Online Allocation with Prediction

• Online allocation with prediction is an approach that integrates machine-learned forecasts into sequential resource assignments to closely approximate offline optimality.
• The methodology blends threshold-based sampling, learning-augmented primal-dual techniques, and hybrid strategies that adapt between prediction trust and worst-case safeguards.
• Empirical results in areas such as ad allocation, healthcare, and cloud scheduling demonstrate robust performance with graceful degradation under increased prediction error.

An online allocation algorithm with prediction is an algorithmic framework for assigning limited resources to sequentially arriving, potentially heterogeneous demands in settings where some auxiliary, potentially imperfect predictions about future demand, rewards, or side information are available prior to or during allocation. This class of algorithms explicitly incorporates these predictions—often generated via machine learning or direct sampling—into the online decision-making process to improve performance compared to pure adversarial or purely stochastic baselines. The core objective is to leverage prediction to approach offline optimality under plausible models, while retaining robust worst-case guarantees when predictions are inaccurate or adversarially misleading.

1. Formal Problem Setting

At its core, the online allocation problem requires irrevocable resource assignment decisions as requests arrive, subject to exogenous constraints (budgets, capacities, fairness), with the offline optimum benchmark defined as the maximum feasible total reward had all demand been revealed in advance. The canonical problem formulations include:

• Multi-unit resource allocation with sampled predictions: Each of N agents (or requests) has an unknown reward $v_t$; $m$ units of a resource are to be allocated across these agents in adversarial or stochastic arrival order. A prediction is obtained via a “test period” that samples each agent/value with some probability $p$, providing direct information about a subset of the rewards before the online allocation phase (Gorlezaei et al., 2022).
• Online matching/budgeted allocation: Ad impressions or items arrive, with each having known or revealed values/bids to a set of buyers/advertisers under offline budget constraints; predicted matching or price vector is provided, either as an explicit dual variable, a suggested matching, or shadow-price vector (Kevi et al., 2024, Spaeh et al., 2023, Golrezaei et al., 2023, An et al., 2024).
• Variants: Generalizations include chance-constrained allocations under stochastic resource consumption predictions (Chen et al., 2023, Chen et al., 2022), fair division and Nash social welfare maximization with predicted agent priorities (Banerjee et al., 2020, Melissourgos et al., 6 Aug 2025), or online VM allocation with aggregate workload demand forecasts (Buchbinder et al., 2020).

Integrating predictions can take the form of direct value or demand forecasts, predicted allocation policies, or machine-generated dual (shadow price) advice.

2. Algorithmic Paradigms and Integration of Predictions

The principal algorithmic templates for online allocation with prediction blend prediction-derived structures with classical online optimization algorithms. The main approaches are:

1. Threshold-based policies based on sampled predictions: Use an initial sample to estimate critical reward quantiles, setting a threshold for acceptance/rejection in the online phase. For instance, set the threshold to the $k$-th largest sample reward (where $k=\lceil m/p \rceil$ if $m$ is the total budget and $p$ the sample rate), and accept any online agent with $v_t \geq \tau$ until depletion (Gorlezaei et al., 2022).
2. Learning-augmented primal-dual algorithms: Embed predictions (e.g., suggested matchings, dual prices, or resource allocations) as advice into the classical primal-dual framework. The online update rules interpolate between following the advice and defaulting to adversarially robust policies. Tuning parameters (e.g., confidence or trust $\eta$) balance consistency (with the prediction) and robustness (to adversarial input) (Kevi et al., 2024, Spaeh et al., 2023, Golrezaei et al., 2023).
3. Hybrid algorithms for mixed stochastic/adversarial models: Fuse prediction-driven routines with fallback adversarial algorithms, adaptively shifting between them according to observed deviations from the forecasted model (such as via statistical distance or error triggers) (Esfandiari et al., 2017, Balseiro et al., 2022, An et al., 2024).
4. Bayesian approaches: Use probabilistic models (e.g., Bayesian neural networks) for predictive estimation of job characteristics (like processing times and next activities) and inject this into combinatorial optimization subproblems (such as min-cost max-flow) for resource allocation, as discussed for business process monitoring (Park et al., 2019).
5. Prediction-corrected online reductions: For online bin-packing and related problems, predictions inform partitioning or load thresholds, improving competitive ratios versus non-clairvoyant policies (Buchbinder et al., 2020).

