Regularization methods such as the Lasso combine shrinkage and variable selection, but strong penalization drives their solutions toward the null model. We introduce Fractional Ridge (Fridge) regression, a penalized regression method that instead shrinks toward models of a prespecified target size. Its penalty vanishes whenever the number of nonzero coefficients does not exceed that target. This behavior connects continuous shrinkage with best subset selection. We develop an efficient recursive penalty-evaluation algorithm, coordinate descent updates, and a predictor-corrector algorithm for tracing solution paths. Under suitable conditions, we establish asymptotic normality, the oracle property, and convergence of global minimizers to best subset solutions under extreme regularization. Simulation studies demonstrate competitive prediction and variable selection across a range of signal strengths and predictor correlations. An application to riboflavin production data illustrates how Fridge selects a small set of predictors whose leading coefficients agree with signals prioritized by a Bayesian method developed specifically for genetic fine-mapping.

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