# Page time and the order parameter for a consciousness state

Alexander Gorsky  Institute for Information Transmission Problems, Moscow
127994, Russia  Laboratory of Complex Networks, Brain and Consciousness
Research Center, Moscow

Abstract

In this Letter using the analogy with the recent resolution of the black hole
information paradox we conjecture the order parameter for the state of
consciousness based on the notion of the Page curve and the Page time. The
entanglement between the state of the brain and time series of neuronal firing
as well as the non-orthogonality of the functional connectomes play a key
role.

##  1 Introduction and motivation

The origin of the consciousness certainly is the interesting scientific
problem. It is desirable to identify the guidance principle behind the
phenomenon and there are several ideas formulated to handle with this issue.
In the integrated information theory(IIT) [1, 2] it is assumed that the
entanglement of the clusters in the times series data for the neuronal firing
could play the role of the order parameter of the consciousness state. The
identification of the clusters can be simplified via the spectral analysis of
the network formed from the time series [3]. The idea of the global neuronal
workspace is based on the version of the self-organized criticality [4, 5]. It
is assumed that interaction between the neuronal groups in nonlinear and
transition to the consciousness state is a kind of phase transition. Another
combination of ideas yields the model of competing cellular assemblies [6].
The neuronal group selection theory focuses at the evolutionary aspects of the
formation of consciousness [7].

More physically motivated thermodynamical approach via a kind of the free
energy principle was suggested in [8]. Some ideas based on the quantum
decoherence were discussed in [9]. The review of the measures of consciousness
can be found in [10] while the recent review of the most popular theories of
consciousness can be found in [11].

Several features are quite common for all models

  * •

The state of consciousness is related to consolidation of effective degrees of
freedom in the brain - functional connectomes

  * •

There is specific timescale attributed to the formation of consciousness state
from the non-conscious state

  * •

Some version of entanglement serves as the order parameter for the
consciousness state

  * •

There is the interplay between the deterministic component and the ensemble
average in the formation of consciousness state

  * •

The cluster structure in the time series is important in some models

  * •

The evolution of the system is unitary

We shall argue that these features can be naturally unified within the
framework of the Page curve and Page time [12, 13] assuming the entanglement
between the brain states and time series of neuron firing.

The Page curve has been invented to keep unitarity in a two-component system
in a pure random state taking into account properly the entanglement of
subsystems. The entanglement is evaluated via the density matrix projected
into the subsystem [13]. The spectrum of projected density matrix before the
Page time obeys the Marchenko-Pastur (MP) law but gets modified after the Page
time. This approach has been applied for the entanglement of a black hole (BH)
and its radiation [12] however the BH example is not unique and in principle
one can consider any other "black box" entangled with its own radiation or
time series of some activity.

The BH information paradox has been known since 70-ths [14] and it claims that
the entropy production of the Hawking BH radiation violates a unitarity at
large times, see [15, 16] for the recent reviews. The resolution of the
information paradox has been found very recently and the island conjecture has
been formulated. It was argued [17, 18] that the state of BH radiation is
encoded in some island subregions behind the BH horizon. This has been
quantified in the simplified model via end of world branes(EOW) [19], each of
them has large number of internal degrees of freedom [20, 21]. The number of
EOW branes grows with time and BH with EOW branes get entangled with the BH
radiation. This is the first ingredient important for the resolution of the
information paradox.

The second crucial ingredient goes as follows. The EOW branes interact and can
develop the wormhole non-perturbative configurations connecting the pairs of
the EOW branes. At the Page time the EOW branes get consolidated into the
single disc geometry since the multiple non-perturbative wormholes start to
dominate [21]. After the Page time the entropy of radiation goes down and the
unitarity gets restored. It is important that after the Page time the Petz map
known in the information theory [22, 23] allows to map the operations inside
the BH Hilbert space into the operations in the Hilbert space of BH radiation
[21]. Such mapping is impossible before the Page time. If the deterministic
component is introduced into the system [24, 25] a kind of perturbative
interaction between the EOW branes is induced.

We conjecture that somewhat similar picture emerges statistically during
formation of consciousness state from the non-conscious one. The Hilbert space
of brain and the Hilbert space of time series together are considered as the
two-component entangled system. We assume that the functional connectomes are
analogue of the EOW branes and encode the information concerning the time
series of neuronal activity. The functional connectomes are non-orthogonal and
interact in two ways. First they interact as multilayer networks and secondly
there is the "contact interaction" since some neurons belong to several
functional connectomes simultaneously. The state of brain with ensemble of
functional networks is entangled with the time series and the number of
functional networks involved grows in time. At time of formation of a
consciousness state the functional networks get consolidated similar to the
consolidation of EOW branes and the entropy attributed to time series stops to
grow.

Hence we assume that the Page curve describes the time evolution of the
entropy production in the time series of neuronal firing and at the Page time
the functional connectomes get consolidated which is necessary condition for
the establishing of a consciousness state. We conjecture that the sign of time
derivative of the entropy production in the time series and clustering of the
networks formed from the time series can serve as the indicators for a state
of consciousness.

##  2 BH toy model

###  2.1 The entanglement of the BH and radiation

Let us recall the notion of the Page curve for a entangled two-component
system. It was shown in [13] that the entanglement spectrum of the system in
pure random state involving two subsystems and projected to one of them
provides the important information concerning unitarity. Let rectangular
matrix $X$ describes uniformly random pure quantum state in $C^{n}\otimes
C^{m}$. Then $Y=XX^{+}$\- reduced density matrix on $C^{n}$ whose eigenvalue
distribution depends on the evolution time. At small times spectral density of
entanglement matrix $Y$ obeys MP distribution [13]. However at some time,
called the Page time the spectrum of the entanglement matrix gets modified and
the entropy production in the subsystem which we have projected on starts to
decrease. The entropy behaves according to Page curve as follows

$S(n,m)=\left(\log n-\frac{n}{2m}\right)\theta(m-n)+\left(\log
m-\frac{m}{2n}\right)\theta(n-m)$ (1)

If the number of degrees of freedom in one subsystem grows with time the
change of regime occurs exactly at the Page time,

The Page curve has been recognized recently in the context of the resolution
of the BH information paradox. In this case the entangled two-component system
is as follows. The one subsystem involves $e^{S}_{BH}$ degrees of freedom
where $S_{BH}=\frac{A}{4G_{N}}$ is the Bekenstein-Hawking BH entropy and $A$
is the area of horizon. The second system describes the ongoing radiation of
the BH and dimension of the corresponding Hilbert space $k$ grows in time.
These two subsystems are entangled and it was argued [17] that there is
quantum extremal surface Q behind the BH horizon [26] encoding the BH
radiation. The island conjecture assumes that the entropy of BH radiation
reads as

$S(\rho_{R})=\min\left[\frac{Area(Q)}{4G_{N}}+S_{\text{bulk}}\right]$ (2)

Here $\rho_{R}$ is the density matrix of the BH radiation in the full theory
coupled to quantum gravity. The second term corresponds to the von Neumann
entropy of the region in the interior of the BH bounded by the quantum
extremal surface Q. At small times the second contribution is small however at
large times the leading saddle in the semi-classical path integral on the
Euclidean black hole gets modified and the different extremal surface starts
to dominate. The key point is that in this new phase the entropy of the
radiation decreases in agreement with a unitarity. The corresponding behaviour
with the Page time is presented in Fig.1

Figure 1: Page curve for the entropy of BH radiation(red)

In the simplified version instead of the extremal surface one considers
multiple EOW branes [21, 20] and analyses what is the saddle point
configuration at the different times. Naively one considers the states

$\sum_{i=1}^{k}\left|\Psi_{i}\right\rangle\otimes\left|i\right\rangle$ (3)

where $\Psi_{i}$ is the state of BH with the non-dynamical EOW branes behind
the horizon while the state $i$ belongs to the auxiliary basis of ongoing
Hawking modes [21]. The states $\Psi_{i}$ are not orthogonal which is the
second important point behind the explanation of the BH information paradox

$\left\langle\Psi_{i}|\Psi_{j}\right\rangle=e^{2S_{BH}}\delta_{ij}+O\left(e^{S_{BH}}\right).$
(4)

The second term corresponds to the contribution from non-perturbative
wormholes or effect of more direct EOW brane interactions and generically is
subject to the ensemble averaging with some measure. Quantitatively the
density matrix projected onto the radiation reads as

$\rho_{R}=\sum_{i,j}^{k}\left\langle\Psi_{j}|\Psi_{i}\right\rangle|i\rangle\langle
j|=\sum_{i,j}^{k}\sum_{a=1}^{e^{S}}C^{*}_{aj}C_{ai}|i\rangle\langle j|$ (5)

In the simplest case the random rectangular $e^{S_{BH}}\times k$ matrices $C$
are averaged with the Gauss measure with unit covariance

$\int dCdC^{+}\exp(-\operatorname{Tr\,}C^{+}C)$ (6)

The spectral density of $\rho_{R}$ has been evaluated in [21] via solution to
the Schwinger-Dyson equation for the resolvent in the planar approximation
which takes into account the replica wormholes between the EOW branes. In
microcanonical ensemble with unit Gaussian distribution it reads as

$\rho_{R}(x)=\frac{ke^{S}_{BH}}{2\pi
x}\sqrt{[x-(k^{-1/2}-e^{S_{BH}/2})^{2}][(k^{-1/2}+e^{S_{BH}/2})^{2}-x)}+\delta(x)(k-e^{S}_{BH})\theta(k-e^{S}_{BH})$
(7)

We see that its smooth part coincides with the MP distribution.

