AdaBoost: Sample Reweighting in Action

Boosting Rounds

Max rounds 12

Label noise 10%

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Round t

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Weak error ε_t

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Learner weight α_t

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Ensemble accuracy

50.0%

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How it works

AdaBoost turns a pool of weak learners (here: axis-aligned decision stumps) into one strong classifier by training them sequentially, each one focused on the previous learner's mistakes:

1. init weights  w_i = 1/N for all N samples
2. fit stump h_t minimizing weighted error ε_t = Σ w_i·[h_t(x_i)≠y_i]
3. learner weight  α_t = ½·ln((1-ε_t)/ε_t)
4. reweight        w_i ← w_i·exp(-α_t·y_i·h_t(x_i)),  then renormalize Σw_i=1
5. final vote      F(x) = sign( Σ_t α_t·h_t(x) )

Sphere size = the sample's current weight — misclassified points grow after every round, forcing the next stump to pay attention to them.
Ring outline marks samples the current ensemble F(x) still gets wrong.
Floor tiles shade the combined decision region sign(F(x)) — a single stump can only cut the plane once, but the weighted vote of several stumps carves out a boundary no single stump could draw (this dataset is XOR-like and not linearly separable by one stump).
A stump with ε_t near 0 gets a large α_t (trusted heavily); one near 0.5 gets α_t near 0 (barely trusted) — exactly the "weighted committee" idea behind Random Forest, Gradient Boosting and stacking.

Real-world relevance: this is the same mechanism behind AdaBoost.M1 and its descendants (LogitBoost, gradient boosting), used for face detection (Viola-Jones), fraud scoring and any setting where combining many weak rules beats hand-tuning one strong one.

Label +1 Label −1 Current stump cut

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