As the budget parameter a decreases in lasso regression, what happens to the flexibility?

In lasso regression, the budget parameter 'a' controls the strength of the penalty applied to the coefficients of the model. When 'a' is large, the penalty is strong, which forces many coefficients to be exactly zero and results in a simpler, less flexible model. As 'a' decreases, the penalty becomes weaker, allowing more coefficients to take on non-zero values.

This increase in the number of non-zero coefficients leads to a more complex model that is capable of capturing more patterns in the data. Therefore, as the budget parameter 'a' decreases, the flexibility of the model increases. However, it is important to note that excessive flexibility can lead to overfitting, where the model performs well on training data but poorly on unseen data.

So, while the flexibility of the model indeed increases as 'a' decreases, in the context of the options provided, the correct interpretation aligns with the adjustment of the penalty leading to a more complex model structure. Thus, a decrease in the budget parameter 'a' does not correspond to a decrease in flexibility; rather, it allows for a more flexible model that can better fit the data.