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A Block-Coordinate Descent EMO Algorithm: Theoretical and Empirical Analysis

Benjamin Doerr, Joshua Knowles, Aneta Neumann, Frank Neumann

TL;DR

This work investigates whether block-coordinate descent can be asymptotically efficient in evolutionary multi-objective optimization and proposes BC-GSEMO, a block-coordinate variant of GSEMO. The authors present a Leading-Ones–based bi-objective benchmark with a Pareto front of size $2^k$ and establish a lower bound of $\\Omega(2^k n \ell)$ for GSEMO, while BC-GSEMO attains an upper bound of $\\O(2^k n \\sqrt{\\ell \\log \\ell})$ for $k \\\le n^{1/3}$, validated by experiments up to $k=4$. Theoretical results are complemented by empirical evidence showing significant speed-ups from block-coordinate updates, attributed to parallel block optimization without destroying existing block structure. The findings suggest that problem decomposition and per-block mutation can yield practical and scalable improvements for multi-objective optimization, with avenues for automatic block identification in complex problems.

Abstract

We consider whether conditions exist under which block-coordinate descent is asymptotically efficient in evolutionary multi-objective optimization, addressing an open problem. Block-coordinate descent, where an optimization problem is decomposed into $k$ blocks of decision variables and each of the blocks is optimized (with the others fixed) in a sequence, is a technique used in some large-scale optimization problems such as airline scheduling, however its use in multi-objective optimization is less studied. We propose a block-coordinate version of GSEMO and compare its running time to the standard GSEMO algorithm. Theoretical and empirical results on a bi-objective test function, a variant of LOTZ, serve to demonstrate the existence of cases where block-coordinate descent is faster. The result may yield wider insights into this class of algorithms.

A Block-Coordinate Descent EMO Algorithm: Theoretical and Empirical Analysis

TL;DR

This work investigates whether block-coordinate descent can be asymptotically efficient in evolutionary multi-objective optimization and proposes BC-GSEMO, a block-coordinate variant of GSEMO. The authors present a Leading-Ones–based bi-objective benchmark with a Pareto front of size and establish a lower bound of for GSEMO, while BC-GSEMO attains an upper bound of for , validated by experiments up to . Theoretical results are complemented by empirical evidence showing significant speed-ups from block-coordinate updates, attributed to parallel block optimization without destroying existing block structure. The findings suggest that problem decomposition and per-block mutation can yield practical and scalable improvements for multi-objective optimization, with avenues for automatic block identification in complex problems.

Abstract

We consider whether conditions exist under which block-coordinate descent is asymptotically efficient in evolutionary multi-objective optimization, addressing an open problem. Block-coordinate descent, where an optimization problem is decomposed into blocks of decision variables and each of the blocks is optimized (with the others fixed) in a sequence, is a technique used in some large-scale optimization problems such as airline scheduling, however its use in multi-objective optimization is less studied. We propose a block-coordinate version of GSEMO and compare its running time to the standard GSEMO algorithm. Theoretical and empirical results on a bi-objective test function, a variant of LOTZ, serve to demonstrate the existence of cases where block-coordinate descent is faster. The result may yield wider insights into this class of algorithms.
Paper Structure (8 sections, 7 theorems, 9 equations, 3 figures, 2 algorithms)

This paper contains 8 sections, 7 theorems, 9 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Block coordinate GSEMO
  3. Benchmark Function and Properties
  4. Runtime Analysis
  5. Lower Bound for the GSEMO
  6. Upper Bound for the BC-GSEMO
  7. Experimental investigations
  8. Discussion and Conclusions

Key Result

Lemma 1

Let $x_B, y_B \in \{0,1\}^\ell$ with $x_B \neq y_B$. Let $i \in [1..\ell]$ be the first bit in which $x_B$ and $y_B$ differ, and let $(x_B)_i = 1$ (and thus $(y_B)_i = 0$). Then the following is true. In particular, if $x_B$ and $y_B$ have equal value under one objective function, then also under the other.

Figures (3)

  • Figure 1: Mean time to reach the optimal population by GSEMO and BC-GSEMO (with $t_{\mathrm{epoch}} =1000$) for $k = 2$ dependent on the bitstring length $n$ for $r = 1, 2, 4$.
  • Figure 2: Mean time to reach the optimal population by GSEMO and BC-GSEMO (with $t_{\mathrm{epoch}} =1000$) for $k = 3$ dependent on the bitstring length $n$ for $r = 1, 2, 4$.
  • Figure 3: Mean time to reach the optimal population by GSEMO and BC-GSEMO (with $t_{\mathrm{epoch}} =1000$) for $k = 4$ dependent on the bitstring length $n$ for $r = 1, 2, 4$.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm:lower']}
  • Lemma 5
  • ...and 3 more