I attacked another child once. Another boy who had been harassing and bullying me for some time (one of many such children, sadly). I went a bit mad and pummeled him with my fists; it lasted about six seconds before I disengaged and more or less ran off crying. I've rarely been so ashamed. I didn't get into any trouble (the teacher knew what was going on and I was clearly contrite). However, I confused the other boy immensely because I was tearfully asking for his forgiveness, and was visibly upset that I had hurt him. I wasn't addressing him as though I were subordinate, mind you, but as an equal. This just didn't fit into his worldview; like many people, he had an unfortunate tendency to see the social world in terms of power struggles. I'd just demonstrated I was "on top" (however briefly), and my upset was confusing him.
The way I see it: If you use violence or aggression against another child, you merely reinforce their understanding that such a system is successful. They haven't stopped seeing the world in terms of control and power and dominance; it's merely that you've proven dominant (or at least not clearly subordinate). You might not be thinking in those terms (you're probably merely thinking "get off my back!!"), but I'm sure they see it as just another power struggle, ending in a revaluation on their part, a revaluation of your suitability to be dominated. It will probably get them off your back, so I understand entirely if a person is unapologetic for it. It solves your problem, so great, but it doesn't solve the problem.
That other boy never bullied or harassed me again. Part of it was no doubt simply because I had hit him. I'm a pacifist, not an idiot; I'm quite aware hitting someone hard will often convince them not to try attacking you again. But I hope that what really affected him was the confusion that after attacking him I should be asking forgiveness. I met him again a year or so back (two-three years since school ended) and he was quite nice, really. And apologetic, genuinely seeking peace. "I was an asshole, wasn't I?" he said. "Yes, pretty much", I replied.
Again, I hope it was more my response to my violence that affected him, rather than my violence itself. But perhaps I just want to believe that, to try and help me deal with the fact that I attacked another being.