Introduction to innsight

In the last decade, there has been a tremendous rush and growth in machine learning. Almost every year, new architectures break records on scientific test data sets, and the number of layers continues to grow through regularization methods making today’s neural networks far more complex than an original linear or logistic model. Nevertheless, the scientific focus is more on the predictive power than on the interpretability and the respective methods that already exist for the interpretation of single predictions or the whole neural networks are only sparsely or rarely implemented in the R programming language. The innsight package addresses this lack of interpretable machine learning methods in R (similar to iNNvestigate for Python) and provides the most popular methods for neural networks trained in R for analysis in one unified interface.

The steps for explaining individual predictions with the provided methods are unified in this package and follow a strict scheme. This will hopefully allow any user a smooth and easy introduction to the possibilities of this package. The steps are:

Step 1: Model Creation and Converting

The innsight package aims to be as flexible as possible and independent of any particular deep learning package in which the passed network was learned or defined. For this reason, there are several ways in this package to pass a neural network and then interpret their predictions.

Package `torch`

Currently, only models created by torch::nn_sequential are accepted. However, the most popular standard layers and activation functions are available:

Linear layers: nn_linear
Convolutional layers: nn_conv1d, nn_conv2d (but only with pading_mode = "zeros" and numerical padding)
Max-pooling layers: nn_max_pool1d, nn_max_pool2d (both only with default arguments for padding (0), dilation (1), return_indices (FALSE)) and ceil_mode (FALSE))
Average-pooling layers: nn_avg_pool1d, nn_avg_pool2d (both only with default arguments for padding (0), return_indices (FALSE)) and ceil_mode (FALSE))
Flatten layer: nn_flatten (see torch issue #716 and use classname = "nn_flatten")
Skipped layers: nn_dropout
Activation functions: nn_relu, nn_leaky_relu, nn_softplus, nn_sigmoid, nn_softmax, nn_tanh (open an issue if you need any more)

If you want to create an instance of the class Converter with a torch model that meets the above conditions, you have to keep the following things in mind:

In a torch model, the shape of the inputs is not stored; hence it must be specified with argument input_dim within the initialization of Converter.
If no further arguments are set for the Converter instance, default labels are generated for the input ('X1', 'X2', …) and output names ('Y1', 'Y2', … ). In torch models this information is not stored or often it is not even present in the training data. In the converter, however, there is the possibility (argument input_names and output_names) to pass the names, which will then be used in all results and plots created by this object.

Example 1: Dense Network

library(torch)
library(innsight)
torch_manual_seed(123)

# Create model
model <- nn_sequential(
  nn_linear(3, 10),
  nn_relu(),
  nn_linear(10, 2, bias = FALSE),
  nn_softmax(2)
)
# Convert the model
conv_dense <- Converter$new(model, input_dim = c(3))
# Convert model with input and output names
conv_dense_with_names <- 
  Converter$new(model, input_dim = c(3),
                input_names = list(c("Price", "Weight", "Height")),
                output_names = list(c("Buy it!", "Don't buy it!")))
# Show output names
conv_dense_with_names$model_dict$output_names
#> [[1]]
#> [1] "Buy it!"       "Don't buy it!"

Example 2: CNN

Unfortunately, at the moment (torch = 0.6.0) it is not possible to use a flatten module in nn_sequential, because it is only implemented as a function (torch_flatten). For this reason we define our own module as described in torch issue #716. To ensure that our package recognizes this module as a flattening layer, classname = "nn_flatten" must be set.

nn_flatten <- nn_module(
    classname = "nn_flatten",
    initialize = function(start_dim = 2, end_dim = -1) {
      self$start_dim <- start_dim
      self$end_dim <- end_dim
    },
    forward = function(x) {
      torch_flatten(x, start_dim = self$start_dim, end_dim = self$end_dim)
    }
  )

# Create CNN for images of size (3, 28, 28)
model <- nn_sequential(
  nn_conv2d(3, 5, c(2, 2)),
  nn_relu(),
  nn_max_pool2d(c(1,2)),
  nn_conv2d(5, 6, c(2, 3), stride = c(1, 2)),
  nn_relu(),
  nn_conv2d(6, 2, c(2, 2), dilation = c(1, 2), padding = c(5,4)),
  nn_relu(),
  nn_avg_pool2d(c(2,2)),
  nn_flatten(),
  nn_linear(48, 5),
  nn_softmax(2)
)

