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We study the problem of adapting to a known sub-rational opponent during online play while remaining robust to rational opponents. We focus on large imperfect-information (zero-sum) games, which makes it impossible to inspect the whole game tree at once and necessitates the use of depth-limited search. However, all existing methods assume rational play beyond the depth-limit, which only allows them to adapt a very limited portion of the opponent's behaviour. We propose an algorithm Adapting Beyond Depth-limit (ABD) that uses a strategy-portfolio approach - which we refer to as matrix-valued states - for depth-limited search. This allows the algorithm to fully utilise all information about the opponent model, making it the first robust-adaptation method to be able to do so in large imperfect-information games. As an additional benefit, the use of matrix-valued states makes the algorithm simpler than traditional methods based on optimal value functions. Our experimental results in poker and battleship show that ABD yields more than a twofold increase in utility when facing opponents who make mistakes beyond the depth limit and also delivers significant improvements in utility and safety against randomly generated opponents.

The accuracy of deployed malware-detection classifiers degrades over time due to changes in data distributions and increasing discrepancies between training and testing data. This phenomenon is known as the concept drift. While the concept drift can be caused by various reasons in general, new malicious files are created by malware authors with a clear intention of avoiding detection. The existence of the intention opens a possibility for predicting such future samples. Including predicted samples in training data should consequently increase the accuracy of the classifiers on new testing data. We compare two methods for predicting future samples: (1) adversarial training and (2) generative adversarial networks (GANs). The first method explicitly seeks for adversarial examples against the classifier that are then used as a part of training data. Similarly, GANs also generate synthetic training data. We use GANs to learn changes in data distributions within different time periods of training data and then apply these changes to generate samples that could be in testing data. We compare these prediction methods on two different datasets: (1) Ember public dataset and (2) the internal dataset of files incoming to Avast. We show that while adversarial training yields more robust classifiers, this method is not a good predictor of future malware in general. This is in contrast with previously reported positive results in different domains (including natural language processing and spam detection). On the other hand, we show that GANs can be successfully used as predictors of future malware. We specifically examine malware families that exhibit significant changes in their data distributions over time and the experimental results confirm that GAN-based predictions can significantly improve the accuracy of the classifier on new, previously unseen data.

Search in test time is often used to improve the performance of reinforcement learning algorithms. Performing theoretically sound search in fully adversarial two-player games with imperfect information is notoriously difficult and requires a complicated training process. We present a method for adding test-time search to an arbitrary policy-gradient algorithm that learns from sampled trajectories. Besides the policy network, the algorithm trains an additional critic network, which estimates the expected values of players following various transformations of the policies given by the policy network. These values are then used for depth-limited search. We show how the values from this critic can create a value function for imperfect information games. Moreover, they can be used to compute the summary statistics necessary to start the search from an arbitrary decision point in the game. The presented algorithm is scalable to very large games since it does not require any search during train time. We evaluate the algorithm's performance when trained along Regularized Nash Dynamics, and we evaluate the benefit of using the search in the standard benchmark game of Leduc hold'em, multiple variants of imperfect information Goofspiel, and Battleships.

Recently, Lad, Patel, and Pratap (LP&P) [Phys. Rev. E 105, 064107 (2022)] revisited a microscopic theory of molecular motion in liquids, proposed by Glass and Rice [Phys. Rev. 176, 239 (1968)]. Coming from this theory, LP&P derived a new equation of motion for the velocity autocorrelation function (VAF) and argued that the friction coefficient of particles in liquids should exponentially depend on time. The numerical solution of this equation was fitted to the results of molecular dynamics simulations on different liquids. In our Comment [Phys. Rev. E 108, 036107 (2023)], we showed that this solution, obtained under the condition of zero derivative of the VAF at time t = 0, is physically incorrect. This was evidenced by our exact analytical solution for the VAF, not found by LP&P, and numerically, by using the same method as in the commented work. In the Reply [Phys. Rev. E 108, 036108 (2023)], Lad, Patel, Pratap, and Pandya claimed that our solution does not satisfy all the necessary boundary conditions and is thus not appropriate for the description of atomic dynamics in liquids. Until and unless proven otherwise they do not find any reason for the reconsideration of their theory. Here we give a rebuttal to this Reply and, returning to the original work by LP&P, show that the presented there equation for the VAF is wrong. Due to errors in its derivation, it is, among other inconsistencies, incompatible precisely with the boundary conditions for the VAF which lie in the basis of their theory., Comment: 11 pages, 4 figures