At some value of k the entropy of the BH and the entropy evaluated from the
radiation density matrix $\rho_{R}$ coincide. This happens at the Page time
and the interpretation from the viewpoint of internal degrees of freedom
behind the horizon has been developed in [21]. It has been argued that the
semiclassical saddle point for EOW branes before the Page time implies that
they can be considered as the separated degrees of freedom which non-
perturbatively interact via wormhole solutions. However upon the summation of
the replica wormholes at large times another saddle point dominates and the
EOW branes form the single disk-like geometry which should be considered as
single consolidated state of multiple EOW branes see Fig 2. The off-diagonal
terms in the matrix elements start to dominate. After the Page time the Petz
map [22, 23] allows to map the operations in the radiation Hilbert space into
the operations in the BH Hilbert space [21].

Figure 2: Islands behind the horizon are entangled with BH radiation. At the
Page time the multiple islands get consolidated into the disc-like geometry.

###  2.2 BBP transition and entanglement spectrum

We have argued above that at the Page time the entanglement spectrum of the
radiation gets modified and some number of the eigenvalues get separated from
continuum after the Page time. This follows from the effect of non-
orthogonality of the internal states of the BH. Recently it has been suggested
[25] that the admixture of the determinism in the averaging over ensemble can
induce the "perturbative" interaction of the EOW branes. Moreover it properly
induces the eigenvalue separation phenomenon in the MP distribution necessary
to get the Page curve.

The phenomenon of the eigenvalue separation is familiar in the probability
theory when we perturb the Gaussian measure for random variables. We are
interested in the deformation of the measure involved into the projected
density matrix in the pure random states that is in the Wishart ensemble. Two
types of measure deformation are known; the shifted Gaussian measure
$\exp(-\operatorname{Tr\,}(H-H_{0})^{2})$ when the expansion around the
nontrivial saddle takes place [27] and the measure with non-null covariance
$\exp(-\operatorname{Tr\,}X^{+}X\Sigma)$ [28, 29]. In both cases there is the
critical value of parameters involved in $H_{0}$ or $\Sigma$ when one or
several eigenvalues get decoupled from the bulk of the spectrum. It is a kind
of phase transition (BBP transition) with own universality class and the
distribution of the largest eigenvalue obeys the so-called spiked Tracy-
Widom(TW) law exactly at the transition point.

To fit with our conjecture we modify the Gaussian measure not by the shift of
mean value as in [25] but by introducing non-null complex covariance which
does the same job [28, 29]. This goes as follows. Let us consider $M$ vectors
$(\vec{y_{1}}\dots\vec{y_{M}})$ each of dimension $N$. The density of all
vectors is Gaussian with covariance $\Sigma$, where $\Sigma$ is positive
$N\times N$ matrix

$P(\vec{y})=\frac{1}{(2\pi)^{N/2}(\det\Sigma)^{1/2}}\exp\left(-\frac{1}{2}\left\langle\vec{y},\Sigma^{-1},\vec{y}\right\rangle\right)$
(8)

Consider the $N\times M$ matrix $X=\left[{\bf y}_{1}-{\bf Y},\dots\bf
y_{M}-{\bf Y}\right]$ and combine it into $N\times N$ covariance matrix
$S=XX^{t}$. Let us assume that $M\to\infty$ and $N\to\infty$ such that their
ratio is finite $\frac{M}{N}=\gamma$. The matrix $S$ belongs to the Laguerre
or Wishart ensemble whose asymptotic properties at $\Sigma=Id$ are known. The
spectral density obeys the MP law

$\rho(x)=\frac{\gamma^{2}}{2x}\sqrt{(b-x)(x-a)},\qquad a<x<b$ (9)

where

$a=\left(\frac{\gamma-1}{\gamma}\right)^{2};\qquad
b=\left(\frac{\gamma+1}{\gamma}\right)^{2}$ (10)

The largest eigenvalue fixing the spectral edge is $\lambda_{max}=b$ and the
spectral fluctuations at the spectral edge obey the TW distribution

$P\left((\lambda_{max}-b)\frac{\gamma}{(1+\gamma)^{4/3}}M^{2/3}\leq
s\right)\rightarrow F_{GOE}(s)$ (11)

where $F_{GOE}$ is the TW distribution for orthogonal ensemble. Now departure
from the identity matrix is introduced and some number of degenerate non-unit
eigenvalues of the covariance matrix $\Sigma$ are selected $l_{1}\dots
l_{k}\neq 1$. The value $l_{1}$ is considered as the control parameter for the
BBP phase transition.

For complex Gaussian samples [28] the critical value of the non-unit
covariance eigenvalue for perturbed case has been found
$l_{crit}=1+\gamma^{-1}$ where $\gamma$ is a parameter of the ensemble.

$P\left((\lambda_{max}-l_{crit}^{2})\frac{\gamma}{(1+\gamma)^{4/3}}N^{2/3}\leq
s\right)\rightarrow F_{k}(s)$ (12)

The function $F_{k}(s)$ is the limiting distribution, where the largest
eigenvalue $l_{1}$ of the covariance matrix has the multiplicity $k$ and is
equal to the critical value, hence we have single perturbation parameter. If
$k$ non-unit eigenvalues of the covariance matrix are above the critical value
$l_{crit}$, i.e. $l_{i}>l_{crit}$, distribution of $\lambda_{max}$ obeys the
distribution in $k\times k$ Gaussian unitary ensemble, $G_{k}(s)$.

$P\left((\lambda_{max}-\left(l_{1}+\frac{l_{1}\gamma^{-2}}{l_{1}-1}\right)\right)N^{1/2}\sqrt{l_{1}^{2}-\frac{l_{1}^{2}\gamma^{-2}}{(l_{1}-1)^{2}}}\leq
s)\rightarrow G_{k}(s)$ (13)

The function $F_{k}(s)$ can be generalized to the function
$F_{k}(s,w_{1},\dots,w_{k})$ if all perturbation parameters are different.
Similarly to the pure TW case, the distribution obeys some solution to the
Painleve II equation with particular monodromy properties [29]. In general
case the determinantal representation of $F_{k}(s,w_{1},\dots,w_{k})$ is
available in terms of the single function of two variables $f(s,w)$ which
satisfy the pair of differential equations depending on the Painleve II
solutions – see [29] for details.

Summarizing, there are three regimes for the spectral density depending on the
relation between $k$ and $e^{S_{BH}}$. The transition at the Page time
indicates that the new saddle in the gravitational path integral corresponding
to connected geometry with multiple wormholes starts to dominate at $k\propto
e^{S_{BH}}$. In terms of the eigenvalues exactly at the Page transition
$k-e^{S_{BH}}$ eigenvalues get escaped from the bulk of distribution and
settle at $\lambda=0$. If somebody would be interested in the question if all
EOW branes get consolidated or not one should look at the sign of the time
derivative for the radiation entropy and at the number of isolated
eigenvalues. This is the order parameter separating the phases.

##  3  Towards the order parameter for a consciousness state

###  3.1 Entropy for the time series

Since we will assume that the time series of the brain activity is the part of
the whole two-component system it is necessary to provide the tool to evaluate
the entropy of the time series similarly to the entropy of the BH radiation.
The effective way goes as follows, first we represent the time series data via
the network and than evaluate the entropy of this network.

There are at least three ways to generate the network from the time series
(see, [30] for the review). The first way involves the introduction of the
effective parameter which quantifies the vicinity in the data set and the
nodes within this effective radius are connected by links. In the second
approach the notion of visability is introduced and the horizontally visable
nodes are connected by links. Finally, in the third approach a kind of Markov
dynamics for the probability is formulated and its Markov operator yields the
network adjacency matrix.

Once we find the network representation for the time series some standard
entropy measured can be applied. The simplest one is the conventional
thermodynamic entropy defined through the density metrics for the network

$\rho=\frac{\exp(-\beta L)}{Z}$ (14)

where $L$ is the graph Laplacian or normalized graph Laplacian, and $\beta$ is
auxiliary parameter which allows to scan the spectrum. The thermodynamical
entropy is defined as

$S=\operatorname{Tr\,}\rho\log\rho$ (15)

and is useful to determine the characteristic time scales for the diffusion in
the network [31].

However we are interested in the entropy production hence we have to consider
the coarse-grained entropy instead of the fine-grained entropy which is
constant in time. The useful coarse-grained information entropy production
measure has been discussed in [32] which is suitable for our purpose,

$\dot{S}=\sum_{ij}P_{ij}log(\frac{P_{ij}}{P_{ji}})$ (16)

where $P_{ij}=Prob[x_{t-1}=i,x_{t}=j]$ and $x_{t}$ corresponds to the state of
the system at time $t$. It is nothing but the Kullback-Leibler divergence
between the forward and reverse transition probabilities. This measure for the
entropy production for different tasks has been investigated experimentally
[32]. It was shown that the entropy production vanishes or is very small in
non-conscious state, is quite substantial for the cognitive task and maximal
at the physical activity. Remark that the entropy production provides the
thermodynamical time arrow and it was noted recently that in the interacting
many-body system the time arrow emerges via the interaction only [33].