# Convert the model
conv_cnn <- Converter$new(model, input_dim = c(3, 10, 10))

Package `keras`

Keras models created by keras_model_sequential or keras_model are accepted. Within these functions, the following layers are allowed to be used:

Linear layers: layer_dense
Convolutional layers: layer_conv_1d, layer_conv_2d
Max-pooling layers: layer_max_pooling_1d, layer_max_pooling_2d
Average-pooling layers: layer_average_pooling_1d, layer_average_pooling_2d
Flatten layer: layer_flatten
Skipped layers: layer_dropout
Activation functions: Activation functions are currently only allowed as a string argument in the individual layers (activation) and not as a standalone layer like e.g. layer_activation_relu. The following are accepted as arguments: "relu", "softplus", "sigmoid", "softmax", "tanh", "linear" (open an issue if you need any more)

Analogous to a torch model, no names of inputs or outputs are stored in a keras model, i.e. if no further arguments are set for the Converter instance, default labels are generated for the input (e.g. 'X1', 'X2', …) and output names ('Y1', 'Y2', … ). In the converter, however, there is the possibility (argument input_names and output_names) to pass the names, which will then be used in all results and plots created by this object.

Example 1: Dense Network

library(keras)
tensorflow::set_random_seed(42)

# Create model
model <- keras_model_sequential()
model <- model %>%
  layer_dense(10, input_shape = c(5), activation = "softplus") %>%
  layer_dense(8, use_bias = FALSE, activation = "tanh") %>%
  layer_dropout(0.2) %>%
  layer_dense(4, activation = "softmax")

# Convert the model
conv_dense <- Converter$new(model)
#> Skipping Dropout ...

Example 2: CNN

library(keras)

# Create model
model <- keras_model_sequential()
model <- model %>%
  layer_conv_2d(4, c(5,4), input_shape = c(10,10,3), activation = "softplus") %>%
  layer_max_pooling_2d(c(2,2), strides = c(1,1)) %>%
  layer_conv_2d(6, c(3,3), activation = "relu", padding = "same") %>%
  layer_max_pooling_2d(c(2,2)) %>%
  layer_conv_2d(4, c(2,2), strides = c(2,1), activation = "relu") %>%
  layer_flatten() %>%
  layer_dense(5, activation = "softmax")

# Convert the model
conv_cnn <- Converter$new(model)

Package `neuralnet`

The usage with nets from the package neuralnet is very simple and straightforward, because the package offers much fewer options than torch or keras. The only thing to note is that no custom activation function can be used. However, the package saves the names of the inputs and outputs, which can of course be overwritten with the arguments input_names and output_names when creating the converter object.

Example 1: Dense Network

library(neuralnet)
data(iris)

# Create model
model <- neuralnet(Species ~ Petal.Length + Petal.Width, iris, 
                   linear.output = FALSE)

# Convert model
conv_dense <- Converter$new(model)
# Show input names
conv_dense$model_dict$input_names
#> [[1]]
#> [1] "Petal.Length" "Petal.Width"
# Show output names
conv_dense$model_dict$output_names
#> [[1]]
#> [1] "setosa"     "versicolor" "virginica"

Step 2: Apply selected Method

The innsight package provides the following tools for analyzing black box neural networks based on dense or convolution layers:

Gradient : Calculation of the model output Gradients with respect to the model inputs including the attribution method Gradient x Input:

\[ \begin{align} \text{Gradient}(x)_i^C &= \frac{\partial f(x)_C}{\partial x_i}\\ \text{Gradient x Input}(x)_i^C &= x_i \cdot \text{Gradient}(x)_i^C \end{align} \]

