scholarly journals Optimal survey schemes for stochastic gradient descent with applications to M-estimation

2019 ◽  
Vol 23 ◽  
pp. 310-337 ◽  
Author(s):  
Stephan Clémençon ◽  
Patrice Bertail ◽  
Emilie Chautru ◽  
Guillaume Papa
Keyword(s):  
Gradient Descent ◽  
A Priori ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Estimation Methods ◽  
Full Data ◽  
Predictive Learning ◽  
Inclusion Probabilities ◽  
Estimation Problems ◽  
The Impact

Iterative stochastic approximation methods are widely used to solve M-estimation problems, in the context of predictive learning in particular. In certain situations that shall be undoubtedly more and more common in the Big Data era, the datasets available are so massive that computing statistics over the full sample is hardly feasible, if not unfeasible. A natural and popular approach to gradient descent in this context consists in substituting the “full data” statistics with their counterparts based on subsamples picked at random of manageable size. It is the main purpose of this paper to investigate the impact of survey sampling with unequal inclusion probabilities on stochastic gradient descent-based M-estimation methods. Precisely, we prove that, in presence of some a priori information, one may significantly increase statistical accuracy in terms of limit variance, when choosing appropriate first order inclusion probabilities. These results are described by asymptotic theorems and are also supported by illustrative numerical experiments.

Online Parameter Estimation of Hydraulic System Based on Stochastic Gradient Descent

2020 ◽  
Author(s):  
Takashi Yamada ◽  
Matthew Howard
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Hydraulic System ◽  
Gradient Descent ◽  
Estimation Error ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Estimation Methods ◽  
Hydraulic Systems ◽  
Parameter Estimation Methods

Abstract In this paper, offline and online parameter estimation methods for hydraulic systems based on stochastic gradient descent are presented. In contrast to conventional approaches, the proposed methods can estimate any parameter in mathematical models based on multi-step prediction error. These advantages are achieved by calculating the gradient of the multi-step error against the estimated parameters using Lagrange multipliers and the calculus of variations, and by forming differentiable models of hydraulic systems. In experiments on a physical hydraulic system, the proposed methods with three different gradient decent methods (normal gradient descent, Nesterov’s Accelerated Gradient (NAG), and Adam) are compared with conventional least squares. In the offline experiment, the proposed method with NAG achieves estimation error about 95% lower than that of least squares. In online estimation, the proposed method with NAG produces predictive models with about 20% lower error than that of the offline method. These results suggest the proposed method is a practical alternative to more conventional parameter estimation methods.

Marrying Stochastic Gradient Descent with Bandits: Learning Algorithms for Inventory Systems with Fixed Costs

2021 ◽  
Author(s):  
Hao Yuan ◽  
Qi Luo ◽  
Cong Shi
Keyword(s):  
Gradient Descent ◽  
Stochastic Systems ◽  
Learning Algorithm ◽  
A Priori ◽  
Performance Measure ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
Fixed Cost ◽  
Periodic Review ◽  
Demand Distribution

We consider a periodic-review single-product inventory system with fixed cost under censored demand. Under full demand distributional information, it is well known that the celebrated (s, S) policy is optimal. In this paper, we assume the firm does not know the demand distribution a priori and makes adaptive inventory ordering decisions in each period based only on the past sales (a.k.a. censored demand). Our performance measure is regret, which is the cost difference between a feasible learning algorithm and the clairvoyant (full-information) benchmark. Compared with prior literature, the key difficulty of this problem lies in the loss of joint convexity of the objective function as a result of the presence of fixed cost. We develop the first learning algorithm, termed the [Formula: see text] policy, that combines the power of stochastic gradient descent, bandit controls, and simulation-based methods in a seamless and nontrivial fashion. We prove that the cumulative regret is [Formula: see text], which is provably tight up to a logarithmic factor. We also develop several technical results that are of independent interest. We believe that the developed framework could be widely applied to learning other important stochastic systems with partial convexity in the objectives. This paper was accepted by Chung Piaw Teo, optimization.

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