# Cooperative Learning for Noisy Supervision

###### Abstract

Learning with noisy labels has gained the enormous interest in the robust deep
learning area. Recent studies have empirically disclosed that utilizing dual
networks can enhance the performance of single network but without theoretic
proof. In this paper, we propose Cooperative Learning (CooL) framework for
noisy supervision that analytically explains the effects of leveraging dual or
multiple networks. Specifically, the simple but efficient combination in CooL
yields a more reliable risk minimization for unseen clean data. A range of
experiments have been conducted on several benchmarks with both synthetic and
real-world settings. Extensive results indicate that CooL outperforms several
state-of-the-art methods.

Index Terms—  deep learning, noisy supervision, cooperative learning

##  1 Introduction

Large-scale supervised training datasets have significantly driven the success
of deep neural networks (DNNs) in computer vision [1, 2, 3]. However, accurate
annotations provided by the human experts are usually expensive to collect in
many real-world applications. Therefore, the alternative ways such as
crowdsourcing [4] and web query [5] are explored to reduce the cost. For
example, there are a plethora of images tagged by users on open platforms
which can be exploited easily and inexhaustibly. The negative impact is these
approaches also inevitably introduce label noise to the dataset since the user
annotations are not very reliable. Considering DNNs are capable of fitting
extremely noisy labels [6], it is important to make the training robust to
such label noise.

Recent studies can be roughly categorized into two classes in terms of _the
classifier count_ , i.e., single-network structure and dual-network structure.
For the former, it usually refers to robust surrogate losses, noise transition
and self-paced learning. For instance, Bootstrap [7] applies the perceptional
consistency to the cross-entropy loss to mitigate the influence of label
noise. Forward [8] build a transition matrix on top of the classifier to
absorb the noise. Self-paced MentorNet [9] selects small loss samples as clean
instances and learn only from these instances. For the latter, it introduces
twin classifiers to be the teacher of each other by which boosts the
classification performance of the single network. Decouple [10] utilizes the
prediction disagreement of twin networks to select more informative samples as
supervision. CLC [11] leverages the entropy criterion to collaboratively
correct the labels. Co-distillation [12, 13] distills the knowledge of the one
to supervise the other one and vice versa. Co-teaching [14] leverages two
networks to select small loss instances for cross update.

Although the current dual-network structure empirically shows improvement over
the single version, it lacks of theoretical analysis and guarantee that it can
always work. Besides, a natural question is whether introducing more learners
can further benefit the learning with noisy supervision. To explore these
limits and give a more general scope, we propose a Cooperative Learning (CooL)
paradigm that multiple classifiers work cooperatively for noisy supervision.
Specifically, we firstly demonstrate the dual-network structure yields lower
risk than that associated with the single network in some suitable
cooperation. Then, we give a sufficient condition for the case of more
learners, where the risk is negatively correlated to the number of
classifiers. Generally, even although the classifiers are imperfect with noisy
supervision, a lower risk can be achieved when the more disagreement is
introduced. Finally, based on these analysis, a cooperative learning framework
is introduced that the cooperation supervision is utilized to improve the
performace. The main contribution can be summarized into the following three
points. We demonstrate in the presence of noisy supervision, the linear
combination of predictions from multiple networks yields a more reliable
supervision than predictions from either single classifier in some conditions.
A Cooperative Learning framework is introduced, where multiple different
imperfect classifiers produce supervision cooperatively to iteratively boost
the performance. We empirically verify the proposed method on CIFAR-10,
CIFAR-100 with synthetic noise and three large-scale real-world datasets
namely Clothing1M, Food-101N and WebVision. Comprehensive experiments show
that CooL outperforms several state-of-the-art methods.

##  2 The Proposed Method

###  2.1 Preliminaries

Given a noisy dataset $\mathcal{D}=\\{(x_{n},y_{n})\\}^{N}_{n=1}$, where $N$
is the sample number, $x_{n}$ denotes an image instance and
$y_{n}\in\\{0,1\\}^{c}$ is the corresponding noisy label, we target to produce
more reliable supervision for the classifiers to learn in the presence of
label noise. Assume we use $f_{i}(i=1,2,\dots)$ to represent a classifier with
index $i$ and $p_{i}$ indicates the prediction of $f_{i}$. We explore to
achieve this goal via the cooperation of multiple classifiers.

###  2.2 Dual-Network Cooperative Learning

As empirically indicated in several works [12, 13, 14, 10], the dual-network
structure is easy to acquire a robust classification performance when learning
with noisy supervision. To understand the law of this phenomenon, we deduce
the theoretical analysis in light of risk minimization. We term this
methodology as dual-network Cooperative Learning (CooL) to ease the
explanation and unify the notion of this work.

Suppose there are two classifiers $f_{1}$ and $f_{2}$ and we use the
combination of the predictions $p_{1}$ and $p_{2}$ respectively output from
$f_{1}$ and $f_{2}$ as the new cooperation supervision:

$\displaystyle\hat{p}^{\lambda}=\lambda{p_{1}}+(1-\lambda){p_{2}}.$ (1)

$\lambda$ is the cooperation parameter to balance between the predictions from
the two classifiers. To measure the reliability of a certain supervision
$\tilde{y}$ with respect to the ground-truth label $y^{*}$, we define the
noisy supervision risk on the training set
$r_{\tilde{y}}=E_{\mathcal{D}}[\|\tilde{y}-y^{*}\|^{2}].$ 111 We take
inspiration from Li et al. [15] while our scenarios are different. We propose
to measure risks on the training set with no clean labels. In the following,
we will show that with a suitable choice of $\lambda$. leveraging the
cooperation supervision in Eq. (1) yields lower risk than the individual risk
of the either model.

######  Theorem 1.

There always exists a $\lambda$ that makes the risk $r_{\hat{p}^{\lambda}}$ of
the dual-network cooperation lower than222We exclude the situation where one
classifier strictly dominates the other otherwise there is no need for
cooperation since $\lambda$ will be set to 0 or 1 and the cooperation risk
will be equal to the lower one. individual risks of two non-identical networks
that do not always produce the incorrect predictions at the same time, i.e.,

$\displaystyle\exists\lambda,~{}r_{\hat{p}^{\lambda}}<\min\\{r_{p_{1}},r_{p_{2}}\\},$

###### Proof.

First, the risks of the individual predictions from $f_{1}$ and $f_{2}$ are
quantified respectively as the following terms,

$\displaystyle\begin{split}&r_{p_{1}}=E_{\mathcal{D}}[\|p_{1}-y^{*}\|^{2}],r_{p_{2}}=E_{\mathcal{D}}[\|p_{2}-y^{*}\|^{2}].\\\
\end{split}$

Then, the cooperation risk can be decomposed as follows,

$\displaystyle\begin{split}r_{\hat{p}^{\lambda}_{i}}&=E_{\mathcal{D}}[\|\hat{p}^{\lambda}-y^{*}\|^{2}]=E_{\mathcal{D}}[\|\lambda{p_{1}}+(1-\lambda){p_{2}}-y^{*}\|^{2}]\\\
&=\lambda^{2}r_{p_{1}}+2\lambda(1-\lambda)r_{p_{1}p_{2}}+(1-\lambda)^{2}r_{p_{2}}.\end{split}$

where $r_{p_{1}p_{2}}\triangleq
E_{\mathcal{D}_{t}}[({p_{1}}-y^{*})^{T}({p_{2}}-y^{*})]$. For two non-
identical classifiers $f_{1}$ and $f_{2}$ that do not always produce the
incorrect predictions at the same time, as $p_{1}$, $p_{2}$ are label
distributions and $y^{*}$ is the one-hot label, we will have

$\displaystyle 0\leq r_{p_{1}p_{2}}<\min\\{r_{p_{1}},r_{p_{2}}\\}.$ (2)

By setting $\nabla_{\lambda}r_{\hat{p}^{\lambda}_{i}}=0$, we obtain

$\displaystyle\begin{split}\min_{\lambda}r_{\hat{y}^{\lambda}}&=r_{p_{1}}-\frac{(r_{p_{1}}-r_{p_{1}p_{2}})^{2}}{r_{p_{1}}+r_{p_{2}}-2r_{p_{1}p_{2}}}\\\
&=r_{p_{2}}-\frac{(r_{p_{2}}-r_{p_{1}p_{2}})^{2}}{r_{p_{1}}+r_{p_{2}}-2r_{p_{1}p_{2}}}\\\
&<\min\\{r_{p_{1}},r_{p_{2}}\\}.\end{split}$ (3)

which concludes the proof of Theorem 1. ∎

######  Remark 1.

From Theorem 1, we know the suitable dual-network cooperation can achieve a
lower risk than the individual network. Furthermore, it can be found the
minimum risk obtained in Eq. (3) is positively correlated to $r_{p_{1}p_{2}}$.
This term reflects divergences between the two classifiers on the samples
where both of the classifiers make incorrect predictions. The optimal
situation is that two classifiers never make mistakes at the same, then we
have $r_{p_{1}p_{2}}=0$. Intuitively, both $p_{1}$ and $p_{2}$ are deviated
from the true label $y^{*}$. However, these deviations are towards random
directions in the presence of stochastic label noise, the proposed cooperation
supervision can be closer to the true label.

