Mixing
A Multiplicative version of McDiarmid’s Inequality like the one of Chernoff-Hoeffding Bounds
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McDiarmid's Inequality basically says the following:
Let $X_1, X_2, X_3, ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the $i$-th variable changes the value of $f$ by at most $c_i$, then
$$Pr(f > E[f]+t), Pr(f < E[f] -t) leq expleft(frac{-2t^2}{sum_i c_i^2}right)$$
Is there a known multiplicative version of this inequality, i.e. we have $Pr(f > (1+epsilon)E[f])$ on the left hand side.
The Chernoff-Hoeffding Bounds actually have the corresponding two versions:
$$Pr[X > E[X] +t] leq exp(-2t^2/n)$$
$$Pr[X > (1+epsilon)E[X]] < exp(-epsilon^2E[X]/3).$$
I am wondering anyone has thought about it for McDiarmid's Inequality. Is it possible to derive the same multiplicative version just by going though the proofs of both of them?
probability inequality
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
$endgroup$
add a comment |
$begingroup$
McDiarmid's Inequality basically says the following:
Let $X_1, X_2, X_3, ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the $i$-th variable changes the value of $f$ by at most $c_i$, then
$$Pr(f > E[f]+t), Pr(f < E[f] -t) leq expleft(frac{-2t^2}{sum_i c_i^2}right)$$
Is there a known multiplicative version of this inequality, i.e. we have $Pr(f > (1+epsilon)E[f])$ on the left hand side.
The Chernoff-Hoeffding Bounds actually have the corresponding two versions:
$$Pr[X > E[X] +t] leq exp(-2t^2/n)$$
$$Pr[X > (1+epsilon)E[X]] < exp(-epsilon^2E[X]/3).$$
I am wondering anyone has thought about it for McDiarmid's Inequality. Is it possible to derive the same multiplicative version just by going though the proofs of both of them?
probability inequality
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
$endgroup$
add a comment |
$begingroup$
McDiarmid's Inequality basically says the following:
Let $X_1, X_2, X_3, ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the $i$-th variable changes the value of $f$ by at most $c_i$, then
$$Pr(f > E[f]+t), Pr(f < E[f] -t) leq expleft(frac{-2t^2}{sum_i c_i^2}right)$$
Is there a known multiplicative version of this inequality, i.e. we have $Pr(f > (1+epsilon)E[f])$ on the left hand side.
The Chernoff-Hoeffding Bounds actually have the corresponding two versions:
$$Pr[X > E[X] +t] leq exp(-2t^2/n)$$
$$Pr[X > (1+epsilon)E[X]] < exp(-epsilon^2E[X]/3).$$
I am wondering anyone has thought about it for McDiarmid's Inequality. Is it possible to derive the same multiplicative version just by going though the proofs of both of them?
probability inequality
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
$endgroup$
McDiarmid's Inequality basically says the following:
Let $X_1, X_2, X_3, ldots, X_n$ denote independent random variables and $f$ is a function of $n$ real arguments. If changing the value of the $i$-th variable changes the value of $f$ by at most $c_i$, then
$$Pr(f > E[f]+t), Pr(f < E[f] -t) leq expleft(frac{-2t^2}{sum_i c_i^2}right)$$
Is there a known multiplicative version of this inequality, i.e. we have $Pr(f > (1+epsilon)E[f])$ on the left hand side.
The Chernoff-Hoeffding Bounds actually have the corresponding two versions:
$$Pr[X > E[X] +t] leq exp(-2t^2/n)$$
$$Pr[X > (1+epsilon)E[X]] < exp(-epsilon^2E[X]/3).$$
I am wondering anyone has thought about it for McDiarmid's Inequality. Is it possible to derive the same multiplicative version just by going though the proofs of both of them?
probability inequality
probability inequality
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
edited Jan 20 at 12:04
Lee David Chung Lin
4,37031241
4,37031241
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
asked Dec 4 '12 at 21:26
Steven WuSteven Wu
111
111
add a comment |
add a comment |
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