Idée - Optimization - # Stochastic Accelerated Gradient Descent

Accelerated Convergence of Stochastic Gradient Descent under Interpolation

Q: How can the generalized stochastic AGD scheme be extended to handle other types of constraints or regularization beyond the unconstrained setting considered in this work

The extension of the generalized stochastic AGD scheme to handle constraints or regularization beyond the unconstrained setting involves incorporating these additional constraints into the optimization problem. One approach is to use proximal operators to handle constraints efficiently. For instance, if the problem involves a simple box constraint where the solution must lie within a certain range, the proximal operator of the indicator function of the feasible set can be used to project the iterates onto the feasible region. This projection step ensures that the iterates remain within the feasible set while still benefiting from the acceleration provided by the AGD scheme. In the case of more complex constraints or regularization terms, such as L1 or L2 regularization, the proximal operator of the corresponding regularizer can be incorporated into the update step of the AGD scheme. This proximal gradient step ensures that the iterates satisfy the constraints or regularization terms while optimizing the objective function. By adapting the update step to include these proximal operators, the generalized stochastic AGD scheme can effectively handle a wide range of constraints and regularization types in optimization problems.

Q: What are the limitations of the interpolation assumption, and how can the analysis be extended to settings with weaker or different assumptions on the stochastic gradients

The interpolation assumption, while powerful in providing fast convergence rates for stochastic optimization algorithms like SGD and AGD, has its limitations. One limitation is that interpolation may not hold in all practical scenarios, especially in high-dimensional or complex optimization problems where the data may not be perfectly separable or linearly independent. In such cases, the assumptions of interpolation may not be realistic, leading to suboptimal convergence rates or even algorithm failure. To address these limitations and extend the analysis to settings with weaker or different assumptions on the stochastic gradients, one approach is to relax the interpolation assumption and consider more general conditions that capture the behavior of the optimization landscape. This could involve exploring conditions beyond interpolation, such as smoothness properties, curvature information, or sparsity patterns in the data. By incorporating these additional properties into the analysis, the optimization algorithms can be adapted to handle a broader range of scenarios and provide robust performance even when strict interpolation assumptions do not hold.

Q: Can the ideas behind the estimating sequences framework be further leveraged to develop new stochastic optimization algorithms that are adaptive to problem-specific properties, beyond just the strong growth constant

The ideas behind the estimating sequences framework can indeed be leveraged to develop new stochastic optimization algorithms that are adaptive to problem-specific properties beyond just the strong growth constant. By utilizing estimating sequences to maintain upper bounds on the objective function and incorporating progress guarantees similar to those used in the analysis of stochastic AGD, new algorithms can be designed to adapt to various problem structures and properties. One potential direction is to incorporate adaptive step sizes or learning rates that adjust based on the estimated progress and upper bounds provided by the estimating sequences. This adaptability can help the algorithm navigate the optimization landscape more efficiently, especially in scenarios where the strong growth condition may not hold or where the interpolation assumption is not applicable. Additionally, exploring different update schemes or acceleration techniques within the estimating sequences framework can lead to the development of novel stochastic optimization algorithms that are tailored to specific problem characteristics and exhibit improved convergence properties.

Concepts de base

The authors prove new convergence rates for a generalized version of stochastic Nesterov acceleration under interpolation conditions. Their approach accelerates any stochastic gradient method that makes sufficient progress in expectation, and the proof applies to both convex and strongly convex functions.

Résumé

The key highlights and insights from the content are:

The authors introduce a generalized version of stochastic Nesterov acceleration that can be applied to any stochastic gradient method making sufficient progress in expectation.
They prove that under interpolation conditions, this generalized stochastic accelerated gradient descent (AGD) scheme can achieve faster convergence rates compared to standard stochastic gradient descent (SGD).
The authors' analysis uses the estimating sequences framework and shows that as long as the primal update (e.g., SGD) satisfies a sufficient progress condition, the generalized stochastic AGD can be accelerated.
Specializing the results to standard stochastic AGD, the authors show an improved dependence on the strong growth constant compared to prior work. This improvement can be larger than the square-root of the condition number.
The authors also extend their analysis to stochastic AGD with preconditioning, demonstrating that preconditioning can further speed up accelerated stochastic optimization when the stochastic gradients are well-conditioned in the preconditioner's norm.
The authors compare their convergence guarantees to existing results in the literature, highlighting settings where their stochastic AGD scheme achieves acceleration over standard SGD.

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Idées clés tirées de

Faster Convergence of Stochastic Accelerated Gradient Descent under Interpolation

by Aaron Mishki... à arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02378.pdf

Faster Convergence of Stochastic Accelerated Gradient Descent under Interpolation

Questions plus approfondies

How can the generalized stochastic AGD scheme be extended to handle other types of constraints or regularization beyond the unconstrained setting considered in this work

The extension of the generalized stochastic AGD scheme to handle constraints or regularization beyond the unconstrained setting involves incorporating these additional constraints into the optimization problem. One approach is to use proximal operators to handle constraints efficiently. For instance, if the problem involves a simple box constraint where the solution must lie within a certain range, the proximal operator of the indicator function of the feasible set can be used to project the iterates onto the feasible region. This projection step ensures that the iterates remain within the feasible set while still benefiting from the acceleration provided by the AGD scheme.
In the case of more complex constraints or regularization terms, such as L1 or L2 regularization, the proximal operator of the corresponding regularizer can be incorporated into the update step of the AGD scheme. This proximal gradient step ensures that the iterates satisfy the constraints or regularization terms while optimizing the objective function. By adapting the update step to include these proximal operators, the generalized stochastic AGD scheme can effectively handle a wide range of constraints and regularization types in optimization problems.

What are the limitations of the interpolation assumption, and how can the analysis be extended to settings with weaker or different assumptions on the stochastic gradients

The interpolation assumption, while powerful in providing fast convergence rates for stochastic optimization algorithms like SGD and AGD, has its limitations. One limitation is that interpolation may not hold in all practical scenarios, especially in high-dimensional or complex optimization problems where the data may not be perfectly separable or linearly independent. In such cases, the assumptions of interpolation may not be realistic, leading to suboptimal convergence rates or even algorithm failure.
To address these limitations and extend the analysis to settings with weaker or different assumptions on the stochastic gradients, one approach is to relax the interpolation assumption and consider more general conditions that capture the behavior of the optimization landscape. This could involve exploring conditions beyond interpolation, such as smoothness properties, curvature information, or sparsity patterns in the data. By incorporating these additional properties into the analysis, the optimization algorithms can be adapted to handle a broader range of scenarios and provide robust performance even when strict interpolation assumptions do not hold.

Can the ideas behind the estimating sequences framework be further leveraged to develop new stochastic optimization algorithms that are adaptive to problem-specific properties, beyond just the strong growth constant

The ideas behind the estimating sequences framework can indeed be leveraged to develop new stochastic optimization algorithms that are adaptive to problem-specific properties beyond just the strong growth constant. By utilizing estimating sequences to maintain upper bounds on the objective function and incorporating progress guarantees similar to those used in the analysis of stochastic AGD, new algorithms can be designed to adapt to various problem structures and properties.
One potential direction is to incorporate adaptive step sizes or learning rates that adjust based on the estimated progress and upper bounds provided by the estimating sequences. This adaptability can help the algorithm navigate the optimization landscape more efficiently, especially in scenarios where the strong growth condition may not hold or where the interpolation assumption is not applicable. Additionally, exploring different update schemes or acceleration techniques within the estimating sequences framework can lead to the development of novel stochastic optimization algorithms that are tailored to specific problem characteristics and exhibit improved convergence properties.