Stochastic gradient descent (SGD) is a popular algorithm for optimization problems arising in high-dimensional inference tasks. Here one produces an estimator of an unknown parameter from independent samples of data by iteratively optimizing a loss function. This loss function is random and often non-convex. We study the performance of the simplest version of SGD, namely online SGD, from a random start in the setting where the parameter space is high-dimensional. We develop nearly sharp thresholds for the number of samples needed for consistent estimation as one varies the dimension. Our thresholds depend only on an intrinsic property of the population loss which we call the information exponent. In particular, our results do not assume uniform control on the loss itself, such as convexity or uniform derivative bounds. The thresholds we obtain are polynomial in the dimension and the precise exponent depends explicitly on the information exponent. As a consequence of our results, we find that except for the simplest tasks, almost all of the data is used simply in the initial search phase to obtain non-trivial correlation with the ground truth. Upon attaining non-trivial correlation, the descent is rapid and exhibits law of large numbers type behavior. We illustrate our approach by applying it to a wide set of inference tasks such as phase retrieval, and parameter estimation for generalized linear models, online PCA, and spiked tensor models, as well as to supervised learning for single-layer networks with general activation functions.
翻译:测深梯度底部( SGD) 是一种在高维推论任务中产生的优化问题流行的算法。 在这里, 我们只能从独立的数据样本中通过迭代优化损失功能来估算一个未知参数。 这个损失函数是随机的, 通常是非碳化的。 我们从参数空间高的环境下随机开始研究SGD的最简单版本的性能, 即在线 SGD 的性能。 我们发现, 除了最简单的任务之外, 我们的数据几乎全部都仅仅用于初始搜索阶段, 以获得人口损失与地面真相的非三重关联, 我们称之为信息源头。 特别是, 我们的结果并不对损失本身进行统一的控制, 例如 共性或统一的衍生界限。 我们获得的阈值是随机的, 而准确的直线性取决于信息。 我们的结果是, 除了最简单的任务之外, 我们几乎所有的数据都仅仅用于初始搜索阶段, 以获得与地面真相的非三重的关联性关系。 在达到非三重的模型中, 我们获得的直线性模型, 我们的直系的直线性模型, 以直系的直系的直系的直系函数, 通过直系的直系的直系的直系的直系的直系模型, 以直系的直系的直系的直系, 直系的直系的直系, 的直系的直系, 直系的直系的直系的直系的直系, 直系的直系, 直系, 直系的直系的直系的直系的直系的直系, 的直系, 直系, 直系的直系, 以直系的直系的直系的直系, 以直系的直系的直系的直系, 的直系的直系的直系的直系的直系的直系, 的直系, 以直系, 直系, 直系, 直系, 直系的直系为直系的直系的直系的直系为直系的直系的直系, 直系的直系的直系的直系的直系的直系的直系的直系的直系的直系的直系, 依。