Current frameworks for training offensive penetration testing agents with deep reinforcement learning struggle to produce agents that perform well in real-world scenarios, due to the reality gap in simulation-based frameworks and the lack of scalability in emulation-based frameworks. Additionally, existing frameworks often use an unrealistic metric that measures the agents' performance on the training data. NASimEmu, a new framework introduced in this paper, addresses these issues by providing both a simulator and an emulator with a shared interface. This approach allows agents to be trained in simulation and deployed in the emulator, thus verifying the realism of the used abstraction. Our framework promotes the development of general agents that can transfer to novel scenarios unseen during their training. For the simulation part, we adopt an existing simulator NASim and enhance its realism. The emulator is implemented with industry-level tools, such as Vagrant, VirtualBox, and Metasploit. Experiments demonstrate that a simulation-trained agent can be deployed in emulation, and we show how to use the framework to train a general agent that transfers into novel, structurally different scenarios. NASimEmu is available as open-source., Comment: NASimEmu is available at https://github.com/jaromiru/NASimEmu and the baseline agents at https://github.com/jaromiru/NASimEmu-agents

There is a limited amount of publicly available data to support research in malware analysis technology. Particularly, there are virtually no publicly available datasets generated from rich sandboxes such as Cuckoo/CAPE. The benefit of using dynamic sandboxes is the realistic simulation of file execution in the target machine and obtaining a log of such execution. The machine can be infected by malware hence there is a good chance of capturing the malicious behavior in the execution logs, thus allowing researchers to study such behavior in detail. Although the subsequent analysis of log information is extensively covered in industrial cybersecurity backends, to our knowledge there has been only limited effort invested in academia to advance such log analysis capabilities using cutting edge techniques. We make this sample dataset available to support designing new machine learning methods for malware detection, especially for automatic detection of generic malicious behavior. The dataset has been collected in cooperation between Avast Software and Czech Technical University - AI Center (AIC).

Learning from raw data input, thus limiting the need for feature engineering, is a component of many successful applications of machine learning methods in various domains. While many problems naturally translate into a vector representation directly usable in standard classifiers, a number of data sources have the natural form of structured data interchange formats (e.g., security logs in JSON/XML format). Existing methods, such as in Hierarchical Multiple Instance Learning (HMIL), allow learning from such data in their raw form. However, the explanation of the classifiers trained on raw structured data remains largely unexplored. By treating these models as sub-set selections problems, we demonstrate how interpretable explanations, with favourable properties, can be generated using computationally efficient algorithms. We compare to an explanation technique adopted from graph neural networks showing an order of magnitude speed-up and higher-quality explanations.

The discussion around the problematic practice of gendering humanoid robots has risen to the foreground in the last few years. To lay the basis for a thorough understanding of how robot's "gender" has been understood within the Human-Robot Interaction (HRI) community - i.e., how it has been manipulated, in which contexts, and which effects it has yield on people's perceptions and interactions with robots - we performed a scoping review of the literature. We identified 553 papers relevant for our review retrieved from 5 different databases. The final sample of reviewed papers included 35 papers written between 2005 and 2021, which involved a total of 3902 participants. In this article, we thoroughly summarize these papers by reporting information about their objectives and assumptions on gender (i.e., definitions and reasons to manipulate gender), their manipulation of robot's "gender" (i.e., gender cues and manipulation checks), their experimental designs (e.g., demographics of participants, employed robots), and their results (i.e., main and interaction effects). The review reveals that robot's "gender" does not affect crucial constructs for the HRI, such as likability and acceptance, but rather bears its strongest effect on stereotyping. We leverage our different epistemological backgrounds in Social Robotics and Gender Studies to provide a comprehensive interdisciplinary perspective on the results of the review and suggest ways to move forward in the field of HRI., Comment: 29 pages, 1 figure, 3 long tables. The present paper has been submitted for publication to the International Journal of Social Robotics and is currently under review

In zero-sum games, the optimal strategy is well-defined by the Nash equilibrium. However, it is overly conservative when playing against suboptimal opponents and it can not exploit their weaknesses. Limited look-ahead game solving in imperfect-information games allows defeating human experts in massive real-world games such as Poker, Liar's Dice, and Scotland Yard. However, since they approximate Nash equilibrium, they tend to only win slightly against weak opponents. We propose methods combining limited look-ahead solving with an opponent model in order to 1) approximate a best response in large games or 2) compute a robust response with control over the robustness of the response. Both methods can compute the response in real time to previously unseen strategies. We present theoretical guarantees of our methods. We show that existing robust response methods do not work combined with limited look-ahead solving of the shelf, and we propose a novel solution for the issue. Our algorithm performs significantly better than multiple baselines in smaller games and outperforms state-of-the-art methods against SlumBot., Comment: 16 pages, 15 figures

