Technically an anti-hero lacks the attributes of the hero, of the “knight in the shining armour” type. I believe that one particular aspect of not being a (classical) hero is the use deception, living double lives, etc.
In The Matrix, Neo lives a double life: he works a dull cubicle job by day, helps his landlady take out the garbage; by night he’s involved in illegal information trade. He is trapped into this double life and he looks for an exit: an answer and a saviour (Morpheus).