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Register a new userExact Optimization of Conformal Predictors via Incremental and Decremental Learning
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Added by
Giovanni Cherubin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Conformal Predictors (CP) are wrappers around ML methods, providing error guarantees under weak assumptions on the data distribution. They are suitable for a wide range of problems, from classification and regression to anomaly detection. Unfortunately, their high computational complexity limits their applicability to large datasets. In this work, we show that it is possible to speed up a CP classifier considerably, by studying it in conjunction with the underlying ML method, and by exploiting incremental&decremental learning. For methods such as k-NN, KDE, and kernel LS-SVM, our approach reduces the running time by one order of magnitude, whilst producing exact solutions. With similar ideas, we also achieve a linear speed up for the harder case of bootstrapping. Finally, we extend these techniques to improve upon an optimization of k-NN CP for regression. We evaluate our findings empirically, and discuss when methods are suitable for CP optimization.
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Abstracting neural networks with constraints they impose on their inputs and outputs can be very useful in the analysis of neural network classifiers and to derive optimization-based algorithms for certification of stability and robustness of feedback systems involving neural networks. In this paper, we propose a convex program, in the form of a Linear Matrix Inequality (LMI), to certify incremental quadratic constraints on the map of neural networks over a region of interest. These certificates can capture several useful properties such as (local) Lipschitz continuity, one-sided Lipschitz continuity, invertibility, and contraction. We illustrate the utility of our approach in two different settings. First, we develop a semidefinite program to compute guaranteed and sharp upper bounds on the local Lipschitz constant of neural networks and illustrate the results on random networks as well as networks trained on MNIST. Second, we consider a linear time-invariant system in feedback with an approximate model predictive controller parameterized by a neural network. We then turn the stability analysis into a semidefinite feasibility program and estimate an ellipsoidal invariant set for the closed-loop system.
Efficiency criteria for conformal prediction, such as emph{observed fuzziness} (i.e., the sum of p-values associated with false labels), are commonly used to emph{evaluate} the performance of given conformal predictors. Here, we investigate whether it is possible to exploit efficiency criteria to emph{learn} classifiers, both conformal predictors and point classifiers, by using such criteria as training objective functions. The proposed idea is implemented for the problem of binary classification of hand-written digits. By choosing a 1-dimensional model class (with one real-valued free parameter), we can solve the optimization problems through an (approximate) exhaustive search over (a discrete version of) the parameter space. Our empirical results suggest that conformal predictors trained by minimizing their observed fuzziness perform better than conformal predictors trained in the traditional way by minimizing the emph{prediction error} of the corresponding point classifier. They also have a reasonable performance in terms of their prediction error on the test set.
In large-scale classification problems, the data set always be faced with frequent updates when a part of the data is added to or removed from the original data set. In this case, conventional incremental learning, which updates an existing classifier by explicitly modeling the data modification, is more efficient than retraining a new classifier from scratch. However, sometimes, we are more interested in determining whether we should update the classifier or performing some sensitivity analysis tasks. To deal with these such tasks, we propose an algorithm to make rational inferences about the updated linear classifier without exactly updating the classifier. Specifically, the proposed algorithm can be used to estimate the upper and lower bounds of the updated classifiers coefficient matrix with a low computational complexity related to the size of the updated dataset. Both theoretical analysis and experiment results show that the proposed approach is superior to existing methods in terms of tightness of coefficients bounds and computational complexity.
We study the rate of change of the multivariate mutual information among a set of random variables when some common randomness is added to or removed from a subset. This is formulated more precisely as two new multiterminal secret key agreement problems which ask how one can increase the secrecy capacity efficiently by adding common randomness to a small subset of users, and how one can simplify the source model by removing redundant common randomness that does not contribute to the secrecy capacity. The combinatorial structure has been clarified along with some meaningful open problems.
Combinatorial optimization problems are typically tackled by the branch-and-bound paradigm. We propose a new graph convolutional neural network model for learning branch-and-bound variable selection policies, which leverages the natural variable-constraint bipartite graph representation of mixed-integer linear programs. We train our model via imitation learning from the strong branching expert rule, and demonstrate on a series of hard problems that our approach produces policies that improve upon state-of-the-art machine-learning methods for branching and generalize to instances significantly larger than seen during training. Moreover, we improve for the first time over expert-designed branching rules implemented in a state-of-the-art solver on large problems. Code for reproducing all the experiments can be found at https://github.com/ds4dm/learn2branch.
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