We show that evolutionarily stable states in general (nonlinear) population games (which can be viewed as continuous vector fields constrained on a polytope) are asymptotically stable under a multiplicative weights dynamic (under appropriate choices of a parameter called the learning rate or step size, which we demonstrate to be crucial to achieve convergence, as otherwise even chaotic behavior is possible to manifest). Our result implies that evolutionary theories based on multiplicative weights are compatible (in principle, more general) with those based on the notion of evolutionary stability. However, our result further establishes multiplicative weights as a nonlinear programming primitive (on par with standard nonlinear programming methods) since various nonlinear optimization problems, such as finding Nash Wardrop equilibria in nonatomic congestion games, which are well-known to be equipped with a convex potential function, and finding strict local maxima of quadratic programming problems, are special cases of the problem of computing evolutionarily stable states in nonlinear population games.
Discrete evolution rules related to multiplicative weights (and online learning primitives) have also been considered, for example, by @cite_3 , however, such evolution rules are customized by design to achieve good performance in selfish routing. Our analysis demonstrates that multiplicative weights may well serve as a primitive of adaptation in a wide variety of environments (an argument that intuitively corresponds with the success that multiplicative weights-based boosting algorithms in machine learning theory, such as AdaBoost, have met practical applications).
@cite_3
1988790447 2023131636 1982492546 1977980414
In the first part of the paper we consider the problem of dynamically apportioning resources among a set of options in a worst-case on-line framework. The model we study can be interpreted as a broad, abstract extension of the well-studied on-line prediction model to a general decision-theoretic setting. We show that the multiplicative weight-update Littlestone?Warmuth rule can be adapted to this model, yielding bounds that are slightly weaker in some cases, but applicable to a considerably more general class of learning problems. We show how the resulting learning algorithm can be applied to a variety of problems, including gambling, multiple-outcome prediction, repeated games, and prediction of points in Rn. In the second part of the paper we apply the multiplicative weight-update technique to derive a new boosting algorithm. This boosting algorithm does not require any prior knowledge about the performance of the weak learning algorithm. We also study generalizations of the new boosting algorithm to the problem of learning functions whose range, rather than being binary, is an arbitrary finite set or a bounded segment of the real line. Evolutionary computation techniques have had limited capabilities in solving large-scale problems due to the large search space demanding large memory and much longer training times. In the work presented here, a genetic programming like rich encoding scheme has been constructed to identify building blocks of knowledge in a learning classifier system. The fitter building blocks from the learning system trained against smaller problems have been utilized in a higher complexity problem in the domain to achieve scalable learning. The proposed system has been examined and evaluated on four different Boolean problem domains: 1) multiplexer, 2) majority-on, 3) carry, and 4) even-parity problems. The major contribution of this paper is to successfully extract useful building blocks from smaller problems and reuse them to learn more complex large-scale problems in the domain, e.g., 135-bit multiplexer problem, where the number of possible instances is 2 135 ≈ 4 × 10 40 , is solved by reusing the extracted knowledge from the learned lower level solutions in the domain. Autonomous scaling is, for the first time, shown to be possible in learning classifier systems. It improves effectiveness and reduces the number of training instances required in large problems, but requires more time due to its sequential build-up of knowledge. This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle. This paper deals with the design of efficiently computable incentive-compatible mechanisms for combinatorial optimization problems with single-minded agents each possibly having multiple private parameters. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone fully polynomial time approximation scheme (FPTAS). This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multiunit auctions. The best previous result for such auctions was a 2-appro-xi-ma-tion. In addition, we present a monotone PTAS for the generalized assignment problem with any constant number of private parameters per agent. The most efficient way to solve packing integer programs (PIPs) is linear programming-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing, and multicast routing. Our primal-dual approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.