{nadir}

nadir (noun): nā-dir

the lowest point.

Fitting with the minimum loss based estimation¹² literature, {nadir} is an implementation of the super learner algorithm with improved support for flexible formula based syntax and which is fond of functional programming techniques such as closures, currying, and function factories.

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{nadir} implements the super learner algorithm³. To quote the Guide to SuperLearner⁴ (a previous implementation):

SuperLearner is an algorithm that uses cross-validation to estimate the performance of multiple machine learning models, or the same model with different settings. It then creates an optimal weighted average of those models, aka an “ensemble”, using the test data performance. This approach has been proven to be asymptotically as accurate as the best possible prediction algorithm that is tested.

Why {nadir} and why reimplement super learner again?

In previous implementations ({SuperLearner}, {sl3}, {mlr3superlearner}), support for flexible formula-based syntax has been limited, instead opting for specifying learners as models on an $X$ matrix and $Y$ outcome vector. Many popular R packages such as lme4 and mgcv (for random effects and generalized additive models) use formulas extensively to specify models using syntax like (age | strata) to specify random effects on age by strata, or s(age, income) to specify a smoothing term on age and income simultaneously.

At present, it is difficult to use these kinds of features in {SuperLearner}, {sl3} and {mlr3superlearner}.

For example, it is easy to imagine the super learner algorithm being appealing to modelers fond of random effects based models because they may want to hedge on the exact nature of the random effects models, not sure if random intercepts are enough or if random slopes should be included, etc., and similar other modeling decisions in other frameworks.

Therefore, the {nadir} package takes as its charges to:

• Implement a syntax in which it is easy to specify different formulas for each of many candidate learners.
• To make it easy to pass new learners to the Super Learner algorithm.

Installation Instructions

At present, {nadir} is only available on GitHub.

devtools::install_github("ctesta01/nadir")

Warning: this package is currently under active development and may be wrong! Do not use this for serious applications until this message has been removed, likely at the time of a future release.

Demonstration

First, let’s start with the simplest possible use case of nadir::super_learner(), which is where the user would like to feed in data, a specification for some regression formula(s), specify a library of learners, and get back a prediction function that is suitable for plugging into downstream analyses, like in Targeted Learning or for pure-prediction applications.

Here is a demo of an extremely simple application of using nadir::super_learner:

library(nadir)

# we'll use a few basic learners
learners <- list(
     glm = lnr_glm,
     rf = lnr_rf,
     glmnet = lnr_glmnet
  )
# more learners are available, see ?learners

sl_model <- super_learner(
  data = mtcars,
  formula = mpg ~ cyl + hp + disp,
  learners = learners)

# the output from super_learner is a prediction function:
# here we are producing predictions based on a weighted combination of the
# trained learners. 
sl_model(mtcars) |> head()

##         Mazda RX4     Mazda RX4 Wag        Datsun 710    Hornet 4 Drive 
##          20.42305          20.42305          24.20789          19.87290 
## Hornet Sportabout           Valiant 
##          16.95804          19.62781

One Step Up: Fancy Formula Features

Continuing with our mtcars example, suppose the user would really like to use random effects or similar types of fancy formula language features. One easy way to do so with nadir::super_learner is using the following syntax:

learners <- list(
     glm = lnr_glm,
     rf = lnr_rf,
     glmnet = lnr_glmnet,
     lmer = lnr_lmer,
     gam = lnr_gam
  )

formulas <- c(
  .default = mpg ~ cyl + hp + disp,   # our first three learners use same formula
  lmer = mpg ~ (1 | cyl) + hp + disp, # both lme4::lmer and mgcv::gam have 
  gam = mpg ~ s(hp) + cyl + disp      # specialized formula syntax
  )

# fit a super_learner
sl_model <- super_learner(
  data = mtcars,
  formulas = formulas,
  learners = learners)
  
sl_model(mtcars) |> head()

##         Mazda RX4     Mazda RX4 Wag        Datsun 710    Hornet 4 Drive 
##          20.46252          20.46252          24.27968          19.85617 
## Hornet Sportabout           Valiant 
##          16.95162          19.63611

How should we assess performance of nadir::super_learner()?

To put the learners and the super learner algorithm on a level playing field, it’s important that learners and super learner both be evaluated on held-out validation/test data that the algorithms have not seen before.

