Fairness, through its many forms and definitions, has become an important issue facing the machine learning community. In this work, we consider how to incorporate group fairness constraints in kernel regression methods. More specifically, we focus on examining the incorporation of these constraints in decision tree regression when cast as a form of kernel regression, with direct applications to random forests and boosted trees amongst other widespread popular inference techniques. We show that order of complexity of memory and computation is preserved for such models and bounds the expected perturbations to the model in terms of the number of leaves of the trees. Importantly, the approach works on trained models and hence can be easily applied to models in current use.

In this study, we propose a novel non-parametric embedded feature selection method based on minimizing the $\ell_0$ norm of the vector of an indicator variable, whose point-wise product of an input selects a subset of features. Our approach relies on the continuous relaxation of Bernoulli distributions, which allows our model to learn the parameters of the approximate Bernoulli distributions via tractable methods. Using these tools we present a general neural network that simultaneously minimizes a loss function while selecting relevant features. We also provide an information-theoretic justification of incorporating Bernoulli distribution into our approach. Finally, we demonstrate the potential of the approach on synthetic and real-life applications.

In the context of recent deep clustering studies, discriminative models dominate the literature and report the most competitive performances. These models learn a deep discriminative neural network classifier in which the labels are latent. Typically, they use multinomial logistic regression posteriors and parameter regularization, as is very common in supervised learning. It is generally acknowledged that discriminative objective functions (e.g., those based on the mutual information or the KL divergence) are more flexible than generative approaches (e.g., K-means) in the sense that they make fewer assumptions about the data distributions and, typically, yield much better unsupervised deep learning results. On the surface, several recent discriminative models may seem unrelated to K-means. This study shows that these models are, in fact, equivalent to K-means under mild conditions and common posterior models and parameter regularization. We prove that, for the commonly used logistic regression posteriors, maximizing the $L_2$ regularized mutual information via an approximate alternating direction method (ADM) is equivalent to a soft and regularized K-means loss. Our theoretical analysis not only connects directly several recent state-of-the-art discriminative models to K-means, but also leads to a new soft and regularized deep K-means algorithm, which yields competitive performance on several image clustering benchmarks.

We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is derived from a suitable formulation of the KKT conditions based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regression coefficients along the solution path with warm start. At each knot in the path, SNAP converges locally superlinearly for the Enet criterion and achieves an optimal local convergence rate for the LASSO criterion, i.e., SNAP converges in one step at the cost of two matrix-vector multiplication per iteration. Under certain regularity conditions on the design matrix and the minimum magnitude of the nonzero elements of the target regression coefficients, we show that SNAP hits a solution with the same signs as the regression coefficients and achieves a sharp estimation error bound in finite steps with high probability. The computational complexity of SNAP is shown to be the same as that of LARS and coordinate descent algorithms per iteration. Simulation studies and real data analysis support our theoretical results and demonstrate that SNAP is faster and accurate than LARS and coordinate descent algorithms.

Following are the various steps we will walk together and try gaining an understanding. In this post, we will consider Linear Regression as the algorithm where the target variable'y' will be explained by 2 features'x1' and'x2' whose coefficients are β1 and β2. First up, lets get some minor prerequisites out of the way in order to understand their use down the line. Optional: Refer Chapter 3 in the link below to gain understanding about Linear Regression. In Fig 1(a) below, Gradient Descent is represented in 3-dim.

