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On the Correctness of the Generalized Isotonic Recursive Partitioning Algorithm

Joong-Ho Won, Jihan Jung

TL;DR

The paper addresses isotonic regression under separable convex losses and shows that, although nonuniqueness can arise and naive GIRP may fail to preserve isotonicity, there exists a correct solution reachable by a minimal modification of GIRP that uses binary partitions and carefully chosen intermediate fits. It introduces Recursive Partitioning (RP) that fixes a reference $b$ and, at each node, selects $b_L$ and $b_U$ closest to $b$ for the lower and upper parts, ensuring the piecewise fits minimize the loss on each subdomain and yield a global isotonic minimizer. A key theoretical contribution is extending the projection-pair framework via maximal upper (or minimal lower) sets and the set-valued derivative $\Sigma_b$ to both differentiable and nondifferentiable losses, together with practical LP/ max-flow formulations and a terminating criterion. The results demonstrate that (i) a globally optimal isotonic solution exists and (ii) RP finds one without backtracking, with the original GIRP recovered when the constant-fit sets are singletons. This yields a robust, general approach for isotonic regression applicable to huber, hinge, and logistic-like losses with guaranteed isotonic intermediate results and correctness.

Abstract

This paper presents an in-depth analysis of the generalized isotonic recursive partitioning (GIRP) algorithm for fitting isotonic models under separable convex losses, proposed by Luss and Rosset [J. Comput. Graph. Statist., 23 (2014), pp. 192--201] for differentiable losses and extended by Painsky and Rosset [IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), pp. 308-321] for nondifferentiable losses. The GIRP algorithm poseses an attractive feature that in each step of the algorithm, the intermediate solution satisfies the isotonicity constraint. The paper begins with an example showing that the GIRP algorithm as described in the literature may fail to produce an isotonic model, suggesting that the existence and uniqueness of the solution to the isotonic regression problem must be carefully addressed. It proceeds with showing that, among possibly many solutions, there indeed exists a solution that can be found by recursive binary partitioning of the set of observed data. A small modification of the GIRP algorithm suffices to obtain a correct solution and preserve the desired property that all the intermediate solutions are isotonic. This proposed modification includes a proper choice of intermediate solutions and a simplification of the partitioning step from ternary to binary.

On the Correctness of the Generalized Isotonic Recursive Partitioning Algorithm

TL;DR

The paper addresses isotonic regression under separable convex losses and shows that, although nonuniqueness can arise and naive GIRP may fail to preserve isotonicity, there exists a correct solution reachable by a minimal modification of GIRP that uses binary partitions and carefully chosen intermediate fits. It introduces Recursive Partitioning (RP) that fixes a reference and, at each node, selects and closest to for the lower and upper parts, ensuring the piecewise fits minimize the loss on each subdomain and yield a global isotonic minimizer. A key theoretical contribution is extending the projection-pair framework via maximal upper (or minimal lower) sets and the set-valued derivative to both differentiable and nondifferentiable losses, together with practical LP/ max-flow formulations and a terminating criterion. The results demonstrate that (i) a globally optimal isotonic solution exists and (ii) RP finds one without backtracking, with the original GIRP recovered when the constant-fit sets are singletons. This yields a robust, general approach for isotonic regression applicable to huber, hinge, and logistic-like losses with guaranteed isotonic intermediate results and correctness.

Abstract

This paper presents an in-depth analysis of the generalized isotonic recursive partitioning (GIRP) algorithm for fitting isotonic models under separable convex losses, proposed by Luss and Rosset [J. Comput. Graph. Statist., 23 (2014), pp. 192--201] for differentiable losses and extended by Painsky and Rosset [IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), pp. 308-321] for nondifferentiable losses. The GIRP algorithm poseses an attractive feature that in each step of the algorithm, the intermediate solution satisfies the isotonicity constraint. The paper begins with an example showing that the GIRP algorithm as described in the literature may fail to produce an isotonic model, suggesting that the existence and uniqueness of the solution to the isotonic regression problem must be carefully addressed. It proceeds with showing that, among possibly many solutions, there indeed exists a solution that can be found by recursive binary partitioning of the set of observed data. A small modification of the GIRP algorithm suffices to obtain a correct solution and preserve the desired property that all the intermediate solutions are isotonic. This proposed modification includes a proper choice of intermediate solutions and a simplification of the partitioning step from ternary to binary.
Paper Structure (9 sections, 10 theorems, 35 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 9 sections, 10 theorems, 35 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recursive binary partitioning
  4. Modified GIRP algorithm
  5. Determining the constant fit
  6. Computing a maximal upper set
  7. Algorithm
  8. Illustrative examples
  9. Conclusion

Key Result

Proposition 1

If $\mathcal{I}_X$ is nonempty and bounded, then problem eqn:isotonic has a solution. If $f = \ell(g(x), \cdot)$ is coercive, i.e., $\lim_{|y|\to\infty} f(y) = \infty$, for each $x$, then problem eqn:isotonic has a solution whether or not $\mathcal{I}_X$ is bounded. If $f$ is strictly convex and coe

Figures (1)

  • Figure 1: Recursive partition tree from \ref{['ex:counter']}. Shown inside each node (box) is the subset of $X$ representing the partition at its own level. On top of each box the fit to the set inside the box is presented in italic. Shaded nodes allow for nonunique fits.

Theorems & Definitions (31)

  • Definition 1: Isotonic function
  • Example 1.1: Constraint violation
  • Example 1.2: Frequency of violation
  • Proposition 1: Existence and uniqueness of solution
  • Definition 2: Constant fit
  • Lemma 1
  • proof
  • Definition 3: Upper set
  • Proposition 2: Theorem 1.4.1 of robertson1988
  • Definition 4: Complementary pair
  • ...and 21 more