  The Extreme Learning Machine (ELM) technique is a machine learning approach
for constructing feed-forward neural networks with a single hidden layer and
their models. The ELM model can be constructed while being trained by
concurrently reducing both the modeling errors and the norm of the output
weights. Usually, the squared loss is widely utilized in the objective function
of ELM, which can be treated as a LASSO problem and hence applying the fast
iterative shrinkage thresholding algorithm (FISTA). However, in this paper, the
minimization problem is solved from a variational inequalities perspective
giving rise to improved algorithms that are more efficient than the FISTA. A
fast general extra-gradient algorithm which is a form of first-order algorithm
is developed with inertial acceleration techniques and variable stepsize which
is updated at every iteration. The strong convergence and linear rate of
convergence of the proposed algorithm are established under mild conditions. In
terms of experiments, two experiments were presented to illustrate the
computational advantage of the proposed method to other algorithms.
