Faster Approximate Fixed Points of $\ell_\infty$-Contractions

We present a new algorithm for finding an $\epsilon$-approximate fixed point of an $\ell_\infty$-contracting function $f : [0, 1]^d \rightarrow [0, 1]^d$. Our algorithm is based on the query-efficient algorithm by Chen, Li, and Yannakakis (STOC 2024), but comes with an improved upper bound of $(\log \frac{1}{\epsilon})^{\mathcal{O}(d \log d)}$ on the overall runtime (while still being query-efficient). By combining this with a recent decomposition theorem for $\ell_\infty$-contracting functions, we then describe a second algorithm that finds an $\epsilon$-approximate fixed point in $(\log \frac{1}{\epsilon})^{\mathcal{O}(\sqrt{d} \log d)}$ queries and time. The key observation here is that decomposition theorems such as the one for $\ell_\infty$-contracting maps often allow a trade-off: If an algorithm's runtime is worse than its query complexity in terms of the dependency on the dimension $d$, then we can improve the runtime at the expense of weakening the query upper bound. By well-known reductions, our results imply a faster algorithm for $\epsilon$-approximately solving Shapley stochastic games.