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Generalized Decision Transformer for Offline Hindsight Information Matching

How to extract as much learning signal from each trajectory data has been a key problem in reinforcement learning (RL), where sample inefficiency has posed serious challenges for practical applications. Recent works have shown that using expressive policy function approximators and conditioning on future trajectory information -- such as future states in hindsight experience replay or returns-to-go in Decision Transformer (DT) -- enables efficient learning of multi-task policies, where at times online RL is fully replaced by offline behavioral cloning, e.g. sequence modeling. We demonstrate that all these approaches are doing hindsight information matching (HIM) -- training policies that can output the rest of trajectory that matches some statistics of future state information. We present Generalized Decision Transformer (GDT) for solving any HIM problem, and show how different choices for the feature function and the anti-causal aggregator not only recover DT as a special case, but also lead to novel Categorical DT (CDT) and Bi-directional DT (BDT) for matching different statistics of the future. For evaluating CDT and BDT, we define offline multi-task state-marginal matching (SMM) and imitation learning (IL) as two generic HIM problems, propose a Wasserstein distance loss as a metric for both, and empirically study them on MuJoCo continuous control benchmarks. CDT, which simply replaces anti-causal summation with anti-causal binning in DT, enables the first effective offline multi-task SMM algorithm that generalizes well to unseen and even synthetic multi-modal state-feature distributions. BDT, which uses an anti-causal second transformer as the aggregator, can learn to model any statistics of the future and outperforms DT variants in offline multi-task IL. Our generalized formulations from HIM and GDT greatly expand the role of powerful sequence modeling architectures in modern RL.

• We introduce hindsight information matching (HIM) (Section 4, Table 1) as a unifying view of existing hindsight-inspired algorithms, and Generalized Decision Transformers (GDT) as a generalization of DT for RL as sequence modeling to solve any HIM problem ( Figure 1). • Inspired by distribution RL (Bellemare et al., 2017;Dabney et al., 2018) and state-marginal matching (SMM) (Lee et al., 2020;Ghasemipour et al., 2020;Gu et al., 2021), we define offline multi-task SMM problems, propose Categorical DT (CDT) (Section 5), validate its empirical performance to match feature distributions (even generalizing to a synthetic bi-modal target distribution at times), and construct the first benchmark tasks for offline multi-task SMM. • Inspired by one-shot imitation learning (Duan et al., 2017;Finn et al., 2017;Dasari & Gupta, 2020), we define offline multi-task imitation learning (IL), propose a Wasserstein-distance evaluation metric, develop Bi-directional DT (BDT) as a fully expressive variant of GDT (Section 5), and demonstrate BDT's competitive performance at offline multi-task IL.  (Chen et al., 2021a) and the table summarizes how it leads to different classes of algorithms with only small architectural changes. If the feature function Φ(s, a) is reward r(s, a) and the anti-causal aggregator is γ-discounted summation, we recover DT for offline RL. If the aggregator is binning, we get Categorical DT (CDT) for offline multi-task state-marginal matching. If the aggregator is a second transformer, we get Bi-directional DT (BDT) for offline multi-task imitation learning (IL), or equivalently one-shot IL. The choices of Φ(s, a) and the aggregator together decide I Φ (τ ) in Hindsight Information Matching (HIM) objective discussed in Section 4 and Table 1, where conversely GDT can essentially solve any HIM problem with proper choices of Φ and aggregator.

RELATED WORK
Hindsight Reinforcement Learning and Behavior Cloning Hindsight techniques (Kaelbling, 1993;Andrychowicz et al., 2017;Pong et al., 2018) have revolutionized off-policy optimization with respect to parameterized reward functions. Two key insights were (1) for off-policy algorithms such as Q-learning (Mnih et al., 2015;Gu et al., 2016) and actor-critic methods (Lillicrap et al., 2016;Haarnoja et al., 2018;Fujimoto et al., 2018;Furuta et al., 2021a), the same transition samples can be used to learn with respect to any reward parameters, as long as the reward function is re-computable, i.e. "relabel"-able, like goal reaching rewards, and (2) if policy or Q-functions are smooth with respect to the reward parameter, generalization can speed up learning even with respect to "unexplored" rewards. In goal-based RL where future states can inform "optimal" reward parameters with respect to the transitions' actions, hindsight methods were applied successfully to enable effective training of goal-based Q-function for sparse rewards (Andrychowicz et al., 2017), derive exact connections between Q-learning and classic model-based RL (Pong et al., 2018), dataefficient off-policy hierarchical RL (Nachum et al., 2018), multi-task RL (Eysenbach et al., 2020;Li et al., 2020), offline RL (Chebotar et al., 2021), and more (Eysenbach et al., 2021;Choi et al., 2021;Ren et al., 2019;Zhao & Tresp, 2018;Ghosh et al., 2021;Nasiriany et al., 2021). Additionally, Lynch et al. (2019) and Gupta et al. (2018) have shown that often BC is sufficient for learning generalizable parameterized policies, due to rich positive examples from future states, and most recently Chen et al. (2021a) and Janner et al. (2021), when combined with powerful transformer architectures (Vaswani et al., 2017), it produced state-of-the-art offline RL and goal-based RL results. Lastly, while motivated from alternative mathematical principles and not for parameterized objectives, future state information was also explored as ways of reducing variance or improving estimations for generic policy gradient methods (Pinto et al., 2017;Guo et al., 2021;Venuto et al., 2021).
Distributional Reinforcement Learning and State-Marginal Matching Modeling the full distribution of returns instead of the averages led to the development of distributional RL algorithms (Bellemare et al., 2017;Dabney et al., 2018;2020;Castro et al., 2018;Barth-Maron et al., 2018) such as Categorical Q-learning (Bellemare et al., 2017). While our work shares techniques such as discretization and binning, these works focus on optimizing a non-conditional reward-maximizing RL policy and therefore our problem definition is closer to that of state-marginal matching algorithms (Hazan et al., 2019;Lee et al., 2020;Ghasemipour et al., 2020;Gu et al., 2021), or equivalently inverse RL algorithms (Ziebart et al., 2008;Ho & Ermon, 2016;Finn et al., 2016;Fu et al., 2018;Ghasemipour et al., 2020) whose connections to feature-expectation matching have been long discussed (Abbeel & Ng, 2004). However, those are often exclusively online algorithms even sample-efficient variants (Kostrikov et al., 2019), since density-ratio estimations with either discriminative (Ghasemipour et al., 2020) or generative (Lee et al., 2020) approach requires on-policy samples, with a rare exception of Kostrikov et al. (2020). Building on the success of DT and brute-force hindsight imitation learning, our Categorical DT is to the best our knowledge the first method that benchmarks offline state-marginal matching problem in the multi-task settings.
RL and Imitation Learning as Sequence Modeling When scaled to the extreme levels of data and computing, sequence models such as transformers (Vaswani et al., 2017) can train models to master an impressive range of capabilities in natural language processing and computer vision (Devlin et al., 2019;Radford et al., 2019;Brown et al., 2020;Radford et al., 2021;Ramesh et al., 2021;Chen et al., 2021b;Bommasani et al., 2021;Dosovitskiy et al., 2020). Comparing to their popularity in other areas, the adoption of transformers or architectural innovations in RL have been slow, partially due the difficulty of using transformers over temporal scales for online RL (Parisotto et al., 2020). Recent successes have focused on processing variable-length per-timestep information such as morphology (Kurin et al., 2021), sensory information (Tang & Ha, 2021), one-shot or few-shot imitation learning (Dasari & Gupta, 2020), or leveraged offline learning (Chen et al., 2021a;Janner et al., 2021). Our formulation enables sequence modeling to solve novel RL problems such as statemarginal matching with minimal architectural modifications to DT, greatly expanding the impacts of transformers and other powerful sequence models in RL.

