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Machine Learning FAQ

Index Let’s assume we mean k-fold cross-validation used for hyperparameter tuning of algorithms for classification, and with “better,” we mean better at estimating the generalization performance. In

And far as computational efficiency is concerned – for example, think of training deep neural nets on large(r) datasets including hyperparameter tuning – I would think carefully about the size of k.

If our dataset is large, I’d therefore recommend choosing smaller values for k, but it is all a balancing act between bias and variance and computational efficiency, and for our final estimate, we still have our independent test set anyway.

Variance and bias in cross-validation: why does leave-one-out CV have higher variance?

The source of confusion though is that when people talk about LOOCV leading to high variability, they aren't talking about the predictions made by the many models built during that loop of cross-validation on the holdout sets.

Imagine that instead of using LOOCV to pick a model, you just had one training set and then you tested a model built using that training data, say, 100 times on 100 single test data points (data points are not part of the training set).

Unfortunately, some part of those associations between the training and test data sets will be noise or spurious associations because, although the test set changes and you can identify noise on this side, the training dataset doesn't and you can't determine what explained variance is due to noise.

In other words, if that particular training set has some spurious correlation with those test points, you're model will have difficulties determining which correlations are real and which are spurious, because even though the test set changes, the training set doesn't.

Machine Learning FAQ

Index Let’s assume we mean k-fold cross-validation used for hyperparameter tuning of algorithms for classification, and with “better,” we mean better at estimating the generalization performance. In

And far as computational efficiency is concerned – for example, think of training deep neural nets on large(r) datasets including hyperparameter tuning – I would think carefully about the size of k.

If our dataset is large, I’d therefore recommend choosing smaller values for k, but it is all a balancing act between bias and variance and computational efficiency, and for our final estimate, we still have our independent test set anyway.

Model evaluation, model selection, and algorithm selection in machine learning

Almost every machine learning algorithm comes with a large number of settings that we, the machine learning researchers and practitioners, need to specify.

These tuning knobs, the so-called hyperparameters, help us control the behavior of machine learning algorithms when optimizing for performance, finding the right balance between bias and variance.

After the machine learning algorithm fit a model to the training set, we evaluated it on the independent test set that we withheld from the machine learning algorithm during model fitting.

In this context, lazy learning (or instance-based learning) means that there is no training or model fitting stage: A k-nearest neighbors model literally stores or memorizes the training data and uses it only at prediction time.

nonparamtric models do not assume that the data follows certain probability distributions unlike parametric methods (exceptions of nonparametric methods that make such assumptions are Bayesian nonparametric methods).

In contrast to k-nearest neighbors, a simple example of a parametric method would be logistic regression, a generalized linear model with a fixed number of model parameters: a weight coefficient for each feature variable in the dataset plus a bias (or intercept) unit.

However, reusing the test set multiple times would introduce a bias and our final performance estimate and likely result in overly optimistic estimates of the generalization performance — we can say that “the test set leaks information.” To avoid this problem, we could use a three-way split, dividing the dataset into a training, validation, and test dataset.

“There ain’t no such thing as a free lunch.” The three-way holdout method for hyperparameter tuning and model selection is not the only — and certainly often not the best — way to approach this task.

However, before we move on to the probably most popular method for model selection, k-fold cross-validation (or sometimes also called “rotation estimation” in older literature), let us have a look at an illustration of the 3-way split holdout method:

We start by splitting our dataset into three parts, a training set for model fitting, a validation set for model selection, and a test set for the final evaluation of the selected model.

(If we use test data for fitting, we do not have data left to evaluate the model, unless we collect new data.) In real-world applications, having the “best possible” model is often desired – or in other words, we do not mind if we slightly underestimated its performance.

In each round, we split the dataset into k parts: one part is used for validation, and the remaining k-1 parts are merged into a training subset for model evaluation as shown in the figure below, which illustrates the process of 5-fold cross-validation:

Just as in the “two-way” holdout method, we use a learning algorithm with fixed hyperparameter settings to fit models to the training folds in each iteration — if we use the k-fold cross-validation method for model evaluation.

The idea behind this approach is to reduce the pessimistic bias by using more training data in contrast to setting aside a relatively large portion of the dataset as test data.

If k is too small, though, we may increase the pessimistic bias of our estimate (since less training data is available for model fitting), and the variance of our estimate may increase as well since the model is more sensitive to how we split the data (later, we will discuss experiments that suggest k=10 as a good choice for k).

