In this article, we will study some important data preprocessing methods. It is a very important step to visualize the data and make it in a suitable form so that the estimators (algorithm) fit well with good accuracy. Standardization is a process that deals with the mean and standard deviation of the data points. As raw data, the values are varying from very low to very high. So, to avoid the low performance in the model we use standardization.

This is a dataset that counts the number of cars passing by during baseball games at the Los Angeles Dodgers home stadium. Let's see how to estimate the traffic. In the preceding lines, the C parameter specifies the penalty for misclassification and epsilon specifies the limit within which no penalty is applied. Output: Mean absolute error 7.28. Output: Predicted traffic is given as 29.

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.

We examine the general problem of inter-domain Gaussian Processes (GPs): problems where the GP realization and the noisy observations of that realization lie on different domains. When the mapping between those domains is linear, such as integration or differentiation, inference is still closed form. However, many of the scaling and approximation techniques that our community has developed do not apply to this setting. In this work, we introduce the hierarchical inducing point GP (HIP-GP), a scalable inter-domain GP inference method that enables us to improve the approximation accuracy by increasing the number of inducing points to the millions. HIP-GP, which relies on inducing points with grid structure and a stationary kernel assumption, is suitable for low-dimensional problems. In developing HIP-GP, we introduce (1) a fast whitening strategy, and (2) a novel preconditioner for conjugate gradients which can be helpful in general GP settings.

Communication efficiency and robustness are two major issues in modern distributed learning framework. This is due to the practical situations where some computing nodes may have limited communication power or may behave adversarial behaviors. To address the two issues simultaneously, this paper develops two communication-efficient and robust distributed learning algorithms for convex problems. Our motivation is based on surrogate likelihood framework and the median and trimmed mean operations. Particularly, the proposed algorithms are provably robust against Byzantine failures, and also achieve optimal statistical rates for strong convex losses and convex (non-smooth) penalties. For typical statistical models such as generalized linear models, our results show that statistical errors dominate optimization errors in finite iterations. Simulated and real data experiments are conducted to demonstrate the numerical performance of our algorithms.

Bayesian optimization (BO) is a powerful paradigm for efficient optimization of black-box objective functions. High-dimensional BO presents a particular challenge, in part because the curse of dimensionality makes it difficult to define as well as do inference over a suitable class of surrogate models. We argue that Gaussian process surrogate models defined on sparse axis-aligned subspaces offer an attractive compromise between flexibility and parsimony. We demonstrate that our approach, which relies on Hamiltonian Monte Carlo for inference, can rapidly identify sparse subspaces relevant to modeling the unknown objective function, enabling sample-efficient high-dimensional BO. In an extensive suite of experiments comparing to existing methods for high-dimensional BO we demonstrate that our algorithm, Sparse Axis-Aligned Subspace BO (SAASBO), achieves excellent performance on several synthetic and real-world problems without the need to set problem-specific hyperparameters.

Causal graphs (CGs) are compact representations of the knowledge of the data generating processes behind the data distributions. When a CG is available, e.g., from the domain knowledge, we can infer the conditional independence (CI) relations that should hold in the data distribution. However, it is not straightforward how to incorporate this knowledge into predictive modeling. In this work, we propose a model-agnostic data augmentation method that allows us to exploit the prior knowledge of the CI encoded in a CG for supervised machine learning. We theoretically justify the proposed method by providing an excess risk bound indicating that the proposed method suppresses overfitting by reducing the apparent complexity of the predictor hypothesis class. Using real-world data with CGs provided by domain experts, we experimentally show that the proposed method is effective in improving the prediction accuracy, especially in the small-data regime.

Designing off-policy reinforcement learning algorithms is typically a very challenging task, because a desirable iteration update often involves an expectation over an on-policy distribution. Prior off-policy actor-critic (AC) algorithms have introduced a new critic that uses the density ratio for adjusting the distribution mismatch in order to stabilize the convergence, but at the cost of potentially introducing high biases due to the estimation errors of both the density ratio and value function. In this paper, we develop a doubly robust off-policy AC (DR-Off-PAC) for discounted MDP, which can take advantage of learned nuisance functions to reduce estimation errors. Moreover, DR-Off-PAC adopts a single timescale structure, in which both actor and critics are updated simultaneously with constant stepsize, and is thus more sample efficient than prior algorithms that adopt either two timescale or nested-loop structure. We study the finite-time convergence rate and characterize the sample complexity for DR-Off-PAC to attain an $\epsilon$-accurate optimal policy. We also show that the overall convergence of DR-Off-PAC is doubly robust to the approximation errors that depend only on the expressive power of approximation functions. To the best of our knowledge, our study establishes the first overall sample complexity analysis for a single time-scale off-policy AC algorithm.

We present a modelling framework for the investigation of supervised learning in non-stationary environments. Specifically, we model two example types of learning systems: prototype-based Learning Vector Quantization (LVQ) for classification and shallow, layered neural networks for regression tasks. We investigate so-called student teacher scenarios in which the systems are trained from a stream of high-dimensional, labeled data. Properties of the target task are considered to be non-stationary due to drift processes while the training is performed. Different types of concept drift are studied, which affect the density of example inputs only, the target rule itself, or both. By applying methods from statistical physics, we develop a modelling framework for the mathematical analysis of the training dynamics in non-stationary environments. Our results show that standard LVQ algorithms are already suitable for the training in non-stationary environments to a certain extent. However, the application of weight decay as an explicit mechanism of forgetting does not improve the performance under the considered drift processes. Furthermore, we investigate gradient-based training of layered neural networks with sigmoidal activation functions and compare with the use of rectified linear units (ReLU). Our findings show that the sensitivity to concept drift and the effectiveness of weight decay differs significantly between the two types of activation function.

Recent years have seen a proliferation of new digital products for the efficient management of indoor spaces, with important applications like emergency management, virtual property showcasing and interior design. These products rely on accurate 3D models of the environments considered, including information on both architectural and non-permanent elements. These models must be created from measured data such as RGB-D images or 3D point clouds, whose capture and consolidation involves lengthy data workflows. This strongly limits the rate at which 3D models can be produced, preventing the adoption of many digital services for indoor space management. We provide an alternative to such data-intensive procedures by presenting Walk2Map, a data-driven approach to generate floor plans only from trajectories of a person walking inside the rooms. Thanks to recent advances in data-driven inertial odometry, such minimalistic input data can be acquired from the IMU readings of consumer-level smartphones, which allows for an effortless and scalable mapping of real-world indoor spaces. Our work is based on learning the latent relation between an indoor walk trajectory and the information represented in a floor plan: interior space footprint, portals, and furniture. We distinguish between recovering area-related (interior footprint, furniture) and wall-related (doors) information and use two different neural architectures for the two tasks: an image-based Encoder-Decoder and a Graph Convolutional Network, respectively. We train our networks using scanned 3D indoor models and apply them in a cascaded fashion on an indoor walk trajectory at inference time. We perform a qualitative and quantitative evaluation using both simulated and measured, real-world trajectories, and compare against a baseline method for image-to-image translation. The experiments confirm the feasibility of our approach.