# Shapley Value Sampling¶

class captum.attr.ShapleyValueSampling(forward_func)[source]

A perturbation based approach to compute attribution, based on the concept of Shapley Values from cooperative game theory. This method involves taking a random permutation of the input features and adding them one-by-one to the given baseline. The output difference after adding each feature corresponds to its attribution, and these difference are averaged when repeating this process n_samples times, each time choosing a new random permutation of the input features.

By default, each scalar value within the input tensors are taken as a feature and added independently. Passing a feature mask, allows grouping features to be added together. This can be used in cases such as images, where an entire segment or region can be grouped together, measuring the importance of the segment (feature group). Each input scalar in the group will be given the same attribution value equal to the change in output as a result of adding back the entire feature group.

More details regarding Shapley Value sampling can be found in these papers: https://www.sciencedirect.com/science/article/pii/S0305054808000804 https://pdfs.semanticscholar.org/7715/bb1070691455d1fcfc6346ff458dbca77b2c.pdf

Parameters

forward_func (callable) – The forward function of the model or any modification of it. The forward function can either return a scalar per example, or a single scalar for the full batch. If a single scalar is returned for the batch, perturbations_per_eval must be 1, and the returned attributions will have first dimension 1, corresponding to feature importance across all examples in the batch.

attribute(inputs, baselines=None, target=None, additional_forward_args=None, feature_mask=None, n_samples=25, perturbations_per_eval=1)[source]

” NOTE: The feature_mask argument differs from other perturbation based methods, since feature indices can overlap across tensors. See the description of the feature_mask argument below for more details.

Parameters
• inputs (tensor or tuple of tensors) – Input for which Shapley value sampling attributions are computed. If forward_func takes a single tensor as input, a single input tensor should be provided. If forward_func takes multiple tensors as input, a tuple of the input tensors should be provided. It is assumed that for all given input tensors, dimension 0 corresponds to the number of examples (aka batch size), and if multiple input tensors are provided, the examples must be aligned appropriately.

• baselines (scalar, tensor, tuple of scalars or tensors, optional) –

Baselines define reference value which replaces each feature when ablated. Baselines can be provided as:

• a single tensor, if inputs is a single tensor, with exactly the same dimensions as inputs or the first dimension is one and the remaining dimensions match with inputs.

• a single scalar, if inputs is a single tensor, which will be broadcasted for each input value in input tensor.

• a tuple of tensors or scalars, the baseline corresponding to each tensor in the inputs’ tuple can be:

• either a tensor with matching dimensions to corresponding tensor in the inputs’ tuple or the first dimension is one and the remaining dimensions match with the corresponding input tensor.

• or a scalar, corresponding to a tensor in the inputs’ tuple. This scalar value is broadcasted for corresponding input tensor.

In the cases when baselines is not provided, we internally use zero scalar corresponding to each input tensor. Default: None

• target (int, tuple, tensor or list, optional) –

Output indices for which difference is computed (for classification cases, this is usually the target class). If the network returns a scalar value per example, no target index is necessary. For general 2D outputs, targets can be either:

• a single integer or a tensor containing a single integer, which is applied to all input examples

• a list of integers or a 1D tensor, with length matching the number of examples in inputs (dim 0). Each integer is applied as the target for the corresponding example.

For outputs with > 2 dimensions, targets can be either:

• A single tuple, which contains #output_dims - 1 elements. This target index is applied to all examples.

• A list of tuples with length equal to the number of examples in inputs (dim 0), and each tuple containing #output_dims - 1 elements. Each tuple is applied as the target for the corresponding example.

Default: None

• additional_forward_args (any, optional) – If the forward function requires additional arguments other than the inputs for which attributions should not be computed, this argument can be provided. It must be either a single additional argument of a Tensor or arbitrary (non-tuple) type or a tuple containing multiple additional arguments including tensors or any arbitrary python types. These arguments are provided to forward_func in order following the arguments in inputs. For a tensor, the first dimension of the tensor must correspond to the number of examples. For all other types, the given argument is used for all forward evaluations. Note that attributions are not computed with respect to these arguments. Default: None

• n_samples (int, optional) – The number of feature permutations tested. Default: 25 if n_samples is not provided.

