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latent class α given the current item parameter estimates and the rest score: 

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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 

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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 

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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 

