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L2C2: Locally Lipschitz Continuous Constraint towards Stable and Smooth Reinforcement Learning
Taisuke Kobayashi
TL;DR
RL policies and value functions need smoothness, but excessive regularization can impair expressiveness and task performance. L2C2 applies local Lipschitz constraints in transition-defined state neighborhoods; in noisy simulations, it achieved equivalent or better performance than conventional methods while smoothing policies. The paper also reports that its suppression of exploration is task-dependent.
Problem
Excessive regularization can degrade RL performance by impairing the expressiveness needed for tasks, while real-robot learning also faces instability with limited data and unpredictable behavior.
Source: §I · p. 1Method
L2C2 applies local Lipschitz regularization in a compact state-space region defined spatio-temporally by each state transition.
Source: §I · p. 1Results
Across four noisy PyBullet simulations, L2C2 achieved equivalent or better task performance than conventional methods and smoothed policies better than no regularization.
Source: §I · p. 1Takeaways & Limitations
The paper concludes that L2C2 smooths policy and value functions without deteriorating task-required expressiveness and contributes to improved learning performance.
Source: B. Result · p. 7Suppression of exploration occurs as a side effect, and its effectiveness is task-dependent.
Source: B. Result · p. 7
Abstract
from arXiv · showhide
This paper proposes a new regularization technique for reinforcement learning (RL) towards making policy and value functions smooth and stable. RL is known for the instability of the learning process and the sensitivity of the acquired policy to noise. Several methods have been proposed to resolve these problems, and in summary, the smoothness of policy and value functions learned mainly in RL contributes to these problems. However, if these functions are extremely smooth, their expressiveness would be lost, resulting in not obtaining the global optimal solution. This paper therefore considers RL under local Lipschitz continuity constraint, so-called L2C2. By designing the spatio-temporal locally compact space for L2C2 from the state transition at each time step, the moderate smoothness can be achieved without loss of expressiveness. Numerical noisy simulations verified that the proposed L2C2 outperforms the task performance while smoothing out the robot action generated from the learned policy.
I. INTRODUCTION
The paper addresses instability, overfitting, and unpredictable behavior in RL by constraining policy and value functions to be moderately smooth without excessive loss of expressiveness. It introduces L2C2, whose local regularization is defined by state transitions, and reports equivalent or better task performance in noisy simulations.
I. INTRODUCTION
- Real-world RL faces sample demands, overfitting with limited robot data, and safety and smoothness concerns about unpredictable behavior.The paper notes that simulation-real gap methods cannot resolve problems when simulators are unavailable.Source: §I · p. 1
- Excessive regularization can degrade performance by suppressing the expressiveness needed for task completion, motivating a design that avoids over-smoothing.The paper notes that mitigating this issue through hyperparameter tuning can be difficult and ad hoc.Source: §I · p. 1
- L2C2 applies local Lipschitz regularization in a transition-defined state-space box, limiting smoothing while retaining expressiveness across adjacent states.The box is centered on the pre-transition state, with the post-transition state as one vertex.Source: §I · p. 1
- Across four noisy PyBullet simulations, L2C2 achieved equivalent or better performance than conventional methods while retaining policy expressiveness and smoothing policies better than no regularization.The paper focuses on smoothness of both policy and value functions, while this reported comparison specifically describes policy smoothness.Source: §I · p. 1
A. Reinforcement learning
- In the RL setup, an agent uses a trainable state-dependent policy to choose actions and receives rewards from state transitions; policy and value functions are optimized to maximize expected return [7].The value function estimates expected return, while policy-gradient optimization uses action goodness estimated from value changes.Source: A. Reinforcement learning · p. 2
- The value network is trained by minimizing temporal-difference error against a target signal, with a slowly updated target network used to stabilize DNN updates.Minimizing the value loss optimizes the network to infer values over states.Source: A. Reinforcement learning · p. 2
B. Conditioning for action policy smoothness: CAPS
CAPS smooths policies with temporal and spatial regularization. Its global reach and excessive regularization can constrain expressiveness and impair learning.
B. Conditioning for action policy smoothness: CAPS
- CAPS adds temporal and spatial regularization terms to the policy objective to make the policy smoother.The method is introduced as a baseline for the proposed approach.Source: B. Conditioning for action policy… · p. 2
- The temporal term suppresses policy variation across the trajectory, while the spatial term compares the policy at a state with a normally sampled state.The sampled state can cover the entire real space, so the spatial term may impose global smoothness when its randomness is not appropriately set.Source: B. Conditioning for action policy… · p. 2
- Excessive CAPS regularization may prevent learning an optimal policy, and Fig. 2 contrasts global smoothness with local regions that can have different smoothness levels.The figure describes large global K as more expressive and small K as smoother; local Lipschitz continuity allows different K values by region.Source: B. Conditioning for action policy… · p. 2, Figure 2 · p. 3
C. Lipschitz continuity
The paper describes global Lipschitz continuity as a uniform bound on function change and local Lipschitz continuity as a neighborhood-specific relaxation. Local bounds allow different smoothness across regions.
