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Partially linear additive models generalize linear ones since they model the relation between a response variable and covariates by assuming that some covariates have a linear relation with the response but each of the others enter through unknown univariate smooth functions. The harmful effect of outliers either in the residuals or in the covariates involved in the linear component has been described in the situation of partially linear models, that is, when only one nonparametric component is involved in the model. When dealing with additive components, the problem of providing reliable estimators when atypical data arise, is of practical importance motivating the need of robust procedures. Hence, we propose a family of robust estimators for partially linear additive models by combining $B-$splines with robust linear regression estimators. We obtain consistency results, rates of convergence and asymptotic normality for the linear components, under mild assumptions. A Monte Carlo study is carried out to compare the performance of the robust proposal with its classical counterpart under different models and contamination schemes. The numerical experiments show the advantage of the proposed methodology for finite samples. We also illustrate the usefulness of the proposed approach on a real data set.

We study the benign overfitting theory in the prediction of the conditional average treatment effect (CATE), with linear regression models. As the development of machine learning for causal inference, a wide range of large-scale models for causality are gaining attention. One problem is that suspicions have been raised that the large-scale models are prone to overfitting to observations with sample selection, hence the large models may not be suitable for causal prediction. In this study, to resolve the suspicious, we investigate on the validity of causal inference methods for overparameterized models, by applying the recent theory of benign overfitting (Bartlett et al., 2020). Specifically, we consider samples whose distribution switches depending on an assignment rule, and study the prediction of CATE with linear models whose dimension diverges to infinity. We focus on two methods: the T-learner, which based on a difference between separately constructed estimators with each treatment group, and the inverse probability weight (IPW)-learner, which solves another regression problem approximated by a propensity score. In both methods, the estimator consists of interpolators that fit the samples perfectly. As a result, we show that the T-learner fails to achieve the consistency except the random assignment, while the IPW-learner converges the risk to zero if the propensity score is known. This difference stems from that the T-learner is unable to preserve eigenspaces of the covariances, which is necessary for benign overfitting in the overparameterized setting. Our result provides new insights into the usage of causal inference methods in the overparameterizated setting, in particular, doubly robust estimators.

Label Ranking (LR) corresponds to the problem of learning a hypothesis that maps features to rankings over a finite set of labels. We adopt a nonparametric regression approach to LR and obtain theoretical performance guarantees for this fundamental practical problem. We introduce a generative model for Label Ranking, in noiseless and noisy nonparametric regression settings, and provide sample complexity bounds for learning algorithms in both cases. In the noiseless setting, we study the LR problem with full rankings and provide computationally efficient algorithms using decision trees and random forests in the high-dimensional regime. In the noisy setting, we consider the more general cases of LR with incomplete and partial rankings from a statistical viewpoint and obtain sample complexity bounds using the One-Versus-One approach of multiclass classification. Finally, we complement our theoretical contributions with experiments, aiming to understand how the input regression noise affects the observed output.

Statistical analysis is increasingly confronted with complex data from general metric spaces, such as symmetric positive definite matrix-valued data and probability distribution functions. [47] and [17] establish a general paradigm of Fr\'echet regression with complex metric space valued responses and Euclidean predictors. However, their proposed local Fr\'echet regression approach involves nonparametric kernel smoothing and suffers from the curse of dimensionality. To address this issue, we in this paper propose a novel random forests weighted local Fr\'echet regression paradigm. The main mechanism of our approach relies on the adaptive kernels generated by random forests. Our first method utilizes these weights as the local average to solve the Fr\'echet mean, while the second method performs local linear Fr\'echet regression, making both methods locally adaptive. Our proposals significantly improve existing Fr\'echet regression methods. Based on the theory of infinite order U-processes and infinite order Mmn-estimator, we establish the consistency, rate of convergence, and asymptotic normality for our proposed random forests weighted Fr\'echet regression estimator, which covers the current large sample theory of random forests with Euclidean responses as a special case. Numerical studies show the superiority of our proposed two methods for Fr\'echet regression with several commonly encountered types of responses such as probability distribution functions, symmetric positive definite matrices, and sphere data. The practical merits of our proposals are also demonstrated through the application to the human mortality distribution data.

