Abstract: We study infinite-horizon average-reward Markov decision processes (AMDPs) in the context of general function approximation. Specifically, we propose a novel algorithmic framework named Fixed-Point Local Optimization (FLOP), which incorporates both model-based and value-based incarnations. In particular, FLOP features a novel construction of confidence sets and a low-switching policy updating scheme, which are tailored to the average-reward and function approximation setting. Moreover, for AMDPs, we propose a novel complexity measure --- average-reward generalized eluder coefficient (AGEC) --- which captures the challenge of exploration in AMDPs with general function approximation. Such a complexity measure encompasses almost all previously known tractable AMDP models, such as linear AMDPs and linear mixture AMDPs, and also includes newly identified cases such as kernel AMDPs and AMDPs with low Bellman eluder dimensions. Using AGEC, we prove that FLOP achieves a sublinear $\tilde{\mathcal{O}}(\mathrm{poly}(d, \mathrm{sp}(v^*)) \sqrt{T \beta })$ regret, where $d$ and $\beta$ correspond to AGEC and the log-covering number of the hypothesis class respectively, $\mathrm{sp}(v^*)$ represents the span of the optimal state bias function, $T$ denotes the number of steps, and $\tilde{\mathcal{O}} (\cdot) $ omits logarithmic factors. When specialized to concrete AMDP models, our regret bounds are comparable to those established by the existing algorithms designed specifically for these special cases. To the best of our knowledge, this paper presents the first comprehensive theoretical framework capable of handling nearly all AMDPs.