Aprendizaje estadístico Gábor Lugosi (Universidad Pompeu Fabra) Kernel methods in a regularization framework Yoonkyung Lee (Ohio State University)
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(35) . Aprendizaje estadístico Gábor Lugosi (Universidad Pompeu Fabra) En este curso se repasarán algunos conceptos de aprendizaje estadístico. Se investigarán los métodos de la minimización empírica del riesgo, la minimización estructural de riesgo, y algunos algoritmos de minimización de funciones convexas de riesgo, como "boosting". Entre las herramientas matemáticas esenciales se discutirán algunas desigualdades de concentración, las nociones de la dimensión Vapnik-Chervonenkis, promedios de Rademacher, etc. En la segunda parte del curso se discutirán algunas ideas básicas de la predicción online de sucesiones "individuales". Este modelo de aprendizaje está estrechamente relacionado con modelos de aprendizaje en la teoría de juegos. Se presentarán algunos resultados sobre algoritmos aleatorizados de predicción.. C2:. Kernel methods in a regularization framework Yoonkyung Lee (Ohio State University)
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(301) P6:. Adaptive estimation of linear functionals by model selection ˜ Carenne LUDENA Instituto Venezolano de Investigaciones Cient´ıficas, Venezuela e-mail: [email protected]. Abstract Consider the following model : Y (t) =< s, t > + √ L (t) , for all t ∈ H, n where H is an Hilbert space with inner product <, > and L is a centered Gaussian process over H such that for all t, u ∈ H, Cov(L(t), L(u)) =< t, u >. Model selection (MS) refers to the problem of selecting a model m leading to a least squares estimator sˆm with close to minimal risk value among a certain collection of possible models Sm ⊂ H. For each m the quadratic risk is Es − sˆm 2 = sm − s2 + dm,n where dm,n is a variance term independent of s , and sm is the projection of s over Sm . Thus the ”correct” choice of model m follows by estimating the bias term in the above bound, which can be done easily, and minimizing the sum. If, however, we consider the problem of estimating T (s), where T is some linear functional, the bias term cannot be simplified nor estimated directly and a new model selection procedure must be developed. We show the procedure is adaptive, up to a constant, and give non asymptotic inequalities for the quadratic risk. An application using a MRA for the problem of estimating a signal or its rth derivative at a given point is developed and minimax rates are seen to hold uniformly over Besov spaces. Simulations are included to illustrate the procedure’s performance.. 1.
(302) P7:. A statistical view at ill posed inverse problems and regularization methods ˜ Carenne LUDENA Instituto Venezolano de Investigaciones Cient´ıficas, Venezuela e-mail: [email protected] We are interested in recovering an unobservable signal x0 based on observations y(ti ) = A(x0 )(ti ) + εi , where A : X → Y is a compact operator with unbounded inverse, X, Y are Hilbert spaces and ti , i = 1, . . . , n is a fixed observation scheme. Although A(x0 ) ∈ Y , we cannot observe this function directly and inference on the unknown function x0 ∈ X will depend on the observed data y(ti ), i = 1, . . . , n. The ill-posedeness of the problem requires the introduction of some kind of regularization procedure, indexed by some regularization parameter α that measures the closeness of the regularized and the original (unregularized) inverse problem. Rules (and algorithms) for the choice of these regularization parameters, with or without prior information on the smoothness properties of the solution are a central issue. Our main interest will be the adaptive choice of α. For this it is necessary to introduce an appropriate penalty function. As examples of regularization procedures we consider linear model selection, Tikhonov methods and iterative procedures such as steepest descent methods. Based on concentration inequalities techniques we derive non asymptotic optimal upper bounds of the mean square error for each regularization method.. 1.
(303) Programa • Lunes 22 – 9:15– 9:30 Inscripci´on – 9:30–11:00 Aprendizaje estad´ıstico Gabor Lugosi – 11:00–11:30 Caf´e y galletas – 11:30–13:00 Kernel methods in a regularization framework Yoonkyung Lee – 13:00–14:00 Desigualdades e identidades de correlaci´on Victor P´ erez–Abreu – 14:00–16:00 Comida – 16:00–17:15 M´etodos Bayesianos para procesamiento de im´agenes Jos´ e Luis Marroqu´ın – 17:15–18:15 Quadratic Markovian probability fields for image binary segmentation. Mariano J. J. Rivera – 19:30–22:00 Taquiza en el CIMATEL • Martes 23 – 9:30–11:00 Aprendizaje estad´ıstico Gabor Lugosi – 11:00–11:30 Caf´e y galletas.
(304) – 11:30–13:00 Kernel methods in a regularization framework Yoonkyung Lee – 13:00–14:00 Adaptive estimation of linear functionals by model selection Carenne Lude˜ na – 14:00–16:00 Comida – 16:00–17:15 Algunas reflexiones sobre tipolog´ıa de modelos con motivo de nichos ecol´ogicos. Miguel Nakamura – 17:15–18:30 A statistical view of ill posed inverse problems and regularization methods. Carenne Lude˜ na • Mi´ercoles 24 – 9:30–11:00 Aprendizaje estad´ıstico Gabor Lugosi – 11:00–11:30 Caf´e y galletas – 11:30–14:00 Regularization and boosting Saharon Rosset – 14:00–16:00 Comida – 16:00–. Tarde libre.. • Jueves 25 – 9:30–11:00 Aprendizaje estad´ıstico Gabor Lugosi – 11:00–11:30 Caf´e y galletas – 11:30–14:00 Regularization and boosting Saharon Rosset – 14:00–16:00 Comida – 16:00–17:30 Kernel methods in a regularization framework Yoonkyung Lee – 17:30–18:00 Estudios de postgrado en Probabilidad y Estad´ıstica en CIMAT. Elo´ısa D´ıaz–Franc´ es.
(305) • Viernes 26 – 9:30–11:00 Aprendizaje estad´ıstico Gabor Lugosi – 11:00–11:30 Caf´e y galletas – 11:30–14:00 Data analytics for marketing decision support Saharon Rosset – 14:00–14:10 Clausura..
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