Aske Plaat

Thomas M. Moerland, Matthias Müller-Brockhausen, Zhao Yang et al.

Due to the empirical success of reinforcement learning, an increasing number of students study the subject. However, from our practical teaching experience, we see students entering the field (bachelor, master and early PhD) often struggle. On the one hand, textbooks and (online) lectures provide the fundamentals, but students find it hard to translate between equations and code. On the other hand, public codebases do provide practical examples, but the implemented algorithms tend to be complex, and the underlying test environments contain multiple reinforcement learning challenges at once. Although this is realistic from a research perspective, it often hinders educational conceptual understanding. To solve this issue we introduce EduGym, a set of educational reinforcement learning environments and associated interactive notebooks tailored for education. Each EduGym environment is specifically designed to illustrate a certain aspect/challenge of reinforcement learning (e.g., exploration, partial observability, stochasticity, etc.), while the associated interactive notebook explains the challenge and its possible solution approaches, connecting equations and code in a single document. An evaluation among RL students and researchers shows 86% of them think EduGym is a useful tool for reinforcement learning education. All notebooks are available from https://www.edugym.org/, while the full software package can be installed from https://github.com/RLG-Leiden/edugym.

Po-Kai Chen, Niki van Stein, Aske Plaat

Mechanistic interpretability of transformers requires identifying not just which components matter but how they compose into the computational route that produced a prediction. Both attention and MLP follow a shared key-value template $ϕ(S)U$. We exploit this structure to develop Unpack, a backward recursion that decomposes credit through both sublayers, producing interaction strengths between any two components, named end-to-end paths with K/Q/V composition labels, and per-token attribution from a single forward pass, without intervention, gradients, or auxiliary training. We evaluate on the indirect object identification task. On GPT-2 small, the method recovers all three composition connections described by Wang et al. (2023), including the mode-specific routing of each connection (K, Q, or V). To test token-level attribution beyond trivial copying, we compare two occurrences of the same name in the same decomposition: the first mention retains strong credit while the duplicate-detection position is suppressed, a pattern absent in matched control prompts. Across the Pythia family from 160M to 6.9B parameters, this suppression pattern is consistently recovered at every scale, demonstrating that the method tracks mechanistic structure without ground-truth circuit labels. Code is available at https://github.com/Fun-Cry/unpacklm.

Aske Plaat, Annie Wong, Suzan Verberne et al.

Large language models (LLMs) with billions of parameters exhibit in-context learning abilities, enabling few-shot learning on tasks that the model was not specifically trained for. Traditional models achieve breakthrough performance on language tasks, but do not perform well on basic reasoning benchmarks. However, a new in-context learning approach, Chain-of-thought, has demonstrated strong multi-step reasoning abilities on these benchmarks. The research on LLM reasoning abilities started with the question whether LLMs can solve grade school math word problems, and has expanded to other tasks in the past few years. This article reviews the field of multi-step reasoning with LLMs. We propose a taxonomy that identifies different ways to generate, evaluate, and control multi-step reasoning. We provide an in-depth coverage of core approaches and open problems, and we propose a research agenda for the near future. We find that multi-step reasoning approaches have progressed beyond math word problems, and can now successfully solve challenges in logic, combinatorial games, and robotics, sometimes by first generating code that is then executed by external tools. Many studies in multi-step methods use reinforcement learning for finetuning, external optimization loops, in-context reinforcement learning, and self-reflection.

Mike Huisman, Thomas M. Moerland, Aske Plaat et al.

Deep learning requires large amounts of data to learn new tasks well, limiting its applicability to domains where such data is available. Meta-learning overcomes this limitation by learning how to learn. In 2001, Hochreiter et al. showed that an LSTM trained with backpropagation across different tasks is capable of meta-learning. Despite promising results of this approach on small problems, and more recently, also on reinforcement learning problems, the approach has received little attention in the supervised few-shot learning setting. We revisit this approach and test it on modern few-shot learning benchmarks. We find that LSTM, surprisingly, outperform the popular meta-learning technique MAML on a simple few-shot sine wave regression benchmark, but that LSTM, expectedly, fall short on more complex few-shot image classification benchmarks. We identify two potential causes and propose a new method called Outer Product LSTM (OP-LSTM) that resolves these issues and displays substantial performance gains over the plain LSTM. Compared to popular meta-learning baselines, OP-LSTM yields competitive performance on within-domain few-shot image classification, and performs better in cross-domain settings by 0.5% to 1.9% in accuracy score. While these results alone do not set a new state-of-the-art, the advances of OP-LSTM are orthogonal to other advances in the field of meta-learning, yield new insights in how LSTM work in image classification, allowing for a whole range of new research directions. For reproducibility purposes, we publish all our research code publicly.

Mike Huisman, Aske Plaat, Jan N. van Rijn

Deep neural networks can yield good performance on various tasks but often require large amounts of data to train them. Meta-learning received considerable attention as one approach to improve the generalization of these networks from a limited amount of data. Whilst meta-learning techniques have been observed to be successful at this in various scenarios, recent results suggest that when evaluated on tasks from a different data distribution than the one used for training, a baseline that simply finetunes a pre-trained network may be more effective than more complicated meta-learning techniques such as MAML, which is one of the most popular meta-learning techniques. This is surprising as the learning behaviour of MAML mimics that of finetuning: both rely on re-using learned features. We investigate the observed performance differences between finetuning, MAML, and another meta-learning technique called Reptile, and show that MAML and Reptile specialize for fast adaptation in low-data regimes of similar data distribution as the one used for training. Our findings show that both the output layer and the noisy training conditions induced by data scarcity play important roles in facilitating this specialization for MAML. Lastly, we show that the pre-trained features as obtained by the finetuning baseline are more diverse and discriminative than those learned by MAML and Reptile. Due to this lack of diversity and distribution specialization, MAML and Reptile may fail to generalize to out-of-distribution tasks whereas finetuning can fall back on the diversity of the learned features.

Zhao Yang, Thomas. M. Moerland, Mike Preuss et al.

While deep reinforcement learning has shown important empirical success, it tends to learn relatively slow due to slow propagation of rewards information and slow update of parametric neural networks. Non-parametric episodic memory, on the other hand, provides a faster learning alternative that does not require representation learning and uses maximum episodic return as state-action values for action selection. Episodic memory and reinforcement learning both have their own strengths and weaknesses. Notably, humans can leverage multiple memory systems concurrently during learning and benefit from all of them. In this work, we propose a method called Two-Memory reinforcement learning agent (2M) that combines episodic memory and reinforcement learning to distill both of their strengths. The 2M agent exploits the speed of the episodic memory part and the optimality and the generalization capacity of the reinforcement learning part to complement each other. Our experiments demonstrate that the 2M agent is more data efficient and outperforms both pure episodic memory and pure reinforcement learning, as well as a state-of-the-art memory-augmented RL agent. Moreover, the proposed approach provides a general framework that can be used to combine any episodic memory agent with other off-policy reinforcement learning algorithms.

Christian M. Fuchs, Nadia M. Murillo, Aske Plaat et al.

Miniaturized satellites are currently not considered suitable for critical, high-priority, and complex multi-phased missions, due to their low reliability. As hardware-side fault tolerance (FT) solutions designed for larger spacecraft can not be adopted aboard very small satellites due to budget, energy, and size constraints, we developed a hybrid FT-approach based upon only COTS components, commodity processor cores, library IP, and standard software. This approach facilitates fault detection, isolation, and recovery in software, and utilizes fault-coverage techniques across the embedded stack within an multiprocessor system-on-chip (MPSoC). This allows our FPGA-based proof-of-concept implementation to deliver strong fault-coverage even for missions with a long duration, but also to adapt to varying performance requirements during the mission. The operator of a spacecraft utilizing this approach can define performance profiles, which allow an on-board computer (OBC) to trade between processing capacity, fault coverage, and energy consumption using simple heuristics. The software-side FT approach developed also offers advantages if deployed aboard larger spacecraft through spare resource pooling, enabling an OBC to more efficiently handle permanent faults. This FT approach in part mimics a critical biological systems's way of tolerating and adjusting to failures, enabling graceful ageing of an MPSoC.

Koen Ponse, Felix Kleuker, Márton Fejér et al.

The transition to sustainable energy is a key challenge of our time, requiring modifications in the entire pipeline of energy production, storage, transmission, and consumption. At every stage, new sequential decision-making challenges emerge, ranging from the operation of wind farms to the management of electrical grids or the scheduling of electric vehicle charging stations. All such problems are well suited for reinforcement learning, the branch of machine learning that learns behavior from data. Therefore, numerous studies have explored the use of reinforcement learning for sustainable energy. This paper surveys this literature with the intention of bridging both the underlying research communities: energy and machine learning. After a brief introduction of both fields, we systematically list relevant sustainability challenges, how they can be modeled as a reinforcement learning problem, and what solution approaches currently exist in the literature. Afterwards, we zoom out and identify overarching reinforcement learning themes that appear throughout sustainability, such as multi-agent, offline, and safe reinforcement learning. Lastly, we also cover standardization of environments, which will be crucial for connecting both research fields, and highlight potential directions for future work. In summary, this survey provides an extensive overview of reinforcement learning methods for sustainable energy, which may play a vital role in the energy transition.

Zhao Yang, Thomas M. Moerland, Mike Preuss et al.

Go-Explore achieved breakthrough performance on challenging reinforcement learning (RL) tasks with sparse rewards. The key insight of Go-Explore was that successful exploration requires an agent to first return to an interesting state ('Go'), and only then explore into unknown terrain ('Explore'). We refer to such exploration after a goal is reached as 'post-exploration'. In this paper, we present a clear ablation study of post-exploration in a general intrinsically motivated goal exploration process (IMGEP) framework, that the Go-Explore paper did not show. We study the isolated potential of post-exploration, by turning it on and off within the same algorithm under both tabular and deep RL settings on both discrete navigation and continuous control tasks. Experiments on a range of MiniGrid and Mujoco environments show that post-exploration indeed helps IMGEP agents reach more diverse states and boosts their performance. In short, our work suggests that RL researchers should consider to use post-exploration in IMGEP when possible since it is effective, method-agnostic and easy to implement.

Mike Huisman, Aske Plaat, Jan N. van Rijn

Gradient-based meta-learning techniques aim to distill useful prior knowledge from a set of training tasks such that new tasks can be learned more efficiently with gradient descent. While these methods have achieved successes in various scenarios, they commonly adapt all parameters of trainable layers when learning new tasks. This neglects potentially more efficient learning strategies for a given task distribution and may be susceptible to overfitting, especially in few-shot learning where tasks must be learned from a limited number of examples. To address these issues, we propose Subspace Adaptation Prior (SAP), a novel gradient-based meta-learning algorithm that jointly learns good initialization parameters (prior knowledge) and layer-wise parameter subspaces in the form of operation subsets that should be adaptable. In this way, SAP can learn which operation subsets to adjust with gradient descent based on the underlying task distribution, simultaneously decreasing the risk of overfitting when learning new tasks. We demonstrate that this ability is helpful as SAP yields superior or competitive performance in few-shot image classification settings (gains between 0.1% and 3.9% in accuracy). Analysis of the learned subspaces demonstrates that low-dimensional operations often yield high activation strengths, indicating that they may be important for achieving good few-shot learning performance. For reproducibility purposes, we publish all our research code publicly.

Matthias Müller-Brockhausen, Aske Plaat, Mike Preuss

Reinforcement Learning (RL) is one of the most dynamic research areas in Game AI and AI as a whole, and a wide variety of games are used as its prominent test problems. However, it is subject to the replicability crisis that currently affects most algorithmic AI research. Benchmarking in Reinforcement Learning could be improved through verifiable results. There are numerous benchmark environments whose scores are used to compare different algorithms, such as Atari. Nevertheless, reviewers must trust that figures represent truthful values, as it is difficult to reproduce an exact training curve. We propose improving this situation by providing access to the original experimental data to validate study results. To that end, we rely on the concept of minimal traces. These allow re-simulation of action sequences in deterministic RL environments and, in turn, enable reviewers to verify, re-use, and manually inspect experimental results without needing large compute clusters. It also permits validation of presented reward graphs, an inspection of individual episodes, and re-use of result data (baselines) for proper comparison in follow-up papers. We offer plug-and-play code that works with Gym so that our measures fit well in the existing RL and reproducibility eco-system. Our approach is freely available, easy to use, and adds minimal overhead, as minimal traces allow a data compression ratio of up to $\approx 10^4:1$ (94GB to 8MB for Atari Pong) compared to a regular MDP trace used in offline RL datasets. The paper presents proof-of-concept results for a variety of games.

Zhao Yang, Thomas M. Moerland, Mike Preuss et al.

Reinforcement learning (RL) is an appealing paradigm for training intelligent agents, enabling policy acquisition from the agent's own autonomously acquired experience. However, the training process of RL is far from automatic, requiring extensive human effort to reset the agent and environments. To tackle the challenging reset-free setting, we first demonstrate the superiority of model-based (MB) RL methods in such setting, showing that a straightforward adaptation of MBRL can outperform all the prior state-of-the-art methods while requiring less supervision. We then identify limitations inherent to this direct extension and propose a solution called model-based reset-free (MoReFree) agent, which further enhances the performance. MoReFree adapts two key mechanisms, exploration and policy learning, to handle reset-free tasks by prioritizing task-relevant states. It exhibits superior data-efficiency across various reset-free tasks without access to environmental reward or demonstrations while significantly outperforming privileged baselines that require supervision. Our findings suggest model-based methods hold significant promise for reducing human effort in RL. Website: https://yangzhao-666.github.io/morefree

Joery A. de Vries, Thomas M. Moerland, Aske Plaat

Hierarchical Reinforcement Learning (HRL) has held longstanding promise to advance reinforcement learning. Yet, it has remained a considerable challenge to develop practical algorithms that exhibit some of these promises. To improve our fundamental understanding of HRL, we investigate hierarchical credit assignment from the perspective of conventional multistep reinforcement learning. We show how e.g., a 1-step `hierarchical backup' can be seen as a conventional multistep backup with $n$ skip connections over time connecting each subsequent state to the first independent of actions inbetween. Furthermore, we find that generalizing hierarchy to multistep return estimation methods requires us to consider how to partition the environment trace, in order to construct backup paths. We leverage these insight to develop a new hierarchical algorithm Hier$Q_k(λ)$, for which we demonstrate that hierarchical credit assignment alone can already boost agent performance (i.e., when eliminating generalization or exploration). Altogether, our work yields fundamental insight into the nature of hierarchical backups and distinguishes this as an additional basis for reinforcement learning research.

Francesca Da Ros, Tarik Začiragić, Aske Plaat et al.

Reproducibility is an important requirement in evolutionary computation, where results largely depend on computational experiments. In practice, reproducibility relies on how algorithms, experimental protocols, and artifacts are documented and shared. Despite growing awareness, there is still limited empirical evidence on the actual reproducibility levels of published work in the field. In this paper, we study the reproducibility practices in papers published in the Evolutionary Combinatorial Optimization and Metaheuristics track of the Genetic and Evolutionary Computation Conference over a ten-year period. We introduce a structured reproducibility checklist and apply it through a systematic manual assessment of the selected corpus. In addition, we propose RECAP (REproducibility Checklist Automation Pipeline), an LLM-based system that automatically evaluates reproducibility signals from paper text and associated code repositories. Our analysis shows that papers achieve an average completeness score of 0.62, and that 36.90% of them provide additional material beyond the manuscript itself. We demonstrate that automated assessment is feasible: RECAP achieves substantial agreement with human evaluators (Cohen's k of 0.67). Together, these results highlight persistent gaps in reproducibility reporting and suggest that automated tools can effectively support large-scale, systematic monitoring of reproducibility practices.

Andreas W. M. Sauter, Nicolò Botteghi, Erman Acar et al.

Causal discovery is the challenging task of inferring causal structure from data. Motivated by Pearl's Causal Hierarchy (PCH), which tells us that passive observations alone are not enough to distinguish correlation from causation, there has been a recent push to incorporate interventions into machine learning research. Reinforcement learning provides a convenient framework for such an active approach to learning. This paper presents CORE, a deep reinforcement learning-based approach for causal discovery and intervention planning. CORE learns to sequentially reconstruct causal graphs from data while learning to perform informative interventions. Our results demonstrate that CORE generalizes to unseen graphs and efficiently uncovers causal structures. Furthermore, CORE scales to larger graphs with up to 10 variables and outperforms existing approaches in structure estimation accuracy and sample efficiency. All relevant code and supplementary material can be found at https://github.com/sa-and/CORE

Andreas W M Sauter, Erman Acar, Aske Plaat

Research on causal effects often relies on synthetic data due to the scarcity of real-world datasets with ground-truth effects. Since current data-generating tools do not always meet all requirements for state-of-the-art research, ad-hoc methods are often employed. This leads to heterogeneity among datasets and delays research progress. We address the shortcomings of current data-generating libraries by introducing CausalPlayground, a Python library that provides a standardized platform for generating, sampling, and sharing structural causal models (SCMs). CausalPlayground offers fine-grained control over SCMs, interventions, and the generation of datasets of SCMs for learning and quantitative research. Furthermore, by integrating with Gymnasium, the standard framework for reinforcement learning (RL) environments, we enable online interaction with the SCMs. Overall, by introducing CausalPlayground we aim to foster more efficient and comparable research in the field. All code and API documentation is available at https://github.com/sa-and/CausalPlayground.

Jonathan Collu, Riccardo Majellaro, Aske Plaat et al.

The ability to perceive and reason about individual objects and their interactions is a goal to be achieved for building intelligent artificial systems. State-of-the-art approaches use a feedforward encoder to extract object embeddings and a latent graph neural network to model the interaction between these object embeddings. However, the feedforward encoder can not extract {\it object-centric} representations, nor can it disentangle multiple objects with similar appearance. To solve these issues, we introduce {\it Slot Structured World Models} (SSWM), a class of world models that combines an {\it object-centric} encoder (based on Slot Attention) with a latent graph-based dynamics model. We evaluate our method in the Spriteworld benchmark with simple rules of physical interaction, where Slot Structured World Models consistently outperform baselines on a range of (multi-step) prediction tasks with action-conditional object interactions. All code to reproduce paper experiments is available from \url{https://github.com/JonathanCollu/Slot-Structured-World-Models}.

Annie Wong, Thomas Bäck, Aske Plaat et al.

Large language models excel on static benchmarks, but their ability as self-learning agents in dynamic environments remains unclear. We evaluate three prompting strategies: self-reflection, heuristic mutation, and planning across dynamic tasks with open-source models. We find that larger models generally outperform smaller ones, but that strategic prompting can close this performance gap. Second, an overly long prompt can negatively impact smaller models on basic reactive tasks, while larger models show more robust behaviour. Third, advanced prompting techniques primarily benefit smaller models on complex games, but offer less improvement for already high-performing large language models. Yet, we find that advanced reasoning methods yield highly variable outcomes: while capable of significantly improving performance when reasoning and decision-making align, they also introduce instability and can lead to big performance drops. Compared to human performance, our findings reveal little evidence of true emergent reasoning. Instead, large language model performance exhibits persistent limitations in areas like planning and spatial coordination, suggesting that large language models still suffer fundamental shortcomings that may not be fully overcome through self-reflective prompting alone. Reasoning is a multi-faceted task, and while methods like Chain-of-thought improve multi-step reasoning on math word problems, our findings using dynamic benchmarks highlight important shortcomings in general reasoning capabilities, indicating a need to move beyond static benchmarks to capture the complexity of reasoning.

