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+# Massive MIMO Transmission for LEO Satellite Communications
+
+Li You, Ke-Xin Li, Jiaheng Wang, Xiqi Gao, Xiang-Gen Xia, and Björn Ottersten
+
+# Abstract
+
+Low earth orbit (LEO) satellite communications are expected to be incorporated in future wireless networks, in particular 5G and beyond networks, to provide global wireless access with enhanced data rates. Massive multiple-input multiple-output (MIMO) techniques, though widely used in terrestrial communication systems, have not been applied to LEO satellite communication systems. In this paper, we propose a massive MIMO transmission scheme with full frequency reuse (FFR) for LEO satellite communication systems and exploit statistical channel state information (sCSI) to address the difficulty of obtaining instantaneous CSI (iCSI) at the transmitter. We first establish the massive MIMO channel model for LEO satellite communications and simplify the transmission designs via performing Doppler and delay compensations at user terminals (UTs). Then, we develop the low-complexity sCSI based downlink (DL) precoder and uplink (UL) receiver in closed-form, aiming to maximize the average signal-to-leakage-plus-noise ratio (ASLNR) and the average signal-to-interference-plus-noise ratio (ASINR), respectively. It is shown that the DL ASLNRs and UL ASINRs of all UTs reach their upper bounds under some channel condition. Motivated by this, we propose a space angle based user grouping (SAUG) algorithm to schedule the served UTs into different groups, where each group of UTs use the same time and frequency resource. The proposed algorithm is asymptotically optimal in the sense that the lower and upper bounds of the achievable rate coincide when the number of satellite antennas or UT groups is sufficiently large. Numerical results demonstrate that the proposed massive MIMO transmission scheme with FFR significantly enhances the data rate of LEO satellite communication systems. Notably, the proposed sCSI based precoder and receiver achieve the similar performance with the iCSI based ones that are often infeasible in practice.
+
+# Index Terms
+
+LEO satellite, massive MIMO, multibeam satellite, full frequency reuse, statistical CSI, user grouping.
+
+This work will be presented in part at the IEEE International Conference on Communications, Dublin, Ireland, Jun. 2020 [1]. L. You, K.-X. Li, J. Wang, and X. Q. Gao are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China, and also with the Purple Mountain Laboratories, Nanjing 211100, China (e-mail: liyou@seu.edu.cn; likexin3488@seu.edu.cn; jhwang@seu.edu.cn; xqgao@seu.edu.cn). X.-G. Xia is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: xxia@ee.udel.edu). B. Ottersten is with the Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, L-2721 Luxembourg City, Luxembourg (e-mail: bjorn.ottersten@uni.lu). (Corresponding author: Xiqi Gao.)
+
+# I. INTRODUCTION
+
+Satellite communication systems can provide seamless wireless coverage so as to complement and extend terrestrial communication networks and, as in recent standardization endeavors [2], are expected to be incorporated in future wireless networks, in particular 5G and beyond networks. Low earth orbit (LEO) satellite communications, with orbits at altitudes of less than 2000 km, have recently gained broad research interests due to the potential in providing global wireless access with enhanced data rates. Compared with the geostationary earth orbit (GEO) counterpart, LEO satellite communication systems impose much less stringent requirements on, e.g., power consumption and transmission signal delays. Recently, several projects, e.g., OneWeb and SpaceX, on LEO satellite communication systems have been launched [3].
+
+In satellite communication systems, multibeam transmission techniques have been widely adopted to increase transmission data rates. As a well-know multibeam solution, a four-color frequency reuse (FR4) scheme where adjacent beams are allocated with non-overlapping frequency spectrum (or different polarizations) is adopted to mitigate the co-channel inter-beam interference [4], [5]. To further enhance the spectral efficiency of satellite communications, the more aggressive full frequency reuse (FFR) schemes [4]–[8], where frequency resources are reused across neighboring beams, have been considered to increase the total available bandwidth in each beam as that has been done in terrestrial cellular systems. Yet, in FFR the inter-beam interference becomes a critical issue, which has to be properly handled. In general, inter-beam interference management can be performed at either the transmitter via precoding or at the receiver via multi-user detection, similar as in terrestrial cellular communication systems [9]. Compared with non-linear dirty paper coding (DPC) precoding and multi-user detection, in practice linear precoding and detection are more preferred in multibeam satellite communication systems due to their low computational complexity and near-optimal performance [10].
+
+It is worth noting that most of the existing works on downlink (DL) precoding in multibeam satellite communications, e.g., [4], [5], rely on precise instantaneous channel state information (iCSI). However, obtaining iCSI at the transmitter sides of satellite communication systems is usually difficult and even infeasible due to a number of practical factors, especially the long propagation delay between a satellite and user terminals (UTs) as well as the mobility of UTs and satellites. In particular, for time-division duplex (TDD) systems, the coherence time of the channel is shorter than the transmission delay, which makes obtaining accurate iCSI via the UL-
+
+DL reciprocity a mission impossible. On the other hand, in more common frequency-division duplex (FDD) systems, obtaining iCSI at the satellite side requires UL feedback from UTs, which inevitably introduces a great among of training and feedback overhead due to mobility of UTs and more importantly could become outdated as a result of the long propagation delay.
+
+In recent years, massive multiple-input multiple-output (MIMO) transmission, where a large number of antennas are equipped at a base station to serve many UTs, has been applied in terrestrial cellular wireless networks, e.g., 5G [11], [12], as an enabling technology. Massive MIMO can substantially increase available degrees of freedom, enhance spectral efficiency, and achieve high data rates. Motivated by this, we propose to exploit massive MIMO along with FFR for LEO satellite communication systems, where a large number of antennas are equipped at the LEO satellite side. Our focus is particularized on the physical layer transmission design for massive MIMO LEO satellite communication systems. We note that it is not necessary to perform predefined multiple beamforming in fully digital-implemented FFR satellite communication systems. Exploiting massive MIMO for satellite communications with FFR can be seen as a technique without predefined beamforming.
+
+Albeit the existence of a large body of literature on massive MIMO in terrestrial cellular communication systems [12], so far massive MIMO has not been applied to satellite communication systems. The performance of massive MIMO systems relies substantially on the available CSI [13]–[15]. As mentioned above, obtaining accurate DL iCSI at the LEO satellite side is generally difficult and even infeasible due to the long propagation delay and the mobility of satellites and UTs, which makes inapplicable of the existing terrestrial massive MIMO transmission approaches relying on iCSI. Meanwhile, the implementation complexity accompanied with massive MIMO becomes a critical concern in satellite communication systems considering the payload limitation on satellites. Consequently, incorporating massive MIMO into LEO satellite communication systems is still an open and challenging task.
+
+For massive MIMO, obtaining iCSI at the transmitter has been a difficult problem even in terrestrial communication systems, especially in high-mobility scenarios. Compared with iCSI, statistical CSI (sCSI) varies much slower and thus can be relatively easily obtained at both the satellite and the UTs with sufficiently high accuracy. Hence, sCSI based DL precoding has been proposed in terrestrial massive MIMO systems [14]–[16]. For massive MIMO satellite communication systems, it is more practical to use sCSI, which can overcome the difficulty of acquiring iCSI and significantly reduce the computational overhead of satellite payloads via the
+
+much less frequent update of transmission strategies including, e.g., DL precoding, UL receiving, and user grouping.
+
+In this paper, we investigate massive MIMO transmission for LEO satellite communication systems using FFR based on sCSI. In particular, we focus on devising DL precoding, UL receiving, and user grouping utilizing sCSI. While this paper focuses on the LEO satellite communications, the proposed massive MIMO transmission schemes can also be extended to other non-terrestrial communication systems, e.g., GEO satellite communication systems, and high-altitude platform (HAP) communication systems. The major contributions of the current work are summarized as follows:
+
+- We introduce massive MIMO into LEO satellite communication systems using FFR and investigate low-complexity and low-overhead transmission strategies based on sCSI.  
+- We establish the massive MIMO channel model for LEO satellite communications by incorporating the LEO satellite signal propagation properties, and simplify the UL/DL transmission designs via performing Doppler and delay compensations at UTs.  
+- We develop the sCSI based DL precoder and UL receiver in closed-form, aiming to maximize the average signal-to-leakage-plus-noise ratio (ASLNR) and the average signal-to-interference-plus-noise ratio (ASINR), respectively, and theoretically prove that the proposed sCSI based scheme asymptotically approaches the iCSI based one.  
+- We propose a space angle based user grouping (SAUG) algorithm using only the channel space angle information, and show that the proposed algorithm is asymptotically optimal in the sense that the lower and upper bounds of the achievable rate coincide when the number of satellite antennas or UT groups is sufficiently large.  
+- Simulation results demonstrate that the proposed massive MIMO transmission scheme with FFR significantly enhances the data rate of LEO satellite communication systems. Notably, the proposed sCSI based precoder and receiver achieve the similar performance with the iCSI based ones that are often infeasible in practice.
+
+The rest of the paper is organized as follows. In Section II, we investigate the channel model and the corresponding transmission signal model for LEO satellite communication systems. Based on the system model, we then investigate the optimal DL and UL transmission strategies for LEO satellite communications in Section III. In Section IV, we further investigate user grouping. We present the numerical results in Section V and conclude the paper in Section VI.
+
+TABLEI VARIABLE LIST  
+
+<table><tr><td>Notation</td><td>Definition</td></tr><tr><td>Mx, My</td><td>Numbers of antennas of UPA at x- and y- axes</td></tr><tr><td>gdlk,p, gulk,p</td><td>Complex channel gains for DL and UL</td></tr><tr><td>Pk</td><td>Number of multipaths</td></tr><tr><td>νk,p, νsatk,p, νsatk,p, νutk,p</td><td>Doppler frequencies</td></tr><tr><td>τk,p, τmink,p, τmaxk,p, τutk,p</td><td>Propagation delays</td></tr><tr><td>vsk,p, vxk,p, vyk,p, vk</td><td>DL array response vectors</td></tr><tr><td>uk, uxk,p, uyk,p</td><td>UL array response vectors</td></tr><tr><td>gd(t,f), gul(k,t,f)</td><td>DL and UL channel vectors</td></tr><tr><td>gd(t,f), gul(k,t,f)</td><td>Complex channel gains after compensation</td></tr><tr><td>θxk,p, θyk,p, θxk,p, θyk,p</td><td>Angles</td></tr><tr><td>φxk,p, φyk,p, φxk,p, φyk,p</td><td>Space angles</td></tr><tr><td>γk</td><td>Channel power</td></tr><tr><td>κk</td><td>Rician factor</td></tr><tr><td>Nus, Ncp</td><td>Numbers of subcarriers and CP</td></tr><tr><td>Ts</td><td>System sampling interval</td></tr><tr><td>Tus, Tcp</td><td>Lengths of OFDM symbol and CP</td></tr><tr><td>gd(k,ℓ,n, gulk,ℓ,n)</td><td>Effective DL and UL frequency domain channel vectors after compensation</td></tr><tr><td>gd(k,ℓ,n, gulk,ℓ,n)</td><td>Effective DL and UL frequency domain channel gains after compensation</td></tr><tr><td>gd(k, gulk,p)</td><td>DL and UL channel vectors</td></tr><tr><td>qdlk,p, qulk,p</td><td>DL and UL transmit power</td></tr><tr><td>bk, wk</td><td>Normalized DL precoder and UL receiver</td></tr><tr><td>ASLNRk, ASINRk</td><td>DL ASLNR and UL ASINR</td></tr><tr><td>Gx, Gy</td><td>Number of groups at x- and y- axes</td></tr><tr><td>Δx, Δy</td><td>Lengths of space angle interval at x- and y- axes</td></tr><tr><td>A(m,n)(g,r)</td><td>Space angle interval</td></tr><tr><td>K(g,r)</td><td>Set of UTs in group (g,r)</td></tr><tr><td>Rdl, Rdil, Rdlb</td><td>DL ergodic rate and its upper bound, lower bound</td></tr></table>
+
+The major variables adopted in the paper is listed in Table I for ease of reference.
+
+# II. SYSTEM MODEL
+
+# A. System Setup
+
+Consider a LEO satellite communication system where a satellite provides services to a number of single-antenna UTs simultaneously. The satellite is equipped with a uniform planar array (UPA) composed of  $M = M_{\mathrm{x}}M_{\mathrm{y}}$  antennas where  $M_{\mathrm{x}}$  and  $M_{\mathrm{y}}$  are the numbers of antennas on the x- and y-axes, respectively. Assume without loss of generality that the antennas are separated by one-half wavelength in both the x- and y-axes, and both  $M_{\mathrm{x}}$  and  $M_{\mathrm{y}}$  are even. The system setup is illustrated in Fig. 1.
+
+![](images/25389a2534b054e8efbb58c89b31c8cfd1586b0dff796ed2177895abc0316563.jpg)  
+Fig. 1. Illustration of the LEO satellite communication system setup.
+
+# B. DL Channel Model
+
+As different UTs are usually spatially separated by a few wavelengths, it is reasonable to assume that the channel realizations between the satellite and different UTs are uncorrelated [17]. We focus on investigating the DL channel between the satellite and UT  $k$ . Using a ray-tracing based channel modeling approach, the complex baseband DL space domain channel response between the LEO satellite and UT  $k$  at instant  $t$  and frequency  $f$  can be represented by [18]–[20]
+
+$$
+\mathbf {g} _ {k} ^ {\mathrm {d l}} (t, f) = \sum_ {p = 0} ^ {P _ {k} - 1} g _ {k, p} ^ {\mathrm {d l}} \cdot \exp \left\{\bar {\jmath} 2 \pi \left[ t \nu_ {k, p} - f \tau_ {k, p} \right] \right\} \cdot \mathbf {v} _ {k, p} \in \mathbb {C} ^ {M \times 1}, \tag {1}
+$$
+
+where  $\mathbb{C}^{M\times N}$  denotes the  $M\times N$  dimensional complex-valued vector space,  $\bar{j} = \sqrt{-1}$ ,  $P_{k}$  denotes the number of channel propagation paths of UT  $k$ , and  $g_{k,p}^{\mathrm{dl}}$ ,  $\nu_{k,p}$ ,  $\tau_{k,p}$ , and  $\mathbf{v}_{k,p}\in \mathbb{C}^{M\times 1}$  are the complex-valued gain, the Doppler shift, the propagation delay, and the DL array response vector associated with path  $p$  of UT  $k$ , respectively. Note that the channel model adopted in (1) is applicable over the time intervals of interest where the relative positions of the LEO satellite and UT  $k$  do not change significantly, and thus the physical channel parameters,  $P_{k}$ ,  $g_{k,p}^{\mathrm{dl}}$ ,  $\nu_{k,p}$ ,  $\tau_{k,p}$ , and  $\mathbf{v}_{k,p}$ , are assumed to be invariant. When the LEO satellite and/or the UT move over
+
+large distances, the above channel parameters will vary and should be updated accordingly [19]. It is worth mentioning that the ray-tracing based channel model in (1) can be applied to different propagation scenarios, and further analysis of the channel model will depend on the parameter properties in the specific scenario. Hereafter, we detail some propagation characteristics of the LEO satellite channels and their impact on the modeling of the channel parameters in (1).
+
+1) Doppler: For LEO satellite communications, assuming that the scatterers are stationary in the considered interval of interest, then the Doppler shift  $\nu_{k,p}$  associated with propagation path  $p$  of UT  $k$  is mainly composed of two independent Doppler shifts,  $\nu_{k,p}^{\mathrm{sat}}$  and  $\nu_{k,p}^{\mathrm{ut}}$ , that are caused by the motions of the LEO satellite and the UT, respectively [21], [22].
+
+It is worth noting that due to the relatively high altitude of the LEO satellite, the Doppler shifts  $\nu_{k,p}^{\mathrm{sat}}$  caused by the motion of the LEO satellite can be assumed to be identical for different propagation paths  $p$  of the same UT  $k$  [21], [22], and different for different UTs. Thus, for notation simplicity, we omit the path index of the Doppler shift  $\nu_{k,p}^{\mathrm{sat}}$  due to the motion of the LEO satellite and rewrite the Doppler shifts as  $\nu_{k,p}^{\mathrm{sat}} = \nu_k^{\mathrm{sat}}$ . On the other hand, the Doppler shifts  $\nu_{k,p}^{\mathrm{ut}}$  due to the motion of the UT are typically different for different propagation paths, which contribute the Doppler spread of the LEO satellite channels [21], [22]. As the scattering characteristics around the UTs mainly determine the Doppler shifts caused by the movement of the UTs, the modeling of the Doppler spread in LEO satellite communications can be similar to that in the traditional terrestrial cellular communications [22].
+
+2) Delay: Due to the relatively large distance between the LEO satellite and the UTs, the propagation delay  $\tau_{k,p}$  associated with path  $p$  of UT  $k$  exhibits a much larger value than that in terrestrial wireless channels. Denote by  $\tau_k^{\mathrm{min}} = \min_p\{\tau_{k,p}\}$  and  $\tau_k^{\mathrm{max}} = \max_p\{\tau_{k,p}\}$  the minimum and maximum values of the propagation delays of UT  $k$ , respectively. The delay spread of the LEO satellite channels  $\tau_k^{\mathrm{max}} - \tau_k^{\mathrm{min}}$  might be much smaller than that of the terrestrial wireless channels as observed in measurement results [22]–[24]. For notational brevity, we define  $\tau_{k,p}^{\mathrm{ut}} \triangleq \tau_{k,p} - \tau_k^{\mathrm{min}}$ . Note that due to, e.g., the long propagation delays in LEO satellite communications, acquiring reliable iCSI at the transmitter sides is usually infeasible, especially when the UTs are in high mobility. Thus, it is more practical to investigate transmission design with, e.g., sCSI, in LEO satellite communications.
+
+3) Angle: The UPA response vector  $\mathbf{v}_{k,p}$  in (1) can be represented by [25], [26]
+
+$$
+\mathbf {v} _ {k, p} \triangleq \mathbf {v} _ {k, p} ^ {\mathrm {x}} \otimes \mathbf {v} _ {k, p} ^ {\mathrm {y}}
+$$
+
+$$
+= \mathbf {v} _ {\mathrm {x}} \left(\vartheta_ {k, p} ^ {\mathrm {x}}\right) \otimes \mathbf {v} _ {\mathrm {y}} \left(\vartheta_ {k, p} ^ {\mathrm {y}}\right) \in \mathbb {C} ^ {M \times 1}, \tag {2}
+$$
+
+where  $\otimes$  denotes the Kronecker product, and  $\mathbf{v}_{k,p}^{d}$  for  $d\in \mathcal{D}\triangleq \{\mathrm{x},\mathrm{y}\}$  is the array response vector of the angle with respect to the x- or y-axis given by
+
+$$
+\begin{array}{l} \mathbf {v} _ {k, p} ^ {d} \triangleq \mathbf {v} _ {d} \left(\vartheta_ {k, p} ^ {d}\right) \\ = \frac {1}{\sqrt {M _ {d}}} \left[ 1 \exp \left\{- \bar {\jmath} \pi \vartheta_ {k, p} ^ {d} \right\} \dots \exp \left\{- \bar {\jmath} \pi \left(M _ {d} - 1\right) \vartheta_ {k, p} ^ {d} \right\} \right] ^ {T} \in \mathbb {C} ^ {M _ {d} \times 1}, \tag {3} \\ \end{array}
+$$
+
+with the superscript  $(\cdot)^T$  denoting the transpose operation. In (3), the parameters  $\vartheta_{k,p}^{\mathrm{x}}$  and  $\vartheta_{k,p}^{\mathrm{y}}$  are related to the physical angles as  $\vartheta_{k,p}^{\mathrm{x}} = \sin \left(\theta_{k,p}^{\mathrm{y}}\right)\cos \left(\theta_{k,p}^{\mathrm{x}}\right)$  and  $\vartheta_{k,p}^{\mathrm{y}} = \cos \left(\theta_{k,p}^{\mathrm{y}}\right)$  where  $\theta_{k,p}^{\mathrm{x}}$  and  $\theta_{k,p}^{\mathrm{y}}$  are the angles with respect to the x- and y-axes associated with the pth propagation path of UT  $k$ , respectively. For satellite communication channels, the angles of all propagation paths associated with the same UT, can be assumed to be identical due to the relatively high altitude of the satellite compared with that of the scatterers located in the vicinity of the UTs [27], i.e.,  $\vartheta_{k,p}^{d} = \vartheta_{k}^{d}$ . Note that the parameters  $\vartheta_{k}^{d}$  can reflect the propagation properties of the LEO satellite channels in the space domain, and we refer to  $\vartheta_{k}^{d}$  as the space angle parameters. Then, the array response vector can be rewritten as
+
+$$
+\begin{array}{l} \mathbf {v} _ {k, p} = \mathbf {v} _ {k} = \mathbf {v} _ {k} ^ {\mathrm {x}} \otimes \mathbf {v} _ {k} ^ {\mathrm {y}} \\ = \mathbf {v} _ {\mathrm {x}} \left(\vartheta_ {k} ^ {\mathrm {x}}\right) \otimes \mathbf {v} _ {\mathrm {y}} \left(\vartheta_ {k} ^ {\mathrm {y}}\right) \in \mathbb {C} ^ {M \times 1}, \tag {4} \\ \end{array}
+$$
+
+which will be referred to as the DL channel direction vector of UT  $k$  that is associated with the space angles  $\vartheta_{k}^{\mathrm{x}}$  and  $\vartheta_{k}^{\mathrm{y}}$ . Note that when the number of antennas  $M_{d}$  for  $d \in \mathcal{D}$  tends to infinity, we can know from (3) and (4) that the channel direction vectors of different UTs are asymptotically orthogonal, i.e.,
+
+$$
+\lim  _ {M _ {d} \rightarrow \infty} \left(\mathbf {v} _ {k} ^ {d}\right) ^ {H} \mathbf {v} _ {k ^ {\prime}} ^ {d} = \delta \left(k - k ^ {\prime}\right), \tag {5}
+$$
+
+where  $(\cdot)^H$  denotes the conjugate-transpose operation.
+
+Based on the above modeling of the propagation properties of LEO satellite communications, we can rewrite the channel response in (1) as follows
+
+$$
+\mathbf {g} _ {k} ^ {\mathrm {d l}} (t, f) = \exp \left\{\bar {\jmath} 2 \pi \left[ t \nu_ {k} ^ {\mathrm {s a t}} - f \tau_ {k} ^ {\min } \right] \right\} \cdot g _ {k} ^ {\mathrm {d l}} (t, f) \cdot \mathbf {v} _ {k}, \tag {6}
+$$
+
+where  $g_{k}^{\mathrm{dl}}(t,f)$  is the DL channel gain of UT  $k$  given by
+
+$$
+\begin{array}{l} g _ {k} ^ {\mathrm {d l}} (t, f) \triangleq \sum_ {p = 0} ^ {P _ {k} - 1} g _ {k, p} ^ {\mathrm {d l}} \cdot \exp \left\{\jmath 2 \pi \left[ t (\nu_ {k, p} - \nu_ {k} ^ {\mathrm {s a t}}) - f (\tau_ {k, p} - \tau_ {k} ^ {\mathrm {m i n}}) \right] \right\} \\ = \sum_ {p = 0} ^ {P _ {k} - 1} g _ {k, p} ^ {\mathrm {d l}} \cdot \exp \left\{\bar {j} 2 \pi \left[ t \nu_ {k, p} ^ {\mathrm {u t}} - f \tau_ {k, p} ^ {\mathrm {u t}} \right] \right\}, \tag {7} \\ \end{array}
+$$
+
+which will be convenient for derivation of the transmission signal model later.
+
+4) Gain: Note that the statistical properties of the fluctuations of the channel gain  $g_{k}^{\mathrm{dl}}(t,f)$  in LEO satellite communications mainly depend on the propagation environment in which the UT is located. Note that LEO satellite communication systems are usually operated under line-of-sight (LOS) propagations and Rician channel model is widely accepted in LOS satellite communication systems. In this work, we focus on the case where both non-shadowed LOS and non-LOS paths of the LEO satellite channels exist [6]. Then, the channel gain  $g_{k}^{\mathrm{dl}}(t,f)$  exhibits the Rician fading distribution with the Rician factor  $\kappa_{k}$  and power  $\mathsf{E}\left\{\left|g_k^{\mathrm{dl}}(t,f)\right|^2\right\} = \gamma_k$ . In other words, the real and imaginary parts of  $g_{k}^{\mathrm{dl}}(t,f)$  are independently and identically real-valued Gaussian distributed with mean  $\sqrt{\frac{\kappa_k\gamma_k}{2(\kappa_k + 1)}}$  and variance  $\frac{\gamma_k}{2(\kappa_k + 1)}$ , respectively.
+
+# C. UL Channel Model
+
+Using the DL channel modeling approach presented in the above subsections, we briefly investigate the UL channel model for LEO satellite communications in this subsection. Note that the UL channel response is the transpose of the DL channel response in TDD systems, and similar channel model can be obtained. Meanwhile, for FDD systems where the relative carrier frequency difference is small, the physical channel parameters,  $P_{k}$ ,  $\nu_{u,p}$ ,  $\tau_{k,p}$ ,  $\vartheta_{k}^{\mathrm{x}}$ , and  $\vartheta_{k}^{\mathrm{y}}$  are almost identical between the UL and DL [28]-[30]. Thus, the major difference between the UL and DL channels lies in the fast fading path gain terms. Similarly as (6), the UL space domain channel response between UT  $k$  and the LEO satellite at time  $t$  and frequency  $f$  can be modeled as
+
+$$
+\mathbf {g} _ {k} ^ {\mathrm {u l}} (t, f) = \exp \left\{\bar {\gamma} 2 \pi \left[ t \nu_ {k} ^ {\mathrm {s a t}} - f \tau_ {k} ^ {\min } \right] \right\} \cdot g _ {k} ^ {\mathrm {u l}} (t, f) \cdot \mathbf {u} _ {k} \in \mathbb {C} ^ {M \times 1}, \tag {8}
+$$
+
+where  $g_k^{\mathrm{ul}}(t,f)$  is the UL channel gain of UT  $k$  given by
+
+$$
+g _ {k} ^ {\mathrm {u l}} (t, f) \triangleq \sum_ {p = 0} ^ {P _ {k} - 1} g _ {k, p} ^ {\mathrm {u l}} \cdot \exp \left\{\bar {\jmath} 2 \pi \left[ t \nu_ {k, p} ^ {\mathrm {u t}} - f \tau_ {k, p} ^ {\mathrm {u t}} \right] \right\}, \tag {9}
+$$
+
+which exhibits the same statistical properties as the DL channel gain  $g_{k}^{\mathrm{dl}}(t,f)$ , i.e., the real and imaginary parts of  $g_{k}^{\mathrm{ul}}(t,f)$  are independently and identically real-valued Gaussian distributed with mean  $\sqrt{\frac{\kappa_k\gamma_k}{2(\kappa_k + 1)}}$  and variance  $\frac{\gamma_k}{2(\kappa_k + 1)}$ , respectively, and  $\mathbf{u}_k$  is the UL channel direction vector given by
+
+$$
+\begin{array}{l} \mathbf {u} _ {k} = \mathbf {u} _ {k} ^ {\mathrm {x}} \otimes \mathbf {u} _ {k} ^ {\mathrm {y}} \\ = \mathbf {u} _ {\mathrm {x}} \left(\vartheta_ {k} ^ {\mathrm {x}}\right) \otimes \mathbf {u} _ {\mathrm {y}} \left(\vartheta_ {k} ^ {\mathrm {y}}\right) \in \mathbb {C} ^ {M \times 1}, \tag {10} \\ \end{array}
+$$
+
+which exhibits a similar structure as the DL channel direction vector  $\mathbf{v}_k$  in (4) but with a center frequency offset for FDD systems, and can be well approximated by  $\mathbf{v}_k$  when the frequency separation between the UL and the DL is not significant [31]. Similarly as the DL case, the LEO satellite UL channel also exhibits the asymptotic orthogonality as
+
+$$
+\lim  _ {M _ {d} \rightarrow \infty} \left(\mathbf {u} _ {k} ^ {d}\right) ^ {H} \mathbf {u} _ {k ^ {\prime}} ^ {d} = \delta \left(k - k ^ {\prime}\right). \tag {11}
+$$
+
+Note that the channel models in (6) and (8) are general in the sense that they take into account the LEO satellite channel propagation properties in the space, time, and frequency domains.
+
+# D. DL/UL Transmission Signal Model
+
+Consider a wideband massive MIMO LEO satellite communication system employing orthogonal frequency division multiplexing (OFDM) modulation [21] with the number of subcarriers,  $N_{\mathrm{us}}$ , and the cyclic prefix (CP),  $N_{\mathrm{cp}}$  samples. Denote by  $T_{\mathrm{s}}$  the system sampling interval. Then, the OFDM symbol length and the CP length are given by  $T_{\mathrm{us}} = N_{\mathrm{us}}T_{\mathrm{s}}$  and  $T_{\mathrm{cp}} = N_{\mathrm{cp}}T_{\mathrm{s}}$ , respectively. Note that with the delay and Doppler properties of the LEO satellite channels taken into account, it is not difficult to select proper OFDM parameters such that the effects of the intersymbol and intercarrier interference can be almost neglected [32].
+
+Let  $\{\mathbf{x}_{\ell,n}^{\mathrm{dl}}\}_{n=0}^{N_{\mathrm{us}}-1}$  be the DL transmit symbols during symbol  $\ell$ . Then, the transmitted signal  $\mathbf{x}_{\ell}^{\mathrm{dl}}(t) \in \mathbb{C}^{M \times 1}$  can be written as [33]
+
+$$
+\mathbf {x} _ {\ell} ^ {\mathrm {d l}} (t) = \sum_ {n = 0} ^ {N _ {\mathrm {u s}} - 1} \mathbf {x} _ {\ell , n} ^ {\mathrm {d l}} \cdot \exp \left\{\bar {j} 2 \pi \frac {n}{T _ {\mathrm {u s}}} t \right\}, - T _ {\mathrm {c p}} \leq t - \ell \left(T _ {\mathrm {c p}} + T _ {\mathrm {u s}}\right) <   T _ {\mathrm {u s}}, \tag {12}
+$$
+
+and the corresponding received signal at UT  $k$  is given by (where the noise is omitted for brevity)
+
+$$
+y _ {k, \ell} ^ {\mathrm {d l}} (t) = \int_ {- \infty} ^ {\infty} \left[ \mathbf {g} _ {k} ^ {\mathrm {d l}} (t, \tau) \right] ^ {T} \cdot \mathbf {x} _ {\ell} ^ {\mathrm {d l}} (t - \tau) \mathrm {d} \tau , \tag {13}
+$$
+
+where  $\mathbf{g}_k^{\mathrm{dl}}(t,\tau)$  is the inverse Fourier transform of  $\mathbf{g}_k^{\mathrm{dl}}(t,f)$  in (6) in terms of  $\tau$ .
+
+Utilizing the Doppler and delay properties of the LEO satellite propagation channels addressed previously, we proceed to perform time and frequency synchronization. In particular, with delay compensation  $\tau_{k}^{\mathrm{syn}} = \tau_{k}^{\mathrm{min}}$  and Doppler compensation  $\nu_{k}^{\mathrm{syn}} = \nu_{k,p}^{\mathrm{sat}}$  applied to the received signal at UT  $k$ , the resultant signal can be represented by
+
+$$
+y _ {k, \ell} ^ {\mathrm {d l}, \text {s y n}} (t) = y _ {k, \ell} ^ {\mathrm {d l}} \left(t + \tau_ {k} ^ {\text {s y n}}\right) \cdot \exp \left\{- \bar {\jmath} 2 \pi \left(t + \tau_ {k} ^ {\text {s y n}}\right) \nu_ {k} ^ {\text {s y n}} \right\}. \tag {14}
+$$
+
+Then, the corresponding signal dispersion in the delay and Doppler domains can be significantly reduced, and it is not difficult to select proper OFDM parameters to mitigate the intersymbol and intercarrier interference [33]. Consequently, the demodulated DL received signal at UT  $k$  over subcarrier  $n$  of OFDM symbol  $\ell$  can be represented by
+
+$$
+y _ {k, \ell , n} ^ {\mathrm {d l}} = \left(\mathbf {g} _ {k, \ell , n} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {x} _ {\ell , n} ^ {\mathrm {d l}}, \tag {15}
+$$
+
+where  $\mathbf{g}_{k,\ell,n}^{\mathrm{dl}}$  is the DL channel of UT  $k$  over symbol  $\ell$  and subcarrier  $n$  given by [33]
+
+$$
+\mathbf {g} _ {k, \ell , n} ^ {\mathrm {d l}} = \mathbf {v} _ {k} \cdot g _ {k, \ell , n} ^ {\mathrm {d l}} \in \mathbb {C} ^ {M \times 1}, \tag {16}
+$$
+
+where  $g_{k,\ell,n}^{\mathrm{dl}} = g_k^{\mathrm{dl}}\left(\ell \left(T_{\mathrm{us}} + T_{\mathrm{cp}}\right), n / T_{\mathrm{us}}\right)$ .
+
+Besides, consider UL transmission employing OFDM modulation with similar parameters as DL transmission. Then, with proper delay and Doppler compensations performed at the UT side, the demodulated UL received signal at the satellite over symbol  $\ell$  and subcarrier  $n$  can be represented as
+
+$$
+\mathbf {y} _ {\ell , n} ^ {\mathrm {u l}} = \sum_ {k} \mathbf {g} _ {k, \ell , n} ^ {\mathrm {u l}} x _ {k, \ell , n} ^ {\mathrm {u l}} \in \mathbb {C} ^ {M \times 1}, \tag {17}
+$$
+
+where  $x_{k,\ell,n}^{\mathrm{ul}}$  is the complex-valued symbols transmitted by UT  $k$ , and  $\mathbf{g}_{k,\ell,n}^{\mathrm{ul}}$  is the UL channel of UT  $k$  over subcarrier  $n$  of OFDM symbol  $\ell$  given by
+
+$$
+\mathbf {g} _ {k, \ell , n} ^ {\mathrm {u l}} = \mathbf {u} _ {k} \cdot g _ {k, \ell , n} ^ {\mathrm {u l}} \in \mathbb {C} ^ {M \times 1}, \tag {18}
+$$
+
+where  $g_{k,\ell,n}^{\mathrm{ul}} = g_k^{\mathrm{ul}} \left( \frac{\ell (T_{\mathrm{us}} + T_{\mathrm{cp}})}{n / T_{\mathrm{us}}} \right)$ . Note that the transmission signal models in (15) and (17) are applicable provided that delay and Doppler compensations are properly performed via exploiting the delay and Doppler properties of the LEO satellite channels described previously.
+
+# III. STATISTICAL CSI BASED DL/UL TRANSMISSIONS
+
+In this section, we investigate DL precoder and UL receiver design for LEO satellite communications based on the channel and signal models established in the above section. Note that the conventional designs of DL precoding vectors and UL receiving vectors in MIMO transmission usually require knowledge of iCSI. However, it is in general infeasible to obtain precise iCSI at the satellite sides for DL of LEO satellite communications. In addition, frequent update of the DL precoding vectors and UL receiving vectors using iCSI will be challenging for implementation on payload of practical satellite communications. Hereafter, we focus on the design of DL precoder and UL receiver utilizing slowly-varying sCSI for satellite communications.
+
+# A. DL Precoder
+
+We first consider DL transmission where  $K$  single antenna UTs are simultaneously served in the same time-frequency blocks, and the served UT set is denoted by  $\mathcal{K} = \{0,1,\dots ,K - 1\}$ . For DL linear precoding performed at the satellite, the signal received by UT  $k\in \mathcal{K}$  in (15) can be rewritten as
+
+$$
+y _ {k} ^ {\mathrm {d l}} = \left(\mathbf {g} _ {k} ^ {\mathrm {d l}}\right) ^ {T} \sum_ {i \in \mathcal {K}} \sqrt {q _ {i} ^ {\mathrm {d l}}} \mathbf {b} _ {i} s _ {i} ^ {\mathrm {d l}} + z _ {k} ^ {\mathrm {d l}}, \tag {19}
+$$
+
+where the subcarrier and symbol indices are omitted for brevity,  $q_{k}^{\mathrm{dl}}$  is the transmit power allocated to UT  $k$ ,  $\mathbf{b}_{k} \in \mathbb{C}^{M \times 1}$  is the normalized transmit precoding vector satisfying the  $\| \mathbf{b}_k\| = \sqrt{\mathbf{b}_k^H\mathbf{b}_k} = 1$ ,  $s_k^{\mathrm{dl}}$  is the signal for UT  $k$  with mean 0 and variance 1, and  $z_{k}^{\mathrm{dl}}$  is the additive circular symmetric complex-valued Gaussian noise with mean 0 and variance  $\sigma_k^{\mathrm{dl}}$ , i.e.,  $z_{k}^{\mathrm{dl}} \sim \mathcal{CN}\left(0,\sigma_{k}^{\mathrm{dl}}\right)$ .
+
+Note that SLNR is a convenient and efficient design metric widely adopted in DL multiuser MIMO transmission, and we first review the SLNR maximization criterion based precoding approach. In particular, the SLNR of UT  $k$  in the DL is given by [34], [35]
+
+$$
+\mathrm {S L N R} _ {k} = \frac {\left| \left(\mathbf {g} _ {k} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {b} _ {k} \right| ^ {2} q _ {k} ^ {\mathrm {d l}}}{\sum_ {i \neq k} \left| \left(\mathbf {g} _ {i} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {b} _ {k} \right| ^ {2} q _ {k} ^ {\mathrm {d l}} + \sigma_ {k} ^ {\mathrm {d l}}} = \frac {\left| \left(\mathbf {g} _ {k} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {b} _ {k} \right| ^ {2}}{\sum_ {i \neq k} \left| \left(\mathbf {g} _ {i} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {b} _ {k} \right| ^ {2} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}}}, \tag {20}
+$$
+
+where  $\rho_{k}^{\mathrm{dl}}\triangleq q_{k}^{\mathrm{dl}} / \sigma_{k}^{\mathrm{dl}}$  is the DL signal-to-noise ratio (SNR) of UT  $k$ . Then the precoder of UT  $k$  that maximizes  $\mathsf{SLNR}_k$  in (20) can be obtained as
+
+$$
+\mathbf {b} _ {k} ^ {\mathrm {s l n r}} = \frac {1}{\eta_ {k} ^ {\mathrm {s l n r}}} \left[ \left(\sum_ {i} \mathbf {g} _ {i} ^ {\mathrm {d l}} \left(\mathbf {g} _ {i} ^ {\mathrm {d l}}\right) ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {g} _ {k} ^ {\mathrm {d l}} \right] ^ {*}, \tag {21}
+$$
+
+where  $(\cdot)^*$  denotes the conjugate operation and  $\eta_k^{\mathrm{slnr}}$  is the power normalization coefficient that is set to satisfy  $\left\| \mathbf{b}_k^{\mathrm{slnr}}\right\| = 1$ . We mention that the SLNR maximization DL precoder in (21) requires knowledge of iCSI  $\mathbf{g}_k^{\mathrm{dl}}$  for all  $k$ . However, it is in general difficult to obtain precise DL iCSI for transmitter at the satellite side.
+
+In the following, we investigate DL precoding for satellite communications using long-term sCSI at the transmitter, including the channel direction vector  $\mathbf{v}_k$  and the statistics of the channel gain  $g_{k,\ell,n}^{\mathrm{dl}}$ . We consider the ASLNR performance metric as follows [36]
+
+$$
+\mathrm {A S L N R} _ {k} \triangleq \frac {\mathsf {E} \left\{\left| \left(\mathbf {g} _ {k} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {b} _ {k} \right| ^ {2} \right\}}{\mathsf {E} \left\{\sum_ {i \neq k} \left| \left(\mathbf {g} _ {i} ^ {\mathrm {d l}}\right) ^ {T} \mathbf {b} _ {k} \right| ^ {2} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \right\}} = \frac {\gamma_ {k} \left| (\mathbf {v} _ {k}) ^ {T} \mathbf {b} _ {k} \right| ^ {2}}{\sum_ {i \neq k} \gamma_ {i} \left| (\mathbf {v} _ {i}) ^ {T} \mathbf {b} _ {k} \right| ^ {2} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}}}, \tag {22}
+$$
+
+where the numerator and the denominator account for the average power of the signal and leakage plus noise, respectively. The sCSI based precoder that maximizes  $\mathrm{ASLNR}_k$  is presented in the following proposition.
+
+Proposition 1: The precoding vector that maximizes  $\mathrm{ASLNR}_k$  in (22) is given by
+
+$$
+\mathbf {b} _ {k} ^ {\text {a s l n r}} = \frac {1}{\eta_ {k} ^ {\text {a s l n r}}} \left[ \left(\sum_ {i} \gamma_ {i} \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {v} _ {k} \right] ^ {*}, \tag {23}
+$$
+
+where  $\eta_k^{\mathrm{aslnr}}$  is the power normalization coefficient that is set to satisfy  $\left\| \mathbf{b}_k^{\mathrm{aslnr}}\right\| = 1$ , and the corresponding maximum ASLNR value is given by
+
+$$
+\mathrm {A S L N R} _ {k} ^ {\max } = \frac {1}{1 - \gamma_ {k} \mathbf {v} _ {k} ^ {H} \left(\sum_ {i} \gamma_ {i} \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {v} _ {k}} - 1. \tag {24}
+$$
+
+Proof: The proof is similar to the iCSI case in [35], and is omitted for brevity.
+
+Proposition 1 provides a sCSI based DL precoder that maximizes the ASLNR in closed-form. Note that the sCSI required in the proposed approach are the channel direction vectors and the average power of all UTs' channels, i.e.,  $\mathbf{v}_k$  and  $\gamma_k$ ,  $\forall k$ . In addition, from the definition of the channel direction vector in (4), only the space angles,  $\vartheta_k^{\mathrm{x}}$  and  $\vartheta_k^{\mathrm{y}}$ , are needed for estimating
+
+$\mathbf{v}_k$ . Thus, the number of parameters in statistical CSI to estimate can be significantly reduced. As the proposed sCSI based DL precoding design is independent of subcarriers and OFDM symbols in transmission interval where the channels statistics do not change significantly and thus is convenient for practical implementation of the satellite payloads. Then, the computational overhead for DL precoding design can be reduced compared with the iCSI based approach.
+
+# B. UL Receiver
+
+In this subsection, we investigate UL receiver design. The UL received signal by the satellite in (17) can be rewritten as
+
+$$
+\mathbf {y} ^ {\mathrm {u l}} = \sum_ {k} \mathbf {g} _ {k} ^ {\mathrm {u l}} \sqrt {q ^ {\mathrm {u l}}} s _ {k} ^ {\mathrm {u l}} + \mathbf {z} ^ {\mathrm {u l}}, \tag {25}
+$$
+
+where the subcarrier and symbol indices are omitted for brevity,  $q^{\mathrm{ul}}$  is the transmit power of one UT,  $s_k^{\mathrm{ul}}$  is the signal sent by UT  $k$  with mean 0 and variance 1, and  $\mathbf{z}^{\mathrm{ul}}$  is the additive Gaussian noise distributed as  $\mathcal{CN}\left(\mathbf{0},\sigma^{\mathrm{ul}}\mathbf{I}_M\right)$ . With a linear receiver at the satellite, the recovered signal of UT  $k$  can be expressed by
+
+$$
+\hat {s} _ {k} ^ {\mathrm {u l}} = \mathbf {w} _ {k} ^ {T} \mathbf {y} ^ {\mathrm {u l}} = \mathbf {w} _ {k} ^ {T} \sum_ {i} \mathbf {g} _ {i} ^ {\mathrm {u l}} \sqrt {q ^ {\mathrm {u l}}} s _ {i} ^ {\mathrm {u l}} + \mathbf {w} _ {k} ^ {T} \mathbf {z} ^ {\mathrm {u l}}, \tag {26}
+$$
+
+where  $\mathbf{w}_k$  is the linear receiving vector of UT  $k$ . Then the SINR of UT  $k$  is given by
+
+$$
+\operatorname {S I N R} _ {k} = \frac {\left| \mathbf {w} _ {k} ^ {T} \mathbf {g} _ {k} ^ {\mathrm {u l}} \right| ^ {2} q ^ {\mathrm {u l}}}{\sum_ {i \neq k} \left| \mathbf {w} _ {k} ^ {T} \mathbf {g} _ {i} ^ {\mathrm {u l}} \right| ^ {2} q ^ {\mathrm {u l}} + \sigma^ {\mathrm {u l}} \| \mathbf {w} _ {k} \| ^ {2}} = \frac {\left| \mathbf {w} _ {k} ^ {T} \mathbf {g} _ {k} ^ {\mathrm {u l}} \right| ^ {2}}{\sum_ {i \neq k} \left| \mathbf {w} _ {k} ^ {T} \mathbf {g} _ {i} ^ {\mathrm {u l}} \right| ^ {2} + \frac {1}{\rho^ {\mathrm {u l}}} \| \mathbf {w} _ {k} \| ^ {2}}, \tag {27}
+$$
+
+where  $\rho^{\mathrm{ul}}\triangleq q^{\mathrm{ul}} / \sigma^{\mathrm{ul}}$  is the UL SNR. It is not difficult to obtain the receiver of UT  $k$  that maximizes SINR in (27) as
+
+$$
+\mathbf {w} _ {k} ^ {\sin \mathrm {r}} = \left[ \left(\sum_ {i} \mathbf {g} _ {i} ^ {\mathrm {u l}} \left(\mathbf {g} _ {i} ^ {\mathrm {u l}}\right) ^ {H} + \frac {1}{\rho^ {\mathrm {u l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {g} _ {k} ^ {\mathrm {u l}} \right] ^ {*}. \tag {28}
+$$
+
+Note that the SINR maximization UL receiving vectors in (28) are in general difficult to be computed in practical satellite communications systems where the payload resource is limited, as they are needed to be updated more frequently in time and frequency.
+
+Similarly as the DL case, we investigate UL receiver design for LEO satellite communications
+
+exploiting sCSI and consider the ASINR performance metric given by
+
+$$
+A S I N R _ {k} \triangleq \frac {E \left\{\left| \mathbf {w} _ {k} ^ {T} \mathbf {g} _ {k} ^ {\mathrm {u l}} \right| ^ {2} \right\}}{E \left\{\sum_ {i \neq k} \left| \mathbf {w} _ {k} ^ {T} \mathbf {g} _ {i} ^ {\mathrm {u l}} \right| ^ {2} + \frac {1}{\rho^ {\mathrm {u l}}} \| \mathbf {w} _ {k} \| ^ {2} \right\}} = \frac {\gamma_ {k} \left| (\mathbf {u} _ {k}) ^ {T} \mathbf {w} _ {k} \right| ^ {2}}{\sum_ {i \neq k} \gamma_ {i} \left| (\mathbf {u} _ {i}) ^ {T} \mathbf {w} _ {k} \right| ^ {2} + \frac {1}{\rho^ {\mathrm {u l}}} \| \mathbf {w} _ {k} \| ^ {2}}. \tag {29}
+$$
+
+Using a similar proof procedure as in Proposition 1, we can obtain that the sCSI based UL receiver that maximizes  $\mathrm{ASINR}_k$  in (29) is given by
+
+$$
+\mathbf {w} _ {k} ^ {\text {a s i n r}} = \left[ \left(\sum_ {i} \gamma_ {i} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {H} + \frac {1}{\rho^ {\mathrm {u l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {u} _ {k} \right] ^ {*}, \tag {30}
+$$
+
+with the corresponding maximum ASINR of UT  $k$  being
+
+$$
+\operatorname {A S I N R} _ {k} ^ {\max } = \frac {1}{1 - \gamma_ {k} \mathbf {u} _ {k} ^ {H} \left(\sum_ {i} \gamma_ {i} \mathbf {u} _ {i} \mathbf {u} _ {i} ^ {H} + \frac {1}{\rho^ {\mathrm {u l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {u} _ {k}} - 1. \tag {31}
+$$
+
+Note that the sCSI based UL receiver in (30) is presented in closed-form, and is based on sCSI, i.e., the channel direction vector  $\mathbf{u}_k$  and the statistics of the channel gain  $g_{k,\ell,n}^{\mathrm{dl}}$ , which can thus mitigate the payload complexity and cost in practical satellite communications.
+
+# C. DL-UL Duality
+
+From (23) and (30), we can obtain the DL-UL duality between the proposed sCSI based DL precoder and UL receiver. Specifically, in the considered transmission interval where the channel statistics do not change significantly, if the DL data transmission  $\mathrm{SNR}\rho_k^{\mathrm{dl}}$  equals the UL data transmission  $\mathrm{SNR}\rho^{\mathrm{ul}}$ , then the sCSI based DL precoding vectors in (23) are equal to the sCSI based UL receiving vectors in (30) with proper power normalization provided that the DL direction vector  $\mathbf{v}_k$  equals the UL direction vector  $\mathbf{u}_k$ , and the transmission complexity can be further reduced. Note that different from the UL-DL duality results based on the perfect iCSI assumption in, e.g., [37] and [38], our result is established using the sCSI at the satellite side.
+
+# D. Upper Bound of ASLNR/ASINR
+
+In this subsection, we investigate the conditions under which the DL ASLNR and UL ASINR metrics considered above can be upper bounded.
+
+Proposition 2: The maximum DL ASLNR value  $\mathrm{ASLNR}_k^{\max}$  in (24) is upper bounded by
+
+$$
+\mathrm {A S L N R} _ {k} ^ {\max } \leq \rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}, \tag {32}
+$$
+
+and the upper bound can be achieved under the condition that
+
+$$
+\left(\mathbf {v} _ {k} ^ {\mathrm {x}}\right) ^ {H} \mathbf {v} _ {i} ^ {\mathrm {x}} = 0 \quad \text {o r} \quad \left(\mathbf {v} _ {k} ^ {\mathrm {y}}\right) ^ {H} \mathbf {v} _ {i} ^ {\mathrm {y}} = 0, \quad \forall k \neq i. \tag {33}
+$$
+
+Besides, the maximum UL ASINR value  $\mathrm{ASINR}_k^{\max}$  in (31) is upper bounded by
+
+$$
+\operatorname {A S I N R} _ {k} ^ {\max } \leq \rho^ {\mathrm {u l}} \gamma_ {k}, \tag {34}
+$$
+
+and the upper bound can be achieved under the condition that
+
+$$
+\left(\mathbf {u} _ {k} ^ {\mathrm {x}}\right) ^ {H} \mathbf {u} _ {i} ^ {\mathrm {x}} = 0 \quad \text {o r} \quad \left(\mathbf {u} _ {k} ^ {\mathrm {y}}\right) ^ {H} \mathbf {u} _ {i} ^ {\mathrm {y}} = 0, \quad \forall k \neq i. \tag {35}
+$$
+
+Proof: Please refer to Appendix A.
+
+Proposition 2 shows that the DL ASLNRs and UL ASINRs of all served UTs with the proposed sCSI based precoder and receiver can reach their upper bounds provided that the corresponding channel direction vectors of different UTs are mutually orthogonal. The result in Proposition 2 is physically intuitive as the DL channel leakage power and the UL inter-user interference can be eliminated provided that the conditions in (33) and (35) are satisfied.
+
+From (5) and (11), we can observe that the optimal conditions obtained in Proposition 2 can be asymptotically satisfied when the number of antennas  $M$  tends to infinity. This corroborates the rationality and potential of exploiting massive MIMO in enhancing the transmission performance of satellite communications.
+
+Remark 1: When the channel direction vectors of the UTs scheduled over the same time-frequency resource blocks satisfy the conditions in (33) and (35) or the number of antennas at the satellite side is sufficiently large, we can obtain from the matrix inversion lemma that the proposed sCSI based precoder/receiver in (23) and (30) will reduce to
+
+$$
+\mathbf {b} _ {k} ^ {\text {a s l n r}} = \mathbf {v} _ {k} ^ {*}, \quad \mathbf {w} _ {k} ^ {\text {a s i n r}} = \mathbf {u} _ {k} ^ {*}. \tag {36}
+$$
+
+Notably, the sCSI based DL precoder and UL receiver presented in (36) approach the ones using iCSI as the number of antennas tends to infinity [11], which demonstrates the asymptotic optimality of the proposed precoder/receiver exploiting sCSI.
+
+Remark 2: Note that for the case with a sufficiently large number of antennas at the satellite side, the precoder/receiver in (36) will asymptotically tend to the discrete Fourier transform
+
+(DFT) based fixed precoder/receiver as follows
+
+$$
+\mathbf {b} _ {k} = \left[ \mathbf {v} _ {\mathrm {x}} \left(\bar {\vartheta} _ {k} ^ {\mathrm {x}}\right) \otimes \mathbf {v} _ {\mathrm {y}} \left(\bar {\vartheta} _ {k} ^ {\mathrm {y}}\right) \right] ^ {*}, \quad \mathbf {w} _ {k} ^ {\text {a s i n r}} = \mathbf {w} _ {k} = \left[ \mathbf {u} _ {\mathrm {x}} \left(\bar {\vartheta} _ {k} ^ {\mathrm {x}}\right) \otimes \mathbf {u} _ {\mathrm {y}} \left(\bar {\vartheta} _ {k} ^ {\mathrm {y}}\right) \right] ^ {*}, \tag {37}
+$$
+
+where  $\overline{\vartheta}_k^d$  is the nearest point of  $\vartheta_k^d$  in the DFT grid satisfying  $\overline{\vartheta}_k^d = -1 + 2n_k^d /M_d$  with  $n_k^d\in [0,M_d - 1]$  being integers and  $\left|\overline{\vartheta}_k^d -\vartheta_k^d\right| < 2 / M_d$  for  $d\in \mathcal{D}$ . In this case, the precoding/receiving vectors for the simultaneously served UTs in the same user group are orthogonal, and can be efficiently implemented with fast Fourier transform (FFT).
+
+# IV. USER GROUPING
+
+From the results in the above section, we can observe that the performance of the proposed sCSI based precoder and receiver in massive MIMO LEO satellite communications will largely depend on the channel statistics of the simultaneously served UTs. As the number of the UTs to be served is usually much larger than that of antennas equipped at the satellites, user grouping is of practical importance. Compared with the terrestrial counterpart, user grouping is of greater interest as the satellite service provider generally aims at serving all UTs in satellite communications. In this section, we investigate user grouping for massive MIMO LEO satellite communications.
+
+# A. Space Angle based User Grouping
+
+Although the conditions in Proposition 2 are desirable for optimizing the performance of DL ASLNRs and UL ASINRs in satellite communications, it is in general difficult to schedule the UTs that rigorously satisfy this condition, and the optimal user grouping pattern can be found through exhaustive search. However, due to the large number of existing UTs in satellite communications, it is usually infeasible to perform an exhaustive search in practical systems.
+
+The optimal user grouping condition presented in Proposition 2 indicates that the channel direction vectors of UTs in the same group should be as orthogonal as possible. From the definitions in (4) and (10), the channel direction vectors are directly related to the channel propagation properties in the space domain, i.e., the channel space angles. Then, the conditions for achieving the upper bounds of ASLNR and ASINR presented in (33) and (35) can be reduced to the condition that the channel space angles should satisfy
+
+$$
+\vartheta_ {k} ^ {\mathrm {x}} - \vartheta_ {i} ^ {\mathrm {x}} = \frac {2}{M _ {\mathrm {x}}} n _ {k, i} ^ {\mathrm {x}} \quad \text {o r} \quad \vartheta_ {k} ^ {\mathrm {y}} - \vartheta_ {i} ^ {\mathrm {y}} = \frac {2}{M _ {\mathrm {y}}} n _ {k, i} ^ {\mathrm {y}}, \quad \forall k \neq i, \tag {38}
+$$
+
+where both  $n_{k,i}^{\mathrm{x}}$  and  $n_{k,i}^{\mathrm{y}}$  are non-zero integers. Motivated by the condition in (38), we propose a space angle based user grouping (SAUG) approach as follows. Specifically, we uniformly divide the space angle range  $[-1,1)$  into  $M_{\mathrm{x}}G_{\mathrm{x}}$  and  $M_{\mathrm{y}}G_{\mathrm{y}}$  equal sectors in the x- and y-axes, respectively, where  $G_{\mathrm{x}}$  and  $G_{\mathrm{y}}$  are both integers and their physical meaning will be clear later. Then, the space angle intervals after division can be represented by
+
+$$
+\mathcal {A} _ {(g, r)} ^ {(m, n)} = \left\{\left(\phi_ {\mathrm {x}}, \phi_ {\mathrm {y}}\right) \mid \phi_ {\mathrm {x}} \in \left[ \phi_ {g, m} ^ {\mathrm {x}} - \frac {\Delta_ {\mathrm {x}}}{2}, \phi_ {g, m} ^ {\mathrm {x}} + \frac {\Delta_ {\mathrm {x}}}{2}\right), \phi_ {\mathrm {y}} \in \left[ \phi_ {r, n} ^ {\mathrm {y}} - \frac {\Delta_ {\mathrm {y}}}{2}, \phi_ {r, n} ^ {\mathrm {y}} + \frac {\Delta_ {\mathrm {y}}}{2}\right) \right\}, \tag {39}
+$$
+
+where  $\phi_{a,b}^{d}$  for  $d\in \mathcal{D}$  is the center space angle of the interval in the x-/y-axis given by
+
+$$
+\phi_ {a, b} ^ {d} = - 1 + \frac {\Delta_ {d}}{2} + (a + b G _ {d}) \Delta_ {d}, \quad 0 \leq a \leq G _ {d} - 1, 0 \leq b \leq M _ {d} - 1, \tag {40}
+$$
+
+with  $\Delta_d = 2 / (M_dG_d)$  being the length of the space angle interval in the x-/y-axis.
+
+With the above definition of the space angle interval division, the UTs can be grouped as follows. A given UT  $k$  is scheduled into the  $(g,r)$ th group if there exist  $0 \leq m \leq M_{\mathrm{x}} - 1$  and  $0 \leq n \leq M_{\mathrm{y}} - 1$  such that the corresponding channel space angles satisfy
+
+$$
+\left(\vartheta_ {k} ^ {\mathrm {x}}, \vartheta_ {k} ^ {\mathrm {y}}\right) \in \mathcal {A} _ {(g, r)} ^ {(m, n)}. \tag {41}
+$$
+
+Denote by  $\mathcal{K}_{(g,r)}^{(m,n)} = \left\{k:(\vartheta_k^{\mathrm{x}},\vartheta_k^{\mathrm{y}})\in \mathcal{A}_{(g,r)}^{(m,n)}\right\}$  the set of UTs whose space angles lie in the interval  $\mathcal{A}_{(g,r)}^{(m,n)}$ . In the proposed SAUG approach, we always require  $\left|\mathcal{K}_{(g,r)}^{(m,n)}\right|\leq 1$  to avoid intra-beam interference. Note that other UTs located in the same space angle interval can be scheduled over different time-frequency resources in a round-robin manner to preserve fairness. Based on the above user grouping procedure, the UTs are scheduled into at most  $G_{\mathrm{x}}G_{\mathrm{y}}$  groups, where the  $(g,r)$ th UT group is defined as
+
+$$
+\mathcal {K} _ {(g, r)} \triangleq \bigcup_ {\substack {0 \leq m \leq M _ {\mathrm {x}} - 1 \\ 0 \leq n \leq M _ {\mathrm {y}} - 1}} \mathcal {K} _ {(g, r)} ^ {(m, n)}. \tag{42}
+$$
+
+Note that the UTs scheduled in the same group will perform transmission over the same time-frequency resources, while UTs in different groups will be allocated with different time-frequency transmission resources.
+
+# B. Achievable Rate Performance
+
+In this subsection, we investigate the achievable rate performance of the proposed SAUG approach. We focus on the DL transmission case, and the UL results can be similarly obtained.
+
+From the DL signal model in (19) and the proposed SAUG approach in the above subsection, the DL achievable ergodic sum rate is given by
+
+$$
+R _ {\mathrm {d l}} = \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \frac {\left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \left| \mathbf {v} _ {k} ^ {T} \mathbf {b} _ {k} ^ {\text {a s l n r}} \right| ^ {2} q _ {k} ^ {\mathrm {d l}}}{\sum_ {i \in \mathcal {K} _ {(g , r)}} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \left| \mathbf {v} _ {k} ^ {T} \mathbf {b} _ {i} ^ {\text {a s l n r}} \right| ^ {2} q _ {i} ^ {\mathrm {d l}} + \sigma_ {k} ^ {\mathrm {d l}}} \right\} \right\}, \tag {43}
+$$
+
+where  $\mathbf{b}_k^{\mathrm{aslnr}}$  is the sCSI based precoder of UT  $k$  presented in (23), and  $\mathcal{K}_{(g,r)}$  is the UT group defined in (42). The ergodic rate expression in (43) is in general difficult to handle. Therefore, we resort to investigate the bounds of the achievable ergodic rate for further analysis. In the following proposition, we first present an upper bound of the DL achievable ergodic sum rate.
+
+Proposition 3: With linear precoder utilizing only sCSI, the DL achievable ergodic sum rate  $R_{\mathrm{dl}}$  in (43) is upper bounded by
+
+$$
+R _ {\mathrm {d l}} \leq R _ {\mathrm {d l}} ^ {\mathrm {u b}} \triangleq \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \rho_ {k} ^ {\mathrm {d l}} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \right\} \right\}, \tag {44}
+$$
+
+where the corresponding upper bound can be achieved provided that the channel direction vectors of the UTs served in the same group satisfy
+
+$$
+\left(\mathbf {v} _ {k} ^ {\mathrm {x}}\right) ^ {H} \mathbf {v} _ {i} ^ {\mathrm {x}} = 0 \quad \text {o r} \quad \left(\mathbf {v} _ {k} ^ {\mathrm {y}}\right) ^ {H} \mathbf {v} _ {i} ^ {\mathrm {y}} = 0, \quad \forall k, i \in \mathcal {K} _ {(g, r)}, k \neq i, \tag {45a}
+$$
+
+$$
+\mathbf {b} _ {k} ^ {\text {a s l n r}} = \left(\mathbf {v} _ {k} ^ {\mathrm {x}} \otimes \mathbf {v} _ {k} ^ {\mathrm {y}}\right) ^ {*}, \quad \forall k. \tag {45b}
+$$
+
+Proof: Please refer to Appendix B.
+
+Proposition 3 provides some insights for optimal DL precoding design with sCSI at the transmitter. In particular, the channel direction vectors of the UTs scheduled to be served over the same time-frequency resources should be as orthogonal as possible. Meanwhile, the beamforming vector of a given UT should be aligned with the corresponding channel direction vector. Note that the previously proposed SAUG approach attempts to schedule the UTs to satisfy the condition in (45a), and the proposed ASLNR based precoding strives to reduce the inter-user interference to approach the condition in (45b) as remarked in (36).
+
+In the following, we further investigate the asymptotic performance of the proposed approach. Before proceeding, we first provide an upper bound of the inner product  $\left|\mathbf{v}_k^H\mathbf{v}_j\right|$  for UTs  $k,j(\forall k\neq j)$  that are scheduled over the same time-frequency transmission resources via the
+
+proposed SAUG approach. From (3), we have
+
+$$
+\left| \mathbf {v} _ {k} ^ {H} \mathbf {v} _ {j} \right| = \left| \frac {\sin \frac {\pi \varphi_ {\mathrm {x}} M _ {\mathrm {x}}}{2}}{M _ {\mathrm {x}} \sin \frac {\pi \varphi_ {\mathrm {x}}}{2}} \right| \left| \frac {\sin \frac {\pi \varphi_ {\mathrm {y}} M _ {\mathrm {y}}}{2}}{M _ {\mathrm {y}} \sin \frac {\pi \varphi_ {\mathrm {y}}}{2}} \right|, \quad \forall k \neq j. \tag {46}
+$$
+
+where  $\varphi_d = \vartheta_k^d -\vartheta_j^d$  for  $d\in \mathcal{D}$ . With the proposed SAUG approach, it is not difficult to show that
+
+$$
+\frac {2}{M _ {d}} m - \Delta_ {d} \leq \varphi_ {d} \leq \frac {2}{M _ {d}} m + \Delta_ {d}, \text {w h e r e} 1 \leq m \leq M _ {d} - 1, d \in \mathcal {D}. \tag {47}
+$$
+
+Then, we can further upper bound the inner product  $\left|\mathbf{v}_k^H\mathbf{v}_j\right|$  as
+
+$$
+\left| \mathbf {v} _ {k} ^ {H} \mathbf {v} _ {j} \right| \leq \left| \frac {\sin \pi \left(1 - \frac {1}{G _ {\mathrm {x}}}\right)}{M _ {\mathrm {x}} \sin \frac {\pi}{M _ {\mathrm {x}}} \left(1 - \frac {1}{G _ {\mathrm {x}}}\right)} \right| \left| \frac {\sin \pi \left(1 - \frac {1}{G _ {\mathrm {y}}}\right)}{M _ {\mathrm {y}} \sin \frac {\pi}{M _ {\mathrm {y}}} \left(1 - \frac {1}{G _ {\mathrm {y}}}\right)} \right|, \quad \forall k \neq j. \tag {48}
+$$
+
+Thus, with sufficiently large numbers of groups  $G_{\mathrm{x}}$  and  $G_{\mathrm{y}}$ , the inner product  $\left|\mathbf{v}_k^H\mathbf{v}_j\right|$  can be sufficiently small. Motivated by this, we present the asymptotic optimality of the proposed SAUG approach combined with sCSI based DL precoder in each UT group in the following proposition.
+
+Proposition 4: The DL achievable ergodic sum rate  $R_{\mathrm{dl}}$  with the proposed sCSI based ASLNR maximization DL precoder and the SAUG approach is lower bounded by
+
+$$
+R _ {\mathrm {d l}} \geq R _ {\mathrm {d l}} ^ {\mathrm {l b}} \triangleq \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \frac {\left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \left(1 - \delta^ {\mathrm {d l}} (\epsilon)\right) q _ {k} ^ {\mathrm {d l}}}{\sum_ {i \in \mathcal {K} _ {(g , r)}} ^ {i \neq k} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \frac {\beta_ {k , i} ^ {\mathrm {d l}} (\epsilon)}{\xi^ {\mathrm {d l}} (\epsilon)} q _ {i} ^ {\mathrm {d l}} + \sigma_ {k} ^ {\mathrm {d l}}} \right\} \right\}, \tag {49}
+$$
+
+where  $\delta^{\mathrm{dl}}(\epsilon),\beta_{k,i}^{\mathrm{dl}}(\epsilon)$  , and  $\xi^{\mathrm{dl}}(\epsilon)$  are given by
+
+$$
+\delta^ {\mathrm {d l}} (\epsilon) = \frac {\left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {2} \left(K _ {\max } - 1\right) ^ {2} \epsilon^ {2}}{\chi^ {\mathrm {d l}} (\epsilon)}, \tag {50a}
+$$
+
+$$
+\chi^ {\mathrm {d l}} (\epsilon) = \frac {1}{\frac {1}{\rho_ {\operatorname* {m i n}} ^ {\mathrm {d l}} \gamma_ {\operatorname* {m i n}}} + 1 + (K _ {\max } - 1) \epsilon}, \tag {50b}
+$$
+
+$$
+\beta_ {k, i} ^ {\mathrm {d l}} (\epsilon) = \frac {\left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {4}}{\left(\rho_ {k} ^ {\mathrm {d l}} \gamma_ {i} \gamma_ {k}\right) ^ {2}} \left(K _ {\max } - 1\right) ^ {2} \epsilon^ {2}, \tag {50c}
+$$
+
+$$
+\xi^ {\mathrm {d l}} (\epsilon) = \frac {1}{\left(1 / \rho_ {\min } ^ {\mathrm {d l}} + \gamma_ {\max } + \gamma_ {\max } \left(K _ {\max } - 1\right) \epsilon\right) ^ {2}}, \tag {50d}
+$$
+
+respectively, with  $K_{\max} = \max_{g,r}\left|\mathcal{K}_{(g,r)}\right|$ ,  $\rho_{\max}^{\mathrm{dl}} = \max_{g,r}\max_{k\in \mathcal{K}_{(g,r)}}\rho_k^{\mathrm{dl}}$ ,  $\rho_{\min}^{\mathrm{dl}} = \min_{g,r}\min_{k\in \mathcal{K}_{(g,r)}}\rho_k^{\mathrm{dl}}$ ,  $\gamma_{\max} = \max_{g,r}\max_{k\in \mathcal{K}_{(g,r)}}\gamma_k$ , and  $\gamma_{\min} = \min_{g,r}\min_{k\in \mathcal{K}_{(g,r)}}\gamma_k$ , provided that the inner product of the channel direction vectors of the UTs scheduled over the same time-frequency resource blocks satisfies  $|\mathbf{v}_k^H\mathbf{v}_j| \leq \epsilon$  for  $\forall k \neq j$  and  $k,j \in \mathcal{K}_{(g,r)}$ . Moreover, when  $\epsilon \to 0$ , the lowed bound of the DL achievable
+
+ergodic rate in (49) asymptotically tends to be equal to the upper bound of the DL achievable ergodic rate in (44), i.e.,
+
+$$
+\lim  _ {\epsilon \rightarrow 0} R _ {\mathrm {d l}} ^ {\mathrm {l b}} = \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \rho_ {k} ^ {\mathrm {d l}} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \right\}\right\} = R _ {\mathrm {d l}} ^ {\mathrm {u b}}. \tag {51}
+$$
+
+Proof: Please refer to Appendix C.
+
+Proposition 4 shows that the proposed approach with SAUG and sCSI based ASLNR maximization DL precoder performed in each UT group is asymptotically optimal when  $\left|\mathbf{v}_k^H\mathbf{v}_j\right| \to 0$ . Note that this condition coincides with the upper bound achieving condition presented in Proposition 2. Therefore, when the number of satellite antennas  $M$  and/or the number of scheduled UT groups is sufficiently large, the proposed approach is asymptotically optimal, which indicates the potential of adopting massive MIMO to serve a large number of UTs in LEO satellite communications. In addition, when the previously derived conditions are not rigorously satisfied (which is the usual case in practice), the proposed sCSI based precoder and receiver can mitigate the inter-user interference and further enhance the transmission performance for satellite communications.
+
+# V. SIMULATION RESULTS
+
+In this section, we provide simulation results to evaluate the performance of the proposed massive MIMO transmission approach for LEO satellite communications. The major simulation setup parameters are listed as follows. The numbers of antennas equipped at the satellite side are set to be  $M_{\mathrm{x}} = M_{\mathrm{y}} = 16$  with half-wavelength antenna spacing in both the x- and y-axes. The channel Rician factor is set to be  $\kappa_{k} = \kappa = 10$  dB, and the channel power is normalized as  $\gamma_{k} = M_{\mathrm{x}}M_{\mathrm{y}}$  for all UT  $k$ . In addition, the channel space angles  $\vartheta_{k,p}^{\mathrm{x}}$  and  $\vartheta_{k,p}^{\mathrm{y}}$  are independently and uniformly distributed in the interval  $[-1,1)$  for all UTs. The numbers of UT groups in the proposed SAUG approach are set to be equal for both x- and y-axes, i.e.,  $G_{\mathrm{x}} = G_{\mathrm{y}} = G$ . The number of UTs to be grouped is set as  $G^{2}M$ .
+
+Note that massive MIMO has not been applied to LEO satellite communications, and we consider and compare the following DL precoding and UL receiving approaches in the simulations:
+
+- IntF: An ideal interference-free (IntF) case where the interference from other scheduled UTs over the same time and frequency resource is "genie-aided" eliminated will be considered as the performance upper bound.
+
+![](images/61ceee804e3182c2d2ad237904daede9a144ed6d8493328e5365a57534012ed9.jpg)  
+(a) DL
+
+![](images/6e1847ca52bb8a20477f74c75b698d8a1d74fde0ba8c86e36765df88042b8b8d.jpg)  
+(b) UL  
+Fig. 2. Sum rate performance comparison between the proposed sCSI (using the true and the estimated sCSI that are obtained via averaging over 50 samples, respectively) and iCSI based precoding/receiving approaches.
+
+- iCSI: Relying on the iCSI, the SLNR maximization DL precoder in (21) and the SINR maximization UL receiver in (28) are adopted, with the assumption that the iCSI can be "genie-aided" obtained.  
+- sCSI: The proposed sCSI based ASLNR maximization DL precoder and ASINR maximization UL receiver in (23) and (30) are adopted, respectively.  
+- Fixed: DFT based fixed DL precoding and UL receiving vectors in (37) are adopted.
+
+In Fig. 2, we evaluate the performance of the proposed sCSI based precoding/receiving approaches, and compare them with the iCSI based ones where UTs are grouped using the proposed SAUG with  $G = 1$ . We consider both cases that utilize the true and estimated sCSI that is obtained via averaging 50 samples. We can observe that in both UL and DL transmissions, the proposed sCSI based precoder and receivers exhibit almost identical performance as the iCSI based ones, while having significantly reduced computational overhead. In addition, the sum rate performance loss utilizing the estimated sCSI can be almost neglected.
+
+In Fig. 3, we evaluate the performance of the proposed SAUG approach with different precoding/receiving approaches versus the number of scheduled groups  $G$  when FFR is adopted across neighboring beams. We can observe that the performance of the proposed sCSI based precoder/receiver can approach that of the interference-free scenario, especially in the case with a large number of scheduled groups, which demonstrates the asymptotic optimality of the proposed transmission approach. In addition, the performance gap between the approach
+
+![](images/4539939dd0d4dfbe6183aa0dd45b28148c95a26cd56d60024077a6e225c25c76.jpg)  
+(a) DL
+
+![](images/6934dc7520ac13e4a0255d33016bcdd5d38b06f4c935c39645cf7f9cf9575288.jpg)  
+(b) UL  
+Fig. 3. Sum rate performance of SAUG with different transmission approaches versus the number of scheduled UT groups for different SNRs when FFR is adopted.
+
+with fixed precoding/receiving vectors and the proposed sCSI based ones becomes smaller as the number of scheduled groups increases, especially in the low SNR regime, which indicates the near-optimality of the approach with fixed precoding/receiving vectors in the case where interference is not dominated.
+
+In Fig. 4, the performance between the proposed transmission approach with FFR and the conventional FR4 approach is compared for different SNRs and channel Rician factors. Similarly as FFR, only one UT is scheduled per beam over the same time and frequency resource in FR4. Note that in the case of FR4, the UTs with the same color are a group of UTs performing transmission over the same time and frequency resource. For FR4, we consider two transmission approaches where "FR4, Conventional" denotes the fixed precoder/receiver in (37) and "FR4, sCSI" denotes the proposed sCSI based precoder/receiver in (23)/(30) applied to the group of UTs over the same time and frequency resource for interference mitigation, respectively. We can observe that the proposed sCSI based precoder/receiver applied to FR4 show sum rate performance gains over the conventional FR4 approach. Moreover, with FFR across neighboring beams, the proposed sCSI based precoder/receiver combined with SAUG can provide significant sum rate performance gains over the conventional FR4 approach, especially in the cases with high SNRs and large Rician factors. Notably, for both UL and DL with an SNR of 20 dB and  $\kappa = 10$  dB, the proposed transmission approach with  $G = 4$  can provide about eight-folded sum rate performance gain over the conventional FR4 approach.
