We address the challenge of online convex optimization where the objective function's gradient exhibits sparsity, indicating that only a small number of dimensions possess non-zero gradients. Our aim is to leverage this sparsity to obtain useful estimates of the objective function's gradient even when the only information available is a limited number of function samples. Our motivation stems from distributed queueing systems like microservices-based applications, characterized by request-response workloads. Here, each request type proceeds through a sequence of microservices to produce a response, and the resource allocation across the collection of microservices is controlled to balance end-to-end latency with resource costs. While the number of microservices is substantial, the latency function primarily reacts to resource changes in a few, rendering the gradient sparse. Our proposed method, CONGO (Compressive Online Gradient Optimization), combines simultaneous perturbation with compressive sensing to estimate gradients. We establish analytical bounds on the requisite number of compressive sensing samples per iteration to maintain bounded bias of gradient estimates, ensuring sub-linear regret. By exploiting sparsity, we reduce the samples required per iteration to match the gradient's sparsity, rather than the problem's original dimensionality. Numerical experiments and real-world microservices benchmarks demonstrate CONGO's superiority over multiple stochastic gradient descent approaches, as it quickly converges to performance comparable to policies pre-trained with workload awareness.

翻译：本文研究在线凸优化问题，其中目标函数的梯度具有稀疏性，即仅有少数维度具有非零梯度。我们的目标是在仅能获取有限数量函数样本的情况下，利用这种稀疏性获得目标函数梯度的有效估计。研究动机源于分布式队列系统（如基于微服务的应用），这类系统以请求-响应型工作负载为特征。在此类系统中，每种请求类型需经过一系列微服务处理以生成响应，通过控制微服务集群的资源分配来实现端到端延迟与资源成本的平衡。虽然微服务数量庞大，但延迟函数仅对少数服务的资源变化敏感，从而导致梯度具有稀疏性。我们提出的CONGO（压缩在线梯度优化）方法将同步扰动与压缩感知相结合以估计梯度。我们通过理论分析确定了每轮迭代所需压缩感知样本数量的界限，以保证梯度估计的偏差有界，从而实现次线性遗憾。通过利用稀疏性，我们将每轮迭代所需样本量降至与梯度稀疏度相匹配的水平，而非原始问题维度。数值实验与真实微服务基准测试表明，CONGO相较于多种随机梯度下降方法具有显著优势，能够快速收敛至与工作负载感知预训练策略相当的性能水平。