We present AccelOpt, a self-improving large language model (LLM) agentic system that autonomously optimizes kernels for emerging AI acclerators, eliminating the need for expert-provided hardware-specific optimization knowledge. AccelOpt explores the kernel optimization space through iterative generation, informed by an optimization memory that curates experiences and insights from previously encountered slow-fast kernel pairs. We build NKIBench, a new benchmark suite of AWS Trainium accelerator kernels with varying complexity extracted from real-world LLM workloads to evaluate the effectiveness of AccelOpt. Our evaluation confirms that AccelOpt's capability improves over iterations, boosting the average percentage of peak throughput from $49\%$ to $61\%$ on Trainium 1 and from $45\%$ to $59\%$ on Trainium 2 for NKIBench kernels. Moreover, AccelOpt is highly cost-effective: using open-source models, it matches the kernel improvements of Claude Sonnet 4 while being $26\times$ cheaper. The code is open-sourced at https://github.com/zhang677/AccelOpt.