PRELIMINARIES
We consider a Markov Decision Process (MDP) defined by the tuple of action space A, state space S, transition probability function p(s |s, a), initial state distribution p(s 0 ), reward function r(s, a), and discount factor γ ∈ (0, 1]. In deep RL, a policy that maps the state space to the action space is parameterized by the function approximators, π θ (a|s) 1 . The RL objective is given by: where p π t (s) = s0:t,a0:t−1 t p(s t |s t−1 , a t−1 )π(a t |s t ) and ρ π (s) = (1 − γ) t γ t p π t (s t = s) are short-hands for time-aligned and time-aggregated state marginal distributions following policy π.

STATE MARGINAL MATCHING
State marginal matching (SMM) (Lee et al., 2020;Hazan et al., 2019;Ghasemipour et al., 2020) has been recently studied as an alternative problem specification in RL, where instead of stationary-reward maximization, the objective is to find a policy minimizing the divergence D between its state marginal distribution ρ π (s) to a given target distribution p * (s) 2 : where D is a divergence measure such as Kullback-Leibler (KL) divergence (Lee et al., 2020;Fu et al., 2018) or, more generally, some f -divergences (Ghasemipour et al., 2020). For the target distribution p * (s), Lee et al. (2020) set a uniform distribution to enhance the exploration over the entire state space; Ghasemipour et al. (2020) and Gu et al. (2021) set through scripted distribution sketches to generate desired behaviors; and adversarial inverse RL methods (Ho & Ermon, 2016;Fu et al., 2018;Ghasemipour et al., 2020;Kostrikov et al., 2020) set as the expert data for imitation learning. Notably, unlike the RL objective in Eq.1, SMM objectives like Eq.2 no longer depend on task rewards and are only functions of state transition dynamics and target state distribution.

PARAMETERIZED RL OBJECTIVES
Lastly, we discuss the basis for methods like HER and TDM (Andrychowicz et al., 2017;Pong et al., 2018), LfP (Lynch et al., 2019), and return-conditioned or upside-down RL (Srivastava et al., 2019;Kumar et al., 2019;Chen et al., 2021a;Janner et al., 2021): parameterized RL objectives. Given parameterized reward functions with parameter z ∈ Z, a conditional policy π(a|s, z) is learned with respect to multiple values of z simultaneously weighted by p(z). As examples, the RL objective in Eq.1 becomes: where the state marginal ρ π z is from rolling out a conditioned policy π(·|·, z). These can be considered as a special case of contextual MDPs (Jiang et al., 2017) and are all multi-task RL problems.

HINDSIGHT INFORMATION MATCHING
We show how HER and TDM (Andrychowicz et al., 2017;Pong et al., 2018), LfP (Lynch et al., 2019), hindsight multi-task RL (Li et al., 2020Eysenbach et al., 2020), and return-conditioned or upside-down RL (Srivastava et al., 2019;Kumar et al., 2019;Chen et al., 2021a;Janner et al., 2021) all belong to hindsight algorithms with a shared idea of using future state information to automatically mine for positive, or "optimal", examples with respect to certain contextual parameter values, where these examples can accelerate RL or be used for behavior cloning (BC), i.e. supervised learning. We start by defining additional notations.
Given a partial trajectory from state s t as τ t = {s t , a t , s t+1 , a t+1 , . . . }, we define its information statistics as I(τ t ). I(τ t ) could be any function of a trajectory that captures some statistical properties in state-space or trajectory-space, such as sufficient statistics of a distribution, like mean, variance or higher-order moments (Wainwright & Jordan, 2008). For convenience, we further define the notion of a feature function Φ(·, ·) : S × A → F 5 , where the trajectory is then noted as   et al., 2017). Φ is the reward function r(s, a) in (2) and (3), an indexing function for state dimensions (e.g. xy-velocities) or a learned function (Nair et al., 2018) in (1), or an identify function in (4). Our CDT introduces a new category, (5) distribution-based I Φ (τ ) = histogram(rt, γ), based on a minimal modification to DT, while our BDT can be considered as DT adapted for (4), the trajectory imitation.
can be an identity function, the reward function r(s, a), sub-dimensions of s (e.g. xy-velocities), or a generic parameterized function (e.g. auto-encoder). Generalizing reward-centric intuitions in DT (Chen et al., 2021a), we define information matching (IM) problems as learning a conditional policy π(a|s, z) whose trajectory rollouts satisfy some desired information statistics value z: An important observation for the IM objective (Eq.4) is that for any given trajectory τ , setting z * = I Φ (τ ) will minimize the inner term divergence D = 0 and therefore τ states and actions are optimal with respect to z = z * and samples of (τ i , z * i ) can be used to accelerate RL or do BC. We call these algorithms hindsight information matching (HIM) algorithms. Table 1, which classifies prior methods into effectively four categories based on I Φ (τ ), leads us to have the following insights around HIM algorithms: • New HIM algorithms can be proposed by simply changing I Φ (τ ), as we did to propose Categorical DT for (5) distribution-based. • Given a choice of I Φ (τ ), new HIM algorithms can be proposed by changing implementation details (Furuta et al., 2021a), such as using "RL" or "BC" as algorithm type, doing "Online" or "Offline" training (Levine et al., 2020), and network architectures. All "Offline" "BC" methods could be adopted easily to "Online" learning through recursive data collections (Ghosh et al., 2021;Kumar et al., 2019;Matsushima et al., 2021). • Only (1) goal-based and (2) multi-task can use "RL" as algorithm type, while all four, plus our (5) distribution-based, can use "BC", because "RL" requires optimizing Eq. 4 with respect to the policy, which gets non-trivial for some choices of I Φ (τ ); e.g. (3-5) return-based, full trajectory imitation, or distribution-based. "BC" bypasses the need to solve Eq. 4 and therefore is universally applicable to any I Φ (τ ) or HIM algorithm 6 .

