  We demonstrate that a machine learning technique with a simple feedforward
neural network can sensitively detect two successive phase transitions
associated with the Berezinskii-Kosterlitz-Thouless (BKT) phase in q-state
clock models simultaneously by analyzing the weight matrix components
connecting the hidden and output layers. We find that the method requires only
a data set of the raw spatial spin configurations for the learning procedure.
This data set is generated by Monte-Carlo thermalizations at selected
temperatures. Neither prior knowledge of, for example, the transition
temperatures, number of phases, and order parameters nor processed data sets
of, for example, the vortex configurations, histograms of spin orientations,
and correlation functions produced from the original spin-configuration data
are needed, in contrast with most of previously proposed machine learning
methods based on supervised learning. Our neural network evaluates the
transition temperatures as T_2/J=0.921 and T_1/J=0.410 for the
paramagnetic-to-BKT transition and BKT-to-ferromagnetic transition in the
eight-state clock model on a square lattice. Both critical temperatures agree
well with those evaluated in the previous numerical studies.
