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Register a new userPiece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning
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Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and noisy data, the quadratic error functionals demonstrated many weaknesses including high sensitivity to contaminating factors and dimensionality curse. Therefore, a lot of recent applications in machine learning exploited properties of non-quadratic error functionals based on $L_1$ norm or even sub-linear potentials corresponding to quasinorms $L_p$ ($0<p<1$). The back side of these approaches is increase in computational cost for optimization. Till so far, no approaches have been suggested to deal with {it arbitrary} error functionals, in a flexible and computationally efficient framework. In this paper, we develop a theory and basic universal data approximation algorithms ($k$-means, principal components, principal manifolds and graphs, regularized and sparse regression), based on piece-wise quadratic error potentials of subquadratic growth (PQSQ potentials). We develop a new and universal framework to minimize {it arbitrary sub-quadratic error potentials} using an algorithm with guaranteed fast convergence to the local or global error minimum. The theory of PQSQ potentials is based on the notion of the cone of minorant functions, and represents a natural approximation formalism based on the application of min-plus algebra. The approach can be applied in most of existing machine learning methods, including methods of data approximation and regularized and sparse regression, leading to the improvement in the computational cost/accuracy trade-off. We demonstrate that on synthetic and real-life datasets PQSQ-based machine learning methods achieve orders of magnitude faster computational performance than the corresponding state-of-the-art methods.
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Although much progress has been made towards robust deep learning, a significant gap in robustness remains between real-world perturbations and more narrowly defined sets typically studied in adversarial defenses. In this paper, we aim to bridge this gap by learning perturbation sets from data, in order to characterize real-world effects for robust training and evaluation. Specifically, we use a conditional generator that defines the perturbation set over a constrained region of the latent space. We formulate desirable properties that measure the quality of a learned perturbation set, and theoretically prove that a conditional variational autoencoder naturally satisfies these criteria. Using this framework, our approach can generate a variety of perturbations at different complexities and scales, ranging from baseline spatial transformations, through common image corruptions, to lighting variations. We measure the quality of our learned perturbation sets both quantitatively and qualitatively, finding that our models are capable of producing a diverse set of meaningful perturbations beyond the limited data seen during training. Finally, we leverage our learned perturbation sets to train models which are empirically and certifiably robust to adversarial image corruptions and adversarial lighting variations, while improving generalization on non-adversarial data. All code and configuration files for reproducing the experiments as well as pretrained model weights can be found at https://github.com/locuslab/perturbation_learning.
The increasing take-up of machine learning techniques requires ever-more application-specific training data. Manually collecting such training data is time-consuming and error-prone process. Data marketplaces represent a compelling alternative, providing an easy way for acquiring data from potential data providers. A key component of such marketplaces is the compensation mechanism for data providers. Classic payoff-allocation methods, such as the Shapley value, can be vulnerable to data-replication attacks, and are infeasible to compute in the absence of efficient approximation algorithms. To address these challenges, we present an extensive theoretical study on the vulnerabilities of game theoretic payoff-allocation schemes to replication attacks. Our insights apply to a wide range of payoff-allocation schemes, and enable the design of customised replication-robust payoff-allocations. Furthermore, we present a novel efficient sampling algorithm for approximating payoff-allocation schemes based on marginal contributions. In our experiments, we validate the replication-robustness of classic payoff-allocation schemes and new payoff-allocation schemes derived from our theoretical insights. We also demonstrate the efficiency of our proposed sampling algorithm on a wide range of machine learning tasks.
We study the statistical limits of Imitation Learning (IL) in episodic Markov Decision Processes (MDPs) with a state space $mathcal{S}$. We focus on the known-transition setting where the learner is provided a dataset of $N$ length-$H$ trajectories from a deterministic expert policy and knows the MDP transition. We establish an upper bound $O(|mathcal{S}|H^{3/2}/N)$ for the suboptimality using the Mimic-MD algorithm in Rajaraman et al (2020) which we prove to be computationally efficient. In contrast, we show the minimax suboptimality grows as $Omega( H^{3/2}/N)$ when $|mathcal{S}|geq 3$ while the unknown-transition setting suffers from a larger sharp rate $Theta(|mathcal{S}|H^2/N)$ (Rajaraman et al (2020)). The lower bound is established by proving a two-way reduction between IL and the value estimation problem of the unknown expert policy under any given reward function, as well as building connections with linear functional estimation with subsampled observations. We further show that under the additional assumption that the expert is optimal for the true reward function, there exists an efficient algorithm, which we term as Mimic-Mixture, that provably achieves suboptimality $O(1/N)$ for arbitrary 3-state MDPs with rewards only at the terminal layer. In contrast, no algorithm can achieve suboptimality $O(sqrt{H}/N)$ with high probability if the expert is not constrained to be optimal. Our work formally establishes the benefit of the expert optimal assumption in the known transition setting, while Rajaraman et al (2020) showed it does not help when transitions are unknown.
Demand forecasting is a central component of the replenishment process for retailers, as it provides crucial input for subsequent decision making like ordering processes. In contrast to point estimates, such as the conditional mean of the underlying probability distribution, or confidence intervals, forecasting complete probability density functions allows to investigate the impact on operational metrics, which are important to define the business strategy, over the full range of the expected demand. Whereas metrics evaluating point estimates are widely used, methods for assessing the accuracy of predicted distributions are rare, and this work proposes new techniques for both qualitative and quantitative evaluation methods. Using the supervised machine learning method Cyclic Boosting, complete individual probability density functions can be predicted such that each prediction is fully explainable. This is of particular importance for practitioners, as it allows to avoid black-box models and understand the contributing factors for each individual prediction. Another crucial aspect in terms of both explainability and generalizability of demand forecasting methods is the limitation of the influence of temporal confounding, which is prevalent in most state of the art approaches.
The idea of unfolding iterative algorithms as deep neural networks has been widely applied in solving sparse coding problems, providing both solid theoretical analysis in convergence rate and superior empirical performance. However, for sparse nonlinear regression problems, a similar idea is rarely exploited due to the complexity of nonlinearity. In this work, we bridge this gap by introducing the Nonlinear Learned Iterative Shrinkage Thresholding Algorithm (NLISTA), which can attain a linear convergence under suitable conditions. Experiments on synthetic data corroborate our theoretical results and show our method outperforms state-of-the-art methods.
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