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Journal ArticleDOI

Shape Constraints in Economics and Operations Research

01 Nov 2018-Statistical Science (Institute of Mathematical Statistics)-Vol. 33, Iss: 4, pp 527-546
TL;DR: This paper briefly reviews an illustrative set of research utilizing shape constraints in the economics and operations research literature and highlights the methodological innovations and applications with a particular emphasis on utility functions, production economics and sequential decision making applications.
Abstract: Shape constraints, motivated by either application-specific assumptions or existing theory, can be imposed during model estimation to restrict the feasible region of the parameters. Although such restrictions may not provide any benefits in an asymptotic analysis, they often improve finite sample performance of statistical estimators and the computational efficiency of finding near-optimal control policies. This paper briefly reviews an illustrative set of research utilizing shape constraints in the economics and operations research literature. We highlight the methodological innovations and applications, with a particular emphasis on utility functions, production economics and sequential decision making applications.
Citations
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Book
01 Jan 1992

222 citations

Posted Content
TL;DR: In this paper, the authors provide a nonparametric characterization of a general collective model for household consumption, which includes externalities and public consumption, and establish testable necessary and sufficient conditions for data consistency with collective rationality that only include observed price and quantity information.
Abstract: We provide a nonparametric characterization of a general collective model for household consumption, which includes externalities and public consumption. Next, we establish testable necessary and sufficient conditions for data consistency with collective rationality that only include observed price and quantity information. These conditions have a similar structure as the generalized axiom of revealed preference for the unitary model, which is convenient from a testing point of view. In addition, we derive the minimum number of goods and observations that enable the rejection of collectively rational household behavior. Copyright The Econometric Society 2007.(This abstract was borrowed from another version of this item.)

19 citations

Proceedings Article
06 Dec 2020
TL;DR: In this paper, the authors prove that hard affine shape constraints on function derivatives can be encoded in kernel machines, which represent one of the most flexible and powerful tools in machine learning and statistics.
Abstract: Shape constraints (such as non-negativity, monotonicity, convexity) play a central role in a large number of applications, as they usually improve performance for small sample size and help interpretability. However enforcing these shape requirements in a hard fashion is an extremely challenging problem. Classically, this task is tackled (i) in a soft way (without out-of-sample guarantees), (ii) by specialized transformation of the variables on a case-by-case basis, or (iii) by using highly restricted function classes, such as polynomials or polynomial splines. In this paper, we prove that hard affine shape constraints on function derivatives can be encoded in kernel machines which represent one of the most flexible and powerful tools in machine learning and statistics. Particularly, we present a tightened second-order cone constrained reformulation, that can be readily implemented in convex solvers. We prove performance guarantees on the solution, and demonstrate the efficiency of the approach in joint quantile regression with applications to economics and to the analysis of aircraft trajectories, among others.

14 citations

Posted Content
TL;DR: This framework can solve instances of the convex regression problem with $n=10^5$ and $d=10$---a QP with 10 billion variables---within minutes; and offers significant computational gains compared to current algorithms.
Abstract: We present new large-scale algorithms for fitting a subgradient regularized multivariate convex regression function to $n$ samples in $d$ dimensions -- a key problem in shape constrained nonparametric regression with widespread applications in statistics, engineering and the applied sciences. The infinite-dimensional learning task can be expressed via a convex quadratic program (QP) with $O(nd)$ decision variables and $O(n^2)$ constraints. While instances with $n$ in the lower thousands can be addressed with current algorithms within reasonable runtimes, solving larger problems (e.g., $n\approx 10^4$ or $10^5$) is computationally challenging. To this end, we present an active set type algorithm on the dual QP. For computational scalability, we perform approximate optimization of the reduced sub-problems; and propose randomized augmentation rules for expanding the active set. Although the dual is not strongly convex, we present a novel linear convergence rate of our algorithm on the dual. We demonstrate that our framework can approximately solve instances of the convex regression problem with $n=10^5$ and $d=10$ within minutes; and offers significant computational gains compared to earlier approaches.

14 citations


Cites background from "Shape Constraints in Economics and ..."

  • ...tions research, statistical learning and engineering applications. In economics applications, for example, convexity/concavity arise in modeling utility and production functions, consumer preferences [37,18], among others. In some stochastic optimization problems, value functions are taken to be convex [35]. See also the works of [16,39,3] for other important applications of convex regression. Statistica...

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References
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Book
01 Jan 1988
TL;DR: This book provides a clear and simple account of the key ideas and algorithms of reinforcement learning, which ranges from the history of the field's intellectual foundations to the most recent developments and applications.
Abstract: Reinforcement learning, one of the most active research areas in artificial intelligence, is a computational approach to learning whereby an agent tries to maximize the total amount of reward it receives when interacting with a complex, uncertain environment. In Reinforcement Learning, Richard Sutton and Andrew Barto provide a clear and simple account of the key ideas and algorithms of reinforcement learning. Their discussion ranges from the history of the field's intellectual foundations to the most recent developments and applications. The only necessary mathematical background is familiarity with elementary concepts of probability. The book is divided into three parts. Part I defines the reinforcement learning problem in terms of Markov decision processes. Part II provides basic solution methods: dynamic programming, Monte Carlo methods, and temporal-difference learning. Part III presents a unified view of the solution methods and incorporates artificial neural networks, eligibility traces, and planning; the two final chapters present case studies and consider the future of reinforcement learning.

37,989 citations


"Shape Constraints in Economics and ..." refers background or methods in this paper

  • ...The SPAR algorithm relies on stochastic approximation theory and bears a strong resemblance to the Q-learning algorithm (Watkins and Dayan, 1992) in the reinforcement learning literature (Sutton and Barto, 1998)....

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  • ...…and reinforcement learning (RL) refer to a set of methodologies and algorithms for approximately solving complex sequential decision problems when the state space is large and/or parts of the system are unknown (Bertsekas and Tsitsiklis, 1996; Sutton and Barto, 1998; Bertsekas, 2012; Powell, 2011)....

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  • ...Approximate dynamic programming (ADP) and reinforcement learning (RL) refer to a set of methodologies and algorithms for approximately solving complex sequential decision problems when the state space is large and/or parts of the system are unknown (Bertsekas and Tsitsiklis, 1996; Sutton and Barto, 1998; Bertsekas, 2012; Powell, 2011)....

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Journal ArticleDOI
26 Feb 2015-Nature
TL;DR: This work bridges the divide between high-dimensional sensory inputs and actions, resulting in the first artificial agent that is capable of learning to excel at a diverse array of challenging tasks.
Abstract: The theory of reinforcement learning provides a normative account, deeply rooted in psychological and neuroscientific perspectives on animal behaviour, of how agents may optimize their control of an environment. To use reinforcement learning successfully in situations approaching real-world complexity, however, agents are confronted with a difficult task: they must derive efficient representations of the environment from high-dimensional sensory inputs, and use these to generalize past experience to new situations. Remarkably, humans and other animals seem to solve this problem through a harmonious combination of reinforcement learning and hierarchical sensory processing systems, the former evidenced by a wealth of neural data revealing notable parallels between the phasic signals emitted by dopaminergic neurons and temporal difference reinforcement learning algorithms. While reinforcement learning agents have achieved some successes in a variety of domains, their applicability has previously been limited to domains in which useful features can be handcrafted, or to domains with fully observed, low-dimensional state spaces. Here we use recent advances in training deep neural networks to develop a novel artificial agent, termed a deep Q-network, that can learn successful policies directly from high-dimensional sensory inputs using end-to-end reinforcement learning. We tested this agent on the challenging domain of classic Atari 2600 games. We demonstrate that the deep Q-network agent, receiving only the pixels and the game score as inputs, was able to surpass the performance of all previous algorithms and achieve a level comparable to that of a professional human games tester across a set of 49 games, using the same algorithm, network architecture and hyperparameters. This work bridges the divide between high-dimensional sensory inputs and actions, resulting in the first artificial agent that is capable of learning to excel at a diverse array of challenging tasks.

23,074 citations


"Shape Constraints in Economics and ..." refers background in this paper

  • ...setting of Nascimento and Powell (2013). However, for a multi-dimensional state, Kunnumkal and Topaloglu (2008a) suggest arbitrarily selecting a dimension in which monotonicity is enforced....

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  • ...Recently, RL with deep neural networks has become popular (Mnih et al., 2015; Silver et al., 2016)....

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Journal ArticleDOI
01 Aug 1996
TL;DR: Tests on real and simulated data sets using classification and regression trees and subset selection in linear regression show that bagging can give substantial gains in accuracy.
Abstract: Bagging predictors is a method for generating multiple versions of a predictor and using these to get an aggregated predictor. The aggregation averages over the versions when predicting a numerical outcome and does a plurality vote when predicting a class. The multiple versions are formed by making bootstrap replicates of the learning set and using these as new learning sets. Tests on real and simulated data sets using classification and regression trees and subset selection in linear regression show that bagging can give substantial gains in accuracy. The vital element is the instability of the prediction method. If perturbing the learning set can cause significant changes in the predictor constructed, then bagging can improve accuracy.

16,118 citations

Journal ArticleDOI
28 Jan 2016-Nature
TL;DR: Using this search algorithm, the program AlphaGo achieved a 99.8% winning rate against other Go programs, and defeated the human European Go champion by 5 games to 0.5, the first time that a computer program has defeated a human professional player in the full-sized game of Go.
Abstract: The game of Go has long been viewed as the most challenging of classic games for artificial intelligence owing to its enormous search space and the difficulty of evaluating board positions and moves. Here we introduce a new approach to computer Go that uses ‘value networks’ to evaluate board positions and ‘policy networks’ to select moves. These deep neural networks are trained by a novel combination of supervised learning from human expert games, and reinforcement learning from games of self-play. Without any lookahead search, the neural networks play Go at the level of stateof-the-art Monte Carlo tree search programs that simulate thousands of random games of self-play. We also introduce a new search algorithm that combines Monte Carlo simulation with value and policy networks. Using this search algorithm, our program AlphaGo achieved a 99.8% winning rate against other Go programs, and defeated the human European Go champion by 5 games to 0. This is the first time that a computer program has defeated a human professional player in the full-sized game of Go, a feat previously thought to be at least a decade away.

14,377 citations

Book
01 Jan 1995

12,671 citations


"Shape Constraints in Economics and ..." refers background or methods in this paper

  • ...The ADMM algorithm has been proven to converge for two groups of variables; however, Mazumder et al. (2015) uses three groups of variables and the converge properties of ADMM with three groups is still an open question (Bertsekas, 1999). imsart-sts ver....

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  • ...three groups is still an open question (Bertsekas, 1999)....

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