$X^\ell = (x_i, y_i)^\ell_{i=1} \subset X \times Y$ — training set, $y_i = y^*(x_i)$

$a(x) = C(b(x))$ — algorithm, where

$b: X \to {R}$ — base algorithm (operator)

$C: {R} \to Y$ — decision rule (composition)

${R}$ — space of estimates

Ensemble of base algorithms \(b_1, \dots, b_T\)

\(a(x) = C(F(b_1(x), \dots, b_T(x)))\)

where \(F: R^T \to R\) — correcting operation

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Why introduce correcting operation \(R\)?

In classification tasks, the set of mappings \(\{F: R^T \to R\}\) is significantly wider than \(\{F: Y^T \to Y\}\)

• Example 1: binary classification, \(Y = \{-1, +1\}\): \[a(x) = \text{sign}(b(x))\] so \(R = \mathbb{R}, b: X \to \mathbb{R}, C(b) = \text{sign}(b)\)
• Example 2: multiclass classification with \(M\) classes \(Y = \{1,\dots, M\}\): \[a(x) = \arg\max_{y \in Y} b_y(x)\] so \(R = \mathbb{R}^M, b: X \to \mathbb{R}^M, C(b_1, \dots, b_M) \equiv \arg \max_{y \in Y} b_y\)

• Example 1: Simple Voting
  \(F(b_1(x),\dots, b_T(x)) = \frac{1}{T} \sum\limits_{t=1}^T b_t(x), x \in X\)
• Example 2: Weighted Voting
  \(F(b_1(x),\dots, b_T(x)) = \sum\limits_{t=1}^T \alpha_tb_t(x), x \in X, \alpha_t \in \mathbb{R}\)
• Example 3: Mixture of Experts
  \(F(b_1(x),\dots, b_T(x)) = \sum\limits_{t=1}^T g_t(x)b_t(x), x \in X, g_t: X \to \mathbb{R}\)

To make sure the algorithms in the composition are distinct:

• They are trained on (random) sub-samples
  — bagging = bootstrap aggregation. [Breiman, 1996]: sub-samples of length \(\ell\) with repetitions. Ratio of objects falling into the sample \(\left( 1 - \frac{1}{e}\right) \approx 0.632\)
• Or on (random) subsets of features
  — RSM = random subspace method, [Ho, 1998]

Let's take \(Y = \{\pm 1\}, b_t: X \to \{-1, {\color{orange}[0]}, +1\}, C(b) = \text{sign}(b)\)

Weighted voting

\(a(x) = \text{sign}\left( \sum\limits_{t=1}^T \alpha_t b_t (x)\right), x \in X\)

Quality functional of composition

Number of errors in \(X^\ell\): \(Q_T = \sum\limits_{i=1}^\ell \left[y_i \sum\limits_{t=1}^T \alpha_t b_t(x_i) < 0 \right]\)

• Fix \(\alpha_1 b_1(x), \dots, \alpha_{t-1} b_{t-1}(x)\) when adding \(\alpha_{t}b_t(x)\)
• Smooth approximation of the threshold loss function \([M < 0]\)

The quality functional \(Q_T\) is estimated from above:

\[Q_T \leq \tilde Q_T = \sum_{i=1}^\ell \underbrace{\exp(-y_i \sum_{t=1}^{T-1} \alpha_t b_t (x_i))}_{w_i} \exp(-y_i \alpha_T b_T (x_i))\]

Normalized weights: \[\tilde W^\ell = (\tilde w_1, \dots, \tilde w_\ell), \tilde w_i = w_i / \sum\limits_{j=1}^\ell w_j\]

Weighted number of erroneous (negative) and correct (positive) classifications for weights vector \(U^\ell = (u_1, \dots, u_\ell)\):

\(N(b, U^\ell) = \sum\limits_{i=1}^\ell u_i [b(x_i) = -y_i]\)

\(P(b, U^\ell) = \sum\limits_{i=1}^\ell u_i [b(x_i) = y_i]\)

\((1 - N - P)\) — weighted number of classification refusals

Assume there are no rejections, \(b_t: X \to \{\pm 1 \}\). Then \(P = 1 − N\)

Theorem (Freund, Schapire, 1995)

Suppose that for any normalized weight vector \(U^\ell\) there exists an algorithm \(b \in B\), classifying the sample at least slightly better than at random: \(N(b, U^\ell) < \frac12\).

Then the minimum of the functional \(\tilde Q_T\) is reached when

\[b_T = \arg\min_{b \in B} N (b, \tilde W^\ell), \quad \alpha_T = \frac12 \ln\frac{1-N(b_T, \tilde W^\ell)}{N(b_T, \tilde W^\ell)}\]

1. Initialize object weights: \(w_i = \frac{1}{\ell}, i = 1, \dots, \ell\)
2. for all \(t = 1, \dots, T\)

Train the basic algorithm: \(b_t = \arg \min\limits_b N(b, W^\ell)\)

\(\alpha_t = \frac12 \ln \frac{1-N(b, W^\ell)}{N(b, W^\ell)}\)

3. Update object weights: \(w_i = w_i \exp(-\alpha_t y_i b_t(x_i)), i = 1, \dots, \ell\)
4. Normalize object weights: \[ w_0 = \sum_{j=1}^\ell w_j ,\ w_i = w_i / w_0, i = 1, \dots, \ell\]

