arXiv:2409.12135v3 Announce Type: replace-cross Abstract: Temporal difference (TD) learning with linear function approximation (linear TD) is a classic and powerful prediction algorithm in reinforcement learning. While it is well-understood that linear TD converges almost surely to a unique point, this convergence traditionally requires the assumption that the features used by the approximator are linearly independent. However, this linear independence assumption does not hold in many practical scenarios. This work is the first to establish the almost sure convergence of linear TD without requiring linearly independent features. We prove that the weight iterates of linear TD converge to a bounded set, and that the value estimates derived from the weights in that set are the same almost everywhere. We also establish a notion of local stability of the weight iterates. Importantly, we do not impose assumptions tailored to feature dependence and do not modify the linear TD algorithm. Key to our analysis is a novel characterization of bounded invariant sets of the mean ODE of linear TD.

arXiv:2510.15862v4 Announce Type: replace Abstract: Large language models (LLMs) augmented with external tools are increasingly deployed as deep research agents that gather, reason over, and synthesize web information to answer complex queries. Although recent open-source systems achieve strong empirical performance via reinforcement learning from web interactions, the impact of key design choices remains under-explored. We formalize deep research as reinforcement learning in an episodic finite Markov decision process and construct a competitive baseline agent grounded in this formulation. Building on this foundation, we systematically examine critical design decisions at both training and inference time and identify four factors that substantially improve performance: replacing rule-based rewards with AI feedback from an LLM judge, fine-tuning with the on-policy RLOO algorithm instead of the off-policy GRPO algorithm, filtering low-quality training samples, and employing an error-tolerant test-time rollout strategy. Together, these design choices yield a deep research agent that establishes state-of-the-art performance among 7B-scale agents when evaluated across ten widely used benchmarks.