Chain derailment from front crank. Fine on large ring won't hold on smaller inner ring
I could be caused because the chain is not running a straight line from the rear cogs to the front chain rings. If you put the bike on the stand and follow the chain from the top of the cogs to the chainrings. You will see if it follows a straight line. It might be slight off so that it won't jump off when on the stand but it does when there is pressure. 

Another reason could the rear cogs wable. If it so, you will need to change the cogs.