GENERALIZED DECISION TRANSFORMER
Following the insights in Section 4, we introduce Generalized Decision Transformer (GDT), which generalizes DT (Chen et al., 2021a) based on different choices of I Φ (τ ), as described in Figure 1 and the last rows of Table 1. We chose DT as the base model since it is a simple model that uses "BC" as the algorithm type and "Transformer" as the architecture. The choice of "BC" is a must, so we can tractably train GDT with respect to any I Φ (τ ) or HIM problem. The choice for the architecture is more flexible; however, we decided to use transformers (Vaswani et al., 2017) in this work due to their enormous scaling successes in language and vision domains (Dosovitskiy et al., 2020;Brown et al., 2020;Ramesh et al., 2021). See Algorithm 1 (in Appendix F) for the full pseudocode. While GDT in Figure 1 can lead to different algorithms depending on different choices of the feature function Φ(s, a) and the anti-causal aggregator (which together determine I Φ (τ )), in this work we focus our empirical studies on the following two variants: Categorical DT (CDT) and Bi-directional DT (BDT).

TASK DEFINITIONS AND METRICS
Before proceeding to define CDT and BDT, we first concretely define the tasks they are designed to solve, namely: offline multi-task state-marginal matching (SMM), and offline multi-task imitation learning (IL). Given the intrinsic connection or equivalence between distribution matching and IL (Ghasemipour et al., 2020), these two separate terminologies may seem redundant. However, inspired by the initial papers studying SMM problems (Lee et al., 2020;Ghasemipour et al., 2020) which qualitatively evaluates distribution matching results in specified state dimensions (e.g. xypositions), we define the imitation task as offline multi-task SMM if specific Φ is given, and as offline multi-task IL if Φ is an identity (i.e. φ = s) or learned (e.g. auto-encoder). We essentially view IL as SMM evaluation on full state.
Given these definition, we also define a single metric for both offline multi-task SMM/IL: typical IL assumes some availability of task reward or success evaluation, and indirectly measure the quality of imitation through it (Ho & Ermon, 2016;Fu et al., 2018). Instead, again grounding on its connection to distribution matching (Ghasemipour et al., 2020), we propose a Wasserstein loss between statemarginal and target distributions as SMM-inspired metrics for evaluating offline multi-task SMM or IL tasks. However, it is often intractable to measure such loss for full state or even for some state dimensions analytically because both state-marginal and target distributions can be non-parametric and we cannot access their densities. In practice, we empirically estimate it employing the binning of the feature space we specified. More discussions are included in Appendix C. , we assume low-dimensional Φ, e.g. rewards or state dimensions like xyz-velocities, and employ the discretization of feature spaces to form categorical approximations of continuous distributions. To compute z * t = I Φ (τ t:T ) for all timesteps t given a trajectory τ 1:T , we use similar recursive Bellman-like computation inspired by Bellemare et al. (2017) (see Appendix F for the details). To the best of our knowledge, this is the first paper to study offline multi-task SMM and propose an effective algorithm for it.

DECISION TRANSFORMER WITH LEARNED Φ
While CDT assumes some low-dimensional Φ is provided for tractable binning and distribution approximation, we also study cases where Φ or r is not provided, and instead Φ is learned through auto-encoding (Hinton & Salakhutdinov, 2006;Bengio et al., 2012) (DT-AE) or contrastive (van den Oord et al., 2018;Srinivas et al., 2020;Yang & Nachum, 2021) (DT-CPC) losses for DT (see Appendix G for the details). In this case, CDT is unnecessary because if Φ learns sufficient features of s, matching their means, i.e. moments, through DT is enough to match any distribution to an arbitrary precision (Wainwright & Jordan, 2008;Li et al., 2015). Since Φ is differentiable with respect to DT's action-prediction losses, we also compare DT-E2E, where we learn Φ through end-to-end differentiation. As Section 5.1 defines, these methods do offline multi-task SMM with full state, or offline multi-task IL, a similar objective to state-marginal matching or adversarial inverse RL (Ho & Ermon, 2016;Ghasemipour et al., 2020) in online RL. To the best of our knowledge, this is the first offline multi-task IL method that explicitly accounts for SMM through architectural bottlenecks.

BI-DIRECTIONAL DECISION TRANSFORMER FOR ONE-SHOT IMITATION LEARNING
The absence of given Φ could be tackled with learning not only parameterized Φ as in Section 5.3, but also a parameterized aggregator. Building on the connection to one-shot imitation learning (Duan et al., 2017), we provide another natural extension of DT under GDT framework called Bi-directional Decision Transformer (BDT), which assumes an identity Φ, and learns representation within the aggregator, in this case a second (anti-causal) transformer (Radford et al., 2018) that takes a reverseorder state sequence as an input. See Algorithm 1 in Appendix F for the pseudocode, and Appendix D for further comments on the connection to one-shot or meta learning. While we found some positive results for even simple unsupervised regularizer approaches (e.g. DT-AE), we observe BDT could achieve substantially better offline multi-task IL results than DT-X variants in Section 5.3.

EXPERIMENTS
We empirically investigate the following questions: • (SMM) Can CDT match unseen reward distributions? • (SMM) Can CDT match and generalize to unseen 1D/2D state-feature distributions? • (SMM) Can CDT match unseen synthesized state-feature distributions? • (IL) Can BDT perform offline one-shot imitation learning in full state?
We experiment on the OpenAI Gym, MuJoCo tasks (HalfCheetah, Hopper, Walker2d, Ant-v3), a common benchmark for continuous control (Brockman et al., 2016;Todorov et al., 2012). Through the experiments, we use medium-expert datasets in D4RL (Fu et al., 2020) to ensure the decent data coverage. We sort all the trajectories by their cumulative rewards, hold out five best trajectories and five 50 percentile trajectories as a test set (10 trajectories in total), and use the rest as a train set. We report the results averaged over 20 rollouts every 4 random seed. We share our implementation to ensure the reproducibility 8 .
As discussed in Section 5.1 and Appendix C, we evaluate CDT/BDT with approximate distribution matching objective: Wasserstein-1 distance between categorical distributions of features in rollouts or target trajectories. We compare CDT to DT, Meta-BC, and FOCAL (Li et al., 2021), a metric-based offline meta RL method, as baselines. While Meta-BC and FOCAL does not solve the offline distribution matching problem directly, they provide decent baseline performance since their offline one-shot adaptation to the given target trajectories could deal with it (see Appendix B for the details).

