Time-Varying Optimization for Streaming Data Via Temporal Weighting
This work addresses the challenge of decision-making in dynamic environments for machine learning and optimization practitioners, but it is incremental as it builds on existing time-varying optimization frameworks with specific weighting strategies.
The paper tackles the problem of learning from streaming data by formulating it as time-varying optimization with weighted average losses, deriving tight bounds on tracking error for uniform and discounted weighting strategies under gradient descent updates, showing that uniform weighting achieves asymptotic convergence with O(1/t) decay while discounted weighting incurs a nonzero error floor.
Classical optimization theory deals with fixed, time-invariant objective functions. However, time-varying optimization has emerged as an important subject for decision-making in dynamic environments. In this work, we study the problem of learning from streaming data through a time-varying optimization lens. Unlike prior works that focus on generic formulations, we introduce a structured, \emph{weight-based} formulation that explicitly captures the streaming-data origin of the time-varying objective, where at each time step, an agent aims to minimize a weighted average loss over all the past data samples. We focus on two specific weighting strategies: (1) uniform weights, which treat all samples equally, and (2) discounted weights, which geometrically decay the influence of older data. For both schemes, we derive tight bounds on the ``tracking error'' (TE), defined as the deviation between the model parameter and the time-varying optimum at a given time step, under gradient descent (GD) updates. We show that under uniform weighting, the TE vanishes asymptotically with a $\mathcal{O}(1/t)$ decay rate, whereas discounted weighting incurs a nonzero error floor controlled by the discount factor and the number of gradient updates performed at each time step. Our theoretical findings are validated through numerical simulations.