[Paper Review] Implicit stochastic gradient descent

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

Stochastic Gradient Optimization Techniques84 references3 citations

TL;DR

This paper proposes implicit stochastic gradient descent (ISGD), a numerically stable and statistically efficient optimization method for large-scale parameter estimation. By implicitly defining iterates that shrink standard updates based on observed Fisher information—without explicit computation—ISGD enhances stability and efficiency, especially when combined with averaging, and achieves optimal asymptotic properties across generalized linear models, Cox models, and M-estimation.

ABSTRACT

Stochastic optimization procedures, such as stochastic gradient descent, have gained popularity for parameter estimation from large data sets. However, standard stochastic optimization procedures cannot effectively combine numerical stability with statistical and computational efficiency. Here, we introduce an implicit stochastic gradient descent procedure, the iterates of which are implicitly defined. Intuitively, implicit iterates shrink the standard iterates. The amount of shrinkage depends on the observed Fisher information matrix, which does not need to be explicitly computed in practice, thus increasing stability without increasing the computational burden. When combined with averaging, the proposed procedure achieves statistical efficiency as well. We derive non-asymptotic bounds and characterize the asymptotic distribution of implicit procedures. Our analysis also reveals the asymptotic variance of a number of existing procedures. We demonstrate implicit stochastic gradient descent by further developing theory for generalized linear models, Cox proportional hazards, and M-estimation problems, and by carrying out extensive experiments. Our results suggest that the implicit stochastic gradient descent procedure is poised to become the workhorse of estimation with large data sets.

Motivation & Objective

• Address the trade-off between numerical stability and statistical efficiency in stochastic optimization for large datasets.
• Develop a stochastic gradient method that maintains stability without increasing computational cost.
• Achieve statistical efficiency through implicit updates that adaptively shrink standard iterates.
• Characterize the asymptotic distribution and non-asymptotic bounds of the proposed procedure.
• Extend the method to generalized linear models, Cox proportional hazards, and M-estimation with theoretical and empirical validation.

Proposed method

• Introduce an implicit update rule where each iterate is defined as the solution to a fixed-point equation involving the gradient and observed Fisher information.
• Use the observed Fisher information matrix implicitly in the update, avoiding explicit computation and preserving computational efficiency.
• Apply averaging to the iterates to improve statistical efficiency and reduce variance.
• Derive non-asymptotic risk bounds and characterize the asymptotic distribution of the estimator.
• Leverage implicit updates to naturally shrink the step size in high-curvature regions, enhancing numerical stability.
• Generalize the framework to exponential family models, survival models (Cox), and M-estimation problems.

Experimental results

Research questions

• RQ1 Can implicit stochastic gradient descent achieve both numerical stability and statistical efficiency in large-scale estimation?
• RQ2 How does the implicit update rule, based on observed Fisher information, affect convergence and variance compared to standard SGD?
• RQ3 What are the non-asymptotic and asymptotic properties of the implicit estimator across different statistical models?
• RQ4 To what extent does averaging improve the efficiency of the implicit procedure?
• RQ5 How does the method perform in practice on generalized linear models, Cox proportional hazards, and M-estimation problems?

Key findings

• The implicit stochastic gradient descent procedure achieves superior numerical stability by implicitly shrinking updates based on local curvature via observed Fisher information.
• The method maintains computational efficiency since the Fisher information is not explicitly computed in the update rule.
• When combined with averaging, ISGD achieves statistical efficiency comparable to full-likelihood estimators.
• Non-asymptotic risk bounds are derived, showing improved convergence behavior under mild regularity conditions.
• The asymptotic distribution of the estimator is characterized, and the asymptotic variance of existing procedures is revealed through the framework.
• Empirical results demonstrate strong performance across generalized linear models, Cox models, and M-estimation, supporting its potential as a standard tool for large-scale inference.