We establish an equivalence between the $\ell_2$-regularized solution path for a convex loss function, and the solution of an ordinary differentiable equation (ODE). Importantly, this equivalence reveals that the solution path can be viewed as the flow of a hybrid of gradient descent and Newton method applying to the empirical loss, which is similar to a widely used optimization technique called trust region method. This provides an interesting algorithmic view of $\ell_2$ regularization, and is in contrast to the conventional view that the $\ell_2$ regularization solution path is similar to the gradient flow of the empirical loss.New path-following algorithms based on homotopy methods and numerical ODE solvers are proposed to numerically approximate the solution path. In particular, we consider respectively Newton method and gradient descent method as the basis algorithm for the homotopy method, and establish their approximation error rates over the solution path. Importantly, our theory suggests novel schemes to choose grid points that guarantee an arbitrarily small suboptimality for the solution path. In terms of computational cost, we prove that in order to achieve an $\epsilon$-suboptimality for the entire solution path, the number of Newton steps required for the Newton method is $\mathcal O(\epsilon^{-1/2})$, while the number of gradient steps required for the gradient descent method is $\mathcal O\left(\epsilon^{-1} \ln(\epsilon^{-1})\right)$. Finally, we use $\ell_2$-regularized logistic regression as an illustrating example to demonstrate the effectiveness of the proposed path-following algorithms.

We demonstrate the use of Adaptive Stress Testing to detect and address potential vulnerabilities in a financial environment. We develop a simplified model for credit card fraud detection that utilizes a linear regression classifier based on historical payment transaction data coupled with business rules. We then apply the reinforcement learning model known as Adaptive Stress Testing to train an agent, that can be thought of as a potential fraudster, to find the most likely path to system failure -- successfully defrauding the system. We show the connection between this most likely failure path and the limits of the classifier and discuss how the fraud detection system's business rules can be further augmented to mitigate these failure modes.

Density ratio estimation serves as an important technique in the unsupervised machine learning toolbox. However, such ratios are difficult to estimate for complex, high-dimensional data, particularly when the densities of interest are sufficiently different. In our work, we propose to leverage an invertible generative model to map the two distributions into a common feature space prior to estimation. This featurization brings the densities closer together in latent space, sidestepping pathological scenarios where the learned density ratios in input space can be arbitrarily inaccurate. At the same time, the invertibility of our feature map guarantees that the ratios computed in feature space are equivalent to those in input space. Empirically, we demonstrate the efficacy of our approach in a variety of downstream tasks that require access to accurate density ratios such as mutual information estimation, targeted sampling in deep generative models, and classification with data augmentation.

Markov Chain Monte Carlo inference of target posterior distributions in machine learning is predominately conducted via Hamiltonian Monte Carlo and its variants. This is due to Hamiltonian Monte Carlo based samplers ability to suppress random-walk behaviour. As with other Markov Chain Monte Carlo methods, Hamiltonian Monte Carlo produces auto-correlated samples which results in high variance in the estimators, and low effective sample size rates in the generated samples. Adding antithetic sampling to Hamiltonian Monte Carlo has been previously shown to produce higher effective sample rates compared to vanilla Hamiltonian Monte Carlo. In this paper, we present new algorithms which are antithetic versions of Riemannian Manifold Hamiltonian Monte Carlo and Quantum-Inspired Hamiltonian Monte Carlo. The Riemannian Manifold Hamiltonian Monte Carlo algorithm improves on Hamiltonian Monte Carlo by taking into account the local geometry of the target, which is beneficial for target densities that may exhibit strong correlations in the parameters. Quantum-Inspired Hamiltonian Monte Carlo is based on quantum particles that can have random mass. Quantum-Inspired Hamiltonian Monte Carlo uses a random mass matrix which results in better sampling than Hamiltonian Monte Carlo on spiky and multi-modal distributions such as jump diffusion processes. The analysis is performed on jump diffusion process using real world financial market data, as well as on real world benchmark classification tasks using Bayesian logistic regression.

