### Abstract

A framework is introduced for sequentially solving convex stochastic minimization problems, where the objective functions change slowly, in the sense that the distance between successive minimizers is bounded. The minimization problems are solved by sequentially applying a selected optimization algorithm, such as stochastic gradient descent, based on drawing a number of samples in order to carry the iterations. Two tracking criteria are introduced to evaluate approximate minimizer quality: one based on being accurate with respect to the mean trajectory, and the other based on being accurate in high probability. An estimate of a bound on the minimizers' change, combined with properties of the chosen optimization algorithm, is used to select the number of samples needed to meet the desired tracking criterion. A technique to estimate the change in minimizers is provided along with analysis to show that eventually the estimate upper bounds the change in minimizers. This estimate of the change in minimizers provides sample size selection rules that guarantee that the tracking criterion is met for sufficiently large number of time steps. Simulations are used to confirm that the estimation approach provides the desired tracking accuracy in practice, while being efficient in terms of number of samples used in each time step.

Original language | English (US) |
---|---|

Article number | 8316928 |

Pages (from-to) | 496-509 |

Number of pages | 14 |

Journal | IEEE Transactions on Automatic Control |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2019 |

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### Keywords

- Gradient methods
- stochastic optimization
- time-varying objective
- tracking problems

### ASJC Scopus subject areas

- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Automatic Control*,

*64*(2), 496-509. [8316928]. https://doi.org/10.1109/TAC.2018.2816168

**Adaptive Sequential Stochastic Optimization.** / Wilson, Craig; Veeravalli, Venugopal V.; Nedic, Angelia.

Research output: Contribution to journal › Article

*IEEE Transactions on Automatic Control*, vol. 64, no. 2, 8316928, pp. 496-509. https://doi.org/10.1109/TAC.2018.2816168

}

TY - JOUR

T1 - Adaptive Sequential Stochastic Optimization

AU - Wilson, Craig

AU - Veeravalli, Venugopal V.

AU - Nedic, Angelia

PY - 2019/2

Y1 - 2019/2

N2 - A framework is introduced for sequentially solving convex stochastic minimization problems, where the objective functions change slowly, in the sense that the distance between successive minimizers is bounded. The minimization problems are solved by sequentially applying a selected optimization algorithm, such as stochastic gradient descent, based on drawing a number of samples in order to carry the iterations. Two tracking criteria are introduced to evaluate approximate minimizer quality: one based on being accurate with respect to the mean trajectory, and the other based on being accurate in high probability. An estimate of a bound on the minimizers' change, combined with properties of the chosen optimization algorithm, is used to select the number of samples needed to meet the desired tracking criterion. A technique to estimate the change in minimizers is provided along with analysis to show that eventually the estimate upper bounds the change in minimizers. This estimate of the change in minimizers provides sample size selection rules that guarantee that the tracking criterion is met for sufficiently large number of time steps. Simulations are used to confirm that the estimation approach provides the desired tracking accuracy in practice, while being efficient in terms of number of samples used in each time step.

AB - A framework is introduced for sequentially solving convex stochastic minimization problems, where the objective functions change slowly, in the sense that the distance between successive minimizers is bounded. The minimization problems are solved by sequentially applying a selected optimization algorithm, such as stochastic gradient descent, based on drawing a number of samples in order to carry the iterations. Two tracking criteria are introduced to evaluate approximate minimizer quality: one based on being accurate with respect to the mean trajectory, and the other based on being accurate in high probability. An estimate of a bound on the minimizers' change, combined with properties of the chosen optimization algorithm, is used to select the number of samples needed to meet the desired tracking criterion. A technique to estimate the change in minimizers is provided along with analysis to show that eventually the estimate upper bounds the change in minimizers. This estimate of the change in minimizers provides sample size selection rules that guarantee that the tracking criterion is met for sufficiently large number of time steps. Simulations are used to confirm that the estimation approach provides the desired tracking accuracy in practice, while being efficient in terms of number of samples used in each time step.

KW - Gradient methods

KW - stochastic optimization

KW - time-varying objective

KW - tracking problems

UR - http://www.scopus.com/inward/record.url?scp=85043775343&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85043775343&partnerID=8YFLogxK

U2 - 10.1109/TAC.2018.2816168

DO - 10.1109/TAC.2018.2816168

M3 - Article

AN - SCOPUS:85043775343

VL - 64

SP - 496

EP - 509

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

SN - 0018-9286

IS - 2

M1 - 8316928

ER -