Koen Ponse, Aske Plaat, Niki van Stein et al.

Accurate economic simulations often require many experimental runs, particularly when combined with reinforcement learning. Unfortunately, training reinforcement learning agents in multi-agent economic environments can be slow. This paper introduces EconoJax, a fast simulated economy, based on the AI economist. EconoJax, and its training pipeline, are completely written in JAX. This allows EconoJax to scale to large population sizes and perform large experiments, while keeping training times within minutes. Through experiments with populations of 100 agents, we show how real-world economic behavior emerges through training within 15 minutes, in contrast to previous work that required several days. We additionally perform experiments in varying sized action spaces to test if some multi-agent methods produce more diverse behavior compared to others. Here, our findings indicate no notable differences in produced behavior with different methods as is sometimes suggested in earlier works. To aid further research, we open-source EconoJax on Github.

Lindsay Spoor, Álvaro Serra-Gómez, Aske Plaat et al.

In safety-critical domains such as robotics, navigation and power systems, constrained optimization problems arise where maximizing performance must be carefully balanced with associated constraints. Safe reinforcement learning provides a framework to address these challenges, with Lagrangian methods being a popular choice. However, the effectiveness of Lagrangian methods crucially depends on the choice of the Lagrange multiplier $λ$, which governs the trade-off between return and constraint cost. A common approach is to update the multiplier automatically during training. Although this is standard in practice, there remains limited empirical evidence on the robustness of an automated update and its influence on overall performance. Therefore, we analyze (i) optimality and (ii) stability of Lagrange multipliers in safe reinforcement learning across a range of tasks. We provide $λ$-profiles that give a complete visualization of the trade-off between return and constraint cost of the optimization problem. These profiles show the highly sensitive nature of $λ$ and moreover confirm the lack of general intuition for choosing the optimal value $λ^*$. Our findings additionally show that automated multiplier updates are able to recover and sometimes even exceed the optimal performance found at $λ^*$ due to the vast difference in their learning trajectories. Furthermore, we show that automated multiplier updates exhibit oscillatory behavior during training, which can be mitigated through PID-controlled updates. However, this method requires careful tuning to achieve consistently better performance across tasks. This highlights the need for further research on stabilizing Lagrangian methods in safe reinforcement learning. The code used to reproduce our results can be found at https://github.com/lindsayspoor/Lagrangian_SafeRL.

Tarik Zaciragic, Aske Plaat, K. Joost Batenburg

In the game of poker, being unpredictable, or bluffing, is an essential skill. When humans play poker, they bluff. However, most works on computer-poker focus on performance metrics such as win rates, while bluffing is overlooked. In this paper we study whether two popular algorithms, DQN (based on reinforcement learning) and CFR (based on game theory), exhibit bluffing behavior in Leduc Hold'em, a simplified version of poker. We designed an experiment where we let the DQN and CFR agent play against each other while we log their actions. We find that both DQN and CFR exhibit bluffing behavior, but they do so in different ways. Although both attempt to perform bluffs at different rates, the percentage of successful bluffs (where the opponent folds) is roughly the same. This suggests that bluffing is an essential aspect of the game, not of the algorithm. Future work should look at different bluffing styles and at the full game of poker. Code at https://github.com/TarikZ03/Bluffing-by-DQN-and-CFR-in-Leduc-Hold-em-Poker-Codebase.

Joery A. de Vries, Ken S. Voskuil, Thomas M. Moerland et al.

MuZero, a model-based reinforcement learning algorithm that uses a value equivalent dynamics model, achieved state-of-the-art performance in Chess, Shogi and the game of Go. In contrast to standard forward dynamics models that predict a full next state, value equivalent models are trained to predict a future value, thereby emphasizing value relevant information in the representations. While value equivalent models have shown strong empirical success, there is no research yet that visualizes and investigates what types of representations these models actually learn. Therefore, in this paper we visualize the latent representation of MuZero agents. We find that action trajectories may diverge between observation embeddings and internal state transition dynamics, which could lead to instability during planning. Based on this insight, we propose two regularization techniques to stabilize MuZero's performance. Additionally, we provide an open-source implementation of MuZero along with an interactive visualizer of learned representations, which may aid further investigation of value equivalent algorithms.

Hui Wang, Michael Emmerich, Mike Preuss et al.

Since AlphaGo and AlphaGo Zero have achieved breakground successes in the game of Go, the programs have been generalized to solve other tasks. Subsequently, AlphaZero was developed to play Go, Chess and Shogi. In the literature, the algorithms are explained well. However, AlphaZero contains many parameters, and for neither AlphaGo, AlphaGo Zero nor AlphaZero, there is sufficient discussion about how to set parameter values in these algorithms. Therefore, in this paper, we choose 12 parameters in AlphaZero and evaluate how these parameters contribute to training. We focus on three objectives~(training loss, time cost and playing strength). For each parameter, we train 3 models using 3 different values~(minimum value, default value, maximum value). We use the game of play 6$\times$6 Othello, on the AlphaZeroGeneral open source re-implementation of AlphaZero. Overall, experimental results show that different values can lead to different training results, proving the importance of such a parameter sweep. We categorize these 12 parameters into time-sensitive parameters and time-friendly parameters. Moreover, through multi-objective analysis, this paper provides an insightful basis for further hyper-parameter optimization.

Hui Wang, Michael Emmerich, Aske Plaat

After the recent groundbreaking results of AlphaGo and AlphaZero, we have seen strong interests in deep reinforcement learning and artificial general intelligence (AGI) in game playing. However, deep learning is resource-intensive and the theory is not yet well developed. For small games, simple classical table-based Q-learning might still be the algorithm of choice. General Game Playing (GGP) provides a good testbed for reinforcement learning to research AGI. Q-learning is one of the canonical reinforcement learning methods, and has been used by (Banerjee $\&$ Stone, IJCAI 2007) in GGP. In this paper we implement Q-learning in GGP for three small-board games (Tic-Tac-Toe, Connect Four, Hex)\footnote{source code: https://github.com/wh1992v/ggp-rl}, to allow comparison to Banerjee et al.. We find that Q-learning converges to a high win rate in GGP. For the $ε$-greedy strategy, we propose a first enhancement, the dynamic $ε$ algorithm. In addition, inspired by (Gelly $\&$ Silver, ICML 2007) we combine online search (Monte Carlo Search) to enhance offline learning, and propose QM-learning for GGP. Both enhancements improve the performance of classical Q-learning. In this work, GGP allows us to show, if augmented by appropriate enhancements, that classical table-based Q-learning can perform well in small games.

