[Paper Review] Implicit stochastic gradient descent for principled estimation with large datasets

Panos Toulis, Edoardo M. Airoldi|arXiv (Cornell University)|Aug 13, 2014

Stochastic Gradient Optimization Techniques59 references6 citations

TL;DR

This paper introduces implicit stochastic gradient descent (ISGD), a robust estimation method for large datasets that implicitly defines iterates using a shrinkage mechanism based on observed Fisher information, eliminating the need for manual learning rate tuning. ISGD achieves superior stability and efficiency compared to explicit stochastic gradient descent, especially in generalized linear and exponential family models, with theoretical guarantees on bias, variance, and efficiency loss.

ABSTRACT

Efficient optimization procedures, such as stochastic gradient descent, have been gaining popularity for estimation tasks with large amounts of data. In this paper, we introduce an implicit stochastic gradient descent estimation procedure that ameliorates the procedures derived from stochastic approximations a la Robbins & Monro (1951), termed explicit for contrast, by using iterates that are implicitly defined. The implicit iterates are shrinked versions of the explicit iterates, and it can be shown that the amount of shrinkage depends on the observed Fisher information, but this latter quantity needs not be directly computed. The implicit procedure is thus robust to the choice of a scalar hyper-parameter in stochastic gradient descent, known as the learning rate, that affects its asymptotic statistical properties. In contrast, the explicit procedure requires the learning rate to agree with the eigenvalues of the Fisher information matrix of the underlying model parameters in order to be stable. In the context of generalized linear models, we derive analytic formulas for the asymptotic bias and variance of both procedures as estimation methods, and quantify their efficiency loss compared to maximum likelihood. We also show how loss in efficiency can be avoided through careful choice of the parameterization. Our analysis naturally extends to exponential family models, and to a general class of estimation methods through Monte-Carlo stochastic gradient descent, in problems where the likelihood is hard to compute but where it is easy to sample from the underlying model. We demonstrate our theory in an extensive set of experiments involving real and simulated data. Implicit stochastic gradient descent compares favorably to other popular estimation methods, and it is a superior form of stochastic gradient descent when it can be implemented efficiently. 1 ar

Motivation & Objective

• Address the instability and sensitivity to learning rate in explicit stochastic gradient descent for large-scale estimation tasks.
• Develop a principled estimation procedure that maintains statistical efficiency without requiring precise tuning of hyperparameters.
• Demonstrate that implicit updates naturally incorporate Fisher information, leading to robustness and improved asymptotic properties.
• Extend the framework to exponential family models and Monte-Carlo stochastic gradient descent for intractable likelihoods.
• Provide theoretical and empirical validation of ISGD’s superiority in bias, variance, and efficiency compared to standard stochastic gradient descent.

Proposed method

• Propose an implicit update rule where each iterate is defined implicitly via a fixed-point equation, avoiding explicit learning rate dependence.
• Use a shrinkage mechanism that scales updates based on observed Fisher information, without explicitly computing it.
• Derive analytic expressions for asymptotic bias and variance of ISGD and explicit SGD in generalized linear models.
• Show that the implicit procedure’s shrinkage effect stabilizes convergence, especially when the learning rate is misspecified.
• Extend the method to models with intractable likelihoods using Monte-Carlo sampling to approximate the score function.
• Demonstrate that parameterization choices can eliminate efficiency loss in ISGD, aligning it with maximum likelihood estimation.

Experimental results

Research questions

• RQ1 How does implicit stochastic gradient descent improve stability compared to explicit stochastic gradient descent in large-scale estimation?
• RQ2 What is the theoretical relationship between implicit updates and Fisher information in the context of stochastic optimization?
• RQ3 To what extent does ISGD reduce bias and variance compared to explicit SGD in generalized linear models?
• RQ4 Can ISGD achieve efficiency close to maximum likelihood estimation, and under what parameterization conditions?
• RQ5 How does ISGD perform in models with intractable likelihoods, and can Monte-Carlo approximation preserve its statistical advantages?

Key findings

• Implicit stochastic gradient descent exhibits superior stability because its updates are inherently scaled by observed Fisher information, reducing sensitivity to learning rate choice.
• The asymptotic bias and variance of ISGD are analytically derived and shown to be more favorable than those of explicit SGD, especially when the learning rate is suboptimal.
• Efficiency loss relative to maximum likelihood estimation can be avoided through appropriate parameterization, making ISGD asymptotically efficient.
• In generalized linear models, ISGD achieves lower mean squared error than explicit SGD under various learning rate settings.
• The method extends naturally to exponential family models and Monte-Carlo stochastic gradient descent, maintaining robustness even when likelihoods are intractable.
• Empirical results on real and simulated data confirm that ISGD outperforms standard stochastic gradient descent and other popular estimation methods in terms of convergence and estimation accuracy.