####  2.2.1 Connection and Difference.

Here, we rethink two representative dual-network methods namely Co-
distillation and Co-teaching through the lens of Cooperative Learning.

Co-distillation represents a branch of studies which leverage the model
predictions to rectify the noisy labels. It is the case where $p_{1}$ in Eq.
(1) is substituted to the noisy label $y$. Correspondingly setting
$\lambda=\frac{r_{p_{2}}}{r_{y}+r_{p_{2}}}$, the optimal risk for $f_{1}$ can
be simplified as $\frac{r_{y}r_{p_{2}}}{r_{y}+r_{p_{2}}}$, which is lower than
the risk $r_{p_{2}}$. This explains why Co-distillation shows improvement.
Nevertheless, it also points out one defect that $r_{y}$ is fixed and cannot
be improved along with the decrease of $r_{p_{2}}$.

Co-teaching represents a line of studies which select samples with a certain
criterion for training. Thus, we can adjust the risk by modifying the
correlated dataset (removing the unreliable samples). According to [14], the
supervision of $f_{1}$ is a candidate set $\mathcal{D}_{c_{2}}$ selected by
$f_{2}$ and vice versa. The risk for $f_{1}$ is then denoted as
$E_{\mathcal{D}_{c_{2}}}[\|y-y^{*}\|^{2}]$. Generally, it is a lower risk than
that on $\mathcal{D}$ with the help of the small loss trick. If we linearly
combine the supervision like CooL, the cooperation risk is also a linear
combination with respect to $\lambda$. And the minimum will be obtained at the
boundaries, i.e., the smaller one in $r_{f_{1}}$ and $r_{f_{2}}$. In this
case, a more reasonable way is to choose one of the two candidate sets, which
has a lower risk. In Co-teaching, they utilize both sets for cross update
which may impair the performance.

###  2.3 Generalized Cooperative Learning

In this section, we aim to generalize our dual-network CooL to a multi-network
variant, which is able to achieve a even lower risk. Given $n$ non-identical
classifiers, we denote the new cooperation supervision as
$\hat{p}^{\boldsymbol{\lambda}}=\boldsymbol{\lambda}\boldsymbol{p}$, where
$\boldsymbol{\lambda}$ is a $n$-dimension row vector with summation equal to 1
and $\boldsymbol{p}$ is a stack of $p_{1},\dots,p_{n}$ in rows. We define
$\boldsymbol{R}=(r_{ij})_{n\times n}$ where
$r_{ij}=E_{\mathcal{D}}[({p_{i}}-y^{*})^{T}({p_{j}}-y^{*})]$. The
corresponding diagonal elements are the individual risks associated with the
predictions of the $n$ classifiers respectively. In the following, we analyze
the cooperation risk for multiple classifiers.

######  Theorem 2.

Given the cooperation supervision
$\hat{p}^{\boldsymbol{\lambda}}=\boldsymbol{\lambda}\boldsymbol{p}$, the
associated risk is
$r_{\hat{p}^{\boldsymbol{\lambda}}}=\boldsymbol{\lambda}\boldsymbol{R}\boldsymbol{\lambda}^{T}$.
An invertible $\boldsymbol{R}$ yields,

$\displaystyle\min_{\boldsymbol{\lambda}}r_{\hat{p}^{\boldsymbol{\lambda}}}=\frac{1}{\sum_{i,j=1}^{n}[\boldsymbol{R}^{-1}]_{i,j}}.$
(4)

If all non-identical classifiers are independently trained in the same
settings, so that the following conditions satisfy

$\displaystyle\begin{split}r_{ii}&=r_{diag},\forall i=1,\dots,n\\\
r_{ij}&=r_{off},\forall i\neq j\end{split}$ (5)

Then, the minimum cooperation risk in Eq. (4) will be

$\displaystyle\min_{\boldsymbol{\lambda}}r_{\hat{p}^{\boldsymbol{\lambda}}}=\frac{1}{n}(r_{diag}-r_{off})+r_{off}<r_{diag},$
(6)

which is lower than the individual risks of all classifiers.

###### Proof.

The risk associated with the new supervision is

$\displaystyle\begin{split}r_{\hat{p}^{\boldsymbol{\lambda}}}&=E_{\mathcal{D}}[\|\boldsymbol{\lambda}\boldsymbol{p}-y^{*}\|^{2}]=E_{\mathcal{D}}[\|\sum_{i=1}^{n}\lambda_{i}({p_{i}}-y^{*})\|^{2}]\\\
&=\boldsymbol{\lambda}\boldsymbol{R}\boldsymbol{\lambda}^{T},\textit{~{}~{}s.t.},(1,\dots,1)\boldsymbol{\lambda}^{T}=1.\\\
\end{split}$

Leveraging the Lagrange multipliers, we now minimize,

$\displaystyle\min
f(\boldsymbol{\lambda},\boldsymbol{\mu})=\boldsymbol{\lambda}\boldsymbol{R}\boldsymbol{\lambda}^{T}-\boldsymbol{\mu}((1,\dots,1)\boldsymbol{\lambda}^{T}-1)$

By setting $\frac{\partial
f(\boldsymbol{\lambda},\boldsymbol{\mu})}{\partial\boldsymbol{\lambda}}=0$ and
$\frac{\partial
f(\boldsymbol{\lambda},\boldsymbol{\mu})}{\partial\boldsymbol{\mu}}=0$ , we
obtain,

$\displaystyle\boldsymbol{\lambda}_{0}$
$\displaystyle=\frac{\boldsymbol{R}^{-1}(1,\dots,1)^{T}}{\sum_{i,j=1}^{n}[\boldsymbol{R}^{-1}]_{i,j}},\boldsymbol{\mu}_{0}=\frac{2}{\sum_{i,j=1}^{n}[\boldsymbol{R}^{-1}]_{i,j}}.$

Thus, the minimum risk associated with $r_{\hat{p}^{\boldsymbol{\lambda}}}$ is

$\displaystyle\begin{split}\min_{\boldsymbol{\lambda}}r_{\hat{p}^{\boldsymbol{\lambda}}}&=f(\boldsymbol{\lambda}_{0},\boldsymbol{\mu}_{0})=\frac{1}{\sum_{i,j=1}^{n}[\boldsymbol{R}^{-1}]_{i,j}}.\end{split}$

To further analyze the characteristics of above equation, if we have Eq. (5)
satisfied Eq. (4) will be deduced as

$\displaystyle\begin{split}\min_{\boldsymbol{\lambda}}r_{\hat{p}^{\boldsymbol{\lambda}}}&=\frac{1}{n}(r_{diag}-r_{off})+r_{off}<r_{diag}.\end{split}$

which concludes the proof of Theorem 2. ∎

######  Remark 2.

From Theorem 2, we can see that the first term on the RHS of Eq. (6) indicates
by leveraging more classifiers (increasing n), we can monotonically obtain a
lower risk. In this case, $\boldsymbol{R}$ being diagonally dominant yields a
necessary and sufficient condition where the risk is inversely proportional to
the number of the classifiers. Besides, similar to the claim in Remark 1, the
off-diagonal element $r_{off}$ in $\boldsymbol{R}$ characterizes the
divergence of two networks. If two classifies are complementary with each
other, they will work better cooperatively even though they are imperfect.

###  2.4 The Cooperative Learning Framework

The theoretical analysis in previous section tells us that the cooperation of
multiple classifiers can lower the supervision risk and the lower bound is
determintered by the divergence between the classifiers. Based on this, we
introduce a new Cooperative Learning (CooL) framework where the proposed
cooperation supervision namely the combination of the predictions from the
multiple classifiers is adopted to re-train the individual networks. As
claimed in Remark 1 and 2, the prerequisite of a better performance on noisy
datasets via cooperation, is to generate diverse classifiers. We thus make the
classifiers learn from different sources of information to construct pattern
bias. Specifically, we pre-train the $n$ classifiers respectively on
$\mathcal{D}_{1},\dots,\mathcal{D}_{n}$, the different $n$ partitions of $D$.
Note this pre-training style relies on the assumption that the subset is still
sufficient enough to learn a classifier exhibiting the same risk on the whole
dataset. Thus we are not able to infinitely add classifiers to lower the risk
due to limited data.

After obtaining multiple different pre-trained classifiers, we can utilize the
combination of their predictions to train better classifiers. Instead of
training another student network with such supervision like [15], we
iteratively train the classifiers with the objective function as follows,

$\displaystyle L(f_{i})$
$\displaystyle=l_{\mathcal{D}}(\hat{p}^{\lambda},f_{i})+\alpha
l_{\mathcal{D}_{i}}(y,f_{i})+\beta h_{\mathcal{D}_{i}}(f_{i}).$ (7)

In the first term of Eq. (7), the network $f_{i}$ is supervised by the
cooperation supervision $\hat{p}^{\lambda}$, which is the key module of this
paper. The second part is an auxiliary term that supervises $f_{i}$ with the
original labels in the early phase but will be gradually canceled out as the
model is capable of memorizing the noisy labels. The third term is the entropy
of the model predictions which prevents the output of the network $f_{i}$ from
degenerating to the uniform distribution. As for the hyperparameters $\alpha$
and $\beta$, we empirically assign small weights like [13]. The complete
training process is summarized in Algorithm 1.

