A Little Is Enough (ALIE)#

class byzfl.ALittleIsEnough(tau=1.5)[source]#

Bases: object

Description#

Execute the A Little Is Enough (ALIE) attack [1]: perturb the mean vector using the coordinate-wise standard deviation of the vectors multiplicatively scaled with the attack factor \(\tau\).

\[\text{ALIE}_{\tau}(x_1, \dots, x_n) = \mu_{x_1, ..., x_n} + \tau \cdot \sigma_{x_1, ..., x_n}\]

where

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

  • \(\mu_{x_1, \dots, x_n} = \frac{1}{n}\sum_{i=1}^{n}x_i\) is the mean vector.

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

  • \(\sigma_{x_1, \dots, x_n}\) is the coordinate-wise standard deviation of \(x_1, \dots, x_n\), i.e., \(\big[\sigma_{x_1, \dots, x_n}\big]_k = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(\big[x_i\big]_k - \big[\mu_{x_1, \dots, x_n}\big]_k)^2}\).

  • \(\tau \in \mathbb{R}\) is the attack factor.

Initialization parameters:

tau (float, optional) – The attack factor \(\tau\) used to adjust the mean vector. Set to 1.5 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 is the same as the input.

Examples

>>> import byzfl
>>> attack = byzfl.ALittleIsEnough(1.5)

Using numpy arrays:

>>> import numpy as np
>>> x = np.array([[1., 2., 3.],       # np.ndarray
>>>               [4., 5., 6.],
>>>               [7., 8., 9.]])
>>> attack(x)
array([ 8.5  9.5 10.5])

Using torch tensors (Warning: We need the tensor to be either a floating point or complex dtype):

>>> import torch
>>> x = torch.tensor([[1., 2., 3.],   # torch.tensor
>>>                   [4., 5., 6.],
>>>                   [7., 8., 9.]])
>>> attack(x)
tensor([ 8.5000,  9.5000, 10.5000])

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.])]
>>> attack(x)
array([ 8.5  9.5 10.5])

Using list of torch tensors (Warning: We need the tensor to be either a floating point or complex dtype):

>>> import torch
>>> x = [torch.tensor([1., 2., 3.]),  # list of torch.tensor
>>>      torch.tensor([4., 5., 6.]),
>>>      torch.tensor([7., 8., 9.])]
>>> attack(x)
tensor([ 8.5000,  9.5000, 10.5000])

References

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