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The optical depth to reionization, $\tau$, is the least constrained parameterof the cosmological $\Lambda$CDM model. To date, its most precise value isinferred from large-scale polarized CMB power spectra from the$\textit{Planck}$ High-Frequency Instrument (HFI). These maps are known tocontain significant contamination by residual non-Gaussian systematic effects,which are hard to model analytically. Therefore, robust constraints on $\tau$are currently obtained through an empirical cross-spectrum likelihood builtfrom simulations. In this paper, we present a likelihood-free inference of$\tau$ from polarized $\textit{Planck}$ HFI maps which, for the first time, isfully based on neural networks (NNs). NNs have the advantage of not requiringan analytical description of the data and can be trained on state-of-the-artsimulations, combining information from multiple channels. By using Gaussiansky simulations and $\textit{Planck}$ $\texttt{SRoll2}$ simulations, includingCMB, noise, and residual instrumental systematic effects, we train, test andvalidate NN models considering different setups. We infer the value of $\tau$directly from $Q$ and $U$ maps at $\sim 4^\circ$ pixel resolution, withoutcomputing power spectra. On $\textit{Planck}$ data, we obtain $\tau_{\rmNN}=0.0579\pm 0.0082$, compatible with current $EE$ cross-spectrum results butwith a $\sim30\%$ larger uncertainty, which can be assigned to the inherentnon-optimality of our estimator and to the retraining procedure applied toavoid biases. While this paper does not improve on current cosmologicalconstraints on $\tau$, our analysis represents a first robust application ofNN-based inference on real data and highlights its potential as a promisingtool for complementary analysis of near-future CMB experiments, also in view ofthe ongoing challenge to achieve a detection of primordial gravitational waves.

Inference of the optical depth to reionization $\tau$ from $\textit{Planck}$ CMB maps with convolutional neural networks