Existing blockchain systems come to consensus on transactions in batches, called blocks. Yet the economic mechanisms those transactions interact with are generally designed to process each individual transaction sequentially, making their behavior reliant on the ordering of transactions within the batch. This abstraction mismatch is the primary source of miner extractible value (MEV), defined as economic value that can be captured by the block proposer (originally the miner) who selects and sequences the transactions to be included in the batch [6]. However, rather than trying to blind the block proposer, or choose a "fair" ordering (which is difficult, if not impossible, to construct in any direct sense on current systems) within a batch, we could alternatively attempt to design economic mechanisms which do not depend on the order of transactions within a block, and instead, process each batch of transactions'all at once'.

We consider a setting where we have a set of data samples that were not uniformly sampled from a population, or where they were sampled from a different population than the one from which we wish to draw some conclusions. A common approach is to assign weights to the samples, so the resulting weighted distribution is representative of the population we wish to study. Here representative means that with the weights, certain expected values or probabilities match or are close to known values for the population we wish to study. A a very simple example, consider a data set where each sample is associated with a person. Our data set is 70% female, whereas we'd like to draw conclusions about a population that is 50% female. A simple solution is to down-weight the female samples, and up-weight the male samples in our data set, so the weighted fraction of females is 50%. As a more sophisticated example, suppose we have multiple groups, for example various combinations of sex, age group, income level, and education, and our goal is to find weights for our samples so the fractions of all these groups matches or approximates known fractions in the population we wish to study. In this case, there will be many possible assignments of weights that match the given fractions, and we need to choose a reasonable one. One approach is to maximize the entropy of the weights, subject to matching the given fractions.

We consider the problem of minimizing a sum of clipped convex functions; applications include clipped empirical risk minimization and clipped control. While the problem of minimizing the sum of clipped convex functions is NP-hard, we present some heuristics for approximately solving instances of these problems. These heuristics can be used to find good, if not global, solutions and appear to work well in practice. We also describe an alternative formulation, based on the perspective transformation, which makes the problem amenable to mixed-integer convex programming and yields computationally tractable lower bounds. We illustrate one of our heuristic methods by applying it to various examples and use the perspective transformation to certify that the solutions are relatively close to the global optimum. This paper is accompanied by an open-source implementation.