  Selecting hyperparameters for unsupervised learning problems is challenging
in general due to the lack of ground truth for validation. Despite the
prevalence of this issue in statistics and machine learning, especially in
clustering problems, there are not many methods for tuning these
hyperparameters with theoretical guarantees. In this paper, we provide a
framework with provable guarantees for selecting hyperparameters in a number of
distinct models. We consider both the subgaussian mixture model and network
models to serve as examples of i.i.d. and non-i.i.d. data. We demonstrate that
the same framework can be used to choose the Lagrange multipliers of penalty
terms in semi-definite programming (SDP) relaxations for community detection,
and the bandwidth parameter for constructing kernel similarity matrices for
spectral clustering. By incorporating a cross-validation procedure, we show the
framework can also do consistent model selection for network models. Using a
variety of simulated and real data examples, we show that our framework
outperforms other widely used tuning procedures in a broad range of parameter
settings.
