Generalization bounds which assess the difference between the true risk and the empirical risk have been studied extensively. However, to obtain bounds, current techniques use strict assumptions such as a uniformly bounded or a Lipschitz loss function. To avoid these assumptions, in this paper, we propose a new generalization bound for Bayesian deep nets by exploiting the contractivity of the Log-Sobolev inequalities. Using these inequalities adds an additional loss-gradient norm term to the generalization bound, which is intuitively a surrogate of the model complexity. Empirically, we analyze the affect of this loss-gradient norm term using different deep nets.

Graph-structured data arise ubiquitously in many application domains. A fundamental problem is to quantify their similarities. Graph kernels are often used for this purpose, which decompose graphs into substructures and compare these substructures. However, most of the existing graph kernels do not have the property of scale-adaptivity, i.e., they cannot compare graphs at multiple levels of granularities. Many real-world graphs such as molecules exhibit structure at varying levels of granularities. To tackle this problem, we propose a new graph kernel called Tree++ in this paper. At the heart of Tree++ is a graph kernel called the path-pattern graph kernel. The path-pattern graph kernel first builds a truncated BFS tree rooted at each vertex and then uses paths from the root to every vertex in the truncated BFS tree as features to represent graphs. The path-pattern graph kernel can only capture graph similarity at fine granularities. In order to capture graph similarity at coarse granularities, we incorporate a new concept called super path into it. The super path contains truncated BFS trees rooted at the vertices in a path. Our evaluation on a variety of real-world graphs demonstrates that Tree++ achieves the best classification accuracy compared with previous graph kernels.

Safety-critical applications require machine learning models that output accurate and calibrated probabilities. While uncalibrated deep networks are known to make over-confident predictions, it is unclear how model confidence is impacted by the variations in the data, such as label noise or class size. In this paper, we investigate the role of the dataset quality by studying the impact of dataset size and the label noise on the model confidence. We theoretically explain and experimentally demonstrate that, surprisingly, label noise in the training data leads to under-confident networks, while reduced dataset size leads to over-confident models. We then study the impact of dataset heterogeneity, where data quality varies across classes, on model confidence. We demonstrate that this leads to heterogenous confidence/accuracy behavior in the test data and is poorly handled by the standard calibration algorithms. To overcome this, we propose an intuitive heterogenous calibration technique and show that the proposed approach leads to improved calibration metrics (both average and worst-case errors) on the CIFAR datasets.

Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general structured principal subspace estimation problem which includes as special cases non-negative PCA/SVD, sparse PCA/SVD, subspace constrained PCA/SVD, and spectral clustering. General minimax lower and upper bounds are established to characterize the interplay between the information-geometric complexity of the structural set for the principal subspaces, the signal-to-noise ratio (SNR), and the dimensionality. The results yield interesting phase transition phenomena concerning the rates of convergence as a function of the SNRs and the fundamental limit for consistent estimation. Applying the general results to the specific settings yields the minimax rates of convergence for those problems, including the previous unknown optimal rates for non-negative PCA/SVD, sparse SVD and subspace constrained PCA/SVD.

In future cell-free (or cell-less) wireless networks, a large number of devices in a geographical area will be served simultaneously in non-orthogonal multiple access scenarios by a large number of distributed access points (APs), which coordinate with a centralized processing pool. For such a centralized cell-free network with static predefined beamforming design, we first derive a closed-form expression of the uplink per-user probability of outage. To significantly reduce the complexity of joint processing of users' signals in presence of a large number of devices and APs, we propose a novel dynamic cell-free network architecture. In this architecture, the distributed APs are partitioned (i.e. clustered) among a set of subgroups with each subgroup acting as a virtual AP equipped with a distributed antenna system (DAS). The conventional static cell-free network is a special case of this dynamic cell-free network when the cluster size is one. For this dynamic cell-free network, we propose a successive interference cancellation (SIC)-enabled signal detection method and an inter-user-interference (IUI)-aware DAS's receive diversity combining scheme. We then formulate the general problem of clustering APs and designing the beamforming vectors with an objective to maximizing the sum rate or maximizing the minimum rate. To this end, we propose a hybrid deep reinforcement learning (DRL) model, namely, a deep deterministic policy gradient (DDPG)-deep double Q-network (DDQN) model, to solve the optimization problem for online implementation with low complexity. The DRL model for sum-rate optimization significantly outperforms that for maximizing the minimum rate in terms of average per-user rate performance. Also, in our system setting, the proposed DDPG-DDQN scheme is found to achieve around $78\%$ of the rate achievable through an exhaustive search-based design.

