29 problems

The procedure uses randomly weighted sums of centered estimated vectors $\widehat{\mathbf U}_{j,t}$ to approximate the distribution of the leading term in the debiasing decompositi…

Let $\widehat{\beta}$ be the estimator based on the quantized data and let $\widehat{\beta}^{o}$ be the ordinary least squares estimator based on the uncompressed data. Write $R_n$…

Let $X_1 \sim \operatorname{Laplace}(0,1/\sqrt{2})$ and $X_2 \sim \operatorname{t}_5(0,\sqrt{3/5})$, as in the third example, and let $\mathcalB_0$ denote the solution set consider…

Consider stochastic gradient descent with exponential moving average, and let the bias error be decomposed into components associated with the eigenspaces of the data-feature covar…

In an infinite-dimensional linear regression model, suppose only $M$-dimensional sketched covariates are observed, a linear predictor with $M$ trainable parameters is trained by on…

Logarithmic-factor conjecture. The dependence on logarithmic factors in these bounds is suboptimal.

Let $\sigma_e$ denote the standard deviation of the dependent-variable noise, and consider the differentially private test for mixtures in the general linear model, together with i…

Let $\sigma_e$ denote the standard deviation of the dependent-variable noise, and consider the differentially private test of a linear relationship, with its noisy estimate of…

The problem concerns learning mixtures of linear regressions in the non-realizable setting, with objective given by the empirical risk minimization formulation discussed above. The…

Let $\pi_0=\pi^{\otimes p}$ with $\pi$ supported on a bounded set, and consider the TAP variational problem for the linear-regression log-partition function, namely the optimizatio…

Let $\beta=(\beta_1,\ldots,\beta_p)$ have product prior $\pi_0=\pi^{\otimes p}$, where $\pi$ is a probability distribution on $[-1,1]$. Let $\mathcal{Z}_p$ denote the linear-regres…

Effective-dimension necessity conjecture. The condition

Let $Hb$ be the data covariance matrix and let $Ab$ range over positive semidefinite matrices in the fourth-moment condition used to prove the SGD guarantees. Two matrices are comm…

Let $xb$ be the data vector, $Hb$ its covariance matrix, and $Ab$ a positive semidefinite matrix. Assume that, for all positive semidefinite $Ab$ and some nonnegative constants $a$…

Consider overparameterized linear regression trained by constant-stepsize stochastic gradient descent with initial iterate $wb_0$, and let $\mathrm{EffectiveVariance}$ denote the v…

Consider overparameterized linear regression trained by constant-stepsize stochastic gradient descent, with initial iterate $wb_0$, population minimizer $wb^*$, data covariance…

Let $N$ be the number of SGD iterations, $gamma$ the constant stepsize, $wb_0$ the initial iterate, $wb^*$ the population-risk minimizer, and $Hb$ the data covariance matrix. Write…

Consider the Gaussian linear regression model with $X\sim \mathcal{N}(0,\sigma^2 I_p)$ and $Y=X^T\beta+z$, where $z\sim \mathcal{N}(0,1)$ is independent of $X$. Let the true parame…

Let $h^*=0$, let $X_i\overset{i.i.d.}{\sim}\mathcal{N}(0,I_p)$, and let $\xi_i\overset{i.i.d.}{\sim}\mathcal{N}(0,\sigma^2)$. For the minimum $\ell_1$-norm interpolated estimator…

The paper considers pNML universal learning and the notion of learnability for model classes, with linear regression as the explicitly solved example. Local learnability conjecture…

The paper compares holistic regression with MISDO and MIQO formulations on real-world datasets. Holistic regression explicitly models significance and multicollinearity constraints…

Let $p\geq n^2$, and consider the general covariance setting of the paper with sparsity level $k_0$ and signal sparsity parameter $\Delta$. In the regime … the squared minimax sepa…

Conjecture on the minimax separation distance. Under these conditions, $\rho^*_{\gamma}[k_0,p]$ is huge.

Consider procedures for the post-selection inference approach discussed in the source, where computational efficiency refers to retaining the computational complexity of approach 1…

Let the normalized observations, error norms, and rates be as described in the preceding setup, and let the conditions in Lemma … should be sufficient for the same rates; in partic…