`L0Learn`

is a fast toolkit for L0-regularized learning. L0 regularization selects the best subset of features and can outperform commonly used feature selection methods (e.g., L1 and MCP) under many sparse learning regimes. The toolkit can (approximately) solve the following three problems \[
\min_{\beta_0, \beta} \sum_{i=1}^{n} \ell(y_i, \beta_0+ \langle x_i, \beta \rangle) + \lambda ||\beta||_0 \quad \quad (L0)
\] \[
\min_{\beta_0, \beta} \sum_{i=1}^{n} \ell(y_i, \beta_0+ \langle x_i, \beta \rangle) + \lambda ||\beta||_0 + \gamma||\beta||_1 \quad (L0L1)
\]

\[ \min_{\beta_0, \beta} \sum_{i=1}^{n} \ell(y_i, \beta_0+ \langle x_i, \beta \rangle) + \lambda ||\beta||_0 + \gamma||\beta||_2^2 \quad (L0L2) \] where \(\ell\) is the loss function, \(\beta_0\) is the intercept, \(\beta\) is the vector of coefficients, and \(||\beta||_0\) denotes the L0 norm of \(\beta\), i.e., the number of non-zeros in \(\beta\). We support both regression and classification using either one of the following loss functions:

- Squared error loss
- Logistic loss (logistic regression)
- Squared hinge loss (smoothed version of SVM).

The parameter \(\lambda\) controls the strength of the L0 regularization (larger \(\lambda\) leads to less non-zeros). The parameter \(\gamma\) controls the strength of the shrinkage component (which is the L1 norm in case of L0L1 or squared L2 norm in case of L0L2); adding a shrinkage term to L0 can be very effective in avoiding overfitting and typically leads to better predictive models. The fitting is done over a grid of \(\lambda\) and \(\gamma\) values to generate a regularization path.

The algorithms provided in `L0Learn`

are based on cyclic coordinate descent and local combinatorial search. Many computational tricks and heuristics are used to speed up the algorithms and improve the solution quality. These heuristics include warm starts, active set convergence, correlation screening, greedy cycling order, and efficient methods for updating the residuals through exploiting sparsity and problem dimensions. Moreover, we employed a new computationally efficient method for dynamically selecting the regularization parameter \(\lambda\) in the path. For more details on the algorithms used, please refer to our paper Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms.

The toolkit is implemented in C++ along with an easy-to-use R interface. In this vignette, we provide a tutorial on using the R interface. Particularly, we will demonstrate how use L0Learnâ€™s main functions for fitting models, cross-validation, and visualization.

L0Learn can be installed directly from CRAN by executing:

If you face installation issues, please refer to the Installation Troubleshooting Wiki. If the issue is not resolved, you can submit an issue on L0Learnâ€™s Github Repo.

To demonstrate how `L0Learn`

works, we will first generate a synthetic dataset and then proceed to fitting L0-regularized models. The synthetic dataset (y,X) will be generated from a sparse linear model as follows:

- X is a 500x1000 design matrix with iid standard normal entries
- B is a 1000x1 vector with the first 10 entries set to 1 and the rest are zeros.
- e is a 500x1 vector with iid standard normal entries
- y is a 500x1 response vector such that y = XB + e

This dataset can be generated in R as follows:

```
set.seed(1) # fix the seed to get a reproducible result
X = matrix(rnorm(500*1000),nrow=500,ncol=1000)
B = c(rep(1,10),rep(0,990))
e = rnorm(500)
y = X%*%B + e
```

We will use L0Learn to estimate B from the data (y,X). First we load L0Learn:

We will start by fitting a simple L0 model and then proceed to the case of L0L2 and L0L1.

To fit a path of solutions for the L0-regularized model with at most 20 non-zeros using coordinate descent (CD), we use the `L0Learn.fit`

function as follows:

This will generate solutions for a sequence of \(\lambda\) values (chosen automatically by the algorithm). To view the sequence of \(\lambda\) along with the associated support sizes (i.e., the number of non-zeros), we use the `print`

method as follows:

```
#> lambda gamma suppSize
#> 1 0.068285500 0 0
#> 2 0.067602600 0 1
#> 3 0.055200200 0 2
#> 4 0.049032300 0 3
#> 5 0.040072500 0 6
#> 6 0.038602900 0 7
#> 7 0.037264900 0 8
#> 8 0.032514200 0 10
#> 9 0.001142920 0 11
#> 10 0.000821227 0 13
#> 11 0.000702292 0 14
#> 12 0.000669520 0 15
#> 13 0.000489938 0 17
```

To extract the estimated B for particular values of \(\lambda\) and \(\gamma\), we use the function `coef(fit,lambda,gamma)`

. For example, the solution at \(\lambda = 0.0325142\) (which corresponds to a support size of 10) can be extracted using

```
#> 1001 x 1 sparse Matrix of class "dgCMatrix"
#>
#> Intercept 0.01052346
#> V1 1.01601401
#> V2 1.01829953
#> V3 1.00606260
#> V4 0.98308999
#> V5 0.97389545
#> V6 0.96148014
#> V7 1.00990709
#> V8 1.08535033
#> V9 1.02685765
#> V10 0.94236638
#> V11 .
#> V12 .
...
```

The output is a sparse vector of type `dgCMatrix`

. The first element in the vector is the intercept and the rest are the B coefficients. Aside from the intercept, the only non-zeros in the above solution are coordinates 1, 2, 3, â€¦, 10, which are the non-zero coordinates in the true support (used to generated the data). Thus, this solution successfully recovers the true support. Note that on some BLAS implementations, the `lambda`

value we used above (i.e., `0.0325142`

) might be slightly different due to the limitations of numerical precision. Moreover, all the solutions in the regularization path can be extracted at once by calling `coef(fit)`

.

The sequence of \(\lambda\) generated by `L0Learn`

is stored in the object `fit`

. Specifically, `fit$lambda`

is a list, where each element of the list is a sequence of \(\lambda\) values corresponding to a single value of \(\gamma\). Since L0 has only one value of \(\gamma\) (i.e., 0), we can access the sequence of \(\lambda\) values using `fit$lambda[[1]]`

. Thus, \(\lambda=0.0325142\) we used previously can be accessed using `fit$lambda[[1]][7]`

(since it is the 7th value in the output of `print`

). So the previous solution can also be extracted using `coef(fit,lambda=fit$lambda[[1]][7], gamma=0)`

.

We can make predictions using a specific solution in the grid using the function `predict(fit,newx,lambda,gamma)`

where newx is a testing sample (vector or matrix). For example, to predict the response for the samples in the data matrix X using the solution with \(\lambda=0.0325142\), we call the prediction function as follows:

```
#> 500 x 1 Matrix of class "dgeMatrix"
#> [,1]
#> [1,] 0.44584683
#> [2,] -1.52213221
#> [3,] -1.11755544
#> [4,] -0.93180249
#> [5,] -4.02095821
#> [6,] 2.02300763
#> [7,] 2.03371819
#> [8,] 0.92234198
#> [9,] 2.33359737
#> [10,] 0.92194909
#> [11,] -4.33165265
#> [12,] -2.13282518
#> [13,] -7.21962511
...
```

We can also visualize the regularization path by plotting the coefficients of the estimated B versus the support size (i.e., the number of non-zeros) using the `plot(fit,gamma)`

method as follows: