Let $\{h_i\}$ be decreasing sequence of $n$ positive reals. Define distribution $p(X=h_i)\propto h_i$ and let $g(s)=E_X[e^{sX}]$ be the moment generating function. For instance, for $h=\{1,\frac{1}{4},\frac{1}{9}\}$, unnormalized distribution over $X$ looks like this
For a given $\epsilon$, I need to find smallest $s$ such that the following is true $$g(-2s)<n\epsilon$$
Is it possible to approximate/bound $s$ by only relying on moments $E[X^i]$ for $i=1,2,\ldots,k$ and small $k$?
Background This problem comes up in average case analysis of gradient descent. $g(-2s)/n$ gives a good approximation to average loss decrease after $s$ steps when minimizing a quadratic with eigenvalues $h_1,h_2,\ldots$ and gradient descent with learning rate 1.
Solving equation above in terms of moments would give a practical way to estimate how many more steps are needed to achieve $\epsilon$ reduction in loss. In practice, $n\approx 10^9$, $s\approx 10^9$, $h_1=1$, $\epsilon \approx 10^{-3}$, $h_i$ probably decay faster than $\frac{1}{i}$
