The premise of boosting is that we're able to find a number of methods (what they call "hypotheses" in AdaBoost) to predict the correct outcomes of the training cases, each method correctly predicting more than 50% of the training cases. Then if we collect a large-ish number of these methods, we can predict the correct outcomes of all the training cases simply by averaging the predictions of these methods. And the other cases will follow the training cases (unless an overfitting happens). Since more than 50% of the cases are correctly predicted by each method, after the averaging more than 50% of the votes for each training case will be correct, and thus the result will be correct too. Of course this depends on the correct predictions being distributed pretty evenly among the cases. If we have a thousand methods that predict correctly the same cases and incorrectly the other cases, obviously after averaging these other cases will still be predicted incorrectly. So the selection of methods must somehow shuffle the preference for the cases, so that the next picked method will predict well the cases that have been predicted poorly by the previous picked methods. That's it, that's the whole basic idea. It's a bit of an oversimplification but it's easy to understand.
I really did mean it as an oversimplification, since AdaBoost uses the Bayesian decisions to do much better than the simple majority counting. Little did I know that there actually is the method of Boost By Majority (BBM) that does just the counting. It has some other differences but more about that later.
The simple averaging can be simulated through the Bayesian means too. Just use the same confidence for each event. Incidentally, that's what the NonAdaBoost algorithm, also known as epsilon-Boost does: it looks for the weak hypotheses that have at least a fixed "edge" gamma (i.e. the probability of the right guess being at least 0.5+gamma) and then always sets the confidence C=0.5+gamma, and uses the same value to adjust the weights of the training cases.
The NonAdaBoost is essentially a version of AdaBoost with a fixed confidence, and suffering from this defect. But Boost By Majority has another big difference: the way it adjusts the weight of the training cases. The formula it uses is pretty complicated, so I won't even try to reproduce it here. But here is the gist: it keeps track of how many rounds are left to the end of the boosting and what is the balance of the votes collected by each training case. If the modulo of the balance is higher than the number of rounds left, it means that the fate of this case can't be changed any more: it's either guaranteed to be guessed right if the balance is positive or guaranteed to be guessed wrong if the balance is negative, so the algorithm gives up on these cases. It gives the most weight to the most undecided cases, and spreads the rest of the weights in the bell shape of the binomial distribution. The result is that unlike AdaBoost and NonAdaBoost, BBM doesn't concentrate on the hardest cases that are likely to be the noise in the data anyway, and thus reduces the overfitting.
The last chapter of the book is about a combination of AdaBoost and BBM called BrownBoost (from the Brownian motion), or also "boosting in continuous time". It starts with the idea that if the returned partial hypothesis has a higher edge than minimally needed, it might still have enough edge after the re-weighting the training cases, then it can be directly reused on the next round too without searching for a new one, and so on until its edge wears off. This gets developed into an algorithm that uses a real number in the range [0,1) instead of the round count, with the actual rounds moving the current point on it by the varying amounts. The speed of the movement is determined by the pre-set desired training error. This training error gets reached when the end of the range is reached. If the target error is set to 0, the algorithm behaves in the same way as AdaBoost.
The downside is that the algorithm is complex, there isn't even a formula for determining the confidence values for each partial hypothesis. Instead you get a system of two equations that connect this confidence value and advancement through the range to be solved numerically. In the great scheme of things it's not a big deal, after all, compared to the finding of the partial hypotheses this shouldn't be such a big overhead. But it's not easy to think of. And the proof of the validity of this algorithm is quite complicated.
I can't help thinking of a couple of simpler ideas.
The first idea, or should I say guess, is that when we do AdaBoost, it fits into a Bayesian model. So if we keep this Bayesian model consistent, the boosting should still be valid. Obviously, I have no proper proof of that but it looks like a reasonable assumption. There is an easy way to take some training case (or a part of its weight) out of rotation and still keep the Bayesian model consistent.
Remember, previously I've described that we start with a Bayesian table that contains each individual training case
CaseId Weight Outcome Ev1 ... EvT 1 1 * true 1 ... 1 ... M 1 * false 0 ... 0
Which then gets conflated into a two-line table, all the cases with the true outcome combined into one line, and the false ones into another line. The conditional probabilities in the table get averaged during conflation but since it's an averaging of all ones (or all zeroes), the result is still one (or zero).
Weight Outcome Ev1 ... EvT 1 * true 1 ... 1 1 * false 0 ... 0
To take a training case out of rotation, we just change its conditional probability in all the following events (that match the AdaBoost's partial hypotheses) to 0.5. This says that no future events will affect it. And accordingly we set the AdaBoost weight of it to 0. Such decisions can be made according to any algorithm, matching any desired curve.
For example, suppose that we decide to take a case out of rotation when its weight relative to the sum of all weights reaches 0.1 (this is probably not a very good rule, since it allows at most 10 cases to be excluded, but simple for the sake of demonstration). Suppose that its a case with the true ("1") outcome. And suppose that all the weights of all the true cases are totaling to the same value as of all the false cases, each of them having the relative weight of 0.5 (not very likely in reality but just as good a number as any other).
After the disablement, the conditional probability of the true cases will become ((0.5*0.1) + (1 * 0.4))/0.5 = 0.9.
Weight Outcome Ev1 ... EvN-1 EvN ... 1 * true 1 ... 1 0.9 ... 1 * false 0 ... 0 0 ...
Once a training case gets disabled, it stays disabled for all the future rounds, and the AdaBoost keeps acting only on the weights of those training cases that still have 1 or 0 for the conditional probability. Obviously, the more you disable, the less will be the effect of the following rounds when the Bayesian computation runs.
Interestingly though the edges of the partial hypotheses will be higher. Remember, the training cases that get thrown away get it for being difficult to predict. So suppose the partial hypothesis EvN would have returned the confidence of 0.8 if that case wasn't thrown away and had guessed that case wrong. When we throw away the stubborn case, that would become 0.8 out of former 0.9, so the confidence becomes 0.8/0.9 = 0.89, an improvement!
However all this throwing-away has no effect on the previous rounds, these are already set in stone. Which begs an easy solution which is my second idea: why not do two passes of AdaBoost? After the first pass look at the final weights of the training cases to determine the most stubborn ones. Throw them away and do the second pass from scratch. After all, BrownBoost requires an adjustment of the target training error which gets done by running it multiple times with the different values and then selecting the best one. Doing two passes of AdaBoost isn't any worse than that.