As you saw in the video, the glmnet model actually fits many models at once (one of the great things about the package). You can exploit this by passing a large number of lambda values, which control the amount of penalization in the model. train() is smart enough to only fit one model per alpha value and pass all of the lambda values at once for simultaneous fitting.

My favorite tuning grid for glmnet models is:

expand.grid(
  alpha = 0:1,
  lambda = seq(0.0001, 1, length = 100)
)

This grid explores a large number of lambda values (100, in fact), from a very small one to a very large one. (You could increase the maximum lambda to 10, but in this exercise 1 is a good upper bound.)

If you want to explore fewer models, you can use a shorter lambda sequence. For example, lambda = seq(0.0001, 1, length = 10) would fit 10 models per value of alpha.

You also look at the two forms of penalized models with this tuneGrid: ridge regression and lasso regression. alpha = 0 is pure ridge regression, and alpha = 1 is pure lasso regression. You can fit a mixture of the two models (i.e. an elastic net) using an alpha between 0 and 1. For example, alpha = 0.05 would be 95% ridge regression and 5% lasso regression.

In this problem you'll just explore the 2 extremes – pure ridge and pure lasso regression – for the purpose of illustrating their differences.

Advantage of glmnet

Machine Learning with caret in R

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