between two steps seem to align into straight lines, with their slopes increasing with $\left<\eta\right>$.[]{data-label="fig:fit-taylor"}](fig5.eps)

Inspired by the appearance of the average loss rate in the lower bound (\[eq:opt-lower-bound\]), we introduce the mean and the variance of the direction dependent losses, $$\left<\eta\right> = \frac1n \sum_{d=0}^{n-1} \eta_d,
\quad
\mbox{and} 
\quad
Q = \frac1n \sum_{d=0}^{n-1} \delta_d^2.$$ By using the Taylor expansion of the success probability function $p_n^{\mathrm{max}}$ around the point $\eta_d=\left<\eta\right>$, the deviations from the uniform loss case can be well estimated at small degrees of non uniformity. Using the permutation symmetry of $p_n^{\mathrm{max}}$ we can express the Taylor series as $$p_n^{\mathrm{max}}(\{\eta\}) = p_n^{\mathrm{max}}(\left<\eta\right>) +
B Q^2 + C W^3 + O(\delta_d^4),
\label{eq:pmax-Taylor}$$ where $W^3 = 1/n \sum_k \delta_k^3$. We notice that $Q$ may be regarded as the mean deviation of $\{\eta\}$ as a distribution, and hence it is a well-defined statistical property of the random noise. In other words, as long as a second order Taylor expansion gives an acceptable approximation, the probability of success depends only on the statistical average and variance ($\left<\eta\right>$, $Q$) of the noise and not on the specific values of $\{\eta\}$. Using numerical simulations, we have determined the values of $B$ up to rank $n=10$, and 