# Metrics¶

## Infidelity¶

Explanation infidelity represents the expected mean-squared error between the explanation multiplied by a meaningful input perturbation and the differences between the predictor function at its input and perturbed input. More details about the measure can be found in the following paper: https://arxiv.org/pdf/1901.09392.pdf

It is derived from the completeness property of well-known attribution algorithms and is a computationally more efficient and generalized notion of Sensitivy-n. The latter measures correlations between the sum of the attributions and the differences of the predictor function at its input and fixed baseline. More details about the Sensitivity-n can be found here: https://arxiv.org/pdf/1711.06104.pdfs

The users can perturb the inputs any desired way by providing any perturbation function that takes the inputs (and optionally baselines) and returns perturbed inputs or perturbed inputs and corresponding perturbations.

This specific implementation is primarily tested for attribution-based explanation methods but the idea can be expanded to use for non attribution-based interpretability methods as well.

Parameters
• forward_func (callable) – The forward function of the model or any modification of it.

• perturb_func (callable) –

The perturbation function of model inputs. This function takes model inputs and optionally baselines as input arguments and returns either a tuple of perturbations and perturbed inputs or just perturbed inputs. For example:

>>> def my_perturb_func(inputs):
>>>   <MY-LOGIC-HERE>
>>>   return perturbations, perturbed_inputs


If we want to only return perturbed inputs and compute perturbations internally then we can wrap perturb_func with infidelity_perturb_func_decorator decorator such as:

>>> from captum.metrics import infidelity_perturb_func_decorator

>>> @infidelity_perturb_func_decorator(<multipy_by_inputs flag>)
>>> def my_perturb_func(inputs):
>>>   <MY-LOGIC-HERE>
>>>   return perturbed_inputs


In case multipy_by_inputs is False we compute perturbations by input - perturbed_input difference and in case multipy_by_inputs flag is True we compute it by dividing (input - perturbed_input) by (input - baselines). The user needs to only return perturbed inputs in perturb_func as described above.

infidelity_perturb_func_decorator needs to be used with multipy_by_inputs flag set to False in case infidelity score is being computed for attribution maps that are local aka that do not factor in inputs in the final attribution score. Such attribution algorithms include Saliency, GradCam, Guided Backprop, or Integrated Gradients and DeepLift attribution scores that are already computed with multipy_by_inputs=False flag.

If there are more than one inputs passed to infidelity function those will be passed to perturb_func as tuples in the same order as they are passed to infidelity function.

If inputs
• is a single tensor, the function needs to return a tuple of perturbations and perturbed input such as: perturb, perturbed_input and only perturbed_input in case infidelity_perturb_func_decorator is used.

• is a tuple of tensors, corresponding perturbations and perturbed inputs must be computed and returned as tuples in the following format:

(perturb1, perturb2, … perturbN), (perturbed_input1, perturbed_input2, … perturbed_inputN)

Similar to previous case here as well we need to return only perturbed inputs in case infidelity_perturb_func_decorator decorates out perturb_func.

It is important to note that for performance reasons perturb_func isn’t called for each example individually but on a batch of input examples that are repeated max_examples_per_batch / batch_size times within the batch.

• inputs (tensor or tuple of tensors) – Input for which 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 values which sometimes represent ablated values and are used to compare with the actual inputs to compute importance scores in attribution algorithms. They can be represented 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.

Default: None

• attributions (tensor or tuple of tensors) –

Attribution scores computed based on an attribution algorithm. This attribution scores can be computed using the implementations provided in the captum.attr package. Some of those attribution approaches are so called global methods, which means that they factor in model inputs’ multiplier, as described in: https://arxiv.org/pdf/1711.06104.pdf Many global attribution algorithms can be used in local modes, meaning that the inputs multiplier isn’t factored in the attribution scores. This can be done duing the definition of the attribution algorithm by passing multipy_by_inputs=False flag. For example in case of Integrated Gradients (IG) we can obtain local attribution scores if we define the constructor of IG as: ig = IntegratedGradients(multipy_by_inputs=False)

Some attribution algorithms are inherently local. Examples of inherently local attribution methods include: Saliency, Guided GradCam, Guided Backprop and Deconvolution.

For local attributions we can use real-valued perturbations whereas for global attributions that perturbation is binary. https://arxiv.org/pdf/1901.09392.pdf

If we want to compute the infidelity of global attributions we can use a binary perturbation matrix that will allow us to select a subset of features from inputs or inputs - baselines space. This will allow us to approximate sensitivity-n for a global attribution algorithm.

infidelity_perturb_func_decorator function decorator is a helper function that computes perturbations under the hood if perturbed inputs are provided.

Attributions have the same shape and dimensionality as the inputs. If inputs is a single tensor then the attributions is a single tensor as well. If inputs is provided as a tuple of tensors then attributions will be tuples of tensors as well.

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. Note that the perturbations are not computed with respect to these arguments. This means that these arguments aren’t being passed to perturb_func as an input argument.

Default: None

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

Indices for selecting predictions from output(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

• n_perturb_samples (int, optional) –

The number of times input tensors are perturbed. Each input example in the inputs tensor is expanded n_perturb_samples times before calling perturb_func function.

Default: 10

• max_examples_per_batch (int, optional) –

The number of maximum input examples that are processed together. In case the number of examples (input batch size * n_perturb_samples) exceeds max_examples_per_batch, they will be sliced into batches of max_examples_per_batch examples and processed in a sequential order. If max_examples_per_batch is None, all examples are processed together. max_examples_per_batch should at least be equal input batch size and at most input batch size * n_perturb_samples.

