  Millimeter wave (mmWave) cell-free MIMO achieves an extremely high rate while
its beam alignment (BA) suffers from excessive overhead due to a large number
of transceivers. Recently, user location and probing measurements are utilized
for BA based on machine learning (ML) models, e.g., deep neural network (DNN).
However, most of these ML models are centralized with high communication and
computational overhead and give no specific consideration to practical issues,
e.g., limited training data and real-time model updates. In this paper, we
study the {probing} beam-based BA for mmWave cell-free MIMO downlink with the
help of broad learning (BL). For channels without and with uplink-downlink
reciprocity, we propose the user-side and base station (BS)-side BL-aided
incremental collaborative BA approaches. Via transforming the centralized BL
into a distributed learning with data and feature splitting respectively, the
user-side and BS-side schemes realize implicit sharing of multiple user data
and multiple BS features. Simulations confirm that the user-side scheme is
applicable to fast time-varying and/or non-stationary channels, while the
BS-side scheme is suitable for systems with low-bandwidth fronthaul links and a
central unit with limited computing power. The advantages of proposed schemes
are also demonstrated compared to traditional and DNN-aided BA schemes.