###  3.2 The conjecture

To formulate our conjecture remind that the brain involves structural and
multiple functional networks. The structural network just reflects the
connections between neurons and is completely known for C’Elegance with 302
nodes only. Each functional network or functional connectome involves all
neurons specialized in some function. They are connected to each other and can
be located in the very different parts of the brain. The functional networks
are assumed to interact and the individual neurons can belong to the several
functional networks simultaneously. It is generally believed that in the state
of consciousness the functional connectomes get consolidated via some
mechanism.

Let us apply the Page’s ideas to the brain and assume that the functional
connectomes play the similar role as EOW branes in the BH example while the
spiking time series play the role of BH radiation. Hence we have the "black
box" with some degrees of freedom inside and its radiation and would like to
investigate the phase of internal state of the brane looking at the properties
of time series only.

Let us follow the same logic as before and assume that the state of the system
involves the state $\Xi_{i}$ which corresponds to the state of the brain with
the functional connectomes in the state i and the states $|i>$ which are the
eigenfunctions of the Laplacian of the graph representing the time series.
Hence we consider the states

$\sum_{i}|\Xi_{i}\rangle\langle i|$ (17)

The density matrix projected to the time series state $\rho_{TS}$ is

$\rho_{TS}=\sum_{i,j}^{k}\langle\Xi_{j}|\Xi_{i}\rangle|i\rangle\langle
j|=\sum_{i,j}^{k}\sum_{a=1}^{S_{brain}}C^{*}_{aj}C_{ai}|i\rangle\langle j|$
(18)

where $S_{brane}$ is the entropy of the brain in the non-conscious state. The
value of $k$ measures the number of functional connectomes which are switched
on.

The projected density matrix yields the entanglement entropy of the spiking
time series $S_{TS}$. To keep the unitarity we assume the existence of the
Page curve and Page time when the internal functional networks get
consolidated, see Fig.3, providing the state of consciousness at the Page
time. After the Page time presumably the Petz map yields the mapping of the
operations within the time series to the operations between the functional
connectomes in the brain.

Figure 3: The functional connectome represents the network of neurons with the
same specialization. There are multiple functional connectomes in the brain
which can overlap due to the multispecialized neurons. We conjecture that
functional connectomes are entangled with the time series of neuron activity
and at the effective Page time when the state of consciousness gets
established they get consolidated.

As in the BH toy example the functional connectomes overlap and are non-
orthogonal. The interaction of the functional connectomes can have
perturbative and non-perturbative contributions. The simplest one is generated
by " contact interaction" we have described above. In the case of a brain this
interaction can be attributed to the neurons with several specializations. Due
to the overlap of functional connectomes the effective brain entropy and the
entropy of the time series become of the same order and the "common space " of
the functional connectomes gets formed at the Page time. Note that the contact
interaction amounts to the non-unit covariance matrix which induces the
isolated eigenvalues in the spectrum corresponding to the clusters in the time
series network [34]. Such clusters in the time series networks are the
essential ingredients of IIT.

Our conjecture fits with the general expected features of the consciousness
state

  * •

The consolidation of the functional networks occurs at the Page time

  * •

There is the timescale when consciousness state gets established

  * •

Order parameter indeed is related to some specific entanglement

  * •

Deterministic component is important. It induces the effective interactions
and simultaneously generates the clusters in the time series

Summarizing the arguments above we suggest that the sign of the time
derivative of the entropy production in the time series serves as the order
parameter $R$ for a consciousness state

$R=\mathrm{sign}\left(\frac{dS_{TS}}{dt}\right).$ (19)

Another important indicator is the increasing of the clusters in the network
formed from the time series. We can not exclude that the entropy production
stop raising at the Page and decreases very slowly at the late times.

Hence it is natural to question how the predictions following from our
conjecture can be measured experimentally and if there are any contradictions
with the existing experimental data. The simplest possibility concerns the
investigation of the entropy production and clusterization of the time series
during the waking up process and the exit from the state of anesthesia.
Hopefully some relevant results concerning the anesthesia state are available.
It was found in [35] that at exit from anesthesia two effects are observed
both for mice and humans (see also [36, 37, 38, 39]). First, the number of
clusters in the networks formed by time series (microstates in notations of
[35]) increases substantially and secondly, consolidation of the functional
connectomes gets increased as well. Moreover both features are reversible
while depth of consolidation depends on the type of anesthesia applied [40].
This is consistent with our conjecture.

Note also that the establishing time of consciousness state is different for
the waking up and the anesthesia exit processes. That is we can assume that
the number of functional networks switched off at anesthesia is larger then at
sleep state. It would be very interesting to measure the entropy production at
the anesthesia process to compare with our conjecture.

##  4 Conclusion

In this Letter we have conjectured using very general statistical arguments
for the unitary two-component entangled many-body system the new order
parameter for the consciousness state which measures if the functional
connectomes are consolidated or not. Certainly it is necessary but
insufficient criteria. Two points are crucial for this conjecture. First,it is
the entanglement between the time series of the neuron activity and the brain
state involving multiple overlapping functional connectomes. Secondly, the
non-orthogonality of the states of functional connectomes which is small at
early time but starts to dominate at the effective Page time. The mechanism
behind the resolution of the information paradox for a BH radiation provides
the proper framework to analyze this problem. The clustering of the time
series density matrix occurs naturally within this approach and is another
indication of the state of consciousness.

The conjecture is a bit speculative but we believe that this framework could
be useful for this research area and suggest the very clear-cut check. We do
not provide the microscopic explanation of the consciousness state restricting
ourselves by the order parameter and do not present the hypothetical
functional whose extremization presumably yields two different saddle points.
However we expect that some notions of the information geometry could be
useful in this respect. In particular the recent observation [41] concerning
the relation between the singularity of the information metric in the simplest
all-to-all disordered Kuramoto model and the synchronization phase transition
provides the example of such relation. Indeed microscopically the
consolidation of the functional connectomes implies the synchronization
phenomena in the multiplex networks. Another approach concerning the quantum
information geometry approach can be found in [42]. Note also that some
analogy between the brain and the BH in a different setting has been mentioned
in [43]. It would be interesting to combine the ideas from that paper with our
study.

I am grateful to A. Alexandrov, K. Anokhin, A. Milekhin, S. Nechaev and N.
Pospelov for the useful discussions and comments. I want to thank IHES and
Nordita where the parts of the work have been done for the hospitality and
support. The work was supported by grant №075-15-2020-801 by Ministry of
Science and Higher Education of Russian Federation.

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# Tree based Progressive Regression Model for Watch-Time Prediction in Short-
video Recommendation

Xiao Lin<EMAIL_ADDRESS>Kuaishou TechnologyBeijingChina ,  Xiaokai
Chen<EMAIL_ADDRESS>Kuaishou TechnologyBeijingChina ,  Linfeng Song
<EMAIL_ADDRESS>Kuaishou TechnologyBeijingChina ,  Jingwei Liu
<EMAIL_ADDRESS>Kuaishou TechnologyBeijingChina ,  Biao Li
<EMAIL_ADDRESS>Kuaishou TechnologyBeijingChina and  Peng Jiang
<EMAIL_ADDRESS>Kuaishou TechnologyBeijingChina

(2018; 2023)

###### Abstract.

An accurate prediction of watch time has been of vital importance to enhance
user engagement in video recommender systems. To achieve this, there are four
properties that a watch time prediction framework should satisfy: first,
despite its continuous value, watch time is also an ordinal variable and the
relative ordering between its values reflects the differences in user
preferences. Therefore the ordinal relations should be reflected in watch time
predictions. Second, the conditional dependence between the video-watching
behaviors should be captured in the model. For instance, one has to watch half
of the video before he/she finishes watching the whole video. Third, modeling
watch time with a point estimation ignores the fact that models might give
results with high uncertainty and this could cause bad cases in recommender
systems. Therefore the framework should be aware of prediction uncertainty.
Forth, the real-life recommender systems suffer from severe bias
amplifications thus an estimation without bias amplification is expected.

How to design a framework that solves these four issues simultaneously remain
unexplored. Therefore we propose TPM (Tree-based Progressive regression Model)
for watch time prediction. Specifically, the ordinal ranks of watch time are
introduced into TPM and the problem is decomposed into a series of conditional
dependent classification tasks which are organized into a tree structure. The
expectation of watch time can be generated by traversing the tree and the
variance of watch time predictions is explicitly introduced into the objective
function as a measurement for uncertainty. Moreover, we illustrate that
backdoor adjustment can be seamlessly incorporated into TPM, which alleviates
bias amplifications.

Extensive offline evaluations have been conducted in public datasets and TPM
have been deployed in a real-world video app Kuaishou with over 300 million
DAUs. The results indicate that TPM outperforms state-of-the-art approaches
and indeed improves video consumption significantly.

