# A Histopathological Image Classification Method Based on Model Fusion in the Weight Space
## 2 Related Works  
The distribution of minima regions in the loss function of deep neural networks has been a subject of study in recent years. [PERSON] et al. [20] observed that along the paths connecting minima in the weight space, which are unrelated to each other, the training and test loss would remain nearly equal to that of the minima, while the test error slightly increases. Their work revealed that the distribution of local minima in the loss function of deep neural networks has certain characteristics. [PERSON] and [PERSON] [23] explored the topology of the loss function of ResNet-56 on the CIFAR-10 dataset by training with CLR and different fixed learning rates. Their work provides theoretical support for finding the optimal region of the loss function. [PERSON] et al. [24] further proposed an ensemble framework that consisted of models along high-accuracy paths in the weight space. They proposed a fast geometric ensemble (FGE) strategy, which first trains a model to about 80%, uses CLR or fixed learning rates for further training, records the model weights for every \(m\) iteration, and ensembles these recorded models for final results. [PERSON] et al. [16] proposed stochastic weight averaging (SWA) by simply averaging the model weights at each recorded point along the trajectory of a stochastic gradient descent (SGD) procedure, with CLR or fixed learning rates. Their work is based on the observation that SGD methods generally converge on the edge of the minima basin. In such cases, weight averaging can lead to a better solution. [PERSON] et al. [25] presented a theoretical analysis of tail-averaging in machine learning models. They presented the generalization error bounds of different weight-averaging strategies in the SGD procedure. Recently, [PERSON] et al. [26] revisited SWA and pointed out that its effectiveness depends on the extent to which the SGD algorithm runs before converging. Their work indicated that fusing models in the weight space can reduce the test variance. [PERSON] et al. [27] proposed that when a pre-trained model is fine-tuned on a new training dataset using different hyperparameters, the model instances would stay in the same loss landscape basin. Moreover, these fine-tuned instances would be similar in feature spaces and close in parameter spaces. Based on their work, [PERSON] et al. [17] proposed model soup, which performs weight fusion over a series of fine-tuned models trained with different hyperparameters. They designed three strategies for weight fusion: average soup, greedy soup, and learned soup. These strategies utilize averaging, selective fusion, and weighted combination to obtain the soup model. For the greedy soup strategy, one would test each ingredient and keep it if its addition improves the performance of the soup model. For the learned soup strategy, one would search for a weighted combination that makes the model perform optimally on the test set. Inspired by their work, in this paper, we introduce the soup model into the histopathological image analysis and propose a novel weight fusion strategy by drawing on the idea of selective ensemble. As an effective strategy for improving the model performance and reducing variance, ensemble learning has been widely used in histopathological image analysis. [PERSON] et al. [28] fine-tuned 4 pre-trained models based on VGG16 and VGG19. They then constructed an ensemble model by averaging the prediction probabilities of the 4 models. The ensemble model outperformed the individual models in terms of test loss. [PERSON] et al. [29] proposed a method based on deep ensemble for the analysis of mitoses in breast histopathology images. First, they selected candidate mitoses via an instance segmentation method based on deep learning. Then a heterogeneous ensemble of deep CNNs was run for mitotic nuclei classification. [PERSON] et al. [30] proposed the stacked generalized ensemble model for breast cancer classification, which uses 6 deep networks as level-0 models and logistic regression as a level-1 model.