Mathematical Sciences Division Research Articles

Good Prophets Know When the End Is Near

We consider a class of online decision-making problems with exchangeable actions, where in each period a controller is presented an input type drawn from some stochastic arrival process. The controller must choose an action, and the final objective depends only on the aggregate type-action counts. Such a framework encapsulates many online stochastic variants of common optimization problems with knapsack, bin packing, and generalized assignment as canonical examples. In such settings, we study a natural model-predictive control algorithm. We introduce general conditions under which this algorithm obtains uniform additive loss (independent of the horizon) compared with an optimal solution with full knowledge of arrivals. Our condition builds on the recently introduced compensated coupling technique, providing a unified view of how uniform additive loss arises as a consequence of the geometry of the underlying decision-making problem. Our characterization allows us to derive uniform loss algorithms for several new settings, including the first such algorithm for online stochastic bin packing. It also lets us study the effect of other modeling assumptions, including choice of horizon, batched decisions, and limited computation. In particular, we show that our condition is fulfilled by the above-mentioned problems when the end of the time horizon is known sufficiently long before the end. In contrast, if at a late stage, there is still uncertainty about the end of the time horizon we show that such uniform loss guarantees are impossible to achieve. We use real and synthetic data to illustrate our findings. This paper was accepted by J. George Shanthikumar, data science. Funding: This work was supported by the Division of Mathematical Sciences [Grant 1839346]; the Division of Electrical, Communications and Cyber Systems [Grant 1847393]; and the Army Research Laboratory [Grant W911NF-17-1-0094]. The authors also gratefully acknowledge support from the NSF [Grants ECCS-1847393, CNS-195599, and DMS-1839346], AFOSR [Grant FA9550-23-1-0068], an ARL award [Grant W911NF-17-1-0094], and ARO MURI [Grant W911NF-19-1-0217]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.04307 .

A Superlinearly Convergent Subgradient Method for Sharp Semismooth Problems

Subgradient methods comprise a fundamental class of nonsmooth optimization algorithms. Classical results show that certain subgradient methods converge sublinearly for general Lipschitz convex functions and converge linearly for convex functions that grow sharply away from solutions. Recent work has moreover extended these results to certain nonconvex problems. In this work, we seek to improve the complexity of these algorithms by asking the following question. Is it possible to design a superlinearly convergent subgradient method? We provide a positive answer to this question for a broad class of sharp semismooth functions. Funding: The research of D. Davis was supported by the Division of Mathematical Sciences [Grant 2047637] and the Alfred P. Sloan Foundation.

SPEECH BY THE VICE PRESIDENT OF THE RAS RAS ACADEMICIAN V.Ya. PANCHENKO

On May 22, 2023, meetings of the Division of Mathematical Sciences of the Russian Academy of Sciences and the Division of Physical Sciences of the Russian Academy of Sciences took place, as well as a ceremonial meeting of the RAS Division of Nanotechnologies and Information Technologies (ONIT RAS), dedicated to the 40th anniversary of the ONIT, at which President of the Russian Academy of Sciences G.Ya. Krasnikov and President of the National Research Center Kurchatov Institute M.V. Koval’chuk delivered congratulations.

Convergence Rates for Regularized Optimal Transport via Quantization

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or Lp regularization, general transport costs, and multimarginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for noncompact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for marginals with a finite [Formula: see text]-moment. Funding: This work was supported by the Alfred P. Sloan Foundation [Grant FG-2016-6282] and the Division of Mathematical Sciences [Grants DMS-1812661 and DMS-2106056].

