Josu Ceberio

Andoni I. Garmendia, Josu Ceberio, Alexander Mendiburu

Neural Combinatorial Optimization attempts to learn good heuristics for solving a set of problems using Neural Network models and Reinforcement Learning. Recently, its good performance has encouraged many practitioners to develop neural architectures for a wide variety of combinatorial problems. However, the incorporation of such algorithms in the conventional optimization framework has raised many questions related to their performance and the experimental comparison with other methods such as exact algorithms, heuristics and metaheuristics. This paper presents a critical analysis on the incorporation of algorithms based on neural networks into the classical combinatorial optimization framework. Subsequently, a comprehensive study is carried out to analyse the fundamental aspects of such algorithms, including performance, transferability, computational cost and generalization to larger-sized instances. To that end, we select the Linear Ordering Problem as a case of study, an NP-hard problem, and develop a Neural Combinatorial Optimization model to optimize it. Finally, we discuss how the analysed aspects apply to a general learning framework, and suggest new directions for future work in the area of Neural Combinatorial Optimization algorithms.

Andoni I. Garmendia, Josu Ceberio, Alexander Mendiburu

Recent advances in graph neural network architectures and increased computation power have revolutionized the field of combinatorial optimization (CO). Among the proposed models for CO problems, Neural Improvement (NI) models have been particularly successful. However, existing NI approaches are limited in their applicability to problems where crucial information is encoded in the edges, as they only consider node features and node-wise positional encodings. To overcome this limitation, we introduce a novel NI model capable of handling graph-based problems where information is encoded in the nodes, edges, or both. The presented model serves as a fundamental component for hill-climbing-based algorithms that guide the selection of neighborhood operations for each iteration. Conducted experiments demonstrate that the proposed model can recommend neighborhood operations that outperform conventional versions for the Preference Ranking Problem with a performance in the 99th percentile. We also extend the proposal to two well-known problems: the Traveling Salesman Problem and the Graph Partitioning Problem, recommending operations in the 98th and 97th percentile, respectively.

Andoni I. Garmendia, Quentin Cappart, Josu Ceberio et al.

Neural Combinatorial Optimization (NCO) is an emerging domain where deep learning techniques are employed to address combinatorial optimization problems as a standalone solver. Despite their potential, existing NCO methods often suffer from inefficient search space exploration, frequently leading to local optima entrapment or redundant exploration of previously visited states. This paper introduces a versatile framework, referred to as Memory-Augmented Reinforcement for Combinatorial Optimization (MARCO), that can be used to enhance both constructive and improvement methods in NCO through an innovative memory module. MARCO stores data collected throughout the optimization trajectory and retrieves contextually relevant information at each state. This way, the search is guided by two competing criteria: making the best decision in terms of the quality of the solution and avoiding revisiting already explored solutions. This approach promotes a more efficient use of the available optimization budget. Moreover, thanks to the parallel nature of NCO models, several search threads can run simultaneously, all sharing the same memory module, enabling an efficient collaborative exploration. Empirical evaluations, carried out on the maximum cut, maximum independent set and travelling salesman problems, reveal that the memory module effectively increases the exploration, enabling the model to discover diverse, higher-quality solutions. MARCO achieves good performance in a low computational cost, establishing a promising new direction in the field of NCO.

Etor Arza, Josu Ceberio, Ekhiñe Irurozki et al.

Non-deterministic measurements are common in real-world scenarios: the performance of a stochastic optimization algorithm or the total reward of a reinforcement learning agent in a chaotic environment are just two examples in which unpredictable outcomes are common. These measures can be modeled as random variables and compared among each other via their expected values or more sophisticated tools such as null hypothesis statistical tests. In this paper, we propose an alternative framework to visually compare two samples according to their estimated cumulative distribution functions. First, we introduce a dominance measure for two random variables that quantifies the proportion in which the cumulative distribution function of one of the random variables stochastically dominates the other one. Then, we present a graphical method that decomposes in quantiles i) the proposed dominance measure and ii) the probability that one of the random variables takes lower values than the other. With illustrative purposes, we re-evaluate the experimentation of an already published work with the proposed methodology and we show that additional conclusions (missed by the rest of the methods) can be inferred. Additionally, the software package RVCompare was created as a convenient way of applying and experimenting with the proposed framework.

Andoni Irazusta Garmendia, Josu Ceberio, Alexander Mendiburu

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.

Mikel Malagón, Josu Ceberio, Jose A. Lozano

Advances in large models, reinforcement learning, and open-endedness have accelerated progress toward autonomous agents that can learn and interact in the real world. To achieve this, flexible tools are needed to create rich, yet computationally efficient, environments. While scalable 2D environments fail to address key real-world challenges like 3D navigation and spatial reasoning, more complex 3D environments are computationally expensive and lack features like customizability and multi-agent support. This paper introduces Craftium, a highly customizable and easy-to-use platform for building rich 3D single- and multi-agent environments. We showcase environments of different complexity and nature: from single- and multi-agent tasks to vast worlds with many creatures and biomes, and customizable procedural task generators. Benchmarking shows that Craftium significantly reduces the computational cost of alternatives of similar richness, achieving +2K steps per second more than Minecraft-based frameworks.