Aske Plaat, Max van Duijn, Niki van Stein et al.

There is great interest in agentic LLMs, large language models that act as agents. We review the growing body of work in this area and provide a research agenda. Agentic LLMs are LLMs that (1) reason, (2) act, and (3) interact. We organize the literature according to these three categories. The research in the first category focuses on reasoning, reflection, and retrieval, aiming to improve decision making; the second category focuses on action models, robots, and tools, aiming for agents that act as useful assistants; the third category focuses on multi-agent systems, aiming for collaborative task solving and simulating interaction to study emergent social behavior. We find that works mutually benefit from results in other categories: retrieval enables tool use, reflection improves multi-agent collaboration, and reasoning benefits all categories. We discuss applications of agentic LLMs and provide an agenda for further research. Important applications are in medical diagnosis, logistics and financial market analysis. Meanwhile, self-reflective agents playing roles and interacting with one another augment the process of scientific research itself. Further, agentic LLMs may provide a solution for the problem of LLMs running out of training data: inference-time behavior generates new training states, such that LLMs can keep learning without needing ever larger datasets. We note that there is risk associated with LLM assistants taking action in the real world, while agentic LLMs are also likely to benefit society.

Zhao Yang, Thomas M. Moerland, Mike Preuss et al.

Non-parametric episodic memory can be used to quickly latch onto high-rewarded experience in reinforcement learning tasks. In contrast to parametric deep reinforcement learning approaches in which reward signals need to be back-propagated slowly, these methods only need to discover the solution once, and may then repeatedly solve the task. However, episodic control solutions are stored in discrete tables, and this approach has so far only been applied to discrete action space problems. Therefore, this paper introduces Continuous Episodic Control (CEC), a novel non-parametric episodic memory algorithm for sequential decision making in problems with a continuous action space. Results on several sparse-reward continuous control environments show that our proposed method learns faster than state-of-the-art model-free RL and memory-augmented RL algorithms, while maintaining good long-run performance as well. In short, CEC can be a fast approach for learning in continuous control tasks.

Michiel van der Meer, Enrico Liscio, Catholijn M. Jonker et al.

Large-scale survey tools enable the collection of citizen feedback in opinion corpora. Extracting the key arguments from a large and noisy set of opinions helps in understanding the opinions quickly and accurately. Fully automated methods can extract arguments but (1) require large labeled datasets that induce large annotation costs and (2) work well for known viewpoints, but not for novel points of view. We propose HyEnA, a hybrid (human + AI) method for extracting arguments from opinionated texts, combining the speed of automated processing with the understanding and reasoning capabilities of humans. We evaluate HyEnA on three citizen feedback corpora. We find that, on the one hand, HyEnA achieves higher coverage and precision than a state-of-the-art automated method when compared to a common set of diverse opinions, justifying the need for human insight. On the other hand, HyEnA requires less human effort and does not compromise quality compared to (fully manual) expert analysis, demonstrating the benefit of combining human and artificial intelligence.

Xi Chen, Aske Plaat, Niki van Stein

Chain-of-thought (CoT) prompting boosts Large Language Models accuracy on multi-step tasks, yet whether the generated "thoughts" reflect the true internal reasoning process is unresolved. We present the first feature-level causal study of CoT faithfulness. Combining sparse autoencoders with activation patching, we extract monosemantic features from Pythia-70M and Pythia-2.8B while they tackle GSM8K math problems under CoT and plain (noCoT) prompting. Swapping a small set of CoT-reasoning features into a noCoT run raises answer log-probabilities significantly in the 2.8B model, but has no reliable effect in 70M, revealing a clear scale threshold. CoT also leads to significantly higher activation sparsity and feature interpretability scores in the larger model, signalling more modular internal computation. For example, the model's confidence in generating correct answers improves from 1.2 to 4.3. We introduce patch-curves and random-feature patching baselines, showing that useful CoT information is not only present in the top-K patches but widely distributed. Overall, our results indicate that CoT can induce more interpretable internal structures in high-capacity LLMs, validating its role as a structured prompting method.

Andreas Sauter, Saber Salehkaleybar, Aske Plaat et al.

Predicting the distribution of outcomes under hypothetical interventions is crucial across healthcare, economics, and policy-making. However, existing methods often require restrictive assumptions, and are typically limited by the lack of amortization across problem instances. We propose ACTIVA, a transformer-based conditional variational autoencoder (VAE) architecture for amortized causal inference, which estimates interventional distributions directly from observational data without. ACTIVA learns a latent representation conditioned on observational inputs and intervention queries, enabling zero-shot inference by amortizing causal knowledge from diverse training scenarios. We provide theoretical insights showing that ACTIVA predicts interventional distributions as mixtures over observationally equivalent causal models. Empirical evaluations on synthetic and semi-synthetic datasets confirm the effectiveness of our amortized approach and highlight promising directions for future real-world applications.

Aske Plaat

Search algorithms are often categorized by their node expansion strategy. One option is the depth-first strategy, a simple backtracking strategy that traverses the search space in the order in which successor nodes are generated. An alternative is the best-first strategy, which was designed to make it possible to use domain-specific heuristic information. By exploring promising parts of the search space first, best-first algorithms are usually more efficient than depth-first algorithms. In programs that play minimax games such as chess and checkers, the efficiency of the search is of crucial importance. Given the success of best-first algorithms in other domains, one would expect them to be used for minimax games too. However, all high-performance game-playing programs are based on a depth-first algorithm. This study takes a closer look at a depth-first algorithm, AB, and a best-first algorithm, SSS. The prevailing opinion on these algorithms is that SSS offers the potential for a more efficient search, but that its complicated formulation and exponential memory requirements render it impractical. The theoretical part of this work shows that there is a surprisingly straightforward link between the two algorithms -- for all practical purposes, SSS is a special case of AB. Subsequent empirical evidence proves the prevailing opinion on SSS to be wrong: it is not a complicated algorithm, it does not need too much memory, and it is also not more efficient than depth-first search.

Koen Ponse, Jan Felix Kleuker, Aske Plaat et al.