Using the verbose = TRUE output from nadir::super_learner(), we can call compare_learners() to see the mean-squared-error (MSE) on the held-out data, also called CV-MSE, for each of the candidate learners specified.

# construct our super learner with verbose = TRUE
sl_model <- super_learner(
  data = mtcars,
  formulas = formulas,
  learners = learners,
  verbose = TRUE)
  
compare_learners(sl_model)

## Inferring the loss metric for learner comparison based on the outcome type:

## outcome_type=continuous -> using mean squared error

## # A tibble: 1 × 5
##     glm    rf glmnet  lmer   gam
##   <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1  11.8  7.70   11.9  10.1  12.9

Plotting code

pacman::p_load('dplyr', 'ggplot2', 'tidyr', 'magrittr')

truth <- sl_model$holdout_predictions$mpg

holdout_var <- sl_model$holdout_predictions |>
  dplyr::group_by(.sl_fold) |> 
  dplyr::summarize(across(everything(), ~ mean((. - mpg)^2))) |> 
  dplyr::summarize(across(everything(), var)) |> 
  select(-mpg, -.sl_fold) |> 
  t() |> 
  as.data.frame() |> 
  tibble::rownames_to_column('learner') |> 
  dplyr::rename(var = V1) |>
  dplyr::mutate(sd = sqrt(var))


jitters <- sl_model$holdout_predictions |> 
  dplyr::mutate(dplyr::across(-.sl_fold, ~ (. - mpg)^2)) |> 
  dplyr::select(-mpg) %>%
  tidyr::pivot_longer(cols = 2:ncol(.), names_to = 'learner', values_to = 'squared_error') |>
  dplyr::group_by(learner, .sl_fold) |> 
  dplyr::summarize(mse = mean(squared_error)) |> 
  ungroup() |> 
  rename(fold = .sl_fold)

## `summarise()` has grouped output by 'learner'. You can override using the
## `.groups` argument.

learner_comparison_df <- sl_model |> 
  compare_learners() |> 
  t() |> 
  as.data.frame() |>
  tibble::rownames_to_column(var = 'learner') |> 
  dplyr::mutate(learner = factor(learner)) |>
  dplyr::rename(mse = V1) |>
  dplyr::left_join(holdout_var) |> 
  dplyr::mutate(
    upper_ci = mse + sd,
    lower_ci = mse - sd) |> 
  dplyr::mutate(learner = forcats::fct_reorder(learner, mse))

## Inferring the loss metric for learner comparison based on the outcome type: 
## outcome_type=continuous -> using mean squared error
## Joining with `by = join_by(learner)`

jitters$learner <- factor(jitters$learner, levels = levels(learner_comparison_df$learner))

learner_comparison_df |> 
  ggplot2::ggplot(ggplot2::aes(y = learner, x = mse, fill = learner)) + 
  ggplot2::geom_col(alpha = 0.5) + 
  ggplot2::geom_jitter(data = jitters, mapping = ggplot2::aes(x = mse), height = .15, shape = 'o') + 
  ggplot2::geom_pointrange(mapping = ggplot2::aes(xmax = upper_ci, xmin = lower_ci),
                           alpha = 0.5) + 
  ggplot2::theme_bw() + 
  ggplot2::ggtitle("Comparison of Candidate Learners") + 
  ggplot2::labs(caption = "Error bars show ±1 standard deviation across the CV estimated MSE for each learner\n
Each open circle represents the hold-out MSE of one fold of the data") + 
  ggplot2::theme(plot.caption.position = 'plot')

Now how should we go about getting the CV-MSE from a super learned model? We will use the cv_super_learner() function that performs another layer of cross-validation in order to assess the specified super learner on folds of held-out data.

If you’d like to read more about how the internals of cv_super_learner() work, please refer to the article Currying, Closures, and Function Factories article

cv_results <- cv_super_learner(
  data = mtcars,
  formulas = formulas,
  learners = learners)

cv_results

## $cv_trained_learners
## # A tibble: 5 × 4
##   split learned_predictor predictions mpg      
##   <int> <list>            <list>      <list>   
## 1     1 <function>        <dbl [7]>   <dbl [7]>
## 2     2 <function>        <dbl [7]>   <dbl [7]>
## 3     3 <function>        <dbl [7]>   <dbl [7]>
## 4     4 <function>        <dbl [6]>   <dbl [6]>
## 5     5 <function>        <dbl [5]>   <dbl [5]>
## 
## $cv_loss
## [1] 8.173704

Plotting code

cv_jitters <- cv_results$cv_trained_learners |> 
  dplyr::select(split, predictions, mpg) |> 
  tidyr::unnest(cols = c('predictions', 'mpg')) |> 
  dplyr::group_by(split) |> 
  dplyr::summarize(mse = mean((mpg - predictions)^2)) |>
  dplyr::bind_cols(learner = 'super_learner')


cv_var <- cv_results$cv_trained_learners |> 
  dplyr::select(split, predictions, mpg) |> 
  tidyr::unnest(cols = c(predictions, mpg)) |> 
  dplyr::mutate(squared_error = (mpg - predictions)^2) |> 
  dplyr::group_by(split) |> 
  dplyr::summarize(mse = mean(squared_error)) |> 
  dplyr::summarize(
    var = var(mse),
    mse = mean(mse),
    sd = sqrt(var),
    upper_ci = mse + sd,
    lower_ci = mse - sd) |> 
  dplyr::bind_cols(learner = 'super_learner')

new_jitters <- bind_rows(jitters, cv_jitters)

learner_comparison_df |> 
  bind_rows(cv_var) |> 
  dplyr::mutate(learner = forcats::fct_reorder(learner, mse)) |> 
  ggplot2::ggplot(ggplot2::aes(y = learner, x = mse, fill = learner)) + 
  ggplot2::geom_col(alpha = 0.5) + 
  ggplot2::geom_jitter(data = new_jitters, mapping = ggplot2::aes(x = mse), height = .15, shape = 'o') + 
  ggplot2::geom_pointrange(mapping = ggplot2::aes(xmax = upper_ci, xmin = lower_ci),
                           alpha = 0.5) + 
  ggplot2::theme_bw() + 
  ggplot2::scale_fill_brewer(palette = 'Set2') + 
  ggplot2::ggtitle("Comparison of Candidate Learners against Super Learner") + 
  ggplot2::labs(caption = "Error bars show ±1 standard deviation across the CV estimated MSE for each learner\n
Each open circle represents the hold-out MSE of one fold of the data") + 
  ggplot2::theme(plot.caption.position = 'plot')

What about model hyperparameters or extra arguments?

Model hyperparameters are easy to handle in {nadir}. Two easy solutions are available to users:

• nadir::super_learner() has an extra_learner_args parameter that can be passed a list of extra arguments for each learner.
• Users can always build new learners (which allows for building in the hyperparameter specification), and using the ... syntax, it’s easy to build new learners from the learners already provided by {nadir}.

Here’s some examples showing each approach.

Using extra_learner_args:

# when using extra_learner_args, it's totally okay to use the 
# same learner multiple times as long as their hyperparameters differ.

sl_model <- nadir::super_learner(
  data = mtcars,
  formula = mpg ~ .,
  learners = c(
    glmnet0 = lnr_glmnet,
    glmnet1 = lnr_glmnet,
    glmnet2 = lnr_glmnet,
    rf0 = lnr_rf,
    rf1 = lnr_rf,
    rf2 = lnr_rf
    ),
  extra_learner_args = list(
    glmnet0 = list(lambda = 0.01),
    glmnet1 = list(lambda = 0.1),
    glmnet2 = list(lambda = 1),
    rf0 = list(ntree = 3),
    rf1 = list(ntree = 10),
    rf2 = list(ntree = 30)
    ),
  verbose = TRUE
)

compare_learners(sl_model)

## Inferring the loss metric for learner comparison based on the outcome type:

## outcome_type=continuous -> using mean squared error

## # A tibble: 1 × 6
##   glmnet0 glmnet1 glmnet2   rf0   rf1   rf2
##     <dbl>   <dbl>   <dbl> <dbl> <dbl> <dbl>
## 1    17.6    12.8    10.2  13.6  8.30  8.12

Building New Learners Programmatically

When does it make more sense to build new learners with the hyperparameters built into them rather than using the extra_learner_args parameter?

One instance when building new learners may make sense is when the user would like to produce a large number of hyperparameterized learners programmatically, for example over a grid of hyperparameter values. Below we show such an example for a 1-d grid of hyperparameters with glmnet.

# produce a "grid" of glmnet learners with lambda set to 
# exp(-1 to 1 in steps of .1)
hyperparameterized_learners <- lapply(
  exp(seq(-1, 1, by = .1)), 
  function(lambda) { 
    return(
      function(data, formula, ...) {
        lnr_glmnet(data, formula, lambda = lambda, ...)
        })
  })
  
# give them names because nadir::super_learner requires that the 
# learners argument be named.
names(hyperparameterized_learners) <- paste0('glmnet', 1:length(hyperparameterized_learners))

# fit the super_learner with 20 glmnets with different lambdas
sl_model_glmnet <- nadir::super_learner(
  data = mtcars,
  learners = hyperparameterized_learners,
  formula = mpg ~ .,
  verbose = TRUE)

compare_learners(sl_model_glmnet)

## Inferring the loss metric for learner comparison based on the outcome type:

## outcome_type=continuous -> using mean squared error

## # A tibble: 1 × 21
##   glmnet1 glmnet2 glmnet3 glmnet4 glmnet5 glmnet6 glmnet7 glmnet8 glmnet9
##     <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>
## 1    8.42    8.27    8.12    8.00    7.89    7.80    7.72    7.70    7.74
## # ℹ 12 more variables: glmnet10 <dbl>, glmnet11 <dbl>, glmnet12 <dbl>,
## #   glmnet13 <dbl>, glmnet14 <dbl>, glmnet15 <dbl>, glmnet16 <dbl>,
## #   glmnet17 <dbl>, glmnet18 <dbl>, glmnet19 <dbl>, glmnet20 <dbl>,
## #   glmnet21 <dbl>

What are currying, closures, and function factories?

R is a functional programming language, which allows for functions to build and return functions just like any other return object.

We refer to functions that create and return another function as a function factory. For an extended reference, see the Advanced R book.

Function factories are so useful in {nadir} because, at their essence, a candidate learner needs to be able to 1) accept training data, and 2) produce a prediction function that can make predictions on heldout validation data. So a typical learner in {nadir} looks like:

lnr_lm <- function(data, formula, ...) {
  model <- stats::lm(formula = formula, data = data, ...)

  predict_from_trained_lm <- function(newdata) {
    predict(model, newdata = newdata, type = 'response')
  }
  return(predict_from_trained_lm)
}

Moreover, given how code-lightweight it is to write a simple learner, this makes it relatively easy for users to write new learners that meet their exact needs.

If you want to implement your own learners, you just need to follow the following pseudocode approach:

lnr_custom <- function(data, formula, ...) {
  model <- # train your model using data, formula, ... 
  
  predict_from_model <- function(newdata) {
    return(...) # return predictions from the trained model 
    # (predictions should be a vector of predictions, one for each row of newdata)
  }
  return(predict_from_model)
}

For more details, read the Currying, Closures, and Function Factories article

Coming Down the Pipe ↩️🚰🔧✨

• (More) Automated tests that try to ensure validity/correctness of the implementation!
• Performance benchmarking (hopefully leading to carefully considered improvements for speed)
• More vignettes/articles soon.
• Better explicit support for binary outcomes and density estimation.

Footnotes

1. van der Laan, Mark J. and Dudoit, Sandrine, “Unified Cross-Validation Methodology For Selection Among Estimators and a General Cross-Validated Adaptive Epsilon-Net Estimator: Finite Sample Oracle Inequalities and Examples” (November 2003). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 130. https://biostats.bepress.com/ucbbiostat/paper130 ↩

2. Zheng, W., & van der Laan, M. J. (2011). Cross-Validated Targeted Minimum-Loss-Based Estimation. In Springer Series in Statistics (pp. 459–474). Springer New York. https://doi.org/10.1007/978-1-4419-9782-1_27 ↩

3. van der Laan, M. J., Polley, E. C., & Hubbard, A. E. (2007). Super Learner. In Statistical Applications in Genetics and Molecular Biology (Vol. 6, Issue 1). Walter de Gruyter GmbH. https://doi.org/10.2202/1544-6115.1309 https://pubmed.ncbi.nlm.nih.gov/17910531/ ↩

4. Guide to {SuperLearner}: https://cran.r-project.org/web/packages/SuperLearner/vignettes/Guide-to-SuperLearner.html ↩