our approach and two baselines (i.e., original and ensemble) at Tab. 3(e) https://arxiv.org/html/2402.13144v1#S3.F3.sf5. We report the comparisons and parameter numbers of ConvNet-3 on CIFAR-10/100 and MLP-3 on CIFAR-10 and MNIST datasets. These experiments demonstrate the effectiveness and generalization of our approach in synthesizing entire model parameters, i.e., achieving similar or even improved performances over baselines. These results suggest the practical applicability of our method. However, we can not synthesize the entire parameters of large architectures, such as ResNet, ViT, and ConvNeXt series. It is mainly constrained by the limitation of the GPU memory. Parameter patterns of original models. Experimental results and ablation studies demonstrate the effectiveness of our method in generating neural network parameters. To explore the intrinsic reason behind this, we use 3 random seeds to train ResNet-18 model from scratch and visualize the parameters in Fig. 3 https://arxiv.org/html/2402.13144v1#S4.F3. We visualize the heat map of parameter distribution via min-max normalization in different layers individually. Based on the visualizations of the parameters of convolutional (Conv.-layer2) and fully connected (FC-layer18) layers, there indeed exist specific parameter patterns among these layers. Based on the learning of these patterns, our approach can generate high-performing neural network parameters. 4 Is P-diff Only Memorizing? In this section, we mainly investigate the difference between original and generated models. We first propose a similarity metric. Then several comparisons and visualizations are conducted to illustrate the characteristics of our approach. Questions and experiment designs. Here, we first ask the following questions: 1) Does p-diff just memorize the samples from the original models in the training set? 2) Is there any difference among adding noise or fine-tuning the original models, and the models generated by our approach? In our paper, we hope that our p-diff can generate some new parameters that perform differently than the original models. To verify this, we design experiments to study the differences between original, noise-added, fine-tuned, and p-diff models by comparing their predictions and visualizations. [image: Refer to caption]Figure 3: Visualizing the parameter distributions of convolutional (Conv.-layer2) and fully connected (FC-layer18) layers. Parameters from different layers show variant patterns while these parameters from the same layer show similar patterns. The index of layer is aligned with the standard ResNet-18. [image: Refer to caption] (a)Similarity comparisons of original and p-diff models. [image: Refer to caption] (b)Similarity comparisons of fine-tuned, noise-added, and p-diff models. [image: Refer to caption] (c)t-SNE of the latent representations of original, p-diff, and adding noise. Figure 4: The similarity represents the Intersection of Union (IoU) over wrong predictions between/among two models (a) shows the comparisons in four cases: similarity among original models and p-diff models, similarity between original and p-diff models, and the maximum similarity (nearest neighbor) between original and p-diff models. (b) displays the accuracy and max similarity of fine-tuned, noise-added, and p-diff models. All the maximum similarities are calculated with the original models. (c) presents the t-SNE (Van der Maaten et al., 2008 https://arxiv.org/html/2402.13144v1#bib.bib61) of latent representations of the original models, p-diff models, and adding noise operation.Similarity metric. We conduct experiments on CIFAR-100 (Krizhevsky et al., 2009 https://arxiv.org/html/2402.13144v1#bib.bib30) with ResNet-18 (He et al., 2016 https://arxiv.org/html/2402.13144v1#bib.bib20) under the default setting, i.e. only generating the parameters of the last two batch normalization layers. We measure the similarity between the two models by calculating the Intersection over Union (IoU) on their wrong predictions. The IoU can be formulated as follows, IoU=|P1wrong∩P2wrong|/|P1wrong∪P2wrong|, (7) where P⋅wrong denotes the indexes of wrong predictions on the validation set, ∩ and ∪ represent union and intersection operations. A higher IoU indicates a greater similarity between the predictions of the two models. the influence of the performance contrasts in experiments, we select models that perform better than 76.5% by default. Similarity of predictions. We evaluate the similarity between the original and p-diff models. For each model, we obtain its similarity by averaging the IoUs with other models. We introduce four comparisons: 1) similarity among original models; 2) similarity among p-diff models; 3) similarity between original and p-diff models; and 4) max similarity (nearest neighbor) between original and p-diff models. We calculate the IoUs for all models in the above four comparisons and report their averaged values in Fig. 4(a) https://arxiv.org/html/2402.13144v1#S4.F4.sf1. One can find that the differences among generated models are much larger than the differences among the original models. Another finding is that even the maximum similarity between the original and generated models is also lower than the similarity among the original models. It shows our p-diff can generate new parameters that perform differently with their training data (i.e. original models). We also compare our approach with the fine-tuned and noise-added models. Specifically, we randomly choose one generated model, and search its nearest neighbor (i.e. max similarity) from the original models. Then, we fine-tune and add random noise from the nearest neighbor to obtain corresponding models. After that, we calculate the similarity of the original with fine-tuned and noise-added models, respectively. Finally, we repeat this operation fifty times and report their average IoUs for analysis. In this experiment, we also constraint the