Before we discuss deviance, we first need to understand what the Saturated, Proposed and Null Models are. A Saturated Model is where the number of parameters/coefficients is equal to the number of data points. This is like a'connect the dots' model where the line or curve passes through each point. This is considered to be the perfect model as it takes into account all the variance in the data and has the maximum achievable likelihood. A Null Model is the opposite with only one parameter, which is the intercept.

Active learning has demonstrated data efficiency in many fields. Existing active learning algorithms, especially in the context of deep Bayesian active models, rely heavily on the quality of uncertainty estimations of the model. However, such uncertainty estimates could be heavily biased, especially with limited and imbalanced training data. In this paper, we propose BALanCe, a Bayesian deep active learning framework that mitigates the effect of such biases. Concretely, BALanCe employs a novel acquisition function which leverages the structure captured by equivalence hypothesis classes and facilitates differentiation among different equivalence classes. Intuitively, each equivalence class consists of instantiations of deep models with similar predictions, and BALanCe adaptively adjusts the size of the equivalence classes as learning progresses. Besides the fully sequential setting, we further propose Batch-BALanCe -- a generalization of the sequential algorithm to the batched setting -- to efficiently select batches of training examples that are jointly effective for model improvement. We show that Batch-BALanCe achieves state-of-the-art performance on several benchmark datasets for active learning, and that both algorithms can effectively handle realistic challenges that often involve multi-class and imbalanced data.

Unrolled computation graphs arise in many scenarios, including training RNNs, tuning hyperparameters through unrolled optimization, and training learned optimizers. Current approaches to optimizing parameters in such computation graphs suffer from high variance gradients, bias, slow updates, or large memory usage. We introduce a method called Persistent Evolution Strategies (PES), which divides the computation graph into a series of truncated unrolls, and performs an evolution strategies-based update step after each unroll. PES eliminates bias from these truncations by accumulating correction terms over the entire sequence of unrolls. PES allows for rapid parameter updates, has low memory usage, is unbiased, and has reasonable variance characteristics. We experimentally demonstrate the advantages of PES compared to several other methods for gradient estimation on synthetic tasks, and show its applicability to training learned optimizers and tuning hyperparameters.

The recently developed matrix based Renyi's entropy enables measurement of information in data simply using the eigenspectrum of symmetric positive semi definite (PSD) matrices in reproducing kernel Hilbert space, without estimation of the underlying data distribution. This intriguing property makes the new information measurement widely adopted in multiple statistical inference and learning tasks. However, the computation of such quantity involves the trace operator on a PSD matrix $G$ to power $\alpha$(i.e., $tr(G^\alpha)$), with a normal complexity of nearly $O(n^3)$, which severely hampers its practical usage when the number of samples (i.e., $n$) is large. In this work, we present computationally efficient approximations to this new entropy functional that can reduce its complexity to even significantly less than $O(n^2)$. To this end, we first develop randomized approximations to $\tr(\G^\alpha)$ that transform the trace estimation into matrix-vector multiplications problem. We extend such strategy for arbitrary values of $\alpha$ (integer or non-integer). We then establish the connection between the matrix-based Renyi's entropy and PSD matrix approximation, which enables us to exploit both clustering and block low-rank structure of $\G$ to further reduce the computational cost. We theoretically provide approximation accuracy guarantees and illustrate the properties of different approximations. Large-scale experimental evaluations on both synthetic and real-world data corroborate our theoretical findings, showing promising speedup with negligible loss in accuracy.

