Powering Pico from Raspberry Pi B+/2 USB

Has anyone had any trouble powering their Pico off a raspberry pi? I'm finding that sometimes when I plug mine in, either directly or through a non-powered hub, the leds all start flashing rapidly and the espruino doesn't run. Or if that doesn't happen, the ESP8266 that's plugged into it never works. Seems like a power stability issue and I'm not sure what to do about it.

Have you tried a Pico without the ESP8266?

Also, do you have a properly powerful USB power supply for the Pi?

I think the Raspberry Pi has a polyfuse that limits the current going to USB. The ESP8266 draws ~400mA peak I think, and that could actually just be a bit too much for the Pi.

I guess putting a massive capacitor across the power rails for the ESP8266 might help though (100uF or so)

Yeah, I think it might be the usb circuitry on the pi limiting the current, I just looked it up and saw that it has its own limits. I did try with a big capacitor but it didn't help. I'll probably just plug it into my nas, which is 50cm from the pi. I've been testing out leaving it running its webserver and sending it commands over that and it had been very stable!