However, this statement would only be true if we perform the holdout method by rotating the training and validation set in two rounds (i.e., using exactly 50% data for training and 50% of the samples for validation in each round, swapping these sets, repeating the training and evaluation procedure, and eventually computing the performance estimate as the arithmetic mean of the two performance estimates on the validation sets).

Unfortunately, there is no Free Lunch though as shown by Yohsua Bengio and Yves Grandvalet in “No unbiased estimator of the variance of k-fold cross-validation.” The main theorem shows that there exists no universal (valid under all distributions) unbiased estimator of the variance of K-fold cross-validation.

For now, let’s conclude this section by looking at an interesting research project where Hawkins and others compared performance estimates via LOOCV to the holdout method and recommend the LOOCV over the latter — if computationally feasible.

The following table summarizes the finding in a comparison of different Ridge Regression models: In rows 1-4, Hawkins and others used 100-sample training sets to compare different methods of model evaluation.

The reported “mean” refers to the averaged difference between the true coefficiants of determination and the coefficients of determination obtained via LOOCV (here called q2) after repeating this procedure on different 100-sample training sets.

In rows 2-4, the researchers used the holdout method for fitting models to 100-sample training sets, and they evaluated the performances on holdout sets of sizes 10, 20, and 50 samples.

For instance, if we repeated a 5-fold cross-validation run 100 times, we would compute the performance estimate for 500 test folds report the cross-validation performance as the arithmetic mean of these 500 folds.

In addition, we can think of the LOOCV estimate as being approximately unbiased: the pessimistic bias of LOOCV (k=n) is intuitively lower compared k<n-fold cross-validation, since almost all (for instance, n-1) training samples are available for model fitting.

Remember that if we use the 0-1 loss function (the prediction is either correct or not), we could consider each prediction as a Bernoulli trial, and the number of correct predictions X is following a binomial distribution X \sim B(n, p), where n \in \mathbb{N} \text{ and } p \in [0,1];

and others, 2009) Or in other words, we can attribute the high variance to the well-known fact that the mean of highly correlated variables has a higher variance than the mean of variables that are not highly correlated.

Maybe, this can intuitively be explained by looking at the relationship between covariance (\text{cov}) and variance (\sigma^2): Proof: \text{Let } \mu = E(X), { then } \quad \text{cov}_{X, X} = E\left[(X - \mu)^2\right] = \sigma^{2}_{X} And the relationship between covariance \text{cov}_{X, Y} and correlation \rho_{X, Y} (X and Y are random variables) is defined as where and The large variance that is often associated with LOOCV has also been observed in empirical studies — for example, I really recommend reading the excellent paper A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection (Kohavi, 1995) by Ron Kohavi.

Before moving on to model selection, let’s summarize this discussion of the bias-variance trade-off by listing the general trends when increasing the number of folds or k: Previously, we used k-fold cross-validation for model evaluation.

Aside from computational efficiency concerns, we only use deep learning algorithms when we have relatively large sample sizes anyway, scenarios where we don’t have to worry about high variance — due to sensitivity of our estimates towards how we split the dataset for training, validation, and testing — so much.

Or to say it with other words, using one of my favorite quotes: “Everything should be made as simple as possible, but not simpler.” — Albert Einstein In model selection practice, we can apply Occam’s razor using the one-standard error method as follows: Although, we may prefer simpler models for several reasons, Pedro Domingos made a good point regarding the performance of “complex” models.

Some of the most powerful learning algorithms output models that seem gratuitously elaborate — sometimes even continuing to add to them after they’ve perfectly fit the data — but that’s how they beat the less powerful ones.

To see how the one-standard error method works in practice, let us apply it to a simple toy dataset: 300 datapoints, concentric circles, and a uniform class distribution (150 samples from class 1 and 150 samples from class 2).

Say we want to optimize the Gamma hyperparameter of a Support Vector Machine (SVM) with a non-linear Radial Basis Function-kernel (RBF-kernel), where \gamma is the free parameter of the Gaussian RBF: (Intuitively, we can think of the Gamma as a parameter that controls the influence of single training samples on the decision boundary.) When I ran the RBF-kernel SVM algorithm with different Gamma values over the training set, using stratified 10-fold cross-validation, I obtained the following performance estimates, where the error bars are the standard errors of the cross-validation estimates:

In fact, \gamma=0.1 seems like a good trade-off between the two aforementioned models — the performance of the corresponding model falls within one standard error of the best performing model with \gamma=0 or \gamma=10.

Assuming that no candidate has a clue about how stocks work, and everyone was guessing randomly, the probability that at least one of the candidates got 8 out of 10 predictions correct is: So, shall we assume that a candidate who got 8 out of 10 predictions correct was not simply guessing randomly?

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