Setup - Installing torch
Before using ‘cito’ make sure that the current version of ‘torch’ is
installed and running.
if(!require(torch)) install.packages("torch")
#> Loading required package: torch
library(torch)
if(!torch_is_installed()) install_torch()
library (cito)
If you have problems installing Torch, check out the installation
help from the torch developer.
Introduction to models and model structures
Loss functions / Likelihoods
Cito can handle many different response types. Common loss functions
from ML but also likelihoods for statistical models are supported:
| Name |
Explanation |
Example / Task |
Meta-Code |
mse |
mean squared error |
Regression, predicting continuous values |
dnn(Sepal.Length~., ..., loss = "mse") |
mae |
mean absolute error |
Regression, predicting continuous values |
dnn(Sepal.Length~., ..., loss = "msa") |
softmax |
categorical cross entropy |
Multi-class, species classification |
dnn(Species~., ..., loss = "softmax") |
cross-entropy |
categorical cross entropy |
Multi-class, species classification |
dnn(Species~., ..., loss = "cross-entropy") |
gaussian |
Normal likelihood |
Regression, residual error is also estimated (similar
to stats::lm()) |
dnn(Sepal.Length~., ..., loss = "gaussian") |
binomial |
Binomial likelihood |
Classification/Logistic regression, mortality (0/1
data) |
dnn(Presence~., ..., loss = "Binomial") |
poisson |
Poisson likelihood |
Regression, count data, e.g. species abundances |
dnn(Abundance~., ..., loss = "Poisson") |
Moreover, all non multilabel losses (all except for softmax or
cross-entropy) can be modeled as multilabel using the cbind syntax
dnn(cbind(Sepal.Length, Sepal.Width)~., …, loss = "mse")
The likelihoods (Gaussian, Binomial, and Poisson) can be also passed
as their stats equivalents:
dnn(Sepal.Length~., ..., loss = stats::gaussian)
Data
In this vignette, we will work with the irirs dataset and build a
regression model.
data <- datasets::iris
head(data)
#> Sepal.Length Sepal.Width Petal.Length Petal.Width Species
#> 1 5.1 3.5 1.4 0.2 setosa
#> 2 4.9 3.0 1.4 0.2 setosa
#> 3 4.7 3.2 1.3 0.2 setosa
#> 4 4.6 3.1 1.5 0.2 setosa
#> 5 5.0 3.6 1.4 0.2 setosa
#> 6 5.4 3.9 1.7 0.4 setosa
#scale dataset
data <- data.frame(scale(data[,-5]),Species = data[,5])
Fitting a simple model
In ‘cito’, neural networks are specified and fitted with the
dnn() function. Models can also be trained on the GPU by
setting device = "cuda"(but only if you have installed the
CUDA dependencies). This is suggested if you are working with large data
sets or networks.
library(cito)
#fitting a regression model to predict Sepal.Length
nn.fit <- dnn(Sepal.Length~. , data = data, epochs = 12, loss = "mse", verbose=FALSE)
You can plot the network structure to give you a visual feedback of
the created object. e aware that this may take some time for large
networks.
The neural network 5 input nodes (3 continoues features, Sepal.Width,
Petal.Length, Petal.Width and the contrasts for the Species variable
(n_classes - 1)) and 1 output node for the response (Sepal.Length).
Baseline loss
At the start of the training we calculate a baseline loss for an an
intercept only model. It allows us to control the training because the
goal is to beat the baseline loss. If we don’t, we need to adjust the
optimization parameters (epochs and lr (learning rate)):
nn.fit <- dnn(Sepal.Length~. , data = data, epochs = 50, lr = 0.6, loss = "mse", verbose = FALSE) # lr too high
#> Error in eval_bare(loop, env): Loss is NA. Bad training, please check learning rate or regularization strength. See vignette('02_Troubleshooting') for help.
vs
nn.fit <- dnn(Sepal.Length~. , data = data, epochs = 50, lr = 0.01, loss = "mse", verbose = FALSE)
See vignette("B-Training_neural_networks") for more
details on how to adjust the optimization procedure and increase the
probability of convergence.
Adding a validation set to the training process
In order to see where your model might suffer from overfitting the
addition of a validation set can be useful. With dnn() you
can put validation = 0.x and define a percentage that will
not be used for training and only for validation after each epoch.
During training, a loss plot will show you how the two losses behave
(see vignette("B-Training_neural_networks") for details on
training NN and guaranteeing their convergence).
