In this subsection, we derive the necessary and sufficient condition in proposition??. Denote B,K be some natural numbers. We introduce the proposition from [8] that proves geometrical convergence of positive concave mapping. Bycorollary 2, g(v(n);Q) is a concave mapping. Wedonotapplyweightdecayanduse cosine scheduled the learning rate.

The objective function is the composition of two expectations of stochastic functions, and ismore challenging tooptimize than vanilla stochastic optimization problems.

In this section, we present some additional experiments. Empirical setup Most of the experimental setups are the same as those in Section 6, except that now we use 5 parties instead of 3 parties. There are 90 dimensions for a single data in YearPredictionMSD dataset, and we let each party hold 18 dimensions. Empirical results We plot the training loss instead of the testing loss since we are comparing differentobjectivefunctions. A.4 Experimentsonotherdatasets In this section, we present the experiment results on another dataset.

Conditional Generative Adversarial Networks (cGANs) are implicit generative models which allow to sample from class-conditional distributions.

We argue that regularizing terms in standard regression methods not only help against overfitting finite data, but sometimes also help in getting bettercausal models.

While the main paper only contains proof sketches or intuitions, all complete proofsarereportedinAppendixA.