资源自适应随机梯度下降用于无需重新求解的在线线性规划

中文摘要 AI 辅助

大型语言模型（LLM）推理和搜索服务的增长增加了在线线性规划问题的规模，促使了计算高效算法的需求。我们开发了用于随机在线线性规划的资源自适应随机梯度下降（RASGD）算法。该算法利用一个请求和当前库存来更新资源价格，每个到达需要O(m)次操作（m为资源数量）和内存，且无需LP或样本平均优化。核心思想是通过一阶SGD更新来表达重新求解的当前资源定价逻辑：每次到达都会刷新对偶目标中的剩余库存配额，而步长在早期学习阶段递减，后期递增以匹配库存调整的速度。在标准的非退化条件下，我们的算法在每条样本路径上都是可行的，并且相对于实现的分数事后最优解实现了O(log T)的期望遗憾，这匹配了下界，即使对于知道分布且具有无限制计算的策略也是如此。分析将固定参考价格周围的曲率转化为库存稳定性，而无需跟踪不同资源水平下的最优价格。数值实验表明，RASGD实现了与每次到达LP重新求解相竞争的遗憾，并改进了测试的一阶基线方法，同时保持了一阶方法的计算效率。这些结果确立了RASGD作为在大规模OLP中实现高分配质量的计算高效方法。

英文摘要

The growth of large language model (LLM) inference and search services increases the scale of online linear programming problems, motivating computationally efficient algorithms. We develop resource-adaptive stochastic gradient descent (RASGD) for stochastic online linear programming. The algorithm uses one request and current inventory to update resource prices, requiring O(m) operations for m resources and memory per arrival and no LP or sample-average optimization. The central idea is to express the current-resource pricing logic of re-solving through a first-order SGD update: each arrival refreshes the remaining-inventory allowance in the dual objective, while the stepsize decreases for early learning and increases later to match the speed of inventory adjustment. Under standard non-degeneracy conditions, our algorithm is feasible on every sample path and achieves O(\log T) expected regret against the realized fractional hindsight optimum, which matches the lower bound, even for policies that know the distribution and have unrestricted computation. The analysis converts curvature around the fixed reference price into inventory stability without tracking optimal prices at changing resource levels. Numerical experiments show that RASGD achieves regret competitive with per-arrival LP re-solving and improves upon the tested first-order baselines, while retaining the computational efficiency of first-order methods. These results establish RASGD as a computationally efficient approach to achieving high allocation quality in large-scale OLP.

发表机构

• Fudan University（复旦大学）

机构由 AI 辅助整理，请以论文原文为准。