# Online Multi-Agent Forecasting with Interpretable Collaborative Graph Neural
Networks

Maosen Li, Siheng Chen, Yanning Shen, Genjia Liu, Ivor Tsang, and Ya Zhang M.
Li and S. Chen are with Cooperative Medianet Innovation Center, Shanghai Jiao
Tong University, Shanghai, 200240, China. E-mail: {maosen_li,
<EMAIL_ADDRESS>Shen is with University of California Irvine,
California, The United States. E-mail<EMAIL_ADDRESS>G. Liu is with
Shanghai Jiao Tong University, Shanghai, 200240, China. E-mail:
<EMAIL_ADDRESS>Tsang is with Australian Artificial Intelligence
Institute, University of Technology Sydney, Sydney, NSW, Australia. E-mail:
<EMAIL_ADDRESS>Zhang is with Cooperative Medianet Innovation Center,
Shanghai Jiao Tong University, Shanghai, 200240, China. E-mail:
<EMAIL_ADDRESS>
###### Abstract

This paper considers predicting future statuses of multiple agents in an
online fashion by exploiting dynamic interactions in the system. We propose a
novel collaborative prediction unit (CoPU), which aggregates the predictions
from multiple collaborative predictors according to a collaborative graph.
Each collaborative predictor is trained to predict the status of an agent by
considering the impact of another agent. The edge weights of the collaborative
graph reflect the importance of each predictor. The collaborative graph is
adjusted online by multiplicative update, which can be motivated by minimizing
an explicit objective. With this objective, we also conduct regret analysis to
indicate that, along with training, our CoPU achieves similar performance with
the best individual collaborative predictor in hindsight. This theoretical
interpretability distinguishes our method from many other graph networks. To
progressively refine predictions, multiple CoPUs are stacked to form a
collaborative graph neural network. Extensive experiments are conducted on
three tasks: online simulated trajectory prediction, online human motion
prediction and online traffic speed prediction, and our methods outperform
state-of-the-art works on the three tasks by $28.6\%$, $17.4\%$ and $21.0\%$
on average, respectively.

###### Index Terms:

Online multi-agent forecasting, collaborative graph, collaborative predictor,
theoretical regret analysis

##  I Introduction

Dynamic multi-agent systems depict a group of co-evolving agents which perform
interactions to activate or constrain the pattern of each other. Dynamic
systems are ubiquitous and critical in many real-world scenarios, such as
transportation networks and smart grids [1, 2, 3]. In many applications, we
need to predict future measurements of a multi-agent system, such as the
traffic on various roads within hours for route planning [4].

To forecast the future status of systems, previous works usually train models
with finite training datasets and evaluate models with unseen test sets in
offline settings. The offline models essentially assume the training and test
samples are stationary and share the same distribution, thus fixed models
could generalize to arbitrary systemic states [5, 6, 7]. We sketch the offline
learning paradigms in Fig. 1 (a).

Figure 1: Comparison of the graph-based predictors in offline and online
settings. (a) For the offline multi-agent forecasting, a training database is
used to tune the predictor, followed by an isolated test phase with a test
database. In the test phase, the model and graph construction mechanism are
fixed; however, the highly dynamic data and possibly outlier test distribution
might cause a generalization gap. We use a dashed box with a graph to
represent trainable models and a solid box with a graph for fixed model and
graph construction. (b) For online learning, the predictor is tested and tuned
simultaneously along with the unlimited iterations. The model could
dynamically adapt to time-varying data for real-time prediction. Note that
predictions are supervised by data flow at delayed iterations.

However, limited training sets are still hard to depict complicated real-world
scenarios [8], and the multi-agent system could be highly dynamic, resulting
in the infinitely possible states changing over time. To make reliable
predictions for the dynamic multi-agent system with streaming states, we need
an online model to adapt to a potential varying environment in real-time.
Taking streaming data captured along time as inputs, the online multi-agent
forecasting model is tested and trained simultaneously step-by-step and never
stops. The online learning methods never fix the model parameters, while they
always tune the models to capture the dynamic even infinite systemic states
along with iteration. The online learning paradigm is sketched in Fig. 1 (b).

Moreover, to learn the pattern of multiple agents, previous methods usually
regard every single agent in an isolated manner, such as autoregressive models
[9], hidden Markov models [10], Kalman filter [11], and deep sequence-to-
sequence models [12], which exhibit increasingly powerful modeling; however,
for multi-agent forecasting, these methods might underestimate potential
interactions among the co-evolving agents in a dynamic system. For example,
traffic flows on neighboring streets are mutually dependent; and human body-
parts might constrain each other during moving. To leverage such relations,
some works build graphs to model the relations for representation learning
[13]. For example, in traffic analysis, [14] builds a road network according
to the urban traffic map. To capture more flexible relations, some works build
trainable graphs as model parameters, whose edge weights are updated in an
end-to-end setting [15, 16]. However, such graphs are typically constructed
through trials and errors, lacking direct guidance and theoretical
interpretation, leading to the difficulties of rigorous understanding and
reasonable improvement. In this work, we propose interpretable update steps to
learn a graph that captures the collaboration among agents. We further provide
the theoretical justification of this graph.

This work aims to design an online forecasting model, which captures the
dynamic interactions among agents for real-time prediction. The core component
of our method is a _collaborative prediction unit_ (CoPU), which automatically
builds a trainable _collaborative graph_ to represent the systematic
interactions. This collaborative graph includes two types of nodes: agents and
collaborative pairs. A collaborative pair, essentially formed by a pair of
agents, refers to the mutual collaboration between the two agents. Fed with
collaborative pairs, a collaborative predictor is built in CoPU to forecast
each agent’s status, aiming to leverage the inter-agent impacts, thus the
collaborative predictors produce the possible measurements for each agent
according to the effects from another agent. Finally, we learn a weight on
each edge of the collaborative graph, aggregating all collaborative predictors
for the final predictions. The trainable edge weights reflect the influence
level of each collaborative pair. Notably, in CoPU, the collaborative
predictor is parameterized and trained through online gradient descent, while,
in a different way, the edge weights are adjusted by an online multiplicative
update step-by-step to reflect the dynamic topology. The update strategy can
be motivated by solving an explicit objective function. To understand CoPU, we
conduct regret analysis and prove that the performance of CoPU can converge to
that of the best single collaborative predictor.

One advantage of the proposed method is its interpretability, which comes from
three aspects. 1) _Algorithm_ : the edge weights of our collaborative graph
are obtained by solving an explicit optimization, while previous works are
based on heuristic designs [5, 17, 18, 19, 20, 13]. 2) _Theory_ : we prove
that the gap between the predictions based on our collaborative graph and the
optimal collaborative pair is within a bounded range. In other words, our
collaborative graph describes the information and importance carried by
different collaborative pairs for prediction. In comparison, previous works
have no theoretical support at all. 3) _Visualization_ : our interpretability
is reflected based on human understanding. In experiments, we visualize the
learned collaborative graphs changing over time, reflecting comprehensible,
reasonable and dynamic graphs in the online setting.

To produce more precise predictions, we further stack a series of CoPUs to
build a _collaborative graph neural network_ (CoGNN), progressively refining
the predictions. In CoGNN, each CoPU with a collaborative graph is directly
supervised by the ground-truth targets, thus all CoPUs predict agents’
statuses in the original measurement space. Skip-connections across CoPUs are
leveraged to stablize prediction. Compared to many spatio-temporal graph
neural networks [17, 21, 14], the advantages of CoGNN include its
interpretability and simplicity, because the information aggregation is
performed according to the importance of collaborative pairs in the clear
output space, while previous methods generate graph features in the hidden
space.

We conduct extensive experiments to validate the proposed CoGNNs on three
important tasks: online simulated trajectory prediction, online human motion
prediction and online traffic speed prediction. We note that we formulate a
new online setup that simultaneously trains and tests models, and we adapt
previous offline state-of-the-art methods to the online setting. According to
the results, CoGNNs significantly outperform state-of-the-art methods.

The main contributions are as follows.

  * •

We propose a novel collaborative prediction unit (CoPU), which predicts the
future statuses of multiple correlated agents based on a collaborative graph.
The collaborative graph contains two types of nodes; that is, individual
agents and collaborative pairs, which represent the agent dynamics and
collaboration status between agents. In this way, an agent could share
customized information with each neighboring agent via the collaborative pair.

  * •

We employ an online multiplicatively update algorithm to learn the edge
weights of the collaborative graph on the fly to solve an explicit objective
function; see Eq. (7). In this way, the graph tuned and tested simultaneously
in an online setting effectively adapts to the changing states of the dynamic
data flows. Moreover, the collaborative graph can be theoretically justified
through regret analysis.

  * •

We propose a novel collaborative graph neural network (CoGNN), which consists
of a sequence of CoPUs and progressively refines multi-agent forecasting.

  * •

We conduct extensive experiments to validate the effectiveness, interpretation
and convergence of our method, which obtains superior and reasonable results.

The rest of this paper is organized as follows. In section II, we introduce
some works related to multi-agent forecasting, graph deep learning and online
learning for prediction. In section III, we define and formulate the problem.
In section IV, we construct the proposed CoPU, introduce the online training
method, and conduct the theoretical regret analysis. In section V, we
construct the entire CoGNN model. Finally, the experiments validating the
advantages of our model and the conclusion of the paper are provided in
Section VI and Section VII.

##  II Related Works

In this section, we introduce critical methods related to our works from
aspects of 1) multi-agent time-series forecasting, 2) graph deep learning and
3) online learning for prediction.

Multi-agent forecasting of time-series. Multi-agent forecasting has attracted
huge attention and brought wide applications. In the early era, state models
were studied [22, 10, 23, 24]. Recently, data-driven models extract deep
representations. DeepState [25], Res-sup. [26], AGED [27] and LSTNet [28]
design recurrent networks for stability. TCN [29] proposes feed-forward
convolution for temporal dependencies. However, the methods do not consider
the informative interactions among agents to benefit forecasting.

Recently, some works construct graphs to explicitly exploit the agent
relations. NRI [5] infers graphs via an autoencoder for prediction. S-RNN [30]
builds factor graphs to propagate agents’ information. DCRNN [18] and ST-GCN
[14] incorporate distance-based graphs for traffic forecasting. GraphWaveNet
[31] and Traj-GCN [15] use adaptive topologies to capture potential
dependencies. DMGNN [17] builds multiscale graphs. However, most graphs are
built with limited human priors or designed through trials and errors, lacking
theoretical guarantees and interpretation. In this work, we develop a novel
framework for multi-agent forecasting, which infers a collaborative graph with
theoretical justification.