REWARD AND STATE-FEATURE MATCHING
First, we evaluate whether CDT could match its rollout to the target distribution. We choose reward and state-feature, such as x-velocity of the agents, as feature spaces to match. To specify the target distributions during the evaluation, we feed the categorical representation of the target to CDT. As the same as the reward case, DT takes the summation of the state-feature over a trajectory as an input.
We quantitatively compare CDT against baselines (Table 2) in x-velocity case, where CDT shows better matching results to the target distributions unseen during training. We provide the reward distribution results and the visualization in Appendix E.1, where CDT performs very well in all cases. To test the generalization additionally, we intervene the target distributions by (1) shifting the target distributions in the test set by constant offsets, and (2) Table 2: Quantitative evaluation of state-feature distribution matching, measuring Wasserstein-1 distance between the rollout and target distributions. We compare Categorical DT against DT, BC, Meta-BC, and FOCAL. CDT achieves better matching than baselines. See Table 9 in Appendix E.1 for the reward distribution results.

GENERALIZATION TO UNSEEN TARGET DISTRIBUTION
The performance of offline methods in RL is often restricted by the coverage or quality of datasets. While we demonstrate CDT can perform offline state-marginal matching to unseen target distributions in Section 6.1, the standard offline datasets might not be diverse enough to observe the generalization since those are collected by single-task reward-maximization policies. To test the generalization to more diverse behaviors, we investigate the following tasks: (1) z-velocity distribution matching with synthesized bi-modal behavior and (2)  To generate diverse behaviors for z-axis, we obtain the expert cheetah that backflips towards -x direction by modifying reward function (see Appendix E.4 for the details). Combining expert backflipping trajectories and expert running forward trajectories from D4RL dataset, we construct a novel dataset with diverse behaviors. We experiment the offline SMM with not only each uni-modal behavior (backflipping or running forward), but also synthesized bi-modal behavior; running forward first, then backflipping during a single rollout, using patchworked target trajectories. Table 3 and Figure 2 (a) show that CDT successfully matches the distribution to both uni-modal (running forward or backflipping) and synthesized bi-modal distributions better than DT and FOCAL, and is comparable to Meta-BC that originally designed to deal with such multi-task settings. Due to the difficulty for RL algorithms to maximize Eq. 4, FOCAL struggles to solve the offline multi-task SMM, even though FOCAL uses the same context embedding as Meta-BC. The bi-modal behavior learned by CDT can be seen at https://sites.google.com/view/generalizeddt.  Table 3: State-feature (z-velocity) distribution matching with uni-modal and (synthesized) bi-modal target trajectories in HalfCheetah environment. Categorical DT matches both uni-and bi-modal trajectories better than DT and FOCAL, and is comparable to Meta-BC that originally aims to solve multi-task problem. In (a), CDT (red) can match not only uni-modal behaviors for both running forward and backflipping, but also bi-modal behaviors; during a single rollout running forward first, then backflipping. DT (yellow) tends to lean backflipping and fails to fit neither uni-modal nor bi-modal ones. In (b), CDT successfully handles the trajectories unseen during training, while DT seems to output covering behaviors over the dataset support.

DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK
Generalization to unknown target demonstrations or tasks has been actively investigated in meta or one-shot RL/IL literature (Duan et al., 2016;Wang et al., 2016). To verify the generalization of CDT to diverse behaviors, we adopt the cheetah-velocity task; a popular task in meta RL/IL (Rakelly et al., 2019;Li et al., 2021;Fakoor et al., 2020), where the cheetah tries to run with the specified velocity. We prepare 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals, and hold out {0.5, 1.5, 2.5} as a test set. See Appendix E.5 for the dataset generation. Table 4 and Figure 2 (b) reveal that CDT outperforms DT or FOCAL, and is slightly better than Meta-BC through the distribution matching evaluation, which implies CDT could solve offline multi-task SMM generalizing to the unknown target trajectories, given sufficiently diverse offline datasets.  Table 4: Generalization to unseen target velocity (x-velocity = 0.5, 1.5, 2.5) among training dataset (x-velocity ∈ [0.0, 3.0]\{0.5, 1.5, 2.5}; uniformly spaced at 0.1 intervals), which is a popular setting in meta RL/IL. CDT successfully deals with unseen target velocities well, outperforming DT and FOCAL, and is slightly better than Meta-BC through the offline multi-task SMM evaluation.

ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE
Lastly, we investigate BDT for offline multi-task IL, where we do not observe rewards nor state features explicitly. Instead, target full state trajectories that the agents are expected to mimic is given. We compare BDT against parameterized Φ variants; DT-AE, -CPC, and -E2E discussed in Section 5.3. We consider three strategies to train the encoder for DT-AE and -CPC: training with only unsupervised loss, training with unsupervised and DT's supervised loss jointly (called as "joint"), and pre-training with only unsupervised loss and freezing the weights during DT training (called as "frozen"). We aggregate the learned Φ by summation within the input sequence (length is 20). We evaluate BDT and DT-X with the same distribution matching evaluation as Section 6.1.
We sweep m-dim learned feature from the encoder Φ(s) with m ∈ {1, 4, 16}. Table 5 presents the offline multi-task IL results with respect to x-velocity distribution, using m = 16 embeddings (see Appendix E.6 for the full results including m = 1, 4, and the reward distribution case). Similar to the discussion in Yang & Nachum (2021), DT-CPC sometimes fails to obtain sufficient representation for imitation. While even simple approaches, DT-AE and DT-AE (frozen), show positive results compared to no-context BC baselines (in Table 2), BDT outperforms all other learned Φ variants or Meta-BC and is comparable to CDT or DT (also in Table 2) with longer input (N = 50). This implies that even though we don't assume the state-feature specification, aggregator choice in GDT with minimal architectural changes may solve the offline distribution matching problem efficiently. We leave more sophisticated objectives or architectural changes as future work.  Table 5: The results of BDT on the state-feature distribution matching problem (m = 16). While even simple auto-encoder regularizers sometimes work well, BDT with longer contexts seems to outperform other strategies and is comparable to CDT or DT (in Table 2). See Appendix E.6 for the full results including m = 1, 4.

CONCLUSION
We provide a unified perspective on a wide range of hindsight algorithms, and generalize the problem formulation as hindsight information matching (HIM). Inspired by recent successes in RL as sequence modeling, we propose Generalized Decision Transformer (GDT) which includes DT, Categorical DT, and Bi-directional DT as special cases, and is applicable to any HIM with proper choices of Φ and aggregator. We show how Categorical DT, a minor modification of DT, enables the first effective offline state-marginal matching algorithm and propose new benchmark tasks for this problem class. We also demonstrate the effectiveness of Bi-directional DT as a one-shot imitation learner, significantly outperforming simple variants based on DT. We hope our proposed HIM and GDT frameworks shed new perspectives on hindsight algorithms and the applicability of sequence modeling to much broader classes of RL problems beyond classic reward-based RL.

