Convex-constrained Sparse Additive Modeling and Its Extensions

Sparse additive modeling is a class of effective methods for performing high-dimensional nonparametric regression. In this work we show how shape constraints such as convexity/concavity and their extensions, can be integrated into additive models. The proposed sparse difference of convex additive models (SDCAM) can estimate most continuous functions without any a priori smoothness assumption. Motivated by a characterization of difference of convex functions, our method incorporates a natural regularization functional to avoid overfitting and to reduce model complexity. Computationally, we develop an efficient backfitting algorithm with linear per-iteration complexity. Experiments on both synthetic and real data verify that our method is competitive against state-of-the-art sparse additive models, with improved performance in most scenarios.Comment: 17 pages, 2 figure

Additive Approximations in High Dimensional Nonparametric Regression via the SALSA

High dimensional nonparametric regression is an inherently difficult problem with known lower bounds depending exponentially in dimension. A popular strategy to alleviate this curse of dimensionality has been to use additive models of \emph{first order}, which model the regression function as a sum of independent functions on each dimension. Though useful in controlling the variance of the estimate, such models are often too restrictive in practical settings. Between non-additive models which often have large variance and first order additive models which have large bias, there has been little work to exploit the trade-off in the middle via additive models of intermediate order. In this work, we propose SALSA, which bridges this gap by allowing interactions between variables, but controls model capacity by limiting the order of interactions. SALSA minimises the residual sum of squares with squared RKHS norm penalties. Algorithmically, it can be viewed as Kernel Ridge Regression with an additive kernel. When the regression function is additive, the excess risk is only polynomial in dimension. Using the Girard-Newton formulae, we efficiently sum over a combinatorial number of terms in the additive expansion. Via a comparison on $15$ real datasets, we show that our method is competitive against $21$ other alternatives.Comment: International Conference on Machine Learning (ICML) 201

Generalized Conditional Gradient for Sparse Estimation

Structured sparsity is an important modeling tool that expands the applicability of convex formulations for data analysis, however it also creates significant challenges for efficient algorithm design. In this paper we investigate the generalized conditional gradient (GCG) algorithm for solving structured sparse optimization problems---demonstrating that, with some enhancements, it can provide a more efficient alternative to current state of the art approaches. After providing a comprehensive overview of the convergence properties of GCG, we develop efficient methods for evaluating polar operators, a subroutine that is required in each GCG iteration. In particular, we show how the polar operator can be efficiently evaluated in two important scenarios: dictionary learning and structured sparse estimation. A further improvement is achieved by interleaving GCG with fixed-rank local subspace optimization. A series of experiments on matrix completion, multi-class classification, multi-view dictionary learning and overlapping group lasso shows that the proposed method can significantly reduce the training cost of current alternatives.Comment: 67 pages, 20 figure

Provably noise-robust, regularised kk-means clustering

We consider the problem of clustering in the presence of noise. That is, when on top of cluster structure, the data also contains a subset of \emph{unstructured} points. Our goal is to detect the clusters despite the presence of many unstructured points. Any algorithm that achieves this goal is noise-robust. We consider a regularisation method which converts any center-based clustering objective into a noise-robust one. We focus on the $k$-means objective and we prove that the regularised version of $k$-means is NP-Hard even for $k=1$. We consider two algorithms based on the convex (sdp and lp) relaxation of the regularised objective and prove robustness guarantees for both. The sdp and lp relaxation of the standard (non-regularised) $k$-means objective has been previously studied by [ABC+15]. Under the stochastic ball model of the data they show that the sdp-based algorithm recovers the underlying structure as long as the balls are separated by $\delta > 2\sqrt{2} + \epsilon$. We improve upon this result in two ways. First, we show recovery even for $\delta > 2 + \epsilon$. Second, our regularised algorithm recovers the balls even in the presence of noise so long as the number of noisy points is not too large. We complement our theoretical analysis with simulations and analyse the effect of various parameters like regularization constant, noise-level etc. on the performance of our algorithm. In the presence of noise, our algorithm performs better than $k$-means++ on MNIST

