  The multi-index model with sparse dimension reduction matrix is a popular
approach to circumvent the curse of dimensionality in a high-dimensional
regression setting. Building on the single-index analysis by Alquier, P. &
Biau, G. (Journal of Machine Learning Research 14 (2013) 243-280), we develop a
PAC-Bayesian estimation method for a possibly misspecified multi-index model
with unknown active dimension and an orthogonal dimension reduction matrix. Our
main result is a non-asymptotic oracle inequality, which shows that the
estimation method adapts to the active dimension of the model, the sparsity of
the dimension reduction matrix and the regularity of the link function. Under a
Sobolev regularity assumption on the link function the estimator achieves the
minimax rate of convergence (up to a logarithmic factor) and no additional
price is paid for the unknown active dimension.