• perturbations_per_eval (int, optional) – Allows multiple ablations to be processed simultaneously in one call to forward_fn. Each forward pass will contain a maximum of perturbations_per_eval * #examples samples. For DataParallel models, each batch is split among the available devices, so evaluations on each available device contain at most (perturbations_per_eval * #examples) / num_devices samples. If the forward function returns a single scalar per batch, perturbations_per_eval must be set to 1. Default: 1

Returns

• attributions (tensor or tuple of tensors):

The attributions with respect to each input feature. If the forward function returns a scalar value per example, attributions will be the same size as the provided inputs, with each value providing the attribution of the corresponding input index. If the forward function returns a scalar per batch, then attribution tensor(s) will have first dimension 1 and the remaining dimensions will match the input. If a single tensor is provided as inputs, a single tensor is returned. If a tuple is provided for inputs, a tuple of corresponding sized tensors is returned.

Return type

tensor or tuple of tensors of attributions

Examples:

>>> # SimpleClassifier takes a single input tensor of size Nx4x4,
>>> # and returns an Nx3 tensor of class probabilities.
>>> net = SimpleClassifier()
>>> # Generating random input with size 2 x 4 x 4
>>> input = torch.randn(2, 4, 4)
>>> # Defining ShapleyValueSampling interpreter
>>> svs = ShapleyValueSampling(net)
>>> # Computes attribution, taking random orderings
>>> # of the 16 features and computing the output change when adding
>>> # each feature. We average over 200 trials (random permutations).
>>> attr = svs.attribute(input, target=1, n_samples=200)

>>> # Alternatively, we may want to add features in groups, e.g.
>>> # grouping each 2x2 square of the inputs and adding them together.
>>> # This can be done by creating a feature mask as follows, which
>>> # defines the feature groups, e.g.:
>>> # +---+---+---+---+
>>> # | 0 | 0 | 1 | 1 |
>>> # +---+---+---+---+
>>> # | 0 | 0 | 1 | 1 |
>>> # +---+---+---+---+
>>> # | 2 | 2 | 3 | 3 |
>>> # +---+---+---+---+
>>> # | 2 | 2 | 3 | 3 |
>>> # +---+---+---+---+
>>> # With this mask, all inputs with the same value are added
>>> # together, and the attribution for each input in the same
>>> # group (0, 1, 2, and 3) per example are the same.
>>> # The attributions can be calculated as follows:
>>> # feature mask has dimensions 1 x 4 x 4
>>> feature_mask = torch.tensor([[[0,0,1,1],[0,0,1,1],
>>>                             [2,2,3,3],[2,2,3,3]]])

class captum.attr.ShapleyValues(forward_func)[source]

A perturbation based approach to compute attribution, based on the concept of Shapley Values from cooperative game theory. This method involves taking each permutation of the input features and adding them one-by-one to the given baseline. The output difference after adding each feature corresponds to its attribution, and these difference are averaged over all possible random permutations of the input features.

By default, each scalar value within the input tensors are taken as a feature and added independently. Passing a feature mask, allows grouping features to be added together. This can be used in cases such as images, where an entire segment or region can be grouped together, measuring the importance of the segment (feature group). Each input scalar in the group will be given the same attribution value equal to the change in output as a result of adding back the entire feature group.

More details regarding Shapley Values can be found in these papers: https://apps.dtic.mil/dtic/tr/fulltext/u2/604084.pdf https://www.sciencedirect.com/science/article/pii/S0305054808000804 https://pdfs.semanticscholar.org/7715/bb1070691455d1fcfc6346ff458dbca77b2c.pdf

NOTE: The method implemented here is very computationally intensive, and should only be used with a very small number of features (e.g. < 7). This implementation simply extends ShapleyValueSampling and evaluates all permutations, leading to a total of n * n! evaluations for n features. Shapley values can alternatively be computed with only 2^n evaluations, and we plan to add this approach in the future.

Parameters

forward_func (callable) – The forward function of the model or any modification of it. The forward function can either return a scalar per example, or a single scalar for the full batch. If a single scalar is returned for the batch, perturbations_per_eval must be 1, and the returned attributions will have first dimension 1, corresponding to feature importance across all examples in the batch.

attribute(inputs, baselines=None, target=None, additional_forward_args=None, feature_mask=None, perturbations_per_eval=1)[source]

” NOTE: The feature_mask argument differs from other perturbation based methods, since feature indices can overlap across tensors. See the description of the feature_mask argument below for more details.

Parameters
• inputs (tensor or tuple of tensors) – Input for which Shapley value sampling attributions are computed. If forward_func takes a single tensor as input, a single input tensor should be provided. If forward_func takes multiple tensors as input, a tuple of the input tensors should be provided. It is assumed that for all given input tensors, dimension 0 corresponds to the number of examples (aka batch size), and if multiple input tensors are provided, the examples must be aligned appropriately.

