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Combinatorial auctions (CAs) are used to allocate multiple items among bidders with complex valuations. Since the value space grows exponentially in the number of items, it is impossible for bidders to report their full value function even in medium-sized settings. Prior work has shown that current designs often fail to elicit the most relevant values of the bidders, thus leading to inefficiencies. We address this problem by introducing a machine learning-based elicitation algorithm to identify which values to query from the bidders. Based on this elicitation paradigm we design a new CA mechanism we call PVM, where payments are determined so that bidders’ incentives are aligned with allocative efficiency. We validate PVM experimentally in several spectrum auction domains, and we show that it achieves high allocative efficiency even when only few values are elicited from the bidders.