Trjtdtk

I'm trying to solve the following problem but I've gotten sort of stuck.

So for adaboost, $err_t = fracsum_i=1^Nw_i Pi (h_t(x^(i)) neq t^(i))sum_i=1^Nw_i$

and $alpha_t = frac12ln(frac1-err_terr_t)$

Weights for the next iteration are $w_i' = w_i exp(-alpha_t t^(i) h_t(x^(i)))$
and this assumes $t$ and $h_t$ takes on a value of either $-1$ or $+1$.

I have to show that the error with respect to the new weights $w_i'$ is $frac12$.
i.e., $err_t' = fracsum_i=1^Nw_i' Pi (h_t(x^(i)) neq t^(i))sum_i=1^Nw_i' = frac12$

i.e., we use the weak learner of iteration t and evaluate it according to the new weights, which will be used to learn the $t+1$-st weak learner.

I simplified it so that $w_i'=w_i sqrtfracerr_t1-err_t$ if $w_i$ was correctly classified and $w_i'=w_i sqrtfrac1-err_terr_t$ if $w_i$ was incorrectly classified. I then tried plugging this into the equation for $err_t'=frac12$ and got $fracerr_t1-err_t fracsum_i=1^Nw_i Pi (h_t(x^(i)) = t^(i))sum_i=1^Nw_i Pi (h_t(x^(i)) neq t^(i)) = 1$ but at this point I sort of ran into a dead end and so I'm wondering how one might show the original question.

Thanks for any help!

You're nearly there. The quantity $err_t/(1-err_t)$ is exactly what you need it to be. It might be easier to see if you think about
$$sum_i=1^N w_i Pi(h_t(x^(i))=t^(i))$$
as
$$sum_i: x^(i)text is correctly classified w_i$$
(just using the indicator function to reduce the summation range).

For simplicity, lets define some variables as follows:

$W_C := sum_i=1^Nw_i Pi (h_t(x^(i)) = t^(i))$

$W_I := sum_i=1^Nw_i Pi (h_t(x^(i)) neq t^(i))$

Therefore, $err_t = W_I/(W_C+W_I)$

$a := sqrtfracerr_t1-err_t = sqrtfracW_IW_C$

Now, for new weights we have

$W'_C := sum_i=1^Nw'_i Pi (h_t(x^(i)) = t^(i)) = aW_C$

$W'_I := sum_i=1^Nw'_i Pi (h_t(x^(i)) neq t^(i)) = (1/a)W_I$

Now, as the final step:

$err'_t = fracW'_IW'_I + W'_C = frac(1/a)W_I(1/a)W_I + aW_C oversettimes a= fracW_IW_I + a^2W_C = fracW_IW_I + fracW_IW_CW_C=frac12$