Random Projections have been widely used to generate embeddings for various graph tasks due to their computational efficiency. The majority of applications have been justified through the Johnson-Lindenstrauss Lemma. In this paper, we take a step further and investigate how well dot product and cosine similarity are preserved by Random Projections. Our analysis provides new theoretical results, identifies pathological cases, and tests them with numerical experiments. We find that, for nodes of lower or higher degrees, the method produces especially unreliable embeddings for the dot product, regardless of whether the adjacency or the (normalized version) transition is used. With respect to the statistical noise introduced by Random Projections, we show that cosine similarity produces remarkably more precise approximations.

$ $The classical theory of statistical estimation aims to estimate a parameter of interest under data generated from a fixed design ("offline estimation"), while the contemporary theory of online learning provides algorithms for estimation under adaptively chosen covariates ("online estimation"). Motivated by connections between estimation and interactive decision making, we ask: is it possible to convert offline estimation algorithms into online estimation algorithms in a black-box fashion? We investigate this question from an information-theoretic perspective by introducing a new framework, Oracle-Efficient Online Estimation (OEOE), where the learner can only interact with the data stream indirectly through a sequence of offline estimators produced by a black-box algorithm operating on the stream. Our main results settle the statistical and computational complexity of online estimation in this framework. $\bullet$ Statistical complexity. We show that information-theoretically, there exist algorithms that achieve near-optimal online estimation error via black-box offline estimation oracles, and give a nearly-tight characterization for minimax rates in the OEOE framework. $\bullet$ Computational complexity. We show that the guarantees above cannot be achieved in a computationally efficient fashion in general, but give a refined characterization for the special case of conditional density estimation: computationally efficient online estimation via black-box offline estimation is possible whenever it is possible via unrestricted algorithms. Finally, we apply our results to give offline oracle-efficient algorithms for interactive decision making.

We study the quadratic prediction error method -- i.e., nonlinear least squares -- for a class of time-varying parametric predictor models satisfying a certain identifiability condition. While this method is known to asymptotically achieve the optimal rate for a wide range of problems, there have been no non-asymptotic results matching these optimal rates outside of a select few, typically linear, model classes. By leveraging modern tools from learning with dependent data, we provide the first rate-optimal non-asymptotic analysis of this method for our more general setting of nonlinearly parametrized model classes. Moreover, we show that our results can be applied to a particular class of identifiable AutoRegressive Moving Average (ARMA) models, resulting in the first optimal non-asymptotic rates for identification of ARMA models.

We consider the task of predicting a response Y from a set of covariates X in settings where the conditional distribution of Y given X changes over time. For this to be feasible, assumptions on how the conditional distribution changes over time are required. Existing approaches assume, for example, that changes occur smoothly over time so that short-term prediction using only the recent past becomes feasible. In this work, we propose a novel invariance-based framework for linear conditionals, called Invariant Subspace Decomposition (ISD), that splits the conditional distribution into a time-invariant and a residual time-dependent component. As we show, this decomposition can be utilized both for zero-shot and time-adaptation prediction tasks, that is, settings where either no or a small amount of training data is available at the time points we want to predict Y at, respectively. We propose a practical estimation procedure, which automatically infers the decomposition using tools from approximate joint matrix diagonalization. Furthermore, we provide finite sample guarantees for the proposed estimator and demonstrate empirically that it indeed improves on approaches that do not use the additional invariant structure.

Supervised machine learning often requires large training sets to train accurate models, yet obtaining large amounts of labeled data is not always feasible. Hence, it becomes crucial to explore active learning methods for reducing the size of training sets while maintaining high accuracy. The aim is to select the optimal subset of data for labeling from an initial unlabeled set, ensuring precise prediction of outcomes. However, conventional active learning approaches are comparable to classical random sampling. This paper proposes a wrapper active learning method for classification, organizing the sampling process into a tree structure, that improves state-of-the-art algorithms. A classification tree constructed on an initial set of labeled samples is considered to decompose the space into low-entropy regions. Input-space based criteria are used thereafter to sub-sample from these regions, the total number of points to be labeled being decomposed into each region. This adaptation proves to be a significant enhancement over existing active learning methods. Through experiments conducted on various benchmark data sets, the paper demonstrates the efficacy of the proposed framework by being effective in constructing accurate classification models, even when provided with a severely restricted labeled data set.

