In the last section, we saw that linear regression hypothesis function needed to be modified inorder to be used for logistic regression problems. Now the obvious question is whether the same cost function can be used here also or is some modification necessary required. The reason that cost function worked for linear regression was because the hypothesis was linear and hence the cost function was convex shaped with a single global minimum. In case of logistic regression, the hypothesis function is no longer linear because of the sigmoid function and using the same definition for cost function would yield a function as shown below. It will be an impossible task to optimize the parameters theta when using such a cost function with so many local minimums.

The second type of Supervised learning algorithm we will look into is called the Logistic Regression. First things first, it is not a regression algorithm as its name suggests, rather it's used classification purposes. But the name comes from the fact that classification is done by considering the linear regression model. As mentioned earlier, Classification methods are used in problems where we want to label or categorize inputs. In this section we will be dealing with binary classification problems (those with 2 possible outcomes).

The second type of Supervised learning algorithm we will look into is called the Logistic Regression. First things first, it is not a regression algorithm as its name suggests, rather it's used classification purposes. But the name comes from the fact that classification is done by considering the linear regression model. As mentioned earlier, Classification methods are used in problems where we want to label or categorize inputs. In this section we will be dealing with binary classification problems (those with 2 possible outcomes).

In this research, an extensive literature review was performed on the recent developments of the ambulance routing problem (ARP) and ambulance location problem (ALP). Both are respective modifications of the vehicle routing problem (VRP) and maximum covering problem (MCP), with modifications to objective functions and constraints. Although alike, a key distinction is emergency service systems (EMS) are considered critical and the optimization of these has become all the more important as a result. Similar to their parent problems, these are NP-hard and must resort to approximations if the space size is too large. Much of the current work has simply been on modifying existing systems through simulation to achieve a more acceptable result. There has been attempts towards using meta-heuristics, though practical experimentation is lacking when compared to VRP or MCP. The contributions of this work are a comprehensive survey of current methodologies, summarized models, and suggested future improvements.

Spin-glasses are universal models that can capture complex behavior of many-body systems at the interface of statistical physics and computer science including discrete optimization, inference in graphical models, and automated reasoning. Computing the underlying structure and dynamics of such complex systems is extremely difficult due to the combinatorial explosion of their state space. Here, we develop deep generative continuous spin-glass distributions with normalizing flows to model correlations in generic discrete problems. We use a self-supervised learning paradigm by automatically generating the data from the spin-glass itself. We demonstrate that key physical and computational properties of the spin-glass phase can be successfully learned, including multi-modal steady-state distributions and topological structures among metastable states. Remarkably, we observe that the learning itself corresponds to a spin-glass phase transition within the layers of the trained normalizing flows. The inverse normalizing flows learns to perform reversible multi-scale coarse-graining operations which are very different from the typical irreversible renormalization group techniques.

Count data take on non-negative integer values and are challenging to properly analyze using standard linear-Gaussian methods such as linear regression and principal components analysis. Generalized linear models enable direct modeling of counts in a regression context using distributions such as the Poisson and negative binomial. When counts contain only relative information, multinomial or Dirichlet-multinomial models can be more appropriate. We review some of the fundamental connections between multinomial and count models from probability theory, providing detailed proofs. These relationships are useful for methods development in applications such as topic modeling of text data and genomics.

Despite its well-known shortcomings, $k$-means remains one of the most widely used approaches to data clustering. Current research continues to tackle its flaws while attempting to preserve its simplicity. Recently, the \textit{power $k$-means} algorithm was proposed to avoid trapping in local minima by annealing through a family of smoother surfaces. However, the approach lacks theoretical justification and fails in high dimensions when many features are irrelevant. This paper addresses these issues by introducing \textit{entropy regularization} to learn feature relevance while annealing. We prove consistency of the proposed approach and derive a scalable majorization-minimization algorithm that enjoys closed-form updates and convergence guarantees. In particular, our method retains the same computational complexity of $k$-means and power $k$-means, but yields significant improvements over both. Its merits are thoroughly assessed on a suite of real and synthetic data experiments.

Document network embedding aims at learning representations for a structured text corpus i.e. when documents are linked to each other. Recent algorithms extend network embedding approaches by incorporating the text content associated with the nodes in their formulations. In most cases, it is hard to interpret the learned representations. Moreover, little importance is given to the generalization to new documents that are not observed within the network. In this paper, we propose an interpretable and inductive document network embedding method. We introduce a novel mechanism, the Topic-Word Attention (TW A), that generates document representations based on the interplay between word and topic representations. We train these word and topic vectors through our general model, Inductive Document Network Embedding (IDNE), by leveraging the connections in the document network. Quantitative evaluations show that our approach achieves state-of-the-art performance on various networks and we qualitatively show that our model produces meaningful and interpretable representations of the words, topics and documents.

The presence of outliers can potentially significantly skew the parameters of machine learning models trained via stochastic gradient descent (SGD). In this paper we propose a simple variant of the simple SGD method: in each step, first choose a set of k samples, then from these choose the one with the smallest current loss, and do an SGD-like update with this chosen sample. Vanilla SGD corresponds to k = 1, i.e. no choice; k >= 2 represents a new algorithm that is however effectively minimizing a non-convex surrogate loss. Our main contribution is a theoretical analysis of the robustness properties of this idea for ML problems which are sums of convex losses; these are backed up with linear regression and small-scale neural network experiments

Data is one of the most important assets of the information age, and its societal impact is undisputed. Yet, rigorous methods of assessing the quality of data are lacking. In this paper, we propose a formal definition for the quality of a given dataset. We assess a dataset's quality by a quantity we call the expected diameter, which measures the expected disagreement between two randomly chosen hypotheses that explain it, and has recently found applications in active learning. We focus on Boolean hyperplanes, and utilize a collection of Fourier analytic, algebraic, and probabilistic methods to come up with theoretical guarantees and practical solutions for the computation of the expected diameter. We also study the behaviour of the expected diameter on algebraically structured datasets, conduct experiments that validate this notion of quality, and demonstrate the feasibility of our techniques.