Conversations with Gábor J. Székely

Gábor J. Székely was born in Budapest, Hungary on February 4, 1947. He graduated from Eötvös Loránd University (ELTE) with an M.S. degree in 1970, and a Ph.D. degree in 1971. He received his Candidate Degree from the Hungarian Academy of Sciences in 1976, and the Doctor of Science Degree (D. Sc.) from the Hungarian Academy of Sciences in 1986. Székely joined the Department of Probability Theory of ELTE in 1970. In 1989, he became the founding chair of the Department of Stochastics of the Budapest Institute of Technology (Technical University of Budapest). In 1995, he moved to the United States as a tenured full professor at Bowling Green State University (BGSU) in Bowling Green, Ohio. Before that, in 1990–1991, he was the first Lukacs Distinguished Professor at BGSU. Székely had several visiting positions, for example, at the University of Amsterdam in 1977 and at Yale University in 1989. Between 2006 and 2022, he served as a Program Director in the Statistics Program of the Division of Mathematical Sciences at the U.S. National Science Foundation. Székely has about 250 publications, including 6 books in several languages. In 1988, he received the Rollo Davidson Prize from Cambridge University, jointly with Imre Z. Ruzsa for their work on algebraic probability theory. In 2010, Székely became an Elected Fellow of the Institute of Mathematical Statistics mostly for his works dealing with physics concepts in statistics like energy statistics and distance correlation. He had the fortune to know and work with world-class mathematicians and statisticians like (in chronological order of their first meetings): P. Erdős, A. Rényi, Y. Linnik, B. de Finetti, A. N. Kolmogorov, H. Robbins, G. Pólya, L. Shepp, G. Wahba, C. R. Rao, B. Efron, P. Bickel and E. Seneta.

Mean Field Contest with Singularity

We formulate a mean field game where each player stops a privately observed Brownian motion with absorption. Players are ranked according to their level of stopping and rewarded as a function of their relative rank. There is a unique mean field equilibrium, and it is shown to be the limit of associated n-player games. Conversely, the mean field strategy induces n-player ε-Nash equilibria for any continuous reward function—but not for discontinuous ones. In a second part, we study the problem of a principal who can choose how to distribute a reward budget over the ranks and aims to maximize the performance of the median player. The optimal reward design (contract) is found in closed form, complementing the merely partial results available in the n-player case. We then analyze the quality of the mean field design when used as a proxy for the optimizer in the n-player game. Surprisingly, the quality deteriorates dramatically as n grows. We explain this with an asymptotic singularity in the induced n-player equilibrium distributions. Funding: M. Nutz is supported by an Alfred P. Sloan Fellowship and the Division of Mathematical Sciences of the National Science Foundation [Grants DMS-1812661 and DMS-2106056]. Y. Zhang is supported in part by the Natural Sciences and Engineering Research Council of Canada [NSERC Discovery Grant RGPIN-2020-06290].

Stochastic Optimization with Decision-Dependent Distributions

Stochastic optimization problems often involve data distributions that change in reaction to the decision variables. This is the case, for example, when members of the population respond to a deployed classifier by manipulating their features so as to improve the likelihood of being positively labeled. Recent works on performative prediction identify an intriguing solution concept for such problems: find the decision that is optimal with respect to the static distribution that the decision induces. Continuing this line of work, we show that, in the strongly convex setting, typical stochastic algorithms—originally designed for static problems—can be applied directly for finding such equilibria with little loss in efficiency. The reason is simple to explain: the main consequence of the distributional shift is that it corrupts algorithms with a bias that decays linearly with the distance to the solution. Using this perspective, we obtain convergence guarantees for popular algorithms, such as stochastic gradient, clipped gradient, prox-point, and dual averaging methods, along with their accelerated and proximal variants. In realistic applications, deployment of a decision rule is often much more expensive than sampling. We show how to modify the aforementioned algorithms so as to maintain their sample efficiency when performing only logarithmically many deployments. Funding: This work was supported by the Division of Computing and Communication Foundations [Grant 1740551] and Division of Mathematical Sciences [Grant 1651851].

Around the World

In science news around the world this week, the InSight mission will probe the innards of Mars; an appeals court has ruled that patents on BRCA genes are valid; a federal court in Brazil has stopped work on the controversial Belo Monte Dam; India opened a $12 million, government-backed laboratory to create a new vaccine against HIV; and a controversial attempt to change the name of the U.S. National Science Foundation's Division of Mathematical Sciences is off the table for now.

