The solution to empirical risk minimization with $f$-divergence
regularization (ERM-$f$DR) is presented under mild conditions on $f$. Under
such conditions, the optimal measure is shown to be unique. Examples of the
solution for particular choices of the function $f$ are presented. Previously
known solutions to common regularization choices are obtained by leveraging the
flexibility of the family of $f$-divergences. These include the unique
solutions to empirical risk minimization with relative entropy regularization
(Type-I and Type-II). The analysis of the solution unveils the following
properties of $f$-divergences when used in the ERM-$f$DR problem: $i\bigl)$
$f$-divergence regularization forces the support of the solution to coincide
with the support of the reference measure, which introduces a strong inductive
bias that dominates the evidence provided by the training data; and $ii\bigl)$
any $f$-divergence regularization is equivalent to a different $f$-divergence
regularization with an appropriate transformation of the empirical risk
function.

给定函数 $f$ 的分裂散度正则化 (Empirical Risk Minimization with $f$-Divergence Regularization) 的解决方案，该解决方案在 $f$ 的条件适当的情况下，得到最优测度是唯一的。通过利用 $f$-divergences 家族的灵活性，得到了对于特定 $f$ 函数选择的解决方案，包括使用相对熵正则化（Type-I 和 Type-II）的唯一解。当在 ERM-$f$DR 问题中使用 $f$-divergence 时，解决方案的分析揭示了以下性质：$i$）$f$-divergence 正则化使得解决方案的支持与参考测度的支持重合，从而引入了强大的归纳偏差，支配着训练数据提供的证据；和 $ii$）任何 $f$-divergence 正则化都等价于使用适当的经验风险函数转换的不同 $f$-divergence 正则化。

经验风险最小化与 f - 散度家族中的正则化的等价性

Equivalence of the Empirical Risk Minimization to Regularization on the  Family of f-Divergences

In deep learning, classification tasks are formalized as optimization
problems solved via the minimization of the cross-entropy. However, recent
advancements in the design of objective functions allow the $f$-divergence
measure to generalize the formulation of the optimization problem for
classification. With this goal in mind, we adopt a Bayesian perspective and
formulate the classification task as a maximum a posteriori probability
problem. We propose a class of objective functions based on the variational
representation of the $f$-divergence, from which we extract a list of five
posterior probability estimators leveraging well-known $f$-divergences. In
addition, driven by the challenge of improving the state-of-the-art approach,
we propose a bottom-up method that leads us to the formulation of a new
objective function (and posterior probability estimator) corresponding to a
novel $f$-divergence referred to as shifted log (SL). First, we theoretically
prove the convergence property of the posterior probability estimators. Then,
we numerically test the set of proposed objective functions in three
application scenarios: toy examples, image data sets, and signal
detection/decoding problems. The analyzed tasks demonstrate the effectiveness
of the proposed estimators and that the SL divergence achieves the highest
classification accuracy in almost all the scenarios.

通过采用贝叶斯观点，将分类任务形式化为最大后验概率问题，并基于变分表示提取了基于 $f$-divergence 的五个后验概率估计器的目标函数类。此外，通过改进现有方法，提出了一种基于自适应 log 转换的新目标函数以及后验概率估计器。理论上证明了后验概率估计器的收敛性，并在玩具示例、图像数据集和信号检测 / 解码问题的三个应用场景中进行了数值测试，结果表明提出的估计器的有效性，而自适应 log 转换具有几乎所有场景中最高的分类准确性。

$f$- 分散基于分类：超越交叉熵的使用

$f$-Divergence Based Classification: Beyond the Use of Cross-Entropy

Quality-Diversity (QD) is a concept from Neuroevolution with some intriguing
applications to Reinforcement Learning. It facilitates learning a population of
agents where each member is optimized to simultaneously accumulate high
task-returns and exhibit behavioral diversity compared to other members. In
this paper, we build on a recent kernel-based method for training a QD policy
ensemble with Stein variational gradient descent. With kernels based on
$f$-divergence between the stationary distributions of policies, we convert the
problem to that of efficient estimation of the ratio of these stationary
distributions. We then study various distribution ratio estimators used
previously for off-policy evaluation and imitation and re-purpose them to
compute the gradients for policies in an ensemble such that the resultant
population is diverse and of high-quality.

本文介绍了一种基于 Kernals 和 $f$-divergence 的 Quality-Diversity 方法，可以用于神经进化和强化学习等领域，在同时获得高任务回报和行为多样性的优化代理的过程中有效地估计各个策略之间的状态分布系数并求解梯度。

利用分布比率估算器来优化具有质量和多样性的学习智能体

Harnessing Distribution Ratio Estimators for Learning Agents with  Quality and Diversity

We propose a general framework to learn deep generative models via
\textbf{V}ariational \textbf{Gr}adient Fl\textbf{ow} (VGrow) on probability
spaces. The evolving distribution that asymptotically converges to the target
distribution is governed by a vector field, which is the negative gradient of
the first variation of the $f$-divergence between them. We prove that the
evolving distribution coincides with the pushforward distribution through the
infinitesimal time composition of residual maps that are perturbations of the
identity map along the vector field. The vector field depends on the density
ratio of the pushforward distribution and the target distribution, which can be
consistently learned from a binary classification problem. Connections of our
proposed VGrow method with other popular methods, such as VAE, GAN and
flow-based methods, have been established in this framework, gaining new
insights of deep generative learning. We also evaluated several commonly used
divergences, including Kullback-Leibler, Jensen-Shannon, Jeffrey divergences as
well as our newly discovered `logD' divergence which serves as the objective
function of the logD-trick GAN. Experimental results on benchmark datasets
demonstrate that VGrow can generate high-fidelity images in a stable and
efficient manner, achieving competitive performance with state-of-the-art GANs.

提出 VGrow 框架来学习深度生成模型，通过 $f$-divergence 进行梯度流，使用 logD-trick GAN 来作为新的目标函数评估不同的散度，结果表明 VGrow 可以生成高保真图像并且表现优异。