Hoeffding inequality for multiple hypothesis

kostya3312 · #1 01-26-2015, 12:20 PM

It's clear for me how inequality works for each hypothesis separately. But I don't understand why we need Hoeffding inequality for multiple hypothesis. If i have training data set of size 'N' then (for fixed tolerance 'e') Hoeffding upper bound is determined for each hypoyhesis. The only thing that remains is to find hypothesis with minimal in-sample rate. Why do we need to consider all hypothesis simultaneously? What information gives us Hoeffding inequality with factor 'M' in it? I undetstand example with coins but I can not relate it to learning problem.

Sorry for my english and thanks.

magdon · #2 01-27-2015, 03:46 PM

Hoeffding for a single hypothesis

tells you that, with high probability,

$|E_{in}(h_1)-E_{out}(h_1)|<\epsilon.$

As you point out, "The only thing that remains is to find hypothesis with minimal in-sample rate." Why would one do this? Because one is confident that Ein is close to Eout for every hypothesis, and so if we find the the hypothesis with minimum Ein, it will likely have minimum Eout. So, to be justified in picking the hypothesis with minimum Ein, we require that

$\forall h_i, |E_{in}(h_i)-E_{out}(h_i)|\le\epsilon.$

Equivalently,

for no $h_i, |E_{in}(h_i)-E_{out}(h_i)|>\epsilon.$

The factor of M comes from using the union bound

$P[for\ no\ h_i, |E_{in}(h_i)-E_{out}(h_i)|>\epsilon]\le$ $P[|E_{in}(h_1)-E_{out}(h_1)|>\epsilon]+P[|E_{in}(h_2)-E_{out}(h_2)|>\epsilon]+\cdots.$

Quote:

Originally Posted by kostya3312

It's clear for me how inequality works for each hypothesis separately. But I don't understand why we need Hoeffding inequality for multiple hypothesis. If i have training data set of size 'N' then (for fixed tolerance 'e') Hoeffding upper bound is determined for each hypoyhesis. The only thing that remains is to find hypothesis with minimal in-sample rate. Why do we need to consider all hypothesis simultaneously? What information gives us Hoeffding inequality with factor 'M' in it? I undetstand example with coins but I can not relate it to learning problem.

Sorry for my english and thanks.

kostya3312 · #3 01-29-2015, 07:18 AM

Thank you, Professor!

I do not quite understand the following:

Quote:

Originally Posted by magdon

The factor of M comes from using the union bound

$P[for\ no\ h_i, |E_{in}(h_i)-E_{out}(h_i)|>\epsilon]\le$ $P[|E_{in}(h_1)-E_{out}(h_1)|>\epsilon]+P[|E_{in}(h_2)-E_{out}(h_2)|>\epsilon]+\cdots.$

I thought that the goal is to get the upper bound for probability of event $A = [for\ AT\ LEAST\ one\ hypothesis\ |E_{in}-E_{out}|>\epsilon]$ . That is, for feasibility of learning the probability of this event should be small. In my opinion two events

(mine) and $B = [for\ no\ h_i, |E_{in}-E_{out}|>\epsilon]$ (yours) are different events. Am I right?

My last question is as follows. The LHS of Hoeffding inequality for M hypothesis is $P[|E_{in}(g)-E_{out}(g)|>\epsilon]$ . It implies that event $C = |E_{in}(g)-E_{out}(g)|>\epsilon$ and event

(event

if you are right) are equal. Though I understand the meaning of event

the meaning of event

isn't so clear for me. What it literally means? I think it means $[absolute\ difference\ between\ E_{in}\ and\ E_{out}\ for\ final\ hypothesis\ g\ is\ greater\ than\ \epsilon]$ . Am I right?

magdon · #4 02-19-2015, 11:38 AM

Sorry, there was a typo in my previous message. Yes they are different events. But they are very related events.

$A = [at\ least\ one\ h_i, |E_{in}-E_{out}|>\epsilon]$

$B = [for\ no\ h_i, |E_{in}-E_{out}|>\epsilon]$

P[B]=1-P[A]>=1-M*...

Quote:

Originally Posted by kostya3312

Thank you, Professor!

I do not quite understand the following:

I thought that the goal is to get the upper bound for probability of event $A = [for\ AT\ LEAST\ one\ hypothesis\ |E_{in}-E_{out}|>\epsilon]$ . That is, for feasibility of learning the probability of this event should be small. In my opinion two events

(mine) and $B = [for\ no\ h_i, |E_{in}-E_{out}|>\epsilon]$ (yours) are different events. Am I right?

My last question is as follows. The LHS of Hoeffding inequality for M hypothesis is $P[|E_{in}(g)-E_{out}(g)|>\epsilon]$ . It implies that event $C = |E_{in}(g)-E_{out}(g)|>\epsilon$ and event

(event

if you are right) are equal. Though I understand the meaning of event

the meaning of event

isn't so clear for me. What it literally means? I think it means $[absolute\ difference\ between\ E_{in}\ and\ E_{out}\ for\ final\ hypothesis\ g\ is\ greater\ than\ \epsilon]$ . Am I right?

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