I had the same issue. It (Echo Dot) would act like it's starting up, the blue ring would circle, then got a light blue bit for a second, then off... and then all over again... over and over. Oddly, and/or coincidentally, I recently put in a bunch of new Philips Hue stuffs (a Bridge, 3 color bulbs, a Bloom, 2 motion sensors), which was indicated as a potential issue in another post regarding these reboots (possibly something about not enough WiFi connections). But... once I replaced the power cord with the original power cord and plug, it seems to be better now. Must have not had enough power.
$.02