Resource-Specific Delay Kernels

• Resource-specific delay kernels are mathematical constructs that map the distribution of feedback over time via parameterized delay functions.
• They employ approaches like Beta distribution discretization and queueing models to capture early, delayed, and peaked outcome profiles.
• Integration into learning and optimization frameworks enables adaptive decision-making under system constraints such as capacity and fairness.

Resource-specific delay kernels provide a rigorous mathematical formalism for capturing the temporal dynamics by which the effect—or feedback—of resource allocations is distributed over time. Emerging in settings from individualized interventions to distributed systems, these kernels formalize the process whereby a unit of allocated resource gives rise to (possibly stochastic) delayed outcomes, enabling algorithms to optimize long-term utility in systems where feedback is neither immediate nor homogeneous across resources. Their explicit incorporation underpins recent advances in adaptive decision-making, particularly in domains facing delayed reward, capacity constraints, resource heterogeneity, and non-trivial fairness requirements.

1. Formal Definitions and Canonical Constructions

A resource-specific delay kernel (henceforth, "delay kernel") maps discrete or continuous time since allocation to the fraction of positive feedback attributed to each lag. For allocation episodes indexed by $t$ and a finite set of resources $R$, a delay kernel for $r\in R$ is

$$K^r:\{0,1,\dots,T-1\}\rightarrow[0,1],\qquad \sum_{\tau=0}^{T-1} K^r(\tau)=1$$

with $K^r(\tau)$ representing the fraction of outcome attributed at lag $\tau$. In the framework of bi-level contextual bandits for individualized resource allocation (Almasi et al., 13 Nov 2025), each $K^r$ is constructed by discretizing the unit interval with a Beta distribution: $$K^r(\tau) = \int_{\frac{\tau}{T}}^{\frac{\tau+1}{T}} \mathrm{Beta}(z;\,\alpha^r,\beta^r) dz$$ where $\mathrm{Beta}(z;\alpha,\beta)$ is the standard beta density. The choice of parameters $(\alpha^r, \beta^r)$ imparts flexibility in modeling early, delayed, or unimodal feedback patterns.

Alternative constructions arise in queueing-theoretic scheduling and service systems, where delay kernels encapsulate steady-state expected waiting time as a nonlinear function of resource parameters, e.g., in distributed M/M/1 or M/G/1 systems the delay kernel is given by $E[W]$ as a function of arrival rate, service rate, and system state (Gamarnik et al., 2017, Spadaccino et al., 24 Sep 2025).

2. Interpretation and Modeling of Feedback Delays

The semantics of a delay kernel are context-dependent:

• In individualized intervention settings, $K^r(\tau)$ is the fraction of total impact from a single resource allocation of type $r$ that arrives exactly $\tau$ rounds later (Almasi et al., 13 Nov 2025). Early, late, and peaked lags are modeled by tuning $(\alpha^r,\beta^r)$.
• In distributed service systems, the delay kernel expresses the asymptotic or steady-state queueing delay per job, as determined by the interplay of memory, messaging rates, and resource capacity (Gamarnik et al., 2017).
• In multiuser wireless scheduling, the effective capacity framework yields a kernel mapping user bandwidth to the exponential decay rate of the delay violation probability (Khalek et al., 2013).

The delay kernel thus generalizes to any mechanism by which resource allocations are "smeared" over time or jobs, analytically capturing the distribution of realized impact.

3. Integration into Learning, Allocation, and Optimization Frameworks

Delay kernels enter optimization objectives as convolutional operators over the allocation sequence: $$y(t) = \sum_{u=1}^t \sum_{i\in\mathcal I_h(u)} \sum_{r\in R} K^r(t-u)\,f(\mathbf x^i(u))\,z_{i,r}(u)$$ where $f(\mathbf x^i(u))$ is a context-based impact predictor and $z_{i,r}(u)$ denotes allocation (Almasi et al., 13 Nov 2025). The cumulative objective, typically

$$\max_{\{z_{i,r}(t)\}} \mathbb{E}\left[\sum_{t=1}^T y(t)\right]$$

enforces constraints (budgets, cooldowns, cohort size) and thereby necessitates explicit modeling of temporal reward propagation.

In queue-based systems, the optimization target is the asymptotic delay kernel as a function of system parameters—memory size, message rates, and queue state: \begin{center} \begin{tabular}{l|l} Memory, Messaging & Delay Kernel $\delta$ \ \hline $\Theta(\log n),\,\omega(n)$ & $0$\ $\omega(\log n),\,\ge\lambda n$ & $0$\

$\gamma\log n,\,\alpha n;\lambda$ & $$\lambda\left[1-\lambda+\sum_{k=1}^\gamma (\alpha/\lambda)^k\right]^{-1}$$

\end{tabular} \end{center} as in (Gamarnik et al., 2017). In AI-intensive distributed applications, delay kernels for "Guaranteed-Resource" and "Shared-Resource" models become nonlinear constraint terms linking resource allocation to violation of end-to-end latency SLAs (Spadaccino et al., 24 Sep 2025).

