Adversarial Training

The ARF algorithm is an iterative procedure. In the first instance, we generate synthetic data by independently sampling from the marginals of each feature and training a random forest (RF) to distinguish original from synthetic samples. If accuracy is greater than \(0.5 + \delta\) (where delta is a user-controlled tolerance parameter, generally set to 0), we create a new dataset by sampling from the marginals within each leaf and training another RF classifier. The procedure repeats until original and synthetic samples cannot be reliably distinguished. With the default verbose = TRUE, the algorithm will print accuracy at each iteration.

# Load libraries
library(arf)
library(data.table)
library(ggplot2)

# Set seed
set.seed(123, "L'Ecuyer-CMRG")

# Train ARF
arf_iris <- adversarial_rf(iris)
#> Iteration: 0, Accuracy: 86.53%
#> Iteration: 1, Accuracy: 40%
#> Warning: executing %dopar% sequentially: no parallel backend registered

The printouts can be turned off by setting verbose = FALSE. Accuracy is still stored within the arf object, so you can evaluate convergence after the fact. The warning appears just once per session. It can be suppressed by setting parallel = FALSE or registering a parallel backend (more on this below).

# Train ARF with no printouts
arf_iris <- adversarial_rf(iris, verbose = FALSE)

# Plot accuracy against iterations (model converges when accuracy <= 0.5)
tmp <- data.frame('Accuracy' = arf_iris$acc, 
                  'Iteration' = seq_len(length(arf_iris$acc)))
ggplot(tmp, aes(Iteration, Accuracy)) + 
  geom_point() + 
  geom_path() +
  geom_hline(yintercept = 0.5, linetype = 'dashed', color = 'red') 

We find a quick drop in accuracy following the resampling procedure, as desired. If the ARF has converged, then resulting splits should form fully factorized leaves, i.e. subregions of the feature space where variables are locally independent.

ARF convergence is asymptotically guaranteed as \(n \rightarrow \infty\) (see Watson et al., 2023, Thm. 1). However, this has no implications for finite sample performance. In practice, we often find that adversarial training completes in just one or two rounds, but this may not hold for some datasets. To avoid infinite loops, users can increase the slack parameter delta or set the max_iters argument (default = 10). In addition to these failsafes, adversarial_rf uses early stopping by default (early_stop = TRUE), which terminates training if factorization does not improve from one round to the next. This is recommended, since discriminator accuracy rarely falls much lower once it has increased.

For density estimation tasks, we recommend increasing the default number of trees. We generally use 100 in our experiments, though this may be suboptimal for some datasets. Likelihood estimates are not very sensitive to this parameter above a certain threshold, but larger models incur extra costs in time and memory. We can speed up computations by registering a parallel backend, in which case ARF training is distributed across cores using the ranger package. Much like with ranger, the default behavior of adversarial_rf is to compute in parallel if possible. How exactly this is done varies across operating systems. The following code works on Unix machines.

# Register cores - Unix
library(doParallel)
registerDoParallel(cores = 2)

Windows requires a different setup.

# Register cores - Windows
library(doParallel)
cl <- makeCluster(2)
registerDoParallel(cl)

In either case, we can now execute in parallel.

# Rerun ARF, now in parallel and with more trees
arf_iris <- adversarial_rf(iris, num_trees = 100)
#> Iteration: 0, Accuracy: 92.33%
#> Iteration: 1, Accuracy: 32%

The result is an object of class ranger, which we can input to downstream functions.

Parameter Learning

The next step is to learn the leaf and distribution parameters using forests for density estimation (FORDE). This function calculates the coverage, bounds, and pdf/pmf parameters for every variable in every leaf. This can be an expensive computation for large datasets, as it requires \(\mathcal{O}\big(B \cdot d \cdot n \cdot \log(n)\big)\) operations, where \(B\) is the number of trees, \(d\) is the data dimensionality, and \(n\) is the sample size. Once again, the process is parallelized by default.

# Compute leaf and distribution parameters
params_iris <- forde(arf_iris, iris)

Default behavior is to use a truncated normal distribution for continuous data (with boundaries given by the tree’s split parameters) and a multinomial distribution for categorical data. We find that this produces stable results in a wide range of settings. You can also use a uniform distribution for continuous features by setting family = 'unif', thereby instantiating a piecewise constant density estimator.