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation emai; June 03–05, 2018; Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††journalyear: 2023††copyright:
rightsretained††conference: Proceedings of the 29th ACM SIGKDD Conference on
Knowledge Discovery and Data Mining; August 6–10, 2023; Long Beach, CA,
USA††booktitle: Proceedings of the 29th ACM SIGKDD Conference on Knowledge
Discovery and Data Mining (KDD ’23), August 6–10, 2023, Long Beach, CA,
USA††doi: 10.1145/3580305.3599919††isbn: 979-8-4007-0103-0/23/08

##  1\. Introduction

Recent years have witnessed the growing popularity of online video services
(e.g. YouTube and Hulu) and video-sharing platforms (TikTok and KuaiShou). And
the amount of time that users spend on watching recommended videos (which is
referred to as watch time) becomes a key metric reflecting user engagement.
Users who get recommendations with higher watch time tend to stay longer in
the platform, which brings a growth of DAU (Daily Active User).

Despite its importance, watch time prediction has not been widely studied in
previous researches (covington2016deep; zhan2022deconfounding). We argue that
there are several special important aspects in watch time modeling:

First, watch time prediction is essentially a regression problem, but the
ordinal relation between watch time predictions is also important in
recommendation. On one hand, watch time is a continuous random variable, the
recommender system needs to get an accurate prediction of watch time for the
usage in downstream phases; on the other hand, watch time is a metric for
video comparison and the ordinal relation of predictions is also important.
For example, given two predictions of watch time for a video: $T_{1}=3.5s$,
$T_{2}=4.5s$ and the ground truth is $T=4s$. The two predictions share same
regression error of $0.5s$ in terms of MAE. However these two predictions lead
to very different consequences in recommender systems. As the system usually
tends to recommend videos with higher predicted watch time, the video is much
more likely to be recommended with a prediction of $4.5s$ compared to the case
with $3.5s$. Therefore, estimating watch time with direct regression fails to
model the ordinal relations between watch time. Ranking losses focus on the
ordinal relations but may lead to predictions that deviate far from the ground
truth. Therefore a good formulation of watch time prediction should satisfy
both requirements simultaneously.

Second, there exists strong conditional dependence in the video-watching
behaviors. For example, one has to watch half of the video before he/she
finishes watching the whole video. This is similar to the case of click and
post-click behaviors (e.g. purchase) in E-commerce platforms (ma2018entire;
wen2020entire). This conditional dependence needs to be considered in watch
time prediction.

Third, to enable a robust prediction of watch time, we expect the prediction
model to be uncertainty-aware about its predictions. For most regression
models, the objective is to get an accurate point estimation by minimizing a
$L_{1}$ or $L_{2}$ loss. Thus the models might produce predictions with high
uncertainty. For real-life recommender systems, this could lead to bad cases
where sub-optimal videos are assigned with high rankings by the model, but
cause unsatisfactory user experiences. However, how to model the uncertainty
in watch prediction is still under-investigated.

Table 1. Comparison between WLR, D2Q and TPM Approaches | WLR | D2Q | TPM
---|---|---|---
Ordinal Relation | $\times$ | $\times$ | $\checkmark$
Conditional Dependence | $\times$ | $\times$ | $\checkmark$
Model Uncertainty | $\times$ | $\times$ | $\checkmark$
Debiasing | $\times$ | $\checkmark$ | $\checkmark$

Forth, most real-life recommender systems suffer from bias amplifications
(e.g. sample selection bias, popularity bias). As the training data for models
is usually collected from the logs in the platform, this causes severe bias
amplification. According to previous studies (zhan2022deconfounding), the
recommendation of video recommender systems can be biased towards videos with
longer durations, which verifies the existence of bias amplification.

Given the four issues, we have reviewed existing studies on watch time
prediction. Although both methods tackle some important limitations and
achieve superior performances in this task, none of them have fully considered
all the four issues. The analyses on two state-of-the-art methods on watch
time prediction (WLR (covington2016deep) and D2Q (zhan2022deconfounding)) can
be found in Table 1.

In WLR, training samples are either positive (the video impression was
clicked) or negative (the impression was un-clicked). And the watch-time
prediction is treated as a binary classification problem, where the positive
samples are weighted with watch time in the cross-entropy loss. Therefore the
odds learned by the classifier equals to the expected watch-time
approximately. Despite its simplicity and effectiveness, WLR still has some
limitations that prevent its direct application in full-sceen video
recommender systems (zhan2022deconfounding), where all video impressions are
watched. Therefore, WLR has to be trained with artificially designated
positive and negative samples and weights, which may cause a poor
approximation of watch-time. Meanwhile the bias amplification effect may get
even more severe in WLR as more weights are assigned to the videos of longer
duration. D2Q (zhan2022deconfounding) alleviates the duration bias by
splitting videos into different groups according to their durations and models
watch time with traditional regression models in each group. Therefore the
ordinal relationships and conditional dependence in watch time values are
neglected; Moreover, both WLR and D2Q treats watch time prediction as a point
estimation problem, thus the uncertainty of predictions is ignored.

$n_{0}:\newline
[0.0,1.0]$$n_{1}:[0.0,0.6]$$n_{3}:[0.0,0.2]$$n_{4}:[0.2,0.6]$$n_{7}:[0.2,0.4]$$n_{8}:[0.4,0.6]$$n_{2}:[0.6,1.0]$$n_{5}:[0.6,0.8]$$n_{6}:[0.8,1.0]$
Figure 1. An example of a tree in TPM. Each node is assigned with a classifier
deciding which interval the prediction of watch time belongs to. And each edge
represents a result of the decision from parent node, which serves as a
condition for current node. For instance, the node $(0.2,0.6)$ represents a
decision whether the prediction of watch time should go to $(0.2,0.4)$ or
$(0.4,0.6)$ given the condition that it belongs to $(0.2,0.6)$

Considering the four aforementioned issues, we propose a new framework TPM for
watch time prediction that solves them simultaneously. Specifically, watch
time is split into multiple ordinal intervals, and watch time prediction is
equivalent to a searching problem deciding which interval the predicted watch
time belongs to. The searching process is modeled as a collection of decision
making problems organized in a tree structure. Each intermediate node in the
tree represents a decision problem which is assigned with a corresponding
classifier. Meanwhile each leaf node represents one of the ordinal intervals
which is assigned with an expected value of watch time. Each edge represents a
possible result of the decision and leads to the next decision. Then the
result from upper level becomes the condition for the decisions in the current
nodes. Therefore the path from root node to a certain leaf node corresponds to
a searching trajectory consisting of a series of decisions. We present a
running example in Fig 1 to illustrate the framework.

We explain how TPM solves the four issues in detail as follows:

  * •

First, we introduce ordinal ranks into the approximation of watch time. The
regression task is decomposed into multiple binary classification tasks whose
labels are associated with the ordinal ranks. This approximation makes use of
both the continuity of watch time and the ordinal relations between the ranks.

  * •

Second, we introduce the conditional dependence into TPM. Each task of a child
node is dependent upon the task from the parent node. In this way, the
conditional dependency between the multiple decomposed classification tasks is
explicitly modeled. Notice that there are multiple decomposition patterns of
the prediction task, we can encode arbitrary conditional dependence into the
model.

  * •

Third, to enable a robust framework for watch time prediction, we introduce
model uncertainty into the objective function. Thanks to the splitting of
watch time into ranks, the predicted watch time can be seen as a random
variable drawn from multinomial distribution, and we can compute the variance
of predicted watch time explicitly, which can be seen as a metric of model
uncertainty. Therefore it is introduced into the objective function so that
TPM gets an accurate estimation of watch time with high confidence.

  * •

Forth, we conduct a causal analysis on the confounding effect of biases and
show that conducting backdoor adjustment is equivalent to a specific
decomposition with multiple classification tasks. We show that this method
applies to different kinds of biases and D2Q (zhan2022deconfounding) can be
seen as one of the special cases of TPM.

Notice that the structure of TPM is similar to decision trees, we argue that
it possesses significant differences with conventional tree models for
regression: First, TPM uses the tree structure to decompose a pure regression
problem into a series of classification problems, thus the decomposition is
conducted on the label space (e.g. the partition of watch time intervals);
while tree models partition the feature space into sub-spaces for prediction.
Second, the tree-alike decomposition in TPM assigns each node with a
corresponding classifier (like neural networks) for decision making; while
conventional tree models directly learn a feature partition rule for labeling.

The contributions of this paper are summarized as follows:

  * •

We propose a new framework TPM for watch-time prediction, transforming
regression problem into a series of multiple classification problems; And both
ordinal relation and conditional dependence are introduced into TPM, which are
neglected in previous studies;

  * •

To enhance the robustness of TPM, we model watch time as a random variable
drawn from a multinomial distribution and introduce variance into the
objective function for uncertainty modeling in training.

  * •

To the best of our knowledge, TPM is the first approach that considers ordinal
relationships, conditional dependence and model uncertainty in watch time
prediction.

  * •

We show that TPM can be easily adapted for debiased recommendation, which
enables easy backdoor adjustment for alleviating bias amplifications;

  * •

Extensive experiments are conducted in both offline datasets and a real-world
short video recommender system i.e. Kuaishou APP. TPM has achieved an
improvement of more than $0.2\%$ in watch time and a significant drop of
negative feedbacks. And TPM has already been deployed online, serving more
than $300$ million DAUs.