3. Performance Guarantees: Consistency and Robustness

A central organizing principle is the explicit trade-off between two guarantees:

• Consistency: Performance relative to the predictor in the case of high prediction accuracy (e.g., achieving $(1-\eta)P(I)$ for predictor value $P(I)$, or $C$ as a consistent ratio).
• Robustness: Competitive ratio relative to the true offline optimum $O(I)$ in the face of prediction error (e.g., maintaining at least $r(\eta)O(I)$, where $r(\eta)$ transitions smoothly from offline optimality to the best-known worst-case competitive ratio).

This balance can be formalized by a continuum of algorithms parameterized by a trust level $\eta$ or consistency target $C$:

Parameter	Consistent Ratio $c(\eta)$	Robustness $r(\eta)$	Description
$\eta=0$	1	$1$	Always follow the prediction
$0<\eta<1$	$1-\eta$	Interpolates	Mix of following prediction and fallback rule
$\eta=1$	best possible w/o pred.	worst-case CR	Ignore prediction; recovers prior CR

For example, the learning-augmented online ad allocation achieves $$\mathrm{ALG} \ge \max\{R(\alpha)\mathrm{OPT},C(\alpha)\mathrm{PRD}\}$$, where $R(\alpha)$ interpolates from $1-1/e$ (worst-case) to 1 (best-case), and $C(\alpha)$ transitions from 0 to 1 as $\alpha$ increases (Spaeh et al., 2023). Similar interpolations are proved in (Kevi et al., 2024, Golrezaei et al., 2023).

4. Representative Algorithmic Schemes and Theoretical Results

4.1. Threshold Algorithm with Prediction (Sampling Model)

Given a sample $\Sample$ of size $\ell$ (from initial test period, sampling rate $p$), sort sample rewards $v_{(1)} \ge ... \ge v_{(\ell)}$. Set $k = \lceil m/p \rceil$ and threshold $\tau = v_{(k)}$. In the online phase, accept agents with $v_t \ge \tau$ until $m$ are chosen. The competitive ratio is $1 - \Theta(1/(p\sqrt{m}))$, with the additive term vanishing as $m\to\infty$ or $p\to1$ (Gorlezaei et al., 2022).

4.2. Learning-Augmented Primal-Dual (Matching/Allocation)

Maintain dual variables and fractional allocations. At each step, compare a prediction-based assignment (from predicted duals or a matching) with the classical primal-dual selection, and assign according to a confidence parameter or empirical success of the prediction so far. The performance guarantee is (for instance): $A(I) \ge \max\{(1-\eta)P(I), r(\eta)O(I)\}$ where $r(\eta)$ interpolates between 1 and the classical competitive ratio (e.g., $(e-1)/e$) as $\eta$ goes from 0 to 1 (Kevi et al., 2024).

4.3. Hybrid Routine for Robust Stochastic/Adversarial Models

For models incorporating both forecastable and adversarial spikes, run a derandomized allocation strategy based on the best forecasted distribution, and divert to a fallback adversarial algorithm (e.g., BALANCE) whenever observed arrivals deviate (in total variation or other distance) from forecast (Esfandiari et al., 2017). The main result: the competitive ratio is at least

$$R(\lambda) = \lambda + (1-\lambda)^2 (1-1/e) - O(\epsilon)$$

where $\lambda$ measures the fraction of offline optimum contributed by the predictable bulk.

4.4. Predictions as Dual (Shadow-Price) Advice

For general convex online allocation, predictions are given as shadow prices $\hat\mu$, and a mirror descent primal-dual algorithm is initialized or guided by $\hat\mu$. If prediction error is small, regret scales as $O(T^{1/2-a})$; otherwise, the algorithm detects divergence and falls back to a (nearly optimal) adversarial mode, ensuring best-of-both-worlds guarantees (An et al., 2024).

