# Mixed Penalties

> This is an experimental feature. There may be bugs; use carefully!

    library(lessSEM)

The `mixedPenalty` function allows you to add multiple penalties to a single model. For instance, you may want to regularize both loadings and regressions in a SEM. In this case, using the same penalty (e.g., lasso) for both types of penalties may actually not be what you want to use because the penalty function is sensitive to the scales of the parameters. Instead, you may want to use two separate lasso penalties for loadings and regressions. Similarly, separate penalties for different parameters have, for instance, been proposed in multi-group models (Geminiani et al., 2021).

> Important: You cannot impose two penalties on the same parameter!

Models are fitted with the glmnet or ista optimizer. Note that the optimizers differ in which penalties they support. The following table provides an overview:

Penalty | Function | glmnet | ista
--- | --- | --- | ---
lasso | addLasso | x | x
elastic net | addElasticNet | x* | -
cappedL1 | addCappedL1 | x | x
lsp | addLsp | x | x
scad | addScad | x | x
mcp | addMcp | x | x

By default, glmnet will be used. Note that the elastic net penalty can only be combined with other elastic net penalties.

## Getting Started

In the following model, we will allow for cross-loadings (`c2`-`c4`). We want to regularize both, cross-loadings and regression coefficients (`r1` - `r3`)

    model <- '
      # latent variable definitions
         ind60 =~ x1 + x2 + x3 + c2*y2 + c3*y3 + c4*y4
         dem60 =~ y1 + y2 + y3 + y4
         dem65 =~ y5 + y6 + y7 + c*y8

      # regressions
        dem60 ~ r1*ind60
        dem65 ~ r2*ind60 + r3*dem60
    '

    lavaanModel <- sem(model,
                       data = PoliticalDemocracy)

Next, we add separate lasso penalties for the loadings and the regressions:

    mp <- lavaanModel |>
      mixedPenalty() |>
      addLasso(regularized = c("c2", "c3", "c4"),
               lambdas = seq(0,1,.1)) |>
      addLasso(regularized = c("r1", "r2", "r3"),
               lambdas = seq(0,1,.2))

Note that we can use the pipe-operator to add multiple penalties. They don’t have to be the same; the following would also work:

    mp <- lavaanModel |>
      mixedPenalty() |>
      addLasso(regularized = c("c2", "c3", "c4"),
               lambdas = seq(0,1,.1)) |>
      addScad(regularized = c("r1", "r2", "r3"),
              lambdas = seq(0,1,.2),
              thetas = 3.7)

To fit the model, we use the `fit`- function:

    fitMp <- fit(mp)

To check which parameter has be regularized with which penalty, we can look at the `penalty` statement in the resulting object:

    fitMp@penalty
    #>    ind60=~x2    ind60=~x3           c2           c3           c4    dem60=~y2    dem60=~y3    dem60=~y4    dem65=~y6
    #>       "none"       "none"      "lasso"      "lasso"      "lasso"       "none"       "none"       "none"       "none"
    #>    dem65=~y7            c           r1           r2           r3       x1~~x1       x2~~x2       x3~~x3       y2~~y2
    #>       "none"       "none"       "scad"       "scad"       "scad"       "none"       "none"       "none"       "none"
    #>       y3~~y3       y4~~y4       y1~~y1       y5~~y5       y6~~y6       y7~~y7       y8~~y8 ind60~~ind60 dem60~~dem60
    #>       "none"       "none"       "none"       "none"       "none"       "none"       "none"       "none"       "none"
    #> dem65~~dem65
    #>       "none"