ETHICS STATEMENT
Since this paper mainly focuses on the reinterpretation of hindsight RL algorithms and experiments on existing benchmark datasets, we believe there are no ethical concerns.

REPRODUCIBILITY STATEMENT
We share our codes to ensure the reproductivity. The details of hyperparameters are described in Appendix A and B. The detailed settings of experiments are described in the Section 6, Appendix E, and G. We report the results averaged over 20 rollouts every 4 random seed.  Meta-BC We also use metric-based Meta-BC (Duan et al., 2017; Dasari & Gupta, 2020) as a strong baseline method for offline multi-task SMM. We adapt deterministic context encoder from FOCAL to infer the task. We train Meta-BC in the same way as FOCAL just replacing BRAC to BC. The objective is mean-squared error minimization.
Throughout our experiments, Meta-BC shows better results than FOCAL, because RL algorithms often struggle to optimize Eq. 4 for distribution matching and FOCAL tries to maximize the task reward, which is not necessarily required for distribution matching problem. In addition, BC often converges faster than offline RL methods.

C DETAILS OF EVALUATION
Throughout the paper, we evaluate both CDT and BDT from a distribution matching perspective by formulating them as offline multi-task SMM or offline multi-task IL problem. However, the current distribution matching research in RL (Lee et al., 2020;Ghasemipour et al., 2020;Gu et al., 2021) has been suffered from the lack of quantitative metrics for evaluation (while standard RL or BC are typically evaluated on the task reward performance). As summarized in Table 7, prior works (Lee et al., 2020; Ghasemipour et al., 2020) evaluate the SMM performance by qualitative density visualization using rollout particles.
Although the distance between state-marginal and target distribution seems the most intuitive and suitable metric to quantify the performance of distribution matching algorithms, it is often intractable to measure such distance analytically because both state-marginal and target distribution can be non-parametric; we cannot access their densities, and only their samples are available. While a concurrent work (Gu et al., 2021) tackles this problem by leveraging sample-based energy distance estimation, we introduce a single SMM-inspired evaluation for both offline multi-task SMM and offline multi-task IL via estimating Wasserstein-1 distance between empirical categorical distributions.
Since the discretization of low-dimensional features, such as reward, have success in many RL methods (Bellemare et al., 2017;Dabney et al., 2018;2020;Furuta et al., 2021b), quantification of distribution matching performance by Wasserstein-1 distance could be reliable evaluations (in addition Wasserstein-1 distance is symmetric, different from KL distance that is asymmetric). We note that due to the equivalence between inverse RL and SMM methods, the performance of distribution matching methods may be measured via task reward when assuming the accessivity to the expert trajectories (Ho & Ermon, 2016;Fu et al., 2018;Kostrikov et al., 2019). However, in our experiments, it is not suitable to evaluate the performance based on task reward since we uses the sub-optimal trajectories (Section 6.1) or multi-task trajectories from different reward functions (Section 6.2).
Offline Meta RL Recently, some works deal with offline meta RL problem, assuming task distribution and offline training with pre-stored transitions collected by the (sub-) optimal or scripted task-conditioned agents (Dorfman et al., 2021;Mitchell et al., 2021;Li et al., 2021;Pong et al., 2021;Zhao et al., 2021). In these works, few test-task trajectories, following the same task distribution but unseen during training phase, are given at test-time (i.e. few-shot), and the agents adapt the given Our work, in contrast, assume no-adaptation at test time and evaluation criteria is a distance between the feature distributions, not the task rewards. Meta IL offline (no-adaptation) one-shot Ours Offline multi-task SMM/IL offline (no-adaptation) one-shot Table 8: Review of problem settings among offline meta RL, meta IL, and ours. In offline meta RL, the agents are trained offline and tested with several demonstrations (few-shot), in a fully-offline manner or allowing online data-collection for adaptation. In meta IL, IRL-based methods train the agents online (

E DETAILS OF EXPERIMENTS
In this section, we provide the details and supplemental quantitative/qualitative results of the experiments in Section 6. Figure 3 visualizes the dataset distributions we use in the experiments. When we sort all the trajectories based on their cumulative rewards, each quality of trajectory (best, middle, worst) shows a different shape of distribution, and the distribution of the whole dataset seems a weighted mixture of those.

E.1 QUANTITATIVE AND QUALITATIVE RESULTS OF REWARD AND STATE-FEATURE MATCHING
We provide the quantitative comaprison of Section 6.1 between Categorical DT and DT in the reward (Table 9) matching problem, computing Wasserstein-1 distance between the discretized rollout and target feature distributions. Generally, similar to the x-velocity case, CDT shows the better results in offline multi-task SMM compared to original DT. We also visualize some of CDT results in Figure 4.  Table 9: Quantitative evaluation of reward distribution matching via measuring Wasserstein-1 distance between the rollout and target distributions. We compare Categorical DT and DT. Since it can capture the multi-modal nature of target distributions, CDT matches the distribution better than the original DT.

E.2 EVALUATION ON TASK PERFORMANCE
While in this paper we focus on the evaluation with the SMM-inspired distribution matching objective, such as Wasserstein-1 distance, we here provide the evaluation on the task rewards. Table 10 shows that DT seems consistently better methods than Categorical DT on the task rewards evaluations, and achieves similar performances to the held-out trajectories. Table 10: Evaluation on the task rewards; conditioning on the held-out trajectories as done in Section 6. We compare the performance between Categorical DT, and DT. DT seems consistently better methods on the task rewards evaluations, and achieves similar performances to the held-out trajectories (averaged over 5 trajectories).

E.3 2D STATE-FEATURE DISTRIBUTION MATCHING
We also consider two-dimensional state-features (xy-velocities) distribution matching in Ant-v3. Same as 1D state-feature distribution matching in HalfCheetah, Hopper, and Walker2d-v3 (Section 6.1), we also use medium-expert(-v2) datasets from D4RL (Fu et al., 2020). We bin the state-features per dimension separately to reduce the dimension of the categorical distribution that CDT takes as input, while in test-time we evaluate the performance with Wasserstein-1 metric on the joint distribution. For DT, we compute the summation of x-and y-velocity each over trajectories and normalize them with the maximum horizon. DT feeds these two scalars as information statistics to match. Table 11 reveals that CDT performs better even in the case of two-dimensional state-features distributions, while DT doesn't generalize to expert-quality trajectories. As shown in Figure 5, while CDT could cope with the distribution shift between expert and medium target distribution, DT always fits the medium one even if the expert trajectory is given as a target. CDT successfully scales to the offline multi-task SMM problem in the multi-dimensional feature spaces.

