10.2 Least Squares and Other Norm Minimization Problems

A frequently occurring problem in statistics and in many other areas of science is a norm minimization problem

(10.13)\[\begin{split}\begin{array}{ll} \mbox{minimize} & \|F x - g\|,\\ \mbox{subject to} & Ax=b, \end{array}\end{split}\]

where \(x\in\real^n\) and of course we can allow other types of constraints. The objective can involve various norms: infinity norm, 1-norm, 2-norm, \(p\)-norms and so on. For instance the most popular case of the 2-norm corresponds to the least squares linear regression, since it is equivalent to minimization of \(\|Fx-g\|_2^2\).

10.2.1 Least squares, 2-norm

In the case of the 2-norm we specify the problem directly in conic quadratic form

(10.14)\[\begin{split}\begin{array}{ll} \mbox{minimize} & t,\\ \mbox{subject to} & (t, Fx-g)\in\Q^{k+1}, \\ & Ax=b. \end{array}\end{split}\]

The first constraint of the problem can be represented as an affine conic constraint. This leads to the following model.

Listing 10.7 Script solving problem (10.14) Click here to download.
# Least squares regression
# minimize \|Fx-g\|_2
norm_lse <- function(F,g,A,b)
{
    n <- dim(F)[2]
    k <- length(g)
    m <- dim(A)[1]

    # Linear constraints in [x; t]
    prob <- list(sense="min")
    prob$A <- cbind(A, rep(0, m))
    prob$bx <- rbind(rep(-Inf, n+1), rep(Inf, n+1))
    prob$bc <- rbind(b, b)
    prob$c <- c(rep(0, n), 1)

    # Affine conic constraint
    prob$F <- rbind( c(rep(0,n), 1),
                     cbind(F, rep(0, k)) )
    prob$g <- c(0, -g)
    prob$cones <- matrix(list("QUAD", k+1, NULL))
    rownames(prob$cones) <- c("type","dim","conepar")

    # Solve
    r <- mosek(prob, list(verbose=1))
    stopifnot(identical(r$response$code, 0))
    r$sol$itr$xx[1:n]
}

10.2.2 Ridge regularisation

Regularisation is classically applied to reduce the impact of outliers and to control overfitting. In the conic version of ridge (Tychonov) regression we consider the problem

(10.15)\[\begin{split}\begin{array}{ll} \mbox{minimize} & \|F x - g\|_2+ \gamma\|x\|_2,\\ \mbox{subject to} & Ax=b, \end{array}\end{split}\]

which can be written explicitly as

(10.16)\[\begin{split}\begin{array}{ll} \mbox{minimize} & t_1+\gamma t_2,\\ \mbox{subject to} & (t_1, Fx-g)\in\Q^{k+1}, \\ & (t_2, x)\in\Q^{n+1},\\ & Ax=b. \end{array}\end{split}\]

The implementation is a small extension of that from the previous section.

Listing 10.8 Script solving problem (10.16) Click here to download.
# Least squares regression with regularization
# minimize \|Fx-g\|_2 + gamma*\|x\|_2
norm_lse_reg <- function(F,g,A,b,gamma)
{
    n <- dim(F)[2]
    k <- length(g)
    m <- dim(A)[1]

    # Linear constraints in [x; t1; t2]
    prob <- list(sense="min")
    prob$A <- cbind(A, rep(0, m), rep(0, m))
    prob$bx <- rbind(rep(-Inf, n+2), rep(Inf, n+2))
    prob$bc <- rbind(b, b)
    prob$c <- c(rep(0, n), 1, gamma)

    # Affine conic constraint
    prob$F <- rbind(     c(rep(0,n), 1,         0),
                     cbind(F,        rep(0, k), rep(0, k)),
                         c(rep(0,n), 0,         1),
                     cbind(diag(n),  rep(0, n), rep(0, n)))
    prob$g <- c(0, -g, rep(0, n+1))
    prob$cones <- cbind(matrix(list("QUAD", k+1, NULL)), 
                        matrix(list("QUAD", n+1, NULL)))
    rownames(prob$cones) <- c("type","dim","conepar")