Graph deep learning. The expressiveness of graphs contributes to various
scenarios such as social networks [32], bioinformatics networks [33] and human
behaviors [19, 20]. Graph neural networks (GNNs) [34, 35, 36, 37], which
expand deep leading to the non-Euclidean domain, have attracted explosive
interests. GNNs can be mainly categorized into the spectral-domain-based [38,
39, 34] and vertex-domain-based [40, 41, 35, 42, 43, 44, 45], which
respectively learn the patterns from the graph Fourier representations or the
raw topologies. In this work, we learn interpretable graphs for multi-agent
forecasting, which captures dynamic and complex correlations for precise
prediction.

Online learning for prediction. A dynamic system evolves with potentially
infinite states over time, performing on online streaming data [46]. Tailored
for the data flows, online prediction models have been designed to update in
real-time [47]. Traditional kernel-based works develop budgeted kernel
learning [48, 49, 50, 51], RF approximations [52, 53, 54] and multi-kernel
learning methods [55]. Recently, some deep-learning-based methods are proposed
for online prediction [56, 57, 58], whose network structures are specifically
designed. In this work, we train graph-based multi-agent forecasting models in
an online setting, and optimize graphs to depict the agents’ interactions in
real-time.

##  III Problem Formulation

With the time-series measurements collected by multiple agents in _real-time_
, the task of online multi-agent forecasting essentially aims to predict the
future statuses of each agent at each online time stamp based on the
corresponding historical states. Mathematically, for the $p$th agent
$v^{(p)}$, at time stamp $t$, let
$\mathbf{x}_{t-\delta:t}^{(p)}\in\mathbb{R}^{\delta\times d}$ be the
measurements of the observed clip with length $\delta$ and feature dimension
$d$, recording $v^{(p)}$’s states within time interval $(t-\delta,t]$; and
$\mathbf{x}_{t:t+\delta}^{(p)}\in\mathbb{R}^{\delta\times d}$ be the
measurements of the corresponding ground-truth clip in the future, reflecting
$v^{(q)}$’s states within $(t,t+\delta]$. We note that the length of
historical and future sequences could be different, while we here use $\delta$
to simplify notation. Our  _online_ predictor $f_{t}(\cdot)$ aims to produce

$\\{\widehat{\mathbf{x}}_{t:t+\delta}^{(p)}\\}_{p=1}^{N}=f_{t}(\\{\mathbf{x}_{t-\delta:t}^{(p)}\\}_{p=1}^{N})$

to approximate the ground-truth
$\\{\mathbf{x}_{t:t+\delta}^{(p)}\\}_{p=1}^{N}$, where the input data
$\mathbf{x}_{t-\delta:t}^{(p)}$ is captured from streaming data flow at time
stamp $t$. Note that the model $f_{t}(\cdot)$ indexed by $t$ is updated
online. The online setting is crucial because the online predictor can
effectively adapt to the unseen, or highly dynamic data distribution.

In contrast, previous works [5, 20, 15, 16, 59] mostly consider an _offline_
setting, which includes training and testing phases. In the training phase, a
predictor is optimized based on a training dataset; in the testing phase, the
predictor is fixed and deployed on a testing dataset. Mathematically, let
$f(\cdot)$ be an offline predictor, $\mathcal{X}_{\rm train}$ be the training
dataset and $\mathcal{X}_{\rm test}$ be the testing dataset, the training and
testing phases work as

$\displaystyle\mathrm{Training}:\\{\widehat{\mathbf{x}}_{0:\delta}^{(p)}\\}_{p=1}^{N}$
$\displaystyle=f(\\{\mathbf{x}_{-\delta:0}^{(p)}\\}_{p=1}^{N}),~{}~{}{\bf
x}\sim\mathcal{X}_{\rm train},$
$\displaystyle\mathrm{Testing}:\\{\widehat{\mathbf{y}}_{0:\delta}^{(p)}\\}_{p=1}^{N}$
$\displaystyle=f(\\{\mathbf{y}_{-\delta:0}^{(p)}\\}_{p=1}^{N}),~{}~{}{\bf\mathbf{y}}\sim\mathcal{X}_{\rm
test}.$

These offline methods assume the training and testing time-series should share
the same data distribution and the resulting offline predictors would be
hardly adapt to domain shift in the testing phase.

To design an online predictor, our core strategy is to exploit the dynamic
mutual relations of agents and capture their evolution status through online
learning. To model pairwise collaborations, we introduce a key concept,
_collaborative pair_ , which groups two agents and can be used to reflect the
effect from one agent to another in a multi-agent system. For example, the
$p$th agent $v^{(p)}$ and the $q$th agent $v^{(q)}$ form a collaborative pair
$(v^{(p)},v^{(q)})$ to indicate the directed effect from $v^{(q)}$ to
$v^{(p)}$. In this way, to predict the future status of the $p$th agent, we
consider
$\widehat{\mathbf{x}}_{t:t+\delta}^{(p)}=f_{t}(\\{\mathbf{x}_{t-\delta:t}^{(p)},\mathbf{x}_{t-\delta:t}^{(q)}\\}_{q=1}^{N})$,
which exploits information from all the collaborative pairs of $v^{(p)}$ to
benefit its prediction. This model considers each collaborative pair as a
basic element, which is different from previous works that regard each
individual agent as a basic element [15, 16].

##  IV Collaborative Prediction Unit

To develop an online multi-agent forecasting model, we propose a collaborative
prediction unit (CoPU) as a basic module, which predicts the status of each
agent from associated collaborative pairs via an collaborative graph.

###  IV-A Collaborative graph

To exploit the interactions among agents, we propose a collaborative graph
that explicitly models the pairwise correlations. Let
$G(\mathcal{V},\mathcal{F},\mathbf{W}_{t})$ be a collaborative graph, where
$\mathcal{V}=\\{v^{(1)},v^{(2)},\cdots,v^{(N)}\\}$ is the agent set with
$v^{(p)}$ modeling the $p$th agent,
$\mathcal{F}=\\{(v^{(p)},v^{(q)})\\}_{p,q=1}^{N}$ is the set of collaborative
pairs with $(v^{(p)},v^{(q)})$ modeling the $(p,q)$th collaborative pair; and
${\bf W}_{t}\in\mathbb{R}^{N\times N}$ is the collaborative graph adjacency
matrix trained online, whose $(p,q)$th element $w_{t}^{(p,q)}$ reflects the
influence level from $v^{(q)}$ to $v^{(p)}$ at time stamp $t$. Note that this
collaborative graph is asymmetric and dynamic; that is, each collaborative
edge represents a directional relationship between agents and each
collaborative edge weight is time-varying during the online training process.

Figure 2: Collaborative prediction unit (CoPU) sketched with three agents for
example. The left plot shows the CoPU based on a collaborative graph to learn
and aggregate information from collaborative pairs, where the circled nodes
represents the agents and the squared nodes represents the collaborative
pairs. The right plot shows the detailed computation, where the collaborative
pairs enables the late aggregation of agents’ predictions.

Different from many ordinary graphs, the collaborative graph has two types of
nodes: agent and collaborative pair. Agents are actual nodes; while
collaborative pairs are virtual nodes as they are naturally obtained when
agents are given, which have directions. To predict the future status of each
agent, we rely on each of its associated collaborative pairs to produce a
possible future status for such an agent; and then, the final status is
obtained by averaging all the possible statuses based on the weighted
collaborative edges.

Different from many previous graph-convolution-based and message-passing-based
models [15, 16, 59, 5, 6, 7], where each node shares the same node feature to
all of its neighbors without considering the demand of each one of its
neighbors, our model, based on a collaborative pair, $v^{(q)}$ helps the
prediction of $v^{(p)}$ by sharing customized information according to
$v^{(p)}$’s status.

###  IV-B Model design

Based on the collaborative graph, to construct the CoPU, we consider two key
operations in series: collaborative-pair-based prediction, which predicts an
agent’s future status based on an associated collaborative pair, which
generate all possible predictions with the collaborative effects between
agents, and weighted collaborative-graph-based aggregation, which obtains the
final prediction. Fig. 2 illustrates the CoPU, where the left plot shows the
modeling thoughts, and the right plot shows the detailed computation.

Collaborative-pair-based prediction. To explicitly form a collaborative pair,
which represent the collaborative states of two agent in history, we propose a
combination function $f_{\rm cb}(\cdot,\cdot)$ to directedly combine the
measurements of two agents. Given the observed measurements
$\mathbf{x}_{t-\delta:t}^{(p)}$,
$\mathbf{x}_{t-\delta:t}^{(q)}\in\mathbb{R}^{\delta\times d}$, the state of
the collaborative pair $(v^{(p)},v^{(q)})$ is

$\mathbf{x}_{t-\delta:t}^{(p,q)}=f_{\rm
cb}(\mathbf{x}_{t-\delta:t}^{(p)},\mathbf{x}_{t-\delta:t}^{(q)})\in\mathbb{R}^{\delta\times
d^{\prime}}.$ (1)

The collaborative state $\mathbf{x}_{t-\delta:t}^{(p,q)}$ carries the directed
effects from the agent $v^{(q)}$ to the agent $v^{(p)}$.

Based on the collaborative pairs and their states, we propose a _collaborative
predictor_ , which embeds any collaborative pair to predict an agent’s future
status under the corresponding effects. Let the collaborative predictor be
$h_{\bm{\theta}_{t}}(\cdot)$, where $\bm{\theta}_{t}$ is the parameter trained
online at time $t$. To predict $v^{(p)}$ under the collaborative effects from
$v^{(q)}$, the predicted status is

$\widehat{\mathbf{x}}^{(p,q)}_{t:t+\delta}=h_{\bm{\theta}_{t}}({\mathbf{x}}_{t-\delta:t}^{(p,q)})\in\mathbb{R}^{\delta\times
d}.$ (2)

The physical meaning of $\widehat{\mathbf{x}}^{(p,q)}_{t:t+\delta}$ is one
possible future trajectory for $v^{(p)}$ by considering the influence from
$v^{(q)}$.

For a collaborative pair, the aggregation function $f_{\rm cb}(\cdot,\cdot)$
and the predictor $h_{\bm{\theta}_{t}}(\cdot)$ could be designed in numerous
forms. Here we specifically consider three typical forms:

  * •

_Autoregressive model._ Let $f_{\rm cb}(\cdot,\cdot)$ concatenate the multi-
order differences of two agents, i.e.,
${\mathbf{x}}_{t-\delta:t}^{(p,q)}=[\mathbf{x}_{t-\delta:t}^{(p)},~{}\mathbf{x}_{t-\delta:t}^{(q)}-\mathbf{x}_{t-\delta:t}^{(p)},~{}\dots,~{}(\mathbf{x}_{t-\delta:t}^{(q)}-\mathbf{x}_{t-\delta:t}^{(p)})^{D}]\in\mathbb{R}^{\delta\times(D+1)d}$,
which provides a polynomial distance feature. We then vectorize
${\mathbf{x}}_{t-\delta:t}^{(p,q)}$ by reshaping it along time, and directly
set a linear $h_{\bm{\theta}_{t}}(\cdot)$ to learn
$\widehat{\mathbf{x}}^{(p,q)}_{t:t+\delta}$.

  * •

_LSTM model._ Let $f_{\rm cb}(\cdot,\cdot)$ concatenate the measurements of
the two agents, i.e.,
${\mathbf{x}}_{t-\delta:t}^{(p,q)}=[\mathbf{x}_{t-\delta:t}^{(p)},\mathbf{x}_{t-\delta:t}^{(q)}]\in\mathbb{R}^{\delta\times
2d}$. The concatenation of the raw measurements reduces the dimensions and
encourages more flexible information. We employ a one-layer LSTM as
$h_{\bm{\theta}_{t}}(\cdot)$ on ${\mathbf{x}}_{t-\delta:t}^{(p,q)}$ to extract
the temporal dynamics.

  * •

_Temporal convolution model._ We also consider $f_{\rm cb}(\cdot,\cdot)$ as
the raw measurement concatenation. We then perform two layers of 1-D
convolution along time to capture sequential dynamics. A ${\tanh(\cdot)}$
function is used as the activation after the first convolution to enhance
flexibility.

Each agent has $N$ collaborative pairs, so we predict $N$ possible future
statuses for one agent. To obtain the final prediction, we need to adaptively
aggregate the $N$ possibilities.

Collaborative-graph-based aggregation. Given the possible future status for an
agent, which is predicted based on each associated collaborative pair, we
aggregate these statuses with different weights on the edge of the
collaborative graph, where the edge weights are updated online and reflect the
influence level from one agent to another. Mathematically, let the graph
adjacency matrix at iteration step $t$ be ${\bf W}_{t}$, whose the $(p,q)$th
element is ${w}_{t}^{(p,q)}$. The final prediction of $v^{(p)}$ under the
effects from all the associated collaborative pairs is formulated as

$\widehat{\mathbf{x}}_{t:t+\delta}^{(p)}=\frac{\sum_{q=1}^{N}{w}_{t}^{(p,q)}\widehat{\mathbf{x}}_{t:t+\delta}^{(p,q)}}{\sum_{q^{\prime}=1}^{N}{w}_{t}^{(p,q^{\prime})}},$
(3)

where the edge weights are normalized for any agent $v^{(p)}$. In all, the
feed-forward operations of the CoPU is formulated as

$\widehat{\mathbf{x}}_{t:t+\delta}^{(p)}=\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}h_{\bm{\theta}_{t}}(f_{\rm
cb}(\mathbf{x}_{t-\delta:t}^{(p)},\mathbf{x}_{t-\delta:t}^{(q)})),$ (4)

where
$\bar{w}_{t}^{(p,q)}={w}_{t}^{(p,q)}/{\sum_{q^{\prime}=1}^{N}{w}_{t}^{(p,q^{\prime})}}$
denotes the normalized collaborative edge weights.

Compared to most common graph-based models, which bridge the same types of
nodes with a scalar edge weight, the proposed CoPU considers the heterogeneous
nodes of agents and collaborative pairs, carrying individual dynamics the
interactive effects in a hybrid manner. The CoPU essentially aggregates the
comprehensive information from the collaborative pairs with different
importance. Moreover, the proposed CoPU has a similar framework with the
online expert mixture algorithm [60, 55], as the different collaborative
predictors associated with an agent could be regarded as experts; however,
each collaborative predictor explicitly exploits the customized information
for the associated agent based on the corresponding collaborative pair,
instead of producing an untargeted prediction independently like most expert
mixture methods.

###  IV-C Optimization

CoPU enables a complete multi-agent forecasting pipeline, and the internal
parameters could be trained given the prediction targets. To train the CoPU
online, at each iteration $t$, we update the model parameters to minimize the
loss function:

$\mathcal{L}=\sum_{t=1}^{T}\sum_{p=1}^{N}\mathcal{L}_{t}\left(\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}h_{\bm{\theta}_{t}}\left(\mathbf{x}_{t-\delta:t}^{(p,q)}\right),\mathbf{x}_{t:t+\delta}^{(p)}\right),$
(5)

where $\mathcal{L}_{t}(\cdot)$ denotes a fixed and convex loss function
evaluated at time $t$; $T$ is the length of iterations before the current
stamp. Since $\mathcal{L}_{t}(\cdot)$ is a convex function (e.g., $\ell_{2}$
loss) w.r.t. $h_{\bm{\theta}_{t}}(\mathbf{x}_{t}^{(p,q)})$, we consider an
upper bound of it,

$\mathcal{L}:=\sum_{t=1}^{T}\sum_{p,q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}\left(\mathbf{x}_{t-\delta:t}^{(p,q)}\right),\mathbf{x}_{t:t+\delta}^{(p)}\right).$
(6)

To minimize Eq. (6), instead of using common online gradient descent to update
$\bar{w}_{t}^{(p,q)}$ and ${\bm{\theta}_{t}}$ together, we consider different
updating mechanisms, which bring theoretical guarantees. We separately
optimize the parameters $\bm{\theta}_{t}$ and $\bar{w}_{t}^{(p,q)}$ with
different steps. First, at time $t$, the parameter $\bm{\theta}_{t}$ could be
trained by online gradient decent; that is,

$\bm{\theta}_{t+1}~{}\leftarrow~{}\bm{\theta}_{t}-\eta\>\nabla{\cal
L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right),$

where we use ${\cal L}_{t}(h_{\bm{\theta}_{t}}^{(p,q)})$ to simplify ${\cal
L}_{t}(h_{\bm{\theta}_{t}}(\mathbf{x}_{t-\delta:t}^{(p,q)}),\mathbf{x}_{t:t+\delta}^{(p)})$;
$\eta\in(0,1)$ is the learning rate. Next, for $\bar{w}_{t}^{(p,q)}$, we
employ a multiplicative exponentiated update, which is inspired from the
randomized weighted majority of online learning [60]. We assume the optimal
edge weight to be $w^{(p,q)}\in[0,1]$, and $\bar{w}_{t}^{(p,q)}$ is updated
through

$\displaystyle w_{t+1}^{(p,q)}\leftarrow$
$\displaystyle\arg\min_{w^{(p,q)}}\,\eta\>{\cal
L}_{t}\left(h_{\bm{\theta}_{t}}\right)\left({w}^{(p,q)}-\bar{w}^{(p,q)}_{t}\right)$
(7) $\displaystyle+{\cal D}_{\rm KL}({w}^{(p,q)}\|\bar{w}^{(p,q)}_{t})$
$\displaystyle=$ $\displaystyle~{}\bar{w}_{t}^{(p,q)}\exp\left(-\eta\>{\cal
L}_{t}\left(h_{\bm{\theta}_{t}}\right)\right),$

where ${\cal D}_{\rm KL}({w}^{(p,q)}\|\bar{w}_{t}^{(p,q)})$ denotes the KL-
divergence between the optimized ${w}^{(p,q)}$ and the collaborative weight at
the last iteration step. The optimization ensures the continuity and
smoothness of the collaborative graph structure by minimizing the element-wise
distances and distribution divergence. We also normalize the new collaborative
edge weights by
$\bar{w}_{t+1}^{(p,q)}=w_{t+1}^{(p,q)}/\sum_{p=1}^{N}w_{t+1}^{(p,q)}$.

In previous graph neural networks [17, 21, 15, 16], graphs are obtained
through either heuristic designs or end-to-end learning in a black-box, which
lack a clear objective and theoretical justification. Compared to these works,
the proposed method obtains the edge weights of the collaborative graph
through optimizing an explicit objective function (7), which could be derived
as an update function based on the input features and historical states in an
online setting.

###  IV-D Comparison with previous works

The proposed CoPU is seemingly similar to spatio-temporal-graph convolution
operations [17, 21, 14] and neural-message-passing operations [5, 6, 7]. Here
we clarify the differences;

Compared to spatio-temporal-graph convolution operations [17, 21, 14], the
proposed CoPU is novel from three aspects:

  * •

Interpretable graph learning. Previous models usually build non-grid
structures by using predefined graphs or adjusting the graph in an intractable
black-box. Each CoPU contains an interpretable graph that optimizes an
explicit objective function for model training and has a theoretical
implication.

  * •

Customized feature propagation. Previous works consider each node shares the
same feature to all of its neighbors without considering the demands of
neighbors. In CoPU, a node shares a customized feature to each neighbor via
the collaborative pairs.

  * •

Interpretable output. Previous methods build deep graph networks to learn the
agent’s dynamics in the hidden high-dimensional spaces, lacking the
interpretation of feature representation; while each CoPU predicts agents’
future statuses, which lie in the original measurement space.

Previous neural-message-passing-based works [5, 6, 7] include two steps to
propagate information: v2e, which transforms the information of a pair of
nodes to an edge, and e2v, which aggregates the information of edges to a
node. Compared to those methods, the proposed CoPU is novel from three
aspects:

  * •

Interpretable graph learning. Previous works either build predefined graphs
based on physical constraints or learn graph structures with errors and trials
in black-boxes; while the edge weights of our collaborative graph are obtained
by optimizing an explicit objective function and has theoretical
justification.

  * •

Meaning of v2e. In the v2e step, previous works update the embedding of each
edge, which is an intermediate-feature-level fusion; while our CoPU directly
generates the prediction of a node conditioned on another node, which is an
output-level fusion.

  * •

Meaning of e2v. In the e2v step, previous works consider the mean aggregation,
which assigns the same weight for each one of the neighbors; while our CoPU
adopts the weighted aggregation, where the weights are the edge weights in the
collaborative graph, obtained through multiplicative updating.

###  IV-E Theoretical interpretation

To understand CoPU, we conduct regret analysis. We consider the best single
collaborative predictor as our regret baseline and analyze the gap from our
CoPU to this predictor. The best predictor has a straightforward meaning and
its corresponding collaborative pair makes the most significant effect in
predicting an agent’s future status. We aim to demonstrate that the regret
tends to be finite in the online setting, which indicates the properties of
our CoPU and the training algorithm from two aspects:

  * •

The CoPU trained online could converge to a statistically stable state during
the training process.

  * •

The performance of CoPU achieves a limited gap from the best single
collaborative pair in statistics, reflecting that CoPU captures significant
and reasonable collaborations for effective and interpretable prediction.

Here we define the collaborative pair in the best collaborative predictor. For
the $p$th agent, it has $N$ collaborative pairs,
$\\{(v^{(p)},v^{(q)})\\}_{q=1}^{N}$. The collaborative pair
$(v^{(p)},v^{(q^{*})})$ for $v^{(p)}$ in the best collaborative predictor is
obtained via

$q^{*}=\arg\min_{q}\sum_{t=1}^{T}\mathcal{L}_{t}\left(h_{\bm{\theta}^{*}}\left({\bf\mathbf{x}}_{t-\delta:t}^{(p,q)}\right)\right),$
(8)

where $\bm{\theta}^{*}$ denotes the parameters of the best collaborative
predictor at iteration step $T$. However, it is hard to implement an online
model by directly determining $(v^{(p)},v^{(q^{*})})$ and meanwhile optimizing
$\bm{\theta}^{*}$; that is why CoPU builds a collaborative graph with
trainable weights to achieve a soft collaborative mechanism; see Eq.(4).

To measure the performance gap between the best collaborative predictor and
CoPU, we define the static regret based on the difference between the
accumulated loss of CoPU and that of the best collaborative predictor in
hindsight,

${\rm
R}_{T}=\frac{1}{T}\sum_{t=1}^{T}\left(\mathcal{L}_{t}\left(\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}h_{\bm{\theta}_{t}}^{(p,q)}\right)-\mathcal{L}_{t}\left(h_{\bm{\theta}^{*}}^{(p,q^{*})}\right)\right),$

where we simplify $h_{\bm{\theta}_{t}}(\mathbf{x}_{t}^{(p,q)})$ as
$h_{\bm{\theta}_{t}}^{(p,q)}$. The regret reflects the gap between the CoPU
and the best collaborative predictor, as well as the convergence rate of
online learning.

To study the CoPU and the performance gap, we aim to calculate an upper bound
of the regret. To facilitate the closed-form theoretical analysis, we consider
a basic predictor (a linear autoregressive model) and propose the following
assumptions that are commonly satisfied by our model:

(AS1) _For the collaborative predictor at time $t$, $h_{\bm{\theta}_{t}}$,
given any collaborative pair, the loss ${\cal L}_{t}(h_{\bm{\theta}_{t}})$ is
convex w.r.t. ${\bm{\theta}_{t}}$._

(AS2) _For any bounded $\bm{\theta}_{t}$, where $\|\bm{\theta}_{t}\|_{2}\leq
C_{\bm{\theta}}$, and any collaborative pair, we have a bounded loss, ${\cal
L}_{t}(h_{\bm{\theta}_{t}})\in[0,1]$, and a bounded gradient, $\|\nabla{\cal
L}_{t}(h_{\bm{\theta}_{t}})\|_{2}\leq L$._

Then, we propose two lemmas, which help to derive the upper bounds of regret.
We first analyze the loss gap between arbitrary collaborative predictors and
the best collaborative predictor.

######  Lemma 1.

Let $h_{\bm{\theta}^{*}}^{(p,q^{*})}$ and $h_{\bm{\theta}_{t}}^{(p,k)}$ be the
predictions of the best collaborative predictor and arbitrary one. Under AS1
and AS2, for any collaborative pair $(v^{(p)},v^{(k)})$, we have

$\frac{1}{T}\sum_{t=1}^{T}\left(\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,k)}\right)-\mathcal{L}_{t}\left(h_{\bm{\theta}^{*}}^{(p,q^{*})}\right)\right)\leq\frac{C_{\bm{\theta}}^{2}}{2\eta
T}+\frac{\eta L^{2}}{2},$

where $\eta$ is the learning rate; $C_{\bm{\theta}}$ is the upper bound of the
norm of parameters; and $L$ is the Lipschitz constant.

###### Proof.

Given any determined parameter $\bm{\theta}$ of the collaborative predictor,
for any collaborative pair $(v^{(p)},v^{(k)})$, we have

$\|\bm{\theta}_{t+1}-\bm{\theta}\|_{2}^{2}=\|\bm{\theta}_{t}-\eta\nabla^{\top}\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})-\bm{\theta}\|_{2}^{2}$
(9)

Considering the convexity property of the loss function (see AS1), the
gradient of the loss function satisfies

$\nabla^{\top}\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})\geq\frac{\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})-\mathcal{L}_{t}(h_{\bm{\theta}}^{(p,k)})}{\bm{\theta}_{t}-\bm{\theta}}$
(10)

Plugging Inequality (10) into Eq. (9), we could easily obtain the following
inequality by rearranging some terms,

$\displaystyle\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})-\mathcal{L}_{t}(h_{\bm{\theta}}^{(p,k)})$
(11) $\displaystyle\leq{}$
$\displaystyle\frac{\eta}{2}\|\nabla\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})\|_{2}^{2}+\frac{\|\bm{\theta}_{t}-\bm{\theta}\|_{2}^{2}-\|\bm{\theta}_{t+1}-\bm{\theta}\|_{2}^{2}}{2\eta}$

We further consider the assumption AS2 that points out the upper bound of
$\|\bm{\theta}_{t}\|_{2}$ and the Lipschitz constant of the gradient
$\nabla\mathcal{L}_{t}(h_{\bm{\theta}_{t}})$; that is, for any collaborative
predictor, we have $\|\bm{\theta}_{t}\|_{2}\leq C_{\bm{\theta}}$ and
$\|\nabla\mathcal{L}_{t}(h_{\bm{\theta}_{t}})\|_{2}\leq L$. We then summing
Inequality (11) over $t=1,\dots,T$, we could easily derive

$\sum_{t=1}^{T}\left(\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})-\mathcal{L}_{t}(h_{\bm{\theta}}^{(p,k)})\right)\leq\frac{C_{\bm{\theta}}^{2}}{2\eta}+\frac{\eta
L^{2}T}{2}$ (12)

where we assume the initial parameter $\bm{\theta}_{1}$ to have a very small
value, leading to
$\|\bm{\theta}_{1}-\bm{\theta}\|_{2}^{2}\approx\|\bm{\theta}\|_{2}^{2}=C_{\bm{\theta}}^{2}$,
and consider $\|\bm{\theta}_{T+1}-\bm{\theta}\|_{2}^{2}$ to be non-negative.

Additionally, we divide by the number of iterations $T$ at the both side of
Inequality (12) to complete the proof.

$\frac{1}{T}\sum_{t=1}^{T}\left(\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,k)})-\mathcal{L}_{t}(h_{\bm{\theta}^{*}}^{(p,q^{*})})\right)\leq\frac{C_{\bm{\theta}}^{2}}{2\eta
T}+\frac{\eta L^{2}}{2}$ (13)

∎

Note that Lemma 1 are hold for any collaborative pair, so that, in hindsight,
we could choose $\bm{\theta}$ to be the parameter at iteration step $T$ and
use the predicted status $h_{\bm{\theta}^{*}}^{(p,q^{*})}$, which is output
from the collaborative predictor corresponding to the single optimal
collaborative pair $(v^{(p)},v^{(q^{*})})$.

We next analyze the gap between CoPU and an arbitrary collaborative predictor.

######  Lemma 2.

Under AS1 and AS2, for any collaborative pair $(v^{(p)},v^{(k)})$ fed into the
predictor $h_{\bm{\theta}_{t}}(\cdot)$, it holds that

$\frac{1}{T}\sum_{t=1}^{T}\left(\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right)-\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,k)}\right)\right)\leq\eta+\frac{\log
N}{\eta T},$

where $N$ is the number of agents.

###### Proof.

We first let ${W}_{t}^{(p)}$ be the sum of edge weights associated with agent
$v^{(p)}$; that is ${W}_{t}^{(p)}=\sum_{q=1}^{N}w_{t}^{(p,q)}$. Therefore,
${W}_{t}^{(p)}$ adjusted through exponentiated update at the time $t+1$ could
be formulated as

$\displaystyle
W_{t+1}^{(p)}=\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\exp\left(-\eta\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right)\right)$
(14)

We represent $\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right)$ as
$\mathcal{L}_{t}$ for simplification. We consider the inequality $\exp(-\eta
x)\leq 1-\eta x+\eta^{2}x^{2}$ for $\forall|\eta|\leq 1$, and Inequality (14)
satisfies

$W_{t+1}^{(p)}\leq
1-\eta\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}+\eta^{2}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}^{2}$

Furthermore, according to $1+x\leq e^{x}$, we have

$W_{t+1}^{(p)}\leq\exp\Big{(}-\eta\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}+\eta^{2}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}^{2}\Big{)}$
(15)

And we then consider the updated situation of Inequality (15) along time when
$t=1,\dots,T$, we then obtain

$W_{T+1}^{(p)}\leq\exp\Big{(}\sum_{t=1}^{T}\eta^{2}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}^{2}-\eta\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}\Big{)}$
(16)

On the other hand, for the agent $v^{(p)}$ and any agent $v^{(k)}$, we have
the inequality,

$\displaystyle W_{T+1}^{(p)}\geq{}$ $\displaystyle w_{T+1}^{(p,k)}$ (17)
$\displaystyle={}$ $\displaystyle
w_{1}^{(p,k)}\exp\left(-\eta\sum_{t=1}^{T}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,k)}\right)\right)$

Combining Inequality (16) and Inequality (17), we arrive at

$\displaystyle\exp\Big{(}-\eta\sum_{t=1}^{T}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}+\eta^{2}\sum_{t=1}^{T}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}^{2}\Big{)}$
(18) $\displaystyle\geq{}$ $\displaystyle
w_{1}^{(p,k)}\exp\left(-\eta\sum_{t=1}^{T}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,k)}\right)\right)$

Taking the logarithm on both side of Inequality (18) and meanwhile considering
$w_{1}^{(p,k)}=1/N$, we could obtain

$\sum_{t=1}^{T}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right)-\sum_{t=1}^{T}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,k)}\right)\leq\eta
T+\frac{\log N}{\eta},$ (19)

where we note that $\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,q)})\in[0,1]$
according to AS2, and Lemma 2 has been proved, where we could divide $T$ on
both side of the Inequality (19). ∎

Lemma 2 shows that the performance gap between an ensemble of collaborative
predictors with the trainable weight $\bar{w}_{t}^{(p,q)}$ and an arbitrary
collaborative predictor fed with any collaborative pairs $(v^{(p)},v^{(k)})$,
including the best predictor, can be upper bounded. Combining Lemma 1 and
Lemma 2, we derive the upper bound of the regret as in Theorem 1.

######  Theorem 1.

Under AS1 and AS2, and with the analysis of Lemma 1 and Lemma 2, the regret of
the proposed CoPU satisfies the following bound.

$\displaystyle\frac{1}{T}\left(\sum_{t=1}^{T}\mathcal{L}_{t}\left(\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}h_{\bm{\theta}_{t}}^{(p,q)}\right)-\sum_{t=1}^{T}\mathcal{L}_{t}\left(h_{\bm{\theta}^{*}}^{(p,q^{*})}\right)\right)$
$\displaystyle\leq\frac{\log N}{\eta T}+\frac{C_{\bm{\theta}}^{2}}{2\eta
T}+\frac{\eta L^{2}}{2}+\eta.$

Setting $\eta=\mathcal{O}(1/\sqrt{T})$, the static regret converges with ${\rm
R}_{T}=\mathcal{O}(1/\sqrt{T})$.

Figure 3: The architecture of the collaborative graph neural network (CoGNN).
The model takes multiple time-series as inputs and aims to generate the future
series. Each graph-temporal prediction unit (CoPU) in the model performs three
operations for progressive prediction. The entire model is trained online for
real-time prediction. Here we show three CoPUs to sketch the model
architecture.

###### Proof.

Combining Inequality (13) and Inequality (18), we could obtain that

$\displaystyle\frac{1}{T}\left(\sum_{t=1}^{T}\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right)-\sum_{t=1}^{T}\mathcal{L}_{t}\left(h_{\bm{\theta}^{*}}^{(p,q^{*})}\right)\right)$
(20) $\displaystyle\leq{}$ $\displaystyle\frac{\log N}{\eta
T}+\frac{C_{\bm{\theta}}^{2}}{2\eta T}+\frac{\eta L^{2}}{2}+\eta.$

Note that the loss function $\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,q)})$ is
convex w.r.t. any $h_{\bm{\theta}_{t}}^{(p,q)}$, thus the loss function
$\mathcal{L}_{t}(\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}h_{\bm{\theta}_{t}}^{(p,q)})$
has an upper bound
$\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}(h_{\bm{\theta}_{t}}^{(p,q)})$;
that is,

$\mathcal{L}_{t}\left(\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}h_{\bm{\theta}_{t}}^{(p,q)}\right)\leq\sum_{q=1}^{N}\bar{w}_{t}^{(p,q)}\mathcal{L}_{t}\left(h_{\bm{\theta}_{t}}^{(p,q)}\right)$
(21)

understanding that Inequality (19) satisfies all the collaborative pairs in
the online system, we could easily prove the Theorem 1 to represent the upper
bound of the static regret. Plugging (21) in (LABEL:eq:Apx13), we could easily
prove the Theorem 1. ∎

Therefore, the regret of CoPU is upper bounded during online training.
Additionally, when $T$ becomes large, the regret converges to a very small
value, reflecting the finite gap between the CoPU and the best single
collaborative predictor. We also validate Theorem 1 by computing the regret
along time in our experiment; see Fig. 6 (a).

##  V Collaborative Graph Neural Networks

Though one collaborative prediction unit (CoPU) can tackle the prediction
task, in the experiments, we find that stacking multiple CoPUs can lead to
even better performance. Therefore, we develop a deep architecture called
_collaborative graph neural networks_ (CoGNN), which employs multiple CoPUs as
computational building blocks.

###  V-A Network architecture

We sketch the architecture of the proposed CoGNN in Fig. 3, where we stack
three CoPUs in cascade for example. Each CoPU in CoGNN progressively fine-
tunes the predictions and approximates the target measurements step-by-step.
According to the examplar Fig. 3, the first and the second CoPUs produce the
immediate prediction results $\widetilde{\mathbf{x}}_{t:t+\delta}$ for all the
agents from the input clips or the first CoPU; the last CoPU generates the
final prediction $\widehat{\mathbf{x}}_{t:t+\delta}$ from the results of the
second CoPU. For each CoPU, we apply a residual connection between the input
and output, reflecting that the internal modules estimate the displacements
from the input data to the target.

To train the network, we apply the same form of loss function on each CoPU,
because each CoPU tackles a complete prediction task and could be trained
directly; see the dash lines in gray in Fig. 3. Moreover, the collaborative
edge weights in each CoPU could be learned to directly optimize an explicit
objective function, Eq. (7), through the exponentiated update.

|  |
---|---|---
(a) Simulated interaction system. | (b) Human motion. | (c) Traffic speed distribution.
Figure 4: Data examples of three scenarios of online multi-agent forecasting.
(a) Simulated interaction systems: we generate systems based on 20 moving
agents with various interactions, and we forecast the trajectory of each agent
given the historical movements in an online setting. (b) Human motion: the
human motion sequences consist of 3D skeleton-based human poses performing a
certain action, e.g., walking, and we generate the future poses given the
historical poses online. (c) Traffic speed distribution in cities: at
different locations, there are sensors recording the traffic speed in real-
time; that is, the traffic speed at each point is a continuous time-series
(see the curve), while the larger sizes of the red circles denote the higher
traffic speed at a certain time step. We aim to predict the traffic speed
given the historical speed online.

###  V-B Comparison with previous works

Compared to previous graph-based prediction networks [17, 21, 14, 5, 6, 7],
our model contributes from two aspects.

First, to our best knowledge, the proposed CoGNN is the first online-learning
method, which is learned and tested along with the streaming data and never
stops. This is crucial because an online method could capture the highly
dynamic patterns of the real-time series and reduce the training/testing gaps
between the training and testing samples in practice. Our CoGNN is scalable
and easy to train, which is suitable for online learning. However, previous
methods only sample the multi-agent systems as sets of independent clips but
neglect the real-time dynamics and complexity.

Second, the proposed CoGNN is more interpretable. The output of each layer in
the proposed CoGNN predicts agents’ future statuses in the original
measurement space, reflecting a clear physical meaning. With a cascaded
structure, our CoGNN can progressively fine-tune the time-series prediction,
achieving effective and stable performance. However, previous methods use
hierarchical graph propagation to extract hidden representations without
interpretable gradual error reduction.

##  VI Experiments

To evaluate the proposed CoGNNs, we conduct extensive experiments on three
important multi-agent forecasting tasks in an online setting; that is, online
simulated trajectory prediction, online human motion prediction and online
traffic speed prediction, where the corresponding scenarios and data examples
are sketched in Fig. 4.

In CoGNNs, we respectively use three forms of collaborative predictors
$h_{\bm{\theta}_{t}}(\cdot)$ in CoPUs; see section IV-B. We note the three
forms of models as ‘CoGNN-AR’, ‘CoGNN-LSTM’ and ‘CoGNN-TC’, respectively. In
CoGNN-AR, to compute the state, ${\bf x}_{t-\delta:t}^{(p,q)}$, of any
collaborative pair $(v^{(p)},v^{(q)})$, the order of measurement difference
$D=10$; for CoGNN-LSTM and CoGNN-AR, the hidden dimension of
$h_{\bm{\theta}_{t}}(\cdot)$ is $64$. The entire CoGNNs are built with $2$
CoPUs, which enable precise and stable forecasting. We train the models on one
GTX-1080Ti GPU with PyTorch 1.4 framework. The learning rates are from $0.01$
to $0.075$ for different forms of CoGNNs. We also apply gradient clipping to
constrain the absolute values of gradients within $10$ for robustness.

###  VI-A Online simulated trajectory prediction

We first evaluate the proposed CoGNNs in online simulated particle systems,
which contains several moving particles whose positions and velocities are
determined by their interactions.

Dataset. In our simulated particle systems, there are $20$ particle agents,
whose initial states (positions and velocities) are set randomly, moving in a
2D space with dynamic mutual interaction. The interactions among the agents
could be represent as a graph structure with symmetric adjacency matrix. Here
we use a simple interaction rule called ’spring’ to construct the complex
systems; that is, each agent has attractions with several other agents that
are randomly selected. One example of our simulated system is illustrated in
Fig. 4 (a). This system was similarly used by [5], where only $5$ agents are
set in a simple system. However, different from the previous work [5] which
generates a set of independent trajectory sequences with fixed and known
interactions, in our simulated systems, agents have randomly changed
interactions with different agents to reflect dynamic and changing states
online. In each interactive simulation, we change the interaction structures
for $20$ times randomly; that is, the interaction adjacency matrix is randomly
adjusted for $20$ times. Moreover, we conduct $10$ times of different
simulation to demonstrate the effectiveness and robustness. In our
experiments, along the streaming data, our CoGNNs are fed with each 10-frame
clip one-by-one along time and generate the next 10-frame clip for prediction.

Baselines and evaluation metrics. We compare our CoGNNs with several methods
on trajectory prediction, including NRI [5], DNRI [6] and EvolveGraph [7].
These methods are designed for the offline scenarios; thus we modify these
works to be run online with the same inputs and outputs as our CoGNNs. We also
adjust their hyper-parameters to be reasonable. Besides being compared to the
state-of-the-art works, CoGNNs are also compared to two degenerated variants:
1) CoGNNs forecasting without collaborative graphs, i.e., ‘CoGNNs (no ${\bf
W}_{t}$)’; and 2) CoGNNs training collaborative graphs via online gradient
descent end-to-end in black-boxes, i.e., ‘CoGNNs (e2e ${\bf W}_{t}$)’. To
evaluate various methods, we employ the average mean square errors (MSE)
between the predicted locations and ground-truth of all the agents over the
training/testing process.

