PAC-Bayesian bounds have proven to be a valuable tool for deriving generalization bounds and for designing new learning algorithms in machine learning. However, it typically focus on providing generalization bounds with respect to a chosen loss function. In classification tasks, due to the non-convex nature of the 0-1 loss, a convex surrogate loss is often used, and thus current PAC-Bayesian bounds are primarily specified for this convex surrogate. This work shifts its focus to providing misclassification excess risk bounds for PAC-Bayesian classification when using a convex surrogate loss. Our key ingredient here is to leverage PAC-Bayesian relative bounds in expectation rather than relying on PAC-Bayesian bounds in probability. We demonstrate our approach in several important applications.

We consider (nonparametric) sparse additive models (SpAM) for classification. The design of a SpAM classifier is based on minimizing the logistic loss with a sparse group Lasso/Slope-type penalties on the coefficients of univariate additive components' expansions in orthonormal series (e.g., Fourier or wavelets). The resulting classifier is inherently adaptive to the unknown sparsity and smoothness. We show that under certain sparse group restricted eigenvalue condition it is nearly-minimax (up to log-factors) simultaneously across the entire range of analytic, Sobolev and Besov classes. The performance of the proposed classifier is illustrated on a simulated and a real-data examples.

This study investigates the misclassification excess risk bound in the context of 1-bit matrix completion, a significant problem in machine learning involving the recovery of an unknown matrix from a limited subset of its entries. Matrix completion has garnered considerable attention in the last two decades due to its diverse applications across various fields. Unlike conventional approaches that deal with real-valued samples, 1-bit matrix completion is concerned with binary observations. While prior research has predominantly focused on the estimation error of proposed estimators, our study shifts attention to the prediction error. This paper offers theoretical analysis regarding the prediction errors of two previous works utilizing the logistic regression model: one employing a max-norm constrained minimization and the other employing nuclear-norm penalization. Significantly, our findings demonstrate that the latter achieves the minimax-optimal rate without the need for an additional logarithmic term. These novel results contribute to a deeper understanding of 1-bit matrix completion by shedding light on the predictive performance of specific methodologies.

Classification is one of the core problems in statistical learning and has been intensively studied in statistical and machine learning literature. Nevertheless, while the theory for binary classification is well developed (see, Devroy, Gyöfri and Lugosi, 1996; Vapnik, 2000; Boucheron, Bousquet and Lugosi, 2005 and references therein for a comprehensive review), its multiclass extensions are much less complete. Consider a general L-class classification with a (high-dimensional) vector of features X X R d and the outcome class label Y {1,..., L}. We can model it as Y (X x) Mult(p 1 (x),..., p L (x)), where p l (x) P (Y l X x), l 1,..., L. A classifier is a measurable function η: X {1,..., L}. The accuracy of a classifier η is defined by a misclassification error R(η) P (Y η(x)). The optimal classifier that minimizes this error is the Bayes classifier η (x) arg max 1 l L p l (x) with R(η) 1 E X max 1 l L p l (x). The probabilities p l (x)'s are, however, unknown and one should derive a classifier η(x) from the available data D: a random sample of n independent observations (X 1, Y 1),..., (X n, Y n) from the joint distribution of (X, Y). The corresponding (conditional) misclassification error of η is R( η) P (Y η(x) D) and the goodness of η w.r.t.

We consider high-dimensional binary classification by sparse logistic regression. We propose a model/feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the non-asymptotic bounds for the resulting misclassification excess risk. The bounds can be reduced under the additional low-noise condition. The proposed complexity penalty is remarkably related to the VC-dimension of a set of sparse linear classifiers. Implementation of any complexity penalty-based criterion, however, requires a combinatorial search over all possible models. To find a model selection procedure computationally feasible for high-dimensional data, we extend the Slope estimator for logistic regression and show that under an additional weighted restricted eigenvalue condition it is rate-optimal in the minimax sense.