##  2\. Related Work

###  2.1. Watch-time Prediction

Watch-time prediction is one of the most-concerned problems in industrial
recommender systems (especially for short-video and movie recommender
systems). However, to the best of our knowledge, only few papers can be
found(covington2016deep; zhan2022deconfounding) in this area. The first work
(covington2016deep) focused on video recommendation in Youtube and proposed
the Weighted Logistic Regression (i.e. WLR) method for watch-time prediction.
And WLR has become a state-of-the-art method in related applications. However,
this method can not be directly applied to full-screen video recommender
systems and WLR may suffer from severe bias issues due to its weighting
mechanism. D2Q (zhan2022deconfounding) alleviates the duration bias by
conducting backdoor adjustments and models watch time with direct watch-time
quantile regression. However the ordinal relationships and dependency between
quantiles are ignored in this method. Moreover, as both methods model watch
time with point estimation, the uncertainty of predictions have not been
considered.

###  2.2. Ordinal regression

Ordinal regression is a technique for predicting ordinal labels, i.e. the
relative order of labels is important. Its application can be found in age
estimation (niu2016ordinal), monocular depth estimation (fu2018deep), head-
pose estimation (hsu2018quatnet). Despite the wide applications of ordinal
regression, it has not been applied on watch time prediction tasks.

Most ordinal regression algorithms are modified from classical classification
algorithms. For instance, SVM has been incorporated with multiple thresholds
and applied on visual classification (NIPS2002_51de85dd); another example is a
combination with Online perceptron algorithm (NIPS2001_5531a583) which is used
for rating prediction. Moreover, ordering information in class attributes is
exploited for transforming ordinal regression into multiple classification
problems (frank2001simple). It is worth noticing that decision tree is used in
this work (frank2001simple). However, the binary classification problems in
ordinal regression are not conditional dependent as those in TPM, which is a
fundamental difference.

###  2.3. Tree based neural networks for recommendation

Tree based models and neural networks are powerful models in various machine
learning applications, especially in recommender systems. Tree based methods
like LambdaMART(burges2010ranknet) are fairly competitive in ranking tasks.
Meanwhile, neural networks achieve state-of-the-art performances in leveraging
sparse and complex features. However, few efforts have been made on combing
the advantages of both methods. An early study (li2019combining) attempts to
combine decision trees with neural networks for searching. Two models are
combined with ensemble learning techniques (e.g. linear combination and
stacking) and the combined model achieves superior performances over single
models. Moreover, tree models have been used for enhancing the embedding
models for explainable recommendation (wang2018tem).

TDM (zhu2018learning; zhu2019joint) is an example of combining tree-based
models and neural networks for recommendation. The idea of TDM is to organize
the candidate retrieval process as a searching process along the tree, so that
most preferred candidates can be retrieved with arbitrary complex models in
logarithmic complexity. TPM differs from TDM in several important aspects:
first, TPM is designed for watch time prediction given users and corresponding
videos, while TDM aims to retrieve relevant candidates from a huge corpus;
second, TPM uses tree structure for problem decomposition while TDM utilizes
tree structure for corpus partition; third, TPM traverse the tree to predict
expected watch time while TDM uses beam search to search for the target leaf
nodes.

###  2.4. Debiased recommendation

Many efforts have been drawn to address the biases in recommendation. Previous
studies on this topic can be roughly divided into three categories:

  * •

inverse propensity scoring: it first computes the propensity score of samples
based on certain assumptions and then the samples are weighted with inverse
propensity scores in the objective function. For example, exposure propensity
has been utilized for solving miss-not-at-random problem(saito2020unbiased);
However the performances of this method are sometimes unstable due to the high
variance of the estimated propensity. And this can be alleviated by propensity
clipping or doubly-robust learning (wang2019doubly) via data imputation.

  * •

causal embedding: this method (bonner2018causal) decomposes related embeddings
into unbiased component and biased component. Both components are used at the
training stage and biased component is discarded at inference stage to get an
unbiased prediction.

  * •

causal intervention: the causes of biases are introduced into the method and
interventions are conducted to eliminate their affections on recommendation.
Randomization and backdoor adjustment are two representative methods for
causal intervention. However, it is costly to conduct randomized experiments
on real-life recommender systems, and backdoor adjustment
(zhan2022deconfounding; wang2021deconfounded) is preferred in practical
scenarios.

Causal intervention method has been used in watch time prediction for
deconfounding duration bias. And we show that backdoor adjustment can be
seamlessly incorporated into TPM and this method applies to other confounding
factors.

##  3\. Tree based Progressive regression Model

We first provide a general formulation for Tree based Progressive regression
Model and introduce how watch time prediction is decomposed into several
conditional dependent classification problems; Then we present the details of
uncertainty modeling for TPM; After that we show that the backdoor adjustment
naturally fits in TPM and how biase amplifications are alleviated in detail.
Before we go deep into the details of the formulation, we provide a list of
notations in Table 2.

Table 2. Notations Notation | Meaning
---|---
$T$ | Watch time
$X$ | Features of user and video
$\mathcal{T}$ | Decomposition tree in TPM
$N_{\mathcal{T}}:\\{n_{i}\\}$ | set of nodes in $\mathcal{T}$
$L_{\mathcal{T}}:\\{l_{k}\\}$ | set of leaf nodes in $\mathcal{T}$
$\mathcal{M}_{i}$ | the classifier assigned to $n_{i}$
$D$ | confounding factor in causal graph
$\phi_{l_{k}}$ | the path from root to leaf node $l_{k}$
$n_{\phi_{l_{k}}(i)}$ | the node at level i along path $\phi_{l_{k}}$
$d(l_{k})$ | the depth of leaf node $l_{k}$
$\gamma_{i},\forall i$ | ordinal ranks of watch time

###  3.1. Formulation for TPM

Given a training instance $(X,T)$ where $X$ represents the features of user
and video, and $T$ represents the watch time, the purpose of watch time
prediction is to find a model $\mathcal{M}$ so that the prediction
$\mathcal{M}(X)$ is close to $T$ under certain metrics.

Instead of treating the problem as direct regression, we first quantize the
scale into ordinal ranks:
$\\{\gamma_{0}\leq\gamma_{1},\ldots,\gamma_{k},\ldots,\leq\gamma_{m}\\}$, and
then cast watch time prediction as the estimation of expected ordinal rank.
And the estimation from ordinal ranks is similar to a searching process with
iterative comparisons.

For instance, if we conduct a linear search along the ranks, the searching
process is as follows: we first decide if $t(x)\leq\gamma_{0}$ or not: if
$t(x)\leq\gamma_{0}$, then the predicted ordinal rank is $\gamma_{0}$;
otherwise we continue by deciding if $t(x)\leq\gamma_{1}$, if it is true, then
the rank is $\gamma_{1}$, otherwise the process continues. And the process
goes on until an interval is finally found.

And if we conduct a binary search, the searching process then becomes: we
first decide if $t(x)\leq\gamma_{m/2}$: if it is true, then $t(x)$ falls into
the set $\\{\gamma_{0},\gamma_{1},\ldots,\gamma_{m/2}\\}$; otherwise it
belongs to a rank from $\\{\gamma_{m/2+1},\ldots,\gamma_{m}\\}$. The process
continues and the searching space is narrowed down to a certain rank.

$n_{0}:[0.0,1.0]$$n_{1}:[0.0,0.25]$$n_{2}:[0.25,1.0]$$n_{3}:[0.25,0.5]$$n_{4}:[0.5,1.0]$$n_{5}:[0.5,0.75]$$n_{6}:[0.75,1.0]$$\mathcal{M}_{0}:p(T\in
n_{2}|n_{0})$$\mathcal{M}_{2}:p(T\in n_{4}|T\in n_{2})$$\mathcal{M}_{4}:p(T\in
n_{6}|T\in n_{4})$ (a) An unbalanced binary tree for linear search

$n_{0}:$$[0.0,1.0]$$n_{1}:$$[0.0,0.5]$$n_{3}:$$[0.0,0.25]$$n_{4}:$$[0.25,0.5]$$n_{2}:$$[0.5,1.0]$$n_{5}:$$[0.5,0.75]$$n_{6}:$$[0.75,1.0]$$\mathcal{M}_{0}:p(T\in
n_{2}|T\in n_{0})$$\mathcal{M}_{1}:p(T\in n_{4}|T\in n_{1})$ (b) A balanced
binary tree for binary search

Figure 2. Two examples of decomposition trees in TPM

Notice that each searching process consists of a sequence of decisions, we
propose to fit the watch time prediction model into a searching process from
root to a leaf node along the tree. (see Fig. 2): for linear search case, the
tree is an unbalanced binary tree; for binary search case, it is a balanced
binary tree.

Therefore we propose a Tree based Progressive Model (TPM) for watch time
prediction. The model consists of a tree for problem decomposition
$\mathcal{T}$ and corresponding classification models
$\\{\mathcal{M}_{i},i\in\\{0,1,\ldots,|N_{\mathcal{T}}|-|L_{\mathcal{T}}|\\}\\}$
where $N_{\mathcal{T}}$ is the set of nodes in $\mathcal{T}$ and
$L_{\mathcal{T}}$ is the set of leaf nodes in $\mathcal{T}$(see Fig 2 as an
overview).

