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Optimization and Regularization Under Arbitrary Objectives

Jared N. Lakhani, Etienne Pienaar

TL;DR

The paper interrogates the practice of applying MH-based two-block MCMC to arbitrary objectives, showing that regularization in this setting is largely dictated by the chosen likelihood form and its sharpness rather than by data alone. By introducing a sharpness parameter $\beta$ and multiple likelihood formulations proportional to the objective, it demonstrates how likelihood curvature concentrates posterior mass and thus drives regularization strength, sometimes reducing the method to a mode-seeking optimizer. The authors apply these ideas to reinforcement-learning-inspired navigation problems, tic-tac-toe, and blackjack, revealing that higher sharpness improves in-sample performance but can hinder out-of-sample generalization and exploration. They further explore hybrids with genetic algorithms and gradient descent, concluding that, at high sharpness, these approaches yield nearly identical in-sample results, highlighting the central claim that excessive likelihood sharpness collapses posterior mass around a dominant mode. The work cautions against interpreting hierarchical MCMC as inherently data-driven regularization and suggests avenues like cooling the sharpness parameter to recover a balance between exploration and exploitation while maintaining robust generalization.

Abstract

This study investigates the limitations of applying Markov Chain Monte Carlo (MCMC) methods to arbitrary objective functions, focusing on a two-block MCMC framework which alternates between Metropolis-Hastings and Gibbs sampling. While such approaches are often considered advantageous for enabling data-driven regularization, we show that their performance critically depends on the sharpness of the employed likelihood form. By introducing a sharpness parameter and exploring alternative likelihood formulations proportional to the target objective function, we demonstrate how likelihood curvature governs both in-sample performance and the degree of regularization inferred by the training data. Empirical applications are conducted on reinforcement learning tasks: including a navigation problem and the game of tic-tac-toe. The study concludes with a separate analysis examining the implications of extreme likelihood sharpness on arbitrary objective functions stemming from the classic game of blackjack, where the first block of the two-block MCMC framework is replaced with an iterative optimization step. The resulting hybrid approach achieves performance nearly identical to the original MCMC framework, indicating that excessive likelihood sharpness effectively collapses posterior mass onto a single dominant mode.

Optimization and Regularization Under Arbitrary Objectives

TL;DR

The paper interrogates the practice of applying MH-based two-block MCMC to arbitrary objectives, showing that regularization in this setting is largely dictated by the chosen likelihood form and its sharpness rather than by data alone. By introducing a sharpness parameter and multiple likelihood formulations proportional to the objective, it demonstrates how likelihood curvature concentrates posterior mass and thus drives regularization strength, sometimes reducing the method to a mode-seeking optimizer. The authors apply these ideas to reinforcement-learning-inspired navigation problems, tic-tac-toe, and blackjack, revealing that higher sharpness improves in-sample performance but can hinder out-of-sample generalization and exploration. They further explore hybrids with genetic algorithms and gradient descent, concluding that, at high sharpness, these approaches yield nearly identical in-sample results, highlighting the central claim that excessive likelihood sharpness collapses posterior mass around a dominant mode. The work cautions against interpreting hierarchical MCMC as inherently data-driven regularization and suggests avenues like cooling the sharpness parameter to recover a balance between exploration and exploitation while maintaining robust generalization.

Abstract

This study investigates the limitations of applying Markov Chain Monte Carlo (MCMC) methods to arbitrary objective functions, focusing on a two-block MCMC framework which alternates between Metropolis-Hastings and Gibbs sampling. While such approaches are often considered advantageous for enabling data-driven regularization, we show that their performance critically depends on the sharpness of the employed likelihood form. By introducing a sharpness parameter and exploring alternative likelihood formulations proportional to the target objective function, we demonstrate how likelihood curvature governs both in-sample performance and the degree of regularization inferred by the training data. Empirical applications are conducted on reinforcement learning tasks: including a navigation problem and the game of tic-tac-toe. The study concludes with a separate analysis examining the implications of extreme likelihood sharpness on arbitrary objective functions stemming from the classic game of blackjack, where the first block of the two-block MCMC framework is replaced with an iterative optimization step. The resulting hybrid approach achieves performance nearly identical to the original MCMC framework, indicating that excessive likelihood sharpness effectively collapses posterior mass onto a single dominant mode.

Paper Structure

This paper contains 59 sections, 2 theorems, 67 equations, 37 figures, 16 tables.

Table of Contents

  1. Background
  2. Introduction
  3. Metropolis-Hastings
  4. $\sigma_\theta^2$: A parameter with a hyperprior
  5. Block 1: Sampling $\boldsymbol{\theta} \mid \sigma_\theta^2, \mathcal{D}$
  6. Block 2: Sampling $\sigma_\theta^2 \mid \boldsymbol{\theta}, \mathcal{D}$
  7. Adaptive Metropolis-Hastings
  8. Adaptive scaling
  9. The Objective $\&$ Regularization
  10. The Likelihood
  11. Likelihood dominance $\&$ implications
  12. The Navigation Problem
  13. Encoding
  14. The game state
  15. Control: Drone movement
  16. ...and 44 more sections

Key Result

Lemma 1

$\frac{1}{n} > \psi^{(1)}(n+1)$ for $n > 0$.

Figures (37)

  • Figure 1: Illustration of the navigation problem (displaying $\boldsymbol{\hat{\theta}}^{\text{GA},\text{(II)}}_{\nu = 4\times 10^{-6}}$ evaluated on an arbitrary out-of-sample initialization).
  • Figure 2: $h(x)$ on interval $[0, 1]$ for $x =k(\boldsymbol{\theta}) \in \{0, 1, \ldots, T\}$.
  • Figure 3: $f(x)$ on interval $[0, 1]$ for $x = k(\boldsymbol{\theta}) \in \{0, 1, \ldots, T\}$
  • Figure 4: $\beta \cdot \log \left[ h(x) \right]$ for $x =k(\boldsymbol{\theta}) \in \{0, 1, \ldots, T\}$ for various $\beta$.
  • Figure 5: $\beta \cdot h\left(\frac{x}{T}\right)$ on interval $[0, \beta]$ for $x =k(\boldsymbol{\theta}) \in \{0, 1, \ldots, T\}$ for various $\beta$.
  • ...and 32 more figures

Theorems & Definitions (5)

  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof