Meamed#

class byzfl.Meamed(f=0)[source]#

Description#

Compute the mean around median along the first axis [1]:

\[\big[\mathrm{Meamed}_{f}(x_1, \ldots, x_n)\big]_k = \frac{1}{n-f} \sum_{j=1}^{n-f} \big[x_{\pi(j)}\big]_k\]

where

  • \(x_1, \dots, x_n\) are the input vectors, which conceptually correspond to gradients submitted by honest and Byzantine participants during a training iteration.

  • \(f\) conceptually represents the expected number of Byzantine vectors.

  • \(\big[\cdot\big]_k\) refers to the \(k\)-th coordinate.

  • \(\mathrm{median}\) refers to the median of \(n\) scalars.

  • \(\pi\) denotes a permutation on \(\big[n\big]\) that sorts the input vectors based on their \(k\)-th coordinate in non-decreasing order of distance to the \(\mathrm{median}\) of the \(k\)-th coordinate across the input vectors. This sorting is expressed as:

\(\Big|\big[x_{\pi_k(1)}\big]_k - \mathrm{median}\big(\big[x_1\big]_k, \ldots, \big[x_n\big]_k\big)\Big| \leq \ldots \leq \Big|\big[x_{\pi_k(n)}\big]_k - \mathrm{median}\big(\big[x_1\big]_k, \ldots, \big[x_n\big]_k\big)\Big|\).

In other words, Meamed computes the average of the \(n-f\) closest elements to the \(\mathrm{median}\) for each dimension \(k\).

Initialization parameters:

f (int, optional) – Number of faulty vectors. Set to 0 by default.

Calling the instance

Input parameters:

vectors (numpy.ndarray, torch.Tensor, list of numpy.ndarray or list of torch.Tensor) – A set of vectors, matrix or tensors.

Returns:

numpy.ndarray or torch.Tensor – The data type of the output will be the same as the input.

Examples

>>> import byzfl
>>> agg = byzfl.Meamed(1)

Using numpy arrays

>>> import numpy as np
>>> x = np.array([[1., 2., 3.],       # np.ndarray
>>>               [4., 5., 6.],
>>>               [7., 8., 9.]])
>>> agg(x)
array([2.5, 3.5, 4.5])

Using torch tensors

>>> import torch
>>> x = torch.tensor([[1., 2., 3.],   # torch.tensor
>>>                   [4., 5., 6.],
>>>                   [7., 8., 9.]])
>>> agg(x)
tensor([2.5000, 3.5000, 4.5000])

Using list of numpy arrays

>>> import numpy as np
>>> x = [np.array([1., 2., 3.]),      # list of np.ndarray
>>>      np.array([4., 5., 6.]),
>>>      np.array([7., 8., 9.])]
>>> agg(x)
array([2.5, 3.5, 4.5])

Using list of torch tensors

>>> import torch
>>> x = [torch.tensor([1., 2., 3.]),  # list of torch.tensor
>>>      torch.tensor([4., 5., 6.]),
>>>      torch.tensor([7., 8., 9.])]
>>> agg(x)
tensor([2.5000, 3.5000, 4.5000])

References

On this page