Lingwei Zhu

LG
h-index5
24papers
119citations
Novelty53%
AI Score51

24 Papers

ROJul 29, 2022Code
Cyclic Policy Distillation: Sample-Efficient Sim-to-Real Reinforcement Learning with Domain Randomization

Yuki Kadokawa, Lingwei Zhu, Yoshihisa Tsurumine et al.

Deep reinforcement learning with domain randomization learns a control policy in various simulations with randomized physical and sensor model parameters to become transferable to the real world in a zero-shot setting. However, a huge number of samples are often required to learn an effective policy when the range of randomized parameters is extensive due to the instability of policy updates. To alleviate this problem, we propose a sample-efficient method named cyclic policy distillation (CPD). CPD divides the range of randomized parameters into several small sub-domains and assigns a local policy to each one. Then local policies are learned while cyclically transitioning to sub-domains. CPD accelerates learning through knowledge transfer based on expected performance improvements. Finally, all of the learned local policies are distilled into a global policy for sim-to-real transfers. CPD's effectiveness and sample efficiency are demonstrated through simulations with four tasks (Pendulum from OpenAIGym and Pusher, Swimmer, and HalfCheetah from Mujoco), and a real-robot, ball-dispersal task. We published code and videos from our experiments at https://github.com/yuki-kadokawa/cyclic-policy-distillation.

LGApr 7, 2022
Automated Sleep Staging via Parallel Frequency-Cut Attention

Zheng Chen, Ziwei Yang, Lingwei Zhu et al.

This paper proposes a novel framework for automatically capturing the time-frequency nature of electroencephalogram (EEG) signals of human sleep based on the authoritative sleep medicine guidance. The framework consists of two parts: the first part extracts informative features by partitioning the input EEG spectrograms into a sequence of time-frequency patches. The second part is constituted by an attention-based architecture to efficiently search for the correlation between partitioned time-frequency patches and defining factors of sleep stages in parallel. The proposed pipeline is validated on the Sleep Heart Health Study dataset with new state-of-the-art results for the stages wake, N2, and N3, obtaining respective F1 scores of 0.93, 0.88, and 0.87, with only EEG signals used. The proposed method also has a high inter-rater reliability of 0.80 kappa. We also visualize the correspondence between sleep staging decisions and features extracted by the proposed method, providing strong interpretability for our model.

AIAug 18, 2024Code
BernGraph: Probabilistic Graph Neural Networks for EHR-based Medication Recommendations

Xihao Piao, Pei Gao, Zheng Chen et al.

The medical community believes binary medical event outcomes in EHR data contain sufficient information for making a sensible recommendation. However, there are two challenges to effectively utilizing such data: (1) modeling the relationship between massive 0,1 event outcomes is difficult, even with expert knowledge; (2) in practice, learning can be stalled by the binary values since the equally important 0 entries propagate no learning signals. Currently, there is a large gap between the assumed sufficient information and the reality that no promising results have been shown by utilizing solely the binary data: visiting or secondary information is often necessary to reach acceptable performance. In this paper, we attempt to build the first successful binary EHR data-oriented drug recommendation system by tackling the two difficulties, making sensible drug recommendations solely using the binary EHR medical records. To this end, we take a statistical perspective to view the EHR data as a sample from its cohorts and transform them into continuous Bernoulli probabilities. The transformed entries not only model a deterministic binary event with a distribution but also allow reflecting \emph{event-event} relationship by conditional probability. A graph neural network is learned on top of the transformation. It captures event-event correlations while emphasizing \emph{event-to-patient} features. Extensive results demonstrate that the proposed method achieves state-of-the-art performance on large-scale databases, outperforming baseline methods that use secondary information by a large margin. The source code is available at \url{https://github.com/chenzRG/BEHRMecom}

LGApr 2, 2022
Cancer Subtyping via Embedded Unsupervised Learning on Transcriptomics Data

Ziwei Yang, Lingwei Zhu, Zheng Chen et al.

