From Relaxed Indexability to Exact Indexability: A $t$-Step Approach for Partially Observable Restless Bandits

Whittle index policies offer a scalable method for restless multi-armed bandits, but under partial observability even determining the indifference subsidy at a single belief requires solving an infinite-horizon belief-state problem with no closed-form value function. Liu [10] addresses this difficulty by linearizing the unknown decision boundary, leading to a linear system and a closed-form approximate Whittle index. However, the resulting threshold uses only a one-step active--passive comparison and does not account for longer-horizon continuation values. We extend this framework to a \emph{$t$-step lookahead threshold policy}. For each subsidy $m$, the threshold is defined by the active-minus-passive advantage under $t$-step finite-horizon value iteration. At $t=1$, the threshold is $m$-independent and recovers the linear threshold of Liu [10]; for $t>1$, it becomes subsidy-dependent through the induced first-crossing structure and tracks the exact decision boundary more closely. The proposed algorithm does not require indexability as an input and includes an indexability verification. Under the original Whittle indexability, we prove that the $t$-step approximate Whittle index converges geometrically to the exact Whittle index, \[ |\widehat W_t(ω)-W(ω)|=O(β^t). \] Numerically, all 2,715 tested three-state instances are verified as indexable according to the proposed criterion. The P95 index error decreases from $2.18\times10^{-2}$ at $t=1$ to $8.93\times10^{-4}$ at $t=8$. In an exact-comparable instance with $β=0.9999$, $t=2$ already recovers the exact Whittle-index ordering. Moderate-depth threshold policies also outperform the one-step baseline and remain close to the optimal dynamic-programming benchmark, while runtime grows mildly with $t$.

Partially Observable Learning for Multi-Platform Dispatch Optimization

Instant delivery platforms have become a critical component of urban logistics, increasingly relying on crowdsourced couriers to fulfill highly dynamic orders. In real-world systems, couriers are not exclusive to a single platform and may concurrently serve multiple platforms, while each platform can only observe its own orders and couriers' interactions due to privacy and operational constraints. This results in a multi-platform dispatch environment with inherent partial observability. However, most existing works on dispatch optimization assume full courier observability and mandatory assignment acceptance, causing substantial performance degradation when deployed in realistic multi-platform settings. In this paper, we propose POLO, a partially observable multi-agent reinforcement learning framework for dispatching optimization in multi-platform instant delivery systems. POLO firstly models each platform-grid pair as an independent agent that learns dispatch policies solely from platform-local observations, aligning the learning process with real-world privacy and operational constraints. To support effective decision-making under incomplete and heterogeneous courier information, POLO introduces a novel attention-based policy representation that selectively aggregates inter-courier information. Moreover, we design a counterfactual reward shaping mechanism to mitigate the non-stationarity induced by joint actions across grids, leading to more stable and scalable learning. We develop a high-fidelity simulator to evaluate dispatch performance under varying numbers of platforms and system scales. Extensive experiments demonstrate that POLO consistently outperforms strong baselines in terms of platform revenue and courier travel efficiency, highlighting its robustness and effectiveness in realistic multi-platform settings.

Learning Suffers More Than the Policy Class Under Partial Observability: A Closed-Form Analysis

When a reinforcement learning agent cannot observe the full state, we usually blame its policies: it cannot see enough to represent a good one. We show that in a solvable case the bigger problem lies elsewhere. Even when a good policy is available and the agent's value function is expressive enough to describe it exactly, learning still ends up somewhere far worse. We study a partially observed linear-quadratic problem in which a standard actor-critic learner can be solved in closed form. At our default setting the best policy the agent can represent is already close to optimal, costing 10.4% more than the ideal controller that observes everything. Learning does not find it. The algorithm instead comes to rest at a policy that is 35% worse than the best one available to it, and we can say exactly where and why. The cause is a bias in what the critic learns rather than a limit on what the actor can express. Because the agent cannot attribute what it sees to the part of the state it cannot observe, the critic misreads that unexplained variation as sharp curvature in its own value estimates, and the actor follows that error away from the optimum. We derive closed-form expressions for the resulting policy, for its cost, and for the one design choice that removes the problem, which is how far the learner looks ahead before trusting its own value estimates. Deep reinforcement learning experiments follow these predictions closely. Notably, giving the agent memory of past observations does not help, while changing how far it looks ahead does.

