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Algorithm Configuration for Structured Pfaffian Settings

Maria-Florina Balcan, Anh Tuan Nguyen, Dravyansh Sharma

TL;DR

This work develops and applies a Pfaffian-enhanced learning framework for data-driven algorithm configuration. By extending the Goldberg–Jerrum framework to Pfaffian functions, the Pfaffian GJ framework enables learning guarantees for parameterized algorithms whose dual utilities exhibit Pfaffian piecewise structure, broadening beyond rational-function routines. A refined Pfaffian piecewise structure ties Pfaffian boundaries and pieces together to yield tighter pseudo-dimension bounds, and the authors demonstrate generalization guarantees for data-driven agglomerative clustering, graph-based semi-supervised learning, and regularized logistic regression, along with online dispersion tools for Pfaffian discontinuities. The framework is complemented by online learning results with dispersion-based no-regret guarantees, and the paper closes with concrete applications, comparisons to prior work, and directions for future research. Overall, the Pfaffian approach broadens the theoretical foundations of data-driven algorithm design and provides principled, scalable guarantees for complex, Pfaffian-structured problems.

Abstract

Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm's hyperparameters using problem instances drawn from the problem distribution of the target application domain. This can be achieved by maximizing empirical utilities that measure the algorithms' performance as a function of their hyperparameters, using problem instances. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piecewise discontinuous structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the \textit{Pfaffian GJ framework}, an extension of the classical \textit{GJ framework}, that is capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a \textit{refined piecewise structure}, which automatically translates to learning guarantees using our proposed framework.

Algorithm Configuration for Structured Pfaffian Settings

TL;DR

This work develops and applies a Pfaffian-enhanced learning framework for data-driven algorithm configuration. By extending the Goldberg–Jerrum framework to Pfaffian functions, the Pfaffian GJ framework enables learning guarantees for parameterized algorithms whose dual utilities exhibit Pfaffian piecewise structure, broadening beyond rational-function routines. A refined Pfaffian piecewise structure ties Pfaffian boundaries and pieces together to yield tighter pseudo-dimension bounds, and the authors demonstrate generalization guarantees for data-driven agglomerative clustering, graph-based semi-supervised learning, and regularized logistic regression, along with online dispersion tools for Pfaffian discontinuities. The framework is complemented by online learning results with dispersion-based no-regret guarantees, and the paper closes with concrete applications, comparisons to prior work, and directions for future research. Overall, the Pfaffian approach broadens the theoretical foundations of data-driven algorithm design and provides principled, scalable guarantees for complex, Pfaffian-structured problems.

Abstract

Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm's hyperparameters using problem instances drawn from the problem distribution of the target application domain. This can be achieved by maximizing empirical utilities that measure the algorithms' performance as a function of their hyperparameters, using problem instances. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piecewise discontinuous structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the \textit{Pfaffian GJ framework}, an extension of the classical \textit{GJ framework}, that is capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a \textit{refined piecewise structure}, which automatically translates to learning guarantees using our proposed framework.
Paper Structure (67 sections, 33 theorems, 97 equations, 3 figures, 2 algorithms)

This paper contains 67 sections, 33 theorems, 97 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Overview of the Pfaffian GJ framework.
  3. Contributions.
  4. Related work
  5. Data-driven algorithm design.
  6. Statistical learning guarantees for data-driven algorithm design.
  7. Online learning guarantees for data-driven algorithm design.
  8. Algorithms with predictions.
  9. Preliminaries
  10. Parameterized algorithms, utility function class, and dual utility function class.
  11. Statistical learning.
  12. Online learning.
  13. Pfaffian GJ framework for data-driven algorithm design
  14. Pfaffian functions
  15. Example 1.
  16. ...and 52 more sections

Key Result

Theorem 3.1

Consider a real-valued function class $\mathcal{U}$, of which each function takes value in $\mathcal{X}$. Assume that $\text{\normalfont Pdim}(\mathcal{U})$ is finite and $\mathcal{U}$ is bounded by $H$. Then given $\epsilon > 0$ and $\delta \in (0, 1)$, for any $m \geq m(\delta, \epsilon)$, where $ Here $\hat{u}_S \in \mathop{\mathrm{arg\,max}}\limits_{u \in \mathcal{U}}\frac{1}{m}\sum_{i = 1}^m

