Weighted-REINFORCE: Adaptive Policy Gradients

Updated 19 September 2025

Weighted-REINFORCE is a class of policy gradient methods that modulate reward, gradient, and experience signals with adaptive weights to manage bias and variance.
Dynamic weight adaptation techniques, such as hypervolume-guided and gradient-based methods, enable efficient multi-objective optimization and faster convergence.
Weighted approaches enhance sample efficiency, exploration control, and distributed learning by refining credit assignment, entropy regularization, and gradient aggregation.

Weighted-REINFORCE encompasses a class of methods that modify the standard REINFORCE algorithm by introducing adaptive or structured weighting into the computation of policy gradients, reward signals, experience prioritization, entropy regularization, or gradient aggregation. These weighting schemes are motivated by the desire to address bias and variance, credit assignment challenges, multi-objective trade-offs, sample efficiency, non-stationarity, and exploration–exploitation balance. Weighted-REINFORCE methods have broad relevance throughout reinforcement learning, including multi-objective RL, offline RL, distributed RL, model-based RL, and biologically motivated learning.

1. Conceptual Foundations and General Formulation

Weighted-REINFORCE refers to policy gradient algorithms in which each component contributing to the learning update is modulated by a weight—be it in rewards, returns, experiences, entropy, or gradients. The canonical REINFORCE update is:

$\Delta\theta \propto \sum_{t} \nabla_\theta \log \pi_\theta(a_t \mid s_t) \cdot G_t$

Weighted variants introduce a weighting vector $w_t$ such that the update takes the form:

$\Delta\theta \propto \sum_{t} w_t \cdot \nabla_\theta \log \pi_\theta(a_t \mid s_t) \cdot G_t$

Here, $w_t$ may be derived from posterior probabilities, objective importance, uncertainty estimates, experience prioritization, or meta-learned signals. In multi-objective RL, weights perform scalarization over objectives; in experience replay, they modulate the impact of samples; for entropy regularization, they modify exploration incentives; and in distributed RL, they adjust gradient aggregation based on actor performance.

2. Multi-Objective Weighted Scalarization and Dynamic Weight Adaptation

When managing multiple objectives with potentially conflicting priorities, a fixed-weight linear scalarization fails to optimally span non-convex Pareto fronts (Lu et al., 14 Sep 2025). Weighted-REINFORCE methods address this by dynamically adapting the objective weights during online training.

Dynamic Weighting Techniques:

Hypervolume-Guided Adaptation: Meta-level reward $r_{\rm pareto}$ is computed as $0.5+1.5\tanh(\Delta{\rm HV})$ and amplifies scalarized rewards when new solutions move the Pareto front.
Gradient-Based Weight Optimization: Objective weights $w_i^{(t)}$ are updated via mirror descent rules using per-objective gradient influences.

These adaptive weight strategies are compatible with canonical policy gradients; the overall objective remains linear in weighted returns:

$\nabla J(\theta) = \sum_{i} w_i \cdot \nabla J_i(\theta)$

Experimental evidence confirms that dynamic weighting achieves Pareto dominance with faster convergence and superior multi-objective alignment compared to static weighting (Lu et al., 14 Sep 2025).

3. Weighted Experience Sampling and Prioritization

Several methods improve sample efficiency and error minimization by weighting the impact of experiences during learning.

Likelihood-Free Importance Weights: Experiences are weighted by the stationary state-action distribution of the current policy ( $w(s,a) = d^\pi(s,a)/d^D(s,a)$ ), with the ratio estimated via classifier-based methods for robust prioritization (Sinha et al., 2020).
Prioritization-Based Weighted Loss (PBWL): TD errors in off-policy RL are directly weighted in the loss function ( $L_W = \frac{1}{N}\sum_j (\omega_j \delta_j)^2$ ), employing normalization, Gaussian filtering, and softmax scaling. This sharpens sample efficiency and speeds up convergence (reported 33–76% reduction in convergence time) (Park et al., 2022).
Diverse Experience Replay (DER): Full trajectories are stored and selected for diversity in the return vector signature, maintaining coverage across the objective space under dynamic weighting (Abels et al., 2018).
Reweighting Imaginary Transitions: In model-based RL, a meta-gradient adjusts weights of synthesized transitions by measuring their effect on subsequent loss computed with real samples, filtering harmful synthetic data (Huang et al., 2021).

Weighted sampling mechanisms enhance sample efficiency, reduce bias, and provide resilience to policy non-stationarity and changing reward regimes.

