THE human brain is the instrument of the high powers of intelligence that distinguish man from all other living creatures. The secret of man's most distinctive attribute is hidden in the texture of his brain, and perhaps will never be fully revealed. Yet from time to time, with the growth of knowledge and the discovery of new methods of approach, we can profitably return to this greatest of all biological problems and get new glimpses of the factors that have made man what he is. Two considerations make the present time appropriate for taking stock of the state of our knowledge of these matters. The emergence of a clearer understanding of the sequence of structural changes in the brain and body of man's ancestors enables us to interpret the physiological factors involved in the widening and deepening of the intellectual powers and to appreciate the conditions essential for the attainment of-such mental growth. In the second place, the new points of view regarding the functions of the cerebral cortex that have emerged from Dr. Henry Head's suggestive clinical investigations prompt one to examine the brain anew and endeavour to integrate the results of his brilliant analysis with those revealed in the study of the evolution of the brain. Whether or not it is yet possible fully to correlate the facts and conclusions of these two disciplines into one coherent body of doctrine, it is well worth while to make the attempt to do so, if for no other reason than to direct attention to new problems that call for solution.

The Human Brain

Proceedings of the American Mathematical Society

The maximum Nash welfare (MNW) solution --- which selects an allocation that maximizes the product of utilities --- is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is unexpectedly, strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy free up to one good --- a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and --- even more so --- in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation, and demonstrate that it scales well on real data. These results lead us to believe that MNW is the ultimate solution for allocating indivisible goods, and underlie its deployment on a popular fair division website.

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The Unreasonable Fairness of Maximum Nash Welfare

We design a new algorithm for batch active learning with deep neural network models. Our algorithm, Batch Active learning by Diverse Gradient Embeddings (BADGE), samples groups of points that are disparate and high-magnitude when represented in a hallucinated gradient space, a strategy designed to incorporate both predictive uncertainty and sample diversity into every selected batch. Crucially, BADGE trades off between diversity and uncertainty without requiring any hand-tuned hyperparameters. We show that while other approaches sometimes succeed for particular batch sizes or architectures, BADGE consistently performs as well or better, making it a versatile option for practical active learning problems.

Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds

This paper is the first of two written by Brad Efron and Ron Thisted studying the frequency distribution of words in the Shakespearean canon. The key idea due to Fisher in the context of sampling of species is simple and elegant. When applied to Shakespeare the idea appears to be preposterous: an author has a personal vocabulary of word species represented by a distribution G, and text is generated by sampling from this distribution. Most results do not require successive words to be sampled independently, which leaves room for individual style and context, but stationarity is needed for prediction and inference. The expected number of words that occur x ≥ 1 times in a large sample of n words is

Estimating the Number of Unseen Species: How Many Words did Shakespeare Know?

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where 0

Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

We give a deterministic polynomial time 2^O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze this is almost the best possible assuming oracle access to independent sets of the matroid. There are two main ingredients in our result: For the first, we build upon recent results of Adiprasito, Huh, and Katz and Huh and Wang on combinatorial hodge theory to derive a connection between matroids and log-concave polynomials. We expect that several new applications in approximation algorithms will be derived from this connection in future. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is log-concave as a function over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the corresponding polynomials.

Log-Concave Polynomials, Entropy, and a Deterministic Approximation Algorithm for Counting Bases of Matroids

Strongly Rayleigh distributions are natural generalizations of product and determinantal probability distributions and satisfy the strongest form of negative dependence properties. We show that the “natural” Monte Carlo Markov Chain (MCMC) algorithm mixes rapidly in the support of a homogeneous strongly Rayleigh distribution. As a byproduct, our proof implies Markov chains can be used to efficiently generate approximate samples of a k-determinantal point process. This answers an open question raised by Deshpande and Rademacher (2010) which was studied recently by Kang (2013); Li et al. (2015); Rebeschini and Karbasi (2015).

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Monte Carlo Markov Chain Algorithms for Sampling Strongly Rayleigh Distributions and Determinantal Point Processes

Recently Cole and Gkatzelis [10] gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash social welfare (NSW). We give constant factor algorithms for a substantial generalization of their problem - to the case of separable, piecewise-linear concave utility functions. We give two such algorithms, the first using market equilibria and the second using the theory of real stable polynomials. Both approaches require new algorithmic ideas.

Nash social welfare for indivisible items under separable, piecewise-linear concave utilities

In this paper we consider a mechanism design problem in the context of large-scale crowdsourcing markets such as Amazon's Mechanical Turk mturk, ClickWorker clickworker, CrowdFlower crowdflower. In these markets, there is a requester who wants to hire workers to accomplish some tasks. Each worker is assumed to give some utility to the requester on getting hired. Moreover each worker has a minimum cost that he wants to get paid for getting hired. This minimum cost is assumed to be private information of the workers. The question then is -- if the requester has a limited budget, how to design a direct revelation mechanism that picks the right set of workers to hire in order to maximize the requester's utility? We note that although the previous work (Singer (2010) chen et al. (2011)) has studied this problem, a crucial difference in which we deviate from earlier work is the notion of large-scale markets that we introduce in our model. Without the large market assumption, it is known that no mechanism can achieve a competitive ratio better than 0.414 and 0.5 for deterministic and randomized mechanisms respectively (while the best known deterministic and randomized mechanisms achieve an approximation ratio of 0.292 and 0.33 respectively). In this paper, we design a budget-feasible mechanism for large markets that achieves a competitive ratio of 1 -- 1/e = 0.63. Our mechanism can be seen as a generalization of an alternate way to look at the proportional share mechanism, which is used in all the previous works so far on this problem. Interestingly, we can also show that our mechanism is optimal by showing that no truthful mechanism can achieve a factor better than 1 -- 1/e, thus, fully resolving this setting. Finally we consider the more general case of submodular utility functions and give new and improved mechanisms for the case when the market is large.

Nima Anari

Papers

Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid

Log-Concave Polynomials, Entropy, and a Deterministic Approximation Algorithm for Counting Bases of Matroids

Monte Carlo Markov Chain Algorithms for Sampling Strongly Rayleigh Distributions and Determinantal Point Processes

Nash social welfare for indivisible items under separable, piecewise-linear concave utilities

Mechanism Design for Crowdsourcing: An Optimal 1-1/e Competitive Budget-Feasible Mechanism for Large Markets