We compare classification and regression tasks in the overparameterized linear model with Gaussian features. On the one hand, we show that with sufficient overparameterization all training points are support vectors: solutions obtained by least-squares minimum-norm interpolation, typically used for regression, are identical to those produced by the hard-margin support vector machine (SVM) that minimizes the hinge loss, typically used for training classifiers. On the other hand, we show that there exist regimes where these solutions are near-optimal when evaluated by the 0 − 1 test loss function, but do not generalize if evaluated by the square loss function, i.e. they achieve the null risk. Our results demonstrate the very different roles and properties of loss functions used at the training phase (optimization) and the testing phase (generalization). * indicates equal contribution. The key results were unveiled at the ITA workshop in San Diego in Feb 2020. 1 See, e.g., Section 8.1.2 in [19] for a representative informal discussion and [47,3] for theoretical analyses.
Regression analysis is extensively used for prediction and prognostication, and its use has substantial overlap with the domain of machine learning. The main objective of this paper is to compare the performance of two regression techniques namely Simple Linear Regression (SLR) and Multiple Linear Regression (MLR) algorithms by two cases: predicting the salary of employees after certain years and predicting the prices of real estates. An employee’s salary depends on numerous factors, such as total employee experience, certifications, and overall experience as a lead and manager. The factors in predicting house prices are the area of land (sqft_living), condition, waterfront, number of bedrooms, and so on. The dataset used in this experiment is an open-source dataset from KaggleInc. The algorithms were compared using parameters like R-squared value, Mean absolute error (MAE), Mean Squared Error (MSE), Median Absolute Error (MDAE), Variance Score, and Root Mean Square Error (RMSE). Results have shown that MLR provides the better efficiency in comparison to SLR.
Learning problems commonly exhibit an interesting feedback mechanism wherein the population data reacts to competing decision makers' actions. This paper formulates a new game theoretic framework for this phenomenon, called multi-player performative prediction. We focus on two distinct solution concepts, namely (i) performatively stable equilibria and (ii) Nash equilibria of the game. The latter equilibria are arguably more informative, but can be found efficiently only when the game is monotone. We show that under mild assumptions, the performatively stable equilibria can be found efficiently by a variety of algorithms, including repeated retraining and repeated (stochastic) gradient play. We then establish transparent sufficient conditions for strong monotonicity of the game and use them to develop algorithms for finding Nash equilibria. We investigate derivative free methods and adaptive gradient algorithms wherein each player alternates between learning a parametric description of their distribution and gradient steps on the empirical risk. Synthetic and semi-synthetic numerical experiments illustrate the results.
An overarching goal in machine learning is to build a generalizable model with few samples. To this end, overparameterization has been the subject of immense interest to explain the generalization ability of deep nets even when the size of the dataset is smaller than that of the model. While the prior literature focuses on the classical supervised setting, this paper aims to demystify overparameterization for meta-learning. Here we have a sequence of linear-regression tasks and we ask:(1) Given earlier tasks, what is the optimal linear representation of features for a new downstream task? and (2) How many samples do we need to build this representation? This work shows that surprisingly, overparameterization arises as a natural answer to these fundamental meta-learning questions. Specifically, for (1), we first show that learning the optimal representation coincides with the problem of designing a task-aware regularization to promote inductive bias. We leverage this inductive bias to explain how the downstream task actually benefits from overparameterization, in contrast to prior works on few-shot learning. For (2), we develop a theory to explain how feature covariance can implicitly help reduce the sample complexity well below the degrees of freedom and lead to small estimation error. We then integrate these findings to obtain an overall performance guarantee for our meta-learning algorithm. Numerical experiments on real and synthetic data verify our insights on overparameterized meta-learning.
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