TABLE I: Prediction results (mean values and standard deviations of MSEs) on
the tasks of online simulated prediction, where we consider 10 interaction
systems with 20 trajectories. We also compare the variants of CoGNNs: CoGNNs
(no ${\bf W}_{t}$), which predict each agent based on this agent itself
without using the collaborative graph; and CoGNNs (e2e ${\bf W}_{t}$), which
train the collaborative graph through standard gradient descent in an end-to-
end setting.

| Prediction steps
---|---
Methods | 1 | 2 | 5 | 8 | 10
Baselines | ZeroV |  13.43 $\pm$ 1.94 |  13.43 $\pm$ 1.95 |  13.48 $\pm$ 2.08 |  13.58 $\pm$ 2.35 |  13.69 $\pm$ 2.64
NRI [5] |  0.56 $\pm$ 0.16 |  0.58 $\pm$ 0.15 |  0.60 $\pm$ 0.18 |  1.03 $\pm$ 0.30 |  1.59 $\pm$ 0.46
DNRI [6] |  0.26 $\pm$ 0.10 |  0.28 $\pm$ 0.12 |  0.31 $\pm$ 0.12 |  0.51 $\pm$ 0.15 |  0.77 $\pm$ 0.19
EvolveGraph [7] |  0.22 $\pm$ 0.10 |  0.24 $\pm$ 0.09 |  0.28 $\pm$ 0.10 |  0.44 $\pm$ 0.13 |  0.68 $\pm$ 0.15
no ${\bf W}_{t}$ |  CoGNN-AR (no ${\bf W}_{t}$) |  0.23 $\pm$ 0.06 |  0.24 $\pm$ 0.07 |  0.41 $\pm$ 0.12 |  0.75 $\pm$ 0.23 |  1.09 $\pm$ 0.32
CoGNN-LSTM (no ${\bf W}_{t}$) |  0.16 $\pm$ 0.04 |  0.16 $\pm$ 0.05 |  0.20 $\pm$ 0.05 |  0.40 $\pm$ 0.14 |  0.61 $\pm$ 0.16
CoGNN-TC (no ${\bf W}_{t}$) |  0.19 $\pm$ 0.08 |  0.23 $\pm$ 0.11 |  0.24 $\pm$ 0.11 |  0.42 $\pm$ 0.16 |  0.63 $\pm$ 0.18
e2e ${\bf W}_{t}$ |  CoGNN-AR (e2e ${\bf W}_{t}$) |  0.20 $\pm$ 0.04 |  0.21 $\pm$ 0.06 |  0.37 $\pm$ 0.15 |  0.74 $\pm$ 0.22 |  1.06 $\pm$ 0.34
CoGNN-LSTM (e2e ${\bf W}_{t}$) |  0.17 $\pm$ 0.06 |  0.16 $\pm$ 0.06 |  0.19 $\pm$ 0.07 |  0.37 $\pm$ 0.13 |  0.60 $\pm$ 0.18
CoGNN-TC (e2e ${\bf W}_{t}$) |  0.15 $\pm$ 0.05 |  0.18 $\pm$ 0.07 |  0.20 $\pm$ 0.07 |  0.36 $\pm$ 0.12 |  0.58 $\pm$ 0.17
Ours | CoGNN-AR |  0.19 $\pm$ 0.06 |  0.20 $\pm$ 0.06 |  0.37 $\pm$ 0.12 |  0.70 $\pm$ 0.21 |  1.01 $\pm$ 0.30
CoGNN-LSTM |  0.15 $\pm$ 0.05 |  0.15 $\pm$ 0.06 |  0.19 $\pm$ 0.05 |  0.35 $\pm$ 0.10 |  0.59 $\pm$ 0.17
CoGNN-TC |  0.16 $\pm$ 0.06 |  0.18 $\pm$ 0.05 |  0.18 $\pm$ 0.06 |  0.35 $\pm$ 0.12 |  0.56 $\pm$ 0.16

Results. For various methods, Table I presents the average forecasting results
over the online learning processes, where we show the prediction performance
at $5$ prediction steps of the generated 10-frame clips, respectively. Since
all the models are run on $10$ different interaction systems, the mean values
and standard deviations of the prediction MSEs are presented for comparison.
We see that, 1) compared to the state-of-the-art methods, our CoGNNs achieve
more precise forecasting and outperform the baselines, especially CoGNN-LSTM
and CoGNN-TC achieve much more effective prediction at various prediction
steps; 2) compared to the model variants, ‘no ${\bf W}_{t}$’ and ‘e2e ${\bf
W}_{t}$’, the proposed CoGNNs consistently outperforms these two variants,
demonstrating the importance and effectiveness of our intepretable
collaborative graphs and the exponentiated update strategy.

###  VI-B Online human motion prediction

We next conduct experiments on online human motion prediction to verify the
effectiveness of our CoGNN.

Dataset. Various models are trained and tested on two motion capture datasets:
Human3.6M (H3.6M) [61] and CMU Mocap111http://mocap.cs.cmu.edu/. H3.6M
contains $15$ classes of activities. There are $32$ body joints with their 3D
coordinates. An examplar motion clip in H3.6M is shown in Fig. 4 (b). CMU
Mocap contains $8$ activities with $38$ joints. Following the previous
settings [26, 62, 15], we downsample all motion sequences along time by two
for both datasets. We take the motion clips of the past $400$ms ($10$ frames)
as input, and forecast the future $400$ms.

TABLE II: Average MPJPEs of CoGNNs and baselines for motion prediction on 4
representative actions of H3.6M. We also compare two types of variants of
CoGNNs, i.e., ‘no ${\bf W}_{t}$’ and ‘e2e ${\bf W}_{t}$’; see the definition
of these two variants in the caption of Table I.

Motions | Walking | Eating | Smoking | Discussion
---|---|---|---|---
Methods | 80 | 160 | 320 | 400 | 80 | 160 | 320 | 400 | 80 | 160 | 320 | 400 | 80 | 160 | 320 | 400
Baselines |  Res-sup [26] | 23.0 | 32.1 | 37.8 | 44.4 | 20.6 | 30.2 | 37.1 | 46.0 | 23.2 | 27.6 | 35.8 | 42.7 | 30.1 | 41.8 | 46.8 | 52.0
NRI [5] | 21.6 | 30.4 | 36.0 | 42.4 | 19.5 | 28.7 | 36.4 | 42.5 | 22.4 | 26.8 | 33.2 | 40.1 | 28.7 | 39.3 | 44.9 | 51.3
CSM [62] | 22.1 | 31.5 | 39.4 | 43.3 | 19.9 | 29.1 | 39.7 | 43.9 | 21.3 | 26.5 | 32.4 | 38.6 | 31.2 | 38.0 | 44.4 | 50.8
Traj-GCN [15] | 22.9 | 30.1 | 38.9 | 43.6 | 19.0 | 24.9 | 35.8 | 38.6 | 19.7 | 25.3 | 28.2 | 35.9 | 24.1 | 36.2 | 43.4 | 47.3
DNRI [6] | 22.3 | 31.8 | 39.2 | 43.7 | 19.3 | 27.6 | 36.7 | 40.5 | 20.5 | 26.3 | 29.1 | 36.7 | 26.8 | 37.2 | 43.8 | 49.0
DMGNN [17] | 22.6 | 31.1 | 39.2 | 44.1 | 18.2 | 24.2 | 32.5 | 39.2 | 19.9 | 24.3 | 27.8 | 35.7 | 24.6 | 35.8 | 43.4 | 49.5
HisRep [63] | 22.6 | 29.0 | 37.3 | 42.1 | 17.3 | 22.5 | 33.6 | 37.3 | 19.2 | 25.4 | 29.9 | 36.3 | 22.8 | 33.6 | 40.7 | 46.5
EvolveGraph [7] | 22.2 | 30.7 | 38.3 | 43.5 | 18.1 | 23.8 | 34.3 | 38.9 | 20.4 | 26.0 | 29.7 | 37.1 | 24.2 | 34.7 | 42.9 | 48.4
no ${\bf W}_{t}$ |  CoGNN-AR (no ${\bf W}_{t}$) | 26.2 | 39.9 | 62.3 | 65.4 | 20.3 | 27.8 | 43.0 | 48.7 | 23.2 | 30.5 | 46.0 | 52.7 | 24.8 | 35.6 | 55.7 | 63.7
CoGNN-LSTM (no ${\bf W}_{t}$) | 23.3 | 23.9 | 29.8 | 40.1 | 19.8 | 20.0 | 26.4 | 35.0 | 20.3 | 20.8 | 28.0 | 33.9 | 29.5 | 30.0 | 40.6 | 48.9
CoGNN-TC (no ${\bf W}_{t}$) | 25.8 | 27.3 | 37.2 | 47.2 | 19.6 | 20.1 | 30.1 | 38.3 | 17.8 | 19.1 | 29.7 | 36.3 | 22.2 | 25.3 | 36.7 | 48.3
e2e ${\bf W}_{t}$ |  CoGNN-AR (e2e ${\bf W}_{t}$) | 24.9 | 35.1 | 50.5 | 57.9 | 17.6 | 26.8 | 38.7 | 40.3 | 22.3 | 28.2 | 43.6 | 48.7 | 24.9 | 34.3 | 51.7 | 58.6
CoGNN-LSTM (e2e ${\bf W}_{t}$) | 23.5 | 23.7 | 33.7 | 40.9 | 18.4 | 17.0 | 27.6 | 34.1 | 19.4 | 20.0 | 27.3 | 24.2 | 28.4 | 29.6 | 40.1 | 47.5
CoGNN-TC (e2e ${\bf W}_{t}$) | 22.6 | 25.2 | 35.3 | 41.7 | 18.3 | 19.9 | 29.2 | 35.7 | 18.0 | 18.8 | 28.4 | 35.5 | 22.0 | 26.7 | 36.1 | 47.5
Ours | CoGNN-AR | 22.3 | 33.7 | 51.5 | 56.8 | 16.6 | 23.0 | 36.5 | 41.5 | 19.5 | 27.7 | 41.6 | 45.4 | 24.1 | 30.1 | 50.4 | 58.0
CoGNN-LSTM | 22.6 | 23.3 | 33.6 | 40.7 | 18.2 | 19.2 | 26.1 | 33.7 | 19.7 | 19.4 | 25.2 | 33.8 | 28.7 | 29.3 | 38.9 | 47.3
CoGNN-TC | 21.5 | 21.4 | 31.0 | 39.7 | 16.8 | 16.2 | 28.7 | 36.0 | 16.4 | 16.6 | 26.6 | 33.6 | 19.9 | 18.5 | 34.1 | 45.2

TABLE III: Average MPJPEs of CoGNNs and baselines for motion prediction on CMU
Mocap. We also compare two types of variants of CoGNNs, i.e., ‘no ${\bf
W}_{t}$’ and ‘e2e ${\bf W}_{t}$’; see the definition of these two variants in
the caption of Table I

