SVD is a supervised learning approach that decompose a matrix into a product of matrices. It does matrix factorization via following formula:
Different to MF, SVD incoporate bias of users as 
and bias of items as 
.
SVD is commonly used in recommendation senario to decompose a user-item rating matrix into user profile and item profile, and to predict unknown user-item pair's rating by compute the dot product of user profile and item profile.
SVD's loss function:
where λ is a small positive values to rescale the penalizer. Denote 
, and the gradients on different parameters are:
Here we simplify participants of the federation process into three parties. Party A represents Guest, party B represents Host. Party C, which is also known as “Arbiter,” is a third party that works as coordinator. Party C is responsible coordinate training process and encrypted data exchange.
Inspired by VFedMF, we can divide the parameters of SVD into item-related (e.g. p) and user-related (e.g. q) ones. Based on the setting of same-user and different-item, we let party A, B share the same user-profile and hold item-profile individually. The rating data is protected by keeping the item-profile unknown to each other.
| User-Related | Item-Related | |
|---|---|---|
| Parameters |  |
 |
| Training mode | Jointly update | Local update |
User-related parameters:
Let:
Then the parameter updates of user-related parameters can be represented as:
Item-related parameters
The item-related parameters can be updated locally by A,B using the same equation as regular SVD.
Compute 
According to equation, we need to compute µ before the training of SVD, where µ is the global average rating score. Intutively, µ can be computed using following equation.
- L2 regularization
- Mini-batch mechanism
- Five optimization methods: a) “sgd”: gradient descent with arbitrary batch size b) “rmsprop”: RMSProp c) “adam”: Adam d) “adagrad”: AdaGrad e) “nesterov_momentum_sgd”: Nesterov Momentum
- Three converge criteria: a) "diff": Use difference of loss between two iterations, not available for multi-host training b) "weight_diff": Use difference of model weights
- Support validation for every arbitrary iterations
- Learning rate decay mechanism.