Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLearning sparse optimal rule fit by safe screening
440
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider linear prediction models in the form of a sparse linear combination of rules, where a rule is an indicator function defined over a hyperrectangle in the input space. Since the number of all possible rules generated from the training dataset becomes extremely large, it has been difficult to consider all of them when fitting a sparse model. In this paper, we propose Safe Optimal Rule Fit (SORF) as an approach to resolve this problem, which is formulated as a convex optimization problem with sparse regularization. The proposed SORF method utilizes the fact that the set of all possible rules can be represented as a tree. By extending a recently popularized convex optimization technique called safe screening, we develop a novel method for pruning the tree such that pruned nodes are guaranteed to be irrelevant to the prediction model. This approach allows us to efficiently learn a prediction model constructed from an exponentially large number of all possible rules. We demonstrate the usefulness of the proposed method by numerical experiments using several benchmark datasets.
rate research
Read More
We present the design and implementation of a custom discrete optimization technique for building rule lists over a categorical feature space. Our algorithm produces rule lists with optimal training performance, according to the regularized empirical risk, with a certificate of optimality. By leveraging algorithmic bounds, efficient data structures, and computational reuse, we achieve several orders of magnitude speedup in time and a massive reduction of memory consumption. We demonstrate that our approach produces optimal rule lists on practical problems in seconds. Our results indicate that it is possible to construct optimal sparse rule lists that are approximately as accurate as the COMPAS proprietary risk prediction tool on data from Broward County, Florida, but that are completely interpretable. This framework is a novel alternative to CART and other decision tree methods for interpretable modeling.
The problem of learning a sparse model is conceptually interpreted as the process of identifying active features/samples and then optimizing the model over them. Recently introduced safe screening allows us to identify a part of non-active features/samples. So far, safe screening has been individually studied either for feature screening or for sample screening. In this paper, we introduce a new approach for safely screening features and samples simultaneously by alternatively iterating feature and sample screening steps. A significant advantage of considering them simultaneously rather than individually is that they have a synergy effect in the sense that the results of the previous safe feature screening can be exploited for improving the next safe sample screening performances, and vice-versa. We first theoretically investigate the synergy effect, and then illustrate the practical advantage through intensive numerical experiments for problems with large numbers of features and samples.
Popular machine learning estimators involve regularization parameters that can be challenging to tune, and standard strategies rely on grid search for this task. In this paper, we revisit the techniques of approximating the regularization path up to predefined tolerance $epsilon$ in a unified framework and show that its complexity is $O(1/sqrt[d]{epsilon})$ for uniformly convex loss of order $d geq 2$ and $O(1/sqrt{epsilon})$ for Generalized Self-Concordant functions. This framework encompasses least-squares but also logistic regression, a case that as far as we know was not handled as precisely in previous works. We leverage our technique to provide refined bounds on the validation error as well as a practical algorithm for hyperparameter tuning. The latter has global convergence guarantee when targeting a prescribed accuracy on the validation set. Last but not least, our approach helps relieving the practitioner from the (often neglected) task of selecting a stopping criterion when optimizing over the training set: our method automatically calibrates this criterion based on the targeted accuracy on the validation set.
We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under the restricted isometry assumption. For a given parametrization yielding a non-convex optimization problem, we show that prescribed choices of initialization, step size and stopping time yield a statistically and computationally optimal algorithm that achieves the minimax rate with the same cost required to read the data up to poly-logarithmic factors. Beyond minimax optimality, we show that our algorithm adapts to instance difficulty and yields a dimension-independent rate when the signal-to-noise ratio is high enough. Key to the computational efficiency of our method is an increasing step size scheme that adapts to refined estimates of the true solution. We validate our findings with numerical experiments and compare our algorithm against explicit $ell_{1}$ penalization. Going from hard instances to easy ones, our algorithm is seen to undergo a phase transition, eventually matching least squares with an oracle knowledge of the true support.
Sparse classifiers such as the support vector machines (SVM) are efficient in test-phases because the classifier is characterized only by a subset of the samples called support vectors (SVs), and the rest of the samples (non SVs) have no influence on the classification result. However, the advantage of the sparsity has not been fully exploited in training phases because it is generally difficult to know which sample turns out to be SV beforehand. In this paper, we introduce a new approach called safe sample screening that enables us to identify a subset of the non-SVs and screen them out prior to the training phase. Our approach is different from existing heuristic approaches in the sense that the screened samples are guaranteed to be non-SVs at the optimal solution. We investigate the advantage of the safe sample screening approach through intensive numerical experiments, and demonstrate that it can substantially decrease the computational cost of the state-of-the-art SVM solvers such as LIBSVM. In the current big data era, we believe that safe sample screening would be of great practical importance since the data size can be reduced without sacrificing the optimality of the final solution.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