A Conversation with Dorothy Gilford

In 1946, Public Law 588 of the 79th Congress established the Office of Naval Research (ONR). Its mission was to plan, foster and encourage scientific research in support of Naval problems. The establishment of ONR predates the National Science Foundation and initiated the refocusing of scientific infrastructure in the United States following World War II. At the time, ONR was the only source for federal support of basic research in the United States. Dorothy Gilford was one of the first Heads of the Probability and Statistics program at the Office of Naval Research (1955 to 1962), and she went on to serve as Director of the Mathematical Sciences Division (1962 to 1968). During her time at ONR, Dorothy influenced many areas of statistics and mathematics and was ahead of her time in promoting interdisciplinary projects. Dorothy continued her career at the National Center for Education Statistics (1969 to 1974). She was active in starting international comparisons of education outcomes in different countries, which has influenced educational policy in the United States. Dorothy went on to serve in many capacities at the National Academy of Sciences, including Director of Human Resources Studies (1975 to 1978), Senior Statistician on the Committee on National Statistics (1978 to 1988) and Director of the Board on International Comparative Studies in Education (1988 to 1994). The following is a conversation we had with Dorothy Gilford in March of 2004. We found her to be an interesting person and a remarkable statistician. We hope you agree.

A coupled level set projection method applied to ink jet simulation

A finite difference level set-projection method on rectangular grid is developed for piezoelectric ink jet simulation. The model is based on the Navier-Stokes equations for incompressible two-phase flows in the presence of surface tension and density jump across the interface separating ink and air, coupled to an electric circuit model which describes the driving mechanism behind the process, and a macroscopic contact model which describes the air-ink-wall dynamics. We simulate the axisymmetric flow using a combination of second order projection methods to solve the fluid equations and level set methods to track the air/ink interface. The numerical method can be used to analyze the motion of the interface, breakoff and formation of satellites, and effect of nozzle geometry on droplet size and motion. We focus on close comparison of our numerical ink jet simulation with experimental data. The corresponding author. This author was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract Number DE-AC03-76SF00098, and the Division of Mathematical Sciences of the National Science Foundation.

NSF to Double Number of Math Institutes

American mathematics just multiplied itself by two. On 1 July, the Division of Mathematical Sciences (DMS) at the National Science Foundation (NSF) announced the creation of three new mathematical sciences research institutes, bringing the total number of such NSF-funded institutes to six. DMS's director, Philippe Tondeur, says he has “incredibly high expectations” for the institutes, which he describes as “vessels for start-up activities.”

Ocean acoustic research initiative announced

A new long‐term research initiative in ocean acoustic signal processing has been announced by the Ocean Acoustic Program (Code 11250A) and the Mathematical Sciences Division (Code 1111) of the Office of Naval Research. Ocean acoustic signal processing is defined as the development of phase coherent signal processing formalisms that incorporate the physics of acoustic wave‐field propagation through and scatter from an anisotropic, inhomogeneous stochastic ocean environment. It is motivated by the problems of detection, classification, and source localization in both passive and active sonar systems.The Ocean Acoustic Program and the Mathematical Sciences Division invite the submission of 2‐ or 3‐page planning letters that outline research. The planning letters should include budget estimates for 1‐ to 3‐year research programs that address the key issues given above.

