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Time-Varying Optimization for Streaming Data Via Temporal Weighting

Muhammad Faraz Ul Abrar, Nicolò Michelusi, Erik G. Larsson

TL;DR

The paper addresses learning from streaming data under time-varying objectives by introducing a weight-based formulation where the objective at time $t$ is a weighted sum $F_t(w)=\sum_{i=1}^t a_i(t) f_i(w)$. It analyzes gradient-descent tracking of the time-varying minimizer under two weighting schemes—uniform and geometrically discounted—deriving sharp tracking-error bounds: a vanishing TE with rate $O(1/t)$ for uniform weights, and a non-vanishing asymptotic error floor for discounted weights that depends on the discount factor $\gamma$ and the per-time-step updates $E$. The results provide explicit expressions for the asymptotic tracking error (ATE) under discounting and show how the update budget $E$ and weight choice trade off responsiveness against accuracy. Numerical experiments with scalar quadratic losses corroborate the theory and illustrate practical implications for streaming and continual learning applications, including guidance on selecting $\gamma$ and the update budget to achieve desired accuracy.

Abstract

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.

Time-Varying Optimization for Streaming Data Via Temporal Weighting

TL;DR

The paper addresses learning from streaming data under time-varying objectives by introducing a weight-based formulation where the objective at time is a weighted sum . It analyzes gradient-descent tracking of the time-varying minimizer under two weighting schemes—uniform and geometrically discounted—deriving sharp tracking-error bounds: a vanishing TE with rate for uniform weights, and a non-vanishing asymptotic error floor for discounted weights that depends on the discount factor and the per-time-step updates . The results provide explicit expressions for the asymptotic tracking error (ATE) under discounting and show how the update budget and weight choice trade off responsiveness against accuracy. Numerical experiments with scalar quadratic losses corroborate the theory and illustrate practical implications for streaming and continual learning applications, including guidance on selecting and the update budget to achieve desired accuracy.

Abstract

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 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.
Paper Structure (7 sections, 6 theorems, 48 equations, 3 figures)

This paper contains 7 sections, 6 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Tracking Error Analysis
  4. Uniform Weights
  5. Discounted Weights
  6. Numerical Results
  7. Conclusion

Key Result

Proposition 1

Define $S(t)\;\triangleq\;\sum_{i=1}^{t-1}\frac{\alpha^{t-i}}{\,i+1\,}\,,$ and let $A=\max\{t_0 S(t_0), \frac{2\alpha}{1-\alpha}\}$, with $t_0 = \lceil \frac{2\alpha}{1-\alpha}\rceil$. Then for all $t\geq t_0$,

Figures (3)

  • Figure 1: Root mean squared (solid lines) and maximum tracking error (dashed lines with markers) vs. iterations for uniform weights, $\mu=0.1, \eta=2$.
  • Figure 2: Root mean squared (solid lines) and maximum tracking error (dashed lines with markers) vs. iterations for discounted weights, $\gamma = 0.7, \mu=0.1, \eta=2.85$.
  • Figure 3: Root mean squared TE vs. iterations for varying discount factor $\gamma$.

Theorems & Definitions (15)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Remark 4
  • Proposition 3
  • Lemma 1
  • ...and 5 more