performances of all models, i.e., only good models are used here for reducing the bias of visualization. We empirically set the amplitude of random noise with the range from 0.01 to 0.1 to prevent substantial performance drops. Based on the results in Fig. 4(b) https://arxiv.org/html/2402.13144v1#S4.F4.sf2, we find that the performances of fine-tuned and noise-added models are hard to outperform the original models. Besides, the similarities between fine-tuned or noise-added and original models are very high, which indicates these two operations can not obtain novel but high-performing models. However, our generated models achieve diverse similarities and superior performances compared to the original models. Comparison of latent representations. In addition to predictions, we assess the distributions of latent representations for the original and generated models using t-SNE (Van der Maaten et al., 2008 https://arxiv.org/html/2402.13144v1#bib.bib61). To identify the differences between our approach and the operation of adding noise to the latent representations of original models, we also include the adding noise operation as a comparison in Fig. 4(c) https://arxiv.org/html/2402.13144v1#S4.F4.sf3. The added noise is random Gaussian noise with an amplitude of 0.1. One can find that p-diff can generate novel latent representations while adding noise just makes interpolation around the latent representations of original models. [image: Refer to caption] (a)Visualization of parameter trajectories of p-diff. [image: Refer to caption](b)IoUs of high-performing (Acc.≥76.5%) generated models. Figure 5: (a) shows the parameter trajectories of our approach and original models distribution via t-SNE. (b) illustrates max IoUs between generated and original models in different K settings. Sim. denotes similarity.The trajectories of p-diff process. We plot the generated parameters of different time steps in the inference stage to form trajectories to explore its generation process. Five trajectories (initialized by 5 random noise) are shown in Fig. 5(a) https://arxiv.org/html/2402.13144v1#S4.F5.sf1. We also plot the average parameters of the original models and their standard deviation (std). As the time step increases, the generated parameters are overall close to the original models. Although we keep a narrow performance range constraint for visualization, there is still a certain distance between the end points (orange triangles) of trajectories and average parameters (five-pointed star). Another finding is that the five trajectories are diverse.
From memorizing to generate new parameters.
From the metrics, we find our method has good generalization among channel-wise normalization operations, such as GN and IN. Table 8: We design ablations about the intensity of input noise ξV and latent noise ξZ, generating variant types of parameters. ‘para.’ denotes
To investigate the impact of the number of original models (K) on the diversity of generated models, we visualize the max similarities between original and generated models with different K in Fig. 5(b) https://arxiv.org/html/2402.13144v1#S4.F5.sf2. Specifically, we continually generate parameters until 50 models perform better than 76.5% in all cases. The generated models almost memorize the original model when K=1, as indicated by the narrow similarity range and high value. The similarity range of these generated models becomes larger as K increases, demonstrating our approach can generate parameters that perform differently from the original models. 5 Related WorkDiffusion models. Diffusion models have achieved remarkable results in visual generation. These methods (Ho et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib24; Dhariwal & Nichol, 2021 https://arxiv.org/html/2402.13144v1#bib.bib12; Ho et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib25; Peebles & Xie, 2022 https://arxiv.org/html/2402.13144v1#bib.bib44; Hertz et al., 2023 https://arxiv.org/html/2402.13144v1#bib.bib23; Li et al., 2023 https://arxiv.org/html/2402.13144v1#bib.bib34) are based on non-equilibrium thermodynamics (Jarzynski, 1997 https://arxiv.org/html/2402.13144v1#bib.bib27; Sohl-Dickstein et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib56), and the its pathway is similar to GAN (Zhu et al., 2017 https://arxiv.org/html/2402.13144v1#bib.bib67; Isola et al., 2017 https://arxiv.org/html/2402.13144v1#bib.bib26; Brock et al., 2018a https://arxiv.org/html/2402.13144v1#bib.bib5), VAE (Kingma & Welling, 2013 https://arxiv.org/html/2402.13144v1#bib.bib28; Razavi et al., 2019 https://arxiv.org/html/2402.13144v1#bib.bib47), and flow-based model (Dinh et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib13; Rezende & Mohamed, 2015 https://arxiv.org/html/2402.13144v1#bib.bib49). Diffusion models can be categorized into three main branches. The first branch focuses on enhancing the synthesis quality of diffusion models, exemplified by models like DALL⋅E 2 (Ramesh et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib46), Imagen (Saharia et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib53), and Stable Diffusion (Rombach et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib51). The second branch aims to improve the sampling speed, including DDIM (Song et al., 2021 https://arxiv.org/html/2402.13144v1#bib.bib58), Analytic-DPM (Bao et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib1), and DPM-Solver (Lu et al., 2022 https://arxiv.org/html/2402.13144v1#bib.bib38). The final branch involves reevaluating diffusion models from a continuous perspective, like score-based models (Song & Ermon, 2019 https://arxiv.org/html/2402.13144v1#bib.bib59; Feng et al., 2023 https://arxiv.org/html/2402.13144v1#bib.bib16). Parameter generation. HyperNet (Ha et al., 2017 https://arxiv.org/html/2402.13144v1#bib.bib19) dynamically generates the weights of a model with variable architecture. Smash (Brock et al., 2018b