5. Applications and Empirical Findings

• Healthcare/capacity allocation: Online hospital bed assignment with predicted disease severity, using test-period sampling to form the acceptance threshold, recovers over 98% of the offline optimum for moderate sample rates and large $m$ (Gorlezaei et al., 2022).
• Ad allocation: Real-world and synthetic data show robust improvement of prediction-augmented primal-dual over pure worst-case for moderate-to-good predictors; performance degrades gracefully as prediction quality decreases (Spaeh et al., 2023, Kevi et al., 2024).
• Cloud scheduling: Average-load and full-load-vector predictions improve competitive ratios—from $(1.69 + o(1))$ in the average case to constant-factor optimality when full load-vector forecast is available and precise (Buchbinder et al., 2020).
• Fair division and Nash social welfare: Prediction-aware allocation algorithms dramatically improve log-factor competitive ratios compared to linear or worse baseline in adversarially difficult settings (Banerjee et al., 2020, Banerjee et al., 2022, Melissourgos et al., 6 Aug 2025).

6. Limits, Hardness, and Robustness to Prediction Error

All frameworks rigorously characterize the smooth degradation from near-offline to worst-case performance as prediction error increases. Notably:

• For any fixed sample rate $p$, the $1/(p\sqrt{m})$ loss in (Gorlezaei et al., 2022) is tight even for the best possible algorithm (adversarial lower bound).
• In primal-dual frameworks, for adversarially bad predictions, the algorithm default to the best known competitive ratio (e.g., $1-1/e$ for AdWords), retaining adversarial robustness (Kevi et al., 2024, Spaeh et al., 2023).
• In fair allocation (e.g., EFX), approximate EFX can never be guaranteed unless prediction error, measured in total variation, is below a sharp threshold; even small error can preclude fairness guarantees for $n\geq 3$ agents in the online setting (Melissourgos et al., 6 Aug 2025).

7. Research Directions and Open Problems

Recent works identify several open problems and extensions:

• Designing allocation algorithms that can interpolate gracefully on Pareto-optimal consistency/robustness frontiers for multi-resource and multi-fare-class settings (Golrezaei et al., 2023).
• Leveraging richer predictions (such as joint distribution forecasts, correlated demand, or full load-vectors) while maintaining rigorous worst-case guarantees.
• Combining prediction quality estimation and robustness dynamically online, particularly as forecast accuracy changes non-stationarily (An et al., 2024).
• Broadening the application scope to other online allocation primitives, including chance-constrained, fairness-constrained, or Nash social welfare objectives with predictions.

• "Online Resource Allocation with Samples" (Gorlezaei et al., 2022)
• "Primal-Dual Algorithms with Predictions for Online Bounded Allocation and Ad-Auctions Problems" (Kevi et al., 2024)
• "Online Ad Allocation with Predictions" (Spaeh et al., 2023)
• "Online Resource Allocation with Convex-set Machine-Learned Advice" (Golrezaei et al., 2023)
• "Online Allocation with Traffic Spikes: Mixing Adversarial and Stochastic Models" (Esfandiari et al., 2017)
• "Online Virtual Machine Allocation with Predictions" (Buchbinder et al., 2020)
• "Online Nash Social Welfare Maximization with Predictions" (Banerjee et al., 2020)
• "Online EFX Allocations with Predictions" (Melissourgos et al., 6 Aug 2025)
• "Best of Many in Both Worlds: Online Resource Allocation with Predictions under Unknown Arrival Model" (An et al., 2024)
• "Online Resource Allocation under Horizon Uncertainty" (Balseiro et al., 2022)
• "Proportionally Fair Online Allocation of Public Goods with Predictions" (Banerjee et al., 2022)
• "Online Algorithm for Chance Constrained Resource Allocation" (Chen et al., 2023)