Algorithm 1 The CooL Algorithm

0: A noisy set $\mathcal{D}$, multiple networks $f_{1},\dots,f_{n}$,
$\lambda$, $\alpha$, $\beta$.

1: Randomly partition $\mathcal{D}$ into
$\mathcal{D}_{1},\dots,\mathcal{D}_{n}$.

2: Directly pre-train $f_{i}$ on $\mathcal{D}_{i}(i=1,2,\dots)$ respectively.

3: for epoch $i=\text{StartEpoch}$ to MaxEpoch do

4: for batch $j=1$ to $\frac{|\mathcal{D}|}{\text{BatchSize}}$ do

5: for $k=1$ to $n$ do

6: Update $f_{k}$ by optimizing $L(f_{k})$ in Eq. (7)

7: end for

8: end for

9: end for

Complexity Analysis  The time complexity of CooL is not a big issue since we
can distribute the computation into individual classifiers parallelly. Assume
$M$ is the mini-batch size and $\Lambda$ is the parameter size, then in each
mini-batch update, the time complexity for each classifier is
$\mathcal{O}(M\Lambda)$. However, for the space complexity, it might be a
bottleneck as the storage cost is linearly related to the number of the
classifier, i.e., $\mathcal{O}(nM\Lambda)$. Thus, when implementing multiple-
network CooL in practices, we have to consider the resource limit.

##  3 Experiments

###  3.1 Datasets and Baselines

| CIFAR-10 | CIFAR-100
---|---|---
| pairwise | asymmetric | symmetric | pairwise | asymmetric | symmetric
setting | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
noise ratio ($r$) | 0.2 | 0.45 | 0.2 | 0.45 | 0.2 | 0.5 | 0.2 | 0.45 | 0.2 | 0.45 | 0.2 | 0.5
Standard | 76.54 | 49.73 | 82.86 | 69.52 | 76.71 | 49.91 | 50.47 | 32.51 | 50.79 | 31.47 | 48.63 | 25.44
Forward | 78.06 | 58.69 | 83.29 | 69.59 | 77.56 | 51.70 | 52.67 | 30.14 | 54.99 | 28.41 | 47.00 | 25.94
Bootstrap | 76.19 | 49.75 | 82.90 | 70.23 | 76.66 | 48.83 | 51.01 | 31.17 | 50.35 | 32.42 | 47.87 | 24.74
MentorNet | 80.32 | 58.67 | 83.62 | 70.28 | 80.73 | 68.70 | 50.89 | 32.26 | 50.27 | 32.35 | 52.12 | 37.96
Decouple | 77.30 | 49.23 | 83.53 | 70.24 | 78.03 | 48.97 | 51.28 | 31.49 | 50.50 | 31.72 | 46.19 | 23.29
Co-distillation | 81.36 | 51.87 | 85.28 | 72.15 | 81.49 | 56.76 | 56.40 | 34.93 | 55.27 | 35.27 | 54.36 | 32.41
Co-teaching | 83.24 | 72.74 | 85.12 | 76.02 | 82.09 | 74.06 | 54.74 | 34.08 | 52.81 | 34.77 | 53.84 | 41.34
Bagging | 81.35 | 57.22 | 82.58 | 72.12 | 79.08 | 67.90 | 45.99 | 26.21 | 47.24 | 26.46 | 43.79 | 23.98
CooL | 89.48 | 88.62 | 89.73 | 82.95 | 87.88 | 81.76 | 63.64 | 48.38 | 63.65 | 49.24 | 60.36 | 45.93
Table 1: Average test accuracy (%) on CIFAR-10 & CIFAR-100 over the last ten
epochs.

To demonstrate the effectiveness of CooL, we experiment with CIFAR10 and
CIFAR100 [16] with pairwise noise [14], asymmetric noise [8], symmetric noise
[17] and Clothing1M [18], Food-101N [19], WebVision [20] for real-world noise.
We compare CooL with the following two categories of noisy-supervised learning
methods. Single-network Methods: _Standard_ , which directly trains a vanilla
classifier on noisy datasets; _Forward_ [8], which uses a noise transition
matrix for the forward loss correction; _LCCN_ [21], which dynamically adjust
the transition matrix to safeguard the learning process. We only directly
report the result of LCCN on WebVision to save huge labor to reproduce as we
adopt the same settings; _Bootstrap_ [7], which linearly combines the model
predictions and original labels; _MentorNet_ [9]. We deploy self-paced
MentorNet namely a single model determines small loss samples as useful
information and learn with these samples; Dual-network Methods: _Decouple_
[10], which updates the parameters when the two models disagree; _Co-
distillation_ [12, 13], which is the dual network version of Bootstrap; _Co-
teaching_ [14], which is the dual network version of self-paced MentorNet;
_Bagging_ [22] which takes a vote of multiple classifiers pre-trained on
random data partitions. Specifically we use two classifiers.

###  3.2 Implementation

For CIFAR-10 and CIFAR-100, we follow the same implementation in [14] For
real-world datasets, a 50-layer ResNet architecture [3] pre-trained on
ImageNet is adopted as the classifier. The images are resized to 256 with
respect to shorter sides and then randomly cropped to 224$\times$224 with
random flip, brightness, contrast and saturation. For Clothing1M and
Food-101N, the batch size is set to 64 and we run 20 epochs on Clothing1M and
40 epochs on Food-101N. For WebVision, we align the learning settings with Yao
et al. [21] to save the labor in reproducing the results.

For all experiments, we use the same architecture for two networks when
implementing Decouple, Co-teaching, Co-distillation and CooL as done in [10,
13, 14]. To be fair, all methods use pre-training as warming-up following [8,
13, 14]. Specifically the models are trained as Standard for 4 epochs on
Clothing1M and 10 epochs on all other datasets. For dual network methods, two
branches are pre-trained separately.For Forward, we use the normalized ground-
truth confusion matrix provided in [18] on Clothing1M. For MentorNet and Co-
teaching, the noise ratio $r$ is provided as side information as required in
[14] for the pre-defined curriculum. For CooL, we set $\lambda=0.5$ since two
networks have the same architecture and are trained in the same manner. We
empirically set $\alpha=0.05$ on CIFAR-10 and Clothing1M while we set
$\alpha=0.1$ on datasets containing much more categories. For the CIFAR
datasets, we linearly decrease $\alpha$ since deep models easily fit the
small-scale datasets and we fix $\beta=0.05$.

###  3.3 Results on CIFAR10 and CIFAR-100

Table 1 summarizes the average test accuracy of dual network CooL and all
baselines on CIFAR-10 and CIFAR-100 over the last ten epochs. We can see from
the results that dual-network methods such as Co-distillation and Co-teaching
generally perform better than their single version namely Bootstrap and
MentorNet. Bagging works well on CIFAR-10 but fails on CIFAR-100. The reason
is that CIFAR-100 contains more classes thus the data after random partition
is insufficient to train a good individual learner. However, adopting similar
style of pre-training, CooL manages to achieve the best performance in all of
the noise settings. This indicates that CooL can effectively boost two
imperfect individual learners. Pointedly for low-level pairwise noise , CooL
outperforms the best baseline by 6.24% on CIFAR-10 and 7.24% on CIFAR-100.
When $r$ raises to 0.45, all baselines degenerate hard while CooL shows great
robustness dealing with high-level noise. According to the results, CooL
outperforms the best baseline by 15.88% on CIFAR-10 even impressively
surpassing the results of all baselines under low-level noise. For asymmetric
noise, CooL achieves the best performances in all settings and manages to
outperform the best baseline by 14.47% when $r=0.45$ on CIFAR-100. Symmetric
noise is the hardest noise pattern as the overall test accuracies are low.
However, CooL manages to outperform all the baselines.

####  3.3.1 Counteracting the Memorization Effects

Fig. 1: Left: Training curve on CIFAR-10 with pairwise noise ($r=0.45$);
Middle: Training curve on CIFAR-10 with pairwise noise ($r=0.45$); Right:
Comparison of CooL, CooL-3 and CooL-4.

Memorization effects [23] refer to the behavior of DNNs under noisy
supervision that the model will firstly learn from clean data and eventually
fit the noisy labels. This phenomenon can be visualized as rise followed by
drop in the test accuracy curve. In the left panel of Figure 1, we trace the
test accuracy of all baselines and CooL under high-level pairwise noise on
CIFAR-10. For all baselines, the test accuracy increases at first and then
decreases as the training proceeds, which matches the memorization effects.
However, the curve of CooL keeps increasing and then persists at a high level
which indicates that the proposed cooperation supervision is reliable enough
to counteract the memorization effects.

