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Latent Exploration for Reinforcement Learning

Alberto Silvio Chiappa, Alessandro Marin Vargas, Ann Zixiang Huang, Alexander Mathis

arXiv:2305.20065v2cs.ROcs.AIcs.LGq-bio.NC

TL;DR

Independent actuator noise can be inefficient for over-actuated motor control because perturbations may cancel or disrupt coordination. Lattice injects temporally correlated noise into the policy’s latent state, inducing full-covariance action perturbations, and improves reward and energy efficiency across locomotion and musculoskeletal tasks. Its main limitation is additional computational overhead.

  • Problem

    Independent exploration noise can be suboptimal for over-actuated systems, while correlation across actuators has received less attention than temporal correlation.

  • Method

    Lattice perturbs the policy network’s latent state to exploit learned actuator synergies and generate temporally correlated actions with a full covariance matrix.

  • Results

    Lattice improves performance in complex locomotion and muscle-control tasks, including higher reward in reaching and manipulation and lower energy consumption.

  • Takeaways & Limitations

    Structured exploration in time and actuator space can discover high-reward, energy-efficient policies for complex motor-control tasks.

  • Takeaways & Limitations

    Modeling full-covariance action distributions adds approximately 20%-30% training overhead over gSDE and PPO/SAC.

Abstract

from arXiv · show

In Reinforcement Learning, agents learn policies by exploring and interacting with the environment. Due to the curse of dimensionality, learning policies that map high-dimensional sensory input to motor output is particularly challenging. During training, state of the art methods (SAC, PPO, etc.) explore the environment by perturbing the actuation with independent Gaussian noise. While this unstructured exploration has proven successful in numerous tasks, it can be suboptimal for overactuated systems. When multiple actuators, such as motors or muscles, drive behavior, uncorrelated perturbations risk diminishing each other's effect, or modifying the behavior in a task-irrelevant way. While solutions to introduce time correlation across action perturbations exist, introducing correlation across actuators has been largely ignored. Here, we propose LATent TIme-Correlated Exploration (Lattice), a method to inject temporally-correlated noise into the latent state of the policy network, which can be seamlessly integrated with on- and off-policy algorithms. We demonstrate that the noisy actions generated by perturbing the network's activations can be modeled as a multivariate Gaussian distribution with a full covariance matrix. In the PyBullet locomotion tasks, Lattice-SAC achieves state of the art results, and reaches 18% higher reward than unstructured exploration in the Humanoid environment. In the musculoskeletal control environments of MyoSuite, Lattice-PPO achieves higher reward in most reaching and object manipulation tasks, while also finding more energy-efficient policies with reductions of 20-60%. Overall, we demonstrate the effectiveness of structured action noise in time and actuator space for complex motor control tasks. The code is available at: https://github.com/amathislab/lattice.

1 Introduction

Independent exploration noise can be inefficient when multiple actuators jointly produce behavior. Lattice addresses this by perturbing policy-network latent states to induce structured, temporally correlated action noise.

  • 1 Introduction: Lattice targets the inefficiency of independent actuator noise by correlating exploration across action components as well as across time.Opposing perturbations can cancel, while independent actuator perturbations can disrupt coordination in over-actuated systems.
  • 1 Introduction: Lattice applies independent noise to the policy network’s latent state, exploiting actuator synergies learned by the network to structure action perturbations.The method extends gSDE with a policy-dependent latent perturbation and can model actions using a full covariance matrix.
  • 1 Introduction: Lattice integrates with both SAC and PPO and improves complex motor-control performance across standard locomotion and musculoskeletal benchmarks.The introduction reports competitive PyBullet performance, improved Humanoid efficiency, and higher MyoSuite rewards with lower energy use.
  • 1 Introduction: In musculoskeletal control, Lattice-PPO outperforms baselines on reaching and object-manipulation tasks while finding more energy-efficient policies.Pose tasks are included among the evaluated settings, but the cited introduction emphasizes reaching and manipulation gains.

2 Related work

Prior exploration methods address temporal coherence, alternative exploratory policies, or over-actuated control through specialized mechanisms. Lattice instead combines time-correlated, state-dependent exploration with correlation across action components in an end-to-end framework.

