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Context: BUIR makes use of two distinct encoder networks that have the same structure: online encoder f_{\theta} and target encoder f_{\xi}.They are parameterized by \theta and \xi, respectively.The key idea of BUIR is to train the online encoder by using outputs of the target encoder as its target, while gradually improving the target encoder as well.The main difference of BUIR from existing end-to-end learning frameworks is that f_{\theta} and f_{\xi} are updated in different ways.The online encoder is trained to minimize the error between its output and the target, whereas the target network is slowly updated based on the momentum update (Heet al., 2020b) so as to keep its output consistent. To sum up, the parameters of the online encoder and target encoder are optimized by(4)\begin{split}\theta&\leftarrow\theta-\eta\cdot\nabla_{\theta}\mathcal{L}_{\theta,\xi}\\\xi&\leftarrow\tau\cdot\xi+(1-\tau)\cdot\theta.\end{split}\eta is the learning rate for stochastic optimization, and \tau\in[0,1] is a momentum coefficient (also called as target decay) for momentum-based moving average.The online encoder f_{\theta} (and the predictor q_{\theta}) is effectively optimized by the gradients back-propagated from the loss (Equation (3)), while the target encoder f_{\xi} is updated as the moving average of the online encoder.By taking a large value of \tau, the target encoder slowly approximates the online encoder.This momentum-based update makes \xi evolve more slowly than \theta, which enables to bootstrap the representations by providing enhanced but consistent targets to the online encoders (Heet al., 2020b; Grill et al., 2020).Figure 1 illustrates the overall framework of BUIR with the simple one-hot encoders. Bypassing the collapsed solution.  It is obvious that the loss in Equation (3) admits the collapsed solution with respect to \theta and \xi, which means both the encoders generate the same representations for all users and items.For this reason, the conventional end-to-end learning strategy, which optimizes both f_{\theta} and f_{\xi} to minimize the loss (i.e., cross-prediction error), may easily lead to such collapsed solution.In contrast, our proposed framework updates each of the encoders in different ways.From Equation (4), the online encoder is optimized to minimize the loss, while the target encoder is updated to slowly approximate the online encoder.That is, the direction of updating the target encoder (\theta-\xi) totally differs from that of updating the online encoder (-\nabla_{\theta}\mathcal{L}_{\theta,\xi}),and this effectively keeps both the encoders from converging to the collapsed solution.Several recent work on bootstrapping-based representation learning (Grill et al., 2020; Chen and He, 2021) empirically demonstrated that this kind of dynamics (i.e., updating two networks differently) allows to avoid the collapsed solution without any explicit term to prevent it. Similarly to Section 3.2, the online encoder is trained by minimizing \mathcal{L}_{\theta,\xi}(\psi(u,\mathcal{V}_{u}),\psi(v,\mathcal{U}_{v})), and the target encoder is slowly updated by the momentum mechanism.After the optimization is finished, the interaction score is inferred by f_{\theta}(u,\mathcal{V}_{u}) and f_{\theta}(v,\mathcal{U}_{v}) (Equation (5)).Figure 2 shows an example of our data augmentation which injects a certain level of perturbations to the neighbors.

Question: In BUIR, how does the online encoder updated compared to the target encoder?
The online encoder is updated to minimize the error between the output and the target and updated by the gradients back-propagated from the loss, but target network is updated based on the momentum update and updated as the moving average of the online encoder .