Understanding Alpha and Lambda in glmnet

What Alpha and Lambda Control

In glmnet, alpha and lambda control different aspects of regularization. Alpha sets the type of penalty: it determines the balance between the lasso’s absolute-value penalty and the ridge penalty. Lambda sets the overall strength of that penalty. A fitted coefficient therefore depends on both values, as well as on the data, the model family, and preprocessing such as predictor scaling.

For Gaussian regression, glmnet uses a penalized least-squares objective. In the expression below, \(y_i\) is response \(i\), \(x_{ij}\) is predictor \(j\) for observation \(i\), \(\beta_0\) is the intercept, and \(\beta_j\) is the coefficient for predictor \(j\). The intercept is not penalized. This objective uses glmnet’s Gaussian loss scaling; changing the loss scaling changes the numerical scale of lambda.

The lasso can set coefficients exactly to zero, while ridge generally shrinks coefficients without setting them exactly to zero. Elastic net combines these behaviors; for \(0<\alpha<1\), it includes both an absolute-value and a squared-coefficient penalty. Alpha is not the percentage of predictors retained, and lambda is not a significance threshold. Neither parameter alone tells us that a predictor is important.

Why the Coefficient Formula Has a Threshold

A simple Gaussian setting makes the roles of the parameters visible. Assume the response and predictors are centered, each predictor has mean square one, and distinct predictors are orthogonal, so their sample cross-products divided by \(n\) are zero. Define \(z_j=(1/n)\sum_i x_{ij}y_i\), the unpenalized least-squares coefficient for predictor \(j\) under these assumptions. With no intercept needed after centering, the objective separates into one term per coefficient.

For one predictor, after dropping terms that do not depend on \(\beta\), the part of the objective involving its coefficient is:

To see why a threshold occurs, consider a positive coefficient. Differentiating the objective for \(\beta>0\) gives \(\beta-z+\lambda\alpha+\lambda(1-\alpha)\beta=0\). Hence a positive solution is possible only when \(z>\lambda\alpha\), and then its value is \((z-\lambda\alpha)/(1+\lambda(1-\alpha))\). For a negative coefficient, the derivative of \(|\beta|\) is \(-1\), giving a negative solution only when \(z<-\lambda\alpha\). If \(-\lambda\alpha\leq z\leq\lambda\alpha\), the minimum is at zero: moving away from zero cannot overcome the penalty. Combining these cases gives the soft-threshold formula.

This formula is a teaching result for the stated orthogonal, standardized Gaussian setup, not a general closed-form solution for glmnet. With correlated predictors, the coefficients are coupled and glmnet solves the full penalized optimization problem. With other model families, the likelihood contribution differs. In practical fits, also check the response family, standardization, and whether reported coefficients have been returned on the original predictor scale.

Reading a Fitted glmnet Model

A glmnet fit is usually evaluated along a sequence of lambda values for a chosen alpha. A coefficient path records how the fitted coefficients change across that sequence. At a fixed lambda, the coefficients describe the model fitted with that alpha and lambda, subject to the specified data and preprocessing. They do not by themselves establish how useful a predictor is in other samples or under a different modeling choice.

In R, a basic Gaussian fit with a chosen mixture might be written as fit <- glmnet(x, y, family = "gaussian", alpha = 0.5). Coefficients at a supplied lambda can be requested with coef(fit, s = 0.2). The requested value should correspond to the fit’s lambda path or be handled according to the function’s interpolation behavior. Check the returned object and coefficient names rather than assuming every coefficient is nonzero or that the display order matches a preferred interpretation.

With the default predictor standardization, glmnet standardizes predictors for fitting and reports coefficients on the original predictor scale. If standardization is changed, or preprocessing is done separately, coefficient magnitudes are not directly comparable without accounting for the scales. For factors, the model matrix contains indicator columns; the printed coefficient for one indicator is not automatically a single overall measure of that factor’s importance.

What a Lambda Change Does—and Does Not—Mean

For a fixed alpha, increasing lambda strengthens the penalty in the objective. In the orthogonal formula, increasing lambda raises the lasso threshold \(\lambda\alpha\), and also changes the elastic net denominator when \(\alpha<1\). Coefficients can therefore shrink, and for a lasso-containing penalty some can become exactly zero. In a general correlated design, individual coefficient paths need not follow the simple scalar formula, so use the fitted path or coefficient table rather than extrapolating from the orthogonal case.

The numerical lambda values should not be compared as if they were universal. They depend on the response and predictor scales, loss definition, sample, and model family. A value such as 0.25 is not intrinsically “small” or “large.” A reported fit should identify alpha and the lambda used, and relevant preprocessing choices should be made clear.

A lambda selected for predictive performance is a modeling choice, not proof that retained variables are more important than omitted ones. Likewise, alpha specifies the penalty mixture; choosing an alpha does not assign relative scientific importance to predictors. Interpret coefficients in context, including their units, coding, conditional nature, and the uncertainty inherent in the model and data.