Accelerated Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods

Core Concepts

Generic optimization methods such as mirror descent and steepest descent can achieve significantly faster margin maximization rates compared to previous results, by transforming the optimization problem into an equivalent regularized bilinear game that can be solved using online learning algorithms. Similarly, adversarial training methods can also attain faster margin maximization rates that match the best known rates for optimization on clean data.

Abstract

The paper presents a series of state-of-the-art implicit bias rates for mirror descent and steepest descent algorithms in the context of linear classification. The key insight is to transform the generic optimization problem into an equivalent regularized bilinear game that can be solved using online learning algorithms. This provides a unified framework for analyzing the implicit bias of various optimization methods.
The main highlights are:

For mirror descent with the squared ℓq-norm potential, the algorithm achieves a faster ∥·∥q-margin maximization rate of O(log n log T / (q-1)T) with an appropriately chosen step size. This can be further improved to O(1/T(q-1) + log n log T / T^2) with a more aggressive step size.

For steepest descent with a strongly convex norm, the margin maximization rate can be improved from O((log n + log T) / √T) to O(log n / T).

Even faster O(log n / T^2(q-1)) ∥·∥q-margin maximization rates can be achieved using either mirror descent with Nesterov acceleration or steepest descent with extra gradient and momentum.

For adversarial training with ℓs-norm perturbations (s ∈ (1, 2]), normalized gradient descent achieves a rate of O(log n / T) towards the (2, s)-mix-norm max-margin classifier.

Equipping adversarial training with Nesterov-style acceleration further improves the rates to O(log n / T^2) for s ∈ (1, 2], and O(log n / T) for s > 2.

The key technical contribution is the identification of the correct online learning algorithms that can solve the regularized bilinear (or multilinear) game, leading to the accelerated margin maximization rates.

Stats

The paper does not contain any explicit numerical data or statistics. The main results are theoretical bounds on the margin maximization rates and directional errors of various optimization algorithms.

Quotes

"First-order optimization methods tend to inherently favor certain solutions over others when minimizing an underdetermined training objective that has multiple global optima. This phenomenon, known as implicit bias, plays a critical role in understanding the generalization capabilities of optimization algorithms."
"Even unregularized first-order optimization methods are observed to converge to solutions that generalize well to test data, as multiple empirical studies have repeatedly confirmed."
"The choice of s, and therefore the choice of the optimization algorithm, affects the entire nature of the eventual solution, thus playing a pivotal role in robustness."

Key Insights Distilled From

Faster Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods

by Guanghui Wan... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2305.17544.pdf

Deeper Inquiries

How can the insights from this work on linear classification be extended to more complex neural network architectures and non-linear classification tasks

The insights gained from this work on linear classification can be extended to more complex neural network architectures and non-linear classification tasks by adapting the concepts of implicit bias and margin maximization. In neural networks, the optimization objective is also underdetermined, and understanding the implicit bias of optimization algorithms can provide valuable insights into which solutions are favored and how quickly the parameters converge to these solutions. By analyzing the implicit bias of optimization methods in neural networks, researchers can gain a better understanding of how different architectures and optimization techniques impact the generalization capabilities of the models. Additionally, the concept of margin maximization can be applied to non-linear classification tasks by considering the decision boundaries and margins in higher-dimensional feature spaces. This can help in improving the robustness and generalization of neural networks in complex classification tasks.

What are the implications of the faster margin maximization rates on the generalization and robustness properties of the learned models in practical applications

The implications of faster margin maximization rates on the generalization and robustness properties of learned models in practical applications are significant. Faster margin maximization rates indicate that the optimization algorithms converge to solutions with larger margins more quickly. Models with larger margins are known to have better generalization properties, as they are less likely to overfit to the training data. Additionally, models with larger margins are often more robust to noise and perturbations in the input data, leading to improved performance on unseen data and in adversarial settings. Therefore, the faster margin maximization rates obtained in this work suggest that the learned models are likely to generalize better and be more robust in practical applications, making them more reliable and effective in real-world scenarios.

Can the regularized multilinear game framework be further generalized to capture other forms of structured optimization problems beyond adversarial training

The regularized multilinear game framework presented in this work can be further generalized to capture other forms of structured optimization problems beyond adversarial training. By extending the framework to accommodate different types of optimization objectives, constraints, and game dynamics, researchers can apply the same principles of implicit bias analysis and margin maximization to a wider range of optimization problems. This generalization can provide a unified framework for analyzing the implicit bias of various optimization methods in different settings, allowing for a deeper understanding of how optimization algorithms behave and converge to solutions in diverse optimization scenarios. By adapting the framework to different structured optimization problems, researchers can uncover new insights into the implicit bias of optimization algorithms and improve the efficiency and effectiveness of optimization techniques in various applications.