+
+![](images/e563143779b20df816c4f15db7d6219a7e87074590574f6d9fe4054d6dd15237.jpg)  
+(a) DL
+
+![](images/8d8e6e171e6c7744c5619da1920d3f16919fc44e0384e11f09f578004dab0261.jpg)  
+(b) UL  
+Fig. 4. Sum rate performance comparison between the proposed approach with FFR and the conventional FR4 approach under different Rician factors.
+
+# VI. CONCLUSION
+
+In this paper, we have investigated massive MIMO transmission for LEO satellite communications exploiting sCSI with FFR. We first established the massive MIMO channel model for LEO satellite communications by taking into account the LEO satellite signal propagation properties and simplified the UL/DL transmission designs via performing Doppler and delay compensations at UTs. Then, we developed the sCSI based DL precoder and UL receiver in closed-form, under the criteria of maximizing the ASLNR and the ASINR, respectively, and revealed the duality between them. We further showed that the DL ASLNRs and UL ASINRs can reach their upper bounds provided that the channel direction vectors of the simultaneously served UTs are orthogonal, and proposed a space angle based user grouping (SAUG) approach motivated by this condition. Besides, we showed the asymptotic optimality of the proposed massive MIMO transmission approach exploiting sCSI. Simulation results showed that the proposed massive MIMO transmission scheme with FFR significantly enhances the data rate of LEO satellite communication systems. Notably, the proposed sCSI based precoder and receiver achieved the similar performance with the iCSI based ones that are often infeasible in practice. Future work includes detailed investigation on low complexity sCSI estimation, transmission designs for the cases with UTs using multiple antenna or directive antennas, low peak-to-average power ratio transmission signal design, and extension to the multiple LEO satellite communication systems, etc.
+
+# APPENDIX A
+
+# PROOF OF PROPOSITION 2
+
+We focus on the proof of the DL case and the proof of the UL case can be similarly obtained. We first show the upper bound of  $\mathrm{ASLNR}_{k}^{\max}$  in (32). From (24), we can obtain that  $\mathrm{ASLNR}_{k}^{\max}$  with the proposed sCSI based precoder can be upper bounded by
+
+$$
+\begin{array}{l} \mathsf {A S L N R} _ {k} ^ {\max} = \frac {1}{1 - \gamma_ {k} \mathbf {v} _ {k} ^ {H} \left(\sum_ {i} \gamma_ {i} \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d i}} \mathbf {I} _ {M}}\right) ^ {- 1} \mathbf {v} _ {k}} - 1 \\ \stackrel {\mathrm {(a)}} {\leq} \frac {1}{1 - \gamma_ {k} \mathbf {v} _ {k} ^ {H} \left(\gamma_ {k} \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d i}} \mathbf {I}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {v} _ {k}} - 1 \\ \stackrel {(b)} {=} \rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}, \tag {52} \\ \end{array}
+$$
+
+where (a) follows from that  $\gamma_{i}\mathbf{v}_{i}\mathbf{v}_{i}^{H}$  is positive semidefinite for  $\forall i$ , and (b) follows from the Sherman-Morrison formula [39, Eq. (15.2b)].
+
+We then show the achievability of the upper bound. From the definition of  $\mathbf{v}_k$  in (4), the condition given in (33) is equivalent to  $\mathbf{v}_k^H\mathbf{v}_i = 0$  for  $\forall k \neq i$ . Thus, we can obtain that
+
+$$
+\begin{array}{l} \left(\sum_ {i} \gamma_ {i} \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} = \left(\gamma_ {k} \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} \\ = \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} \left(\gamma_ {k} \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right), \tag {53} \\ \end{array}
+$$
+
+which yields
+
+$$
+\left(\sum_ {i} \gamma_ {i} \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} = \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} \left(\gamma_ {k} \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1}. \tag {54}
+$$
+
+Taking the traces of both sides of (54), we can further obtain
+
+$$
+\mathbf {v} _ {k} ^ {H} \left(\sum_ {i} \gamma_ {i} \mathbf {v} _ {i} \mathbf {v} _ {i} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {v} _ {k} = \mathbf {v} _ {k} ^ {H} \left(\gamma_ {k} \mathbf {v} _ {k} \mathbf {v} _ {k} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {v} _ {k}. \tag {55}
+$$
+
+Thus, the inequality in (a) of (52) can be obtained when the condition in (33) is satisfied. This concludes the proof.
+
+# APPENDIX B
+
+# PROOF OF PROPOSITION 3
+
+The achievable ergodic rate in (43) can be upper bounded by
+
+$$
+\begin{array}{l} R _ {\mathrm {d l}} \stackrel {\mathrm {(a)}} {\leq} \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \rho_ {k} ^ {\mathrm {d l}} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \left| \mathbf {v} _ {k} ^ {T} \mathbf {b} _ {k} ^ {\mathrm {a s l n r}} \right| ^ {2} \right\} \right\} \\ \stackrel {(b)} {\leq} \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \rho_ {k} ^ {\mathrm {d l}} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \right\} \right\}, \tag {56} \\ \end{array}
+$$
+
+where (a) follows from  $|\cdot|^2 \geq 0$ , and (b) follows from the Cauchy-Schwarz inequality.
+
+We then examine the condition under which the upper bound in (44) can be achieved. The inequality (a) in (56) becomes tight when  $\mathbf{b}_i^{\mathrm{aslnr}}$  is orthogonal to  $(\mathbf{v}_k^{\mathrm{x}}\otimes \mathbf{v}_k^{\mathrm{y}})^*$  for  $i\neq k$ . In addition, as the equality in (b) can be achieved when  $\mathbf{b}_k^{\mathrm{aslnr}} = (\mathbf{v}_k^{\mathrm{x}}\otimes \mathbf{v}_k^{\mathrm{y}})^*$  [40], [41], we can obtain that for  $\forall k\neq i\in \mathcal{K}_{(g,r)}$ , the channel direction vectors should satisfy  $(\mathbf{v}_k^{\mathrm{x}}\otimes \mathbf{v}_k^{\mathrm{y}})^H (\mathbf{v}_i^{\mathrm{x}}\otimes \mathbf{v}_i^{\mathrm{y}}) = (\mathbf{v}_k^{\mathrm{x}})^H\mathbf{v}_i^{\mathrm{x}}(\mathbf{v}_k^{\mathrm{y}})^H\mathbf{v}_i^{\mathrm{y}} = 0$ , i.e.,  $(\mathbf{v}_k^{\mathrm{x}})^H\mathbf{v}_i^{\mathrm{x}} = 0$  or  $(\mathbf{v}_k^{\mathrm{y}})^H\mathbf{v}_i^{\mathrm{y}} = 0$ . This concludes the proof.
+
+# APPENDIX C
+
+# PROOF OF PROPOSITION 4
+
+We first define some auxiliary variables for clarity of further proof. For notational brevity, we focus on a specific UT group, namely, the  $(g,r)$ th UT group  $\mathcal{K}_{(g,r)}$ , and omit the group index as the UTs in different groups are scheduled over different time-frequency transmission resources. For a given UT  $k \in \mathcal{K}_{(g,r)}$ , we define  $\underline{\mathbf{b}}_k^{\mathrm{aslnr}} \triangleq \left(\sum_i \gamma_i \mathbf{v}_i \mathbf{v}_i^H + \frac{1}{\rho_k^{\mathrm{dl}}} \mathbf{I}_M\right)^{-1} \mathbf{v}_k$ . From (23), we can have  $\mathbf{b}_k^{\mathrm{aslnr}} = \left(\underline{\mathbf{b}}_k^{\mathrm{aslnr}} / \left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}} \right\|\right)^*$ . Then, the DL sum rate in (43) can be rewritten as
+
+$$
+R _ {\mathrm {d l}} = \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \frac {\left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \frac {\left| \mathbf {v} _ {k} ^ {H} \underline {{\mathbf {b}}} _ {k} ^ {\text {a s l n r}} \right| ^ {2}}{\left\| \underline {{\mathbf {b}}} _ {k} ^ {\text {a s l n r}} \right\| ^ {2}} q _ {k} ^ {\mathrm {d l}}}{\sum_ {i \in \mathcal {K} _ {(g , r)}} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \frac {\left| \mathbf {v} _ {k} ^ {H} \underline {{\mathbf {b}}} _ {i} ^ {\text {a s l n r}} \right| ^ {2}}{\left\| \underline {{\mathbf {b}}} _ {i} ^ {\text {a s l n r}} \right\| ^ {2}} q _ {i} ^ {\mathrm {d l}} + \sigma_ {k} ^ {\mathrm {d l}}} \right\} \right\}. \tag {57}
+$$
+
+In order to obtain a lower bound of  $R_{\mathrm{dl}}$  in (57), we provide an upper bound of  $\left|\mathbf{v}_k^H\underline{\mathbf{b}}_i^{\mathrm{aslnr}}\right|^2$  for  $\forall k \neq i$ , a lower bound of  $\left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right\|^2$ , and a lower bound of  $\left|\mathbf{v}_k^H\underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right|^2 / \left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right\|^2$ , respectively, in the following.
+
+A. Upper Bound of  $\left|\mathbf{v}_k^H\underline{\mathbf{b}}_i^{\mathrm{aslnr}}\right|^2$  for  $\forall k \neq i$
+
+Denote by  $K \triangleq |\mathcal{K}_{(g,r)}|$ ,  $\mathbf{V} \triangleq [\mathbf{v}_1, \ldots, \mathbf{v}_K]$ , and  $\Gamma \triangleq \operatorname{diag}\left\{[\gamma_1, \ldots, \gamma_K]^T\right\}$ . Then,  $\mathbf{v}_k^H \underline{\mathbf{b}}_i^{\mathrm{aslnr}}$  can be expressed by the  $(k,i)$ th element of the following matrix
+
+$$
+\mathbf {A} \triangleq \mathbf {V} ^ {H} \left(\mathbf {V} \boldsymbol {\Gamma} \mathbf {V} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {V} \triangleq \boldsymbol {\Gamma} ^ {- 1} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \boldsymbol {\Gamma} ^ {- 1} \left(\frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \boldsymbol {\Gamma} ^ {- 1} + \mathbf {V} ^ {H} \mathbf {V}\right) ^ {- 1} \boldsymbol {\Gamma} ^ {- 1}, \tag {58}
+$$
+
+where (a) follows from the matrix inversion lemma. Let  $\mathbf{B} \triangleq \frac{1}{\rho_k^{\mathrm{dI}}}\pmb{\Gamma}^{-1} + \mathbf{V}^H\mathbf{V}$ . Then,  $\mathbf{v}_k^H\underline{\mathbf{b}}_i^{\mathrm{aslnr}}$  can be further written as
+
+$$
+\mathbf {v} _ {k} ^ {H} \underline {{\mathbf {b}}} _ {i} ^ {\text {a s l n r}} = [ \mathbf {A} ] _ {k, i} = \left\{ \begin{array}{l l} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {i} \gamma_ {k}} [ \mathbf {B} ^ {- 1} ] _ {k, i}, & \text {i f} i \neq k \\ \frac {1}{\gamma_ {k}} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {k} ^ {2}} [ \mathbf {B} ^ {- 1} ] _ {k, k}, & \text {i f} i = k \end{array} . \right. \tag {59}
+$$
+
+Denote by  $b_{k,j}$  the  $(k,j)$ th element of  $\mathbf{B}$ , and  $D_k^\prime (\mathbf{B}) \triangleq \sum_{j\neq k}|b_{k,j}| = \sum_{j\neq k}\left|\mathbf{v}_k^H\mathbf{v}_j\right|$ . Then, according to Gersgorin disc theorem [42, Theorem 6.1.1], for an arbitrary eigenvalue  $\lambda$  of  $\mathbf{B}$ , there exists an integer  $1\leq p\leq K$  such that
+
+$$
+\left| \lambda - b _ {p, p} \right| \leq D _ {p} ^ {\prime} (\mathbf {B}) \stackrel {(a)} {\leq} (K - 1) \epsilon^ {(b)} \leq (K _ {\max } - 1) \epsilon , \tag {60}
+$$
+
+where (a) follows from  $\left|\mathbf{v}_p^H\mathbf{v}_j\right| \leq \epsilon$  for all  $j \neq p$ , and (b) follows from  $K \leq K_{\max}$ . Denote by  $\lambda_{\max}$  and  $\lambda_{\min}$  the largest and smallest eigenvalues of  $\mathbf{B}$ , respectively. For  $\forall i \neq k$ , we can have the following inequality
+
+$$
+\begin{array}{l} \left| \left[ \mathbf {B} ^ {- 1} \right] _ {k, i} \right| = \left| \mathbf {e} _ {k} ^ {T} \mathbf {B} ^ {- 1} \mathbf {e} _ {i} \right| \stackrel {\mathrm {(a)}} {\leq} \frac {1 / \lambda_ {\min} - 1 / \lambda_ {\max}}{1 / \lambda_ {\min} + 1 / \lambda_ {\max}} \sqrt {\mathbf {e} _ {k} ^ {T} \mathbf {B} ^ {- 1} \mathbf {e} _ {k}} \sqrt {\mathbf {e} _ {i} ^ {T} \mathbf {B} ^ {- 1} \mathbf {e} _ {i}} \\ = \frac {\lambda_ {\mathrm {m a x}} - \lambda_ {\mathrm {m i n}}}{\lambda_ {\mathrm {m a x}} + \lambda_ {\mathrm {m i n}}} \sqrt {[ \mathbf {B} ^ {- 1} ] _ {k , k}} \sqrt {[ \mathbf {B} ^ {- 1} ] _ {i , i}} \overset {(\mathrm {b})} {\leq} \frac {2 (K _ {\mathrm {m a x}} - 1) \epsilon}{\lambda_ {\mathrm {m a x}} + \lambda_ {\mathrm {m i n}}} \sqrt {[ \mathbf {B} ^ {- 1} ] _ {k , k}} \sqrt {[ \mathbf {B} ^ {- 1} ] _ {i , i}} \\ \stackrel {\mathrm {(c)}} {\leq} \rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max } \left(K _ {\max } - 1\right) \epsilon \sqrt {\left[ \mathbf {B} ^ {- 1} \right] _ {k , k}} \sqrt {\left[ \mathbf {B} ^ {- 1} \right] _ {i , i}} \\ \stackrel {\mathrm {(d)}} {\leq} \left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {2} \left(K _ {\max } - 1\right) \epsilon , \tag {61} \\ \end{array}
+$$
+
+where  $\mathbf{e}_k$  is the  $k$ th column of identity matrix, (a) follows from Wielandt's inequality [42, Eq. (7.4.12.2)], (b) follows from (60), (c) follows from Weyl's inequality [42, Corollary 4.3.15], and (d) follows from Rayleigh quotient theorem [42, Theorem 4.2.2(c)]. From (59) and the inequality in (61), we can obtain
+
+$$
+\left| \mathbf {v} _ {k} ^ {H} \underline {{\mathbf {b}}} _ {i} ^ {\text {a s l n r}} \right| ^ {2} \leq \frac {\left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {4}}{\left(\rho_ {k} ^ {\mathrm {d l}} \gamma_ {i} \gamma_ {k}\right) ^ {2}} \left(K _ {\max } - 1\right) ^ {2} \epsilon^ {2} \triangleq \beta_ {k, i} ^ {\mathrm {d l}} (\epsilon), \quad \forall i \neq k. \tag {62}
+$$
+
+B. Lower Bound of  $\left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right\|$
+
+Defining  $\zeta_{\mathrm{max}}(\mathbf{X})$  as the maximum eigenvalue of matrix  $\mathbf{X}$ . Then, a lower bound of  $\left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}} \right\|^2$  can be obtained as
+
+$$
+\begin{array}{l} \left\| \underline {{\mathbf {b}}} _ {k} ^ {\mathrm {a s l n r}} \right\| ^ {2} = \mathbf {v} _ {k} ^ {H} \left(\mathbf {V} \boldsymbol {\Gamma} \mathbf {V} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 2} \mathbf {v} _ {k} \overset {\mathrm {(a)}} {\geq} \frac {1}{\zeta_ {\max} ^ {2} \left(\mathbf {V} \boldsymbol {\Gamma} \mathbf {V} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right)} \\ \stackrel {(b)} {\geq} \frac {1}{\left(1 / \rho_ {k} ^ {\mathrm {d l}} + \zeta_ {\max } \left(\mathbf {V} \boldsymbol {\Gamma} \mathbf {V} ^ {H}\right)\right) ^ {2}} \stackrel {(c)} {\geq} \frac {1}{\left(1 / \rho_ {\min } ^ {\mathrm {d l}} + \gamma_ {\max } + \gamma_ {\max } \left(K _ {\max } - 1\right) \epsilon\right) ^ {2}} \triangleq \xi^ {\mathrm {d l}} (\epsilon), \tag {63} \\ \end{array}
+$$
+
+where (a) follows from Rayleigh quotient theorem [42, Theorem 4.2.2(c)], (b) follows from Weyl's inequality [42, Corollary 4.3.15], and (c) holds by applying Gersgorin disc theorem [42, Theorem 6.1.1] to matrix  $\mathbf{V}\mathbf{T}\mathbf{V}^H$ .
+
+C. Lower Bound of  $\left|\mathbf{v}_k^H\underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right|^2 / \left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right\|^2$
+
+Before proceeding, we first present some preliminary results. An upper bound of  $\left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right\|^2$  can be obtained as
+
+$$
+\begin{array}{l} \left\| \underline {{\mathbf {b}}} _ {k} ^ {\mathrm {a s l n r}} \right\| ^ {2} = \left\| \left(\mathbf {V} \boldsymbol {\Gamma} \mathbf {V} ^ {H} + \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \mathbf {I} _ {M}\right) ^ {- 1} \mathbf {V} \mathbf {e} _ {k} \right\| ^ {2} \stackrel {(a)} {=} \left\| \mathbf {V} \mathbf {B} ^ {- 1} \boldsymbol {\Gamma} ^ {- 1} \mathbf {e} _ {k} \right\| ^ {2} = \mathbf {e} _ {k} ^ {T} \boldsymbol {\Gamma} ^ {- 1} \mathbf {B} ^ {- 1} \mathbf {V} ^ {H} \mathbf {V} \mathbf {B} ^ {- 1} \boldsymbol {\Gamma} ^ {- 1} \mathbf {e} _ {k} \\ \stackrel {\mathrm {(b)}} {=} \frac {1}{\gamma_ {k} ^ {2}} \left[ \mathbf {B} ^ {- 1} \left(\mathbf {B} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \boldsymbol {\Gamma} ^ {- 1}\right) \mathbf {B} ^ {- 1} \right] _ {k, k} = \frac {1}{\gamma_ {k} ^ {2}} \left(\left[ \mathbf {B} ^ {- 1} \right] _ {k, k} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}}} \left[ \mathbf {B} ^ {- 1} \boldsymbol {\Gamma} ^ {- 1} \mathbf {B} ^ {- 1} \right] _ {k, k}\right) \\ = \frac {1}{\gamma_ {k} ^ {2}} \left(\left[ \mathbf {B} ^ {- 1} \right] _ {k, k} - \sum_ {j = 1} ^ {K} \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {j}} \left| \left[ \mathbf {B} ^ {- 1} \right] _ {k, j} \right| ^ {2}\right) \\ \leq \frac {1}{\gamma_ {k} ^ {2}} \left[ \mathbf {B} ^ {- 1} \right] _ {k, k} \left(1 - \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}} \left[ \mathbf {B} ^ {- 1} \right] _ {k, k}\right) \triangleq \eta_ {k} ^ {\mathrm {d l}}, \tag {64} \\ \end{array}
+$$
+
+where (a) follows from the matrix inversion lemma, and (b) follows from the definition of  $\Gamma$  and  $\mathbf{B}$ . In addition,  $[\mathbf{B}^{-1}]_{k,k}$  can be lower bounded by
+
+$$
+\left[ \mathbf {B} ^ {- 1} \right] _ {k, k} \overset {\mathrm {(a)}} {\geq} \frac {1}{\lambda_ {\operatorname* {m a x}}} \overset {\mathrm {(b)}} {\geq} \frac {1}{b _ {p , p} + \left(K _ {\operatorname* {m a x}} - 1\right) \epsilon} \geq \frac {1}{\frac {1}{\rho_ {\operatorname* {m i n}} ^ {\mathrm {d l}} \gamma_ {\operatorname* {m i n}}} + 1 + \left(K _ {\operatorname* {m a x}} - 1\right) \epsilon} \triangleq \chi^ {\mathrm {d l}} (\epsilon), \tag {65}
+$$
+
+where (a) follows from Rayleigh quotient theorem [42, Theorem 4.2.2(c)], (b) follows from Gersgorin disc theorem [42, Theorem 6.1.1]. Moreover, the following inequality can be obtained
+
+$$
+1 = \left| \left[ \mathbf {B} \mathbf {B} ^ {- 1} \right] _ {k, k} \right| = \left| \sum_ {j = 1} ^ {K} b _ {k, j} \left[ \mathbf {B} ^ {- 1} \right] _ {j, k} \right|
+$$
+
+$$
+\begin{array}{l} \stackrel {\mathrm {(a)}} {\geq} b _ {k, k} \left[ \mathbf {B} ^ {- 1} \right] _ {k, k} - \sum_ {j \neq k} | b _ {k, j} | \left| \left[ \mathbf {B} ^ {- 1} \right] _ {j, k} \right| \stackrel {\mathrm {(b)}} {\geq} b _ {k, k} \left[ \mathbf {B} ^ {- 1} \right] _ {k, k} - \sum_ {j \neq k} \left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {2} (K _ {\max } - 1) \epsilon^ {2} \\ = \left(\frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}} + 1\right) \left[ \mathbf {B} ^ {- 1} \right] _ {k, k} - \left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {2} \left(K _ {\max } - 1\right) ^ {2} \epsilon^ {2}, \tag {66} \\ \end{array}
+$$
+
+where (a) follows from the triangle inequality, and (b) follows from  $|b_{k,j}| = |\mathbf{v}_k^H\mathbf{v}_j|\leq \epsilon$  for  $\forall j\neq k$  and the inequality in (61). Then, we can obtain
+
+$$
+\begin{array}{l} \frac {1}{[ \mathbf {B} ^ {- 1} ] _ {k , k}} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}} \overset {\mathrm {(a)}} {\geq} 1 - \frac {1}{[ \mathbf {B} ^ {- 1} ] _ {k , k}} \left(\rho_ {\max} ^ {\mathrm {d l}} \gamma_ {\max}\right) ^ {2} (K _ {\max} - 1) ^ {2} \epsilon^ {2} \\ \stackrel {(b)} {\geq} 1 - \frac {\left(\rho_ {\max } ^ {\mathrm {d l}} \gamma_ {\max }\right) ^ {2} \left(K _ {\max } - 1\right) ^ {2} \epsilon^ {2}}{\chi^ {\mathrm {d l}} (\epsilon)} \triangleq 1 - \delta^ {\mathrm {d l}} (\epsilon), \tag {67} \\ \end{array}
+$$
+
+where (a) follows from (66), and (b) follows from (65).
+
+Consequently,  $\left|\mathbf{v}_k^H\underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right|^2 / \left\| \underline{\mathbf{b}}_k^{\mathrm{aslnr}}\right\|^2$  can be lower bounded by
+
+$$
+\frac {\left| \mathbf {v} _ {k} ^ {H} \underline {{\mathbf {b}}} _ {k} ^ {\text {a s l n r}} \right| ^ {2}}{\left\| \underline {{\mathbf {b}}} _ {k} ^ {\text {a s l n r}} \right\| ^ {2}} \overset {(a)} {\geq} \frac {\left| \mathbf {v} _ {k} ^ {H} \underline {{\mathbf {b}}} _ {k} ^ {\text {a s l n r}} \right| ^ {2}}{\eta_ {k} ^ {\mathrm {d l}}} \overset {(b)} {=} \frac {\left(1 - \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}} [ \mathbf {B} ^ {- 1} ] _ {k , k}\right) ^ {2}}{\gamma_ {k} ^ {2} \eta_ {k} ^ {\mathrm {d l}}} \overset {(c)} {=} \frac {1}{[ \mathbf {B} ^ {- 1} ] _ {k , k}} - \frac {1}{\rho_ {k} ^ {\mathrm {d l}} \gamma_ {k}} \overset {(d)} {\geq} 1 - \delta^ {\mathrm {d l}} (\epsilon), \tag {68}
+$$
+
+where (a) follows from the inequality in (64), (b) follows from (59), (c) follows from the definition of  $\eta_k^{\mathrm{dl}}$  in (64), and (d) follows from (67).
+
+Combining (57), (62), (63), and (68), we can obtain a lower bound of  $R_{\mathrm{dl}}$  as
+
+$$
+R _ {\mathrm {d l}} \geq \frac {1}{G _ {\mathrm {x}} G _ {\mathrm {y}}} \sum_ {g = 0} ^ {G _ {\mathrm {x}} - 1} \sum_ {r = 0} ^ {G _ {\mathrm {y}} - 1} \sum_ {k \in \mathcal {K} _ {(g, r)}} \mathsf {E} \left\{\log_ {2} \left\{1 + \frac {\left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \left(1 - \delta^ {\mathrm {d l}} (\epsilon)\right) q _ {k} ^ {\mathrm {d l}}}{\sum_ {i \in \mathcal {K} _ {(g , r)}} ^ {i \neq k} \left| g _ {k} ^ {\mathrm {d l}} \right| ^ {2} \frac {\beta_ {k , i} ^ {\mathrm {d l}} (\epsilon)}{\xi^ {\mathrm {d l}} (\epsilon)} q _ {i} ^ {\mathrm {d l}} + \sigma_ {k} ^ {\mathrm {d l}}} \right\} \right\} \triangleq R _ {\mathrm {d l}} ^ {\mathrm {l b}}. \tag {69}
+$$
+
+Note that  $\lim_{\epsilon \to 0} \delta^{\mathrm{dl}}(\epsilon) = 0$ ,  $\lim_{\epsilon \to 0} \beta_{k,i}^{\mathrm{dl}}(\epsilon) = 0$ , and  $\xi^{\mathrm{dl}}(\epsilon) > 0$ , thus we can obtain  $\lim_{\epsilon \to 0} R_{\mathrm{dl}}^{\mathrm{lb}} = R_{\mathrm{dl}}^{\mathrm{ub}}$ . This concludes the proof.
+
+# REFERENCES
+
+[1] L. You, K.-X. Li, J. Wang, X. Q. Gao, X.-G. Xia, and B. Ottersten, “LEO satellite communications with massive MIMO,” in Proc. IEEE ICC, Dublin, Ireland, 2020, pp. 1-6.  
+[2] A. Guidotti, A. Vanelli-Coralli, M. Conti, S. Andreacci, S. Chatzinotas, N. Maturo, B. Evans, A. Awoseyila, A. Ugolini, T. Foggi, L. Gaudio, N. Alagha, and S. Cioni, "Architectures and key technical challenges for 5G systems incorporating satellites," IEEE Trans. Veh. Technol., vol. 68, no. 3, pp. 2624-2639, Mar. 2019.  
+[3] B. Di, L. Song, Y. Li, and H. V. Poor, "Ultra-dense LEO: Integration of satellite access networks into 5G and beyond," IEEE Wireless Commun., vol. 26, no. 2, pp. 62-69, Apr. 2019.  
+[4] R. T. Schwarz, T. Delamotte, K.-U. Storek, and A. Knopp, “MIMO applications for multibeam satellites,” IEEE Trans. Broadcast., vol. 65, no. 4, pp. 664–681, Dec. 2019.
+
+[5] M. Á. Vázquez, A. Pérez-Neira, D. Christopoulos, S. Chatzinotas, B. Ottersten, P.-D. Arapoglou, A. Ginesi, and G. Tarocco, “Precoding in multibeam satellite communications: Present and future challenges,” IEEE Wireless Commun., vol. 23, no. 6, pp. 88–95, Dec. 2016.  
+[6] N. Letzepis and A. J. Grant, "Capacity of the multiple spot beam satellite channel with Rician fading," IEEE Trans. Inf. Theory, vol. 54, no. 11, pp. 5210-5222, Nov. 2008.  
+[7] W. Wang, A. Liu, Q. Zhang, L. You, X. Q. Gao, and G. Zheng, "Robust multigroup multicast transmission for frame-based multi-beam satellite systems," IEEE Access, vol. 6, pp. 46074-46083, Aug. 2018.  
+[8] L. You, A. Liu, W. Wang, and X. Q. Gao, "Outage constrained robust multigroup multicast beamforming for multi-beam satellite communication systems," IEEE Wireless Commun. Lett., vol. 8, no. 2, pp. 352-355, Apr. 2019.  
+[9] T. Yoo, G. J. Foschini, R. A. Valenzuela, and A. J. Goldsmith, "Common rate support in multi-antenna downlink channels using semi-orthogonal user selection," IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3449-3461, Jun. 2011.  
+[10] J. Lee and N. Jindal, "High SNR analysis for MIMO broadcast channels: Dirty paper coding versus linear precoding," IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4787-4792, Dec. 2007.  
+[11] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.  
+[12] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An overview of massive MIMO: Benefits and challenges,” IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 742–758, Oct. 2014.  
+[13] L. You, X. Q. Gao, X.-G. Xia, N. Ma, and Y. Peng, "Pilot reuse for massive MIMO transmission over spatially correlated Rayleigh fading channels," IEEE Trans. Wireless Commun., vol. 14, no. 6, pp. 3352-3366, Jun. 2015.  
+[14] C. Sun, X. Q. Gao, S. Jin, M. Matthaiou, Z. Ding, and C. Xiao, “Beam division multiple access transmission for massive MIMO communications,” IEEE Trans. Commun., vol. 63, no. 6, pp. 2170–2184, Jun. 2015.  
+[15] A.-A. Lu, X. Q. Gao, W. Zhong, C. Xiao, and X. Meng, "Robust transmission for massive MIMO downlink with imperfect CSI," IEEE Trans. Commun., vol. 67, no. 8, pp. 5362-5376, Aug. 2019.  
+[16] L. You, J. Xiong, K.-X. Li, W. Wang, and X. Q. Gao, “Non-orthogonal unicast and multicast transmission for massive MIMO with statistical channel state information,” IEEE Access, vol. 6, pp. 66841-66849, Nov. 2018.  
+[17] A.adhikary, J.Nam, J.-Y.Ahn, and G.Caire, "Joint spatial division and multiplexing--The large-scale array regime," IEEE Trans. Inf. Theory, vol. 59, no. 10, pp. 6441-6463, Oct. 2013.  
+[18] A. G. Kanatas and A. D. Panagopoulos, Radio Wave Propagation and Channel Modeling for Earth-Space Systems. New York, NY, USA: CRC Press, 2016.  
+[19] B. Clerckx and C. Oestges, MIMO Wireless Networks: Channels, Techniques and Standards for Multi-Antenna, Multi-User and Multi-Cell Systems, 2nd ed. Oxford, U.K.: Academic Press, 2013.  
+[20] L. You, X. Q. Gao, A. L. Swindlehurst, and W. Zhong, "Channel acquisition for massive MIMO-OFDM with adjustable phase shift pilots," IEEE Trans. Signal Process., vol. 64, no. 6, pp. 1461-1476, Mar. 2016.  
+[21] A. Papathanassiou, A. K. Salkintzis, and P. T. Mathiopoulos, “A comparison study of the uplink performance of W-CDMA and OFDM for mobile multimedia communications via LEO satellites,” IEEE Personal Commun., vol. 8, no. 3, pp. 35–43, Jun. 2001.  
+[22] F. Pérez Fontán, M. Vázquez-Castro, C. Enjamio Cabado, J. Pita García, and E. Kubista, “Statistical modeling of the LMS channel,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1549–1567, Nov. 2001.  
+[23] B. R. Vojcic, R. L. Pickholtz, and L. B. Milstein, “Performance of DS-CDMA with imperfect power control operating over a low earth orbiting satellite link,” IEEE J. Sel. Areas Commun., vol. 12, no. 4, pp. 560–567, May 1994.  
+[24] 3GPP TR 38.811 V15.0.0, "3rd Generation Partnership Project; Technical Specification Group Radio Access Network; Study on New Radio (NR) to support non terrestrial networks (Release 15)," Tech. Rep., Jun. 2018.
+
+[25] C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. Hoboken, NJ, USA: John Wiley & Sons, 2016.  
+[26] R. Shafin, L. Liu, Y. Li, A. Wang, and J. Zhang, “Angle and delay estimation for 3-D massive MIMO/FD-MIMO systems based on parametric channel modeling,” IEEE Trans. Wireless Commun., vol. 16, no. 8, pp. 5370–5383, Aug. 2017.  
+[27] S. Jaeckel, L. Raschkowski, K. Borner, L. Thiele, F. Burkhardt, and E. Eberlein, “QuaDRiGa - Quasi Deterministic Radio Channel Generator, User Manual and Documentation, v2.0.0,” Tech. Rep., Aug. 2017.  
+[28] H. Xu, D. Chizhik, H. Huang, and R. Valenzuela, “A generalized space-time multiple-input multiple-output (MIMO) channel model,” IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 966–975, May 2004.  
+[29] U. Ugurlu, R. Wichman, C. B. Ribeiro, and C. Wijting, “A multipath extraction-based CSI acquisition method for FDD cellular networks with massive antenna arrays,” IEEE Trans. Wireless Commun., vol. 15, no. 4, pp. 2940–2953, Apr. 2016.  
+[30] G. Barriac and U. Madhow, "Space-time communication for OFDM with implicit channel feedback," IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3111-3129, Dec. 2004.  
+[31] M. Barzegar Khalilsarai, S. Haghhighatshoar, X. Yi, and G. Caire, "FDD massive MIMO via UL/DL channel covariance extrapolation and active channel sparsification," IEEE Trans. Wireless Commun., vol. 18, no. 1, pp. 121-135, Jan. 2019.  
+[32] M. Morelli, C.-C. J. Kuo, and M.-O. Pun, "Synchronization techniques for orthogonal frequency division multiple access (OFDMA): A tutorial review," Proc. IEEE, vol. 95, no. 7, pp. 1394-1427, Jul. 2007.  
+[33] T. Hwang, C. Yang, G. Wu, S. Li, and G. Y. Li, “OFDM and its wireless applications: A survey,” IEEE Trans. Veh. Technol., vol. 58, no. 4, pp. 1673–1694, May 2009.  
+[34] M. Sadek, A. Tarighat, and A. H. Sayed, “A leakage-based precoding scheme for downlink multi-user MIMO channels,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1711–1721, May 2007.  
+[35] P. Patcharamaneepakorn, S. Armour, and A. Doufexi, “On the equivalence between SLNR and MMSE precoding schemes with single-antenna receivers,” IEEE Commun. Lett., vol. 16, no. 7, pp. 1034–1037, Jul. 2012.  
+[36] J. Joung and A. H. Sayed, “Relay selection for grouped-relay networks using the average SLNR measure,” in Proc. IEEE CAMSAP, Aruba, Dutch Antilles, 2009, pp. 273–279.  
+[37] P. Viswanath and D. N. C. Tse, "Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality," IEEE Trans. Inf. Theory, vol. 49, no. 8, pp. 1912-1921, Aug. 2003.  
+[38] S. Shi, M. Schubert, and H. Boche, "Downlink MMSE transceiver optimization for multiuser MIMO systems: Duality and sum-MSE minimization," IEEE Trans. Signal Process., vol. 55, no. 11, pp. 5436-5446, Nov. 2007.  
+[39] G. A. F. Seber, A Matrix Handbook for Statisticians. Hoboken, NJ, USA: Wiley, 2008.  
+[40] A. M. Tulino, A. Lozano, and S. Verdu, "Capacity-achieving input covariance for single-user multi-antenna channels," IEEE Trans. Wireless Commun., vol. 5, no. 3, pp. 662-671, Mar. 2006.  
+[41] X. Q. Gao, B. Jiang, X. Li, A. B. Gershman, and M. R. McKay, "Statistical eigenmode transmission over jointly correlated MIMO channels," IEEE Trans. Inf. Theory, vol. 55, no. 8, pp. 3735-3750, Aug. 2009.  
+[42] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed. New York, NY, USA: Cambridge Univ. Press, 2012.

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+        "type": "text",
+        "text": "CodeBERT: A Pre-Trained Model for Programming and Natural Languages",
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+        "text": "Zhangyin Feng $^{1*}$ , Daya Guo $^{2*}$ , Duyu Tang $^{3}$ , Nan Duan $^{3}$ , Xiaocheng Feng $^{1}$ , Ming Gong $^{4}$ , Linjun Shou $^{4}$ , Bing Qin $^{1}$ , Ting Liu $^{1}$ , Daxin Jiang $^{4}$ , Ming Zhou $^{3}$",
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+        "text": "<sup>1</sup> Research Center for Social Computing and Information Retrieval, Harbin Institute of Technology, China",
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+        "type": "text",
+        "text": "2 The School of Data and Computer Science, Sun Yat-sen University, China",
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+        "type": "text",
+        "text": "<sup>3</sup> Microsoft Research Asia, Beijing, China",
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+        "text": "<sup>4</sup> Microsoft Search Technology Center Asia, Beijing, China",
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+        "type": "text",
+        "text": "{zyfeng,xcfeng,qinb,tliu}@ir.hit.edu.cn",
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+        "text": "{dutang,nanduan,migon,lisho,djiang,mingzhou}@microsoft.com",
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+    {
+        "type": "text",
+        "text": "Abstract",
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+    },
+    {
+        "type": "text",
+        "text": "We present CodeBERT, a bimodal pre-trained model for programming language (PL) and natural language (NL). CodeBERT learns general-purpose representations that support downstream NL-PL applications such as natural language code search, code documentation generation, etc. We develop CodeBERT with Transformer-based neural architecture, and train it with a hybrid objective function that incorporates the pre-training task of replaced token detection, which is to detect plausible alternatives sampled from generators. This enables us to utilize both \"bimodal\" data of NL-PL pairs and \"unimodal\" data, where the former provides input tokens for model training while the latter helps to learn better generators. We evaluate CodeBERT on two NL-PL applications by fine-tuning model parameters. Results show that CodeBERT achieves state-of-the-art performance on both natural language code search and code documentation generation. Furthermore, to investigate what type of knowledge is learned in CodeBERT, we construct a dataset for NL-PL probing, and evaluate in a zero-shot setting where parameters of pre-trained models are fixed. Results show that CodeBERT performs better than previous pre-trained models on NL-PL probing.<sup>1</sup>",
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+    {
+        "type": "text",
+        "text": "1 Introduction",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Large pre-trained models such as ELMo (Peters et al., 2018), GPT (Radford et al., 2018), BERT (Devlin et al., 2018), XLNet (Yang et al., 2019)",
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+        "text": "and RoBERTa (Liu et al., 2019) have dramatically improved the state-of-the-art on a variety of natural language processing (NLP) tasks. These pre-trained models learn effective contextual representations from massive unlabeled text optimized by self-supervised objectives, such as masked language modeling, which predicts the original masked word from an artificially masked input sequence. The success of pre-trained models in NLP also drives a surge of multi-modal pre-trained models, such as ViBERT (Lu et al., 2019) for language-image and VideoBERT (Sun et al., 2019) for language-video, which are learned from bimodal data such as language-image pairs with bimodal self-supervised objectives.",
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+    {
+        "type": "text",
+        "text": "In this work, we present CodeBERT, a bimodal pre-trained model for natural language (NL) and programming language (PL) like Python, Java, JavaScript, etc. CodeBERT captures the semantic connection between natural language and programming language, and produces general-purpose representations that can broadly support NL-PL understanding tasks (e.g. natural language code search) and generation tasks (e.g. code documentation generation). It is developed with the multilayer Transformer (Vaswani et al., 2017), which is adopted in a majority of large pre-trained models. In order to make use of both bimodal instances of NL-PL pairs and large amount of available unimodal codes, we train CodeBERT with a hybrid objective function, including standard masked language modeling (Devlin et al., 2018) and replaced token detection (Clark et al., 2020), where unimodal codes help to learn better generators for producing better alternative tokens for the latter objective.",
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+    {
+        "type": "text",
+        "text": "We train CodeBERT from Github code reposito",
+        "bbox": [
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+    },
+    {
+        "type": "aside_text",
+        "text": "arXiv:2002.08155v4 [cs.CL] 18 Sep 2020",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "*Work done while this author was an intern at Microsoft Research Asia.",
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+    },
+    {
+        "type": "page_footnote",
+        "text": "1 All the codes and data are available at https://github.com/microsoft/CodeBERT",
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+    {
+        "type": "text",
+        "text": "ries in 6 programming languages, where bimodal datapoints are codes that pair with function-level natural language documentations (Husain et al., 2019). Training is conducted in a setting similar to that of multilingual BERT (Pires et al., 2019), in which case one pre-trained model is learned for 6 programming languages with no explicit markers used to denote the input programming language. We evaluate CodeBERT on two downstream NL-PL tasks, including natural language code search and code documentation generation. Results show that fine-tuning the parameters of CodeBERT achieves state-of-the-art performance on both tasks. To further investigate what type of knowledge is learned in CodeBERT, we construct a dataset for NL-PL probing, and test CodeBERT in a zero-shot scenario, i.e. without fine-tuning the parameters of CodeBERT. We find that CodeBERT consistently outperforms RoBERTa, a purely natural language-based pre-trained model. The contributions of this work are as follows:",
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+    {
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+            "- CodeBERT is the first large NL-PL pretrained model for multiple programming languages.",
+            "- Empirical results show that CodeBERT is effective in both code search and code-to-text generation tasks.",
+            "- We further created a dataset which is the first one to investigate the probing ability of the code-based pre-trained models."
+        ],
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+    {
+        "type": "text",
+        "text": "2 Background",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Pre-Trained Models in NLP",
+        "text_level": 1,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Large pre-trained models (Peters et al., 2018; Radford et al., 2018; Devlin et al., 2018; Yang et al., 2019; Liu et al., 2019; Raffel et al., 2019) have brought dramatic empirical improvements on almost every NLP task in the past few years. Successful approaches train deep neural networks on large-scale plain texts with self-supervised learning objectives. One of the most representative neural architectures is the Transformer (Vaswani et al., 2017), which is also the one used in this work. It contains multiple self-attention layers, and can be conventionally learned with gradient decent in an end-to-end manner as every component is differentiable. The terminology \"self-supervised\" means that supervisions used for pre-training are automatically collected from raw data without manual",
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+    {
+        "type": "text",
+        "text": "annotation. Dominant learning objectives are language modeling and its variations. For example, in GPT (Radford et al., 2018), the learning objective is language modeling, namely predicting the next word  $w_{k}$  given the preceding context words  $\\{w_{1}, w_{2}, \\dots, w_{k-1}\\}$ . As the ultimate goal of pretraining is not to train a good language model, it is desirable to consider both preceding and following contexts to learn better general-purpose contextual representations. This leads us to the masked language modeling objective used in BERT (Devlin et al., 2018), which learns to predict the masked words of a randomly masked word sequence given surrounding contexts. Masked language modeling is also used as one of the two learning objectives for training CodeBERT.",
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+    {
+        "type": "text",
+        "text": "2.2 Multi-Modal Pre-Trained Models",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The remarkable success of the pre-trained model in NLP has driven the development of multi-modal pre-trained model that learns implicit alignment between inputs of different modalities. These models are typically learned from bimodal data, such as pairs of language-image or pairs of language-video. For example, ViLBERT (Lu et al., 2019) learns from image caption data, where the model learns by reconstructing categories of masked image region or masked words given the observed inputs, and meanwhile predicting whether the caption describes the image content or not. Similarly, VideoBERT (Sun et al., 2019) learns from language-video data and is trained by video and text masked token prediction. Our work belongs to this line of research as we regard NL and PL as different modalities. Our method differs from previous works in that the fuels for model training include not only bimodal data of NL-PL pairs, but larger amounts of unimodal data such as codes without paired documentations.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A concurrent work (Kanade et al., 2019) uses masked language modeling and next sentence prediction as the objective to train a BERT model on Python source codes, where a sentence is a logical code line as defined by the Python standard. In terms of the pre-training process, CodeBERT differs from their work in that (1) CodeBERT is trained in a cross-modal style and leverages both bimodal NL-PL data and unimodal PL/NL data, (2) CodeBERT is pre-trained over six programming languages, and (3) CodeBERT is trained with a new learning objective based on replaced token",
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+    {
+        "type": "text",
+        "text": "detection.",
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+    {
+        "type": "text",
+        "text": "3 CodeBERT",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We describe the details about CodeBERT in this section, including the model architecture, the input and output representations, the objectives and data used for training CodeBERT, and how to fine-tune CodeBERT when it is applied to downstream tasks.",
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+    {
+        "type": "text",
+        "text": "3.1 Model Architecture",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We follow BERT (Devlin et al., 2018) and RoBERTa (Liu et al., 2019), and use multi-layer bidirectional Transformer (Vaswani et al., 2017) as the model architecture of CodeBERT. We will not review the ubiquitous Transformer architecture in detail. We develop CodeBERT by using exactly the same model architecture as RoBERTa-base. The total number of model parameters is 125M.",
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+    {
+        "type": "text",
+        "text": "3.2 Input/Output Representations",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the pre-training phase, we set the input as the concatenation of two segments with a special separator token, namely [CLS],  $w_{1}, w_{2}, \\ldots, w_{n}$ , [SEP],  $c_{1}, c_{2}, \\ldots, c_{m}$ , [EOS]. One segment is natural language text, and another is code from a certain programming language. [CLS] is a special token in front of the two segments, whose final hidden representation is considered as the aggregated sequence representation for classification or ranking. Following the standard way of processing text in Transformer, we regard a natural language text as a sequence of words, and split it as WordPiece (Wu et al., 2016). We regard a piece of code as a sequence of tokens.",
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+    {
+        "type": "text",
+        "text": "The output of CodeBERT includes (1) contextual vector representation of each token, for both natural language and code, and (2) the representation of [CLS], which works as the aggregated sequence representation.",
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+    {
+        "type": "text",
+        "text": "3.3 Pre-Training Data",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We train CodeBERT with both bimodal data, which refers to parallel data of natural language-code pairs, and unimodal data, which stands for codes without paired natural language texts and natural language without paired codes.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use datapoints from Github repositories, where each bimodal datapoint is an individual function with paired documentation, and each uni-modal code is a function without paired documentation. Specifically, we use a recent large dataset",
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+    {
+        "type": "table",
+        "img_path": "images/a002adbdf3ce799370b3c77ff030eaea2cbdaf9bf405a4bc1f94c14936f0a3c8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>TRAINING DATA</td><td>bimodal DATA</td><td>unimodal CODES</td></tr><tr><td>GO</td><td>319,256</td><td>726,768</td></tr><tr><td>JAVA</td><td>500,754</td><td>1,569,889</td></tr><tr><td>JAVAscript</td><td>143,252</td><td>1,857,835</td></tr><tr><td>PHP</td><td>662,907</td><td>977,821</td></tr><tr><td>PYTHON</td><td>458,219</td><td>1,156,085</td></tr><tr><td>RUBY</td><td>52,905</td><td>164,048</td></tr><tr><td>ALL</td><td>2,137,293</td><td>6,452,446</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Statistics of the dataset used for training CodeBERT.",
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+    },
+    {
+        "type": "text",
+        "text": "provided by Husain et al. (2019), which includes 2.1M bimodal datapoints and 6.4M unimodal codes across six programming languages (Python, Java, JavaScript, PHP, Ruby, and Go). Data statistics is shown in Table 1. $^{2}$",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The data comes from publicly available open-source non-fork GitHub repositories and are filtered with a set of constraints and rules. For example, (1) each project should be used by at least one other project, (2) each documentation is truncated to the first paragraph, (3) documentations shorter than three tokens are removed, (4) functions shorter than three lines are removed, and (5) function names with substring \"test\" are removed. An example of the data is given in Figure 1<sup>3</sup>.",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/dfb9fdee528028c1c0c0eb591cfa11540dc07eb3ad1fb80615b6a3322ebad838.jpg",
+        "image_caption": [
+            "Figure 1: An example of the NL-PL pair, where NL is the first paragraph (filled in red) from the documentation (dashed line in black) of a function."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.4 Pre-Training CodeBERT",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We describe the two objectives used for training CodeBERT here. The first objective is masked language modeling (MLM), which has proven effective in literature (Devlin et al., 2018; Liu et al.,",
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+    {
+        "type": "page_footnote",
+        "text": "<sup>2</sup>Since we will evaluate on the natural language code search task, we only use the training data of Husain et al. (2019) to train CodeBERT with no access to the dev and testing data.",
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+    {
+        "type": "page_footnote",
+        "text": "3The source of the illustrating example comes from https://github.com/apache/spark/blob/618d6bfff71073c8c93501ab7392c3cc579730f0b/python/pyspark/rdd.py#L125-L138",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/3e79d5432408f9c7354ee138d1a252282bbef5871530c68ea0ba9fc7e14a7ad9.jpg",
+        "image_caption": [
+            "Figure 2: An illustration about the replaced token detection objective. Both NL and code generators are language models, which generate plausible tokens for masked positions based on surrounding contexts. NL-Code discriminator is the targeted pre-trained model, which is trained via detecting plausible alternatives tokens sampled from NL and PL generators. NL-Code discriminator is used for producing general-purpose representations in the fine-tuning step. Both NL and code generators are thrown out in the fine-tuning step."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2019; Sun et al., 2019). We apply masked language modeling on bimodal data of NL-PL pairs. The second objective is replaced token detection (RTD), which further uses a large amount of unimodal data, such as codes without paired natural language texts. Detailed hyper-parameters for model pre-training are given in Appendix B.1.",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Objective #1: Masked Language Modeling (MLM) Given a datapoint of NL-PL pair  $(x = \\{w, c\\})$  as input, where  $w$  is a sequence of NL words and  $c$  is a sequence of PL tokens, we first select a random set of positions for both NL and PL to mask out (i.e.  $m^w$  and  $m^c$ , respectively), and then replace the selected positions with a special [MASK] token. Following Devlin et al. (2018),  $15\\%$  of the tokens from  $x$  are masked out.",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {i} ^ {w} \\sim \\operatorname {u n i f} \\{1, | \\boldsymbol {w} | \\} \\text {f o r} i = 1 \\text {t o} | \\boldsymbol {w} | \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nm _ {i} ^ {c} \\sim \\operatorname {u n i f} \\{1, | c | \\} \\text {f o r} i = 1 \\text {t o} | c | \\tag {2}\n$$\n",
+        "text_format": "latex",
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+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} ^ {\\text {m a s k e d}} = \\operatorname {R E P L A C E} \\left(\\boldsymbol {w}, \\boldsymbol {m} ^ {\\boldsymbol {w}}, [ M A S K ]\\right) \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {c} ^ {\\text {m a s k e d}} = \\operatorname {R E P L A C E} (\\boldsymbol {c}, \\boldsymbol {m} ^ {\\boldsymbol {c}}, [ M A S K ]) \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {x} = \\boldsymbol {w} + \\boldsymbol {c} \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The MLM objective is to predict the original tokens which are masked out, formulated as follows, where  $p^{D_1}$  is the discriminator which predicts a token from a large vocabulary.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {L} _ {\\mathrm {M L M}} (\\theta) = \\sum_ {i \\in \\boldsymbol {m} ^ {\\boldsymbol {w}} \\cup \\boldsymbol {m} ^ {\\boldsymbol {c}}} - \\log p ^ {D _ {1}} \\left(x _ {i} \\mid \\boldsymbol {w} ^ {\\text {m a x k e d}}, \\boldsymbol {c} ^ {\\text {m a x k e d}}\\right) \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Objective #2: Replaced Token Detection (RTD) In the MLM objective, only bimodal data (i.e. datapoints of NL-PL pairs) is used for training. Here we present the objective of replaced token detection. The RTD objective (Clark et al., 2020) is originally developed for efficiently learning pre-trained model for natural language. We adapt it in our scenario, with the advantage of using both bimodal and unimodal data for training. Specifically, there are two data generators here, an NL generator  $p^{G_w}$  and a PL generator  $p^{G_c}$ , both for generating plausible alternatives for the set of randomly masked positions.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {w} _ {i} \\sim p ^ {G _ {w}} \\left(w _ {i} \\mid \\boldsymbol {w} ^ {\\text {m a x k e d}}\\right) \\text {f o r} i \\in \\boldsymbol {m} ^ {\\boldsymbol {w}} \\tag {7}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {c} _ {i} \\sim p ^ {G _ {c}} \\left(c _ {i} \\mid c ^ {\\text {m a x k e d}}\\right) \\text {f o r} i \\in \\boldsymbol {m} ^ {\\boldsymbol {c}} \\tag {8}\n$$\n",
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+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} ^ {\\text {c o r r u p t}} = \\operatorname {R E P L A C E} (\\boldsymbol {w}, \\boldsymbol {m} ^ {\\boldsymbol {w}}, \\hat {\\boldsymbol {w}}) \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {c} ^ {\\text {c o r r u p t}} = \\operatorname {R E P L A C E} (\\boldsymbol {c}, \\boldsymbol {m} ^ {\\boldsymbol {c}}, \\hat {\\boldsymbol {c}}) \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {x} ^ {\\text {c o r r u p t}} = \\boldsymbol {w} ^ {\\text {c o r r u p t}} + \\boldsymbol {c} ^ {\\text {c o r r u p t}} \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The discriminator is trained to determine whether a word is the original one or not, which is a binary classification problem. It is worth noting that the RTD objective is applied to every position in the input, and it differs from GAN (generative adversarial network) in that if a generator happens to produce the correct token, the label of that token is \"real\" instead of \"fake\" (Clark et al., 2020). The loss function of RTD with regard to the discriminator parameterized by  $\\theta$  is given below, where  $\\delta(i)$  is",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "an indicator function and  $p^{D_2}$  is the discriminator that predicts the probability of the  $i$ -th word being original.",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathcal {L} _ {\\mathrm {R T D}} (\\theta) = \\sum_ {i = 1} ^ {| \\boldsymbol {w} | + | \\boldsymbol {c} |} \\left(\\delta (i) \\log p ^ {D _ {2}} \\left(\\boldsymbol {x} ^ {\\text {c o r r u p t}}, i\\right) + \\right. \\\\ \\left. \\left(1 - \\delta (i)\\right) \\left(1 - \\log p ^ {D _ {2}} \\left(\\boldsymbol {x} ^ {\\text {c o r r u p t}}, i\\right)\\right)\\right) \\tag {12} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\delta (i) = \\left\\{ \\begin{array}{l l} 1, & \\text {i f} x _ {i} ^ {\\text {c o r r u p t}} = x _ {i}. \\\\ 0, & \\text {o t h e r w i s e .} \\end{array} \\right. \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "There are many different ways to implement the generators. In this work, we implement two efficient n-gram language models (Jurafsky, 2000) with bidirectional contexts, one for NL and one for PL, and learn them from corresponding unimodel datapoints, respectively. The approach is easily generalized to learn bimodal generators or use more complicated generators like Transformer-based neural architecture learned in a joint manner. We leave these to future work. The PL training data is the unimodal codes as shown in Table 1, and the NL training data comes from the documentations from bimodal data. One could easily extend these two training datasets to larger amount. The final loss function are given below.",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\theta} \\mathcal {L} _ {\\mathrm {M L M}} (\\theta) + \\mathcal {L} _ {\\mathrm {R T D}} (\\theta) \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5 Fine-Tuning CodeBERT",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We have different settings to use CodeBERT in downstream NL-PL tasks. For example, in natural language code search, we feed the input as the same way as the pre-training phase and use the representation of [CLS] to measure the semantic relevance between code and natural language query, while in code-to-text generation, we use an encoder-decoder framework and initialize the encoder of a generative model with CodeBERT. Details are given in the experiment section.",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiment",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We present empirical results in this section to verify the effectiveness of CodeBERT. We first describe the use of CodeBERT in natural language code search (§4.1), in a way that model parameters of CodeBERT are fine-tuned. After that, we present the NL-PL probing task (§4.2), and evaluate CodeBERT in a zero-shot setting where the parameters",
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+    {
+        "type": "text",
+        "text": "of CodeBERT are fixed. Finally, we evaluate CodeBERT on a generation problem, i.e. code documentation generation (§4.3), and further evaluate on a programming language which is never seen in the training phase (§4.4).",
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+    {
+        "type": "text",
+        "text": "4.1 Natural Language Code Search",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given a natural language as the input, the objective of code search is to find the most semantically related code from a collection of codes. We conduct experiments on the CodeSearchNet corpus (Husain et al., 2019). We follow the official evaluation metric to calculate the Mean Reciprocal Rank (MRR) for each pair of test data  $(c, w)$  over a fixed set of 999 distractor codes. We further calculate the macro-average MRR for all languages as an overall evaluation metric. It is helpful to note that this metric differs from the AVG metric in the original paper, where the answer is retrieved from candidates from all six languages. We fine-tune a language-specific model for each programming language. We train each model with a binary classification loss function, where a softmax layer is connected to the representation of [CLS]. Both training and validation datasets are created in a way that positive and negative samples are balanced. Negative samples consist of balanced number of instances with randomly replaced NL (i.e.  $(c, \\hat{w})$ ) and PL (i.e.  $(\\hat{c}, w)$ ). Detailed hyper-parameters for model fine-tuning are given in Appendix B.2.",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Model Comparisons Table 2 shows the results of different approaches on the CodeSearchNet corpus. The first four rows are reported by Husain et al. (2019), which are joint embeddings of NL and PL (Gu et al., 2018; Mitra et al., 2018). NBOw represents neural bag-of-words. CNN, BIRNN and SELFATT stand for 1D convolutional neural network (Kim, 2014), bidirectional GRU-based recurrent neural network (Cho et al., 2014), and multi-head attention (Vaswani et al., 2017), respectively.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We report the remaining numbers in Table 2. We train all these pre-trained models by regarding codes as a sequence of tokens. We also continuously train RoBERTa only on codes from CodeSearchNet with masked language modeling. Results show that CodeBERT consistently performs",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "4More details about the dataset are given in Appendix A.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "page_footnote",
+        "text": "5We have fine-tuned a multi-lingual model for six programming languages, but find that it performs worse that fine-tuning a language-specific model for each programming language.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/77c785dec4d5142c066d6c044a04d83665c3edeb214f104ce16205c3ec4d84a7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>RUBY</td><td>JAVASCIPT</td><td>GO</td><td>PYTHON</td><td>JAVA</td><td>PHP</td><td>MA-AVG</td></tr><tr><td>NBOW</td><td>0.4285</td><td>0.4607</td><td>0.6409</td><td>0.5809</td><td>0.5140</td><td>0.4835</td><td>0.5181</td></tr><tr><td>CNN</td><td>0.2450</td><td>0.3523</td><td>0.6274</td><td>0.5708</td><td>0.5270</td><td>0.5294</td><td>0.4753</td></tr><tr><td>BiRNN</td><td>0.0835</td><td>0.1530</td><td>0.4524</td><td>0.3213</td><td>0.2865</td><td>0.2512</td><td>0.2580</td></tr><tr><td>SELFATT</td><td>0.3651</td><td>0.4506</td><td>0.6809</td><td>0.6922</td><td>0.5866</td><td>0.6011</td><td>0.5628</td></tr><tr><td>ROBERTA</td><td>0.6245</td><td>0.6060</td><td>0.8204</td><td>0.8087</td><td>0.6659</td><td>0.6576</td><td>0.6972</td></tr><tr><td>PT w/ CODE ONLY (INIT=s)</td><td>0.5712</td><td>0.5557</td><td>0.7929</td><td>0.7855</td><td>0.6567</td><td>0.6172</td><td>0.6632</td></tr><tr><td>PT w/ CODE ONLY (INIT=R)</td><td>0.6612</td><td>0.6402</td><td>0.8191</td><td>0.8438</td><td>0.7213</td><td>0.6706</td><td>0.7260</td></tr><tr><td>CODEBERT (MLM, INIT=s)</td><td>0.5695</td><td>0.6029</td><td>0.8304</td><td>0.8261</td><td>0.7142</td><td>0.6556</td><td>0.6998</td></tr><tr><td>CODEBERT (MLM, INIT=R)</td><td>0.6898</td><td>0.6997</td><td>0.8383</td><td>0.8647</td><td>0.7476</td><td>0.6893</td><td>0.7549</td></tr><tr><td>CODEBERT (RTD, INIT=R)</td><td>0.6414</td><td>0.6512</td><td>0.8285</td><td>0.8263</td><td>0.7150</td><td>0.6774</td><td>0.7233</td></tr><tr><td>CODEBERT (MLM+RTD, INIT=R)</td><td>0.6926</td><td>0.7059</td><td>0.8400</td><td>0.8685</td><td>0.7484</td><td>0.7062</td><td>0.7603</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Results on natural language code retrieval. Baselines include four joint embeddings (first group) of NL and PL, RoBERTa, and RoBERTa which is continuously trained with masked language modeling on codes only (second group). PT stands for pre-training. We train CodeBERT (third group) with different settings, including using different initialization (from scratch (INIT=S) or initialized with the parameters of RoBERTa (INIT=R)) and using different learning objectives (MLM, RTD, or the combination of both).",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "better than RoBERTa and the model pre-trained with code only. CodeBERT (MLM) learned from scratch performs better than RoBERTa. Unsurprisingly, initializing CodeBERT with RoBERTa improves the performance  $^{6}$ .",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 NL-PL Probing",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the previous subsection, we show the empirical effectiveness of CodeBERT in a setting that the parameters of CodeBERT are fine-tuned in downstream tasks. In this subsection, we further investigate what type of knowledge is learned in CodeBERT without modifying the parameters.",
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+    {
+        "type": "text",
+        "text": "Task Formulation and Data Construction Following the probing experiments in NLP (Petroni et al., 2019; Talmor et al., 2019), we study NL-PL probing here. Since there is no existing work towards this goal, we formulate the problem of NL-PL probing and create the dataset by ourselves. Given an NL-PL pair  $(c, w)$ , the goal of NL-PL probing is to test model's ability to correctly predict/recover the masked token of interest (either a code token  $c_i$  or word token  $w_j$ ) among distractors. There are two major types of distractors: one is the whole target vocabulary used for the masked language modeling objective (Petroni et al., 2019), and another one has fewer candidates which are filter or curated based on experts' understanding about the ability to be tested (Talmor et al., 2019). We follow the second direction and formulate NL-PL probing as a multi-choice question answering task, where the question is cloze-style in which a certain token",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "is replaced by  $[MASK]$  and distractor candidate answers are curated based on our expertise.",
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+    {
+        "type": "text",
+        "text": "Specifically, we evaluate on the NL side and PL side, respectively. To ease the effort of data collection, we collect data automatically from NL-PL pairs in both validation and testing sets of CodeSearchNet, both of which are unseen in the pretraining phase. To evaluate on the NL side, we select NL-PL pairs whose NL documentations include one of the six keywords (max, maximize, min, minimize, less, greater), and group them to four candidates by merging first two keywords and the middle two keywords. The task is to ask pre-trained models to select the correct one instead of three other distractors. That is to say, the input in this setting includes the complete code and a masked NL documentation. The goal is to select the correct answer from four candidates. For the PL side, we select codes containing keywords max and min, and formulate the task as a two-choice answer selection problem. Here, the input includes complete NL documentation and a masked PL code, and the goal is to select the correct answer from two candidates. Since code completion is an important scenario, we would like to test model's ability in predicting the correct token merely based on preceding PL contexts. Therefore, we add an additional setting for PL side, where the input includes the complete NL documentation and preceding PL codes. Data statistics is given in the top two rows in Table 3.",
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+    {
+        "type": "text",
+        "text": "Model Comparisons Results are given in Table 3. We report accuracy, namely the number of correctly predicted instances over the number of all instances, for each programming language. Since",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "page_footnote",
+        "text": "<sup>6</sup>We further give a learning curve of different pre-trained models in the fine-tuning process in Appendix C.",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4e21a6a58fb92d927272dd3e8aa7baf7789615195c2cd594d87db3c1e6d27283.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>RUBY</td><td>JAVASCIPT</td><td>GO</td><td>PYTHON</td><td>JAVA</td><td>PHP</td><td>ALL</td></tr><tr><td colspan=\"8\">NUMBER OF DATAPoints FOR PROBING</td></tr><tr><td>PL (2 CHOICES)</td><td>38</td><td>272</td><td>152</td><td>1,264</td><td>482</td><td>407</td><td>2,615</td></tr><tr><td>NL (4 CHOICES)</td><td>20</td><td>65</td><td>159</td><td>216</td><td>323</td><td>73</td><td>856</td></tr><tr><td colspan=\"8\">PL PROBING</td></tr><tr><td>ROBERTA</td><td>73.68</td><td>63.97</td><td>72.37</td><td>59.18</td><td>59.96</td><td>69.78</td><td>62.45</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>71.05</td><td>77.94</td><td>89.47</td><td>70.41</td><td>70.12</td><td>82.31</td><td>74.11</td></tr><tr><td>CODEBERT (MLM)</td><td>86.84</td><td>86.40</td><td>90.79</td><td>82.20</td><td>90.46</td><td>88.21</td><td>85.66</td></tr><tr><td colspan=\"8\">PL PROBING WITH PRECEDED CONTEXT ONLY</td></tr><tr><td>ROBERTA</td><td>73.68</td><td>53.31</td><td>51.32</td><td>55.14</td><td>42.32</td><td>52.58</td><td>52.24</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>63.16</td><td>48.53</td><td>61.84</td><td>56.25</td><td>58.51</td><td>58.97</td><td>56.71</td></tr><tr><td>CODEBERT (MLM)</td><td>65.79</td><td>50.74</td><td>59.21</td><td>62.03</td><td>54.98</td><td>59.95</td><td>59.12</td></tr><tr><td colspan=\"8\">NL PROBING</td></tr><tr><td>ROBERTA</td><td>50.00</td><td>72.31</td><td>54.72</td><td>61.57</td><td>61.61</td><td>65.75</td><td>61.21</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>55.00</td><td>67.69</td><td>60.38</td><td>68.06</td><td>65.02</td><td>68.49</td><td>65.19</td></tr><tr><td>CODEBERT (MLM)</td><td>65.00</td><td>89.23</td><td>66.67</td><td>76.85</td><td>73.37</td><td>79.45</td><td>74.53</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "datasets in different programming languages are extremely unbalanced, we report the accumulated metric with the same way. We use CodeBERT (MLM) here because its output layer naturally fits for probing. Results show that CodeBERT performs better than baselines on almost all languages on both NL and PL probing. The numbers with only preceding contexts are lower than that with bidirectional contexts, which suggests that code completion is challenging. We leave it as a future work.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We further give a case study on PL-NL probing. We mask NL token and PL token separately, and report the predicted probabilities of RoBERTa and CodeBERT. Figure 3 illustrates the example of a python code<sup>7</sup>. We can see that RoBERTa fails in both cases, whereas CodeBERT makes the correct prediction in both NL and PL settings.",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Code Documentation Generation",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although the pre-training objective of CodeBERT does not include generation-based objectives (Lewis et al., 2019), we would like to investigate to what extent does CodeBERT perform on generation tasks. Specifically, we study code-to-NL generation, and report results for the documentation generation task on CodeSearchNet Corpus in six programming languages. Since the generated documentations are short and higher order n-grams may not overlap, we remedy this problem by using smoothed BLEU score (Lin and Och, 2004).",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e64b87002281d5456c0e8f63166cf4b1450b10633676d16bd7887e1d67514c52.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/95ffa6293d51f95e332e4d63b29adb35fbdbae9cceccbc9a1471bf73efe7deab.jpg",
+        "table_caption": [
+            "Table 3: Statistics of the data for NL-PL probing and the performance of different pre-trained models. Accuracies  $(\\%)$  are reported. Best results in each group are in bold."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td>max</td><td>min</td><td>less</td><td>greater</td></tr><tr><td rowspan=\"2\">NL</td><td>Roberta</td><td>96.24%</td><td>3.73%</td><td>0.02%</td><td>0.01%</td></tr><tr><td>CodeBERT (MLM)</td><td>39.38%</td><td>60.60%</td><td>0.02%</td><td>0.0003%</td></tr><tr><td rowspan=\"2\">PL</td><td>Roberta</td><td>95.85%</td><td>4.15%</td><td>-</td><td>-</td></tr><tr><td>CodeBERT (MLM)</td><td>0.001%</td><td>99.999%</td><td>-</td><td>-</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 3: Case study on python language. Masked tokens in NL (in blue) and PL (in yellow) are separately applied. Predicted probabilities of RoBERTa and CodeBERT are given.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Model Comparisons We compare our model with several baselines, including a RNN-based model with attention mechanism (Sutskever et al., 2014), the Transformer (Vaswani et al., 2017), RoBERTa and the model pre-trained on code only. To demonstrate the effectiveness of CodeBERT on code-to-NL generation tasks, we adopt various pre-trained models as encoders and keep the hyperparameters consistent. Detailed hyper-parameters are given in Appendix B.3.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows the results with different models for the code-to-documentation generation task. As we can see, models pre-trained on programming language outperform RoBERTa, which illustrates that pre-training models on programming",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "page_footnote",
+        "text": "The example comes from https://github.com/peri-source/peri/blob/61beed5deaaf978ab31ed716e8470d86ba639867/peri/comp/psfcalc.py#L994-L1002",
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/14e5d194cead93abaa64c4b1865c1be73b8430e26a11879bb0210ae88aebf97e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>RUBY</td><td>JAVASCIPT</td><td>GO</td><td>PYTHON</td><td>JAVA</td><td>PHP</td><td>OVERALL</td></tr><tr><td>SEQ2SEQ</td><td>9.64</td><td>10.21</td><td>13.98</td><td>15.93</td><td>15.09</td><td>21.08</td><td>14.32</td></tr><tr><td>TRANSFORMER</td><td>11.18</td><td>11.59</td><td>16.38</td><td>15.81</td><td>16.26</td><td>22.12</td><td>15.56</td></tr><tr><td>ROBERTA</td><td>11.17</td><td>11.90</td><td>17.72</td><td>18.14</td><td>16.47</td><td>24.02</td><td>16.57</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>11.91</td><td>13.99</td><td>17.78</td><td>18.58</td><td>17.50</td><td>24.34</td><td>17.35</td></tr><tr><td>CODEBERT (RTD)</td><td>11.42</td><td>13.27</td><td>17.53</td><td>18.29</td><td>17.35</td><td>24.10</td><td>17.00</td></tr><tr><td>CODEBERT (MLM)</td><td>11.57</td><td>14.41</td><td>17.78</td><td>18.77</td><td>17.38</td><td>24.85</td><td>17.46</td></tr><tr><td>CODEBERT (RTD+MLM)</td><td>12.16</td><td>14.90</td><td>18.07</td><td>19.06</td><td>17.65</td><td>25.16</td><td>17.83</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "language could improve code-to-NL generation. Besides, results in the Table 4 show that CodeBERT pre-trained with RTD and MLM objectives brings a gain of 1.3 BLEU score over RoBERTa overall and achieve the state-of-the-art performance $^8$ .",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Generalization to Programming Languages NOT in Pre-training",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We would like to evaluate CodeBERT on the programming language which is never seen in the pretraining step. To this end, we study the task of generating a natural language summary of a C# code snippet. We conduct experiments on the dataset of CodeNN (Iyer et al., 2016)<sup>9</sup>, which consists of 66,015 pairs of questions and answers automatically collected from StackOverflow. This dataset is challenging since the scale of dataset is orders of magnitude smaller than CodeSearchNet Corpus. We evaluate models using smoothed BLEU-4 score and use the same evaluation scripts as Iyer et al. (2016).",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/3622b7d2ef855755e33f25a2c4ae90d6bb132674728c1cd704908b044a093ed2.jpg",
+        "table_caption": [
+            "Table 4: Results on Code-to-Documentation generation, evaluated with smoothed BLEU-4 score."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>BLEU</td></tr><tr><td>MOSES (KOEHN ET AL., 2007)</td><td>11.57</td></tr><tr><td>IR</td><td>13.66</td></tr><tr><td>SUM-NN (RUSH ET AL., 2015)</td><td>19.31</td></tr><tr><td>2-LAYER BILSTM</td><td>19.78</td></tr><tr><td>TRANSFORMER (VASWANI ET AL., 2017)</td><td>19.68</td></tr><tr><td>TREELSTM (TAI ET AL., 2015)</td><td>20.11</td></tr><tr><td>CODENN (IYER ET AL., 2016)</td><td>20.53</td></tr><tr><td>CODE2SEQ (ALON ET AL., 2019)</td><td>23.04</td></tr><tr><td>ROBERTA</td><td>19.81</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>20.65</td></tr><tr><td>CODEBERT (RTD)</td><td>22.14</td></tr><tr><td>CODEBERT (MLM)</td><td>22.32</td></tr><tr><td>CODEBERT (MLM+RTD)</td><td>22.36</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Code-to-NL generation on C# language.",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Model Comparisons Table 5 shows that our model with MLM and RTD pre-training objectives achieves 22.36 BLEU score and improves by 2.55 points over RoBERTa, which illustrates CodeBERT",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "could generalize better to other programming language which is never seen in the pre-training step. However, our model achieve slightly lower results than code2seq (Alon et al., 2019). The main reason could be that code2seq makes use of compositional paths in its abstract syntax tree (AST) while CodeBERT only takes original code as the input. We have trained a version of CodeBERT by traversing the tree structure of AST following a certain order, but applying that model does not bring improvements on generation tasks. This shows a potential direction to improve CodeBERT by incorporating AST.",
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+    {
+        "type": "text",
+        "text": "5 Conclusion",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we present CodeBERT, which to the best of our knowledge is the first large bimodal pre-trained model for natural language and programming language. We train CodeBERT on both bimodal and unimodal data, and show that finetuning CodeBERT achieves state-of-the-art performance on downstream tasks including natural language code search and code-to-documentation generation. To further investigate the knowledge embodied in pre-trained models, we formulate the task of NL-PL probing and create a dataset for probing. We regard the probing task as a cloze-style answer selection problem, and curate distractors for both NL and PL parts. Results show that, with model parameters fixed, CodeBERT performs better than RoBERTa and a continuously trained model using codes only.",
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+        "type": "text",
+        "text": "There are many potential directions for further research on this field. First, one could learn better generators with bimodal evidence or more complicated neural architecture to improve the replaced token detection objective. Second, the loss functions of CodeBERT mainly target on NL-PL understanding tasks. Although CodeBERT achieves strong BLEU scores on code-to-documentation generation, the CodeBERT itself could be further improved by generation-related learning objectives.",
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+        "type": "page_footnote",
+        "text": "<sup>8</sup>We further give some output examples in Appendix E.",
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+    {
+        "type": "page_footnote",
+        "text": "$^{9}$ https://github.com/sriniyer/codenn",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "How to successfully incorporate AST into the pretraining step is also an attractive direction. Third, we plan to apply CodeBERT to more NL-PL related tasks, and extend it to more programming languages. Flexible and powerful domain/language adaptation methods are necessary to generalize well.",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Xiaocheng Feng is the corresponding author of this work. We thank the anonymous reviewers for their insightful comments. Zhangyin Feng, Xiaocheng Feng, Bing Qin and Ting Liu are supported by the National Key R&D Program of China via grant 2018YFB1005103 and National Natural Science Foundation of China (NSFC) via grant 61632011 and 61772156.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            115,
+            382,
+            215,
+            398
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Uri Alon, Shaked Brody, Omer Levy, and Eran Yahav. 2019. code2seq: Generating sequences from structured representations of code. International Conference on Learning Representations.",
+            "Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. 2014. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078.",
+            "Kevin Clark, Minh-Thang Luong, Quoc V. Le, and Christopher D. Manning. 2020. {ELECTRA}: Pretraining text encoders as discriminators rather than generators. In International Conference on Learning Representations.",
+            "Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805.",
+            "Xiaodong Gu, Hongyu Zhang, and Sunghun Kim. 2018. Deep code search. In 2018 IEEE/ACM 40th International Conference on Software Engineering (ICSE), pages 933-944. IEEE.",
+            "Hamel Husain, Ho-Hsiang Wu, Tiferet Gazit, Miltiadis Allamanis, and Marc Brockschmidt. 2019. Code-searchnet challenge: Evaluating the state of semantic code search. arXiv preprint arXiv:1909.09436.",
+            "Srinivasan Iyer, Ioannis Konstas, Alvin Cheung, and Luke Zettlemoyer. 2016. Summarizing source code using a neural attention model. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2073-2083."
