The AI community is increasingly focused on merging logic with deep learning
to create Neuro-Symbolic (NeSy) paradigms and assist neural approaches with
symbolic knowledge. A significant trend in the literature involves integrating
axioms and facts in loss functions by grounding logical symbols with neural
networks and operators with fuzzy semantics. Logic Tensor Networks (LTN) is one
of the main representatives in this category, known for its simplicity,
efficiency, and versatility. However, it has been previously shown that not all
fuzzy operators perform equally when applied in a differentiable setting.
Researchers have proposed several configurations of operators, trading off
between effectiveness, numerical stability, and generalization to different
formulas. This paper presents a configuration of fuzzy operators for grounding
formulas end-to-end in the logarithm space. Our goal is to develop a
configuration that is more effective than previous proposals, able to handle
any formula, and numerically stable. To achieve this, we propose semantics that
are best suited for the logarithm space and introduce novel simplifications and
improvements that are crucial for optimization via gradient-descent. We use LTN
as the framework for our experiments, but the conclusions of our work apply to
any similar NeSy framework. Our findings, both formal and empirical, show that
the proposed configuration outperforms the state-of-the-art and that each of
our modifications is essential in achieving these results.

本文介绍了一种模糊算子的配置方案，用于在对数空间中端到端地完成基础公式的基础工作，并通过梯度下降进行优化，从而优于现有技术的现状，并表明新的修正对于实现这些结果是至关重要的。

logLTN：对数空间中的可微模糊逻辑

logLTN: Differentiable Fuzzy Logic in the Logarithm Space

We introduce FIFA, a fast approximate inference method for action
segmentation and alignment. Unlike previous approaches, FIFA does not rely on
expensive dynamic programming for inference. Instead, it uses an approximate
differentiable energy function that can be minimized using gradient-descent.
FIFA is a general approach that can replace exact inference improving its speed
by more than 5 times while maintaining its performance. FIFA is an anytime
inference algorithm that provides a better speed vs. accuracy trade-off
compared to exact inference. We apply FIFA on top of state-of-the-art
approaches for weakly supervised action segmentation and alignment as well as
fully supervised action segmentation. FIFA achieves state-of-the-art results on
most metrics on two action segmentation datasets.

介绍了 FIFA，一种用于动作分割和对齐的快速近似推理方法。使用可微分近似能量函数进行最小化，提高了速度超过 5 倍，同时保持良好性能并在两个动作分割数据集上实现了最先进的结果。

FIFA: 快速推理逼近动作分割

FIFA: Fast Inference Approximation for Action Segmentation

The strategy of early stopping is a regularization technique based on
choosing a stopping time for an iterative algorithm. Focusing on non-parametric
regression in a reproducing kernel Hilbert space, we analyze the early stopping
strategy for a form of gradient-descent applied to the least-squares loss
function. We propose a data-dependent stopping rule that does not involve
hold-out or cross-validation data, and we prove upper bounds on the squared
error of the resulting function estimate, measured in either the $L^2(P)$ and
$L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various
kernel classes, including Sobolev smoothness classes and other forms of
reproducing kernel Hilbert spaces. We show through simulation that our stopping
rule compares favorably to two other stopping rules, one based on hold-out data
and the other based on Stein's unbiased risk estimate. We also establish a
tight connection between our early stopping strategy and the solution path of a
kernel ridge regression estimator.

本文针对非参数回归问题中的一种梯度下降算法，提出了一种基于数据的提前停止策略，不需要保留数据或交叉验证，同时证明了该策略有较好的性能表现，可应用于 Sobolev 平滑性类等多种核函数类中。此外，本文还展现了该策略与核岭回归估计器的解路径之间的紧密联系。