Cancer is one of the deadliest diseases worldwide. Accurate diagnosis and classification of cancer subtypes are indispensable for effective clinical treatment. Promising results on automatic cancer subtyping systems have been published recently with the emergence of various deep learning methods. However, such automatic systems often overfit the data due to the high dimensionality and scarcity. In this paper, we propose to investigate automatic subtyping from an unsupervised learning perspective by directly constructing the underlying data distribution itself, hence sufficient data can be generated to alleviate the issue of overfitting. Specifically, we bypass the strong Gaussianity assumption that typically exists but fails in the unsupervised learning subtyping literature due to small-sized samples by vector quantization. Our proposed method better captures the latent space features and models the cancer subtype manifestation on a molecular basis, as demonstrated by the extensive experimental results.

LGAug 14, 2024Code
q-exponential family for policy optimization

Lingwei Zhu, Haseeb Shah, Han Wang et al.

Policy optimization methods benefit from a simple and tractable policy parametrization, usually the Gaussian for continuous action spaces. In this paper, we consider a broader policy family that remains tractable: the $q$-exponential family. This family of policies is flexible, allowing the specification of both heavy-tailed policies ($q>1$) and light-tailed policies ($q<1$). This paper examines the interplay between $q$-exponential policies for several actor-critic algorithms conducted on both online and offline problems. We find that heavy-tailed policies are more effective in general and can consistently improve on Gaussian. In particular, we find the Student's t-distribution to be more stable than the Gaussian across settings and that a heavy-tailed $q$-Gaussian for Tsallis Advantage Weighted Actor-Critic consistently performs well in offline benchmark problems. Our code is available at \url{https://github.com/lingweizhu/qexp}.

LGJan 27, 2023
Generalized Munchausen Reinforcement Learning using Tsallis KL Divergence

Lingwei Zhu, Zheng Chen, Matthew Schlegel et al.

Many policy optimization approaches in reinforcement learning incorporate a Kullback-Leilbler (KL) divergence to the previous policy, to prevent the policy from changing too quickly. This idea was initially proposed in a seminal paper on Conservative Policy Iteration, with approximations given by algorithms like TRPO and Munchausen Value Iteration (MVI). We continue this line of work by investigating a generalized KL divergence -- called the Tsallis KL divergence -- which use the $q$-logarithm in the definition. The approach is a strict generalization, as $q = 1$ corresponds to the standard KL divergence; $q > 1$ provides a range of new options. We characterize the types of policies learned under the Tsallis KL, and motivate when $q >1$ could be beneficial. To obtain a practical algorithm that incorporates Tsallis KL regularization, we extend MVI, which is one of the simplest approaches to incorporate KL regularization. We show that this generalized MVI($q$) obtains significant improvements over the standard MVI($q = 1$) across 35 Atari games.

LGJul 20, 2022
Cancer Subtyping by Improved Transcriptomic Features Using Vector Quantized Variational Autoencoder

Zheng Chen, Ziwei Yang, Lingwei Zhu et al.

Defining and separating cancer subtypes is essential for facilitating personalized therapy modality and prognosis of patients. The definition of subtypes has been constantly recalibrated as a result of our deepened understanding. During this recalibration, researchers often rely on clustering of cancer data to provide an intuitive visual reference that could reveal the intrinsic characteristics of subtypes. The data being clustered are often omics data such as transcriptomics that have strong correlations to the underlying biological mechanism. However, while existing studies have shown promising results, they suffer from issues associated with omics data: sample scarcity and high dimensionality. As such, existing methods often impose unrealistic assumptions to extract useful features from the data while avoiding overfitting to spurious correlations. In this paper, we propose to leverage a recent strong generative model, Vector Quantized Variational AutoEncoder (VQ-VAE), to tackle the data issues and extract informative latent features that are crucial to the quality of subsequent clustering by retaining only information relevant to reconstructing the input. VQ-VAE does not impose strict assumptions and hence its latent features are better representations of the input, capable of yielding superior clustering performance with any mainstream clustering method. Extensive experiments and medical analysis on multiple datasets comprising 10 distinct cancers demonstrate the VQ-VAE clustering results can significantly and robustly improve prognosis over prevalent subtyping systems.

GNSep 2, 2024
MLOmics: Cancer Multi-Omics Database for Machine Learning

Ziwei Yang, Rikuto Kotoge, Xihao Piao et al.