Recent advancements in algorithms for sequential decision-making under imperfect information have shown remarkable success in large games such as limit- and no-limit poker. These algorithms traditionally formalize the games using the extensive-form game formalism, which, as we show, while theoretically sound, is memory-inefficient and computationally intensive in practice. To mitigate these challenges, a popular workaround involves using a specialized representation based on player specific information-state trees. However, as we show, this alternative significantly narrows the set of games that can be represented efficiently. In this study, we identify the set of large games on which modern algorithms have been benchmarked as being naturally represented by Sequential Bayesian Games. We elucidate the critical differences between extensive-form game and sequential Bayesian game representations, both theoretically and empirically. We further argue that the impressive experimental results often cited in the literature may be skewed, as they frequently stem from testing these algorithms only on this restricted class of games. By understanding these nuances, we aim to guide future research in developing more universally applicable and efficient algorithms for sequential decision-making under imperfect information.

In malware behavioral analysis, the list of accessed and created files very often indicates whether the examined file is malicious or benign. However, malware authors are trying to avoid detection by generating random filenames and/or modifying used filenames with new versions of the malware. These changes represent real-world adversarial examples. The goal of this work is to generate realistic adversarial examples and improve the classifier's robustness against these attacks. Our approach learns latent representations of input strings in an unsupervised fashion and uses gradient-based adversarial attack methods in the latent domain to generate adversarial examples in the input domain. We use these examples to improve the classifier's robustness by training on the generated adversarial set of strings. Compared to classifiers trained only on perturbed latent vectors, our approach produces classifiers that are significantly more robust without a large trade-off in standard accuracy., Comment: 35th Conference on Neural Information Processing Systems (NeurIPS 2021), Strategic ML Workshop

Learning from raw data input, thus limiting the need for manual feature engineering, is one of the key components of many successful applications of machine learning methods. While machine learning problems are often formulated on data that naturally translate into a vector representation suitable for classifiers, there are data sources, for example in cybersecurity, that are naturally represented in diverse files with a unifying hierarchical structure, such as XML, JSON, and Protocol Buffers. Converting this data to vector (tensor) representation is generally done by manual feature engineering, which is laborious, lossy, and prone to human bias about the importance of particular features. Mill and JsonGrinder is a tandem of libraries, which fully automates the conversion. Starting with an arbitrary set of JSON samples, they create a differentiable machine learning model capable of infer from further JSON samples in their raw form., Comment: 5 pages, 2 figures, 1 table, submitted to section on one-source software of Journal of Machine Learning Research

In this report, we present results reproductions for several core algorithms implemented in the OpenSpiel framework for learning in games. The primary contribution of this work is a validation of OpenSpiel's re-implemented search and Reinforcement Learning algorithms against the results reported in their respective originating works. Additionally, we provide complete documentation of hyperparameters and source code required to reproduce these experiments easily and exactly.

Solution concepts of traditional game theory assume entirely rational players; therefore, their ability to exploit subrational opponents is limited. One type of subrationality that describes human behavior well is the quantal response. While there exist algorithms for computing solutions against quantal opponents, they either do not scale or may provide strategies that are even worse than the entirely-rational Nash strategies. This paper aims to analyze and propose scalable algorithms for computing effective and robust strategies against a quantal opponent in normal-form and extensive-form games. Our contributions are: (1) we define two different solution concepts related to exploiting quantal opponents and analyze their properties; (2) we prove that computing these solutions is computationally hard; (3) therefore, we evaluate several heuristic approximations based on scalable counterfactual regret minimization (CFR); and (4) we identify a CFR variant that exploits the bounded opponents better than the previously used variants while being less exploitable by the worst-case perfectly-rational opponent., Comment: 15 pages, 11 figures, submitted to AAAI 2021

We focus on reinforcement learning (RL) in relational problems that are naturally defined in terms of objects, their relations, and object-centric actions. These problems are characterized by variable state and action spaces, and finding a fixed-length representation, required by most existing RL methods, is difficult, if not impossible. We present a deep RL framework based on graph neural networks and auto-regressive policy decomposition that naturally works with these problems and is completely domain-independent. We demonstrate the framework's broad applicability in three distinct domains and show impressive zero-shot generalization over different problem sizes., Comment: code available at https://github.com/jaromiru/sr-drl