Deep Reinforcement Learning can play a key role in addressing sustainable energy challenges. For instance, many grid systems are heavily congested, highlighting the urgent need to enhance operational efficiency. However, reinforcement learning approaches have traditionally been slow due to the high sample complexity and expensive simulation requirements. While recent works have effectively used GPUs to accelerate data generation by converting environments to JAX, these works have largely focussed on classical toy problems. This paper introduces Chargax, a JAX-based environment for realistic simulation of electric vehicle charging stations designed for accelerated training of RL agents. We validate our environment in a variety of scenarios based on real data, comparing reinforcement learning agents against baselines. Chargax delivers substantial computational performance improvements of over 100x-1000x over existing environments. Additionally, Chargax' modular architecture enables the representation of diverse real-world charging station configurations.

Fien van Wetten, Aske Plaat, Max van Duijn

Large language models (LLMs) are known to perform well on language tasks, but struggle with reasoning tasks. This paper explores the ability of LLMs to play the 2D puzzle game Baba is You, in which players manipulate rules by rearranging text blocks that define object properties. Given that this rule-manipulation relies on language abilities and reasoning, it is a compelling challenge for LLMs. Six LLMs are evaluated using different prompt types, including (1) simple, (2) rule-extended and (3) action-extended prompts. In addition, two models (Mistral, OLMo) are finetuned using textual and structural data from the game. Results show that while larger models (particularly GPT-4o) perform better in reasoning and puzzle solving, smaller unadapted models struggle to recognize game mechanics or apply rule changes. Finetuning improves the ability to analyze the game levels, but does not significantly improve solution formulation. We conclude that even for state-of-the-art and finetuned LLMs, reasoning about dynamic rule changes is difficult (specifically, understanding the use-mention distinction). The results provide insights into the applicability of LLMs to complex problem-solving tasks and highlight the suitability of games with dynamically changing rules for testing reasoning and reflection by LLMs.

Annie Wong, Jacob de Nobel, Thomas Bäck et al.

Although deep reinforcement learning methods can learn effective policies for challenging problems such as Atari games and robotics tasks, algorithms are complex, and training times are often long. This study investigates how Evolution Strategies perform compared to gradient-based deep reinforcement learning methods. We use Evolution Strategies to optimize the weights of a neural network via neuroevolution, performing direct policy search. We benchmark both deep policy networks and networks consisting of a single linear layer from observations to actions for three gradient-based methods, such as Proximal Policy Optimization. These methods are evaluated against three classical Evolution Strategies and Augmented Random Search, which all use linear policy networks. Our results reveal that Evolution Strategies can find effective linear policies for many reinforcement learning benchmark tasks, unlike deep reinforcement learning methods that can only find successful policies using much larger networks, suggesting that current benchmarks are easier to solve than previously assumed. Interestingly, Evolution Strategies also achieve results comparable to gradient-based deep reinforcement learning algorithms for higher-complexity tasks. Furthermore, we find that by directly accessing the memory state of the game, Evolution Strategies can find successful policies in Atari that outperform the policies found by Deep Q-Learning. Evolution Strategies also outperform Augmented Random Search in most benchmarks, demonstrating superior sample efficiency and robustness in training linear policy networks.

Zhao Yang, Thomas M. Moerland, Mike Preuss et al.

Learning diverse skills without hand-crafted reward functions could accelerate reinforcement learning in downstream tasks. However, existing skill discovery methods focus solely on maximizing the diversity of skills without considering human preferences, which leads to undesirable behaviors and possibly dangerous skills. For instance, a cheetah robot trained using previous methods learns to roll in all directions to maximize skill diversity, whereas we would prefer it to run without flipping or entering hazardous areas. In this work, we propose a Foundation model Guided (FoG) skill discovery method, which incorporates human intentions into skill discovery through foundation models. Specifically, FoG extracts a score function from foundation models to evaluate states based on human intentions, assigning higher values to desirable states and lower to undesirable ones. These scores are then used to re-weight the rewards of skill discovery algorithms. By optimizing the re-weighted skill discovery rewards, FoG successfully learns to eliminate undesirable behaviors, such as flipping or rolling, and to avoid hazardous areas in both state-based and pixel-based tasks. Interestingly, we show that FoG can discover skills involving behaviors that are difficult to define. Interactive visualisations are available from https://sites.google.com/view/submission-fog.

Jacopo Di Ventura, Jan Felix Kleuker, Aske Plaat et al.

Zero-shot reinforcement learning (RL) has emerged as a setting for developing general agents in an unsupervised manner, capable of solving downstream tasks without additional training or planning at test-time. Unlike conventional RL, which optimizes policies for a fixed reward, zero-shot RL requires agents to encode representations rich enough to support immediate adaptation to any objective, drawing parallels to vision and language foundation models. Despite growing interest, the field lacks a common analytical lens. We present the first unified framework for zero-shot RL. Our formulation introduces a consistent notation and taxonomy that organizes existing approaches and allows direct comparison between them. Central to our framework is the classification of algorithms into two families: direct representations, which learn end-to-end mappings from rewards to policies, and compositional representations, which decompose the representation leveraging the substructure of the value function. Within this framework, we highlight shared principles and key differences across methods, and we derive an extended bound for successor-feature methods, offering a new perspective on their performance in the zero-shot regime. By consolidating existing work under a common lens, our framework provides a principled foundation for future research in zero-shot RL and outlines a clear path toward developing more general agents.

Yunqi Huang, Nishith Chennakeshava, Alexis Carras et al.

Container stowage planning (CSPP) is a critical component of maritime transportation and terminal operations, directly affecting supply chain efficiency. Owing to its complexity, CSPP has traditionally relied on human expertise. While reinforcement learning (RL) has recently been applied to CSPP, systematic benchmark comparisons across different algorithms remain limited. To address this gap, we develop a Gym environment that captures the fundamental features of CSPP and extend it to include crane scheduling in both multi-agent and single-agent formulations. Within this framework, we evaluate five RL algorithms: DQN, QR-DQN, A2C, PPO, and TRPO under multiple scenarios of varying complexity. The results reveal distinct performance gaps with increasing complexity, underscoring the importance of algorithm choice and problem formulation for CSPP. Overall, this paper benchmarks multiple RL methods for CSPP while providing a reusable Gym environment with crane scheduling, thus offering a foundation for future research and practical deployment in maritime logistics.

Peiwen Xing, Aske Plaat, Niki van Stein

Existing AI Music composition tools are limited in generation duration, musical quality, and controllability. We introduce CoComposer, a multi-agent system that consists of five collaborating agents, each with a task based on the traditional music composition workflow. Using the AudioBox-Aesthetics system, we experimentally evaluate CoComposer on four compositional criteria. We test with three LLMs (GPT-4o, DeepSeek-V3-0324, Gemini-2.5-Flash), and find (1) that CoComposer outperforms existing multi-agent LLM-based systems in music quality, and (2) compared to a single-agent system, in production complexity. Compared to non- LLM MusicLM, CoComposer has better interpretability and editability, although MusicLM still produces better music.