Learning rate schedules are ubiquitously used to speed up and improve optimisation. Many different policies have been introduced on an empirical basis, and theoretical analyses have been developed for convex settings. However, in many realistic problems the loss-landscape is high-dimensional and non convex -- a case for which results are scarce. In this paper we present a first analytical study of the role of learning rate scheduling in this setting, focusing on Langevin optimization with a learning rate decaying as $\eta(t)=t^{-\beta}$. We begin by considering models where the loss is a Gaussian random function on the $N$-dimensional sphere ($N\rightarrow \infty$), featuring an extensive number of critical points. We find that to speed up optimization without getting stuck in saddles, one must choose a decay rate $\beta<1$, contrary to convex setups where $\beta=1$ is generally optimal. We then add to the problem a signal to be recovered. In this setting, the dynamics decompose into two phases: an \emph{exploration} phase where the dynamics navigates through rough parts of the landscape, followed by a \emph{convergence} phase where the signal is detected and the dynamics enter a convex basin. In this case, it is optimal to keep a large learning rate during the exploration phase to escape the non-convex region as quickly as possible, then use the convex criterion $\beta=1$ to converge rapidly to the solution. Finally, we demonstrate that our conclusions hold in a common regression task involving neural networks.

In semi-supervised classification, one is given access both to labeled and unlabeled data. As unlabeled data is typically cheaper to acquire than labeled data, this setup becomes advantageous as soon as one can exploit the unlabeled data in order to produce a better classifier than with labeled data alone. However, the conditions under which such an improvement is possible are not fully understood yet. Our analysis focuses on improvements in the minimax learning rate in terms of the number of labeled examples (with the number of unlabeled examples being allowed to depend on the number of labeled ones). We argue that for such improvements to be realistic and indisputable, certain specific conditions should be satisfied and previous analyses have failed to meet those conditions. We then demonstrate examples where these conditions can be met, in particular showing rate changes from $1/\sqrt{\ell}$ to $e^{-c\ell}$ and from $1/\sqrt{\ell}$ to $1/\ell$. These results improve our understanding of what is and isn't possible in semi-supervised learning.

We study regression adjustments with additional covariates in randomized experiments under covariate-adaptive randomizations (CARs) when subject compliance is imperfect. We develop a regression-adjusted local average treatment effect (LATE) estimator that is proven to improve efficiency in the estimation of LATEs under CARs. Our adjustments can be parametric in linear and nonlinear forms, nonparametric, and high-dimensional. Even when the adjustments are misspecified, our proposed estimator is still consistent and asymptotically normal, and their inference method still achieves the exact asymptotic size under the null. When the adjustments are correctly specified, our estimator achieves the minimum asymptotic variance. When the adjustments are parametrically misspecified, we construct a new estimator which is weakly more efficient than linearly and nonlinearly adjusted estimators, as well as the one without any adjustments. Simulation evidence and empirical application confirm efficiency gains achieved by regression adjustments relative to both the estimator without adjustment and the standard two-stage least squares estimator.

The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

We consider the exploration-exploitation trade-off in reinforcement learning and we show that an agent imbued with a risk-seeking utility function is able to explore efficiently, as measured by regret. The parameter that controls how risk-seeking the agent is can be optimized exactly, or annealed according to a schedule. We call the resulting algorithm K-learning and show that the corresponding K-values are optimistic for the expected Q-values at each state-action pair. The K-values induce a natural Boltzmann exploration policy for which the `temperature' parameter is equal to the risk-seeking parameter. This policy achieves an expected regret bound of $\tilde O(L^{3/2} \sqrt{S A T})$, where $L$ is the time horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the total number of elapsed time-steps. This bound is only a factor of $L$ larger than the established lower bound. K-learning can be interpreted as mirror descent in the policy space, and it is similar to other well-known methods in the literature, including Q-learning, soft-Q-learning, and maximum entropy policy gradient, and is closely related to optimism and count based exploration methods. K-learning is simple to implement, as it only requires adding a bonus to the reward at each state-action and then solving a Bellman equation. We conclude with a numerical example demonstrating that K-learning is competitive with other state-of-the-art algorithms in practice.