The Brownian motion of a particle immersed in a medium of charged particles is considered when the system is placed in magnetic or electric fields. Coming from the Zwanzig-Caldeira-Legget particle-bath model, we modify it so that not only the charged Brownian particle (BP) but also the bath particles respond to the external fields. For stationary systems the generalized Langevin equations are derived. Arbitrarily time-dependent electric fields do not affect the memory functions, the thermal noise force, and the BP velocity correlation functions. In the case of a constant magnetic field two equations with different memory functions are obtained for the BP motion in the plane perpendicular to the field. As distinct from the previous theories, the random thermal force depends on the field magnitude. Its time correlation function is connected with one of the found memory functions through the familiar second fluctuation-dissipation theorem., Comment: 8 pages, no figures

Anomaly detection is a method for discovering unusual and suspicious behavior. In many real-world scenarios, the examined events can be directly linked to the actions of an adversary, such as attacks on computer networks or frauds in financial operations. While the defender wants to discover such malicious behavior, the attacker seeks to accomplish their goal (e.g., exfiltrating data) while avoiding the detection. To this end, anomaly detectors have been used in a game-theoretic framework that captures these goals of a two-player competition. We extend the existing models to more realistic settings by (1) allowing both players to have continuous action spaces and by assuming that (2) the defender cannot perfectly observe the action of the attacker. We propose two algorithms for solving such games -- a direct extension of existing algorithms based on discretizing the feature space and linear programming and the second algorithm based on constrained learning. Experiments show that both algorithms are applicable for cases with low feature space dimensions but the learning-based method produces less exploitable strategies and it is scalable to higher dimensions. Moreover, we use real-world data to compare our approaches with existing classifiers in a data-exfiltration scenario via the DNS channel. The results show that our models are significantly less exploitable by an informed attacker., Comment: 9 pages, 3 figures, 3 tables. Extended Abstract of this paper is accepted to AAMAS 2020

Classification with Costly Features (CwCF) is a classification problem that includes the cost of features in the optimization criteria. Individually for each sample, its features are sequentially acquired to maximize accuracy while minimizing the acquired features' cost. However, existing approaches can only process data that can be expressed as vectors of fixed length. In real life, the data often possesses rich and complex structure, which can be more precisely described with formats such as XML or JSON. The data is hierarchical and often contains nested lists of objects. In this work, we extend an existing deep reinforcement learning-based algorithm with hierarchical deep sets and hierarchical softmax, so that it can directly process this data. The extended method has greater control over which features it can acquire and, in experiments with seven datasets, we show that this leads to superior performance. To showcase the real usage of the new method, we apply it to a real-life problem of classifying malicious web domains, using an online service., Comment: formerly Hierarchical Multiple-Instance Data Classification with Costly Features; RL4RealLife @ ICML2021; code available at https://github.com/jaromiru/rcwcf

This work focuses on a specific classification problem, where the information about a sample is not readily available, but has to be acquired for a cost, and there is a per-sample budget. Inspired by real-world use-cases, we analyze average and hard variations of a directly specified budget. We postulate the problem in its explicit formulation and then convert it into an equivalent MDP, that can be solved with deep reinforcement learning. Also, we evaluate a real-world inspired setting with sparse training dataset with missing features. The presented method performs robustly well in all settings across several distinct datasets, outperforming other prior-art algorithms. The method is flexible, as showcased with all mentioned modifications and can be improved with any domain independent advancement in RL.

Multiagent decision-making in partially observable environments is usually modelled as either an extensive-form game (EFG) in game theory or a partially observable stochastic game (POSG) in multiagent reinforcement learning (MARL). One issue with the current situation is that while most practical problems can be modelled in both formalisms, the relationship of the two models is unclear, which hinders the transfer of ideas between the two communities. A second issue is that while EFGs have recently seen significant algorithmic progress, their classical formalization is unsuitable for efficient presentation of the underlying ideas, such as those around decomposition. To solve the first issue, we introduce factored-observation stochastic games (FOSGs), a minor modification of the POSG formalism which distinguishes between private and public observation and thereby greatly simplifies decomposition. To remedy the second issue, we show that FOSGs and POSGs are naturally connected to EFGs: by "unrolling" a FOSG into its tree form, we obtain an EFG. Conversely, any perfect-recall timeable EFG corresponds to some underlying FOSG in this manner. Moreover, this relationship justifies several minor modifications to the classical EFG formalization that recently appeared as an implicit response to the model's issues with decomposition. Finally, we illustrate the transfer of ideas between EFGs and MARL by presenting three key EFG techniques -- counterfactual regret minimization, sequence form, and decomposition -- in the FOSG framework., Comment: A 2020 update of the original 2019 version of the paper. (Rewrote the main text and clarified the relationship between FOSGs/POSGs and EFGs. Some of the technical results are now presented in the appendix.)