The tree in TPM consists of a set of nodes:
$\mathcal{T(X)}=\\{N_{\mathcal{T}}\bigcup L_{\mathcal{T}}\\}$. Each non-root
node represents an interval consisting of consecutive ordinal ranks, i.e.
$n_{i}:[\gamma_{s_{i}},\gamma_{e_{i}}],e_{i}-s_{i}>1$. Without loss of
generality, root node is assumed to be the full space:
$T\in[\gamma_{0},\gamma_{m}]$. Each leaf node is assigned with a specific
ordinal rank i.e. $l_{i}:[\gamma_{i},\gamma_{i+1}]$. And the subspace of a
parent node is the union of sub-spaces of its children. And a path from root
to a leaf $l_{k}$ is denoted as an ordered node set
$\phi_{l_{k}}=\\{\hat{n}_{\phi_{l_{k}}}(0),\ldots,\hat{n}_{\phi_{l_{k}}}(d(l_{k}))\\}$,
where $\hat{n}_{\phi_{l_{k}}}(i)$ is the node at level $i$ along the path
$\phi_{l_{k}}$ and $d(l_{k})$ is the depth of leaf node $l_{k}$. The following
equation always holds:

$\hat{n}_{\phi_{l_{k}}}(i)\subseteq\hat{n}_{\phi_{l_{k}}}(j),\forall i\geqq j$

In TPM, each non-leaf node is assigned with a classifier, and its outputs
indicate the conditional probabilities that watch time belongs to the
corresponding ordinal ranks of the child nodes given the probability from its
parent’s output. Therefore given an instance $X$ and a tree $\mathcal{T}$, its
predicted watch time $T$ follows a multinomial distribution as follows:

(1) $\displaystyle p(T\in l_{k}|X,\mathcal{T})$
$\displaystyle=p(T\in\hat{n}_{\phi_{l_{k}}}(i),\forall i\leq
d(l_{k})|X,\mathcal{T})$
$\displaystyle=p(T\in\hat{n}_{\phi_{l_{k}}}(d(l_{k}))|X,\mathcal{T},T\in\hat{n}_{\phi_{l_{k}}}(i),\forall
i<d(l_{k}))$ $\displaystyle\cdot p(T\in\hat{n}_{\phi_{l_{k}}}(i),\forall
i<d(l_{k})|X,\mathcal{T})$ $\displaystyle=\prod_{1\leq i\leq
d(l_{k})}p(T\in\hat{n}_{\phi_{l_{k}}}(i)|X,\mathcal{T},T\in\hat{n}_{\phi_{l_{k}}}(i-1))$

where
$p(T\in\hat{n}_{\phi_{l_{k}}}(i)|X,\mathcal{T},T\in\hat{n}_{\phi_{l_{k}}}(i-1))$
is parameterized by the classifier $\mathcal{M}_{\hat{n}_{\phi_{l_{k}}}(i-1)}$
assigned to node $\hat{n}_{\phi_{l_{k}}}(i-1)$. Based on the derivations, the
expectation of watch time given a tree $\mathcal{T}$ can be computed as
follows:

(2) $E(T|X,\mathcal{T})=\sum_{l_{k}\in L_{\mathcal{T}}}E(T|T\in
l_{k},X,\mathcal{T})p(T\in l_{k}|X,\mathcal{T})$

Notice that the expectation involves the term $E(T|T\in l_{k},X,\mathcal{T})$,
this can be estimated by any predictive model. In this paper, we employ a
simple method for estimation:

$E(T|T\in l_{k},X,\mathcal{T})=(\gamma_{l_{k}}+\gamma_{l_{k+1}})/2$

By building multiple trees for watch time estimation, the expectation of watch
time can be computed by incorporating the distribution of trees as a prior:

(3)
$E(T|X)=\sum_{\mathcal{T}}E(T|X,\mathcal{T})p(\mathcal{T}|X)=\sum_{\mathcal{T}}E(T|X,\mathcal{T})p(\mathcal{T})$

Despite that TPM allows a bagging scheme for prediction as Eqn. 3, we restrict
the number of trees to be one for simplicity.

###  3.2. Tree Construction

Notice that there is no limit on the type of trees in TPM, the structure of
the trees can be designed according to the tasks and dataset. In TPM, each
tree corresponds to a decomposition of the ordinal ranks and each non-leaf
node inside the tree corresponds to a classifier. As revealed in previous
studies (kaur2019systematic; johnson2019survey), label imbalance adds
difficulty to the predictive modeling, we try to construct the tree with
balanced label distribution for each node.

Therefore, we compute the quantiles of watch time and set them as the ordinal
ranks for discretion. Then we split the ordinal ranks into halves iteratively
and set them to the leaf nodes of a complete binary tree. The tree is
constructed by merging two child nodes to a parent node repeatedly. Therefore
the classifier for each node is a binary classifier and the label distribution
is balanced for each classifier. For example, when the scale of watch time is
split into 4 intervals uniformly, the constructed tree corresponds to the
balanced binary tree in Fig 2.

###  3.3. Uncertainty Modeling

Previous methods (covington2016deep; zhan2022deconfounding) focus on modeling
watch time with point estimation, however it is unknown how much confidence
should be placed on the predictions. And we show that TPM does not only model
the error of expected watch time but also attempts to minimize the uncertainty
of its predictions.

123456780.10.20.3$\gamma$$P(T=\gamma|X,\mathcal{T})$ (a) Prediction from
$\mathcal{M}_{a}$

123456780.10.20.3$\gamma$$P(T=\gamma|X,\mathcal{T})$ (b) Prediction from
$\mathcal{M}_{b}$

Figure 3. An example of watch time predictions as a distribution

Notice that given a tree for problem decomposition, TPM predicts the
probabilities that watch time belongs to the ordinal ranks respectively.
Therefore watch time becomes a random variable following a multinomial
distribution of $p(T\in l_{k}|X,\mathcal{T}),\forall l_{k}\in
L_{\mathcal{T}}$. This property of predicting watch time with a distribution
is very helpful, as it enables an approximate estimation of watch time
variance:

(4) $Var(T|X,\mathcal{T})=E(T^{2}|X,\mathcal{T})-E(T|X,\mathcal{T})^{2}$

Notice that $p(T\in l_{k}|X,\mathcal{T})$ can be computed with Eqn 1, the
variances can be computed easily under the assumption $T\sim p(T\in
l_{k}|X,\mathcal{T}),\forall l_{k}\in L_{\mathcal{T}}$.

A simple example is depicted in Fig 3 to illustrate the idea: assuming that
the scale of watch time is split into eight ordinal ranks, the predictions of
two models $\mathcal{M}_{a}$ and $\mathcal{M}_{b}$ have same expectation of
watch time: $E(T)=4.5$. However, it is easy to verify that these two
predictions have distinct variances:
$Var_{\mathcal{M}_{a}}(T)>Var_{\mathcal{M}_{b}}(T)$. This indicates that
$\mathcal{M}_{a}$ is more uncertain about its prediction. And it is expected
that a model is able to get correct estimation of watch time with high
certainty. Therefore we explicitly add variance of predicted watch time into
the objective function of TPM.

###  3.4. Training with TPM

Now we present the training process of TPM. Given a training sample $(X,T)$
and the tree $\mathcal{T}$ in TPM, we first identify the ordinal rank of the
sample and the corresponding leaf node $l_{k}(T)\in L_{\mathcal{T}}$. Then the
path from root to the leaf node is identified and the sample is associated to
the classifiers along the path.

Each classifier takes $X$ as input, and the label is identified with the child
node along the path. In this paper, $\mathcal{T}$ is a balanced binary tree,
and each non-leaf node is assigned with a binary classification task. For each
classifier, samples belonging to the right-hand child node is seen as a
positive sample.

Consider an example in Fig 2, given a sample $(X,T)$ and $T=0.8$, this sample
is associated to classifiers $\mathcal{M}_{0}$ and $\mathcal{M}_{2}$, and for
both classifiers, it is a positive sample.

Algorithm 1 Tree based Progressive-regression Model:

1: Input: Training data: $(X_{i},T_{i}),\forall i$, A decomposition tree:
$\mathcal{T}$;

2: Output: The classifiers of nodes $\mathcal{M}_{j},\forall j\in
N_{\mathcal{T}}\setminus L_{\mathcal{T}}$;

3: for each batch do

4: Assign each training sample into the leaf nodes of $\mathcal{T}$ by fitting
$T_{i}$ to the ordinal ranks of $l_{k},\forall k$;

5: Assign $(X_{i},T_{i})$ to the classifiers along corresponding path $l_{k}$;

6: Compute the loglikelihood of $(X_{i},T_{i})$ belonging to path $l_{k}$ as
Eqn. 1;

7: Compute $E(T|X,\mathcal{T})$ and $Var(T|X,\mathcal{T})$ as Eqn. 2 and Eqn.
4;

8: Compute the final objective function $\mathcal{L}(T,X,\mathcal{T})$ as Eqn.
5;

9: Update $\mathcal{M}_{j},\forall j$ by minimizing
$\mathcal{L}(T,X,\mathcal{T})$;

10: end for

The objective function of TPM consists of three components:

  * •

Classification error of classifiers along the path: TPM attempts to maximize
the likelihood w.r.t. $p(\hat{T}\in l_{k}(T)|X,\mathcal{T})$, where $\hat{T}$
is the predicted watch.

