Bayesian optimization (BO) is a popular approach to optimize
expensive-to-evaluate black-box functions. A significant challenge in BO is to
scale to high-dimensional parameter spaces while retaining sample efficiency. A
solution considered in existing literature is to embed the high-dimensional
space in a lower-dimensional manifold, often via a random linear embedding. In
this paper, we identify several crucial issues and misconceptions about the use
of linear embeddings for BO. We study the properties of linear embeddings from
the literature and show that some of the design choices in current approaches
adversely impact their performance. We show empirically that properly
addressing these issues significantly improves the efficacy of linear
embeddings for BO on a range of problems, including learning a gait policy for
robot locomotion.

本文通过对现有线性嵌入的文献进行研究后，识别出其中的一些关键问题和误解，并在解决这些问题的基础上，极大地提高了利用线性嵌入进行 Bayesian 优化的效能，包括机器人运动中学习步态策略等方面。

高维贝叶斯优化中线性嵌入的重新审视

Re-Examining Linear Embeddings for High-Dimensional Bayesian  Optimization

We propose learning deep models that are monotonic with respect to a
user-specified set of inputs by alternating layers of linear embeddings,
ensembles of lattices, and calibrators (piecewise linear functions), with
appropriate constraints for monotonicity, and jointly training the resulting
network. We implement the layers and projections with new computational graph
nodes in TensorFlow and use the ADAM optimizer and batched stochastic
gradients. Experiments on benchmark and real-world datasets show that six-layer
monotonic deep lattice networks achieve state-of-the art performance for
classification and regression with monotonicity guarantees.

通过交替使用线性嵌入层、格子集合和校准器（分段线性函数）构建具有单调性的深度模型，使用 TensorFlow 实现，并使用 ADAM 优化器和随机梯度下降方法，实验结果表明，六层的单调深度格子网络在分类和回归问题上的性能都达到了最先进的水平。

深层格网络和局部单调函数

Deep Lattice Networks and Partial Monotonic Functions

We address the problem of communicating domain knowledge from a user to the
designer of a clustering algorithm. We propose a protocol in which the user
provides a clustering of a relatively small random sample of a data set. The
algorithm designer then uses that sample to come up with a data representation
under which $k$-means clustering results in a clustering (of the full data set)
that is aligned with the user's clustering. We provide a formal statistical
model for analyzing the sample complexity of learning a clustering
representation with this paradigm. We then introduce a notion of capacity of a
class of possible representations, in the spirit of the VC-dimension, showing
that classes of representations that have finite such dimension can be
successfully learned with sample size error bounds, and end our discussion with
an analysis of that dimension for classes of representations induced by linear
embeddings.

本文提出一种协议，将用户提供的较小的数据样本进行聚类，并在此基础上建立一个数据表示方法，通过此方法学习聚类表征，并分析其统计样本复杂度，以及线性嵌入诱导的表征类的 VC 维，从而可以学习成功地学习具有有限 VC 维的表征类。