C. Lipschitz continuity
- Global Lipschitz continuity requires the inequality to hold for every pair of points in the domain, with K serving as the Lipschitz constant.Smaller K corresponds to a smoother function.Source: C. Lipschitz continuity · p. 3
- For DNNs, global Lipschitz continuity is guaranteed with continuously differentiable activations, but K is unclear and existing regularization is difficult to limit locally.The paper identifies local control as the goal of its approach.Source: C. Lipschitz continuity · p. 3
- Local Lipschitz continuity requires the bound only within a neighborhood Ui of xi, allowing each subset to have its own constant Ki.The paper illustrates this as allowing different smoothness in different subsets.Source: C. Lipschitz continuity · p. 3
III. PROPOSAL
L2C2 applies local Lipschitz constraints to policy and value functions, seeking smooth behavior within state neighborhoods without restricting behavior outside them. The hard-constrained formulation would require per-state Lipschitz constants that are difficult to tune and DNN constants that are computationally intractable to calculate.
A. Theoretical formulation of hard-constrained problem
- Local Lipschitz constraints preserve adaptability outside each neighborhood while seeking robustness to small state changes within it.The paper contrasts this with CAPS’s global policy constraint, which it says can cause loss of expressiveness.Source: A. Theoretical formulation of… · p. 3
- The hard-constrained formulation requires neighborhood-specific Lipschitz constants for both policy and value functions.The constants should be small where high smoothness is needed and larger where small state changes may require function jumps.Source: A. Theoretical formulation of… · p. 3
- Tuning those constants for every state is intractable, and calculating actual DNN Lipschitz constants is NP-hard.Source: A. Theoretical formulation of… · p. 3
B. Practical formulation with soft regularization
The practical formulation replaces hard constraints with soft regularization, using sampled neighborhood comparisons and Lagrange multipliers. Constant regularization gains can still yield implicitly adaptive Lipschitz constants as the balance between task losses and regularization changes.
B. Practical formulation with soft regularization
- Monte Carlo approximation makes neighborhood expectations tractable, while reusing the already-computed state s reduces computation to state pairs (s, ˜s).Source: B. Practical formulation with soft… · p. 4
- The practical objective converts policy and value constraints into soft regularization terms using Lagrange multipliers.The respective gains are λπ and λV.Source: B. Practical formulation with soft… · p. 4
- With constant gains, the effective Lipschitz constants can adjust implicitly with the task-loss-to-regularization balance.Smaller task losses leave room for stronger constraints; larger losses correspond to weaker constraints.Source: B. Practical formulation with soft… · p. 4
C. Implementation
The implementation defines neighborhoods from each state transition and uses normalized L∞ distance to set local regularization strength. It also chooses distance measures and sampling procedures intended to stabilize updates and ease implementation.
C. Implementation
- The implementation choices prioritize ease of implementation and numerical stabilization, though other designs may perform better.Source: C. Implementation · p. 4
- The subset size σ is relative to the transition distance from s to s′, setting a noise scale for expected smoothness along that transition.Source: C. Implementation · p. 4
- Normalized L∞ distance avoids cross-dimension dependencies during sampling and scales distance relative to the traveled transition.An ϵ term prevents division by zero; sampling uses an independent uniform distribution for each dimension.Source: C. Implementation · p. 4
- The distance-dependent regularization gain varies with position in the neighborhood, while constant gains can produce implicit adaptation to the loss-to-regularization ratio.Source: Figure 3 · p. 4, Figure 4 · p. 5
- Policy and value distances use squared error forms to support continuous gradients during gradient-based updates.The policy distance is squared Hellinger distance; the value distance accommodates vector-valued ensemble predictions, and action-value distance can use squared error over actions.Source: C. Implementation · p. 5
D. Difference from CAPS
L2C2 differs from CAPS by adapting regularization to state-transition neighborhoods and distance. This design aims to preserve local smoothness without sacrificing global expressiveness, while also regularizing the value function.
