As shown in the main text, under the assumption that the influence network is unbiased, our factor baselines are indeed valid control variates. We prove this result below, repeating the statement itself for posterity and providing a supplementary lemma on control variates as a restatement of known results. Let X, Y and Zbe random variables where the law of Xconditional on Z is denoted Pθ(X|Z), and Y is independent of X conditioned on Z; i.e. Then, we have that E[Y θln Pθ(X)] = 0. Proof. Factor baselines are valid control variates if GΣ is true to the MDP (i.e.

A.1 Proof of Monotonicity and Submodularity In Equation (3a), we stated the objective of the knapsack cover to be Remark 1. f+M is monotonically increasing. A.2 Knapsack Cover To find a solution to problem 3, we use the greedy algorithm proposed by Badanidiyuru and Vondrák [2], which deals with submodular maximization subject to a system of lknapsack constraints and with pmatroid constraints. We present an adapted version of the algorithm in Algorithm 2 where l = 1. Theparameter allows us to 16 trade-off solution time and solution quality. In this work, we set = 0.2.

We consider semidefinite programs (SDPs) of size nwith equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n ksuch that X = YY is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem.

However, an emerging issue in allocation problems is data privacy.

We compute the convergence rates of the RSVB approximate posterior and the corresponding optimal value.

Inprior work, ithas been shown that, when the constraints on the factorized variable regularly define a smooth manifold, providedk is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading tothequestion: aresuch points also approximately optimal?

Assume the conditions in Proposition 1 hold.