Analyzing Regularization in Linear Models with Reparametrization of Gradient Flow

Core Concepts

The author explores the impact of reparametrizations on implicit bias in linear models, providing conditions for convergence and specific regularization.

Abstract

The content delves into the relationship between reparametrizations of gradient flow and implicit bias in linear models. It discusses conditions for convergence and specific regularization functions that induce implicit biases closely connected to certain regularizers like ℓp or trigonometric ones. The analysis highlights the importance of model parameters, loss functions, and link functions on the behavior of gradient flow. Theoretical results are presented to characterize the implicit bias and convergence behavior under different reparametrizations. The study unifies various findings on implicit bias in regression models, offering insights into designing reparametrizations that encode specific regularizers implicitly into gradient flow/descent.

Stats

For sufficiently small step-sizes, the gradient descent trajectory converges to the least-square fit w∞ := limt→∞ w(t) = A†y.
Recent works proved that solving an overparametrized but unregularized least-squares problem via vanilla gradient flow leads to approximate ℓ1-minimization among all possible solutions.
If all z(k) are initialized with the same vector w0 and gradient flow is applied to (4), then the iterates w(k) stay identical over time.
The formulation in (4) is equivalent to training deep linear networks with diagonal weight matrices by viewing z(ℓ) as the diagonal of the ℓ-th layer weight matrix.
The assumption that ew is regular will be critical when characterizing the implicit bias of gradient flow.
For small α, the reparametrized flow ρrp(w(t)) approximately minimizes the ℓp-norm among all global minimizers ez = ρrp(z) of L(ρlink(Aez), y).
Mirror descent with spectral hypentropy mirror map previously appeared in [50].
An intriguing connection regarding implicit regularization induced by large step sizes coupled with SGD noise has been discussed in [3, 15].

Quotes

"The tendency of (stochastic) gradient descent to prefer certain minimizers over others is commonly called implicit bias/regularization."
"Many existing works on implicit regularization explicitly assume T = ∞ and convergence of w."
"If A is generated at random with iid Gaussian entries, this additional requirement holds with probability one."

Key Insights Distilled From

How to induce regularization in linear models

by Hung... at arxiv.org 03-07-2024

https://arxiv.org/pdf/2308.04921.pdf

Deeper Inquiries

How does reparametrization influence convergence behavior compared to traditional methods

Reparametrization plays a crucial role in influencing the convergence behavior of gradient flow compared to traditional methods. By reparametrizing the input space using functions like ρrp, we can induce specific implicit biases that guide the optimization process towards solutions with desired properties. This allows us to design reparametrizations that encode regularization constraints into the optimization landscape, leading to improved convergence rates and better generalization performance. The choice of reparametrization function directly impacts how gradient flow navigates through the parameter space, affecting both the speed and quality of convergence.

What are potential drawbacks or limitations of relying on implicit biases for model optimization

While leveraging implicit biases for model optimization can offer several advantages such as improved generalization and faster convergence, there are potential drawbacks and limitations to consider. One limitation is that relying solely on implicit biases may not always guarantee optimal solutions or robust performance across different datasets or tasks. Implicit biases are inherently dependent on various factors such as initialization, data distribution, and network architecture, which can make them sensitive to changes in these parameters. Additionally, complex neural network architectures may require more sophisticated techniques beyond simple reparametrizations to effectively leverage implicit biases for optimization.

How can these findings be applied to more complex neural network architectures beyond linear models

The findings from analyzing reparametrized gradient flow in linear models can be applied to more complex neural network architectures by extending the concepts of overparameterization and implicit bias. For instance:

Regularizers: Reparameterizing deep neural networks with specific functions could induce regularizers similar to ℓp norms or trigonometric functions.
Optimization: Understanding how different choices of loss functions influence gradient flow can help optimize training procedures for deep learning models.
Implicit Bias: By studying simplified settings first (such as overparameterized linear models), researchers can gain insights into how implicit bias affects training dynamics in deeper architectures.
Overall, applying these insights to more complex neural networks could lead to improved training strategies, better generalization capabilities, and enhanced interpretability of model behavior.