A New Bound on the Cumulant Generating Function of Dirichlet Processes (2409.18621v1)

Abstract: In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X \sim \text{DP}(\alpha \nu_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $\alpha \mapsto \log \mathbb{E}{X \sim \text{DP}(\alpha \nu_0)}[\exp( \mathbb{E}_X[\alpha f])]$, where $\mathbb{E}_X[f] = \int f dX$. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF $ \log\mathbb{E}{X\sim \text{DP}(\alpha\nu_0)}{\exp(\mathbb{E}_{X}{[f]})} $ for any $\alpha > 0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $\alpha\mathrm{KL}(\nu_0\Vert \cdot)$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.

• The paper establishes a novel non-asymptotic bound on the cumulant generating function of Dirichlet Processes using superadditivity.
• It leverages Varadhan’s integral lemma and reversed KL-divergence to convert asymptotic results into practical concentration bounds.
• The research provides actionable methods for constructing confidence regions and improving policies in nonparametric Bayesian and machine learning applications.

A New Bound on the Cumulant Generating Function of Dirichlet Processes

This paper presents a novel approach to bounding the Cumulant Generating Function (CGF) of Dirichlet Processes (DPs), a critical element in nonparametric Bayesian statistics and numerous applications, such as machine learning and reinforcement learning. The main contribution lies in demonstrating the superadditivity of the function related to the CGF of the DP, and leveraging this property to derive practical concentration bounds.

Key Technical Contributions

• Superadditivity Demonstration: The paper establishes the superadditivity of the function $\alpha \mapsto \log \mathbb{E}_{X \sim \DP(\alpha \nu_0)}[\exp(\mathbb{E}_X[\alpha f])]$, where $f$ is a continuous function over a compact metric space $\Omega$. This property is leveraged using Fekete’s lemma to derive bounds applicable in various contexts.
• Cumulant Generating Function Bound: By employing Varadhan's integral lemma and other theoretical frameworks, the paper converts a known asymptotic large deviation principle into a non-asymptotic upper bound on the CGF for any $\alpha > 0$. The bound is formulated using the scaled reversed Kullback-Leibler divergence, $\alpha \KL{\nu_0}{\cdot}$.

Practical Implications and Applications

The derived bounds are applicable to the construction of confidence regions for sums of independent DPs. This has significant implications in fields requiring nonparametric modeling and analysis, such as topic modeling and statistical learning.

The paper further explores the use of these bounds in the stochastic semi-bandit problem, particularly for the Combinatorial Thompson Sampling (CTS) policy. It shows that the derived bounds can help confirm the statistical efficiency of CTS, aligning its expected regret with known efficient policies like ESCB in specific configurations.

Theoretical Implications

From a theoretical standpoint, this work enriches the understanding of concentration phenomena for DPs by providing a non-asymptotic bound that complements existing asymptotic results. The linkage between the CGF and the reversed KL-divergence adds depth to the mathematical characterization of DPs in Bayesian frameworks.

Future Research Directions

The results open several avenues for future exploration. Extending these bounds to more complex and high-dimensional settings could enhance their applicability. Additionally, exploring the potential for integrating these findings with other nonparametric models could further advance the methodology in probabilistic modeling and inference.

Conclusion

This paper provides a significant step in understanding and leveraging the behavior of Dirichlet Processes through novel bounding techniques on their CGF. The combination of superadditivity, large deviation principles, and information-theoretic measures yields powerful results that can inform both theory and practice in statistical methodologies involving DPs. The implications for effective and efficient statistical inference are profound, with potential to influence a range of applications in machine learning and beyond.