Mikel Malagón, Josu Ceberio, Jose A. Lozano

This work introduces a growable and modular neural network architecture that naturally avoids catastrophic forgetting and interference in continual reinforcement learning. The structure of each module allows the selective combination of previous policies along with its internal policy, accelerating the learning process on the current task. Unlike previous growing neural network approaches, we show that the number of parameters of the proposed approach grows linearly with respect to the number of tasks, and does not sacrifice plasticity to scale. Experiments conducted in benchmark continuous control and visual problems reveal that the proposed approach achieves greater knowledge transfer and performance than alternative methods.

Judith Echevarrieta, Etor Arza, Aritz Pérez et al.

In certain real-world optimization scenarios, practitioners are not interested in solving multiple problems but rather in finding the best solution to a single, specific problem. When the computational budget is large relative to the cost of evaluating a candidate solution, multiple heuristic alternatives can be tried to solve the same given problem, each possibly with a different algorithm, parameter configuration, initialization, or stopping criterion. The sequential selection of which alternative to try next is crucial for efficiently identifying the one that provides the best possible solution across multiple attempts. Despite the relevance of this problem in practice, it has not yet been the exclusive focus of any existing review. Several sequential alternative selection strategies have been proposed in different research topics, but they have not been comprehensively and systematically unified under a common perspective. This work presents a focused review of single-problem multi-attempt heuristic optimization. It brings together suitable strategies to this problem that have been studied separately through algorithm selection, parameter tuning, multi-start and resource allocation. These strategies are explained using a unified terminology within a common framework, which supports the development of a taxonomy for systematically organizing and classifying them.

Josu Ceberio, Borja Calvo

Experimentation is an intrinsic part of research in artificial intelligence since it allows for collecting quantitative observations, validating hypotheses, and providing evidence for their reformulation. For that reason, experimentation must be coherent with the purposes of the research, properly addressing the relevant questions in each case. Unfortunately, the literature is full of works whose experimentation is neither rigorous nor convincing, oftentimes designed to support prior beliefs rather than answering the relevant research questions. In this paper, we focus on the field of metaheuristic optimization, since it is our main field of work, and it is where we have observed the misconduct that has motivated this letter. Even if we limit the focus of this manuscript to the experimental part of the research, our main goal is to sew the seed of sincere critical assessment of our work, sparking a reflection process both at the individual and the community level. Such a reflection process is too complex and extensive to be tackled as a whole. Therefore, to bring our feet to the ground, we will include in this document our reflections about the role of experimentation in our work, discussing topics such as the use of benchmark instances vs instance generators, or the statistical assessment of empirical results. That is, all the statements included in this document are personal views and opinions, which can be shared by others or not. Certainly, having different points of view is the basis to establish a good discussion process.

Ruben Solozabal, Josu Ceberio, Martin Takáč

This paper presents a framework to tackle constrained combinatorial optimization problems using deep Reinforcement Learning (RL). To this end, we extend the Neural Combinatorial Optimization (NCO) theory in order to deal with constraints in its formulation. Notably, we propose defining constrained combinatorial problems as fully observable Constrained Markov Decision Processes (CMDP). In that context, the solution is iteratively constructed based on interactions with the environment. The model, in addition to the reward signal, relies on penalty signals generated from constraint dissatisfaction to infer a policy that acts as a heuristic algorithm. Moreover, having access to the complete state representation during the optimization process allows us to rely on memory-less architectures, enhancing the results obtained in previous sequence-to-sequence approaches. Conducted experiments on the constrained Job Shop and Resource Allocation problems prove the superiority of the proposal for computing rapid solutions when compared to classical heuristic, metaheuristic, and Constraint Programming (CP) solvers.

Etor Arza, Aritz Perez, Ekhine Irurozki et al.

The Quadratic Assignment Problem (QAP) is a well-known permutation-based combinatorial optimization problem with real applications in industrial and logistics environments. Motivated by the challenge that this NP-hard problem represents, it has captured the attention of the optimization community for decades. As a result, a large number of algorithms have been proposed to tackle this problem. Among these, exact methods are only able to solve instances of size $n<40$. To overcome this limitation, many metaheuristic methods have been applied to the QAP. In this work, we follow this direction by approaching the QAP through Estimation of Distribution Algorithms (EDAs). Particularly, a non-parametric distance-based exponential probabilistic model is used. Based on the analysis of the characteristics of the QAP, and previous work in the area, we introduce Kernels of Mallows Model under the Hamming distance to the context of EDAs. Conducted experiments point out that the performance of the proposed algorithm in the QAP is superior to (i) the classical EDAs adapted to deal with the QAP, and also (ii) to the specific EDAs proposed in the literature to deal with permutation problems.

Mikel Malagon, Josu Ceberio

Neural networks are gaining popularity in the reinforcement learning field due to the vast number of successfully solved complex benchmark problems. In fact, artificial intelligence algorithms are, in some cases, able to overcome human professionals. Usually, neural networks have more than a couple of hidden layers, and thus, they involve a large quantity of parameters that need to be optimized. Commonly, numeric approaches are used to optimize the inner parameters of neural networks, such as the stochastic gradient descent. However, these techniques tend to be computationally very expensive, and for some tasks, where effectiveness is crucial, high computational costs are not acceptable. Along these research lines, in this paper we propose to optimize the parameters of neural networks by means of estimation of distribution algorithms. More precisely, the univariate marginal distribution algorithm is used for evolving neural networks in various reinforcement learning tasks. For the sake of validating our idea, we run the proposed algorithm on four OpenAI Gym benchmark problems. In addition, the obtained results were compared with a standard genetic algorithm. Revealing, that optimizing with UMDAc provides better results than the genetic algorithm in most of the cases.