We can access the best parameters according to the BIC with:

    coef(fitMp, criterion = "BIC")
    #>
    #>                        Tuning ||--||  Estimates
    #>  ---------------------------- ||--|| ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
    #>  tuningParameterConfiguration ||--||  ind60=~x2  ind60=~x3         c2         c3         c4  dem60=~y2  dem60=~y3  dem60=~y4
    #>  ============================ ||--|| ========== ========== ========== ========== ========== ========== ========== ==========
    #>                       11.0000 ||--||     2.1817     1.8188          .          .          .     1.3540     1.0440     1.2995
    #>
    #>
    #>  ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
    #>   dem65=~y6  dem65=~y7          c         r1         r2         r3     x1~~x1     x2~~x2     x3~~x3     y2~~y2     y3~~y3
    #>  ========== ========== ========== ========== ========== ========== ========== ========== ========== ========== ==========
    #>      1.2585     1.2825     1.3098     1.4738     0.4533     0.8644     0.0818     0.1184     0.4673     6.4896     5.3399
    #>
    #>
    #>  ---------- ---------- ---------- ---------- ---------- ---------- ------------ ------------ ------------
    #>      y4~~y4     y1~~y1     y5~~y5     y6~~y6     y7~~y7     y8~~y8 ind60~~ind60 dem60~~dem60 dem65~~dem65
    #>  ========== ========== ========== ========== ========== ========== ============ ============ ============
    #>      2.8871     1.9419     2.3901     4.3428     3.5096     2.9403       0.4482       3.8717       0.1149

The `tuningParameterConfiguration` refers to the rows in the lambda, theta, and alpha matrices that resulted in the best fit:

    getTuningParameterConfiguration(regularizedSEMMixedPenalty = fitMp,
                                    tuningParameterConfiguration = 11)
    #>                 parameter penalty lambda alpha
    #> ind60=~x2       ind60=~x2    none      0     0
    #> ind60=~x3       ind60=~x3    none      0     0
    #> c2                     c2   lasso      1     1
    #> c3                     c3   lasso      1     1
    #> c4                     c4   lasso      1     1
    #> dem60=~y2       dem60=~y2    none      0     0
    #> dem60=~y3       dem60=~y3    none      0     0
    #> dem60=~y4       dem60=~y4    none      0     0
    #> dem65=~y6       dem65=~y6    none      0     0
    #> dem65=~y7       dem65=~y7    none      0     0
    #> c                       c    none      0     0
    #> r1                     r1    scad      0     0
    #> r2                     r2    scad      0     0
    #> r3                     r3    scad      0     0
    #> x1~~x1             x1~~x1    none      0     0
    #> x2~~x2             x2~~x2    none      0     0
    #> x3~~x3             x3~~x3    none      0     0
    #> y2~~y2             y2~~y2    none      0     0
    #> y3~~y3             y3~~y3    none      0     0
    #> y4~~y4             y4~~y4    none      0     0
    #> y1~~y1             y1~~y1    none      0     0
    #> y5~~y5             y5~~y5    none      0     0
    #> y6~~y6             y6~~y6    none      0     0
    #> y7~~y7             y7~~y7    none      0     0
    #> y8~~y8             y8~~y8    none      0     0
    #> ind60~~ind60 ind60~~ind60    none      0     0
    #> dem60~~dem60 dem60~~dem60    none      0     0
    #> dem65~~dem65 dem65~~dem65    none      0     0

In this case, the best model has no cross-loadings, but the regressions remained unregularized: The lambda for the cross-loadings is large (1), while the lambda for the regressions is 0 (no regularization).

## Using ista

The glmnet optimizer is typically considerably faster than ista. However, sometimes glmnet may run into issues. In that case, it can help to switch to ista:

    mp <- lavaanModel |>
      # Change the optimizer and the control object:
      mixedPenalty(method = "ista",
                   control = controlIsta()) |>
      addLasso(regularized = c("c2", "c3", "c4"),
                    lambdas = seq(0,1,.1)) |>
      addLasso(regularized = c("r1", "r2", "r3"),
                    lambdas = seq(0,1,.2))