https://arxiv.org/html/2402.13144v1#bib.bib6) introduces a flexible scheme based on memory read-writes that can define a diverse range of architectures. (Peebles et al., 2023 https://arxiv.org/html/2402.13144v1#bib.bib45) collect 23 million checkpoints and train a conditional generator via a transformer-based diffusion model. MetaDiff (Zhang & Yu, 2023 https://arxiv.org/html/2402.13144v1#bib.bib66) introduces a diffusion-based meta-learning method for few-shot learning, where a layer is replaced by a diffusion U-Net (Ronneberger et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib52). HyperDiffusion (Erkoç et al., 2023 https://arxiv.org/html/2402.13144v1#bib.bib15) directly utilizes a diffusion model on MLPs to generate new neural implicit fields. Different from them, we analyze the intrinsic differences between images and parameters and design corresponding modules to learn the distributions of the high-performing parameters. Stochastic and Bayesian neural networks. Our approach could be viewed as learning a prior over network parameters, represented by the trained diffusion model. Learning parameter priors for neural networks has been studied in classical literature. Stochastic neural networks (SNNs) (Sompolinsky et al., 1988 https://arxiv.org/html/2402.13144v1#bib.bib57; Bottou et al., 1991 https://arxiv.org/html/2402.13144v1#bib.bib4; Wong, 1991 https://arxiv.org/html/2402.13144v1#bib.bib64; Schmidt et al., 1992 https://arxiv.org/html/2402.13144v1#bib.bib54; Murata et al., 1994 https://arxiv.org/html/2402.13144v1#bib.bib39) also learn such priors by introducing randomness to improve the robustness and generalization of neural networks. The Bayesian neural networks (Neal, 2012 https://arxiv.org/html/2402.13144v1#bib.bib40; Kingma & Welling, 2013 https://arxiv.org/html/2402.13144v1#bib.bib28; Rezende et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib50; Kingma et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib29; Gal & Ghahramani, 2016 https://arxiv.org/html/2402.13144v1#bib.bib17) aims to model a probability distribution over neural networks to mitigate overfitting, learn from small datasets, and asses the uncertainty of model predictions. (Graves, 2011 https://arxiv.org/html/2402.13144v1#bib.bib18) propose an easily implementable stochastic variational method as a practical approximation to Bayesian inference for neural networks. They introduce a heuristic pruner to reduce the number of network weights, resulting in improved generalization. (Welling & Teh, 2011 https://arxiv.org/html/2402.13144v1#bib.bib62) combine Langevin dynamics with SGD to incorporate a Gaussian prior into the gradient. This transforms SGD optimization into a sampling process.Bayes by Backprop (Blundell et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib2) learns a probability distribution prior over the weights of a neural network. These methods mostly operate in small-scale settings, while p-diff shows its effectiveness in real-world architectures. 6 Discussion and Conclusion Neural networks have several popular learning paradigms, such as supervised learning (Krizhevsky et al., 2012 https://arxiv.org/html/2402.13144v1#bib.bib31; Simonyan & Zisserman, 2014 https://arxiv.org/html/2402.13144v1#bib.bib55; He et al., 2016 https://arxiv.org/html/2402.13144v1#bib.bib20; Dosovitskiy et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib14), self-supervised learning (Devlin et al., 2018 https://arxiv.org/html/2402.13144v1#bib.bib11; Brown et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib7; He et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib21, 2022 https://arxiv.org/html/2402.13144v1#bib.bib22), and more. In this study, we observe that diffusion models can be employed to generate high-performing and novel neural network parameters, demonstrating their superiority. Using diffusion steps for neural network parameter updates shows a potentially novel paradigm in deep learning. However, we acknowledge that images/videos and parameters are signals of different natures, and this distinction must be handled with care. Additionally, even though diffusion models have achieved considerable success in image/video generation, their application to parameters remains relatively underexplored. These pose a series of challenges for neural network diffusion. We propose an initial approach to address some of these challenges. Nevertheless, there are still unresolved challenges, including memory constraints for generating the entire parameters of large architectures, the efficiency of structure designs, and performance stability. Acknowledgments. We thank Kaiming He, Dianbo Liu, Mingjia Shi, Zheng Zhu, Bo Zhao, Jiawei Liu, Yong Liu, Ziheng Qin, Zangwei Zheng, Yifan Zhang, Xiangyu Peng, Hongyan Chang, David Yin, Dave Zhenyu Chen, Ahmad Sajedi, and George Cazenavette for valuable discussions and feedbacks. References - Bao et al. (2022)Bao, F., Li, C., Zhu, J., and Zhang, B.Analytic-DPM: an analytic estimate of the optimal reverse variance in diffusion probabilistic models.In *ICLR*, 2022.URL https://openreview.net/forum?id=0xiJLKH-ufZ. - Blundell et al. (2015)Blundell, C., Cornebise, J., Kavukcuoglu, K., and Wierstra, D.Weight uncertainty in neural network.In *ICML*. PMLR, 2015. - Bossard et al. (2014)Bossard, L., Guillaumin, M., and Van Gool, L.Food-101–mining discriminative components with random forests.In *ECCV*. Springer, 2014. - Bottou et al. (1991)Bottou, L. et al.Stochastic gradient learning in neural networks.*Proceedings of Neuro-Nımes*, 91(8), 1991. - Brock et al. (2018a)Brock, A., Donahue, J., and Simonyan, K.Large scale gan training for high fidelity natural image synthesis.*arXiv preprint arXiv:1809.11096*, 2018a. - Brock et al. (2018b)Brock, A., Lim, T., Ritchie, J., and Weston, N.SMASH: One-shot model architecture search through hypernetworks.In *ICLR*, 2018b.URL https://openreview.net/forum?id=rydeCEhs-. - Brown et al. (2020)Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J. D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al.Language models are few-shot learners.