Jan Felix Kleuker, Aske Plaat, Thomas Moerland

Policy Mirror Descent (PMD) has emerged as a unifying framework in reinforcement learning (RL) by linking policy gradient methods with a first-order optimization method known as mirror descent. At its core, PMD incorporates two key regularization components: (i) a distance term that enforces a trust region for stable policy updates and (ii) an MDP regularizer that augments the reward function to promote structure and robustness. While PMD has been extensively studied in theory, empirical investigations remain scarce. This work provides a large-scale empirical analysis of the interplay between these two regularization techniques, running over 500k training seeds on small RL environments. Our results demonstrate that, although the two regularizers can partially substitute each other, their precise combination is critical for achieving robust performance. These findings highlight the potential for advancing research on more robust algorithms in RL, particularly with respect to hyperparameter sensitivity.

Riccardo Majellaro, Jonathan Collu, Aske Plaat et al.

Extracting structured representations from raw visual data is an important and long-standing challenge in machine learning. Recently, techniques for unsupervised learning of object-centric representations have raised growing interest. In this context, enhancing the robustness of the latent features can improve the efficiency and effectiveness of the training of downstream tasks. A promising step in this direction is to disentangle the factors that cause variation in the data. Previously, Invariant Slot Attention disentangled position, scale, and orientation from the remaining features. Extending this approach, we focus on separating the shape and texture components. In particular, we propose a novel architecture that biases object-centric models toward disentangling shape and texture components into two non-overlapping subsets of the latent space dimensions. These subsets are known a priori, hence before the training process. Experiments on a range of object-centric benchmarks reveal that our approach achieves the desired disentanglement while also numerically improving baseline performance in most cases. In addition, we show that our method can generate novel textures for a specific object or transfer textures between objects with distinct shapes.

Joost Broekens, Bernhard Hilpert, Suzan Verberne et al.

Large language models, in particular generative pre-trained transformers (GPTs), show impressive results on a wide variety of language-related tasks. In this paper, we explore ChatGPT's zero-shot ability to perform affective computing tasks using prompting alone. We show that ChatGPT a) performs meaningful sentiment analysis in the Valence, Arousal and Dominance dimensions, b) has meaningful emotion representations in terms of emotion categories and these affective dimensions, and c) can perform basic appraisal-based emotion elicitation of situations based on a prompt-based computational implementation of the OCC appraisal model. These findings are highly relevant: First, they show that the ability to solve complex affect processing tasks emerges from language-based token prediction trained on extensive data sets. Second, they show the potential of large language models for simulating, processing and analyzing human emotions, which has important implications for various applications such as sentiment analysis, socially interactive agents, and social robotics.

Zhao Yang, Thomas M. Moerland, Mike Preuss et al.

Go-Explore achieved breakthrough performance on challenging reinforcement learning (RL) tasks with sparse rewards. The key insight of Go-Explore was that successful exploration requires an agent to first return to an interesting state ('Go'), and only then explore into unknown terrain ('Explore'). We refer to such exploration after a goal is reached as 'post-exploration'. In this paper we present a systematic study of post-exploration, answering open questions that the Go-Explore paper did not answer yet. First, we study the isolated potential of post-exploration, by turning it on and off within the same algorithm. Subsequently, we introduce new methodology to adaptively decide when to post-explore and for how long to post-explore. Experiments on a range of MiniGrid environments show that post-exploration indeed boosts performance (with a bigger impact than tuning regular exploration parameters), and this effect is further enhanced by adaptively deciding when and for how long to post-explore. In short, our work identifies adaptive post-exploration as a promising direction for RL exploration research.

Aske Plaat

Deep reinforcement learning has gathered much attention recently. Impressive results were achieved in activities as diverse as autonomous driving, game playing, molecular recombination, and robotics. In all these fields, computer programs have taught themselves to solve difficult problems. They have learned to fly model helicopters and perform aerobatic manoeuvers such as loops and rolls. In some applications they have even become better than the best humans, such as in Atari, Go, poker and StarCraft. The way in which deep reinforcement learning explores complex environments reminds us of how children learn, by playfully trying out things, getting feedback, and trying again. The computer seems to truly possess aspects of human learning; this goes to the heart of the dream of artificial intelligence. The successes in research have not gone unnoticed by educators, and universities have started to offer courses on the subject. The aim of this book is to provide a comprehensive overview of the field of deep reinforcement learning. The book is written for graduate students of artificial intelligence, and for researchers and practitioners who wish to better understand deep reinforcement learning methods and their challenges. We assume an undergraduate-level of understanding of computer science and artificial intelligence; the programming language of this book is Python. We describe the foundations, the algorithms and the applications of deep reinforcement learning. We cover the established model-free and model-based methods that form the basis of the field. Developments go quickly, and we also cover advanced topics: deep multi-agent reinforcement learning, deep hierarchical reinforcement learning, and deep meta learning.

Zhao Yang, Mike Preuss, Aske Plaat

Learning to solve sparse-reward reinforcement learning problems is difficult, due to the lack of guidance towards the goal. But in some problems, prior knowledge can be used to augment the learning process. Reward shaping is a way to incorporate prior knowledge into the original reward function in order to speed up the learning. While previous work has investigated the use of expert knowledge to generate potential functions, in this work, we study whether we can use a search algorithm(A*) to automatically generate a potential function for reward shaping in Sokoban, a well-known planning task. The results showed that learning with shaped reward function is faster than learning from scratch. Our results indicate that distance functions could be a suitable function for Sokoban. This work demonstrates the possibility of solving multiple instances with the help of reward shaping. The result can be compressed into a single policy, which can be seen as the first phrase towards training a general policy that is able to solve unseen instances.

Aske Plaat, Walter Kosters, Mike Preuss

Deep reinforcement learning has shown remarkable success in the past few years. Highly complex sequential decision making problems from game playing and robotics have been solved with deep model-free methods. Unfortunately, the sample complexity of model-free methods is often high. To reduce the number of environment samples, model-based reinforcement learning creates an explicit model of the environment dynamics. Achieving high model accuracy is a challenge in high-dimensional problems. In recent years, a diverse landscape of model-based methods has been introduced to improve model accuracy, using methods such as uncertainty modeling, model-predictive control, latent models, and end-to-end learning and planning. Some of these methods succeed in achieving high accuracy at low sample complexity, most do so either in a robotics or in a games context. In this paper, we survey these methods; we explain in detail how they work and what their strengths and weaknesses are. We conclude with a research agenda for future work to make the methods more robust and more widely applicable to other applications.

