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We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form 
$o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where 
$L(n)$ is a slowly varying function and 
$\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed 
$O(n^{1/4})$. To break the 
$O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.
