Accurate chart comprehension represents a critical challenge in advancing multimodal learning systems, as extensive information is compressed into structured visual representations. However, existing vision-language models (VLMs) frequently struggle to generalize on unseen charts because it requires abstract, symbolic, and quantitative reasoning over structured visual representations. In this work, we introduce Chart-RL, an effective reinforcement learning (RL) method that employs mathematically verifiable rewards to enhance chart question answering in VLMs. Our experiments demonstrate that Chart-RL consistently outperforms supervised fine-tuning (SFT) across different chart understanding benchmarks, achieving relative improvements of 16.7% on MultiChartQA, and 11.5% on ChartInsights. We conduct robustness analysis, where Chart-RL achieves enhanced performance in 18 of 25 perturbed chart categories, demonstrating strong consistency and reasoning capability across visual variations. Furthermore, we demonstrate that task difficulty and inherent complexity are more critical than data quantity in RL training. For instance, Chart-RL trained on merely 10 complex chart-query examples significantly outperforms models trained on over 6,000 simple examples. Additionally, training on challenging reasoning tasks not only improves in-domain generalization relative to simpler tasks, but also facilitate strong transfer to out-of-domain visual mathematical problems.

Multi-hop question answering (MHQA) is a practical bottleneck in industry applications such as enterprise assistants, customer-support copilots, and compliance analysis, where systems must combine evidence across multiple documents before answering. Large language models (LLMs) remain brittle in this setting: iterative retrieval can commit too early to low-recall trajectories, while planning-only approaches can produce static query sets that fail to adapt when intermediate evidence changes. We propose Planned Active Retrieval and Reasoning RAG (PAR²-RAG), a training-free two-stage framework that separates coverage from commitment. PAR²-RAG first performs breadth-first anchoring to build a high-recall evidence frontier, then applies depth-first refinement with evidence sufficiency control in an iterative loop. This design targets deployment constraints by avoiding retraining cycles, reducing maintenance overhead under changing corpora, and improving scalability across domains. Across four MHQA benchmarks, PAR²-RAG consistently outperforms strong baselines: compared with IRCoT, it achieves up to 23.5% higher answer accuracy and up to 10.5% NDCG gains in retrieval quality.