Controlled variable (CV) selection from available measurements through exhaustive search is computationally forbidding for large-scale problems. We have recently proposed novel bidirectional branch and bound (B3) approaches for CV selection using the minimum singular value (MSV) rule and the local worst-case loss criterion in the framework of self-optimizing control. However, the MSV rule is approximate and worst-case scenario may not occur frequently in practice. In this work, the B3 approach is extended to CV selection based on the recently developed local average loss metric, which represents the expected loss incurred over the long-term operation of the plant. Lower bounds on local average loss and fast pruning algorithms are derived for the efficient B3 algorithm. Numerical tests and binary distillation column case study are used to demonstrate the computational efficiency of the proposed method.