  • 2 Related work: Lattice extends time-correlated exploration by pairing state-dependent noise with correlation across action components.Earlier methods use colored noise, autoregressive processes, Markov chains, or periodically fixed perturbations to smooth exploration over time.
  • 2 Related work: Alternative exploration methods encourage rare-state visitation, diverse skills, or sequential exploration through counts, curiosity, random networks, value ensembles, or auxiliary policies.These approaches target exploration through state visitation, skill diversity, uncertainty, or goal-conditioned behavior rather than latent action coordination.
  • 2 Related work: Lattice is presented as complementary to curriculum learning and exploratory policies rather than as a replacement for them.The method proposes implicit monolithic exploration through the policy network and can be combined with other approaches.
  • 2 Related work: Over-actuated systems have been addressed with curricula, feature engineering, demonstrations, coordination networks, energy functions, and state-space-covering controllers.The related methods often require specialized or ad-hoc solutions, whereas Lattice is trained end-to-end with on-policy and off-policy algorithms.

3 Motivating example: The case of a flexor-extensor, single joint arm

A flexor-extensor elbow example shows why latent perturbations can produce more behaviorally effective exploration than independent action noise. The result is supported by both an analytical toy model and a realistic musculoskeletal-arm experiment.

  • 3 Motivating example: The case of a flexor-extensor, single joint arm: Latent noise produces angular acceleration with variance 4α^2σ^2, compared with 2α^2σ^2 under independent action noise with matched marginal action variance.The latent perturbation correlates flexor and extensor activations while preserving their marginal distributions.
  • 3 Motivating example: The case of a flexor-extensor, single joint arm: Independent perturbations of opposing muscles can cancel, reducing their effect on observed kinematics and limiting behavior-space exploration.Perturbing the latent state instead increases variance in the behavior space.
  • 3 Motivating example: The case of a flexor-extensor, single joint arm: In a realistic arm model with three flexors and three extensors, latent perturbation yields a full action covariance matrix with positive within-group correlations.The learned policy distinguishes agonist and antagonist muscles through the covariance structure.
  • 3 Motivating example: The case of a flexor-extensor, single joint arm: With matched action-variance magnitudes, latent perturbations produce higher joint-angle variance than independent muscle-action perturbations.The experiment compares paired actions in cloned environments and interprets the higher variance as wider kinematic exploration.

4 Methods

Lattice introduces time- and actuator-correlated exploration by perturbing the policy’s latent state, while retaining an analytically tractable multivariate Gaussian action distribution. Its implementation extends gSDE with configurable temporal correlation, variance controls, and covariance regularization.

  • 4.1 LATent TIme-Correlated Exploration (Lattice): The experience-collection procedure applies the sampled perturbations to latent and action pathways before stepping the environment, unlike standard independent action noise.The algorithm descriptions distinguish standard exploration from time- and actuator-correlated perturbations.
  • 4.1 LATent TIme-Correlated Exploration (Lattice): Lattice perturbs the last latent policy state before its linear action mapping, producing actions with a full covariance matrix WΣxW^T + Σa and an analytical log π(at|st).The latent perturbation uses diagonal Gaussian covariance Σx, while Σa represents independent action noise.
  • 4.1 LATent TIme-Correlated Exploration (Lattice): In PyBullet locomotion, Lattice, gSDE, and Pink Noise perform comparably except in Humanoid, where Lattice is more efficient and achieves higher average reward.Figure 3 reports cumulative episode reward as mean ± s.e.m. across random seeds; T denotes exploration-noise correlation period.
  • 4.2 Lattice generalizes time-correlated noise: Lattice samples latent and action noise matrices every T steps, so larger T produces stronger temporal correlation in the applied exploration noise.The parameter α turns correlated action noise on or off, with α = 0 recovering gSDE.
  • 4.2 Lattice generalizes time-correlated noise: Lattice extends gSDE by learning noise parameters Sx and Sa, with network weights determining the action-distribution mean and covariance.The expected action can train the policy, while the standard-deviation component regulates the learned noise magnitude.
  • 4.3 Implementation details of Lattice: Implementation controls include learnable noise magnitudes, variance clipping, latent-size rescaling, and diagonal regularization to prevent singular action covariance.The rescaling correction removes dependence of noise magnitude on latent-state size, while regularization ensures positive definiteness.

5 Experiments

The experiments evaluate Lattice with SAC in PyBullet locomotion and PPO with an LSTM in MyoSuite muscle control. Lattice is strongest for higher-dimensional locomotion and complex reaching or manipulation, where it improves reward and often reduces energy use, but standard exploration can be better for pose tasks.

