This study innovates upon the field of dialog-driven query resolution (DDQR) utilizing knowledge graphs (KGs). Traditional methods in DDQR predominantly depend on fully supervised signals that presuppose the existence of perfect logical forms for queries. This reliance on gold logical forms for answer extraction is impractical in diverse real-world applications. When such forms are absent, contemporary methods, which lean on weak supervisory signals or employ heuristic and reinforcement strategies, recast DDQR as a knowledge graph path optimization problem. Despite the non-availability of gold logical forms, the rich conversational context provided by comprehensive dialog histories and domain-specific knowledge can be leveraged to optimize path selection in KGs effectively. We introduce an advanced method, termed CONVEX (CONVersational EXploration), which utilizes contrastive learning for path ranking. CONVEX addresses critical challenges by enabling learning under weak supervision and integrating the conversational context to enhance the representation quality for more effective path discrimination. Extensive evaluations of CONVEX on established benchmarks demonstrate its superiority across various metrics over current methods, notably improving Mean Reciprocal Rank (MRR) and Hit@5 by up to 20 and 36 percentage points respectively.