4. Weighted Entropy, Policy Regularization, and Exploration Control

Weighted entropy regularization provides state–action-specific control over exploration pressure, replacing the uniform entropy bonus in standard formulations.

Weighted Entropy in REINFORCE: The entropy term is modified as

$H^w(\pi(\cdot|s)) = -\sum_a w(s,a) \pi(a|s) \log \pi(a|s)$

where the weight function $w(s,a)$ encodes historical visitation, policy uncertainty, or expert priors (Zhao et al., 2020, Bui et al., 2022).

State-Dependent Entropy in IRL: In Weighted Maximum Entropy IRL, the entropy coefficient μ(s) is optimized per state to match expert behavioral stochasticity:

$\max_\pi \ \mathbb{E}_{\tau \sim \pi}\left[ \sum_t \gamma^t (r(a_t|s_t) - \mu(s_t) \ln \pi(a_t|s_t)) \right]$

The resulting policy $\pi^*(a|s)$ is a softmax with temperature proportional to μ(s) (Bui et al., 2022).

Such weighted entropy regularization allows for context-sensitive exploration, rapid adaptation, and learned exploitation–exploration balance.

5. Weighted Gradient Aggregation in Distributed and Ensemble RL

Distributed RL and ensemble methods utilize weighting schemes to optimize the aggregation of gradients or predictions from multiple agents or models.

Reward-Weighted and Loss-Weighted Gradient Merger: In distributed RL, each agent’s gradient is scaled by its episodic reward or loss relative to its cohort:

$w_i = \frac{r_i ~~({\rm or}~L_i)}{\sum_j r_j ~~({\rm or}~L_j)} + \frac{1}{h}$

These weights emphasize learning signals from richer (high-reward) or more challenging (high-loss) environments (Holen et al., 2023).

Online Weighted Q-Ensembles: Multiple RL agents (e.g., DDPG critics) are aggregated by weighting their Q-value outputs according to normalized TD error minimization. Agents with higher TD errors are assigned lower weight, reducing the impact of poorly tuned models and accelerating hyperparameter search (Garcia et al., 2022).

Such weighted aggregation protocols increase robustness, reduce variance, and enable practical scaling to complex RL workloads.

6. Theoretical and Biologically Motivated Weighting Mechanisms

Extending Weighted-REINFORCE to neural credit assignment, biologically plausible learning rules employ local weighting signals to overcome high variance and poor scaling:

Weight Maximization: Each hidden unit in a neural network maximizes the change in its outgoing weight norm ( $v^{(l)} \cdot \Delta v_t^{(l)}$ ), providing local reinforcement signals (Chung, 2020, Chung, 2023).
Unbiased Weight Maximization: For Bernoulli-logistic units, unbiased gradient estimates are constructed using Monte Carlo integration over the continuum [0, 1], yielding updates:

$\Delta_{\rm UWM} b = \hat r'(U) \cdot H \cdot (H - \sigma(b))$

This rule achieves unbiasedness and correct structural credit assignment even in deep or discrete-activation networks (Chung, 2023).

These approaches bring principled, local, and unbiased weighting to the training dynamics, often outperforming classical algorithms in learning speed and scalability.

7. Practical Development and Extensions

Weighted-REINFORCE concepts appear across recent developments:

Adaptive Q-Value Weighting in Offline RL: Q-Value Weighted Regression (QWR) uses weighted regression based on policy improvement signals, outperforming standard regression when data is limited (Kozakowski et al., 2021).
Policy Gradient with Second-Order Momentum: Curvature-adaptive weighting using diagonal Hessian estimates and exponential moving averages enhances sample efficiency and variance reduction in policy optimization (Sun, 16 May 2025).
Application to Sequence Modeling and Drug Design: Weighted variants of REINFORCE are used to optimize chemical LLMs for drug discovery, with additional reward shaping, experience replay, and hill-climbing mechanisms to balance chemical validity and reward maximization (Thomas et al., 27 Jan 2025).

Weighted approaches are modular and generally compatible with policy gradient, actor-critic, model-based, and ensemble RL architectures.

Weighted-REINFORCE is defined by the insertion of principled weighting schemes into one or more stages of the RL optimization process, improving sample efficiency, multi-objective trade-offs, structural credit assignment, exploration strategy, and distributed learning convergence. Its methods span dynamic scalarization, experience prioritization, entropy regularization, gradient aggregation, and biologically motivated learning rules, with empirical validation across a wide range of RL domains and architectures.