You need to have food (rations) in your inventory. Servants will eat two lowest quality food, you and Livia highest quality food. If you don't have food, servants will get learning penalties (eventually they will learn nothing during training) and may run away. You will also lose any Livia's progress, if she is starving. Feeding higher quality food to servants will give them learning bonuses (they will learn faster).
Servant's own inventory is only for changing their clothes (or stripping them naked, if you wish).