• baselines (scalar, tensor, tuple of scalars or tensors, optional) –

Baselines define reference value which replaces each feature when ablated. Baselines can be provided as:

• a single tensor, if inputs is a single tensor, with exactly the same dimensions as inputs or the first dimension is one and the remaining dimensions match with inputs.

• a single scalar, if inputs is a single tensor, which will be broadcasted for each input value in input tensor.

• a tuple of tensors or scalars, the baseline corresponding to each tensor in the inputs’ tuple can be:

• either a tensor with matching dimensions to corresponding tensor in the inputs’ tuple or the first dimension is one and the remaining dimensions match with the corresponding input tensor.

• or a scalar, corresponding to a tensor in the inputs’ tuple. This scalar value is broadcasted for corresponding input tensor.

In the cases when baselines is not provided, we internally use zero scalar corresponding to each input tensor. Default: None

• target (int, tuple, tensor or list, optional) –

Output indices for which difference is computed (for classification cases, this is usually the target class). If the network returns a scalar value per example, no target index is necessary. For general 2D outputs, targets can be either:

• a single integer or a tensor containing a single integer, which is applied to all input examples

• a list of integers or a 1D tensor, with length matching the number of examples in inputs (dim 0). Each integer is applied as the target for the corresponding example.

For outputs with > 2 dimensions, targets can be either:

• A single tuple, which contains #output_dims - 1 elements. This target index is applied to all examples.

• A list of tuples with length equal to the number of examples in inputs (dim 0), and each tuple containing #output_dims - 1 elements. Each tuple is applied as the target for the corresponding example.

Default: None

• additional_forward_args (any, optional) – If the forward function requires additional arguments other than the inputs for which attributions should not be computed, this argument can be provided. It must be either a single additional argument of a Tensor or arbitrary (non-tuple) type or a tuple containing multiple additional arguments including tensors or any arbitrary python types. These arguments are provided to forward_func in order following the arguments in inputs. For a tensor, the first dimension of the tensor must correspond to the number of examples. For all other types, the given argument is used for all forward evaluations. Note that attributions are not computed with respect to these arguments. Default: None

• perturbations_per_eval (int, optional) – Allows multiple ablations to be processed simultaneously in one call to forward_fn. Each forward pass will contain a maximum of perturbations_per_eval * #examples samples. For DataParallel models, each batch is split among the available devices, so evaluations on each available device contain at most (perturbations_per_eval * #examples) / num_devices samples. If the forward function returns a single scalar per batch, perturbations_per_eval must be set to 1. Default: 1

Returns

• attributions (tensor or tuple of tensors):

The attributions with respect to each input feature. If the forward function returns a scalar value per example, attributions will be the same size as the provided inputs, with each value providing the attribution of the corresponding input index. If the forward function returns a scalar per batch, then attribution tensor(s) will have first dimension 1 and the remaining dimensions will match the input. If a single tensor is provided as inputs, a single tensor is returned. If a tuple is provided for inputs, a tuple of corresponding sized tensors is returned.

Return type

tensor or tuple of tensors of attributions

Examples:

>>> # SimpleClassifier takes a single input tensor of size Nx4x4,
>>> # and returns an Nx3 tensor of class probabilities.
>>> net = SimpleClassifier()
>>> # Generating random input with size 2 x 4 x 4
>>> input = torch.randn(2, 4, 4)

>>> # We may want to add features in groups, e.g.
>>> # grouping each 2x2 square of the inputs and adding them together.
>>> # This can be done by creating a feature mask as follows, which
>>> # defines the feature groups, e.g.:
>>> # +---+---+---+---+
>>> # | 0 | 0 | 1 | 1 |
>>> # +---+---+---+---+
>>> # | 0 | 0 | 1 | 1 |
>>> # +---+---+---+---+
>>> # | 2 | 2 | 3 | 3 |
>>> # +---+---+---+---+
>>> # | 2 | 2 | 3 | 3 |
>>> # +---+---+---+---+
>>> # With this mask, all inputs with the same value are added
>>> # together, and the attribution for each input in the same
>>> # group (0, 1, 2, and 3) per example are the same.
>>> # The attributions can be calculated as follows:
>>> # feature mask has dimensions 1 x 4 x 4
>>> feature_mask = torch.tensor([[[0,0,1,1],[0,0,1,1],
>>>                             [2,2,3,3],[2,2,3,3]]])

>>> # With only 4 features, it is feasible to compute exact
>>> # Shapley Values. These can be computed as follows:
>>> sv = ShapleyValues(net)