Confused on question 6.

[butterscotch]

Originally Posted by dobrokot

"all 3 points" are points 101,110,111 outside D ? "2 points" are 2 points of given three?

You are right. 2 points of the remaining points in X.

Originally Posted by dobrokot

May be matches outside these 3 points (matches inside D) should be counted too?

The 3 points used in the final score are the 3 points in X outside of D.

[tom.mancino]

Originally Posted by yaser

A possible target function is any function that could have generated the 5 data points in this problem, i.e., any function whose values on these five points all agree with the data.

There are points in the input space here, which are all binary combinations of the 3 input variables from to . For each of these points, a Boolean function may return 0 or 1; hence two possibilities. Therefore, for all 8 points, a Boolean function may return (8 times) possibilities, which gives us the number of different Boolean functions .

Dr. Yaser or anyone, I am a little lost still on this problem set, I think due to a fundamental lack of mathematical knowledge (i.e my fault - I am completely self taught). I am able to visualize the entire boolean set of 8 possible points (000 - 111) - i.e. the total Xn set, after a little google assistance on boolean number theory but then I get lost in attempting to understand how to compare the other Boolean functions not in X to derive the T/F values. I am assuming it is something fundamental I am missing in boolean mathematics, but am not sure.

[butterscotch]

no worries!

x_n is a vector of 3 values. x_n = [x_n1, x_n2, x_n3]. Each value can be 0 or 1. So there could 8 (2 * 2 * 2) distinct set of x_n vector values.

In digital logic, the boolean values true/false are represented as 1 and 0. 1 is true and 0 is false.

in 6c) The problem defines g as XOR: "if the number of 1's in x is odd, g returns 1; if it is even g returns 0".
Consider x_n = [1,0,0] then, the number of 1 in this example is 1, which is odd, so g returns 1. so g(x_n) = 1.

[lhamilton]

Sorry, I am also a bit confused on question #6.

Specifically, I want to understand 6(d). It says g returns the opposite of the XOR function: if the number of 1s is odd, it returns 0, otherwise it returns 1.

Here is what I am unclear on. Is the meaning of 6(d) that the hypothesis on set D is simply D, and then outside of D it is this opposite of XOR function? Or is 6(d) trying to define a hypothesis for the entire dataset?

Clearly, 6(d) does the exact wrong thing on D, so by definition there are no target functions that satisfy 6(d) if that's the function defined on the whole dataset. But if he's only describe what g does outside of D, then it's a totally valid target function.

Let me ask my question a different way; perhaps that will be clearer.

For 6(d), is g[0,0,0] = 0 or is g[0,0,0]=1?

[butterscotch]

"We want to determine the hypothesis that agrees the most with the possible target functions." and we are measuring this by counting how many of the 3 points not in D, agree with the hypothesis for each of the 8 target functions. In 6(a) & 6(b), the hypothesis is only defined on the last three points. Although 6(c) & (d) are not, it is known to us what f(x) is for points in D. So I think you are more interested in g[1,1,1], g[1,1,0], g[1,0,1].

[kumarpiyush]

Originally Posted by noahdavis

Thank you - I understand now. For some reason it took me a leap to figure out how to build the "target function" such that it could be measured as stated in the problem. Originally, I had a list of 8 "functions" - but each function was just simply one of the 8 permutations where a permutation was an input point and a possible output.

Does this mean the combinations of (000) ,(001) upto (111) are the 8 target functions?

[kumarpiyush]

Originally Posted by yaser

Possible target function is a notion introduced in this problem in order to make a point about learning. In general, there is one target function, albeit unknown. Here we spell out "unkown" by considering all the possibilities the target function can assume. We can afford to do that here because there is only a finite number of possibilities.

Hypotheses are the products of learning that try to approximate the target function. In this problem, we prescribe different learning scenarios that result in different hypotheses, then attempt to grade these hypotheses. We grade them according to how well each of them approximates the target function. The twist is that we consider all possible target functions and grade the hypothesis according to how well it approximates each of these possible targets.

I understood it now :-)

[lhamilton]

I still don't think I'm interpreting this question correctly.

For 6(d), the function described does not match the data set D.

So, given that, am I correct in thinking that for hypothesis 6(d) there are zero target functions that match the hypothesis?

Because, by definition, a target function must agree with the given data set D. Right?

[butterscotch]

Yes. the target function agrees with D and there are 8 of them.

Now we want to determine the hypothesis that agrees the most with the possible target functions.

Problem 6 defines this measurement as counting how many of the target functions match with each hypothesis on the three points.

[Manny]

I'm confused, were we supposed to work this out by hand or were we supposed to code it out?