+        ],
+        "bbox": [
+            117,
+            406,
+            492,
+            910
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Dan Jurafsky. 2000. Speech & language processing. Pearson Education India.",
+            "Aditya Kanade, Petros Maniatis, Gogul Balakrishnan, and Kensen Shi. 2019. Pre-trained contextual embedding of source code. arXiv preprint arXiv:2001.00059.",
+            "Yoon Kim. 2014. Convolutional neural networks for sentence classification. arXiv preprint arXiv:1408.5882.",
+            "Philipp Koehn, Hieu Hoang, Alexandra Birch, Chris Callison-Burch, Marcello Federico, Nicola Bertoldi, Brooke Cowan, Wade Shen, Christine Moran, Richard Zens, et al. 2007. Moses: Open source toolkit for statistical machine translation. In Proceedings of the 45th annual meeting of the association for computational linguistics companion volume proceedings of the demo and poster sessions, pages 177-180.",
+            "Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Ves Stoyanov, and Luke Zettlemoyer. 2019. Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension. arXiv preprint arXiv:1910.13461.",
+            "Chin-Yew Lin and Franz Josef Och. 2004. Orange: a method for evaluating automatic evaluation metrics for machine translation. In Proceedings of the 20th international conference on Computational Linguistics, page 501. Association for Computational Linguistics.",
+            "Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692.",
+            "Jiasen Lu, Dhruv Batra, Devi Parikh, and Stefan Lee. 2019. Vilbert: Pretraining task-agnostic visi-olinguistic representations for vision-and-language tasks. In Advances in Neural Information Processing Systems, pages 13-23.",
+            "Bhaskar Mitra, Nick Craswell, et al. 2018. An introduction to neural information retrieval. Foundations and Trends in Information Retrieval, 13(1):1-126.",
+            "Matthew E Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. 2018. Deep contextualized word representations. arXiv preprint arXiv:1802.05365.",
+            "Fabio Petroni, Tim Roktaschel, Patrick Lewis, Anton Bakhtin, Yuxiang Wu, Alexander H Miller, and Sebastian Riedel. 2019. Language models as knowledge bases? arXiv preprint arXiv:1909.01066.",
+            "Telmo Pires, Eva Schlinger, and Dan Garrette. 2019. How multilingual is multilingual bert? arXiv preprint arXiv:1906.01502."
+        ],
+        "bbox": [
+            512,
+            76,
+            884,
+            909
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. 2018. Improving language understanding by generative pre-training. URL https://s3-us-west-2. amazonaws. com/openai-assetss/researchcovers/languageunsupervised/language understanding paper.pdf.",
+        "bbox": [
+            117,
+            74,
+            505,
+            156
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2019. Exploring the limits of transfer learning with a unified text-to-text transformer. arXiv preprint arXiv:1910.10683.",
+        "bbox": [
+            115,
+            166,
+            490,
+            233
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alexander M Rush, Sumit Chopra, and Jason Weston. 2015. A neural attention model for abstractive sentence summarization. arXiv preprint arXiv:1509.00685.",
+        "bbox": [
+            115,
+            244,
+            490,
+            298
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chen Sun, Austin Myers, Carl Vondrick, Kevin Murphy, and Cordelia Schmid. 2019. Videobert: A joint model for video and language representation learning. arXiv preprint arXiv:1904.01766.",
+        "bbox": [
+            115,
+            309,
+            490,
+            363
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ilya Sutskever, Oriol Vinyals, and Quoc V Le. 2014. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pages 3104-3112.",
+        "bbox": [
+            115,
+            374,
+            490,
+            429
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kai Sheng Tai, Richard Socher, and Christopher D Manning. 2015. Improved semantic representations from tree-structured long short-term memory networks. arXiv preprint arXiv:1503.00075.",
+        "bbox": [
+            115,
+            439,
+            490,
+            494
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alon Talmor, Yanai Elazar, Yoav Goldberg, and Jonathan Berant. 2019. olmpics-on what language model pre-training captures. arXiv preprint arXiv:1912.13283.",
+        "bbox": [
+            115,
+            505,
+            490,
+            558
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In Advances in neural information processing systems, pages 5998-6008.",
+        "bbox": [
+            115,
+            570,
+            490,
+            638
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. 2016. Google's neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144.",
+        "bbox": [
+            115,
+            648,
+            490,
+            740
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Ruslan Salakhutdinov, and Quoc V Le. 2019. Xlnet: Generalized autoregressive pretraining for language understanding. arXiv preprint arXiv:1906.08237.",
+        "bbox": [
+            115,
+            752,
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+            819
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A Data Statistic",
+        "text_level": 1,
+        "bbox": [
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+            834,
+            272,
+            850
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Data statistics of the training/validation/testing data splits for six programming languages are given in Table 6.",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/d61832405592abb2341be1cdb3cd0b0393b5d764cb3de70c977453d04e0433d8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CODE SEARCH</td><td>TRAINING</td><td>DEV</td><td>TESTING</td></tr><tr><td>GO</td><td>635,635</td><td>28,483</td><td>14,291</td></tr><tr><td>JAVA</td><td>908,886</td><td>30,655</td><td>26,909</td></tr><tr><td>JAVAscript</td><td>247,773</td><td>16,505</td><td>6,483</td></tr><tr><td>PHP</td><td>1,047,406</td><td>52,029</td><td>28,391</td></tr><tr><td>PYTHON</td><td>824,342</td><td>46,213</td><td>22,176</td></tr><tr><td>RUBY</td><td>97,580</td><td>4,417</td><td>2,279</td></tr></table>",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Data statistics about the CodeSearchNet Corpus for natural language code search.",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B Train Details",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B.1 Pre-training",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We train CodeBERT on one NVIDIA DGX-2 machine using FP16. It combines 16 interconnected NVIDIA Tesla V100 with 32GB memory. We use the following set of hyper-parameters to train models: batchsize is 2,048 and learning rate is 5e-4. We use Adam to update the parameters and set the number of warmup steps as 10K. We set the max length as 512 and the max training step is 100K. Training 1,000 batches of data costs 600 minutes with MLM objective, 120 minutes with RTD objective.",
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+    },
+    {
+        "type": "text",
+        "text": "B.2 CodeSearch",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the fine-turning step, we set the learning rate as 1e-5, the batch size as 64, the max sequence length as 200 and the max fine-tuning epoch as 8. As the same with pre-training, We use Adam to update the parameters. We choose the model performed best on the development set, and use that to evaluate on the test set.",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B.3 Code Summarization on Six Programming Languages",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We use Transformer with 6 layers, 768 dimensional hidden states and 12 attention heads as our decoder in all settings. We set the max length of input and inference as 256 and 64, respectively. We use the Adam optimizer to update model parameters. The learning rate and the batch size are 5e-5 and 64, respectively. We tune hyperparameters and perform early stopping on the development set.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "B.4 Code Summarization on C#",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Since state-of-the-art methods use RNN as their decoder, we choose a 2-layer GRU with an attention mechanism as our decoder for a comparison. We fine-tune models using a grid search with the following set of hyper-parameters: batchsize is in \\{32, 64\\} and learning rate is in \\{2e-5, 5e-5\\}. We report",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "the number when models achieve best performance on the development set.",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C Learning Curve of CodeSearch",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "From Figure 4, we can see that CodeBERT performs better at the early stage, which reflects that CodeBERT provides good initialization for learning downstream tasks.",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/db6357b1d8c49497159e567595a575c4f04bc6f76a6f358063976b8a202bf0e3.jpg",
+        "image_caption": [
+            "Figure 4: Learning curve of different pre-trained models in the fine-tuning step. We show results on Python and Java."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/556ef6f6bb3c91bfb2d76dac20414ec4366486c6b2cb0ee0cbd706798c5ef43f.jpg",
+        "image_caption": [],
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+    },
+    {
+        "type": "text",
+        "text": "D Late Fusion",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In section §4.1, we show that CodeBERT performs well in the setting where natural languages and codes have early interactions. Here, we investigate whether CodeBERT is good at working as a unified encoder. We apply CodeBERT for natural language code search in a later fusion setting, where CodeBERT first encodes NL and PL separately, and then calculates the similarity by dot-product. In this way, code search is equivalent to find the nearest codes in the shared vector space. This scenario also facilitates the use of CodeBERT in an online system, where the representations of codes are calculated in advance. In the runtime, a system only needs to compute the representation of NL and vector-based dot-product.",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We fine-tune CodeBERT with the following objective, which maximizes the dot-product of the ground truth while minimizing the dot-product of distractors.",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n- \\frac {1}{N} \\sum_ {i} \\log \\left(\\frac {\\exp \\left(E n c \\left(c _ {i}\\right) ^ {\\intercal} E n c \\left(w _ {i}\\right)\\right)}{\\sum_ {j} \\exp \\left(E n c \\left(c _ {j}\\right) ^ {\\intercal} E n c \\left(w _ {i}\\right)\\right)}\\right) \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Results are given in Table 7. We just do this setting on two languages with a relatively small amount of data.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We can see that CodeBERT performs better than RoBERTa and the model pre-trained with codes",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/0e8033c6c9e05c656ae76855c1b4fb81fc3e40848531274b4db7a86001c4c3ae.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>RUBY</td><td>GO</td></tr><tr><td>ROBERTA</td><td>0.0043</td><td>0.0030</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>0.1648</td><td>0.4179</td></tr><tr><td>CODEBERT</td><td>0.6870</td><td>0.8372</td></tr></table>",
+        "bbox": [
+            531,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 7: Results on natural language code search by late fusion.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "only. And late fusion performs comparable with the standard way. What's more, late fusion is more efficient and this setting could be used in an online system.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "E Case Study",
+        "text_level": 1,
+        "bbox": [
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+            282,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To qualitatively analyze the effectiveness of CodeBERT, we give some cases for code search and code documentation generation tasks.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Considering the limited space, we only give the top2 results of the query for python programming language. As show in Figure 5, search results are very relevant with query.",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 and Figure 7 show the outputs with different models for the code documentation generation task. As we can see, CodeBERT performs better than all baselines.",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [
+            "Query"
+        ],
+        "code_body": "create file and write something",
+        "guess_lang": "txt",
+        "bbox": [
+            152,
+            93,
+            327,
+            105
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [
+            "Search Results (top2)"
+        ],
+        "code_body": "https://github.com/darknessomi/musicbox/blob/master/NEMbox/util.py#L37-L40   \ndef create_file(path, default  $=$  \"\\\\n\"): if not os.path.exists(path): with open(path, \"w\") as f: f.write(default)",
+        "guess_lang": "python",
+        "bbox": [
+            152,
+            130,
+            631,
+            216
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [],
+        "code_body": "https://github.com/datakortet/yamldirs/blob/master/yamldirs/filemaker.py#L114-L118",
+        "guess_lang": "txt",
+        "bbox": [
+            154,
+            237,
+            631,
+            250
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [],
+        "code_body": "def make_file(self, filename, content):\n    '''Create a new file with name``filename``and content``content``.\n    '''with open(filename,'w') as fp:\n        fp.write(content)",
+        "guess_lang": "python",
+        "bbox": [
+            154,
+            260,
+            618,
+            319
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 5: Python CodeSearch example. The results are searched from 1,156,085 python code data. We only give the top2 results because space is limited.",
+        "bbox": [
+            114,
+            338,
+            882,
+            367
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [
+            "Gold: Add a write error result"
+        ],
+        "code_body": "public void addWriteErrorResult(final BulkWriteError writeError, final IndexMap indexMap) {\n    notNull(\"writeError\", writeError);\n    mergeWriteErrors(asList(writeError), indexMap);\n}",
+        "guess_lang": "java",
+        "bbox": [
+            149,
+            387,
+            505,
+            445
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [],
+        "code_body": "CodeBERT: Add a write error result.  \nPRE-TRAIN W/ CODEONLY : Merges the given write error.  \nRoberta: Add a write operation to the map.  \nTransformer: Adds an error to the write map.",
+        "guess_lang": "txt",
+        "bbox": [
+            151,
+            481,
+            482,
+            543
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [],
+        "code_body": "RNN: Add an error map.",
+        "guess_lang": "txt",
+        "bbox": [
+            152,
+            546,
+            287,
+            558
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 6: Java code documentation generation output example.",
+        "bbox": [
+            285,
+            576,
+            712,
+            590
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [],
+        "code_body": "def create_or_update(self, list_id, subscriber_hash, data):  \n    subscriber_hash = check subscribed_hash(subscriber_hash)  \n    self.list_id = list_id  \n    self subscribers_hash = subscriber_hash  \n    if 'email_address' not in data:  \n        raise ValueError('The list member must have an email_address')  \n    check_email(data['email_address'])  \n    if 'status_if_new' not in data:  \n        raise ValueError('The list member must have a status_if_new')  \n    if data['status_if_new'] not in [' subscribed', 'un subscribed', 'cleaned', 'pending', 'transactional']:  \n        raise ValueError('The list member status_if_new must be one of \" subscribed\", \"un subscribed\", \"cleaned\", \"pending\", or \"transactional\"))  \n    return self._mc_client._put(url= self._build_path(list_id, 'members', subscriber_hash), data=data)",
+        "guess_lang": "python",
+        "bbox": [
+            152,
+            612,
+            684,
+            757
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [
+            "Gold: Add or update a list member."
+        ],
+        "code_body": "CodeBERT: Create or update a list member.  \nPRE-TRAIN W/ CODEONLY: Create or update a subscriber.  \nRoberta: Create or update an existing record.  \nTransformer: Create or update a subscription.  \nRNN: Creates or updates an email address.",
+        "guess_lang": "txt",
+        "bbox": [
+            154,
+            790,
+            477,
+            868
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 7: Python code documentation generation output example.",
+        "bbox": [
+            275,
+            883,
+            722,
+            898
+        ],
+        "page_idx": 11
+    }
+]

2002.08xxx/2002.08155/9ddac694-f769-4a0a-acb8-3d4e0835ba7b_origin.pdf

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2002.08xxx/2002.08155/full.md

@@ -0,0 +1,428 @@
+# CodeBERT: A Pre-Trained Model for Programming and Natural Languages
+
+Zhangyin Feng $^{1*}$ , Daya Guo $^{2*}$ , Duyu Tang $^{3}$ , Nan Duan $^{3}$ , Xiaocheng Feng $^{1}$ , Ming Gong $^{4}$ , Linjun Shou $^{4}$ , Bing Qin $^{1}$ , Ting Liu $^{1}$ , Daxin Jiang $^{4}$ , Ming Zhou $^{3}$
+
+<sup>1</sup> Research Center for Social Computing and Information Retrieval, Harbin Institute of Technology, China
+
+2 The School of Data and Computer Science, Sun Yat-sen University, China
+
+<sup>3</sup> Microsoft Research Asia, Beijing, China
+
+<sup>4</sup> Microsoft Search Technology Center Asia, Beijing, China
+
+{zyfeng,xcfeng,qinb,tliu}@ir.hit.edu.cn
+
+guody5@mail2.sysu.edu.cn
+
+{dutang,nanduan,migon,lisho,djiang,mingzhou}@microsoft.com
+
+# Abstract
+
+We present CodeBERT, a bimodal pre-trained model for programming language (PL) and natural language (NL). CodeBERT learns general-purpose representations that support downstream NL-PL applications such as natural language code search, code documentation generation, etc. We develop CodeBERT with Transformer-based neural architecture, and train it with a hybrid objective function that incorporates the pre-training task of replaced token detection, which is to detect plausible alternatives sampled from generators. This enables us to utilize both "bimodal" data of NL-PL pairs and "unimodal" data, where the former provides input tokens for model training while the latter helps to learn better generators. We evaluate CodeBERT on two NL-PL applications by fine-tuning model parameters. Results show that CodeBERT achieves state-of-the-art performance on both natural language code search and code documentation generation. Furthermore, to investigate what type of knowledge is learned in CodeBERT, we construct a dataset for NL-PL probing, and evaluate in a zero-shot setting where parameters of pre-trained models are fixed. Results show that CodeBERT performs better than previous pre-trained models on NL-PL probing.<sup>1</sup>
+
+# 1 Introduction
+
+Large pre-trained models such as ELMo (Peters et al., 2018), GPT (Radford et al., 2018), BERT (Devlin et al., 2018), XLNet (Yang et al., 2019)
+
+and RoBERTa (Liu et al., 2019) have dramatically improved the state-of-the-art on a variety of natural language processing (NLP) tasks. These pre-trained models learn effective contextual representations from massive unlabeled text optimized by self-supervised objectives, such as masked language modeling, which predicts the original masked word from an artificially masked input sequence. The success of pre-trained models in NLP also drives a surge of multi-modal pre-trained models, such as ViBERT (Lu et al., 2019) for language-image and VideoBERT (Sun et al., 2019) for language-video, which are learned from bimodal data such as language-image pairs with bimodal self-supervised objectives.
+
+In this work, we present CodeBERT, a bimodal pre-trained model for natural language (NL) and programming language (PL) like Python, Java, JavaScript, etc. CodeBERT captures the semantic connection between natural language and programming language, and produces general-purpose representations that can broadly support NL-PL understanding tasks (e.g. natural language code search) and generation tasks (e.g. code documentation generation). It is developed with the multilayer Transformer (Vaswani et al., 2017), which is adopted in a majority of large pre-trained models. In order to make use of both bimodal instances of NL-PL pairs and large amount of available unimodal codes, we train CodeBERT with a hybrid objective function, including standard masked language modeling (Devlin et al., 2018) and replaced token detection (Clark et al., 2020), where unimodal codes help to learn better generators for producing better alternative tokens for the latter objective.
+
+We train CodeBERT from Github code reposito
+
+ries in 6 programming languages, where bimodal datapoints are codes that pair with function-level natural language documentations (Husain et al., 2019). Training is conducted in a setting similar to that of multilingual BERT (Pires et al., 2019), in which case one pre-trained model is learned for 6 programming languages with no explicit markers used to denote the input programming language. We evaluate CodeBERT on two downstream NL-PL tasks, including natural language code search and code documentation generation. Results show that fine-tuning the parameters of CodeBERT achieves state-of-the-art performance on both tasks. To further investigate what type of knowledge is learned in CodeBERT, we construct a dataset for NL-PL probing, and test CodeBERT in a zero-shot scenario, i.e. without fine-tuning the parameters of CodeBERT. We find that CodeBERT consistently outperforms RoBERTa, a purely natural language-based pre-trained model. The contributions of this work are as follows:
+
+- CodeBERT is the first large NL-PL pretrained model for multiple programming languages.  
+- Empirical results show that CodeBERT is effective in both code search and code-to-text generation tasks.  
+- We further created a dataset which is the first one to investigate the probing ability of the code-based pre-trained models.
+
+# 2 Background
+
+# 2.1 Pre-Trained Models in NLP
+
+Large pre-trained models (Peters et al., 2018; Radford et al., 2018; Devlin et al., 2018; Yang et al., 2019; Liu et al., 2019; Raffel et al., 2019) have brought dramatic empirical improvements on almost every NLP task in the past few years. Successful approaches train deep neural networks on large-scale plain texts with self-supervised learning objectives. One of the most representative neural architectures is the Transformer (Vaswani et al., 2017), which is also the one used in this work. It contains multiple self-attention layers, and can be conventionally learned with gradient decent in an end-to-end manner as every component is differentiable. The terminology "self-supervised" means that supervisions used for pre-training are automatically collected from raw data without manual
+
+annotation. Dominant learning objectives are language modeling and its variations. For example, in GPT (Radford et al., 2018), the learning objective is language modeling, namely predicting the next word  $w_{k}$  given the preceding context words  $\{w_{1}, w_{2}, \dots, w_{k-1}\}$ . As the ultimate goal of pretraining is not to train a good language model, it is desirable to consider both preceding and following contexts to learn better general-purpose contextual representations. This leads us to the masked language modeling objective used in BERT (Devlin et al., 2018), which learns to predict the masked words of a randomly masked word sequence given surrounding contexts. Masked language modeling is also used as one of the two learning objectives for training CodeBERT.
+
+# 2.2 Multi-Modal Pre-Trained Models
+
+The remarkable success of the pre-trained model in NLP has driven the development of multi-modal pre-trained model that learns implicit alignment between inputs of different modalities. These models are typically learned from bimodal data, such as pairs of language-image or pairs of language-video. For example, ViLBERT (Lu et al., 2019) learns from image caption data, where the model learns by reconstructing categories of masked image region or masked words given the observed inputs, and meanwhile predicting whether the caption describes the image content or not. Similarly, VideoBERT (Sun et al., 2019) learns from language-video data and is trained by video and text masked token prediction. Our work belongs to this line of research as we regard NL and PL as different modalities. Our method differs from previous works in that the fuels for model training include not only bimodal data of NL-PL pairs, but larger amounts of unimodal data such as codes without paired documentations.
+
+A concurrent work (Kanade et al., 2019) uses masked language modeling and next sentence prediction as the objective to train a BERT model on Python source codes, where a sentence is a logical code line as defined by the Python standard. In terms of the pre-training process, CodeBERT differs from their work in that (1) CodeBERT is trained in a cross-modal style and leverages both bimodal NL-PL data and unimodal PL/NL data, (2) CodeBERT is pre-trained over six programming languages, and (3) CodeBERT is trained with a new learning objective based on replaced token
+
+detection.
+
+# 3 CodeBERT
+
+We describe the details about CodeBERT in this section, including the model architecture, the input and output representations, the objectives and data used for training CodeBERT, and how to fine-tune CodeBERT when it is applied to downstream tasks.
+
+# 3.1 Model Architecture
+
+We follow BERT (Devlin et al., 2018) and RoBERTa (Liu et al., 2019), and use multi-layer bidirectional Transformer (Vaswani et al., 2017) as the model architecture of CodeBERT. We will not review the ubiquitous Transformer architecture in detail. We develop CodeBERT by using exactly the same model architecture as RoBERTa-base. The total number of model parameters is 125M.
+
+# 3.2 Input/Output Representations
+
+In the pre-training phase, we set the input as the concatenation of two segments with a special separator token, namely [CLS],  $w_{1}, w_{2}, \ldots, w_{n}$ , [SEP],  $c_{1}, c_{2}, \ldots, c_{m}$ , [EOS]. One segment is natural language text, and another is code from a certain programming language. [CLS] is a special token in front of the two segments, whose final hidden representation is considered as the aggregated sequence representation for classification or ranking. Following the standard way of processing text in Transformer, we regard a natural language text as a sequence of words, and split it as WordPiece (Wu et al., 2016). We regard a piece of code as a sequence of tokens.
+
+The output of CodeBERT includes (1) contextual vector representation of each token, for both natural language and code, and (2) the representation of [CLS], which works as the aggregated sequence representation.
+
+# 3.3 Pre-Training Data
+
+We train CodeBERT with both bimodal data, which refers to parallel data of natural language-code pairs, and unimodal data, which stands for codes without paired natural language texts and natural language without paired codes.
+
+We use datapoints from Github repositories, where each bimodal datapoint is an individual function with paired documentation, and each uni-modal code is a function without paired documentation. Specifically, we use a recent large dataset
+
+<table><tr><td>TRAINING DATA</td><td>bimodal DATA</td><td>unimodal CODES</td></tr><tr><td>GO</td><td>319,256</td><td>726,768</td></tr><tr><td>JAVA</td><td>500,754</td><td>1,569,889</td></tr><tr><td>JAVAscript</td><td>143,252</td><td>1,857,835</td></tr><tr><td>PHP</td><td>662,907</td><td>977,821</td></tr><tr><td>PYTHON</td><td>458,219</td><td>1,156,085</td></tr><tr><td>RUBY</td><td>52,905</td><td>164,048</td></tr><tr><td>ALL</td><td>2,137,293</td><td>6,452,446</td></tr></table>
+
+Table 1: Statistics of the dataset used for training CodeBERT.
+
+provided by Husain et al. (2019), which includes 2.1M bimodal datapoints and 6.4M unimodal codes across six programming languages (Python, Java, JavaScript, PHP, Ruby, and Go). Data statistics is shown in Table 1. $^{2}$
+
+The data comes from publicly available open-source non-fork GitHub repositories and are filtered with a set of constraints and rules. For example, (1) each project should be used by at least one other project, (2) each documentation is truncated to the first paragraph, (3) documentations shorter than three tokens are removed, (4) functions shorter than three lines are removed, and (5) function names with substring "test" are removed. An example of the data is given in Figure 1<sup>3</sup>.
+
+![](images/dfb9fdee528028c1c0c0eb591cfa11540dc07eb3ad1fb80615b6a3322ebad838.jpg)  
+Figure 1: An example of the NL-PL pair, where NL is the first paragraph (filled in red) from the documentation (dashed line in black) of a function.
+
+# 3.4 Pre-Training CodeBERT
+
+We describe the two objectives used for training CodeBERT here. The first objective is masked language modeling (MLM), which has proven effective in literature (Devlin et al., 2018; Liu et al.,
+
+![](images/3e79d5432408f9c7354ee138d1a252282bbef5871530c68ea0ba9fc7e14a7ad9.jpg)  
+Figure 2: An illustration about the replaced token detection objective. Both NL and code generators are language models, which generate plausible tokens for masked positions based on surrounding contexts. NL-Code discriminator is the targeted pre-trained model, which is trained via detecting plausible alternatives tokens sampled from NL and PL generators. NL-Code discriminator is used for producing general-purpose representations in the fine-tuning step. Both NL and code generators are thrown out in the fine-tuning step.
+
+2019; Sun et al., 2019). We apply masked language modeling on bimodal data of NL-PL pairs. The second objective is replaced token detection (RTD), which further uses a large amount of unimodal data, such as codes without paired natural language texts. Detailed hyper-parameters for model pre-training are given in Appendix B.1.
+
+Objective #1: Masked Language Modeling (MLM) Given a datapoint of NL-PL pair  $(x = \{w, c\})$  as input, where  $w$  is a sequence of NL words and  $c$  is a sequence of PL tokens, we first select a random set of positions for both NL and PL to mask out (i.e.  $m^w$  and  $m^c$ , respectively), and then replace the selected positions with a special [MASK] token. Following Devlin et al. (2018),  $15\%$  of the tokens from  $x$  are masked out.
+
+$$
+m _ {i} ^ {w} \sim \operatorname {u n i f} \{1, | \boldsymbol {w} | \} \text {f o r} i = 1 \text {t o} | \boldsymbol {w} | \tag {1}
+$$
+
+$$
+m _ {i} ^ {c} \sim \operatorname {u n i f} \{1, | c | \} \text {f o r} i = 1 \text {t o} | c | \tag {2}
+$$
+
+$$
+\boldsymbol {w} ^ {\text {m a s k e d}} = \operatorname {R E P L A C E} \left(\boldsymbol {w}, \boldsymbol {m} ^ {\boldsymbol {w}}, [ M A S K ]\right) \tag {3}
+$$
+
+$$
+\boldsymbol {c} ^ {\text {m a s k e d}} = \operatorname {R E P L A C E} (\boldsymbol {c}, \boldsymbol {m} ^ {\boldsymbol {c}}, [ M A S K ]) \tag {4}
+$$
+
+$$
+\boldsymbol {x} = \boldsymbol {w} + \boldsymbol {c} \tag {5}
+$$
+
+The MLM objective is to predict the original tokens which are masked out, formulated as follows, where  $p^{D_1}$  is the discriminator which predicts a token from a large vocabulary.
+
+$$
+\mathcal {L} _ {\mathrm {M L M}} (\theta) = \sum_ {i \in \boldsymbol {m} ^ {\boldsymbol {w}} \cup \boldsymbol {m} ^ {\boldsymbol {c}}} - \log p ^ {D _ {1}} \left(x _ {i} \mid \boldsymbol {w} ^ {\text {m a x k e d}}, \boldsymbol {c} ^ {\text {m a x k e d}}\right) \tag {6}
+$$
+
+Objective #2: Replaced Token Detection (RTD) In the MLM objective, only bimodal data (i.e. datapoints of NL-PL pairs) is used for training. Here we present the objective of replaced token detection. The RTD objective (Clark et al., 2020) is originally developed for efficiently learning pre-trained model for natural language. We adapt it in our scenario, with the advantage of using both bimodal and unimodal data for training. Specifically, there are two data generators here, an NL generator  $p^{G_w}$  and a PL generator  $p^{G_c}$ , both for generating plausible alternatives for the set of randomly masked positions.
+
+$$
+\hat {w} _ {i} \sim p ^ {G _ {w}} \left(w _ {i} \mid \boldsymbol {w} ^ {\text {m a x k e d}}\right) \text {f o r} i \in \boldsymbol {m} ^ {\boldsymbol {w}} \tag {7}
+$$
+
+$$
+\hat {c} _ {i} \sim p ^ {G _ {c}} \left(c _ {i} \mid c ^ {\text {m a x k e d}}\right) \text {f o r} i \in \boldsymbol {m} ^ {\boldsymbol {c}} \tag {8}
+$$
+
+$$
+\boldsymbol {w} ^ {\text {c o r r u p t}} = \operatorname {R E P L A C E} (\boldsymbol {w}, \boldsymbol {m} ^ {\boldsymbol {w}}, \hat {\boldsymbol {w}}) \tag {9}
+$$
+
+$$
+\boldsymbol {c} ^ {\text {c o r r u p t}} = \operatorname {R E P L A C E} (\boldsymbol {c}, \boldsymbol {m} ^ {\boldsymbol {c}}, \hat {\boldsymbol {c}}) \tag {10}
+$$
+
+$$
+\boldsymbol {x} ^ {\text {c o r r u p t}} = \boldsymbol {w} ^ {\text {c o r r u p t}} + \boldsymbol {c} ^ {\text {c o r r u p t}} \tag {11}
+$$
+
+The discriminator is trained to determine whether a word is the original one or not, which is a binary classification problem. It is worth noting that the RTD objective is applied to every position in the input, and it differs from GAN (generative adversarial network) in that if a generator happens to produce the correct token, the label of that token is "real" instead of "fake" (Clark et al., 2020). The loss function of RTD with regard to the discriminator parameterized by  $\theta$  is given below, where  $\delta(i)$  is
+
+an indicator function and  $p^{D_2}$  is the discriminator that predicts the probability of the  $i$ -th word being original.
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {R T D}} (\theta) = \sum_ {i = 1} ^ {| \boldsymbol {w} | + | \boldsymbol {c} |} \left(\delta (i) \log p ^ {D _ {2}} \left(\boldsymbol {x} ^ {\text {c o r r u p t}}, i\right) + \right. \\ \left. \left(1 - \delta (i)\right) \left(1 - \log p ^ {D _ {2}} \left(\boldsymbol {x} ^ {\text {c o r r u p t}}, i\right)\right)\right) \tag {12} \\ \end{array}
+$$
+
+$$
+\delta (i) = \left\{ \begin{array}{l l} 1, & \text {i f} x _ {i} ^ {\text {c o r r u p t}} = x _ {i}. \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {13}
+$$
+
+There are many different ways to implement the generators. In this work, we implement two efficient n-gram language models (Jurafsky, 2000) with bidirectional contexts, one for NL and one for PL, and learn them from corresponding unimodel datapoints, respectively. The approach is easily generalized to learn bimodal generators or use more complicated generators like Transformer-based neural architecture learned in a joint manner. We leave these to future work. The PL training data is the unimodal codes as shown in Table 1, and the NL training data comes from the documentations from bimodal data. One could easily extend these two training datasets to larger amount. The final loss function are given below.
+
+$$
+\min  _ {\theta} \mathcal {L} _ {\mathrm {M L M}} (\theta) + \mathcal {L} _ {\mathrm {R T D}} (\theta) \tag {14}
+$$
+
+# 3.5 Fine-Tuning CodeBERT
+
+We have different settings to use CodeBERT in downstream NL-PL tasks. For example, in natural language code search, we feed the input as the same way as the pre-training phase and use the representation of [CLS] to measure the semantic relevance between code and natural language query, while in code-to-text generation, we use an encoder-decoder framework and initialize the encoder of a generative model with CodeBERT. Details are given in the experiment section.
+
+# 4 Experiment
+
+We present empirical results in this section to verify the effectiveness of CodeBERT. We first describe the use of CodeBERT in natural language code search (§4.1), in a way that model parameters of CodeBERT are fine-tuned. After that, we present the NL-PL probing task (§4.2), and evaluate CodeBERT in a zero-shot setting where the parameters
+
+of CodeBERT are fixed. Finally, we evaluate CodeBERT on a generation problem, i.e. code documentation generation (§4.3), and further evaluate on a programming language which is never seen in the training phase (§4.4).
+
+# 4.1 Natural Language Code Search
+
+Given a natural language as the input, the objective of code search is to find the most semantically related code from a collection of codes. We conduct experiments on the CodeSearchNet corpus (Husain et al., 2019). We follow the official evaluation metric to calculate the Mean Reciprocal Rank (MRR) for each pair of test data  $(c, w)$  over a fixed set of 999 distractor codes. We further calculate the macro-average MRR for all languages as an overall evaluation metric. It is helpful to note that this metric differs from the AVG metric in the original paper, where the answer is retrieved from candidates from all six languages. We fine-tune a language-specific model for each programming language. We train each model with a binary classification loss function, where a softmax layer is connected to the representation of [CLS]. Both training and validation datasets are created in a way that positive and negative samples are balanced. Negative samples consist of balanced number of instances with randomly replaced NL (i.e.  $(c, \hat{w})$ ) and PL (i.e.  $(\hat{c}, w)$ ). Detailed hyper-parameters for model fine-tuning are given in Appendix B.2.
+
+Model Comparisons Table 2 shows the results of different approaches on the CodeSearchNet corpus. The first four rows are reported by Husain et al. (2019), which are joint embeddings of NL and PL (Gu et al., 2018; Mitra et al., 2018). NBOw represents neural bag-of-words. CNN, BIRNN and SELFATT stand for 1D convolutional neural network (Kim, 2014), bidirectional GRU-based recurrent neural network (Cho et al., 2014), and multi-head attention (Vaswani et al., 2017), respectively.
+
+We report the remaining numbers in Table 2. We train all these pre-trained models by regarding codes as a sequence of tokens. We also continuously train RoBERTa only on codes from CodeSearchNet with masked language modeling. Results show that CodeBERT consistently performs
+
+<table><tr><td>MODEL</td><td>RUBY</td><td>JAVASCIPT</td><td>GO</td><td>PYTHON</td><td>JAVA</td><td>PHP</td><td>MA-AVG</td></tr><tr><td>NBOW</td><td>0.4285</td><td>0.4607</td><td>0.6409</td><td>0.5809</td><td>0.5140</td><td>0.4835</td><td>0.5181</td></tr><tr><td>CNN</td><td>0.2450</td><td>0.3523</td><td>0.6274</td><td>0.5708</td><td>0.5270</td><td>0.5294</td><td>0.4753</td></tr><tr><td>BiRNN</td><td>0.0835</td><td>0.1530</td><td>0.4524</td><td>0.3213</td><td>0.2865</td><td>0.2512</td><td>0.2580</td></tr><tr><td>SELFATT</td><td>0.3651</td><td>0.4506</td><td>0.6809</td><td>0.6922</td><td>0.5866</td><td>0.6011</td><td>0.5628</td></tr><tr><td>ROBERTA</td><td>0.6245</td><td>0.6060</td><td>0.8204</td><td>0.8087</td><td>0.6659</td><td>0.6576</td><td>0.6972</td></tr><tr><td>PT w/ CODE ONLY (INIT=s)</td><td>0.5712</td><td>0.5557</td><td>0.7929</td><td>0.7855</td><td>0.6567</td><td>0.6172</td><td>0.6632</td></tr><tr><td>PT w/ CODE ONLY (INIT=R)</td><td>0.6612</td><td>0.6402</td><td>0.8191</td><td>0.8438</td><td>0.7213</td><td>0.6706</td><td>0.7260</td></tr><tr><td>CODEBERT (MLM, INIT=s)</td><td>0.5695</td><td>0.6029</td><td>0.8304</td><td>0.8261</td><td>0.7142</td><td>0.6556</td><td>0.6998</td></tr><tr><td>CODEBERT (MLM, INIT=R)</td><td>0.6898</td><td>0.6997</td><td>0.8383</td><td>0.8647</td><td>0.7476</td><td>0.6893</td><td>0.7549</td></tr><tr><td>CODEBERT (RTD, INIT=R)</td><td>0.6414</td><td>0.6512</td><td>0.8285</td><td>0.8263</td><td>0.7150</td><td>0.6774</td><td>0.7233</td></tr><tr><td>CODEBERT (MLM+RTD, INIT=R)</td><td>0.6926</td><td>0.7059</td><td>0.8400</td><td>0.8685</td><td>0.7484</td><td>0.7062</td><td>0.7603</td></tr></table>
+
+Table 2: Results on natural language code retrieval. Baselines include four joint embeddings (first group) of NL and PL, RoBERTa, and RoBERTa which is continuously trained with masked language modeling on codes only (second group). PT stands for pre-training. We train CodeBERT (third group) with different settings, including using different initialization (from scratch (INIT=S) or initialized with the parameters of RoBERTa (INIT=R)) and using different learning objectives (MLM, RTD, or the combination of both).
+
+better than RoBERTa and the model pre-trained with code only. CodeBERT (MLM) learned from scratch performs better than RoBERTa. Unsurprisingly, initializing CodeBERT with RoBERTa improves the performance  $^{6}$ .
+
+# 4.2 NL-PL Probing
+
+In the previous subsection, we show the empirical effectiveness of CodeBERT in a setting that the parameters of CodeBERT are fine-tuned in downstream tasks. In this subsection, we further investigate what type of knowledge is learned in CodeBERT without modifying the parameters.
+
+Task Formulation and Data Construction Following the probing experiments in NLP (Petroni et al., 2019; Talmor et al., 2019), we study NL-PL probing here. Since there is no existing work towards this goal, we formulate the problem of NL-PL probing and create the dataset by ourselves. Given an NL-PL pair  $(c, w)$ , the goal of NL-PL probing is to test model's ability to correctly predict/recover the masked token of interest (either a code token  $c_i$  or word token  $w_j$ ) among distractors. There are two major types of distractors: one is the whole target vocabulary used for the masked language modeling objective (Petroni et al., 2019), and another one has fewer candidates which are filter or curated based on experts' understanding about the ability to be tested (Talmor et al., 2019). We follow the second direction and formulate NL-PL probing as a multi-choice question answering task, where the question is cloze-style in which a certain token
+
+is replaced by  $[MASK]$  and distractor candidate answers are curated based on our expertise.
+
+Specifically, we evaluate on the NL side and PL side, respectively. To ease the effort of data collection, we collect data automatically from NL-PL pairs in both validation and testing sets of CodeSearchNet, both of which are unseen in the pretraining phase. To evaluate on the NL side, we select NL-PL pairs whose NL documentations include one of the six keywords (max, maximize, min, minimize, less, greater), and group them to four candidates by merging first two keywords and the middle two keywords. The task is to ask pre-trained models to select the correct one instead of three other distractors. That is to say, the input in this setting includes the complete code and a masked NL documentation. The goal is to select the correct answer from four candidates. For the PL side, we select codes containing keywords max and min, and formulate the task as a two-choice answer selection problem. Here, the input includes complete NL documentation and a masked PL code, and the goal is to select the correct answer from two candidates. Since code completion is an important scenario, we would like to test model's ability in predicting the correct token merely based on preceding PL contexts. Therefore, we add an additional setting for PL side, where the input includes the complete NL documentation and preceding PL codes. Data statistics is given in the top two rows in Table 3.
+
+Model Comparisons Results are given in Table 3. We report accuracy, namely the number of correctly predicted instances over the number of all instances, for each programming language. Since
+
+<table><tr><td></td><td>RUBY</td><td>JAVASCIPT</td><td>GO</td><td>PYTHON</td><td>JAVA</td><td>PHP</td><td>ALL</td></tr><tr><td colspan="8">NUMBER OF DATAPoints FOR PROBING</td></tr><tr><td>PL (2 CHOICES)</td><td>38</td><td>272</td><td>152</td><td>1,264</td><td>482</td><td>407</td><td>2,615</td></tr><tr><td>NL (4 CHOICES)</td><td>20</td><td>65</td><td>159</td><td>216</td><td>323</td><td>73</td><td>856</td></tr><tr><td colspan="8">PL PROBING</td></tr><tr><td>ROBERTA</td><td>73.68</td><td>63.97</td><td>72.37</td><td>59.18</td><td>59.96</td><td>69.78</td><td>62.45</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>71.05</td><td>77.94</td><td>89.47</td><td>70.41</td><td>70.12</td><td>82.31</td><td>74.11</td></tr><tr><td>CODEBERT (MLM)</td><td>86.84</td><td>86.40</td><td>90.79</td><td>82.20</td><td>90.46</td><td>88.21</td><td>85.66</td></tr><tr><td colspan="8">PL PROBING WITH PRECEDED CONTEXT ONLY</td></tr><tr><td>ROBERTA</td><td>73.68</td><td>53.31</td><td>51.32</td><td>55.14</td><td>42.32</td><td>52.58</td><td>52.24</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>63.16</td><td>48.53</td><td>61.84</td><td>56.25</td><td>58.51</td><td>58.97</td><td>56.71</td></tr><tr><td>CODEBERT (MLM)</td><td>65.79</td><td>50.74</td><td>59.21</td><td>62.03</td><td>54.98</td><td>59.95</td><td>59.12</td></tr><tr><td colspan="8">NL PROBING</td></tr><tr><td>ROBERTA</td><td>50.00</td><td>72.31</td><td>54.72</td><td>61.57</td><td>61.61</td><td>65.75</td><td>61.21</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>55.00</td><td>67.69</td><td>60.38</td><td>68.06</td><td>65.02</td><td>68.49</td><td>65.19</td></tr><tr><td>CODEBERT (MLM)</td><td>65.00</td><td>89.23</td><td>66.67</td><td>76.85</td><td>73.37</td><td>79.45</td><td>74.53</td></tr></table>
+
+datasets in different programming languages are extremely unbalanced, we report the accumulated metric with the same way. We use CodeBERT (MLM) here because its output layer naturally fits for probing. Results show that CodeBERT performs better than baselines on almost all languages on both NL and PL probing. The numbers with only preceding contexts are lower than that with bidirectional contexts, which suggests that code completion is challenging. We leave it as a future work.
+
+We further give a case study on PL-NL probing. We mask NL token and PL token separately, and report the predicted probabilities of RoBERTa and CodeBERT. Figure 3 illustrates the example of a python code<sup>7</sup>. We can see that RoBERTa fails in both cases, whereas CodeBERT makes the correct prediction in both NL and PL settings.
+
+# 4.3 Code Documentation Generation
+
+Although the pre-training objective of CodeBERT does not include generation-based objectives (Lewis et al., 2019), we would like to investigate to what extent does CodeBERT perform on generation tasks. Specifically, we study code-to-NL generation, and report results for the documentation generation task on CodeSearchNet Corpus in six programming languages. Since the generated documentations are short and higher order n-grams may not overlap, we remedy this problem by using smoothed BLEU score (Lin and Och, 2004).
+
+![](images/e64b87002281d5456c0e8f63166cf4b1450b10633676d16bd7887e1d67514c52.jpg)
+
+Table 3: Statistics of the data for NL-PL probing and the performance of different pre-trained models. Accuracies  $(\%)$  are reported. Best results in each group are in bold.  
+
+<table><tr><td colspan="2"></td><td>max</td><td>min</td><td>less</td><td>greater</td></tr><tr><td rowspan="2">NL</td><td>Roberta</td><td>96.24%</td><td>3.73%</td><td>0.02%</td><td>0.01%</td></tr><tr><td>CodeBERT (MLM)</td><td>39.38%</td><td>60.60%</td><td>0.02%</td><td>0.0003%</td></tr><tr><td rowspan="2">PL</td><td>Roberta</td><td>95.85%</td><td>4.15%</td><td>-</td><td>-</td></tr><tr><td>CodeBERT (MLM)</td><td>0.001%</td><td>99.999%</td><td>-</td><td>-</td></tr></table>
+
+Figure 3: Case study on python language. Masked tokens in NL (in blue) and PL (in yellow) are separately applied. Predicted probabilities of RoBERTa and CodeBERT are given.
+
+Model Comparisons We compare our model with several baselines, including a RNN-based model with attention mechanism (Sutskever et al., 2014), the Transformer (Vaswani et al., 2017), RoBERTa and the model pre-trained on code only. To demonstrate the effectiveness of CodeBERT on code-to-NL generation tasks, we adopt various pre-trained models as encoders and keep the hyperparameters consistent. Detailed hyper-parameters are given in Appendix B.3.
+
+Table 4 shows the results with different models for the code-to-documentation generation task. As we can see, models pre-trained on programming language outperform RoBERTa, which illustrates that pre-training models on programming
+
+<table><tr><td>MODEL</td><td>RUBY</td><td>JAVASCIPT</td><td>GO</td><td>PYTHON</td><td>JAVA</td><td>PHP</td><td>OVERALL</td></tr><tr><td>SEQ2SEQ</td><td>9.64</td><td>10.21</td><td>13.98</td><td>15.93</td><td>15.09</td><td>21.08</td><td>14.32</td></tr><tr><td>TRANSFORMER</td><td>11.18</td><td>11.59</td><td>16.38</td><td>15.81</td><td>16.26</td><td>22.12</td><td>15.56</td></tr><tr><td>ROBERTA</td><td>11.17</td><td>11.90</td><td>17.72</td><td>18.14</td><td>16.47</td><td>24.02</td><td>16.57</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>11.91</td><td>13.99</td><td>17.78</td><td>18.58</td><td>17.50</td><td>24.34</td><td>17.35</td></tr><tr><td>CODEBERT (RTD)</td><td>11.42</td><td>13.27</td><td>17.53</td><td>18.29</td><td>17.35</td><td>24.10</td><td>17.00</td></tr><tr><td>CODEBERT (MLM)</td><td>11.57</td><td>14.41</td><td>17.78</td><td>18.77</td><td>17.38</td><td>24.85</td><td>17.46</td></tr><tr><td>CODEBERT (RTD+MLM)</td><td>12.16</td><td>14.90</td><td>18.07</td><td>19.06</td><td>17.65</td><td>25.16</td><td>17.83</td></tr></table>
+
+language could improve code-to-NL generation. Besides, results in the Table 4 show that CodeBERT pre-trained with RTD and MLM objectives brings a gain of 1.3 BLEU score over RoBERTa overall and achieve the state-of-the-art performance $^8$ .
+
+# 4.4 Generalization to Programming Languages NOT in Pre-training
+
+We would like to evaluate CodeBERT on the programming language which is never seen in the pretraining step. To this end, we study the task of generating a natural language summary of a C# code snippet. We conduct experiments on the dataset of CodeNN (Iyer et al., 2016)<sup>9</sup>, which consists of 66,015 pairs of questions and answers automatically collected from StackOverflow. This dataset is challenging since the scale of dataset is orders of magnitude smaller than CodeSearchNet Corpus. We evaluate models using smoothed BLEU-4 score and use the same evaluation scripts as Iyer et al. (2016).
+
+Table 4: Results on Code-to-Documentation generation, evaluated with smoothed BLEU-4 score.  
+
+<table><tr><td>MODEL</td><td>BLEU</td></tr><tr><td>MOSES (KOEHN ET AL., 2007)</td><td>11.57</td></tr><tr><td>IR</td><td>13.66</td></tr><tr><td>SUM-NN (RUSH ET AL., 2015)</td><td>19.31</td></tr><tr><td>2-LAYER BILSTM</td><td>19.78</td></tr><tr><td>TRANSFORMER (VASWANI ET AL., 2017)</td><td>19.68</td></tr><tr><td>TREELSTM (TAI ET AL., 2015)</td><td>20.11</td></tr><tr><td>CODENN (IYER ET AL., 2016)</td><td>20.53</td></tr><tr><td>CODE2SEQ (ALON ET AL., 2019)</td><td>23.04</td></tr><tr><td>ROBERTA</td><td>19.81</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>20.65</td></tr><tr><td>CODEBERT (RTD)</td><td>22.14</td></tr><tr><td>CODEBERT (MLM)</td><td>22.32</td></tr><tr><td>CODEBERT (MLM+RTD)</td><td>22.36</td></tr></table>
+
+Table 5: Code-to-NL generation on C# language.
+
+Model Comparisons Table 5 shows that our model with MLM and RTD pre-training objectives achieves 22.36 BLEU score and improves by 2.55 points over RoBERTa, which illustrates CodeBERT
+
+could generalize better to other programming language which is never seen in the pre-training step. However, our model achieve slightly lower results than code2seq (Alon et al., 2019). The main reason could be that code2seq makes use of compositional paths in its abstract syntax tree (AST) while CodeBERT only takes original code as the input. We have trained a version of CodeBERT by traversing the tree structure of AST following a certain order, but applying that model does not bring improvements on generation tasks. This shows a potential direction to improve CodeBERT by incorporating AST.
+
+# 5 Conclusion
+
+In this paper, we present CodeBERT, which to the best of our knowledge is the first large bimodal pre-trained model for natural language and programming language. We train CodeBERT on both bimodal and unimodal data, and show that finetuning CodeBERT achieves state-of-the-art performance on downstream tasks including natural language code search and code-to-documentation generation. To further investigate the knowledge embodied in pre-trained models, we formulate the task of NL-PL probing and create a dataset for probing. We regard the probing task as a cloze-style answer selection problem, and curate distractors for both NL and PL parts. Results show that, with model parameters fixed, CodeBERT performs better than RoBERTa and a continuously trained model using codes only.
+
+There are many potential directions for further research on this field. First, one could learn better generators with bimodal evidence or more complicated neural architecture to improve the replaced token detection objective. Second, the loss functions of CodeBERT mainly target on NL-PL understanding tasks. Although CodeBERT achieves strong BLEU scores on code-to-documentation generation, the CodeBERT itself could be further improved by generation-related learning objectives.
+
+How to successfully incorporate AST into the pretraining step is also an attractive direction. Third, we plan to apply CodeBERT to more NL-PL related tasks, and extend it to more programming languages. Flexible and powerful domain/language adaptation methods are necessary to generalize well.
+
+# Acknowledgments
+
+Xiaocheng Feng is the corresponding author of this work. We thank the anonymous reviewers for their insightful comments. Zhangyin Feng, Xiaocheng Feng, Bing Qin and Ting Liu are supported by the National Key R&D Program of China via grant 2018YFB1005103 and National Natural Science Foundation of China (NSFC) via grant 61632011 and 61772156.
+
+# References
+
+Uri Alon, Shaked Brody, Omer Levy, and Eran Yahav. 2019. code2seq: Generating sequences from structured representations of code. International Conference on Learning Representations.  
+Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. 2014. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078.  
+Kevin Clark, Minh-Thang Luong, Quoc V. Le, and Christopher D. Manning. 2020. {ELECTRA}: Pretraining text encoders as discriminators rather than generators. In International Conference on Learning Representations.  
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805.  
+Xiaodong Gu, Hongyu Zhang, and Sunghun Kim. 2018. Deep code search. In 2018 IEEE/ACM 40th International Conference on Software Engineering (ICSE), pages 933-944. IEEE.  
+Hamel Husain, Ho-Hsiang Wu, Tiferet Gazit, Miltiadis Allamanis, and Marc Brockschmidt. 2019. Code-searchnet challenge: Evaluating the state of semantic code search. arXiv preprint arXiv:1909.09436.  
+Srinivasan Iyer, Ioannis Konstas, Alvin Cheung, and Luke Zettlemoyer. 2016. Summarizing source code using a neural attention model. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2073-2083.
+
+Dan Jurafsky. 2000. Speech & language processing. Pearson Education India.  
+Aditya Kanade, Petros Maniatis, Gogul Balakrishnan, and Kensen Shi. 2019. Pre-trained contextual embedding of source code. arXiv preprint arXiv:2001.00059.  
+Yoon Kim. 2014. Convolutional neural networks for sentence classification. arXiv preprint arXiv:1408.5882.  
+Philipp Koehn, Hieu Hoang, Alexandra Birch, Chris Callison-Burch, Marcello Federico, Nicola Bertoldi, Brooke Cowan, Wade Shen, Christine Moran, Richard Zens, et al. 2007. Moses: Open source toolkit for statistical machine translation. In Proceedings of the 45th annual meeting of the association for computational linguistics companion volume proceedings of the demo and poster sessions, pages 177-180.  
+Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Ves Stoyanov, and Luke Zettlemoyer. 2019. Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension. arXiv preprint arXiv:1910.13461.  
+Chin-Yew Lin and Franz Josef Och. 2004. Orange: a method for evaluating automatic evaluation metrics for machine translation. In Proceedings of the 20th international conference on Computational Linguistics, page 501. Association for Computational Linguistics.  
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692.  
+Jiasen Lu, Dhruv Batra, Devi Parikh, and Stefan Lee. 2019. Vilbert: Pretraining task-agnostic visi-olinguistic representations for vision-and-language tasks. In Advances in Neural Information Processing Systems, pages 13-23.  
+Bhaskar Mitra, Nick Craswell, et al. 2018. An introduction to neural information retrieval. Foundations and Trends in Information Retrieval, 13(1):1-126.  
+Matthew E Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. 2018. Deep contextualized word representations. arXiv preprint arXiv:1802.05365.  
+Fabio Petroni, Tim Roktaschel, Patrick Lewis, Anton Bakhtin, Yuxiang Wu, Alexander H Miller, and Sebastian Riedel. 2019. Language models as knowledge bases? arXiv preprint arXiv:1909.01066.  
+Telmo Pires, Eva Schlinger, and Dan Garrette. 2019. How multilingual is multilingual bert? arXiv preprint arXiv:1906.01502.
+
+Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. 2018. Improving language understanding by generative pre-training. URL https://s3-us-west-2. amazonaws. com/openai-assetss/researchcovers/languageunsupervised/language understanding paper.pdf.
+
+Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2019. Exploring the limits of transfer learning with a unified text-to-text transformer. arXiv preprint arXiv:1910.10683.
+
+Alexander M Rush, Sumit Chopra, and Jason Weston. 2015. A neural attention model for abstractive sentence summarization. arXiv preprint arXiv:1509.00685.
+
+Chen Sun, Austin Myers, Carl Vondrick, Kevin Murphy, and Cordelia Schmid. 2019. Videobert: A joint model for video and language representation learning. arXiv preprint arXiv:1904.01766.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. 2014. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pages 3104-3112.
+
+Kai Sheng Tai, Richard Socher, and Christopher D Manning. 2015. Improved semantic representations from tree-structured long short-term memory networks. arXiv preprint arXiv:1503.00075.
+
+Alon Talmor, Yanai Elazar, Yoav Goldberg, and Jonathan Berant. 2019. olmpics-on what language model pre-training captures. arXiv preprint arXiv:1912.13283.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In Advances in neural information processing systems, pages 5998-6008.
+
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. 2016. Google's neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144.
+
+Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Ruslan Salakhutdinov, and Quoc V Le. 2019. Xlnet: Generalized autoregressive pretraining for language understanding. arXiv preprint arXiv:1906.08237.
+
+# A Data Statistic
+
+Data statistics of the training/validation/testing data splits for six programming languages are given in Table 6.
+
+<table><tr><td>CODE SEARCH</td><td>TRAINING</td><td>DEV</td><td>TESTING</td></tr><tr><td>GO</td><td>635,635</td><td>28,483</td><td>14,291</td></tr><tr><td>JAVA</td><td>908,886</td><td>30,655</td><td>26,909</td></tr><tr><td>JAVAscript</td><td>247,773</td><td>16,505</td><td>6,483</td></tr><tr><td>PHP</td><td>1,047,406</td><td>52,029</td><td>28,391</td></tr><tr><td>PYTHON</td><td>824,342</td><td>46,213</td><td>22,176</td></tr><tr><td>RUBY</td><td>97,580</td><td>4,417</td><td>2,279</td></tr></table>
+
+Table 6: Data statistics about the CodeSearchNet Corpus for natural language code search.
+
+# B Train Details
+
+# B.1 Pre-training
+
+We train CodeBERT on one NVIDIA DGX-2 machine using FP16. It combines 16 interconnected NVIDIA Tesla V100 with 32GB memory. We use the following set of hyper-parameters to train models: batchsize is 2,048 and learning rate is 5e-4. We use Adam to update the parameters and set the number of warmup steps as 10K. We set the max length as 512 and the max training step is 100K. Training 1,000 batches of data costs 600 minutes with MLM objective, 120 minutes with RTD objective.
+
+# B.2 CodeSearch
+
+In the fine-turning step, we set the learning rate as 1e-5, the batch size as 64, the max sequence length as 200 and the max fine-tuning epoch as 8. As the same with pre-training, We use Adam to update the parameters. We choose the model performed best on the development set, and use that to evaluate on the test set.
+
+# B.3 Code Summarization on Six Programming Languages
+
+We use Transformer with 6 layers, 768 dimensional hidden states and 12 attention heads as our decoder in all settings. We set the max length of input and inference as 256 and 64, respectively. We use the Adam optimizer to update model parameters. The learning rate and the batch size are 5e-5 and 64, respectively. We tune hyperparameters and perform early stopping on the development set.
+
+# B.4 Code Summarization on C#
+
+Since state-of-the-art methods use RNN as their decoder, we choose a 2-layer GRU with an attention mechanism as our decoder for a comparison. We fine-tune models using a grid search with the following set of hyper-parameters: batchsize is in \{32, 64\} and learning rate is in \{2e-5, 5e-5\}. We report
+
+the number when models achieve best performance on the development set.
+
+# C Learning Curve of CodeSearch
+
+From Figure 4, we can see that CodeBERT performs better at the early stage, which reflects that CodeBERT provides good initialization for learning downstream tasks.
+
+![](images/db6357b1d8c49497159e567595a575c4f04bc6f76a6f358063976b8a202bf0e3.jpg)  
+Figure 4: Learning curve of different pre-trained models in the fine-tuning step. We show results on Python and Java.
+
+![](images/556ef6f6bb3c91bfb2d76dac20414ec4366486c6b2cb0ee0cbd706798c5ef43f.jpg)
+
+# D Late Fusion
+
+In section §4.1, we show that CodeBERT performs well in the setting where natural languages and codes have early interactions. Here, we investigate whether CodeBERT is good at working as a unified encoder. We apply CodeBERT for natural language code search in a later fusion setting, where CodeBERT first encodes NL and PL separately, and then calculates the similarity by dot-product. In this way, code search is equivalent to find the nearest codes in the shared vector space. This scenario also facilitates the use of CodeBERT in an online system, where the representations of codes are calculated in advance. In the runtime, a system only needs to compute the representation of NL and vector-based dot-product.
+
+We fine-tune CodeBERT with the following objective, which maximizes the dot-product of the ground truth while minimizing the dot-product of distractors.
+
+$$
+- \frac {1}{N} \sum_ {i} \log \left(\frac {\exp \left(E n c \left(c _ {i}\right) ^ {\intercal} E n c \left(w _ {i}\right)\right)}{\sum_ {j} \exp \left(E n c \left(c _ {j}\right) ^ {\intercal} E n c \left(w _ {i}\right)\right)}\right) \tag {15}
+$$
+
+Results are given in Table 7. We just do this setting on two languages with a relatively small amount of data.
+
+We can see that CodeBERT performs better than RoBERTa and the model pre-trained with codes
+
+<table><tr><td>MODEL</td><td>RUBY</td><td>GO</td></tr><tr><td>ROBERTA</td><td>0.0043</td><td>0.0030</td></tr><tr><td>PRE-TRAIN W/ CODE ONLY</td><td>0.1648</td><td>0.4179</td></tr><tr><td>CODEBERT</td><td>0.6870</td><td>0.8372</td></tr></table>
+
+Table 7: Results on natural language code search by late fusion.
+
+only. And late fusion performs comparable with the standard way. What's more, late fusion is more efficient and this setting could be used in an online system.
+
+# E Case Study
+
+To qualitatively analyze the effectiveness of CodeBERT, we give some cases for code search and code documentation generation tasks.
+
+Considering the limited space, we only give the top2 results of the query for python programming language. As show in Figure 5, search results are very relevant with query.
+
+Figure 6 and Figure 7 show the outputs with different models for the code documentation generation task. As we can see, CodeBERT performs better than all baselines.
+
+Query  
+```txt
+create file and write something
+```
+
+Search Results (top2)  
+```python
+https://github.com/darknessomi/musicbox/blob/master/NEMbox/util.py#L37-L40   
+def create_file(path, default  $=$  "\\n"): if not os.path.exists(path): with open(path, "w") as f: f.write(default)
+```
+
+```txt
+https://github.com/datakortet/yamldirs/blob/master/yamldirs/filemaker.py#L114-L118
+```
+
+```python
+def make_file(self, filename, content):
+    '''Create a new file with name``filename``and content``content``.
+    '''with open(filename,'w') as fp:
+        fp.write(content)
+```
+
+Figure 5: Python CodeSearch example. The results are searched from 1,156,085 python code data. We only give the top2 results because space is limited.
+
+Gold: Add a write error result  
+```java
+public void addWriteErrorResult(final BulkWriteError writeError, final IndexMap indexMap) {
+    notNull("writeError", writeError);
+    mergeWriteErrors(asList(writeError), indexMap);
+}
+```
+
+```txt
+CodeBERT: Add a write error result.  
+PRE-TRAIN W/ CODEONLY : Merges the given write error.  
+Roberta: Add a write operation to the map.  
+Transformer: Adds an error to the write map.
+```
+
+```txt
+RNN: Add an error map.
+```
+
+Figure 6: Java code documentation generation output example.
+
+```python
+def create_or_update(self, list_id, subscriber_hash, data):  
+    subscriber_hash = check subscribed_hash(subscriber_hash)  
+    self.list_id = list_id  
+    self subscribers_hash = subscriber_hash  
+    if 'email_address' not in data:  
+        raise ValueError('The list member must have an email_address')  
+    check_email(data['email_address'])  
+    if 'status_if_new' not in data:  
+        raise ValueError('The list member must have a status_if_new')  
+    if data['status_if_new'] not in [' subscribed', 'un subscribed', 'cleaned', 'pending', 'transactional']:  
+        raise ValueError('The list member status_if_new must be one of " subscribed", "un subscribed", "cleaned", "pending", or "transactional"))  
+    return self._mc_client._put(url= self._build_path(list_id, 'members', subscriber_hash), data=data)
+```
+
+Gold: Add or update a list member.  
+```txt
+CodeBERT: Create or update a list member.  
+PRE-TRAIN W/ CODEONLY: Create or update a subscriber.  
+Roberta: Create or update an existing record.  
+Transformer: Create or update a subscription.  
+RNN: Creates or updates an email address.
+```
+
+Figure 7: Python code documentation generation output example.