Annie Wong, Thomas Bäck, Anna V. Kononova et al.

This paper surveys the field of deep multiagent reinforcement learning. The combination of deep neural networks with reinforcement learning has gained increased traction in recent years and is slowly shifting the focus from single-agent to multiagent environments. Dealing with multiple agents is inherently more complex as (a) the future rewards depend on multiple players' joint actions and (b) the computational complexity increases. We present the most common multiagent problem representations and their main challenges, and identify five research areas that address one or more of these challenges: centralised training and decentralised execution, opponent modelling, communication, efficient coordination, and reward shaping. We find that many computational studies rely on unrealistic assumptions or are not generalisable to other settings; they struggle to overcome the curse of dimensionality or nonstationarity. Approaches from psychology and sociology capture promising relevant behaviours, such as communication and coordination, to help agents achieve better performance in multiagent settings. We suggest that, for multiagent reinforcement learning to be successful, future research should address these challenges with an interdisciplinary approach to open up new possibilities in multiagent reinforcement learning.

Matthias Müller-Brockhausen, Mike Preuss, Aske Plaat

The idea of transfer in reinforcement learning (TRL) is intriguing: being able to transfer knowledge from one problem to another problem without learning everything from scratch. This promises quicker learning and learning more complex methods. To gain an insight into the field and to detect emerging trends, we performed a database search. We note a surprisingly late adoption of deep learning that starts in 2018. The introduction of deep learning has not yet solved the greatest challenge of TRL: generalization. Transfer between different domains works well when domains have strong similarities (e.g. MountainCar to Cartpole), and most TRL publications focus on different tasks within the same domain that have few differences. Most TRL applications we encountered compare their improvements against self-defined baselines, and the field is still missing unified benchmarks. We consider this to be a disappointing situation. For the future, we note that: (1) A clear measure of task similarity is needed. (2) Generalization needs to improve. Promising approaches merge deep learning with planning via MCTS or introduce memory through LSTMs. (3) The lack of benchmarking tools will be remedied to enable meaningful comparison and measure progress. Already Alchemy and Meta-World are emerging as interesting benchmark suites. We note that another development, the increase in procedural content generation (PCG), can improve both benchmarking and generalization in TRL.

Zhao Yang, Mike Preuss, Aske Plaat

Transfer learning can speed up training in machine learning and is regularly used in classification tasks. It reuses prior knowledge from other tasks to pre-train networks for new tasks. In reinforcement learning, learning actions for a behavior policy that can be applied to new environments is still a challenge, especially for tasks that involve much planning. Sokoban is a challenging puzzle game. It has been used widely as a benchmark in planning-based reinforcement learning. In this paper, we show how prior knowledge improves learning in Sokoban tasks. We find that reusing feature representations learned previously can accelerate learning new, more complex, instances. In effect, we show how curriculum learning, from simple to complex tasks, works in Sokoban. Furthermore, feature representations learned in simpler instances are more general, and thus lead to positive transfers towards more complex tasks, but not vice versa. We have also studied which part of the knowledge is most important for transfer to succeed, and identify which layers should be used for pre-training.

Hui Wang, Mike Preuss, Aske Plaat

AlphaZero has achieved impressive performance in deep reinforcement learning by utilizing an architecture that combines search and training of a neural network in self-play. Many researchers are looking for ways to reproduce and improve results for other games/tasks. However, the architecture is designed to learn from scratch, tabula rasa, accepting a cold-start problem in self-play. Recently, a warm-start enhancement method for Monte Carlo Tree Search was proposed to improve the self-play starting phase. It employs a fixed parameter $I^\prime$ to control the warm-start length. Improved performance was reported in small board games. In this paper we present results with an adaptive switch method. Experiments show that our approach works better than the fixed $I^\prime$, especially for "deep," tactical, games (Othello and Connect Four). We conjecture that the adaptive value for $I^\prime$ is also influenced by the size of the game, and that on average $I^\prime$ will increase with game size. We conclude that AlphaZero-like deep reinforcement learning benefits from adaptive rollout based warm-start, as Rapid Action Value Estimate did for rollout-based reinforcement learning 15 years ago.

Mike Huisman, Aske Plaat, Jan N. van Rijn

Deep learning typically requires large data sets and much compute power for each new problem that is learned. Meta-learning can be used to learn a good prior that facilitates quick learning, thereby relaxing these requirements so that new tasks can be learned quicker; two popular approaches are MAML and the meta-learner LSTM. In this work, we compare the two and formally show that the meta-learner LSTM subsumes MAML. Combining this insight with recent empirical findings, we construct a new algorithm (dubbed TURTLE) which is simpler than the meta-learner LSTM yet more expressive than MAML. TURTLE outperforms both techniques at few-shot sine wave regression and image classification on miniImageNet and CUB without any additional hyperparameter tuning, at a computational cost that is comparable with second-order MAML. The key to TURTLE's success lies in the use of second-order gradients, which also significantly increases the performance of the meta-learner LSTM by 1-6% accuracy.

Michael Emmerich, Joost Nibbeling, Marios Kefalas et al.

The general problem in this paper is vertex (node) subset selection with the goal to contain an infection that spreads in a network. Instead of selecting the single most important node, this paper deals with the problem of selecting multiple nodes for removal. As compared to previous work on multiple-node selection, the trade-off between cost and benefit is considered. The benefit is measured in terms of increasing the epidemic threshold which is a measure of how difficult it is for an infection to spread in a network. The cost is measured in terms of the number and size of nodes to be removed or controlled. Already in its single-objective instance with a fixed number of $k$ nodes to be removed, the multiple vertex immunisation problems have been proven to be NP-hard. Several heuristics have been developed to approximate the problem. In this work, we compare meta-heuristic techniques with exact methods on the Shield-value, which is a sub-modular proxy for the maximal eigenvalue and used in the current state-of-the-art greedy node-removal strategies. We generalise it to the multi-objective case and replace the greedy algorithm by a quadratic program (QP), which then can be solved with exact QP solvers. The main contribution of this paper is the insight that, if time permits, exact and problem-specific methods approximation should be used, which are often far better than Pareto front approximations obtained by general meta-heuristics. Based on these, it will be more effective to develop strategies for controlling real-world networks when the goal is to prevent or contain epidemic outbreaks. This paper is supported by ready to use Python implementation of the optimization methods and datasets.