# Recompute with uniform density
params_unif <- forde(arf_iris, iris, family = 'unif')

This method tends to perform poorly in practice, and we do not recommend it. The option is implemented primarily for benchmarking purposes. Alternative families, e.g. truncated Poisson or beta distributions, may be useful for certain problems. Future releases will expand the range of options for the family argument.

The alpha and epsilon arguments allow for optional regularization of multinomial and uniform distributions, respectively. These help prevent zero likelihood samples when test data fall outside the support of training data. The former is a pseudocount parameter that applies Laplace smoothing within leaves, preventing unobserved values from being assigned zero probability unless splits explicitly rule them out. In other words, we impose a flat Dirichlet prior and report posterior probabilities rather than maximum likelihood estimates. The latter is a slack parameter on empirical bounds that expands the estimated extrema for continuous features by a factor of \(1 + \epsilon\).

Compare the results of our original probability estimates for the Species variable with those obtained by adding a pseudocount of \(\alpha = 0.1\).

# Recompute with additive smoothing
params_alpha <- forde(arf_iris, iris, alpha = 0.1)

# Compare results
head(params_iris$cat)
#>    f_idx variable       val prob
#> 1:     1  Species virginica    1
#> 2:     2  Species virginica    1
#> 3:     3  Species virginica    1
#> 4:     4  Species virginica    1
#> 5:     5  Species virginica    1
#> 6:     6  Species    setosa    1
head(params_alpha$cat)
#>    f_idx variable        val       prob
#> 1:     1  Species  virginica 0.93939394
#> 2:     1  Species     setosa 0.03030303
#> 3:     1  Species versicolor 0.03030303
#> 4:     2  Species  virginica 0.96825397
#> 5:     2  Species     setosa 0.01587302
#> 6:     2  Species versicolor 0.01587302

Under Laplace smoothing, extreme probabilities only occur when the splits explicitly demand it. Otherwise, all values shrink toward a uniform prior. Note that these two data tables may not have exactly the same rows, as we omit zero probability events to conserve memory. However, we can verify that probabilities sum to unity for each leaf-variable combination.

# Sum probabilities
summary(params_iris$cat[, sum(prob), by = .(f_idx, variable)]$V1)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>       1       1       1       1       1       1
summary(params_alpha$cat[, sum(prob), by = .(f_idx, variable)]$V1)
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>       1       1       1       1       1       1

The forde function outputs a list of length 6, with entries for (1) continuous features; (2) categorical features; (3) leaf parameters; (4) variable metadata; (5) factor levels; and (6) data input class.

params_iris
#> $cnt
#>       f_idx     variable  min  max        mu      sigma
#>    1:     1 Petal.Length -Inf  Inf 5.9333333 0.20816660
#>    2:     1  Petal.Width 2.45  Inf 2.5000000 0.01041469
#>    3:     1 Sepal.Length -Inf  Inf 6.7333333 0.45092498
#>    4:     1  Sepal.Width -Inf  Inf 3.4000000 0.17320508
#>    5:     2 Petal.Length 5.65  Inf 6.1166667 0.34302575
#>   ---                                                  
#> 7144:  1786  Sepal.Width -Inf 2.85 2.4666667 0.05773503
#> 7145:  1787 Petal.Length -Inf 1.45 1.3833333 0.04082483
#> 7146:  1787  Petal.Width -Inf 0.35 0.2166667 0.07527727
#> 7147:  1787 Sepal.Length 4.45 4.95 4.7333333 0.12110601
#> 7148:  1787  Sepal.Width 2.85 3.55 3.1333333 0.16329932
#> 
#> $cat
#>       f_idx variable        val  prob
#>    1:     1  Species  virginica 1.000
#>    2:     2  Species  virginica 1.000
#>    3:     3  Species  virginica 1.000
#>    4:     4  Species  virginica 1.000
#>    5:     5  Species  virginica 1.000
#>   ---                                
#> 2056:  1784  Species versicolor 1.000
#> 2057:  1785  Species     setosa 0.125
#> 2058:  1785  Species versicolor 0.875
#> 2059:  1786  Species versicolor 1.000
#> 2060:  1787  Species     setosa 1.000
#> 
#> $forest
#>       f_idx tree leaf        cvg
#>    1:     1    1    7 0.02000000
#>    2:     2    1   21 0.04000000
#>    3:     3    1   27 0.02000000
#>    4:     4    1   35 0.07333333
#>    5:     5    1   37 0.01333333
#>   ---                           
#> 1783:  1783  100   83 0.04666667
#> 1784:  1784  100   85 0.04000000
#> 1785:  1785  100   86 0.05333333
#> 1786:  1786  100   87 0.02000000
#> 1787:  1787  100   88 0.04000000
#> 
#> $meta
#>        variable   class    family decimals
#> 1: Sepal.Length numeric truncnorm        1
#> 2:  Sepal.Width numeric truncnorm        1
#> 3: Petal.Length numeric truncnorm        1
#> 4:  Petal.Width numeric truncnorm        1
#> 5:      Species  factor  multinom       NA
#> 
#> $levels
#>    variable        val
#> 1:  Species  virginica
#> 2:  Species     setosa
#> 3:  Species versicolor
#> 
#> $input_class
#> [1] "data.frame"