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+[
+    {
+        "type": "text",
+        "text": "Using Hindsight to Anchor Past Knowledge in Continual Learning",
+        "text_level": 1,
+        "bbox": [
+            163,
+            119,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Arslan Chaudhry $^{1,*}$ , Albert Gordo $^{2}$ , Puneet K. Dokania $^{1}$ , Philip Torr $^{1}$ , David Lopez-Paz $^{2}$",
+        "bbox": [
+            127,
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+            176
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "<sup>1</sup>University of Oxford, <sup>2</sup>Facebook AI",
+        "bbox": [
+            372,
+            176,
+            620,
+            191
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
+        "bbox": [
+            250,
+            273,
+            313,
+            286
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In continual learning, the learner faces a stream of data whose distribution changes over time. Modern neural networks are known to suffer under this setting, as they quickly forget previously acquired knowledge. To address such catastrophic forgetting, many continual learning methods implement different types of experience replay, re-learning on past data stored in a small buffer known as episodic memory. In this work, we complement experience replay with a new objective that we call \"anchoring\", where the learner uses bilevel optimization to update its knowledge on the current task, while keeping intact predictions on some anchor points of past tasks. These anchor points are learned using gradient-based optimization to maximize forgetting, which is approximated by fine-tuning the currently trained model on the episodic memory of past tasks. Experiments on several supervised learning benchmarks for continual learning demonstrate that our approach improves the standard experience replay in terms of both accuracy and forgetting metrics and for various sizes of episodic memory.",
+        "bbox": [
+            101,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction",
+        "text_level": 1,
+        "bbox": [
+            212,
+            558,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We study the problem of continual learning, where a machine learning model experiences a sequence of tasks. Each of these tasks is presented as a stream of input-output pairs, where each pair is drawn identically and independently (iid) from the corresponding task probability distribution. Since the length of the learning experience is not specified a priori, the learner can only assume a single pass over the data and, due to space constraints, store nothing but a few examples in a small episodic memory. At all times during the lifetime of the model, predictions on examples from any task may be requested. Addressing continual learning is an important research problem, since it would enable the community to move past the assumption of \"identically and independently distributed data\", and allow a better deployment of machine learning in-the-wild. However, continual learning presents one major challenge, catastrophic forgetting (McCloskey and Cohen 1989). That is, as the learner experiences new tasks, it quickly forgets previously acquired knowledge. This is a hindrance especially for state-of-the-art deep learn",
+        "bbox": [
+            86,
+            578,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ing models, where all parameters are updated after observing each example.",
+        "bbox": [
+            519,
+            273,
+            910,
+            300
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Continual learning has received increasing attention from the scientific community during the last decade. The state of the art algorithms for continual learning fall into three categories. First, regularization-based approaches reduce forgetting by restricting the updates in model parameters that were important for previous tasks (Kirkpatrick et al. 2016; Rebuffi, Kolesnikov, and Lampert 2017; Aljundi et al. 2018; Chaudhry et al. 2018; Nguyen et al. 2018). However, when the number of tasks are large, the regularization of past tasks becomes obsolete, leading to the representation drift (Titsias et al. 2019). Second, modular approaches (Rusu et al. 2016; Lee et al. 2017) add new modules to the learner as new tasks are learned. While modular architectures overcome forgetting by design, the memory complexity of these approaches scales with the number of tasks. Third, the memory-based methods (Lopez-Paz and Ranzato 2017; Hayes, Cahill, and Kanan 2018; Isele and Cosgun 2018; Riemer et al. 2019; Chaudhry et al. 2019a) store a few examples from past tasks in an \"episodic memory\", to be revisited when training for a new task. Contrary to modular approaches, memory-based methods add a very small memory overhead for each new task. Memory-based methods are the reigning state-of-the-art, but their performance remains a far cry from a simple oracle accessing all the data at once, hence turning the continual learning experience back into a normal supervised learning task. Despite intense research efforts, such gap in performance renders the problem of continual learning an open research question.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Contribution We propose Hindsight Anchor Learning (HAL), a continual learning approach to improve the performance of memory-based continual learning algorithms. HAL leverages bilevel optimization to regularize the training objective with one representational point per class per task, called anchors. These anchors are constructed via gradient ascent in the image space, by maximizing one approximation to the forgetting loss for the current task throughout the entire continual learning experience. We estimate the amount of forgetting that the learner would suffer on these anchors if it were to be trained on future tasks in hindsight: that is, by measuring forgetting on a temporary predictor that has been fine-tuned on the episodic memory of past tasks.",
+        "bbox": [
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+    },
+    {
+        "type": "aside_text",
+        "text": "arXiv:2002.08165v2 [cs.LG] 2 Mar 2021",
+        "bbox": [
+            22,
+            268,
+            57,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "page_footnote",
+        "text": "*Correspondence to arslan.chaudhry@eng.ox.ac.uk. Copyright © 2021, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.",
+        "bbox": [
+            84,
+            851,
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+            888
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anchors learned in such a way lie close to the classifier's decision boundary, as visualized in Figure 2. Since points near the decision boundary are the easiest to forget when updating the learner on future tasks, keeping prediction invariant on such anchors preserves the performance of previous tasks effectively. In sum, the overall parameter update of HAL uses nested optimization to minimize the loss of the current mini-batch, while keeping the predictions of all anchors invariant.",
+        "bbox": [
+            81,
+            68,
+            480,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Results We compare HAL to EWC (Kirkpatrick et al. 2016), ICARL (Rebuffi, Kolesnikov, and Lampert 2017), VCL (Nguyen et al. 2018), AGEM (Chaudhry et al. 2019a), experience replay (Hayes, Cahill, and Kanan 2018; Riemer et al. 2019), MER (Riemer et al. 2019), and MIR (Aljundi et al. 2019a) across four commonly used benchmarks in supervised continual learning (MNIST permutations, MNIST rotations, split CIFAR-100, and split miniImageNet). In these experiments, HAL achieves state-of-the-art performance, improving accuracy by up to  $7.5\\%$  and reducing forgetting by almost  $23\\%$  over the experience replay baseline. We show that these results hold for various sizes of episodic memory (between 1 and 5 examples per class per task).",
+        "bbox": [
+            81,
+            210,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We now begin our exposition by reviewing the continual learning setup. The rest of the manuscript then presents our new algorithm HAL (Section 3), showcases its empirical performance (Section 4), surveys the related literature (Section 5), and offers some concluding remarks (Section 6).",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Continual learning setup",
+        "text_level": 1,
+        "bbox": [
+            158,
+            472,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In continual learning, a learner experiences a stream of data triplets  $(x_{i},y_{i},t_{i})$  containing an input  $x_{i}$ , a target  $y_{i}$ , and a task identifier  $t_i\\in \\mathcal{T} = \\{1,\\ldots ,T\\}$ . Each input-target pair  $(x_{i},y_{i})\\in \\mathcal{X}\\times \\mathcal{Y}_{t_{i}}$  is an identical and independently distributed example drawn from some unknown distribution  $P_{t_i}(X,Y)$ , representing the  $t_i$ -th learning task. We assume that the tasks are experienced in order ( $t_i\\leq t_j$  for all  $i\\leq j$ ), and that the total number of tasks  $T$  is not known a priori. Under this setup, our goal is to estimate a predictor  $f_{\\theta} = (w\\circ \\phi):\\mathcal{X}\\times \\mathcal{T}\\to \\mathcal{Y}$ , parameterized by  $\\theta \\in \\mathbb{R}^P$ , and composed of a feature extractor  $\\phi :\\mathcal{X}\\rightarrow \\mathcal{H}$  and a classifier  $w:\\mathcal{H}\\to \\mathcal{V}$ , that minimizes the multi-task error",
+        "bbox": [
+            81,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\frac {1}{T} \\sum_ {t = 1} ^ {T} \\mathbb {E} _ {(x, y) \\sim P _ {t}} [ \\ell (f (x, t), y) ], \\tag {1}\n$$\n",
+        "text_format": "latex",
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where  $\\mathcal{Y} = \\cup_{t\\in \\mathcal{T}}\\mathcal{Y}_t$  , and  $\\ell :\\mathcal{Y}\\times \\mathcal{Y}\\to \\mathbb{R}$  is a loss function.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inspired by prior literature in continual learning, (Lopez-Paz and Ranzato 2017; Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Chaudhry et al. 2019a), we consider streams of data that are experienced only once. Therefore, the learner cannot revisit any but a small number of data triplets chosen to be stored in a small episodic memory  $\\mathcal{M}$ . More specifically, we consider tiny \"ring\" episodic memories, which contain the last  $m$  observed examples per class for each of the experienced tasks, where  $m \\in \\{1,3,5\\}$ . That is, considering as variables the number of experienced tasks  $t$  and examples  $n$ , we study continual learning algorithms with a  $O(t)$  memory footprint.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Following Lopez-Paz and Ranzato (2017) and Chaudhry et al. (2018), we monitor two statistics to evaluate the quality of continual learning algorithms: final average accuracy, and final maximum forgetting. First, the final average accuracy of a predictor is defined as",
+        "bbox": [
+            514,
+            68,
+            911,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\text {A c c u r a c y} = \\frac {1}{T} \\sum_ {j = 1} ^ {T} a _ {T, j}, \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            630,
+            148,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where  $a_{i,j}$  denotes the test accuracy on task  $j$  after the model has finished experiencing task  $i$ . That is, the final average accuracy measures the test performance of the model at every task after the continual learning experience has finished. Second, the final maximum forgetting is defined as",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\text {F o r g e t t i n g} = \\frac {1}{T - 1} \\sum_ {j = 1} ^ {T - 1} \\max  _ {l \\in \\{1, \\dots , T - 1 \\}} \\left(a _ {l, j} - a _ {T, j}\\right), \\tag {3}\n$$\n",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "that is, the decrease in performance for each of the tasks between their peak accuracy and their accuracy after the continual learning experience has finished.",
+        "bbox": [
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+            330,
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+        ],
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+    {
+        "type": "text",
+        "text": "Finally, following Chaudhry et al. (2019a), we use the first  $k < T$  tasks to cross validate the hyper-parameters of each of the considered continual learning algorithms. These first  $k$  tasks are not considered when computing the final average accuracy and maximum forgetting metrics.",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "3 Hindsight Anchor Learning (HAL)",
+        "text_level": 1,
+        "bbox": [
+            549,
+            460,
+            879,
+            477
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The current state of the art algorithms for continual learning are based on experience replay (Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Chaudhry et al. 2019b). These methods update the model  $f_{\\theta}$  while storing a small amount of past observed triplets in an episodic memory  $\\mathcal{M} = \\{(x', y', t')\\}$ . For a new minibatch of observations  $\\mathcal{B} := \\{(x, y, t)\\}$  from task  $t$ , the learner samples a minibatch  $\\mathcal{B}_{\\mathcal{M}}$  from  $\\mathcal{M}$  at random, and employ the rule  $\\theta \\gets \\theta - \\alpha \\cdot \\nabla_{\\theta} \\ell(\\mathcal{B} \\cup \\mathcal{B}_{\\mathcal{M}})$  to update its parameters, where",
+        "bbox": [
+            514,
+            481,
+            913,
+            609
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\ell (\\mathcal {A}) = \\frac {1}{| \\mathcal {A} |} \\sum_ {(x, y, t) \\in \\mathcal {A}} \\ell (f _ {\\theta} (x, t), y) \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            596,
+            617,
+            911,
+            655
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "denotes the average loss across a collection of triplets  $\\mathcal{A} = \\mathcal{B} \\cup \\mathcal{B}_{\\mathcal{M}}$ . In general,  $\\mathcal{B}_{\\mathcal{M}}$  is constructed to have the same size as  $\\mathcal{B}$ , but it can be smaller if the episodic memory  $\\mathcal{M}$  does not yet contain enough samples.",
+        "bbox": [
+            514,
+            664,
+            913,
+            719
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The episodic memory  $\\mathcal{M}$  reminds the predictor about how to perform past tasks using only a small amount of data. As such, the behaviour of the predictor on past tasks outside the data stored in  $\\mathcal{M}$  is not guaranteed. Moreover, since  $\\mathcal{M}$  is usually very small, the performance of the predictor becomes sensitive to the choice of samples stored in the episodic memory. Because of this reason, we propose to further fix the behaviour of the predictor at a collection of carefully constructed anchor points  $e_{t'}$ , one per class per past task  $t'$ , at each parameter update.",
+        "bbox": [
+            514,
+            720,
+            913,
+            859
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Let us assume that the anchor points  $e_{t'}$  are given—we will see later how to construct them in practice. To constrain",
+        "bbox": [
+            516,
+            859,
+            911,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "the change of the predictor at these anchor points, we propose a two-step parameter update rule:",
+        "bbox": [
+            81,
+            68,
+            478,
+            97
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\tilde {\\theta} \\leftarrow \\theta - \\alpha \\nabla_ {\\theta} \\ell (\\mathcal {B} \\cup \\mathcal {B} _ {\\mathcal {M}}), \\\\ \\theta \\leftarrow \\theta - \\alpha \\nabla_ {\\theta} \\left(\\ell \\left(\\mathcal {B} \\cup \\mathcal {B} _ {\\mathcal {M}}\\right) + \\lambda \\sum_ {t ^ {\\prime} <   t} \\left(f _ {\\theta} \\left(e _ {t ^ {\\prime}}, t ^ {\\prime}\\right) - f _ {\\tilde {\\theta}} \\left(e _ {t ^ {\\prime}}, t ^ {\\prime}\\right)\\right) ^ {2}\\right). \\tag {5} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            86,
+            104,
+            480,
+            171
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The first step computes a temporary parameter vector  $\\tilde{\\theta}$  by minimizing the loss at a minibatch from the current task  $t$ , and the episodic memory of past tasks (this is the usual experience replay parameter update). The second step employs a nested optimization to perform the actual update of the parameter  $\\theta$ , which trades-off the minimization of  $(a)$  the loss value at the current minibatch and the episodic memory, as well as  $(b)$  changes in predictions at the anchor points for all past tasks. The proposed rule not only updates the predictor conservatively, thereby reducing forgetting, but also, as shown analytically in Appendix A, improves the forward transfer by maximizing the inner product between the gradients on  $\\mathcal{B} \\cup \\mathcal{B}_{\\mathcal{M}}$  and anchor points. In this respect, it bears similarity to gradient-based meta-learning approaches (Finn, Abbeel, and Levine 2017; Nichol and Schulman 2018; Riemer et al. 2019).",
+        "bbox": [
+            81,
+            181,
+            478,
+            402
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next, let us discuss how to choose the anchor points,  $e_t$  (one per class per task) as to preserve the performance of the current task throughout the entire learning experience. Ideally, the anchor points should attempt to minimize the forgetting on the current task as the learner is updated with future tasks. One could achieve this by letting  $e_t$  to be an example from the task  $t$  that would undergo maximum forgetting during the entire continual learning experience. Then, requiring the predictions to remain invariant at  $e_t$ , by using Eq. 5, could effectively reduce forgetting on the current task. Mathematically, the desirable  $e_t$  for the label  $y_t$  is obtained by maximizing the following Forgetting loss:",
+        "bbox": [
+            81,
+            402,
+            478,
+            570
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n(e _ {t}, y _ {t}) \\leftarrow \\underset {(x, y) \\sim P _ {t}} {\\arg \\max } \\ell \\left(f _ {\\theta_ {T}} (x, t), y _ {t}\\right) - \\ell \\left(f _ {\\theta_ {t}} (x, t), y _ {t}\\right), \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\theta_{t}$  is the parameter vector obtained after training on task  $t$  and  $\\theta_{T}$  is the final parameter vector obtained after the entire learning experience. Thus, keeping the predictions intact on the pair  $(e_t,y_t)$  above can effectively preserve the performance of task  $t$ . However, the idealistic Eq. 6 requires access to  $(a)$  the entire distribution  $P_{t}$  to compute the maximization, and  $(b)$  access to all future distributions  $t^{\\prime} > t$  to compute the final parameter vector  $\\theta_{T}$ . Both are unrealistic assumptions under the continual learning setup described in Section 2, as the former requires storing the entire dataset of task  $t$ , and the latter needs access to future tasks.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To circumvent  $(a)$ , we can recast Eq. 6 as an optimization problem and learn the desired  $e_t$  by initializing it at random and using  $k$  gradient ascent updates for a given label  $y_t$  in the image space  $(\\mathcal{X} \\in \\mathbb{R}^D)$ . The proposed optimization objective is given by:",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\max  _ {e _ {t} \\in \\mathbb {R} ^ {D}} \\left(\\underbrace {\\ell \\left(f _ {\\theta_ {T}} \\left(e _ {t} , t\\right) , y _ {t}\\right) - \\ell \\left(f _ {\\theta_ {t}} \\left(e _ {t} , t\\right) , y _ {t}\\right)} _ {\\text {F o r g e t t i n g l o s s}} - \\gamma \\underbrace {\\left(\\phi \\left(e _ {t}\\right) - \\phi_ {t}\\right) ^ {2}} _ {\\text {M e a n e m b e d d i n g l o s s}}\\right), \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the regularizer, given by the mean embedding loss, constrains the search space by trying to push the anchor point embedding towards the mean data embedding. We recall that  $\\phi$  denotes the feature extractor of the predictor, and  $\\phi_t$  is the neural mean embedding (Smola et al. 2007) of all observed examples from task  $t$ . Since the feature extractor is updated after experiencing each data point, the mean embedding  $\\phi_t$  are computed as running averages. That is, after observing a minibatch  $\\mathcal{B} = \\{(x,y,t)\\}$  of task  $t$ , we update:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\phi_ {t} \\leftarrow \\beta \\cdot \\phi_ {t} + (1 - \\beta) \\frac {1}{| \\mathcal {B} |} \\sum_ {x \\in \\mathcal {B}} \\phi (x), \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\phi_t$  is initialized to zero at the beginning of the learning experience. In our experiments, we learn one  $e_t$  per class for each task. We fix the  $y_t$  to the corresponding class label, and discard  $\\phi_t$  after training on task  $t$ . Learning  $e_t$  in this manner circumvents the requirement of storing the entire distribution  $P_t$  for the current task  $t$ .",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Still, Eq. 7 requires the parameter vector  $\\theta_T$ , to be obtained in the distant future after all learning tasks have been experienced. To waive this impossible requirement, we propose to approximate the future by simulating the past. That is, instead of measuring the forgetting that would happen after the model is trained for future tasks, we measure the forgetting that happens when the model is fine-tuned for past tasks. In this way, we say that forgetting is estimated in hind-sight, using past experiences. More concretely, after training on task  $t$  and obtaining the parameter vector  $\\theta_t$ , we minimize the loss during one epoch on the episodic memory  $\\mathcal{M}$  to obtain a temporary parameter vector  $\\theta_{\\mathcal{M}}$  that approximates  $\\theta_T$ , and update  $e_t$  as:",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} e _ {t} \\leftarrow e _ {t} + \\alpha \\nabla_ {e _ {t}} \\left(\\ell \\left(f _ {\\theta_ {\\mathcal {M}}} \\left(e _ {t}, t\\right), y _ {t}\\right) - \\ell \\left(f _ {\\theta_ {t}} \\left(e _ {t}, t\\right), y _ {t}\\right) \\right. \\\\ \\left. - \\gamma \\left(\\phi \\left(e _ {t}\\right) - \\phi_ {t}\\right) ^ {2}\\right). \\tag {9} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This completes the description of our proposed algorithm for continual learning, which combines experience replay with anchors learned in hindsight. We call our approach Hind-sight Anchor Learning (HAL) and summarize the entire learning process as follows:",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hindsight Anchor Learning (HAL)",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Initialize  $\\theta \\sim P(\\theta)$  and  $\\{e_t \\sim P(e)\\}_{t=1}^T$  from normal distributions  $P(\\theta)$  and  $P(e)$ ,  $\\mathcal{M} = \\{\\}$ .",
+            "- Initialize  $\\mathcal{M} = \\{\\}$",
+            "- For each task  $t = 1, \\dots, T$ :"
+        ],
+        "bbox": [
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+            681,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "- For each minibatch  $\\mathcal{B}$  from task  $t$ :",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "* Sample  $\\mathcal{B}_{\\mathcal{M}}$  from  $\\mathcal{M}$",
+            "* Update  $\\theta$  using Eq. 5",
+            "* Update  ${\\phi }_{t}$  using Eq. 8",
+            "* Update  $\\mathcal{M}$  by adding  $\\mathcal{B}$  in a FIFO ring buffer"
+        ],
+        "bbox": [
+            560,
+            763,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Fine-tune on  $\\mathcal{M}$  to obtain  $\\theta_{\\mathcal{M}}$",
+            "Build  $e_t$  using Eq. 9 k times",
+            "-Discard  $\\phi_t$"
+        ],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "- Return  $\\theta$ .",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We contrast HAL with another idea, Maximally Interfered Retrieval (MIR) (Aljundi et al. 2019a) in that MIR selects a minibatch from an already populated replay buffer at each training step, whereas HAL writes to the replay buffer at the end of each task and samples randomly from the replay buffer. Furthermore, MIR selects the minibatch by measuring a one-step increase in loss incurred by the minibatch whereas HAL uses an approximate forgetting loss Eq. 9 that measures an increase over multiple training steps.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now report experiments on standard image classification benchmarks for continual learning.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Datasets and tasks",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We perform experiments on four supervised classification benchmarks for continual learning. Permuted MNIST is a variant of the MNIST dataset of handwritten digits (LeCun 1998) where each task applies a fixed random pixel permutation to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. Rotated MNIST is another variant of MNIST, where each task applies a fixed random image rotation (between 0 and 180 degrees) to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. Split CIFAR is a variant of the CIFAR-100 dataset (Krizhevsky and Hinton 2009; Zenke, Poole, and Ganguli 2017), where each task contains the data pertaining 5 random classes (without replacement) out of the total 100 classes. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes. Split miniImageNet is a variant of the ImageNet dataset (Russakovsky et al. 2015; Vinyals et al. 2016), containing a subset of images and classes from the original dataset. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For all datasets, the first 3 tasks are used for hyperparameter optimization (grids available in Appendix C). The learner can perform multiple epochs on these three initial tasks that are later discarded for evaluation.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Baselines",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We compare our proposed model HAL to the following baselines. Finetune is a single model trained on a stream of data, without any regularization or episodic memory. ICARL (Rebuffi, Kolesnikov, and Lampert 2017) uses nearest-mean-of-exemplar rule for classification and avoids catastrophic forgetting by regularizing over the feature representations of previous tasks using knowledge distillation loss (Hinton, Vinyals, and Dean 2014). EWC (Kirkpatrick et al. 2016) is a continual learning method that limits changes to parameters critical to past tasks, as measured by the Fisher information matrix. VCL (Nguyen et al. 2018) is a continual learning method that uses online variational inference for approximating the posterior distribution, which is then used to regularize the model. AGEM (Chaudhry et al. 2019a) is a continual learning method improving on (Lopez-Paz and Ranzato 2017), which uses an episodic memory of",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "parameter gradients to limit forgetting. MER (Riemer et al. 2019) is a continual learning method that combines episodic memory with meta-learning to limit forgetting. ER-Ring (Chaudhry et al. 2019b) is a continual learning method that uses a ring buffer as episodic memory. MIR (Aljundi et al. 2019a) is a continual learning method based on experience replay that selects a minibatch from the episodic memory that incurs the maximum change in loss. Multitask is an oracle baseline that has access to all data to optimize Eq. 1, useful to estimate an upper bound on the obtainable accuracy (Eq. 2). Clone-and-finetune is an oracle baseline training one independent model per task, where the model for task  $t'$  is initialized by cloning the parameters of the model for task  $t' - 1$ .",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "All baselines use the same neural network architecture: a perceptron with two hidden layers of 256 ReLU neurons in the MNIST experiments, and a ResNet18, with three times less feature maps across all layers, similar to Lopez-Paz and Ranzato (2017), in CIFAR and ImageNet experiments. The task identifiers are used to select the output head in the CIFAR and ImageNet experiments, while ignored in the MNIST experiments. Batch size is set to 10 for both the stream of data and episodic memory across experiments and models. The size of episodic memory is set between 1 and 5 examples per class per task. The results of VCL are complied by running the official implementation<sup>1</sup>, that only works for fully-connected networks, in our continual learning setup. All the other baselines use our unified code base which is available here<sup>2</sup>. All experiments are averaged over five runs using different random seeds, where each seed corresponds to a different dataset ordering among tasks.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Results",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes the main results of our experiments when the episodic memory of only one example per class per task is used. First, our proposed HAL is the method achieving maximum Accuracy (Eq. 2) and minimal Forgetting (Eq. 3) for all benchmarks. This does not include Oracle baselines Multitask (which has access to all data simultaneously) and Clone-and-finetune (which trains a separate model per task). Second, the relative gains from ER-Ring to HAL are substantial, confirming that the anchoring objective (Eq. 5) allows experience-replay methods to generalize better with the same amount of episodic memory.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Third, regularization based approaches, such as EWC (Kirkpatrick et al. 2016) and VCL (Nguyen et al. 2018), suffer under the single epoch setup. As noted by Chaudhry et al. (2019a), EWC requires multiple passes over the samples of each task to perform well. The poor performance of VCL is attributed to the noisy posterior estimation in the single pass setup. Note that approaches making use of memory (MER, ER, MIR, and HAL) work substantially better in this setup.",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fourth, ICARL (Rebuffi, Kolesnikov, and Lampert 2017), another method making use of episodic memory, performs poorly in our setup. From Table 1, it can be argued that di",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{1}$ https://github.com/nvcuong/variational-continual-learning",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "page_footnote",
+        "text": "$^{2}$ https://github.com/arslan-chaudhry/HindsightAnchor",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/8747c22ff9cf262042d449736d1c5b6cd22e28615c036c7b199bd23e83cdb1e5.jpg",
+        "table_caption": [
+            "Table 1: Accuracy (Eq. 2) and Forgetting (Eq. 3) results of continual learning experiments. Averages and standard deviations are computed over five runs using different random seeds. When used, episodic memories contain up to one example per class per task. Last two rows are oracle baselines."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">PERMUTED MNIST</td><td colspan=\"2\">ROTATED MNIST</td></tr><tr><td></td><td>ACCURACY</td><td>FORGETTING</td><td>ACCURACY</td><td>FORGETTING</td></tr><tr><td>FINETUNE</td><td>53.5 (±1.46)</td><td>0.29 (±0.01)</td><td>41.9 (±1.37)</td><td>0.50 (±0.01)</td></tr><tr><td>EWC (KIRKPATRICK ET AL. 2016)</td><td>63.1 (±1.40)</td><td>0.18 (±0.01)</td><td>44.1 (±0.99)</td><td>0.47 (±0.01)</td></tr><tr><td>VCL (NGUYEN ET AL. 2018)</td><td>51.8 (±1.54)</td><td>0.44 (±0.01)</td><td>48.2 (±0.99)</td><td>0.50 (±0.01)</td></tr><tr><td>VCL-RANDOM (NGUYEN ET AL. 2018)</td><td>52.3 (±0.66)</td><td>0.43 (±0.01)</td><td>54.4 (±1.44)</td><td>0.44 (±0.01)</td></tr><tr><td>AGEM (CHAUDHRY ET AL. 2019A)</td><td>62.1 (±1.39)</td><td>0.21 (±0.01)</td><td>50.9 (±0.92)</td><td>0.40 (±0.01)</td></tr><tr><td>MER (RIEMER ET AL. 2019)</td><td>69.9 (±0.40)</td><td>0.14 (±0.01)</td><td>66.0 (±2.04)</td><td>0.23 (±0.01)</td></tr><tr><td>ER-RING (CHAUDHRY ET AL. 2019B)</td><td>70.2 (±0.56)</td><td>0.12 (±0.01)</td><td>65.9 (±0.41)</td><td>0.24 (±0.01)</td></tr><tr><td>MIR (ALJUNDI ET AL. 2019A)</td><td>71.1 (±0.41)</td><td>0.11 (±0.01)</td><td>-</td><td>-</td></tr><tr><td>HAL (OURS)</td><td>73.6 (±0.31)</td><td>0.09 (±0.01)</td><td>68.4 (±0.72)</td><td>0.21 (±0.01)</td></tr><tr><td>CLONE-AND-FINETUNE</td><td>81.4 (±0.35)</td><td>0.0</td><td>87.5 (±0.11)</td><td>0.0</td></tr><tr><td>MULTITASK</td><td>83.0</td><td>0.0</td><td>83.3</td><td>0.0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/bf1a9f6048e5c0bb148bd85ed900e3d89cd5b99f7a54c600e70ddcbcb764e6ab.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">SPLIT CIFAR</td><td colspan=\"2\">SPLIT MINIIMAGENET</td></tr><tr><td></td><td>ACCURACY</td><td>FORGETTING</td><td>ACCURACY</td><td>FORGETTING</td></tr><tr><td>FINETUNE</td><td>42.9 (±2.07)</td><td>0.25 (±0.03)</td><td>34.7 (±2.69)</td><td>0.26 (±0.03)</td></tr><tr><td>EWC (KIRKPATRICK ET AL. 2016)</td><td>42.4 (±3.02)</td><td>0.26 (±0.02)</td><td>37.7 (±3.29)</td><td>0.21 (±0.03)</td></tr><tr><td>ICARL (REBUFFI, KOLESNIKOV, AND LAMPERT 2017)</td><td>46.4 (±1.21)</td><td>0.16 (±0.01)</td><td>-</td><td>-</td></tr><tr><td>AGEM (CHAUDHRY ET AL. 2019A)</td><td>54.9 (±2.92)</td><td>0.14 (±0.03)</td><td>48.2 (±2.49)</td><td>0.13 (±0.02)</td></tr><tr><td>MER (RIEMER ET AL. 2019)</td><td>49.7 (±2.97)</td><td>0.19 (±0.03)</td><td>45.5 (±1.49)</td><td>0.15 (±0.01)</td></tr><tr><td>ER-RING (CHAUDHRY ET AL. 2019B)</td><td>56.2 (±1.93)</td><td>0.13 (±0.01)</td><td>49.0 (±2.61)</td><td>0.12 (±0.02)</td></tr><tr><td>MIR (ALJUNDI ET AL. 2019A)</td><td>57.1 (±1.81)</td><td>0.12 (±0.01)</td><td>49.3 (±2.15)</td><td>0.12 (±0.01)</td></tr><tr><td>HAL (OURS)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td><td>51.6 (±2.02)</td><td>0.10 (±0.01)</td></tr><tr><td>CLONE-AND-FINETUNE</td><td>60.3 (±0.55)</td><td>0.0</td><td>50.3 (±1.00)</td><td>0.0</td></tr><tr><td>MULTITASK</td><td>68.3</td><td>0.0</td><td>63.5</td><td>0.0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "rect training on a very small episodic memory, as done in experience replay, allows the method to generalize better compared to when the same memory is used indirectly in the knowledge distillation loss (Hinton, Vinyals, and Dean 2014) as done in ICARL.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Fig 1 shows the accuracy (Eq. 2) of methods employing episodic memory when the size of memory is increased. We use 1 to 5 examples per class per task, resulting in a total memory size from 200 to 1000 for MNIST experiments, and from 85 to 425 for CIFAR and ImageNet experiments. The corresponding numbers for Forgetting are given in Appendix B. HAL consistently improves on ER-Ring and other baselines.",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 in Appendix B provides the training time of the continual learning baselines on MNIST benchmarks. Although HAL adds an overhead on top of the experience replay baseline, it is substantially faster than MER —another approach that makes use of nested optimization to reduce forgetting. However, HAL requires extra memory to store task anchors that, as we will show next, are more effective than additional data samples one can store for experience replay. Overall, we conclude that HAL provides the best tradeoff in terms of efficiency and performance.",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Ablation Study",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now turn our attention towards two questions; (1) whether for the same episodic memory size in bytes HAL improves over the experience replay baseline, (2) whether fine-tuning on the replay buffer is a good approximation of forgetting when the learner is updated on future tasks.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To answer the first question, let  $|\\mathcal{M}|$  be the total size of episodic memory for all tasks when one example per class per task is stored in the replay buffer. We then run the experience replay with double the size of episodic memory (i.e.) storing two examples per class per task instead of one. The episodic memory size in HAL, on the other hand, is kept at  $|\\mathcal{M}|$ . This effectively makes the size of memory in bytes taken by experience replay and that of HAL equal as the latter requires extra memory to store anchors. Table 2 summarizes the results of this study. For the same memory size in bytes, HAL performs better than experience replay when additional real data samples are stored in the episodic memory. It is surprising that the anchors learned by HAL, initialized from random noise and learned using gradient-based optimization, perform better compared to randomly sampled real data. To understand this, in Figure 2 we visualize HAL's anchors along with the task data in the image",
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+    {
+        "type": "image",
+        "img_path": "images/b16bbdadfe1c5da6c83772c008e6f251fe08f7fde54f68cdf5acd8f356a913bf.jpg",
+        "image_caption": [
+            "(a) Permuted MNIST"
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+        "image_caption": [
+            "(b) Rotated MNIST"
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+    {
+        "type": "image",
+        "img_path": "images/fb61212077920520fcde77e0b54495ecc38b61c9c8e1eb7784470e93cf2e6312.jpg",
+        "image_caption": [
+            "(c) Split CIFAR"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/84392707214066eb4c8750af688f73b2909a3fc2ce9efc97417bd0bde891fd21.jpg",
+        "image_caption": [
+            "(d) Split miniImageNet",
+            "Figure 1: Accuracy (Eq. 2) results for different episodic memory sizes."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/3e9a29d8957939e87d4e06f987282a9bcc8bd50462ff6c46e64a12bb8d2a3610.jpg",
+        "table_caption": [
+            "Table 2: Comparison of HAL with experience replay. ER-Ring and HAL use one example per class per task in the episodic memory, whereas ER-Ring-2  $|\\mathcal{M}|$  uses two examples per class per task in the memory. Averages and standard deviations are computed over five runs using different random seeds."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"2\">Permuted MNIST</td><td colspan=\"2\">Split CIFAR</td></tr><tr><td></td><td>Accuracy</td><td>Forgetting</td><td>Accuracy</td><td>Forgetting</td></tr><tr><td>ER-Ring-|M|</td><td>70.2 ±(0.56)</td><td>0.12 (±0.01)</td><td>56.2 (±1.93)</td><td>0.13 (±0.01)</td></tr><tr><td>ER-Ring-2|M|</td><td>71.9 (±0.31)</td><td>0.11 (±0.01)</td><td>58.6 (±2.68)</td><td>0.12 (±0.01)</td></tr><tr><td>HAL-|M|</td><td>73.6 (±0.31)</td><td>0.09 (±0.01)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/68415b86c41310a86b5e8e136a0e95314268093c0a9b7856ff0f319558e9ebb5.jpg",
+        "table_caption": [
+            "Table 3: Performance comparison of HAL with Oracle where the learner has access to all the future tasks to exactly quantify forgetting of an anchor. Averages and standard deviations are computed over five runs using different random seeds."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Anchor type</td><td colspan=\"2\">Permuted MNIST</td><td colspan=\"2\">Split CIFAR</td></tr><tr><td></td><td>Accuracy</td><td>Forgetting</td><td>Accuracy</td><td>Forgetting</td></tr><tr><td>HAL</td><td>73.6 (±0.31)</td><td>0.09 (±0.01)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td></tr><tr><td>Oracle</td><td>73.9 (±0.41)</td><td>0.09 (±0.01)</td><td>61.1 (±0.94)</td><td>0.09 (±0.01)</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and feature space on Permuted MNIST benchmark. From the left of the figure, it can be seen that HAL anchors lie in the data cluster of a class in the image space, suggesting that the mean embedding loss in Eq. 9 effectively regularizes against outliers. More interestingly, the figure on the right shows that these anchors lie at or close to the cluster edges in the feature space. In other words, the anchor points learned by HAL lie close to the classifier decision boundary. This can explain their effectiveness compared to the real data samples that can lie anywhere in the data cluster in feature space.",
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+    {
+        "type": "text",
+        "text": "Finally, to answer the second part, we assume a non-continual setup where at each step the learner has an oracle access to all future tasks. After training on task  $t$ , the learner is fine-tuned on all future tasks and anchor points are subsequently learned by optimizing idealistic Eq. 7. The results",
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+    },
+    {
+        "type": "text",
+        "text": "are reported in Table 3. It can be seen from the table that the proposed HAL performs very close to the noncontinual oracle baseline. This suggests that HAL's approximation of forgetting when the learner is updated on future tasks by replaying past data is effective in many existing continual learning benchmarks.",
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+    {
+        "type": "text",
+        "text": "5 Related work",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In continual learning (Ring 1997), also called lifelong learning (Thrun 1998), a learner addresses a sequence of changing tasks without storing the complete datasets of these tasks. This contrasts with multitask learning (Caruana 1997), where the learner assumes simultaneous access to data from all tasks. The main challenge in continual learning is to avoid catastrophic interference (McCloskey and Cohen 1989; McClelland, McNaughton, and O'Reilly 1995; Goodfellow et al. 2013), that is, the learner forgetting previously acquired knowledge when learning new tasks. The state-of-the-art methods in continual learning can be categorized into three classes.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "First, regularization approaches discourage updating parameters important for past tasks (Kirkpatrick et al. 2016; Aljundi et al. 2018; Nguyen et al. 2018; Zenke, Poole, and Ganguli 2017). While efficient in terms of memory and computation, these approaches suffer from brittleness due to feature drift for large number of tasks (Titsias et al. 2019). Additionally, these approaches are only effective when the learner can perform multiple passes over each task (Chaudhry et al. 2019a), a case deemed unrealistic in this work.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Second, modular approaches use different parts of the prediction function for each new task (Fernando et al. 2017; Aljundi, Chakravarty, and Tuytelaars 2017; Rosenbaum, Klinger, and Riemer 2018; Chang et al. 2018; Xu and Zhu 2018; Ferran Alet 2018). Modular approaches do not scale to a large number of tasks, as they require searching over the combinatorial space of module architectures. Another modular approach (Rusu et al. 2016; Lee et al. 2017) adds new parts to the prediction function as new tasks are learned. By construction, modular approaches have zero forgetting, but their memory requirements increase with the number of tasks.",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fcd9ebc526b8ddc695d1470ce158cf495e0e9ba11c0ea6529c19dbea2ebb548.jpg",
+        "image_caption": [
+            "Image Space"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3f5b18590290746f3c8ce7d93ba6504df65b7f0a28e9ad23f412940fabf846b6.jpg",
+        "image_caption": [
+            "Feature Space",
+            "Figure 2: t-SNE visualization of images and anchors (HAL) in the image space (left) and the feature space (right) on Permuted MNIST benchmark for a single task. Anchor points are exaggerated in size for the purpose of better visualization. The left plot shows that anchor points lie with in the data cluster of a class, whereas the right plot shows that, in the feature space, anchor points lie close to the edge of the cluster of a class or near decision boundaries."
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Third, episodic memory approaches maintain and revisit a small episodic memory of data from past tasks. In some of these methods (Li and Hoiem 2016; Rebuffi, Kolesnikov, and Lampert 2017), examples in the episodic memory are replayed and predictions are kept invariant by means of distillation (Hinton, Vinyals, and Dean 2014). In other approaches (Lopez-Paz and Ranzato 2017; Chaudhry et al. 2019a; Aljundi et al. 2019b) the episodic memory is used as an optimization constraint that discourages increases in loss of past tasks. More recently, several works (Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Rolnick et al. 2018; Chaudhry et al. 2019b) have shown that directly optimizing the loss of episodic memory, also known as experience replay, is cheaper than constraint-based approaches and improves the prediction performance. Our contribution in this paper has been to improve experience replay methods with task anchors learned in hindsight.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "There are other definitions of continual learning, such as the one of task-free continual learning. The task-free formulation does not consider the notion of tasks, and instead works on undivided data streams (Aljundi, Kelchtermans, and Tuytelaars 2019; Aljundi et al. 2019b). We have focused on the task-based definition of continual learning and, similar to many recent works (Lopez-Paz and Ranzato 2017; Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Chaudhry et al. 2019a), assumed that only a single pass through the data was possible.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, our gradient-based learning of anchors bears a similarity to (Simonyan, Vedaldi, and Zisserman 2014) and (Wang et al. 2018). In Simonyan, Vedaldi, and Zisserman (2014), the authors use gradient ascent on class scores to find saliency maps of a classification model. Contrary to",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "them, our proposed hindsight learning objective optimizes for the forgetting metric, as reducing it is necessary for continual learning. Dataset distillation (Wang et al. 2018) proposes to encode the entire dataset in a few synthetic points at a given parameter vector by a gradient-based optimization process. Their method requires access to the entire dataset of a task for optimization purposes. We, instead, learn anchors in hindsight from the replay buffer of past tasks after training is finished for the current task. While Wang et al. (2018) aim to replicate the performance of the entire dataset from the synthetic points, we focus on reducing forgetting of an already learned task.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We introduced a bilevel optimization objective, dubbed anchoring, for continual learning. In our approach, we learned one \"anchor point\" per class per task, where predictions are requested to remain invariant by means of nested optimization. These anchors are learned using gradient-based optimization and represent points that would maximize the forgetting of the current task throughout the entire learning experience. We simulate the forgetting that would happen during the learning of future tasks in hindsight, that is, by taking temporary gradient steps across a small episodic memory of past tasks. We call our approach Hindsight Anchor Learning (HAL). As shown in our experiments, anchoring in hindsight complements and improves the performance of continual learning methods based on experience replay, achieving a new state of the art on four standard continual learning benchmarks.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement",
+        "text_level": 1,
+        "bbox": [
+            200,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Marc'Aurelio Ranzato for helpful discussions. This work was supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1 and EPSRC/MURI grant EP/N019474/1. We would also like to acknowledge the Royal Academy of Engineering and FiveAI. AC is funded by Amazon Research award.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            233,
+            196,
+            330,
+            212
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Aljundi, R.; Babiloni, F.; Elhoseiny, M.; Rohrbach, M.; and Tuytelaars, T. 2018. Memory Aware Synapses: Learning what (not) to forget. In ECCV.",
+            "Aljundi, R.; Belilovsky, E.; Tuytelaars, T.; Charlin, L.; Caccia, M.; Lin, M.; and Page-Caccia, L. 2019a. Online Continual Learning with Maximal Interfered Retrieval. In Wallach, H.; Larochelle, H.; Beygelzimer, A.; d'Alché-Buc, F.; Fox, E.; and Garnett, R., eds., Advances in Neural Information Processing Systems 32, 11849-11860. Curran Associates, Inc. URL http://papers.nips.cc/paper/9357-online-continual-learning-with-maximal-interfered-retrieval.pdf.",
+            "Aljundi, R.; Chakravarty, P.; and Tuytelaars, T. 2017. Expert Gate: Lifelong Learning with a Network of Experts. In CVPR, 7120-7129.",
+            "Aljundi, R.; Kelchtermans, K.; and Tuytelaars, T. 2019. Task-free continual learning. In CVPR, 11254-11263.",
+            "Aljundi, R.; Lin, M.; Goujaud, B.; and Bengio, Y. 2019b. Online continual learning with no task boundaries. arXiv preprint arXiv:1903.08671.",
+            "Caruana, R. 1997. Multitask learning. Machine learning 28(1): 41-75.",
+            "Chang, M.; Gupta, A.; Levine, S.; and Griffiths, T. L. 2018. Automatically Composing Representation Transformations as a Means for Generalization. In ICML workshop Neural Abstract Machines and Program Induction v2.",
+            "Chaudhry, A.; Dokania, P. K.; Ajanthan, T.; and Torr, P. H. 2018. Riemannian Walk for Incremental Learning: Understanding Forgetting and Intransigence. In ECCV.",
+            "Chaudhry, A.; Ranzato, M.; Rohrbach, M.; and Elhoseiny, M. 2019a. Efficient Lifelong Learning with A-GEM. In ICLR.",
+            "Chaudhry, A.; Rohrbach, M.; Elhoseiny, M.; Ajanthan, T.; Dokania, P. K.; Torr, P. H.; and Ranzato, M. 2019b. Continual Learning with Tiny Episodic Memories. arXiv preprint arXiv:1902.10486.",
+            "Fernando, C.; Banarse, D.; Blundell, C.; Zwols, Y.; Ha, D.; Rusu, A. A.; Pritzel, A.; and Wierstra, D. 2017. Pathnet: Evolution channels gradient descent in super neural networks. arXiv preprint arXiv:1701.08734.",
+            "Ferran Alet, Tomas Lozano-Perez, L. P. K. 2018. Modular Meta-Learning. arXiv preprint arXiv:1806.10166v1.",
+            "Finn, C.; Abbeel, P.; and Levine, S. 2017. Model-agnostic meta-learning for fast adaptation of deep networks. In ICML-Volume 70, 1126-1135. JMLR.org."
+        ],
+        "bbox": [
+            84,
+            215,
+            478,
+            888
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Goodfellow, I. J.; Mirza, M.; Xiao, D.; Courville, A.; and Bengio, Y. 2013. An empirical investigation of catastrophic forgetting in gradient-based neural networks. arXiv preprint arXiv:1312.6211.",
+            "Hayes, T. L.; Cahill, N. D.; and Kanan, C. 2018. Memory Efficient Experience Replay for Streaming Learning. arXiv preprint arXiv:1809.05922.",
+            "Hinton, G.; Vinyals, O.; and Dean, J. 2014. Distilling the knowledge in a neural network. In NIPS.",
+            "Isele, D.; and Cosgun, A. 2018. Selective Experience Replay for Lifelong Learning. arXiv preprint arXiv:1802.10269.",
+            "Kirkpatrick, J.; Pascanu, R.; Rabinowitz, N. C.; Veness, J.; Desjardins, G.; Rusu, A. A.; Milan, K.; Quan, J.; Ramalho, T.; Grabska-Barwinska, A.; Hassabis, D.; Clopath, C.; Kumaran, D.; and Hadsell, R. 2016. Overcoming catastrophic forgetting in neural networks. *PNAS*.",
+            "Krizhevsky, A.; and Hinton, G. 2009. Learning multiple layers of features from tiny images. https://www.cs.toronto.edu/kriz/cifar.html.",
+            "LeCun, Y. 1998. The MNIST database of handwritten digits. http://yann.lecun.com/exdb/mnist/.",
+            "Lee, J.; Yun, J.; Hwang, S.; and Yang, E. 2017. Lifelong Learning with Dynamically Expandable Networks. arXiv preprint arXiv:1708.01547.",
+            "Li, Z.; and Hoiem, D. 2016. Learning without forgetting. In ECCV, 614-629.",
+            "Lopez-Paz, D.; and Ranzato, M. 2017. Gradient Episodic Memory for Continuum Learning. In NIPS.",
+            "McClelland, J. L.; McNaughton, B. L.; and O'Reilly, R. C. 1995. Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory. Psychological review 102(3): 419.",
+            "McCloskey, M.; and Cohen, N. J. 1989. Catastrophic interference in connectionist networks: The sequential learning problem. Psychology of learning and motivation 24: 109-165.",
+            "Nguyen, C. V.; Li, Y.; Bui, T. D.; and Turner, R. E. 2018. Variational continual learning. *ICLR*.",
+            "Nichol, A.; and Schulman, J. 2018. Reptile: a scalable met-learning algorithm. arXiv preprint arXiv:1803.02999, 2018",
+            "Rebuffi, S.-V.; Kolesnikov, A.; and Lampert, C. H. 2017. iCaRL: Incremental Classifier and Representation Learning. In CVPR.",
+            "Riemer, M.; Cases, I.; Ajemian, R.; Liu, M.; Rish, I.; Tu, Y.; and Tesauro, G. 2019. Learning to Learn without Forgetting By Maximizing Transfer and Minimizing Interference. In ICLR.",
+            "Ring, M. B. 1997. CHILD: A first step towards continual learning. Machine Learning 28(1): 77-104."
+        ],
+        "bbox": [
+            517,
+            68,
+            911,
+            888
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Rolnick, D.; Ahuja, A.; Schwarz, J.; Lillicrap, T. P.; and Wayne, G. 2018. Experience Replay for Continual Learning. CoRR abs/1811.11682. URL http://arxiv.org/abs/1811.11682.",
+            "Rosenbaum, C.; Klinger, T.; and Riemer, M. 2018. Routing Networks: Adaptive Selection of Non-Linear Functions for Multi-Task Learning. In ICLR.",
+            "Russakovsky, O.; Deng, J.; Su, H.; Krause, J.; Satheesh, S.; Ma, S.; Huang, Z.; Karpathy, A.; Khosla, A.; Bernstein, M.; Berg, A. C.; and Fei-Fei, L. 2015. ImageNet Large Scale Visual Recognition Challenge. IJCV 115(3): 211-252.",
+            "Rusu, A. A.; Rabinowitz, N. C.; Desjardins, G.; Soyer, H.; Kirkpatrick, J.; Kavukcuoglu, K.; Pascanu, R.; and Hadsell, R. 2016. Progressive neural networks. arXiv preprint arXiv:1606.04671.",
+            "Simonyan, K.; Vedaldi, A.; and Zisserman, A. 2014. Deep inside convolutional networks: Visualising image classification models and saliency maps. In ICLR.",
+            "Smola, A.; Gretton, A.; Song, L.; and Scholkopf, B. 2007. A Hilbert space embedding for distributions. In ALT, 13-31. Springer.",
+            "Thrun, S. 1998. Lifelong learning algorithms. In Learning to learn, 181-209. Springer.",
+            "Titsias, M. K.; Schwarz, J.; Matthews, A. G. d. G.; Pascanu, R.; and Teh, Y. W. 2019. Functional Regularisation for Continual Learning using Gaussian Processes. arXiv preprint arXiv:1901.11356.",
+            "Vinyals, O.; Blundell, C.; Lillicrap, T.; Wierstra, D.; et al. 2016. Matching networks for one shot learning. In NIPS, 3630-3638.",
+            "Wang, T.; Zhu, J.-Y.; Torralba, A.; and Efros, A. A. 2018. Dataset Distillation. arXiv preprint arXiv:1811.10959.",
+            "Xu, J.; and Zhu, Z. 2018. Reinforced Continual Learning. In arXiv preprint arXiv:1805.12369v1.",
+            "Zenke, F.; Poole, B.; and Ganguli, S. 2017. Continual Learning Through Synaptic Intelligence. In ICML."
+        ],
+        "bbox": [
+            84,
+            68,
+            478,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Appendix",
+        "text_level": 1,
+        "bbox": [
+            455,
+            66,
+            540,
+            85
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Section A describes the approximate update performed by the anchoring objective (Eq. 5 in the main paper). Section B reports more experimental results. Section C provides the grid considered for hyper-parameters. Section E gives the pseudocode for HAL.",
+        "bbox": [
+            81,
+            85,
+            915,
+            128
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A Approximate Update of Anchoring Objective",
+        "text_level": 1,
+        "bbox": [
+            290,
+            141,
+            705,
+            159
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Here we will use a Taylor series expansion to approximate the update performed by the anchoring objective (Eq. 5 in the main paper). In particular, we are interested in the regularization part of the anchoring objective that involves a nested update. We refer to this gradient as  $g_{anc}$ . We follow similar arguments as (Nichol and Schulman 2018).",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Let  $\\theta_0$  be the parameter vector before the temporary update of the anchoring objective (Eq. 5). Moreover, let  $\\ell_{ce}$  and  $\\ell_{L2}$  be the cross-entropy and L2 losses, respectively. We use the following definitions:",
+        "bbox": [
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+            202,
+            913,
+            231
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {g} _ {0} = \\ell_ {c e} ^ {\\prime} (\\theta_ {0}) \\quad \\left(\\text {g r a d i e n t o f c r o s s - e n t r o p y l o s s a t i n i t i a l p o i n t o n} \\mathcal {B} \\cup \\mathcal {B} _ {\\mathcal {M}}\\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            237,
+            753,
+            253
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {H} _ {0} = \\ell_ {c e} ^ {\\prime \\prime} (\\theta_ {0}) \\quad \\left(\\text {H e s s i a n o f c r o s s - e n t r o p y l o s s a t i n i t i a l p o i n t o n} \\mathcal {B} \\cup \\mathcal {B} _ {\\mathcal {M}}\\right)\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            256,
+            751,
+            273
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {g} _ {1} = \\ell_ {L 2} ^ {\\prime} \\left(\\theta_ {0}\\right) \\quad (\\text {g r a d i e n t o f L 2 l o s s a t i n i t i a l p o i n t o n a n c h o r s})\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            277,
+            679,
+            294
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\bar {H} _ {1} = \\ell_ {L 2} ^ {\\prime \\prime} (\\theta_ {0}) \\quad \\text {(g r a d i e n t o f L 2 l o s s a t i n i t i a l p o i n t o n a n c h o r s)}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            296,
+            679,
+            314
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Let  $U_0 = \\theta_0 - \\alpha \\overline{g}_0$  be the operator giving a temporary update in the two-step process of (Eq. 5), and let  $\\theta_1$  be the temporary update itself (i.e.)  $\\theta_1 \\coloneqq U_0$  (note that  $\\tilde{\\theta}$  is used in the main paper instead of  $\\theta_1$ ). The  $g_{anc}$  is given by:",
+        "bbox": [
+            81,
+            337,
+            913,
+            371
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} g _ {a n c} = \\frac {\\partial}{\\partial \\theta_ {0}} \\ell_ {L 2} (U _ {0}) \\\\ = U _ {0} ^ {\\prime} \\cdot \\ell_ {L 2} ^ {\\prime} (\\theta_ {1}) \\\\ = \\left(I - \\alpha \\bar {H} _ {0}\\right) \\cdot \\ell_ {L 2} ^ {\\prime} \\left(\\theta_ {1}\\right), \\tag {10} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            386,
+            911,
+            457
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where the second step is obtained by using the chain rule. Now, if we calculate the first order Taylor series approximation of  $\\ell_{L2}^{\\prime}(\\theta_1)$ ,",
+        "bbox": [
+            81,
+            463,
+            915,
+            494
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\ell_ {L 2} ^ {\\prime} \\left(\\theta_ {1}\\right) = \\ell_ {L 2} ^ {\\prime} \\left(\\theta_ {0}\\right) + \\ell_ {L 2} ^ {\\prime \\prime} \\left(\\theta_ {0}\\right) \\cdot \\left(\\theta_ {1} - \\theta_ {0}\\right) + O \\left(\\left| \\left| \\theta_ {1} - \\theta_ {0} \\right| \\right| ^ {2}\\right) \\\\ = \\bar {g} _ {1} + \\bar {H} _ {1} \\cdot \\left(\\theta_ {0} - \\alpha \\bar {g} _ {0} - \\theta_ {0}\\right) + O \\left(\\alpha^ {2}\\right) \\\\ = \\bar {g} _ {1} - \\alpha \\bar {H} _ {1} \\cdot \\bar {g} _ {0} + O \\left(\\alpha^ {2}\\right), \\tag {11} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            511,
+            911,
+            570
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where in the second step we substituted the value of  $\\theta_{1}$ . By putting Eq. 11 in Eq. 10 and after some simplification we get:",
+        "bbox": [
+            98,
+            575,
+            893,
+            592
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ng _ {a n c} = \\bar {g} _ {1} - \\alpha \\left(\\bar {H} _ {1} \\cdot \\bar {g} _ {0} + \\bar {H} _ {0} \\cdot \\bar {g} _ {1}\\right) + O \\left(\\alpha^ {2}\\right). \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            602,
+            911,
+            619
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This form is very similar to the second-order MAML gradient formulation, Eq. 25 in (Nichol and Schulman 2018). Further simplification of the inner product terms between Hessian and gradient yields the inner product between the gradients  $\\overline{g_0}$  and  $\\overline{g_1}$ . This shows that similar to MAML (Finn, Abbeel, and Levine 2017), Reptile (Nichol and Schulman 2018) and MER (Riemer et al. 2019), the anchoring objective, as described in Eq. 5 of the main paper, maximizes the inner product between the gradients. However, unlike the other meta-learning approaches, in the anchoring objective, these gradients correspond to different loss functions, cross-entropy and L2 losses on data from the current task and episodic memory, and HAL anchors, respectively.",
+        "bbox": [
+            81,
+            622,
+            913,
+            708
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B More Results",
+        "text_level": 1,
+        "bbox": [
+            421,
+            719,
+            573,
+            734
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows a more fine-grained analysis of average accuracy as new tasks are learned on Permuted MNIST and Split CIFAR. HAL preserves the performance of a predictor more effectively than other baselines.",
+        "bbox": [
+            81,
+            738,
+            913,
+            766
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tables 5 and 6 show the accuracy and forgetting of methods employing episodic memory when the size of memory is increased. We use 3 to 5 examples per class per task, resulting in a total memory size from 600 to 1000 for MNIST experiments, and from 255 to 425 for CIFAR and ImageNet experiments.",
+        "bbox": [
+            81,
+            766,
+            913,
+            808
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "C Hyper-parameter Selection",
+        "text_level": 1,
+        "bbox": [
+            362,
+            821,
+            630,
+            838
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we report the hyper-parameters grid considered for experiments. The best values for different benchmarks are given in parentheses.",
+        "bbox": [
+            81,
+            840,
+            913,
+            869
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "- Multitask",
+        "bbox": [
+            84,
+            875,
+            166,
+            887
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/03c605b5355952321432ebafbebab2455ea6634b777752c15b7d6111f01bc6e3.jpg",
+        "image_caption": [
+            "Permuted MNIST"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            125,
+            68,
+            421,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9eafc4c0a015aa8f6d274a5c523990b5f63941959c47917321a54a29ba240a9.jpg",
+        "image_caption": [
+            "Split CIFAR",
+            "Figure 3: Evolution of Accuracy (Eq. 2) as new tasks are learned. When used, episodic memories contain up to one example per class per task."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
+            66,
+            867,
+            169
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/e9350121d7bad21648c37272c4921285045e555487fa02c8c3b1327933ed0a55.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Anchor type</td><td colspan=\"2\">Split CIFAR</td></tr><tr><td></td><td>Accuracy</td><td>Forgetting</td></tr><tr><td>Real Data Anchor</td><td>58.0 (±0.15)</td><td>0.12 (±0.01)</td></tr><tr><td>HAL (ours)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td></tr></table>",
+        "bbox": [
+            326,
+            289,
+            671,
+            368
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf498d4189b19aa297454450490a5004bf92fd45f69a82e85f73ba7778314394.jpg",
+        "image_caption": [
+            "Table 4: Impact of anchor selection, where we compare a randomly chosen data point as an anchor (Real Data Anchor) with our optimized anchor selection (HAL).",
+            "Figure 4: Training time (s) of MNIST experiments for the entire continual learning experience. MER and HAL both use meta-learning objectives to reduce forgetting."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            334,
+            383,
+            661,
+            549
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "- Clone-and-finetune",
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "- Finetune",
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "EWC",
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "- regularization: [0.1, 1, 10 (MNIST perm, rot, CIFAR, miniImageNet), 100, 1000]",
+            "- AGEM",
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "- MER"
+        ],
+        "bbox": [
+            86,
+            614,
+            888,
+            887
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/62de80cd73e9ea1a98889b86e0da15497995b341816f861b0eb46aedf0901f25.jpg",
+        "table_caption": [
+            "Table 5: Accuracy (Eq. 2) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Metrics are averaged over five runs using different random seeds."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">PERMUTED MNIST</td><td colspan=\"2\">ROTATED MNIST</td></tr><tr><td></td><td>|M| = 600</td><td>|M| = 1000</td><td>|M| = 600</td><td>|M| = 1000</td></tr><tr><td>VCL-RANDOM</td><td>55.8 (±1.29)</td><td>58.5 (±1.21)</td><td>61.2 (±0.12)</td><td>64.4 (±0.16)</td></tr><tr><td>AGEM</td><td>63.2 (±1.47)</td><td>64.1 (±0.74)</td><td>49.9 (±1.49)</td><td>53.0 (±1.52)</td></tr><tr><td>MER</td><td>74.9 (±0.49)</td><td>78.3 (±0.19)</td><td>76.5 (±0.30)</td><td>77.3 (±1.13)</td></tr><tr><td>ER-RING</td><td>73.5 (±0.43)</td><td>75.8 (±0.24)</td><td>74.7 (±0.56)</td><td>76.5 (±0.48)</td></tr><tr><td>HAL (OURS)</td><td>76.2 (±0.52)</td><td>78.4 (±0.27)</td><td>77.0 (±0.66)</td><td>78.7 (±0.97)</td></tr></table>",
+        "bbox": [
+            238,
+            104,
+            759,
+            218
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/9234d1c0f36bf0cbf27b6c531f18863adb5161b9842017bc8ba849cb196e61b6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">SPLIT CIFAR</td><td colspan=\"2\">SPLIT MINIIMAGENET</td></tr><tr><td></td><td>|M| = 255</td><td>|M| = 425</td><td>|M| = 255</td><td>|M| = 425</td></tr><tr><td>ICARL</td><td>51.7 (±1.41)</td><td>51.2 (±1.32)</td><td>-</td><td>-</td></tr><tr><td>AGEM</td><td>56.9 (±3.45)</td><td>59.9 (±2.64)</td><td>51.6 (±2.69)</td><td>54.3 (±1.56)</td></tr><tr><td>MER</td><td>57.7 (±2.59)</td><td>60.6 (±2.09)</td><td>49.4 (±3.43)</td><td>54.8 (±1.79)</td></tr><tr><td>ER-RING</td><td>60.9 (±1.44)</td><td>62.6 (±1.77)</td><td>53.5 (±1.42)</td><td>54.2 (±3.23)</td></tr><tr><td>HAL (OURS)</td><td>62.9 (±1.49)</td><td>64.4 (±2.15)</td><td>56.5 (±0.87)</td><td>57.2 (±1.54)</td></tr></table>",
+        "bbox": [
+            241,
+            233,
+            751,
+            345
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/39e86ee5265bf8d6bd0eca63a60bf1e77f44ab71810252d7e0e724d49a7991a0.jpg",
+        "table_caption": [
+            "Table 6: Forgetting (Eq. 3) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Averages and standard deviations are computed over five runs using different random seeds."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">PERMUTED MNIST</td><td colspan=\"2\">ROTATED MNIST</td></tr><tr><td></td><td>|M| = 600</td><td>|M| = 1000</td><td>|M| = 600</td><td>|M| = 1000</td></tr><tr><td>VCL-RANDOM</td><td>0.39 (±0.01)</td><td>0.36 (±0.01)</td><td>0.37 (±0.01)</td><td>0.33 (±0.01)</td></tr><tr><td>AGEM</td><td>0.20 (±0.01)</td><td>0.19 (±0.01)</td><td>0.41 (±0.01)</td><td>0.38 (±0.01)</td></tr><tr><td>MER</td><td>0.14 (±0.01)</td><td>0.09 (±0.01)</td><td>0.12 (±0.01)</td><td>0.11 (±0.01)</td></tr><tr><td>ER-RING</td><td>0.09 (±0.01)</td><td>0.07 (±0.01)</td><td>0.15 (±0.01)</td><td>0.13 (±0.01)</td></tr><tr><td>HAL (OURS)</td><td>0.07 (±0.01)</td><td>0.05 (±0.01)</td><td>0.12 (±0.01)</td><td>0.11 (±0.01)</td></tr></table>",
+        "bbox": [
+            238,
+            416,
+            759,
+            527
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/237360b2dbdc6e72654f42e7702f5e6e8bdf283e7b553cb3929e05477c6adb99.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">SPLIT CIFAR</td><td colspan=\"2\">SPLIT MINIIMAGENET</td></tr><tr><td></td><td>|M| = 255</td><td>|M| = 425</td><td>|M| = 255</td><td>|M| = 425</td></tr><tr><td>ICARL</td><td>0.13 (±0.02)</td><td>0.13 (±0.02)</td><td>-</td><td>-</td></tr><tr><td>AGEM</td><td>0.13 (±0.03)</td><td>0.10 (±0.02)</td><td>0.10 (±0.02)</td><td>0.08 (±0.01)</td></tr><tr><td>MER</td><td>0.11 (±0.01)</td><td>0.09 (±0.02)</td><td>0.12 (±0.02)</td><td>0.07 (±0.01)</td></tr><tr><td>ER-RING</td><td>0.09 (±0.01)</td><td>0.06 (±0.01)</td><td>0.07 (±0.02)</td><td>0.08 (±0.02)</td></tr><tr><td>HAL (OURS)</td><td>0.08 (±0.01)</td><td>0.06 (±0.01)</td><td>0.06 (±0.01)</td><td>0.06 (±0.01)</td></tr></table>",
+        "bbox": [
+            241,
+            542,
+            751,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- learning rate: [0.003, 0.01, 0.03 (MNIST, CIFAR, miniImageNet), 0.1, 0.3, 1.0]",
+            "- within batch meta-learning rate: [0.01, 0.03, 0.1 (MNIST, CIFAR, miniImageNet), 0.3, 1.0]",
+            "- current batch learning rate multiplier: [1, 2, 5 (CIFAR, miniImageNet), 10 (MNIST)]"
+        ],
+        "bbox": [
+            98,
+            684,
+            888,
+            758
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- ER-Ring",
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "HAL",
+            "- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]",
+            "- regularization  $(\\lambda)$ : [0.01, 0.03, 0.1 (MNIST perm, rot), 0.3 (miniImageNet), 1 (CIFAR), 3, 10]"
+        ],
+        "bbox": [
+            86,
+            762,
+            888,
+            888
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- mean embedding strength  $(\\gamma)$ : [0.01, 0.03, 0.1 (MNIST perm, rot, CIFAR, miniImageNet), 0.3, 1, 3, 10]",
+            "- decay rate  $(\\beta)$ : 0.5",
+            "gradient steps on anchors  $(k)$  : 100"
+        ],
+        "bbox": [
+            98,
+            68,
+            800,
+            130
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D Hyperparameter Sensitivity",
+        "text_level": 1,
+        "bbox": [
+            359,
+            142,
+            633,
+            159
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Table 7, we report the performance of HAL against a range of hyperparameters. For a given hyperparameter in the table, all the other hyperparameters are set to their optimal values found in Sec C of the appendix. HAL is not sensitive to the choice of hyperparameters.",
+        "bbox": [
+            81,
+            161,
+            915,
+            205
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/a80e3f1dacbb2aefe74473a816f4b6c530f0a0ec46239fd70a533addd40747ec.jpg",
+        "table_caption": [
+            "Table 7: Average Accuracy of HAL on different values of hyperparameters. For a given hyperparameter in the table, all the other hyperparameters are set to their optimal values found in Sec C of the appendix."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td>λ</td><td>Acc</td><td>γ</td><td>Acc</td><td>β</td><td>Acc</td></tr><tr><td rowspan=\"3\">PERMUTED MNIST</td><td>0.01</td><td>72.8 ±(0.52)</td><td>0.01</td><td>73.1 ±(0.20)</td><td>0.1</td><td>72.5 ±(0.95)</td></tr><tr><td>0.1</td><td>73.6 ±(0.31)</td><td>0.1</td><td>73.6 ±(0.31)</td><td>0.5</td><td>73.6 ±(0.31)</td></tr><tr><td>1.0</td><td>73.2 ±(0.85)</td><td>1.0</td><td>73.4 ±(0.41)</td><td>0.9</td><td>72.9 ±(0.39)</td></tr><tr><td rowspan=\"3\">SPLIT CIFAR100</td><td>0.01</td><td>58.5 ±(1.25)</td><td>0.01</td><td>59.8 ±(0.65)</td><td>0.1</td><td>58.7 ±(1.17)</td></tr><tr><td>0.1</td><td>59.2 ±(0.91)</td><td>0.1</td><td>60.4 ±(0.54)</td><td>0.5</td><td>60.4 ±(0.54)</td></tr><tr><td>1.0</td><td>60.4 ±(0.54)</td><td>1.0</td><td>60.2 ±(1.21)</td><td>0.9</td><td>59.6 ±(1.05)</td></tr></table>",
+        "bbox": [
+            204,
+            255,
+            789,
+            368
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E HAL Algorithm",
+        "text_level": 1,
+        "bbox": [
+            408,
+            400,
+            586,
+            419
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 provides a pseudocode for HAL.",
+        "bbox": [
+            83,
+            420,
+            385,
+            435
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "code",
+        "sub_type": "code",
+        "code_caption": [
+            "Algorithm 1 Training of HAL on sequential data  $\\mathcal{D} = \\{\\mathcal{D}_1,\\dots ,\\mathcal{D}_T\\}$ , with total replay buffer size 'mem sz', learning rate  $\\alpha$  regularization strength  $\\lambda$  mean embedding decay  $\\beta$  mean embedding strength  $\\eta$"
+        ],
+        "code_body": "1: procedure HAL(D, mem sz,  $\\alpha ,\\lambda ,\\beta$    \n2:  $\\mathcal{M}\\gets \\{\\} *\\mathrm{mem\\_sz}$    \n3:  $\\{e_1,\\dots ,e_T\\} \\leftarrow \\{\\}$    \n4: for  $t\\in \\{1,\\dots ,T\\}$  do   \n5:  $\\phi_t\\gets \\vec{0}$    \n6: for  $\\mathcal{B}\\sim \\mathcal{D}_t$  do ▷ Sample a batch from current task   \n7:  $\\mathcal{B}_{\\mathcal{M}}\\sim \\mathcal{M}$  ▷ Sample a batch from episodic memory   \n8:  $\\tilde{\\theta}\\gets \\theta -\\alpha \\cdot \\nabla_{\\theta}\\ell (\\mathcal{B}\\cup \\mathcal{B}_{\\mathcal{M}})$  ▷ Temporary parameter update   \n9:  $\\theta \\gets \\theta -\\alpha \\cdot \\nabla_{\\theta}\\left(\\ell (\\mathcal{B}\\cup \\mathcal{B}_{\\mathcal{M}}) + \\lambda \\cdot \\sum_{t^{\\prime} <   t}(f_{\\theta}(e_{t^{\\prime}},t^{\\prime}) - f_{\\tilde{\\theta}}(e_{t^{\\prime}},t^{\\prime}))^{2}\\right)$  ▷ Anchoring objective (Eq. 5)   \n10:  $\\phi_t\\gets \\beta \\cdot \\phi_t + (1 - \\beta)\\cdot \\phi (\\mathcal{B})$  ▷ Running average of mean embedding   \n11:  $\\mathcal{M}\\gets$  UpdateMemory(M,B) ▷ Add samples to a ring buffer   \n12: end for   \n13:  $e_t,\\theta \\gets$  GetAnchors(M,  $\\theta ,\\phi_t,\\eta)$  ▷ Get anchors for current task   \n14: end for   \n15: return  $\\theta ,\\mathcal{M}$    \n16: end procedure   \n1: procedure GETANCHORS(M,  $\\theta_t,\\phi_t,\\gamma)$    \n2:  $\\theta \\gets \\theta_t$    \n3: for  $\\mathcal{B}_{\\mathcal{M}}\\sim \\mathcal{M}$  do   \n4:  $\\theta \\gets \\theta -\\alpha \\cdot \\nabla_{\\theta}\\ell (\\mathcal{B}_{\\mathcal{M}})$  ▷ Finetune  $\\theta_t$  by taking SGD steps on the episodic memory   \n5: end for   \n6:  $\\theta_{\\mathcal{M}}\\gets \\theta$  ▷ Store the updated parameter   \n7:  $e_t\\gets$  rand() ▷ Initialize the task anchors   \n8: for  $1,\\dots ,k$  do   \n9:  $e_t\\gets e_t + \\alpha \\cdot \\nabla_{e_t}\\left(\\ell (f_{\\theta_{\\mathcal{M}}} (e_t,t),y_t) - \\ell (f_{\\theta_t}(e_t,t),y_t) - \\gamma (\\phi (e_t) - \\phi_t)^2\\right)$  ▷ Maximize forgetting (Eq. 9)   \n10: end for   \n11: return  $e_t,\\theta_t$    \n12: end procedure",
+        "guess_lang": "latex",
+        "bbox": [
+            84,
+            482,
+            913,
+            872
+        ],
+        "page_idx": 12
+    }
+]

2002.08xxx/2002.08165/a4f0c2ad-a173-4b99-8972-7a973b7b598f_origin.pdf

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2002.08xxx/2002.08165/full.md

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+# Using Hindsight to Anchor Past Knowledge in Continual Learning
+
+Arslan Chaudhry $^{1,*}$ , Albert Gordo $^{2}$ , Puneet K. Dokania $^{1}$ , Philip Torr $^{1}$ , David Lopez-Paz $^{2}$
+
+<sup>1</sup>University of Oxford, <sup>2</sup>Facebook AI
+
+# Abstract
+
+In continual learning, the learner faces a stream of data whose distribution changes over time. Modern neural networks are known to suffer under this setting, as they quickly forget previously acquired knowledge. To address such catastrophic forgetting, many continual learning methods implement different types of experience replay, re-learning on past data stored in a small buffer known as episodic memory. In this work, we complement experience replay with a new objective that we call "anchoring", where the learner uses bilevel optimization to update its knowledge on the current task, while keeping intact predictions on some anchor points of past tasks. These anchor points are learned using gradient-based optimization to maximize forgetting, which is approximated by fine-tuning the currently trained model on the episodic memory of past tasks. Experiments on several supervised learning benchmarks for continual learning demonstrate that our approach improves the standard experience replay in terms of both accuracy and forgetting metrics and for various sizes of episodic memory.
+
+# 1 Introduction
+
+We study the problem of continual learning, where a machine learning model experiences a sequence of tasks. Each of these tasks is presented as a stream of input-output pairs, where each pair is drawn identically and independently (iid) from the corresponding task probability distribution. Since the length of the learning experience is not specified a priori, the learner can only assume a single pass over the data and, due to space constraints, store nothing but a few examples in a small episodic memory. At all times during the lifetime of the model, predictions on examples from any task may be requested. Addressing continual learning is an important research problem, since it would enable the community to move past the assumption of "identically and independently distributed data", and allow a better deployment of machine learning in-the-wild. However, continual learning presents one major challenge, catastrophic forgetting (McCloskey and Cohen 1989). That is, as the learner experiences new tasks, it quickly forgets previously acquired knowledge. This is a hindrance especially for state-of-the-art deep learn
+
+ing models, where all parameters are updated after observing each example.
+
+Continual learning has received increasing attention from the scientific community during the last decade. The state of the art algorithms for continual learning fall into three categories. First, regularization-based approaches reduce forgetting by restricting the updates in model parameters that were important for previous tasks (Kirkpatrick et al. 2016; Rebuffi, Kolesnikov, and Lampert 2017; Aljundi et al. 2018; Chaudhry et al. 2018; Nguyen et al. 2018). However, when the number of tasks are large, the regularization of past tasks becomes obsolete, leading to the representation drift (Titsias et al. 2019). Second, modular approaches (Rusu et al. 2016; Lee et al. 2017) add new modules to the learner as new tasks are learned. While modular architectures overcome forgetting by design, the memory complexity of these approaches scales with the number of tasks. Third, the memory-based methods (Lopez-Paz and Ranzato 2017; Hayes, Cahill, and Kanan 2018; Isele and Cosgun 2018; Riemer et al. 2019; Chaudhry et al. 2019a) store a few examples from past tasks in an "episodic memory", to be revisited when training for a new task. Contrary to modular approaches, memory-based methods add a very small memory overhead for each new task. Memory-based methods are the reigning state-of-the-art, but their performance remains a far cry from a simple oracle accessing all the data at once, hence turning the continual learning experience back into a normal supervised learning task. Despite intense research efforts, such gap in performance renders the problem of continual learning an open research question.
+
+Contribution We propose Hindsight Anchor Learning (HAL), a continual learning approach to improve the performance of memory-based continual learning algorithms. HAL leverages bilevel optimization to regularize the training objective with one representational point per class per task, called anchors. These anchors are constructed via gradient ascent in the image space, by maximizing one approximation to the forgetting loss for the current task throughout the entire continual learning experience. We estimate the amount of forgetting that the learner would suffer on these anchors if it were to be trained on future tasks in hindsight: that is, by measuring forgetting on a temporary predictor that has been fine-tuned on the episodic memory of past tasks.
+
+Anchors learned in such a way lie close to the classifier's decision boundary, as visualized in Figure 2. Since points near the decision boundary are the easiest to forget when updating the learner on future tasks, keeping prediction invariant on such anchors preserves the performance of previous tasks effectively. In sum, the overall parameter update of HAL uses nested optimization to minimize the loss of the current mini-batch, while keeping the predictions of all anchors invariant.
+
+Results We compare HAL to EWC (Kirkpatrick et al. 2016), ICARL (Rebuffi, Kolesnikov, and Lampert 2017), VCL (Nguyen et al. 2018), AGEM (Chaudhry et al. 2019a), experience replay (Hayes, Cahill, and Kanan 2018; Riemer et al. 2019), MER (Riemer et al. 2019), and MIR (Aljundi et al. 2019a) across four commonly used benchmarks in supervised continual learning (MNIST permutations, MNIST rotations, split CIFAR-100, and split miniImageNet). In these experiments, HAL achieves state-of-the-art performance, improving accuracy by up to  $7.5\%$  and reducing forgetting by almost  $23\%$  over the experience replay baseline. We show that these results hold for various sizes of episodic memory (between 1 and 5 examples per class per task).
+
+We now begin our exposition by reviewing the continual learning setup. The rest of the manuscript then presents our new algorithm HAL (Section 3), showcases its empirical performance (Section 4), surveys the related literature (Section 5), and offers some concluding remarks (Section 6).
+
+# 2 Continual learning setup
+
+In continual learning, a learner experiences a stream of data triplets  $(x_{i},y_{i},t_{i})$  containing an input  $x_{i}$ , a target  $y_{i}$ , and a task identifier  $t_i\in \mathcal{T} = \{1,\ldots ,T\}$ . Each input-target pair  $(x_{i},y_{i})\in \mathcal{X}\times \mathcal{Y}_{t_{i}}$  is an identical and independently distributed example drawn from some unknown distribution  $P_{t_i}(X,Y)$ , representing the  $t_i$ -th learning task. We assume that the tasks are experienced in order ( $t_i\leq t_j$  for all  $i\leq j$ ), and that the total number of tasks  $T$  is not known a priori. Under this setup, our goal is to estimate a predictor  $f_{\theta} = (w\circ \phi):\mathcal{X}\times \mathcal{T}\to \mathcal{Y}$ , parameterized by  $\theta \in \mathbb{R}^P$ , and composed of a feature extractor  $\phi :\mathcal{X}\rightarrow \mathcal{H}$  and a classifier  $w:\mathcal{H}\to \mathcal{V}$ , that minimizes the multi-task error
+
+$$
+\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbb {E} _ {(x, y) \sim P _ {t}} [ \ell (f (x, t), y) ], \tag {1}
+$$
+
+where  $\mathcal{Y} = \cup_{t\in \mathcal{T}}\mathcal{Y}_t$  , and  $\ell :\mathcal{Y}\times \mathcal{Y}\to \mathbb{R}$  is a loss function.
+
+Inspired by prior literature in continual learning, (Lopez-Paz and Ranzato 2017; Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Chaudhry et al. 2019a), we consider streams of data that are experienced only once. Therefore, the learner cannot revisit any but a small number of data triplets chosen to be stored in a small episodic memory  $\mathcal{M}$ . More specifically, we consider tiny "ring" episodic memories, which contain the last  $m$  observed examples per class for each of the experienced tasks, where  $m \in \{1,3,5\}$ . That is, considering as variables the number of experienced tasks  $t$  and examples  $n$ , we study continual learning algorithms with a  $O(t)$  memory footprint.
+
+Following Lopez-Paz and Ranzato (2017) and Chaudhry et al. (2018), we monitor two statistics to evaluate the quality of continual learning algorithms: final average accuracy, and final maximum forgetting. First, the final average accuracy of a predictor is defined as
+
+$$
+\text {A c c u r a c y} = \frac {1}{T} \sum_ {j = 1} ^ {T} a _ {T, j}, \tag {2}
+$$
+
+where  $a_{i,j}$  denotes the test accuracy on task  $j$  after the model has finished experiencing task  $i$ . That is, the final average accuracy measures the test performance of the model at every task after the continual learning experience has finished. Second, the final maximum forgetting is defined as
+
+$$
+\text {F o r g e t t i n g} = \frac {1}{T - 1} \sum_ {j = 1} ^ {T - 1} \max  _ {l \in \{1, \dots , T - 1 \}} \left(a _ {l, j} - a _ {T, j}\right), \tag {3}
+$$
+
+that is, the decrease in performance for each of the tasks between their peak accuracy and their accuracy after the continual learning experience has finished.
+
+Finally, following Chaudhry et al. (2019a), we use the first  $k < T$  tasks to cross validate the hyper-parameters of each of the considered continual learning algorithms. These first  $k$  tasks are not considered when computing the final average accuracy and maximum forgetting metrics.
+
+# 3 Hindsight Anchor Learning (HAL)
+
+The current state of the art algorithms for continual learning are based on experience replay (Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Chaudhry et al. 2019b). These methods update the model  $f_{\theta}$  while storing a small amount of past observed triplets in an episodic memory  $\mathcal{M} = \{(x', y', t')\}$ . For a new minibatch of observations  $\mathcal{B} := \{(x, y, t)\}$  from task  $t$ , the learner samples a minibatch  $\mathcal{B}_{\mathcal{M}}$  from  $\mathcal{M}$  at random, and employ the rule  $\theta \gets \theta - \alpha \cdot \nabla_{\theta} \ell(\mathcal{B} \cup \mathcal{B}_{\mathcal{M}})$  to update its parameters, where
+
+$$
+\ell (\mathcal {A}) = \frac {1}{| \mathcal {A} |} \sum_ {(x, y, t) \in \mathcal {A}} \ell (f _ {\theta} (x, t), y) \tag {4}
+$$
+
+denotes the average loss across a collection of triplets  $\mathcal{A} = \mathcal{B} \cup \mathcal{B}_{\mathcal{M}}$ . In general,  $\mathcal{B}_{\mathcal{M}}$  is constructed to have the same size as  $\mathcal{B}$ , but it can be smaller if the episodic memory  $\mathcal{M}$  does not yet contain enough samples.
+
+The episodic memory  $\mathcal{M}$  reminds the predictor about how to perform past tasks using only a small amount of data. As such, the behaviour of the predictor on past tasks outside the data stored in  $\mathcal{M}$  is not guaranteed. Moreover, since  $\mathcal{M}$  is usually very small, the performance of the predictor becomes sensitive to the choice of samples stored in the episodic memory. Because of this reason, we propose to further fix the behaviour of the predictor at a collection of carefully constructed anchor points  $e_{t'}$ , one per class per past task  $t'$ , at each parameter update.
+
+Let us assume that the anchor points  $e_{t'}$  are given—we will see later how to construct them in practice. To constrain
+
+the change of the predictor at these anchor points, we propose a two-step parameter update rule:
+
+$$
+\begin{array}{l} \tilde {\theta} \leftarrow \theta - \alpha \nabla_ {\theta} \ell (\mathcal {B} \cup \mathcal {B} _ {\mathcal {M}}), \\ \theta \leftarrow \theta - \alpha \nabla_ {\theta} \left(\ell \left(\mathcal {B} \cup \mathcal {B} _ {\mathcal {M}}\right) + \lambda \sum_ {t ^ {\prime} <   t} \left(f _ {\theta} \left(e _ {t ^ {\prime}}, t ^ {\prime}\right) - f _ {\tilde {\theta}} \left(e _ {t ^ {\prime}}, t ^ {\prime}\right)\right) ^ {2}\right). \tag {5} \\ \end{array}
+$$
+
+The first step computes a temporary parameter vector  $\tilde{\theta}$  by minimizing the loss at a minibatch from the current task  $t$ , and the episodic memory of past tasks (this is the usual experience replay parameter update). The second step employs a nested optimization to perform the actual update of the parameter  $\theta$ , which trades-off the minimization of  $(a)$  the loss value at the current minibatch and the episodic memory, as well as  $(b)$  changes in predictions at the anchor points for all past tasks. The proposed rule not only updates the predictor conservatively, thereby reducing forgetting, but also, as shown analytically in Appendix A, improves the forward transfer by maximizing the inner product between the gradients on  $\mathcal{B} \cup \mathcal{B}_{\mathcal{M}}$  and anchor points. In this respect, it bears similarity to gradient-based meta-learning approaches (Finn, Abbeel, and Levine 2017; Nichol and Schulman 2018; Riemer et al. 2019).
+
+Next, let us discuss how to choose the anchor points,  $e_t$  (one per class per task) as to preserve the performance of the current task throughout the entire learning experience. Ideally, the anchor points should attempt to minimize the forgetting on the current task as the learner is updated with future tasks. One could achieve this by letting  $e_t$  to be an example from the task  $t$  that would undergo maximum forgetting during the entire continual learning experience. Then, requiring the predictions to remain invariant at  $e_t$ , by using Eq. 5, could effectively reduce forgetting on the current task. Mathematically, the desirable  $e_t$  for the label  $y_t$  is obtained by maximizing the following Forgetting loss:
+
+$$
+(e _ {t}, y _ {t}) \leftarrow \underset {(x, y) \sim P _ {t}} {\arg \max } \ell \left(f _ {\theta_ {T}} (x, t), y _ {t}\right) - \ell \left(f _ {\theta_ {t}} (x, t), y _ {t}\right), \tag {6}
+$$
+
+where  $\theta_{t}$  is the parameter vector obtained after training on task  $t$  and  $\theta_{T}$  is the final parameter vector obtained after the entire learning experience. Thus, keeping the predictions intact on the pair  $(e_t,y_t)$  above can effectively preserve the performance of task  $t$ . However, the idealistic Eq. 6 requires access to  $(a)$  the entire distribution  $P_{t}$  to compute the maximization, and  $(b)$  access to all future distributions  $t^{\prime} > t$  to compute the final parameter vector  $\theta_{T}$ . Both are unrealistic assumptions under the continual learning setup described in Section 2, as the former requires storing the entire dataset of task  $t$ , and the latter needs access to future tasks.
