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Over the last few years, many studies have found an empirical relationshipbetween the abundance of a star and its age. Here we estimate spectroscopicstellar ages for 178 825 red-giant stars observed by the APOGEE survey with amedian statistical uncertainty of 17%. To this end, we use the supervisedmachine learning technique XGBoost, trained on a high-quality dataset of 3060red-giant and red-clump stars with asteroseismic ages observed by both APOGEEand Kepler. After verifying the obtained age estimates with independentcatalogues, we investigate some of the classical chemical, positional, andkinematic relationships of the stars as a function of their age. We find a veryclear imprint of the outer-disc flare in the age maps and confirm the recentlyfound split in the local age-metallicity relation. We present new and precisemeasurements of the Galactic radial metallicity gradient in small age binsbetween 0.5 and 12 Gyr, confirming a steeper metallicity gradient for $\sim2-5$Gyr old populations and a subsequent flattening for older populations mostlyproduced by radial migration. In addition, we analyse the dispersion about theabundance gradient as a function of age. We find a clear power-law trend (withan exponent $\beta\approx0.15$) for this relation, indicating a relativelysmooth radial migration history in the Galactic disc over the past $7-9$ Gyr.Departures from this power law may possibly be related to the Gaia Enceladusmerger and passages of the Sagittarius dSph galaxy. Finally, we confirmprevious measurements showing a steepening in the age-velocity dispersionrelation at around $\sim9$ Gyr, but now extending it over a large extent of theGalactic disc (5 kpc $<R_{\rm Gal}<13$ kpc). [Abridged]

Spectroscopic age estimates for APOGEE red-giant stars: Precise spatial and kinematic trends with age in the Galactic disc