Framing the investigation of diverse cancers as a machine learning problem has recently shown significant potential in multi-omics analysis and cancer research. Empowering these successful machine learning models are the high-quality training datasets with sufficient data volume and adequate preprocessing. However, while there exist several public data portals, including The Cancer Genome Atlas (TCGA) multi-omics initiative or open-bases such as the LinkedOmics, these databases are not off-the-shelf for existing machine learning models. In this paper, we introduce MLOmics, an open cancer multi-omics database aiming at serving better the development and evaluation of bioinformatics and machine learning models. MLOmics contains 8,314 patient samples covering all 32 cancer types with four omics types, stratified features, and extensive baselines. Complementary support for downstream analysis and bio-knowledge linking are also included to support interdisciplinary analysis.

LGJun 22, 2022
Automated Cancer Subtyping via Vector Quantization Mutual Information Maximization

Zheng Chen, Lingwei Zhu, Ziwei Yang et al.

Cancer subtyping is crucial for understanding the nature of tumors and providing suitable therapy. However, existing labelling methods are medically controversial, and have driven the process of subtyping away from teaching signals. Moreover, cancer genetic expression profiles are high-dimensional, scarce, and have complicated dependence, thereby posing a serious challenge to existing subtyping models for outputting sensible clustering. In this study, we propose a novel clustering method for exploiting genetic expression profiles and distinguishing subtypes in an unsupervised manner. The proposed method adaptively learns categorical correspondence from latent representations of expression profiles to the subtypes output by the model. By maximizing the problem -- agnostic mutual information between input expression profiles and output subtypes, our method can automatically decide a suitable number of subtypes. Through experiments, we demonstrate that our proposed method can refine existing controversial labels, and, by further medical analysis, this refinement is proven to have a high correlation with cancer survival rates.

SPApr 21, 2022
Multi-Tier Platform for Cognizing Massive Electroencephalogram

Zheng Chen, Lingwei Zhu, Ziwei Yang et al.

An end-to-end platform assembling multiple tiers is built for precisely cognizing brain activities. Being fed massive electroencephalogram (EEG) data, the time-frequency spectrograms are conventionally projected into the episode-wise feature matrices (seen as tier-1). A spiking neural network (SNN) based tier is designed to distill the principle information in terms of spike-streams from the rare features, which maintains the temporal implication in the nature of EEGs. The proposed tier-3 transposes time- and space-domain of spike patterns from the SNN; and feeds the transposed pattern-matrices into an artificial neural network (ANN, Transformer specifically) known as tier-4, where a special spanning topology is proposed to match the two-dimensional input form. In this manner, cognition such as classification is conducted with high accuracy. For proof-of-concept, the sleep stage scoring problem is demonstrated by introducing multiple EEG datasets with the largest comprising 42,560 hours recorded from 5,793 subjects. From experiment results, our platform achieves the general cognition overall accuracy of 87% by leveraging sole EEG, which is 2% superior to the state-of-the-art. Moreover, our developed multi-tier methodology offers visible and graphical interpretations of the temporal characteristics of EEG by identifying the critical episodes, which is demanded in neurodynamics but hardly appears in conventional cognition scenarios.

LGMay 16, 2022
Enforcing KL Regularization in General Tsallis Entropy Reinforcement Learning via Advantage Learning

Lingwei Zhu, Zheng Chen, Eiji Uchibe et al.

Maximum Tsallis entropy (MTE) framework in reinforcement learning has gained popularity recently by virtue of its flexible modeling choices including the widely used Shannon entropy and sparse entropy. However, non-Shannon entropies suffer from approximation error and subsequent underperformance either due to its sensitivity or the lack of closed-form policy expression. To improve the tradeoff between flexibility and empirical performance, we propose to strengthen their error-robustness by enforcing implicit Kullback-Leibler (KL) regularization in MTE motivated by Munchausen DQN (MDQN). We do so by drawing connection between MDQN and advantage learning, by which MDQN is shown to fail on generalizing to the MTE framework. The proposed method Tsallis Advantage Learning (TAL) is verified on extensive experiments to not only significantly improve upon Tsallis-DQN for various non-closed-form Tsallis entropies, but also exhibits comparable performance to state-of-the-art maximum Shannon entropy algorithms.

LGMay 16, 2022
$q$-Munchausen Reinforcement Learning

Lingwei Zhu, Zheng Chen, Eiji Uchibe et al.

