Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises

Recently, several studies consider the stochastic optimization problem but in a heavy-tailed noise regime, i.e., the difference between the stochastic gradient and the true gradient is assumed to have a finite $p$-th moment (say being upper bounded by $\sigma^{p}$ for some $\sigma\geq0$) where $p\in(1,2]$, which not only generalizes the traditional finite variance assumption ($p=2$) but also has been observed in practice for several different tasks. Under this challenging assumption, lots of new progress has been made for either convex or nonconvex problems, however, most of which only consider smooth objectives. In contrast, people have not fully explored and well understood this problem when functions are nonsmooth. This paper aims to fill this crucial gap by providing a comprehensive analysis of stochastic nonsmooth convex optimization with heavy-tailed noises. We revisit a simple clipping-based algorithm, whereas, which is only proved to converge in expectation but under the additional strong convexity assumption. Under appropriate choices of parameters, for both convex and strongly convex functions, we not only establish the first high-probability rates but also give refined in-expectation bounds compared with existing works. Remarkably, all of our results are optimal (or nearly optimal up to logarithmic factors) with respect to the time horizon $T$ even when $T$ is unknown in advance. Additionally, we show how to make the algorithm parameter-free with respect to $\sigma$, in other words, the algorithm can still guarantee convergence without any prior knowledge of $\sigma$

Effects of Seawater Corrosion and Freeze-Thaw Cycles on Mechanical Properties of Fatigue Damaged Reinforced Concrete Beams

The effects of seawater corrosion and freeze-thaw cycles on the structural behavior of fatigue damaged reinforced concrete (FDRC) beams were experimentally studied. Results show that the residual strength of FDRC beams reduces as the fatigue load level (the ratio of maximum fatigue load to the ultimate static load) increases. The reduction in the loading capacity of FDRC beams in atmosphere environment was about 6.5% and 17.8% for given fatigue load levels of 0.2 and 0.3, respectively. However, if FDRC beams are exposed to the environment of seawater wet-dry cycles or to the environment of alternating actions of freeze-thaw and seawater immersion, as expected during the service life of RC bridge structures in coastal regions or in cold coastal regions, a more rapid reduction in the strength and stiffness of the beams is observed. The significance of an accurate simulation of working load and service condition RC bridge structures in coastal regions and cold coastal regions is highlighted

Near-Optimal Non-Convex Stochastic Optimization under Generalized Smoothness

The generalized smooth condition, $(L_{0},L_{1})$-smoothness, has triggered people's interest since it is more realistic in many optimization problems shown by both empirical and theoretical evidence. Two recent works established the $O(\epsilon^{-3})$ sample complexity to obtain an $O(\epsilon)$-stationary point. However, both require a large batch size on the order of $\mathrm{ploy}(\epsilon^{-1})$, which is not only computationally burdensome but also unsuitable for streaming applications. Additionally, these existing convergence bounds are established only for the expected rate, which is inadequate as they do not supply a useful performance guarantee on a single run. In this work, we solve the prior two problems simultaneously by revisiting a simple variant of the STORM algorithm. Specifically, under the $(L_{0},L_{1})$-smoothness and affine-type noises, we establish the first near-optimal $O(\log(1/(\delta\epsilon))\epsilon^{-3})$ high-probability sample complexity where $\delta\in(0,1)$ is the failure probability. Besides, for the same algorithm, we also recover the optimal $O(\epsilon^{-3})$ sample complexity for the expected convergence with improved dependence on the problem-dependent parameter. More importantly, our convergence results only require a constant batch size in contrast to the previous works.Comment: The whole paper is rewritten with new results in V

Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise

We consider the stochastic optimization problem with smooth but not necessarily convex objectives in the heavy-tailed noise regime, where the stochastic gradient's noise is assumed to have bounded $p$th moment ($p\in(1,2]$). Zhang et al. (2020) is the first to prove the $\Omega(T^{\frac{1-p}{3p-2}})$ lower bound for convergence (in expectation) and provides a simple clipping algorithm that matches this optimal rate. Cutkosky and Mehta (2021) proposes another algorithm, which is shown to achieve the nearly optimal high-probability convergence guarantee $O(\log(T/\delta)T^{\frac{1-p}{3p-2}})$, where $\delta$ is the probability of failure. However, this desirable guarantee is only established under the additional assumption that the stochastic gradient itself is bounded in $p$th moment, which fails to hold even for quadratic objectives and centered Gaussian noise. In this work, we first improve the analysis of the algorithm in Cutkosky and Mehta (2021) to obtain the same nearly optimal high-probability convergence rate $O(\log(T/\delta)T^{\frac{1-p}{3p-2}})$, without the above-mentioned restrictive assumption. Next, and curiously, we show that one can achieve a faster rate than that dictated by the lower bound $\Omega(T^{\frac{1-p}{3p-2}})$ with only a tiny bit of structure, i.e., when the objective function $F(x)$ is assumed to be in the form of $\mathbb{E}_{\Xi\sim\mathcal{D}}[f(x,\Xi)]$, arguably the most widely applicable class of stochastic optimization problems. For this class of problems, we propose the first variance-reduced accelerated algorithm and establish that it guarantees a high-probability convergence rate of $O(\log(T/\delta)T^{\frac{1-p}{2p-1}})$ under a mild condition, which is faster than $\Omega(T^{\frac{1-p}{3p-2}})$. Notably, even when specialized to the finite-variance case, our result yields the (near-)optimal high-probability rate $O(\log(T/\delta)T^{-1/3})$

Efficient Modeling of Surrogates to Improve Multi-source High-dimensional Biobank Studies

Surrogate variables in electronic health records (EHR) and biobank data play an important role in biomedical studies due to the scarcity or absence of chart-reviewed gold standard labels. We develop a novel approach named SASH for {\bf S}urrogate-{\bf A}ssisted and data-{\bf S}hielding {\bf H}igh-dimensional integrative regression. It is a semi-supervised approach that efficiently leverages sizable unlabeled samples with error-prone EHR surrogate outcomes from multiple local sites, to improve the learning accuracy of the small gold-labeled data. {To facilitate stable and efficient knowledge extraction from the surrogates, our method first obtains a preliminary supervised estimator, and then uses it to assist training a regularized single index model (SIM) for the surrogates. Interestingly, through a chain of convex and properly penalized sparse regressions that approximate the SIM loss with bias-correction, our method avoids the local minima issue of the SIM training, and fully eliminates the impact of the preliminary estimator's large error. In addition, it protects individual-level information through summary-statistics-based data aggregation across the local sites, leveraging a similar idea of bias-corrected approximation for SIM.} Through simulation studies, we demonstrate that our method outperforms existing approaches on finite samples. Finally, we apply our method to develop a high dimensional genetic risk model for type II diabetes using large-scale data sets from UK and Mass General Brigham biobanks, where only a small fraction of subjects in one site has been labeled via chart reviewing

How Does Nuclear Wastewater Discharge Affect Fishery and Marine Environment: A Case Study of Japan

With the increasing use of nuclear energy, human lives have benefited from a variety of aspects since nuclear energy can produce carbon-free electricity. Nevertheless, governments must be cautious about the waste nuclear energy produces for it’s extremely harmful to the environment and has detrimental impacts on human health. Since the nuclear water at the Fukushima plant was released in the following years after 2011, both Japan and its neighboring countries were seriously affected. Some other coastal areas also have varying degrees of pollution depending on the ocean current. The extent of the impact of nuclear wastewater namely the affected areas and the diffusion of elements in nuclear wastewater will be shown in the paper. Additionally, this paper will analyze and elaborate on how nuclear wastewater can affect the marine environment due to the structure of the marine environment and the properties of nuclear wastewater. Lastly, the impact of nuclear wastewater on the fishery in Japan and neighboring countries will be discussed by showing data from relevant research papers. This paper will focus on the impact of nuclear wastewater on the marine environment and the vicinity fishing industry