• Base classifiers (weak classifiers)
  • Decision trees — most commonly used
  • Threshold rules (data stumps)
    \(B = \{b(x) = [f_j(x) \lessgtr\theta] | j = 1, \dots, n, \theta \in \mathbb{R}\}\)
  • For strong classifiers (as SVM for instance), boosting is usually not effective
• Noise rejection: dispose of objects with the largest \(w_i\)
• Additional stop criterion: increase in error rate on the validation set

• Over-sensitivity to outliers due to $e^M$
• AdaBoost creates "black boxes" — cumbersome, not interpretable compositions of hundreds of algorithms
• It requires fairly large training sets (Bagging can manage with shorter ones)

Ways to eliminate:

• Other approximations to the threshold loss function
• Continuous real base algorithms $b_t: X \to \mathbb{R}$
• Explicit optimization of margins, without approximation
• Less greedy strategies for growing the composition

Linear (convex) combination of base algorithms:

\[a(x) = \sum\limits_{t=1}^T \alpha_t b_t(x), \ x \in X, \ \alpha_t \in \mathbb{R}_+\]

Quality functional with arbitrary loss function $\mathcal{L}(a, y)$:

\[Q(\alpha, b, X^\ell) = \sum\limits_{i=1}^\ell \mathcal{L} \underbrace{\overbrace{(\sum\limits_{t=1}^{T-1} \alpha_t b_t(x_i)}^{f_{T-1,i}} + \alpha b(x_i)}_{f_{T,i}} , y_i) \to \min\limits_{\alpha, b}\]

$f_{T-1, i}$ — current approximation, $f_{T, i}$ — next approximation

Gradient method of minimization $Q(f) \to \min, f \in \mathbb{R}^\ell$:

$f_0 =$ initial approximation

$f_{T,i} = f_{T-1,i} - \alpha g_i, \ i = 1, \dots, \ell$

$g_i = \mathcal{L}^\prime (f_{T-1, i}, y_i)$ — components of the gradient vector, $\alpha$ — gradient step

Observation: this is very similar to one boosting iteration!

$f_{T,i} = f_{T-1,i} + \alpha b(x_i), \ i = 1, \dots, \ell$

Idea: we will look for such a basic algorithm $b_T$ that the vector $(b_T(x_i))_{i=1}^\ell$ will approximate the antigradient vector $(-g_i)_{i=1}^\ell$ $$b_T = \arg\min\limits_b \sum\limits_{i=1}^\ell (b(x_i) + g_i)^2$$

1. Initialization: $f_i = 0, i = 1, \dots, \ell$
2. for all $t = 1, \dots, \color{orange}{T}$

Basic algorithm approximating antigradient:

$$b_t = \arg \min\limits_b \sum\limits_{i=1}^\ell (b(x_i) + \mathcal{L}^\prime(f_i, y_i))^2$$
3. One-dimensional minimization problem: $\alpha_t = \arg \min\limits_{\alpha > 0} \sum\limits_{i=1}^\ell \mathcal{L}(f_i + \alpha b_t(x_i), y_i)$
4. Updating the vector of values on the sample objects: $f_i = f_i + \alpha_t b_t(x_i)), i = 1, \dots, \ell$

Advantages:

• Improved quality
• Enhanced convergence
• Reduced training time

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Friedman G. Stochastic Gradient Boosting, 1999

Regression: \(\mathcal{L}(a,y) = (a-y)^2\)

• \(b_T(x)\) is trained on differences \(y_i - \sum\limits_{t=1}^{T-1} \alpha_t b_t (x_i)\)
• If regression is linear, then \(\alpha_t\) doesn't need to be trained

Classification: \(\mathcal{L}(a,y) = e^{-ay}, b_t \in \{-1, 0 , +1\}\)

• GB exactly matches AdaBoost

Regression and classification trees (CART):

$$b(x) = \sum\limits_{j=1}^J w_j [x \in R_j]$$

where $R_j$ is the area of space covered by leaf $j$, $w_j$ is the leaf weights, and $J$ is the number of leaves in the tree.

• Ensembles allow solving complex tasks that are poorly solved by individual algorithms
• It's too difficult to train the ensemble as a whole. Therefore, we train basic algorithms one by one
• An important discovery in the mid-90s: boosting's generalizing ability doesn't worsen with the increasing complexity $T$
• Gradient boosting — the most general of all boostings:
  • an arbitrary loss function
  • an arbitrary space of estimates $R$
  • suitable for regression, classification, ranking
• Its stochastic version SGB — better and faster
• Most often, GB is applied to decision trees
• Gradient boosting over Oblivious Decision Trees with categorical features = Catboost

• Boosting is better for large training datasets and classes with complex shape borders
• Bagging and RSM are better for small training datasets
• RSM is better in cases where there are more features than objects, or when there are many non-informative features
• Bagging and RSM can be effectively parallelized, boosting is performed strictly sequentially

Training a random forest:

• Bagging over decision trees, without pruning
• The feature in each tree node is chosen from a random subset of $k$ out of $n$ features

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The number of trees $T$ can be selected based on the out-of-bag criterion — the number of errors on objects $x_i$, if we don't consider the votes of trees for which $x_i$ was training:

$$\text{out-of-bag}(a) = \sum\limits_{i=1}^\ell \left[\text{sign}(\sum\limits_{t=1}^T [x_i \notin U_t] b_t(x_i)) \neq y_i \right] \to \min$$

This is an unbiased estimate of generalization ability.

Main conclusion

The estimate depends on $|B|$, but not on $T$. Voting does not increase the complexity of the effectively used set of algorithms.