A* Sampling (1411.0030v2)

Abstract: The problem of drawing samples from a discrete distribution can be converted into a discrete optimization problem. In this work, we show how sampling from a continuous distribution can be converted into an optimization problem over continuous space. Central to the method is a stochastic process recently described in mathematical statistics that we call the Gumbel process. We present a new construction of the Gumbel process and A* sampling, a practical generic sampling algorithm that searches for the maximum of a Gumbel process using A* search. We analyze the correctness and convergence time of A* sampling and demonstrate empirically that it makes more efficient use of bound and likelihood evaluations than the most closely related adaptive rejection sampling-based algorithms.

• The paper introduces A* Sampling, which converts sampling from discrete and continuous distributions into an optimization problem.
• It employs a top-down Gumbel process and A* search to ensure exact, efficient sampling by adaptively refining likelihood bounds.
• The method outperforms traditional adaptive rejection sampling by focusing computational resources on pertinent regions and reducing unnecessary evaluations.

Analysis of \OurAlgorithm: A Generic Sampling Tool

This paper discusses a novel sampling algorithm called \OurAlgorithm, which competently converts the problem of sampling from both discrete and continuous probability distributions into an optimization problem, harnessing a concept termed the \ourprocess. This algorithm is pivotal for probabilistic modeling applications where guaranteed inference quality is crucial. By utilizing the top-down construction of a Gumbel process and employing \astar search, the method successfully draws exact samples, presenting potentially pronounced efficiency benefits over existing adaptive rejection sampling techniques.

Overview of Contributions

One of the primary contributions of this work is the introduction and formalization of a stochastic process, identified here as the \ourprocess, which models perturbation over space. This process is detailed as having particularly useful theoretical properties that generalize previous discrete sampling methods, such as the Gumbel-Max trick—an established technique employed for obtaining samples from discrete distributions by calculating the argmax of perturbed energies.

Theoretical Insights

The paper significantly elaborates on constructing and leveraging the \ourprocess to achieve a comprehensive understanding for sampling from a continuous distribution. It outlines how to manage perturbations in continuous domains without necessitating the initialization of infinite random variables. This is achieved through a novel algorithmic technique that draws parallels to the stick-breaking construction used in Dirichlet processes, rather than relying heavily on endless I.D. random variables.

Critically, the paper explores the linearity of the Gumbel process in accommodating bounded differences between two given log densities, improving the computational tractability of \ouralgorithm when sampling complex models that are decomposable into a tractable component ($i(x)$) and a boundable component ($o(x)$). This approach shows promise as it allows the algorithm to adaptively refine regions of interest based on likelihoods and bounds.

Practical Implications and Comparative Analysis

The empirical insights in the paper demonstrate \OurAlgorithm's capability to solve a variety of illustrative and challenging sampling problems with efficiency gains, especially in the situation of tighter bound evaluations. Notably, the comparison with adaptive rejection sampling methods, like \osstar, highlights \OurAlgorithm's strengths in focusing computational resources on pertinent regions—thereby reducing unnecessary evaluations. Moreover, unlike traditional rejection sampling methods, \OurAlgorithm makes effective use of tighter bounds, potentially reducing computational resource needs as the complexity of the problem increases.

Future Directions

The implications of this work suggest various exciting avenues for future research. Notably, the exploration of correlated perturbations in both continuous and discrete dimensions offers an opportunity for developing approximation techniques that retain the algorithm's rigorous sampling properties while improving computational efficiency. Addressing high-dimensional spaces remains another challenging frontier. Solutions incorporating conditional independence structures or dimensionality-reducing techniques might further enhance scalability.

Conclusion

Overall, this research positions \OurAlgorithm as a versatile tool within the probabilistic inference community, yielding both theoretical novelty and practical performance enhancements over existing techniques. It reflects a paradigm shift in sampling methodologies, where optimization strategies can significantly streamline achieving exact, independent samples from complex distributions. As the landscape of probabilistic modeling broadens, methods like \OurAlgorithm have the potential to be integral components of future inference algorithms, especially those embedded in systems requiring robust and exact probabilistic predictions.