## 2 Related Works
### Previous Work
#### 2.4.2 The Previous Method
#### 3.4.3 Expansion to Multidimensional Data

Including the proposed TT-VFDL-ST, current vertical federated learning methods assume the problem state in which multivariate time-series data are distributed in each guest party. To apply to real-world scenarios, we might consider the case of the input data being expanded to multidimensional. There are three cases of the input data being expanded to multidimensional as follows:

* _A case where the pattern length of the raw input data held in each guest party is expanded_ In the proposed TT-VFDL-ST, the pattern length of the feature extracted and sent from each guest party affects the communication costs, not the pattern length of the raw input data of each guest party. Therefore, even if the pattern length of the raw input data is increased, the communication costs might not be much affected by designing the local model of each guest party to have a short feature dimension while preserving high performances. * _A case where the dimension of the input data of each guest party is expanded by adding other stream data held in the same guest party to the existing input_ This is the case for example when utilizing only one ECG stream for vertical federated learning of the target task before, but trying to add other ECG streams held in the same guest party to the input, for better partial training performance. However, even in this case, the pattern length of the feature affects the communication costs, not the dimension of the input. Therefore, if the input dimension is expanded by taking only streams held in the same guest party, the communication cost may not be significantly affected by designing the local model to have a short length of feature vectors while maintaining high performance. * _A case where the dimension of the input data of each guest party is expanded by adding other stream data held in other guest parties to the existing input_ The case where utilizing other guest parties' raw time-series data for expanding the input dimension of each guest party is not generally dealt with by current vertical federated learning methods including TT-VFDL-ST. This is because the exposure of raw data of each guest party is inevitable in that case. However, applying our task-driven approach, the vertical federated learning method to deal with that case can be derived. We are now working on that method and achieving some meaningful performances from the working method. This will be our next publication after some refining. #### 3.4.4 Training in Each Party

Including the proposed TT-VFDL-ST, existing methods utilizing transfer learning for their distributed learning such as our previous work [7] and existing studies [8; 9; 10; 11] have separate training loops on each party. This is compared to [12; 13; 14; 15], which distributes the single training loop for the target task model to each party. The first advantage of training each party's local model with a separate training loop is that the communication costs could be efficiently decreased since only one feature dataset needs to be sent to the host party from each guest party. Moreover, with a separate training loop in each party, the design, training, and performance tuning of each local model could be performed independently regardless of other parties. Therefore, in TT-VFDL-ST, each local model and its training strategies should be designed considering each party's existing resource or computing capacity. In real-world scenarios, each party's capacities are quite different. Therefore, when applying TT-VFDL-ST to real-world scenarios, some limitations might arise regarding a party in the worst conditions in practice. For example, the feature vector size for designing the local model in each guest party might be affected by the host party's computing capacity. Also, the overall training time of the proposed TT-VFDL-ST is affected by not only the host party's training time but also the training time of the guest party that finished last.