Actor-critic (AC) algorithms, empowered by neural networks, have had significant empirical success in recent years. However, most of the existing theoretical support for AC algorithms focuses on the case of linear function approximations, or linearized neural networks, where the feature representation is fixed throughout training. Such a limitation fails to capture the key aspect of representation learning in neural AC, which is pivotal in practical problems. In this work, we take a mean-field perspective on the evolution and convergence of feature-based neural AC. Specifically, we consider a version of AC where the actor and critic are represented by overparameterized two-layer neural networks and are updated with two-timescale learning rates. The critic is updated by temporal-difference (TD) learning with a larger stepsize while the actor is updated via proximal policy optimization (PPO) with a smaller stepsize. In the continuous-time and infinite-width limiting regime, when the timescales are properly separated, we prove that neural AC finds the globally optimal policy at a sublinear rate. Additionally, we prove that the feature representation induced by the critic network is allowed to evolve within a neighborhood of the initial one.

For supervised classification problems, this paper considers estimating the query's label probability through local regression using observed covariates. Well-known nonparametric kernel smoother and $k$-nearest neighbor ($k$-NN) estimator, which take label average over a ball around the query, are consistent but asymptotically biased particularly for a large radius of the ball. To eradicate such bias, local polynomial regression (LPoR) and multiscale $k$-NN (MS-$k$-NN) learn the bias term by local regression around the query and extrapolate it to the query itself. However, their theoretical optimality has been shown for the limit of the infinite number of training samples. For correcting the asymptotic bias with fewer observations, this paper proposes a local radial regression (LRR) and its logistic regression variant called local radial logistic regression (LRLR), by combining the advantages of LPoR and MS-$k$-NN. The idea is simple: we fit the local regression to observed labels by taking the radial distance as the explanatory variable and then extrapolate the estimated label probability to zero distance. Our numerical experiments, including real-world datasets of daily stock indices, demonstrate that LRLR outperforms LPoR and MS-$k$-NN.

Handling clustering problems are important in data statistics, pattern recognition and image processing. The mean-shift algorithm, a common unsupervised algorithms, is widely used to solve clustering problems. However, the mean-shift algorithm is restricted by its huge computational resource cost. In previous research[10], we proposed a novel GPU-accelerated Faster Mean-shift algorithm, which greatly speed up the cosine-embedding clustering problem. In this study, we extend and improve the previous algorithm to handle Euclidean distance metrics. Different from conventional GPU-based mean-shift algorithms, our algorithm adopts novel Seed Selection & Early Stopping approaches, which greatly increase computing speed and reduce GPU memory consumption. In the simulation testing, when processing a 200K points clustering problem, our algorithm achieved around 3 times speedup compared to the state-of-the-art GPU-based mean-shift algorithms with optimized GPU memory consumption. Moreover, in this study, we implemented a plug-and-play model for faster mean-shift algorithm, which can be easily deployed. (Plug-and-play model is available: https://github.com/masqm/Faster-Mean-Shift-Euc)

First, to train parameters \(w \) and \(b \) of a logistic regression model we need to define a cost function. Given a training set of \(m\) training examples, we want to find parameters \(w\) and \(b \), so that \(\hat{y}\) is as close to \(y \) (ground truth). Here, we will use \((i) \) superscript to index different training examples. Henceforth, we will use loss (error) function \(\mathcal{L}\) to measure how well our algorithm is doing. In logistic regression squared error loss function is not an optimal choice.

When we use labeled data to train a machine-learning algorithm (e.g. When we use unlabeled data to train a machine-learning algorithm and allow it to find patterns in the data(e.g. Using dimensionality reduction to transform raw data into numerical features that can be processed with machine learning that contain information about the original data (e.g. An unsupervised learning task where the algorithm learns patterns in input data to generate new examples (fake data) that would appear to have been drawn from the original dataset. The part of the GAN that generates fake data.

This function is responsible for connecting to the database to extract the data, as some target format depends on the storage we decide on. We have multiple options, such as tradition, Hadoop data lake, cloud storage bucket (s3, ADLS gen2, GCP storage). For the simplicity of this project, and it seems the data volume is minimal, let's keep it in the csv format. Another exciting decision point will be how to get the data consistently and efficiently, which meets one of our NFR, such as incremental or batch load. However, instead of the batch load (SQL select *) and then write to file in a for loop, we will consider using the COPY out for performance needed, why?