+
+To circumvent  $(a)$ , we can recast Eq. 6 as an optimization problem and learn the desired  $e_t$  by initializing it at random and using  $k$  gradient ascent updates for a given label  $y_t$  in the image space  $(\mathcal{X} \in \mathbb{R}^D)$ . The proposed optimization objective is given by:
+
+$$
+\max  _ {e _ {t} \in \mathbb {R} ^ {D}} \left(\underbrace {\ell \left(f _ {\theta_ {T}} \left(e _ {t} , t\right) , y _ {t}\right) - \ell \left(f _ {\theta_ {t}} \left(e _ {t} , t\right) , y _ {t}\right)} _ {\text {F o r g e t t i n g l o s s}} - \gamma \underbrace {\left(\phi \left(e _ {t}\right) - \phi_ {t}\right) ^ {2}} _ {\text {M e a n e m b e d d i n g l o s s}}\right), \tag {7}
+$$
+
+where the regularizer, given by the mean embedding loss, constrains the search space by trying to push the anchor point embedding towards the mean data embedding. We recall that  $\phi$  denotes the feature extractor of the predictor, and  $\phi_t$  is the neural mean embedding (Smola et al. 2007) of all observed examples from task  $t$ . Since the feature extractor is updated after experiencing each data point, the mean embedding  $\phi_t$  are computed as running averages. That is, after observing a minibatch  $\mathcal{B} = \{(x,y,t)\}$  of task  $t$ , we update:
+
+$$
+\phi_ {t} \leftarrow \beta \cdot \phi_ {t} + (1 - \beta) \frac {1}{| \mathcal {B} |} \sum_ {x \in \mathcal {B}} \phi (x), \tag {8}
+$$
+
+where  $\phi_t$  is initialized to zero at the beginning of the learning experience. In our experiments, we learn one  $e_t$  per class for each task. We fix the  $y_t$  to the corresponding class label, and discard  $\phi_t$  after training on task  $t$ . Learning  $e_t$  in this manner circumvents the requirement of storing the entire distribution  $P_t$  for the current task  $t$ .
+
+Still, Eq. 7 requires the parameter vector  $\theta_T$ , to be obtained in the distant future after all learning tasks have been experienced. To waive this impossible requirement, we propose to approximate the future by simulating the past. That is, instead of measuring the forgetting that would happen after the model is trained for future tasks, we measure the forgetting that happens when the model is fine-tuned for past tasks. In this way, we say that forgetting is estimated in hind-sight, using past experiences. More concretely, after training on task  $t$  and obtaining the parameter vector  $\theta_t$ , we minimize the loss during one epoch on the episodic memory  $\mathcal{M}$  to obtain a temporary parameter vector  $\theta_{\mathcal{M}}$  that approximates  $\theta_T$ , and update  $e_t$  as:
+
+$$
+\begin{array}{l} e _ {t} \leftarrow e _ {t} + \alpha \nabla_ {e _ {t}} \left(\ell \left(f _ {\theta_ {\mathcal {M}}} \left(e _ {t}, t\right), y _ {t}\right) - \ell \left(f _ {\theta_ {t}} \left(e _ {t}, t\right), y _ {t}\right) \right. \\ \left. - \gamma \left(\phi \left(e _ {t}\right) - \phi_ {t}\right) ^ {2}\right). \tag {9} \\ \end{array}
+$$
+
+This completes the description of our proposed algorithm for continual learning, which combines experience replay with anchors learned in hindsight. We call our approach Hind-sight Anchor Learning (HAL) and summarize the entire learning process as follows:
+
+# Hindsight Anchor Learning (HAL)
+
+- Initialize  $\theta \sim P(\theta)$  and  $\{e_t \sim P(e)\}_{t=1}^T$  from normal distributions  $P(\theta)$  and  $P(e)$ ,  $\mathcal{M} = \{\}$ .  
+- Initialize  $\mathcal{M} = \{\}$  
+- For each task  $t = 1, \dots, T$ :
+
+- For each minibatch  $\mathcal{B}$  from task  $t$ :
+
+* Sample  $\mathcal{B}_{\mathcal{M}}$  from  $\mathcal{M}$  
+* Update  $\theta$  using Eq. 5  
+* Update  ${\phi }_{t}$  using Eq. 8  
+* Update  $\mathcal{M}$  by adding  $\mathcal{B}$  in a FIFO ring buffer
+
+- Fine-tune on  $\mathcal{M}$  to obtain  $\theta_{\mathcal{M}}$  
+Build  $e_t$  using Eq. 9 k times  
+-Discard  $\phi_t$
+
+- Return  $\theta$ .
+
+We contrast HAL with another idea, Maximally Interfered Retrieval (MIR) (Aljundi et al. 2019a) in that MIR selects a minibatch from an already populated replay buffer at each training step, whereas HAL writes to the replay buffer at the end of each task and samples randomly from the replay buffer. Furthermore, MIR selects the minibatch by measuring a one-step increase in loss incurred by the minibatch whereas HAL uses an approximate forgetting loss Eq. 9 that measures an increase over multiple training steps.
+
+# 4 Experiments
+
+We now report experiments on standard image classification benchmarks for continual learning.
+
+# Datasets and tasks
+
+We perform experiments on four supervised classification benchmarks for continual learning. Permuted MNIST is a variant of the MNIST dataset of handwritten digits (LeCun 1998) where each task applies a fixed random pixel permutation to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. Rotated MNIST is another variant of MNIST, where each task applies a fixed random image rotation (between 0 and 180 degrees) to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. Split CIFAR is a variant of the CIFAR-100 dataset (Krizhevsky and Hinton 2009; Zenke, Poole, and Ganguli 2017), where each task contains the data pertaining 5 random classes (without replacement) out of the total 100 classes. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes. Split miniImageNet is a variant of the ImageNet dataset (Russakovsky et al. 2015; Vinyals et al. 2016), containing a subset of images and classes from the original dataset. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes.
+
+For all datasets, the first 3 tasks are used for hyperparameter optimization (grids available in Appendix C). The learner can perform multiple epochs on these three initial tasks that are later discarded for evaluation.
+
+# Baselines
+
+We compare our proposed model HAL to the following baselines. Finetune is a single model trained on a stream of data, without any regularization or episodic memory. ICARL (Rebuffi, Kolesnikov, and Lampert 2017) uses nearest-mean-of-exemplar rule for classification and avoids catastrophic forgetting by regularizing over the feature representations of previous tasks using knowledge distillation loss (Hinton, Vinyals, and Dean 2014). EWC (Kirkpatrick et al. 2016) is a continual learning method that limits changes to parameters critical to past tasks, as measured by the Fisher information matrix. VCL (Nguyen et al. 2018) is a continual learning method that uses online variational inference for approximating the posterior distribution, which is then used to regularize the model. AGEM (Chaudhry et al. 2019a) is a continual learning method improving on (Lopez-Paz and Ranzato 2017), which uses an episodic memory of
+
+parameter gradients to limit forgetting. MER (Riemer et al. 2019) is a continual learning method that combines episodic memory with meta-learning to limit forgetting. ER-Ring (Chaudhry et al. 2019b) is a continual learning method that uses a ring buffer as episodic memory. MIR (Aljundi et al. 2019a) is a continual learning method based on experience replay that selects a minibatch from the episodic memory that incurs the maximum change in loss. Multitask is an oracle baseline that has access to all data to optimize Eq. 1, useful to estimate an upper bound on the obtainable accuracy (Eq. 2). Clone-and-finetune is an oracle baseline training one independent model per task, where the model for task  $t'$  is initialized by cloning the parameters of the model for task  $t' - 1$ .
+
+All baselines use the same neural network architecture: a perceptron with two hidden layers of 256 ReLU neurons in the MNIST experiments, and a ResNet18, with three times less feature maps across all layers, similar to Lopez-Paz and Ranzato (2017), in CIFAR and ImageNet experiments. The task identifiers are used to select the output head in the CIFAR and ImageNet experiments, while ignored in the MNIST experiments. Batch size is set to 10 for both the stream of data and episodic memory across experiments and models. The size of episodic memory is set between 1 and 5 examples per class per task. The results of VCL are complied by running the official implementation<sup>1</sup>, that only works for fully-connected networks, in our continual learning setup. All the other baselines use our unified code base which is available here<sup>2</sup>. All experiments are averaged over five runs using different random seeds, where each seed corresponds to a different dataset ordering among tasks.
+
+# Results
+
+Table 1 summarizes the main results of our experiments when the episodic memory of only one example per class per task is used. First, our proposed HAL is the method achieving maximum Accuracy (Eq. 2) and minimal Forgetting (Eq. 3) for all benchmarks. This does not include Oracle baselines Multitask (which has access to all data simultaneously) and Clone-and-finetune (which trains a separate model per task). Second, the relative gains from ER-Ring to HAL are substantial, confirming that the anchoring objective (Eq. 5) allows experience-replay methods to generalize better with the same amount of episodic memory.
+
+Third, regularization based approaches, such as EWC (Kirkpatrick et al. 2016) and VCL (Nguyen et al. 2018), suffer under the single epoch setup. As noted by Chaudhry et al. (2019a), EWC requires multiple passes over the samples of each task to perform well. The poor performance of VCL is attributed to the noisy posterior estimation in the single pass setup. Note that approaches making use of memory (MER, ER, MIR, and HAL) work substantially better in this setup.
+
+Fourth, ICARL (Rebuffi, Kolesnikov, and Lampert 2017), another method making use of episodic memory, performs poorly in our setup. From Table 1, it can be argued that di
+
+Table 1: Accuracy (Eq. 2) and Forgetting (Eq. 3) results of continual learning experiments. Averages and standard deviations are computed over five runs using different random seeds. When used, episodic memories contain up to one example per class per task. Last two rows are oracle baselines.  
+
+<table><tr><td>METHOD</td><td colspan="2">PERMUTED MNIST</td><td colspan="2">ROTATED MNIST</td></tr><tr><td></td><td>ACCURACY</td><td>FORGETTING</td><td>ACCURACY</td><td>FORGETTING</td></tr><tr><td>FINETUNE</td><td>53.5 (±1.46)</td><td>0.29 (±0.01)</td><td>41.9 (±1.37)</td><td>0.50 (±0.01)</td></tr><tr><td>EWC (KIRKPATRICK ET AL. 2016)</td><td>63.1 (±1.40)</td><td>0.18 (±0.01)</td><td>44.1 (±0.99)</td><td>0.47 (±0.01)</td></tr><tr><td>VCL (NGUYEN ET AL. 2018)</td><td>51.8 (±1.54)</td><td>0.44 (±0.01)</td><td>48.2 (±0.99)</td><td>0.50 (±0.01)</td></tr><tr><td>VCL-RANDOM (NGUYEN ET AL. 2018)</td><td>52.3 (±0.66)</td><td>0.43 (±0.01)</td><td>54.4 (±1.44)</td><td>0.44 (±0.01)</td></tr><tr><td>AGEM (CHAUDHRY ET AL. 2019A)</td><td>62.1 (±1.39)</td><td>0.21 (±0.01)</td><td>50.9 (±0.92)</td><td>0.40 (±0.01)</td></tr><tr><td>MER (RIEMER ET AL. 2019)</td><td>69.9 (±0.40)</td><td>0.14 (±0.01)</td><td>66.0 (±2.04)</td><td>0.23 (±0.01)</td></tr><tr><td>ER-RING (CHAUDHRY ET AL. 2019B)</td><td>70.2 (±0.56)</td><td>0.12 (±0.01)</td><td>65.9 (±0.41)</td><td>0.24 (±0.01)</td></tr><tr><td>MIR (ALJUNDI ET AL. 2019A)</td><td>71.1 (±0.41)</td><td>0.11 (±0.01)</td><td>-</td><td>-</td></tr><tr><td>HAL (OURS)</td><td>73.6 (±0.31)</td><td>0.09 (±0.01)</td><td>68.4 (±0.72)</td><td>0.21 (±0.01)</td></tr><tr><td>CLONE-AND-FINETUNE</td><td>81.4 (±0.35)</td><td>0.0</td><td>87.5 (±0.11)</td><td>0.0</td></tr><tr><td>MULTITASK</td><td>83.0</td><td>0.0</td><td>83.3</td><td>0.0</td></tr></table>
+
+<table><tr><td>METHOD</td><td colspan="2">SPLIT CIFAR</td><td colspan="2">SPLIT MINIIMAGENET</td></tr><tr><td></td><td>ACCURACY</td><td>FORGETTING</td><td>ACCURACY</td><td>FORGETTING</td></tr><tr><td>FINETUNE</td><td>42.9 (±2.07)</td><td>0.25 (±0.03)</td><td>34.7 (±2.69)</td><td>0.26 (±0.03)</td></tr><tr><td>EWC (KIRKPATRICK ET AL. 2016)</td><td>42.4 (±3.02)</td><td>0.26 (±0.02)</td><td>37.7 (±3.29)</td><td>0.21 (±0.03)</td></tr><tr><td>ICARL (REBUFFI, KOLESNIKOV, AND LAMPERT 2017)</td><td>46.4 (±1.21)</td><td>0.16 (±0.01)</td><td>-</td><td>-</td></tr><tr><td>AGEM (CHAUDHRY ET AL. 2019A)</td><td>54.9 (±2.92)</td><td>0.14 (±0.03)</td><td>48.2 (±2.49)</td><td>0.13 (±0.02)</td></tr><tr><td>MER (RIEMER ET AL. 2019)</td><td>49.7 (±2.97)</td><td>0.19 (±0.03)</td><td>45.5 (±1.49)</td><td>0.15 (±0.01)</td></tr><tr><td>ER-RING (CHAUDHRY ET AL. 2019B)</td><td>56.2 (±1.93)</td><td>0.13 (±0.01)</td><td>49.0 (±2.61)</td><td>0.12 (±0.02)</td></tr><tr><td>MIR (ALJUNDI ET AL. 2019A)</td><td>57.1 (±1.81)</td><td>0.12 (±0.01)</td><td>49.3 (±2.15)</td><td>0.12 (±0.01)</td></tr><tr><td>HAL (OURS)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td><td>51.6 (±2.02)</td><td>0.10 (±0.01)</td></tr><tr><td>CLONE-AND-FINETUNE</td><td>60.3 (±0.55)</td><td>0.0</td><td>50.3 (±1.00)</td><td>0.0</td></tr><tr><td>MULTITASK</td><td>68.3</td><td>0.0</td><td>63.5</td><td>0.0</td></tr></table>
+
+rect training on a very small episodic memory, as done in experience replay, allows the method to generalize better compared to when the same memory is used indirectly in the knowledge distillation loss (Hinton, Vinyals, and Dean 2014) as done in ICARL.
+
+Fig 1 shows the accuracy (Eq. 2) of methods employing episodic memory when the size of memory is increased. We use 1 to 5 examples per class per task, resulting in a total memory size from 200 to 1000 for MNIST experiments, and from 85 to 425 for CIFAR and ImageNet experiments. The corresponding numbers for Forgetting are given in Appendix B. HAL consistently improves on ER-Ring and other baselines.
+
+Figure 4 in Appendix B provides the training time of the continual learning baselines on MNIST benchmarks. Although HAL adds an overhead on top of the experience replay baseline, it is substantially faster than MER —another approach that makes use of nested optimization to reduce forgetting. However, HAL requires extra memory to store task anchors that, as we will show next, are more effective than additional data samples one can store for experience replay. Overall, we conclude that HAL provides the best tradeoff in terms of efficiency and performance.
+
+# Ablation Study
+
+We now turn our attention towards two questions; (1) whether for the same episodic memory size in bytes HAL improves over the experience replay baseline, (2) whether fine-tuning on the replay buffer is a good approximation of forgetting when the learner is updated on future tasks.
+
+To answer the first question, let  $|\mathcal{M}|$  be the total size of episodic memory for all tasks when one example per class per task is stored in the replay buffer. We then run the experience replay with double the size of episodic memory (i.e.) storing two examples per class per task instead of one. The episodic memory size in HAL, on the other hand, is kept at  $|\mathcal{M}|$ . This effectively makes the size of memory in bytes taken by experience replay and that of HAL equal as the latter requires extra memory to store anchors. Table 2 summarizes the results of this study. For the same memory size in bytes, HAL performs better than experience replay when additional real data samples are stored in the episodic memory. It is surprising that the anchors learned by HAL, initialized from random noise and learned using gradient-based optimization, perform better compared to randomly sampled real data. To understand this, in Figure 2 we visualize HAL's anchors along with the task data in the image
+
+![](images/b16bbdadfe1c5da6c83772c008e6f251fe08f7fde54f68cdf5acd8f356a913bf.jpg)  
+(a) Permuted MNIST
+
+![](images/8a321a270dd3f54e1386d08d0964307612a50d7290ec7f380c8f66493a8f202a.jpg)  
+(b) Rotated MNIST
+
+![](images/fb61212077920520fcde77e0b54495ecc38b61c9c8e1eb7784470e93cf2e6312.jpg)  
+(c) Split CIFAR
+
+![](images/84392707214066eb4c8750af688f73b2909a3fc2ce9efc97417bd0bde891fd21.jpg)  
+(d) Split miniImageNet  
+Figure 1: Accuracy (Eq. 2) results for different episodic memory sizes.
+
+Table 2: Comparison of HAL with experience replay. ER-Ring and HAL use one example per class per task in the episodic memory, whereas ER-Ring-2  $|\mathcal{M}|$  uses two examples per class per task in the memory. Averages and standard deviations are computed over five runs using different random seeds.  
+
+<table><tr><td>Method</td><td colspan="2">Permuted MNIST</td><td colspan="2">Split CIFAR</td></tr><tr><td></td><td>Accuracy</td><td>Forgetting</td><td>Accuracy</td><td>Forgetting</td></tr><tr><td>ER-Ring-|M|</td><td>70.2 ±(0.56)</td><td>0.12 (±0.01)</td><td>56.2 (±1.93)</td><td>0.13 (±0.01)</td></tr><tr><td>ER-Ring-2|M|</td><td>71.9 (±0.31)</td><td>0.11 (±0.01)</td><td>58.6 (±2.68)</td><td>0.12 (±0.01)</td></tr><tr><td>HAL-|M|</td><td>73.6 (±0.31)</td><td>0.09 (±0.01)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td></tr></table>
+
+Table 3: Performance comparison of HAL with Oracle where the learner has access to all the future tasks to exactly quantify forgetting of an anchor. Averages and standard deviations are computed over five runs using different random seeds.  
+
+<table><tr><td>Anchor type</td><td colspan="2">Permuted MNIST</td><td colspan="2">Split CIFAR</td></tr><tr><td></td><td>Accuracy</td><td>Forgetting</td><td>Accuracy</td><td>Forgetting</td></tr><tr><td>HAL</td><td>73.6 (±0.31)</td><td>0.09 (±0.01)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td></tr><tr><td>Oracle</td><td>73.9 (±0.41)</td><td>0.09 (±0.01)</td><td>61.1 (±0.94)</td><td>0.09 (±0.01)</td></tr></table>
+
+and feature space on Permuted MNIST benchmark. From the left of the figure, it can be seen that HAL anchors lie in the data cluster of a class in the image space, suggesting that the mean embedding loss in Eq. 9 effectively regularizes against outliers. More interestingly, the figure on the right shows that these anchors lie at or close to the cluster edges in the feature space. In other words, the anchor points learned by HAL lie close to the classifier decision boundary. This can explain their effectiveness compared to the real data samples that can lie anywhere in the data cluster in feature space.
+
+Finally, to answer the second part, we assume a non-continual setup where at each step the learner has an oracle access to all future tasks. After training on task  $t$ , the learner is fine-tuned on all future tasks and anchor points are subsequently learned by optimizing idealistic Eq. 7. The results
+
+are reported in Table 3. It can be seen from the table that the proposed HAL performs very close to the noncontinual oracle baseline. This suggests that HAL's approximation of forgetting when the learner is updated on future tasks by replaying past data is effective in many existing continual learning benchmarks.
+
+# 5 Related work
+
+In continual learning (Ring 1997), also called lifelong learning (Thrun 1998), a learner addresses a sequence of changing tasks without storing the complete datasets of these tasks. This contrasts with multitask learning (Caruana 1997), where the learner assumes simultaneous access to data from all tasks. The main challenge in continual learning is to avoid catastrophic interference (McCloskey and Cohen 1989; McClelland, McNaughton, and O'Reilly 1995; Goodfellow et al. 2013), that is, the learner forgetting previously acquired knowledge when learning new tasks. The state-of-the-art methods in continual learning can be categorized into three classes.
+
+First, regularization approaches discourage updating parameters important for past tasks (Kirkpatrick et al. 2016; Aljundi et al. 2018; Nguyen et al. 2018; Zenke, Poole, and Ganguli 2017). While efficient in terms of memory and computation, these approaches suffer from brittleness due to feature drift for large number of tasks (Titsias et al. 2019). Additionally, these approaches are only effective when the learner can perform multiple passes over each task (Chaudhry et al. 2019a), a case deemed unrealistic in this work.
+
+Second, modular approaches use different parts of the prediction function for each new task (Fernando et al. 2017; Aljundi, Chakravarty, and Tuytelaars 2017; Rosenbaum, Klinger, and Riemer 2018; Chang et al. 2018; Xu and Zhu 2018; Ferran Alet 2018). Modular approaches do not scale to a large number of tasks, as they require searching over the combinatorial space of module architectures. Another modular approach (Rusu et al. 2016; Lee et al. 2017) adds new parts to the prediction function as new tasks are learned. By construction, modular approaches have zero forgetting, but their memory requirements increase with the number of tasks.
+
+![](images/1fcd9ebc526b8ddc695d1470ce158cf495e0e9ba11c0ea6529c19dbea2ebb548.jpg)  
+Image Space
+
+![](images/3f5b18590290746f3c8ce7d93ba6504df65b7f0a28e9ad23f412940fabf846b6.jpg)  
+Feature Space  
+Figure 2: t-SNE visualization of images and anchors (HAL) in the image space (left) and the feature space (right) on Permuted MNIST benchmark for a single task. Anchor points are exaggerated in size for the purpose of better visualization. The left plot shows that anchor points lie with in the data cluster of a class, whereas the right plot shows that, in the feature space, anchor points lie close to the edge of the cluster of a class or near decision boundaries.
+
+Third, episodic memory approaches maintain and revisit a small episodic memory of data from past tasks. In some of these methods (Li and Hoiem 2016; Rebuffi, Kolesnikov, and Lampert 2017), examples in the episodic memory are replayed and predictions are kept invariant by means of distillation (Hinton, Vinyals, and Dean 2014). In other approaches (Lopez-Paz and Ranzato 2017; Chaudhry et al. 2019a; Aljundi et al. 2019b) the episodic memory is used as an optimization constraint that discourages increases in loss of past tasks. More recently, several works (Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Rolnick et al. 2018; Chaudhry et al. 2019b) have shown that directly optimizing the loss of episodic memory, also known as experience replay, is cheaper than constraint-based approaches and improves the prediction performance. Our contribution in this paper has been to improve experience replay methods with task anchors learned in hindsight.
+
+There are other definitions of continual learning, such as the one of task-free continual learning. The task-free formulation does not consider the notion of tasks, and instead works on undivided data streams (Aljundi, Kelchtermans, and Tuytelaars 2019; Aljundi et al. 2019b). We have focused on the task-based definition of continual learning and, similar to many recent works (Lopez-Paz and Ranzato 2017; Hayes, Cahill, and Kanan 2018; Riemer et al. 2019; Chaudhry et al. 2019a), assumed that only a single pass through the data was possible.
+
+Finally, our gradient-based learning of anchors bears a similarity to (Simonyan, Vedaldi, and Zisserman 2014) and (Wang et al. 2018). In Simonyan, Vedaldi, and Zisserman (2014), the authors use gradient ascent on class scores to find saliency maps of a classification model. Contrary to
+
+them, our proposed hindsight learning objective optimizes for the forgetting metric, as reducing it is necessary for continual learning. Dataset distillation (Wang et al. 2018) proposes to encode the entire dataset in a few synthetic points at a given parameter vector by a gradient-based optimization process. Their method requires access to the entire dataset of a task for optimization purposes. We, instead, learn anchors in hindsight from the replay buffer of past tasks after training is finished for the current task. While Wang et al. (2018) aim to replicate the performance of the entire dataset from the synthetic points, we focus on reducing forgetting of an already learned task.
+
+# 6 Conclusion
+
+We introduced a bilevel optimization objective, dubbed anchoring, for continual learning. In our approach, we learned one "anchor point" per class per task, where predictions are requested to remain invariant by means of nested optimization. These anchors are learned using gradient-based optimization and represent points that would maximize the forgetting of the current task throughout the entire learning experience. We simulate the forgetting that would happen during the learning of future tasks in hindsight, that is, by taking temporary gradient steps across a small episodic memory of past tasks. We call our approach Hindsight Anchor Learning (HAL). As shown in our experiments, anchoring in hindsight complements and improves the performance of continual learning methods based on experience replay, achieving a new state of the art on four standard continual learning benchmarks.
+
+# Acknowledgement
+
+The authors would like to thank Marc'Aurelio Ranzato for helpful discussions. This work was supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1 and EPSRC/MURI grant EP/N019474/1. We would also like to acknowledge the Royal Academy of Engineering and FiveAI. AC is funded by Amazon Research award.
+
+# References
+
+Aljundi, R.; Babiloni, F.; Elhoseiny, M.; Rohrbach, M.; and Tuytelaars, T. 2018. Memory Aware Synapses: Learning what (not) to forget. In ECCV.  
+Aljundi, R.; Belilovsky, E.; Tuytelaars, T.; Charlin, L.; Caccia, M.; Lin, M.; and Page-Caccia, L. 2019a. Online Continual Learning with Maximal Interfered Retrieval. In Wallach, H.; Larochelle, H.; Beygelzimer, A.; d'Alché-Buc, F.; Fox, E.; and Garnett, R., eds., Advances in Neural Information Processing Systems 32, 11849-11860. Curran Associates, Inc. URL http://papers.nips.cc/paper/9357-online-continual-learning-with-maximal-interfered-retrieval.pdf.  
+Aljundi, R.; Chakravarty, P.; and Tuytelaars, T. 2017. Expert Gate: Lifelong Learning with a Network of Experts. In CVPR, 7120-7129.  
+Aljundi, R.; Kelchtermans, K.; and Tuytelaars, T. 2019. Task-free continual learning. In CVPR, 11254-11263.  
+Aljundi, R.; Lin, M.; Goujaud, B.; and Bengio, Y. 2019b. Online continual learning with no task boundaries. arXiv preprint arXiv:1903.08671.  
+Caruana, R. 1997. Multitask learning. Machine learning 28(1): 41-75.  
+Chang, M.; Gupta, A.; Levine, S.; and Griffiths, T. L. 2018. Automatically Composing Representation Transformations as a Means for Generalization. In ICML workshop Neural Abstract Machines and Program Induction v2.  
+Chaudhry, A.; Dokania, P. K.; Ajanthan, T.; and Torr, P. H. 2018. Riemannian Walk for Incremental Learning: Understanding Forgetting and Intransigence. In ECCV.  
+Chaudhry, A.; Ranzato, M.; Rohrbach, M.; and Elhoseiny, M. 2019a. Efficient Lifelong Learning with A-GEM. In ICLR.  
+Chaudhry, A.; Rohrbach, M.; Elhoseiny, M.; Ajanthan, T.; Dokania, P. K.; Torr, P. H.; and Ranzato, M. 2019b. Continual Learning with Tiny Episodic Memories. arXiv preprint arXiv:1902.10486.  
+Fernando, C.; Banarse, D.; Blundell, C.; Zwols, Y.; Ha, D.; Rusu, A. A.; Pritzel, A.; and Wierstra, D. 2017. Pathnet: Evolution channels gradient descent in super neural networks. arXiv preprint arXiv:1701.08734.  
+Ferran Alet, Tomas Lozano-Perez, L. P. K. 2018. Modular Meta-Learning. arXiv preprint arXiv:1806.10166v1.  
+Finn, C.; Abbeel, P.; and Levine, S. 2017. Model-agnostic meta-learning for fast adaptation of deep networks. In ICML-Volume 70, 1126-1135. JMLR.org.
+
+Goodfellow, I. J.; Mirza, M.; Xiao, D.; Courville, A.; and Bengio, Y. 2013. An empirical investigation of catastrophic forgetting in gradient-based neural networks. arXiv preprint arXiv:1312.6211.  
+Hayes, T. L.; Cahill, N. D.; and Kanan, C. 2018. Memory Efficient Experience Replay for Streaming Learning. arXiv preprint arXiv:1809.05922.  
+Hinton, G.; Vinyals, O.; and Dean, J. 2014. Distilling the knowledge in a neural network. In NIPS.  
+Isele, D.; and Cosgun, A. 2018. Selective Experience Replay for Lifelong Learning. arXiv preprint arXiv:1802.10269.  
+Kirkpatrick, J.; Pascanu, R.; Rabinowitz, N. C.; Veness, J.; Desjardins, G.; Rusu, A. A.; Milan, K.; Quan, J.; Ramalho, T.; Grabska-Barwinska, A.; Hassabis, D.; Clopath, C.; Kumaran, D.; and Hadsell, R. 2016. Overcoming catastrophic forgetting in neural networks. *PNAS*.  
+Krizhevsky, A.; and Hinton, G. 2009. Learning multiple layers of features from tiny images. https://www.cs.toronto.edu/kriz/cifar.html.  
+LeCun, Y. 1998. The MNIST database of handwritten digits. http://yann.lecun.com/exdb/mnist/.  
+Lee, J.; Yun, J.; Hwang, S.; and Yang, E. 2017. Lifelong Learning with Dynamically Expandable Networks. arXiv preprint arXiv:1708.01547.  
+Li, Z.; and Hoiem, D. 2016. Learning without forgetting. In ECCV, 614-629.  
+Lopez-Paz, D.; and Ranzato, M. 2017. Gradient Episodic Memory for Continuum Learning. In NIPS.  
+McClelland, J. L.; McNaughton, B. L.; and O'Reilly, R. C. 1995. Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory. Psychological review 102(3): 419.  
+McCloskey, M.; and Cohen, N. J. 1989. Catastrophic interference in connectionist networks: The sequential learning problem. Psychology of learning and motivation 24: 109-165.  
+Nguyen, C. V.; Li, Y.; Bui, T. D.; and Turner, R. E. 2018. Variational continual learning. *ICLR*.  
+Nichol, A.; and Schulman, J. 2018. Reptile: a scalable met-learning algorithm. arXiv preprint arXiv:1803.02999, 2018  
+Rebuffi, S.-V.; Kolesnikov, A.; and Lampert, C. H. 2017. iCaRL: Incremental Classifier and Representation Learning. In CVPR.  
+Riemer, M.; Cases, I.; Ajemian, R.; Liu, M.; Rish, I.; Tu, Y.; and Tesauro, G. 2019. Learning to Learn without Forgetting By Maximizing Transfer and Minimizing Interference. In ICLR.  
+Ring, M. B. 1997. CHILD: A first step towards continual learning. Machine Learning 28(1): 77-104.
+
+Rolnick, D.; Ahuja, A.; Schwarz, J.; Lillicrap, T. P.; and Wayne, G. 2018. Experience Replay for Continual Learning. CoRR abs/1811.11682. URL http://arxiv.org/abs/1811.11682.  
+Rosenbaum, C.; Klinger, T.; and Riemer, M. 2018. Routing Networks: Adaptive Selection of Non-Linear Functions for Multi-Task Learning. In ICLR.  
+Russakovsky, O.; Deng, J.; Su, H.; Krause, J.; Satheesh, S.; Ma, S.; Huang, Z.; Karpathy, A.; Khosla, A.; Bernstein, M.; Berg, A. C.; and Fei-Fei, L. 2015. ImageNet Large Scale Visual Recognition Challenge. IJCV 115(3): 211-252.  
+Rusu, A. A.; Rabinowitz, N. C.; Desjardins, G.; Soyer, H.; Kirkpatrick, J.; Kavukcuoglu, K.; Pascanu, R.; and Hadsell, R. 2016. Progressive neural networks. arXiv preprint arXiv:1606.04671.  
+Simonyan, K.; Vedaldi, A.; and Zisserman, A. 2014. Deep inside convolutional networks: Visualising image classification models and saliency maps. In ICLR.  
+Smola, A.; Gretton, A.; Song, L.; and Scholkopf, B. 2007. A Hilbert space embedding for distributions. In ALT, 13-31. Springer.  
+Thrun, S. 1998. Lifelong learning algorithms. In Learning to learn, 181-209. Springer.  
+Titsias, M. K.; Schwarz, J.; Matthews, A. G. d. G.; Pascanu, R.; and Teh, Y. W. 2019. Functional Regularisation for Continual Learning using Gaussian Processes. arXiv preprint arXiv:1901.11356.  
+Vinyals, O.; Blundell, C.; Lillicrap, T.; Wierstra, D.; et al. 2016. Matching networks for one shot learning. In NIPS, 3630-3638.  
+Wang, T.; Zhu, J.-Y.; Torralba, A.; and Efros, A. A. 2018. Dataset Distillation. arXiv preprint arXiv:1811.10959.  
+Xu, J.; and Zhu, Z. 2018. Reinforced Continual Learning. In arXiv preprint arXiv:1805.12369v1.  
+Zenke, F.; Poole, B.; and Ganguli, S. 2017. Continual Learning Through Synaptic Intelligence. In ICML.
+
+# Appendix
+
+Section A describes the approximate update performed by the anchoring objective (Eq. 5 in the main paper). Section B reports more experimental results. Section C provides the grid considered for hyper-parameters. Section E gives the pseudocode for HAL.
+
+# A Approximate Update of Anchoring Objective
+
+Here we will use a Taylor series expansion to approximate the update performed by the anchoring objective (Eq. 5 in the main paper). In particular, we are interested in the regularization part of the anchoring objective that involves a nested update. We refer to this gradient as  $g_{anc}$ . We follow similar arguments as (Nichol and Schulman 2018).
+
+Let  $\theta_0$  be the parameter vector before the temporary update of the anchoring objective (Eq. 5). Moreover, let  $\ell_{ce}$  and  $\ell_{L2}$  be the cross-entropy and L2 losses, respectively. We use the following definitions:
+
+$$
+\bar {g} _ {0} = \ell_ {c e} ^ {\prime} (\theta_ {0}) \quad \left(\text {g r a d i e n t o f c r o s s - e n t r o p y l o s s a t i n i t i a l p o i n t o n} \mathcal {B} \cup \mathcal {B} _ {\mathcal {M}}\right)
+$$
+
+$$
+\bar {H} _ {0} = \ell_ {c e} ^ {\prime \prime} (\theta_ {0}) \quad \left(\text {H e s s i a n o f c r o s s - e n t r o p y l o s s a t i n i t i a l p o i n t o n} \mathcal {B} \cup \mathcal {B} _ {\mathcal {M}}\right)
+$$
+
+$$
+\bar {g} _ {1} = \ell_ {L 2} ^ {\prime} \left(\theta_ {0}\right) \quad (\text {g r a d i e n t o f L 2 l o s s a t i n i t i a l p o i n t o n a n c h o r s})
+$$
+
+$$
+\bar {H} _ {1} = \ell_ {L 2} ^ {\prime \prime} (\theta_ {0}) \quad \text {(g r a d i e n t o f L 2 l o s s a t i n i t i a l p o i n t o n a n c h o r s)}
+$$
+
+Let  $U_0 = \theta_0 - \alpha \overline{g}_0$  be the operator giving a temporary update in the two-step process of (Eq. 5), and let  $\theta_1$  be the temporary update itself (i.e.)  $\theta_1 \coloneqq U_0$  (note that  $\tilde{\theta}$  is used in the main paper instead of  $\theta_1$ ). The  $g_{anc}$  is given by:
+
+$$
+\begin{array}{l} g _ {a n c} = \frac {\partial}{\partial \theta_ {0}} \ell_ {L 2} (U _ {0}) \\ = U _ {0} ^ {\prime} \cdot \ell_ {L 2} ^ {\prime} (\theta_ {1}) \\ = \left(I - \alpha \bar {H} _ {0}\right) \cdot \ell_ {L 2} ^ {\prime} \left(\theta_ {1}\right), \tag {10} \\ \end{array}
+$$
+
+where the second step is obtained by using the chain rule. Now, if we calculate the first order Taylor series approximation of  $\ell_{L2}^{\prime}(\theta_1)$ ,
+
+$$
+\begin{array}{l} \ell_ {L 2} ^ {\prime} \left(\theta_ {1}\right) = \ell_ {L 2} ^ {\prime} \left(\theta_ {0}\right) + \ell_ {L 2} ^ {\prime \prime} \left(\theta_ {0}\right) \cdot \left(\theta_ {1} - \theta_ {0}\right) + O \left(\left| \left| \theta_ {1} - \theta_ {0} \right| \right| ^ {2}\right) \\ = \bar {g} _ {1} + \bar {H} _ {1} \cdot \left(\theta_ {0} - \alpha \bar {g} _ {0} - \theta_ {0}\right) + O \left(\alpha^ {2}\right) \\ = \bar {g} _ {1} - \alpha \bar {H} _ {1} \cdot \bar {g} _ {0} + O \left(\alpha^ {2}\right), \tag {11} \\ \end{array}
+$$
+
+where in the second step we substituted the value of  $\theta_{1}$ . By putting Eq. 11 in Eq. 10 and after some simplification we get:
+
+$$
+g _ {a n c} = \bar {g} _ {1} - \alpha \left(\bar {H} _ {1} \cdot \bar {g} _ {0} + \bar {H} _ {0} \cdot \bar {g} _ {1}\right) + O \left(\alpha^ {2}\right). \tag {12}
+$$
+
+This form is very similar to the second-order MAML gradient formulation, Eq. 25 in (Nichol and Schulman 2018). Further simplification of the inner product terms between Hessian and gradient yields the inner product between the gradients  $\overline{g_0}$  and  $\overline{g_1}$ . This shows that similar to MAML (Finn, Abbeel, and Levine 2017), Reptile (Nichol and Schulman 2018) and MER (Riemer et al. 2019), the anchoring objective, as described in Eq. 5 of the main paper, maximizes the inner product between the gradients. However, unlike the other meta-learning approaches, in the anchoring objective, these gradients correspond to different loss functions, cross-entropy and L2 losses on data from the current task and episodic memory, and HAL anchors, respectively.
+
+# B More Results
+
+Figure 3 shows a more fine-grained analysis of average accuracy as new tasks are learned on Permuted MNIST and Split CIFAR. HAL preserves the performance of a predictor more effectively than other baselines.
+
+Tables 5 and 6 show the accuracy and forgetting of methods employing episodic memory when the size of memory is increased. We use 3 to 5 examples per class per task, resulting in a total memory size from 600 to 1000 for MNIST experiments, and from 255 to 425 for CIFAR and ImageNet experiments.
+
+# C Hyper-parameter Selection
+
+In this section, we report the hyper-parameters grid considered for experiments. The best values for different benchmarks are given in parentheses.
+
+- Multitask
+
+![](images/03c605b5355952321432ebafbebab2455ea6634b777752c15b7d6111f01bc6e3.jpg)  
+Permuted MNIST
+
+![](images/d9eafc4c0a015aa8f6d274a5c523990b5f63941959c47917321a54a29ba240a9.jpg)  
+Split CIFAR  
+Figure 3: Evolution of Accuracy (Eq. 2) as new tasks are learned. When used, episodic memories contain up to one example per class per task.
+
+<table><tr><td>Anchor type</td><td colspan="2">Split CIFAR</td></tr><tr><td></td><td>Accuracy</td><td>Forgetting</td></tr><tr><td>Real Data Anchor</td><td>58.0 (±0.15)</td><td>0.12 (±0.01)</td></tr><tr><td>HAL (ours)</td><td>60.4 (±0.54)</td><td>0.10 (±0.01)</td></tr></table>
+
+![](images/cf498d4189b19aa297454450490a5004bf92fd45f69a82e85f73ba7778314394.jpg)  
+Table 4: Impact of anchor selection, where we compare a randomly chosen data point as an anchor (Real Data Anchor) with our optimized anchor selection (HAL).  
+Figure 4: Training time (s) of MNIST experiments for the entire continual learning experience. MER and HAL both use meta-learning objectives to reduce forgetting.
+
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+- Clone-and-finetune  
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+- Finetune  
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+EWC  
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+- regularization: [0.1, 1, 10 (MNIST perm, rot, CIFAR, miniImageNet), 100, 1000]  
+- AGEM  
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+- MER
+
+Table 5: Accuracy (Eq. 2) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Metrics are averaged over five runs using different random seeds.  
+
+<table><tr><td>METHOD</td><td colspan="2">PERMUTED MNIST</td><td colspan="2">ROTATED MNIST</td></tr><tr><td></td><td>|M| = 600</td><td>|M| = 1000</td><td>|M| = 600</td><td>|M| = 1000</td></tr><tr><td>VCL-RANDOM</td><td>55.8 (±1.29)</td><td>58.5 (±1.21)</td><td>61.2 (±0.12)</td><td>64.4 (±0.16)</td></tr><tr><td>AGEM</td><td>63.2 (±1.47)</td><td>64.1 (±0.74)</td><td>49.9 (±1.49)</td><td>53.0 (±1.52)</td></tr><tr><td>MER</td><td>74.9 (±0.49)</td><td>78.3 (±0.19)</td><td>76.5 (±0.30)</td><td>77.3 (±1.13)</td></tr><tr><td>ER-RING</td><td>73.5 (±0.43)</td><td>75.8 (±0.24)</td><td>74.7 (±0.56)</td><td>76.5 (±0.48)</td></tr><tr><td>HAL (OURS)</td><td>76.2 (±0.52)</td><td>78.4 (±0.27)</td><td>77.0 (±0.66)</td><td>78.7 (±0.97)</td></tr></table>
+
+<table><tr><td>METHOD</td><td colspan="2">SPLIT CIFAR</td><td colspan="2">SPLIT MINIIMAGENET</td></tr><tr><td></td><td>|M| = 255</td><td>|M| = 425</td><td>|M| = 255</td><td>|M| = 425</td></tr><tr><td>ICARL</td><td>51.7 (±1.41)</td><td>51.2 (±1.32)</td><td>-</td><td>-</td></tr><tr><td>AGEM</td><td>56.9 (±3.45)</td><td>59.9 (±2.64)</td><td>51.6 (±2.69)</td><td>54.3 (±1.56)</td></tr><tr><td>MER</td><td>57.7 (±2.59)</td><td>60.6 (±2.09)</td><td>49.4 (±3.43)</td><td>54.8 (±1.79)</td></tr><tr><td>ER-RING</td><td>60.9 (±1.44)</td><td>62.6 (±1.77)</td><td>53.5 (±1.42)</td><td>54.2 (±3.23)</td></tr><tr><td>HAL (OURS)</td><td>62.9 (±1.49)</td><td>64.4 (±2.15)</td><td>56.5 (±0.87)</td><td>57.2 (±1.54)</td></tr></table>
+
+Table 6: Forgetting (Eq. 3) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Averages and standard deviations are computed over five runs using different random seeds.  
+
+<table><tr><td>METHOD</td><td colspan="2">PERMUTED MNIST</td><td colspan="2">ROTATED MNIST</td></tr><tr><td></td><td>|M| = 600</td><td>|M| = 1000</td><td>|M| = 600</td><td>|M| = 1000</td></tr><tr><td>VCL-RANDOM</td><td>0.39 (±0.01)</td><td>0.36 (±0.01)</td><td>0.37 (±0.01)</td><td>0.33 (±0.01)</td></tr><tr><td>AGEM</td><td>0.20 (±0.01)</td><td>0.19 (±0.01)</td><td>0.41 (±0.01)</td><td>0.38 (±0.01)</td></tr><tr><td>MER</td><td>0.14 (±0.01)</td><td>0.09 (±0.01)</td><td>0.12 (±0.01)</td><td>0.11 (±0.01)</td></tr><tr><td>ER-RING</td><td>0.09 (±0.01)</td><td>0.07 (±0.01)</td><td>0.15 (±0.01)</td><td>0.13 (±0.01)</td></tr><tr><td>HAL (OURS)</td><td>0.07 (±0.01)</td><td>0.05 (±0.01)</td><td>0.12 (±0.01)</td><td>0.11 (±0.01)</td></tr></table>
+
+<table><tr><td>METHOD</td><td colspan="2">SPLIT CIFAR</td><td colspan="2">SPLIT MINIIMAGENET</td></tr><tr><td></td><td>|M| = 255</td><td>|M| = 425</td><td>|M| = 255</td><td>|M| = 425</td></tr><tr><td>ICARL</td><td>0.13 (±0.02)</td><td>0.13 (±0.02)</td><td>-</td><td>-</td></tr><tr><td>AGEM</td><td>0.13 (±0.03)</td><td>0.10 (±0.02)</td><td>0.10 (±0.02)</td><td>0.08 (±0.01)</td></tr><tr><td>MER</td><td>0.11 (±0.01)</td><td>0.09 (±0.02)</td><td>0.12 (±0.02)</td><td>0.07 (±0.01)</td></tr><tr><td>ER-RING</td><td>0.09 (±0.01)</td><td>0.06 (±0.01)</td><td>0.07 (±0.02)</td><td>0.08 (±0.02)</td></tr><tr><td>HAL (OURS)</td><td>0.08 (±0.01)</td><td>0.06 (±0.01)</td><td>0.06 (±0.01)</td><td>0.06 (±0.01)</td></tr></table>
+
+- learning rate: [0.003, 0.01, 0.03 (MNIST, CIFAR, miniImageNet), 0.1, 0.3, 1.0]  
+- within batch meta-learning rate: [0.01, 0.03, 0.1 (MNIST, CIFAR, miniImageNet), 0.3, 1.0]  
+- current batch learning rate multiplier: [1, 2, 5 (CIFAR, miniImageNet), 10 (MNIST)]
+
+- ER-Ring  
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+HAL  
+- learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0]  
+- regularization  $(\lambda)$ : [0.01, 0.03, 0.1 (MNIST perm, rot), 0.3 (miniImageNet), 1 (CIFAR), 3, 10]
+
+- mean embedding strength  $(\gamma)$ : [0.01, 0.03, 0.1 (MNIST perm, rot, CIFAR, miniImageNet), 0.3, 1, 3, 10]  
+- decay rate  $(\beta)$ : 0.5  
+gradient steps on anchors  $(k)$  : 100
+
+# D Hyperparameter Sensitivity
+
+In Table 7, we report the performance of HAL against a range of hyperparameters. For a given hyperparameter in the table, all the other hyperparameters are set to their optimal values found in Sec C of the appendix. HAL is not sensitive to the choice of hyperparameters.
+
+Table 7: Average Accuracy of HAL on different values of hyperparameters. For a given hyperparameter in the table, all the other hyperparameters are set to their optimal values found in Sec C of the appendix.  
+
+<table><tr><td>DATASET</td><td>λ</td><td>Acc</td><td>γ</td><td>Acc</td><td>β</td><td>Acc</td></tr><tr><td rowspan="3">PERMUTED MNIST</td><td>0.01</td><td>72.8 ±(0.52)</td><td>0.01</td><td>73.1 ±(0.20)</td><td>0.1</td><td>72.5 ±(0.95)</td></tr><tr><td>0.1</td><td>73.6 ±(0.31)</td><td>0.1</td><td>73.6 ±(0.31)</td><td>0.5</td><td>73.6 ±(0.31)</td></tr><tr><td>1.0</td><td>73.2 ±(0.85)</td><td>1.0</td><td>73.4 ±(0.41)</td><td>0.9</td><td>72.9 ±(0.39)</td></tr><tr><td rowspan="3">SPLIT CIFAR100</td><td>0.01</td><td>58.5 ±(1.25)</td><td>0.01</td><td>59.8 ±(0.65)</td><td>0.1</td><td>58.7 ±(1.17)</td></tr><tr><td>0.1</td><td>59.2 ±(0.91)</td><td>0.1</td><td>60.4 ±(0.54)</td><td>0.5</td><td>60.4 ±(0.54)</td></tr><tr><td>1.0</td><td>60.4 ±(0.54)</td><td>1.0</td><td>60.2 ±(1.21)</td><td>0.9</td><td>59.6 ±(1.05)</td></tr></table>
+
+# E HAL Algorithm
+
+Algorithm 1 provides a pseudocode for HAL.
+
+Algorithm 1 Training of HAL on sequential data  $\mathcal{D} = \{\mathcal{D}_1,\dots ,\mathcal{D}_T\}$ , with total replay buffer size 'mem sz', learning rate  $\alpha$  regularization strength  $\lambda$  mean embedding decay  $\beta$  mean embedding strength  $\eta$  
+```latex
+1: procedure HAL(D, mem sz,  $\alpha ,\lambda ,\beta$    
+2:  $\mathcal{M}\gets \{\} *\mathrm{mem\_sz}$    
+3:  $\{e_1,\dots ,e_T\} \leftarrow \{\}$    
+4: for  $t\in \{1,\dots ,T\}$  do   
+5:  $\phi_t\gets \vec{0}$    
+6: for  $\mathcal{B}\sim \mathcal{D}_t$  do ▷ Sample a batch from current task   
+7:  $\mathcal{B}_{\mathcal{M}}\sim \mathcal{M}$  ▷ Sample a batch from episodic memory   
+8:  $\tilde{\theta}\gets \theta -\alpha \cdot \nabla_{\theta}\ell (\mathcal{B}\cup \mathcal{B}_{\mathcal{M}})$  ▷ Temporary parameter update   
+9:  $\theta \gets \theta -\alpha \cdot \nabla_{\theta}\left(\ell (\mathcal{B}\cup \mathcal{B}_{\mathcal{M}}) + \lambda \cdot \sum_{t^{\prime} <   t}(f_{\theta}(e_{t^{\prime}},t^{\prime}) - f_{\tilde{\theta}}(e_{t^{\prime}},t^{\prime}))^{2}\right)$  ▷ Anchoring objective (Eq. 5)   
+10:  $\phi_t\gets \beta \cdot \phi_t + (1 - \beta)\cdot \phi (\mathcal{B})$  ▷ Running average of mean embedding   
+11:  $\mathcal{M}\gets$  UpdateMemory(M,B) ▷ Add samples to a ring buffer   
+12: end for   
+13:  $e_t,\theta \gets$  GetAnchors(M,  $\theta ,\phi_t,\eta)$  ▷ Get anchors for current task   
+14: end for   
+15: return  $\theta ,\mathcal{M}$    
+16: end procedure   
+1: procedure GETANCHORS(M,  $\theta_t,\phi_t,\gamma)$    
+2:  $\theta \gets \theta_t$    
+3: for  $\mathcal{B}_{\mathcal{M}}\sim \mathcal{M}$  do   
+4:  $\theta \gets \theta -\alpha \cdot \nabla_{\theta}\ell (\mathcal{B}_{\mathcal{M}})$  ▷ Finetune  $\theta_t$  by taking SGD steps on the episodic memory   
+5: end for   
+6:  $\theta_{\mathcal{M}}\gets \theta$  ▷ Store the updated parameter   
+7:  $e_t\gets$  rand() ▷ Initialize the task anchors   
+8: for  $1,\dots ,k$  do   
+9:  $e_t\gets e_t + \alpha \cdot \nabla_{e_t}\left(\ell (f_{\theta_{\mathcal{M}}} (e_t,t),y_t) - \ell (f_{\theta_t}(e_t,t),y_t) - \gamma (\phi (e_t) - \phi_t)^2\right)$  ▷ Maximize forgetting (Eq. 9)   
+10: end for   
+11: return  $e_t,\theta_t$    
+12: end procedure
+```

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+    {
+        "type": "text",
+        "text": "Federated Learning in the Sky: Joint Power Allocation and Scheduling with UAV Swarms",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Tengchan Zeng, Omid Semiari, Mohammad Mozaffari, Mingzhe Chen, Walid Saad, and Mehdi Bennis",
+        "bbox": [
+            114,
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+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract—Unmanned aerial vehicle (UAV) swarms must exploit machine learning (ML) in order to execute various tasks ranging from coordinated trajectory planning to cooperative target recognition. However, due to the lack of continuous connections between the UAV swarm and ground base stations (BSs), using centralized ML will be challenging, particularly when dealing with a large volume of data. In this paper, a novel framework is proposed to implement distributed federated learning (FL) algorithms within a UAV swarm that consists of a leading UAV and several following UAVs. Each following UAV trains a local FL model based on its collected data and then sends this trained local model to the leading UAV who will aggregate the received models, generate a global FL model, and transmit it to followers over the intra-swarm network. To identify how wireless factors, like fading, transmission delay, and UAV antenna angle deviations resulting from wind and mechanical vibrations, impact the performance of FL, a rigorous convergence analysis for FL is performed. Then, a joint power allocation and scheduling design is proposed to optimize the convergence rate of FL while taking into account the energy consumption during convergence and the delay requirement imposed by the swarm's control system. Simulation results validate the effectiveness of the FL convergence analysis and show that the joint design strategy can reduce the number of communication rounds needed for convergence by as much as  $35\\%$  compared with the baseline design.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "I. INTRODUCTION",
+        "text_level": 1,
+        "bbox": [
+            214,
+            527,
+            346,
+            541
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Swarms of unmanned aerial vehicles (UAVs) will play an important role in various services ranging from delivery of goods to monitoring [1] and [2]. To deliver those services, UAV swarms will employ machine learning (ML) for executing various tasks such as consensus trajectory planning, target recognition, and localization. However, due to the high altitude and mobility of UAVs, continuous connections between UAVs and ground base stations (BSs) cannot be guaranteed. Hence, using centralized ML approaches to execute learning-related tasks will be challenging, particularly when transmitting a large",
+        "bbox": [
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+            546,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This research was supported, in part, by the U.S. National Science Foundation under Grants CNS-1739642 and CNS-1941348, and by the Academy of Finland Project CARMA, by the Academy of Finland Project MISSION, by the Academy of Finland Project SMARTER, as well as by the INFOTECH Project NOOR.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "T. Zeng and W. Saad are with Wireless@VT, Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, 24061 USA (e-mail: tengchan@vt.edu; walids@vt.edu).",
+            "O. Semiari is with Department of Electrical and Computer Engineering, University of Colorado Colorado Springs, Colorado Springs, CO, 80918 USA (e-mail: osemiari@uccs.edu).",
+            "M. Mozaffari is with Ericsson Research, Santa Clara, CA, 95054 USA (e-mail: mohammad.mozaffari@ericsson.com).",
+            "M. Chen is with Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544 USA (e-mail: mingzhec@princeton.edu).",
+            "M. Dennis is with the Centre for Wireless Communications, University of Oulu, 90014 Oulu, Finland (e-mail:mehdi.bennis@oulu.fi)."
