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When Cross-Validation is More Powerful than Regularization

Regularization is a way of avoiding overfit by restricting the magnitude of model coefficients (or in deep learning, node weights). A simple example of regularization is the use of ridge or lasso regression to fit linear models in the presence of collinear variables or (quasi-)separation. The intuition is that smaller coefficients are less sensitive to idiosyncracies in the training data, and hence, less likely to overfit.

Cross-validation is a way to safely reuse training data in nested model situations. This includes both the case of setting hyperparameters before fitting a model, and the case of fitting models (let’s call them base learners) that are then used as variables in downstream models, as shown in Figure 1. In either situation, using the same data twice can lead to models that are overtuned to idiosyncracies in the training data, and more likely to overfit.

Figure 1 Properly nesting models with cross-validation
Figure 1 Properly nesting models with cross-validation

In general, if any stage of your modeling pipeline involves looking at the outcome (we’ll call that a y-aware stage), you cannot directly use the same data in the following stage of the pipeline. If you have enough data, you can use separate data in each stage of the modeling process (for example, one set of data to learn hyperparameters, another set of data to train the model that uses those hyperparameters). Otherwise, you should use cross-validation to reduce the nested model bias.

Cross-validation is relatively computationally expensive; regularization is relatively cheap. Can you mitigate nested model bias by using regularization techniques instead of cross-validation?

The short answer: no, you shouldn’t. But as, we’ve written before, demonstrating this is more memorable than simply saying “Don’t do that.”

A simple example

Suppose you have a system with two categorical variables. The variable x_s has 10 levels, and the variable x_n has 100 levels. The outcome y is a function of x_s, but not of x_n (but you, the analyst building the model, don’t know this). Here’s the head of the data.

##     x_s  x_n           y
## 2  s_10 n_72  0.34228110
## 3  s_01 n_09 -0.03805102
## 4  s_03 n_18 -0.92145960
## 9  s_08 n_43  1.77069352
## 10 s_08 n_17  0.51992928
## 11 s_01 n_78  1.04714355

With most modeling techniques, a categorical variable with K levels is equivalent to K or K-1 numerical (indicator or dummy) variables, so this system actually has around 110 variables. In real life situations where a data scientist is working with high-cardinality categorical variables, or with a lot of categorical variables, the number of actual variables can begin to swamp the size of training data, and/or bog down the machine learning algorithm.

One way to deal with these issues is to represent each categorical variable by a single variable model (or base learner), and then use the predictions of those base learners as the inputs to a bigger model. So instead of fitting a model with 110 indicator variables, you can fit a model with two numerical variables. This is a simple example of nested models.

Figure 2 Impact coding as an example of nested models
Figure 2 Impact coding as an example of nested model

We refer to this procedure as “impact coding,” and it is one of the data treatments available in the vtreat package, specifically for dealing with high-cardinality categorical variables. But for now, let’s go back to the original problem.

The naive way

For this simple example, you might try representing each variable as the expected value of y - mean(y) in the training data, conditioned on the variable’s level. So the ith “coefficient” of the one-variable model would be given by:

vi = E[y|x = si] − E[y]

Where si is the ith level. Let’s show this with the variable x_s (the code for all the examples in this article is here):

##     x_s      meany      coeff
## 1  s_01  0.7998263  0.8503282
## 2  s_02 -1.3815640 -1.3310621
## 3  s_03 -0.7928449 -0.7423430
## 4  s_04 -0.8245088 -0.7740069
## 5  s_05  0.7547054  0.8052073
## 6  s_06  0.1564710  0.2069728
## 7  s_07 -1.1747557 -1.1242539
## 8  s_08  1.3520153  1.4025171
## 9  s_09  1.5789785  1.6294804
## 10 s_10 -0.7313895 -0.6808876

In other words, whenever the value of x_s is s_01, the one variable model vs returns the value 0.8503282, and so on. If you do this for both variables, you get a training set that looks like this:

##     x_s  x_n           y         vs         vn
## 2  s_10 n_72  0.34228110 -0.6808876 0.64754957
## 3  s_01 n_09 -0.03805102  0.8503282 0.54991135
## 4  s_03 n_18 -0.92145960 -0.7423430 0.01923877
## 9  s_08 n_43  1.77069352  1.4025171 1.90394159
## 10 s_08 n_17  0.51992928  1.4025171 0.26448341
## 11 s_01 n_78  1.04714355  0.8503282 0.70342961

Now fit a linear model for y as a function of vs and vn.

model_raw = lm(y ~ vs + vn,
               data=dtrain_treated)
summary(model_raw)
## 
## Call:
## lm(formula = y ~ vs + vn, data = dtrain_treated)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.33068 -0.57106  0.00342  0.52488  2.25472 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.05050    0.05597  -0.902    0.368    
## vs           0.77259    0.05940  13.006   <2e-16 ***
## vn           0.61201    0.06906   8.862   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8761 on 242 degrees of freedom
## Multiple R-squared:  0.6382, Adjusted R-squared:  0.6352 
## F-statistic: 213.5 on 2 and 242 DF,  p-value: < 2.2e-16

Note that this model gives significant coefficients to both vs and vn, even though y is not a function of x_n (or vn). Because you used the same data to fit the one variable base learners and to fit the larger model, you have overfit.

The right way: cross-validation

The correct way to impact code (or to nest models in general) is to use cross-validation techniques. Impact coding with cross-validation is already implemented in vtreat; note the similarity between this diagram and Figure 1 above.

Figure 3 Cross-validated data preparation with vtreat
Figure 3 Cross-validated data preparation with vtreat

The training data is used both to fit the base learners (as we did above) and to also to create a data frame of cross-validated base learner predictions (called a cross-frame in vtreat). This cross-frame is used to train the overall model. Let’s fit the correct nested model, using vtreat.

library(vtreat)
library(wrapr)
xframeResults = mkCrossFrameNExperiment(dtrain, 
                                  qc(x_s, x_n), "y",
                                  codeRestriction = qc(catN), 
                                  verbose = FALSE)
# the plan uses the one-variable models to treat data
treatmentPlan = xframeResults$treatments
# the cross-frame
dtrain_treated = xframeResults$crossFrame

head(dtrain_treated)
##     x_s_catN   x_n_catN           y
## 1 -0.6337889 0.91241547  0.34228110
## 2  0.8342227 0.82874089 -0.03805102
## 3 -0.7020597 0.18198634 -0.92145960
## 4  1.3983175 1.99197404  1.77069352
## 5  1.3983175 0.11679580  0.51992928
## 6  0.8342227 0.06421659  1.04714355
variables = setdiff(colnames(dtrain_treated), "y")

model_X = lm(mk_formula("y", variables), 
             data=dtrain_treated)
summary(model_X)
## 
## Call:
## lm(formula = mk_formula("y", variables), data = dtrain_treated)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2157 -0.7343  0.0225  0.7483  2.9639 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.04169    0.06745  -0.618    0.537    
## x_s_catN     0.92968    0.06344  14.656   <2e-16 ***
## x_n_catN     0.10204    0.06654   1.533    0.126    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.055 on 242 degrees of freedom
## Multiple R-squared:  0.4753, Adjusted R-squared:  0.471 
## F-statistic: 109.6 on 2 and 242 DF,  p-value: < 2.2e-16

This model correctly determines that x_n (and its one-variable model x_n_catN) do not affect the outcome. We can compare the performance of this model to the naive model on holdout data.

rmse rsquared
ypred_naive 1.303778 0.2311538
ypred_crossval 1.093955 0.4587089

The correct model has a much smaller root-mean-squared error and a much larger R-squared than the naive model when applied to new data.

An attempted alternative: regularized models.

But cross-validation is so complicated. Can’t we just regularize? As we’ll show in the appendix of this article, for a one-variable model, L2-regularization is simply Laplace smoothing. Again, we’ll represent each “coefficient” of the one-variable model as the Laplace smoothed value minus the grand mean.

vi = ∑xj = si yi/(counti + λ) − E[yi]

Where counti is the frequency of si in the training data, and λ is the smoothing parameter (usually 1). If λ = 1 then the first term on the right is just adding one to the frequency of the level and then taking the “adjusted conditional mean” of y.

Again, let’s show this for the variable x_s.

##     x_s      sum_y count_y   grandmean         vs
## 1  s_01  20.795484      26 -0.05050187  0.8207050
## 2  s_02 -37.302227      27 -0.05050187 -1.2817205
## 3  s_03 -22.199656      28 -0.05050187 -0.7150035
## 4  s_04 -14.016649      17 -0.05050187 -0.7282009
## 5  s_05  19.622340      26 -0.05050187  0.7772552
## 6  s_06   3.129419      20 -0.05050187  0.1995218
## 7  s_07 -35.242672      30 -0.05050187 -1.0863585
## 8  s_08  36.504412      27 -0.05050187  1.3542309
## 9  s_09  33.158549      21 -0.05050187  1.5577086
## 10 s_10 -16.821957      23 -0.05050187 -0.6504130

After applying the one variable models for x_s and x_n to the data, the head of the resulting treated data looks like this:

##     x_s  x_n           y         vs         vn
## 2  s_10 n_72  0.34228110 -0.6504130 0.44853367
## 3  s_01 n_09 -0.03805102  0.8207050 0.42505898
## 4  s_03 n_18 -0.92145960 -0.7150035 0.02370493
## 9  s_08 n_43  1.77069352  1.3542309 1.28612835
## 10 s_08 n_17  0.51992928  1.3542309 0.21098803
## 11 s_01 n_78  1.04714355  0.8207050 0.61015422

Now fit the overall model:

## 
## Call:
## lm(formula = y ~ vs + vn, data = dtrain_treated)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.30354 -0.57688 -0.02224  0.56799  2.25723 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.06665    0.05637  -1.182    0.238    
## vs           0.81142    0.06203  13.082  < 2e-16 ***
## vn           0.85393    0.09905   8.621  8.8e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8819 on 242 degrees of freedom
## Multiple R-squared:  0.6334, Adjusted R-squared:  0.6304 
## F-statistic: 209.1 on 2 and 242 DF,  p-value: < 2.2e-16

Again, both variables look significant. Even with regularization, the model is still overfit. Comparing the performance of the models on holdout data, you see that the regularized model does a little better than the naive model, but not as well as the correctly cross-validated model.

rmse rsquared
ypred_naive 1.303778 0.2311538
ypred_crossval 1.093955 0.4587089
ypred_reg 1.267648 0.2731756

The Moral of the Story

Unfortunately, regularization is not enough to overcome nested model bias. Whenever you apply a y-aware process to your data, you have to use cross-validation methods (or a separate data set) at the next stage of your modeling pipeline.

Appendix: Derivation of Laplace Smoothing as L2-Regularization

Without regularization, the optimal one-variable model for y in terms of a categorical variable with K levels {sj} is a set of K coefficients v such that

f(\mathbf{v}) := \sum\limits_{i=1}^N (y_i - v_i)^2

is minimized (N is the number of data points). L2-regularization adds a penalty to the magnitude of v, so that the goal is to minimize

f(\mathbf{v}) := \sum\limits_{i=1}^N (y_i - v_i)^2 + \lambda \sum\limits_{j=1}^K {v_j}^2

where λ is a known smoothing hyperparameter, usually set (in this case) to 1.

To minimize the above expression for a single coefficient vj, take the deriviative with respect to vj and set it to zero:

\sum\nolimits_{x_i = s_j} -2 (y_i - v_j) + 2 \lambda v_j = 0\\ \sum\nolimits_{x_i = s_j }-y_i + \sum\nolimits_{x_i = s_j} v_j + \lambda v_j = 0\\ \sum\nolimits_{x_i = s_j }-y_i + \text{count}_j v_j + \lambda v_j = 0

Where countj is the number of times the level sj appears in the training data. Now solve for vj:

v_j (\text{count}_j + \lambda) = \sum\nolimits_{x_i = s_j} y_i\\ v_j = \sum\nolimits_{x_i = s_i} y_i / (\text{count}_j + \lambda)

This is Laplace smoothing. Note that it is also the one-variable equivalent of ridge regression.

