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KhEdz K& Z ^/ E W ^ Æ Ç r Æ ~ ò ò } µ v Z À } Ks/ r í õ Z X } µ v Ç } µ v Ç W } µ o } v í Z î u í ì î U ô í í î ó X î o o P Z v Ç í U î í ô U ð ñ î ï ð X í u } v P ò ñ U î ò ï ï î X î. Û r U ú p z E « C ?. Where S(z;r) = sign(z)maxfjzj r;0gis the softthresholding function, a = W2 ij W iiW jj, b = S ij W ij wTi Dw j, and c = X ij D ij The main cost is in computing wT i Dw j Instead, we maintain U = DW after each coordinate updates, and then compute wT i u j only O(p) ops per updates.

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R ` t @ N g I C X g A. ٧ Y XW V U T S R Q P O N M L f e d c b a ` _ ^ \ Z q p on m l k j i h g ٨ g f e d c b a ` _ ^ \ r qp o n m l k j i h ~ } { z y x wv u ts. Where S(z;r) = sign(z)maxfjzj r;0gis the softthresholding function The key computational bottleneck here is in computing the terms wT i Dw j, which take O(p2) time when implemented naively To address this, 9 proposed to store and maintain U = DW, which reduced the cost to O(p) flops per update However, this is not a strategy we can use when.

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Author Xiaoxi Liu Created Date Z. 1 Prove that VCER, Ja E R and functions f and g such that lim f(x) = 0, lim g(x) = 0, lim f(x) g(x) = C xa Xa xa 0 This is how you show that 0 is an indeterminate form ю 00, and o ♡ are also 1 2 Prove the same way that indeterminate forms 3 Prove that 1º,0°, and 00° are indeterminate forms (You will only get c > 0 this time). Dominating the GOP nomination process, CRA has racked up dozens of big primary victories, including that of US Rep Richard Pombo (RCalif) in 1992, and last week’s special election primary.

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R ∞ −∞ g(x,θ) dx < ∞, then, the following equality holds, d dθ Z ∞ −∞ f(x,θ) dx = Z ∞ −∞ ∂ ∂θ f(x,θ) dx The conditions say that the first derivative of the function must be bounded by another function whose integral is finite Now, we are ready to prove the following theorem. Theorem 6 (Constrained optimization) Let f A ⊂ Rn → R be a continuously differentiable function, where A is an open set Let g A ⊂ Rn → R also be a continuously differentiable function For c ∈ R, let S = g−1(c) = {x g(x) = c} If, among all points in S, the function f has an extreme point at p 0, then ∇f(p. ³ µ Â Ü x K U C ?.

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R í ï } u u µ v À o Ç >/ À v P / v v v & > } v P o D Z'/E' õ X ì ñ 9 ¨ ô ñ U ì ì ì ¨ í î í U ò õ ó ¨ î ì ò U ò õ ó r í ð í î P v Ç } v P v P D } Z > } ~ u } U o À U d µ } o µ u v. BASIC STATISTICS 1 SAMPLES,RANDOMSAMPLING ANDSAMPLESTATISTICS 11 Random Sample The random variables X1,X2,, are called a random sample of size n fromthe populationf(x)if X1,X2,, are mutuallyindependent random variablesand themar ginal probability density function of each Xi is the same function of f(x) Alternatively, X1,X2,, are called independent and identically. ٢ Á À¿ ¾ ½ ¼ » º ¹ ¸ ¶ µ ÌË Ê É È Ç Æ Å Ä Ã Â ٣ É È Ç Æ Å Ä Ã Â Á c b a Ð Ï Î Í Ì Ë Ê o n m l k j i h g f e d.

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R } v v À ( } u Z yD> E/^d ^W ô ì ì r ñ ï } v } o ~ v Æ & v ' U o } Z E/^d E } v o s µ o v o Ç ~Es W Z W l l v À X v X P } À l ô ì ì r ñ ï. CS174 Lecture 10 John Canny Chernoff Bounds Chernoff bounds are another kind of tail bound Like Markoff and Chebyshev, they bound the total amount of probability of some random variable Y that is in the “tail”, ie far from the mean Recall that Markov bounds apply to any nonnegative random variableY and have the form PrY ≥ t ≤Y. P R I O R I T Y D a t e 0 2 / 2 1 / 2 0 0 7.