*NeurIPS*, 33, 2020. - Coates et al. (2011)Coates, A., Ng, A., and Lee, H.An analysis of single-layer networks in unsupervised feature learning.In *Proceedings of the fourteenth international conference on artificial intelligence and statistics*. JMLR Workshop and Conference Proceedings, 2011. - Cristianini et al. (2000)Cristianini, N., Shawe-Taylor, J., et al.*An introduction to support vector machines and other kernel-based learning methods*.Cambridge university press, 2000. - Deng et al. (2009)Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L.Imagenet: A large-scale hierarchical image database.In *CVPR*. Ieee, 2009. - Devlin et al. (2018)Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K.Bert: Pre-training of deep bidirectional transformers for language understanding.*arXiv preprint arXiv:1810.04805*, 2018. - Dhariwal & Nichol (2021)Dhariwal, P. and Nichol, A.Diffusion models beat gans on image synthesis.*NeurIPS*, 34, 2021. - Dinh et al. (2014)Dinh, L., Krueger, D., and Bengio, Y.Nice: Non-linear independent components estimation.*arXiv preprint arXiv:1410.8516*, 2014. - Dosovitskiy et al. (2020)Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., et al.An image is worth 16x16 words: Transformers for image recognition at scale.*arXiv preprint arXiv:2010.11929*, 2020. - Erkoç et al. (2023)Erkoç, Z., Ma, F., Shan, Q., Nießner, M., and Dai, A.Hyperdiffusion: Generating implicit neural fields with weight-space diffusion.*arXiv preprint arXiv:2303.17015*, 2023. - Feng et al. (2023)Feng, B. T., Smith, J., Rubinstein, M., Chang, H., Bouman, K. L., and Freeman, W. T.Score-based diffusion models as principled priors for inverse imaging.*arXiv preprint arXiv:2304.11751*, 2023. - Gal & Ghahramani (2016)Gal, Y. and Ghahramani, Z.Dropout as a bayesian approximation: Representing model uncertainty in deep learning.In *ICML*. PMLR, 2016. - Graves (2011)Graves, A.Practical variational inference for neural networks.*NeurIPS*, 24, 2011. - Ha et al. (2017)Ha, D., Dai, A. M., and Le, Q. V.Hypernetworks.In *ICLR*, 2017.URL https://openreview.net/forum?id=rkpACe1lx. - He et al. (2016)He, K., Zhang, X., Ren, S., and Sun, J.Deep residual learning for image recognition.In *CVPR*, 2016. - He et al. (2020)He, K., Fan, H., Wu, Y., Xie, S., and Girshick, R.Momentum contrast for unsupervised visual representation learning.In *CVPR*, 2020. - He et al. (2022)He, K., Chen, X., Xie, S., Li, Y., Dollár, P., and Girshick, R.Masked autoencoders are scalable vision learners.In *CVPR*, 2022. - Hertz et al. (2023)Hertz, A., Mokady, R., Tenenbaum, J., Aberman, K., Pritch, Y., and Cohen-or, D.Prompt-to-prompt image editing with cross-attention control.In *ICLR*, 2023.URL https://openreview.net/forum?id=_CDixzkzeyb. - Ho et al. (2020)Ho, J., Jain, A., and Abbeel, P.Denoising diffusion probabilistic models.*NeurIPS*, 33, 2020. - Ho et al. (2022)Ho, J., Chan, W., Saharia, C., Whang, J., Gao, R., Gritsenko, A., Kingma, D. P., Poole, B., Norouzi, M., Fleet, D. J., et al.Imagen video: High definition video generation with diffusion models.*arXiv preprint arXiv:2210.02303*, 2022. - Isola et al. (2017)Isola, P., Zhu, J.-Y., Zhou, T., and Efros, A. A.Image-to-image translation with conditional adversarial networks.In *CVPR*, 2017. - Jarzynski (1997)Jarzynski, C.Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach.*Physical Review E*, 56(5), 1997. - Kingma & Welling (2013)Kingma, D. P. and Welling, M.Auto-encoding variational bayes.*arXiv preprint arXiv:1312.6114*, 2013. - Kingma et al. (2015)Kingma, D. P., Salimans, T., and Welling, M.Variational dropout and the local reparameterization trick.*NeurIPS*, 28, 2015. - Krizhevsky et al. (2009)Krizhevsky, A., Hinton, G., et al.Learning multiple layers of features from tiny images.2009. - Krizhevsky et al. (2012)Krizhevsky, A., Sutskever, I., and Hinton, G. E.Imagenet classification with deep convolutional neural networks. *NeurIPS*, 25, 2012. - Krogh & Vedelsby (1994)Krogh, A. and Vedelsby, J.Neural network ensembles, cross validation, and active learning.*NeurIPS*, 7, 1994. - LeCun et al. (1998)LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P.Gradient-based learning applied to document recognition.*Proceedings of the IEEE*, 86(11), 1998. - Li et al. (2023)Li, A. C., Prabhudesai, M., Duggal, S., Brown, E., and Pathak, D.Your diffusion model is secretly a zero-shot classifier.*arXiv preprint arXiv:2303.16203*, 2023. - Lin et al. (2014)Lin, T.-Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Dollár, P., and Zitnick, C. L.Microsoft coco: Common objects in context.In *Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13*, pp. 740–755. Springer, 2014. - Liu et al. (2022)Liu, Z., Mao, H., Wu, C.-Y., Feichtenhofer, C., Darrell, T., and Xie, S.A convnet for the 2020s.In *CVPR*, 2022. - Long et al. (2015)Long, J., Shelhamer, E., and Darrell, T.Fully convolutional networks for semantic segmentation.In *Proceedings of the IEEE conference on computer vision and pattern recognition*, pp. 3431–3440, 2015. - Lu et al. (2022)Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J.DPM-solver: A fast ODE solver for diffusion probabilistic model sampling in around 10 steps.In Oh, A. H., Agarwal, A., Belgrave, D., and Cho, K. (eds.), *NeurIPS*, 2022.URL https://openreview.net/forum?id=2uAaGwlP_V. - Murata et al. (1994)Murata, N., Yoshizawa, S., and Amari, S.-i.Network information criterion-determining the number of hidden units for an artificial neural network model.*IEEE transactions on neural networks*, 5(6), 1994. - Neal (2012)Neal, R. M.*Bayesian learning for neural networks*, volume 118.Springer Science & Business Media, 2012. - Nichol et al. (2021)Nichol, A., Dhariwal, P., Ramesh, A., Shyam, P., Mishkin, P., McGrew, B., Sutskever, I., and Chen, M.Glide: Towards photorealistic image generation and editing with text-guided diffusion models.