The recently successful Munchausen Reinforcement Learning (M-RL) features implicit Kullback-Leibler (KL) regularization by augmenting the reward function with logarithm of the current stochastic policy. Though significant improvement has been shown with the Boltzmann softmax policy, when the Tsallis sparsemax policy is considered, the augmentation leads to a flat learning curve for almost every problem considered. We show that it is due to the mismatch between the conventional logarithm and the non-logarithmic (generalized) nature of Tsallis entropy. Drawing inspiration from the Tsallis statistics literature, we propose to correct the mismatch of M-RL with the help of $q$-logarithm/exponential functions. The proposed formulation leads to implicit Tsallis KL regularization under the maximum Tsallis entropy framework. We show such formulation of M-RL again achieves superior performance on benchmark problems and sheds light on more general M-RL with various entropic indices $q$.

SPApr 2, 2022
Adaptive Spike-Like Representation of EEG Signals for Sleep Stages Scoring

Lingwei Zhu, Koki Odani, Ziwei Yang et al.

Recently there has seen promising results on automatic stage scoring by extracting spatio-temporal features from electroencephalogram (EEG). Such methods entail laborious manual feature engineering and domain knowledge. In this study, we propose an adaptive scheme to probabilistically encode, filter and accumulate the input signals and weight the resultant features by the half-Gaussian probabilities of signal intensities. The adaptive representations are subsequently fed into a transformer model to automatically mine the relevance between features and corresponding stages. Extensive experiments on the largest public dataset against state-of-the-art methods validate the effectiveness of our proposed method and reveal promising future directions.

AIFeb 26
ODEBrain: Continuous-Time EEG Graph for Modeling Dynamic Brain Networks

Haohui Jia, Zheng Chen, Lingwei Zhu et al.

Modeling neural population dynamics is crucial for foundational neuroscientific research and various clinical applications. Conventional latent variable methods typically model continuous brain dynamics through discretizing time with recurrent architecture, which necessarily results in compounded cumulative prediction errors and failure of capturing instantaneous, nonlinear characteristics of EEGs. We propose ODEBRAIN, a Neural ODE latent dynamic forecasting framework to overcome these challenges by integrating spatio-temporal-frequency features into spectral graph nodes, followed by a Neural ODE modeling the continuous latent dynamics. Our design ensures that latent representations can capture stochastic variations of complex brain states at any given time point. Extensive experiments verify that ODEBRAIN can improve significantly over existing methods in forecasting EEG dynamics with enhanced robustness and generalization capabilities.

LGFeb 19
TIFO: Time-Invariant Frequency Operator for Stationarity-Aware Representation Learning in Time Series

Xihao Piao, Zheng Chen, Lingwei Zhu et al.

Nonstationary time series forecasting suffers from the distribution shift issue due to the different distributions that produce the training and test data. Existing methods attempt to alleviate the dependence by, e.g., removing low-order moments from each individual sample. These solutions fail to capture the underlying time-evolving structure across samples and do not model the complex time structure. In this paper, we aim to address the distribution shift in the frequency space by considering all possible time structures. To this end, we propose a Time-Invariant Frequency Operator (TIFO), which learns stationarity-aware weights over the frequency spectrum across the entire dataset. The weight representation highlights stationary frequency components while suppressing non-stationary ones, thereby mitigating the distribution shift issue in time series. To justify our method, we show that the Fourier transform of time series data implicitly induces eigen-decomposition in the frequency space. TIFO is a plug-and-play approach that can be seamlessly integrated into various forecasting models. Experiments demonstrate our method achieves 18 top-1 and 6 top-2 results out of 28 forecasting settings. Notably, it yields 33.3% and 55.3% improvements in average MSE on the ETTm2 dataset. In addition, TIFO reduces computational costs by 60% -70% compared to baseline methods, demonstrating strong scalability across diverse forecasting models.

AIFeb 26
RepSPD: Enhancing SPD Manifold Representation in EEGs via Dynamic Graphs

Haohui Jia, Zheng Chen, Lingwei Zhu et al.

Decoding brain activity from electroencephalography (EEG) is crucial for neuroscience and clinical applications. Among recent advances in deep learning for EEG, geometric learning stands out as its theoretical underpinnings on symmetric positive definite (SPD) allows revealing structural connectivity analysis in a physics-grounded manner. However, current SPD-based methods focus predominantly on statistical aggregation of EEGs, with frequency-specific synchronization and local topological structures of brain regions neglected. Given this, we propose RepSPD, a novel geometric deep learning (GDL)-based model. RepSPD implements a cross-attention mechanism on the Riemannian manifold to modulate the geometric attributes of SPD with graph-derived functional connectivity features. On top of this, we introduce a global bidirectional alignment strategy to reshape tangent-space embeddings, mitigating geometric distortions caused by curvature and thereby enhancing geometric consistency. Extensive experiments demonstrate that our proposed framework significantly outperforms existing EEG representation methods, exhibiting superior robustness and generalization capabilities.

