latent class α given the current item parameter estimates and the rest score: The β ( t ) j and φ ( t ) jj ′ are the provisional estimates of β$_{j}$ and φ$_{jj}$ ′ obtained at the t -th iteration. The local quadratic approximation to log f ( x$_{ij}$ | x$_{i,}$$_{−}$$_{j}$ , α , B , Φ ) is obtained as where and Recall that the key assumption of the GDCM is that the latent class accounts for most variation of response data and only a fraction of variance is attributed to interdependency of items. This assumption encourages sparsity and simplicity in the design matrix such that many of the coeffi- cients in Φ are close to zero and a small subset of the coefficients have larger and nonzero values. Common practice for enforcing sparsity on the Markov network is to impose L$_{1}$ regularization on the log pseudo-likelihood function (H¨ ofling & Tibshirani, 2009), and this approach is employed in the current study. Let λ be a regularization parameter that controls the sparsity (or complexity) of Φ . The objective function to be optimized with respect to Φ is then given by subject to symmetry φ$_{jj}$ ′ = φ$_{j}$ $^{′}$j (1 ≤ j < j ′ ≤ J ). The L$_{1}$ penalty in (9) performs both the variable selection and regularization on the parameters in that it induces a solution with many zero P ( t ) ij ( α ) = exp$^{}$ $_{}$β ( t ) j $^{⊤}$α + ∑ j $^{′}$̸ = j φ ( t ) jj $_{′}$x$_{ij}$ ′   1 + exp$^{}$ $_{}$β ( t ) j $^{⊤}$α + ∑ j $^{′}$̸ = j φ ( t ) jj $_{′}$x$_{ij}$ ′   . log f ( x$_{ij}$ | x$_{i,}$$_{−}$$_{j}$ , α , B , Φ ) ≈ − 1 2 ω ( t ) ij  $_{}$β ⊤ j α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{ij}$ ′ − y ( t ) ij   2 , ω ( t ) ij = P ( t ) ij ( α ) $^{(}$1 − P ( t ) ij ( α ) $^{)}$, y ( t ) ij = β ( t ) j $^{⊤}$α + ∑ j $^{′}$̸ = j φ ( t ) jj $_{′}$x$_{j}$ ′ + $^{(}$x$_{ij}$ − P ( t ) ij ( α ) $^{)}$ω ( t ) ij − $^{1}$. arg max Φ N ∑ i =1 ∑ α f ( α | x$_{i}$ , θ ( t $^{)}$) J ∑ j =1 − 1 2 ω ( t ) ij  $_{}$β ⊤ j α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{ij}$ ′ − y ( t ) ij   2 − N ∑ 1 ≤ j<j $^{′}$≤ J λ | φ$_{jj}$ ′ | (9) 11