4. Estimation and Adaptation of Delay Kernels

Parameterization of delay kernels follows two approaches:

• Domain-driven selection: Kernel shapes are fixed a priori using historical or expert knowledge about expected impact timing, as in (Almasi et al., 13 Nov 2025), where $(\alpha^r,\beta^r)$ are not learned but rather imposed to reflect plausible reward dynamics.
• Statistical estimation: While no explicit online procedure is implemented in (Almasi et al., 13 Nov 2025), a canonical likelihood-based estimator for kernel parameters is

$$\widehat{\theta}^r = \arg\min_{\alpha,\beta} -\sum_{\Delta t\;\text{observed}} \log K^r_{\alpha,\beta}(\Delta t)$$

reflecting maximum-likelihood estimation from observed delay lags. Adaptive or meta-learned kernel estimation is cited as possible via related meta-learning methodologies.

In queueing systems, kernel parameters arise directly from measurable physical or protocol characteristics (e.g., arrival rates, memory or message rates, user bandwidth), and are thus inferred from these exogenous variables (Gamarnik et al., 2017, Spadaccino et al., 24 Sep 2025, Khalek et al., 2013).

5. Algorithmic Roles and Policy Updates

Delay kernels enter algorithmic workflows both explicitly and implicitly:

• In bi-level contextual bandits (Almasi et al., 13 Nov 2025), the kernel enters reward aggregation, affecting all feedback received by the predictive model $f$, and thus altering UCB-based allocation and subsequent policy refinement in constrained bandit procedures.
• In distributed resource systems (Gamarnik et al., 2017), the kernel's analytic form directly determines the optimal regime for vanishing delay, shapes the phase transition under resource constraints, and differentiates among policies (e.g., pull-based, random assignment, power-of-d-choices).
• In networked AI application orchestration (Spadaccino et al., 24 Sep 2025), the kernels are embedded as nonlinear constraints coupling allocation variables with system-level latency, driving the need for iterative, bi-convex alternation and convexification in optimization (e.g., upper-bounding utilization to admit convex surrogates for non-convex delay terms).

6. Empirical Findings and Impact of Kernel Shapes

Empirical results highlight pronounced effects of kernel shape:

• Peaked (Type I) kernels: Early-accumulating, unimodal kernels improve adaptation rate and reduce regret in delayed-reward bandits; even delay-agnostic algorithms perform moderately well if most feedback arrives quickly, but delay-aware models outperform via explicit knowledge of $K^r$ (Almasi et al., 13 Nov 2025).
• Long-tailed (Type II) kernels: Spread-out or heavy-tailed kernels slow learning and degrade performance in delay-agnostic methods (UCB, EXP3), whereas delay-aware policies (such as MetaCUB) maintain robustness and comparatively low regret.
• In queueing and distributed scheduling, sharp threshold and blowup phenomena are analytically predicted by the kernel, yielding phase transitions in steady-state delay as load, memory, or message rate cross critical thresholds (Gamarnik et al., 2017).

Empirical analysis in SPARQ (Spadaccino et al., 24 Sep 2025) further demonstrates that incorporating nonlinear kernels for "shared resource" leads to lower operational cost and SLA adherence compared to policies that assume linear or "private queue" delays.

7. Broader Applications and Theoretical Implications

Resource-specific delay kernels unify frameworks for handling delayed feedback across several domains:

• Personalized intervention and social resource allocation: Algorithms learn to optimize policies under complex, temporally nonlocal reward structures, accommodating heterogeneity in individual responsiveness and system delay (Almasi et al., 13 Nov 2025).
• Distributed cloud and edge computing: The contrast between guaranteed and shared resource models reveals fundamental trade-offs in provisioning, queuing delay, and statistical multiplexing, with the kernel capturing the risk of nonlinear latency blowup (Spadaccino et al., 24 Sep 2025).
• Wireless multiuser scheduling: The effective-capacity-based delay kernel links bandwidth, delay violation probability, and video quality, directly informing both rate allocation and admission control (Khalek et al., 2013).

By formalizing the propagation of resource impact over time, delay kernels facilitate design, analysis, and optimization of algorithms sensitive to delayed, distributed, or non-immediate feedback in complex stochastic systems.