D. Difference from CAPS
- L2C2 weakens regularization for distant states, whereas CAPS applies equal strength, making balanced gain tuning difficult.Near the current state, L2C2 can provide more smoothness without increasing the regularization gain.Source: D. Difference from CAPS · p. 5
- L2C2 combines spatial and temporal regularization through neighborhoods based on state transitions, unlike CAPS's independent regularizers.The design leaves room for policy change near a transition while strengthening regularization around the next state, avoiding convergence to a stationary policy in the described extreme case.Source: D. Difference from CAPS · p. 5
- L2C2 sets its relative regularization range from each transition distance, easing tuning across problems with different state spaces.CAPS specifies the noise scale for perturbed states absolutely, which requires problem-specific fine-tuning.Source: D. Difference from CAPS · p. 5
- Unlike CAPS, L2C2 also regularizes the value function, whose fluctuations can make TD error noisy and learning unstable.The paper notes that target networks are used to stabilize TD error and cites prior work reporting improved learning stability from value-function smoothness.Source: D. Difference from CAPS · p. 5
- Together, these differences aim to make policy and value functions locally smooth without losing global expressiveness.The paper proposes this moderate smoothness as a route to learning stability and noise robustness; numerical verification is deferred to later sections.Source: D. Difference from CAPS · p. 5
A. Setup
The setup evaluates three methods on four PyBullet tasks with noisy observations, measuring episode rewards and temporal action changes. The implementations use PPO-RPE with prioritized replay and shared policy/value network components.
A. Setup
- Four noisy PyBullet tasks are evaluated over 18 seeds, using episode reward and temporal action change to assess performance and policy smoothness.The tasks are Swingup, Reacher, Hopper, and Ant; noise scales are 0.01 for Swingup and Reacher and 0.001 for Hopper and Ant, and final scores use 100 post-training runs.Source: A. Setup · p. 6
- Policy and value functions use two-layer fully connected networks with 100-neuron hidden layers, Layer Normalization, and Squish activation.The policy is a diagonal normal distribution; the ensemble value is the mean of five outputs from networks sharing layers up to the output.Source: A. Setup · p. 6
- The comparison includes vanilla RL, CAPS, and L2C2, with PPO-RPE, prioritized replay, and t-soft-updated target networks used for learning.CAPS settings follow its official implementation because recommended values were unavailable; L2C2's policy regularization matches CAPS's minimum gain, while its value-function gain is smaller.Source: A. Setup · p. 6
B. Result
L2C2 generally matched or exceeded Vanilla in learning performance and improved policy smoothness, while CAPS over-smoothed policies and reached local solutions. L2C2’s effects on exploration and smoothness varied by task.
B. Result
- CAPS fell into local solutions on every task as excessive smoothing impaired policy expressiveness, and avoiding this may require problem-dependent gains and noise scales.The paper gives switching from swing-up to inverted balance as an example of a behavior CAPS could not handle.Source: B. Result · p. 6
- Hopper performance improved as smoothing suppressed risky exploration, while in Ant, the highest-dimensional-action task, exploration suppression sometimes delayed or prevented learning.The paper notes that reduced excessive exploration may be useful for real robots because exploration is risky and consumes energy.Source: B. Result · p. 6, Figure 5 · p. 7
- L2C2 basically improved policy smoothness from early learning, but Hopper smoothness is not comparable because L2C2 and Vanilla learned different optimal policies.The paper describes the learned Hopper behaviors as globally optimal for L2C2 and locally optimal for Vanilla.Source: B. Result · p. 6, Figure 6 · p. 7
- L2C2 achieved equivalent or better learning performance than Vanilla, and outperformed it in test results.Fig. 5 reports the learning-curve comparison; Table III summarizes test results.Source: B. Result · p. 6, Figure 5 · p. 7, Table III · p. 7
- With τ increased from 0.1 to 0.5, scores were 376 for L2C2 with β = 0.1 and 284 for β = 0; only β = 0.1 exceeded 800 in Swingup.The β = 0 ablation showed little difference from β = 0.1 in the reported test results, suggesting value-function regularization was ineffective in this implementation.Source: B. Result · pp. 6–7
V. SUMMARY AND FUTURE WORK
L2C2 converts a hard local Lipschitz constraint into regularization terms and defines neighborhoods from state transitions. In four noisy simulations, it matched or exceeded conventional methods while smoothing policies, though its implementation still has limitations.
V. SUMMARY AND FUTURE WORK
- L2C2 converts a hard-constrained minimization problem into Lagrange-multiplier regularization, with transition-distance neighborhoods making spatial regularization spatio-temporal.The hard constraint cannot be handled directly because it is NP-hard.Source: §V · p. 7
- Across four noisy simulations, L2C2 achieved equivalent or better task performance than conventional methods without losing policy expressiveness, and smoothed policies better than no regularization.Source: §V · pp. 7–8
- The implementation replaces the upper bound with an expected value and hard constraints with a linear weighted sum, limiting its formulation and motivating alternatives.The paper notes that adversarially generating ˜s could use the upper bound directly but would increase adversarial-network training cost.Source: §V · p. 8