Motions | Average
---|---
Methods | 80 | 160 | 320 | 400
Baselines |  Res-sup [26] | 22.9 | 30.7 | 40.4 | 47.4
CSM [62] | 21.0 | 28.5 | 35.6 | 41.7
NRI [5] | 19.7 | 26.7 | 33.6 | 41.1
Traj-GCN [15] | 17.3 | 24.5 | 31.8 | 37.9
DNRI [6] | 18.3 | 25.5 | 32.4 | 38.9
DMGNN [17] | 16.8 | 23.8 | 30.6 | 37.7
HisRep [63] | 15.6 | 21.9 | 28.7 | 35.2
EvolveGraph [7] | 17.0 | 24.1 | 33.4 | 38.6
no ${\bf W}_{t}$ |  CoGNN-AR (no ${\bf W}_{t}$) | 27.1 | 34.7 | 52.7 | 57.0
CoGNN-LSTM (no ${\bf W}_{t}$) | 18.6 | 21.0 | 30.0 | 37.2
CoGNN-TC (no ${\bf W}_{t}$) | 17.0 | 19.1 | 28.4 | 34.7
e2e ${\bf W}_{t}$ |  CoGNN-AR (e2e ${\bf W}_{t}$) | 23.1 | 30.6 | 45.4 | 51.2
CoGNN-LSTM (e2e ${\bf W}_{t}$) | 16.9 | 18.7 | 24.6 | 30.2
CoGNN-TC (e2e ${\bf W}_{t}$) | 16.4 | 18.2 | 25.4 | 31.4
Ours | CoGNN-AR | 21.6 | 28.5 | 43.5 | 50.3
CoGNN-LSTM | 16.7 | 17.7 | 23.3 | 29.0
CoGNN-TC | 15.4 | 17.2 | 23.8 | 28.5

Baselines and evaluation metrics. We compare our model to state-of-the-art
works, including two non-graph methods, Res-sup [26] and CSM [62], and six
graph-based methods, NRI [5], Traj-GCN [15], DNRI [6], DMGNN [17], HisRep [63]
and EvolveGraph [7]. For these offline methods, we also adjust them to take
the same streaming data as our CoGNNs and generate future poses as effectively
as possible in an online setting. For evaluation, we calculate the Mean Per
Joint Postion Error (MPJPE) between each predicted pose and the corresponding
ground-truth.

Results. We first evaluate various methods on H3.6M. Besides being compared to
the state-of-the-art works, CoGNNs are also compared to two degenerated
variants: i.e., ‘CoGNN (no ${\bf W}_{t}$)’ and ‘CoGNN (e2e ${\bf W}_{t}$)’;
see introduction in the experiments of online simulated trajectory prediction.
Table II shows the forecasting MPJPEs on 4 representative actions: ‘Walking’,
‘Eating’, ‘Smoking’ and ‘Discussion’, presenting the average MPJPEs at the
future $80$ms, $160$ms, $320$ms and $400$ms. We see that, 1) compared to the
state-of-the-art methods, our CoGNNs achieve more precise forecasting and
outperform the baselines; 2) compared to the model variants, ‘no ${\bf
W}_{t}$’ and ‘e2e ${\bf W}_{t}$’, the proposed CoGNNs consistently outperforms
these two variants, demonstrating the importance and effectiveness of our
intepretable collaborative graphs and the exponentiated update strategy. See
the forecasting results of another 11 actions in Appendix.

To further qualitatively evaluate our CoGNNs, we visualize the future poses
produced by various methods for human motion prediction on H3.6M dataset. We
illustrate the motions of a clip of ‘Walking’ within the future 400 ms, which
are predicted by DMGNN [17], CoGNN-LSTM and CoGNN-TC; see Fig. 5. We see that,
for the baseline model, DMGNN, there are large errors after the 160th ms of
the prediction time, where the left arm on the human body cannot bend
flexibly. As for the CoGNN-LSTM and CoGNN-TC, the generated poses are much
closet to the ground-truths than the DMGNN model. Moreover, we note that, the
illustrated human poses is the predicted samples at one certain iteration
step. To evaluate the online prediction system, we should focus more on the
continuous performance and the prediction errors during the whole online
learning process.

Additionally, We also various methods on another large-scale datatset, CMU
Mocap, for online motion prediction. The average prediction results (MPJPE)
across all the actions at the future $80$ms $160$ms $320$ms and $400$ms are
presented in Table III. We see that, 1) the proposed CoGNN models outperform
the state-of-the-art methods with a large margin; 2) the proposed versions of
CoGNNs achieve consistent better performances than the other two types of
model variants to show the effectiveness of our collaborative mechanism. The
detailed forecasting results of each class of activities are presented in
Appendix.

Figure 5: Predicted samples for human motion prediction on the action of
‘Walking’ on H3.6M dataset. TABLE IV: Comparisons between our CoGNNs and
state-of-the-art methods for online traffic speed forecasting on four datasets
based on three metrics. We also present the degenerated variants of CoGNNs
which abandon the collaborative graphs, i.e., ‘no ${\bf W}_{t}$’.

Datasets | METR-LA | PeMS-BAY | PeMS-D4 | PeMS-D8
---|---|---|---|---
Methods | MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAPE | MAE | RMSE | MAP
Baselines |  ST-GCN [14] | 5.68 | 8.92 | 13.0% | 3.09 | 6.57 | 7.7% | 37.28 | 46.44 | 30.5% | 30.49 | 39.57 | 24.8%
DCRNN [18] | 5.45 | 8.76 | 12.6% | 3.07 | 6.69 | 7.8% | 37.67 | 46.95 | 31.1% | 31.31 | 39.35 | 25.4%
ASTGCN [13] | 4.66 | 7.43 | 11.7% | 2.51 | 4.29 | 6.0% | 32.15 | 43.72 | 28.8% | 28.85 | 36.97 | 23.0%
G-WaveNet [31] | 4.42 | 7.10 | 11.2% | 2.31 | 4.07 | 5.3% | 28.58 | 39.59 | 25.6% | 27.81 | 36.03 | 19.8%
STSGCN [4] | 4.73 | 7.62 | 11.6% | 2.44 | 4.31 | 6.8% | 23.82 | 30.85 | 18.9% | 24.47 | 33.11 | 16.4%
AGCRN [64] | 4.19 | 6.40 | 8.3% | 2.16 | 3.74 | 4.7% | 19.65 | 28.54 | 16.0% | 22.72 | 29.03 | 13.4%
no ${\bf W}_{t}$ |  CoGNN-AR (no ${\bf W}_{t}$) | 4.42 | 6.61 | 8.8% | 1.72 | 2.65 | 4.1% | 19.37 | 28.02 | 17.3% | 16.92 | 26.61 | 12.7%
CoGNN-LSTM (no ${\bf W}_{t}$) | 6.08 | 8.35 | 14.6% | 2.74 | 3.68 | 4.4% | 27.45 | 35.14 | 19.3% | 20.12 | 29.73 | 13.7%
CoGNN-TC (no ${\bf W}_{t}$) | 3.61 | 5.89 | 8.0% | 1.43 | 2.41 | 3.0% | 20.27 | 29.48 | 16.1% | 16.99 | 24.14 | 11.9%
Ours | CoGNN-AR | 4.09 | 6.22 | 8.3% | 1.59 | 2.42 | 3.8% | 19.07 | 26.94 | 15.3% | 15.75 | 23.74 | 11.4%
CoGNN-LSTM | 5.04 | 7.28 | 13.4% | 2.21 | 3.52 | 4.2% | 26.95 | 32.31 | 18.9% | 19.62 | 28.87 | 13.1%
CoGNN-TC | 3.46 | 5.75 | 7.8% | 1.36 | 2.24 | 2.8% | 18.93 | 27.71 | 14.7% | 15.51 | 22.35 | 10.8%

###  VI-C Online traffic speed prediction

Furthermore, the proposed CoGNNs are also evaluated on the tasks of online
traffic speed prediction.

Datasets. We conduct experiments on four large-scale traffic datasets in the
real world, including the traffic data of Los Angeles county [65] or different
regions of the California highway [66]:

$\bullet$ METR-LA contains traffic speed collected by 207 sensors around Los
Angeles from March to June in 2012.

$\bullet$ PeMS-BAY contains traffic speed at Bay Area collected by 325 sensors
on CalTrans Performance Measurement System (PeMS) from Jan. to May in 2017.

$\bullet$ PeMS-D4 contains traffic speed collected by 307 sensors at San
Francisco from Jan. to Feb. in 2018.

$\bullet$ PeMS-D8 contains traffic speed collected by 170 sensors in San
Bernardino from July to Aug. in 2016.

We illustrate the traffic speed distribution of METR-LA in Fig. 4 (c) as an
example, where we denote that each sensor captures the dynamic traffic speed
along time. For these datasets, we aggregate the data sequence into 5-minute
intervals from 30-second frequency. We take the past $12$-frame-length ($1$
hour) sequence as the input and forecast the next $12$ frames.

Baselines and evaluation metrics. To evaluate the effectiveness our CoGNNs on
the online traffic speed prediction, we introduce several state-of-the-art
models for comparison,including ST-GCN [14], DCRNN [18], ASTGCN [13],
G-WaveNet [31], STSGCN [4] and AGCRN [64]. To test various methods, we utilize
three metrics, which are mean absolute error (MAE), root mean square error
(RMSE) and mean absolute percentage error (MAPE) over the whole iterative
training and test process.

Results. We compare CoGNNs to various algorithms on four datasets. We also
test the degenerated variants of CoGNNs without collaborative graphs, i.e.,
CoGNNs (no ${\bf W}_{t}$). We evaluate on the mean MAE, mean RMSE and mean
MAPE as shown in Table IV. We see that, 1) compared to the previous works, our
CoGNNs obtain the lowest errors on all the datasets; 2) compared to the model
variants without collaborative graphs, the complete CoGNNs also achieve the
much better prediction performances. The detailed traffic forecasting at
different prediction time are presented in Appendix.

###  VI-D Model analysis

Regrets of various forms of CoGNNs.

|
---|---
(a) Regret curves. | (b) Effects of learning rates.
Figure 6: The regret curves and the performance with various learning rates of
CoGNNs for online human motion prediction.

According to our theoretical regret analysis, the static regrets converge with
${\rm R}_{T}=\mathcal{O}(1/\sqrt{T})$. Here we compute and visualize the
static regrets of different forms of CoGNNs to verify our analysis. For online
motion prediction on H3.6M, we illustrate the regret curves along time within
$4500$ iterations; see Fig. 6 (a). The regrets of three forms of CoGNNs could
converge to very low values within approximately $2500$ iterations.
Additionally, the regret curves reflect that CoGNNs quickly close the
performance gaps with the models that only consider the best collaborative
pairs, showing the effectiveness and interpretation of our CoPUs.

