Efficient and Numerically Stable Sparse Learning Sihong Xie 1, Wei Fan 2, Olivier Verscheure 2, and Jiangtao Ren 3 1 University of Illinois at Chicago, USA 2 IBM T.J. Watson Research Center, New York, USA 3 Sun Yat-Sen University, Guangzhou, China

Sparse Linear Model Input: Output: sparse linear model Learning formulation Sparse regularization Large Scale Contest

Objectives Sparsity Accuracy Numerical Stability limited precision friendly Scalability Large scale training data (rows and columns)

Outline “Numerical un-stability” of two popular approaches Propose sparse linear model online numerically stable parallelizable good sparcity – don’t take features unless necesseary Experiments results

Stability in Sparse learning Numerical Problems of Direct Iterative Methods Numerical Problems of Mirror Descent

Stability in Sparse learning Iterative Hard Thresholding (IHT) Solve the following optimization problem Linear model Label vector Data Matrix Sparse Degree The error to minimize L-0 regularization

Stability in Sparse learning Iterative Hard Thresholding (IHT) Incorporating gradient descent with hard thresholding At each iteration: Hard Thresholding: Keep s top significant elements Negative of Gradient 1 2

Iterative Hard Thresholding (IHT) Advantages: Simple and scalable Stability in Sparse learning Convergence of IHT

Stability in Sparse learning For IHT algorithm to converge, the iteration matrix should have its spectral radius less than 1 Spectral radius Iterative Hard Thresholding (IHT)

Experiments Divergence of IHT

Stability in Sparse learning Example of error growth of IHT

Stability in Sparse learning Numerical Problems of Mirror Descent Numerical Problems of Direct Iterative Methods Numerical Problems of Mirror Descent

Stability in Sparse learning Mirror Descent Algorithm (MDA) Solve the L-1 regularized formulation Maintain two vectors: primal and dual

Stability in Sparse learning Dual space Primal space Illustration adopted from Peter Bartlett’s lecture slide link function soft-thresholding Sparse Dual vector p is a parameter for MDA

Stability in Sparse learning MDA link function Floating number system significant digits base exponent × Example: A computer with only 4 significant digits = 0.1

Stability in Sparse learning The difference between elements is amplified via the link function, when comparing elements in dual and primal vectors, respectively

Experiments Numerical problem of MDA Experimental settings Train models with 40% density. Parameter p is set to 2ln(d) (p=33) and 0.5 ln(d) respectively [ ST2009] [ST2009] Shai S. Shwartz and Ambuj Tewari. Stochastic methods for ℓ1 regularized loss minimization. In ICML, pages 929–936. ACM, 2009.

Performance Criteria Percentage of elements that are truncated during prediction Dynamical range Experiments Numerical problem of MDA

Objectives of a Simple Approach Numerically Stable Computationally efficient Online, parallelizable Accurate models with higher sparsity Costly to obtain too many features (e.g. medical diagnostics) For an excellent theoretical treatment of trading off between accuracy and sparsity see S. Shalev-Shwartz, N. Srebro, and T. Zhang. Trading accuracy for sparsity. Technical report, TTIC, May 2009.

The proposed method algorithm SVM like margin

Numerical Stability and Scalability Considerations numerical stability Less conditions on data matrix such as spectral radius and no change of scales Less precision demanding (works under limited precision, theorem 1) Under mild conditions, the proposed method converges even for a large number of iterations proportional to Machine precision

Numerical Stability and Scalability Considerations Online fashion: one example at a time Parallelization for intensive data access Data can be distributed to computers, where parts of the inner product can be obtained. Small network communication (only parts of inner product and signals to update model)

The proposed method properties Soft-thresholding L1-regularization for sparse model Perceptron: avoids updates when the current features are able to predict well – sparcity Convergence under soft-thresholding and limited precision (Lemma 2 and Theorem 1) – numerical stability Generalization error bound (Theorem 3) Don’t complicate the model when unnecessary

The proposed method A toy example f1f1 f2f2 f3f3 label A B C w1w1 w2w2 w3w w1w1 w2w2 w3w w1w1 w2w2 w3w Relatively dense model The proposed method: a sparse model is enough to predict well (margin indicates good-enough model, so enough features) TG: truncated descent 1 st update 2 nd update 3 rd update

Experiments Overall comparison The proposed algorithm + 3 baseline sparse learning algorithms ( all with logistic loss function ) SMIDAS (MDA based [ST2009]): p = 0.5log(d) (cannot run with bigger p due to numerical problem) TG (Truncated Gradient [LLZ2009]) SCD (Stochastic Coordinate Descent [ST2009]) [ST2009] Shai Shalev-Shwartz and Ambuj Tewari, Stochastic methods for l1 regularized loss minimization. Proceedings of the 26th International Conference on Machine Learning, pages , [LLZ2009] John Langford, Lihong Li, and Tong Zhang. Sparse online learning via truncated gradient. Journal of Machine Learning Research, 10:777–801, 2009.

Experiments Overall comparison Accuracy under the same model density First 7 datasets: maximum 40% of features Webspam: select maximum 0.1% of features Stop running the program when maximum percentage of features are selected

Experiments Overall comparison Accuracy vs. sparsity The proposed algorithm works consistently better than other baselines. On 5 out of 8 tasks, stopped updating model before reaching the maximum density (40% of features) On task 1, outperforms others with 10% features On task 3, ties with the best baseline using 20% features Sparse Convergence

Conclusion Numerical Stability of Sparse Learning Gradient Descent using matrix iteration may diverge without the spectral radius assumption. When dimensionality is high, MDA produces many infinitesimal elements. Trading off Sparsity and Accuracy Other methods (TG, SCD) are unable to train accurate models with high sparsity. Proposed approach is numerically stable, online parallelizable and converges. Controlled by margin L-1 regularization and soft threshold Experimental codes are available