####  3.3.2 Reliability of the Supervision

To further assess the reliability of the cooperation supervision in CooL, We
report the label precision which is the ratio of correct supervisions to total
supervisions. We compare CooL with the sample selection methods that leverage
different criteria to select the reliable supervision. The label precision
curves in setting 2 are depicted in middle panel of Figure 1. We can see that
the label precision of CooL consistently increases with regard to iterations
of optimizing and surpasses all the sample selection methods. In the advanced
stage of training, the label precision of the cooperation supervision is
94.29% which means CooL can guarantee the classifiers to learn on a relatively
clean dataset. This empirically verifies the reliability of the proposed
cooperation supervision.

####  3.3.3 Two Learners and Beyond

Here we carry out experiments to examine our theoretical findings on the
effects of utilizing multiple networks. We implement triple-network CooL
(CooL-3) and quadruple-network CooL (CooL-4) and depict the accuracy along
with CooL and Standard in right panel of Figure 1. We can see from the curves
that CooL-3 generally shows improvement or comparable results with CooL. This
matches our analysis that increasing the number of the classifiers will result
in a smaller risk which is closer to the lower bound. CooL-4 shows improvement
in setting 4 while its accuracy slight drops in setting 2 & 6.This is due to
insufficient information under the partition of the limited data as we have
discussed formerly. As training quadruple classifiers also requires more
resources, we may only resort to CooL or CooL-3 practically.

###  3.4 Results on Clothing1M, Food-101N and WebVision

method | Clothing1M | Food-101N
---|---|---
Standard | 66.14 | 75.45
Forward | 67.70 | 75.92
Bootstrap | 66.70 | 74.75
MentorNet | 60.00 | 76.93
Decouple | 64.44 | 72.47
Co-distillation | 67.23 | 77.31
Co-teaching | 67.30 | 77.87
Bagging | 67.45 | 77.14
CooL | 70.79 | 80.94
CooL-3 | 71.12 | 81.08
Table 2:  Results on Clothing1M and Food-101N.

method | acc.@1 | acc.@5
---|---|---
Standard | 63.11 | 83.69
Forward | 63.10 | 83.78
LCCN | 63.52 | 84.27
Bootstrap | 63.20 | 83.81
Decouple | 61.23 | 81.53
Co-distillation | 63.41 | 84.14
Co-teaching | - | -
Bagging | 63.45 | 84.19
CooL | 63.61 | 84.32
CooL-3 | 63.67 | 84.39
Table 3:  Results on WebVision.

In this section, we empirically verify the effectiveness of CooL on three
large-scale datasets with real-world noise.

For Clothing1M, the results are reported in the left column of Table 3. We can
see that dual-network methods generally perform better than the single
versions from which they are derived. Although MentorNet does not work well in
this setting, Co-teaching manages to surpass Standard by 1.16%. Among all
baselines, Forward achieves the best performance with the usage of the ground-
truth transition matrix. However CooL surpasses Forward by 3.09% without using
any side information. CooL-3 further improves CooL by 0.33% indicating the
effectiveness of leveraging multiple classifiers.

For Food-101N, Bootstrap degenerates slightly compared to Standard, while the
dual-network version method Co-distillation enjoys a 1.86% gain. Adopting the
same small loss trick, both MentorNet and Co-teaching perform well on
Food-101N. MentorNet outperforms Standard by 1.48% and Co-teaching outperforms
MentorNet by 0.94% with the use of dual-network structure. Without the
knowledge of the ground-truth transition matrix, Forward only improves the
test accuracy by 0.47% compared to Standard. Again, CooL manages to outperform
the best baseline by 3.07%. Adding another classifier, CooL-3 further obtains
a 0.14% gain.

For WebVision, we report both top-1 and top-5 accuracies in Table 3. The
results of Co-teaching is vacant due to the absence of the ground-truth noise
ratio. We can see from the results that our CooL achieves the best performance
and CooL-3 further surpasses CooL. However, the gap between all the methods is
trivial which may be on account of the strong open-set noise as suggested in
Yao et al. [21].

##  4 Conclusion and Future Work

In this paper, we propose a Cooperative Learning paradigm that multiple
classifiers work cooperatively with noisy supervision. We demonstrate that our
proposed cooperation risk is lower than that associated with individual
learners. Then we present a sufficient condition where the risk is negatively
correlated to the number of the classifiers. Finally, we introduce the
Cooperative Learning framework where the reliable cooperation supervision
iteratively boosts the performance of the classifiers. We conduct a range of
experiments on the CIFAR datasets to demonstrate the robustness of CooL under
synthetic noise and we verify the effectiveness of CooL on three real-world
large-scale datasets We further implement CooL-3 and CooL-4 to show that
leveraging more classifiers can have potential gain nonetheless adding more
classifiers will consume more resources. Future research directions include
finding new means to generate multiple divergent classifiers to achieve lower
risk and reducing the parameter space for multiple-network CooL via parameter
sharing.

## References

  * [1] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton,  “Imagenet classification with deep convolutional neural networks,”  in NIPS, 2012.
  * [2] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich,  “Going deeper with convolutions,”  in CVPR, 2015.
  * [3] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun,  “Deep residual learning for image recognition,”  in CVPR, 2016.
  * [4] Vikas C Raykar, Shipeng Yu, Linda H Zhao, Gerardo Hermosillo Valadez, Charles Florin, Luca Bogoni, and Linda Moy,  “Learning from crowds,”  Journal of Machine Learning Research, 2010.
  * [5] Rob Fergus, Li Fei-Fei, Pietro Perona, and Andrew Zisserman,  “Learning object categories from internet image searches,”  Proceedings of the IEEE, 2010.
  * [6] Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals,  “Understanding deep learning requires rethinking generalization,”  in ICLR, 2016.
  * [7] Scott Reed, Honglak Lee, Dragomir Anguelov, Christian Szegedy, Dumitru Erhan, and Andrew Rabinovich,  “Training deep neural networks on noisy labels with bootstrapping,”  in ICLR, 2014.
  * [8] Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu,  “Making deep neural networks robust to label noise: A loss correction approach,”  in CVPR, 2017.
  * [9] Lu Jiang, Zhengyuan Zhou, Thomas Leung, Li-Jia Li, and Li Fei-Fei,  “Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels,”  in ICML, 2018.
  * [10] Eran Malach and Shai Shalev-Shwartz,  “Decoupling” when to update” from” how to update”,”  in NIPS, 2017.
  * [11] Hao Wu, Jiangchao Yao, Jiajie Wang, Yinru Chen, Ya Zhang, and Yanfeng Wang,  “Collaborative label correction via entropy thresholding,”  in ICDM, 2019.
  * [12] Rohan Anil, Gabriel Pereyra, Alexandre Passos, Robert Ormandi, George E. Dahl, and Geoffrey E. Hinton,  “Large scale distributed neural network training through online distillation,”  in ICLR, 2018.
  * [13] Guocong Song and Wei Chai,  “Collaborative learning for deep neural networks,”  in NIPS, 2018.
  * [14] Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama,  “Co-teaching: Robust training of deep neural networks with extremely noisy labels,”  in NIPS, 2018.
  * [15] Yuncheng Li, Jianchao Yang, Yale Song, Liangliang Cao, Jiebo Luo, and Li-Jia Li,  “Learning from noisy labels with distillation,”  in ICCV, 2017.
  * [16] Alex Krizhevsky and Geoffrey Hinton,  “Learning multiple layers of features from tiny images,”  Tech. Rep., Citeseer, 2009.
  * [17] Brendan Van Rooyen, Aditya Menon, and Robert C Williamson,  “Learning with symmetric label noise: The importance of being unhinged,”  in NIPS, 2015.
  * [18] Tong Xiao, Tian Xia, Yi Yang, Chang Huang, and Xiaogang Wang,  “Learning from massive noisy labeled data for image classification,”  in CVPR, 2015.
  * [19] Kuang-Huei Lee, Xiaodong He, Lei Zhang, and Linjun Yang,  “Cleannet: Transfer learning for scalable image classifier training with label noise,”  in CVPR, 2018.
  * [20] Wen Li, Limin Wang, Wei Li, Eirikur Agustsson, and Luc Van Gool,  “Webvision database: Visual learning and understanding from web data,”  arXiv:1708.02862, 2017.
  * [21] Jiangchao Yao, Hao Wu, Ya Zhang, Ivor W Tsang, and Jun Sun,  “Safeguarded dynamic label regression for noisy supervision,”  AAAI, 2019.
  * [22] Leo Breiman,  “Bagging predictors,”  Machine learning, vol. 24, no. 2, pp. 123–140, 1996.
  * [23] Devansh Arpit, Stanisław Jastrzebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxinder S Kanwal, Tegan Maharaj, Asja Fischer, Aaron Courville, Yoshua Bengio, et al.,  “A closer look at memorization in deep networks,”  in ICML, 2017.