Effect of learning rate $\eta$. The learning rate affects the training of the
collaborative predictor and the collaborative graph in each CoPU. Here we vary
the learning rate $\eta$ from $0.0075$ to $0.125$ to test the model
performance for online motion prediction on H3.6M. The mean MPJPEs of
different forms of CoGNNs are illustrated in Fig. 6 (b). We see that, CoGNN-
AR, CoGNN-LSTM and CoGNN-TC achieve their best performance when $\eta$ is
$0.075$, $0.05$ and $0.01$, respectively. CoGNN-AR tends to be more stable
with various $\eta$ than others; CoGNN-LSTM and CoGNN-TC obtain better
performance given the reasonable $\eta$.

Effect of number of CoPUs in CoGNNs.

TABLE V: The avergae MPJPE of CoGNNs with different numbers of CoPUs for
online human motion prediction.

Methods | 1 CoPU | 2 CoPUs | 3 CoPUs | 4 CoPUs
---|---|---|---|---
CoGNN-AR | 38.16 | 34.42 | 34.46 | 35.15
CoGNN-LSTM | 28.48 | 26.73 | 27.47 | 26.92
CoGNN-TC | 26.07 | 24.39 | 24.74 | 25.03

Here we analyze how different numbers of CoPUs in the CoGNN architectures
affect the performance. We test different forms of CoGNNs with $1$ to $4$
CoPUs for online human motion prediction on H3.6M. The mean MPJPEs are
presented in Table V. For three forms of CoGNNs, two CoPUs help to achieve the
lowest MPJPEs, which outperform the models with only one CoPU to different
extents, showing the effectiveness of the progressively refinement. For $3$ or
$4$ CoPUs, the model performance shows sub-par results due to the slight over-
smoothing, but it still converges to stable values.

Visualization of the learned collaborative graphs. To show that our model
effectively captures the implicit systematic dependencies by learning
collaborative graphs, here we visualize the learned collaborative graphs on
the various tasks of online simulated trajectory prediction, human motion
prediction and traffic speed prediction, respectively.

We first visualize the collaborative graphs for online simulated trajectory
prediction. We compare the learned collaborative graphs and the predefined
interactions that are used to simulate the particle interaction systems. We
illustrate the adjacency matrices of both simulated graphs and learned graphs.
To reflect that our model could adapt to the changing states of the
interaction systems, we show the learned graphs at different iteration steps
during the online learning process. The adjacency matrices are illustrated in
Fig. 7. We see that, at one iteration step, $t_{1}$ or $t_{2}$, the learned
collaborative graph captures similar topologies to the simulated graphs with
weighted edges to different extents. At different iteration steps, the
collaborative graphs could adapt to the changing simulated interaction
structures. Note that the element values of the graph adjacency matrices are
not similar due to the normalization in our weight update algorithm, while we
focus more on the visual similarity of topologies. In this way, we verify that
the collaborative graph could reasonably depict the implicit interactions in
the complex systems.

Figure 7: Visualization of the adjacency matrices of the collaborative graph
on the task of online simulated trajectory prediction. The two rows denotes
the different interactions at different time, while the first and the second
columns denotes the adjacency matrices of the simulated graphs and the learned
graphs, respectively. Figure 8: Visualization of the collaborative graphs on
the task of online human motion prediction. The edges whose weights are larger
than $0.2$ are plotted in red. (a) The learned graphs for different actions,
showing distinct interactions. (b) The graphs evolves along time, showing
dynamic interactions.

Here we visualize the learned collaborative graphs on H3.6M dataset. We use
red segments to plot the graph edges whose weights are larger than $0.2$ on
human poses. Since any edge weights range from $0$ to $1$, and most of them
are lower than $0.2$, thus the plotted edges show the important collaborative
pairs, i.e., the important influence between two agents. The graphs are
illustrated in Fig. 8. In plot (a), different actions have different graphs,
indicating distinct relations on specific actions; for example, arms and legs
affect across left and right for walking, while hands are highly related to
other joints for eating and smoking. In plot (b), we show the dynamic graphs
changing along time during discussion.

Figure 9: The learned collaborative graph for online traffic speed prediction
on METR-LA dataset. The edges whose collaborative weights are larger than
$0.1$ are illustrated, reflecting the most important collaborative effects
among the $207$ nodes.

We also show the learned collaborative graph for online traffic speed
prediction. As an example, we show the learned graph on METR-LA dataset. We
illustrate the edges whose collaborative weights are larger than $0.1$ in Fig.
9, which reflects the most important collaborative effects among the $207$
nodes to some extent. We see that, the traffic speed data collected by
different sensors are mainly affected by their geographically nearby traffic.
For example, a certain node on a single long high-way is mainly affected by
other nodes on the same high-way, because a single long high-way could be
regarded as an isolated systems, in which the the vehicles move and stop
collaboratively. Moreover, near the intersections or the traffic circles, one
node tends to be affected by the traffic around across roads, since many
vehicles perform agglomeration effects in these local regions. Fig. 9 shows a
reasonable structure to depict the collaborative effects. The collaborative
graphs for online human motion prediction have been illustrated in Figure 3 of
our main text. Different from the human motion data, in a traffic scenario,
the interactions and relations between two nodes with a relatively long
distance is usually weak. That is because traffic data has very strong spatial
location attributes, and higher synergy tends to appear in more local areas,
such as local congestion during peak hours; and two nodes farther apart are
difficult to affect each other because of the decay of collaborative effects
along distances.

##  VII Conclusion

We develop a novel method to predict the future statuses of a multi-agent
system online. We propose a novel CoPU, which uses a collaborative graph to
aggregate multiple collaborative predictors that learn dynamics from
collaborative pairs. The collaborative graph depicts the influence level of
collaborative pairs, which are adjusted with the guidance from an explicit
objective. The regret analysis show that our method achieves the similar
performance with the best single collaborative predictor. Multiple CoPUs are
stacked as a CoGNN. Experiments demonstrate the effectiveness of our method
for various multi-agent forecasting tasks.

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|  Maosen Li recieved the B.E. degree in optical engineering from University
of Electronic Science and Technology of China (UESTC), Chengdu, China, in
2017. He is working toward the Ph.D. degree at Cooperative Meidianet
Innovation Center in Shanghai Jiao Tong University since 2017. His research
interests include computer vision, machine learning, graph representation
learning, and video analysis. He is the reviewer of some prestigious
international journals and conferences, including IEEE-TPAMI, IEEE-TNNLS,
IJCV, IEEE-TMM, PR, ICML, NeurIPS and AAAI. He is a student member of the
IEEE.
---|---
|  Siheng Chen is an associate professor at Shanghai Jiao Tong University.
Before that, he was a research scientist at Mitsubishi Electric Research
Laboratories (MERL). Before joining MERL, he was an autonomy engineer at Uber
Advanced Technologies Group, working on the perception and prediction systems
of self-driving cars. Before joining Uber, he was a postdoctoral research
associate at Carnegie Mellon University. Chen received the doctorate in
Electrical and Computer Engineering from Carnegie Mellon University in 2016,
where he also received two masters degrees in Electrical and Computer
Engineering and Machine Learning, respectively. He received his bachelor’s
degree in Electronics Engineering in 2011 from Beijing Institute of
Technology, China. Chen was the recipient of the 2018 IEEE Signal Processing
Society Young Author Best Paper Award. His coauthored paper received the Best
Student Paper Award at IEEE GlobalSIP 2018. He organized the special session
”Bridging graph signal processing and graph neural networks” at ICASSP 2020.
His research interests include graph signal processing, graph neural networks
and autonomous driving. He is a member of IEEE.
---|---
|  Yanning Shen received her Ph.D. degree from the University of Minnesota
(UMN) in 2019. She was a finalist for the Best Student Paper Award at the 2017
IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive
Processing, and the 2017 Asilomar Conference on Signals, Systems, and
Computers. She was selected as a Rising Star in EECS by Stanford University in
2017, and received the UMN Doctoral Dissertation Fellowship in 2018. Her
research interests span the areas of machine learning, network science, data
science and statistical-signal processing.
---|---
|  Genjia Liu is an undergraduate student at Shanghai Jiao Tong University
since 2017. He is going to work toward the Ph.D. degree at Cooperative
Medianet Innovation Center in Shanghai Jiao Tong University from 2021. His
research interests include graph signal processing and graph representation
learning.
---|---
|  Ivor Tsang is Professor of Artificial Intelligence with the University of
Technology Sydney. He is also the Research Director of the Australian
Artificial Intelligence Institute. His research interests include transfer
learning, deep generative models, and big data analytics. In 2013, Prof Tsang
received his prestigious ARC Future Fellowship for his research regarding
Machine Learning on Big Data. In 2019, his JMLR paper titled ”Towards
ultrahigh dimensional feature selection for big data” received the
International Consortium of Chinese Mathematicians Best Paper Award. In 2020,
Prof Tsang was recognized as the AI 2000 AAAI/IJCAI Most Influential Scholar
in Australia for his outstanding contributions to the field of AAAI/IJCAI
between 2009 and 2019. His research on transfer learning granted him the Best
Student Paper Award at CVPR 2010 and the 2014 IEEE TMM Prize Paper Award. In
addition, he received the IEEE TNN Outstanding 2004 Paper Award in 2007. He
serves as a Senior Area Chair/Area Chair for NeurIPS, ICML, AISTATS, AAAI and
IJCAI, and the Editorial Board for JMLR, MLJ, and IEEE TPAMI.
---|---
|  Ya Zhang is currently a professor at the Cooperative Medianet Innovation
Center, Shanghai Jiao Tong University. Her research interest is mainly in
machine learning with applications to multimedia and healthcare. Dr. Zhang
holds a Ph.D. degree in Information Sciences and Technology from Pennsylvania
State University and a bachelor’s degree from Tsinghua University in China.
Before joining Shanghai Jiao Tong University, Dr. Zhang was a research manager
at Yahoo! Labs, where she led an R&D team of researchers with strong
backgrounds in data mining and machine learning to improve the web search
quality of Yahoo international markets. Prior to joining Yahoo, Dr. Zhang was
an assistant professor at the University of Kansas with a research focus on
machine learning applications in bioinformatics and information retrieval. Dr.
Zhang has published more than 70 refereed papers in prestigious international
conferences and journals, including TPAMI, TIP, TNNLS, ICDM, CVPR, ICCV, ECCV,
and ECML. She currently holds 5 US patents and 4 Chinese patents and has 9
pending patents in the areas of multimedia analysis. She was appointed the
Chief Expert for the ’Research of Key Technologies and Demonstration for
Digital Media Self-organizing’ project under the 863 program by the Ministry
of Science and Technology of China. She is a member of IEEE.
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