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Wilmington, North Carolina once had a thriving Black middle class and an elected interracial government. In 1898 white supremacists used violence to overthrow that government, kill Black residents and drive many others from the city.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In machine learning and computational learning theory, LogitBoost is a boosting algorithm formulated by Jerome Friedman, Trevor Hastie, and Robert Tibshirani.
The original paper casts the AdaBoost algorithm into a statistical framework.[1] Specifically, if one considers AdaBoost as a generalized additive model and then applies the cost function of logistic regression, one can derive the LogitBoost algorithm.[2]
LogitBoost can be seen as a convex optimization. Specifically, given that we seek an additive model of the form
the LogitBoost algorithm minimizes the logistic loss:
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In machine learning and computational learning theory, LogitBoost is a boosting algorithm formulated by Jerome Friedman, Trevor Hastie, and Robert Tibshirani. The original paper casts the AdaBoost algorithm into a statistical framework. Specifically, if one considers AdaBoost as a generalized additive model and then applies the cost function of logistic regression, one can derive the LogitBoost algorithm.
In computer science, a logistic model tree (LMT) is a classification model with an associated supervised training algorithm that combines logistic regression (LR) and decision tree learning. Logistic model trees are based on the earlier idea of a model tree: a decision tree that has linear regression models at its leaves to provide a piecewise linear regression model (where ordinary decision trees with constants at their leaves would produce a piecewise constant model). In the logistic variant, the LogitBoost algorithm is used to produce an LR model at every node in the tree; the node is then split using the C4.5 criterion. Each LogitBoost invocation is warm-started from its results in the parent node. Finally, the tree is pruned. The basic LMT induction algorithm uses cross-validation to find a number of LogitBoost iterations that does not overfit the training data. A faster version has been proposed that uses the Akaike information criterion to control LogitBoost stopping.
Mariusz Ziółko (born September 18, 1946, in Hamburg) is a Polish Professor of automatics and signal processing. He graduated in 1970 from AGH University of Science and Technology, where he also received PhD in 1973 and habilitation in 1990. In 2001 he received a title of a professor of technical studies from a president of Poland. He was a DAAD stipendist at University of Wuppertal. He was a visiting professor at Fachhochule St. Pölten at Austria in 2005. He is a professor in AGH Department of Electronics, where he leads a Digital Signal Processing Group. He was a member of several scientific organisations including IEEE, SIAM, EURASIP, ISCA and a chair of several conferences. He is a coauthor of pioneer publications from mathematical optimization models in biology. His works had also an important impact on Polish speech recognition technologies.
In machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). Given X {\displaystyle {\mathcal {X}}} as the space of all possible inputs (usually X ⊂ R d {\displaystyle {\mathcal {X}}\subset \mathbb {R} ^{d}} ), and Y = { − 1 , 1 } {\displaystyle {\mathcal {Y}}=\{-1,1\}} as the set of labels (possible outputs), a typical goal of classification algorithms is to find a function f : X → Y {\displaystyle f:{\mathcal {X}}\to {\mathcal {Y}}} which best predicts a label y {\displaystyle y} for a given input x → {\displaystyle {\vec {x}}} . However, because of incomplete information, noise in the measurement, or probabilistic components in the underlying process, it is possible for the same x → {\displaystyle {\vec {x}}} to generate different y {\displaystyle y} . As a result, the goal of the learning problem is to minimize expected loss (also known as the risk), defined as I [ f ] = ∫ X × Y V ( f ( x → ) , y ) p ( x → , y ) d x → d y {\displaystyle I[f]=\displaystyle \int _{{\mathcal {X}}\times {\mathcal {Y}}}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy} where V ( f ( x → ) , y ) {\displaystyle V(f({\vec {x}}),y)} is a given loss function, and p ( x → , y ) {\displaystyle p({\vec {x}},y)} is the probability density function of the process that generated the data, which can equivalently be written as p ( x → , y ) = p ( y ∣ x → ) p ( x → ) . {\displaystyle p({\vec {x}},y)=p(y\mid {\vec {x}})p({\vec {x}}).