  • 5 Experiments: Experiments benchmark continuous-control tasks in PyBullet and MyoSuite, implement Lattice as a Stable Baselines 3 gSDE extension, and average results over five random seeds.MyoSuite experiments use PPO with an LSTM and keep the architecture and PPO hyperparameters identical across methods.
  • 5.1 Pybullet locomotion environments: Lattice-SAC matches gSDE-SAC and SAC-pink on lower-dimensional PyBullet morphologies but substantially outperforms them on Humanoid.The experiments use identical SAC architecture and hyperparameters across methods and average results over five random seeds.
  • 5.2 Musculoskeletal control: MyoSuite environments: The MyoSuite evaluation covers three pose, two reach, and three object-manipulation tasks with targets defined over joints, fingertips, or manipulated objects.The task set includes Elbow, Finger, and Hand Pose; Finger and Hand Reach; and Reorient, Pend, and Baoding.
  • 5.2 Musculoskeletal control: MyoSuite environments: In MyoSuite, Lattice consistently outperforms baselines on reaching and challenging object-manipulation tasks involving a 39-muscle hand and observations exceeding 100 dimensions.Pose-task performance is comparable or better for standard PPO, possibly because dense target states make exploration and coordination less important.
  • 5.2 Musculoskeletal control: MyoSuite environments: Lattice-PPO achieves higher reward at lower energy cost in reaching and object-manipulation tasks, with energy savings of 20% to 60% versus independent exploration.Figure 5 contrasts reward gain and energy saving; pose tasks favor independent exploration with similar energy consumption.

6 How does Lattice explore?

Lattice shifts exploration toward coordinated, task-relevant actuator patterns and produces lower-dimensional, more correlated actions. In Humanoid and musculoskeletal control, this structure is associated with improved energy efficiency and performance.

  • Lattice finds more energy-efficient control strategies in most musculoskeletal environments without compromising performance, and Reorient improves cumulative reward at a fraction of the energy cost.
  • The authors hypothesize that coordinated noise prevents energy-consuming co-activations that the agent would otherwise need to compensate for.
  • Lattice-SAC redirects Humanoid exploration noise toward task-relevant leg motors, allocating 50% to legs versus 40% for SAC.SAC allocates 40% to legs, 45% to arms, and 15% to body; Lattice-SAC allocates 50%, 32%, and 18%, respectively.
  • Lattice policies consistently require fewer principal components to explain action variance, indicating actions lie on a lower-dimensional manifold across the evaluated tasks.
  • Lattice increases cross-actuator coordination: its action correlations resemble the structured off-diagonal noise covariance, unlike nearly uncorrelated standard noise.The authors relate this structure to the covariance transformation Cov(a_i, a_j) = WCov(x)W^T.

7 Discussion and Limitations

The paper presents Lattice as a broadly integrable latent-exploration method that discovers high-reward, energy-efficient motor policies. Its main practical limitation is computational overhead from modeling full-covariance action distributions.

  • Lattice discovers high-reward and energy-efficient policies in complex locomotion and muscle-control tasks, especially reach and object-manipulation tasks with reduced energy consumption.
  • Perturbing the policy network's last layer yields an analytical action distribution, enabling Lattice to enhance on-policy and off-policy continuous-control algorithms.
  • Lattice introduces approximately 20%-30% training overhead over gSDE and PPO/SAC because estimating action probabilities requires additional matrix multiplications.The authors suggest diagonal or low-rank distribution matrices as a future direction for reducing this cost.
  • The authors frame energy-efficient learned policies as potentially useful for energy-sensitive robotics applications in Artificial Intelligence, Robotics, and Neuroscience.

A.1 Singe-joint arm: detailed calculations

The single-joint arm derivation contrasts independent action noise with correlated latent noise. With matched marginal action variance, latent perturbations double the angular-acceleration variance and are evaluated in a MyoSuite elbow task.