E.4 DETAILS OF SYNTHESIZING UNSEEN BI-MODAL DISTRIBUTION
To construct the dataset, we modified the original reward function in HalfCheetah-v3, adding absolute z-velocity term (such as +np.abs(z_vel)), where the expert cheetah backflips towards -x direction.
We trained SAC agent until convergence (3 million gradient steps), using pytorch implementation released by Furuta et al. (2021a), and then collected 500 trajectories × 1000 time steps. We combined them with 500 trajectories × 1000 time steps from halfcheetah-expert-v2 dataset in D4RL, which consists of both backflipping and running forward behaviors.
For the evaluation, we prepared additional 5 trajectories of backflipping and 5 of running forward as uni-modal behaviors (10 test trajectories in total). In addition, we synthesized a bi-modal behavior by dividing each 1000-step trajectories into 500-step sub-trajectories, and concatenating them across different behaviors, which results in the patchworked trajectories of first 500-step running forward and next 500-step backflipping (also 10 trajectories in total).

E.5 DETAILS OF DIVERSE UNSEEN DISTRIBUTION FROM META LEARNING TASK
Following prior meta RL/IL works (Rakelly et al., 2019;Ghasemipour et al., 2019;Pong et al., 2021;Li et al., 2021;Fakoor et al., 2020), we modified the reward function for the cheetah to run with specified velocity (such as -np.abs(x_vel -target_vel)), and set the horizon to 200 steps.
We prepared 31 target x-velocities; taken from [0.0, 3.0], uniformly spaced at 0.1 intervals. We also trained the SAC agents until convergence (3 million gradient steps), using pytorch implementation released by Furuta et al. (2021a), and collected 250 trajectories × 200 time steps each. To simplify the problem than meta learning settings, we held out the 10 trajectories whose x-velocity is {0.5, 1.5, 2.5} as a test set, and used the rest as a train data. Table 12 and Table 13 show the one-shot reward and state-feature distribution matching results respectively. In addition to the embedding size m, we also sweep the different context window N = 20, 50, 100 for BDT (m = 16). While even simple auto-encoder regularizer (DT-AE or DT-AE (frozen)) sometimes works well compared to no-context BC baselines presented in Table 2 and Table 9, BDT, using (second) anti-causal transformer as an encoder Φ and its aggregator, seems consistently better than other strategies or Meta-BC baseline for offline multi-task SMM and is comparable to CDT or DT results (also presented in Table 2 and Table 9) with longer context length (N = 50). Through the experiment we observe that while there are no clear trends in DT-AE, -CPC or -E2E as the size of context embedding m grows, BDT improves its performance with larger size of embedding. Such intuitive properties might a good features to design the architectures.

E.6 QUANTITATIVE AND QUALITATIVE RESULTS OF ONE-SHOT DISTRIBUTION MATCHING IN FULL STATE
We visualize the qualitative results of BDT (m = 16, N = 20) in Figure 6.

E.7 SHIFTING THE TARGET DISTRIBUTION
To generate the unseen but manageable generalization-test trajectories within the support of dataset distribution, we make the reward and velocities values of trajectories in the test set shift with a constant offset: bin_size ×{−3.0, −2.0, −1.0, 0.0, +1.0, +2.0, +3.0}. Table 14 shows the quantitative comparison between CDT and DT, based on Wasserstein-1 distance between two distributions. CDT successfully handles the distribution shifts (especially in hopper) better than DT. We also provide the state-feature results (Table 15) and qualitative visualizations (Figure 7 and Figure 8), which reveals that CDT can match the rollouts to the shifted target distributions when they are within the support of dataset distributions.     (Table 14). Categorical DT (red) can match the rollouts to the shifted target distributions (blue) when the shifted targets are within the support of dataset distributions. For DT (yellow, captioned as Deterministic), we only visualize the delta function at the means of rollouts.
Published as a conference paper at ICLR 2022  (Table 15). For DT (yellow, captioned as Deterministic), we only visualize the delta function at the means of rollouts.

E.8 SYNTHESIZING UNREALISTIC TARGET DISTRIBUTION
We synthesize six target distributions manually generating x-velocity samples from Gaussian distributions via python scripts as done in prior works (Ghasemipour et al., 2020;Gu et al., 2021). Since we consider the one-dimensional feature space (x-velocity), we simply specify the mean and standard deviation of Gaussian distributions referring each dataset distribution, as shown in Figure 3, and then generate the 1000 samples per trajectory. While these targets are designed at least within the support of the dataset, we do not consider physical realizability. For the last two sets, that have different distributional parameters {(µ 1 , σ 1 ), (µ 2 , σ 2 )}, we sampled from two gaussian distributions 500 samples each, and marge them as one trajectory that has multiple modes.
Although they might be unrealistic, Figure 9 implies that CDT tends to match the target distributions, even in the cases of bi-modal target distributions. Such generalization to synthesized distribution is an important benefit of distribution-conditioned training. We also quantify the performance of CDT against DT from distributional matching perspective in Table 16. We synthesize the each target distribution from Gaussian distributions. While the results are worse than for realistic test target distributions in Figure 4, considering that many of these arbitrary synthetic targets could be unrealizable, seeing bi-modal matching results show that indeed CDT has learned to generalize something non-trivial.

F DETAILS FOR CATEGORICAL DECISION TRANSFORMER
Categorical Decision Transformer (CDT) takes histograms of categorical distribution (i.e. discrete approximations of feature distributions; B-dim vector) as the inputs of the transformer. Here we describe how to compute the distributions for all timesteps given a trajectory t ∈ [0, 1, . . . , T ]. For simplicity, we explain the case of one-dimensional feature (e.g. scalar reward), but for n-dimensional features, we can adopt following procedure per each dimension and arrive at n categorical features. This essentially approximates the full joints with a product of independent marginal distribution per dimension, and ensures that number of samples for getting reasonably discretized approximations do not need to grow exponentially as the dimension grows.
At first, we discretize the feature space F using B bins (per dimension), and convert the feature φ t into the one-hot representationφ t . The range of feature space [φ min , φ max ] is pre-defined from the given offline datasets. z Φhist (t), the categorical feature distribution at time step t, can be computed recursively following Bellman-like equation: z Φhist (t) ∝φ t + γ(1 − 1[t = T ])z Φhist (t + 1), where 1 is the indicator function. We compute a series of z Φhist in a backward manner starting from T .
After we obtain the desired information statistics for all trajectories, we feed them to the categorical transformer during training/test time. We describe the python-like pseudocode in Algorithm 1, coloring the changes from the original Decision Transformer (Chen et al., 2021a).

