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Does the expected cross validated error/NLL from regularisation have a one minimum?

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Does the expected cross validated error/NLL from regularisation have a one minimum?

When trying to choose the value of the regularisation parameter(s) in lasso, ridge regression, or elasticnet, one generally computes the cross validated error or negative log-likelihood as explained by this answer. Further down the page, another answer shows an example of a plot of the cross validated error against the regularisation parameter $\lambda$, reproduced here:

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My question is whether the expected value of this error (or NLL as relevant), in any regression setting, has one and only one minimum over the possible values of the regularisation parameters?