Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOptimal PAC-Bayesian Posteriors for Stochastic Classifiers and their use for Choice of SVM Regularization Parameter
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
PAC-Bayesian set up involves a stochastic classifier characterized by a posterior distribution on a classifier set, offers a high probability bound on its averaged true risk and is robust to the training sample used. For a given posterior, this bound captures the trade off between averaged empirical risk and KL-divergence based model complexity term. Our goal is to identify an optimal posterior with the least PAC-Bayesian bound. We consider a finite classifier set and 5 distance functions: KL-divergence, its Pinskers and a sixth degree polynomial approximations; linear and squared distances. Linear distance based model results in a convex optimization problem. We obtain closed form expression for its optimal posterior. For uniform prior, this posterior has full support with weights negative-exponentially proportional to number of misclassifications. Squared distance and Pinskers approximation bounds are possibly quasi-convex and are observed to have single local minimum. We derive fixed point equations (FPEs) using partial KKT system with strict positivity constraints. This obviates the combinatorial search for subset support of the optimal posterior. For uniform prior, exponential search on a full-dimensional simplex can be limited to an ordered subset of classifiers with increasing empirical risk values. These FPEs converge rapidly to a stationary point, even for a large classifier set when a solver fails. We apply these approaches to SVMs generated using a finite set of SVM regularization parameter values on 9 UCI datasets. These posteriors yield stochastic SVM classifiers with tight bounds. KL-divergence based bound is the tightest, but is computationally expensive due to non-convexity and multiple calls to a root finding algorithm. Optimal posteriors for all 5 distance functions have lowest 10% test error values on most datasets, with linear distance being the easiest to obtain.
rate research
Read More
A recent technique of randomized smoothing has shown that the worst-case (adversarial) $ell_2$-robustness can be transformed into the average-case Gaussian-robustness by smoothing a classifier, i.e., by considering the averaged prediction over Gaussian noise. In this paradigm, one should rethink the notion of adversarial robustness in terms of generalization ability of a classifier under noisy observations. We found that the trade-off between accuracy and certified robustness of smoothed classifiers can be greatly controlled by simply regularizing the prediction consistency over noise. This relationship allows us to design a robust training objective without approximating a non-existing smoothed classifier, e.g., via soft smoothing. Our experiments under various deep neural network architectures and datasets show that the certified $ell_2$-robustness can be dramatically improved with the proposed regularization, even achieving better or comparable results to the state-of-the-art approaches with significantly less training costs and hyperparameters.
We investigate optimal posteriors for recently introduced cite{begin2016pac} chi-squared divergence based PAC-Bayesian bounds in terms of nature of their distribution, scalability of computations, and test set performance. For a finite classifier set, we deduce bounds for three distance functions: KL-divergence, linear and squared distances. Optimal posterior weights are proportional to deviations of empirical risks, usually with subset support. For uniform prior, it is sufficient to search among posteriors on classifier subsets ordered by these risks. We show the bound minimization for linear distance as a convex program and obtain a closed-form expression for its optimal posterior. Whereas that for squared distance is a quasi-convex program under a specific condition, and the one for KL-divergence is non-convex optimization (a difference of convex functions). To compute such optimal posteriors, we derive fast converging fixed point (FP) equations. We apply these approaches to a finite set of SVM regularization parameter values to yield stochastic SVMs with tight bounds. We perform a comprehensive performance comparison between our optimal posteriors and known KL-divergence based posteriors on a variety of UCI datasets with varying ranges and variances in risk values, etc. Chi-squared divergence based posteriors have weaker bounds and worse test errors, hinting at an underlying regularization by KL-divergence based posteriors. Our study highlights the impact of divergence function on the performance of PAC-Bayesian classifiers. We compare our stochastic classifiers with cross-validation based deterministic classifier. The latter has better test errors, but ours is more sample robust, has quantifiable generalization guarantees, and is computationally much faster.
In this paper, we improve the PAC-Bayesian error bound for linear regression derived in Germain et al. [10]. The improvements are twofold. First, the proposed error bound is tighter, and converges to the generalization loss with a well-chosen temperature parameter. Second, the error bound also holds for training data that are not independently sampled. In particular, the error bound applies to certain time series generated by well-known classes of dynamical models, such as ARX models.
Existing guarantees in terms of rigorous upper bounds on the generalization error for the original random forest algorithm, one of the most frequently used machine learning methods, are unsatisfying. We discuss and evaluate various PAC-Bayesian approaches to derive such bounds. The bounds do not require additional hold-out data, because the out-of-bag samples from the bagging in the training process can be exploited. A random forest predicts by taking a majority vote of an ensemble of decision trees. The first approach is to bound the error of the vote by twice the error of the corresponding Gibbs classifier (classifying with a single member of the ensemble selected at random). However, this approach does not take into account the effect of averaging out of errors of individual classifiers when taking the majority vote. This effect provides a significant boost in performance when the errors are independent or negatively correlated, but when the correlations are strong the advantage from taking the majority vote is small. The second approach based on PAC-Bayesian C-bounds takes dependencies between ensemble members into account, but it requires estimating correlations between the errors of the individual classifiers. When the correlations are high or the estimation is poor, the bounds degrade. In our experiments, we compute generalization bounds for random forests on various benchmark data sets. Because the individual decision trees already perform well, their predictions are highly correlated and the C-bounds do not lead to satisfactory results. For the same reason, the bounds based on the analysis of Gibbs classifiers are typically superior and often reasonably tight. Bounds based on a validation set coming at the cost of a smaller training set gave better performance guarantees, but worse performance in most experiments.
The dynamic ensemble selection of classifiers is an effective approach for processing label-imbalanced data classifications. However, such a technique is prone to overfitting, owing to the lack of regularization methods and the dependence of the aforementioned technique on local geometry. In this study, focusing on binary imbalanced data classification, a novel dynamic ensemble method, namely adaptive ensemble of classifiers with regularization (AER), is proposed, to overcome the stated limitations. The method solves the overfitting problem through implicit regularization. Specifically, it leverages the properties of stochastic gradient descent to obtain the solution with the minimum norm, thereby achieving regularization; furthermore, it interpolates the ensemble weights by exploiting the global geometry of data to further prevent overfitting. According to our theoretical proofs, the seemingly complicated AER paradigm, in addition to its regularization capabilities, can actually reduce the asymptotic time and memory complexities of several other algorithms. We evaluate the proposed AER method on seven benchmark imbalanced datasets from the UCI machine learning repository and one artificially generated GMM-based dataset with five variations. The results show that the proposed algorithm outperforms the major existing algorithms based on multiple metrics in most cases, and two hypothesis tests (McNemars and Wilcoxon tests) verify the statistical significance further. In addition, the proposed method has other preferred properties such as special advantages in dealing with highly imbalanced data, and it pioneers the research on the regularization for dynamic ensemble methods.
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
