Which one of the following linear discriminants is most prone to overfitting a training data set?

Question

Which one of the following linear discriminants is most prone to overfitting a training data set?

BlackTom AI · Accepted Answer

We start by restating the setup: the question asks which linear discriminant is most prone to overfitting a training data set.
Option 1 proposes f_x = w0 + w1 x1 + w2 x2 + w3 x3, a relatively small, purely linear model with four terms.
Option 2 proposes f_x = w0 + w1 x1 + w2 x2 + w3 x3 + w4 x4 + w5 x5 + w6 x6, a linear model with seven terms, still purely linear but with more features than Option 1.
Option 3 proposes f_x = w0 + w1 x1 + w2 x2 + w3 x3 + w4 x4 + w5 x5 + w6 x6 + w7 * x2/x3, which adds a nonlinear interaction term x2/x3 on top of the same linear terms, dramatically increasing model capacity.
Option 4 proposes f_x = w0 + w1 x1 + w2 x2 + w3 x3 + w4 x4 + w5 x5, a linear model with six terms, between the sizes of Options 1 and 2 without nonlinearities.

Now, evaluating each option: 
- Option 1 is quite compact; with only four features, it has limited capacity and is less flexible, so it is less likely to overfit compared to models with more parameters, all else equal.
- Option 2 expands the linear model to seven terms, which raises the risk of overfitting relative to Option 1, but since it remains linear, its overfitting tendency is constrained by the data-to-parameter ratio.
- Option 4 is also linear and includes six terms. It offers more complexity than Option 1 but still lacks nonlinear shaping; its overfitting risk is present but not extreme, assuming a reasonable dataset size.
- Option 3 stands out because it combines the linear terms with a nonlinear feature x2/x3. This nonlinear interaction increases the hypothesis space significantly and allows the model to fit complex patterns, including noise, which substantially heightens the likelihood of overfitting when training data are limited or noisy.

In machine learning, overfitting is strongly associated with higher model capacity relative to the amount and noisiness of training data. Nonlinear features like x2/x3 provide the model with more flexibility to memorize training examples rather than generalize, making Option 3 the most prone to overfitting among the given choices.

Therefore, the reasoning points to Option 3 as the most overfitting-prone discriminant, because it adds a nonlinear interaction term that expands capacity beyond purely linear models, increasing the risk of capturing noise in the training data. The other options, while varying in the number of linear terms, do not introduce such nonlinear capacity, so their propensity to overfit is comparatively lower.

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Which one of the following linear discriminants is most prone to overfitting a training data set?

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