G DETAILS OF DECISION TRANSFORMER WITH LEARNED Φ
While any unsupervised regularizer for an encoder could be combined into the action MSE loss of Decision Transformer to obtain the learned Φ efficiently, we observe that even simple objectives, such as auto-encoder and contrastive loss, sometimes perform well. For auto-encoder regularization (DT-AE), we train the MLP encoder (parameterized by ψ) and decoder with MSE loss of current state reconstruction: min ψ E s∼D s − decoder ψ (encoder ψ (s)) 2 .
Then, we use the output of the encoder as a learned information statistics. In addition, for contrastive loss (DT-CPC), we adopt CURL objective (Srinivas et al., 2020) for state input while adding gaussian perturbation ∼ N (µ = 0.0, σ = 0.1) as data argumentation (following Sinha et al. (2021)). We train the MLP encoder as a query, use its momentum encoder as a key: , where W is a learned parameter matrix for the bi-linear inner-product, andψ is the weights of the momentum encoder, updated asψ ← mψ + (1 − m)ψ, and m = 0.95. We treat its trained encoder output as a learned information statistics. It remains as future work to combine more advanced, temporary-extended objectives, such as attentive contrastive learning approach proposed in Yang & Nachum (2021). As described in Section 6.3, we consider three strategies to train the encoder for DT-AE and -CPC: training with only unsupervised loss, training with unsupervised and DT's supervised loss jointly (called as "joint"), and pre-training with only unsupervised loss and freezing the weights during DT training (called as "frozen"). We train DT-E2E with only DT's supervised loss as in Algorithm 1 for CDT and BDT.


Dynamic optimized cooperation in multi-source multi-relay wireless networks with random linear network coding

The authors present a strategy that improves the reliability of data trans- mission from multiple sources towards a destination D via multiple relays over wireless channels. After the broadcast phase, the destination selects the best cooperative direct link sources with the highest instantaneous signal-to-noise ratio γ , then generates and sends the random lin- ear network coding coefﬁcient matrix to the relays. After receiving the random linear network coding–coded symbols (packets) from the re- lays, D completes the reception and decoding of all source nodes packets. Monte Carlo simulations are conducted with Galois ﬁeld (GF) ( q ) symbols ( q = 2, 4 and 8) and the results are compared with the scenario of static cooperative direct links. An important decrease of symbol error rate (SER) is achieved. Furthermore, the efﬁciency of this approach is conﬁrmed for the whole range of possible values of path loss exponent on direct links, that is, all types of environment between source nodes and destination D .