  * •

Predicition Variance: $Var(\hat{T}|X,\mathcal{T})$. For easier optimization,
we use standard deviation in the loss function:
$Var(\hat{T}|X,\mathcal{T})^{0.5}$

  * •

Regression error: a loss function evaluating the difference between the final
prediction of watch time and groundtruth: $|T-E(\hat{T})|$.

The final objective function is a weighted sum of the three components:

(5) $max.\mathcal{L}=\alpha_{1}log(p(\hat{T}\in
l_{k}(T)|X,\mathcal{T}))-\alpha_{2}Var(\hat{T}|X,\mathcal{T})^{0.5}-\alpha_{3}\|E(\hat{T})-T\|_{2}$

The training process is illustrated in Alg. 1.

###  3.5. Combined with Backdoor Adjustment

Now we present how backdoor adjustment seamlessly adapts to TPM for debiasing
recommendation. First, we present the causal graph in Fig 4 to illustrate the
bias effects to watch time prediction. Denote the confounding factor as $D$,
the feature representations as $X$ and the watch time as $T$, the effect
between variables are reflected in the edges:

  * •

$D\rightarrow T$: Confounding factors affect watch time directly. This should
be captured by models for an accurate estimation (wang2021deconfounded;
zhan2022deconfounding).

  * •

$D\rightarrow X$: Confounding factors affect feature representations
implicitly. This should be eliminated so that bias amplifications can be
avoided.

  * •

$X\rightarrow T$: Feature representations directly affect watch time,
including the effects of user preferences and video contents, etc.

Figure 4. Causal Graph illustrating the confounding effect in watch time
prediction. $D$, $X$, $T$ represent the confounding factor, the input features
and watch time respectively.

.

$n_{0}:$$n_{1}\cup n_{2}$$n_{1}:$$n_{3}\cup
n_{4}$$n_{3}:$$(T=1,D=1)$$(T=2,D=2)$$n_{4}:$$(T=2,D=1)$$(T=4,D=2)$$n_{2}:$$n_{5}\cup
n_{6}$$n_{5}:$$(T=3,D=1)$$(T=6,D=2)$$n_{6}:$$(T=4,D=1)$$(T=8,D=2)$$\mathcal{M}_{0}:p(T\in
n_{2}|T\in n_{0},D)$$\mathcal{M}_{1}:p(T\in n_{4}|T\in n_{1},D)$ Figure 5. An
examples of decomposition tree in TPM when backdoor adjustment is conducted.
Each node is asscoiated with both watch time ($T$) and the condounding factor
$D$.

Without loss of generality, we assume $D$ follows a multinomial distribution:
$D\sim P(D=d),\forall d$. The deconfounded estimation of watch time can be
derived by blocking the edge $D\rightarrow X$ (wang2021deconfounded;
zhan2022deconfounding):

(6) $\displaystyle E(T|do(X))$
$\displaystyle=\sum_{d}P(D=d|do(X))E(T|do(X),D=d)$ (7)
$\displaystyle=\sum_{d}P(D=d|X)E(T|X,D=d)$ (8)
$\displaystyle=\sum_{d}P(D=d)E(T|X,D=d)$

Notice that

$E(T|X)=\sum_{\mathcal{T}}E(T|X,\mathcal{T})p(\mathcal{T}|X)$
$E(T|X,D=d)=\sum_{\mathcal{T}}E(T|X,\mathcal{T},D=d)p(\mathcal{T}|X,D=d)$

we have:

(9) $\displaystyle E(T|do(X))$ (10) $\displaystyle=\sum_{d}P(D=d)E(T|X,D=d)$
(11)
$\displaystyle=\sum_{d}P(D=d)\sum_{\mathcal{T}}E(T|X,\mathcal{T},D=d)p(\mathcal{T}|X,D=d)$
(12)
$\displaystyle=\sum_{\mathcal{T}}\sum_{d}P(D=d)E(T|X,\mathcal{T},D=d)p(\mathcal{T}|X,D=d)$

Specifically, this indicates that we can conduct backdoor adjustment by
constructing trees according to the distribution of confounding factors and
train the classifiers by splitting samples to the corresponding trees. This
can be achieved by splitting the scale of confounding factors into groups and
construct the tree accordingly. Meanwhile, the training data should be split
according to the groups and the classifiers in each group is trained with the
split data respectively (See Fig. 5 for example).

Specifically, we can inject $D$ into TPM as follows:

(13) $\displaystyle\mathcal{L}_{1}$ $\displaystyle=p(T\in
l_{k}|X,D,\mathcal{T})$ $\displaystyle=\prod_{1\leq i\leq
d(l_{k})}p(T\in\hat{n}_{\phi_{l_{k}}}(i)|X,D,\mathcal{T},T\in\hat{n}_{\phi_{l_{k}}}(i-1))$
(14) $E(T|X,D,\mathcal{T})=\sum_{l_{k}\in L_{\mathcal{T}}}E(T|T\in
l_{k},X,D,\mathcal{T})p(T\in l_{k}|X,D,\mathcal{T})$ (15)
$\mathcal{L}_{2}=Var(T|X,D,\mathcal{T})=E(T^{2}|X,D,\mathcal{T})-E(T|X,D,\mathcal{T})^{2}$
(16)
$\mathcal{L}(T,X,D,\mathcal{T})=\alpha_{1}\mathcal{L}_{1}-\alpha_{2}\mathcal{L}_{2}-\alpha_{3}\|E(\hat{T})-T\|_{2}$

Algorithm 2 Training TPM with Backdoor Adjustment

1: Input: Training data: $(X_{i},T_{i}),\forall i$, A confounding factor $D$,
A decomposition tree $\mathcal{T}$;

2: Output: The classifiers of nodes $\mathcal{M}_{j},\forall j\in
N_{\mathcal{T}}\setminus L_{\mathcal{T}}$;

3: for each batch do

4: Assign each training sample into the leaf nodes of $\mathcal{T}$ by
matching $(T_{i},D_{i})$ to the ordinal ranks of $l_{k},\forall k$;

5: Assign $(X_{i},D_{i},T_{i})$ to classifiers along corresponding path
$l_{k}$;

6: Compute the loglikelihood of $(X_{i},T_{i})$ belonging to path $l_{k}$ by
adding $D_{i}$ to Eqn. 13

7: Compute $E(T|X,D,\mathcal{T})$ and $Var(T|X,D,\mathcal{T})$ as Eqn. 14 and
Eqn. 15;

8: Compute the final objective function $\mathcal{L}(T,X,\mathcal{T})$ as Eqn.
16;

9: Update $\mathcal{M}_{j},\forall j$ by minimizing
$\mathcal{L}(T,X,D,\mathcal{T})$;

10: end for

The training process is illustrated in Alg. 2.

###  3.6. Model Architecture

Notice that TPM does not limit the architecture of classifiers, any
architecture for binary classifier applies to TPM. Therefore we adopt a
multiple layer perceptron as the backbone structure for the classifiers. The
architecture is presented in Fig 6.

Since each non-leaf node in a tree corresponds to a binary classification
task, a naive design is to build one classifier for each node where the
classifiers are trained independently. However this would cause a considerable
large model size thus does not apply to the real-life environment. Therefore
we design a single model for all classification tasks by sharing parameters of
hidden layers across tasks. Meanwhile, task-specific output layers are
introduced into the network to produce outputs for each node.

Figure 6. Network Architectures of the Classifier in TPM, where $o_{n_{j}}$ is
the output for the task assigned to node $n_{j}$

##  4\. Experiments

We conduct extensive experiments in both offline and online environments to
demonstrate the effectiveness of TPM. Three research questions are
investigated in the experiments:

  * •

First, how does TPM perform in comparison with state-of-the-art methods for
watch time prediction in terms of recommendation accuracy?

  * •

Second, how does TPM perform when combined with backdoor adjustment?

  * •

Third, how do tree construction in TPM and variance modeling affect its
performance?

###  4.1. Experiment Setup

Now we provide an introduction to the experiment setup, including dataset,
methods for comparison and metrics for evaluation.

####  4.1.1. Datasets

We adopt two public datasets Kuaishou (collected from Kuaishou App
111https://kuairec.com/) and CIKM16 (from CIKM16 Cup) for offline experiments.
Note that CIKM16 aims to predict the dwell time for each session in online
search results. We use each item in the session as a single feature for input.
Kuaishou dataset contains $7,176$ users, $10,728$ items, $12,530,806$
impressions; and CIKM16 dataset contains $310,302$ sessions, $122,991$ items
and the average length of each session is $3.981$.

####  4.1.2. Methods

Two state-of-the-art methods for watch time prediction are selected for
comparison, including WLR (Weighted Logistic Regression) and D2Q (Duration-
Deconfounded Quantile). Moreover, ordinal regression is a method for
transforming regression to classification, and it is also selected for
comparison. The details of these methods are presented as follows:

  * •

WLR (Weighted Logistic Regression) (covington2016deep): This method treats
watch time regression as binary classification problem, where impressed and
clicked videos are positive samples and impressed but unclicked ones are
negative samples. The losses of positive samples are weighted with watch time
and the learned odds are used as approximate watch time. However there is no
explicit negative samples in full-screen video streaming apps since all videos
are impressed and played. We follow the implementation in D2Q
(zhan2022deconfounding) by treating short-played samples as negative ones.

