I was going to make a quick reply flippantly supporting Alcaro, but now I'm not sure... I don't know if it really works out as well as he thinks.
Typing while I think...
Suppose it''s would cost 30,000 troops to fight or 30,000 coins to cater all 20 province (you don't know this ahead of time) and you only have 20,000 troops.
Suppose you fight until 0. You will lose all 20,000 troops. You will have won n provinces, lost x troops in province (n+1), and then catered (20-n) provinces (including province (n+1)). In total, you will have spent (30,000+x) in troops and coins. You will have lost 20,000 troops, (20,000-x) in won provinces and x troops in the lost province. You will have catered (10,000+x) coins. The best strategy is to minimize x. And the best way to minimize x is to lose the battle in the province where x will be smallest.
If you are very weak (less than 15,000 troops in my above example), it's better to start small and lose early. If you are very strong (more than 15,000), it's better to start large and lose late. Also, this transition point is not between the 10th and 11th province, but somewhere higher, as province costs increase.
At this point, the stumbling block is that, strictly speaking, you don't know how you stand at the start. You don't know if you have enough and it doesn't matter which end to start from, or if you are short but strong and should start at province 20, or if you are short and weak and should start at province 1. Ultimately, one has to make a guess, educated or otherwise, at what you think your initial situation is. The better your guess is to the underlying reality, the more likely you are to optimize your strategy.
Bottom line, there is no universal strategy. The stronger your initial troop strength, the better the odds that going with a top-down vs. bottom up strategy is the optimal fighting strategy, but you don't know where that transition point is.