Reward Structure Shapes the Interaction Between Episodic Exploration and Neural Memory in Reinforcement Learning

In partially observable reinforcement learning, agents face a dual bottleneck: they must explore to encounter rewarding states and retain that experience in memory to optimize their policies. Exploration bonuses and memory architectures are traditionally evaluated in isolation, leaving their interaction unmeasured, and standard notions of sparse reward conflate temporal signal density with what the reward actually supervises. We present a controlled study crossing episodic exploration bonuses with diverse neural memory architectures across three environments that vary how the content of memory is acquired. An identical bonus signal yields three distinct interaction patterns: it amplifies architectural capacity differences where memory content must be actively discovered and retained unsupervised; equalizes architectures to a shared ceiling where the content, once sought out, is a single reward-supervised cue; and is null where the observation stream is purely scheduled. Controlled reward manipulations verify that these patterns track reward structure rather than density: a dense reward neutralizes a bonus only if it directly supervises the required latent memory, and a small avoidable penalty on exploratory actions (leaving the optimum unchanged) induces policy convergence to suboptimal stationary states, which either bonus resolves. We then formalize reward sparsity with observation-anchored reward machines, separating structural sparsity (an automaton reproduces the return without the task-required history) from potential sparsity (the one-step reward misprices local exploratory actions); the resulting vocabulary organizes the three regimes by the retention burden each task exposes. Together, these results show exploration and memory are complements, not substitutes: a bonus induces exposure, and only memory converts exposure into return.

Minimal Markovization via Stable Quotients in Holonomy-Cover Decision Processes

An agent acting under partial observability must retain a recursively updateable statistic of history that restores the Markov property, but the smallest such statistic is generally unknown. We characterize this minimal Markov sufficient statistic for holonomy-cover decision processes, a structured POMDP class in which the visible dynamics are Markov and every realized visible transition applies a fixed permutation to a hidden mode. In particular, we construct the stable quotient, the coarsest observation-wise abstraction preserving one-step rewards and quotient successors, and prove that the pair of the current observation and stable class forms an exact finite Markov state. When the current class is correctly initialized, exact class tracking requires exactly the minimal memory symbols, in the sense that under reachability and pairwise decision separation at a maximizing observation, no arbitrary finite-memory controller can use fewer. Under resettable diagnostics, nearest-prototype class inference has exponentially decaying error, and a calibrate-then-restart reduction transfers finite-MDP guarantees to the recovered state. The results enable \emph{Holonomy Memory Reinforcement Learning}. It represents memory by the current stable class, updates it through ordered edge transports, identifies local class coordinates when diagnostics are available, and applies a standard finite-MDP RL backbone after synchronization. Experiments recover an exact compression from raw states to quotient states and achieve perfect paired-order accuracy with three decision-time memory states, matching the quotient oracle and outperforming the non-oracle baselines.

Neuro-Symbolic Meta-Policies for Temporal Knowledge-Graph Memory under Partial Observability

Partially observable reinforcement learning requires deciding what to retain, retrieve, and forget over time. We introduce a neuro-symbolic meta-policy that learns which symbolic memory heuristic to apply at each decision point while keeping execution symbolic. Our setting uses temporal knowledge-graph memory in RoomKG, where hidden state and observations are represented as Resource Description Framework (RDF) graphs and memory is augmented with temporal RDF triple annotations. The model combines knowledge-graph encoding of memory contents with value heads for question answering, exploration, and forgetting, yielding a controller that is both adaptive and inspectable. This gives the work a direct Semantic Web grounding through RDF-based representation, annotation-compatible graph semantics, and graph-based symbolic operations over explicit memory state. On train/test room splits at long-term memory capacity of 512, the qualifier-aware StarE-GNN configuration achieves the best held-out performance among the compared symbolic, neural, and neuro-symbolic systems while preserving step-level traceability of memory-management decisions.

Expected Free Energy as Belief-Dependent Utility for rho-POMDPs

An agent acting under partial observability must decide when to gather information and which observations are worth their cost. Standard POMDPs value information only through its eventual effect on reward. The $ρ$-POMDP framework instead rewards uncertainty reduction directly, through a belief-dependent utility $ρ$, but in practice both the choice of $ρ$ and the weight placed on it are tuned by hand for every task. We show that active inference removes this tuning entirely. Minimizing Expected Free Energy (EFE) is exactly equivalent to solving a $ρ$-POMDP whose utility is expected information gain, and the exploration weight is fixed at $w=1$ because the variational bound expresses pragmatic and epistemic value in the same units (nats). We prove this equivalence for observe-then-commit POMDPs and extend it to factored observation POMDPs, a broader class that covers interleaved observe-act problems such as non-destructive testing and mobile sensing, where gathering information leaves the hidden state unchanged. Experiments support the theory. Across environments ranging from the classic Tiger problem to RockSample and a new Structural Inspection benchmark with over 65,000 states, the untuned weight matches or outperforms reward-only planning at the same horizon, avoids the over-exploration of bonuses tuned per task, and sits near the reward-maximizing knee of the success-reward Pareto frontier. The practical payoff is an exploration objective that works out of the box. In applications such as fault detection and medical screening, where every test has a price and every missed fault has a cost, EFE supplies a belief-dependent utility that is derived rather than tuned.

