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Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysis

Seunghoon Paik, Kangjie Zhou, Matus Telgarsky, Ryan J. Tibshirani

TL;DR

This work develops a unified framework of basic inequalities that tie the iteration count of first-order optimization methods to implicit regularization, enabling direct comparisons with explicit penalties. By deriving bounds that relate $f( heta_T)-f(z)$ to accumulated step sizes and geometric distances, the authors connect finite-time dynamics to regularization strength across gradient descent, mirror descent, and KL-based schemes, and extend the analysis to generalized linear models and model-aggregation tasks. They provide training envelopes, asymptotic training behavior, and high-probability risk bounds showing that early-stopped or KL-/KL-based methods can match the statistical performance of traditional explicit regularization under broad design assumptions, including misspecification. The paper also offers empirical validation on GLMs, demonstrates equivalence results in randomized-predictor settings, and discusses extensions to proximal-gradient and NoLips algorithms, underscoring the framework’s versatility for both theory and practice in statistical risk analysis.

Abstract

We introduce \textit{basic inequalities} for first-order iterative optimization algorithms, forming a simple and versatile framework that connects implicit and explicit regularization. While related inequalities appear in the literature, we isolate and highlight a specific form and develop it as a well-rounded tool for statistical analysis. Let $f$ denote the objective function to be optimized. Given a first-order iterative algorithm initialized at $θ_0$ with current iterate $θ_T$, the basic inequality upper bounds $f(θ_T)-f(z)$ for any reference point $z$ in terms of the accumulated step sizes and the distances between $θ_0$, $θ_T$, and $z$. The bound translates the number of iterations into an effective regularization coefficient in the loss function. We demonstrate this framework through analyses of training dynamics and prediction risk bounds. In addition to revisiting and refining known results on gradient descent, we provide new results for mirror descent with Bregman divergence projection, for generalized linear models trained by gradient descent and exponentiated gradient descent, and for randomized predictors. We illustrate and supplement these theoretical findings with experiments on generalized linear models.

Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysis

TL;DR

This work develops a unified framework of basic inequalities that tie the iteration count of first-order optimization methods to implicit regularization, enabling direct comparisons with explicit penalties. By deriving bounds that relate to accumulated step sizes and geometric distances, the authors connect finite-time dynamics to regularization strength across gradient descent, mirror descent, and KL-based schemes, and extend the analysis to generalized linear models and model-aggregation tasks. They provide training envelopes, asymptotic training behavior, and high-probability risk bounds showing that early-stopped or KL-/KL-based methods can match the statistical performance of traditional explicit regularization under broad design assumptions, including misspecification. The paper also offers empirical validation on GLMs, demonstrates equivalence results in randomized-predictor settings, and discusses extensions to proximal-gradient and NoLips algorithms, underscoring the framework’s versatility for both theory and practice in statistical risk analysis.

Abstract

We introduce \textit{basic inequalities} for first-order iterative optimization algorithms, forming a simple and versatile framework that connects implicit and explicit regularization. While related inequalities appear in the literature, we isolate and highlight a specific form and develop it as a well-rounded tool for statistical analysis. Let denote the objective function to be optimized. Given a first-order iterative algorithm initialized at with current iterate , the basic inequality upper bounds for any reference point in terms of the accumulated step sizes and the distances between , , and . The bound translates the number of iterations into an effective regularization coefficient in the loss function. We demonstrate this framework through analyses of training dynamics and prediction risk bounds. In addition to revisiting and refining known results on gradient descent, we provide new results for mirror descent with Bregman divergence projection, for generalized linear models trained by gradient descent and exponentiated gradient descent, and for randomized predictors. We illustrate and supplement these theoretical findings with experiments on generalized linear models.
Paper Structure (88 sections, 39 theorems, 212 equations, 3 figures, 2 tables)

This paper contains 88 sections, 39 theorems, 212 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of contributions.
  3. Definitions, notations, and proofs.
  4. Related work
  5. Implicit regularization.
  6. Explicit regularization.
  7. Early stopping.
  8. KL-divergence regularization.
  9. Basic inequalities
  10. Gradient descent
  11. Mirror descent
  12. Generalized linear models and ridge regularity
  13. Risk analysis: ridge-regularized GLMs
  14. Comparison with existing literature.
  15. Risk analysis: gradient descent on GLMs
  16. ...and 73 more sections

Key Result

Theorem 1

Let a function $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be convex, differentiable, and $L$-smooth for some $L>0$. Consider gradient descent with iterates eq:gd-iterate with step sizes $\eta_t\in(0,1/L]$. Then, for any reference point $z\in\mathbb{R}^d$ and any stopping time $T\in\mathbb{N}$, it holds In particular, for a constant step size $\eta_t = \eta$, this simplifies to

Figures (3)

  • Figure 1: Training envelopes. Rows and columns of each subfigure correspond to three GLM tasks (rows) and two $(n,d)$ regimes (columns). The $x$-axis represents the total elapsed time $\tau$ on a $\log_{10}$ scale. The $y$-axis represents the regularized objective, i.e., the loss plus penalty. There are three lines for GD and four lines for EGD: the red line plots the implicit objective $f(\theta_T) + \frac{1}{4\tau}\|\theta_T\|_2^2$; the green, blue, and orange lines plot the explicit objective $f({\hat{\theta}_\lambda})+\lambda\| {\hat{\theta}_\lambda} \|_2^2$ with $\lambda = 1/(4\tau)$, $\lambda = 1/\tau$, and $\lambda = (d+1)/(2\tau)$.
  • Figure 2: Prediction risk. Rows and columns of each subfigure correspond to three GLM tasks (rows) and two $(n,d)$ regimes (columns). The $x$-axis represents the total elapsed time $\tau$ on a $\log_{10}$ scale. The $y$-axis represents the prediction risk. There are three lines in each subfigure: the red line plots the risk curve of $\theta_T$; the green and blue lines plot the risk curve of ${\hat{\theta}_\lambda}$ with $\lambda = 1/(4\tau)$ and $\lambda = 1/\tau$.
  • Figure 3: Solution paths. Rows represent three GLM tasks. Columns show two parameterization regimes, with $\theta_T$ and ${\hat{\theta}_\lambda}$ in adjacent columns within each regime. The $x$-axis represents the total elapsed time $\tau$ or inverse regularization parameter $1/\lambda$ on a $\log_{10}$ scale. The $y$-axis represents the values of individual components of the estimators. We plot all $d=20$ components for underparameterized regimes and the first $40$ components for overparameterized regimes. Component colors are consistent between $\theta_T$ and ${\hat{\theta}_\lambda}$ for better comparison.

Theorems & Definitions (55)

  • Theorem 1: Basic inequality; gradient descent
  • proof : Proof of Theorem \ref{['thm:gd-basicineq']}
  • Corollary 1: Training envelope; gradient descent
  • Corollary 2: Training dynamics; gradient descent
  • Theorem 2: Basic inequality; mirror descent
  • proof : Proof of Theorem \ref{['thm:md-basicineq']}
  • Corollary 3: Training envelope; mirror descent
  • Corollary 4: Training envelope; exponentiated gradient descent
  • Corollary 5: Training dynamics; mirror descent
  • Proposition 1: Risk bound; ridge-regularized GLM
  • ...and 45 more