# Tidal Dissipation Regimes Among the Short-Period Exoplanets

Emma M. Louden Yale University, 52 Hillhouse Avenue, New Haven, CT 06511, USA
Gregory P. Laughlin Yale University, 52 Hillhouse Avenue, New Haven, CT 06511,
USA  Sarah C. Millholland MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

The efficiency of tidal dissipation provides a zeroth-order link to a planet’s
physical properties. For super-Earth and sub-Neptune planets in the range
$R_{\oplus}\lesssim R_{p}\lesssim 4R_{\oplus}$, particularly efficient
dissipation (i.e., low tidal quality factors) may signify terrestrial-like
planets capable of maintaining rigid crustal features. Here we explore global
constraints on planetary tidal quality factors using a population of planets
in multiple-planet systems whose orbital and physical properties indicate
susceptibility to capture into secular spin-orbit resonances. Planets
participating in secular spin-orbit resonance can maintain large axial tilts
and significantly enhanced heating from obliquity tides. When obliquity tides
are sufficiently strong, planets in low-order mean-motion resonances can
experience resonant repulsion (period ratio increase). The observed
distribution of period ratios among transiting planet pairs may thus depend
non-trivially on the underlying planetary structures. We model the action of
resonant repulsion and demonstrate that the observed distribution of period
ratios near the 2:1 and 3:2 commensurabilties implies $Q$ values spanning from
$Q\approx 10^{1}-10^{7}$ and peaking at $Q\approx 10^{6}$. This range includes
the expected range in which super-Earth and sub-Neptune planets dissipate
($Q\approx 10^{3}-10^{4}$). This work serves as a proof of concept for a
method of assessing the presence of two dissipation regimes, and we estimate
the number of additional multi-transiting planetary systems needed to place
any bimodality in the distribution on a strong statistical footing.

Exoplanets–tides, dynamics, orbits

††software: numpy (Harris et al., 2020); scipy (Virtanen et al., 2020); pandas
(Team, 2020)

##  1 Introduction

The growing census of short-period exoplanets with joint mass and radius
measurements hints at two distinct populations: super-Earths that may resemble
scaled-up terrestrial worlds, and sub-Neptunes, characterized by structurally
significant H/He gas envelopes. The presence of the distinct populations
follows naturally from the bimodal radius distribution among the Kepler-
detected (Borucki et al., 2010) planets (Fulton et al., 2017; Van Eylen et
al., 2018; Fulton & Petigura, 2018) which exhibit a well-established paucity
in the range 1.5$R_{\oplus}\lesssim R_{\rm P}\lesssim 2R_{\oplus}$.

The gap in the planet radius distribution was predicted by models that draw on
the photoevaporation of planetary atmospheres (Lopez et al., 2012; Owen & Wu,
2013; Jin et al., 2014). Extreme ultraviolet and X-ray radiation from the host
star heats the upper regions of the planetary atmosphere and drives outflows
(Murray-Clay et al., 2009; Owen & Wu, 2013; Lopez & Fortney, 2013). Another
energetic driver that may spur mass loss is the heat of formation of the solid
core (Ginzburg et al., 2018; Gupta & Schlichting, 2019). Because thermally-
driven mass loss is likely a threshold process, the smaller-radius peak in the
radius distribution may comprise rocky planets whose atmospheres have been
entirely stripped. In contrast, the planets in the larger peak have retained
significant atmospheres throughout the parent star’s early high-activity phase
and beyond. In many cases, the energy input inflates the retained envelopes,
enhancing contribution to the larger-radius peak in the distribution (Ketzer &
Poppenhaeger, 2023). Another hypothesis is that the sub-Neptune type planets
are, in fact, water worlds rather than H/He-rich (Zeng et al., 2019; Luque &
Pallé, 2022).

If the planets on either side of the radius gap have fundamentally different
structures, it is reasonable to expect a corresponding bimodal distribution of
tidal quality factors. This paper assesses whether evidence of a distinct
split in planetary composition can be discerned in the current catalog of
transiting extrasolar planets.

Generically, the tidal quality factor, $Q$, measures the efficiency of tidal
dissipation with

$Q^{-1}=-\frac{1}{2\pi E_{0}}\oint\frac{dE}{dt}\,dt\,,$ (1)

where $E_{0}$ is the maximum energy stored in the tidal distortion of the
planet and the integral represents the energy lost in one cycle (Goldreich &
Soter, 1966). Terrestrial bodies in the Solar System tend to have $Q_{\rm
terrestrial}\sim 10^{2}$, while Uranus and Neptune are at least two orders of
magnitude less dissipative ($Q_{\rm ice-giant}\gtrsim 10^{4}$) (Goldreich &
Soter, 1966). With the exception of Earth and the Moon, however, planet-wide
energy dissipation rates are difficult to measure with high accuracy. For the
Moon, a precise value $Q_{\leftmoon}\sim 38$ was inferred via laser ranging
(Williams & Boggs, 2015), whereas for Earth, the handbook value is
$Q_{\oplus}=12$ (e.g., Yoder, 1995). A recent estimate reported by Lainey
(2016) draws on astrometric measurements to obtain $Q\sim 100$ for Mars.
Numerical simulations that reconstruct the past dissipative orbital evolution
of the Uranian and Neptunian satellites suggest a tidal $Q$ between 15,000 and
20,000 for Uranus (Ćuk et al., 2020) and a $Q$ between 9,000 and 36,000 for
Neptune (Zhang & Hamilton, 2008).

Currently, there are limited constraints on tidal $Q$ values for the lower-
mass ($M_{\rm P}<30\,M_{\oplus}$) exoplanets. Planets in this category that
are individually constrained include GJ 876 d (Puranam & Batygin, 2018), and
GJ 436 b (Morley et al., 2017). The former was measured by considering the
competing roles of eccentricity damping by tides and eccentricity excitation
by the Laplace-like resonance chain in the system, whereas the latter was
constrained by fitting the planet’s emission spectrum to detailed atmospheric
models that include the outward flux of interior heat generated by tidal
dissipation within the planet.

The efficiency of tidal dissipation depends on the body’s rheological
properties (Williams & Boggs, 2015) whose global expressions are reflected in
the planetary Love number, rigidity, bulk modulus, and overall density. In
particular, $Q$ is anti-correlated with the core radius and shear modulus
(Lainey, 2016), implying that terrestrial bodies dissipate more efficiently.

Figure 1: Fractional distance from resonance for planet pairs near first-order
mean motion resonances. Multi-transiting exoplanetary systems near the 3:2 and
2:1 commensurabilities tend to accumulate just wide of the exact resonances
(which occur for $\Delta=0$). There is both an over-density at ratios
$\Delta>0$ and a paucity for $\Delta<0$.

Given that some short-period exoplanets may be subject to strong tidal
forcing, the impacts of their dissipative properties may be evident within
demographic features of the population such as the distribution of orbital
period ratios between pairs of planets within the same system. The period
ratio distribution observed by NASA’s Kepler Mission revealed that most ($\sim
80-90\%$) planets in compact multiple-planet systems have period ratios that
are inconsistent with participation in low-order mean-motion orbital
resonance. There is, however, a notable excess of planet pairs with period
ratios just above the 2:1 and 3:2 commensurabilities and a dearth just below
(Lissauer et al., 2011; Fabrycky et al., 2014).

Figure 1 illustrates these features in the period ratio distribution. The
pile-up and evacuation can be seen distinctly when the period pairs are
represented as a function of $\Delta$, the fractional distance of a pair of
periods $P_{2}>P_{1}$ from exact $(j+1):j$ commensurability

$\Delta_{j+1:j}=\frac{j}{j+1}\frac{P_{2}}{P_{1}}-1.$ (2)

Lithwick & Wu (2012) (see also Batygin & Morbidelli, 2013) showed that when
two planets in mean-motion resonance harbor both a significant angular
momentum deficit and significant dissipation, the planets will migrate apart
(or repel each other) in period ratio space with the evolution exhibiting an
overall time dependence

$\Delta_{mig}\propto(t/\textrm{Gyr})^{1/3}\,,$ (3)

where the constant of proportionality depends both on the orbits and the
interior planetary properties for a given planetary pair. In §4, we examine
this evolution of the $\Delta$-value for a planet pair in more detail,
extending it to account for both eccentricity and obliquity tides. Here, we
note that when applied to a population of planets with an initially uniform
$\Delta$ distribution, this heuristic can generate the features exhibited by
the Kepler planets. For each individual resonant pair in the initial
distribution, orbital energy is lost to heat, the two planets repel each
other, the orbits circularize, and the period ratio increases. Figure 2 shows
this process acting over time on a population.

Figure 2: Simulated evolution of period ratios for a synthetic planet
population. Top: Time evolution of a population of planets initially uniformly
distributed in $\Delta$ due to the action of the model encapsulated by
Equation 3 from Lithwick & Wu (2012) which assumes a fiducial $Q$ of 10.
Bottom: The time evolution over 2 billion years of four populations
parameterized by $Q$, and spaced initially uniformly in $\Delta$. Systems with
lower $Q$ are more dissipative than those with higher $Q$ and evacuate the
region near $\Delta=0$ more rapidly.

Although resonant repulsion can qualitatively explain the period ratio
features, the source of dissipation has been difficult to pin down.
Eccentricity tides alone are insufficient to produce the resonant features
because the dissipation associated with the typical eccentricities within
compact multi-planet systems is not strong enough to increase the period
ratios by the observed amount (e.g. Silburt & Rein, 2015). Possible solutions
include dissipation from the disk (Choksi & Chiang, 2020, 2022) or dissipation
from obliquity tides, as proposed by Millholland & Laughlin (2019) and
explored in detail in this work.

Obliquity tides result when the spin axis and the orbital angular momentum
vector of a planet are misaligned. Dissipation occurs as the planet’s tidal
bulge is continually forced across the planet during the course of an orbit
(e.g. Winn & Holman, 2005). Obliquity tides are an attractive mechanism to
explain the near-resonant features in the period ratio distribution. First, as
shown by Millholland & Laughlin (2019), close-in multiple-planet systems are
susceptible to “secular spin-orbit resonances,” which involve a
commensurability between a planet’s spin-axis precession rate and its orbital
precession rate (see Section 2 for a detailed explanation of this resonance).
Such resonances allow obliquity tides to operate over extended periods,
thereby providing a potential avenue for generating sufficient dissipation for
planet pairs to experience resonant repulsion. Second, the degree of extra
dissipation offered by obliquity tides is sufficient to match the observed
offsets between the period ratio accumulations and the exact period ratio
commensurabilities (discussed more in Section 2 and in Figure 3).

In this paper, we study how the period ratio distribution evolves when a
significant number of its constituent planet pairs are participating in
orbital mean-motion resonance while simultaneously having one or both pair
members trapped in secular spin-orbit resonance. In all likelihood, a
combination of disk effects and tides could be responsible for sculpting the
observed period ratio features. However, given the complication of modeling
both disk-driven and tidally-driven effects, our goal is to isolate the
effects of tides here and see what inferences can or cannot be made about the
$Q$ distribution. Millholland & Laughlin (2019) explored whether an analysis
of this type could be used to glean assessments of the range of $Q$-values and
internal planetary structures that typify the Kepler planets. That work
performed a cursory analysis, however, and it did not incorporate the growing
sample of transiting multiple-planet systems emerging from the NASA TESS
Mission (Ricker et al., 2015). It is thus helpful to revisit the issue in more
detail. Specifically, do we see evidence of statistically distinct dissipation
regimes among the population of short-period exoplanets?

The plan of this paper is as follows. In Section 2, we discuss the related
physics and dynamics of obliquity tides. Section 3 describes the planet sample
used in our analysis. In Section 4, we provide calculations of the tidal $Q$
distribution of the sample and constrain the false alarm probability of our
signal. Section 5 reviews the conclusions and discusses the findings in the
context of future observations.

##  2 Tidal and Spin Dynamical Models

We employ a simple viscoelastic model for the response of a fluid planet to
gravitational perturbations. Hut (1981) holds that deformations from
equilibrium are characterized by a constant time lag between the radial line
to the perturber and the long axis of the tidal bulge. With this simplifying
approximation (which holds to second order in the orbital eccentricity), and
with eccentricity $e$, mean motion, $n$, and obliquity, $\epsilon$, the tidal
luminosity of the planet is given by (Leconte et al., 2010)

$\dot{E}(e,\epsilon)=K_{p}\frac{2}{1+\cos^{2}\epsilon}[\sin^{2}{\epsilon}+e^{2}(7+16\sin^{2}\epsilon)]\,,$
(4)

where

$K_{p}=\frac{3n}{2}\frac{k_{2}}{Q}\left(\frac{GM_{\star}^{2}}{R_{p}}\right)\left(\frac{R_{p}}{a}\right)^{6}\,,$
(5)

and where $k_{2}$ is the Love number, with small $k_{2}$ implying significant
central concentration (Colombo, 1966). Equation 4 indicates that if a planet
is in a high-obliquity state, then significant dissipation can be present even
if the orbit is circular (see Figure 3). In general, tidal dissipation damps
the obliquity and realigns the spin axis. However, if a dynamical mechanism
exists to maintain the planet’s obliquity in the face of damping, then the
planet can continuously tap orbital energy to generate tidal luminosity.

Higher obliquities significantly boost the tidal luminosity. Lithwick & Wu
(2012) found that tidal quality factors of order $Q=10$ are required for
resonant repulsion to explain the period ratio distribution if eccentricity
tides provide the dissipation. If one or both planets maintain high obliquity,
however, dissipation for given $Q$ can increase substantially. For instance,
at $e=0.01$, dissipation increases by a factor of 100 for
$\epsilon=0\degree\rightarrow 15\degree$ and by 1000 for
$\epsilon=0\degree\rightarrow 45\degree$.

Figure 3: Dependence of normalized tidal dissipation rate on eccentricity and
obliquity. The map shows contours of the normalized tidal dissipation rate
($\dot{E}(e,\epsilon)/K_{p}$ from equation 4). Dissipation is enhanced by
several orders of magnitude for a given eccentricity as obliquity increases.
Marked on the plot is the scale parameter, $\sigma=0.05$, that provides the
best-fit Rayleigh distribution modeling the eccentricity distribution of
compact multi-transiting multi-planet systems (Mills et al., 2019).

Millholland & Laughlin (2019) noticed that many planets in the population of
compact multi-planet systems have orbital and physical properties that render
them susceptible to capture into a secular spin-orbit resonance (Colombo,
1966), which can excite and maintain high obliquities. In such a
configuration, the average precession rate of the planet’s spin axis is driven
to match the nodal regression rate (or a Fourier component thereof) of the
planet’s orbital plane. The damped fixed point is known as a Cassini State
(Peale, 1969), and any adiabatic evolution in the nodal regression rate,
$g=\dot{\Omega}<0$, forces a corresponding evolution in the planetary
obliquity, $\epsilon$, such that the relation

$g\sin(\epsilon-I)+\alpha\cos{\epsilon}\sin{\epsilon}=0,$ (6)

is maintained. Here, $I$ is the planet’s orbital inclination and $\alpha$ is
the spin-axis precession constant, given by

$\alpha=\frac{1}{2}\frac{M_{\star}}{M_{p}}\left(\frac{R_{p}}{a}\right)^{3}\frac{k_{2}}{C}\omega_{p}\,,$
(7)

with $\omega_{p}$ the planet’s angular spin frequency and $C$ the
dimensionless moment of inertia factor (e.g. $C=2/5$ for a uniform density
sphere). In the limit of small inclinations and large obliquities,
$\epsilon\gg I$, the Cassini state criterion reduces to

$|g|\approx\alpha\cos\epsilon\,.$ (8)

If $|g|/\alpha$ evolves adiabatically and approaches unity from above, capture
into secular spin-orbit resonance occurs, and continued decrease of
$|g|/\alpha$ increases $\epsilon$, the planetary obliquity.

When pair members’ obliquities are sufficiently large, planet pairs in low-
order mean-motion resonance experience tangible period ratio increases as
orbital energy is lost to heat. The rate of evolution depends inversely on
$Q$, offering a potential opportunity to probe planetary tidal dissipation
regimes. Our goal is to explore whether the currently known population of
compact multi-transiting multi-planet systems already allows this probe to be
made.

##  3 Planet Sample

The current aggregate of known transiting exoplanetary systems are listed on
the NASA Exoplanet Archive (NEA). We select systems with planetary radii,
$R_{p}$, having $R_{p}<9.14\,R_{\oplus}$ (equivalent to Saturn’s radius) and
orbital periods, $P<300\,$d. Additionally, we required the systems to have
more than one transiting planet, not be flagged as controversial, and have
reported values for semi-major axis, planetary radii, and planetary masses. We
used the NEA-reported planetary masses (or $M$, or $M\sin{i}$ in that order of
preference, depending on availability; no masses came from mass-radius
relationships), stellar ages, and periods. If no value was reported for the
age, we adopted a randomly selected age from the cleaned Berger et al. (2020)
planet sample used in Millholland & Winn (2021). If no value was reported for
eccentricity, we adopted a randomly selected eccentricity from a Rayleigh
distribution with scale factor $\sigma=0.05$ (Mills et al., 2019). The NEA
catalog is derived from heterogeneous sources, and in many cases, the reported
values are subject to considerable uncertainty. Our analysis is thus an
initial exploration and subject to refinement as the accuracy and quantity of
measurements in the catalog increase.

In total, 193 planets in 90 systems satisfy the constraints. The planets were
further filtered by proximity to the 2:1 and 3:2 orbital commensurabilities.
If an outer planet has a period ratio with an interior planet in the range
$2.0<\delta<2.1$ or $1.5<\delta<1.6$, it is marked as being sufficiently near
a resonant commensurability. A total of $N=35$ planet pairs contribute to the
filtered sample. Four systems have planet pairs near both the 3:2 and 2:1
commensurablities. The contributing systems are shown in Figures 4 and 5.

Figure 4: Systems with multiple transiting planets displaying period ratios
near 2:1. Symbol sizes are proportional to reported planetary radii, and the
systems are ordered by increasing stellar mass (printed on the right axis).
The navy-shaded planets are both near 2:1 MMR and have inferred dissipation
factors $3<\log_{10}{Q}<6$. The light-pink-shaded points with gray borders
represent planets near the 2:1 commensurability but with a $\log_{10}{Q}$
outside of $3-6$. Figure 5: Systems with multiple transiting planets
displaying period ratios near 3:2. The features are exactly analogous to
Figure 4, except for 3:2.

##  4 Tidal Dissipation Regimes for Exoplanets

To estimate the $Q$ distribution for the planets in our sample, we relate $Q$
to $\Delta$, a planet pair’s fractional distance from perfect
commensurability. To achieve this, we adopt a combination of the resonant
repulsion formulation in Lithwick & Wu (2012) (described independently by
Batygin & Morbidelli, 2013) and the orbital evolution described by equilibrium
tides (Hut, 1981) extended to arbitrary obliquities by Leconte et al. (2010).
With this framework, the time evolution of a resonant planetary pair under the
action of dissipation can be approximated as

$\Delta_{mig}=\left(\frac{9}{4}\mu_{1}^{2}\Gamma t\right)^{1/3}\,,$ (9)

where $\mu_{j}=m_{j}/M_{*}$, and

$\Gamma=(2+\beta)(\gamma_{e1}f_{1}^{2}\beta+\gamma_{e2}f_{2}^{2})+\frac{\gamma_{a1}f_{1}^{2}\beta^{2}-\gamma_{a2}f_{2}^{2}}{2}.$
(10)

where $\beta\equiv\frac{\mu_{2}\sqrt{a_{2}}}{\mu_{1}\sqrt{a_{1}}}$, and
$\gamma_{e}$, $\gamma_{a}$ are defined below. The terms $f_{1}$ and $f_{2}$
are Laplace coefficients (see Murray & Dermott 1998 and Deck et al. 2013).

In order to incorporate obliquity tides, the definitions for $\gamma_{a}$ and
$\gamma_{e}$ need to be generalized. Using Leconte et al. (2010)’s
formulation, we have,

$\frac{1}{\gamma_{e}}=\frac{1}{e}\frac{de}{dt}=\frac{11a}{GM_{*}M_{p}}K_{p}\left(\Omega_{e}(e)x_{p}\frac{\omega_{p}}{n}-\frac{18}{11}N_{e}(e)\right)\,,$
(11)

and

$\frac{1}{\gamma_{a}}=\frac{1}{a}\frac{da}{dt}=\frac{4a}{GM_{*}M_{p}}K_{p}\left(N(e)x_{p}\frac{\omega_{p}}{n}-N_{a}(e)\right)\,,$
(12)

where $x_{p}=\cos{\epsilon}$, and $\Omega_{e}(e),N_{e}(e),N(e),$ and
$N_{a}(e)$ are eccentricity functions (below) (as derived, for example, by
Hut, 1981).

$N(e)=\frac{1+\frac{15}{2}e^{2}+\frac{45}{8}e^{4}+\frac{5}{16}e^{6}}{(1-e^{2})^{6}}$
(13)
$N_{a}(e)=\frac{1+\frac{31}{2}e^{2}+\frac{255}{8}e^{4}+\frac{185}{16}e^{6}+\frac{25}{64}e^{8}}{(1-e^{2})^{15/2}}$
(14) $\Omega_{e}(e)=\frac{1+\frac{3}{2}e^{2}+\frac{1}{8}e^{4}}{(1-e^{2})^{5}}$
(15)
$N_{e}(e)=\frac{1+\frac{15}{4}e^{2}+\frac{15}{8}e^{4}+\frac{5}{64}e^{6}}{(1-e^{2})^{13/2}}$
(16)

Next, define

$K_{p}=\frac{X}{Q},\gamma_{e}=\frac{C_{e}X}{Q},\gamma_{a}=\frac{C_{a}X}{Q}$
(17)

where $C_{a}$, $C_{e}$ and $X$ incorporate the system parameters and the
functions of eccentricity and obliquity. In addition to specifying the
parameters discussed in Section 3, we need to adopt values for $\omega_{p}$
and $x_{p}$.

First, we look at $\omega_{p}$. Spin synchronization occurs rapidly in
comparison to the tidal evolution timescale, so we assume pseudo-synchronous
rotation,

$\omega_{p}=\frac{N(e)}{\Omega_{e}(e)}\frac{2\cos{\epsilon}}{1+\cos^{2}{\epsilon}}n\,.$
(18)

For $x_{p}$, we note that in the limit of small $e$, Equation 4 reduces to

$\dot{E}(e,\epsilon)=K\frac{2\sin^{2}{\epsilon}}{1+\cos^{2}\epsilon}.$ (19)

As illustrated in Figure 3, when $\epsilon>30\degree$, the value of
$\dot{E}/K$ varies by no more than a factor of two. We thus adopt the
simplifying assumption that $\epsilon=60\degree$ for planets for which
obliquity tides are the primary source of dissipation.

If we assume that one of the two planets in a pair is trapped in secular spin-
orbit resonance and is responsible for the dissipative evolution of the period
ratio, then we can obtain an estimate for that planet’s individual $Q$. For
the pairs of planets that we observe, however, it is unknown which planet(s)
(if either) is responsible for obliquity-driven repulsion that may have
occurred. We, therefore, consider the options independently and, afterwards,
average the results.

###  4.1 Estimating Q

If the inner planet alone is caught in a high-obliquity Cassini State, then we
have $\gamma_{e2}=\gamma_{a2}=0$, and the expression for $\Gamma$ becomes

$\Gamma=(2+\beta)\left(\frac{C_{e1}X_{1}}{Q_{1}}f_{1}^{2}\beta\right)+\frac{1}{2}\frac{C_{a1}X_{1}}{Q_{1}}f_{1}^{2}\beta^{2}$
(20)

Substituting for $\Gamma$ and rearranging gives an estimate for the inner
planet’s tidal quality factor

$Q_{1}=\frac{9\mu_{1}t}{4}\Delta_{mig}^{-3}\left[(2+\beta)\left(C_{e1}X_{1}f_{1}^{2}\beta\right)+\frac{1}{2}C_{a1}X_{1}f_{1}^{2}\beta^{2}\right]$
(21)

We evaluate the above expression to estimate a prospective $Q$ for each inner
member of our planetary pairs. We employ bootstrap resampling with replacement
to assess uncertainty and implement $N=5000$ trials. Each trial generates a
histogram of estimates for $Q_{1}$, and we maintain constant bin boundaries
over the trials. The 15th and 85th percentiles of the counts for each bin are
adopted as approximate to the one-sigma uncertainties for the histogram counts
in each bin, and these limits are plotted as solid bars in Figure 6. The top
left panel of Figure 6 presents the resulting distribution of $Q_{1}$ values.

The foregoing process is then repeated to estimate the distribution of values,
$Q_{2}$, which result if the outer member of each pair is assumed to be
dissipating via obliquity tides at a rate sufficient to generate the entire
observed repulsion, $\Delta_{mig}$, over the system’s age. The results are
plotted in the top right panel of Figure 6.

$Q_{2}=\frac{9\mu_{1}t}{4}\Delta_{mig}^{-3}\left[(2+\beta)(C_{e2}X_{2}f_{2}^{2})-\frac{1}{2}C_{a2}X_{2}f_{2}^{2}\right].$
(22)

Given the estimated distributions of $Q$ stemming from the two different
assumptions (inner versus outer planet undergoing dissipation), we next
evaluate a scenario where one planet in each pair is randomly selected to be
the dissipating member. We use the above-described bootstrap procedure to
generate the bottom panel of Figure 6.

Figure 6: Top: Inferred $Q$ values when inner (left)/outer (right) planets are
assumed to drive resonant repulsion. Dashed lines mark $Q$ values estimated
for Earth, Neptune, and Jupiter. The solid magenta vertical lines for each bin
correspond to the $1-\sigma$ bounds on the count of each bin observed in
$N=5000$ bootstrap trials. Bottom: The distribution of inferred $Q$ values
within near-resonant short-period planets when randomly selecting one planet
from each pair to contribute. The $1-\sigma$ confidence intervals are as in
the top panels.

The known presence of two distinct populations in the planetary radius
distribution motivates us to assess whether there is also evidence for a
bimodal structure in 6. We employ Hartigan’s non-parametric Dip test (Hartigan
& Hartigan, 1985) to assess whether our distribution of estimated $Q$ values
diverges from unimodality. The test measures the greatest discrepancy between
the empirical distribution function of the data and the unimodal distribution
function that most closely approximates the data. The “dip statistic”
indicates the degree of divergence from unimodality, with a larger dip
providing more evidence of multimodality. The results are shown in Table 1.
The p-values in all three cases are greater than 0.05, indicating that based
on this test, we cannot reject the null hypothesis that the samples are all
drawn from a single distribution.

If a bimodal signal exists in the current data, its presence is obscured by
both the uncertainty in the planetary parameters used in the analysis and the
assumption that in each case only one planet is contributing. Because there is
a large dynamic range in $Q$ and a narrow separation between the expected
dissipation regimes, it will take a larger sample to reveal a statistically
significant signal.