#20% of data set is used as validation set
nn.fit <- dnn(Sepal.Length~., data = data, epochs = 32,
loss= "mse", validation = 0.2)
Weights oft the last and the epoch with the lowest validation loss
are saved:
length(nn.fit$weights)
#> [1] 2
The default is to use the weights of the last epoch. But we can also
tell the model to use the weights with the lowest validation loss:
nn.fit$use_model_epoch = 1 # Default, use last epoch
nn.fit$use_model_epoch = 2 # Use weights from epoch with lowest validation loss
Methods
cito supports many of the well-known methods from other statistical
packages:
predict(nn.fit)
coef(nn.fit)
print(nn.fit)
Explainable AI - Understanding your model
xAI can produce outputs that are similar to known outputs from
statistical models:
Feature importance (returned by summary(nn.fit)) |
Feature importance based on permutations. See Fisher, Rudin, and Dominici (2018) Similar to how much
variance in the data is explained by the features. |
anova(model) |
Average conditional effects (ACE) (returned by
summary(nn.fit)) |
Average of local derivatives, approximation of linear effects. See
Pichler and Hartig
(2023) |
lm(Y~X) |
Standard Deviation of Conditional Effects (SDCE) (returned by
summary(nn.fit)) |
Standard deviation of the average conditional effects. Correlates
with the non-linearity of the effects. |
|
Partial dependency plots (PDP(nn.fit)) |
Visualization of the response-effect curve. |
plot(allEffects(model)) |
Accumulated local effect plots (ALE(nn.fit)) |
Visualization of the response-effect curve. More robust against
collinearity compared to PDPs |
plot(allEffects(model)) |
The summary() returns feature importance, ACE and
SDCE:
# Calculate and return feature importance
summary(nn.fit)
#> Summary of Deep Neural Network Model
#>
#> Feature Importance:
#> variable importance_1
#> 1 Sepal.Width 1.133617
#> 2 Petal.Length 2.286806
#> 3 Petal.Width 2.783383
#> 4 Species 1.053672
#>
#> Average Conditional Effects:
#> Response_1
#> Sepal.Width 0.06250046
#> Petal.Length 0.35557117
#> Petal.Width 0.39397602
#>
#> Standard Deviation of Conditional Effects:
#> Response_1
#> Sepal.Width 0.08405767
#> Petal.Length 0.08007347
#> Petal.Width 0.06509943
#returns weights of neural network
coef(nn.fit)
Uncertainties/p-Values
We can use bootstrapping to obtain uncertainties for the xAI metrics
(and also for the predictions). For that we have to retrain our model
with enabled bootstrapping:
df = data
df[,2:4] = scale(df[,2:4]) # scaling can help the NN to convergence faster
nn.fit <- dnn(Sepal.Length~., data = df,
epochs = 100,
verbose = FALSE,
loss= "mse",
bootstrap = 30L
)
summary(nn.fit)
#> Summary of Deep Neural Network Model
#>
#> ── Feature Importance
#> Importance Std.Err Z value Pr(>|z|)
#> Sepal.Width → Sepal.Length 1.084 0.353 3.07 0.0022 **
#> Petal.Length → Sepal.Length 6.778 2.649 2.56 0.0105 *
#> Petal.Width → Sepal.Length 1.075 0.496 2.17 0.0301 *
#> Species → Sepal.Length 0.209 0.119 1.76 0.0790 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ── Average Conditional Effects
#> ACE Std.Err Z value Pr(>|z|)
#> Sepal.Width → Sepal.Length 0.2761 0.0578 4.77 1.8e-06 ***
#> Petal.Length → Sepal.Length 0.7587 0.1142 6.64 3.1e-11 ***
#> Petal.Width → Sepal.Length 0.2448 0.0857 2.86 0.0043 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ── Standard Deviation of Conditional Effects
#> ACE Std.Err Z value Pr(>|z|)
#> Sepal.Width → Sepal.Length 0.1161 0.0260 4.46 8.1e-06 ***
#> Petal.Length → Sepal.Length 0.2134 0.0823 2.59 0.00953 **
#> Petal.Width → Sepal.Length 0.1021 0.0269 3.80 0.00015 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
We find that Petal.Length and Sepal.Width are significant
(categorical features are not supported yet for average conditional
effects).
Let’s compare the output to statistical outputs:
anova(lm(Sepal.Length~., data = df))
#> Analysis of Variance Table
#>
#> Response: Sepal.Length
#> Df Sum Sq Mean Sq F value Pr(>F)
#> Sepal.Width 1 2.060 2.060 15.0011 0.0001625 ***
#> Petal.Length 1 123.127 123.127 896.8059 < 2.2e-16 ***
#> Petal.Width 1 2.747 2.747 20.0055 1.556e-05 ***
#> Species 2 1.296 0.648 4.7212 0.0103288 *
#> Residuals 144 19.770 0.137
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Feature importance and the anova report Petal.Length as the most
important feature.
Visualization of the effects:
library(effects)
#> Loading required package: carData
#> lattice theme set by effectsTheme()
#> See ?effectsTheme for details.
plot(allEffects(lm(Sepal.Length~., data = df)))
There are some differences between the statistical model and the NN -
which is to be expected because the NN can fit the data more flexible.
But at the same time the differences have large confidence intervals
(e.g. the effect of Petal.Width)
Architecture
The architecture in NN usually refers to the width and depth of the
hidden layers (the layers between the input and the output layer) and
their activation functions. You can increase the complexity of the NN by
adding layers and/or making them wider:
# "simple NN" - low complexity
nn.fit <- dnn(Sepal.Length~., data = data, epochs = 100,
loss= "mse", validation = 0.2,
hidden = c(5L), verbose=FALSE)
# "large NN" - high complexity
nn.fit <- dnn(Sepal.Length~., data = data, epochs = 100,
loss= "mse", validation = 0.2,
hidden = c(100L, 100), verbose=FALSE)
There is no definitive guide to choosing the right architecture for
the right task. However, there are some general rules/recommendations:
In general, wider, and deeper neural networks can improve generalization
- but this is a double-edged sword because it also increases the risk of
overfitting. So, if you increase the width and depth of the network, you
should also add regularization (e.g., by increasing the lambda
parameter, which corresponds to the regularization strength).