+        ],
+        "bbox": [
+            68,
+            765,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "volume of data over aerial links. Instead, a distributed learning approach would be more apropos [3]. In particular, one can use federated learning (FL) to enable each UAV to perform distributed ML tasks without relying on any centralized BSs [4]. In this case, UAVs do not need to send any raw data to BSs when training learning models.",
+        "bbox": [
+            501,
+            202,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In essence, FL allows each UAV in a swarm to train its learning model based on its own collected data, and it can use the intra-swarm network to share FL parameters related to the learned models with other UAVs. As the learning process proceeds, UAVs in the swarm can reach a consensus on their collective learning tasks, e.g., trajectory planning or target recognition. However, since the updates of the learning models in FL are transmitted over a wireless network, the FL convergence and task consensus for the UAV swarm will inevitably be affected by wireless factors such as transmission delay. Also, due to the high mobility of UAVs, other factors (like wind and mechanical vibrations) can increase the uncertainty of wireless channels by affecting the UAVs' antenna angles which, in turn, will impact the FL convergence.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A number of recent works have investigated how wireless communication impacts FL [5]–[7]. For instance, in [5], the authors solve the joint learning, wireless resource allocation, and user selection problem to minimize the FL convergence time while optimizing the FL performance. Also, the work in [6] proposes a strategy for bandwidth allocation and device scheduling to improve the energy efficiency for networks implementing FL. Moreover, [7] studies the impact of different scheduling policies on the performance of FL. While interesting, none of these works in [5]–[7] considers the role of FL in a UAV swarm. Also, due to the high mobility of UAVs and their limited energy, the analysis in [5]–[7] cannot be directly applied for UAV swarms.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The main contribution of this paper is a novel framework for enabling FL within a swarm of wireless-connected UAVs. In particular, we first conduct a convergence analysis for FL to show how wireless factors within the UAV swarm impact the convergence of FL. We then determine the convergence round, defined as the minimum number of communication rounds needed to achieve FL convergence. Using this key insight, we formulate an optimization problem that jointly designs the power allocation and scheduling for the UAV swarm network to reduce the FL convergence round. In particular, due to the stringent energy limitations of UAVs, we consider the constraint of the energy consumed by learning, communications, and flying during FL convergence. We also take into account",
+        "bbox": [
+            501,
+            700,
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+        "page_idx": 0
+    },
+    {
+        "type": "aside_text",
+        "text": "arXiv:2002.08196v2 [cs.LG] 10 Jun 2020",
+        "bbox": [
+            22,
+            267,
+            57,
+            707
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "the delay constraint imposed by the control system to guarantee the stability of the UAV swarm. To solve the joint design problem, we use a sample average approximation approach from stochastic programming along with a dual method from convex optimization. To the best of our knowledge, this is the first work that implements FL for the UAV swarm, studies the impact of wireless factors on the convergence of FL, and optimizes the FL convergence by jointly designing power allocation and scheduling of the UAV network. Simulation results validate the convergence analysis of FL and show that the joint design can reduce the convergence round by as much as  $35\\%$  compared with baselines without the joint design.",
+        "bbox": [
+            68,
+            66,
+            491,
+            247
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The rest of the paper is organized as follows. Section II presents the system model for the UAV swarm. Section III analyzes the FL convergence and shows the joint system design. Section IV provides simulation results, and conclusions are drawn in Section V.",
+        "bbox": [
+            70,
+            247,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "II. SYSTEM MODEL",
+        "text_level": 1,
+        "bbox": [
+            207,
+            330,
+            354,
+            344
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Consider a swarm of wirelessly connected autonomous UAVs flying at the same altitude, as shown in Fig. 1(a). The UAV swarm consists of a leader  $L$  and a set  $\\mathcal{I}$  of  $I$  followers. Every follower keeps a target distance and speed with the leader. While flying, the UAV swarm collects data and performs FL for data analysis and inference tasks like trajectory planning and cooperative target recognition. Using FL, each follower uses its collected data to train a local FL model and send the parameters related to the learned model to the leading UAV in the uplink, as shown in Fig. 1(a). The leading UAV will integrate all received information to generate a global FL model, and, then, transmit the parameters of the global model to following UAVs over the downlink. Moreover, to guarantee that the followers fly with the same speed while keeping a safe distance, the leading UAV will also broadcast the target spacing information and its speed and heading direction.",
+        "bbox": [
+            68,
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+            491,
+            592
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A. Federated learning model",
+        "text_level": 1,
+        "bbox": [
+            70,
+            599,
+            272,
+            616
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the learning model, we assume that UAV  $i \\in \\mathcal{I}$  collects a set  $\\{\\pmb{x}_{i1},\\pmb{x}_{i2},\\dots,\\pmb{x}_{iN_i}\\}$  of input data where each collected sample is represented by a vector  $\\pmb{x}_{in}$ ,  $n \\in \\{1,\\dots,N_i\\}$  that captures the input features and  $N_{i}$  is the number of collected samples. We also assume the input sample  $\\pmb{x}_{in}$ ,  $n \\in \\{1,\\dots,N_i\\}$ , corresponds to a single output  $y_{in}$  [4]. The output vector is thereby  $\\{y_{i1},\\dots,y_{iN_i}\\}$  for UAV  $i$ . We define a vector  $\\pmb{w}_i$  as the parameters related to the local FL model that is trained by  $\\{\\pmb{x}_{i1},\\pmb{x}_{i2},\\dots,\\pmb{x}_{iN_i}\\}$  and  $\\{y_{i1},\\dots,y_{iN_i}\\}$  at UAV  $i$ . The convergence of the FL training processes requires each local learning vector to converge to a vector  $\\pmb{w}^*$  which solves the following problem:",
+        "bbox": [
+            68,
+            619,
+            490,
+            800
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\underset {\\boldsymbol {w} \\in \\mathbb {R} ^ {d}} {\\arg \\min } F (\\boldsymbol {w}) = \\frac {1}{N} \\sum_ {i} ^ {I} \\sum_ {n = 1} ^ {N _ {i}} f (\\boldsymbol {w}, \\boldsymbol {x} _ {i n}, y _ {i n}), \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            130,
+            799,
+            488,
+            839
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where  $N = \\sum_{i}^{I}N_{i}$  is the total number of the collected samples by all followers, and  $f(\\pmb {w},\\pmb{x}_{in},y_{in})$  captures the loss function when using learning vector  $\\pmb{w}$  for dataset  $\\{\\pmb {x}_{in},y_{in}\\}$ . Note that, the loss function  $f(\\pmb {w},\\pmb{x}_{in},y_{in}),i\\in \\mathcal{I},0\\leq n\\leq N_i,$",
+        "bbox": [
+            68,
+            839,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/56d45f43e204b171ef87a2bb14d6dbde9f1ed2fd7efef56ae10faca062d78ebf.jpg",
+        "image_caption": [
+            "(a) Communication and learning models."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            547,
+            66,
+            870,
+            218
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/532aa106c947b40349baa42189c3b1f0f2989912d8c82bd93117fbbc95e83a61.jpg",
+        "image_caption": [
+            "(b) Angle deviations and control system.",
+            "Fig. 1. Illustration of our system model."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            542,
+            247,
+            879,
+            316
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "plays a pivotal role in determining the FL performance, and the expression of the loss function is application-specific. For example, for a simple linear regression FL algorithm,  $f(\\boldsymbol{w}, \\boldsymbol{x}_{in}, y_{in}) = (\\boldsymbol{w}^T \\boldsymbol{x}_{in} - y_{in})^2$ .",
+        "bbox": [
+            503,
+            369,
+            924,
+            431
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To solve (1), the FL framework uses an iterative update scheme [4]. In particular, the leading UAV will first generate an initial global FL model represented by vector  $\\boldsymbol{w}^{(0)}$  and send the initial vector to all followers. Hence, in the first communication round, follower  $i\\in \\mathcal{I}$  will first use  $\\boldsymbol{w}^{(0)}$  for its own data to train the local model and, then, it sends the vector of the trained model to the leader. Next, the leading UAV will aggregate all received local FL vectors and update the global FL model vector which will be later transmitted to the followers. Each communication round will be followed by another round, and the same process will repeat among leader and followers in each round. In this case, as FL proceeds, the local and global models are sequentially updated, and the total loss  $F(\\boldsymbol {w})$  for the updated global model with vector  $\\boldsymbol{w}$  will continuously decrease [4]. To identify whether the optimal solution is found for (1), one must analyze the convergence of the loss function  $F(\\boldsymbol {w})$  to  $F(\\boldsymbol {w}^{*})$ . That is, when the gap between the current loss  $F(\\boldsymbol {w})$  and the minimal loss  $F(\\boldsymbol {w}^{*})$  is below a threshold  $\\varepsilon$ , the FL optimization problem is solved [8]. Therefore, we can use the convergence of  $F(\\boldsymbol {w})$  to  $F(\\boldsymbol {w}^{*})$  to quantify the FL performance.",
+        "bbox": [
+            503,
+            431,
+            926,
+            748
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Moreover, for each communication round, we can divide the total time duration  $T_{r}$  into two periods: Uplink and downlink transmission. In particular, to guarantee that the leading UAV has enough time to process all received models from its followers, all uplink transmissions should be completed within a target time  $T_{u}(\\beta) = \\beta T_{r}$ , where  $\\beta \\in \\{0,1\\}$  is a scheduling parameter to schedule uplink-downlink traffic in time. Also, to receive the global FL model update from the leading UAV successfully, the time constraint for downlink transmissions is thereby  $T_{d}(\\beta) = (1 - \\beta)T_{r}$ . In this case, if the communication",
+        "bbox": [
+            501,
+            748,
+            924,
+            900
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "link between follower  $i \\in \\mathcal{I}$  and leader  $L$  fails to meet the time constraints  $T_{d}(\\beta)$  and  $T_{u}(\\beta)$ , the global FL model cannot use the corresponding FL model for the aggregation. At the same time, for the local FL model, the following UAV cannot use the recently updated global vector to train its local data. In other words, the transmission delay of the uplink and downlink links will impact the update of the global and local FL models thus having a major impact on FL convergence.",
+        "bbox": [
+            68,
+            65,
+            491,
+            186
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition, when training the global FL model, we can calculate the energy consumption for the UAV  $L$  as  $E_{L} = \\kappa C\\phi^{2}\\sum_{i=1}^{I}S(\\pmb{w}_{i})$ , where  $\\kappa$  captures the energy consumption coefficient depending on the computing system and  $C$  is the number of computing cycles needed per data bit [9].  $\\phi$  is the frequency of the CPU clock of UAVs, and  $S(\\pmb{w}_{i})$  is the packet size of  $\\pmb{w}_{i}$ , transmitted from UAV  $i \\in \\mathcal{I}$ , in bits. Similarly, we can determine the training energy consumption for follower  $i \\in \\mathcal{I}$  as  $E_{i} = \\kappa C\\phi^{2}\\sum_{n=1}^{N_{i}}S(\\pmb{x}_{in})$ .",
+        "bbox": [
+            70,
+            186,
+            491,
+            325
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "B. Communication model",
+        "text_level": 1,
+        "bbox": [
+            71,
+            330,
+            250,
+            345
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To minimize the interference from other UAVs located outside of the swarm, we assume that all UAVs use directional antennas, as shown in Fig. 1(a), However, as shown in Fig. 1(b), due to the impact of wind, payload, and non-ideal mechanical and control systems, the angle of the UAVs will randomly fluctuate and deviate from the initial angle setting. Based on the central limit theorem, we model the angle deviation for each UAV as a Gaussian random variable [10]. Moreover, we consider a squared cosine function to capture the antenna aperture of UAV  $j \\in \\mathcal{I} \\cup \\{L\\}$  when communicating with UAV  $l \\in \\mathcal{I} \\cup \\{L\\} / j$  as follows [11]:",
+        "bbox": [
+            68,
+            349,
+            490,
+            515
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG _ {j l} \\left(\\theta_ {j l} + \\vartheta_ {j}\\right) = \\left\\{ \\begin{array}{c c} \\cos^ {2} \\left(\\frac {\\pi}{2} \\left(\\theta_ {j l} + \\vartheta_ {j}\\right)\\right), & \\text {i f} | \\theta_ {j l} + \\vartheta_ {j} | \\leq 1, \\\\ G _ {\\min }, & \\text {o t h e r w i s e}, \\end{array} \\right. \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            75,
+            515,
+            488,
+            556
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $\\theta_{jl}$  is the initial angle setting for UAV  $j$  when communicating with UAV  $l$ ,  $\\vartheta_{j} \\sim \\mathcal{N}(0, \\sigma_{j}^{2})$  is the angle deviation with variance  $\\sigma_{j}^{2}$ , and  $G_{\\mathrm{min}}$  captures the antenna gain at the side lobes. Also, similar to [10], we can approximate (2) by using a sectionalized expression:",
+        "bbox": [
+            70,
+            558,
+            491,
+            632
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nG _ {j l} \\left(\\theta_ {j l} + \\vartheta_ {j}, M\\right) = \\left\\{ \\begin{array}{c c} \\cos^ {2} \\left(\\frac {\\pi m}{2 M}\\right), & \\text {i f} \\frac {m}{M} \\leq \\left| \\theta_ {j l} + \\vartheta_ {j} \\right| \\leq \\frac {m + 1}{M}, \\\\ G _ {\\min }, & \\text {o t h e r w i s e}, \\end{array} \\right. \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            76,
+            631,
+            488,
+            674
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $m\\in \\{1,\\dots,M\\}$",
+        "bbox": [
+            70,
+            675,
+            230,
+            690
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To reduce the interference over the uplink transmissions, we assume that uplinks do not share the wireless resource with each other. Hence, the transmission delay of the uplink between follower  $i \\in \\mathcal{I}$  and leader  $L$  can be calculated as",
+        "bbox": [
+            70,
+            691,
+            491,
+            748
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nT _ {i L} = \\frac {S (\\boldsymbol {w} _ {i})}{B _ {u} \\log_ {2} \\left(1 + \\frac {p _ {i} h _ {i L} d _ {i L} ^ {- \\alpha} G _ {i L} G _ {L i}}{\\sum_ {i ^ {\\prime} \\in \\Phi_ {i}} p _ {i ^ {\\prime}} h _ {i ^ {\\prime} L} d _ {i ^ {\\prime} L} ^ {- \\alpha} G _ {i ^ {\\prime} L} G _ {L i ^ {\\prime}} + B _ {u} \\gamma_ {0}}\\right)}, \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            84,
+            748,
+            488,
+            787
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{u}$  is the bandwidth used by each subchannel in the uplink,  $p_i\\in (0,p_{\\mathrm{max}})$  is the transmission power of UAV  $i$  with maximum power as  $p_{\\mathrm{max}}$  and  $\\alpha$  is the path-loss exponent.  $h_{iL}$  is the channel gain of the Rician fading channel between UAVs  $i$  and  $L$  ,and  $\\gamma_0$  is the noise power spectral density. Note that, despite the use of directional antenna, the swarm still experiences uplink interference generated by UAVs located",
+        "bbox": [
+            70,
+            787,
+            491,
+            893
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "outside of the swarm. In particular, these interfering UAVs share the same channel resource and exist in the main lobe of the UAV  $L$ , and we define  $\\Phi_{i}$  as the set of UAVs that generates interference to the uplink from UAV  $i$  to UAV  $L$ .",
+        "bbox": [
+            501,
+            65,
+            924,
+            126
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Similarly, we can derive the transmission delay  $T_{Li}$  for the downlink from UAV  $L$  to UAV  $i \\in \\mathcal{I}$  as:",
+        "bbox": [
+            503,
+            126,
+            924,
+            155
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nT _ {L i} = \\frac {S (\\boldsymbol {w})}{B _ {d} \\log_ {2} \\left(1 + \\frac {p _ {L} h _ {L i} d _ {L i} ^ {- \\alpha} G _ {L i} G _ {i L}}{\\sum_ {i ^ {\\prime} \\in \\Phi_ {L}} p _ {i ^ {\\prime}} h _ {i ^ {\\prime} i} d _ {i ^ {\\prime} i} ^ {- \\alpha} G _ {i ^ {\\prime} i} G _ {i i ^ {\\prime}} + B _ {d} \\gamma_ {0}}\\right)}, \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            522,
+            152,
+            924,
+            196
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $B_{d}$  is the downlink bandwidth,  $p_L \\in (0, p_{\\mathrm{max}})$  is the transmission power of UAV  $L$ , and  $\\Phi_L$  refers to the set of UAVs that will generate interference at the downlink.",
+        "bbox": [
+            503,
+            194,
+            926,
+            239
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "C. Control model",
+        "text_level": 1,
+        "bbox": [
+            504,
+            246,
+            630,
+            258
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To guarantee constant speed and altitude and avoid collisions between UAVs within the swarm, the leading UAV will broadcast its speed and heading direction to the followers in the downlink. Here, the control system of each follower will use both its sensor data (e.g. location) and information received from the wireless links to coordinate its movement and achieve a target spacing and speed. Note that the target distance between the UAV leader and each follower is predefined such that there will be no collision between two nearby UAVs.",
+        "bbox": [
+            501,
+            263,
+            924,
+            400
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Similar to our previous work in [12], we can build a Cartesian coordinate system to capture the locations of UAVs in the swarm, and, then, we decompose the velocity of each UAV into two components, as shown in Fig. 1(b). We can also define the control law of each UAV the same way as the one provided in [12]. Since the transmission delay will have a negative impact on the stability control of the UAV swarm, we must consider the delay requirement imposed by the control system when designing the UAV network.",
+        "bbox": [
+            501,
+            401,
+            926,
+            535
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition, in order to fly with a constant speed and maintain a stable flying motion, each UAV must spend energy to overcome the gravity and the air drag forces due to the wind and forward motions. For a forward speed  $v \\in (0, v_{\\max})$  with  $v_{\\max}$  as the maximum speed, the minimum flying power of UAV  $j \\in \\mathcal{I} \\cup \\{L\\}$  is  $\\bar{p}_{j,\\min}(v) = \\hat{v}_j A_j$ , where  $\\hat{v}_j$  is the induced velocity required for constant speed  $v$  and given thrust  $A_j = mg$  with  $m$  being the UAV mass and  $g$  being the gravitational constant [13]. Also, the induced velocity  $\\hat{v}_j$  can be obtained by solving the following equation [13]:",
+        "bbox": [
+            503,
+            536,
+            926,
+            686
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\hat {v} _ {j} = \\frac {2 A _ {j}}{q r ^ {2} \\pi \\varrho \\sqrt {v ^ {2} + \\hat {v} _ {j} ^ {2}}}, \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            633,
+            685,
+            924,
+            724
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where  $q$  and  $r$  capture, respectively, the number and diameter of the UAV rotors, and  $\\varrho$  is the air density. Moreover, we can further correct the theoretical minimum motion power consumption by the overall power efficiency  $\\eta$  of the UAV in order to obtain the actual power consumption as  $\\bar{p}_j(v) = \\bar{p}_{j,\\min}(v) / \\eta$ . Since the control of a UAV's dynamic motion consumes the most energy [13], we must consider the flying energy consumption when designing the swarm of UAVs. In particular, the flying energy consumption can be calculated as  $\\bar{p}_j(v)T$  during the flying time  $T$ .",
+        "bbox": [
+            501,
+            722,
+            926,
+            872
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To guarantee the convergence of FL and the stable operation of the control system in the UAV swarm, we need to properly",
+        "bbox": [
+            503,
+            872,
+            924,
+            902
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "design the wireless communication network. At the same time, to guarantee that the energy spent on learning, communication, and flying will not exceed the energy limitation of each UAV, we need to consider the energy consumption during the FL convergence. Next, we first conduct the convergence analysis for the FL algorithm and derive the number of communication rounds needed to achieve the FL convergence. Then, we formulate an optimization problem that jointly designs the power allocation and scheduling policy to minimize the convergence round of FL while considering the delay requirement from the control system and energy consumption during the FL convergence.",
+        "bbox": [
+            70,
+            66,
+            491,
+            247
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "III. CONVERGENCE ANALYSIS AND JOINT DESIGN",
+        "text_level": 1,
+        "bbox": [
+            99,
+            253,
+            460,
+            266
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A. FL convergence analysis",
+        "text_level": 1,
+        "bbox": [
+            70,
+            271,
+            264,
+            286
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to guarantee FL convergence, we assume that the following UAVs adopt a standard gradient descent method to update their local FL models [4]. Thus, for following UAV  $i \\in \\mathcal{I}$ , the local model  $\\boldsymbol{w}_i^{(t)}$  at communication round  $t$  is given by",
+        "bbox": [
+            70,
+            290,
+            491,
+            364
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} _ {i} ^ {(t)} = \\boldsymbol {w} ^ {(t - 1)} - \\frac {\\bar {\\lambda}}{N _ {i}} \\nabla F _ {i} (\\boldsymbol {w} ^ {(t - 1)}), \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            161,
+            364,
+            488,
+            393
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\pmb{w}^{(t - 1)}$  is the global FL model at communication round  $t - 1$ ,  $\\bar{\\lambda}$  is the learning rate, and  $F_{i}(\\pmb{w}^{(t - 1)}) = \\sum_{n = 1}^{N_{i}}f(\\pmb{w}^{(t - 1)},\\pmb{x}_{in},y_{in})$ . After the leading UAV collects local vectors  $\\pmb{w}_i^{(t)}, i\\in \\mathcal{I}$ , the global FL model can be updated:",
+        "bbox": [
+            70,
+            393,
+            491,
+            458
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)}}{\\sum_ {i = 1} ^ {I} N _ {i}}. \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            204,
+            455,
+            488,
+            494
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, for ensuring successful updates of both global and local FL models as shown in (7) and (8), the transmission delay of uplink and downlink should be within, respectively,  $T_{u}(\\beta)$  and  $T_{d}(\\beta)$ . Hence, after considering the impact of the transmission delays, we can rewrite the global FL model update as",
+        "bbox": [
+            70,
+            491,
+            491,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)} C _ {i , t}}{\\sum_ {i = 1} ^ {I} N _ {i} C _ {i , t}}, \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            577,
+            488,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with",
+        "bbox": [
+            71,
+            613,
+            106,
+            625
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with  $C_{i,t} = \\left\\{ \\begin{array}{ll}1, & \\mathrm{with~probability~}\\mathbb{P}(T_{iL,t}\\leq T_u(\\beta),T_{Li,t}\\leq T_d(\\beta)),\\\\ 0, & \\mathrm{otherwise.} \\end{array} \\right.$",
+        "bbox": [
+            71,
+            622,
+            496,
+            651
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the aim of quantifying the convergence of FL, we use the notion of a convergence round, defined as the minimum number of communication rounds needed to achieve a target difference  $\\varepsilon$  of the expected gap between current loss and the minimal loss, i.e.,  $\\mathbb{E}(F(\\boldsymbol{w}) - F(\\boldsymbol{w}^{*})) \\leq \\varepsilon$ . Moreover, to determine the convergence round, we make the following two standard assumptions: Function  $F(\\boldsymbol{w})$ :  $\\mathbb{R}^n \\to \\mathbb{R}$  is continuously differentiable, and the gradient of  $F(\\boldsymbol{w})$  is uniformly Lipschitz continuous with positive parameter  $U$ . We also consider the function  $F$  to be strongly convex with positive parameter  $\\mu$ , and these exists constants  $\\zeta_1 \\geq 0$  and  $\\zeta_2 \\geq 1$ , meeting  $||\\nabla F_i(\\boldsymbol{w})||^2 \\leq \\zeta_1 + \\zeta_2||\\nabla F(\\boldsymbol{w})||^2$  [14]. Given the above assumptions, we can derive the convergence round.",
+        "bbox": [
+            70,
+            651,
+            491,
+            847
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. To realize an expected convergence of  $F(\\boldsymbol{w})$  under an accuracy threshold  $\\varepsilon$ , i.e.,  $\\mathbb{E}(F(\\boldsymbol{w}) - F(\\boldsymbol{w}^{*})) \\leq \\varepsilon$ ,",
+        "bbox": [
+            70,
+            854,
+            491,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "the convergence round is given by:",
+        "bbox": [
+            504,
+            66,
+            746,
+            80
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\varphi = \\left[ \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\sum_ {i = 1} ^ {I} \\sum_ {n = 1} ^ {N _ {i}} f (\\boldsymbol {w} ^ {(0)} , \\boldsymbol {x} _ {i n} , y _ {i n})} \\right], \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            540,
+            79,
+            924,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where  $\\lceil \\cdot \\rceil$  is the ceiling function, and  $\\rho$  captures the convergence speed given as follows",
+        "bbox": [
+            503,
+            116,
+            924,
+            145
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\rho = \\frac {\\sum_ {i = 1} ^ {T} N _ {i} \\mathbb {P} \\left(T _ {i L , t} \\leq T _ {u} (\\beta) , T _ {L i , t} \\leq T _ {d} (\\beta)\\right) \\mu}{N U}. \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            540,
+            145,
+            924,
+            176
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof: Due to the space limitation, the proof is included in Appendix A.",
+        "bbox": [
+            503,
+            180,
+            924,
+            210
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As shown in Theorem 1, the convergence performance of FL depends on the transmission delay of both uplink and downlink in the network. In particular, to increase the convergence speed, we need to maximize the probability that both uplink and downlink meet the corresponding delay requirements of FL. Thus, Theorem 1 provides a concrete characterization of the interplay between wireless communications and FL performance in a UAV swarm.",
+        "bbox": [
+            501,
+            217,
+            926,
+            337
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the stability analysis of the control system, we will follow the method provided by our previous work in [12]. That is, we first build the augmented error state vector. Then, we use Lyapunov-Razumikhin theorem to derive the control system delay requirements  $\\tau_{i}, i \\in \\mathcal{I}$ , for downlink that can guarantee the stability of the UAV swarm.",
+        "bbox": [
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+            338,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "B. Problem formulation and solution concept",
+        "text_level": 1,
+        "bbox": [
+            504,
+            436,
+            815,
+            452
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, we formulate an optimization problem to minimize the convergence round by jointly designing the power allocation and scheduling for the UAV network, as follows:",
+        "bbox": [
+            503,
+            455,
+            924,
+            501
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\left\\{\\boldsymbol {p}, p _ {L}, \\beta , v \\right\\}} \\varphi \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            544,
+            502,
+            924,
+            526
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathrm {s . t .} \\mathbb {P} \\left[ \\varphi E _ {L} + \\varphi p _ {L} T _ {d} (\\beta) + \\varphi \\bar {p} _ {L} (v) T _ {r} \\leq \\bar {E} \\right] \\geq \\xi_ {L}, \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            544,
+            527,
+            924,
+            551
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {P} \\left[ \\varphi E _ {i} + \\varphi p _ {i} T _ {i L} + \\varphi \\bar {p} _ {i} (v) T _ {r} \\leq \\bar {E} \\right] \\geq \\xi_ {i}, i \\in \\mathcal {I}, \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            566,
+            553,
+            924,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathbb {P} \\left(T _ {L i} \\leq \\tau_ {i}\\right) \\geq \\xi_ {C}, i \\in \\mathcal {I}, \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            566,
+            580,
+            924,
+            597
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np _ {L} \\in (0, p _ {L, \\max }, p _ {i} \\in (0, p _ {i, \\max }), i \\in \\mathcal {I}, \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            565,
+            599,
+            924,
+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\beta \\in (0, 1), v \\in (0, v _ {\\max }) \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            566,
+            619,
+            924,
+            635
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where vector  $\\pmb{p} = [p_1, \\dots, p_I]$ . Constraint (13) guarantees that the probability of total energy consumption for the leading UAV being less than a threshold  $\\bar{E}$  will be greater than  $\\xi_L \\in (0, 1)$ . Similarly to (13), constraint (14) represents the constraint on energy consumption of each follower  $i \\in \\mathcal{I}$ . Constraint (15) guarantees that the UAV communication network is reliable to support the stability of the swarm with probability  $\\xi_C$ . Constraints (16) and (17) ensure that the optimization variables, i.e., the transmission power, scheduling parameter, and velocity, are chosen within reasonable ranges. Note that, in the optimization problem, we also optimize the operation speed of the UAV swarm to minimize the motion energy consumption and relax the energy constraints in (13) and (14).",
+        "bbox": [
+            501,
+            633,
+            926,
+            829
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since both exponent and base in the logarithm function (10) are less than 1, minimizing the logarithm function in (12) is",
+        "bbox": [
+            503,
+            830,
+            924,
+            859
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "equivalent to minimizing the base for the constant exponent. Also, according to (11), we can simplify (12) as",
+        "bbox": [
+            70,
+            66,
+            488,
+            95
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\max  _ {\\left\\{\\boldsymbol {p}, p _ {L}, \\beta , v \\right\\}} \\sum_ {i = 1} ^ {I} N _ {i} \\mathbb {P} \\left(T _ {i L, t} \\leq T _ {u} (\\beta), T _ {L i, t} \\leq T _ {d} (\\beta)\\right). \\tag {18}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            104,
+            95,
+            488,
+            133
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We observe that, after simplifications, both objective function and constraints are represented by probability terms. In this case, directly deriving the probability terms will be challenging since it requires multidimensional integrations. Also, as the optimization problem is not convex, employing convex approximations to simplify the optimization problem will be impossible. Instead, we use a sample average approximation approach where the probability terms in the objective function and constraints are replaced by an empirical distribution found by random samples [15]. In particular, we first generate  $K$  independent samples of the random parameters, i.e., wireless channel gains and angle deviations, and we calculate the corresponding transmission delay and convergence round. Then, we can reformulate the optimization problem as",
+        "bbox": [
+            70,
+            132,
+            490,
+            343
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\max  _ {\\left\\{\\boldsymbol {p}, p _ {L}, \\beta , v \\right\\}} \\sum_ {i = 1} ^ {I} \\sum_ {k = 1} ^ {K} N _ {i} \\mathbb {1} \\left(T _ {u} (\\beta) - T _ {i L, k}\\right) \\mathbb {1} \\left(T _ {d} (\\beta) - T _ {L i, k}\\right) \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            84,
+            343,
+            488,
+            395
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\text {s . t .} \\sum_ {k = 1} ^ {K} \\mathbb {1} \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {L} + \\varphi_ {k} p _ {L} T _ {d} (\\beta) + \\varphi_ {k} \\bar {p} _ {L} (v) T _ {r}\\right)\\right) \\geq K \\xi_ {L}, \\tag {20}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            81,
+            398,
+            488,
+            454
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {K} \\mathbb {1} \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {i} + \\varphi_ {k} p _ {i} T _ {i L, k} + \\varphi_ {k} \\bar {p} _ {i} (v) T _ {r}\\right)\\right) \\geq K \\xi_ {i}, i \\in \\mathcal {I}, \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            98,
+            457,
+            488,
+            512
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {k = 1} ^ {K} \\mathbb {1} \\left(\\tau_ {i} - T _ {L i, k}\\right) \\geq K \\xi_ {C}, i \\in \\mathcal {I}, \\tag {22}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            98,
+            515,
+            488,
+            556
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(16) and (17),",
+        "bbox": [
+            98,
+            559,
+            199,
+            574
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the indicator function  $\\mathbb{1}(r) = 1$ , once  $r \\geq 0$ ; otherwise, we have  $\\mathbb{1}(r) = 0$ . Due to the presence of the indicator function, the reformulated problem is non-smooth. To obtain a smooth problem, we can further replace the indicator functions with modified sigmoid functions, i.e.,  $\\Gamma(r) = \\frac{1}{1 + \\exp(-\\bar{c}r)}$ , where  $\\bar{c}$  determines how quickly the modified sigmoid function changes near 0. To obtain a sub-optimal solution to the reformulated optimization problem with the indicator functions replaced by the modified sigmoid functions, we can use the dual method [16]. In particular, the Lagrangian function is",
+        "bbox": [
+            70,
+            574,
+            490,
+            726
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\mathcal {J} (\\boldsymbol {\\lambda}, \\boldsymbol {p}, p _ {L}, \\beta , v) = \\sum_ {i = 1} ^ {I} \\sum_ {k = 1} ^ {K} N _ {i} \\Gamma (T _ {u} (\\beta) - T _ {i L, k}) \\Gamma (T _ {d} (\\beta) - T _ {L i, k}) +\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            73,
+            732,
+            488,
+            770
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\lambda_ {1} \\left(\\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {L} + \\varphi_ {k} p _ {L} T _ {d} (\\beta) + \\varphi_ {k} \\bar {p} _ {L} (v) T _ {r}\\right)\\right) - K \\xi_ {L}\\right) +\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            73,
+            772,
+            467,
+            808
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {i = 1} ^ {I} \\lambda_ {i + 1} \\left(\\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {i} + \\varphi_ {k} p _ {i} T _ {i L, k} + \\varphi_ {k} \\bar {p} _ {i} (v) T _ {r}\\right)\\right) - K \\xi_ {i}\\right) +\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            73,
+            810,
+            486,
+            847
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\sum_ {i = 1} ^ {I} \\lambda_ {I + 1 + i} \\left(\\sum_ {k = 1} ^ {K} \\Gamma \\left(\\tau_ {i} - T _ {L i, k}\\right) - K \\xi_ {C}\\right), \\tag {23}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            75,
+            849,
+            488,
+            885
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where vector  $\\pmb{\\lambda} = [\\lambda_1, \\dots, \\lambda_{2I + 1}] \\succeq \\mathbf{0}_{1 \\times (2I + 1)}$  is the vector",
+        "bbox": [
+            73,
+            885,
+            490,
+            901
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/dcd06c5890b02b0232808c32ea9dfd5e4cc89f2b45f5d9f65f2af3eb8f0feed3.jpg",
+        "table_caption": [
+            "Table. I. Simulation parameters."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameters</td><td>Values</td></tr><tr><td>Number of followers I</td><td>5</td></tr><tr><td>Transmission power threshold pmax</td><td>0.5 W</td></tr><tr><td>Maximum speed vmax</td><td>20 m/s [18]</td></tr><tr><td>Energy consumption efficient κ</td><td>10-28 [18]</td></tr><tr><td>Number of cycles needed per bit C</td><td>103 [18]</td></tr><tr><td>Frequency of the CPU φ</td><td>109cycle/s</td></tr><tr><td>Time for each communication round Tr</td><td>0.1 s</td></tr><tr><td>Side lobe gain Gmin, path loss exponent α</td><td>-2 dB, 2.5</td></tr><tr><td>Noise spectral density γ0</td><td>-174 dBm/Hz</td></tr><tr><td>Packet size Sw and Swi</td><td>10 kB</td></tr><tr><td>Number of rotors q and the diameter r</td><td>4, 0.254 m [4]</td></tr><tr><td>Power efficiency η and density ρ of the air</td><td>70 %, 1.225 kg/m3 [4]</td></tr><tr><td>Number of samples K, Energy limits E</td><td>1,000, 7,000 J</td></tr></table>",
+        "bbox": [
+            542,
+            73,
+            890,
+            231
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "of Lagrangian multipliers, and the dual objective function can be defined as  $\\mathcal{D}(\\pmb{\\lambda}) = \\max_{\\pmb{p}, p_L, v, \\beta} \\mathcal{J}(\\pmb{\\lambda}, \\pmb{p}, p_L, \\beta, v)$ . The corresponding dual optimization problem is",
+        "bbox": [
+            503,
+            243,
+            923,
+            287
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\min  _ {\\boldsymbol {\\lambda}} \\mathcal {D} (\\boldsymbol {\\lambda}) \\quad \\text {s . t .} \\boldsymbol {\\lambda} \\geq \\mathbf {0}. \\tag {24}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            633,
+            289,
+            923,
+            309
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although the dual problem in (24) is always convex [17],  $\\mathcal{D}(\\lambda)$  is not differentiable. Instead, we can use subgradients given by",
+        "bbox": [
+            503,
+            306,
+            923,
+            335
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta \\lambda_ {1} = \\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} ^ {*} E _ {L} + \\varphi_ {k} ^ {*} p _ {L} T _ {d} ^ {*} + \\varphi_ {k} ^ {*} \\bar {p} _ {L} ^ {*} T _ {r}\\right)\\right) - K \\xi_ {L},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            506,
+            335,
+            877,
+            376
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta \\lambda_ {i + 1} = \\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} ^ {*} E _ {i} + \\varphi_ {k} ^ {*} p _ {i} ^ {*} T _ {i L, k} ^ {*} + \\varphi_ {k} ^ {*} \\bar {p} _ {i} ^ {*} T _ {r}\\right)\\right) - K \\xi_ {i}, i \\in \\mathcal {I},\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            508,
+            378,
+            921,
+            419
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\Delta \\lambda_ {I + 1 + i} = \\sum_ {k = 1} ^ {K} \\Gamma \\left(\\tau_ {i} - T _ {L i, k} ^ {*}\\right) - K \\xi_ {C}, i \\in \\mathcal {I}, \\tag {25}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            506,
+            421,
+            923,
+            460
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the terms  $\\varphi_k^*, T_d^*, T_{iL,k}^*, T_{Li,k}^*, \\bar{p}_L^*, \\bar{p}_i^*$  are expressed by optimized variables  $p^*, p_L^*, \\beta^*, v^*$ . The proof of subgradients is similar to the one provided in [16], and is omitted here. Thereby, we can solve the problem in (24) by either the subgradient method or the ellipsoid method, and their complexities are, respectively,  $\\mathcal{O}\\left(\\frac{2I + 1}{\\epsilon^2}\\right)$  and  $\\mathcal{O}\\left((2I + 1)^2 \\ln \\frac{1}{\\epsilon}\\right)$  with accuracy  $\\epsilon$  [17]. Then, the sub-optimal solution of  $\\{\\pmb{p}, p_L, \\beta, v\\}$  can be obtained by solving dual objective function  $\\mathcal{D}(\\lambda)$ . In particular, similar to [16], we use the iterative method to sequentially derive the sub-optimal value of each element in  $\\{\\pmb{p}, p_L, \\beta, v\\}$  (the details are omitted here due to space limitations). Note that, we assume that all these steps of solving the optimization problem are done by a central unit (e.g., cloud or BS), before the swarm starts training their learning models in FL. In particular, there is no need for the central unit to collect any information from UAVs, since all samples of wireless channel gains and antenna deviations are randomly generated by the central unit itself. Also, since the number of UAVs in the swarm is usually small, the complexity of using sample average approximation and dual approach will be low. As a result, the central unit can readily obtain the sub-optimal solution to the joint design problem and later send the power allocation and scheduling parameters to UAVs in the swarm.",
+        "bbox": [
+            501,
+            460,
+            923,
+            806
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "IV. SIMULATION RESULTS AND ANALYSIS",
+        "text_level": 1,
+        "bbox": [
+            563,
+            815,
+            864,
+            829
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For our simulations, we first validate the theoretical analysis in Theorem 1. Then, we show the impact of angle deviations on the convergence of FL, and we compare our joint design with baseline schemes that optimize power allocation and schedul",
+        "bbox": [
+            503,
+            834,
+            923,
+            893
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/d48332a15ab16294fc429fd8db6df0fd8d11ab440b33f32abda4c4400b076a11.jpg",
+        "image_caption": [
+            "Fig. 2. Validation of Theorem 1."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            138,
+            68,
+            421,
+            229
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ing separately. In particular, we consider two baselines. The first baseline is a system with optimized power allocation (same power allocation in the joint design) and randomized scheduling parameters. The second baseline is a system with optimized scheduling (same scheduling used by the joint design) and randomized power allocation. We also assume equal uplink and downlink bandwidths, i.e.,  $B_{u} = B_{d} = 1 \\mathrm{MHz}$ , and equal angle deviation variance for each UAV, i.e.,  $\\sigma_j^2 = \\sigma^2$ ,  $j \\in \\mathcal{I} \\cup \\{L\\}$ . All simulation parameters are summarized in Table I.",
+        "bbox": [
+            70,
+            279,
+            491,
+            415
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 shows the convergence round versus the difference threshold  $\\varepsilon$ . Note that, in Fig. 2, we choose the range of  $\\varepsilon \\in (5, 25)$  based on the value of  $\\sum_{i=1}^{I} \\sum_{n=1}^{N_i} f(\\boldsymbol{w}_0, \\boldsymbol{x}_{in}, y_{in})$  and the range of  $\\varepsilon$  will be varied for different settings of data and initial global FL model and the accuracy requirement. As observed from Fig. 2, the theoretical analysis derived in Theorem 1 is aligned with the simulation results with less than  $5\\%$  difference, thus corroborating the validity of Theorem 1. Moreover, Fig. 2 shows that, when the difference threshold increases, the convergence round decreases. This is because, with a larger difference threshold, the requirement of convergence becomes less stringent. In this case, FL requires fewer communication rounds to converge.",
+        "bbox": [
+            70,
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+            611
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fig. 3 shows the convergence round when the variance of angle deviations changes. From Fig. 3, we observe that, when the variance of angle deviations increases, FL needs more communication rounds to converge. This is due to the fact that, when the angle deviation variance increases, the antennas at transmitter and receiver in the network will be less aligned, leading to a drop in the antenna gains' product between transmitter and receiver in (4) and (5). As a result, the transmission delay of wireless links will increase, and the probability of meeting the delay requirements, i.e.,  $\\mathbb{P}(T_{iL,t} \\leq T_u, T_{Li,t} \\leq T_d)$ , decreases. Therefore, more communication rounds are needed to achieve the FL convergence. Moreover, as shown in Fig. 3, when the bandwidth allocated to uplink and downlink increases, the FL algorithm requires fewer communication rounds to achieve convergence. This stems from the fact that, a large bandwidth improves the probability of meeting the delay requirements, yielding a fast FL convergence.",
+        "bbox": [
+            70,
+            611,
+            491,
+            867
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4 compares our proposed joint power allocation and scheduling design with the baselines without a joint design. It",
+        "bbox": [
+            70,
+            869,
+            491,
+            898
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9354308eb292f13f5eb1cbce4825b37a3fc2f50ff64a3d3ee2fef09a4608e1b9.jpg",
+        "image_caption": [
+            "Fig. 3. Impact of angle deviations on the FL convergence."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            562,
+            66,
+            864,
+            227
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/98d46815a2e58e129133ac172afe0dc66d6f03182f512840f7e8d36b11a7477b.jpg",
+        "image_caption": [
+            "Fig. 4. Comparisons between systems with and without joint design."
+        ],
+        "image_footnote": [],
+        "bbox": [
+            555,
+            266,
+            859,
+            425
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "is shown that, for the same network setting, the convergence round for a network with joint design is always less than its counterparts of baselines. In particular, when the bandwidth is  $1\\mathrm{MHz}$ , the system with a joint design reduces the convergence round by as much as  $35\\%$  compared with the baseline system with optimized scheduling and randomized power allocation design. Moreover, as shown in Fig. 4, when the bandwidth assigned to uplink and downlink increases, the performance gap between the system with the proposed joint design and the baselines decreases. That is because, as we increase the bandwidth, it becomes more probable for all three systems to meet the delay constraints at uplink and downlink. Therefore, the impact of communications delay on the FL convergence will be minimized.",
+        "bbox": [
+            501,
+            473,
+            926,
+            684
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "V. CONCLUSIONS",
+        "text_level": 1,
+        "bbox": [
+            648,
+            694,
+            779,
+            707
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have studied the possibility of implementing FL over a swarm of UAVs. In particular, we have carried out a convergence analysis to study the impact of wireless factors, such as transmission delay and antenna angle deviations, on the convergence of FL. Using the derived insight, we have jointly designed the power allocation and scheduling policy for the UAV swarm to optimize the convergence performance of FL while guaranteeing the stability of control system and controlling the energy consumption. Simulation results have corroborated the convergence analysis of FL and showed the merits of the proposed joint design.",
+        "bbox": [
+            501,
+            712,
+            924,
+            878
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX",
+        "text_level": 1,
+        "bbox": [
+            243,
+            66,
+            320,
+            79
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A. Proof of Theorem 1",
+        "text_level": 1,
+        "bbox": [
+            70,
+            85,
+            230,
+            99
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "According to the assumptions about function  $F(\\pmb{w}) : \\mathbb{R}^n \\to \\mathbb{R}$  made in Section III, we know that function  $F(\\pmb{w})$  is continuously differentiable, and the gradient of  $F(\\pmb{w})$  is uniformly Lipschitz continuous, i.e., for some positive parameter  $U$ ,  $||\\nabla F(\\pmb{w}^{(t + 1)}) - \\nabla F(\\pmb{w}^{(t)})|| \\leq U||\\pmb{w}^{(t + 1)} - \\pmb{w}^{(t)}||$ ; the function  $F$  is strongly convex with positive parameter  $\\mu$ :  $F(\\pmb{w}^{(t + 1)}) \\geq F(\\pmb{w}^{(t)}) + (\\pmb{w}^{(t + 1)} - \\pmb{w}^{(t)})^T\\nabla F(\\pmb{w}^{(t)}) + \\frac{1}{2}\\mu ||\\pmb{w}^{(t + 1)} - \\pmb{w}^{(t)}||$ . If  $F$  is twice-continuously differentiable, these two assumptions are equivalent to  $\\mu I \\leq \\nabla^2 F(\\pmb{w}) \\leq UI$ . Also, following a standard assumption in stochastic optimization, we consider that there exists constants  $\\zeta_1 \\geq 0$  and  $\\zeta_2 \\geq 1$ , meeting  $||\\nabla F_i(\\pmb{w})||^2 \\leq \\zeta_1 + \\zeta_2||\\nabla F(\\pmb{w})||^2$  [14].",
+        "bbox": [
+            70,
+            104,
+            491,
+            285
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this case, since the global FL model is the aggregation of all local FL models, the global FL model without the impact of the transmission delay can be given as",
+        "bbox": [
+            70,
+            286,
+            491,
+            330
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)}}{\\sum_ {i = 1} ^ {I} N _ {i}} = \\boldsymbol {w} ^ {(t - 1)} - \\lambda \\nabla F (\\boldsymbol {w} ^ {(t - 1)}). \\tag {26}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            99,
+            329,
+            488,
+            366
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "After taking into account the impact of transmission delays, we can rewrite the global FL model update as",
+        "bbox": [
+            70,
+            364,
+            488,
+            393
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)} C _ {i , t}}{\\sum_ {i = 1} ^ {I} N _ {i} C _ {i , t}} = \\boldsymbol {w} ^ {(t - 1)} - \\lambda (\\nabla F (\\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)}),\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            71,
+            391,
+            488,
+            430
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where  $e^{(t)} = -\\nabla F(\\pmb{w}^{(t-1)}) + \\frac{\\sum_{i=1}^{I} N_i \\nabla F_i(\\pmb{w}^{(t-1)}) C_{i,t}}{\\sum_{i=1}^{I} N_i C_{i,t}}$ . Based on the assumption on the uniform Lipschitz continuity and strong convexity, we can have the following inequalities:",
+        "bbox": [
+            70,
+            429,
+            488,
+            478
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} F (\\boldsymbol {w} ^ {(t)}) \\leq F (\\boldsymbol {w} ^ {(t - 1)}) + (\\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)}) ^ {T} \\nabla \\tilde {F} (\\boldsymbol {w} ^ {(t - 1)}) \\\\ + \\frac {U}{2} \\| \\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)} \\| ^ {2}, \\tag {27} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            91,
+            478,
+            488,
+            526
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} F (\\boldsymbol {w} ^ {(t)}) \\geq F (\\boldsymbol {w} ^ {(t - 1)}) + (\\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)}) ^ {T} \\nabla F (\\boldsymbol {w} ^ {(t - 1)}) \\\\ + \\frac {\\mu}{2} \\left\\| \\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)} \\right\\| ^ {2}. \\tag {28} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            91,
+            527,
+            488,
+            571
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Since  $\\pmb{w}^{(t)} = \\overline{\\pmb{w}}^{(t - 1)} - \\lambda (\\nabla F(\\pmb{w}^{(t - 1)}) + e^{(t)})$ , we can simplify (27) when the learning rate is  $\\lambda = \\frac{1}{U}$  as",
+        "bbox": [
+            70,
+            571,
+            488,
+            602
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} F (\\boldsymbol {w} ^ {(t)}) \\leq F (\\boldsymbol {w} ^ {(t - 1)}) - \\frac {1}{U} \\left(\\nabla F (\\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)}\\right) ^ {T} \\nabla F (\\boldsymbol {w} ^ {(t - 1)}) \\\\ + \\frac {1}{2 U} | | \\nabla F (\\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)} | | ^ {2} \\\\ = F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - \\frac {1}{2 U} \\left\\| \\nabla F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) \\right\\| ^ {2} + \\frac {1}{2 U} \\left\\| e ^ {(t)} \\right\\| ^ {2}. \\tag {29} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            71,
+            602,
+            488,
+            699
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To find a lower bound on the norm of  $\\nabla F(\\pmb{w}^{(t)})$ , we can minimize both sides of (28) with respect  $\\pmb{w}^{(t)}$ . The minimal value of the left-hand side of (28) is achieved when  $\\pmb{w}^{(t)} = \\pmb{w}^*$ , and the minimal value of the right-hand side of (28) is realized when  $\\pmb{w}^{(t)} = \\pmb{w}^{(t - 1)} - \\frac{1}{\\mu}\\nabla F(\\pmb{w}^{(t - 1)})$ . Particularly, we have",
+        "bbox": [
+            70,
+            700,
+            488,
+            777
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF \\left(\\boldsymbol {w} ^ {*}\\right) \\geq F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - \\frac {1}{2 \\mu} | | \\nabla F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) | | ^ {2}. \\tag {30}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            130,
+            777,
+            488,
+            806
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "When replacing  $\\pmb{w}^{(t - 1)}$  with  $\\pmb{w}^{(t)}$  in (30), we can obtain a lower bound for the norm of  $\\nabla F(\\pmb{w}^{(t)})$  as",
+        "bbox": [
+            70,
+            805,
+            488,
+            835
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\left| \\left| \\nabla F \\left(\\boldsymbol {w} ^ {(t)}\\right) \\right| \\right| ^ {2} \\geq 2 \\mu \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right). \\tag {31}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            143,
+            835,
+            488,
+            852
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Combining (29) and (31), we can obtain an upper bound of the current loss and the minimal loss given by",
+        "bbox": [
+            70,
+            851,
+            491,
+            880
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nF \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right) \\leq \\left(1 - \\frac {\\mu}{U}\\right) \\left[ F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right) \\right] + \\frac {1}{2 U} \\left\\| e ^ {(t)} \\right\\| ^ {2}.\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            71,
+            880,
+            491,
+            906
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "According to [19], when  $\\mathbb{E}[||e^{(t)}||^2] \\leq 2U\\left(\\frac{\\mu}{U} - \\rho^{(t)}\\right)\\mathbb{E}(F(\\boldsymbol{w}^{(t)}) - F(\\boldsymbol{w}^*))$ , we can achieve the strong expected linear convergence, i.e.,",
+        "bbox": [
+            503,
+            80,
+            924,
+            126
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\overline {{\\mathbb {E}}} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\leq (1 - \\rho^ {(t)}) \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            506,
+            125,
+            916,
+            143
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "According to the strong expected linear convergence requirement, we know that the convergence rate satisfies",
+        "bbox": [
+            503,
+            143,
+            921,
+            170
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\rho^ {(t)} \\leq \\frac {\\mu}{U} - \\frac {\\mathbb {E} [ | | e ^ {(t)} | | ^ {2} ]}{2 U \\mathbb {E} (F (\\boldsymbol {w} ^ {(t)}) - F (\\boldsymbol {w} ^ {*}))}. \\tag {32}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            581,
+            170,
+            923,
+            203
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By using the results in [20], we have the following inequality:",
+        "bbox": [
+            504,
+            200,
+            923,
+            215
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left(\\left\\| e ^ {(t)} \\right\\| ^ {2}\\right) \\leq \\frac {1}{N} \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} \\mathbb {E} \\left(\\nabla F \\left(\\boldsymbol {w} ^ {(t)}\\right)\\right)\\right) \\times \\\\ (1 - \\mathbb {P} (T _ {i L} \\leq T _ {u} (\\beta), T _ {L i} \\leq T _ {d} (\\beta))). \\tag {33} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            537,
+            214,
+            923,
+            271
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The right-hand side of (32) will meet the following inequality:",
+        "bbox": [
+            504,
+            272,
+            923,
+            286
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\frac {\\mu}{U} - \\frac {\\mathbb {E} [ | | e ^ {(t)} | | ^ {2} ]}{2 U \\mathbb {E} (F (\\boldsymbol {w} ^ {(t)}) - F (\\boldsymbol {w} ^ {*}))} \\geq \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} \\mathbb {E} \\left(\\nabla F (\\boldsymbol {w} ^ {(t)})\\right)\\right) \\\\ \\times \\frac {\\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{2 N U \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)}. \\tag {34} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            506,
+            285,
+            923,
+            358
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Therefore, to guarantee that (32) always exists, we have",
+        "bbox": [
+            504,
+            356,
+            885,
+            369
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\rho^ {(t)} \\leq \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} \\mathbb {E} \\left(\\nabla F \\left(\\boldsymbol {w} ^ {(t)}\\right)\\right)\\right) \\\\ \\times \\frac {(1 - \\mathbb {P} (T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)))}{2 N U \\mathbb {E} (F (\\boldsymbol {w} ^ {(t)}) - F (\\boldsymbol {w} ^ {*}))} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            517,
+            369,
+            828,
+            444
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\stackrel {(a)} {\\leq} \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} 2 \\mu \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)\\right) \\\\ \\times \\frac {\\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{2 N U \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            524,
+            445,
+            825,
+            520
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\stackrel {(b)} {\\leq} \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(2 \\mu \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)\\right) \\\\ \\times \\frac {\\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{2 N U \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            524,
+            522,
+            828,
+            597
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n= \\frac {\\mu}{U} - \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\mu \\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{N U}, \\tag {35}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            522,
+            598,
+            923,
+            630
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where in (a), we use the results derived in (31), and the derivation in (b) is based on the fact that  $\\zeta_1 \\geq 0$  and  $\\zeta_2 \\geq 1$ . Assume  $\\rho = \\frac{\\mu}{U} - \\frac{\\sum_{i=1}^{I} N_i \\mu(1 - \\mathbb{P}(T_{iL} \\leq T_u(\\beta), T_{Li} \\leq T_d(\\beta)))}{NU} = \\frac{\\sum_{i=1}^{I} N_i \\mu(\\mathbb{P}(T_{iL} \\leq T_u(\\beta), T_{Li} \\leq T_d(\\beta)))}{NU}$ , then, we can have",
+        "bbox": [
+            504,
+            627,
+            923,
+            693
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\leq (1 - \\rho) \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\\\ \\leq (1 - \\rho) ^ {2} \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t - 2)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            519,
+            691,
+            908,
+            729
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "··",
+        "bbox": [
+            666,
+            739,
+            681,
+            746
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\leq (1 - \\rho) ^ {t} \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(0)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right).\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            669,
+            751,
+            895,
+            768
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We can further determine the convergence round needed to achieve a target difference threshold, i.e.,  $\\mathbb{E}(F(\\boldsymbol{w}) - F(\\boldsymbol{w}^{*}))\\leq \\varepsilon$ , as follows:",
+        "bbox": [
+            503,
+            768,
+            923,
+            811
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{l} t \\geq \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\mathbb {E} (F (\\boldsymbol {w} ^ {(0)}) - F (\\boldsymbol {w} ^ {*}))} \\\\ \\stackrel {(a)} {\\geq} \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\mathbb {E} (F (\\boldsymbol {w} ^ {(0)}))} \\\\ = \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\sum_ {i = 1} ^ {I} \\sum_ {n = 1} ^ {N _ {i}} f \\left(\\boldsymbol {w} ^ {(0)}, \\boldsymbol {x} _ {i n} , y _ {i n}\\right)}, \\tag {36} \\\\ \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            568,
+            810,
+            923,
+            906
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where in (a), we use the fact that  $1 - \\rho \\leq 1$ . Since the convergence round must be integral, we can have the results in Theorem 1.",
+        "bbox": [
+            70,
+            66,
+            491,
+            111
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES",
+        "text_level": 1,
+        "bbox": [
+            233,
+            119,
+            330,
+            133
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "[1] M. Mozaffari, W. Saad, M. Dennis, Y. Nam, and M. Debbah, \"A tutorial on UAVs for wireless networks: Applications, challenges, and open problems,\" IEEE Communications Surveys Tutorials, vol. 21, no. 3, pp. 2334-2360, Third Quarter, 2019.",
+            "[2] M. Mozaffari, A. Kasgari, W. Saad, M. Bennis, and M. Debbah, \"Beyond 5G with UAVs: Foundations of a 3D wireless cellular network,\" IEEE Transactions on Wireless Communications, vol. 18, no. 1, pp. 357-372, Jan 2019.",
+            "[3] J. Park, S. Samarakoon, M. Bennis, and M. Debbah, \"Wireless network intelligence at the edge,\" Proceeding of the IEEE, to appear, 2019.",
+            "[4] J. Konecný, H. B. McMahan, D. Ramage, and P. Richtárik, \"Federated optimization: Distributed machine learning for on-device intelligence,\" arXiv preprint http://arxiv.org/abs/1610.02527 arXiv:1610.02527, 2016.",
+            "[5] M. Chen, H. V. Poor, W. Saad, and S. Cui, \"Convergence time optimization for federated learning over wireless networks,\" arXiv preprint http://arxiv.org/abs/2001.07845 arXiv:2001.07845.",
+            "[6] Q. Zeng, Y. Du, K. K. Leung, and K. Huang, \"Energy-efficient radio resource allocation for federated edge learning,\" arXiv preprint http://arxiv.org/abs/1907.06040 arXiv:1907.06040, 2019.",
+            "[7] H. H. Yang, Z. Liu, T. Q. S. Quek, and H. V. Poor, \"Scheduling policies for federated learning in wireless networks,\" IEEE Transactions on Communications, to appear, 2019.",
+            "[8] N. H. Tran, W. Bao, A. Zomaya, N. Minh N.H., and C. S. Hong, \"Federated learning over wireless networks: Optimization model design and analysis,\" in Proc. of IEEE International Conference on Computer Communications, Paris, France, Apr. 2019.",
+            "[9] Y. Pan, C. Pan, Z. Yang, and M. Chen, \"Resource allocation for D2D communications underlaying a NOMA-based cellular network,\" IEEE Wireless Communications Letters, vol. 7, no. 1, pp. 130-133, Feb. 2018.",
+            "[10] M. T. Dabiri, H. Safi, S. Parsaeefard, and W. Saad, \"Analytical channel models for millimeter wave UAV networks under hovering fluctuations,\" arXiv preprint http://arxiv.org/abs/1905.01477 arXiv:1905.01477, 2019.",
+            "[11] ITU-R, Mathematical models for radiodetermination radar systems antenna patterns for use in interference analyses, Recommendation ITU-R M.1851-1, Jan. 2018.",
+            "[12] T. Zeng, M. Mozaffari, O. Semiari, W. Saad, M. Bennis, and M. Debbah, \"Wireless communications and control for swarms of cellular-connected UAVs,\" in Proc. of IEEE Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, Oct. 2018.",
+            "[13] J. K. Stolaroff, C. Samaras, E. R. O'Neill, A. S. Mitchell, and D. Ceperley, \"Energy use and life cycle greenhouse gas emissions of drones for commercial package delivery,\" Nature Communications, vol. 9, no. 409, pp. 1-13, Feb. 2018.",
+            "[14] D. P. Bertsekas and J. N. Tsitsiklis, Neuro-dynamic programming. Athena Scientific Belmont, MA, 1996, vol. 5.",
+            "[15] B. K. Pagnoncelli, S. Ahmed, and A. Shapiro, \"Sample average approximation method for chance constrained programming: theory and applications,\" Journal of optimization theory and applications, vol. 142, no. 2, pp. 399-416, 2009.",
+            "[16] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Transactions on Communications, vol. 54, no. 7, pp. 1310–1322, July 2006.",
+            "[17] Y. Nesterov, Lectures on convex optimization. Springer, 2018, vol. 137.",
+            "[18] F. Zhou, Y. Wu, R. Q. Hu, and Y. Qian, \"Computation rate maximization in UAV-enabled wireless-powered mobile-edge computing systems,\" IEEE Journal on Selected Areas in Communications, vol. 36, no. 9, pp. 1927-1941, Sep. 2018.",
+            "[19] M. P. Friedlander and M. Schmidt, “Hybrid deterministic-stochastic methods for data fitting,” SIAM Journal on Scientific Computing, vol. 34, no. 3, pp. A1380-A1405, 2012.",
+            "[20] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, \"A joint learning and communications framework for federated learning over wireless networks,\" arXiv preprint http://arxiv.org/abs/1909.07972 arXiv:1909.07972, 2019."
+        ],
+        "bbox": [
+            73,
+            141,
+            491,
+            866
+        ],
+        "page_idx": 7
+    }
+]

2002.08xxx/2002.08196/36a27757-1c49-4da1-b5a3-ec1408b5d759_model.json

@@ -0,0 +1,2031 @@
+[
+    [
+        {
+            "type": "aside_text",
+            "bbox": [
+                0.023,
+                0.268,
+                0.058,
+                0.708
+            ],
+            "angle": 270,
+            "content": "arXiv:2002.08196v2 [cs.LG] 10 Jun 2020"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.128,
+                0.068,
+                0.87,
+                0.138
+            ],
+            "angle": 0,
+            "content": "Federated Learning in the Sky: Joint Power Allocation and Scheduling with UAV Swarms"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.116,
+                0.156,
+                0.882,
+                0.173
+            ],
+            "angle": 0,
+            "content": "Tengchan Zeng, Omid Semiari, Mohammad Mozaffari, Mingzhe Chen, Walid Saad, and Mehdi Bennis"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.204,
+                0.493,
+                0.52
+            ],
+            "angle": 0,
+            "content": "Abstract—Unmanned aerial vehicle (UAV) swarms must exploit machine learning (ML) in order to execute various tasks ranging from coordinated trajectory planning to cooperative target recognition. However, due to the lack of continuous connections between the UAV swarm and ground base stations (BSs), using centralized ML will be challenging, particularly when dealing with a large volume of data. In this paper, a novel framework is proposed to implement distributed federated learning (FL) algorithms within a UAV swarm that consists of a leading UAV and several following UAVs. Each following UAV trains a local FL model based on its collected data and then sends this trained local model to the leading UAV who will aggregate the received models, generate a global FL model, and transmit it to followers over the intra-swarm network. To identify how wireless factors, like fading, transmission delay, and UAV antenna angle deviations resulting from wind and mechanical vibrations, impact the performance of FL, a rigorous convergence analysis for FL is performed. Then, a joint power allocation and scheduling design is proposed to optimize the convergence rate of FL while taking into account the energy consumption during convergence and the delay requirement imposed by the swarm's control system. Simulation results validate the effectiveness of the FL convergence analysis and show that the joint design strategy can reduce the number of communication rounds needed for convergence by as much as \\(35\\%\\) compared with the baseline design."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.215,
+                0.529,
+                0.348,
+                0.542
+            ],
+            "angle": 0,
+            "content": "I. INTRODUCTION"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.547,
+                0.493,
+                0.7
+            ],
+            "angle": 0,
+            "content": "Swarms of unmanned aerial vehicles (UAVs) will play an important role in various services ranging from delivery of goods to monitoring [1] and [2]. To deliver those services, UAV swarms will employ machine learning (ML) for executing various tasks such as consensus trajectory planning, target recognition, and localization. However, due to the high altitude and mobility of UAVs, continuous connections between UAVs and ground base stations (BSs) cannot be guaranteed. Hence, using centralized ML approaches to execute learning-related tasks will be challenging, particularly when transmitting a large"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.709,
+                0.493,
+                0.766
+            ],
+            "angle": 0,
+            "content": "This research was supported, in part, by the U.S. National Science Foundation under Grants CNS-1739642 and CNS-1941348, and by the Academy of Finland Project CARMA, by the Academy of Finland Project MISSION, by the Academy of Finland Project SMARTER, as well as by the INFOTECH Project NOOR."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.766,
+                0.493,
+                0.8
+            ],
+            "angle": 0,
+            "content": "T. Zeng and W. Saad are with Wireless@VT, Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, 24061 USA (e-mail: tengchan@vt.edu; walids@vt.edu)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.8,
+                0.493,
+                0.834
+            ],
+            "angle": 0,
+            "content": "O. Semiari is with Department of Electrical and Computer Engineering, University of Colorado Colorado Springs, Colorado Springs, CO, 80918 USA (e-mail: osemiari@uccs.edu)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.834,
+                0.493,
+                0.857
+            ],
+            "angle": 0,
+            "content": "M. Mozaffari is with Ericsson Research, Santa Clara, CA, 95054 USA (e-mail: mohammad.mozaffari@ericsson.com)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.857,
+                0.493,
+                0.879
+            ],
+            "angle": 0,
+            "content": "M. Chen is with Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544 USA (e-mail: mingzhec@princeton.edu)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.879,
+                0.493,
+                0.902
+            ],
+            "angle": 0,
+            "content": "M. Dennis is with the Centre for Wireless Communications, University of Oulu, 90014 Oulu, Finland (e-mail:mehdi.bennis@oulu.fi)."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.07,
+                0.766,
+                0.493,
+                0.902
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.203,
+                0.925,
+                0.293
+            ],
+            "angle": 0,
+            "content": "volume of data over aerial links. Instead, a distributed learning approach would be more apropos [3]. In particular, one can use federated learning (FL) to enable each UAV to perform distributed ML tasks without relying on any centralized BSs [4]. In this case, UAVs do not need to send any raw data to BSs when training learning models."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.293,
+                0.927,
+                0.504
+            ],
+            "angle": 0,
+            "content": "In essence, FL allows each UAV in a swarm to train its learning model based on its own collected data, and it can use the intra-swarm network to share FL parameters related to the learned models with other UAVs. As the learning process proceeds, UAVs in the swarm can reach a consensus on their collective learning tasks, e.g., trajectory planning or target recognition. However, since the updates of the learning models in FL are transmitted over a wireless network, the FL convergence and task consensus for the UAV swarm will inevitably be affected by wireless factors such as transmission delay. Also, due to the high mobility of UAVs, other factors (like wind and mechanical vibrations) can increase the uncertainty of wireless channels by affecting the UAVs' antenna angles which, in turn, will impact the FL convergence."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.505,
+                0.927,
+                0.7
+            ],
+            "angle": 0,
+            "content": "A number of recent works have investigated how wireless communication impacts FL [5]–[7]. For instance, in [5], the authors solve the joint learning, wireless resource allocation, and user selection problem to minimize the FL convergence time while optimizing the FL performance. Also, the work in [6] proposes a strategy for bandwidth allocation and device scheduling to improve the energy efficiency for networks implementing FL. Moreover, [7] studies the impact of different scheduling policies on the performance of FL. While interesting, none of these works in [5]–[7] considers the role of FL in a UAV swarm. Also, due to the high mobility of UAVs and their limited energy, the analysis in [5]–[7] cannot be directly applied for UAV swarms."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.701,
+                0.927,
+                0.897
+            ],
+            "angle": 0,
+            "content": "The main contribution of this paper is a novel framework for enabling FL within a swarm of wireless-connected UAVs. In particular, we first conduct a convergence analysis for FL to show how wireless factors within the UAV swarm impact the convergence of FL. We then determine the convergence round, defined as the minimum number of communication rounds needed to achieve FL convergence. Using this key insight, we formulate an optimization problem that jointly designs the power allocation and scheduling for the UAV swarm network to reduce the FL convergence round. In particular, due to the stringent energy limitations of UAVs, we consider the constraint of the energy consumed by learning, communications, and flying during FL convergence. We also take into account"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.067,
+                0.493,
+                0.248
+            ],
+            "angle": 0,
+            "content": "the delay constraint imposed by the control system to guarantee the stability of the UAV swarm. To solve the joint design problem, we use a sample average approximation approach from stochastic programming along with a dual method from convex optimization. To the best of our knowledge, this is the first work that implements FL for the UAV swarm, studies the impact of wireless factors on the convergence of FL, and optimizes the FL convergence by jointly designing power allocation and scheduling of the UAV network. Simulation results validate the convergence analysis of FL and show that the joint design can reduce the convergence round by as much as \\(35\\%\\) compared with baselines without the joint design."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.248,
+                0.492,
+                0.324
+            ],
+            "angle": 0,
+            "content": "The rest of the paper is organized as follows. Section II presents the system model for the UAV swarm. Section III analyzes the FL convergence and shows the joint system design. Section IV provides simulation results, and conclusions are drawn in Section V."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.208,
+                0.332,
+                0.356,
+                0.345
+            ],
+            "angle": 0,
+            "content": "II. SYSTEM MODEL"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.351,
+                0.493,
+                0.593
+            ],
+            "angle": 0,
+            "content": "Consider a swarm of wirelessly connected autonomous UAVs flying at the same altitude, as shown in Fig. 1(a). The UAV swarm consists of a leader \\( L \\) and a set \\( \\mathcal{I} \\) of \\( I \\) followers. Every follower keeps a target distance and speed with the leader. While flying, the UAV swarm collects data and performs FL for data analysis and inference tasks like trajectory planning and cooperative target recognition. Using FL, each follower uses its collected data to train a local FL model and send the parameters related to the learned model to the leading UAV in the uplink, as shown in Fig. 1(a). The leading UAV will integrate all received information to generate a global FL model, and, then, transmit the parameters of the global model to following UAVs over the downlink. Moreover, to guarantee that the followers fly with the same speed while keeping a safe distance, the leading UAV will also broadcast the target spacing information and its speed and heading direction."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.071,
+                0.601,
+                0.273,
+                0.617
+            ],
+            "angle": 0,
+            "content": "A. Federated learning model"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.62,
+                0.491,
+                0.801
+            ],
+            "angle": 0,
+            "content": "In the learning model, we assume that UAV \\(i \\in \\mathcal{I}\\) collects a set \\(\\{\\pmb{x}_{i1},\\pmb{x}_{i2},\\dots,\\pmb{x}_{iN_i}\\}\\) of input data where each collected sample is represented by a vector \\(\\pmb{x}_{in}\\), \\(n \\in \\{1,\\dots,N_i\\}\\) that captures the input features and \\(N_{i}\\) is the number of collected samples. We also assume the input sample \\(\\pmb{x}_{in}\\), \\(n \\in \\{1,\\dots,N_i\\}\\), corresponds to a single output \\(y_{in}\\) [4]. The output vector is thereby \\(\\{y_{i1},\\dots,y_{iN_i}\\}\\) for UAV \\(i\\). We define a vector \\(\\pmb{w}_i\\) as the parameters related to the local FL model that is trained by \\(\\{\\pmb{x}_{i1},\\pmb{x}_{i2},\\dots,\\pmb{x}_{iN_i}\\}\\) and \\(\\{y_{i1},\\dots,y_{iN_i}\\}\\) at UAV \\(i\\). The convergence of the FL training processes requires each local learning vector to converge to a vector \\(\\pmb{w}^*\\) which solves the following problem:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.131,
+                0.8,
+                0.49,
+                0.84
+            ],
+            "angle": 0,
+            "content": "\\[\n\\underset {\\boldsymbol {w} \\in \\mathbb {R} ^ {d}} {\\arg \\min } F (\\boldsymbol {w}) = \\frac {1}{N} \\sum_ {i} ^ {I} \\sum_ {n = 1} ^ {N _ {i}} f (\\boldsymbol {w}, \\boldsymbol {x} _ {i n}, y _ {i n}), \\tag {1}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.84,
+                0.492,
+                0.902
+            ],
+            "angle": 0,
+            "content": "where \\(N = \\sum_{i}^{I}N_{i}\\) is the total number of the collected samples by all followers, and \\(f(\\pmb {w},\\pmb{x}_{in},y_{in})\\) captures the loss function when using learning vector \\(\\pmb{w}\\) for dataset \\(\\{\\pmb {x}_{in},y_{in}\\}\\). Note that, the loss function \\(f(\\pmb {w},\\pmb{x}_{in},y_{in}),i\\in \\mathcal{I},0\\leq n\\leq N_i,\\)"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.548,
+                0.068,
+                0.872,
+                0.219
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.605,
+                0.223,
+                0.826,
+                0.235
+            ],
+            "angle": 0,
+            "content": "(a) Communication and learning models."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.544,
+                0.248,
+                0.88,
+                0.318
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.605,
+                0.32,
+                0.825,
+                0.332
+            ],
+            "angle": 0,
+            "content": "(b) Angle deviations and control system."
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.59,
+                0.339,
+                0.838,
+                0.353
+            ],
+            "angle": 0,
+            "content": "Fig. 1. Illustration of our system model."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.371,
+                0.925,
+                0.432
+            ],
+            "angle": 0,
+            "content": "plays a pivotal role in determining the FL performance, and the expression of the loss function is application-specific. For example, for a simple linear regression FL algorithm, \\( f(\\boldsymbol{w}, \\boldsymbol{x}_{in}, y_{in}) = (\\boldsymbol{w}^T \\boldsymbol{x}_{in} - y_{in})^2 \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.432,
+                0.927,
+                0.749
+            ],
+            "angle": 0,
+            "content": "To solve (1), the FL framework uses an iterative update scheme [4]. In particular, the leading UAV will first generate an initial global FL model represented by vector \\(\\boldsymbol{w}^{(0)}\\) and send the initial vector to all followers. Hence, in the first communication round, follower \\(i\\in \\mathcal{I}\\) will first use \\(\\boldsymbol{w}^{(0)}\\) for its own data to train the local model and, then, it sends the vector of the trained model to the leader. Next, the leading UAV will aggregate all received local FL vectors and update the global FL model vector which will be later transmitted to the followers. Each communication round will be followed by another round, and the same process will repeat among leader and followers in each round. In this case, as FL proceeds, the local and global models are sequentially updated, and the total loss \\(F(\\boldsymbol {w})\\) for the updated global model with vector \\(\\boldsymbol{w}\\) will continuously decrease [4]. To identify whether the optimal solution is found for (1), one must analyze the convergence of the loss function \\(F(\\boldsymbol {w})\\) to \\(F(\\boldsymbol {w}^{*})\\). That is, when the gap between the current loss \\(F(\\boldsymbol {w})\\) and the minimal loss \\(F(\\boldsymbol {w}^{*})\\) is below a threshold \\(\\varepsilon\\), the FL optimization problem is solved [8]. Therefore, we can use the convergence of \\(F(\\boldsymbol {w})\\) to \\(F(\\boldsymbol {w}^{*})\\) to quantify the FL performance."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.749,
+                0.925,
+                0.901
+            ],
+            "angle": 0,
+            "content": "Moreover, for each communication round, we can divide the total time duration \\( T_{r} \\) into two periods: Uplink and downlink transmission. In particular, to guarantee that the leading UAV has enough time to process all received models from its followers, all uplink transmissions should be completed within a target time \\( T_{u}(\\beta) = \\beta T_{r} \\), where \\( \\beta \\in \\{0,1\\} \\) is a scheduling parameter to schedule uplink-downlink traffic in time. Also, to receive the global FL model update from the leading UAV successfully, the time constraint for downlink transmissions is thereby \\( T_{d}(\\beta) = (1 - \\beta)T_{r} \\). In this case, if the communication"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.066,
+                0.493,
+                0.188
+            ],
+            "angle": 0,
+            "content": "link between follower \\( i \\in \\mathcal{I} \\) and leader \\( L \\) fails to meet the time constraints \\( T_{d}(\\beta) \\) and \\( T_{u}(\\beta) \\), the global FL model cannot use the corresponding FL model for the aggregation. At the same time, for the local FL model, the following UAV cannot use the recently updated global vector to train its local data. In other words, the transmission delay of the uplink and downlink links will impact the update of the global and local FL models thus having a major impact on FL convergence."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.188,
+                0.493,
+                0.326
+            ],
+            "angle": 0,
+            "content": "In addition, when training the global FL model, we can calculate the energy consumption for the UAV \\( L \\) as \\( E_{L} = \\kappa C\\phi^{2}\\sum_{i=1}^{I}S(\\pmb{w}_{i}) \\), where \\( \\kappa \\) captures the energy consumption coefficient depending on the computing system and \\( C \\) is the number of computing cycles needed per data bit [9]. \\( \\phi \\) is the frequency of the CPU clock of UAVs, and \\( S(\\pmb{w}_{i}) \\) is the packet size of \\( \\pmb{w}_{i} \\), transmitted from UAV \\( i \\in \\mathcal{I} \\), in bits. Similarly, we can determine the training energy consumption for follower \\( i \\in \\mathcal{I} \\) as \\( E_{i} = \\kappa C\\phi^{2}\\sum_{n=1}^{N_{i}}S(\\pmb{x}_{in}) \\)."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.072,
+                0.332,
+                0.251,
+                0.346
+            ],
+            "angle": 0,
+            "content": "B. Communication model"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.07,
+                0.351,
+                0.491,
+                0.516
+            ],
+            "angle": 0,
+            "content": "To minimize the interference from other UAVs located outside of the swarm, we assume that all UAVs use directional antennas, as shown in Fig. 1(a), However, as shown in Fig. 1(b), due to the impact of wind, payload, and non-ideal mechanical and control systems, the angle of the UAVs will randomly fluctuate and deviate from the initial angle setting. Based on the central limit theorem, we model the angle deviation for each UAV as a Gaussian random variable [10]. Moreover, we consider a squared cosine function to capture the antenna aperture of UAV \\( j \\in \\mathcal{I} \\cup \\{L\\} \\) when communicating with UAV \\( l \\in \\mathcal{I} \\cup \\{L\\} / j \\) as follows [11]:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.076,
+                0.516,
+                0.49,
+                0.558
+            ],
+            "angle": 0,
+            "content": "\\[\nG _ {j l} \\left(\\theta_ {j l} + \\vartheta_ {j}\\right) = \\left\\{ \\begin{array}{c c} \\cos^ {2} \\left(\\frac {\\pi}{2} \\left(\\theta_ {j l} + \\vartheta_ {j}\\right)\\right), & \\text {i f} | \\theta_ {j l} + \\vartheta_ {j} | \\leq 1, \\\\ G _ {\\min }, & \\text {o t h e r w i s e}, \\end{array} \\right. \\tag {2}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.559,
+                0.493,
+                0.633
+            ],
+            "angle": 0,
+            "content": "where \\(\\theta_{jl}\\) is the initial angle setting for UAV \\(j\\) when communicating with UAV \\(l\\), \\(\\vartheta_{j} \\sim \\mathcal{N}(0, \\sigma_{j}^{2})\\) is the angle deviation with variance \\(\\sigma_{j}^{2}\\), and \\(G_{\\mathrm{min}}\\) captures the antenna gain at the side lobes. Also, similar to [10], we can approximate (2) by using a sectionalized expression:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.078,
+                0.632,
+                0.49,
+                0.675
+            ],
+            "angle": 0,
+            "content": "\\[\nG _ {j l} \\left(\\theta_ {j l} + \\vartheta_ {j}, M\\right) = \\left\\{ \\begin{array}{c c} \\cos^ {2} \\left(\\frac {\\pi m}{2 M}\\right), & \\text {i f} \\frac {m}{M} \\leq \\left| \\theta_ {j l} + \\vartheta_ {j} \\right| \\leq \\frac {m + 1}{M}, \\\\ G _ {\\min }, & \\text {o t h e r w i s e}, \\end{array} \\right. \\tag {3}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.676,
+                0.231,
+                0.691
+            ],
+            "angle": 0,
+            "content": "where \\(m\\in \\{1,\\dots,M\\}\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.692,
+                0.492,
+                0.749
+            ],
+            "angle": 0,
+            "content": "To reduce the interference over the uplink transmissions, we assume that uplinks do not share the wireless resource with each other. Hence, the transmission delay of the uplink between follower \\( i \\in \\mathcal{I} \\) and leader \\( L \\) can be calculated as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.085,
+                0.749,
+                0.49,
+                0.789
+            ],
+            "angle": 0,
+            "content": "\\[\nT _ {i L} = \\frac {S (\\boldsymbol {w} _ {i})}{B _ {u} \\log_ {2} \\left(1 + \\frac {p _ {i} h _ {i L} d _ {i L} ^ {- \\alpha} G _ {i L} G _ {L i}}{\\sum_ {i ^ {\\prime} \\in \\Phi_ {i}} p _ {i ^ {\\prime}} h _ {i ^ {\\prime} L} d _ {i ^ {\\prime} L} ^ {- \\alpha} G _ {i ^ {\\prime} L} G _ {L i ^ {\\prime}} + B _ {u} \\gamma_ {0}}\\right)}, \\tag {4}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.789,
+                0.493,
+                0.894
+            ],
+            "angle": 0,
+            "content": "where \\(B_{u}\\) is the bandwidth used by each subchannel in the uplink, \\(p_i\\in (0,p_{\\mathrm{max}})\\) is the transmission power of UAV \\(i\\) with maximum power as \\(p_{\\mathrm{max}}\\) and \\(\\alpha\\) is the path-loss exponent. \\(h_{iL}\\) is the channel gain of the Rician fading channel between UAVs \\(i\\) and \\(L\\) ,and \\(\\gamma_0\\) is the noise power spectral density. Note that, despite the use of directional antenna, the swarm still experiences uplink interference generated by UAVs located"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.066,
+                0.925,
+                0.127
+            ],
+            "angle": 0,
+            "content": "outside of the swarm. In particular, these interfering UAVs share the same channel resource and exist in the main lobe of the UAV \\( L \\), and we define \\( \\Phi_{i} \\) as the set of UAVs that generates interference to the uplink from UAV \\( i \\) to UAV \\( L \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.127,
+                0.925,
+                0.156
+            ],
+            "angle": 0,
+            "content": "Similarly, we can derive the transmission delay \\( T_{Li} \\) for the downlink from UAV \\( L \\) to UAV \\( i \\in \\mathcal{I} \\) as:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.523,
+                0.154,
+                0.925,
+                0.197
+            ],
+            "angle": 0,
+            "content": "\\[\nT _ {L i} = \\frac {S (\\boldsymbol {w})}{B _ {d} \\log_ {2} \\left(1 + \\frac {p _ {L} h _ {L i} d _ {L i} ^ {- \\alpha} G _ {L i} G _ {i L}}{\\sum_ {i ^ {\\prime} \\in \\Phi_ {L}} p _ {i ^ {\\prime}} h _ {i ^ {\\prime} i} d _ {i ^ {\\prime} i} ^ {- \\alpha} G _ {i ^ {\\prime} i} G _ {i i ^ {\\prime}} + B _ {d} \\gamma_ {0}}\\right)}, \\tag {5}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.195,
+                0.927,
+                0.24
+            ],
+            "angle": 0,
+            "content": "where \\(B_{d}\\) is the downlink bandwidth, \\(p_L \\in (0, p_{\\mathrm{max}})\\) is the transmission power of UAV \\(L\\), and \\(\\Phi_L\\) refers to the set of UAVs that will generate interference at the downlink."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.505,
+                0.247,
+                0.631,
+                0.26
+            ],
+            "angle": 0,
+            "content": "C. Control model"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.265,
+                0.925,
+                0.401
+            ],
+            "angle": 0,
+            "content": "To guarantee constant speed and altitude and avoid collisions between UAVs within the swarm, the leading UAV will broadcast its speed and heading direction to the followers in the downlink. Here, the control system of each follower will use both its sensor data (e.g. location) and information received from the wireless links to coordinate its movement and achieve a target spacing and speed. Note that the target distance between the UAV leader and each follower is predefined such that there will be no collision between two nearby UAVs."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.402,
+                0.927,
+                0.536
+            ],
+            "angle": 0,
+            "content": "Similar to our previous work in [12], we can build a Cartesian coordinate system to capture the locations of UAVs in the swarm, and, then, we decompose the velocity of each UAV into two components, as shown in Fig. 1(b). We can also define the control law of each UAV the same way as the one provided in [12]. Since the transmission delay will have a negative impact on the stability control of the UAV swarm, we must consider the delay requirement imposed by the control system when designing the UAV network."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.537,
+                0.927,
+                0.687
+            ],
+            "angle": 0,
+            "content": "In addition, in order to fly with a constant speed and maintain a stable flying motion, each UAV must spend energy to overcome the gravity and the air drag forces due to the wind and forward motions. For a forward speed \\( v \\in (0, v_{\\max}) \\) with \\( v_{\\max} \\) as the maximum speed, the minimum flying power of UAV \\( j \\in \\mathcal{I} \\cup \\{L\\} \\) is \\( \\bar{p}_{j,\\min}(v) = \\hat{v}_j A_j \\), where \\( \\hat{v}_j \\) is the induced velocity required for constant speed \\( v \\) and given thrust \\( A_j = mg \\) with \\( m \\) being the UAV mass and \\( g \\) being the gravitational constant [13]. Also, the induced velocity \\( \\hat{v}_j \\) can be obtained by solving the following equation [13]:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.635,
+                0.686,
+                0.925,
+                0.725
+            ],
+            "angle": 0,
+            "content": "\\[\n\\hat {v} _ {j} = \\frac {2 A _ {j}}{q r ^ {2} \\pi \\varrho \\sqrt {v ^ {2} + \\hat {v} _ {j} ^ {2}}}, \\tag {6}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.723,
+                0.927,
+                0.873
+            ],
+            "angle": 0,
+            "content": "where \\( q \\) and \\( r \\) capture, respectively, the number and diameter of the UAV rotors, and \\( \\varrho \\) is the air density. Moreover, we can further correct the theoretical minimum motion power consumption by the overall power efficiency \\( \\eta \\) of the UAV in order to obtain the actual power consumption as \\( \\bar{p}_j(v) = \\bar{p}_{j,\\min}(v) / \\eta \\). Since the control of a UAV's dynamic motion consumes the most energy [13], we must consider the flying energy consumption when designing the swarm of UAVs. In particular, the flying energy consumption can be calculated as \\( \\bar{p}_j(v)T \\) during the flying time \\( T \\)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.873,
+                0.925,
+                0.904
+            ],
+            "angle": 0,
+            "content": "To guarantee the convergence of FL and the stable operation of the control system in the UAV swarm, we need to properly"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.067,
+                0.493,
+                0.248
+            ],
+            "angle": 0,
+            "content": "design the wireless communication network. At the same time, to guarantee that the energy spent on learning, communication, and flying will not exceed the energy limitation of each UAV, we need to consider the energy consumption during the FL convergence. Next, we first conduct the convergence analysis for the FL algorithm and derive the number of communication rounds needed to achieve the FL convergence. Then, we formulate an optimization problem that jointly designs the power allocation and scheduling policy to minimize the convergence round of FL while considering the delay requirement from the control system and energy consumption during the FL convergence."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.101,
+                0.254,
+                0.462,
+                0.267
+            ],
+            "angle": 0,
+            "content": "III. CONVERGENCE ANALYSIS AND JOINT DESIGN"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.071,
+                0.272,
+                0.266,
+                0.287
+            ],
+            "angle": 0,
+            "content": "A. FL convergence analysis"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.291,
+                0.492,
+                0.366
+            ],
+            "angle": 0,
+            "content": "In order to guarantee FL convergence, we assume that the following UAVs adopt a standard gradient descent method to update their local FL models [4]. Thus, for following UAV \\( i \\in \\mathcal{I} \\), the local model \\( \\boldsymbol{w}_i^{(t)} \\) at communication round \\( t \\) is given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.163,
+                0.365,
+                0.49,
+                0.395
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {w} _ {i} ^ {(t)} = \\boldsymbol {w} ^ {(t - 1)} - \\frac {\\bar {\\lambda}}{N _ {i}} \\nabla F _ {i} (\\boldsymbol {w} ^ {(t - 1)}), \\tag {7}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.395,
+                0.492,
+                0.459
+            ],
+            "angle": 0,
+            "content": "where \\(\\pmb{w}^{(t - 1)}\\) is the global FL model at communication round \\(t - 1\\), \\(\\bar{\\lambda}\\) is the learning rate, and \\(F_{i}(\\pmb{w}^{(t - 1)}) = \\sum_{n = 1}^{N_{i}}f(\\pmb{w}^{(t - 1)},\\pmb{x}_{in},y_{in})\\). After the leading UAV collects local vectors \\(\\pmb{w}_i^{(t)}, i\\in \\mathcal{I}\\), the global FL model can be updated:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.205,
+                0.457,
+                0.49,
+                0.495
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)}}{\\sum_ {i = 1} ^ {I} N _ {i}}. \\tag {8}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.492,
+                0.492,
+                0.58
+            ],
+            "angle": 0,
+            "content": "However, for ensuring successful updates of both global and local FL models as shown in (7) and (8), the transmission delay of uplink and downlink should be within, respectively, \\( T_{u}(\\beta) \\) and \\( T_{d}(\\beta) \\). Hence, after considering the impact of the transmission delays, we can rewrite the global FL model update as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.191,
+                0.578,
+                0.49,
+                0.617
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)} C _ {i , t}}{\\sum_ {i = 1} ^ {I} N _ {i} C _ {i , t}}, \\tag {9}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.073,
+                0.614,
+                0.107,
+                0.625
+            ],
+            "angle": 0,
+            "content": "with"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.073,
+                0.623,
+                0.497,
+                0.652
+            ],
+            "angle": 0,
+            "content": "with \\(C_{i,t} = \\left\\{ \\begin{array}{ll}1, & \\mathrm{with~probability~}\\mathbb{P}(T_{iL,t}\\leq T_u(\\beta),T_{Li,t}\\leq T_d(\\beta)),\\\\ 0, & \\mathrm{otherwise.} \\end{array} \\right.\\)"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.652,
+                0.493,
+                0.848
+            ],
+            "angle": 0,
+            "content": "With the aim of quantifying the convergence of FL, we use the notion of a convergence round, defined as the minimum number of communication rounds needed to achieve a target difference \\(\\varepsilon\\) of the expected gap between current loss and the minimal loss, i.e., \\(\\mathbb{E}(F(\\boldsymbol{w}) - F(\\boldsymbol{w}^{*})) \\leq \\varepsilon\\). Moreover, to determine the convergence round, we make the following two standard assumptions: Function \\(F(\\boldsymbol{w})\\): \\(\\mathbb{R}^n \\to \\mathbb{R}\\) is continuously differentiable, and the gradient of \\(F(\\boldsymbol{w})\\) is uniformly Lipschitz continuous with positive parameter \\(U\\). We also consider the function \\(F\\) to be strongly convex with positive parameter \\(\\mu\\), and these exists constants \\(\\zeta_1 \\geq 0\\) and \\(\\zeta_2 \\geq 1\\), meeting \\(||\\nabla F_i(\\boldsymbol{w})||^2 \\leq \\zeta_1 + \\zeta_2||\\nabla F(\\boldsymbol{w})||^2\\) [14]. Given the above assumptions, we can derive the convergence round."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.855,
+                0.492,
+                0.886
+            ],
+            "angle": 0,
+            "content": "Theorem 1. To realize an expected convergence of \\( F(\\boldsymbol{w}) \\) under an accuracy threshold \\( \\varepsilon \\), i.e., \\( \\mathbb{E}(F(\\boldsymbol{w}) - F(\\boldsymbol{w}^{*})) \\leq \\varepsilon \\),"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.067,
+                0.747,
+                0.081
+            ],
+            "angle": 0,
+            "content": "the convergence round is given by:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.541,
+                0.08,
+                0.925,
+                0.121
+            ],
+            "angle": 0,
+            "content": "\\[\n\\varphi = \\left[ \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\sum_ {i = 1} ^ {I} \\sum_ {n = 1} ^ {N _ {i}} f (\\boldsymbol {w} ^ {(0)} , \\boldsymbol {x} _ {i n} , y _ {i n})} \\right], \\tag {10}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.117,
+                0.925,
+                0.146
+            ],
+            "angle": 0,
+            "content": "where \\( \\lceil \\cdot \\rceil \\) is the ceiling function, and \\( \\rho \\) captures the convergence speed given as follows"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.542,
+                0.146,
+                0.925,
+                0.178
+            ],
+            "angle": 0,
+            "content": "\\[\n\\rho = \\frac {\\sum_ {i = 1} ^ {T} N _ {i} \\mathbb {P} \\left(T _ {i L , t} \\leq T _ {u} (\\beta) , T _ {L i , t} \\leq T _ {d} (\\beta)\\right) \\mu}{N U}. \\tag {11}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.181,
+                0.925,
+                0.211
+            ],
+            "angle": 0,
+            "content": "Proof: Due to the space limitation, the proof is included in Appendix A."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.218,
+                0.927,
+                0.338
+            ],
+            "angle": 0,
+            "content": "As shown in Theorem 1, the convergence performance of FL depends on the transmission delay of both uplink and downlink in the network. In particular, to increase the convergence speed, we need to maximize the probability that both uplink and downlink meet the corresponding delay requirements of FL. Thus, Theorem 1 provides a concrete characterization of the interplay between wireless communications and FL performance in a UAV swarm."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.339,
+                0.926,
+                0.43
+            ],
+            "angle": 0,
+            "content": "For the stability analysis of the control system, we will follow the method provided by our previous work in [12]. That is, we first build the augmented error state vector. Then, we use Lyapunov-Razumikhin theorem to derive the control system delay requirements \\(\\tau_{i}, i \\in \\mathcal{I}\\), for downlink that can guarantee the stability of the UAV swarm."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.505,
+                0.438,
+                0.816,
+                0.453
+            ],
+            "angle": 0,
+            "content": "B. Problem formulation and solution concept"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.457,
+                0.925,
+                0.502
+            ],
+            "angle": 0,
+            "content": "Here, we formulate an optimization problem to minimize the convergence round by jointly designing the power allocation and scheduling for the UAV network, as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.545,
+                0.503,
+                0.925,
+                0.527
+            ],
+            "angle": 0,
+            "content": "\\[\n\\min  _ {\\left\\{\\boldsymbol {p}, p _ {L}, \\beta , v \\right\\}} \\varphi \\tag {12}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.545,
+                0.528,
+                0.925,
+                0.553
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathrm {s . t .} \\mathbb {P} \\left[ \\varphi E _ {L} + \\varphi p _ {L} T _ {d} (\\beta) + \\varphi \\bar {p} _ {L} (v) T _ {r} \\leq \\bar {E} \\right] \\geq \\xi_ {L}, \\tag {13}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.567,
+                0.554,
+                0.925,
+                0.58
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {P} \\left[ \\varphi E _ {i} + \\varphi p _ {i} T _ {i L} + \\varphi \\bar {p} _ {i} (v) T _ {r} \\leq \\bar {E} \\right] \\geq \\xi_ {i}, i \\in \\mathcal {I}, \\tag {14}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.567,
+                0.581,
+                0.925,
+                0.598
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathbb {P} \\left(T _ {L i} \\leq \\tau_ {i}\\right) \\geq \\xi_ {C}, i \\in \\mathcal {I}, \\tag {15}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.566,
+                0.6,
+                0.925,
+                0.617
+            ],
+            "angle": 0,
+            "content": "\\[\np _ {L} \\in (0, p _ {L, \\max }, p _ {i} \\in (0, p _ {i, \\max }), i \\in \\mathcal {I}, \\tag {16}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.567,
+                0.62,
+                0.925,
+                0.636
+            ],
+            "angle": 0,
+            "content": "\\[\n\\beta \\in (0, 1), v \\in (0, v _ {\\max }) \\tag {17}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.635,
+                0.927,
+                0.83
+            ],
+            "angle": 0,
+            "content": "where vector \\(\\pmb{p} = [p_1, \\dots, p_I]\\). Constraint (13) guarantees that the probability of total energy consumption for the leading UAV being less than a threshold \\(\\bar{E}\\) will be greater than \\(\\xi_L \\in (0, 1)\\). Similarly to (13), constraint (14) represents the constraint on energy consumption of each follower \\(i \\in \\mathcal{I}\\). Constraint (15) guarantees that the UAV communication network is reliable to support the stability of the swarm with probability \\(\\xi_C\\). Constraints (16) and (17) ensure that the optimization variables, i.e., the transmission power, scheduling parameter, and velocity, are chosen within reasonable ranges. Note that, in the optimization problem, we also optimize the operation speed of the UAV swarm to minimize the motion energy consumption and relax the energy constraints in (13) and (14)."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.831,
+                0.926,
+                0.861
+            ],
+            "angle": 0,
+            "content": "Since both exponent and base in the logarithm function (10) are less than 1, minimizing the logarithm function in (12) is"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.067,
+                0.49,
+                0.097
+            ],
+            "angle": 0,
+            "content": "equivalent to minimizing the base for the constant exponent. Also, according to (11), we can simplify (12) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.106,
+                0.097,
+                0.49,
+                0.134
+            ],
+            "angle": 0,
+            "content": "\\[\n\\max  _ {\\left\\{\\boldsymbol {p}, p _ {L}, \\beta , v \\right\\}} \\sum_ {i = 1} ^ {I} N _ {i} \\mathbb {P} \\left(T _ {i L, t} \\leq T _ {u} (\\beta), T _ {L i, t} \\leq T _ {d} (\\beta)\\right). \\tag {18}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.133,
+                0.491,
+                0.344
+            ],
+            "angle": 0,
+            "content": "We observe that, after simplifications, both objective function and constraints are represented by probability terms. In this case, directly deriving the probability terms will be challenging since it requires multidimensional integrations. Also, as the optimization problem is not convex, employing convex approximations to simplify the optimization problem will be impossible. Instead, we use a sample average approximation approach where the probability terms in the objective function and constraints are replaced by an empirical distribution found by random samples [15]. In particular, we first generate \\(K\\) independent samples of the random parameters, i.e., wireless channel gains and angle deviations, and we calculate the corresponding transmission delay and convergence round. Then, we can reformulate the optimization problem as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.085,
+                0.344,
+                0.49,
+                0.396
+            ],
+            "angle": 0,
+            "content": "\\[\n\\max  _ {\\left\\{\\boldsymbol {p}, p _ {L}, \\beta , v \\right\\}} \\sum_ {i = 1} ^ {I} \\sum_ {k = 1} ^ {K} N _ {i} \\mathbb {1} \\left(T _ {u} (\\beta) - T _ {i L, k}\\right) \\mathbb {1} \\left(T _ {d} (\\beta) - T _ {L i, k}\\right) \\tag {19}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.082,
+                0.4,
+                0.49,
+                0.455
+            ],
+            "angle": 0,
+            "content": "\\[\n\\text {s . t .} \\sum_ {k = 1} ^ {K} \\mathbb {1} \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {L} + \\varphi_ {k} p _ {L} T _ {d} (\\beta) + \\varphi_ {k} \\bar {p} _ {L} (v) T _ {r}\\right)\\right) \\geq K \\xi_ {L}, \\tag {20}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.099,
+                0.458,
+                0.49,
+                0.513
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {K} \\mathbb {1} \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {i} + \\varphi_ {k} p _ {i} T _ {i L, k} + \\varphi_ {k} \\bar {p} _ {i} (v) T _ {r}\\right)\\right) \\geq K \\xi_ {i}, i \\in \\mathcal {I}, \\tag {21}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.099,
+                0.516,
+                0.49,
+                0.558
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {k = 1} ^ {K} \\mathbb {1} \\left(\\tau_ {i} - T _ {L i, k}\\right) \\geq K \\xi_ {C}, i \\in \\mathcal {I}, \\tag {22}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.099,
+                0.56,
+                0.2,
+                0.575
+            ],
+            "angle": 0,
+            "content": "(16) and (17),"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.575,
+                0.491,
+                0.727
+            ],
+            "angle": 0,
+            "content": "where the indicator function \\(\\mathbb{1}(r) = 1\\), once \\(r \\geq 0\\); otherwise, we have \\(\\mathbb{1}(r) = 0\\). Due to the presence of the indicator function, the reformulated problem is non-smooth. To obtain a smooth problem, we can further replace the indicator functions with modified sigmoid functions, i.e., \\(\\Gamma(r) = \\frac{1}{1 + \\exp(-\\bar{c}r)}\\), where \\(\\bar{c}\\) determines how quickly the modified sigmoid function changes near 0. To obtain a sub-optimal solution to the reformulated optimization problem with the indicator functions replaced by the modified sigmoid functions, we can use the dual method [16]. In particular, the Lagrangian function is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.074,
+                0.733,
+                0.489,
+                0.771
+            ],
+            "angle": 0,
+            "content": "\\[\n\\mathcal {J} (\\boldsymbol {\\lambda}, \\boldsymbol {p}, p _ {L}, \\beta , v) = \\sum_ {i = 1} ^ {I} \\sum_ {k = 1} ^ {K} N _ {i} \\Gamma (T _ {u} (\\beta) - T _ {i L, k}) \\Gamma (T _ {d} (\\beta) - T _ {L i, k}) +\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.075,
+                0.773,
+                0.468,
+                0.809
+            ],
+            "angle": 0,
+            "content": "\\[\n\\lambda_ {1} \\left(\\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {L} + \\varphi_ {k} p _ {L} T _ {d} (\\beta) + \\varphi_ {k} \\bar {p} _ {L} (v) T _ {r}\\right)\\right) - K \\xi_ {L}\\right) +\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.075,
+                0.811,
+                0.488,
+                0.848
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {i = 1} ^ {I} \\lambda_ {i + 1} \\left(\\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} E _ {i} + \\varphi_ {k} p _ {i} T _ {i L, k} + \\varphi_ {k} \\bar {p} _ {i} (v) T _ {r}\\right)\\right) - K \\xi_ {i}\\right) +\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.076,
+                0.851,
+                0.49,
+                0.886
+            ],
+            "angle": 0,
+            "content": "\\[\n\\sum_ {i = 1} ^ {I} \\lambda_ {I + 1 + i} \\left(\\sum_ {k = 1} ^ {K} \\Gamma \\left(\\tau_ {i} - T _ {L i, k}\\right) - K \\xi_ {C}\\right), \\tag {23}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.074,
+                0.886,
+                0.491,
+                0.902
+            ],
+            "angle": 0,
+            "content": "where vector \\(\\pmb{\\lambda} = [\\lambda_1, \\dots, \\lambda_{2I + 1}] \\succeq \\mathbf{0}_{1 \\times (2I + 1)}\\) is the vector"
+        },
+        {
+            "type": "table_caption",
+            "bbox": [
+                0.616,
+                0.062,
+                0.812,
+                0.074
+            ],
+            "angle": 0,
+            "content": "Table. I. Simulation parameters."