SmoothGrad : Calculation of the smoothed model output gradients (SmoothGrad) with respect to the model inputs by averaging the gradients over number of inputs with added noise (including SmoothGrad x Input): \[ \text{SmoothGrad}(x)_i^C = \mathbb{E}_{\varepsilon \sim \mathcal{N}(0, \sigma^2)}\left[\frac{\partial f(x + \varepsilon)_C}{\partial (x + \varepsilon)_i}\right] \approx \frac{1}{n} \sum_{k=1}^n \frac{\partial f(x + \varepsilon_k)_C}{\partial (x + \varepsilon_k)_i} \] with $\varepsilon_1, \ldots \varepsilon_n \sim \mathcal{N}(0, \sigma^2)$.
LRP : Back-propagating the model output to the model input neurons to obtain relevance scores for the model prediction which is known as Layer-wise Relevance Propagation: \[ f(x)_C \approx \sum_{i=1}^d R_i \] with $R_i$ relevance score for input neuron $i$.
DeepLift : Calculation of a decomposition of the model output with respect to the model inputs and a reference input which is known as Deep Learning Important FeaTures (DeepLift): \[ \Delta f(x)_C = f(x)_C - f(x_\text{ref}) = \sum_{i=1}^d C_i \] with $C_i$ contribution score for input neuron $i$ to the difference-from-reference model output.
ConnectionWeights : This is a naive and old approach by calculating the product of all weights from an input to an output neuron and then adding them up (see Connection Weights).

Notes

By default, the last activation function is not taken into account for all data-based methods. Because often this is the sigmoid/logistic or the softmax function, which has increasingly smaller gradients with growing distance from 0, which leads to the so-called saturation problem. But if you still want to consider the last activation function, use the argument ignore_last_act = FALSE.
For data with channels, it is not possible to determine exactly on which axis the channels are located. Internally, all data and the converted model are in the data format ‘channels first’, i.e. directly after the batch dimension (batch_size, channels, input_dim). In case you want to pass data with ‘channels last’ (e.g. for MNIST-data (batch_size, 28, 28, 3)), you have to indicate that with argument channels_first in the applied method.
It can happen with very large and deep neural networks that the calculation for all outputs requires both the entire memory and takes a very long time. But often, the results are needed only for certain output nodes. By default, only the results for the first 10 outputs are calculated, which can be adjusted individually with the argument output_idx by passing the relevant output indices.

Example 1: Gradient and Gradient x Input

Make sure that Example 1 from the section on neuralnet and Example 2 from the section on keras were run last.

# Apply method 'Gradient' for the dense network
grad_dense <- Gradient$new(conv_dense, iris[-c(1,2,5)])

# Apply method 'Gradient x Input' for CNN
x <- torch_randn(c(10,3,10,10))
grad_cnn <- Gradient$new(conv_cnn, x, times_input = TRUE)

Example 2: SmoothGrad and SmoothGrad x Input

Make sure that Example 1 from the section on neuralnet and Example 2 from the section on keras were run last.

# Apply method 'SmoothGrad' for the dense network
smooth_dense <- SmoothGrad$new(conv_dense, iris[-c(1,2,5)])

# Apply method 'SmoothGrad x Input' for CNN
x <- torch_randn(c(10,3,10,10))
smooth_cnn <- SmoothGrad$new(conv_cnn, x, times_input = TRUE)

Example 3: LRP

Make sure that Example 1 from the section on neuralnet and Example 2 from the section on keras were run last.

# Apply method 'LRP' for the dense network
lrp_dense <- LRP$new(conv_dense, iris[-c(1,2,5)])

# Apply method 'LRP' for CNN with alpha-beta-rule
x <- torch_randn(c(10,10,10,3))
lrp_cnn <- LRP$new(conv_cnn, x, rule_name = "alpha_beta", rule_param = 1,
                   channels_first = FALSE)

Example 4: DeepLift

Make sure that Example 1 from the section on neuralnet and Example 2 from the section on keras were run last.

# Define reference value
x_ref <- array(colMeans(iris[-c(1,2,5)]), dim = c(1,2))
# Apply method 'DeepLift' for the dense network
deeplift_dense <- DeepLift$new(conv_dense, iris[-c(1,2,5)], x_ref = x_ref)

# Apply method 'DeepLift' for CNN
x <- torch_randn(c(10,3,10,10))
deeplift_cnn <- DeepLift$new(conv_cnn, x)

Step 3: Show and Plot the Results

Once a method object has been created, the results can be returned as an array, data.frame, or torch_tensor, and can be further processed as desired. In addition, for each of the three sizes of the inputs (1D, 2D, or 3D) suitable plot and boxplot functions based on ggplot2 are implemented. Due to the complexity of higher dimensional inputs, these plots and boxplots can also be displayed as an interactive plotly plots by using the argument as_plotly.

Get Results

Each instance of the interpretability methods have the class method get_result, which is used to return the results. You can choose between the data formats array, data.frame or torch_tensor by passing the name as an character for argument type.

Array (Default)

# Get result (make sure 'grad_dense' is defined!)
result_array <- grad_dense$get_result()

# Show for datapoint 1 and 71 the result
result_array[c(1,71),,]
#> , , setosa
#> 
#>    Petal.Length Petal.Width
#> 1    -17.705021  -37.914467
#> 71    -1.752118   -3.752078
#> 
#> , , versicolor
#> 
#>    Petal.Length Petal.Width
#> 1    0.30512863  0.65341860
#> 71   0.03019602  0.06466337
#> 
#> , , virginica
#> 
#>    Petal.Length Petal.Width
#> 1     22.329956   47.818550
#> 71     2.209809    4.732201

Data Frame

# Get result as data.frame (make sure 'lrp_cnn' is defined!)
result_data.frame <- lrp_cnn$get_result("data.frame")

# Show the first 5 rows
head(result_data.frame, 5)
#>     data feature_h feature_w channel class        value
#> 1 data_1        H1        W1      C1    Y1 1.408816e-05
#> 2 data_2        H1        W1      C1    Y1 1.124055e-05
#> 3 data_3        H1        W1      C1    Y1 3.590281e-06
#> 4 data_4        H1        W1      C1    Y1 1.057909e-05
#> 5 data_5        H1        W1      C1    Y1 1.460757e-05

Torch Tensor

# Get result (make sure 'deeplift_dense' is defined!)
result_torch <- deeplift_dense$get_result("torch_tensor")

# Show for datapoint 1 and 71 the result
result_torch[c(1,71),,]
#> torch_tensor
#> (1,.,.) = 
#>   60.1352  -1.0364 -75.8438
#>   54.5763  -0.9406 -68.8328
#> 
#> (2,.,.) = 
#>  -5.5242  0.0952  6.9673
#>  -6.8194  0.1175  8.6008
#> [ CPUFloatType{2,2,3} ]

Plot Results

The package innsight also provides methods for visualizing the results. By default a ggplot2-plot is created, but it can also be rendered as an interactive plotly plot with the as_plotly argument. You can use the argument output_idx to select the indices of the output nodes for the plot. In addition, if the results have channels, the aggr_channels argument can be used to determine how the channels are aggregated.

Note: If you want to make a change to the results before plotting, you can get the results with Method$result (torch tensor!), change it accordingly, and then save it back to the attribute Method$result as a torch tensor.

Plot Function

Plot result for the first data point and all outputs: (make sure smooth_dense and lrp_cnn are defined!)

plot(smooth_dense, output_idx = 1:3)

# You can plot several data points at once
plot(smooth_dense, data_idx = c(1,71), output_idx = 1:3)

# Plot result for the first data point and first and fourth output
plot(lrp_cnn, aggr_channels = 'norm', output_idx = c(1,4))

# Create a plotly plot for the first output
plot(lrp_cnn, aggr_channels = 'norm', output_idx = c(1), as_plotly = TRUE)

Boxplot Function

Plot result for the first data point and first two outputs:
(make sure smooth_dense is defined!)

boxplot(smooth_dense, output_idx = 1:2)

# Use no preprocess function (default: abs) and plot reference data point
boxplot(smooth_dense, output_idx = 1:3, preprocess_FUN = identity,
        ref_data_idx = c(55))

Same as plotly.
In addition, you have a drop-down menu to select other data points you can define the indices for the drop-down menu with individual_data_idx.

boxplot(smooth_dense, output_idx = 1:3, preprocess_FUN = identity,
        ref_data_idx = c(55), as_plotly = TRUE, individual_data_idx = c(1))

Introduction to innsight

Step 1: Model Creation and Converting

Package torch

Example 1: Dense Network

Example 2: CNN

Package keras

Example 1: Dense Network

Example 2: CNN

Package neuralnet

Example 1: Dense Network

Step 2: Apply selected Method

Notes

Example 1: Gradient and Gradient x Input

Example 2: SmoothGrad and SmoothGrad x Input

Example 3: LRP

Example 4: DeepLift

Step 3: Show and Plot the Results

Get Results

Array (Default)

Data Frame

Torch Tensor

Plot Results

Plot Function

Boxplot Function

Package `torch`

Package `keras`

Package `neuralnet`