Sum-of-Squares Polynomial Flow

Triangular map is a recent construct in probability theory that allows one to transform any source probability density function to any target density function. Based on triangular maps, we propose a general framework for high-dimensional density estimation, by specifying one-dimensional transformations (equivalently conditional densities) and appropriate conditioner networks. This framework (a) reveals the commonalities and differences of existing autoregressive and flow based methods, (b) allows a unified understanding of the limitations and representation power of these recent approaches and, (c) motivates us to uncover a new Sum-of-Squares (SOS) flow that is interpretable, universal, and easy to train. We perform several synthetic experiments on various density geometries to demonstrate the benefits (and short-comings) of such transformations. SOS flows achieve competitive results in simulations and several real-world datasets.Comment: 13 pages, ICML'201

Convergence of Gradient Methods on Bilinear Zero-Sum Games

Min-max formulations have attracted great attention in the ML community due to the rise of deep generative models and adversarial methods, while understanding the dynamics of gradient algorithms for solving such formulations has remained a grand challenge. As a first step, we restrict to bilinear zero-sum games and give a systematic analysis of popular gradient updates, for both simultaneous and alternating versions. We provide exact conditions for their convergence and find the optimal parameter setup and convergence rates. In particular, our results offer formal evidence that alternating updates converge "better" than simultaneous ones

Regularizers versus Losses for Nonlinear Dimensionality Reduction: A Factored View with New Convex Relaxations

We demonstrate that almost all non-parametric dimensionality reduction methods can be expressed by a simple procedure: regularized loss minimization plus singular value truncation. By distinguishing the role of the loss and regularizer in such a process, we recover a factored perspective that reveals some gaps in the current literature. Beyond identifying a useful new loss for manifold unfolding, a key contribution is to derive new convex regularizers that combine distance maximization with rank reduction. These regularizers can be applied to any loss.Comment: Appears in Proceedings of the 29th International Conference on Machine Learning (ICML 2012

Distributional Reinforcement Learning for Efficient Exploration

In distributional reinforcement learning (RL), the estimated distribution of value function models both the parametric and intrinsic uncertainties. We propose a novel and efficient exploration method for deep RL that has two components. The first is a decaying schedule to suppress the intrinsic uncertainty. The second is an exploration bonus calculated from the upper quantiles of the learned distribution. In Atari 2600 games, our method outperforms QR-DQN in 12 out of 14 hard games (achieving 483 \% average gain across 49 games in cumulative rewards over QR-DQN with a big win in Venture). We also compared our algorithm with QR-DQN in a challenging 3D driving simulator (CARLA). Results show that our algorithm achieves near-optimal safety rewards twice faster than QRDQN

Indiscriminate Data Poisoning Attacks on Neural Networks

Data poisoning attacks, in which a malicious adversary aims to influence a model by injecting "poisoned" data into the training process, have attracted significant recent attention. In this work, we take a closer look at existing poisoning attacks and connect them with old and new algorithms for solving sequential Stackelberg games. By choosing an appropriate loss function for the attacker and optimizing with algorithms that exploit second-order information, we design poisoning attacks that are effective on neural networks. We present efficient implementations that exploit modern auto-differentiation packages and allow simultaneous and coordinated generation of tens of thousands of poisoned points, in contrast to existing methods that generate poisoned points one by one. We further perform extensive experiments that empirically explore the effect of data poisoning attacks on deep neural networks

Robust Multiple Kernel k-means Clustering using Min-Max Optimization

Multiple kernel learning is a type of multiview learning that combines different data modalities by capturing view-specific patterns using kernels. Although supervised multiple kernel learning has been extensively studied, until recently, only a few unsupervised approaches have been proposed. In the meanwhile, adversarial learning has recently received much attention. Many works have been proposed to defend against adversarial examples. However, little is known about the effect of adversarial perturbation in the context of multiview learning, and even less in the unsupervised case. In this study, we show that adversarial features added to a view can make the existing approaches with the min-max formulation in multiple kernel clustering yield unfavorable clusters. To address this problem and inspired by recent works in adversarial learning, we propose a multiple kernel clustering method with the min-max framework that aims to be robust to such adversarial perturbation. We evaluate the robustness of our method on simulation data under different types of adversarial perturbations and show that it outperforms several compared existing methods. In the real data analysis, We demonstrate the utility of our method on a real-world problem.Comment: R package is available at https://github.com/SeojinBang/MKK