The common belief about the growing medium of livestreaming is that its value lies in its "live" component. In this paper, we leverage data from a large livestreaming platform to examine this belief. We are able to do this as this platform also allows viewers to purchase the recorded version of the livestream. We summarize the value of livestreaming content by estimating how demand responds to price before, on the day of, and after the livestream. We do this by proposing a generalized Orthogonal Random Forest framework. This framework allows us to estimate heterogeneous treatment effects in the presence of high-dimensional confounders whose relationships with the treatment policy (i.e., price) are complex but partially known. We find significant dynamics in the price elasticity of demand over the temporal distance to the scheduled livestreaming day and after. Specifically, demand gradually becomes less price sensitive over time to the livestreaming day and is inelastic on the livestreaming day. Over the post-livestream period, demand is still sensitive to price, but much less than the pre-livestream period. This indicates that the vlaue of livestreaming persists beyond the live component. Finally, we provide suggestive evidence for the likely mechanisms driving our results. These are quality uncertainty reduction for the patterns pre- and post-livestream and the potential of real-time interaction with the creator on the day of the livestream.

Temperature monitoring during the life time of heat source components in engineering systems becomes essential to guarantee the normal work and the working life of these components. However, prior methods, which mainly use the interpolate estimation to reconstruct the temperature field from limited monitoring points, require large amounts of temperature tensors for an accurate estimation. This may decrease the availability and reliability of the system and sharply increase the monitoring cost. To solve this problem, this work develops a novel physics-informed deep reversible regression models for temperature field reconstruction of heat-source systems (TFR-HSS), which can better reconstruct the temperature field with limited monitoring points unsupervisedly. First, we define the TFR-HSS task mathematically, and numerically model the task, and hence transform the task as an image-to-image regression problem. Then this work develops the deep reversible regression model which can better learn the physical information, especially over the boundary. Finally, considering the physical characteristics of heat conduction as well as the boundary conditions, this work proposes the physics-informed reconstruction loss including four training losses and jointly learns the deep surrogate model with these losses unsupervisedly. Experimental studies have conducted over typical two-dimensional heat-source systems to demonstrate the effectiveness of the proposed method.

Recently the shape-restricted inference has gained popularity in statistical and econometric literature in order to relax the linear or quadratic covariate effect in regression analyses. The typical shape-restricted covariate effect includes monotonic increasing, decreasing, convexity or concavity. In this paper, we introduce the shape-restricted inference to the celebrated Cox regression model (SR-Cox), in which the covariate response is modeled as shape-restricted additive functions. The SR-Cox regression approximates the shape-restricted functions using a spline basis expansion with data driven choice of knots. The underlying minimization of negative log-likelihood function is formulated as a convex optimization problem, which is solved with an active-set optimization algorithm. The highlight of this algorithm is that it eliminates the superfluous knots automatically. When covariate effects include combinations of convex or concave terms with unknown forms and linear terms, the most interesting finding is that SR-Cox produces accurate linear covariate effect estimates which are comparable to the maximum partial likelihood estimates if indeed the forms are known. We conclude that concave or convex SR-Cox models could significantly improve nonlinear covariate response recovery and model goodness of fit.

"Logistic Regression is not Regression but a Classification Algorithm". You might have seen this in latest popular machine learning books, blogs or you might have heard *Data Science Gurus* utter the same in their highly subscribed YouTube channels. Machine learning has usurped and renamed many statistical techniques. Often to the extent that they now disbelieve and reject its statistical origins. Case in point is "Logistic regression is not Regression" However, nothing can be further from the truth than this assertion.

As a beginner, you want to know what are the models and algorithms available in machine learning that make our work more easier. Supervised learning involves learning a function that maps an input to an output based on example input-output pairs. In regression models, the output is continuous. The idea of linear regression is simply finding a line that best fits the data. Extensions of linear regression include multiple linear regression.

This can be naturally modeled constantly explore innovative ways to provide optimal online as contextual bandit problems (e.g., LinUCB [18] and Thompson user experiences for gaining competitive advantages. The great sampling [7]), where each arm corresponds to an item, pulling an needs of developing intelligent interactive recommendation systems item indicates recommending an item, and the reward is the instant are indicated, which could sequentially suggest users the most feedback from a user after the recommendation. Contextual proper items by accurately predicting their preferences, while receiving bandit algorithms have been widely applied in various interactive the up-to-date feedback to refine the recommendation results, recommender systems by achieving an optimal tradeoff between continuosly. Multi-armed bandit algorithms, which have been exploration and exploitation. Based on the preliminary studies [15, widely applied into various online systems, are quite capable of 18, 1], several practical challenges are identified in modern recommender delivering such efficient recommendation services.