The only family of algorithms that I could think of being scale-invariant are tree-based methods. Let’s take the general CART decision tree algorithm. Without going into much depth regarding information gain and impurity measures, we can think of the decision as “is feature x_i >= some_val?” Intuitively, we can see that it really doesn’t matter on which scale this feature is (centimeters, Fahrenheit, a standardized scale – it really doesn’t matter).

Some examples of algorithms where feature scaling matters are:

  • k-nearest neighbors with an Euclidean distance measure if want all features to contribute equally
  • k-means (see k-nearest neighbors)
  • logistic regression, SVMs, perceptrons, neural networks etc. if you are using gradient descent/ascent-based optimization, otherwise some weights will update much faster than others
  • linear discriminant analysis, principal component analysis, kernel principal component analysis since you want to find directions of maximizing the variance (under the constraints that those directions/eigenvectors/principal components are orthogonal); you want to have features on the same scale since you’d emphasize variables on “larger measurement scales” more.

There are many more cases than I can possibly list here … I always recommend you to think about the algorithm and what it’s doing, and then it typically becomes obvious whether we want to scale your features or not.

In addition, we’d also want to think about whether we want to “standardize” or “normalize” (here: scaling to [0, 1] range) our data. Some algorithms assume that our data is centered at 0. For example, if we initialize the weights of a small multi-layer perceptron with tanh activation units to 0 or small random values centered around zero, we want to update the model weights “equally.” As a rule of thumb I’d say: When in doubt, just standardize the data, it shouldn’t hurt.



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