What breed, or mix is he? It will be possible to train him to stop but you will have to put in quite a bit of effort as he has learnt that begging is extremely rewarding, and dogs constantly repeat behaviours that get them a reward, that is how they are programmed.
What you need to do is make not begging more rewarding than begging. As soon as we sit down at the table to eat my dog will instantly lie down, that's because while he stays in this position every now and again someone will get up from the table and give him something to eat, it might be just a sliver of sausage, once in a blue moon it is a huge piece of chicken. From his point of view it is in his best interests to offer the downstay behaviour as it can be immensely rewarding.
This can feel confusing as you are told never to feed the dog from the table as you will make him beg, but dogs have a very sensitive nose and you are eating something good, they are bound to be interested, you can teach them not to beg by teaching them that something bad will happen if they do (punishment) but personally I would never train a dog this way. Fast better to teach the dog to offer a more acceptable behaviour and reward out, thus changing the way he is wired to behave. You only ever reward them for offering the desired behaviour, never for begging.
This means you now have to make not begging more rewarding than begging, bearing in mind that begging has been exceedingly rewarding in the past. If he were mine I would first train a reliable downstay in the house, with a clicker and a high reward foodstuff. Once achieved I would begin putting him in a downstay at meal times, this can be in a bed in the room, but it sounds like you eat out a lot? So maybe the best place would be under the table, so his pleading eyes are kept away from well meaning strangers. At first during meals you need to literally rain treats down on him for maintaining the stay while you are eating, so he gets just how rewarding it is to remain motionless. Then over time gradually decrease, and make sure the rewards vary in value, if you always just offer a small dog treat of the same type, then he may well decide to go back to his old behaviour. So change it up and occasionally make it something amazing, like a whole chicken breast, in his mind this will make offering this behaviour totally addictive.
My dog only gets something maybe twice a week or so, but nothing will move him from the spot, as maybe just maybe this will be the time he wins the jackpot 
Sorry for the essay, hope it is of use. I am awake early, lying in bed nervous and exited as I am going to pick up our new rescue dog at midday :)