To fit the model, we use the `fit`- function:

    fitMp <- fit(mp)

    coef(fitMp, criterion = "BIC")
    #>
    #>                        Tuning ||--||  Estimates
    #>  ---------------------------- ||--|| ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
    #>  tuningParameterConfiguration ||--||  ind60=~x2  ind60=~x3         c2         c3         c4  dem60=~y2  dem60=~y3  dem60=~y4
    #>  ============================ ||--|| ========== ========== ========== ========== ========== ========== ========== ==========
    #>                       11.0000 ||--||     2.1818     1.8188          .          .          .     1.3541     1.0441     1.2997
    #>
    #>
    #>  ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
    #>   dem65=~y6  dem65=~y7          c         r1         r2         r3     x1~~x1     x2~~x2     x3~~x3     y2~~y2     y3~~y3
    #>  ========== ========== ========== ========== ========== ========== ========== ========== ========== ========== ==========
    #>      1.2586     1.2825     1.3099     1.4740     0.4532     0.8643     0.0818     0.1184     0.4672     6.4892     5.3396
    #>
    #>
    #>  ---------- ---------- ---------- ---------- ---------- ---------- ------------ ------------ ------------
    #>      y4~~y4     y1~~y1     y5~~y5     y6~~y6     y7~~y7     y8~~y8 ind60~~ind60 dem60~~dem60 dem65~~dem65
    #>  ========== ========== ========== ========== ========== ========== ============ ============ ============
    #>      2.8861     1.9420     2.3904     4.3421     3.5090     2.9392       0.4482       3.8710       0.1159

The `tuningParameterConfiguration` refers to the rows in the lambda, theta, and alpha matrices that resulted in the best fit:

    getTuningParameterConfiguration(regularizedSEMMixedPenalty = fitMp,
                                    tuningParameterConfiguration = 11)
    #>                 parameter penalty lambda theta alpha
    #> ind60=~x2       ind60=~x2    none      0     0     0
    #> ind60=~x3       ind60=~x3    none      0     0     0
    #> c2                     c2   lasso      1     0     1
    #> c3                     c3   lasso      1     0     1
    #> c4                     c4   lasso      1     0     1
    #> dem60=~y2       dem60=~y2    none      0     0     0
    #> dem60=~y3       dem60=~y3    none      0     0     0
    #> dem60=~y4       dem60=~y4    none      0     0     0
    #> dem65=~y6       dem65=~y6    none      0     0     0
    #> dem65=~y7       dem65=~y7    none      0     0     0
    #> c                       c    none      0     0     0
    #> r1                     r1   lasso      0     0     1
    #> r2                     r2   lasso      0     0     1
    #> r3                     r3   lasso      0     0     1
    #> x1~~x1             x1~~x1    none      0     0     0
    #> x2~~x2             x2~~x2    none      0     0     0
    #> x3~~x3             x3~~x3    none      0     0     0
    #> y2~~y2             y2~~y2    none      0     0     0
    #> y3~~y3             y3~~y3    none      0     0     0
    #> y4~~y4             y4~~y4    none      0     0     0
    #> y1~~y1             y1~~y1    none      0     0     0
    #> y5~~y5             y5~~y5    none      0     0     0
    #> y6~~y6             y6~~y6    none      0     0     0
    #> y7~~y7             y7~~y7    none      0     0     0
    #> y8~~y8             y8~~y8    none      0     0     0
    #> ind60~~ind60 ind60~~ind60    none      0     0     0
    #> dem60~~dem60 dem60~~dem60    none      0     0     0
    #> dem65~~dem65 dem65~~dem65    none      0     0     0

Here is a short run-time comparison of ista and glmnet with the lasso-regularized model from above: Five repetitions using ista took 43.788 seconds, while glmnet took 1.733 seconds. That is, if you can use glmnet with your model, we recommend that you do.

## Bibliography

- Geminiani, E., Marra, G., & Moustaki, I. (2021). Single- and multiple-group penalized factor analysis: A trust-region algorithm approach with integrated automatic multiple tuning parameter selection. Psychometrika, 86(1), 65–95. https://doi.org/10.1007/s11336-021-09751-8