Windowed Pooling for Stable Routing

• Windowed pooling is a method that groups contiguous inputs within fixed or adaptive windows to achieve stable routing in both network and neural systems.
• It leverages adaptive window sizing and prioritization techniques to smooth local fluctuations, control congestion, and optimize resource allocation.
• Empirical results show improved accuracy, reduced layer execution, and faster convergence, validating its efficiency in environments ranging from adversarial queuing to dynamic layer routing in LLMs.

Windowed pooling for stable routing is a methodological framework in networked and neural systems wherein inputs—whether they are data packets in adversarial queuing, tokens in Mixture-of-Experts (MoE) Transformers, or hidden-state vectors in LLMs—are processed, grouped, and routed using operations over fixed or adaptive “windows” of data. This approach is central to enhancing stability in routing policies by smoothing out local fluctuations, controlling congestion, and supporting robust allocation of resources across diverse architectures. The design and effects of windowed pooling strategies have been explored across distinct research threads, including adversarial network routing with feedback (Chlebus et al., 2018), stable expert assignment in MoE models (Dai et al., 2022), and dynamic layer routing in transformer-based LLMs (Heakl et al., 14 Oct 2025).

1. Foundational Concepts of Windowed Pooling

Windowed pooling denotes a procedure in which sets of contiguous inputs—be they packets or tokens—are aggregated within non-overlapping or sliding windows. Instead of routing each element independently, the system pools representations over each window, typically by averaging, max-pooling, or other summary operations. In the context of dynamic layer routing for LLMs (Heakl et al., 14 Oct 2025), for a sequence of T tokens, hidden states are partitioned into W windows, and each window $S_i$ yields a mean-pooled vector $m_i = (1/|S_i|) \sum_{t \in S_i} H_t^{(\ell-1)}$ that summarizes local context for a router’s decision. In adversarial packet routing (Chlebus et al., 2018), windowed pooling can correspond to grouping transmission opportunities, adapting the window size to reflect network feedback and congestion, and enforcing stability by bounding the number of packets transmitted per window to satisfy admissibility constraints.

The underlying rationale for windowed pooling is that local aggregation ameliorates the instability arising from fine-grained, noisy, or transient input variations. This is particularly critical in long sequences or large-scale systems, where individual token or packet dynamics may be volatile, but their windowed aggregates show smoother, more predictable behavior.

2. Windowed Pooling in Adversarial Routing with Feedback

In adversarial queuing theory, the “delayed-feedback” adversarial model (Chlebus et al., 2018) extends the leaky-bucket scheme by introducing antitokens—objects that reflect the impact of transient transmission failures. When a packet is stalled, an antitoken is created and, on expiration, annihilates a token in the injection bucket, reducing the effective injection rate. Stability is defined via admissibility conditions: for any queue $q$ and time interval $T$,

$$\sum_{t \in T} I_{(q)}^{(\delta)}(t) \leq r \cdot \left( \sum_{t \in T} (1 - s_{(q)}^{(\delta)}(t)) \right) + b$$

where $I_{(q)}^{(\delta)}(t)$ is the injection at time $t$ and $s_{(q)}^{(\delta)}(t)$ encodes rounds when delayed feedback is active.

Windowed pooling arises as a natural extension for managing injection opportunities—by grouping packets within windows and modulating window size in response to feedback (i.e., antitoken arrivals), the system dynamically enforces congestion control akin to TCP’s window reduction. The model shows universally stable policies (FTG, NFS, SIS) retain stability under such feedback, while strategies lacking prioritization (FIFO, NTG) remain unstable. Windowed pooling further enables budgeted transmission allocations per window, and may utilize priority-based subgroupings to prioritize critical traffic, ensuring bounded queue sizes and robustness despite adversarial injection patterns.

3. Stable Routing in Mixture-of-Experts Transformers

StableMoE (Dai et al., 2022) addresses routing fluctuation in MoE Transformers, where learning-to-route methods may assign the same input to different experts during training, but only one is activated in inference. This dilutes update signals, slows convergence, and hampers sample efficiency. The solution involves a two-stage procedure: stage 1 learns a balanced routing strategy using a greedy assignment score $s_{t,i} = E_i^T h_t^{(l-1)}$ (where $E_i$ is the expert centroid) coupled to a balance loss to spread tokens evenly, and a distilled lightweight router trained to mimic these assignments. Stage 2 freezes the router, creating a fixed mapping from tokens to experts for the remainder of training and inference.

