Hung Yi Lee | Searcharxiv

arXiv2021

High-Dimensional Learning under ApproximateSparsity with Applications to Nonsmooth Estimation and Regularized Neural Networks

High-dimensional statistical learning (HDSL) has wide applications in data analysis, operations research, and decision-making. Despite the availability of multiple theoretical frameworks, most existing HDSL schemes stipulate the following two conditions: (a) the sparsity, and (b) the restricted strong convexity (RSC). This paper generalizes both conditions via the use of the folded concave penalty (FCP). More specifically, we consider an M-estimation problem where (i) the (conventional) sparsity is relaxed into the approximate sparsity and (ii) the RSC is completely absent. We show that the FCP-based regularization leads to poly-logarithmic sample complexity; the training data size is only required to be poly-logarithmic in the problem dimensionality. This finding can facilitate the analysis of two important classes of models that are currently less understood: the high-dimensional nonsmooth learning and the (deep) neural networks (NN). For both problems, we show that the poly-logarithmic sample complexity can be maintained. In particular, our results indicate that the generalizability of NNs under over-parameterization can be theoretically ensured with the aid of regularization.

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arXiv2019

Regularized Sample Average Approximation for High-Dimensional Stochastic Optimization Under Low-Rankness

This paper concerns a high-dimensional stochastic programming problem of minimizing a function of expected cost with a matrix argument. To this problem, one of the most widely applied solution paradigms is the sample average approximation (SAA), which uses the average cost over sampled scenarios as a surrogate to approximate the expected cost. Traditional SAA theories require the sample size to grow rapidly when the problem dimensionality increases. Indeed, for a problem of optimizing over a $p$-by-$p$ matrix, the sample complexity of the SAA is given by $\tilde O(1)\cdot \frac{p^2}{ε^2}\cdot{polylog}(\frac{1}ε)$ to achieve an $ε$-suboptimality gap, for some poly-logarithmic function ${polylog}(\,\cdot\,)$ and some quantity $\tilde O(1)$ independent of dimensionality $p$ and sample size $n$. In contrast, this paper considers a regularized SAA (RSAA) with a low-rankness-inducing penalty. We demonstrate that the sample complexity of RSAA is $\tilde O(1)\cdot \frac{p}{ε^3}\cdot {polylog}(p,\,\frac{1}ε)$, which is almost linear in $p$ and thus indicates a substantially lower dependence on dimensionality. Therefore, RSAA can be more advantageous than SAA especially for larger scale and higher dimensional problems. Due to the close correspondence between stochastic programming and statistical learning, our results also indicate that high-dimensional low-rank matrix recovery is possible generally beyond a linear model, even if the common assumption of restricted strong convexity is completely absent.

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