A Course in Bayesian Statistics

A Course in Bayesian Statistics

The Work of the Committee on National Statistics

The Committee on National Statistics was established by the National Research Council (the operating arm of the National Academy of Sciences) in response to growing concern about the adequacy, validity, timeliness, and utility of statistical procedures and information central to major decisions. Although the NRC has had a long-standing interest in the quality of the statistical information basic to public policy formation, the immediate impetus to establish the Committee came from a major recommendation of the President's Commission on Federal Statistics, that an independent, continuing group be established to review the statistical system and recommend improvements [1]. The Commission reported: are convinced that ... a need exists for continuous review of statistical activities, on a selective basis, by a group of broadly representative professionals without direct relationships with the and urged . . that an NAS-NRC committee be established to provide an outside review of statistical activities [2]. The Committee, first appointed in January, 1972, is attached to the Division of Mathematical Sciences in the National Research Council. Members are appointed for three-year terms and serve without recompense. Current members of the Committee are William Kruskal, chairman, Morris Hansen, Stanley Lebergott, Rupert G. Miller, Jr., Frederick Mosteller, I. Richard Savage, Elizabeth Scott, William Shaw, and Conrad Taeuber. Former members include: Douglas Chapman, Cuthbert Daniel and Bernard Greenberg. In addition, the Committee has a small professional staff, all serving part-time: Executive Director, Margaret E. Martin; Research Director, Edward Tufte; Consultants, Hyman B. Kaitz, Walt R. Simmons. The Committee and its staff represent a broad spectrum of disciplines in which statistics are applied, as well as a variety of experience in statistical methodology, data collection, and data analysis. As a group sponsored by the National Academy of Sciences, the Committee is primarily oriented toward giving scientific advice to the government. Yet even its name indicates a somewhat broader interest national statistics rather than federal statistics is intended to connote any statistics of important public concern, whether collected by the government or not. Coming to the Committee months after it had been established, I searched for a detailed statement of its functions or a frame of reference as an aid in guiding future planning. The most direct statement was in the President's Commission's report and might be summarized in three words as, Carry on, chaps. In these circumstances one might think that the early. attention of the Committee would have concentrated on building such a framework and establishing priorities within it. However, it early became apparent, with the wide range of backgrounds and experience represented on the Committee, its infrequent meetings, and the multitude of urgent statistical problems pressing for attention, that we might better proceed with a few critical projects immediately. We were in danger of bogging down in discussions of generalities. Partly, the functions of the Committee on National Statistics may be viewed in terms of what the Committee is not. It is not a group representing the profession, as does the American Statistical Association; it is not a group representing users, as does the Federal Statistics Users' Conference; it is not intended to duplicate the functions of the statistical agencies, nor of academic research, nor of commercial or nonprofit contracting organizations. Rather, it is expected, by selecting significant, broadly applicable projects and approaching them in a creative, multidisciplinary fashion, to focus the expertise of specialists outside the government on important statistical issues. At the same time, these projects must be such that most of them can be funded by the specific agencies which will be the recipients of the advice; and the process of developing the projects must be reasonably compatible with the style of operation of the National Research Council. The NRC studies problems of significance which require scientific or technical competence and for which the resources of the NRC are considered particularly suitable. Despite these constraints, there is no lack of suitable issues to begin on. We are constrained, rather, by the smallness of our staff, the limits of our expertise and the interests of our members and possible sponsors. The I This is a revision of an invited address to the Social Statistics Section, American Statistical Association, New York City, 30 December 1973. * Executive Director, Committee on National Statistics, National Academy of Sciences-National Research Council, 2101 Constitution Ave., N.W., Washington, D.C. 20418.

Numerical Analysis and Partial Differential Equations

By G. Forsythe and P. Rosenbloom London : Chapman & Hall Ltd. Pp. x + 204. Price 60s These two books are numbers II and V in a series forming a joint project of the (U.S.) Office of Naval Research and Applied Mechanics Reviews. A preface common to both volumes, by the Director of the Mathematical Sciences Division, O.N.R., includes the following (condensation by the reviewer): The present articles are aimed at a mathematically literate audience looking for contemporary information on important problems for use in classroom or seminar, in other fields of science and engineering, or for personal interest.

Mathematical Aspects of Subsonic and Transonic Gas Dynamics

By Lipman Bers London : Chapman and Hall Ltd. Pp. xv + 164. Price 62s. In the preface to this, the third volume in Surveys in Applied Mathematics, F. J. Weyl the Director of the Mathematical Sciences Division of the Office of Naval Research gives broadly speaking two aims as the basis for the series. These are (i) the need to make readily available up-to-date results in selected areas of research and (ii) the need to relate contributions from both sides of the iron curtain in order to produce a balanced appreciation of the present state of knowledge in any such area.