Textual Stochastic Gradient Descent: Discrete Optimization of External Memory for Reasoning Language Agents

While Large Language Models (LLMs) possess strong reasoning capabilities, enabling them to learn continuously from experience without parametric retraining remains an open challenge. Existing Retrieval-Augmented Generation (RAG) approaches typically treat memory as a static or append-only corpus, leading to "memory saturation''---where accumulating noise and redundant information degrade performance over time. To address this, we propose an Experience Risk Minimization (ERM) framework that formalizes the experience library as a learnable parameter under an explicit capacity budget. We introduce Textual Stochastic Gradient Descent (TSGD), a discrete optimization algorithm that refines this library via failure-driven Add, Edit, and Delete operations. TSGD estimates ``textual gradients'' through self-reflection and employs a dual-verification mechanism to ensure generalization, effectively preventing overfitting to local errors. Empirical results on MATH and AIME benchmarks demonstrate that TSGD achieves state-of-the-art performance, improving accuracy by up to 18.7\% over zero-shot baselines and significantly outperforming static RAG, all while maintaining a compact memory footprint (compressing hundreds of experiences into $\approx$30 high-utility rules).