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New vtreat Documentation (Starting with Multinomial Classification)

Nina Zumel finished some great new documentation showing how to use Python vtreat to prepare data for multinomial classification mode. And I have finally finished porting the documentation to R vtreat. So we now have good introductions on how to use vtreat to prepare data for the common tasks of:

That is now 8 introductions to start with. To use vtreat you only have to work through one introduction (the one helping with the task you have at hand in the language you are using).

As I have said before:

  • vtreat helps with project blocking issues commonly seen in real world data: missing values, re-coding categorical variables, and dealing high cardinality categorical variables.
  • If you aren’t using a tool like vtreat in your data science projects: you are really missing out (and making more work for yourself).
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Preparing Data for Supervised Classification

Nina Zumel has been polishing up new vtreat for Python documentation and tutorials. They are coming out so good that I find to be fair to the R community I must start to back-port this new documentation to vtreat for R.

Continue reading Preparing Data for Supervised Classification

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New Getting Started with vtreat Documentation

Win Vector LLC‘s Dr. Nina Zumel has just released some new vtreat documentation.

vtreat is a an all-in one step data preparation system that helps defend your machine learning algorithms from:

  • Missing values
  • Large cardinality categorical variables
  • Novel levels from categorical variables

I hoped she could get the Python vtreat documentation up to parity with the R vtreat documentation. But I think she really hit the ball out of the park, and went way past that.

The new documentation is 3 “getting started” guides. These guides deliberately overlap, so you don’t have to read them all. Just read the one suited to your problem and go.

The new guides:

Perhaps we can back-port the new guides to the R version at some point.

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A Kind Note That We Really Appreciate

The following really made my day.

I tell every data scientist I know about vtreat and urge them to read the paper.
Jason Wolosonovich

Jason, thanks for your support and thank you so much for taking the time to say this (and for your permission to quote you on this).

For those interested the R version of vtreat can be found here, the paper can be found here, and the in-development Python/Pandas version of vtreat can be found (with examples) here.

Chapter of 8 Zumel, Mount, Practical Data Science with R, 2nd Edition, Manning 2019 has a more operational discussion of vtreat (which itself uses concepts developed in chapter 4).

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Big News: Porting vtreat to Python

We at Win-Vector LLC have some big news.

We are finally porting a streamlined version of our R vtreat variable preparation package to Python.

vtreat is a great system for preparing messy data for supervised machine learning.

The new implementation is based on Pandas, and we are experimenting with pushing the sklearn.pipeline.Pipeline APIs to their limit. In particular we have found the .fit_transform() pattern is a great way to express building up a cross-frame to avoid nested model bias (in this case .fit_transform() != .fit().transform()). There is a bit of difference in how object oriented APIs compose versus how functional APIs compose. We are making an effort to research how to make this an advantage, and not a liability.

The new repository is here. And we have a non-trivial worked classification example. Next up is multinomial classification. After that a few validation suites to prove the two vtreat systems work similarly. And then we have some exciting new capabilities.

The first application is going to be a shortening and streamlining of our current 4 day data science in Python course (while allowing more concrete examples!).

This also means data scientists who use both R and Python will have a few more tools that present similarly in each language.

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vtreat Variable Importance

vtreat‘s purpose is to produce pure numeric R data.frames that are ready for supervised predictive modeling (predicting a value from other values). By ready we mean: a purely numeric data frame with no missing values and a reasonable number of columns (missing-values re-encoded with indicators, and high-degree categorical re-encode by effects codes or impact codes).

In this note we will discuss a small aspect of the vtreat package: variable screening.

Continue reading vtreat Variable Importance

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Sharing Modeling Pipelines in R

Reusable modeling pipelines are a practical idea that gets re-developed many times in many contexts. wrapr supplies a particularly powerful pipeline notation, and a pipe-stage re-use system (notes here). We will demonstrate this with the vtreat data preparation system.

Continue reading Sharing Modeling Pipelines in R

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Modeling multi-category Outcomes With vtreat

vtreat is a powerful R package for preparing messy real-world data for machine learning. We have further extended the package with a number of features including rquery/rqdatatable integration (allowing vtreat application at scale on Apache Spark or data.table!).

In addition vtreat and can now effectively prepare data for multi-class classification or multinomial modeling.

Continue reading Modeling multi-category Outcomes With vtreat