Default: None

• normalize (bool, optional) –

Normalize the dot product of the input perturbation and the attribution so the infidelity value is invariant to constant scaling of the attribution values. The normalization factor beta is defined as the ratio of two mean values:

$\beta = \frac{ \mathbb{E}_{I \sim \mu_I} [ I^T \Phi(f, x) (f(x) - f(x - I)) ] }{ \mathbb{E}_{I \sim \mu_I} [ (I^T \Phi(f, x))^2 ] }$

Please refer the original paper for the meaning of the symbols. Same normalization can be found in the paper’s official implementation https://github.com/chihkuanyeh/saliency_evaluation

Default: False

Returns

A tensor of scalar infidelity scores per

input example. The first dimension is equal to the number of examples in the input batch and the second dimension is one.

Return type

infidelities (tensor)

Examples::
>>> # ImageClassifier takes a single input tensor of images Nx3x32x32,
>>> # and returns an Nx10 tensor of class probabilities.
>>> net = ImageClassifier()
>>> saliency = Saliency(net)
>>> input = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> # Computes saliency maps for class 3.
>>> # define a perturbation function for the input
>>> def perturb_fn(inputs):
>>>    noise = torch.tensor(np.random.normal(0, 0.003, inputs.shape)).float()
>>>    return noise, inputs - noise
>>> # Computes infidelity score for saliency maps
>>> infid = infidelity(net, perturb_fn, input, attribution)


## Sensitivity¶

class captum.metrics.sensitivity_max(explanation_func, inputs, perturb_func=<function default_perturb_func>, perturb_radius=0.02, n_perturb_samples=10, norm_ord='fro', max_examples_per_batch=None, **kwargs)[source]

Explanation sensitivity measures the extent of explanation change when the input is slightly perturbed. It has been shown that the models that have high explanation sensitivity are prone to adversarial attacks: Interpretation of Neural Networks is Fragile https://www.aaai.org/ojs/index.php/AAAI/article/view/4252

sensitivity_max metric measures maximum sensitivity of an explanation using Monte Carlo sampling-based approximation. By default in order to do so it samples multiple data points from a sub-space of an L-Infinity ball that has a perturb_radius radius using default_perturb_func default perturbation function. In a general case users can use any L_p ball or any other custom sampling technique that they prefer by providing a custom perturb_func.

Note that max sensitivity is similar to Lipschitz Continuity metric however it is more robust and easier to estimate. Since the explanation, for instance an attribution function, may not always be continuous, can lead to unbounded Lipschitz continuity. Therefore the latter isn’t always appropriate.

More about the Lipschitz Continuity Metric can also be found here On the Robustness of Interpretability Methods https://arxiv.org/pdf/1806.08049.pdf and Towards Robust Interpretability with Self-Explaining Neural Networks https://papers.nips.cc/paper8003-towards-robust-interpretability- with-self-explaining-neural-networks.pdf

More details about sensitivity max can be found here: On the (In)fidelity and Sensitivity of Explanations https://arxiv.org/pdf/1901.09392.pdf

Parameters
• explanation_func (callable) – This function can be the attribute method of an attribution algorithm or any other explanation method that returns the explanations.

• inputs (tensor or tuple of tensors) – Input for which explanations are computed. If explanation_func takes a single tensor as input, a single input tensor should be provided. If explanation_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.

• perturb_func (callable) –

The perturbation function of model inputs. This function takes

model inputs and optionally perturb_radius if the function takes more than one argument and returns perturbed inputs.

If there are more than one inputs passed to sensitivity function those will be passed to perturb_func as tuples in the same order as they are passed to sensitivity function.

It is important to note that for performance reasons perturb_func isn’t called for each example individually but on a batch of input examples that are repeated max_examples_per_batch / batch_size times within the batch.

Default: default_perturb_func

The epsilon radius used for sampling. In the default_perturb_func it is used as the radius of the L-Infinity ball. In a general case it can serve as a radius of any L_p nom. This argument is passed to perturb_func if it takes more than one argument.

Default: 0.02

• n_perturb_samples (int, optional) –

The number of times input tensors are perturbed. Each input example in the inputs tensor is expanded n_perturb_samples times before calling perturb_func function.

Default: 10

• norm_ord (int, float, inf, -inf, 'fro', 'nuc', optional) –

The type of norm that is used to compute the norm of the sensitivity matrix which is defined as the difference between the explanation function at its input and perturbed input.

Default: ‘fro’

• max_examples_per_batch (int, optional) –

The number of maximum input

examples that are processed together. In case the number of examples (input batch size * n_perturb_samples) exceeds max_examples_per_batch, they will be sliced into batches of max_examples_per_batch examples and processed in a sequential order. If max_examples_per_batch is None, all examples are processed together. max_examples_per_batch should at least be equal input batch size and at most input batch size * n_perturb_samples.

Default: None

**kwargs (Any, optional): Contains a list of arguments that are passed

to explanation_func explanation function which in some cases could be the attribute function of an attribution algorithm. Any additional arguments that need be passed to the explanation function should be included here. For instance, such arguments include: additional_forward_args, baselines and target.

Returns

A tensor of scalar sensitivity scores per

input example. The first dimension is equal to the number of examples in the input batch and the second dimension is one. Returned sensitivities are normalized by the magnitudes of the input explanations.

Return type

sensitivities (tensor)

Examples::
>>> # ImageClassifier takes a single input tensor of images Nx3x32x32,
>>> # and returns an Nx10 tensor of class probabilities.
>>> net = ImageClassifier()
>>> saliency = Saliency(net)
>>> input = torch.randn(2, 3, 32, 32, requires_grad=True)
>>> # Computes sensitivity score for saliency maps of class 3
>>> sens = sensitivity_max(saliency.attribute, input, target = 3)