+        },
+        {
+            "type": "table",
+            "bbox": [
+                0.544,
+                0.074,
+                0.892,
+                0.232
+            ],
+            "angle": 0,
+            "content": "<table><tr><td>Parameters</td><td>Values</td></tr><tr><td>Number of followers I</td><td>5</td></tr><tr><td>Transmission power threshold pmax</td><td>0.5 W</td></tr><tr><td>Maximum speed vmax</td><td>20 m/s [18]</td></tr><tr><td>Energy consumption efficient κ</td><td>10-28 [18]</td></tr><tr><td>Number of cycles needed per bit C</td><td>103 [18]</td></tr><tr><td>Frequency of the CPU φ</td><td>109cycle/s</td></tr><tr><td>Time for each communication round Tr</td><td>0.1 s</td></tr><tr><td>Side lobe gain Gmin, path loss exponent α</td><td>-2 dB, 2.5</td></tr><tr><td>Noise spectral density γ0</td><td>-174 dBm/Hz</td></tr><tr><td>Packet size Sw and Swi</td><td>10 kB</td></tr><tr><td>Number of rotors q and the diameter r</td><td>4, 0.254 m [4]</td></tr><tr><td>Power efficiency η and density ρ of the air</td><td>70 %, 1.225 kg/m3 [4]</td></tr><tr><td>Number of samples K, Energy limits E</td><td>1,000, 7,000 J</td></tr></table>"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.244,
+                0.924,
+                0.289
+            ],
+            "angle": 0,
+            "content": "of Lagrangian multipliers, and the dual objective function can be defined as \\(\\mathcal{D}(\\pmb{\\lambda}) = \\max_{\\pmb{p}, p_L, v, \\beta} \\mathcal{J}(\\pmb{\\lambda}, \\pmb{p}, p_L, \\beta, v)\\). The corresponding dual optimization problem is"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.634,
+                0.29,
+                0.924,
+                0.31
+            ],
+            "angle": 0,
+            "content": "\\[\n\\min  _ {\\boldsymbol {\\lambda}} \\mathcal {D} (\\boldsymbol {\\lambda}) \\quad \\text {s . t .} \\boldsymbol {\\lambda} \\geq \\mathbf {0}. \\tag {24}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.308,
+                0.924,
+                0.337
+            ],
+            "angle": 0,
+            "content": "Although the dual problem in (24) is always convex [17], \\(\\mathcal{D}(\\lambda)\\) is not differentiable. Instead, we can use subgradients given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.508,
+                0.337,
+                0.878,
+                0.377
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta \\lambda_ {1} = \\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} ^ {*} E _ {L} + \\varphi_ {k} ^ {*} p _ {L} T _ {d} ^ {*} + \\varphi_ {k} ^ {*} \\bar {p} _ {L} ^ {*} T _ {r}\\right)\\right) - K \\xi_ {L},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.509,
+                0.379,
+                0.922,
+                0.42
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta \\lambda_ {i + 1} = \\sum_ {k = 1} ^ {K} \\Gamma \\left(\\bar {E} - \\left(\\varphi_ {k} ^ {*} E _ {i} + \\varphi_ {k} ^ {*} p _ {i} ^ {*} T _ {i L, k} ^ {*} + \\varphi_ {k} ^ {*} \\bar {p} _ {i} ^ {*} T _ {r}\\right)\\right) - K \\xi_ {i}, i \\in \\mathcal {I},\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.508,
+                0.422,
+                0.924,
+                0.462
+            ],
+            "angle": 0,
+            "content": "\\[\n\\Delta \\lambda_ {I + 1 + i} = \\sum_ {k = 1} ^ {K} \\Gamma \\left(\\tau_ {i} - T _ {L i, k} ^ {*}\\right) - K \\xi_ {C}, i \\in \\mathcal {I}, \\tag {25}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.461,
+                0.924,
+                0.807
+            ],
+            "angle": 0,
+            "content": "where the terms \\(\\varphi_k^*, T_d^*, T_{iL,k}^*, T_{Li,k}^*, \\bar{p}_L^*, \\bar{p}_i^*\\) are expressed by optimized variables \\(p^*, p_L^*, \\beta^*, v^*\\). The proof of subgradients is similar to the one provided in [16], and is omitted here. Thereby, we can solve the problem in (24) by either the subgradient method or the ellipsoid method, and their complexities are, respectively, \\(\\mathcal{O}\\left(\\frac{2I + 1}{\\epsilon^2}\\right)\\) and \\(\\mathcal{O}\\left((2I + 1)^2 \\ln \\frac{1}{\\epsilon}\\right)\\) with accuracy \\(\\epsilon\\) [17]. Then, the sub-optimal solution of \\(\\{\\pmb{p}, p_L, \\beta, v\\}\\) can be obtained by solving dual objective function \\(\\mathcal{D}(\\lambda)\\). In particular, similar to [16], we use the iterative method to sequentially derive the sub-optimal value of each element in \\(\\{\\pmb{p}, p_L, \\beta, v\\}\\) (the details are omitted here due to space limitations). Note that, we assume that all these steps of solving the optimization problem are done by a central unit (e.g., cloud or BS), before the swarm starts training their learning models in FL. In particular, there is no need for the central unit to collect any information from UAVs, since all samples of wireless channel gains and antenna deviations are randomly generated by the central unit itself. Also, since the number of UAVs in the swarm is usually small, the complexity of using sample average approximation and dual approach will be low. As a result, the central unit can readily obtain the sub-optimal solution to the joint design problem and later send the power allocation and scheduling parameters to UAVs in the swarm."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.564,
+                0.816,
+                0.866,
+                0.83
+            ],
+            "angle": 0,
+            "content": "IV. SIMULATION RESULTS AND ANALYSIS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.835,
+                0.924,
+                0.895
+            ],
+            "angle": 0,
+            "content": "For our simulations, we first validate the theoretical analysis in Theorem 1. Then, we show the impact of angle deviations on the convergence of FL, and we compare our joint design with baseline schemes that optimize power allocation and schedul"
+        }
+    ],
+    [
+        {
+            "type": "image",
+            "bbox": [
+                0.139,
+                0.069,
+                0.422,
+                0.23
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.18,
+                0.239,
+                0.383,
+                0.253
+            ],
+            "angle": 0,
+            "content": "Fig. 2. Validation of Theorem 1."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.28,
+                0.493,
+                0.416
+            ],
+            "angle": 0,
+            "content": "ing separately. In particular, we consider two baselines. The first baseline is a system with optimized power allocation (same power allocation in the joint design) and randomized scheduling parameters. The second baseline is a system with optimized scheduling (same scheduling used by the joint design) and randomized power allocation. We also assume equal uplink and downlink bandwidths, i.e., \\( B_{u} = B_{d} = 1 \\mathrm{MHz} \\), and equal angle deviation variance for each UAV, i.e., \\( \\sigma_j^2 = \\sigma^2 \\), \\( j \\in \\mathcal{I} \\cup \\{L\\} \\). All simulation parameters are summarized in Table I."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.417,
+                0.493,
+                0.612
+            ],
+            "angle": 0,
+            "content": "Fig. 2 shows the convergence round versus the difference threshold \\(\\varepsilon\\). Note that, in Fig. 2, we choose the range of \\(\\varepsilon \\in (5, 25)\\) based on the value of \\(\\sum_{i=1}^{I} \\sum_{n=1}^{N_i} f(\\boldsymbol{w}_0, \\boldsymbol{x}_{in}, y_{in})\\) and the range of \\(\\varepsilon\\) will be varied for different settings of data and initial global FL model and the accuracy requirement. As observed from Fig. 2, the theoretical analysis derived in Theorem 1 is aligned with the simulation results with less than \\(5\\%\\) difference, thus corroborating the validity of Theorem 1. Moreover, Fig. 2 shows that, when the difference threshold increases, the convergence round decreases. This is because, with a larger difference threshold, the requirement of convergence becomes less stringent. In this case, FL requires fewer communication rounds to converge."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.612,
+                0.493,
+                0.868
+            ],
+            "angle": 0,
+            "content": "Fig. 3 shows the convergence round when the variance of angle deviations changes. From Fig. 3, we observe that, when the variance of angle deviations increases, FL needs more communication rounds to converge. This is due to the fact that, when the angle deviation variance increases, the antennas at transmitter and receiver in the network will be less aligned, leading to a drop in the antenna gains' product between transmitter and receiver in (4) and (5). As a result, the transmission delay of wireless links will increase, and the probability of meeting the delay requirements, i.e., \\(\\mathbb{P}(T_{iL,t} \\leq T_u, T_{Li,t} \\leq T_d)\\), decreases. Therefore, more communication rounds are needed to achieve the FL convergence. Moreover, as shown in Fig. 3, when the bandwidth allocated to uplink and downlink increases, the FL algorithm requires fewer communication rounds to achieve convergence. This stems from the fact that, a large bandwidth improves the probability of meeting the delay requirements, yielding a fast FL convergence."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.87,
+                0.492,
+                0.899
+            ],
+            "angle": 0,
+            "content": "Fig. 4 compares our proposed joint power allocation and scheduling design with the baselines without a joint design. It"
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.563,
+                0.068,
+                0.865,
+                0.228
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.536,
+                0.234,
+                0.894,
+                0.25
+            ],
+            "angle": 0,
+            "content": "Fig. 3. Impact of angle deviations on the FL convergence."
+        },
+        {
+            "type": "image",
+            "bbox": [
+                0.557,
+                0.267,
+                0.861,
+                0.426
+            ],
+            "angle": 0,
+            "content": null
+        },
+        {
+            "type": "image_caption",
+            "bbox": [
+                0.505,
+                0.433,
+                0.925,
+                0.448
+            ],
+            "angle": 0,
+            "content": "Fig. 4. Comparisons between systems with and without joint design."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.474,
+                0.927,
+                0.685
+            ],
+            "angle": 0,
+            "content": "is shown that, for the same network setting, the convergence round for a network with joint design is always less than its counterparts of baselines. In particular, when the bandwidth is \\(1\\mathrm{MHz}\\), the system with a joint design reduces the convergence round by as much as \\(35\\%\\) compared with the baseline system with optimized scheduling and randomized power allocation design. Moreover, as shown in Fig. 4, when the bandwidth assigned to uplink and downlink increases, the performance gap between the system with the proposed joint design and the baselines decreases. That is because, as we increase the bandwidth, it becomes more probable for all three systems to meet the delay constraints at uplink and downlink. Therefore, the impact of communications delay on the FL convergence will be minimized."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.65,
+                0.695,
+                0.78,
+                0.708
+            ],
+            "angle": 0,
+            "content": "V. CONCLUSIONS"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.503,
+                0.713,
+                0.925,
+                0.88
+            ],
+            "angle": 0,
+            "content": "In this paper, we have studied the possibility of implementing FL over a swarm of UAVs. In particular, we have carried out a convergence analysis to study the impact of wireless factors, such as transmission delay and antenna angle deviations, on the convergence of FL. Using the derived insight, we have jointly designed the power allocation and scheduling policy for the UAV swarm to optimize the convergence performance of FL while guaranteeing the stability of control system and controlling the energy consumption. Simulation results have corroborated the convergence analysis of FL and showed the merits of the proposed joint design."
+        }
+    ],
+    [
+        {
+            "type": "title",
+            "bbox": [
+                0.244,
+                0.068,
+                0.321,
+                0.08
+            ],
+            "angle": 0,
+            "content": "APPENDIX"
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.071,
+                0.086,
+                0.232,
+                0.101
+            ],
+            "angle": 0,
+            "content": "A. Proof of Theorem 1"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.105,
+                0.493,
+                0.286
+            ],
+            "angle": 0,
+            "content": "According to the assumptions about function \\( F(\\pmb{w}) : \\mathbb{R}^n \\to \\mathbb{R} \\) made in Section III, we know that function \\( F(\\pmb{w}) \\) is continuously differentiable, and the gradient of \\( F(\\pmb{w}) \\) is uniformly Lipschitz continuous, i.e., for some positive parameter \\( U \\), \\( ||\\nabla F(\\pmb{w}^{(t + 1)}) - \\nabla F(\\pmb{w}^{(t)})|| \\leq U||\\pmb{w}^{(t + 1)} - \\pmb{w}^{(t)}|| \\); the function \\( F \\) is strongly convex with positive parameter \\( \\mu \\): \\( F(\\pmb{w}^{(t + 1)}) \\geq F(\\pmb{w}^{(t)}) + (\\pmb{w}^{(t + 1)} - \\pmb{w}^{(t)})^T\\nabla F(\\pmb{w}^{(t)}) + \\frac{1}{2}\\mu ||\\pmb{w}^{(t + 1)} - \\pmb{w}^{(t)}|| \\). If \\( F \\) is twice-continuously differentiable, these two assumptions are equivalent to \\( \\mu I \\leq \\nabla^2 F(\\pmb{w}) \\leq UI \\). Also, following a standard assumption in stochastic optimization, we consider that there exists constants \\( \\zeta_1 \\geq 0 \\) and \\( \\zeta_2 \\geq 1 \\), meeting \\( ||\\nabla F_i(\\pmb{w})||^2 \\leq \\zeta_1 + \\zeta_2||\\nabla F(\\pmb{w})||^2 \\) [14]."
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.287,
+                0.493,
+                0.331
+            ],
+            "angle": 0,
+            "content": "In this case, since the global FL model is the aggregation of all local FL models, the global FL model without the impact of the transmission delay can be given as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.1,
+                0.33,
+                0.49,
+                0.367
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)}}{\\sum_ {i = 1} ^ {I} N _ {i}} = \\boldsymbol {w} ^ {(t - 1)} - \\lambda \\nabla F (\\boldsymbol {w} ^ {(t - 1)}). \\tag {26}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.365,
+                0.49,
+                0.394
+            ],
+            "angle": 0,
+            "content": "After taking into account the impact of transmission delays, we can rewrite the global FL model update as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.073,
+                0.392,
+                0.49,
+                0.431
+            ],
+            "angle": 0,
+            "content": "\\[\n\\boldsymbol {w} ^ {(t)} = \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\boldsymbol {w} _ {i} ^ {(t)} C _ {i , t}}{\\sum_ {i = 1} ^ {I} N _ {i} C _ {i , t}} = \\boldsymbol {w} ^ {(t - 1)} - \\lambda (\\nabla F (\\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)}),\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.43,
+                0.49,
+                0.479
+            ],
+            "angle": 0,
+            "content": "where \\(e^{(t)} = -\\nabla F(\\pmb{w}^{(t-1)}) + \\frac{\\sum_{i=1}^{I} N_i \\nabla F_i(\\pmb{w}^{(t-1)}) C_{i,t}}{\\sum_{i=1}^{I} N_i C_{i,t}}\\). Based on the assumption on the uniform Lipschitz continuity and strong convexity, we can have the following inequalities:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.093,
+                0.479,
+                0.49,
+                0.527
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} F (\\boldsymbol {w} ^ {(t)}) \\leq F (\\boldsymbol {w} ^ {(t - 1)}) + (\\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)}) ^ {T} \\nabla \\tilde {F} (\\boldsymbol {w} ^ {(t - 1)}) \\\\ + \\frac {U}{2} \\| \\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)} \\| ^ {2}, \\tag {27} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.093,
+                0.528,
+                0.49,
+                0.572
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} F (\\boldsymbol {w} ^ {(t)}) \\geq F (\\boldsymbol {w} ^ {(t - 1)}) + (\\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)}) ^ {T} \\nabla F (\\boldsymbol {w} ^ {(t - 1)}) \\\\ + \\frac {\\mu}{2} \\left\\| \\boldsymbol {w} ^ {(t)} - \\boldsymbol {w} ^ {(t - 1)} \\right\\| ^ {2}. \\tag {28} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.572,
+                0.49,
+                0.603
+            ],
+            "angle": 0,
+            "content": "Since \\(\\pmb{w}^{(t)} = \\overline{\\pmb{w}}^{(t - 1)} - \\lambda (\\nabla F(\\pmb{w}^{(t - 1)}) + e^{(t)})\\), we can simplify (27) when the learning rate is \\(\\lambda = \\frac{1}{U}\\) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.073,
+                0.603,
+                0.49,
+                0.7
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} F (\\boldsymbol {w} ^ {(t)}) \\leq F (\\boldsymbol {w} ^ {(t - 1)}) - \\frac {1}{U} \\left(\\nabla F (\\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)}\\right) ^ {T} \\nabla F (\\boldsymbol {w} ^ {(t - 1)}) \\\\ + \\frac {1}{2 U} | | \\nabla F (\\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)} | | ^ {2} \\\\ = F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - \\frac {1}{2 U} \\left\\| \\nabla F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) \\right\\| ^ {2} + \\frac {1}{2 U} \\left\\| e ^ {(t)} \\right\\| ^ {2}. \\tag {29} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.701,
+                0.49,
+                0.779
+            ],
+            "angle": 0,
+            "content": "To find a lower bound on the norm of \\(\\nabla F(\\pmb{w}^{(t)})\\), we can minimize both sides of (28) with respect \\(\\pmb{w}^{(t)}\\). The minimal value of the left-hand side of (28) is achieved when \\(\\pmb{w}^{(t)} = \\pmb{w}^*\\), and the minimal value of the right-hand side of (28) is realized when \\(\\pmb{w}^{(t)} = \\pmb{w}^{(t - 1)} - \\frac{1}{\\mu}\\nabla F(\\pmb{w}^{(t - 1)})\\). Particularly, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.132,
+                0.778,
+                0.49,
+                0.807
+            ],
+            "angle": 0,
+            "content": "\\[\nF \\left(\\boldsymbol {w} ^ {*}\\right) \\geq F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - \\frac {1}{2 \\mu} | | \\nabla F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) | | ^ {2}. \\tag {30}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.806,
+                0.49,
+                0.836
+            ],
+            "angle": 0,
+            "content": "When replacing \\(\\pmb{w}^{(t - 1)}\\) with \\(\\pmb{w}^{(t)}\\) in (30), we can obtain a lower bound for the norm of \\(\\nabla F(\\pmb{w}^{(t)})\\) as"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.144,
+                0.836,
+                0.49,
+                0.853
+            ],
+            "angle": 0,
+            "content": "\\[\n\\left| \\left| \\nabla F \\left(\\boldsymbol {w} ^ {(t)}\\right) \\right| \\right| ^ {2} \\geq 2 \\mu \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right). \\tag {31}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.852,
+                0.492,
+                0.881
+            ],
+            "angle": 0,
+            "content": "Combining (29) and (31), we can obtain an upper bound of the current loss and the minimal loss given by"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.073,
+                0.881,
+                0.492,
+                0.907
+            ],
+            "angle": 0,
+            "content": "\\[\nF \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right) \\leq \\left(1 - \\frac {\\mu}{U}\\right) \\left[ F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right) \\right] + \\frac {1}{2 U} \\left\\| e ^ {(t)} \\right\\| ^ {2}.\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.081,
+                0.925,
+                0.127
+            ],
+            "angle": 0,
+            "content": "According to [19], when \\(\\mathbb{E}[||e^{(t)}||^2] \\leq 2U\\left(\\frac{\\mu}{U} - \\rho^{(t)}\\right)\\mathbb{E}(F(\\boldsymbol{w}^{(t)}) - F(\\boldsymbol{w}^*))\\), we can achieve the strong expected linear convergence, i.e.,"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.508,
+                0.126,
+                0.917,
+                0.144
+            ],
+            "angle": 0,
+            "content": "\\[\n\\overline {{\\mathbb {E}}} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\leq (1 - \\rho^ {(t)}) \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.144,
+                0.923,
+                0.171
+            ],
+            "angle": 0,
+            "content": "According to the strong expected linear convergence requirement, we know that the convergence rate satisfies"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.583,
+                0.171,
+                0.924,
+                0.204
+            ],
+            "angle": 0,
+            "content": "\\[\n\\rho^ {(t)} \\leq \\frac {\\mu}{U} - \\frac {\\mathbb {E} [ | | e ^ {(t)} | | ^ {2} ]}{2 U \\mathbb {E} (F (\\boldsymbol {w} ^ {(t)}) - F (\\boldsymbol {w} ^ {*}))}. \\tag {32}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.202,
+                0.924,
+                0.216
+            ],
+            "angle": 0,
+            "content": "By using the results in [20], we have the following inequality:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.538,
+                0.215,
+                0.924,
+                0.272
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left(\\left\\| e ^ {(t)} \\right\\| ^ {2}\\right) \\leq \\frac {1}{N} \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} \\mathbb {E} \\left(\\nabla F \\left(\\boldsymbol {w} ^ {(t)}\\right)\\right)\\right) \\times \\\\ (1 - \\mathbb {P} (T _ {i L} \\leq T _ {u} (\\beta), T _ {L i} \\leq T _ {d} (\\beta))). \\tag {33} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.273,
+                0.924,
+                0.287
+            ],
+            "angle": 0,
+            "content": "The right-hand side of (32) will meet the following inequality:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.508,
+                0.286,
+                0.924,
+                0.359
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\frac {\\mu}{U} - \\frac {\\mathbb {E} [ | | e ^ {(t)} | | ^ {2} ]}{2 U \\mathbb {E} (F (\\boldsymbol {w} ^ {(t)}) - F (\\boldsymbol {w} ^ {*}))} \\geq \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} \\mathbb {E} \\left(\\nabla F (\\boldsymbol {w} ^ {(t)})\\right)\\right) \\\\ \\times \\frac {\\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{2 N U \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)}. \\tag {34} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.357,
+                0.887,
+                0.371
+            ],
+            "angle": 0,
+            "content": "Therefore, to guarantee that (32) always exists, we have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.518,
+                0.371,
+                0.83,
+                0.445
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\rho^ {(t)} \\leq \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} \\mathbb {E} \\left(\\nabla F \\left(\\boldsymbol {w} ^ {(t)}\\right)\\right)\\right) \\\\ \\times \\frac {(1 - \\mathbb {P} (T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)))}{2 N U \\mathbb {E} (F (\\boldsymbol {w} ^ {(t)}) - F (\\boldsymbol {w} ^ {*}))} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.525,
+                0.446,
+                0.826,
+                0.521
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\stackrel {(a)} {\\leq} \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(\\zeta_ {1} + \\zeta_ {2} 2 \\mu \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)\\right) \\\\ \\times \\frac {\\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{2 N U \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.525,
+                0.523,
+                0.83,
+                0.598
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\stackrel {(b)} {\\leq} \\frac {\\mu}{U} - \\sum_ {i = 1} ^ {I} N _ {i} \\left(2 \\mu \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)\\right) \\\\ \\times \\frac {\\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{2 N U \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right)} \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.523,
+                0.599,
+                0.924,
+                0.631
+            ],
+            "angle": 0,
+            "content": "\\[\n= \\frac {\\mu}{U} - \\frac {\\sum_ {i = 1} ^ {I} N _ {i} \\mu \\left(1 - \\mathbb {P} \\left(T _ {i L} \\leq T _ {u} (\\beta) , T _ {L i} \\leq T _ {d} (\\beta)\\right)\\right)}{N U}, \\tag {35}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.505,
+                0.628,
+                0.924,
+                0.694
+            ],
+            "angle": 0,
+            "content": "where in (a), we use the results derived in (31), and the derivation in (b) is based on the fact that \\(\\zeta_1 \\geq 0\\) and \\(\\zeta_2 \\geq 1\\). Assume \\(\\rho = \\frac{\\mu}{U} - \\frac{\\sum_{i=1}^{I} N_i \\mu(1 - \\mathbb{P}(T_{iL} \\leq T_u(\\beta), T_{Li} \\leq T_d(\\beta)))}{NU} = \\frac{\\sum_{i=1}^{I} N_i \\mu(\\mathbb{P}(T_{iL} \\leq T_u(\\beta), T_{Li} \\leq T_d(\\beta)))}{NU}\\), then, we can have"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.52,
+                0.693,
+                0.909,
+                0.731
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\leq (1 - \\rho) \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t - 1)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\\\ \\leq (1 - \\rho) ^ {2} \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(t - 2)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right) \\\\ \\end{array}\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.667,
+                0.74,
+                0.683,
+                0.747
+            ],
+            "angle": 0,
+            "content": "··"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.671,
+                0.752,
+                0.897,
+                0.769
+            ],
+            "angle": 0,
+            "content": "\\[\n\\leq (1 - \\rho) ^ {t} \\mathbb {E} \\left(F \\left(\\boldsymbol {w} ^ {(0)}\\right) - F \\left(\\boldsymbol {w} ^ {*}\\right)\\right).\n\\]"
+        },
+        {
+            "type": "text",
+            "bbox": [
+                0.504,
+                0.769,
+                0.924,
+                0.812
+            ],
+            "angle": 0,
+            "content": "We can further determine the convergence round needed to achieve a target difference threshold, i.e., \\(\\mathbb{E}(F(\\boldsymbol{w}) - F(\\boldsymbol{w}^{*}))\\leq \\varepsilon\\), as follows:"
+        },
+        {
+            "type": "equation",
+            "bbox": [
+                0.57,
+                0.811,
+                0.924,
+                0.907
+            ],
+            "angle": 0,
+            "content": "\\[\n\\begin{array}{l} t \\geq \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\mathbb {E} (F (\\boldsymbol {w} ^ {(0)}) - F (\\boldsymbol {w} ^ {*}))} \\\\ \\stackrel {(a)} {\\geq} \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\mathbb {E} (F (\\boldsymbol {w} ^ {(0)}))} \\\\ = \\log_ {1 - \\rho} \\frac {\\varepsilon}{\\sum_ {i = 1} ^ {I} \\sum_ {n = 1} ^ {N _ {i}} f \\left(\\boldsymbol {w} ^ {(0)}, \\boldsymbol {x} _ {i n} , y _ {i n}\\right)}, \\tag {36} \\\\ \\end{array}\n\\]"
+        }
+    ],
+    [
+        {
+            "type": "text",
+            "bbox": [
+                0.071,
+                0.067,
+                0.493,
+                0.112
+            ],
+            "angle": 0,
+            "content": "where in (a), we use the fact that \\(1 - \\rho \\leq 1\\). Since the convergence round must be integral, we can have the results in Theorem 1."
+        },
+        {
+            "type": "title",
+            "bbox": [
+                0.234,
+                0.121,
+                0.331,
+                0.134
+            ],
+            "angle": 0,
+            "content": "REFERENCES"
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.082,
+                0.142,
+                0.49,
+                0.187
+            ],
+            "angle": 0,
+            "content": "[1] M. Mozaffari, W. Saad, M. Dennis, Y. Nam, and M. Debbah, \"A tutorial on UAVs for wireless networks: Applications, challenges, and open problems,\" IEEE Communications Surveys Tutorials, vol. 21, no. 3, pp. 2334-2360, Third Quarter, 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.082,
+                0.187,
+                0.492,
+                0.232
+            ],
+            "angle": 0,
+            "content": "[2] M. Mozaffari, A. Kasgari, W. Saad, M. Bennis, and M. Debbah, \"Beyond 5G with UAVs: Foundations of a 3D wireless cellular network,\" IEEE Transactions on Wireless Communications, vol. 18, no. 1, pp. 357-372, Jan 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.233,
+                0.492,
+                0.255
+            ],
+            "angle": 0,
+            "content": "[3] J. Park, S. Samarakoon, M. Bennis, and M. Debbah, \"Wireless network intelligence at the edge,\" Proceeding of the IEEE, to appear, 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.256,
+                0.492,
+                0.289
+            ],
+            "angle": 0,
+            "content": "[4] J. Konecný, H. B. McMahan, D. Ramage, and P. Richtárik, \"Federated optimization: Distributed machine learning for on-device intelligence,\" arXiv preprint http://arxiv.org/abs/1610.02527 arXiv:1610.02527, 2016."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.289,
+                0.492,
+                0.323
+            ],
+            "angle": 0,
+            "content": "[5] M. Chen, H. V. Poor, W. Saad, and S. Cui, \"Convergence time optimization for federated learning over wireless networks,\" arXiv preprint http://arxiv.org/abs/2001.07845 arXiv:2001.07845."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.323,
+                0.492,
+                0.357
+            ],
+            "angle": 0,
+            "content": "[6] Q. Zeng, Y. Du, K. K. Leung, and K. Huang, \"Energy-efficient radio resource allocation for federated edge learning,\" arXiv preprint http://arxiv.org/abs/1907.06040 arXiv:1907.06040, 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.357,
+                0.492,
+                0.391
+            ],
+            "angle": 0,
+            "content": "[7] H. H. Yang, Z. Liu, T. Q. S. Quek, and H. V. Poor, \"Scheduling policies for federated learning in wireless networks,\" IEEE Transactions on Communications, to appear, 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.391,
+                0.492,
+                0.436
+            ],
+            "angle": 0,
+            "content": "[8] N. H. Tran, W. Bao, A. Zomaya, N. Minh N.H., and C. S. Hong, \"Federated learning over wireless networks: Optimization model design and analysis,\" in Proc. of IEEE International Conference on Computer Communications, Paris, France, Apr. 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.081,
+                0.436,
+                0.492,
+                0.47
+            ],
+            "angle": 0,
+            "content": "[9] Y. Pan, C. Pan, Z. Yang, and M. Chen, \"Resource allocation for D2D communications underlaying a NOMA-based cellular network,\" IEEE Wireless Communications Letters, vol. 7, no. 1, pp. 130-133, Feb. 2018."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.47,
+                0.491,
+                0.504
+            ],
+            "angle": 0,
+            "content": "[10] M. T. Dabiri, H. Safi, S. Parsaeefard, and W. Saad, \"Analytical channel models for millimeter wave UAV networks under hovering fluctuations,\" arXiv preprint http://arxiv.org/abs/1905.01477 arXiv:1905.01477, 2019."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.505,
+                0.491,
+                0.537
+            ],
+            "angle": 0,
+            "content": "[11] ITU-R, Mathematical models for radiodetermination radar systems antenna patterns for use in interference analyses, Recommendation ITU-R M.1851-1, Jan. 2018."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.537,
+                0.492,
+                0.583
+            ],
+            "angle": 0,
+            "content": "[12] T. Zeng, M. Mozaffari, O. Semiari, W. Saad, M. Bennis, and M. Debbah, \"Wireless communications and control for swarms of cellular-connected UAVs,\" in Proc. of IEEE Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, Oct. 2018."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.583,
+                0.492,
+                0.628
+            ],
+            "angle": 0,
+            "content": "[13] J. K. Stolaroff, C. Samaras, E. R. O'Neill, A. S. Mitchell, and D. Ceperley, \"Energy use and life cycle greenhouse gas emissions of drones for commercial package delivery,\" Nature Communications, vol. 9, no. 409, pp. 1-13, Feb. 2018."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.629,
+                0.492,
+                0.652
+            ],
+            "angle": 0,
+            "content": "[14] D. P. Bertsekas and J. N. Tsitsiklis, Neuro-dynamic programming. Athena Scientific Belmont, MA, 1996, vol. 5."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.652,
+                0.492,
+                0.697
+            ],
+            "angle": 0,
+            "content": "[15] B. K. Pagnoncelli, S. Ahmed, and A. Shapiro, \"Sample average approximation method for chance constrained programming: theory and applications,\" Journal of optimization theory and applications, vol. 142, no. 2, pp. 399-416, 2009."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.697,
+                0.492,
+                0.731
+            ],
+            "angle": 0,
+            "content": "[16] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Transactions on Communications, vol. 54, no. 7, pp. 1310–1322, July 2006."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.731,
+                0.492,
+                0.742
+            ],
+            "angle": 0,
+            "content": "[17] Y. Nesterov, Lectures on convex optimization. Springer, 2018, vol. 137."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.742,
+                0.492,
+                0.787
+            ],
+            "angle": 0,
+            "content": "[18] F. Zhou, Y. Wu, R. Q. Hu, and Y. Qian, \"Computation rate maximization in UAV-enabled wireless-powered mobile-edge computing systems,\" IEEE Journal on Selected Areas in Communications, vol. 36, no. 9, pp. 1927-1941, Sep. 2018."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.787,
+                0.492,
+                0.821
+            ],
+            "angle": 0,
+            "content": "[19] M. P. Friedlander and M. Schmidt, “Hybrid deterministic-stochastic methods for data fitting,” SIAM Journal on Scientific Computing, vol. 34, no. 3, pp. A1380-A1405, 2012."
+        },
+        {
+            "type": "ref_text",
+            "bbox": [
+                0.074,
+                0.821,
+                0.492,
+                0.867
+            ],
+            "angle": 0,
+            "content": "[20] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, \"A joint learning and communications framework for federated learning over wireless networks,\" arXiv preprint http://arxiv.org/abs/1909.07972 arXiv:1909.07972, 2019."
+        },
+        {
+            "type": "list",
+            "bbox": [
+                0.074,
+                0.142,
+                0.492,
+                0.867
+            ],
+            "angle": 0,
+            "content": null
+        }
+    ]
+]

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+# Federated Learning in the Sky: Joint Power Allocation and Scheduling with UAV Swarms
+
+Tengchan Zeng, Omid Semiari, Mohammad Mozaffari, Mingzhe Chen, Walid Saad, and Mehdi Bennis
+
+Abstract—Unmanned aerial vehicle (UAV) swarms must exploit machine learning (ML) in order to execute various tasks ranging from coordinated trajectory planning to cooperative target recognition. However, due to the lack of continuous connections between the UAV swarm and ground base stations (BSs), using centralized ML will be challenging, particularly when dealing with a large volume of data. In this paper, a novel framework is proposed to implement distributed federated learning (FL) algorithms within a UAV swarm that consists of a leading UAV and several following UAVs. Each following UAV trains a local FL model based on its collected data and then sends this trained local model to the leading UAV who will aggregate the received models, generate a global FL model, and transmit it to followers over the intra-swarm network. To identify how wireless factors, like fading, transmission delay, and UAV antenna angle deviations resulting from wind and mechanical vibrations, impact the performance of FL, a rigorous convergence analysis for FL is performed. Then, a joint power allocation and scheduling design is proposed to optimize the convergence rate of FL while taking into account the energy consumption during convergence and the delay requirement imposed by the swarm's control system. Simulation results validate the effectiveness of the FL convergence analysis and show that the joint design strategy can reduce the number of communication rounds needed for convergence by as much as  $35\%$  compared with the baseline design.
+
+# I. INTRODUCTION
+
+Swarms of unmanned aerial vehicles (UAVs) will play an important role in various services ranging from delivery of goods to monitoring [1] and [2]. To deliver those services, UAV swarms will employ machine learning (ML) for executing various tasks such as consensus trajectory planning, target recognition, and localization. However, due to the high altitude and mobility of UAVs, continuous connections between UAVs and ground base stations (BSs) cannot be guaranteed. Hence, using centralized ML approaches to execute learning-related tasks will be challenging, particularly when transmitting a large
+
+This research was supported, in part, by the U.S. National Science Foundation under Grants CNS-1739642 and CNS-1941348, and by the Academy of Finland Project CARMA, by the Academy of Finland Project MISSION, by the Academy of Finland Project SMARTER, as well as by the INFOTECH Project NOOR.
+
+T. Zeng and W. Saad are with Wireless@VT, Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, 24061 USA (e-mail: tengchan@vt.edu; walids@vt.edu).  
+O. Semiari is with Department of Electrical and Computer Engineering, University of Colorado Colorado Springs, Colorado Springs, CO, 80918 USA (e-mail: osemiari@uccs.edu).  
+M. Mozaffari is with Ericsson Research, Santa Clara, CA, 95054 USA (e-mail: mohammad.mozaffari@ericsson.com).  
+M. Chen is with Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544 USA (e-mail: mingzhec@princeton.edu).  
+M. Dennis is with the Centre for Wireless Communications, University of Oulu, 90014 Oulu, Finland (e-mail:mehdi.bennis@oulu.fi).
+
+volume of data over aerial links. Instead, a distributed learning approach would be more apropos [3]. In particular, one can use federated learning (FL) to enable each UAV to perform distributed ML tasks without relying on any centralized BSs [4]. In this case, UAVs do not need to send any raw data to BSs when training learning models.
+
+In essence, FL allows each UAV in a swarm to train its learning model based on its own collected data, and it can use the intra-swarm network to share FL parameters related to the learned models with other UAVs. As the learning process proceeds, UAVs in the swarm can reach a consensus on their collective learning tasks, e.g., trajectory planning or target recognition. However, since the updates of the learning models in FL are transmitted over a wireless network, the FL convergence and task consensus for the UAV swarm will inevitably be affected by wireless factors such as transmission delay. Also, due to the high mobility of UAVs, other factors (like wind and mechanical vibrations) can increase the uncertainty of wireless channels by affecting the UAVs' antenna angles which, in turn, will impact the FL convergence.
+
+A number of recent works have investigated how wireless communication impacts FL [5]–[7]. For instance, in [5], the authors solve the joint learning, wireless resource allocation, and user selection problem to minimize the FL convergence time while optimizing the FL performance. Also, the work in [6] proposes a strategy for bandwidth allocation and device scheduling to improve the energy efficiency for networks implementing FL. Moreover, [7] studies the impact of different scheduling policies on the performance of FL. While interesting, none of these works in [5]–[7] considers the role of FL in a UAV swarm. Also, due to the high mobility of UAVs and their limited energy, the analysis in [5]–[7] cannot be directly applied for UAV swarms.
+
+The main contribution of this paper is a novel framework for enabling FL within a swarm of wireless-connected UAVs. In particular, we first conduct a convergence analysis for FL to show how wireless factors within the UAV swarm impact the convergence of FL. We then determine the convergence round, defined as the minimum number of communication rounds needed to achieve FL convergence. Using this key insight, we formulate an optimization problem that jointly designs the power allocation and scheduling for the UAV swarm network to reduce the FL convergence round. In particular, due to the stringent energy limitations of UAVs, we consider the constraint of the energy consumed by learning, communications, and flying during FL convergence. We also take into account
+
+the delay constraint imposed by the control system to guarantee the stability of the UAV swarm. To solve the joint design problem, we use a sample average approximation approach from stochastic programming along with a dual method from convex optimization. To the best of our knowledge, this is the first work that implements FL for the UAV swarm, studies the impact of wireless factors on the convergence of FL, and optimizes the FL convergence by jointly designing power allocation and scheduling of the UAV network. Simulation results validate the convergence analysis of FL and show that the joint design can reduce the convergence round by as much as  $35\%$  compared with baselines without the joint design.
+
+The rest of the paper is organized as follows. Section II presents the system model for the UAV swarm. Section III analyzes the FL convergence and shows the joint system design. Section IV provides simulation results, and conclusions are drawn in Section V.
+
+# II. SYSTEM MODEL
+
+Consider a swarm of wirelessly connected autonomous UAVs flying at the same altitude, as shown in Fig. 1(a). The UAV swarm consists of a leader  $L$  and a set  $\mathcal{I}$  of  $I$  followers. Every follower keeps a target distance and speed with the leader. While flying, the UAV swarm collects data and performs FL for data analysis and inference tasks like trajectory planning and cooperative target recognition. Using FL, each follower uses its collected data to train a local FL model and send the parameters related to the learned model to the leading UAV in the uplink, as shown in Fig. 1(a). The leading UAV will integrate all received information to generate a global FL model, and, then, transmit the parameters of the global model to following UAVs over the downlink. Moreover, to guarantee that the followers fly with the same speed while keeping a safe distance, the leading UAV will also broadcast the target spacing information and its speed and heading direction.
+
+# A. Federated learning model
+
+In the learning model, we assume that UAV  $i \in \mathcal{I}$  collects a set  $\{\pmb{x}_{i1},\pmb{x}_{i2},\dots,\pmb{x}_{iN_i}\}$  of input data where each collected sample is represented by a vector  $\pmb{x}_{in}$ ,  $n \in \{1,\dots,N_i\}$  that captures the input features and  $N_{i}$  is the number of collected samples. We also assume the input sample  $\pmb{x}_{in}$ ,  $n \in \{1,\dots,N_i\}$ , corresponds to a single output  $y_{in}$  [4]. The output vector is thereby  $\{y_{i1},\dots,y_{iN_i}\}$  for UAV  $i$ . We define a vector  $\pmb{w}_i$  as the parameters related to the local FL model that is trained by  $\{\pmb{x}_{i1},\pmb{x}_{i2},\dots,\pmb{x}_{iN_i}\}$  and  $\{y_{i1},\dots,y_{iN_i}\}$  at UAV  $i$ . The convergence of the FL training processes requires each local learning vector to converge to a vector  $\pmb{w}^*$  which solves the following problem:
+
+$$
+\underset {\boldsymbol {w} \in \mathbb {R} ^ {d}} {\arg \min } F (\boldsymbol {w}) = \frac {1}{N} \sum_ {i} ^ {I} \sum_ {n = 1} ^ {N _ {i}} f (\boldsymbol {w}, \boldsymbol {x} _ {i n}, y _ {i n}), \tag {1}
+$$
+
+where  $N = \sum_{i}^{I}N_{i}$  is the total number of the collected samples by all followers, and  $f(\pmb {w},\pmb{x}_{in},y_{in})$  captures the loss function when using learning vector  $\pmb{w}$  for dataset  $\{\pmb {x}_{in},y_{in}\}$ . Note that, the loss function  $f(\pmb {w},\pmb{x}_{in},y_{in}),i\in \mathcal{I},0\leq n\leq N_i,$
+
+![](images/56d45f43e204b171ef87a2bb14d6dbde9f1ed2fd7efef56ae10faca062d78ebf.jpg)  
+(a) Communication and learning models.
+
+![](images/532aa106c947b40349baa42189c3b1f0f2989912d8c82bd93117fbbc95e83a61.jpg)  
+(b) Angle deviations and control system.  
+Fig. 1. Illustration of our system model.
+
+plays a pivotal role in determining the FL performance, and the expression of the loss function is application-specific. For example, for a simple linear regression FL algorithm,  $f(\boldsymbol{w}, \boldsymbol{x}_{in}, y_{in}) = (\boldsymbol{w}^T \boldsymbol{x}_{in} - y_{in})^2$ .
+
+To solve (1), the FL framework uses an iterative update scheme [4]. In particular, the leading UAV will first generate an initial global FL model represented by vector  $\boldsymbol{w}^{(0)}$  and send the initial vector to all followers. Hence, in the first communication round, follower  $i\in \mathcal{I}$  will first use  $\boldsymbol{w}^{(0)}$  for its own data to train the local model and, then, it sends the vector of the trained model to the leader. Next, the leading UAV will aggregate all received local FL vectors and update the global FL model vector which will be later transmitted to the followers. Each communication round will be followed by another round, and the same process will repeat among leader and followers in each round. In this case, as FL proceeds, the local and global models are sequentially updated, and the total loss  $F(\boldsymbol {w})$  for the updated global model with vector  $\boldsymbol{w}$  will continuously decrease [4]. To identify whether the optimal solution is found for (1), one must analyze the convergence of the loss function  $F(\boldsymbol {w})$  to  $F(\boldsymbol {w}^{*})$ . That is, when the gap between the current loss  $F(\boldsymbol {w})$  and the minimal loss  $F(\boldsymbol {w}^{*})$  is below a threshold  $\varepsilon$ , the FL optimization problem is solved [8]. Therefore, we can use the convergence of  $F(\boldsymbol {w})$  to  $F(\boldsymbol {w}^{*})$  to quantify the FL performance.
+
+Moreover, for each communication round, we can divide the total time duration  $T_{r}$  into two periods: Uplink and downlink transmission. In particular, to guarantee that the leading UAV has enough time to process all received models from its followers, all uplink transmissions should be completed within a target time  $T_{u}(\beta) = \beta T_{r}$ , where  $\beta \in \{0,1\}$  is a scheduling parameter to schedule uplink-downlink traffic in time. Also, to receive the global FL model update from the leading UAV successfully, the time constraint for downlink transmissions is thereby  $T_{d}(\beta) = (1 - \beta)T_{r}$ . In this case, if the communication
+
+link between follower  $i \in \mathcal{I}$  and leader  $L$  fails to meet the time constraints  $T_{d}(\beta)$  and  $T_{u}(\beta)$ , the global FL model cannot use the corresponding FL model for the aggregation. At the same time, for the local FL model, the following UAV cannot use the recently updated global vector to train its local data. In other words, the transmission delay of the uplink and downlink links will impact the update of the global and local FL models thus having a major impact on FL convergence.
+
+In addition, when training the global FL model, we can calculate the energy consumption for the UAV  $L$  as  $E_{L} = \kappa C\phi^{2}\sum_{i=1}^{I}S(\pmb{w}_{i})$ , where  $\kappa$  captures the energy consumption coefficient depending on the computing system and  $C$  is the number of computing cycles needed per data bit [9].  $\phi$  is the frequency of the CPU clock of UAVs, and  $S(\pmb{w}_{i})$  is the packet size of  $\pmb{w}_{i}$ , transmitted from UAV  $i \in \mathcal{I}$ , in bits. Similarly, we can determine the training energy consumption for follower  $i \in \mathcal{I}$  as  $E_{i} = \kappa C\phi^{2}\sum_{n=1}^{N_{i}}S(\pmb{x}_{in})$ .
+
+# B. Communication model
+
+To minimize the interference from other UAVs located outside of the swarm, we assume that all UAVs use directional antennas, as shown in Fig. 1(a), However, as shown in Fig. 1(b), due to the impact of wind, payload, and non-ideal mechanical and control systems, the angle of the UAVs will randomly fluctuate and deviate from the initial angle setting. Based on the central limit theorem, we model the angle deviation for each UAV as a Gaussian random variable [10]. Moreover, we consider a squared cosine function to capture the antenna aperture of UAV  $j \in \mathcal{I} \cup \{L\}$  when communicating with UAV  $l \in \mathcal{I} \cup \{L\} / j$  as follows [11]:
+
+$$
+G _ {j l} \left(\theta_ {j l} + \vartheta_ {j}\right) = \left\{ \begin{array}{c c} \cos^ {2} \left(\frac {\pi}{2} \left(\theta_ {j l} + \vartheta_ {j}\right)\right), & \text {i f} | \theta_ {j l} + \vartheta_ {j} | \leq 1, \\ G _ {\min }, & \text {o t h e r w i s e}, \end{array} \right. \tag {2}
+$$
+
+where  $\theta_{jl}$  is the initial angle setting for UAV  $j$  when communicating with UAV  $l$ ,  $\vartheta_{j} \sim \mathcal{N}(0, \sigma_{j}^{2})$  is the angle deviation with variance  $\sigma_{j}^{2}$ , and  $G_{\mathrm{min}}$  captures the antenna gain at the side lobes. Also, similar to [10], we can approximate (2) by using a sectionalized expression:
+
+$$
+G _ {j l} \left(\theta_ {j l} + \vartheta_ {j}, M\right) = \left\{ \begin{array}{c c} \cos^ {2} \left(\frac {\pi m}{2 M}\right), & \text {i f} \frac {m}{M} \leq \left| \theta_ {j l} + \vartheta_ {j} \right| \leq \frac {m + 1}{M}, \\ G _ {\min }, & \text {o t h e r w i s e}, \end{array} \right. \tag {3}
+$$
+
+where  $m\in \{1,\dots,M\}$
+
+To reduce the interference over the uplink transmissions, we assume that uplinks do not share the wireless resource with each other. Hence, the transmission delay of the uplink between follower  $i \in \mathcal{I}$  and leader  $L$  can be calculated as
+
+$$
+T _ {i L} = \frac {S (\boldsymbol {w} _ {i})}{B _ {u} \log_ {2} \left(1 + \frac {p _ {i} h _ {i L} d _ {i L} ^ {- \alpha} G _ {i L} G _ {L i}}{\sum_ {i ^ {\prime} \in \Phi_ {i}} p _ {i ^ {\prime}} h _ {i ^ {\prime} L} d _ {i ^ {\prime} L} ^ {- \alpha} G _ {i ^ {\prime} L} G _ {L i ^ {\prime}} + B _ {u} \gamma_ {0}}\right)}, \tag {4}
+$$
+
+where  $B_{u}$  is the bandwidth used by each subchannel in the uplink,  $p_i\in (0,p_{\mathrm{max}})$  is the transmission power of UAV  $i$  with maximum power as  $p_{\mathrm{max}}$  and  $\alpha$  is the path-loss exponent.  $h_{iL}$  is the channel gain of the Rician fading channel between UAVs  $i$  and  $L$  ,and  $\gamma_0$  is the noise power spectral density. Note that, despite the use of directional antenna, the swarm still experiences uplink interference generated by UAVs located
+
+outside of the swarm. In particular, these interfering UAVs share the same channel resource and exist in the main lobe of the UAV  $L$ , and we define  $\Phi_{i}$  as the set of UAVs that generates interference to the uplink from UAV  $i$  to UAV  $L$ .
+
+Similarly, we can derive the transmission delay  $T_{Li}$  for the downlink from UAV  $L$  to UAV  $i \in \mathcal{I}$  as:
+
+$$
+T _ {L i} = \frac {S (\boldsymbol {w})}{B _ {d} \log_ {2} \left(1 + \frac {p _ {L} h _ {L i} d _ {L i} ^ {- \alpha} G _ {L i} G _ {i L}}{\sum_ {i ^ {\prime} \in \Phi_ {L}} p _ {i ^ {\prime}} h _ {i ^ {\prime} i} d _ {i ^ {\prime} i} ^ {- \alpha} G _ {i ^ {\prime} i} G _ {i i ^ {\prime}} + B _ {d} \gamma_ {0}}\right)}, \tag {5}
+$$
+
+where  $B_{d}$  is the downlink bandwidth,  $p_L \in (0, p_{\mathrm{max}})$  is the transmission power of UAV  $L$ , and  $\Phi_L$  refers to the set of UAVs that will generate interference at the downlink.
+
+# C. Control model
+
+To guarantee constant speed and altitude and avoid collisions between UAVs within the swarm, the leading UAV will broadcast its speed and heading direction to the followers in the downlink. Here, the control system of each follower will use both its sensor data (e.g. location) and information received from the wireless links to coordinate its movement and achieve a target spacing and speed. Note that the target distance between the UAV leader and each follower is predefined such that there will be no collision between two nearby UAVs.
+
+Similar to our previous work in [12], we can build a Cartesian coordinate system to capture the locations of UAVs in the swarm, and, then, we decompose the velocity of each UAV into two components, as shown in Fig. 1(b). We can also define the control law of each UAV the same way as the one provided in [12]. Since the transmission delay will have a negative impact on the stability control of the UAV swarm, we must consider the delay requirement imposed by the control system when designing the UAV network.
+
+In addition, in order to fly with a constant speed and maintain a stable flying motion, each UAV must spend energy to overcome the gravity and the air drag forces due to the wind and forward motions. For a forward speed  $v \in (0, v_{\max})$  with  $v_{\max}$  as the maximum speed, the minimum flying power of UAV  $j \in \mathcal{I} \cup \{L\}$  is  $\bar{p}_{j,\min}(v) = \hat{v}_j A_j$ , where  $\hat{v}_j$  is the induced velocity required for constant speed  $v$  and given thrust  $A_j = mg$  with  $m$  being the UAV mass and  $g$  being the gravitational constant [13]. Also, the induced velocity  $\hat{v}_j$  can be obtained by solving the following equation [13]:
+
+$$
+\hat {v} _ {j} = \frac {2 A _ {j}}{q r ^ {2} \pi \varrho \sqrt {v ^ {2} + \hat {v} _ {j} ^ {2}}}, \tag {6}
+$$
+
+where  $q$  and  $r$  capture, respectively, the number and diameter of the UAV rotors, and  $\varrho$  is the air density. Moreover, we can further correct the theoretical minimum motion power consumption by the overall power efficiency  $\eta$  of the UAV in order to obtain the actual power consumption as  $\bar{p}_j(v) = \bar{p}_{j,\min}(v) / \eta$ . Since the control of a UAV's dynamic motion consumes the most energy [13], we must consider the flying energy consumption when designing the swarm of UAVs. In particular, the flying energy consumption can be calculated as  $\bar{p}_j(v)T$  during the flying time  $T$ .
+
+To guarantee the convergence of FL and the stable operation of the control system in the UAV swarm, we need to properly
+
+design the wireless communication network. At the same time, to guarantee that the energy spent on learning, communication, and flying will not exceed the energy limitation of each UAV, we need to consider the energy consumption during the FL convergence. Next, we first conduct the convergence analysis for the FL algorithm and derive the number of communication rounds needed to achieve the FL convergence. Then, we formulate an optimization problem that jointly designs the power allocation and scheduling policy to minimize the convergence round of FL while considering the delay requirement from the control system and energy consumption during the FL convergence.
+
+# III. CONVERGENCE ANALYSIS AND JOINT DESIGN
+
+# A. FL convergence analysis
+
+In order to guarantee FL convergence, we assume that the following UAVs adopt a standard gradient descent method to update their local FL models [4]. Thus, for following UAV  $i \in \mathcal{I}$ , the local model  $\boldsymbol{w}_i^{(t)}$  at communication round  $t$  is given by
+
+$$
+\boldsymbol {w} _ {i} ^ {(t)} = \boldsymbol {w} ^ {(t - 1)} - \frac {\bar {\lambda}}{N _ {i}} \nabla F _ {i} (\boldsymbol {w} ^ {(t - 1)}), \tag {7}
+$$
+
+where  $\pmb{w}^{(t - 1)}$  is the global FL model at communication round  $t - 1$ ,  $\bar{\lambda}$  is the learning rate, and  $F_{i}(\pmb{w}^{(t - 1)}) = \sum_{n = 1}^{N_{i}}f(\pmb{w}^{(t - 1)},\pmb{x}_{in},y_{in})$ . After the leading UAV collects local vectors  $\pmb{w}_i^{(t)}, i\in \mathcal{I}$ , the global FL model can be updated:
+
+$$
+\boldsymbol {w} ^ {(t)} = \frac {\sum_ {i = 1} ^ {I} N _ {i} \boldsymbol {w} _ {i} ^ {(t)}}{\sum_ {i = 1} ^ {I} N _ {i}}. \tag {8}
+$$
+
+However, for ensuring successful updates of both global and local FL models as shown in (7) and (8), the transmission delay of uplink and downlink should be within, respectively,  $T_{u}(\beta)$  and  $T_{d}(\beta)$ . Hence, after considering the impact of the transmission delays, we can rewrite the global FL model update as
+
+$$
+\boldsymbol {w} ^ {(t)} = \frac {\sum_ {i = 1} ^ {I} N _ {i} \boldsymbol {w} _ {i} ^ {(t)} C _ {i , t}}{\sum_ {i = 1} ^ {I} N _ {i} C _ {i , t}}, \tag {9}
+$$
+
+with
+
+with  $C_{i,t} = \left\{ \begin{array}{ll}1, & \mathrm{with~probability~}\mathbb{P}(T_{iL,t}\leq T_u(\beta),T_{Li,t}\leq T_d(\beta)),\\ 0, & \mathrm{otherwise.} \end{array} \right.$
+
+With the aim of quantifying the convergence of FL, we use the notion of a convergence round, defined as the minimum number of communication rounds needed to achieve a target difference  $\varepsilon$  of the expected gap between current loss and the minimal loss, i.e.,  $\mathbb{E}(F(\boldsymbol{w}) - F(\boldsymbol{w}^{*})) \leq \varepsilon$ . Moreover, to determine the convergence round, we make the following two standard assumptions: Function  $F(\boldsymbol{w})$ :  $\mathbb{R}^n \to \mathbb{R}$  is continuously differentiable, and the gradient of  $F(\boldsymbol{w})$  is uniformly Lipschitz continuous with positive parameter  $U$ . We also consider the function  $F$  to be strongly convex with positive parameter  $\mu$ , and these exists constants  $\zeta_1 \geq 0$  and  $\zeta_2 \geq 1$ , meeting  $||\nabla F_i(\boldsymbol{w})||^2 \leq \zeta_1 + \zeta_2||\nabla F(\boldsymbol{w})||^2$  [14]. Given the above assumptions, we can derive the convergence round.
+
+Theorem 1. To realize an expected convergence of  $F(\boldsymbol{w})$  under an accuracy threshold  $\varepsilon$ , i.e.,  $\mathbb{E}(F(\boldsymbol{w}) - F(\boldsymbol{w}^{*})) \leq \varepsilon$ ,
+
+the convergence round is given by:
+
+$$
+\varphi = \left[ \log_ {1 - \rho} \frac {\varepsilon}{\sum_ {i = 1} ^ {I} \sum_ {n = 1} ^ {N _ {i}} f (\boldsymbol {w} ^ {(0)} , \boldsymbol {x} _ {i n} , y _ {i n})} \right], \tag {10}
+$$
+
+where  $\lceil \cdot \rceil$  is the ceiling function, and  $\rho$  captures the convergence speed given as follows
+
+$$
+\rho = \frac {\sum_ {i = 1} ^ {T} N _ {i} \mathbb {P} \left(T _ {i L , t} \leq T _ {u} (\beta) , T _ {L i , t} \leq T _ {d} (\beta)\right) \mu}{N U}. \tag {11}
+$$
+
+Proof: Due to the space limitation, the proof is included in Appendix A.
+
+As shown in Theorem 1, the convergence performance of FL depends on the transmission delay of both uplink and downlink in the network. In particular, to increase the convergence speed, we need to maximize the probability that both uplink and downlink meet the corresponding delay requirements of FL. Thus, Theorem 1 provides a concrete characterization of the interplay between wireless communications and FL performance in a UAV swarm.
+
+For the stability analysis of the control system, we will follow the method provided by our previous work in [12]. That is, we first build the augmented error state vector. Then, we use Lyapunov-Razumikhin theorem to derive the control system delay requirements  $\tau_{i}, i \in \mathcal{I}$ , for downlink that can guarantee the stability of the UAV swarm.
+
+# B. Problem formulation and solution concept
+
+Here, we formulate an optimization problem to minimize the convergence round by jointly designing the power allocation and scheduling for the UAV network, as follows:
+
+$$
+\min  _ {\left\{\boldsymbol {p}, p _ {L}, \beta , v \right\}} \varphi \tag {12}
+$$
+
+$$
+\mathrm {s . t .} \mathbb {P} \left[ \varphi E _ {L} + \varphi p _ {L} T _ {d} (\beta) + \varphi \bar {p} _ {L} (v) T _ {r} \leq \bar {E} \right] \geq \xi_ {L}, \tag {13}
+$$
+
+$$
+\mathbb {P} \left[ \varphi E _ {i} + \varphi p _ {i} T _ {i L} + \varphi \bar {p} _ {i} (v) T _ {r} \leq \bar {E} \right] \geq \xi_ {i}, i \in \mathcal {I}, \tag {14}
+$$
+
+$$
+\mathbb {P} \left(T _ {L i} \leq \tau_ {i}\right) \geq \xi_ {C}, i \in \mathcal {I}, \tag {15}
+$$
+
+$$
+p _ {L} \in (0, p _ {L, \max }, p _ {i} \in (0, p _ {i, \max }), i \in \mathcal {I}, \tag {16}
+$$
+
+$$
+\beta \in (0, 1), v \in (0, v _ {\max }) \tag {17}
+$$
+
+where vector  $\pmb{p} = [p_1, \dots, p_I]$ . Constraint (13) guarantees that the probability of total energy consumption for the leading UAV being less than a threshold  $\bar{E}$  will be greater than  $\xi_L \in (0, 1)$ . Similarly to (13), constraint (14) represents the constraint on energy consumption of each follower  $i \in \mathcal{I}$ . Constraint (15) guarantees that the UAV communication network is reliable to support the stability of the swarm with probability  $\xi_C$ . Constraints (16) and (17) ensure that the optimization variables, i.e., the transmission power, scheduling parameter, and velocity, are chosen within reasonable ranges. Note that, in the optimization problem, we also optimize the operation speed of the UAV swarm to minimize the motion energy consumption and relax the energy constraints in (13) and (14).
+
+Since both exponent and base in the logarithm function (10) are less than 1, minimizing the logarithm function in (12) is
+
+equivalent to minimizing the base for the constant exponent. Also, according to (11), we can simplify (12) as
+
+$$
+\max  _ {\left\{\boldsymbol {p}, p _ {L}, \beta , v \right\}} \sum_ {i = 1} ^ {I} N _ {i} \mathbb {P} \left(T _ {i L, t} \leq T _ {u} (\beta), T _ {L i, t} \leq T _ {d} (\beta)\right). \tag {18}
+$$
+
+We observe that, after simplifications, both objective function and constraints are represented by probability terms. In this case, directly deriving the probability terms will be challenging since it requires multidimensional integrations. Also, as the optimization problem is not convex, employing convex approximations to simplify the optimization problem will be impossible. Instead, we use a sample average approximation approach where the probability terms in the objective function and constraints are replaced by an empirical distribution found by random samples [15]. In particular, we first generate  $K$  independent samples of the random parameters, i.e., wireless channel gains and angle deviations, and we calculate the corresponding transmission delay and convergence round. Then, we can reformulate the optimization problem as
+
+$$
+\max  _ {\left\{\boldsymbol {p}, p _ {L}, \beta , v \right\}} \sum_ {i = 1} ^ {I} \sum_ {k = 1} ^ {K} N _ {i} \mathbb {1} \left(T _ {u} (\beta) - T _ {i L, k}\right) \mathbb {1} \left(T _ {d} (\beta) - T _ {L i, k}\right) \tag {19}
+$$
+
+$$
+\text {s . t .} \sum_ {k = 1} ^ {K} \mathbb {1} \left(\bar {E} - \left(\varphi_ {k} E _ {L} + \varphi_ {k} p _ {L} T _ {d} (\beta) + \varphi_ {k} \bar {p} _ {L} (v) T _ {r}\right)\right) \geq K \xi_ {L}, \tag {20}
+$$
+
+$$
+\sum_ {k = 1} ^ {K} \mathbb {1} \left(\bar {E} - \left(\varphi_ {k} E _ {i} + \varphi_ {k} p _ {i} T _ {i L, k} + \varphi_ {k} \bar {p} _ {i} (v) T _ {r}\right)\right) \geq K \xi_ {i}, i \in \mathcal {I}, \tag {21}
+$$
+
+$$
+\sum_ {k = 1} ^ {K} \mathbb {1} \left(\tau_ {i} - T _ {L i, k}\right) \geq K \xi_ {C}, i \in \mathcal {I}, \tag {22}
+$$
+
+(16) and (17),
+
+where the indicator function  $\mathbb{1}(r) = 1$ , once  $r \geq 0$ ; otherwise, we have  $\mathbb{1}(r) = 0$ . Due to the presence of the indicator function, the reformulated problem is non-smooth. To obtain a smooth problem, we can further replace the indicator functions with modified sigmoid functions, i.e.,  $\Gamma(r) = \frac{1}{1 + \exp(-\bar{c}r)}$ , where  $\bar{c}$  determines how quickly the modified sigmoid function changes near 0. To obtain a sub-optimal solution to the reformulated optimization problem with the indicator functions replaced by the modified sigmoid functions, we can use the dual method [16]. In particular, the Lagrangian function is
+
+$$
+\mathcal {J} (\boldsymbol {\lambda}, \boldsymbol {p}, p _ {L}, \beta , v) = \sum_ {i = 1} ^ {I} \sum_ {k = 1} ^ {K} N _ {i} \Gamma (T _ {u} (\beta) - T _ {i L, k}) \Gamma (T _ {d} (\beta) - T _ {L i, k}) +
+$$
+
+$$
+\lambda_ {1} \left(\sum_ {k = 1} ^ {K} \Gamma \left(\bar {E} - \left(\varphi_ {k} E _ {L} + \varphi_ {k} p _ {L} T _ {d} (\beta) + \varphi_ {k} \bar {p} _ {L} (v) T _ {r}\right)\right) - K \xi_ {L}\right) +
+$$
+
+$$
+\sum_ {i = 1} ^ {I} \lambda_ {i + 1} \left(\sum_ {k = 1} ^ {K} \Gamma \left(\bar {E} - \left(\varphi_ {k} E _ {i} + \varphi_ {k} p _ {i} T _ {i L, k} + \varphi_ {k} \bar {p} _ {i} (v) T _ {r}\right)\right) - K \xi_ {i}\right) +
+$$
+
+$$
+\sum_ {i = 1} ^ {I} \lambda_ {I + 1 + i} \left(\sum_ {k = 1} ^ {K} \Gamma \left(\tau_ {i} - T _ {L i, k}\right) - K \xi_ {C}\right), \tag {23}
+$$
+
+where vector  $\pmb{\lambda} = [\lambda_1, \dots, \lambda_{2I + 1}] \succeq \mathbf{0}_{1 \times (2I + 1)}$  is the vector
+
+Table. I. Simulation parameters.  
+
+<table><tr><td>Parameters</td><td>Values</td></tr><tr><td>Number of followers I</td><td>5</td></tr><tr><td>Transmission power threshold pmax</td><td>0.5 W</td></tr><tr><td>Maximum speed vmax</td><td>20 m/s [18]</td></tr><tr><td>Energy consumption efficient κ</td><td>10-28 [18]</td></tr><tr><td>Number of cycles needed per bit C</td><td>103 [18]</td></tr><tr><td>Frequency of the CPU φ</td><td>109cycle/s</td></tr><tr><td>Time for each communication round Tr</td><td>0.1 s</td></tr><tr><td>Side lobe gain Gmin, path loss exponent α</td><td>-2 dB, 2.5</td></tr><tr><td>Noise spectral density γ0</td><td>-174 dBm/Hz</td></tr><tr><td>Packet size Sw and Swi</td><td>10 kB</td></tr><tr><td>Number of rotors q and the diameter r</td><td>4, 0.254 m [4]</td></tr><tr><td>Power efficiency η and density ρ of the air</td><td>70 %, 1.225 kg/m3 [4]</td></tr><tr><td>Number of samples K, Energy limits E</td><td>1,000, 7,000 J</td></tr></table>
+
+of Lagrangian multipliers, and the dual objective function can be defined as  $\mathcal{D}(\pmb{\lambda}) = \max_{\pmb{p}, p_L, v, \beta} \mathcal{J}(\pmb{\lambda}, \pmb{p}, p_L, \beta, v)$ . The corresponding dual optimization problem is
+
+$$
+\min  _ {\boldsymbol {\lambda}} \mathcal {D} (\boldsymbol {\lambda}) \quad \text {s . t .} \boldsymbol {\lambda} \geq \mathbf {0}. \tag {24}
+$$
+
+Although the dual problem in (24) is always convex [17],  $\mathcal{D}(\lambda)$  is not differentiable. Instead, we can use subgradients given by
+
+$$
+\Delta \lambda_ {1} = \sum_ {k = 1} ^ {K} \Gamma \left(\bar {E} - \left(\varphi_ {k} ^ {*} E _ {L} + \varphi_ {k} ^ {*} p _ {L} T _ {d} ^ {*} + \varphi_ {k} ^ {*} \bar {p} _ {L} ^ {*} T _ {r}\right)\right) - K \xi_ {L},
+$$
+
+$$
+\Delta \lambda_ {i + 1} = \sum_ {k = 1} ^ {K} \Gamma \left(\bar {E} - \left(\varphi_ {k} ^ {*} E _ {i} + \varphi_ {k} ^ {*} p _ {i} ^ {*} T _ {i L, k} ^ {*} + \varphi_ {k} ^ {*} \bar {p} _ {i} ^ {*} T _ {r}\right)\right) - K \xi_ {i}, i \in \mathcal {I},
+$$
+
+$$
+\Delta \lambda_ {I + 1 + i} = \sum_ {k = 1} ^ {K} \Gamma \left(\tau_ {i} - T _ {L i, k} ^ {*}\right) - K \xi_ {C}, i \in \mathcal {I}, \tag {25}
+$$
+
+where the terms  $\varphi_k^*, T_d^*, T_{iL,k}^*, T_{Li,k}^*, \bar{p}_L^*, \bar{p}_i^*$  are expressed by optimized variables  $p^*, p_L^*, \beta^*, v^*$ . The proof of subgradients is similar to the one provided in [16], and is omitted here. Thereby, we can solve the problem in (24) by either the subgradient method or the ellipsoid method, and their complexities are, respectively,  $\mathcal{O}\left(\frac{2I + 1}{\epsilon^2}\right)$  and  $\mathcal{O}\left((2I + 1)^2 \ln \frac{1}{\epsilon}\right)$  with accuracy  $\epsilon$  [17]. Then, the sub-optimal solution of  $\{\pmb{p}, p_L, \beta, v\}$  can be obtained by solving dual objective function  $\mathcal{D}(\lambda)$ . In particular, similar to [16], we use the iterative method to sequentially derive the sub-optimal value of each element in  $\{\pmb{p}, p_L, \beta, v\}$  (the details are omitted here due to space limitations). Note that, we assume that all these steps of solving the optimization problem are done by a central unit (e.g., cloud or BS), before the swarm starts training their learning models in FL. In particular, there is no need for the central unit to collect any information from UAVs, since all samples of wireless channel gains and antenna deviations are randomly generated by the central unit itself. Also, since the number of UAVs in the swarm is usually small, the complexity of using sample average approximation and dual approach will be low. As a result, the central unit can readily obtain the sub-optimal solution to the joint design problem and later send the power allocation and scheduling parameters to UAVs in the swarm.