AIFeb 22, 2023
Drugs Resistance Analysis from Scarce Health Records via Multi-task Graph Representation

Honglin Shu, Pei Gao, Lingwei Zhu et al.

Clinicians prescribe antibiotics by looking at the patient's health record with an experienced eye. However, the therapy might be rendered futile if the patient has drug resistance. Determining drug resistance requires time-consuming laboratory-level testing while applying clinicians' heuristics in an automated way is difficult due to the categorical or binary medical events that constitute health records. In this paper, we propose a novel framework for rapid clinical intervention by viewing health records as graphs whose nodes are mapped from medical events and edges as correspondence between events in given a time window. A novel graph-based model is then proposed to extract informative features and yield automated drug resistance analysis from those high-dimensional and scarce graphs. The proposed method integrates multi-task learning into a common feature extracting graph encoder for simultaneous analyses of multiple drugs as well as stabilizing learning. On a massive dataset comprising over 110,000 patients with urinary tract infections, we verify the proposed method is capable of attaining superior performance on the drug resistance prediction problem. Furthermore, automated drug recommendations resemblant to laboratory-level testing can also be made based on the model resistance analysis.

LGJan 24, 2025Code
Fat-to-Thin Policy Optimization: Offline RL with Sparse Policies

Lingwei Zhu, Han Wang, Yukie Nagai

Sparse continuous policies are distributions that can choose some actions at random yet keep strictly zero probability for the other actions, which are radically different from the Gaussian. They have important real-world implications, e.g. in modeling safety-critical tasks like medicine. The combination of offline reinforcement learning and sparse policies provides a novel paradigm that enables learning completely from logged datasets a safety-aware sparse policy. However, sparse policies can cause difficulty with the existing offline algorithms which require evaluating actions that fall outside of the current support. In this paper, we propose the first offline policy optimization algorithm that tackles this challenge: Fat-to-Thin Policy Optimization (FtTPO). Specifically, we maintain a fat (heavy-tailed) proposal policy that effectively learns from the dataset and injects knowledge to a thin (sparse) policy, which is responsible for interacting with the environment. We instantiate FtTPO with the general $q$-Gaussian family that encompasses both heavy-tailed and sparse policies and verify that it performs favorably in a safety-critical treatment simulation and the standard MuJoCo suite. Our code is available at \url{https://github.com/lingweizhu/fat2thin}.

LGJan 31, 2025
Towards Physiologically Sensible Predictions via the Rule-based Reinforcement Learning Layer

Lingwei Zhu, Zheng Chen, Yukie Nagai et al.

This paper adds to the growing literature of reinforcement learning (RL) for healthcare by proposing a novel paradigm: augmenting any predictor with Rule-based RL Layer (RRLL) that corrects the model's physiologically impossible predictions. Specifically, RRLL takes as input states predicted labels and outputs corrected labels as actions. The reward of the state-action pair is evaluated by a set of general rules. RRLL is efficient, general and lightweight: it does not require heavy expert knowledge like prior work but only a set of impossible transitions. This set is much smaller than all possible transitions; yet it can effectively reduce physiologically impossible mistakes made by the state-of-the-art predictor models. We verify the utility of RRLL on a variety of important healthcare classification problems and observe significant improvements using the same setup, with only the domain-specific set of impossibility changed. In-depth analysis shows that RRLL indeed improves accuracy by effectively reducing the presence of physiologically impossible predictions.

LGAug 6, 2025
Symmetric Behavior Regularization via Taylor Expansion of Symmetry

Lingwei Zhu, Zheng Chen, Han Wang et al.