Furthermore, in Pichler &
Hartig, 2023, we investigated the effects of the hyperparameters on
the prediction performance as a function of the data size. For example,
we found that the selu activation function outperforms
relu for small data sizes (<100 observations).
We recommend starting with moderate sizes (like the defaults), and if
the model doesn’t generalize/converge, try larger networks along with a
regularization that helps to minimize the risk of overfitting (see
vignette("B-Training_neural_networks") ).
Activation functions
By default, all layers are fitted with SeLU as activation function.
\[
relu(x) = max (0,x)
\]You can also adjust the activation function of each layer
individually to build exactly the network you want. In this case you
have to provide a vector the same length as there are hidden layers. The
activation function of the output layer is chosen with the loss argument
and does not have to be provided.
#selu as activation function for all layers:
nn.fit <- dnn(Sepal.Length~., data = data, hidden = c(10,10,10,10), activation= "relu")
#layer specific activation functions:
nn.fit <- dnn(Sepal.Length~., data = data,
hidden = c(10,10,10,10), activation= c("relu","selu","tanh","sigmoid"))
Note: The default activation function should be adequate for most
tasks. We don’t recommend tuning it.
Training hyperparameters
Regularization
Elastic net regularization
If elastic net is used, ‘cito’ will produce a sparse, generalized
neural network. The L1/L2 loss can be controlled with the arguments
alpha and lambda.
\[
loss = \lambda * [ (1 - \alpha) * |weights| + \alpha |weights|^2 ]
\]
#elastic net penalty in all layers:
nn.fit <- dnn(Species~., data = data, alpha = 0.5, lambda = 0.01, verbose=FALSE, loss = "softmax")
Dropout Regularization
Dropout regularization as proposed in Srivastava
et al. can be controlled similar to elastic net regularization. In
this approach, a percentage of different nodes gets left during each
epoch.
#dropout of 35% on all layers:
nn.fit <- dnn(Species~., data = data, loss = "softmax", dropout = 0.35, verbose=FALSE)
#dropout of 35% only on last 2 layers:
nn.fit <- dnn(Species~., data = data, loss = "softmax", dropout = c(0, 0, 0.35, 0.35), verbose=FALSE)
Learning rate
Learning rate scheduler
Learning rate scheduler allow you to start with a high learning rate
and decrease it during the training process. This leads to an overall
faster training. You can choose between different types of schedulers.
Namely, lambda, multiplicative, one_cycle and step.
The function config_lr_scheduler() helps you setup such a scheduler.
See ?config_lr_scheduler() for more information
# Step Learning rate scheduler that reduces learning rate every 16 steps by a factor of 0.5
scheduler <- config_lr_scheduler(type = "step",
step_size = 16,
gamma = 0.5)
nn.fit <- dnn(Sepal.Length~., data = data,lr = 0.01, lr_scheduler= scheduler, verbose = FALSE)
Optimizer
Optimizer are responsible for fitting the neural network. The
optimizer tries to minimize the loss function. As default the stochastic
gradient descent is used. Custom optimizers can be used with
config_optimizer().
See ?config_optimizer() for more information.
# adam optimizer with learning rate 0.002, betas to 0.95, 0.999 and eps to 1.5e-08
opt <- config_optimizer(
type = "sgd")
nn.fit <- dnn(Species~., data = data, optimizer = opt, lr=0.002, verbose=FALSE, loss = "softmax")
Early Stopping
Adding early stopping criteria helps you save time by stopping the
training process early, if the validation loss of the current epoch is
bigger than the validation loss n epochs early. The n can be defined by
the early_stopping argument. It is required to set validation >
0.
# Stops training if validation loss at current epoch is bigger than that 15 epochs earlier
nn.fit <- dnn(Sepal.Length~., data = data, epochs = 1000,
validation = 0.2, early_stopping = 15, verbose=FALSE)
Continue training process
You can continue the training process of an existing model with
continue_training().
# simple example, simply adding another 12 epochs to the training process
nn.fit <- continue_training(nn.fit, epochs = 12, verbose=FALSE)
head(predict(nn.fit))
#> [,1]
#> [1,] -0.9836558
#> [2,] -1.4395623
#> [3,] -1.3351157
#> [4,] -1.3388002
#> [5,] -0.8847700
#> [6,] -0.4809422
It also allows you to change any training parameters, for example the
learning rate. You can analyze the training process with
analyze_training().
# picking the model with the smalles validation loss
# with changed parameters, in this case a smaller learning rate and a smaller batchsize
nn.fit <- continue_training(nn.fit,
epochs = 32,
changed_params = list(lr = 0.001, batchsize = 16),
verbose = FALSE)
