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A Correction of Pseudo Log-Likelihood Method

Shi Feng, Nuoya Xiong, Zhijie Zhang, Wei Chen

TL;DR

This paper gives a counterexample that the maximum pseudo log-likelihood estimation fails and provides a solution to correct the algorithms in \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}.

Abstract

Pseudo log-likelihood is a type of maximum likelihood estimation (MLE) method used in various fields including contextual bandits, influence maximization of social networks, and causal bandits. However, in previous literature \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}, the log-likelihood function may not be bounded, which may result in the algorithm they proposed not well-defined. In this paper, we give a counterexample that the maximum pseudo log-likelihood estimation fails and then provide a solution to correct the algorithms in \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}.

A Correction of Pseudo Log-Likelihood Method

TL;DR

This paper gives a counterexample that the maximum pseudo log-likelihood estimation fails and provides a solution to correct the algorithms in \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}.

Abstract

Pseudo log-likelihood is a type of maximum likelihood estimation (MLE) method used in various fields including contextual bandits, influence maximization of social networks, and causal bandits. However, in previous literature \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}, the log-likelihood function may not be bounded, which may result in the algorithm they proposed not well-defined. In this paper, we give a counterexample that the maximum pseudo log-likelihood estimation fails and then provide a solution to correct the algorithms in \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}.
Paper Structure (4 sections, 4 theorems, 19 equations)

This paper contains 4 sections, 4 theorems, 19 equations.

Table of Contents

  1. Problem Description
  2. Counter-Examples
  3. The Solution
  4. Conclusion

Key Result

Lemma 1

By doing the conversion above, we can replace function $\mu$ by $g$ such that $\lim_{x\rightarrow +\infty}g(x)=+\infty$, $g$ is monotone increasing and twice differentiable, $g$ satisfies Assumptions asm.1 and asm.2, and when $x$ is in the range $\left[-\sum_{i=1}^d\mathsf{ReLU}(-\theta^*_i),\sum_{i

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1: Main Theorem
  • proof