  • Case 1: action space noise: With independent action noise, elbow angular acceleration has variance 2α^2σ^2.
  • Case 2: latent space noise: With correlated latent noise and unchanged marginal activation distributions, angular-acceleration variance becomes 4α^2σ^2.The latent perturbation makes flexor and extensor activations dependent while preserving the same expectation.
  • Latent-space noise therefore produces twice the variance of action-space noise in the simplified arm model.
  • Simulation test: The MyoSuite elbow experiment synchronizes noisy and deterministic simulations after every step to isolate each perturbation's immediate effect on elbow angle.
  • Simulation test: The comparison matches each action component's variance across latent and action perturbations, isolating covariance structure rather than scale as the experimental difference.

A.2 Lattice’s Action Distribution Parameterization

Lattice perturbs the latent state and independently adds action noise before producing actions through the policy's linear output map. Independent Gaussian parameter elements yield Gaussian noise vectors with diagonal covariance, enabling analytical action-distribution computation.

  • Lattice computes the action as a linear transformation of the perturbed latent vector plus independent action noise: a = W(x + ϵ_x) + ϵ_a.
  • The latent and action perturbation matrices contain independently distributed Gaussian elements whose learned standard deviations are stored in S(x) and S(a).
  • Each noise-vector element is a sum of independent Gaussian random variables, so its distribution can be computed analytically from its mean and variance.
  • Noise elements at different indices have zero covariance, giving the joint latent-and-action noise a diagonal covariance matrix.
  • The action distribution follows directly from the linear transformation rule for multivariate Gaussian variables, using independence of the latent and action perturbation parameters.

A.3 Noise rescaling across networks of different size

Lattice rescales latent and action noise so its variance remains comparable across policy networks with different sizes. The derivation attributes this need to latent-noise variance scaling with latent-state dimensionality.

  • A.3 Noise rescaling across networks of different size: The rescaling is designed to make S_x and S_a invariant to network size.The section explicitly states that both latent and action noise scales are rescaled for this purpose.
  • A.3 Noise rescaling across networks of different size: Latent-noise variance scales with the size of the latent state, motivating network-size-dependent rescaling.The derivation starts from the standard deviation of each latent-noise element and concludes that its variance scales with N_x.
  • A.3 Noise rescaling across networks of different size: The correction keeps the initialization of log S̃_x consistent across networks with different latent-state sizes.The same rescaling argument applies to the action-noise parameter S_a.

A.4 Conditions on the covariance matrix of the action distribution

The appendix establishes covariance-matrix conditions, documents the experimental settings, and reports detailed results and policy-dimensionality analyses for Lattice against standard exploration. Across the supplied experiments, Lattice generally produces lower-dimensional and more energy-efficient policies, while its reward advantage is task-dependent.

  • A.4 Conditions on the covariance matrix of the action distribution: The action covariance is constructed as WΣ_xW^⊤ + Σ_a and is positive semidefinite by construction.The appendix separately notes that regularization prevents the minimum eigenvalue from falling below γ.
  • A.4 Conditions on the covariance matrix of the action distribution: Without regularization, sparse latent representations such as ReLU can produce zero covariance eigenvalues, including a null covariance matrix when x is zero.The regularization term addresses this degeneracy by adding a multiple of the identity matrix.
  • A.6 Parameters of the PyBullet and MyoSuite environments: The experiments use default PyBullet settings and common reward components across MyoSuite algorithms, with results averaged over five seeds.The reported appendix tables list parameters and detailed reward and energy results for SAC, gSDE-SAC, Lattice-SAC, PPO, gSDE-PPO, and Lattice-PPO.
  • A.6 Parameters of the PyBullet and MyoSuite environments: The appendix reduces the Hand Pose target-pose range because no algorithm solved the full-range environment without a curriculum.This change makes the environment suitable for comparing standard and latent exploration under the stated setup.
  • A.7 Hyperparameters of SAC, PPO, gSDE and Lattice: The initial log σ value was assessed for Hand Pose and Reorient, with the default value of 0 reported as performing optimally.The corresponding reward-versus-initial-standard-deviation analysis appears in Figure F1.
  • A.9 Evolution of the dimensionality of the policy during training: Lattice policies converge to lower-dimensional action manifolds than SAC/PPO across environments and random seeds.Relevant action components are similar initially, then diverge during training; the final dimensionality is consistently lower with Lattice.
  • A.8 Detailed reward and energy results of all experiments: Lattice finds more energy-efficient control strategies in most musculoskeletal environments without compromising performance.Energy consumption is initially similar across methods, then diverges during training, with Lattice achieving lower consumption consistently across seeds.
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