[Paper Review] Primal-Dual Active-Set Methods for Isotonic Regression and Trend Filtering

Zheng Han, Frank E. Curtis|arXiv (Cornell University)|Aug 10, 2015

Control Systems and Identification23 references3 citations

TL;DR

This paper proposes primal-dual active-set (PDAS) algorithms for isotonic regression (IR) and trend filtering (TF), offering a warm-startable, parallelizable alternative to the traditional Pool Adjacent Violators (PAV) method. The PDAS method achieves $Ø(n)$ work complexity and convergence like PAV, while demonstrating superior performance in numerical experiments and enabling efficient solution of online and large-scale problems.

ABSTRACT

Isotonic regression (IR) is a non-parametric calibration method used in supervised learning. For performing large-scale IR, we propose a primal-dual active-set (PDAS) algorithm which, in contrast to the state-of-the-art Pool Adjacent Violators (PAV) algorithm, can be parallized and is easily warm-started thus well-suited in the online settings. We prove that, like the PAV algorithm, our PDAS algorithm for IR is convergent and has a work complexity of O(n), though our numerical experiments suggest that our PDAS algorithm is often faster than PAV. In addition, we propose PDAS variants (with safeguarding to ensure convergence) for solving related trend filtering (TF) problems, providing the results of experiments to illustrate their effectiveness.

Motivation & Objective

• Address the limitations of the PAV algorithm in large-scale and online settings, where it lacks warm-starting capability and parallelization support.
• Develop a primal-dual active-set (PDAS) method for isotonic regression that maintains the theoretical guarantees of PAV while enabling warm-starting and parallel execution.
• Extend the PDAS framework to trend filtering problems with non-smooth regularization, ensuring convergence through safeguarding strategies.
• Demonstrate the practical superiority of PDAS and its variants over PAV, interior-point methods, and other solvers in terms of speed, reliability, and scalability.

Proposed method

• Formulate isotonic regression as a convex optimization problem with monotonicity constraints: minimize $\frac{1}{2}\sum \omega_i(y_i - \theta_i)^2$ subject to $\theta_1 \leq \cdots \leq \theta_n$.
• Develop a primal-dual active-set method that iteratively identifies active constraints (violations of monotonicity) and updates the solution via dual ascent and primal correction.
• Introduce a safeguarding strategy in the PDAS variant for trend filtering to ensure convergence when using aggressive update rules.
• Apply the PDAS framework to trend filtering problems with $\ell_1$-norm regularization on first- or second-order differences, using $g(\theta) = \|D\theta\|_1$ or $\|(D\theta)_+\|_1$.
• Leverage the structure of the problem to enable efficient computation of active sets and dual updates, supporting warm-starting and parallelization.
• Implement and test the algorithm with a dynamic block partitioning strategy that maintains monotonicity and allows for efficient updates of the solution and dual variables.

Experimental results

Research questions

• RQ1 Can a primal-dual active-set method be designed for isotonic regression that matches the theoretical complexity of PAV while enabling warm-starting and parallelization?
• RQ2 How does the performance of the proposed PDAS algorithm compare to PAV in terms of convergence speed and scalability on large-scale datasets?
• RQ3 Can PDAS be effectively extended to trend filtering problems with non-smooth regularization, and what safeguards are needed to ensure convergence?
• RQ4 How effective is the proposed safeguarding strategy in improving the reliability and efficiency of PDAS for second-order trend filtering compared to unconstrained or conservatively safeguarded variants?
• RQ5 To what extent does warm-starting improve the performance of PDAS for sequential or online isotonic regression and trend filtering problems?

Key findings

• The PDAS algorithm for isotonic regression achieves the same $\mathcal{O}(n)$ work complexity and convergence guarantee as PAV, but is significantly faster in numerical experiments.
• PDAS can be warm-started, enabling dramatic performance improvements in online and sequential settings, where PAV cannot be efficiently restarted.
• For trend filtering with first-order differences, all PDAS variants (PDAS, SF1, SF2) achieved 100% success rate across all tested problem sizes ($n \in \{10^4, 1.7\times10^5, 3.3\times10^5\}$).
• When using second-order differences, PDAS and SF1 failed on 100% of instances due to divergence, while SF2 with the proposed safeguard achieved 100% success rate.
• In terms of runtime and iteration count, SF2 outperformed both PDAS and SF1 on second-order problems, demonstrating that the proposed safeguard enables more aggressive and effective updates.
• Warm-starting SF2 reduced the number of iterations and runtime dramatically compared to cold-start, while interior-point methods showed minimal benefit from good initial guesses, highlighting the advantage of PDAS in dynamic settings.