*arXiv preprint arXiv:2112.10741*, 2021. - Nilsback & Zisserman (2008)Nilsback, M.-E. and Zisserman, A.Automated flower classification over a large number of classes.In *2008 Sixth Indian conference on computer vision, graphics & image processing*. IEEE, 2008. - Parkhi et al. (2012)Parkhi, O. M., Vedaldi, A., Zisserman, A., and Jawahar, C.Cats and dogs.In *CVPR*. IEEE, 2012. - Peebles & Xie (2022)Peebles, W. and Xie, S.Scalable diffusion models with transformers.*arXiv preprint arXiv:2212.09748*, 2022. - Peebles et al. (2023)Peebles, W., Radosavovic, I., Brooks, T., Efros, A. A., and Malik, J.Learning to learn with generative models of neural network checkpoints, 2023.URL https://openreview.net/forum?id=JXkz3zm8gJ. - Ramesh et al. (2022)Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., and Chen, M.Hierarchical text-conditional image generation with clip latents.*arXiv preprint arXiv:2204.06125*, 1(2), 2022. - Razavi et al. (2019)Razavi, A., Van den Oord, A., and Vinyals, O.Generating diverse high-fidelity images with vq-vae-2.*NeurIPS*, 32, 2019. - Ren et al. (2015)Ren, S., He, K., Girshick, R., and Sun, J.Faster r-cnn: Towards real-time object detection with region proposal networks.*Advances in neural information processing systems*, 28, 2015. - Rezende & Mohamed (2015)Rezende, D. and Mohamed, S.Variational inference with normalizing flows.In *ICML*. PMLR, 2015. - Rezende et al. (2014)Rezende, D. J., Mohamed, S., and Wierstra, D.Stochastic backpropagation and approximate inference in deep generative models.In *ICML*. PMLR, 2014. - Rombach et al. (2022)Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B.High-resolution image synthesis with latent diffusion models.In *CVPR*, 2022. - Ronneberger et al. (2015)Ronneberger, O., Fischer, P., and Brox, T.U-net: Convolutional networks for biomedical image segmentation.In *Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18*, pp. 234–241. Springer, 2015. - Saharia et al. (2022)Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E. L., Ghasemipour, K., Gontijo Lopes, R., Karagol Ayan, B., Salimans, T., et al.Photorealistic text-to-image diffusion models with deep language understanding.*NeurIPS*, 35, 2022. - Schmidt et al. (1992)Schmidt, W. F., Kraaijveld, M. A., Duin, R. P., et al.Feed forward neural networks with random weights.In *ICPR*. IEEE Computer Society Press, 1992. - Simonyan & Zisserman (2014)Simonyan, K. and Zisserman, A.Very deep convolutional networks for large-scale image recognition.*arXiv preprint arXiv:1409.1556*, 2014. - Sohl-Dickstein et al. (2015)Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S.Deep unsupervised learning using nonequilibrium thermodynamics.In *ICML*. PMLR, 2015. - Sompolinsky et al. (1988)Sompolinsky, H., Crisanti, A., and Sommers, H.-J.Chaos in random neural networks.*Physical review letters*, 61(3), 1988. - Song et al. (2021)Song, J., Meng, C., and Ermon, S.Denoising diffusion implicit models.In *ICLR*, 2021.URL https://openreview.net/forum?id=St1giarCHLP. - Song & Ermon (2019)Song, Y. and Ermon, S.Generative modeling by estimating gradients of the data distribution.*NeurIPS*, 32, 2019. - Tian et al. (2020)Tian, Z., Shen, C., Chen, H., and He, T.Fcos: A simple and strong anchor-free object detector.*IEEE T-PAMI*, 44(4):1922–1933, 2020. - Van der Maaten et al. (2008)Van der Maaten, L., Hinton, G., and Van der Maaten, L.Visualizing data using t-sne.*JMLR*, 9(11), 2008. - Welling & Teh (2011)Welling, M. and Teh, Y. W.Bayesian learning via stochastic gradient langevin dynamics.In *ICML*, 2011. - Wightman (2019)Wightman, R.Pytorch image models. https://github.com/rwightman/pytorch-image-models, 2019. - Wong (1991)Wong, E.Stochastic neural networks.*Algorithmica*, 6(1-6), 1991. - Wortsman et al. (2022)Wortsman, M., Ilharco, G., Gadre, S. Y., Roelofs, R., Gontijo-Lopes, R., Morcos, A. S., Namkoong, H., Farhadi, A., Carmon, Y., Kornblith, S., et al.Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time.In *ICML*, pp. 23965–23998. PMLR, 2022. - Zhang & Yu (2023)Zhang, B. and Yu, D.Metadiff: Meta-learning with conditional diffusion for few-shot learning.*arXiv preprint arXiv:2307.16424*, 2023. - Zhu et al. (2017)Zhu, J.-Y., Park, T., Isola, P., and Efros, A. A.Unpaired image-to-image translation using cycle-consistent adversarial networks. In *ICCV*, 2017. Appendix A Experimental Settings In this section, we introduce detailed experiment settings, datasets, and instructions of code for reproducing. A.1 Training recipe We provide our basic training recipe with specific details in Tab. 4 https://arxiv.org/html/2402.13144v1#A1.T4. This recipe is based on the setting of ResNet-18 with CIFAR-100 dataset. We introduce these details of general training hyperparameters, autoencoder, and latent diffusion model, respectively. It may be necessary to make adjustments to the learning rate and the training iterations for other datasets. Training Setting Configuration K, i.e., the number of original models 200 batch size 200 Autoencoder optimizer AdamW learning rate 1e-3 training iterations 30, 000 optimizer momentum betas=(0.9, 0.999) weight decay 2e-6 ξV, i.e., noise added on the input parameters 0.001 ξZ, i.e., noise added on the latent representations 0.1 Diffusion optimizer AdamW learning rate 1e-3 training iterations 30, 000 optimizer momentum betas=(0.9, 0.999) weight decay 2e-6 ema β 0.9999 betas start 1e-4 betas end 2e-2 betas schedule linear T, i.e., maximum time steps in the training stage 1000Table 4: Our basic training recipe based on CIFAR100 dataset and ResNet-18 backbone.A.2 Datasets We evaluate the effectiveness of p-diff on 8 datasets. To be specific, CIFAR-10/100 (Krizhevsky et al., 2009 https://arxiv.org/html/2402.13144v1#bib.bib30). The CIFAR datasets comprise colored natural images of dimensions 32×32, categorized into 10 and 100 classes, respectively. Each dataset consists of 50,000 images for training and 10,000 images for testing.ImageNet-1K (Deng et al., 2009 https://arxiv.org/html/2402.13144v1#bib.bib10) derived from the larger ImageNet-21K dataset, ImageNet-1K is a curated subset featuring 1,000 categories. It encompasses 1,281,167 training images and 50,000 validation images. STL-10 (Coates et al., 2011 https://arxiv.org/html/2402.13144v1#bib.bib8) comprises 96×96 color images, spanning 10 different object categories. It serves as a versatile resource for various computer vision tasks, including image classification and object recognition. Flowers (Nilsback & Zisserman, 2008 https://arxiv.org/html/2402.13144v1#bib.bib42) is a dataset comprising 102 distinct flower categories, with each category representing a commonly occurring flower species found in the United Kingdom. Pets (Parkhi et al., 2012 https://arxiv.org/html/2402.13144v1#bib.bib43) includes around 7000 images with 37 categories. The images have large variations in scale, pose, and lighting. F-101 (Bossard et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib3) consists of 365K images that are crawled from Google, Bing, Yelp, and TripAdvisor using the Food-101 taxonomy. In the appendix, we extend our p-diff in object detection, semantic segmentation, and image generation tasks. Therefore, we also introduce the extra-used datasets in the following. COCO (Lin et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib35) consists of over 200,000 images featuring complex scenes with 80 object categories. It is widely used for object detection and segmentation tasks. We implement image generation task on CIFAR-10. A.3 Instructions for code We have submitted the source code as the supplementary materials in a zipped file named as ‘p-diff.zip’ for reproduction. A README is also included for the instructions for running the code. Appendix B Explorations of Designs and Strategies In this section, we introduce the reasons for the designs and strategies of our approach. B.1 Why 1D CNNs? Considering the great differences between visual data and neural network parameters, we default to using 1D CNNs in parameter autoencoder and generation. The detailed designs of 1D CNNs can be found in the following. Each layer in 1D CNNs includes two 1D convolutional layers with a normalization layer and an activation layer. More details of the 1D CNNs can be found at core/module/modules in our code zip file. Here naturally raises a question: are there alternatives to 1D CNNs? We can use pure fully connected (FC) layers as an alternative. To answer this question, we compare the performance of FC layers and 1D CNNs. The experiments are conducted on MNIST with ConvNet-3 as the backbone. Based on our experimental results in Tab. 5 https://arxiv.org/html/2402.13144v1#A2.T5, 1D CNNs consistently outperform FC in all architectures. Meanwhile, the memory occupancy of 1D CNNs is smaller than FC. Table 5: Comparison of using 1D CNNs and fully connected (FC) layers. 1D CNNs perform better than FC layers, especially in memory and time. Arch. Method Dataset Time (s)↓ Best↑ Average↑ Median↑ Worst↑ Memory (MB)↓ ConvNet-3 FC MNIST 17 98.0 90.1 93.6 70.2 1375 ConvNet-3 1D CNNs MNIST 16 99.2 92.1 94.2 73.6 1244Table 6: Comparison of using batch normalization, group normalization, and instance normalization in our approach. We also report the results without normalization. ‘norm.’ denotes normalization. Default settings are marked in gray. Bold entries are best results. norm. best avg. med. original 94.3 - - no norm. 94.0 82.8 80.1 BN 88.7 84.3 88.2 GN 94.3 89.8 93.9 IN 94.4 88.5 94.2 (a)Results on CIFAR-10. norm. best avg. med. original 99.6 - - no norm. 99.5 84.1 98.4 BN 99.3 86.7 99.1 GN 99.6 93.2 99.3 IN 99.6 92.7 99.4 (b)Results on MNIST. norm. best avg. med. original 76.7 - - no norm. 76.1 67.4 69.9 BN 75.9 70.7 73.3 GN 76.8 72.1 75.8 IN 76.9 72.4 75.6 (c)Results on CIFAR-100. Table 7: Comparisons between VAE and our proposed p-diff. VAE performs worse than our approach, especially on the metric of average and medium accuracy. num. of original models best avg. med. 1 75.6 61.2 70.4 10 76.5 65.8 71.5 50 76.5 63.0 71.8 200 76.7 62.7 70.8 500 76.7 62.6 71.9 (d)Result of VAE num. of original models best avg. med. 1 76.6 (+1.0) 70.7 (+9.5) 73.2 (+2.8) 10 76.5 (+0.0) 71.2 (+5.4) 73.8 (+2.3) 50 76.7 (+0.2) 71.3 (+8.3) 74.3 (+2.5) 200 76.9 (+0.2) 72.4 (+9.7) 75.6 (+4.8) 500 76.8 (+0.1) 72.3 (+9.7) 75.4 (+3.5) (e)P-diff vs VAE, improvements are reported in (). B.2 Is variational autoencoder an alternative to our approach? Variational autoencoder (VAE) (Cristianini et al., 2000 https://arxiv.org/html/2402.13144v1#bib.bib9) can be regarded as a probabilistic generative model and achieve many remarkable results in the generation area. We also implement VAE to generate neural network parameters. We first introduce the details of VAE in our experiment. We implement vanilla VAE using the same backbone of the autoencoder in p-diff for a fair comparison. We evaluate the VAE generator in the case of different K and compare its best, average, and medium performances with p-diff generated models. Based on the results in Tab. 7 https://arxiv.org/html/2402.13144v1#A2.T7, our approach outperforms VAE by a large margin in all cases. Another interesting finding is that the average performance of VAE generated models goes down as the number of original models increases. B.3 Which normalization strategy is suitable? Considering the intrinsic difference between images and neural network parameters, we explore the influence of different normalization strategies. We ablate batch normalization (BN), group normalization (GN), and instance normalization (IN) on CIFAR-10, MNIST, and CIFAR-100, respectively. We also implement our method without normalization for additional comparison. Their best, average, and medium performances of 100 generated models are reported in Tab. 6(c) https://arxiv.org/html/2402.13144v1#A2.F6.sf3. Based on the results, we have the following observations: 1) BN obtains the worst overall performance