Scalable Policy Gradient Algorithms

• Scalable policy gradient algorithms are reinforcement learning methods that optimize parameterized policies in high-dimensional and large-scale settings.
• They employ techniques such as series expansion, eligibility traces, and Rao-Blackwellisation to reduce variance and complexity in POMDPs and multi-agent systems.
• These approaches enable efficient distributed implementations and robust convergence as state, action, and agent dimensions grow.

Scalable policy gradient algorithms constitute a class of methods in reinforcement learning (RL) and control designed for efficient optimization of parameterized policies in high-dimensional and large-scale environments, including partially observable Markov decision processes (POMDPs), multi-agent networked systems, deep structured teams, and modern LLMs. The focus is on algorithmic constructions, complexity reduction, variance control, and distributed implementation, which enable tractable learning as the state, action, agent, or parameter dimensions grow. Core advances in this area include model-based series expansion, model-free eligibility tracing, variance reduction, distributed consensus, direct Newton-style updates, regularization strategies, and the use of scalable surrogate losses.

1. Foundations: Policy Gradient Theorems and Parameterization

Policy gradient algorithms optimize a differentiable performance metric—typically the average or discounted return—by ascending its gradient with respect to a parameterized policy $\pi_\theta$. For infinite-horizon POMDPs, such as those addressed by Aberdeen & Baxter, the agent uses a stochastic finite-state controller (FSC) to provide memory embedding, with policy parameters $(\theta,\phi)$ governing both action selection $\mu(u|\theta,g,y)$ and memory transition $\omega(h|\phi,g,y)$; the joint process over world state and controller state is Markov with stationary distribution $\pi(\theta, \phi)$. The long-run average reward $\eta(\theta,\phi)$ possesses a gradient

$$\nabla_{(\theta, \phi)} \eta = \pi'(∇P)[I - P + e\pi']^{-1}r,$$

where $P$ is the joint transition matrix, $r$ is the reward, $e$ is the all-ones vector, and $∇P$ collects partial derivatives with respect to $(\theta, \phi)$ (Aberdeen et al., 2 Dec 2025). A discounted, expectation-based form generalizes standard REINFORCE/GPOMDP, yielding scalable Monte Carlo estimators.

For distributed multi-agent settings (REC-MARL), the global network reward decomposes as sums of local rewards depending on state-action tuples of neighbors, and policies are decomposed into products over agents. The actor-critic variant leverages softmax parameterization per agent, allowing gradient computation via local surrogates and neighbor exchanges, with sample/iteration complexity driven only by local state-action space dimensions (Liu et al., 2022).

2. Scalable Algorithmic Constructions

Model-Based Series Expansion and Sparse Computation

The GAMP algorithm exploits full POMDP models, replacing expensive matrix inversion $[I-P+e\pi']^{-1}$ with a series expansion $x_N=\sum_{n=0}^N P^n r$, and computing $\pi$ by repeated sparse matrix–vector multiplication; per-iteration complexity is $\mathcal{O}(c |S| |G| N)$, where $c$ is the sparsity per row (Aberdeen et al., 2 Dec 2025). This approach ensures zero-variance policy-gradient estimates and avoids sampling noise.

Model-Free Simulation: Eligibility Traces

IState-GPOMDP circumvents the need for explicit models by using trajectory sampling and two eligibility traces for $\theta$ and $\phi$. The trace updates are

$$z^{\theta}_{t+1} = \beta z^{\theta}_t + \nabla_{\theta} \log \mu(u_t|\theta,g_t,y_t),$$

with analogous updates for $\phi$, and moving average computation for the gradient. This yields per-step constant computational cost, with overall complexity $O(T(n_\theta + n_\phi))$ (Aberdeen et al., 2 Dec 2025).

Variance Reduction via Rao-Blackwellisation

Exp-GPOMDP analytically marginalizes over FSC internal states using belief recursion, forming a Rao-Blackwellised trace that reduces sampling variance and improves reliability, particularly for large $|Q|$. At each time, beliefs are propagated deterministically,

$$\alpha_{t+1}(h) = \sum_{g} \alpha_t(g)\,\omega(h|\phi,g,y_t).$$

The gradient runs over a marginal policy $\hat{\mu}$ derived from $\alpha_{t+1}$, incurring complexity $O(|Q|(n_\theta + n_\phi))$ per step (Aberdeen et al., 2 Dec 2025).

Distributed Actor-Critic for Multi-Agent Systems

TD-RDAC decomposes the policy gradient into local gradients per agent, leveraging only local state-action dimensions and neighbor communication. The actor step uses local and neighbor TD-errors, updating $\theta_n$,

$$\theta_n \gets \theta_n + \eta \left[ \frac{1}{1-\gamma}\sum_{h=0}^{H-1} \gamma^h \frac{1}{N} \sum_{k \in N(n)} \delta_k(h) \nabla_{\theta_n} \log \pi_n(a_n(h) | s_n(h)) + \text{regularization} \right].$$

Complexity scales as $O(NS_\max A_\max)$ per iteration (Liu et al., 2022).