The notion of applying windowed pooling, while not explicit in the source, is suggested as a plausible extension. Pooling features across local windows of tokens, e.g., $$s_{t,i} = E_i^T \operatorname{Pool}(h_{t-w/2}, ..., h_{t+w/2})$$, would allow the balanced and cohesive routing strategy to capitalize on context-specific relationships among neighboring tokens. Windowed pooling may reduce fluctuations in routing decisions, produce smoother expert assignments, and optimize expert utilization, particularly for long-form language modeling or translation tasks where local sequence context is informative.

4. Dynamic Layer Routing in LLMs

Dr.LLM (Heakl et al., 14 Oct 2025) introduces a dynamic routing framework for transformer-based LLMs, in which lightweight per-layer routers select among skip, execute, or repeat actions for each block. Windowed pooling is integral to router input: for each layer, tokens’ hidden states are partitioned into windows ($W$ typically set to 8), and each window’s mean-pooled vector is processed by a two-layer Linear-GELU-Linear MLP router. The average router output across windows yields logits for decision-making, $z_\ell = (1/W) \sum_{i=1}^{W} r_\ell(m_i)$, translated via softmax into per-layer action probabilities.

This mechanism ensures the router bases its decisions on stable, low-dimensional summaries, mitigating volatility present in long contexts and improving robustness to local noise. Training leverages explicit supervision, using Monte Carlo Tree Search (MCTS) to discover layer execution paths ($y_\ell^\star \in \{\text{skip, execute, repeat}\}$) that optimize task accuracy under a compute budget. Routers are trained with teacher forcing, and only router parameters are updated—the base LLM remains frozen.

Empirical results indicate Dr.LLM improves accuracy (up to +3.4 percentage points), saves layers per example (average 4–11 reduction), and generalizes routing policies across out-of-domain tasks with negligible performance degradation (0.85% accuracy loss), outperforming prior methods by significant margins. Windowed pooling is central to achieving this efficiency-stability tradeoff.

5. Scheduling Disciplines and Priority in Windowed Strategies

The stability of windowed pooling for routing depends heavily on the underlying scheduling discipline and use of priorities. In adversarial routing (Chlebus et al., 2018), policies capable of handling multiple priority levels (e.g., FTG, NFS, SIS) are universally stable, as they can respond to delayed feedback by prioritizing urgent packets, ensuring the admissibility condition is met. Incorporating packet or token priority within pooling windows—that is, prioritizing transmission or computation for high-importance items within each window—reinforces stability by reducing delay for critical executions.

Lemma 2 of (Chlebus et al., 2018) provides a bounding principle:

$$\sum_{t \in T} w_{(q)}(t) \leq \sum_{t \in T} s_{(q)}^{(\delta)}(t) + \delta$$

indicating the cumulative delay is controlled by the feedback schedule and reaction delay $\delta$. A plausible implication is that, when tuning windowed pooling hyperparameters (window size, burstiness, reaction delay), adherence to similar bounds is instrumental in ensuring system stability under adversarial or fluctuating conditions.

6. Efficiency, Accuracy, and Generalization

Windowed pooling for stable routing yields measurable improvements in system efficiency and accuracy across both networking and computational models. In Dr.LLM, routers utilizing windowed pooled inputs reduce the number of executed layers without accuracy loss or retraining of backbone parameters (Heakl et al., 14 Oct 2025). In StableMoE, fixed routing assignments post-distillation produce faster convergence and lower perplexity/bleu scores compared to alternative MoE methods (Dai et al., 2022). Furthermore, these systems generalize across tasks: Dr.LLM’s routing policies, learned on ARC and DART, continue to provide both efficiency and stable accuracy on MMLU, GSM8k, AIME, TruthfulQA, SQuADv2, GPQA, PIQA, and AGIEval.

This suggests a broad applicability—windowed pooling strategies, when properly parameterized and coupled to prioritized scheduling or expert balancing, furnish robust frameworks for stable routing in various adversarial, sequential, and deep learning domains.

7. Implications and Future Directions

Advances in windowed pooling for stable routing establish a principled approach for mitigating instability from input fluctuations, adversarial conditions, and unbalanced resource allocation. Robustness is achieved not by fine-grained per-element decision-making but through controlled aggregation and prioritization within windows, guided by feedback mechanisms in networks or explicit supervision in learning systems. This synthesis of adversarial queuing theory, efficient transformer routing, and supervised dynamic computation allocation is a significant step toward scalable, stable, and resource-aware architectures.

Possible future directions include developing adaptive window sizing based on observed delays or task complexity, incorporating more sophisticated pooling operations (e.g., attention-weighted pooling), and extending principles to multi-agent systems, distributed learning, and high-performance networking environments. The rigorous frameworks detailed in (Chlebus et al., 2018, Dai et al., 2022), and (Heakl et al., 14 Oct 2025) provide precise criteria for designing, tuning, and evaluating windowed pooling protocols to guarantee stability, efficiency, and transferability of routing behaviors across domains.