Studying conditional independence among many variables with few observations is a challenging task.Gaussian Graphical Models (GGMs) tackle this problem by encouraging sparsity in the precision matrix through $l_q$ regularization with $q\leq1$.However, most GMMs rely on the $l_1$ norm because the objective is highly non-convex for sub-$l_1$ pseudo-norms.In the frequentist formulation, the $l_1$ norm relaxation provides the solution path as a function of the shrinkage parameter $\lambda$.In the Bayesian formulation, sparsity is instead encouraged through a Laplace prior, but posterior inference for different $\lambda$ requires repeated runs of expensive Gibbs samplers.Here we propose a general framework for variational inference with matrix-variate Normalizing Flow in GGMs, which unifies the benefits of frequentist and Bayesian frameworks.As a key improvement on previous work, we train with one flow a continuum of sparse regression models jointly for all regularization parameters $\lambda$ and all $l_q$ norms, including non-convex sub-$l_1$ pseudo-norms.Within one model we thus have access to (i) the evolution of the posterior for any $\lambda$ and any $l_q$ (pseudo-) norm, (ii) the marginal log-likelihood for model selection, and (iii) the frequentist solution paths through simulated annealing in the MAP limit.

Many modern machine learning algorithms are formulated as regularized M-estimation problems, in which a regularization (tuning) parameter controls a trade-off between model fit to the training data and model complexity. To select the ``best'' tuning parameter value that achieves a good trade-off, an approximated solution path needs to be computed. In practice, this is often done through selecting a grid of tuning parameter values and solving the regularized problem at the selected grid points. However, given any desired level of accuracy, it is often not clear how to choose the grid points and also how accurately one should solve the regularized problems at the selected gird points, both of which can greatly impact the overall amount of computation. In the context of $\ell_2$-regularized $M$-estimation problem, we propose a novel grid point selection scheme and an adaptive stopping criterion for any given optimization algorithm that produces an approximated solution path with approximation error guarantee. Theoretically, we prove that the proposed solution path can approximate the exact solution path to arbitrary level of accuracy, while saving the overall computation as much as possible. Numerical results also corroborate with our theoretical analysis.

We study the energy-optimal shortest path problem for electric vehicles (EVs) in large-scale road networks, where recuperated energy along downhill segments introduces negative energy costs. While traditional point-to-point pathfinding algorithms for EVs assume a known initial energy level, many real-world scenarios involving uncertainty in available energy require planning optimal paths for all possible initial energy levels, a task known as energy-optimal profile search. Existing solutions typically rely on specialized profile-merging procedures within a label-correcting framework that results in searching over complex profiles. In this paper, we propose a simple yet effective label-setting approach based on multi-objective A* search, which employs a novel profile dominance rule to avoid generating and handling complex profiles. We develop four variants of our method and evaluate them on real-world road networks enriched with realistic energy consumption data. Experimental results demonstrate that our energy profile A* search achieves performance comparable to energy-optimal A* with a known initial energy level.

We then provide sufficient conditions under which PSM always outputs sparse solutions such that its computational performance can be significantly boosted.

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It enjoys model-free coverage guarantee regardless of the underlying distribution of the data.

These works optimize the heuristic function to solve regression problems .

In this paper, we present an explanation of this phenomenon.