    # Solve
    r <- mosek(prob, list(verbose=1))
    stopifnot(identical(r$response$code, 0))
    r$sol$itr$xx[1:n]
}

Note that classically least squares problems are formulated as quadratic problems and then the objective function would be written as

\[\|Fx-g\|_2^2+\gamma\|x\|_2^2.\]

This version can easily be obtained by replacing the quadratic cone with an appropriate rotated quadratic cone in (10.16). Then they core of the implementation would change as follows:

Listing 10.9 Script solving classical quadratic ridge regression Click here to download.
    # Affine conic constraint
    prob$F <- rbind(     c(rep(0,n), 1,         0),
                         c(rep(0, n+2)),
                     cbind(F,        rep(0, k), rep(0, k)),
                         c(rep(0,n), 0,         1),
                         c(rep(0, n+2)),
                     cbind(diag(n),  rep(0, n), rep(0, n)))
    prob$g <- c(0, 0.5, -g, 0, 0.5, rep(0, n))
    prob$cones <- cbind(matrix(list("RQUAD", k+2, NULL)), 
                        matrix(list("RQUAD", n+2, NULL)))
    rownames(prob$cones) <- c("type","dim","conepar")

Fig. 10.2 shows the solution to a polynomial fitting problem for a few variants of least squares regression with and without ridge regularization.

_images/fit_poly1.png

Fig. 10.2 Three fits to a dataset at various levels of regularization.

10.2.3 Lasso regularization

In lasso or least absolute shrinkage and selection operator the regularization term is the 1-norm of the solution

(10.17)\[\begin{split}\begin{array}{ll} \mbox{minimize} & \|F x - g\|_2+ \gamma\|x\|_1,\\ \mbox{subject to} & Ax=b. \end{array}\end{split}\]

This variant typically tends to give preference to sparser solutions, i.e. solutions where only a few elements of \(x\) are nonzero, and therefore it is used as an efficient approximation to the cardinality constrained problem with an upper bound on the 0-norm of \(x\). To see how it works we first implement (10.17) adding the constraint \(t\geq \|x\|_1\) as a series of linear constraints

\[u_i\geq -x_i,\ u_i\geq x_i,\ t\geq \sum u_i,\]

so that eventually the problem becomes

\[\begin{split}\begin{array}{ll} \mbox{minimize} & t_1+\gamma t_2,\\ \mbox{subject to} & u+x\geq 0,\\ & u-x\geq 0,\\ & t_2-e^Tu \geq 0,\\ & Ax=b,\\ & (t_1, Fx-g) \in \Q^{k+1}. \end{array}\end{split}\]
Listing 10.10 Script solving problem (10.17) Click here to download.
# Least squares regression with lasso regularization
# minimize \|Fx-g\|_2 + gamma*\|x\|_1
norm_lse_lasso <- function(F,g,A,b,gamma)
{
    n <- dim(F)[2]
    k <- length(g)
    m <- dim(A)[1]

    # Linear constraints in [x; u; t1; t2]
    prob <- list(sense="min")
    prob$A <- rbind(cbind(A,    matrix(0, m, n+2)),
                    cbind(diag(n), diag(n), matrix(0, n, 2)),
                    cbind(-diag(n), diag(n), matrix(0, n, 2)),
                    c(rep(0, n), rep(-1, n), 0, 1))
    prob$bx <- rbind(rep(-Inf, 2*n+2), rep(Inf, 2*n+2))
    prob$bc <- rbind(c(b, rep(0, 2*n+1)), c(b, rep(Inf, 2*n+1)))
    prob$c <- c(rep(0, 2*n), 1, gamma)

    # Affine conic constraint
    prob$F <- rbind(c(rep(0, 2*n), 1, 0),
                    cbind(F, matrix(0, k, n+2)))
    prob$g <- c(0, -g)
    prob$cones <- matrix(list("QUAD", k+1, NULL))
    rownames(prob$cones) <- c("type","dim","conepar")

    # Solve
    r <- mosek(prob, list(verbose=1))
    stopifnot(identical(r$response$code, 0))
    r$sol$itr$xx[1:n]
}

The sparsity pattern of the solution of a large random regression problem can look for example as follows:

Lasso regularization
Gamma 0.0100  density 99%   |Fx-g|_2: 54.3722
Gamma 0.1000  density 87%   |Fx-g|_2: 54.3939
Gamma 0.3000  density 67%   |Fx-g|_2: 54.5319
Gamma 0.6000  density 40%   |Fx-g|_2: 54.8379
Gamma 0.9000  density 26%   |Fx-g|_2: 55.0720
Gamma 1.3000  density 12%   |Fx-g|_2: 55.1903

10.2.4 p-norm minimization

Now we consider the minimization of the \(p\)-norm defined for \(p>1\) as

(10.18)\[\|y\|_p = \left(\sum_i |y_i|^p \right)^{1/p}.\]

We have the optimization problem:

(10.19)\[\begin{split}\begin{array}{ll} \mbox{minimize} & \|F x - g\|_p,\\ \mbox{subject to} & Ax=b. \end{array}\end{split}\]

Increasing the value of \(p\) forces stronger penalization of outliers as ultimately, when \(p\to\infty\), the \(p\)-norm \(\|y\|_p\) converges to the infinity norm \(\|y\|_\infty\) of \(y\). According to the Modeling Cookbook the \(p\)-norm bound \(t\geq \|Fx-g\|_p\) can be added to the model using a sequence of three-dimensional power cones and we obtain an equivalent problem

(10.20)\[\begin{split}\begin{array}{ll} \mbox{minimize} & t \\ \mbox{subject to} & (r_i,t,(Fx-g)_i)\in\POW_3^{1/p,1-1/p}, \\ & e^Tr = t, \\ & Ax = b. \end{array}\end{split}\]

The power cones can be added one by one to the structure representing affine conic constraints. Each power cone will require one \(r_i\), one copy of \(t\) and one row from \(F\) and \(g\). An alternative solution is to create the vector

\[[r_1;\ldots;r_k;t;\ldots;t;Fx-g]\]

and then reshuffle its elements into

\[[r_1;t;F_1x-g_1;\ldots;r_k;t;F_kx-g_k]\]

using an appropriate permutation matrix. This approach is demonstrated in the code below.

Listing 10.11 Script solving problem (10.20) Click here to download.
# P-norm minimization
# minimize \|Fx-g\|_p
norm_p_norm <- function(F,g,A,b,p)
{
    n <- dim(F)[2]
    k <- length(g)
    m <- dim(A)[1]

    # Linear constraints in [x; r; t]
    prob <- list(sense="min")
    prob$A <- rbind(cbind(A,    matrix(0, m, k+1)),
                    c(rep(0, n), rep(1, k), -1))
    prob$bx <- rbind(rep(-Inf, n+k+1), rep(Inf, n+k+1))
    prob$bc <- rbind(c(b, 0), c(b, 0))
    prob$c <- c(rep(0, n+k), 1)

    # Permutation matrix which picks triples (r_i, t, F_ix-g_i)
    M <- cbind(sparseMatrix(seq(1,3*k,3), seq(1,k), x=rep(1,k), dims=c(3*k,k)),
               sparseMatrix(seq(2,3*k,3), seq(1,k), x=rep(1,k), dims=c(3*k,k)),
               sparseMatrix(seq(3,3*k,3), seq(1,k), x=rep(1,k), dims=c(3*k,k)))

    # Affine conic constraint
    prob$F <- M %*% rbind(cbind(matrix(0, k, n), diag(k), rep(0, k)),
                          cbind(matrix(0, k, n+k), rep(1, k)),
                          cbind(F, matrix(0, k, k+1)))
    prob$g <- as.numeric(M %*% c(rep(0, 2*k), -g))
    prob$cones <- matrix(list("PPOW", 3, c(1, p-1)), nrow=3, ncol=k)
    rownames(prob$cones) <- c("type","dim","conepar")

    # Solve
    r <- mosek(prob, list(verbose=1))
    stopifnot(identical(r$sol$itr$prosta, "PRIMAL_AND_DUAL_FEASIBLE"))
    r$sol$itr$xx[1:n]
}
_images/fit_pnorms.png

Fig. 10.3 \(p\)-norm minimizing fits of a polynomial of degree at most 5 to the data for various values of \(p\).