Fenchel-Young losses are a family of convex loss functions, encompassing the
squared, logistic and sparsemax losses, among others. Each Fenchel-Young loss
is implicitly associated with a link function, for mapping model outputs to
predictions. For instance, the logistic loss is associated with the soft argmax
link function. Can we build new loss functions associated with the same link
function as Fenchel-Young losses? In this paper, we introduce Fitzpatrick
losses, a new family of convex loss functions based on the Fitzpatrick
function. A well-known theoretical tool in maximal monotone operator theory,
the Fitzpatrick function naturally leads to a refined Fenchel-Young inequality,
making Fitzpatrick losses tighter than Fenchel-Young losses, while maintaining
the same link function for prediction. As an example, we introduce the
Fitzpatrick logistic loss and the Fitzpatrick sparsemax loss, counterparts of
the logistic and the sparsemax losses. This yields two new tighter losses
associated with the soft argmax and the sparse argmax, two of the most
ubiquitous output layers used in machine learning. We study in details the
properties of Fitzpatrick losses and in particular, we show that they can be
seen as Fenchel-Young losses using a modified, target-dependent generating
function. We demonstrate the effectiveness of Fitzpatrick losses for label
proportion estimation.

引入了 Fitzpatrick 损失函数，它是一种基于 Fitzpatrick 函数的新型凸损失函数，与 Fenchel-Young 损失函数具有相同的链接函数，但更紧致，同时保持预测的链接函数。

使用 Fitzpatrick Losses 学习

Learning with Fitzpatrick Losses

Longitudinal binary relational data can be better understood by implementing
a latent space model for dynamic networks. This approach can be broadly
extended to many types of weighted edges by using a link function to model the
mean of the dyads, or by employing a similar strategy via data augmentation. To
demonstrate this, we propose models for count dyads and for non-negative real
dyads, analyzing simulated data and also both mobile phone data and world
export/import data. The model parameters and latent actors' trajectories,
estimated by Markov chain Monte Carlo algorithms, provide insight into the
network dynamics.

采用潜空间模型，利用链接函数建模平均耦合缔结及数据扩充进行广泛扩展，应用该方法于计数耦合缔结和非负实耦合缔结的模型分析模拟数据以及移动电话数据与国际贸易数据，通过马尔科夫蒙特卡罗算法评估模型参数和潜在演员轨迹获得对网络动力学的深入洞见。

加权边动态网络的潜空间模型

Latent Space Models for Dynamic Networks with Weighted Edges

We introduce algorithms for learning nonlinear dynamical systems of the form
$x_{t+1}=\sigma(\Theta^{\star}x_t)+\varepsilon_t$, where $\Theta^{\star}$ is a
weight matrix, $\sigma$ is a nonlinear link function, and $\varepsilon_t$ is a
mean-zero noise process. We give an algorithm that recovers the weight matrix
$\Theta^{\star}$ from a single trajectory with optimal sample complexity and
linear running time. The algorithm succeeds under weaker statistical
assumptions than in previous work, and in particular i) does not require a
bound on the spectral norm of the weight matrix $\Theta^{\star}$ (rather, it
depends on a generalization of the spectral radius) and ii) enjoys guarantees
for non-strictly-increasing link functions such as the ReLU. Our analysis has
two key components: i) we give a general recipe whereby global stability for
nonlinear dynamical systems can be used to certify that the state-vector
covariance is well-conditioned, and ii) using these tools, we extend well-known
algorithms for efficiently learning generalized linear models to the dependent
setting.

该研究介绍了一种算法用于从单个轨迹中恢复权重矩阵，并成功应用于非线性动力系统，无需对其进行谱范数的限制。该算法的计算复杂度为线性，并具有较低的采样复杂度。

从单个轨迹中学习非线性动态系统

Learning nonlinear dynamical systems from a single trajectory

We study losses for binary classification and class probability estimation
and extend the understanding of them from margin losses to general composite
losses which are the composition of a proper loss with a link function. We
characterise when margin losses can be proper composite losses, explicitly show
how to determine a symmetric loss in full from half of one of its partial
losses, introduce an intrinsic parametrisation of composite binary losses and
give a complete characterisation of the relationship between proper losses and
``classification calibrated'' losses. We also consider the question of the
``best'' surrogate binary loss. We introduce a precise notion of ``best'' and
show there exist situations where two convex surrogate losses are
incommensurable. We provide a complete explicit characterisation of the
convexity of composite binary losses in terms of the link function and the
weight function associated with the proper loss which make up the composite
loss. This characterisation suggests new ways of ``surrogate tuning''. Finally,
in an appendix we present some new algorithm-independent results on the
relationship between properness, convexity and robustness to misclassification
noise for binary losses and show that all convex proper losses are non-robust
to misclassification noise.

本文研究了二元分类和类概率估计中的损失，将对它们的理解从边缘损失扩展到了一般的组合损失，文中表征了何时边缘损失可以成为适当的组合损失，明确地展示了如何从部分损失的一半中完整地确定对称损失，并介绍了组合二元损失的内在参数化的概念，并给出了适当损失和 “分类校准” 损失之间关系的完整的特征。本文还探讨了 “最佳” 替代二元损失的问题，并介绍了精确的 “最佳” 的概念，以及存在两种凸代理损失不可比较的情况。最后，文中在附录中提供了一些新的算法无关的关于二元损失适当性、凸性和对于误分类噪声的鲁棒性之间关系的结果，并且证明了所有凸适当损失都对误分类噪声不鲁棒。