Deep reinforcement learning is successful in decision making for sophisticated games, such as Atari, Go, etc. However, real-world decision making often requires reasoning with partial information extracted from complex visual observations. This paper presents Discriminative Particle Filter Reinforcement Learning (DPFRL), a new reinforcement learning framework for complex partial observations. DPFRL encodes a differentiable particle filter in the neural network policy for explicit reasoning with partial observations over time. The particle filter maintains a belief using learned discriminative update, which is trained end-to-end for decision making. We show that using the discriminative update instead of standard generative models results in significantly improved performance, especially for tasks with complex visual observations, because they circumvent the difficulty of modeling complex observations that are irrelevant to decision making. In addition, to extract features from the particle belief, we propose a new type of belief feature based on the moment generating function. DPFRL outperforms state-of-the-art POMDP RL models in Flickering Atari Games, an existing POMDP RL benchmark, and in Natural Flickering Atari Games, a new, more challenging POMDP RL benchmark introduced in this paper. Further, DPFRL performs well for visual navigation with real-world data in the Habitat environment.

In a supervised learning setting, we have a yardstick or plumbline to judge how well we are doing: the response itself. A frequent question in biological and biomedical applications is whether a property of interest (say, disease type, cell type, the prognosis of a patient) can be "predicted", given one or more other properties, called the predictors. Often we are motivated by a situation in which the property to be predicted is unknown (it lies in the future, or is hard to measure), while the predictors are known. The crucial point is that we learn the prediction rule from a set of training data in which the property of interest is also known. Once we have the rule, we can either apply it to new data, and make actual predictions of unknown outcomes; or we can dissect the rule with the aim of better understanding the underlying biology. Compared to unsupervised learning and what we have seen in Chapters 5, 7 and 9, where we do not know what we are looking for or how to decide whether our result is "right", we are on much more solid ground with supervised learning: the objective is clearly stated, and there are straightforward criteria to measure how well we are doing. The central issues in supervised learning151151 Sometimes the term statistical learning is used, more or less exchangeably. Or did our rule indeed pick up some of the pertinent patterns in the system being studied, which will also apply to yet unseen new data? An example for overfitting: two regression lines are fit to data in the \((x, y)\)-plane (black points). We can think of such a line as a rule that predicts the \(y\)-value, given an \(x\)-value. Both lines are smooth, but the fits differ in what is called their bandwidth, which intuitively can be interpreted their stiffness. The blue line seems overly keen to follow minor wiggles in the data, while the orange line captures the general trend but is less detailed. The effective number of parameters needed to describe the blue line is much higher than for the orange line. Also, if we were to obtain additional data, it is likely that the blue line would do a worse job than the orange line in modeling the new data. We'll formalize these concepts –training error and test set error– later in this chapter. Although exemplified here with line fitting, the concept applies more generally to prediction models. See exemplary applications that motivate the use of supervised learning methods.

Despite its almost ubiquitous use in the business industry and social sciences, time series analysis and by extension time series forecasting is one of the least understood machine learning methods new data scientists and machine learning engineers are undertaking. The purpose of this blog is to provide an overview of this lesser-known but incredibly important machine learning technique. To answer this question, let's take a step back to discuss the types of data that we use for typical regression and classification tasks. When we make a prediction about a new observation, that model is built from hundreds or thousands of previous observations that are either all captured at a single point in time, or from data points in which time does not matter. This is known as cross-sectional data.

Many imbalanced classification tasks require a skillful model that predicts a crisp class label, where both classes are equally important. An example of an imbalanced classification problem where a class label is required and both classes are equally important is the detection of oil spills or slicks in satellite images. The detection of a spill requires mobilizing an expensive response, and missing an event is equally expensive, causing damage to the environment. One way to evaluate imbalanced classification models that predict crisp labels is to calculate the separate accuracy on the positive class and the negative class, referred to as sensitivity and specificity. These two measures can then be averaged using the geometric mean, referred to as the G-mean, that is insensitive to the skewed class distribution and correctly reports on the skill of the model on both classes. In this tutorial, you will discover how to develop a model to predict the presence of an oil spill in satellite images and evaluate it using the G-mean metric. Develop an Imbalanced Classification Model to Detect Oil Spills Photo by Lenny K Photography, some rights reserved. In this project, we will use a standard imbalanced machine learning dataset referred to as the "oil spill" dataset, "oil slicks" dataset or simply "oil."

Learning the cost function for optimal transport from observed transport plan or its samples has been cast as a bi-level optimization problem. In this paper, we derive an unconstrained convex optimization formulation for the problem which can be further augmented by any customizable regularization. This novel framework avoids repeatedly solving a forward optimal transport problem in each iteration which has been a thorny computational bottleneck for the bi-level optimization approach. To validate the effectiveness of this framework, we develop two numerical algorithms, one is a fast matrix scaling method based on the Sinkhorn-Knopp algorithm for the discrete case, and the other is a supervised learning algorithm that realizes the cost function as a deep neural network in the continuous case. Numerical results demonstrate promising efficiency and accuracy advantages of the proposed algorithms over existing state of the art methods.