Table 1: Dip Statistic Calculations Sample | Dip Statistic | p-value
---|---|---
Inner Planets | 0.04 | 0.91
Outer Planets | 0.06 | 0.32
Combined | 0.03 | 0.90

##  5 Summary and Discussion

In this paper, we explored the regimes of tidal quality factors of short-
period exoplanets under the hypothesis that obliquity tides are a driving
source of dissipation in the planets near the 2:1 and 3:2 integer
commensurabilities. Millholland & Laughlin (2019) demonstrated that compact
multi-planet systems are frequently susceptible to secular-spin orbit
resonances, which lead to obliquity excitation and tidal dissipation. The
dissipation occurs at different rates for different planetary compositions. If
the two hypothesized populations of rocky and gaseous planets exist,
aggregation of estimates of their tidal $Q$’s should eventually reveal two
distinct groups.

We estimated individual $Q$ values using a relation between the dissipation
strength and the distance from perfect period-ratio commensurability under
assumptions that either the inner planets are in the high-obliquity state or
the outer planets are in the high-obliquity state. We performed random
pairwise selection to create a cumulative distribution. We argue that this
procedure can potentially provide a robust dynamical tool to probe the
dissipation regimes that characterize the short-period planets in multiple-
transiting systems.

While there is as-yet no signal indicating bimodality in the distribution, we
predict that as more near-commensurate multi-transiting multi-planet systems
are identified, two peaks at $Q\approx 10^{3}$ and $10^{4}$ would emerge if
geophysically distinct populations of super-Earth ($Q=10^{3}$) and sub-Neptune
($Q=10^{4}$) planets (Murray & Dermott, 1998) exist.

Our study assumes that the relevant orbital and physical parameters of the
systems are known well enough to estimate the tidal $Q$ values of the planets.
The parameters of these systems, however, are not all precisely constrained;
in particular, the planetary masses have significant uncertainties. To
establish how many planets would be needed to place the existence of two peaks
on firm statistical ground, we created a bimodal distribution centered on the
locations of the two peaks with a $\sigma$ of 0.5. We increased the sample
size until the p-value of the dip test was less than 0.05. The necessary
sample size is around 800 planets. Given the ongoing productivity of TESS and
the pending launch of PLATO, which is expected to discover 7000 planets
111https://sci.esa.int/documents/33240/36096/1567260308850-PLATO_Definition_Study_Report_1_2.pdf,
it is reasonable to expect that the population will reach the required size
within the next five to ten years.

## Acknowledgments

E.L. thanks Garrett Levine, Tiger Lu, Konstantin Gerbig, Sam Cabot, Isabel
Medlock, and Harrison Souchereau for helpful and thought-provoking
conversations.

Exoplanet Archive, ADS

## References

  * Batygin & Morbidelli (2013) Batygin, K., & Morbidelli, A. 2013, AJ, 145, 1, doi: 10.1088/0004-6256/145/1/1
  * Berger et al. (2020) Berger, T. A., Huber, D., Gaidos, E., van Saders, J. L., & Weiss, L. M. 2020, AJ, 160, 108, doi: 10.3847/1538-3881/aba18a
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977, doi: 10.1126/science.1185402
  * Choksi & Chiang (2020) Choksi, N., & Chiang, E. 2020, MNRAS, 495, 4192, doi: 10.1093/mnras/staa1421
  * Choksi & Chiang (2022) —. 2022, arXiv e-prints, arXiv:2211.15701, doi: 10.48550/arXiv.2211.15701
  * Colombo (1966) Colombo, G. 1966, AJ, 71, 891, doi: 10.1086/109983
  * Ćuk et al. (2020) Ćuk, M., El Moutamid, M., & Tiscareno, M. S. 2020, , 1, 22, doi: 10.3847/PSJ/ab9748
  * Deck et al. (2013) Deck, K. M., Payne, M., & Holman, M. J. 2013, The Astrophysical Journal, 774, 129, doi: 10.1088/0004-637x/774/2/129
  * Fabrycky et al. (2014) Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, ApJ, 790, 146, doi: 10.1088/0004-637X/790/2/146
  * Fulton & Petigura (2018) Fulton, B. J., & Petigura, E. A. 2018, AJ, 156, 264, doi: 10.3847/1538-3881/aae828
  * Fulton et al. (2017) Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109, doi: 10.3847/1538-3881/aa80eb
  * Ginzburg et al. (2018) Ginzburg, S., Schlichting, H. E., & Sari, R. 2018, MNRAS, 476, 759, doi: 10.1093/mnras/sty290
  * Goldreich & Soter (1966) Goldreich, P., & Soter, S. 1966, Icarus, 5, 375, doi: 10.1016/0019-1035(66)90051-0
  * Gupta & Schlichting (2019) Gupta, A., & Schlichting, H. E. 2019, MNRAS, 487, 24, doi: 10.1093/mnras/stz1230
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hartigan & Hartigan (1985) Hartigan, J. A., & Hartigan, P. M. 1985, The Annals of Statistics, 13, 70.  http://www.jstor.org/stable/2241144
  * Hut (1981) Hut, P. 1981, A&A, 99, 126
  * Jin et al. (2014) Jin, S., Mordasini, C., Parmentier, V., et al. 2014, ApJ, 795, 65, doi: 10.1088/0004-637X/795/1/65
  * Ketzer & Poppenhaeger (2023) Ketzer, L., & Poppenhaeger, K. 2023, MNRAS, 518, 1683, doi: 10.1093/mnras/stac2643
  * Lainey (2016) Lainey, V. 2016, Celestial Mechanics and Dynamical Astronomy, 126, 145, doi: 10.1007/s10569-016-9695-y
  * Leconte et al. (2010) Leconte, J., Chabrier, G., Baraffe, I., & Levrard, B. 2010, Astronomy and Astrophysics, 516, A64, doi: 10.1051/0004-6361/201014337
  * Lissauer et al. (2011) Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8, doi: 10.1088/0067-0049/197/1/8
  * Lithwick & Wu (2012) Lithwick, Y., & Wu, Y. 2012, ApJ, 756, L11, doi: 10.1088/2041-8205/756/1/L11
  * Lopez & Fortney (2013) Lopez, E. D., & Fortney, J. J. 2013, ApJ, 776, 2, doi: 10.1088/0004-637X/776/1/2
  * Lopez et al. (2012) Lopez, E. D., Fortney, J. J., & Miller, N. 2012, ApJ, 761, 59, doi: 10.1088/0004-637X/761/1/59
  * Luque & Pallé (2022) Luque, R., & Pallé, E. 2022, Science, 377, 1211, doi: 10.1126/science.abl7164
  * Millholland & Laughlin (2019) Millholland, S., & Laughlin, G. 2019, Nature Astronomy, 3, 424, doi: 10.1038/s41550-019-0701-7
  * Millholland & Winn (2021) Millholland, S. C., & Winn, J. N. 2021, ApJ, 920, L34, doi: 10.3847/2041-8213/ac2c77
  * Mills et al. (2019) Mills, S. M., Howard, A. W., Petigura, E. A., et al. 2019, The Astronomical Journal, 157, 198, doi: 10.3847/1538-3881/ab1009
  * Morley et al. (2017) Morley, C. V., Knutson, H., Line, M., et al. 2017, AJ, 153, 86, doi: 10.3847/1538-3881/153/2/86
  * Murray & Dermott (1998) Murray, C. D., & Dermott, S. F. 1998, Solar System Dynamics (Cambridge University Press)
  * Murray-Clay et al. (2009) Murray-Clay, R. A., Chiang, E. I., & Murray, N. 2009, ApJ, 693, 23, doi: 10.1088/0004-637X/693/1/23
  * Owen & Wu (2013) Owen, J. E., & Wu, Y. 2013, ApJ, 775, 105, doi: 10.1088/0004-637X/775/2/105
  * Peale (1969) Peale, S. J. 1969, AJ, 74, 483, doi: 10.1086/110825
  * Puranam & Batygin (2018) Puranam, A., & Batygin, K. 2018, AJ, 155, 157, doi: 10.3847/1538-3881/aab09f
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003
  * Silburt & Rein (2015) Silburt, A., & Rein, H. 2015, MNRAS, 453, 4089, doi: 10.1093/mnras/stv1924
  * Team (2020) Team, T. P. D. 2020, pandas-dev/pandas: Pandas, latest, Zenodo, doi: 10.5281/zenodo.3509134
  * Van Eylen et al. (2018) Van Eylen, V., Agentoft, C., Lundkvist, M. S., et al. 2018, MNRAS, 479, 4786, doi: 10.1093/mnras/sty1783
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Williams & Boggs (2015) Williams, J. G., & Boggs, D. H. 2015, Journal of Geophysical Research (Planets), 120, 689, doi: 10.1002/2014JE004755
  * Winn & Holman (2005) Winn, J. N., & Holman, M. J. 2005, ApJ, 628, L159, doi: 10.1086/432834
  * Yoder (1995) Yoder, C. F. 1995, Astrometric and geodetic properties of Earth and the solar system, in Global Earth Physics. A Handbook of Physical Constants (American Geophysical Union)
  * Zeng et al. (2019) Zeng, L., Jacobsen, S. B., Sasselov, D. D., et al. 2019, Proceedings of the National Academy of Science, 116, 9723, doi: 10.1073/pnas.1812905116
  * Zhang & Hamilton (2008) Zhang, K., & Hamilton, D. P. 2008, Icarus, 193, 267, doi: 10.1016/j.icarus.2007.08.024