Stochastic Linear Bandits with Partially Observed Actions

The stochastic linear bandit, where actions are represented as vectors and rewards are linear, is a central paradigm for sequential decision making. We study a partially observed variant of this problem in which the learning agent only sees a random subset of coordinates for each action. Such partial observability arises naturally in settings like recommendation and healthcare, where full action descriptions can be expensive or even impossible to obtain. In general, this makes sublinear regret information-theoretically impossible. However, we show that this barrier can be overcome when the action vectors have low intrinsic dimension. We propose an algorithm, TOFU-POV, that estimates the latent action subspace using the masked actions, imputes current actions using an epoch-wise frozen representation, and runs OFUL in the resulting low-dimensional coordinates. Our theory shows that TOFU-POV enjoys a $\sqrt{T}$ regret that scales with the intrinsic action subspace dimension as opposed to the ambient dimension and quantifies the interaction between these quantities and the missingness, decision set size, and subspace conditioning. We also devise a rank-adaptive algorithm that does not require the knowledge of the intrinsic dimension. We complement these guarantees with a lower bound based on a novel product construction that separates usual reward-learning uncertainty from a missingness-dependent cost intrinsic to partial observation. Synthetic and real data experiments support our theory and show that TOFU-POV can substantially improve upon natural baselines in this challenging problem.

ASK in the Dark: Uncertainty-Gated LLM Assistance under Partial Observability

Reinforcement learning agents operating under partial observability must act on incomplete information, making them natural candidates for guidance from small language models (SLMs) that carry broad reasoning priors. Yet integrating SLM guidance into this setting has proven difficult: across all test environments, vanilla uncertainty-gated approaches achieve an overwrite rate at or near zero, meaning the SLM almost never contributes an independent action. We trace this failure to the bare egocentric prompt, which provides insufficient context for genuine reasoning, and identify it as a context problem rather than a capacity problem. We propose ASK+, which supplies the SLM with trajectory-aware context (a partially revealed map, visited positions, and action history) and structured chain-of-thought reasoning, converting it from a passive redundancy check into a more informative consultant that occasionally corrects the policy. We further establish that the predictive entropy signal used for selective querying measures action uncertainty rather than state uncertainty and remains informative in POMDPs, making uncertainty-gated assistance viable beyond fully observable settings. The stateful prompt drives substantial gains: on DoorKey, where vanilla ASK matches PPO (both 89%), ASK+ reaches 93% success; on FourRooms, success climbs from 53% to 70%; on HigherLower, accuracy reaches 73.7%, matching the SLM-only upper bound. Across all environments, Qwen3.5-2B matches or exceeds Qwen3.5-4B, confirming that prompt design and selective gating dominate the impact of model scale, enabling guidance without large models.

HyPOLE: Hyperproperty-Guided Multi-Agent Reinforcement Learning under Partial Observation

Formal specification is a powerful tool to guide the learning process and provides significant advantages over reward shaping: (1) mathematical rigor; (2) expressiveness to specify objectives and constraints, and (3) the ability to define tactics to achieve objectives. However, these benefits remain largely unexplored in the context of Multi-Agent Reinforcement Learning (MARL). This paper introduces HyPOLE, a novel framework for MARL under partial observability, where learning is guided by the expressive power of the so-called hyperproperties and, in particular, the temporal logic HyperLTL. We integrate Centralized Training for Decentralized Execution (CTDE) techniques with HyPOLE to synthesize decentralized policies, and our evaluation on SMAC, MessySMAC, and WildFire benchmark demonstrates clear advantages over baselines.

Direct Advantage Estimation for Scalable and Sample-efficient Deep Reinforcement Learning

Direct Advantage Estimation (DAE) has been shown to improve the sample efficiency of deep reinforcement learning algorithms. However, its reliance on full environment observability limits its applicability in realistic settings, and its requirement to model transition probabilities incurs substantial computational overhead for high-dimensional observations. In the present work, we address both limitations. First, we extend the theoretical framework of DAE to partially observable domains with minimal modifications. Second, we reduce its computational complexity by introducing discrete latent dynamics models that efficiently approximate transition probabilities. We evaluate our approach on the Arcade Learning Environment and find that DAE scales effectively with function approximator capacity while retaining high sample efficiency.

Short-Term-to-Long-Term Memory Transfer for Knowledge Graphs under Partial Observability

Reinforcement learning under partial observability requires deciding what information to retain, yet most memory-based approaches do not explicitly model short-term-to-long-term transfer of symbolic observations. We study this transfer process in a temporal knowledge-graph memory setting and cast it as a neuro-symbolic value-based decision problem: for each observed triple, the agent chooses whether to keep or drop it before long-term insertion. To handle variable-sized short-term buffers, we use a per-item Q-learning design with shared parameters and a practical temporal-difference update over matched items across consecutive steps. On the RoomKG benchmark at long-term memory capacity 128, learned transfer decisions outperform symbolic and neural baselines, including symbolic baselines with temporal annotations and history-based LSTM/Transformer baselines. Across transfer-policy ablations, a lightweight local short-term-only variant performs best, and step-level behavior shows that the policy keeps navigation- and query-relevant facts while discarding lower-value candidate facts, supporting explicit and interpretable memory decisions under memory constraints.

Efficient learning by implicit exploration in bandit problems with side observations

We consider online learning problems under a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.