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

Source: arXiv - 2512.24999v1

Overview

The paper introduces a clean, unified way to reason about the behavior of first‑order optimization algorithms (gradient descent, mirror descent, exponentiated gradient, etc.) through what the authors call basic inequalities. These inequalities bound the suboptimality f(θ_T) – f(z) in terms of the algorithm’s step‑size schedule and the geometry of the iterates, turning the number of iterations into an effective regularization term. By making this connection explicit, the authors bridge the gap between optimization dynamics and statistical risk analysis, offering a tool that can be applied across many machine learning pipelines.

Key Contributions

• Basic inequality framework – a simple, general bound that applies to any first‑order method and any reference point z, linking iteration count to an implicit regularizer.
• Unified treatment of implicit vs. explicit regularization – shows how early stopping (implicit) can be interpreted as adding a regularization term to the loss.
• New theoretical results for:
  • Mirror descent with Bregman‑divergence projections.
  • Gradient descent on generalized linear models (GLMs).
  • Exponentiated gradient descent on GLMs.
  • Randomized predictors (e.g., stochastic ensembles).
• Refined analyses of classic gradient descent – tighter constants and clearer dependence on step‑size schedules.
• Empirical validation – experiments on GLMs illustrate how the bounds predict actual training dynamics and test‑time risk.

Methodology

1. Setup – Consider an objective f(θ) (e.g., empirical risk) and a first‑order algorithm that updates θ_t using only gradients (or sub‑gradients).

2. Deriving the basic inequality – Starting from the algorithm’s update rule, the authors manipulate the recursion to obtain

   $$
   f(\theta_T) - f(z) \le \frac{1}{\sum_{t=0}^{T-1}\eta_t}\Bigl( D(z,\theta_0) - D(z,\theta_T) + \sum_{t=0}^{T-1}\eta_t^2 G_t^2 \Bigr),
   $$

   where η_t are step sizes, D is a distance‑generating function (Euclidean distance for GD, Bregman divergence for mirror descent), and G_t bounds the gradient norm.

3. Interpretation as regularization – The denominator ∑ η_t behaves like a regularization strength: larger total step‑size → smaller effective penalty, mirroring early‑stopping effects.

4. Specializing the inequality – By plugging in specific D and η_t choices, the authors recover known results (e.g., GD on strongly convex losses) and derive new bounds for mirror descent and exponentiated gradient.

5. Statistical risk analysis – The inequality is combined with standard concentration tools to translate optimization error into prediction risk guarantees for GLMs and randomized predictors.

The derivation stays at a level that requires only basic calculus and convex analysis, making it approachable for engineers familiar with gradient‑based training loops.

Results & Findings

Empirically, the authors train logistic regression and Poisson regression models on synthetic and real datasets. The observed test error follows the predicted decay as a function of total step‑size, confirming that the basic inequality captures the practical trade‑off between iteration count and regularization strength.

Practical Implications

• Early‑stopping as a design knob – Instead of hand‑crafting ℓ₂/ℓ₁ penalties, practitioners can tune the total learning‑rate budget (∑η_t) to achieve a desired regularization effect. This is especially handy for large‑scale deep learning where adding explicit regularizers can be costly.
• Algorithm choice guided by geometry – The MD bound clarifies when mirror descent (e.g., using KL divergence for probability vectors) will give tighter guarantees than plain GD, informing the selection of optimizers for constrained or simplex‑structured parameters (e.g., attention weights, topic models).
• Hyper‑parameter budgeting – The inequality provides a theoretically grounded way to allocate learning‑rate schedules across epochs, helping to avoid over‑training while still exploiting the full data.
• Randomized ensembles – The result on randomized predictors justifies simple tricks like snapshot ensembling (averaging checkpoints) or Monte‑Carlo dropout as statistically sound regularization mechanisms.
• Diagnostics – By monitoring ‖θ_t‑θ_{t‑1}‖ and the accumulated step‑size, engineers can estimate the implicit regularization strength on‑the‑fly, enabling adaptive early‑stopping criteria.

Overall, the framework equips ML engineers with a principled lens to interpret training dynamics, replace ad‑hoc regularization heuristics, and make more informed optimizer/hyper‑parameter decisions.

Limitations & Future Work

• Assumptions on convexity – The core inequalities are derived for convex (or strongly convex) objectives. Extending the theory to non‑convex deep nets remains an open challenge.
• Gradient norm bounds – The bounds rely on a uniform upper bound G_t for gradient norms, which can be loose in practice, especially for poorly scaled data.
• Step‑size schedules – While the analysis accommodates arbitrary η_t, the tightest results assume diminishing or constant step sizes; adaptive methods (Adam, RMSProp) are not directly covered.
• Statistical models – Experiments focus on generalized linear models; applying the framework to more complex models (e.g., neural networks, structured prediction) would test its robustness.

Future directions suggested by the authors include:

1. Developing analogous basic inequalities for stochastic variance‑reduced methods.
2. Exploring non‑convex extensions via local curvature measures.
3. Integrating the framework with automated hyper‑parameter optimization pipelines.

Authors

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

Paper Information

• arXiv ID: 2512.24999v1
• Categories: math.ST, cs.LG, math.NA, math.OC, stat.ML
• Published: December 31, 2025
• PDF: Download PDF