# Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions

@article{Agarwal2014StochasticOA, title={Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions}, author={Alekh Agarwal and Sahand N. Negahban and Martin J. Wainwright}, journal={2014 48th Annual Conference on Information Sciences and Systems (CISS)}, year={2014}, pages={1-2} }

Summary form only given. Stochastic optimization algorithms have many desirable features for large-scale machine learning, and accordingly have been the focus of renewed and intensive study in the last several years (e.g., see the papers [2], [5], [14] and references therein). The empirical efficiency of these methods is backed with strong theoretical guarantees, providing sharp bounds on their convergence rates. These convergence rates are known to depend on the structure of the underlying…

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## References

SHOWING 1-10 OF 38 REFERENCES

### Fast global convergence rates of gradient methods for high-dimensional statistical recovery

- Computer ScienceNIPS
- 2010

The theory guarantees that Nesterov's first-order method has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical Euclidean distance between the true unknown parameter θ* and the optimal solution ^θ.

### Fast global convergence of gradient methods for high-dimensional statistical recovery

- Computer ScienceArXiv
- 2011

The theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical distance between the true unknown parameter $\theta^*$ and an optimal solution $\hat{\theta}$.

### High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity

- Computer ScienceNIPS
- 2011

This work is able to both analyze the statistical error associated with any global optimum, and prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers.

### Restricted Eigenvalue Properties for Correlated Gaussian Designs

- Computer ScienceJ. Mach. Learn. Res.
- 2010

This paper proves directly that the restricted nullspace and eigenvalue conditions hold with high probability for quite general classes of Gaussian matrices for which the predictors may be highly dependent, and hence restricted isometry conditions can be violated with high probabilities.

### A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers

- Computer Science, MathematicsNIPS
- 2009

A unified framework for establishing consistency and convergence rates for regularized M-estimators under high-dimensional scaling is provided and one main theorem is state and shown how it can be used to re-derive several existing results, and also to obtain several new results.

### Efficient Online and Batch Learning Using Forward Backward Splitting

- Computer ScienceJ. Mach. Learn. Res.
- 2009

The two phase approach enables sparse solutions when used in conjunction with regularization functions that promote sparsity, such as l1, l2, l22, and l∞ regularization, and is extended and given efficient implementations for very high-dimensional data with sparsity.

### Optimal Stochastic Approximation Algorithms for Strongly Convex Stochastic Composite Optimization, II: Shrinking Procedures and Optimal Algorithms

- Computer Science, MathematicsSIAM J. Optim.
- 2013

A multistage AC-SA algorithm is introduced, which possesses an optimal rate of convergence for solving strongly convex SCO problems in terms of the dependence on not only the target accuracy, but also a number of problem parameters and the selection of initial points.

### Gradient methods for minimizing composite objective function

- Computer Science, Mathematics
- 2007

In this paper we analyze several new methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a black-box oracle, and…

### Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling

- Computer ScienceIEEE Transactions on Automatic Control
- 2012

This work develops and analyze distributed algorithms based on dual subgradient averaging and provides sharp bounds on their convergence rates as a function of the network size and topology, and shows that the number of iterations required by the algorithm scales inversely in the spectral gap of thenetwork.

### Robust Stochastic Approximation Approach to Stochastic Programming

- Computer Science, MathematicsSIAM J. Optim.
- 2009

It is intended to demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems.