Indeed, this is a central question of recent workstudyinggeneralized lowrankmodels.

In addition, we present an alternative design that permits a near-optimal sublinear decoding timeofO(klog2k+klogn).

Leveraging this result, our work provides the first provably efficient implementation of PseudoPML.

Ineachstep, A's output distribution is within ζ of the true distribution N(t,1;S). Consider a hypothetical sampling algorithm A0 in which A is run, and then the output is altered by rejection to match the true distribution.

Ourresult follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaptation of the LASSO optimization problem that accommodates truncation.

Regarding computational aspects of PML, [CSS19a] provided the first efficient algorithm with a non-trivialapproximation guarantee ofexp( n2/3logn),which further implied asample optimal universal estimator forn 1/6.

Regarding computational aspects of PML, [CSS19a] provided the first efficient algorithm with a non-trivialapproximation guarantee ofexp( n2/3logn),which further implied asample optimal universal estimator forn 1/6.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+\epsilon$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.