on all three metrics. Since BN operates in the batch dimension and introduces undesired correlations among model parameters 2) GN and IN perform better than without normalization, i.e. ‘no norm.’ in the Tab. 6(c) https://arxiv.org/html/2402.13144v1#A2.F6.sf3. That could be explained by some outlier parameters affecting the performance a lot. 3) parameter. Default settings are marked in gray. Bold entries are best results. para. noise best avg. med. 1e-4 76.7 72.1 75.6 1e-3 76.9 72.4 75.6 1e-2 76.3 70.4 74.4 1e-1 76.8 71.4 75.1 (f)Ablation of input noise ξV. latent noise best avg. med. 1e-3 76.7 67.3 73.2 1e-2 76.6 70.1 74.7 1e-1 76.9 72.6 75.6 1e-0 76.7 74.0 75.0 (g)Ablation of latent noise ξZ. para. type original best avg. med. linear 76.6 76.6 47.3 71.1 conv 76.2 76.2 71.3 76.1 shortcut 75.9 76.0 73.6 75.7 bn 76.7 76.9 72.4 75.6 (h)Ablation of types of parameters. Appendix C More Ablations In this section, we introduce more ablation studies of our method. Same as the main paper, if not otherwise stated, we default to training ResNet-18 on CIFAR-100 and report the best, average, and medium accuracy. C.1 The intensity of noise added into input parameters In the main paper, we ablate the effectiveness of the added noise into input parameters. Here, we study the impact of the intensity of this noise. Specifically, we explore four levels of noise intensity and report their best, average, and medium results in Tab. 6(f) https://arxiv.org/html/2402.13144v1#A2.F6.sf6. One can find that, our default intensity achieves the best overall performance. Both too-large and too-small noise intensities fail to obtain good results. That can be explained by that the too-large noise may destroy the original distribution of parameters while too-small noise can not provide enough effectiveness of augmentation. C.2 The intensity of noise added into latent representations Similar to Sec. C.1 https://arxiv.org/html/2402.13144v1#A3.SS1, we also ablate the noise intensity added into latent representations. As shown in Tab. 6(g) https://arxiv.org/html/2402.13144v1#A2.F6.sf7, the performance stability of generated models becomes better as the noise intensity increases. However, too-large noise also breaks the distribution of the original latent representations. C.3 The generalization on other types of parameters In the main paper, we investigate the effectiveness of our approach in generating normalization parameters. We also evaluate our approach on other types of parameters, such as linear, convolutional, and shortcut layers. Here, we show the details of the above three type layers as follows: 1) linear layer: the last linear layer of ResNet-18. 2) convolutional layer: first convolutional layer of ResNet-18. 3) shortcut layer: the shortcut layer between 7th and 8th layer of ResNet-18. The training data preparation is the same as we mentioned in the main paper. As illustrated in Tab. 6(h) https://arxiv.org/html/2402.13144v1#A2.F6.sf8, we find our approach consistently achieves similar or improved performance compared to the original models. Appendix D Open ExplorationsD.1 Do we need to train 1000-step diffusion model? We default to training the latent diffusion model via random sampling from 1000 time steps. Can we reduce the number of time steps in the training stage? To study the impact of the time steps, we conduct an ablation and report the results in Tab. 6(k) https://arxiv.org/html/2402.13144v1#A4.F6.sf11. Several findings can be summarized as follows: 1) Too small time steps might not be strong enough to generate high-performing models with good stability. 2) The best stability performances are obtained by setting the maximum time steps as 100. 3). Increasing the maximum time steps from 1000 to 2000 can not improve the performance. We will further upgrade our design based on this exploration. Table 9: Exploring the influence of maximum time steps in the training stage. We conduct experiments on CIFAR-10, MNIST, and CIFAR-100 datasets, respectively. Bold entries are best results. maximum step best avg. med. 10 94.4 82.0 93.8 100 94.3 94.3 94.3 1000 94.4 88.5 94.2 2000 94.3 85.8 94.2 (i)Results on CIFAR-10. maximum step best avg. med. 10 99.6 89.9 98.9 100 99.6 99.6 99.6 1000 99.6 92.7 99.4 2000 99.6 94.1 99.5 (j)Results on MNIST. maximum step best avg. med. 10 76.6 70.6 74.9 100 76.8 75.9 76.5 1000 76.9 72.4 75.6 2000 76.8 73.1 75.1 (k)Results on CIFAR-100. D.2 Potential applications Neural network diffusion can be utilized or help the following potential research areas. 1) Parameters initialization: our approach can generate high-performing initialized parameters. Therefore, that would speed up the optimization and reduce the overall cost of training. 2) Domain adaptation: our approach may have three benefits in the domain adaptation area. First, we can directly use the diffusion process to learn the well-performed models trained by different domain data. Second, some hard adaptations can be achieved by our approach. Third, the adaptation efficiency might be improved largely. Appendix E Other Finding and Comparison ResultsE.1 How to select generated parameters? P-diff can rapidly generate numerous high-performance models. How do we evaluate these models? There are two primary strategies. The first one is to directly test them on the validation set and select the best-performing model. The second one is to compute the loss of model outputs compared to the ground truth on the training set to choose a model. We generated hundred model parameters with performance distributions in different intervals and displayed their accuracy curves on both the training and validation sets in Fig. 6(l) https://arxiv.org/html/2402.13144v1#A5.F6.sf12. The experimental results indicate that p-diff exhibits a high level of consistency between the training and validation sets. To provide a fair comparison with baseline methods, we default to choose the model that performs the best results on the training