Consider an optimization problem with n binary variables and $d+1$ linear objective functions. Each valid solution in ${0,1}^n$ gives rise to an objective vector in $R^{d+1}$, and one often wants  to enumerate the Pareto optima among them. In the worst case there may be exponentially many Pareto optima; however, it was recently shown that in (a generalization of) the smoothed analysis framework, the expected number is polynomial in $n$ (Roeglin, Teng, FOCS 2009).  Unfortunately, the bound obtained had a rather bad dependence on $d$; roughly $n^{d^d}$. In this paper we show a significantly improved  bound of $n^{2d}$.Our proof is based on analyzing two algorithms. The first algorithm, on input a Pareto optimal $x$, outputs a "testimony" containing clues about $x$'s objective vector, $x$'s coordinates, and the region of  space $B$ in which $x$'s objective vector lies. The second algorithm can be regarded as a SPECULATIVE execution of the first -- it can uniquely reconstruct $x$ from the testimony's clues and just SOME of  the probability space's outcomes. The remainder of the probability space's outcomes are just enough to bound the probability that $x$'s objective vector falls into the region $B$.
This is joint work with Ryan O'Donnell.