  * •

D2Q (Duration-Deconfounded Quantile): This approach first splits samples into
ten groups based on the duration of videos. Then a regression model is trained
to predict the watch time quantile for each group. Finally a prediction of
watch time is retrieved by mapping predicted quantile to the watch time
domain. In our experiments, the duration group number is set to 32, which
achieves the best performances in predictive accuracy.

  * •

OR (Ordinal Regression) (NIPS2001_5531a583): Ordinal regression transforms
labels into K ranks and each rank is assigned with a classifier predicting
whether the prediction is greater than the rank. As no existing studies on
watch time prediction have ever adopted ordinal regression for modeling, we
build a baseline for comparison by applying ordinal regression to watch time
prediction directly. Meanwhile, we introduce deconfounding factors into this
method in a same way with D2Q and TPM, for fair comparison.

  * •

TPM222https://github.com/jackielinxiao/TPM: The proposed approach in this
paper. Since D2Q focuses on duration bias in recommendation, we also set
confounding factor to video duration for comparison.

For fair comparison, the model structures of these approaches are the same
except the output layers and corresponding loss functions.

####  4.1.3. Metrics

As we concern with an accurate prediction of watch time as well as its ranking
capability, we adopt two metrics for evaluation, including MAE (Mean Absolute
Error) and XAUC (zhan2022deconfounding):

  * •

MAE (Mean Average Error): This metric is a typical measurement for evaluating
regression accuracy. Denote the predition as $\hat{y}$ and the true watch time
as $y$,

$MAE=\frac{1}{N}\sum_{i=1}^{N}|\hat{y}_{i}-y|$

  * •

XAUC(zhan2022deconfounding): this metric evaluates if the predictions of two
samples are in the same order with their true watch time. Such pairs are
uniformly sampled and the percentile of samples that are correctly ordered by
predictions is XAUC.

###  4.2. Offline Experiments

####  4.2.1. Comparison with other methods

Table 3. Comparison between TPM and other approaches Approaches | Kuaishou | CIKM16
---|---|---
MAE | XAUC | MAE | XAUC
WLR | 6.047 | 0.525 | 0.998 | 0.672
D2Q | 5.426 | 0.565 | 0.899 | 0.661
OR | 5.321 | 0.558 | 0.918 | 0.664
TPM | $\mathbf{4.741}$ | $\mathbf{0.599}$ | $\mathbf{0.884}$ | $\mathbf{0.676}$

We compare the performances of different approaches and the results are listed
in Table 3. Notice that TPM achieves the superior performances over other
approaches significantly, this verifies the advantage of TPM in predictive
accuracy. The comparison between TPM, OR and D2Q indicates the effectiveness
of introducing ordinal relationships into watch time prediction. Meanwhile the
comparison between TPM and OR further verifies the benefits of modeling
conditional dependence in watch time prediction.

####  4.2.2. Ablation Studies

Table 4. TPM with different components on KuaiRec Approaches | Watch Time | Watch Time Ratio
---|---|---
MAE | XAUC | MAE | XAUC
TPM | 4.741 | 0.599 | 0.467 | 0.712
TPM w.t.o. mse | 4.887 | 0.595 | 0.482 | 0.711
TPM w.t.o. var | 4.875 | 0.592 | 0.480 | 0.706
TPM w.t.o. deconfounding | ${4.984}$ | ${0.576}$ | ${0.497}$ | ${0.691}$
OR w.t.o. deconfounding | ${5.312}$ | ${0.549}$ | ${0.517}$ | ${0.673}$

We conduct ablation studies on TPM and the results are listed in Table 4. The
comparison between TPMs with/without mse loss indicate that adding mse loss
helps to improve the metric of MAE without sacrificing too much on the ranking
metric. The comparison between TPMs with/without var loss indicate that adding
variance constraints helps to improve the accuracy. And TPMs with/without
deconfounding factors indicate that video watching behaviors are indeed easily
affected by the factors, and TPM without deconfounding factors is still
competitive because of its considering of the ordinal relationships, the
conditional dependency and the variances.

481632$0.5$$0.52$$0.54$$0.56$$0.58$$0.6$$\\#$Duration
GroupXAUCTPMWLR481632$4$$4.5$$5$$5.5$$6$$\\#$ Duration GroupMAETPMWLR Figure
7. The performances of TPM with various numbers of groups when the numer of
leafs is 32 481632$0.5$$0.52$$0.54$$0.56$$0.58$$0.6$$\\#$Nodes in
TPMXAUCTPMWLR481632$4$$4.5$$5$$5.5$$6$$6.5$$\\#$ nodes in
$\mathcal{T}$MAETPMWLR Figure 8. The performances of TPM with various numbers
of nodes in the decomposition tree.

We alter the number of duration groups and the performances of TPM are
depicted in Fig. 7. And the results indicate that splitting samples by
duration indeed helps and TPM can seamlessly accommodate with backdoor
adjustments. We also conduct experiments of TPM with various number of nodes
in the tree, the results are illustrated in Fig.8. As depicted in the figure,
there is a proper number of nodes for tree construction. This coincides with
the intuition that the tree should be constructed according to the task and
dataset. Meanwhile, to illustrate the effects of variance for uncertainty
modeling, we alter the weights for variance in the loss function and present
the results in Fig. 9. The results reveal that there is a proper weight of
variance which leads to lower uncertainty and satisfactory accuracy.

0.10.51.010.0$0.5$$0.52$$0.54$$0.56$$0.58$$0.6$Weight of uncertainty
lossXAUCTPM0.10.51.010.0$0$$0.02$$0.04$$0.06$$0.08$$0.1$Weight of uncertainty
lossSqrt of VarianceTPM Figure 9. The performances of TPM with different
uncertainty weight in training when the num of leafs is 32.

###  4.3. Online Experiments

We also conduct online A/B experiments on a real-world short-video recommender
system in KuaiShou APP. As D2Q is a state-of-the-art method for watch time
prediction, it is adopted as a baseline for comparison. For TPM, the
decomposition tree is design as a complete binary tree with 32 leaf nodes, and
the number of duration groups is set to 32.

####  4.3.1. Experiment Setup

In online A/B experiments, the traffic is split into ten buckets uniformly.
Two buckets of traffic are assigned to baseline while the other two are
assigned to TPM. As revealed in (zhan2022deconfounding), Kuaishou serves over
320 million users daily and the results collected from $20\%$ of traffic is
very convincing.

The real-life recommender systems are usually complicated. However, most of
the systems follow a two-stage framework where a set of candidate items are
retrieved in the first stage and the top-ranking items are selected from the
candidates in the ranking stage. Watch time prediction serves as one component
in the ranking stage. The items are ranked with multiple predictions
(including watch time predictions) and those with higher watch time
predictions are more likely to be recommended.

####  4.3.2. Experiment Results

The experiments have been launched on the system for 4 days, and the results
are listed in Table 5.

Table 5. Comparison between TPM and baseline online, all values are the relative improvements of TPM over the baseline. Watch Time and Forward are positive metrics where higher values are better; Short View is a negative metric where lower values are better. Meanwhile Forward is a constraint metric, and an experiment with more than $1\%$ drop of constraint metrics is not acceptable. For Online A/B tests, an improvement of $0.1\%$ in watch time is very significant. Days | Watch Time | Forward | Short View
---|---|---|---
Day 1 | $\mathbf{0.246\%}$ | $-0.002\%$ | $\mathbf{-0.312\%}$
Day 2 | $\mathbf{0.210\%}$ | $0.040\%$ | $\mathbf{-0.139\%}$
Day 3 | $\mathbf{0.234\%}$ | $0.001\%$ | $\mathbf{-0.110\%}$
Day 4 | $\mathbf{0.265\%}$ | $0.225\%$ | $\mathbf{-0.220\%}$

The metrics for online experiments include accumulated watch time, forward
counts (forward the video to friends) and short view counts (watch time is
short in respect to the video duration). In online experiments, watch time is
a core metric while forward is a constrained metric. Notice that TPM
outperforms the baseline in watch time related metrics which verifies the
advantage of TPM in predictive accuracy. Moreover the number of negative feed-
backs are significantly lower in TPM. And this coincides with the idea of
modeling uncertainty in TPM, aiming to produce both accurate and confident
predictions. Meanwhile, the differences between TPM and baseline on metrics of
interaction are insignificant, thus can be neglected safely.

##  5\. Conclusion

Watch time prediction is one of the core problems in short-video
recommendation, as its accuracy affects the quality of videos recommended to
users, thus impacting user engagement to the platform. We point out that four
issues should be addressed in a real-world watch time prediction framework:
first, the ordinal differences between watch time values should be considered;
second, the conditional dependence between the video-watching behaviors should
be modeled; third, the uncertainty of predictions should be involved in the
framework; forth, the framework should take bias amplification into
consideration.

To solve these issues simultaneously, we propose TPM (Tree-based Progressive
regression Model) for watch time prediction. We reveal that watch time
prediction can be decomposed into several conditional dependent classification
problems that are organized into a tree structure. Meanwhile, the variance of
watch time predictions is introduced into the objective function as model
uncertainty. And the bias amplification problem is addressed by incorporating
backdoor adjustment into TPM seamlessly.

Extensive offline evaluations and online experiments in real-life recommender
systems have been conducted and the results validate the effectiveness of TPM.
Moreover, TPM has already been deployed in Kuaishou APP, serving over $300$
million users daily.