set and compare it with the baseline. [image: Refer to caption](l)Accuracy distribution in p-diff models. [image: Refer to caption] (m)Visualization of initial, SGD-trained, p-diff generated model. Figure 6: P-diff can generate models with great consistency on both training and validation sets contrast compared to the original model. (a) shows the accuracy distribution of training and validation sets in hundred p-diff models. (b) displays a heat map of initial, SGD-trained, p-diff generated parameters of the normalization layer in ResNet-18.E.2 Parameter visualization To provide a more intuitive understanding, we compare the parameters generated by our approach, SGD optimization (original), and randomly initialized. Taking ResNet-18 as an example, we report the mean, std, accuracy (acc.), and IoU of the normalization layer parameters of training on CIFAR-100 in Fig. 6(m) https://arxiv.org/html/2402.13144v1#A5.F6.sf13. There is a significant difference between the parameters generated by our approach and the randomly initialized parameters, mean: 0.37 vs 0.36, std: 0.22 vs 0.21 The IoU between ours and SGD is 0.87. This visualization and results confirm that the diffusion process can learn the patterns of high-performance parameters and generate new good models from random noise. More importantly, our generated model has a great behavior contrast compared to the original model, which is reflected in the low IoU value. E.3 Efficiency of parameter generation [image: Refer to caption](a)Acc. of R-18. [image: Refer to caption](b)Acc. of ViT-Base. Figure 7: We compare the accuracy curves of our method and SGD under three cases. (a): ResNet-18 on CIFAR-100. (b): ViT-Base on ImageNet-1K. Our approach speeds up at least 15 × than standard SGD process. To evaluate the generation efficiency of our method, we compare the validation accuracy curves of our method and SGD training among the following cases: 1) parameter diffusion with ResNet-18 on CIFAR-100; 2) parameter diffusion with ViT-Base on ImageNet-1K. We utilize the random initialized parameters for our method and SGD to make a fair comparison. As illustrated in Fig. 7 https://arxiv.org/html/2402.13144v1#A5.F7, our method can speed up at least 15 ×compared to the SGD without performance drops. On ImageNet-1K, we can speed up by 44 × when compared to the vanilla SGD optimization, which illustrates the more significant potential when applying our approach to large training datasets. Appendix F Generalization on Other Tasks We implement our method for other visual tasks, i.e., object detection, semantic segmentation, image generation. Experimental results illustrate the ability of our method to generalize to various tasks. F.1 Object detection Faster R-CNN (Ren et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib48) utilizes a region proposal network (RPN) which shares full-image convolutional features with the detection network to improve Fast R-CNN on object detection task. The FCOS (Fully Convolutional One-Stage) (Tian et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib60) model is a single-stage object detection model that simplifies the detection process by eliminating the need for anchor boxes. In the object detection task, We implement Faster R-CNN (Ren et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib48) and FCOS (Tian et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib60) with ResNet-50 backbone on the COCO (Lin et al., 2014 https://arxiv.org/html/2402.13144v1#bib.bib35)dataset based on torch/torchvision https://pytorch.org/vision/stable/models.html. Considering the time cost of data for p-diff, we directly use the pre-trained parameters as our first training data, then fine-tune it to obtain other training data. The parameters of the boxing predictor layer are generated by p-diff. We report the results in Tab. 10 https://arxiv.org/html/2402.13144v1#A6.T10. Our method can get models with similar or even better performance than the original model in seconds. model/performance best original mAP best p-diff mAP Faster R-CNN 36.9 37.0 FCOS 39.1 39.1Table 10: P-diff in object detection task. We report the mAP of best original model and best p-diff generated model.F.2 Semantic segmentation Fully Convolutional Network (FCN) (Long et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib37) was designed to efficiently process and analyze images at the pixel level, allowing for the semantic segmentation of objects within an image. Following the approach in object detection, we implement semantic segmentation task using FCN (Long et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib37) with ResNet-50 backbone to evaluate a subset of COCO val2017, on the 20 categories that are present in the Pascal VOC dataset. We generate a subset of the parameters of backbone and report the results in Tab. 11 https://arxiv.org/html/2402.13144v1#A6.T11. Our approach can generate high-performing neural network parameters in semantic segmentation task. model/performance original p-diff mean IoU pixelwise acc. mean IoU pixelwise acc. FCN 60.5 91.4 60.7 91.5Table 11: P-diff in semantic segmentation task. We report mean IoU and pixelwise accuracy of best original model and best p-diff model.F.3 Image generation model/performance original FID p-diff FID DDPM UNet 3.17 3.19 Table 12: P-diff in image generation task. We report the FID score on the CIFAR-10 dataset. DDPM (Ho et al., 2020 https://arxiv.org/html/2402.13144v1#bib.bib24) is a diffusion-based method in image generation, where UNet (Ronneberger et al., 2015 https://arxiv.org/html/2402.13144v1#bib.bib52) is used to model the noise. In the image generation task, we use p-diff to generate a subset of model parameters of UNet. For comparison, we evaluate the p-diff model’s FID score on the CIFAR-10 dataset and report the results in Tab. 12 https://arxiv.org/html/2402.13144v1#A6.T12. The best p-diff generated UNet get similar performance to the original model.