} Within classification, several commonly used loss functions are written solely in terms of the product of the true label y {\displaystyle y} and the predicted label f ( x → ) {\displaystyle f({\vec {x}})} . Therefore, they can be defined as functions of only one variable υ = y f ( x → ) {\displaystyle \upsilon =yf({\vec {x}})} , so that V ( f ( x → ) , y ) = ϕ ( y f ( x → ) ) = ϕ ( υ ) {\displaystyle V(f({\vec {x}}),y)=\phi (yf({\vec {x}}))=\phi (\upsilon )} with a suitably chosen function ϕ : R → R {\displaystyle \phi :\mathbb {R} \to \mathbb {R} } . These are called margin-based loss functions. Choosing a margin-based loss function amounts to choosing ϕ {\displaystyle \phi } . Selection of a loss function within this framework impacts the optimal f ϕ ∗ {\displaystyle f_{\phi }^{*}} which minimizes the expected risk, see empirical risk minimization. In the case of binary classification, it is possible to simplify the calculation of expected risk from the integral specified above. Specifically, I [ f ] = ∫ X × Y V ( f ( x → ) , y ) p ( x → , y ) d x → d y = ∫ X ∫ Y ϕ ( y f ( x → ) ) p ( y ∣ x → ) p ( x → ) d y d x → = ∫ X [ ϕ ( f ( x → ) ) p ( 1 ∣ x → ) + ϕ ( − f ( x → ) ) p ( − 1 ∣ x → ) ] p ( x → ) d x → = ∫ X [ ϕ ( f ( x → ) ) p ( 1 ∣ x → ) + ϕ ( − f ( x → ) ) ( 1 − p ( 1 ∣ x → ) ) ] p ( x → ) d x → {\displaystyle {\begin{aligned}I[f]&=\int _{{\mathcal {X}}\times {\mathcal {Y}}}V(f({\vec {x}}),y)\,p({\vec {x}},y)\,d{\vec {x}}\,dy\\[6pt]&=\int _{\mathcal {X}}\int _{\mathcal {Y}}\phi (yf({\vec {x}}))\,p(y\mid {\vec {x}})\,p({\vec {x}})\,dy\,d{\vec {x}}\\[6pt]&=\int _{\mathcal {X}}[\phi (f({\vec {x}}))\,p(1\mid {\vec {x}})+\phi (-f({\vec {x}}))\,p(-1\mid {\vec {x}})]\,p({\vec {x}})\,d{\vec {x}}\\[6pt]&=\int _{\mathcal {X}}[\phi (f({\vec {x}}))\,p(1\mid {\vec {x}})+\phi (-f({\vec {x}}))\,(1-p(1\mid {\vec {x}}))]\,p({\vec {x}})\,d{\vec {x}}\end{aligned}}} The second equality follows from the properties described above. The third equality follows from the fact that 1 and −1 are the only possible values for y {\displaystyle y} , and the fourth because p ( − 1 ∣ x ) = 1 − p ( 1 ∣ x ) {\displaystyle p(-1\mid x)=1-p(1\mid x)} . The term within brackets [ ϕ ( f ( x → ) ) p ( 1 ∣ x → ) + ϕ ( − f ( x → ) ) ( 1 − p ( 1 ∣ x → ) ) ] {\displaystyle [\phi (f({\vec {x}}))p(1\mid {\vec {x}})+\phi (-f({\vec {x}}))(1-p(1\mid {\vec {x}}))]} is known as the conditional risk. One can solve for the minimizer of I [ f ] {\displaystyle I[f]} by taking the functional derivative of the last equality with respect to f {\displaystyle f} and setting the derivative equal to 0. This will result in the following equation ∂ ϕ ( f ) ∂ f η + ∂ ϕ ( − f ) ∂ f ( 1 − η ) = 0 , ( 1 ) {\displaystyle {\frac {\partial \phi (f)}{\partial f}}\eta +{\frac {\partial \phi (-f)}{\partial f}}(1-\eta )=0,\;\;\;\;\;(1)} where η = p ( y = 1 | x → ) {\displaystyle \eta =p(y=1|{\vec {x}})} , which is also equivalent to setting the derivative of the conditional risk equal to zero. Given the binary nature of classification, a natural selection for a loss function (assuming equal cost for false positives and false negatives) would be the 0-1 loss function (0–1 indicator function), which takes the value of 0 if the predicted classification equals that of the true class or a 1 if the predicted classification does not match the true class. This selection is modeled by V ( f ( x → ) , y ) = H ( − y f ( x → ) ) {\displaystyle V(f({\vec {x}}),y)=H(-yf({\vec {x}}))} where H {\displaystyle H} indicates the Heaviside step function. However, this loss function is non-convex and non-smooth, and solving for the optimal solution is an NP-hard combinatorial optimization problem. As a result, it is better to substitute loss function surrogates which are tractable for commonly used learning algorithms, as they have convenient properties such as being convex and smooth. In addition to their computational tractability, one can show that the solutions to the learning problem using these loss surrogates allow for the recovery of the actual solution to the original classification problem. Some of these surrogates are described below. In practice, the probability distribution p ( x → , y ) {\displaystyle p({\vec {x}},y)} is unknown. Consequently, utilizing a training set of n {\displaystyle n} independently and identically distributed sample points S = { ( x → 1 , y 1 ) , … , ( x → n , y n ) } {\displaystyle S=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}} drawn from the data sample space, one seeks to minimize empirical risk I S [ f ] = 1 n ∑ i = 1 n V ( f ( x → i ) , y i ) {\displaystyle I_{S}[f]={\frac {1}{n}}\sum _{i=1}^{n}V(f({\vec {x}}_{i}),y_{i})} as a proxy for expected risk. (See statistical learning theory for a more detailed description.)
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →The Greenwood District.
