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Unformatted text preview: Homework6 Solutions Question 1 As discussed in class the concept of a medium built man can be expressed as an axis parallel rectangle. Therefore, it can be defined in terms of 4 real numbers: a, b, c, d such that: a ≤ height ≤ b, c ≤ weight ≤ d We assume that a consistent algorithm is available. Part I In this part we take as hypotheses 4 floating point numbers, each represented in terms of 4 bytes (32 bits). a How many hypotheses are in the hypotheses class? Answer: The number of hypotheses in the hypotheses class is: r = 2 32 · 4 = 2 128 b How many randomly obtained examples are needed for PAC learning with accuracy parameter epsilon1 = 0 . 1 and confidence parameter δ = 0 . 05? Answer: At least 918 randomly obtained examples are needed. m ≥ 1 epsilon1 ln( r/δ ) = 10 ln(2 128 / . 05) = 917 . 2 c What is the confidence level (what is δ ) if it is known that 1000 randomly chosen examples were used, and the desired accuracy is level is epsilon1 = 0 . 1? Answer: δ ≥ r (1 epsilon1 ) m = 2 128 × . 9 1000 = 5 . 95 E 8 Answer can also be calculated based on formula m ≥ 1 /epsilon1 ln( r/δ ), which gives us δ ≥ 1 . 266 E 5. This value is not as sharp as the one above. d What is the accuracy level (what is epsilon1 ) if it is known that 1000 randomly chosen examples were used, and the desired confidence level is δ = 0 . 05? Answer: δ ≥ r (1 epsilon1 ) m ⇒ (1 epsilon1 ) m ≤ δ/r ⇒ m ln(1 epsilon1 ) ≤ (ln δ ln r ) ⇒ 1 epsilon1 ≤ e (ln δ ln r ) /m ⇒ epsilon1 ≥ 1 e (ln δ ln r ) /m ⇒ epsilon1 ≥ 1 e (ln 0 . 05 ln 2 128 ) / 1000 = 0 . 0876 Answer can also be calculated based on formula m ≥ 1 /epsilon1 ln( r/δ ), which gives us epsilon1 ≥ . 0917. This value is not as sharp as the one above. Part II In this part we assume that we are able to represent arbitrarily accurate axis parallel rectangles....
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 Spring '09
 Dr.mohammednizari

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