High-dimensional stochastic optimization with the generalized Dantzig estimator

We propose a generalized version of the Dantzig selector. We show that it satisfies sparsity oracle inequalities in prediction and estimation. We consider then the particular case of high-dimensional linear regression model selection with the Huber loss function. In this case we derive the sup-norm convergence rate and the sign concentration property of the Dantzig estimators under a mutual coherence assumption on the dictionary

Asymptotics and Concentration Bounds for Bilinear Forms of Spectral Projectors of Sample Covariance

Let $X,X_1,\dots, X_n$ be i.i.d. Gaussian random variables with zero mean and covariance operator $\Sigma={\mathbb E}(X\otimes X)$ taking values in a separable Hilbert space ${\mathbb H}.$ Let $${\bf r}(\Sigma):=\frac{{\rm tr}(\Sigma)}{\|\Sigma\|_{\infty}}$$ be the effective rank of $\Sigma,$ ${\rm tr}(\Sigma)$ being the trace of $\Sigma$ and $\|\Sigma\|_{\infty}$ being its operator norm. Let $$\hat \Sigma_n:=n^{-1}\sum_{j=1}^n (X_j\otimes X_j)$$ be the sample (empirical) covariance operator based on $(X_1,\dots, X_n).$ The paper deals with a problem of estimation of spectral projectors of the covariance operator $\Sigma$ by their empirical counterparts, the spectral projectors of $\hat \Sigma_n$ (empirical spectral projectors). The focus is on the problems where both the sample size $n$ and the effective rank ${\bf r}(\Sigma)$ are large. This framework includes and generalizes well known high-dimensional spiked covariance models. Given a spectral projector $P_r$ corresponding to an eigenvalue $\mu_r$ of covariance operator $\Sigma$ and its empirical counterpart $\hat P_r,$ we derive sharp concentration bounds for bilinear forms of empirical spectral projector $\hat P_r$ in terms of sample size $n$ and effective dimension ${\bf r}(\Sigma).$ Building upon these concentration bounds, we prove the asymptotic normality of bilinear forms of random operators $\hat P_r -{\mathbb E}\hat P_r$ under the assumptions that $n\to \infty$ and ${\bf r}(\Sigma)=o(n).$ In a special case of eigenvalues of multiplicity one, these results are rephrased as concentration bounds and asymptotic normality for linear forms of empirical eigenvectors. Other results include bounds on the bias ${\mathbb E}\hat P_r-P_r$ and a method of bias reduction as well as a discussion of possible applications to statistical inference in high-dimensional principal component analysis

Pac-bayesian bounds for sparse regression estimation with exponential weights

We consider the sparse regression model where the number of parameters $p$ is larger than the sample size $n$. The difficulty when considering high-dimensional problems is to propose estimators achieving a good compromise between statistical and computational performances. The BIC estimator for instance performs well from the statistical point of view \cite{BTW07} but can only be computed for values of $p$ of at most a few tens. The Lasso estimator is solution of a convex minimization problem, hence computable for large value of $p$. However stringent conditions on the design are required to establish fast rates of convergence for this estimator. Dalalyan and Tsybakov \cite{arnak} propose a method achieving a good compromise between the statistical and computational aspects of the problem. Their estimator can be computed for reasonably large $p$ and satisfies nice statistical properties under weak assumptions on the design. However, \cite{arnak} proposes sparsity oracle inequalities in expectation for the empirical excess risk only. In this paper, we propose an aggregation procedure similar to that of \cite{arnak} but with improved statistical performances. Our main theoretical result is a sparsity oracle inequality in probability for the true excess risk for a version of exponential weight estimator. We also propose a MCMC method to compute our estimator for reasonably large values of $p$.Comment: 19 page