We argue that the extensive-form game (EFG) model isn't powerful enough to express all important aspects of imperfect information games, such as those related to decomposition and online game solving. We present a principled attempt to fix the formalism by considering information partitions that correspond to observations. We show that EFGs cannot be "fixed" without additional knowledge about the original task, and show how to extend the EFG model under this assumption. However, during our investigation, we ultimately concluded that a better solution is to abandon EFGs entirely and instead adopt a model based on partially observable stochastic games. The primary contribution of the text thus lies in exposing the problems with EFGs and presenting a detailed study on introducing imperfect information by adding observations into an underlying perfect-information model.

We provide a formal definition of depth-limited games together with an accessible and rigorous explanation of the underlying concepts, both of which were previously missing in imperfect-information games. The definition works for an arbitrary extensive-form game and is not tied to any specific game-solving algorithm. Moreover, this framework unifies and significantly extends three approaches to depth-limited solving that previously existed in extensive-form games and multiagent reinforcement learning but were not known to be compatible. A key ingredient of these depth-limited games are value functions. Focusing on two-player zero-sum imperfect-information games, we show how to obtain optimal value functions and prove that public information provides both necessary and sufficient context for computing them. We provide a domain-independent encoding of the domains that allows for approximating value functions even by simple feed-forward neural networks, which are then able to generalize to unseen parts of the game. We use the resulting value network to implement a depth-limited version of counterfactual regret minimization. In three distinct domains, we show that the algorithm's exploitability is roughly linearly dependent on the value network's quality and that it is not difficult to train a value network with which depth-limited CFR's performance is as good as that of CFR with access to the full game., Comment: The first two authors contributed equally

Online game playing algorithms produce high-quality strategies with a fraction of memory and computation required by their offline alternatives. Continual Resolving (CR) is a recent theoretically sound approach to online game playing that has been used to outperform human professionals in poker. However, parts of the algorithm were specific to poker, which enjoys many properties not shared by other imperfect information games. We present a domain-independent formulation of CR applicable to any two-player zero-sum extensive-form games that works with an abstract resolving algorithm. We further describe and implement its Monte Carlo variant (MCCR) which uses Monte Carlo Counterfactual Regret Minimization (MCCFR) as a resolver. We prove the correctness of CR and show an $O(T^{-1/2})$-dependence of MCCR's exploitability on the computation time. Furthermore, we present an empirical comparison of MCCR with incremental tree building to Online Outcome Sampling and Information-set MCTS on several domains.

In the present work, by using the method of accumulation of phase shifts in the rotating frame, the attenuation function S(t) of the NMR signal from an ensemble of spin-bearing particles in a magnetic-field gradient is expressed through the particle mean square displacement in a form applicable for any kind of stationary stochastic dynamics of spins and for any times. S(t) is evaluated providing that the random motion of particles can be modeled by the generalized Langevin equation (GLE) with a colored random force driving the particles. The memory integral in this equation is the convolution of the particle velocity or its acceleration with a memory kernel related to the random force by the fluctuation-dissipation theorem. We consider three popular models of the BM with memory: the model of viscoelastic (Maxwell) fluids with the memory exponentially decaying in time, the fractional BM model, and the model of the hydrodynamic BM. In all the cases the solutions of the GLEs are obtained in an exceedingly simple way. The corresponding attenuation functions are then found for the free-induction NMR signal and the pulsed and steady-gradient spin-echo experiments. The results for the free-particle fractional BM compare favorably with experiments acquired in human neuronal tissues and with the observed subdiffusion dynamics in proteins., Comment: arXiv admin note: substantial text overlap with arXiv:1701.02486, arXiv:1803.01544, arXiv:1803.01554