These parameters can be used for a variety of downstream tasks, such as likelihood estimation and data synthesis.

Likelihood Estimation

To calculate log-likelihoods, we pass params on to the lik function, along with the data whose likelihood we wish to evaluate. For total evidence queries (i.e., those spanning all variables and no conditioning events), it is faster to also include arf in the function call.

# Compute likelihood under truncated normal and uniform distributions
ll <- lik(params_iris, iris, arf = arf_iris)
ll_unif <- lik(params_unif, iris, arf = arf_iris)

# Compare average negative log-likelihood (lower is better)
-mean(ll)
#> [1] 0.6762925
-mean(ll_unif)
#> [1] 3.861917

Note that the piecewise constant estimator does considerably worse in this experiment.

The lik function can also be used to compute the likelihood of some partial state, i.e. a setting in which some but not all variable values are specified. Let’s take a look at that iris dataset:

head(iris)
#>   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

Say we want to calculate sample likelihoods using only continuous data. That is, we provide values for the first four variables but exclude the fifth. In this case, the model will have to marginalize over Species:

# Compute likelihoods after marginalizing over Species
iris_without_species <- iris[, -5]
ll_cnt <- lik(params_iris, iris_without_species)

# Compare results
tmp <- data.frame(Total = ll, Partial = ll_cnt)
ggplot(tmp, aes(Total, Partial)) + 
  geom_point() + 
  geom_abline(slope = 1, intercept = 0, color = 'red')

We find that likelihoods are almost identical, with slightly higher likelihood on average for partial samples. This is expected, since they have less variation to model.

In this example, we have used the same data throughout. This may lead to overfitting. With sufficient data, it is preferable to use a training set for adversarial_rf, a validation set for forde, and a test set for lik. Alternatively, we can set the oob argument to TRUE for either of the latter two functions, in which case computations are performed only on out-of-bag (OOB) data. These are samples that are randomly excluded from a given tree due to the bootstrapping subroutine of the RF classifier. Note that this only works when the dataset x passed to forde or lik is the same one used to train the arf. Recall that a sample’s probability of being excluded from a single tree is \(\exp(-1) \approx 0.368\). When using oob = TRUE, be sure to include enough trees so that every observation is likely to be OOB at least a few times.

Data Synthesis

For this experiment, we use the smiley simulation from the mlbench package, which allows for easy visual assessment. We draw a training set of \(n = 1000\) and simulate \(1000\) synthetic datapoints. Resulting data are plotted side by side.

# Simulate training data
library(mlbench)
x <- mlbench.smiley(1000)
x <- data.frame(x$x, x$classes)
colnames(x) <- c('X', 'Y', 'Class')

# Fit ARF
arf_smiley <- adversarial_rf(x, mtry = 2)
#> Iteration: 0, Accuracy: 90.17%
#> Iteration: 1, Accuracy: 37.84%

# Estimate parameters
params_smiley <- forde(arf_smiley, x)

# Simulate data
synth <- forge(params_smiley, n_synth = 1000)

# Compare structure
str(x)
#> 'data.frame':    1000 obs. of  3 variables:
#>  $ X    : num  -0.841 -0.911 -0.91 -0.743 -0.863 ...
#>  $ Y    : num  0.874 0.926 1.051 0.918 1.157 ...
#>  $ Class: Factor w/ 4 levels "1","2","3","4": 1 1 1 1 1 1 1 1 1 1 ...
str(synth)
#> 'data.frame':    1000 obs. of  3 variables:
#>  $ X    : num  -0.89219 -0.74509 -0.00148 -0.11204 -0.39182 ...
#>  $ Y    : num  -0.119 0.742 0.617 0.183 -0.926 ...
#>  $ Class: Factor w/ 4 levels "1","3","2","4": 4 1 2 2 4 4 1 4 3 1 ...

