Coactive learning is an online problem solving setting where the solutions provided by a solver are interactively improved by a domain expert, which in turn drives learning. In this paper we extend the study of coactive learning to problems where obtaining a globally optimal or near-optimal solution may be intractable or where an expert can only be expected to make small, local improvements to a candidate solution. The goal of learning in this new setting is to minimize the cost as measured by the expert effort over time. We first establish theoretical bounds on the average cost of the existing coactive Perceptron algorithm. In addition, we consider new online algorithms that use cost-sensitive and Passive-Aggressive (PA) updates, showing similar or improved theoretical bounds. We provide an empirical evaluation of the learners in various domains, which show that the Perceptron based algorithms are quite effective and that unlike the case for online classification, the PA algorithms do not yield significant performance gains.