Figures (3)

  • Figure 1: A simple illustration of the general idea of the Pfaffian GJ framework. Given a problem instance $\boldsymbol{x}$ and a real-valued threshold $r$, a Pfaffian GJ algorithm $\Gamma_{\boldsymbol{x}, r}$ takes as the inputs any possible parameters $\boldsymbol{a}=(a_{1} ,\ \dotsc ,a_{d})$ and outputs if $u_{\boldsymbol{a}}(\boldsymbol{x}) \geq r$, by combining basic arithmetic operators, Pfaffian functions, and conditional statements. If we can bound the complexity of $\Gamma_{\boldsymbol{x}, r}$, our results imply that we can bound the pseudo-dimension of the utility function class $\mathcal{U}$.
  • Figure 3: An example of the original piecewise structure (Definition \ref{['def:piecewise-structure']}) and our proposed Pfaffian piecewise structure (Definition \ref{['def:pfaffian-piecewise-structure']}). Here, (a) demonstrates the sheer view of the piecewise structure of a specific dual utility function $u^*_{\boldsymbol{x}}$, while (b) shows the corresponding top view for better illustration of regions and their boundaries. As can be seen, there are three boundary functions $g_{\boldsymbol{x}, 1}(\boldsymbol{a}) = \frac{1}{2}a_1^2 - a_2$, $g_{\boldsymbol{x}, 2}(\boldsymbol{a}) = (a_1 - 5)^2 + (a_2 - 5)^2 - 16$, and $g_{\boldsymbol{x}, 3}(\boldsymbol{a}) = a_1 - e^{a_2}$, partitioning the domain $\mathcal{A}$ into $7$ regions. In each region, the function $u^*_{\boldsymbol{x}}(\boldsymbol{a})$ takes the form of a Pfaffian function. What is not captured by the original piecewise structure Definition \ref{['def:piecewise-structure']} is that, in this example, there are only 4 forms that $u^*_{\boldsymbol{x}}(\boldsymbol{a})$ can take, which is either $a_1 + \frac{a_2}{2}$ (blue region), $e^{-0.2(a_1^2 + a_2^2)}$ (red regions), $\log(a_2) + 2$ (green region), and $\sqrt{a_1^2 + a_2^2 + \exp(0.1\sqrt{a_1})}$, (yellow region). It can be verified that all the piece and boundary functions are Pfaffian function from the Pfaffian chain $\mathcal{C}_{\boldsymbol{x}}(\boldsymbol{a}, e^{a_2}, e^{-0.2(a_1^2 + a_2^2)}, \frac{1}{\sqrt{a_1}}, e^{0.1\sqrt{a_1}}, \frac{1}{\sqrt{a_1^2 + a_2^2 + \exp(0.1\sqrt{a_1})}})$.
  • Figure 4: A demonstration of how the computation of a dual utility function satisfying Pfaffian piecewise structure can be described by the Pfaffian GJ algorithm. Given an input $\boldsymbol{x} \in \mathcal{X}$ and a threshold $r \in \mathbb{R}$, the function $u^*_{\boldsymbol{x}}$ is piecewise structured with boundary functions $g^{(i)}_{\boldsymbol{x}}$ (for $i = 1, \dots k$), and piece functions $f_{h, \mathbf{b}}$ ($\mathbf{b} \in \{0, 1\}^{k}$). Note that, the piece functions $f_{h, \mathbf{b}}$ can take at most $k_\mathcal{F}$ forms and all the piece and boundary functions are Pfaffian functions from the chain $\mathcal{C}$.

Theorems & Definitions (69)

  • Definition 1: Pseudo-dimension, pollard1984convergence
  • Theorem 3.1: pollard1984convergence
  • Definition 2: Pfaffian Chain, khovanskiui1991fewnomials
  • Definition 3: Complexity of Pfaffian chain
  • Definition 4: Pfaffian functions, khovanskiui1991fewnomials
  • Definition 5: Pfaffian GJ algorithm
  • Definition 6: Pfaffian chain associated with Pfaffian GJ algorithm
  • Lemma 4.1
  • Remark 1
  • Theorem 4.2
  • ...and 59 more