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Maximizing Generalized Relative Entropy for Count Vectors under Linear Constraints and its Concentration Property


Core Concepts
The paper introduces a generalization of relative entropy to non-negative count vectors, and shows that under linear constraints, this generalized relative entropy exhibits a concentration phenomenon around its maximum value and the vector that maximizes it.
Abstract
The paper presents a new generalization of relative entropy, called the generalized relative entropy G(x||y), for non-negative vectors x and y. This extends the previously introduced generalized entropy G(x) for non-negative vectors. The key points are: The generalized relative entropy G(x||y) is defined and its basic properties are established, such as non-negativity, concavity, and relationship to ordinary relative entropy. The generalization is motivated by a combinatorial setting involving the allocation of balls to bins of varying sizes, which provides an interpretation for G(x||y) in terms of the number of realizations of a count vector x under a "prior" vector y. The problem of maximizing G(x||y) under linear constraints is studied, including the dualization of the optimization problem. The paper shows that the generalized relative entropy G(x||y) exhibits a concentration phenomenon around its maximum value, as well as around the vector x* that maximizes it. This concentration becomes exponentially stronger as the problem size increases through scaling. A probabilistic formulation is also presented, where the concentration results are extended to the probability of the count vector with maximum generalized relative entropy under a given prior. Several examples are provided to illustrate the key results on concentration. The paper provides a comprehensive theoretical analysis of this generalized relative entropy measure and its concentration properties, with applications in areas like information theory and combinatorial optimization.
Stats
The sum of the elements in the optimal count vector x* is bounded as |n* - s*| ≤ m/2, where n* is the sum of the elements in x* and s* is the sum of the elements in the real-valued optimal solution. The L-infinity distance between the optimal count vector x* and the real-valued optimal solution x* is bounded as ||x* - x*||_∞ ≤ 1/2. The L1 distance between the optimal count vector x* and the real-valued optimal solution x* is bounded as ||x* - x*||_1 ≤ m/2. The L1 distance between the density vector f* corresponding to the optimal count vector x* and the density vector χ* corresponding to the real-valued optimal solution x* is bounded as ||f* - χ*||_1 ≤ m/n*.
Quotes
"The logical strength of the concentration property is that it follows from a purely combinatorial argument, with minimal assumptions. Notably, it need not involve any notion of probability or randomness." "The generalized relative entropy presented here contains the generalized entropy of [Oik17] as a simple special case, so we point out some important new or improved results."

Deeper Inquiries

How can the concentration results be extended to other types of constraints beyond linear constraints

To extend the concentration results to constraints beyond linear constraints, we can consider incorporating non-linear constraints into the optimization problem. This can be achieved by formulating the constraints using non-linear functions of the decision variables. By doing so, we can analyze the concentration phenomenon for a broader range of optimization problems that involve non-linear constraints. Additionally, techniques such as convex relaxation or approximation methods can be used to handle non-linear constraints and still derive concentration properties for the generalized relative entropy.

What are the potential applications of the generalized relative entropy and its concentration properties in areas like information theory, combinatorial optimization, or machine learning

The generalized relative entropy and its concentration properties have various potential applications in different fields: Information Theory: In information theory, the concentration properties of relative entropy can be utilized to analyze the efficiency of information encoding and decoding processes. Understanding how the relative entropy concentrates around optimal solutions can lead to improvements in data compression and transmission. Combinatorial Optimization: In combinatorial optimization, the concentration of relative entropy for count vectors can be applied to problems such as graph theory, network flow optimization, and scheduling. By leveraging the concentration properties, more efficient algorithms can be developed to solve combinatorial optimization problems. Machine Learning: In machine learning, the generalized relative entropy can be used as a measure of dissimilarity between probability distributions. The concentration properties can help in model selection, feature selection, and clustering tasks by providing insights into the stability and robustness of the solutions obtained. Overall, the applications of the generalized relative entropy and its concentration properties span across various domains where optimization, information theory, and probabilistic modeling play a crucial role.

How can the theoretical insights from this work inform the design of more efficient algorithms for solving optimization problems involving count vectors under constraints

The theoretical insights from the work on generalized relative entropy and its concentration properties can inform the design of more efficient algorithms for solving optimization problems involving count vectors under constraints in the following ways: Algorithm Design: By understanding the concentration properties of the relative entropy, algorithms can be designed to exploit the concentration phenomenon to converge faster to optimal solutions. Techniques such as stochastic optimization, convex relaxation, and dualization can be employed to leverage the concentration properties effectively. Constraint Handling: The insights from the theoretical analysis can guide the development of algorithms that can handle complex constraints efficiently. By incorporating tolerances and error bounds into the optimization process, algorithms can be more robust and capable of finding near-optimal solutions even in the presence of constraints. Scalability and Performance: The theoretical insights can help in designing algorithms that scale well to large optimization problems involving count vectors. By considering the concentration thresholds and scaling factors, algorithms can be optimized for performance and efficiency in handling high-dimensional data. In conclusion, the theoretical insights from the study of generalized relative entropy and its concentration properties can pave the way for the development of advanced algorithms that are tailored to efficiently solve optimization problems with count vectors under constraints.
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