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What can humans compute in their heads? We are thinking of a variety of cryptographic protocols, games like sudoku, crossword puzzles, speed chess, and so on. For example, can a person compute a function in his or her head so that an eavesdropper with a powerful computer—who sees the responses to random inputs—still cannot infer responses to new inputs? To address such questions, we propose a rigorous model of human computation and associated measures of complexity. We apply the model and measures first and foremost to the problem of 1) humanly computable password generation and then, consider related problems of 2) humanly computable “oneway functions” and 3) humanly computable “pseudorandom generators.” The theory of human computability developed here plays by different rules than standard computability; the polynomial vs. exponential time divide of modern computability theory is irrelevant to human computation. In human computability, the step counts for both humans and computers must be more concrete. As an application and running example, password generation schemas are humanly computable algorithms based on private keys. Humanly computable and/or humanly usable mean, roughly speaking, that any human needing—and capable of using—passwords can if sufficiently motivated generate and memorize a secret key in less thanmore »

Our expanding understanding of the brain at the level of neurons and synapses, and the level of cognitive phenomena such as language, leaves a formidable gap between these two scales. Here we introduce a computational system which promises to bridge this gap: the Assembly Calculus. It encompasses operations on assemblies of neurons, such as project, associate, and merge, which appear to be implicated in cognitive phenomena, and can be shown, analytically as well as through simulations, to be plausibly realizable at the level of neurons and synapses. We demonstrate the reach of this system by proposing a brain architecture for syntactic processing in the production of language, compatible with recent experimental results.

We consider the communication complexity of a number of distributed optimization problems. We start with the problem of solving a linear system. Suppose there is a coordinator together with s servers P1, …, Ps, the ith of which holds a subset A(i) x = b(i) of ni constraints of a linear system in d variables, and the coordinator would like to output an x ϵ ℝd for which A(i) x = b(i) for i = 1, …, s. We assume each coefficient of each constraint is specified using L bits. We first resolve the randomized and deterministic communication complexity in the pointtopoint model of communication, showing it is (d2 L + sd) and (sd2L), respectively. We obtain similar results for the blackboard communication model. As a result of independent interest, we show the probability a random matrix with integer entries in {–2L, …, 2L} is invertible is 1–2−Θ(dL), whereas previously only 1 – 2−Θ(d) was known. When there is no solution to the linear system, a natural alternative is to find the solution minimizing the ℓp loss, which is the ℓp regression problem. While this problem has been studied, we give improved upper or lower bounds for every value ofmore »

Dimensionality reduction is a classical technique widely used for data analysis. One foundational instantiation is Principal Component Analysis (PCA), which minimizes the average reconstruction error. In this paper, we introduce the multicriteria dimensionality reduction problem where we are given multiple objectives that need to be optimized simultaneously. As an application, our model captures several fairness criteria for dimensionality reduction such as the FairPCA problem introduced by Samadi et al. [NeurIPS18] and the Nash Social Welfare (NSW) problem. In the FairPCA problem, the input data is divided into k groups, and the goal is to find a single ddimensional representation for all groups for which the maximum reconstruction error of any one group is minimized. In NSW the goal is to maximize the product of the individual variances of the groups achieved by the common lowdimensinal space.
Our main result is an exact polynomialtime algorithm for the twocriteria dimensionality reduction problem when the two criteria are increasing concave functions. As an application of this result, we obtain a polynomial time algorithm for FairPCA for k=2 groups, resolving an open problem of Samadi et al.[NeurIPS18], and a polynomial time algorithm for NSW objective for k=2 groups. We also give approximation algorithms formore »