Machine Learning (stat.ML)

Abstract: A scaling law refers to the observation that the test performance of a model improves as the number of training data increases. A fast scaling law implies that one can solve machine learning problems by simply boosting the data and the model sizes. Yet, in many cases, the benefit of adding more data can be negligible. In this work, we study the rate of scaling laws of nearest neighbor classifiers. We show that a scaling law can have two phases: in the first phase, the generalization error depends polynomially on the data dimension and decreases fast; whereas in the second phase, the error depends exponentially on the data dimension and decreases slowly. Our analysis highlights the complexity of the data distribution in determining the generalization error. When the data distributes benignly, our result suggests that nearest neighbor classifier can achieve a generalization error that depends polynomially, instead of exponentially, on the data dimension.

Abstract: We consider the fundamental task of optimizing a real-valued function defined in a potentially high-dimensional Euclidean space, such as the loss function in many machine-learning tasks or the logarithm of the probability distribution in statistical inference. We use the warped Riemannian geometry notions to redefine the optimisation problem of a function on Euclidean space to a Riemannian manifold with a warped metric, and then find the function's optimum along this manifold. The warped metric chosen for the search domain induces a computational friendly metric-tensor for which optimal search directions associate with geodesic curves on the manifold becomes easier to compute. Performing optimization along geodesics is known to be generally infeasible, yet we show that in this specific manifold we can analytically derive Taylor approximations up to third-order. In general these approximations to the geodesic curve will not lie on the manifold, however we construct suitable retraction maps to pull them back onto the manifold. Therefore, we can efficiently optimize along the approximate geodesic curves. We cover the related theory, describe a practical optimization algorithm and empirically evaluate it on a collection of challenging optimisation benchmarks. Our proposed algorithm, using third-order approximation of geodesics, outperforms standard Euclidean gradient-based counterparts in term of number of iterations until convergence and an alternative method for Hessian-based optimisation routines.

Abstract: Quantification is a supervised machine learning task, focused on estimating the class prevalence of a dataset rather than labeling its individual observations. We introduce Continuous Sweep, a new parametric binary quantifier inspired by the well-performing Median Sweep. Median Sweep is currently one of the best binary quantifiers, but we have changed this quantifier on three points, namely 1) using parametric class distributions instead of empirical distributions, 2) optimizing decision boundaries instead of applying discrete decision rules, and 3) calculating the mean instead of the median. We derive analytic expressions for the bias and variance of Continuous Sweep under general model assumptions. This is one of the first theoretical contributions in the field of quantification learning. Moreover, these derivations enable us to find the optimal decision boundaries. Finally, our simulation study shows that Continuous Sweep outperforms Median Sweep in a wide range of situations.

Abstract: This paper proposes a novel framework for identifying an agent's risk aversion using interactive questioning. Our study is conducted in two scenarios: a one-period case and an infinite horizon case. In the one-period case, we assume that the agent's risk aversion is characterized by a cost function of the state and a distortion risk measure. In the infinite horizon case, we model risk aversion with an additional component, a discount factor. Assuming the access to a finite set of candidates containing the agent's true risk aversion, we show that asking the agent to demonstrate her optimal policies in various environment, which may depend on their previous answers, is an effective means of identifying the agent's risk aversion. Specifically, we prove that the agent's risk aversion can be identified as the number of questions tends to infinity, and the questions are randomly designed. We also develop an algorithm for designing optimal questions and provide empirical evidence that our method learns risk aversion significantly faster than randomly designed questions in simulations. Our framework has important applications in robo-advising and provides a new approach for identifying an agent's risk preferences.