Elastic net regularization: mean square error monotonically increases with lambda

This is quite coincidental as my question is nearly identical to this one asked shortly before, but I am also using elastic net regularization with R's glmnet library as a method of variable selection (but in my case it is for a Gaussian and not binomial family).

I have an instance where there does not appear to be any value of lambda which reduces the deviance of the fit. In the left figure below, the $\lambda$ values are selected automatically by cv.glmnet(); in the right figure below I have specified the values to be $\exp(\{-11\ldots-1\})$ (using 250 points evenly spaced between -11 and -1), but in either case it appears there the deviance or mean square error (MSE) is monotonically increasing with $\lambda$.

Does this say something useful about my system, and is there any value of $\lambda$ for which a meaningful interpretation of the selected variables can be extracted?

I would additionally appreciate reference to peer-reviewed literature or conference proceedings which discuss similar cases - I have come up empty-handed in my search.

Many thanks, community.

(For non-R users, the left dotted vertical line in each figure indicates the minimum MSE value - which in my problem corresponds to the minimum value of $\lambda$ selected in both cases - and the right dotted vertical line corresponds to the $\lambda$ of the minimum MSE + 1 standard error solution.)

• The cross-validated MSE seems exactly zero, suggesting zero irreducible error, which is unlikely in real-world data problems. Also, are the MSE values for log(Lambda) from -11 to -5 precisely zero or just very close? Can you perhaps share code and/or data? Feb 26, 2021 at 1:47

The interpretation is exactly the same as I discussed in the Q & A you link to. All your right-hand plot has done is extend the penalty ($\log(\lambda)$) into a region of exponentially decreasing values, i.e. exponentially decreasing amounts of shrinkage.
The CV deviance of the model is flat over a range of values for $\log(\lambda)$ yet there is shrinkage being applied; sufficient shrinkage to remove some variables from the model completely. I'm not sure you can trust the values on the upper axis for your automatic plot - do you really have 1636 covariates in the model? - but for the left-hand plot the simplest/smaller model within 1 SE of the best model has somewhere between 100 and 87 covariates in it.
• @crippledlambda then, in that case, you are seeing huge shrinkage for effectively no impact on the model's ability to explain variance in your sample. That is good! I think at low values of $\lambda$ you are overfitting so go with the 1SE mode fit. Oct 29, 2014 at 3:48
• @crippledlambda your setting is sparse; you have remove ~ 1800 of the predictors from the model by shrinking their coefficients to zero. The CV is a stochastic procedure; samples are assigned to folds at random hence one would expect some variation between runs unless you set the random seed the same each time. If you want to hone in the best 1SE value of $\lambda$ you'll need to use a lot more points over range say -6 to -3. but really it isn't going to matter much in terms of prediction. Oct 29, 2014 at 4:11
• Alternatively, run the CV quite a few times and record the $\lambda$ for the 1SE solution from each of these. Use a representative value for your chosen $\lambda$. Oct 29, 2014 at 4:13