This paper introduces symmetric divergences to behavior regularization policy optimization (BRPO) to establish a novel offline RL framework. Existing methods focus on asymmetric divergences such as KL to obtain analytic regularized policies and a practical minimization objective. We show that symmetric divergences do not permit an analytic policy as regularization and can incur numerical issues as loss. We tackle these challenges by the Taylor series of $f$-divergence. Specifically, we prove that an analytic policy can be obtained with a finite series. For loss, we observe that symmetric divergences can be decomposed into an asymmetry and a conditional symmetry term, Taylor-expanding the latter alleviates numerical issues. Summing together, we propose Symmetric $f$ Actor-Critic (S$f$-AC), the first practical BRPO algorithm with symmetric divergences. Experimental results on distribution approximation and MuJoCo verify that S$f$-AC performs competitively.

LGJul 16, 2021
Geometric Value Iteration: Dynamic Error-Aware KL Regularization for Reinforcement Learning

Toshinori Kitamura, Lingwei Zhu, Takamitsu Matsubara

The recent boom in the literature on entropy-regularized reinforcement learning (RL) approaches reveals that Kullback-Leibler (KL) regularization brings advantages to RL algorithms by canceling out errors under mild assumptions. However, existing analyses focus on fixed regularization with a constant weighting coefficient and do not consider cases where the coefficient is allowed to change dynamically. In this paper, we study the dynamic coefficient scheme and present the first asymptotic error bound. Based on the dynamic coefficient error bound, we propose an effective scheme to tune the coefficient according to the magnitude of error in favor of more robust learning. Complementing this development, we propose a novel algorithm, Geometric Value Iteration (GVI), that features a dynamic error-aware KL coefficient design with the aim of mitigating the impact of errors on performance. Our experiments demonstrate that GVI can effectively exploit the trade-off between learning speed and robustness over uniform averaging of a constant KL coefficient. The combination of GVI and deep networks shows stable learning behavior even in the absence of a target network, where algorithms with a constant KL coefficient would greatly oscillate or even fail to converge.

LGJul 13, 2021
Cautious Policy Programming: Exploiting KL Regularization in Monotonic Policy Improvement for Reinforcement Learning

Lingwei Zhu, Toshinori Kitamura, Takamitsu Matsubara

In this paper, we propose cautious policy programming (CPP), a novel value-based reinforcement learning (RL) algorithm that can ensure monotonic policy improvement during learning. Based on the nature of entropy-regularized RL, we derive a new entropy regularization-aware lower bound of policy improvement that only requires estimating the expected policy advantage function. CPP leverages this lower bound as a criterion for adjusting the degree of a policy update for alleviating policy oscillation. Different from similar algorithms that are mostly theory-oriented, we also propose a novel interpolation scheme that makes CPP better scale in high dimensional control problems. We demonstrate that the proposed algorithm can trade o? performance and stability in both didactic classic control problems and challenging high-dimensional Atari games.

LGJul 12, 2021
Cautious Actor-Critic

Lingwei Zhu, Toshinori Kitamura, Takamitsu Matsubara

The oscillating performance of off-policy learning and persisting errors in the actor-critic (AC) setting call for algorithms that can conservatively learn to suit the stability-critical applications better. In this paper, we propose a novel off-policy AC algorithm cautious actor-critic (CAC). The name cautious comes from the doubly conservative nature that we exploit the classic policy interpolation from conservative policy iteration for the actor and the entropy-regularization of conservative value iteration for the critic. Our key observation is the entropy-regularized critic facilitates and simplifies the unwieldy interpolated actor update while still ensuring robust policy improvement. We compare CAC to state-of-the-art AC methods on a set of challenging continuous control problems and demonstrate that CAC achieves comparable performance while significantly stabilizes learning.

LGAug 25, 2020
Ensuring Monotonic Policy Improvement in Entropy-regularized Value-based Reinforcement Learning

Lingwei Zhu, Takamitsu Matsubara

This paper aims to establish an entropy-regularized value-based reinforcement learning method that can ensure the monotonic improvement of policies at each policy update. Unlike previously proposed lower-bounds on policy improvement in general infinite-horizon MDPs, we derive an entropy-regularization aware lower bound. Since our bound only requires the expected policy advantage function to be estimated, it is scalable to large-scale (continuous) state-space problems. We propose a novel reinforcement learning algorithm that exploits this lower-bound as a criterion for adjusting the degree of a policy update for alleviating policy oscillation. We demonstrate the effectiveness of our approach in both discrete-state maze and continuous-state inverted pendulum tasks using a linear function approximator for value estimation.