Abstract--Azure (the cloud service provided by Microsoft) is composed of physical computing units which are called nodes. These nodes are controlled by a software component called Fabric Controller (FC), which can consider the nodes to be in one of many different states such as Ready, Unhealthy, Booting, etc. Some of these states correspond to a node being unresponsive to FCs requests. When a node goes unresponsive for more than a set threshold, FC intervenes and reboots the node. We minimized the downtime caused by the intervention threshold when a node switches to Unhealthy state by fitting various heavy-tail probability distributions. We consider using features of the node to customize the organic recovery model to the individual nodes that go unhealthy. This regression approach allows us to use information about the node like hardware, software versions, historical performance indicators, etc. to inform the organic recovery model and hence the optimal threshold. In another direction, we consider generalizing this to an arbitrary number of thresholds within the node state machine (or Markov chain). When the states become intertwined in ways that different thresholds start affecting each other, we cant simply optimize each of them in isolation. For best results, we must consider this as an optimization problem in many variables (the number of thresholds). We no longer have a nice closed form solution for this more complex problem like we did with one threshold, but we can still use numerical techniques (gradient descent) to solve it. Section 2 will briefly go over the architecture of Azure and the data we use for modeling. Sections 3 and 4 will formulate the problem mathematically. In section 5, we will discuss extending the model to multiple thresholds and in section 6, we will explore ways to use regression, so we can leverage features to customize our recovery models.

Sparse model selection is ubiquitous from linear regression to graphical models where regularization paths, as a family of estimators upon the regularization parameter varying, are computed when the regularization parameter is unknown or decided data-adaptively. Traditional computational methods rely on solving a set of optimization problems where the regularization parameters are fixed on a grid that might be inefficient. In this paper, we introduce a simple iterative regularization path, which follows the dynamics of a sparse Mirror Descent algorithm or a generalization of Linearized Bregman Iterations with nonlinear loss. Its performance is competitive to \texttt{glmnet} with a further bias reduction. A path consistency theory is presented that under the Restricted Strong Convexity (RSC) and the Irrepresentable Condition (IRR), the path will first evolve in a subspace with no false positives and reach an estimator that is sign-consistent or of minimax optimal $\ell_2$ error rate. Early stopping regularization is required to prevent overfitting. Application examples are given in sparse logistic regression and Ising models for NIPS coauthorship.

This paper proposes a two-stage scoring approach to help lenders decide their fund allocations in the peer-to-peer (P2P) lending market. The existing scoring approaches focus on only either probability of default (PD) prediction, known as credit scoring, or profitability prediction, known as profit scoring, to identify the best loans for investment. Credit scoring fails to deliver the main need of lenders on how much profit they may obtain through their investment. On the other hand, profit scoring can satisfy that need by predicting the investment profitability. However, profit scoring completely ignores the class imbalance problem where most of the past loans are non-default. Consequently, ignorance of the class imbalance problem significantly affects the accuracy of profitability prediction. Our proposed two-stage scoring approach is an integration of credit scoring and profit scoring to address the above challenges. More specifically, stage 1 is designed as credit scoring to identify non-default loans while the imbalanced nature of loan status is considered in PD prediction. The loans identified as non-default are then moved to stage 2 for prediction of profitability, measured by internal rate of return. Wide and deep learning is used to build the predictive models in both stages to achieve both memorization and generalization. Extensive numerical studies are conducted based on real-world data to verify the effectiveness of the proposed approach. The numerical studies indicate our two-stage scoring approach outperforms the existing credit scoring and profit scoring approaches.

Robust estimation under Huber's $\epsilon$-contamination model has become an important topic in statistics and theoretical computer science. Rate-optimal procedures such as Tukey's median and other estimators based on statistical depth functions are impractical because of their computational intractability. In this paper, we establish an intriguing connection between f-GANs and various depth functions through the lens of f-Learning. Similar to the derivation of f-GAN, we show that these depth functions that lead to rate-optimal robust estimators can all be viewed as variational lower bounds of the total variation distance in the framework of f-Learning. This connection opens the door of computing robust estimators using tools developed for training GANs. In particular, we show that a JS-GAN that uses a neural network discriminator with at least one hidden layer is able to achieve the minimax rate of robust mean estimation under Huber's $\epsilon$-contamination model. Interestingly, the hidden layers for the neural net structure in the discriminator class is shown to be necessary for robust estimation.

One of the definitional debates that bedevils the artificial intelligence (AI) field is whether machine learning is an AI-based method or technology. Or is it just an analytics-based activity? After all, it is statistical in nature, and attempts--as virtually analytical methods do--to fit a line or curve to a set of data points. And what difference does it make? Basic machine learning is practically indistinguishable from predictive analytics.