Constrained learning has become increasingly important, especially in the realm of algorithmic fairness and machine learning. In these settings, predictive models are developed specifically to satisfy pre-defined notions of fairness. Here, we study the general problem of constrained statistical machine learning through a statistical functional lens. We consider learning a function-valued parameter of interest under the constraint that one or several pre-specified real-valued functional parameters equal zero or are otherwise bounded. We characterize the constrained functional parameter as the minimizer of a penalized risk criterion using a Lagrange multiplier formulation. We show that closed-form solutions for the optimal constrained parameter are often available, providing insight into mechanisms that drive fairness in predictive models. Our results also suggest natural estimators of the constrained parameter that can be constructed by combining estimates of unconstrained parameters of the data generating distribution. Thus, our estimation procedure for constructing fair machine learning algorithms can be applied in conjunction with any statistical learning approach and off-the-shelf software. We demonstrate the generality of our method by explicitly considering a number of examples of statistical fairness constraints and implementing the approach using several popular learning approaches.

Symbolic regression has excelled in uncovering equations from physics, chemistry, biology, and related disciplines. However, its effectiveness becomes less certain when applied to experimental data lacking inherent closed-form expressions. Empirically derived relationships, such as entire stress-strain curves, may defy concise closed-form representation, compelling us to explore more adaptive modeling approaches that balance flexibility with interpretability. In our pursuit, we turn to Generalized Additive Models (GAMs), a widely used class of models known for their versatility across various domains. Although GAMs can capture non-linear relationships between variables and targets, they cannot capture intricate feature interactions. In this work, we investigate both of these challenges and propose a novel class of models, Shape Arithmetic Expressions (SHAREs), that fuses GAM's flexible shape functions with the complex feature interactions found in mathematical expressions. SHAREs also provide a unifying framework for both of these approaches. We also design a set of rules for constructing SHAREs that guarantee transparency of the found expressions beyond the standard constraints based on the model's size.

A sharp asymptotics of the under-bagging (UB) method, which is a popular ensemble learning method for training classifiers from an imbalanced data, is derived and used to compare with several other standard methods for learning from imbalanced data, in the scenario where a linear classifier is trained from a binary mixture data. The methods compared include the under-sampling (US) method, which trains a model using a single realization of the subsampled dataset, and the simple weighting (SW) method, which trains a model with a weighted loss on the entire data. It is shown that the performance of UB is improved by increasing the size of the majority class, even if the class imbalance can be large, especially when the size of the minority class is small. This is in contrast to US, whose performance does not change as the size of the majority class increases, and SW, whose performance decreases as the imbalance increases. These results are different from the case of the naive bagging in training generalized linear models without considering the structure of class imbalance, indicating the intrinsic difference between the ensembling and the direct regularization on the parameters.

Machine learning is now used in many applications thanks to its ability to predict, generate, or discover patterns from large quantities of data. However, the process of collecting and transforming data for practical use is intricate. Even in today's digital era, where substantial data is generated daily, it is uncommon for it to be readily usable; most often, it necessitates meticulous manual data preparation. The haste in developing new models can frequently result in various shortcomings, potentially posing risks when deployed in real-world scenarios (e.g., social discrimination, critical failures), leading to the failure or substantial escalation of costs in AI-based projects. This chapter provides a comprehensive overview of established methodological tools, enriched by our practical experience, in the development of datasets for machine learning. Initially, we develop the tasks involved in dataset development and offer insights into their effective management (including requirements, design, implementation, evaluation, distribution, and maintenance). Then, we provide more details about the implementation process which includes data collection, transformation, and quality evaluation. Finally, we address practical considerations regarding dataset distribution and maintenance.

One of the objectives of continual learning is to prevent catastrophic forgetting in learning multiple tasks sequentially, and the existing solutions have been driven by the conceptualization of the plasticity-stability dilemma. However, the convergence of continual learning for each sequential task is less studied so far. In this paper, we provide a convergence analysis of memory-based continual learning with stochastic gradient descent and empirical evidence that training current tasks causes the cumulative degradation of previous tasks. We propose an adaptive method for nonconvex continual learning (NCCL), which adjusts step sizes of both previous and current tasks with the gradients. The proposed method can achieve the same convergence rate as the SGD method when the catastrophic forgetting term which we define in the paper is suppressed at each iteration. Further, we demonstrate that the proposed algorithm improves the performance of continual learning over existing methods for several image classification tasks.