# Put it all together
x$Data <- 'Original'
synth$Data <- 'Synthetic'
df <- rbind(x, synth)

# Plot results
ggplot(df, aes(X, Y, color = Class, shape = Class)) + 
  geom_point() + 
  facet_wrap(~ Data)

The general shape is clearly recognizable, even if some stray samples are evident and borders are not always crisp. This can be improved with more training data.

Conditioning

ARFs can also be used to compute likelihoods and synthesize data under conditioning events that specify values or ranges for input features. For instance, say we want to evaluate the likelihood of samples from the iris dataset under the condition that Species = 'setosa'. There are several ways to encode evidence, but the simplest is to pass a partial observation.

# Compute conditional likelihoods
evi <- data.frame(Species = 'setosa')
ll_conditional <- lik(params_iris, query = iris_without_species, evidence = evi)

# Compare NLL across species (shifting to positive range for visualization)
tmp <- iris
tmp$NLL <- -ll_conditional + max(ll_conditional) + 1
ggplot(tmp, aes(Species, NLL, fill = Species)) + 
  geom_boxplot() + 
  scale_y_log10() + 
  ylab('Negative Log-Likelihood') + 
  theme(legend.position = 'none')
#> Warning: Removed 3 rows containing non-finite values (`stat_boxplot()`).

As expected, measurements for non-setosa samples appear much less likely under this conditioning event.

The partial observation method of passing evidence requires users to specify unique feature values for each conditioning variable. A more flexible alternative is to construct a data frame of conditioning events, potentially including inequalities. For example, we may want to calculate the likelihood of samples given that Species = 'setosa' and Petal.Width > 0.3.

# Data frame of conditioning events
evi <- data.frame(variable = c('Species', 'Petal.Width'),
                  relation = c('==', '>'), 
                  value = c('setosa', 0.3))
evi
#>      variable relation  value
#> 1     Species       == setosa
#> 2 Petal.Width        >    0.3

Each row is treated as an extra condition, so we can define intervals by putting multiple constraints on a single variable.

evi <- data.frame(variable = c('Species', 'Petal.Width', 'Petal.Width'),
                  relation = c('==', '>', '<='), 
                  value = c('setosa', 0.3, 0.5))
evi
#>      variable relation  value
#> 1     Species       == setosa
#> 2 Petal.Width        >    0.3
#> 3 Petal.Width       <=    0.5

Resulting likelihoods will be computed on the condition that all queries are drawn from setosa flowers with petal width on the interval \((0.3, 0.5]\).

A final method for passing evidence is to directly compute a posterior distribution on leaves. This could be useful for particularly complex conditioning events for which there is currently no inbuilt interface, such as polynomial constraints or arbitrary propositions in disjunctive normal form. In this case, we just require a data frame with columns for f_idx and wt. If the latter does not sum to unity, the distribution will be normalized with a warning.

# Drawing random weights
evi <- data.frame(f_idx = params_iris$forest$f_idx,
                  wt = rexp(nrow(params_iris$forest)))
evi$wt <- evi$wt / sum(evi$wt)
head(evi)
#>   f_idx           wt
#> 1     1 1.794477e-05
#> 2     2 2.983682e-04
#> 3     3 2.302738e-06
#> 4     4 1.078006e-03
#> 5     5 4.394211e-04
#> 6     6 4.283289e-04

Each of these methods can be used for conditional sampling as well.

# Simulate class-conditional data for smiley example
evi <- data.frame(Class = 4)
synth2 <- forge(params_smiley, n_synth = 250, evidence = evi)

# Put it all together
synth2$Data <- 'Synthetic'
df <- rbind(x, synth2)

# Plot results
ggplot(df, aes(X, Y, color = Class, shape = Class)) + 
  geom_point() + 
  facet_wrap(~ Data)

By conditioning on Class = 4, we restrict our sampling to the smile itself, rather than eyes or nose.

Computing conditional expectations is similarly straightforward.

expct(params_smiley, evidence = evi)
#>             X          Y
#> 1 -0.01207206 -0.6734144

These are the average \(X, Y\) coordinates for the smile above.