Supervised learning with missing data aims at building the best prediction of a target output based on partially-observed inputs. Major approaches to address this problem can be decomposed into $(i)$ impute-then-predict strategies, which first fill in the empty input components and then apply a unique predictor and $(ii)$ Pattern-by-Pattern (P-b-P) approaches, where a predictor is built on each missing pattern. In this paper, we theoretically analyze how three classical linear classifiers, namely perceptron, logistic regression and linear discriminant analysis (LDA), behave with Missing Completely At Random (MCAR) data, depending on the strategy (imputation or P-b-P) to handle missing values. We prove that both imputation and P-b-P approaches are ill-specified in a logistic regression framework, thus questioning the relevance of such approaches to handle missing data. The most favorable auspices to perform classification with missing data concern P-b-P LDA methods. We provide finite-sample bounds for the excess risk in this framework, even for high-dimensional settings or MNAR data. Experiments illustrate our theoretical findings.

Constant (naive) imputation is still widely used in practice as this is a first easy-to-use technique to deal with missing data. Yet, this simple method could be expected to induce a large bias for prediction purposes, as the imputed input may strongly differ from the true underlying data. However, recent works suggest that this bias is low in the context of high-dimensional linear predictors when data is supposed to be missing completely at random (MCAR). This paper completes the picture for linear predictors by confirming the intuition that the bias is negligible and that surprisingly naive imputation also remains relevant in very low dimension. To this aim, we consider a unique underlying random features model, which offers a rigorous framework for studying predictive performances, whilst the dimension of the observed features varies. Building on these theoretical results, we establish finite-sample bounds on stochastic gradient (SGD) predictors applied to zero-imputed data, a strategy particularly well suited for large-scale learning. If the MCAR assumption appears to be strong, we show that similar favorable behaviors occur for more complex missing data scenarios.

Two different approaches exist to handle missing values for prediction: either imputation, prior to fitting any predictive algorithms, or dedicated methods able to natively incorporate missing values. While imputation is widely (and easily) use, it is unfortunately biased when low-capacity predictors (such as linear models) are applied afterward. However, in practice, naive imputation exhibits good predictive performance. In this paper, we study the impact of imputation in a high-dimensional linear model with MCAR missing data. We prove that zero imputation performs an implicit regularization closely related to the ridge method, often used in high-dimensional problems. Leveraging on this connection, we establish that the imputation bias is controlled by a ridge bias, which vanishes in high dimension. As a predictor, we argue in favor of the averaged SGD strategy, applied to zero-imputed data. We establish an upper bound on its generalization error, highlighting that imputation is benign in the $d \gg \sqrt{n}$ regime. Experiments illustrate our findings.

Missing values arise in most real-world data sets due to the aggregation of multiple sources and intrinsically missing information (sensor failure, unanswered questions in surveys...). In fact, the very nature of missing values usually prevents us from running standard learning algorithms. In this paper, we focus on the extensively-studied linear models, but in presence of missing values, which turns out to be quite a challenging task. Indeed, the Bayes predictor can be decomposed as a sum of predictors corresponding to each missing pattern. This eventually requires to solve a number of learning tasks, exponential in the number of input features, which makes predictions impossible for current real-world datasets. First, we propose a rigorous setting to analyze a least-square type estimator and establish a bound on the excess risk which increases exponentially in the dimension. Consequently, we leverage the missing data distribution to propose a new algorithm, and derive associated adaptive risk bounds that turn out to be minimax optimal. Numerical experiments highlight the benefits of our method compared to state-of-the-art algorithms used for predictions with missing values.