3. Regularization, Natural and Second-Order Methods

KL-Regularized Policy Gradients and RPG

KL regularization stabilizes policy gradients, especially for off-policy and LLM settings. Four algorithmic regimes—forward/reverse, normalized/unnormalized KL—are unified via regularized policy gradient (RPG) objectives, with surrogates and clipped importance weights for variance control. For example, the URKL surrogate: $$\nabla_{\theta} J_{URKL} = Z_{old} \mathbb{E}_{\tilde\pi_{old}} [ w\, (R - \beta \log w)\, \nabla_{\theta} \log \pi_{\theta} ],$$ is implemented efficiently via RPG-Style Clip, enabling robust and stable large-scale training (Zhang et al., 23 May 2025).

Approximate Newton and Natural Policy Gradients

The approximate Newton method preconditions the gradient using a diagonal proxy for the Hessian induced by entropy regularization (Shannon or general $f$-divergence),

$\Lambda_{(s,a)} = -\tau \frac{w_\pi(s)}{\pi_s^a}.$

Discrete updates in parameter potentials yield locally quadratic convergence, unattainable via standard first-order approaches; complexity is dictated by sparse linear system solves over the state space (Li et al., 2021). For generic policies, natural policy-gradient descent (NPG) corresponds to Shannon-entropy regularization, with full step convergence and scalable matrix-vector multiplication.

4. Scalability in Deep Structured Teams and Multi-Agent RL

Linear Quadratic Deep Structured Teams (LQ-DST) admit scalable policy-gradient and natural-gradient methods by leveraging a low-dimensional parameterization independent of the total agent count. For jointly coupled subpopulations, global convergence to Riccati-optimal feedbacks is assured under stabilizability, detectability, and sufficient regularization. Computational complexity per iteration is $O(\sum_s d^s_x d^s_u + d_{\text{deep}}^2)$, dominated by per-population Riccati solves (Fathi et al., 2020).

Centralized multi-agent deep RL achieves scalability via distributed consensus and a single shared policy parameter $\theta$ among a large pool of agents. Rollout and gradient steps are fully parallelizable, and communication is limited to exchanging parameter updates, maintaining sample efficiency and stability for populations exceeding several hundred agents (Khan et al., 2018).

5. Exploration-Efficient and Variance-Controlled Extensions

Policy Cover–Policy Gradient (PC-PG/EPOC) introduces an ensemble of past policies to define a policy cover, ensuring global coverage and overcoming local minima traps. Empirical covariance of feature embeddings defines exploration bonuses. The inner actor step in each episode is a natural policy-gradient (NPG) update on the surrogate MDP with reward augmented by bonus, yielding polynomial sample and runtime complexity even under state-aggregation or misspecification (Agarwal et al., 2020).

Stein Variational Policy Gradient (SVPG) offers scalable parallel variance reduction by maintaining a set of policy particles whose updates combine standard policy-gradient attraction with repulsive kernel interactions. This approach efficiently explores policy space and achieves higher returns with reduced sample complexity per update (Liu et al., 2017).

6. Complexity Analysis and Empirical Findings

The following table summarizes the complexity profiles for representative scalable policy-gradient algorithms:

Algorithm	Per-step Complexity	Scalability Dimension
GAMP (model-based)	$O(c\,|S|\,|G|\,N)$	Sparsity, POMDP size
IState-GPOMDP	$O(T(n_{\theta}+n_{\phi}))$	Params, trajectory steps
Exp-GPOMDP	$O(T\,|Q|\,(n_{\theta}+n_{\phi}))$	Internal state number
TD-RDAC	$O(N\,S_{max}A_{max})$	Number of agents
Approx-Newton (MDP)	$O(\text{iters}\cdot \text{nnz}(P))$	State, action space
LQ-DST PG/NPG	$O(\sum_s d_x^sd_u^s+d_{\text{deep}}^2)$	Local/feature dims
DiMA-PG (Centralized MA)	$O(N)$ per iteration	Population size
PC-PG (EPOC)	Poly($|S|,|A|,H,1/\epsilon$)	Tabular, RKHS, function approx
SVPG	$O(N \cdot C_{env} + N^2\,d)$	Particle number, param dim

Under proper design (series expansion, sparse representation, distributed communication, Rao-Blackwellisation), scalability is maintained with zero to minimal loss in convergence rate or sample-efficiency as state/action/agent/parameter dimensions grow.

7. Limitations, Insights, and Future Directions

Scalability in policy gradient algorithms is contingent on structural sparsity, low-dimensional parameterization, series-based approximations, and robust variance reduction. Key limitations include slow mixing and bias as dimensionality increases, local optima trapping in high-parameter space, and possible explosion in FSC states in POMDPs. Future work includes hierarchical/factored controller designs, advanced Krylov-subspace solvers for expansion acceleration, actor-critic combinations for further variance reduction, and the development of reliably tunable parametric surrogate families for improved convergence (Aberdeen et al., 2 Dec 2025, Gummadi et al., 2022). For distributed systems, per-agent local complexity remains the primary driver, and further work in efficient neighbor communication and hybrid consensus is ongoing (Liu et al., 2022, Khan et al., 2018).

Scalable policy gradient algorithms now underpin cutting-edge RL and reasoning systems, including those for mathematical reasoning in LLMs—where KL-correct surrogates, truncated importance sampling, and iterative policy references have set new empirical and theoretical performance benchmarks (Zhang et al., 23 May 2025).