Thanks for the information, Yannick.
As of today 2017-12-07 I still experience the issue and it become noticeable. About 2% of all requests to my site are dropped due to this error. I did optimize the website to use less reads from DB and use Memcache more. I did clean the database from outdated entries to make it lighter. I also thought that probably I was reached the limits of my Cloud SQL level and I've upgraded to higher level, but none of these measures helped. Still the same percentage of dropped requests. Few observations:
1. The rate of "Lost connection" errors obviously has correlation with the amount of request per second to my website. When the request per second is less than 1/s I see quite few errors. When it gets higher to 4/s I see a lot of errors.
2. Errors have correlation with the latency. Almost every error entry in the log shows that request processing time was from 5s to 15s while cpu usage was quite small, so I believe all of this time it waits for connection to Cloud SQL.
3. Higher latency leads to creation of additional instances which actually doesn't solve the problem, but increases the cost for me.
4. Errors are started to appear in early November and most of the days 1-2% of requests were dropped. However, there were few days when it was just few errors of this kind while the amount of request were the same as for other days. Few months before website was working perfectly even with higher load.