Zeroth-Order Hard-Thresholding: Sparse Optimization Algorithm Analysis

Główne pojęcia

Zeroth-Order Hard-Thresholding is a novel algorithm for sparse optimization with dimension-independent convergence.

Streszczenie

Zeroth-Order Hard-Thresholding (SZOHT) is introduced as a stochastic zeroth-order gradient hard-thresholding algorithm for ℓ0 constrained black-box stochastic optimization problems. The algorithm addresses the challenge of working with zeroth-order gradients and hard-thresholding operators, providing convergence analysis under standard assumptions. SZOHT reveals a conflict between ZO estimators' deviation and the expansivity of the hard-thresholding operator, offering insights into the minimal number of random directions required in ZO gradients. The query complexity of SZOHT is shown to be independent or weakly dependent on dimensionality, contrasting existing ZO algorithms. Experimental results demonstrate competitive performance in portfolio optimization and adversarial attacks.

Statystyki

O(κ log( 1 ε))
O(d ε2)
O(s log(d) log( 1 ε))
O((k + d s^2 )κ^2 log( 1 ε))
O(kκ^2 log( 1 ε))

Cytaty

"SZOHT provides a dimension-independent convergence rate in the smooth case."
"Experimental results show SZOHT's competitive performance in portfolio optimization and adversarial attacks."
"The query complexity of SZOHT contrasts existing zeroth-order algorithms by being independent or weakly dependent on dimensionality."

Kluczowe wnioski z

Zeroth-Order Hard-Thresholding

by William de V... o arxiv.org 03-19-2024

https://arxiv.org/pdf/2210.05279.pdf

Głębsze pytania

How can the condition number of f impact the convergence of SZOHT?

The condition number of a function, denoted as κ, plays a crucial role in determining the convergence properties of optimization algorithms like SZOHT. In the context of SZOHT, a high condition number implies that the function is ill-conditioned, making it more challenging for iterative optimization algorithms to converge efficiently.
Specifically, in SZOHT, a higher condition number can lead to slower convergence rates and potentially require more iterations to reach an optimal solution. This is because a high condition number indicates that small changes in input parameters (perturbations) can result in large variations in output values. As such, when dealing with functions with high condition numbers using SZOHT, careful tuning of hyperparameters becomes essential to ensure convergence.
To mitigate the impact of a high condition number on the convergence of SZOHT, one approach could be to regularize or precondition the objective function before applying the algorithm. By incorporating techniques like regularization or preconditioning methods tailored to handle ill-conditioned problems into SZOHT, it may be possible to improve its performance and accelerate convergence.

How might SZOHT be extended to handle different types of sparse structures beyond ℓ0 constraints?

SZOHT's applicability can be extended beyond ℓ0 constraints by adapting its methodology to accommodate various types of sparse structures commonly encountered in optimization problems. Some potential extensions include:

Low-rank Approximations: Modify SZOTH's gradient estimation technique and hard-thresholding operator to incorporate low-rank approximations for matrices or tensors where sparsity arises from low-rank structures rather than individual element sparsity.

Graph Sparsity: Develop specialized versions of SZOTH that leverage graph-based sparsity patterns commonly found in network analysis or social network applications. The algorithm would need modifications in how gradients are estimated and thresholds applied based on graph connectivity.

Group Sparsity: Extend SZOTH to handle group sparsity where variables are grouped together and enforced jointly as either active or inactive simultaneously during optimization iterations.

Non-Convex Constraints: Adaptation for handling non-convex constraints while maintaining sparsity requirements by integrating appropriate penalty terms or constraint formulations into the algorithm design.

By customizing aspects such as gradient estimation strategies and thresholding mechanisms according to specific sparse structures present in diverse optimization scenarios, SZOTH can effectively address a broader range of real-world problems requiring sparse solutions.

What are potential improvements to reduce query complexity further?

Reducing query complexity further is essential for enhancing efficiency and scalability across various applications utilizing zeroth-order optimization methods like ${Context:Zeroth-Order Hard-Thresholding}. Here are some potential improvements:

Adaptive Learning Rates: Implement adaptive learning rate schemes within ${Context:SZOT}H that dynamically adjust learning rates based on historical information about gradients' behavior at each iteration.

2 .Acceleration Techniques: Incorporate acceleration techniques such as momentum updates or Nesterov acceleration into ${Context:SZOT}Hto expedite convergence speed without compromising accuracy.
3 .Variance Reduction Methods: Integrate variance reduction techniques like control variates or stochastic average gradient (SAGA) approaches within ${Context:SZOT}Hto reduce noise-induced fluctuations during gradient estimations.
4 .Regularization Strategies: Apply regularization methods tailored towards zeroth-order settings within${Context:SZOT}Hto stabilize training process by preventing overfitting due limited query access.
5 .Parallelization: Explore parallel computing paradigms like distributed computing frameworksor GPU accelerationsto enable concurrent processingof queriesand enhance overall computational efficiencyof${Context:SZOT}Halgorithm
By implementing these enhancements strategicallyintothe existing frameworkof${Context:SZOT},itispossibletofurtherreducequerycomplexityandimproveoverallperformanceacrossdiverseoptimizationalgorithmscenarios