Algorithm 1.

FedSLD.

Input: Initialized model parameter weights w0, number of clients N, number of local epochs E, batch size B, is the batch size, learning rate η, number of rounds R.

1: ∀i ∈ [N], c ∈ [C] acquire ni,c client i’s numbers of samples of each class c.

2: ∀c ∈ [C], $\widetilde{𝒫}(y=c)=\frac{{\sum }_{i=1}^N{n}_{i,c}}{{\sum }_{i=1}^N{n}_i}$// compute estimated prior label distribution.

3: for r ← 1, 2, …, R do

4:  $\forall i\in [N]w_i^r=w^{r-1}$ // broadcast model parameters.

5:  for i ← 1, 2, …, N in parallel do

6:   for ${\left\{x_k,y_k\right\}}_{k=1}^B$ in all minibatches do

7:    ∀c, $p_b(y=c)\leftarrow {\sum }_{k=1}^B〚y_k=c〛/B$

8:    Compute loss ${ℒ}_b$ by Equation 3.

9:    $w_i^r\leftarrow w_i^r-\eta {\nabla }_w{ℒ}_b$

10:   end for

11:  end for

12:  $w^r={\sum }_{i=1}^N\frac{n_i}{n}{w}_I^r$ // aggregate model updates

13: end for

14: return wR