Introduction: Recent advances in wireless networks exploit the concepts of cooperative communications (CC) [1,2] and network coding (NC) [3][4][5] to enhance wireless systems' performance by increasing system throughput, diversity order, spectral efficiency and robustness in particular when CC and NC are jointy used [6][7][8]. In multi-source multirelay and one destination wireless networks [9], the contribution of direct links between source nodes and the destination are not negligible. The right choice of cooperative direct links is crucial for achieving optimal system performance by reducing error probability and system energy and increasing diversity order [10]. The authors in [11] derived closed forms of outage probability and diversity gain of NCC-based multi-source multi-relay wireless networks where they proposed and applied a generalized relay selection (RS) method. They showed that the full diversity gain is only achieved in the scenario where the number of arbitrarily selected relays is greater or equal to N (number of sources) with N highest signal-to-noise ratio (SNR) relays. A new smart decodeand-forward (DF) RS scheme was proposed in [12] which considers the source-destination links outage status. Relays use linear NC with coefficients from a generator matrix of maximum distance separable (MDS) code. The proposed scheme outperforms the conventional min-max RS scheme. In [13], the authors showed how the introduction of multipleinput multiple-output (MIMO) strategy can improve the performance of NCC systems. They introduced a new RS strategy which utilizes only the local channel state information (CSI), thus allowing to reduce the signaling overhead. By equipping the relays and destination with multiple antennas, they proved the superiority of their proposed scheme in improving the performance of MIMO-NCC systems over the min-max traditional RS criterion scheme. RS [14,15] is the main goal of almost all the previously mentioned works. We assume the relays as fixed network nodes along with the destination (base station, BS). Relay nodes are capable of performing random linear network coding (RLNC) [16][17][18] on their received signals with the assistance of destination D which has computational capabilities to execute the optimization process. This later relies on the mobility of source nodes (mobile users) to dynamically select the best cooperative ones in every transmission round.
System model: In order to highlight the main features and improvements offered by our proposed strategy, we use a multi-source multirelay and single destination wireless system as illustrated in Figure 1. The N S source nodes (mobile users) S i (i = 1, 2, . . . , N S ) are communicating with the destination D or BS via N R relays R j ( j = 1, 2, . . . , N R ) with the assumption N R ≤ N S . All source nodes are inside the coverage area of the BS, that is, have direct links with the destination. Assumptions on the distances d SiR j (from source node S i to relay node R j ), d SiD (from source node S i to destination D) and d R j D (from relay node R j to destination D) are as follows: We assume all nodes equipped with a single antenna and relays work in half-duplex mode. Transmissions are performed in non-overlapping time slots in time division multiple access (TDMA) scheme. The destination and the relays are fixed while the source nodes are mobile. Direct links S i → D are further weakened by the path loss due to their longer distances. Our proposed approach relies on this particular fading and the fact that source nodes are moving mobile users, hence, distances d SiD are permanently changing. According to the number of source nodes N S , the number of relays N R being constant, two scenarios are possible. The first one, when N R = N S , does not make use of cooperation but still requires the reception of all source nodes signals at destination D in the broadcast phase. The second scenario that is most optimized by our approach is when N R < N S since it exploits the cooperation of the remaining N S − N R direct links to destination. The N R relays apply RLNC in GF (q), where q = 2 b and b denotes the number of bits per symbol (packet), on the N S packets received in both scenarios.
Transmitted symbols are elements of finite fields GF (q) (q = 2, 4, 8, . . .) and are modulated using q-ary PSK. Receiving nodes have perfect CSI of all their links that are subject to the Nakagami-m fading model. The transmission round of all N S symbols lasts N S + N R + 1 time slots and is divided into three stages, namely, the broadcast phase, the optimization phase and the relaying phase.
Broadcast phase: The broadcast phase lasts N S time slots where each source node S i (i = 1, 2, . . . , N S ) modulates, using q-ary PSK, its symbol m Si into x Si and broadcasts the latter to destination D and all N R relays. The received signal at each relay R j ( j = 1, 2, . . . , N R ) from each source node S i (i = 1, 2, . . . , N S ) is given by where E Si is the average symbol energy of source S i , h SiR j is the Nakagami-m channel fading coefficient on link S i → R j and n SiR j is the complex additive white Gaussian noise (AWGN) at the input of R j . The AWGN component n SiR j is independent identically distributed (i.i.d.) random variable (RV) with zero mean and variance σ 2 SiR j , that is, n SiR j ∼ CN (0, σ 2 SiR j ). Upon reception, the relay applies the following equalization operation where h * SiR j is the complex conjugate of h SiR j . Then, by applying q-PSK demodulation onx SiR j , we obtainm SiR j the estimate of m Si at relay R j .
The key to our optimization process is to account for the effect of distance and path loss of cooperative direct links S i → D (i = 1, 2, . . . , N S ). Therefore, the received signal at D from S i in the broadcast phase has the form whereh SiD = PL SiD h SiD represents the actual channel fading coefficient of link S i → D and PL SiD represents its path loss which is evaluated using the simplified model commonly used for system design [19] where K is a unitless constant depending on the antenna characteristics and average channel attenuation, d 0 is a reference distance and α is the path loss exponent. Parameters K and d 0 are usually adjusted to approximate either an analytical or empirical model. Path loss exponent α ranges from 2 to 6 depending on the carrier frequency, type of environment and obstructions. For urban areas, α is approximately equal to 3. Upon reception of the N S signals, destination D first applies the following optimization processing.
then |h SI 1 D | 2 is replaced by 0 in (5) to find I 2 and so on until we find I NS −NR . Only signals y SI k D (k = 1, 2, . . . , N S − N R ) are decoded at destination as follows: for k = 1, 2, . . . , N S − N R . Then, eachx SI k D is demapped intom SI k D using q-PSK demodulation for use in the next relaying phase. Afterward, destination D generates the N R × N R full-rank RLNC matrix A with elements in GF (q) For instance, if q = 8, N S = 6, N R = 4, the selected cooperative sources are S 2 and S 4 , that is, I 1 = 2 and I 2 = 4 and the generated fullrank 4 × 4 RLNC matrix in GF (8)  is the augmented 4 × 6 matrix that is sent to the relays.
Relaying phase: Each relay R j ( j = 1, 2, . . . , N R ) has N S decoded symbolsm SiR j from the broadcast phase and the augmented N R × N S RLNC coefficients matrixÃ from the optimization phase. Each relay index j is equal to the number of its corresponding row of coefficients in matrixÃ. That is, each relay R j derives its network coded symbol equal to the linear combination ofã jk (k = 1, 2, . . . , N S ) andm SiR j (i = 1, 2, . . . , N S ). Then, the N R RLNC-coded symbols are given by for j = 1, 2, . . . , N R . The symbol ⊕ designates the bitwise exclusive OR (XOR) operator in GF (q) and hence the resulting symbols m R j are elements of GF (q) as well. After mapping them into x R j ( j = 1, 2, . . . , N R ) with q-ary PSK modulation, they are transmitted in N R time slots to destination D. In each time slot N S + 1 + j ( j = 1, 2, . . . , N R ), D receives one RLNC signal given by After equalization intox R j followed by demodulation intom R j , destination D applies XOR between eachm R j and all the previously selected cooperative symbols (in optimization phase), that is, for j = 1, 2, . . . , N R : Procedure of expression (9) serves to eliminate all the cooperative direct link symbols from the combinations (i.e. RLNC network coded symbols) m R j . The last step is to recover the remaining N R source nodes symbols by solving, using Gaussian elimination in GF (q), the following system of linear equations where a i j (i, j = 1, 2, . . . , N R ) are elements of the N R × N R full-rank (invertible) RLNC matrix A generated randomly at D in the optimization phase. The vector of elementsm S j ( j = 1, 2, . . . , N R ) solutions of (10) represents the set of the estimate values of the remaining source nodes symbols (those that were not decoded at destination D in the optimization phase). Finally, the set of all the N S source nodes messages in the initial order is obtained by injecting the cooperative links messagesm Ik D at the corresponding indices I k (k = 1, 2, . . . , N S − N R ) into the setm S j ( j = 1, 2, . . . , N R ).
Performance evaluation: In this section, the performance of our proposed optimization strategy is analysed by simulation and comparison with the scenario where the cooperative source nodes are fixed and numerical results are provided. The performance metrics used are symbol error rate (SER) versus SNR for a fixed value of path loss exponent α and SER versus α for fixed values of SNR. In our simulations, the network configuration is set as follows: Relay nodes are fixed at a distance of 5 km from the BS (destination D) and source nodes are mobile users moving between the relays and the maximum radio range of the BS set to 20 km, that is, 5 km ≤ d SiD ≤ 20 km (i = 1, 2, . . . , N S ). Hence, the path loss parameters K and d 0 are adjusted to fit this configuration as K = 1 and d 0 = 5 km. All transmitting nodes have the same average SNR per symbol (same average symbol energy). In every round (N S + N R + 1 time slots), a set of N S GF(q) symbols, a set of N S × N R fading coefficients h SiR j , a set of N S fading coefficients h SiD and a set of N R fading coefficients h R j D are randomly generated. With every coefficient h kl , a corresponding AWGN component n kl is randomly generated. Also, a set of N S random distances d SiD ranging from 5 km to 20 km is generated each round. In our simulations, N S = 3 and N R = 2. All fading coefficients are Nakagami-m i.i.d. The results are averaged over 10 simulations with 1000 symbols in each simulation. The SER (or error probability) is evaluated for the transmission of one of the source nodes symbols that are recovered after Gaussian elimination at destination D (not a direct link to destination). Figure 2 shows how the proposed approach outperforms the fixed cooperative sources scenario in terms of SER versus SNR for orders of Galois field (GF) q = 2, 4 and 8. Nakagami-m shape factor is set to 2 and the path loss exponent equals 3. For transmission of binary phase shift keying (BPSK) symbols at an error rate of 10 −3 , the SNR required in fixed cooperation scenario is SNR = 16 dB, while in dynamic optimized cooperation, we need only SNR = 12 dB which demonstrates that our approach allows to achieve a gain of 4 dB. Almost the same gain is produced in case of GF symbols of higher orders q = 4 and q = 8.
In Figure 3, we extend the results to the whole range of path loss exponent possible values to prove the superiority and outperformance of our strategy over the fixed cooperation scenario in reducing the symbol error probability in GF (2). The gap of SER between our proposed dynamic optimized cooperation and fixed cooperation increases with SNR, that is, with the average symbol transmit energy, as we can see it clearly illustrated in Figure 4(a) for GF (4) symbols and for all values of path loss exponent α. Figure 4(b) reenforces the validity of these results with GF (8) symbols transmission.

Conclusion:
We proposed and detailed a new strategy for optimizing the performance of NCC-based multi-source multi-relay wireless networks. We introduced a cooperative source nodes selection technique with fixed relay nodes equipped with RLNC computation capability. Numerical results obtained through Monte Carlo simulations show significant system performance enhancement in terms of SER in comparison with fixed cooperative source nodes scenario. Furthermore, the efficiency of this new approach is consolidated with results showing important SER decrease for the whole range of possible values of path loss exponent α. In addition, the proposed approach requires no retransmission because of the