Hannan consistency, or no external regret, is a~key concept for learning in games. An action selection algorithm is Hannan consistent (HC) if its performance is eventually as good as selecting the~best fixed action in hindsight. If both players in a~zero-sum normal form game use a~Hannan consistent algorithm, their average behavior converges to a~Nash equilibrium (NE) of the~game. A similar result is known about extensive form games, but the~played strategies need to be Hannan consistent with respect to the~counterfactual values, which are often difficult to obtain. We study zero-sum extensive form games with simultaneous moves, but otherwise perfect information. These games generalize normal form games and they are a special case of extensive form games. We study whether applying HC algorithms in each decision point of these games directly to the~observed payoffs leads to convergence to a~Nash equilibrium. This learning process corresponds to a~class of Monte Carlo Tree Search algorithms, which are popular for playing simultaneous-move games but do not have any known performance guarantees. We show that using HC algorithms directly on the~observed payoffs is not sufficient to guarantee the~convergence. With an~additional averaging over joint actions, the~convergence is guaranteed, but empirically slower. We further define an~additional property of HC algorithms, which is sufficient to guarantee the~convergence without the~averaging and we empirically show that commonly used HC algorithms have this property., Comment: arXiv admin note: substantial text overlap with arXiv:1509.00149

Extensive-form games (EFGs) model finite sequential interactions between players. The amount of memory required to represent these games is the main bottleneck of algorithms for computing optimal strategies and the size of these strategies is often impractical for real-world applications. A common approach to tackle the memory bottleneck is to use information abstraction that removes parts of information available to players thus reducing the number of decision points in the game. However, existing information-abstraction techniques are either specific for a particular domain, they do not provide any quality guarantees, or they are applicable to very small subclasses of EFGs. We present domain-independent abstraction methods for creating imperfect recall abstractions in extensive-form games that allow computing strategies that are (near) optimal in the original game. To this end, we introduce two novel algorithms, FPIRA and CFR+IRA, based on fictitious play and counterfactual regret minimization. These algorithms can start with an arbitrary domain specific, or the coarsest possible, abstraction of the original game. The algorithms iteratively detect the missing information they require for computing a strategy for the abstract game that is (near) optimal in the original game. This information is then included back into the abstract game. Moreover, our algorithms are able to exploit imperfect-recall abstractions that allow players to forget even history of their own actions. However, the algorithms require traversing the complete unabstracted game tree. We experimentally show that our algorithms can closely approximate Nash equilibrium of large games using abstraction with as little as 0.9% of information sets of the original game. Moreover, the results suggest that memory savings increase with the increasing size of the original games.

Nuclear magnetic resonance (NMR) is a widely used nondestructive method to study random motion of spin-bearing particles in different systems. In the long-time limit the theoretical description of the NMR experiments is well developed and allows proper interpretation of measurements of normal and anomalous diffusion. The traditional description becomes, however, insufficient for the shorter-time dynamics of the particles. In the present paper, the all-time attenuation function of the NMR signal in a magnetic-field gradient due to the Brownian motion (BM) of particles in incompressible liquids is calculated by using the method of accumulation of phases by a precessing magnetic moment, without reference to a concrete model of the stochastic dynamics. The obtained expressions are then used to evaluate the attenuation within the hydrodynamic theory of the BM. It is shown that the well-known time behavior of the formulas corresponding to the Einstein theory of diffusion in the case of steady gradient and Hahn's echo experiments is reached at times much larger than the characteristic time of the loss of memory in the particle dynamics. At shorter times the attenuation function significantly differs from the classical formulas used to interpret these experiments.

The methods of Nuclear Magnetic Resonance belong to the best developed and often used tools for studying random motion of particles in different systems, including soft biological tissues. In the long-time limit the current mathematical description of the experiments allows proper interpretation of measurements of normal and anomalous diffusion. The shorter-time dynamics is however correctly considered only in a few works that do not go beyond the standard memoryless Langevin description of the Brownian motion (BM). In the present work, the attenuation function S(t) for an ensemble of spin-bearing particles in a magnetic-field gradient, expressed in a form applicable for any kind of stationary stochastic dynamics of spins with or without a memory, is calculated in the frame of the model of fractional BM. The solution of the model for particles trapped in a harmonic potential is obtained in an exceedingly simple way and used for the calculation of S(t). In the limit of free particles coupled to a fractal heat bath, the results compare favorably with experiments acquired in human neuronal tissues. The effect of the trap is demonstrated by introducing a simple model for the generalized diffusion coefficient of the particle.