+
+# IV. SIMULATION RESULTS AND ANALYSIS
+
+For our simulations, we first validate the theoretical analysis in Theorem 1. Then, we show the impact of angle deviations on the convergence of FL, and we compare our joint design with baseline schemes that optimize power allocation and schedul
+
+![](images/d48332a15ab16294fc429fd8db6df0fd8d11ab440b33f32abda4c4400b076a11.jpg)  
+Fig. 2. Validation of Theorem 1.
+
+ing separately. In particular, we consider two baselines. The first baseline is a system with optimized power allocation (same power allocation in the joint design) and randomized scheduling parameters. The second baseline is a system with optimized scheduling (same scheduling used by the joint design) and randomized power allocation. We also assume equal uplink and downlink bandwidths, i.e.,  $B_{u} = B_{d} = 1 \mathrm{MHz}$ , and equal angle deviation variance for each UAV, i.e.,  $\sigma_j^2 = \sigma^2$ ,  $j \in \mathcal{I} \cup \{L\}$ . All simulation parameters are summarized in Table I.
+
+Fig. 2 shows the convergence round versus the difference threshold  $\varepsilon$ . Note that, in Fig. 2, we choose the range of  $\varepsilon \in (5, 25)$  based on the value of  $\sum_{i=1}^{I} \sum_{n=1}^{N_i} f(\boldsymbol{w}_0, \boldsymbol{x}_{in}, y_{in})$  and the range of  $\varepsilon$  will be varied for different settings of data and initial global FL model and the accuracy requirement. As observed from Fig. 2, the theoretical analysis derived in Theorem 1 is aligned with the simulation results with less than  $5\%$  difference, thus corroborating the validity of Theorem 1. Moreover, Fig. 2 shows that, when the difference threshold increases, the convergence round decreases. This is because, with a larger difference threshold, the requirement of convergence becomes less stringent. In this case, FL requires fewer communication rounds to converge.
+
+Fig. 3 shows the convergence round when the variance of angle deviations changes. From Fig. 3, we observe that, when the variance of angle deviations increases, FL needs more communication rounds to converge. This is due to the fact that, when the angle deviation variance increases, the antennas at transmitter and receiver in the network will be less aligned, leading to a drop in the antenna gains' product between transmitter and receiver in (4) and (5). As a result, the transmission delay of wireless links will increase, and the probability of meeting the delay requirements, i.e.,  $\mathbb{P}(T_{iL,t} \leq T_u, T_{Li,t} \leq T_d)$ , decreases. Therefore, more communication rounds are needed to achieve the FL convergence. Moreover, as shown in Fig. 3, when the bandwidth allocated to uplink and downlink increases, the FL algorithm requires fewer communication rounds to achieve convergence. This stems from the fact that, a large bandwidth improves the probability of meeting the delay requirements, yielding a fast FL convergence.
+
+Fig. 4 compares our proposed joint power allocation and scheduling design with the baselines without a joint design. It
+
+![](images/9354308eb292f13f5eb1cbce4825b37a3fc2f50ff64a3d3ee2fef09a4608e1b9.jpg)  
+Fig. 3. Impact of angle deviations on the FL convergence.
+
+![](images/98d46815a2e58e129133ac172afe0dc66d6f03182f512840f7e8d36b11a7477b.jpg)  
+Fig. 4. Comparisons between systems with and without joint design.
+
+is shown that, for the same network setting, the convergence round for a network with joint design is always less than its counterparts of baselines. In particular, when the bandwidth is  $1\mathrm{MHz}$ , the system with a joint design reduces the convergence round by as much as  $35\%$  compared with the baseline system with optimized scheduling and randomized power allocation design. Moreover, as shown in Fig. 4, when the bandwidth assigned to uplink and downlink increases, the performance gap between the system with the proposed joint design and the baselines decreases. That is because, as we increase the bandwidth, it becomes more probable for all three systems to meet the delay constraints at uplink and downlink. Therefore, the impact of communications delay on the FL convergence will be minimized.
+
+# V. CONCLUSIONS
+
+In this paper, we have studied the possibility of implementing FL over a swarm of UAVs. In particular, we have carried out a convergence analysis to study the impact of wireless factors, such as transmission delay and antenna angle deviations, on the convergence of FL. Using the derived insight, we have jointly designed the power allocation and scheduling policy for the UAV swarm to optimize the convergence performance of FL while guaranteeing the stability of control system and controlling the energy consumption. Simulation results have corroborated the convergence analysis of FL and showed the merits of the proposed joint design.
+
+# APPENDIX
+
+# A. Proof of Theorem 1
+
+According to the assumptions about function  $F(\pmb{w}) : \mathbb{R}^n \to \mathbb{R}$  made in Section III, we know that function  $F(\pmb{w})$  is continuously differentiable, and the gradient of  $F(\pmb{w})$  is uniformly Lipschitz continuous, i.e., for some positive parameter  $U$ ,  $||\nabla F(\pmb{w}^{(t + 1)}) - \nabla F(\pmb{w}^{(t)})|| \leq U||\pmb{w}^{(t + 1)} - \pmb{w}^{(t)}||$ ; the function  $F$  is strongly convex with positive parameter  $\mu$ :  $F(\pmb{w}^{(t + 1)}) \geq F(\pmb{w}^{(t)}) + (\pmb{w}^{(t + 1)} - \pmb{w}^{(t)})^T\nabla F(\pmb{w}^{(t)}) + \frac{1}{2}\mu ||\pmb{w}^{(t + 1)} - \pmb{w}^{(t)}||$ . If  $F$  is twice-continuously differentiable, these two assumptions are equivalent to  $\mu I \leq \nabla^2 F(\pmb{w}) \leq UI$ . Also, following a standard assumption in stochastic optimization, we consider that there exists constants  $\zeta_1 \geq 0$  and  $\zeta_2 \geq 1$ , meeting  $||\nabla F_i(\pmb{w})||^2 \leq \zeta_1 + \zeta_2||\nabla F(\pmb{w})||^2$  [14].
+
+In this case, since the global FL model is the aggregation of all local FL models, the global FL model without the impact of the transmission delay can be given as
+
+$$
+\boldsymbol {w} ^ {(t)} = \frac {\sum_ {i = 1} ^ {I} N _ {i} \boldsymbol {w} _ {i} ^ {(t)}}{\sum_ {i = 1} ^ {I} N _ {i}} = \boldsymbol {w} ^ {(t - 1)} - \lambda \nabla F (\boldsymbol {w} ^ {(t - 1)}). \tag {26}
+$$
+
+After taking into account the impact of transmission delays, we can rewrite the global FL model update as
+
+$$
+\boldsymbol {w} ^ {(t)} = \frac {\sum_ {i = 1} ^ {I} N _ {i} \boldsymbol {w} _ {i} ^ {(t)} C _ {i , t}}{\sum_ {i = 1} ^ {I} N _ {i} C _ {i , t}} = \boldsymbol {w} ^ {(t - 1)} - \lambda (\nabla F (\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)}),
+$$
+
+where  $e^{(t)} = -\nabla F(\pmb{w}^{(t-1)}) + \frac{\sum_{i=1}^{I} N_i \nabla F_i(\pmb{w}^{(t-1)}) C_{i,t}}{\sum_{i=1}^{I} N_i C_{i,t}}$ . Based on the assumption on the uniform Lipschitz continuity and strong convexity, we can have the following inequalities:
+
+$$
+\begin{array}{l} F (\boldsymbol {w} ^ {(t)}) \leq F (\boldsymbol {w} ^ {(t - 1)}) + (\boldsymbol {w} ^ {(t)} - \boldsymbol {w} ^ {(t - 1)}) ^ {T} \nabla \tilde {F} (\boldsymbol {w} ^ {(t - 1)}) \\ + \frac {U}{2} \| \boldsymbol {w} ^ {(t)} - \boldsymbol {w} ^ {(t - 1)} \| ^ {2}, \tag {27} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} F (\boldsymbol {w} ^ {(t)}) \geq F (\boldsymbol {w} ^ {(t - 1)}) + (\boldsymbol {w} ^ {(t)} - \boldsymbol {w} ^ {(t - 1)}) ^ {T} \nabla F (\boldsymbol {w} ^ {(t - 1)}) \\ + \frac {\mu}{2} \left\| \boldsymbol {w} ^ {(t)} - \boldsymbol {w} ^ {(t - 1)} \right\| ^ {2}. \tag {28} \\ \end{array}
+$$
+
+Since  $\pmb{w}^{(t)} = \overline{\pmb{w}}^{(t - 1)} - \lambda (\nabla F(\pmb{w}^{(t - 1)}) + e^{(t)})$ , we can simplify (27) when the learning rate is  $\lambda = \frac{1}{U}$  as
+
+$$
+\begin{array}{l} F (\boldsymbol {w} ^ {(t)}) \leq F (\boldsymbol {w} ^ {(t - 1)}) - \frac {1}{U} \left(\nabla F (\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)}\right) ^ {T} \nabla F (\boldsymbol {w} ^ {(t - 1)}) \\ + \frac {1}{2 U} | | \nabla F (\boldsymbol {w} ^ {(t - 1)}) + e ^ {(t)} | | ^ {2} \\ = F \left(\boldsymbol {w} ^ {(t - 1)}\right) - \frac {1}{2 U} \left\| \nabla F \left(\boldsymbol {w} ^ {(t - 1)}\right) \right\| ^ {2} + \frac {1}{2 U} \left\| e ^ {(t)} \right\| ^ {2}. \tag {29} \\ \end{array}
+$$
+
+To find a lower bound on the norm of  $\nabla F(\pmb{w}^{(t)})$ , we can minimize both sides of (28) with respect  $\pmb{w}^{(t)}$ . The minimal value of the left-hand side of (28) is achieved when  $\pmb{w}^{(t)} = \pmb{w}^*$ , and the minimal value of the right-hand side of (28) is realized when  $\pmb{w}^{(t)} = \pmb{w}^{(t - 1)} - \frac{1}{\mu}\nabla F(\pmb{w}^{(t - 1)})$ . Particularly, we have
+
+$$
+F \left(\boldsymbol {w} ^ {*}\right) \geq F \left(\boldsymbol {w} ^ {(t - 1)}\right) - \frac {1}{2 \mu} | | \nabla F \left(\boldsymbol {w} ^ {(t - 1)}\right) | | ^ {2}. \tag {30}
+$$
+
+When replacing  $\pmb{w}^{(t - 1)}$  with  $\pmb{w}^{(t)}$  in (30), we can obtain a lower bound for the norm of  $\nabla F(\pmb{w}^{(t)})$  as
+
+$$
+\left| \left| \nabla F \left(\boldsymbol {w} ^ {(t)}\right) \right| \right| ^ {2} \geq 2 \mu \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right). \tag {31}
+$$
+
+Combining (29) and (31), we can obtain an upper bound of the current loss and the minimal loss given by
+
+$$
+F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right) \leq \left(1 - \frac {\mu}{U}\right) \left[ F \left(\boldsymbol {w} ^ {(t - 1)}\right) - F \left(\boldsymbol {w} ^ {*}\right) \right] + \frac {1}{2 U} \left\| e ^ {(t)} \right\| ^ {2}.
+$$
+
+According to [19], when  $\mathbb{E}[||e^{(t)}||^2] \leq 2U\left(\frac{\mu}{U} - \rho^{(t)}\right)\mathbb{E}(F(\boldsymbol{w}^{(t)}) - F(\boldsymbol{w}^*))$ , we can achieve the strong expected linear convergence, i.e.,
+
+$$
+\overline {{\mathbb {E}}} \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right) \leq (1 - \rho^ {(t)}) \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t - 1)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right).
+$$
+
+According to the strong expected linear convergence requirement, we know that the convergence rate satisfies
+
+$$
+\rho^ {(t)} \leq \frac {\mu}{U} - \frac {\mathbb {E} [ | | e ^ {(t)} | | ^ {2} ]}{2 U \mathbb {E} (F (\boldsymbol {w} ^ {(t)}) - F (\boldsymbol {w} ^ {*}))}. \tag {32}
+$$
+
+By using the results in [20], we have the following inequality:
+
+$$
+\begin{array}{l} \mathbb {E} \left(\left\| e ^ {(t)} \right\| ^ {2}\right) \leq \frac {1}{N} \sum_ {i = 1} ^ {I} N _ {i} \left(\zeta_ {1} + \zeta_ {2} \mathbb {E} \left(\nabla F \left(\boldsymbol {w} ^ {(t)}\right)\right)\right) \times \\ (1 - \mathbb {P} (T _ {i L} \leq T _ {u} (\beta), T _ {L i} \leq T _ {d} (\beta))). \tag {33} \\ \end{array}
+$$
+
+The right-hand side of (32) will meet the following inequality:
+
+$$
+\begin{array}{l} \frac {\mu}{U} - \frac {\mathbb {E} [ | | e ^ {(t)} | | ^ {2} ]}{2 U \mathbb {E} (F (\boldsymbol {w} ^ {(t)}) - F (\boldsymbol {w} ^ {*}))} \geq \frac {\mu}{U} - \sum_ {i = 1} ^ {I} N _ {i} \left(\zeta_ {1} + \zeta_ {2} \mathbb {E} \left(\nabla F (\boldsymbol {w} ^ {(t)})\right)\right) \\ \times \frac {\left(1 - \mathbb {P} \left(T _ {i L} \leq T _ {u} (\beta) , T _ {L i} \leq T _ {d} (\beta)\right)\right)}{2 N U \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right)}. \tag {34} \\ \end{array}
+$$
+
+Therefore, to guarantee that (32) always exists, we have
+
+$$
+\begin{array}{l} \rho^ {(t)} \leq \frac {\mu}{U} - \sum_ {i = 1} ^ {I} N _ {i} \left(\zeta_ {1} + \zeta_ {2} \mathbb {E} \left(\nabla F \left(\boldsymbol {w} ^ {(t)}\right)\right)\right) \\ \times \frac {(1 - \mathbb {P} (T _ {i L} \leq T _ {u} (\beta) , T _ {L i} \leq T _ {d} (\beta)))}{2 N U \mathbb {E} (F (\boldsymbol {w} ^ {(t)}) - F (\boldsymbol {w} ^ {*}))} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \stackrel {(a)} {\leq} \frac {\mu}{U} - \sum_ {i = 1} ^ {I} N _ {i} \left(\zeta_ {1} + \zeta_ {2} 2 \mu \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right)\right) \\ \times \frac {\left(1 - \mathbb {P} \left(T _ {i L} \leq T _ {u} (\beta) , T _ {L i} \leq T _ {d} (\beta)\right)\right)}{2 N U \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right)} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \stackrel {(b)} {\leq} \frac {\mu}{U} - \sum_ {i = 1} ^ {I} N _ {i} \left(2 \mu \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right)\right) \\ \times \frac {\left(1 - \mathbb {P} \left(T _ {i L} \leq T _ {u} (\beta) , T _ {L i} \leq T _ {d} (\beta)\right)\right)}{2 N U \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right)} \\ \end{array}
+$$
+
+$$
+= \frac {\mu}{U} - \frac {\sum_ {i = 1} ^ {I} N _ {i} \mu \left(1 - \mathbb {P} \left(T _ {i L} \leq T _ {u} (\beta) , T _ {L i} \leq T _ {d} (\beta)\right)\right)}{N U}, \tag {35}
+$$
+
+where in (a), we use the results derived in (31), and the derivation in (b) is based on the fact that  $\zeta_1 \geq 0$  and  $\zeta_2 \geq 1$ . Assume  $\rho = \frac{\mu}{U} - \frac{\sum_{i=1}^{I} N_i \mu(1 - \mathbb{P}(T_{iL} \leq T_u(\beta), T_{Li} \leq T_d(\beta)))}{NU} = \frac{\sum_{i=1}^{I} N_i \mu(\mathbb{P}(T_{iL} \leq T_u(\beta), T_{Li} \leq T_d(\beta)))}{NU}$ , then, we can have
+
+$$
+\begin{array}{l} \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right) \leq (1 - \rho) \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t - 1)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right) \\ \leq (1 - \rho) ^ {2} \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(t - 2)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right) \\ \end{array}
+$$
+
+··
+
+$$
+\leq (1 - \rho) ^ {t} \mathbb {E} \left(F \left(\boldsymbol {w} ^ {(0)}\right) - F \left(\boldsymbol {w} ^ {*}\right)\right).
+$$
+
+We can further determine the convergence round needed to achieve a target difference threshold, i.e.,  $\mathbb{E}(F(\boldsymbol{w}) - F(\boldsymbol{w}^{*}))\leq \varepsilon$ , as follows:
+
+$$
+\begin{array}{l} t \geq \log_ {1 - \rho} \frac {\varepsilon}{\mathbb {E} (F (\boldsymbol {w} ^ {(0)}) - F (\boldsymbol {w} ^ {*}))} \\ \stackrel {(a)} {\geq} \log_ {1 - \rho} \frac {\varepsilon}{\mathbb {E} (F (\boldsymbol {w} ^ {(0)}))} \\ = \log_ {1 - \rho} \frac {\varepsilon}{\sum_ {i = 1} ^ {I} \sum_ {n = 1} ^ {N _ {i}} f \left(\boldsymbol {w} ^ {(0)}, \boldsymbol {x} _ {i n} , y _ {i n}\right)}, \tag {36} \\ \end{array}
+$$
+
+where in (a), we use the fact that  $1 - \rho \leq 1$ . Since the convergence round must be integral, we can have the results in Theorem 1.
+
+# REFERENCES
+
+[1] M. Mozaffari, W. Saad, M. Dennis, Y. Nam, and M. Debbah, "A tutorial on UAVs for wireless networks: Applications, challenges, and open problems," IEEE Communications Surveys Tutorials, vol. 21, no. 3, pp. 2334-2360, Third Quarter, 2019.  
+[2] M. Mozaffari, A. Kasgari, W. Saad, M. Bennis, and M. Debbah, "Beyond 5G with UAVs: Foundations of a 3D wireless cellular network," IEEE Transactions on Wireless Communications, vol. 18, no. 1, pp. 357-372, Jan 2019.  
+[3] J. Park, S. Samarakoon, M. Bennis, and M. Debbah, "Wireless network intelligence at the edge," Proceeding of the IEEE, to appear, 2019.  
+[4] J. Konecný, H. B. McMahan, D. Ramage, and P. Richtárik, "Federated optimization: Distributed machine learning for on-device intelligence," arXiv preprint http://arxiv.org/abs/1610.02527 arXiv:1610.02527, 2016.  
+[5] M. Chen, H. V. Poor, W. Saad, and S. Cui, "Convergence time optimization for federated learning over wireless networks," arXiv preprint http://arxiv.org/abs/2001.07845 arXiv:2001.07845.  
+[6] Q. Zeng, Y. Du, K. K. Leung, and K. Huang, "Energy-efficient radio resource allocation for federated edge learning," arXiv preprint http://arxiv.org/abs/1907.06040 arXiv:1907.06040, 2019.  
+[7] H. H. Yang, Z. Liu, T. Q. S. Quek, and H. V. Poor, "Scheduling policies for federated learning in wireless networks," IEEE Transactions on Communications, to appear, 2019.  
+[8] N. H. Tran, W. Bao, A. Zomaya, N. Minh N.H., and C. S. Hong, "Federated learning over wireless networks: Optimization model design and analysis," in Proc. of IEEE International Conference on Computer Communications, Paris, France, Apr. 2019.  
+[9] Y. Pan, C. Pan, Z. Yang, and M. Chen, "Resource allocation for D2D communications underlaying a NOMA-based cellular network," IEEE Wireless Communications Letters, vol. 7, no. 1, pp. 130-133, Feb. 2018.  
+[10] M. T. Dabiri, H. Safi, S. Parsaeefard, and W. Saad, "Analytical channel models for millimeter wave UAV networks under hovering fluctuations," arXiv preprint http://arxiv.org/abs/1905.01477 arXiv:1905.01477, 2019.  
+[11] ITU-R, Mathematical models for radiodetermination radar systems antenna patterns for use in interference analyses, Recommendation ITU-R M.1851-1, Jan. 2018.  
+[12] T. Zeng, M. Mozaffari, O. Semiari, W. Saad, M. Bennis, and M. Debbah, "Wireless communications and control for swarms of cellular-connected UAVs," in Proc. of IEEE Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, USA, Oct. 2018.  
+[13] J. K. Stolaroff, C. Samaras, E. R. O'Neill, A. S. Mitchell, and D. Ceperley, "Energy use and life cycle greenhouse gas emissions of drones for commercial package delivery," Nature Communications, vol. 9, no. 409, pp. 1-13, Feb. 2018.  
+[14] D. P. Bertsekas and J. N. Tsitsiklis, Neuro-dynamic programming. Athena Scientific Belmont, MA, 1996, vol. 5.  
+[15] B. K. Pagnoncelli, S. Ahmed, and A. Shapiro, "Sample average approximation method for chance constrained programming: theory and applications," Journal of optimization theory and applications, vol. 142, no. 2, pp. 399-416, 2009.  
+[16] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Transactions on Communications, vol. 54, no. 7, pp. 1310–1322, July 2006.  
+[17] Y. Nesterov, Lectures on convex optimization. Springer, 2018, vol. 137.  
+[18] F. Zhou, Y. Wu, R. Q. Hu, and Y. Qian, "Computation rate maximization in UAV-enabled wireless-powered mobile-edge computing systems," IEEE Journal on Selected Areas in Communications, vol. 36, no. 9, pp. 1927-1941, Sep. 2018.  
+[19] M. P. Friedlander and M. Schmidt, “Hybrid deterministic-stochastic methods for data fitting,” SIAM Journal on Scientific Computing, vol. 34, no. 3, pp. A1380-A1405, 2012.  
+[20] M. Chen, Z. Yang, W. Saad, C. Yin, H. V. Poor, and S. Cui, "A joint learning and communications framework for federated learning over wireless networks," arXiv preprint http://arxiv.org/abs/1909.07972 arXiv:1909.07972, 2019.

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+# PHYSICS-INFORMED NEURAL NETWORKS FOR SOLVING NONLINEAR DIFFUSIVITY AND BIOT'S EQUATIONS
+
+A PREPRINT
+
+# Teeratorn Kadeethum
+
+Department of Applied Mathematics and Computer Science
+
+Technical University of Denmark
+
+Lyngby, Denmark
+
+teekad@dtu.dk
+
+# Thomas M Jørgensen
+
+Department of Applied Mathematics and Computer Science
+
+Technical University of Denmark
+
+Lyngby, Denmark
+
+tmjq@dtu.dk
+
+# Hamidreza M Nick
+
+The Danish Hydrocarbon Research and Technology Centre
+
+Technical University of Denmark
+
+Lyngby, Denmark
+
+hamid@dtu.dk
+
+February 20, 2020
+
+# ABSTRACT
+
+This paper presents the potential of applying physics-informed neural networks for solving nonlinear multiphysics problems, which are essential to many fields such as biomedical engineering, earthquake prediction, and underground energy harvesting. Specifically, we investigate how to extend the methodology of physics-informed neural networks to solve both the forward and inverse problems in relation to the nonlinear diffusivity and Biot's equations. We explore the accuracy of the physics-informed neural networks with different training example sizes and choices of hyperparameters. The impacts of the stochastic variations between various training realizations are also investigated. In the inverse case, we also study the effects of noisy measurements. Furthermore, we address the challenge of selecting the hyperparameters of the inverse model and illustrate how this challenge is linked to the hyperparameters selection performed for the forward one.
+
+Keywords physics-informed neural networks  $\cdot$  nonlinear Biot's equations  $\cdot$  deep learning  $\cdot$  inverse problem
+
+# Introduction
+
+The volumetric displacement of a porous medium caused by the changes in fluid pressure inside the pore spaces is essential for many applications, including groundwater flow, underground heat mining, fossil fuel production, earthquake mechanics, and biomedical engineering [1-5]. Such volumetric deformation may impact the hydraulic storability and permeability of porous material, which influences the fluid flow behavior. This multiphysics problem, i.e., the coupling between fluid flow and solid deformation, can be captured through the application of Biot's equations of poroelasticity [6, 7]. The Biot's equations can be solved by analytical solutions for simple cases [8, 9]. In complex cases, the finite difference approximation [10, 11], finite volume discretization [12, 13], or more commonly finite element
+
+methods such as the mixed formulation or discontinuous/enriched Galerkin, [14-21], can be used. These numerical methods, however, require significant computational resources, and the accuracy of the solution depends heavily on the quality of the generated mesh. Hence, these methods may not be suitable to handle an inverse problem [22, 23] or a forward problem with complex geometries [24-26].
+
+Recent proposals have speculated that neural-network-based approaches such as deep learning might be an appealing alternative in solving physical problems, which are governed by partial differential equations since they operate without a mesh and are scalable on multithreaded systems [25-29]. Moreover, deep learning has been successfully applied to many applications [30-32] because of its capability to handle highly nonlinear problems [33]. As this technique, in general, requires a significantly large data set to reach a reasonable accuracy [34], its applicability to many scientific and industrial problems can be a challenge [35]. Use of a priori knowledge, however, can reduce the amount of examples needed. For instance, the idea of encoding physical information into the architectures and loss functions of the deep neural networks has been successfully applied to computational fluid dynamics problems [24, 26-29, 36]. The published results illustrate that by incorporating the physical information in the form of regularization terms of the loss functions, the neural networks can be trained to provide good accuracy with a reasonably sized data set.
+
+Since the coupled fluid and solid mechanics process is highly nonlinear and generally involves complex geometries [19,37,38], it seems to fit well into the context of physics-informed neural networks (PINN). For this reason, we propose to apply PINN for solving the nonlinear diffusivity and Biot's equations, both concerning forward and inverse modeling. The rest of the paper is organized as follows. The governing equations of the coupled solid and fluid mechanics are presented in the methodology section. Subsequently, the PINN architecture and its loss functions are defined. We then present forward and inverse modeling results for both the nonlinear diffusivity equation as well as Biot's equations. We also study the impact of the stochastic variations of the training procedures on the accuracy of the predicted primary variables and estimated parameters. Finally, we conclude the findings and describe the possibilities that can enhance the capability of this model, which should be addressed in future works.
+
+# Methodology
+
+# Governing equations
+
+We are interested in solving the nonlinear Biot's equations on the closed domain  $(\Omega)$  which amounts to a time-dependent multiphysics problem coupling solid deformation with fluid flow. Let  $\Omega \subset \mathbb{R}^d$  be the domain of interest in  $d$ -dimensional space where  $d = 1,2$ , or 3 and bounded by boundary,  $\partial \Omega$ .  $\partial \Omega$  can be decomposed into displacement and traction boundaries,  $\partial \Omega_u$  and  $\partial \Omega_\sigma$ , respectively, for the solid deformation problem. For the fluid flow problem,  $\partial \Omega$  is decomposed into pressure and flux boundaries,  $\partial \Omega_p$  and  $\partial \Omega_q$ , respectively. In short,  $\partial \Omega_u$  and  $\partial \Omega_p$  represent the first-kind boundary condition or Dirichlet boundary condition  $(\partial \Omega_D)$ . The  $\partial \Omega_\sigma$  and  $\partial \Omega_q$ , on the other hand, represent the second-kind boundary condition or Neumann boundary condition  $(\partial \Omega_N)$ . The time domain is denoted by  $\mathbb{T} = (0,\tau]$  with  $\tau > 0$ .
+
+As just stated, the coupling between the fluid flow and solid deformation can be captured through the application of Biot's equations of poroelasticity, which is composed of linear momentum and mass balance equations [6]. The linear momentum balance equation can be written as follows:
+
+$$
+\nabla \cdot \boldsymbol {\sigma} (\boldsymbol {u}, p) = \boldsymbol {f}, \tag {1}
+$$
+
+where  $\pmb{u}$  is displacement,  $p$  is fluid pressure,  $\pmb{f}$  is body force. The bold-face letters or symbols denote tensors and vectors, and the normal letters or symbols denote scalar values. Here,  $\sigma$  is the total stress, which is defined as:
+
+$$
+\boldsymbol {\sigma} := \boldsymbol {\sigma} ^ {\prime} (\boldsymbol {u}) - \alpha p \boldsymbol {I}, \tag {2}
+$$
+
+where  $I$  is the identity tensor and  $\alpha$  is Biot's coefficient defined as [39]:
+
+$$
+\alpha := 1 - \frac {K}{K _ {s}}, \tag {3}
+$$
+
+with the bulk modulus of a rock matrix  $K$  and the solid grains modulus  $K_{s}$ . In addition,  $\sigma^{\prime}$  is an effective stress defined as:
+
+$$
+\boldsymbol {\sigma} ^ {\prime} (\boldsymbol {u}) := 2 \mu_ {l} \varepsilon (\boldsymbol {u}) + \lambda_ {l} \boldsymbol {u} \boldsymbol {I}, \tag {4}
+$$
+
+where  $\lambda_{l}$  and  $\mu_{l}$  are Lamé constants,  $\varepsilon (\pmb {u})$  is strain assuming infinitesimal displacements defined as:
+
+$$
+\varepsilon (\boldsymbol {u}) := \frac {1}{2} \left(\nabla \boldsymbol {u} + \nabla^ {T} \boldsymbol {u}\right). \tag {5}
+$$
+
+We can write the linear momentum balance and its boundary conditions as:
+
+$$
+\nabla \cdot \boldsymbol {\sigma} ^ {\prime} (\boldsymbol {u}) - \alpha \nabla \cdot p \boldsymbol {I} = \boldsymbol {f} \text {i n} \Omega \times \mathbb {T}, \tag {6}
+$$
+
+$$
+\boldsymbol {u} = \boldsymbol {u} _ {D} \text {o n} \partial \Omega_ {u} \times \mathbb {T}, \tag {7}
+$$
+
+$$
+\boldsymbol {\sigma} \cdot \boldsymbol {n} = \boldsymbol {\sigma} _ {D} \text {o n} \partial \Omega_ {\sigma} \times \mathbb {T}, \tag {8}
+$$
+
+$$
+\boldsymbol {u} = \boldsymbol {u} _ {0} \text {i n} \Omega \text {a t} t = 0, \tag {9}
+$$
+
+where  $\boldsymbol{u}_D$  and  $\sigma_D$  are prescribed displacement and traction at boundaries, respectively,  $n$  is a normal unit vector, and  $t$  is time.
+
+The mass balance equation is written as [38, 40]:
+
+$$
+\left(\phi c _ {f} + \frac {\alpha - \phi}{K _ {s}}\right) \frac {\partial p}{\partial t} + \alpha \frac {\partial \nabla \cdot \boldsymbol {u}}{\partial t} - \nabla \cdot \mathcal {N} [ \boldsymbol {\kappa} ] (\nabla p - \rho \mathbf {g}) = g \text {i n} \Omega \times \mathbb {T}, \tag {10}
+$$
+
+$$
+p = p _ {D} \text {o n} \partial \Omega_ {p} \times \mathbb {T}, \tag {11}
+$$
+
+$$
+- \mathcal {N} [ \boldsymbol {\kappa} ] (\nabla p - \rho \mathbf {g}) \cdot \boldsymbol {n} = q _ {D} \text {o n} \partial \Omega_ {q} \times \mathbb {T}, \tag {12}
+$$
+
+$$
+p = p _ {0} \text {i n} \Omega \text {a t} t = 0, \tag {13}
+$$
+
+where  $\rho$  is fluid density,  $\phi$  is initial porosity and remains constant throughout the simulation (the volumetric deformation is represented by  $\partial \nabla \cdot \boldsymbol{u} / \partial t$ ),  $c_{f}$  is fluid compressibility,  $\mathbf{g}$  is a gravitational vector,  $g$  is sink/source,  $p_{D}$  and  $q_{D}$  are specified pressure and flux, respectively,  $\mathcal{N}[\cdot]$  represents a nonlinear operator, and  $\kappa$  is hydraulic conductivity defined as:
+
+$$
+\kappa := \left[ \begin{array}{l l l} \kappa^ {x x} & \kappa^ {x y} & \kappa^ {x z} \\ \kappa^ {y x} & \kappa^ {y y} & \kappa^ {y z} \\ \kappa^ {z x} & \kappa^ {z y} & \kappa^ {z z} \end{array} \right], \tag {14}
+$$
+
+where the tensor components characterize the transformation of the components of the gradient of fluid pressure into the components of the velocity vector. The  $\kappa^{xx}$ ,  $\kappa^{yy}$ , and  $\kappa^{zz}$  represent the matrix permeability in x-, y-, and z-direction, respectively. In this study, all off-diagonal terms are zero because we assume that a porous media is isotropic and homogeneous [41, 42]. Note that the diffusivity equation is a specific case of the Biot's equations since Eqs (6) and (10) decouple when  $\alpha = 0$ . The details of this equation are presented in the results and discussion section.
+
+# Physics-informed neural networks model
+
+# Neural network architecture
+
+The neural network architecture used in this study is presented in Fig 1 [33,43,44]. The number of input and output nodes in the neural networks are determined from the problem formulation; for example, if the problem is time-dependent poroelasticity (as discussed in the previous section) and bounded by  $\Omega = [0,1]^1$ , we have two input nodes  $(x$  and  $t)$  and two output nodes  $(u$  and  $p)$  where  $x$  is coordinate in x-direction,  $t$  is time,  $u$  is the displacement in x-direction, and  $p$  is fluid pressure. The number of hidden layers  $(N_{hl})$  and the number of neurons  $(N_{n})$  act as so-called hyperparameters [45]. Each neuron (e.g.,  $\mathrm{H}_{1,1}\ldots \mathrm{H}_{1,\mathrm{N}_n}$ ) is connected to the nodes of the previous layer with adjustable
+
+weights and also has an adjustable bias. We denote the set of weights and biases as (W) and (b), respectively. These variables are learned during a training phase [44,45]. In this paper, we define all hidden layers to have the same number of neurons.
+
+Physics-informed neural networks encode the information given by the differential operators as specific regularizing terms of the loss functions used when training the networks (see the section below). Because the training examples that are used to evaluate these extra regularizing terms, in general, are different from those used to train the network shown in Fig 1, one conceptually introduces an additional neural network - denoted the physical informed neural network [26]. This additional neural network is dependent on all the W and b of the first neural network, but it introduces some extra variables to be learned for the inverse modeling case. This is elaborated in more detail in the below section on training the PINN. Note that while the neural network shown in Fig 1 can be seen as point-to-point learning, the physics-informed regularization indirectly contributes to the local interactions (the stencil). The neural networks are built on the Tensorflow platform [46]. The results produced using either the rectified linear unit (ReLU) or the hyperbolic tangent (tanh) were comparable. Hence, we only present the results using the tanh activation function in this paper.
+
+![](images/78fe112e3aa3aca0013531170de5c6e6db0aff61826ee4141f40a3ab7ea5e089.jpg)  
+Figure 1: General neural network architecture used in this study [33,43,44]. The input layer contains up to  $i$  input nodes, and the output layer is composed of  $1,\dots,k$  output nodes.  $N_{hl}$  refers to the number of hidden layers, and each hidden layer is composed of  $N_{n}$  neurons. Each neuron (e.g.,  $\mathrm{H}_{1,1}\ldots \mathrm{H}_{1,\mathrm{N}_n}$ ) is connected to the nodes of the previous layer with adjustable weights and also has an adjustable bias.
+
+# Physics-informed function
+
+We encode the underlying physical information to the neural networks through the so-called physics-informed function (II), acting as additional regularizing terms in the loss function defined below.
+
+For the linear momentum balance equation Eq (6), we define  $\Pi_{\pmb{u}}$  as follows:
+
+$$
+\Pi_ {\boldsymbol {u}} := \nabla \cdot \boldsymbol {\sigma} ^ {\prime} (\boldsymbol {u}) - \alpha \nabla \cdot p \boldsymbol {I} - \boldsymbol {f} \text {i n} \Omega \times \mathbb {T}, \tag {15}
+$$
+
+and with reference to Eq (8) for its  $\partial \Omega_{\sigma}$ :
+
+$$
+\Pi_ {u _ {\sigma}} := \boldsymbol {\sigma} \cdot \boldsymbol {n} - \boldsymbol {\sigma} _ {D} \text {o n} \partial \Omega_ {\sigma} \times \mathbb {T}, \tag {16}
+$$
+
+and for the mass balance equation Eq (10), we define the  $\Pi_p$  as:
+
+$$
+\Pi_ {p} := \left(\phi c _ {f} + \frac {\alpha - \phi}{K _ {s}}\right) \frac {\partial p}{\partial t} + \alpha \frac {\partial \nabla \cdot \boldsymbol {u}}{\partial t} - \nabla \cdot \mathcal {N} [ \boldsymbol {\kappa} ] (\nabla p - \rho \mathbf {g}) - g \text {i n} \Omega \times \mathbb {T}, \tag {17}
+$$
+
+and for its  $\partial \Omega_q$  according to Eq (12)
+
+$$
+\Pi_ {p _ {q}} := - \mathcal {N} [ \boldsymbol {\kappa} ] (\nabla p - \rho \mathbf {g}) \cdot \boldsymbol {n} - q _ {D} \text {o n} \partial \Omega_ {q} \times \mathbb {T}. \tag {18}
+$$
+
+Demanding the  $\Pi$  terms above to be as close to zero as possible corresponds to fulfilling Eqs. (6), (8), (10), and (12).
+
+# Loss function definition
+
+The loss function applied with the PINN scheme is composed of two parts (here we use a mean squared error -  $MSE$  as the metric). The error on the training data  $(MSE_{tr})$  and the mean square value of the regularization term given by the physics-informed function  $(MSE_{\Pi})$ :
+
+$$
+M S E = M S E _ {t r} + M S E _ {\Pi}, \tag {19}
+$$
+
+where
+
+$$
+M S E _ {\Pi} = M S E _ {\Pi_ {\Omega}} + M S E _ {\Pi_ {\partial \Omega_ {N}}}, \tag {20}
+$$
+
+$$
+M S E _ {\Pi_ {\Omega}} = M S E _ {\Pi_ {u}} + M S E _ {\Pi_ {p}}, \tag {21}
+$$
+
+and
+
+$$
+M S E _ {\Pi_ {\partial \Omega_ {N}}} = M S E _ {\Pi_ {u _ {\sigma}}} + M S E _ {\Pi_ {p q}}. \tag {22}
+$$
+
+where  $MSE_{\Pi_u}, MSE_{\Pi_{u_\sigma}}, MSE_{\Pi_p}$ , and  $MSE_{\Pi_{p_q}}$  correspond to the loss function of Eqs (15), (16), (17), and (18), respectively. The Dirichlet boundary conditions given on  $\partial \Omega_D$  with respect to the linear momentum Eq (7) and mass balance Eq (11) equations are automatically incorporated into the  $MSE_{tr}$ .
+
+A graphical presentation of how boundary points, initial points, as well as domain data, are used for training the neural networks is provided in Fig 2. For the forward model, the set of points that constitutes  $\partial \Omega_D \times \mathbb{T}$  and  $\Omega$  at  $t = 0$  contributes to the  $MSE_{tr}$  term of the loss function. We denote this set of training data as  $N_b$ . Besides, we have a set of collocation points that are sampled from  $\partial \Omega_N \times \mathbb{T}$  and  $\Omega \times \mathbb{T}$ . These training data are used to minimize the  $\Pi$  parts of the loss function, and we denote these data points as  $N_{\Pi}$ .
+
+For the inverse model, we will have a set of known/measured data points that can be sampled from the whole domain, i.e.,  $\Omega \times \mathbb{T}$ . We refer to this set of training data as  $N_{tr}$ . All this data will contribute to both the  $MSE_{tr}$  and the  $MSE_{\Pi_{\Omega}}$  terms of the loss function. Also, we may include an extra set of collocation points (i.e., data where we would have no measured values), which would contribute only to the  $MSE_{\Pi_{\Omega}}$  term.
+
+# Training the PINN
+
+The structure of a PINN architecture is shown in Fig 3. We assume a simple case of two inputs,  $\mathrm{I}_1$  and  $\mathrm{I}_2$ , one output,  $\mathrm{O}$ , and one hidden layer with two neurons. Moreover, we here assume that  $\Pi$  is a function of  $\mathrm{O}$ , the first derivative of  $\mathrm{O}$  with respect to the set of inputs, and the set of physical parameters,  $\theta$ . With  $(\cdot)_{\mathrm{tr}}$  we represent a training set, where the  $\mathrm{O}$  values are known for given  $\mathrm{I}_{1,\mathrm{tr}}$  and  $\mathrm{I}_{2,\mathrm{tr}}$ . This set is used to evaluate the  $MSE_{tr}$  term of Eq (19) at each training cycle. The  $MSE_{\Pi}$  part of Eq (19) is obtained by calculating  $\Pi$  for the collocation points  $(\cdot)_{\Pi}$ ; see bottom part of Fig 3. Calculating  $\Pi$  involves knowing  $\mathrm{O}$  and its derivatives with respect to  $\mathrm{I}_1$  and  $\mathrm{I}_2$ . As these values are unknown for the collocation points, they are estimated using the current approximation of  $\mathrm{O}(\mathrm{W},\mathrm{b},\mathrm{I}_1,\mathrm{I}_2)$  and its derivatives, which can be obtained using automatic differentiation [47,48]. In this way, the regularization term  $MSE_{\Pi}$  indirectly depends on the weights,  $\mathrm{W}$ , and biases,  $\mathrm{b}$ , of the neural architecture shown at the top of Fig 3. As a result,  $\Pi$  can be written as a function of  $\mathrm{W}$ ,  $\mathrm{b}$ ,  $\mathrm{I}_{1,\Pi}$ ,  $\mathrm{I}_{2,\Pi}$ , and  $\theta$ . If we denote this function,  $\gamma$ , we have  $\gamma (\mathrm{W},\mathrm{b},\mathrm{I}_{1,\Pi},\mathrm{I}_{2,\Pi},\theta) = \Pi\left(O(\mathrm{W},\mathrm{b},\mathrm{I}_{1,\Pi},\mathrm{I}_{2,\Pi}),\frac{\partial\mathrm{O}}{\partial\mathrm{I}_1},\frac{\partial\mathrm{O}}{\partial\mathrm{I}_2},\theta\right)$ . Note that the specific mapping of  $\mathrm{I}_{1,\Pi}$  and  $\mathrm{I}_{2,\Pi}$  to  $\Pi$  has been defined as the physical informed neural network in previous work [26] as opposed to the neural network shown in the top of Fig 3.
+
+For both the forward and the inverse modeling cases, one trains the neural networks to establish a mapping from the input space given by  $\mathrm{I}_1$  and  $\mathrm{I}_2$  to the output space, O, by minimizing  $MSE_{tr}$  and  $MSE_{\Pi}$ . The essential differences between the two cases come down to the type of training examples being available to train the networks, as discussed in Fig 2, and whether the physical parameters  $(\theta)$  are known or not. For the forward modeling, we apply  $(\cdot)_{\mathrm{tr}}$  to evaluate the  $MSE_{tr}$  term and  $(\cdot)_{\Pi}$  to evaluate the  $MSE_{\Pi}$  term. The W and b are then adjusted to minimize the sum of these
+
+![](images/a2ee06213ad211268cbf3f570afd3d215b25d20700c558967deef639ee86b147.jpg)  
+Figure 2: Illustration of the parts of input space used for training the PINN: (a) forward model and (b) inverse model. The collocation  $(\Omega \times \mathbb{T})$ , boundary and initial points  $(\partial \Omega_{D} \times \mathbb{T}$  and  $\Omega$  at  $t = 0$ ) are utilized in the forward model. However, only the training points  $(\Omega \times \mathbb{T})$  are employed in the inverse model. X represents a set of spatial coordinates  $(x, y, \text{and } z)$ ,  $t$  is a set of coordinates in the time domain, and S is a set of solution values  $(\boldsymbol{u}$  and  $p$ ) corresponding to X and  $t$ .
+
+![](images/8f4c097b72c037011ac27c8d87b9f7e7f15d8ce5e1802c1159d9a1b4ff53e4b8.jpg)
+
+terms. One should emphasize that for the forward modeling, all the variables to be learned belong to the neural network depicted in the top of Fig 3.
+
+For the inverse modeling, the aim is to estimate  $\theta$ . We still, however, train a neural network to predict O (similar to the forward case). That is, we are not using a loss function that involves measuring a distance between estimated values of  $\theta$  and their ground truth values. Instead, the reasoning behind solving the inverse problem is that we expect the unknown  $\theta$  to converge towards their true values during training because we also allow  $\theta$  to be adjusted along with W and b. During a training phase, these variables are learned to minimize the combined sum of  $MSE_{tr}$  and  $MSE_{\Pi}$ . Specifically, the variables are adjusted by backpropagating the errors as calculated by Eq. (19) [44, 45]. Unlike the forward problem, the boundary and initial conditions are unknown and cannot be used to generate training examples. Instead, we provide training examples that, in real cases, would be obtained from measurements, which ideally correspond to solution points of the forward problem inside  $\Omega$  (see Fig 2b).
+
+As discussed above, the solution of the inverse problem is based on training the neural network to establish a mapping from  $\mathrm{I}_1$  and  $\mathrm{I}_2$  to  $\mathrm{O}$  as we do in the forward case. This means that the hyperparameters estimated from the forward modeling may act as qualified estimates for the hyperparameters in the inverse case, assuming the number of training examples in both cases are similar. By definition of the inverse problem, we do not know the true values of  $\theta$ , so we would have to assume that the exact values of  $\theta$  have a limited influence on the hyperparameters. A more direct way of estimating the hyperparameters in the inverse case would be to divide the data into training and validation sets and then select the hyperparameters that minimize  $MSE$  with respect to learning the mapping from  $\mathrm{I}_1$  and  $\mathrm{I}_2$  to  $\mathrm{O}$ . The downside of this is that we could then not spend all the measurement data on training the neural networks.
+
+# Results and Discussion
+
+We first consider the results of forward and inverse models of the nonlinear diffusivity equation. Subsequently, we present the results of the nonlinear Biot's equations for both the forward and inverse models.
+
+# Nonlinear diffusivity equation
+
+We assume  $\alpha = 0$  to decouple the momentum and mass balance equations and focus on Eq (10), which is reduced to
+
+$$
+\phi c _ {t} \frac {\partial p}{\partial t} - \nabla \cdot \mathcal {N} [ \kappa ] (\nabla p - \rho \mathbf {g}) = g \text {i n} \Omega \times \mathbb {T}, \tag {23}
+$$
+
+![](images/87471125d011f1bfb078f608555c2c8c3dd34cb00d82abbe1f05a34084d291b1.jpg)  
+Figure 3: Graphical illustration of how a traditional neural network is linked to a physics-informed neural network. The set of  $\theta$  represents unknown physical parameters that we want to estimate.
+
+where  $c_{t}$  is a total compressibility. The same boundary and initial conditions, Eqs (11) to (13), are still valid.
+
+We take  $\Omega = [0,1]^1$ ,  $\mathbb{T} = [0,1]$ , and choose the exact solution in  $\Omega$  as:
+
+$$
+p (x, t) := \sin (x + t), \tag {24}
+$$
+
+and  $\mathcal{N}[\kappa ]$  as:
+
+$$
+\mathcal {N} [ \kappa ] := \kappa_ {0} p ^ {2}, \tag {25}
+$$
+
+where  $x$  and  $t$  represent points in  $x$ -direction and time domain, respectively. The  $\kappa_0$  is assumed to be a scalar in this case, i.e.,  $\kappa_0 = \kappa_0$ . All the physical constants are set to 1.0; and subsequently,  $g$  is chosen as:
+
+$$
+g (x, t) := \cos (x + t) + \sin (x + t) ^ {3} - 2 \cos (x + t) ^ {2} \sin (x + t), \tag {26}
+$$
+
+to satisfy the exact solution. Furthermore, the homogeneous boundary conditions are applied to all boundaries and initial conditions using Eq (24). Combining Eqs (23) to (26), the physics-informed function (II) for the nonlinear diffusivity equation is defined:
+
+$$
+\Pi (x, t) := \phi c _ {t} \frac {\partial p}{\partial t} - \kappa_ {0} \frac {\partial}{\partial x} p ^ {2} \left(\frac {\partial}{\partial x} p\right) - g. \tag {27}
+$$
+
+We generate the solution based on an interval mesh having 2559 equidistant spatial intervals and 99 temporal ones; hence, in total, we have 256000 solution points, including the points on boundaries. Subsequently, we randomly draw  $n$  training examples. Half of the remaining points are randomly selected as a validation set, and the remaining ones are used for testing. As an illustration, if we have 256000 solution points and use 100 examples to train the model. We then use 127950 examples for the validation and 127950 examples for the test set.
+
+For the nonlinear diffusivity equation, the forward modeling with a neural network should calculate the  $p$  at any given  $x$  and  $t$  from provided values of  $\phi$ ,  $c_t$ , and  $\kappa_0$ . For the inverse case, the aim is to infer  $\phi$ ,  $c_t$ , and  $\kappa_0$  from observed values of  $x$ ,  $t$ , and  $p$ . The architecture of the neural network corresponding to the top of Fig 3 is illustrated in Fig 4. We have two input nodes and one output node for this case. We use L-BFGS [49]; a quasi-Newton, fullbatch gradient-based optimization algorithm to minimize the loss function with stop criteria as  $\frac{|MSE^k - MSE^{k + 1}|}{\max(|MSE^k|, |MSE^{k + 1}|, 1.0)} < = 10^{-16}$  where  $(\cdot)^k$  and  $(\cdot)^{k + 1}$  are previous and current iteration, respectively. The L-BFGS algorithm has several advantages that are suitable for this study; for example, it is more stable compared to stochastic gradient descent (SGD) and can handle large batch sizes well [49, 50]. This setting is used for all the simulations presented in this paper unless stated otherwise.
+
+![](images/2a3168c38adb25fc94e711da36eadb60db9d95211699766b7c218512b7f2e421.jpg)  
+Figure 4: Neural network architecture used for nonlinear diffusivity equation. This network corresponds to the top part of Fig 3. There are two input nodes,  $x$  and  $t$ , and one output node,  $p$ . The number of hidden layers,  $N_{hl}$ , and the number of neurons for each hidden layer,  $N_{n}$ , denote the hyperparameters.
+
+To reiterate, the sampling strategies of the forward and inverse modelings are illustrated in Fig 2. To be specific, in the forward case, we must determine the solution of the partial differential equation based on known boundary values for the time interval of  $\mathbb{T} = (0,\tau ]$  combined with the initial values at  $t = 0$ . These boundary and initial points are used to calculate the  $MSE_{tr}$  term of Eq. 19. The inner points act as collocation points and are used to calculate the  $MSE_{\Pi}$  term of Eq. 19. Note that for the forward case, we can pick as many points as we wish to calculate for both terms of the loss function. For the inverse model, we know a set of  $x$  and  $t$  and their corresponding values of  $p$  in advance. These examples are used to calculate the  $MSE_{tr}$  and  $MSE_{\Pi}$  terms of Eq. 19. In a real setting, the amount of training examples is limited by the available measurements.
+
+# Verification of the forward model
+
+Following Eq (19) we obtain:
+
+$$
+M S E = M S E _ {b} + M S E _ {\Pi}, \tag {28}
+$$
+
+where
+
+$$
+M S E _ {b} = \frac {1}{N _ {b}} \sum_ {i = 1} ^ {N _ {b}} \left| p \left(x _ {b} ^ {i}, t _ {b} ^ {i}\right) - p ^ {i} \right| ^ {2} \tag {29}
+$$
+
+and
+
+$$
+M S E _ {\Pi} = \frac {1}{N _ {\Pi}} \sum_ {i = 1} ^ {N _ {\Pi}} \left| \Pi \left(x _ {\Pi} ^ {i}, t _ {\Pi} ^ {i}\right) \right| ^ {2}, \tag {30}
+$$
+
+where  $\{x_b^i, t_b^i, p^i\}_{i=1}^{N_b}$  refer to the initial and boundary training data on  $p(x,t)$  while  $\{x_{\Pi}^i, t_{\Pi}^i\}_{i=1}^{N_{\Pi}}$  denote the collocation points for  $\Pi$ , which are sampled using the Latin Hypercube method provided within the pyDOE package [51]. The  $N_b$  is the combined number of initial and boundary training data, and  $N_{\Pi}$  is the number of collocation points.
+
+In Table 1, we explore the accuracy of the PINN as a function of the number of training examples, i.e., different numbers of  $N_{b}$  and  $N_{\Pi}$  for a given size of the network;  $N_{hl}$  and  $N_{n}$  are fixed to four and ten, respectively. Due to the stochastic behavior of the training procedure of the neural networks, we calculate the results as an average over many realizations as it otherwise might be difficult to observe a clear trend of the relative  $\mathcal{L}^2$  error as a function of the number of training examples (see Panel A of Table 1). From Panels B and C of Table 1, we observe that the average accuracy of the PINN is improved as  $N_{b}$  and  $N_{\Pi}$  are increased. Moreover, the combined number of  $N_{b}$  and  $N_{\Pi}$  to obtain an average value of the  $\mathcal{L}^2$  error below  $1.0 \times 10^{-2}$  is in the order of 100s. To illustrate the impact of  $\Pi$  terms, we consider the case
+
+where we simply train a neural network to interpolate a feed-forward solution based on a known set of solution points. Without  $\Pi$  regularization we obtain an average relative  $\mathcal{L}^2$  error of  $5.6\times 10^{-2}$  based on 96 training examples while the average  $\mathcal{L}^2$  error becomes less than  $1.0\times 10^{-2}$  when including  $\Pi$ .
+
+Table 1: Diffusivity equation - forward modeling: PINN performance as function of training set size. Relative  $\mathcal{L}^2$  errors between the exact and predicted values of  $p$  for the validation set. The table shows the dependency on the number of the initial and boundary training data,  $N_{b}$ , and on the number of collocation points,  $N_{\Pi}$ . Here, the network architecture is fixed to 4 layers with 10 neurons per hidden layer.  
+Panel A: Average over 1 realization  
+
+<table><tr><td>NII
+Nb</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td><td>160</td><td>320</td><td>640</td></tr><tr><td>2</td><td>0.1010</td><td>0.1150</td><td>0.4060</td><td>0.2570</td><td>0.1530</td><td>0.0108</td><td>0.1930</td><td>0.0236</td><td>0.0216</td></tr><tr><td>3</td><td>0.0339</td><td>0.0665</td><td>0.1050</td><td>0.0122</td><td>0.0020</td><td>0.0973</td><td>0.0025</td><td>0.0314</td><td>0.0074</td></tr><tr><td>6</td><td>0.0248</td><td>0.0401</td><td>0.1110</td><td>0.0807</td><td>0.0012</td><td>0.0057</td><td>0.0014</td><td>0.0116</td><td>0.0046</td></tr><tr><td>12</td><td>0.0072</td><td>0.1230</td><td>0.0126</td><td>0.0296</td><td>0.0012</td><td>0.0119</td><td>0.0010</td><td>0.0015</td><td>0.0076</td></tr><tr><td>24</td><td>0.0152</td><td>0.0018</td><td>0.0008</td><td>0.0008</td><td>0.0006</td><td>0.0016</td><td>0.0008</td><td>0.0006</td><td>0.0009</td></tr><tr><td>48</td><td>0.0124</td><td>0.0127</td><td>0.0015</td><td>0.0037</td><td>0.0044</td><td>0.0013</td><td>0.0035</td><td>0.0002</td><td>0.0026</td></tr><tr><td>96</td><td>0.0076</td><td>0.0010</td><td>0.0023</td><td>0.0015</td><td>0.0026</td><td>0.0017</td><td>0.0009</td><td>0.0007</td><td>0.0010</td></tr><tr><td>192</td><td>0.0036</td><td>0.0006</td><td>0.0205</td><td>0.0025</td><td>0.0010</td><td>0.0051</td><td>0.0005</td><td>0.0008</td><td>0.0002</td></tr><tr><td>384</td><td>0.0020</td><td>0.0019</td><td>0.0009</td><td>0.0004</td><td>0.0008</td><td>0.0001</td><td>0.0005</td><td>0.0005</td><td>0.0003</td></tr></table>
+
+Panel B: Average over 3 realizations  
+
+<table><tr><td>NII
+Nb</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td><td>160</td><td>320</td><td>640</td></tr><tr><td>2</td><td>0.0972</td><td>0.0596</td><td>0.0417</td><td>0.0859</td><td>0.0380</td><td>0.0086</td><td>0.0041</td><td>0.0105</td><td>0.0050</td></tr><tr><td>3</td><td>0.0744</td><td>0.0652</td><td>0.0969</td><td>0.0415</td><td>0.0173</td><td>0.0268</td><td>0.0261</td><td>0.0084</td><td>0.0077</td></tr><tr><td>6</td><td>0.1020</td><td>0.1160</td><td>0.0841</td><td>0.0133</td><td>0.0071</td><td>0.0756</td><td>0.0217</td><td>0.0078</td><td>0.0064</td></tr><tr><td>12</td><td>0.0853</td><td>0.0099</td><td>0.0751</td><td>0.0088</td><td>0.0061</td><td>0.0456</td><td>0.0036</td><td>0.0059</td><td>0.0033</td></tr><tr><td>24</td><td>0.0447</td><td>0.0272</td><td>0.0104</td><td>0.0214</td><td>0.0290</td><td>0.0015</td><td>0.0041</td><td>0.0017</td><td>0.0060</td></tr><tr><td>48</td><td>0.0201</td><td>0.0037</td><td>0.0012</td><td>0.0230</td><td>0.0040</td><td>0.0072</td><td>0.0042</td><td>0.0060</td><td>0.0025</td></tr><tr><td>96</td><td>0.0104</td><td>0.0020</td><td>0.0013</td><td>0.0024</td><td>0.0048</td><td>0.0011</td><td>0.0009</td><td>0.0019</td><td>0.0020</td></tr><tr><td>192</td><td>0.0186</td><td>0.0014</td><td>0.0022</td><td>0.0006</td><td>0.0007</td><td>0.0011</td><td>0.0006</td><td>0.0004</td><td>0.0008</td></tr><tr><td>384</td><td>0.0015</td><td>0.0011</td><td>0.0010</td><td>0.0009</td><td>0.0005</td><td>0.0007</td><td>0.0010</td><td>0.0005</td><td>0.0005</td></tr></table>
+
+Panel C: Average over 24 realizations  
+
+<table><tr><td>NII
+Nb</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td><td>160</td><td>320</td><td>640</td></tr><tr><td>2</td><td>0.1002</td><td>0.0965</td><td>0.0904</td><td>0.0732</td><td>0.0849</td><td>0.0514</td><td>0.0139</td><td>0.0194</td><td>0.0188</td></tr><tr><td>3</td><td>0.0879</td><td>0.0775</td><td>0.0861</td><td>0.0513</td><td>0.0352</td><td>0.0354</td><td>0.0260</td><td>0.0111</td><td>0.0117</td></tr><tr><td>6</td><td>0.0710</td><td>0.0537</td><td>0.0529</td><td>0.0241</td><td>0.0144</td><td>0.0265</td><td>0.0153</td><td>0.0056</td><td>0.0107</td></tr><tr><td>12</td><td>0.0541</td><td>0.0351</td><td>0.0376</td><td>0.0302</td><td>0.0104</td><td>0.0235</td><td>0.0111</td><td>0.0135</td><td>0.0058</td></tr><tr><td>24</td><td>0.0320</td><td>0.0233</td><td>0.0141</td><td>0.0311</td><td>0.0160</td><td>0.0052</td><td>0.0061</td><td>0.0043</td><td>0.0039</td></tr><tr><td>48</td><td>0.0178</td><td>0.0050</td><td>0.0162</td><td>0.0111</td><td>0.0040</td><td>0.0024</td><td>0.0044</td><td>0.0041</td><td>0.0026</td></tr><tr><td>96</td><td>0.0087</td><td>0.0037</td><td>0.0031</td><td>0.0034</td><td>0.0022</td><td>0.0015</td><td>0.0021</td><td>0.0010</td><td>0.0012</td></tr><tr><td>192</td><td>0.0064</td><td>0.0049</td><td>0.0017</td><td>0.0010</td><td>0.0010</td><td>0.0011</td><td>0.0012</td><td>0.0009</td><td>0.0009</td></tr><tr><td>384</td><td>0.0046</td><td>0.0026</td><td>0.0011</td><td>0.0011</td><td>0.0010</td><td>0.0007</td><td>0.0008</td><td>0.0007</td><td>0.0006</td></tr></table>
+
+Next, we perform a sensitivity analysis to explore how the PINN performance depends on the hyperparameters,  $N_{hl}$  and  $N_{n}$ . We do this using a fixed size of the training set with  $N_{b}$  and  $N_{\Pi}$  equal to 96 and 160, respectively. Ideally, the explored space of the hyperparameters should reveal a parameter set that produces a minimum of the loss function on a validation set. To deal with the stochastic nature of the training procedure, again, we calculate average values obtained over many realizations. The results are presented in Table 2. From panel C, we observe that selecting  $N_{hl} = 6$  and
+
+$N_{n} = 5$  corresponds to the best accuracy in the explored space. For a smaller and larger size of the architecture, the accuracy decreases, corresponding to underfitting and overfitting, respectively [52]. We now apply the trained PINN model using  $N_{b} = 96$ ,  $N_{\Pi} = 160$ ,  $N_{hl} = 6$ , and  $N_{n} = 5$  to the test set. The relative  $\mathcal{L}^2$  error is  $1.43 \times 10^{-3}$ , which is comparable to that of the validation set (see Table 2 - Panel C).
+
+Table 2: Diffusivity equation - forward modeling: PINN performance as function of hyperparameters. Relative  $\mathcal{L}^2$  errors between the exact and predicted values of  $p$  for the validation set. The table shows the dependency on the different number of hidden layers  $N_{hl}$  and different number of neurons per layer  $N_{n}$ . Here, the total number of training and collocation points is fixed to  $N_{b} = 96$  and  $N_{\Pi} = 160$ , respectively.
+
+Panel A: Average over 1 realization  
+
+<table><tr><td>Nn
+Nhl</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td></tr><tr><td>2</td><td>0.0009</td><td>0.0015</td><td>0.0062</td><td>0.0003</td><td>0.0003</td><td>0.0088</td></tr><tr><td>4</td><td>0.0033</td><td>0.0036</td><td>0.0098</td><td>0.0009</td><td>0.0144</td><td>0.0007</td></tr><tr><td>6</td><td>0.0003</td><td>0.0038</td><td>0.0017</td><td>0.0175</td><td>0.0268</td><td>0.0009</td></tr><tr><td>8</td><td>0.0021</td><td>0.0010</td><td>0.0005</td><td>0.0005</td><td>0.0006</td><td>0.0058</td></tr><tr><td>16</td><td>0.0411</td><td>0.0003</td><td>0.0105</td><td>0.0007</td><td>0.0175</td><td>0.0019</td></tr><tr><td>32</td><td>0.0735</td><td>0.0019</td><td>0.0073</td><td>0.1970</td><td>0.0024</td><td>0.0081</td></tr></table>
+
+Panel B: Average over 3 realizations  
+
+<table><tr><td>Nn
+Nhl</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td></tr><tr><td>2</td><td>0.0027</td><td>0.0015</td><td>0.0021</td><td>0.0042</td><td>0.0025</td><td>0.0026</td></tr><tr><td>4</td><td>0.0004</td><td>0.0025</td><td>0.0013</td><td>0.0024</td><td>0.0014</td><td>0.0025</td></tr><tr><td>6</td><td>0.0023</td><td>0.0021</td><td>0.0020</td><td>0.0028</td><td>0.0026</td><td>0.0025</td></tr><tr><td>8</td><td>0.0049</td><td>0.0010</td><td>0.0023</td><td>0.0017</td><td>0.0016</td><td>0.0020</td></tr><tr><td>16</td><td>0.5580</td><td>0.0029</td><td>0.0012</td><td>0.0030</td><td>0.0019</td><td>0.0027</td></tr><tr><td>32</td><td>0.1150</td><td>0.0328</td><td>0.0036</td><td>0.0034</td><td>0.0107</td><td>0.0022</td></tr></table>
+
+Panel C: Average over 24 realizations  
+
+<table><tr><td>Nn
+Nhl</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td></tr><tr><td>2</td><td>0.1660</td><td>0.0022</td><td>0.0032</td><td>0.0025</td><td>0.0025</td><td>0.0020</td></tr><tr><td>4</td><td>0.0171</td><td>0.0024</td><td>0.0018</td><td>0.0021</td><td>0.0018</td><td>0.0022</td></tr><tr><td>6</td><td>0.0032</td><td>0.0014</td><td>0.0018</td><td>0.0024</td><td>0.0020</td><td>0.0023</td></tr><tr><td>8</td><td>0.0253</td><td>0.0016</td><td>0.0019</td><td>0.0020</td><td>0.0021</td><td>0.0025</td></tr><tr><td>16</td><td>0.0912</td><td>0.0019</td><td>0.0019</td><td>0.0019</td><td>0.0025</td><td>0.0033</td></tr><tr><td>32</td><td>0.0462</td><td>0.0090</td><td>0.0037</td><td>0.0027</td><td>0.0036</td><td>0.0031</td></tr></table>
+
+For the selected model ( $N_{hl} = 6$ ,  $N_{n} = 5$ ,  $N_{b} = 96$ , and  $N_{\Pi} = 160$ ), we illustrate the behavior of the mean square training error of this model as function of the training iterations in Fig 5. One can observe that  $MSE_{b}$  is higher than  $MSE_{\Pi}$  in the beginning, but later it approaches zero faster than  $MSE_{\Pi}$ . The  $MSE$  converges steadily without any oscillations.
+
+# Inverse model of diffusivity equation
+
+We rewrite Eq (27) in the parametrized form [23] as presented below:
+
+$$
+\Pi (x, t) := \theta_ {1} \frac {\partial p}{\partial t} - \theta_ {2} \frac {\partial}{\partial x} p ^ {2} \left(\frac {\partial}{\partial x} p\right) - g, \tag {31}
+$$
+
+where
+
+$$
+\theta_ {1} = \phi c _ {t} \quad \text {a n d} \quad \theta_ {2} = \kappa_ {0}. \tag {32}
+$$
+
+![](images/02edd3a3781935b718db5419237166060d6bb7fa006433069341463f539e654a.jpg)  
+Figure 5: Diffusivity equation - forward modeling: mean square training error plot. This model uses  $N_{hl} = 6$ ,  $N_{n} = 5$ ,  $N_{b} = 96$ , and  $N_{\Pi} = 160$ . MSE,  $MSE_{b}$ , and  $MSE_{\Pi}$  are calculated using Eqs. 28, 29, and 30, respectively.
+
+Unlike the forward mode, where the weights and biases are the unknown parameters to be learned, we now have two more unknowns,  $\theta_{1}$  and  $\theta_{2}$ . Using Eq (19) we have:
+
+$$
+M S E _ {t r} = \frac {1}{N _ {t r}} \sum_ {i = 1} ^ {N _ {t r}} \left| p \left(x _ {t r} ^ {i} t _ {t r} ^ {i}\right) - p ^ {i} \right| ^ {2} \tag {33}
+$$
+
+and
+
+$$
+M S E _ {\Pi} = \frac {1}{N _ {t r}} \sum_ {i = 1} ^ {N _ {t r}} \left| \Pi \left(x _ {t r} ^ {i}, t _ {t r} ^ {i}\right) \right| ^ {2}, \tag {34}
+$$
+
+where  $\left\{x_{tr}^i,t_{tr}^i,p^i\right\}_{i = 1}^{N_{tr}}$  refers to the set of training data and  $N_{tr}$  is the number of training data. In contrast to the forward model, we do not need to specify specific collocation points to activate the II dependent loss term; here we can just use the training examples used to calculate  $MSE_{tr}$ . All physical constants are set to one; hence,  $\theta_{1} = 1.0$  and  $\theta_{2} = 1.0$ . Note that  $\theta_{1}$  and  $\theta_{2}$  are considered constant throughout the domain.
+
+We use the hyperparameters obtained from the sensitivity analysis of the forward model (see Panel C of Table 2), i.e.,  $N_{hl} = 6$  and  $N_{n} = 5$ . In Table 3, we illustrate that this choice also yields the least  $\mathcal{L}^2$  error of  $p$  and percentage errors of  $\theta_{1}$  and  $\theta_{2}$  for the inverse problem, supporting the heuristic arguments presented above in the section "Training the PINN". Moreover, we find that the  $\mathcal{L}^2$  error of  $p$  and percentage error of  $\theta_{1}$  and  $\theta_{2}$  are not much different with different combinations of the hyperparameters. Note that the  $\mathcal{L}^2$  error of  $p$  and percentage errors of  $\theta_{1}$  and  $\theta_{2}$  presented in Table 3 represent average values over 24 realizations.
+
+We depict the performance of the PINN model for solving the inverse problem as a function of  $N_{tr}$  in Fig 6. The reported error values of  $\theta_{1}$  and  $\theta_{2}$  are percentage errors, while the relative  $\mathcal{L}^2$  error is shown for  $p$ . The error bars show the standard derivation (±1 SD) based on 24 realizations. We observe that a minimum of  $N_{tr} = 200$  is required by the PINN model to avoid substantial stochastic fluctuations in the estimated values of  $\theta_{1}$  and  $\theta_{2}$ . Moreover, we observe that the PINN model with  $N_{tr} > 200$  provides average percentage errors of  $\theta_{1}$  and  $\theta_{2}$  less than  $1\%$ , but with a large standard deviation. Over 24 realizations of the trained PINN with  $N_{tr} = 200$ , the minimum percentage errors are  $2.31 \times 10^{-2}$  and  $7.33 \times 10^{-2}$  and the maximum are  $1.93 \times 10^{-1}$  and  $4.92 \times 10^{-1}$ , for  $\theta_{1}$  and  $\theta_{2}$ , respectively. To obtain 200 training examples in a real setting, i.e., lab experiments or field observations, is realistic; hence, the results illustrate the feasibility of PINN to solve the inverse problem based on a reasonably sized data set.
+
+Table 3: Diffusivity equation - inverse modeling: PINN performance as function of hyperparameters. Relative  $\mathcal{L}^2$  error of  $p$  and percentage error of  $\theta_{1}$  and  $\theta_{2}$  for different number of hidden layers,  $N_{hl}$ , and different number of neurons per layer,  $N_{n}$ . The  $N_{tr}$  is fixed at 250. Note that we pick the optimal hyperparameters, i.e.,  $N_{hl} = 6$  and  $N_{n} = 5$  from the sensitivity analysis of the forward model. Results shown in this table are an average over 24 realizations.  
+
+<table><tr><td></td><td>\(N_{n}\) \(N_{hl}\)</td><td>2</td><td>5</td><td>10</td></tr><tr><td rowspan="3">p</td><td>4</td><td>0.0004</td><td>0.0003</td><td>0.0003</td></tr><tr><td>6</td><td>0.0007</td><td>0.0002</td><td>0.0004</td></tr><tr><td>8</td><td>0.0010</td><td>0.0002</td><td>0.0003</td></tr><tr><td rowspan="3">\(\theta_{1}\)</td><td>4</td><td>0.1888</td><td>0.1357</td><td>0.1969</td></tr><tr><td>6</td><td>0.3200</td><td>0.1065</td><td>0.3261</td></tr><tr><td>8</td><td>0.5195</td><td>0.1272</td><td>0.1125</td></tr><tr><td rowspan="3">\(\theta_{2}\)</td><td>4</td><td>0.3443</td><td>0.3562</td><td>0.3289</td></tr><tr><td>6</td><td>0.7121</td><td>0.1912</td><td>0.6946</td></tr><tr><td>8</td><td>0.7949</td><td>0.3088</td><td>0.2504</td></tr></table>
+
+![](images/1a6c0e478f956100123b74325ff51b9c1c8b8141f93270a358542a0ebef96bf9.jpg)  
+Figure 6: Diffusivity equation - inverse modeling: PINN performance as function of training set size. This figure shows the estimated error dependency on the amount of training data,  $N_{tr}$ . The error bars show mean and standard derivation (±1 SD) based on 24 realizations. Note that there is no noise added in this investigation, and all physical constants are set to one, i.e.,  $\theta_{1} = \theta_{2} = 1.0$ . The reported error values of  $\theta_{1}$  and  $\theta_{2}$  are percentage errors while the relative  $\mathcal{L}^2$  error is shown for  $p$ .
+
+Next, we perform a systematic study of the effect of additive noise in data, which is created from the true data as follows [26]:
+
+$$
+\boldsymbol {X} _ {\text {n o i s e}} = \boldsymbol {X} _ {\text {t r u e}} + \epsilon \mathcal {S} \left(\boldsymbol {X} _ {\text {t r u e}}\right) \mathcal {G} (0, 1), \tag {35}
+$$
+
+where  $X_{noise}$  and  $X_{true}$  is the vector of the data with and without noise, respectively. The  $\epsilon$  determines the noise level,  $S(\cdot)$  represents a standard deviation operator,  $\mathcal{G}(0,1)$  is a random value, which is sampled from the Gaussian distribution with mean and standard deviation of zero and one, respectively. The noise generated from this procedure is fully random and uncorrelated. The results are presented in Table 4 as average values over 10 training realizations. We can observe that the error increases with the noise level  $(\epsilon)$  while the error decreases as the  $N_{tr}$  is increased. As expected, the PINN model requires more data to accurately approximate the unknown physical parameters when the noise level is high.
+
+Table 4: Diffusivity equation - inverse modeling: PINN performance as function of noise. This figure shows the average percentage errors of  $\theta_{1}$  and  $\theta_{2}$  for different numbers of training data,  $N_{tr}$ , as function of the noise levels. Here, the neural network architecture is kept fixed to 6 layers and 5 neurons per layer. The results are averages over 10 realizations.  
+
+<table><tr><td colspan="2">Noise (ε)</td><td rowspan="2">0%</td><td rowspan="2">1%</td><td rowspan="2">5%</td><td rowspan="2">10%</td></tr><tr><td colspan="2">Ntr</td></tr><tr><td rowspan="4">θ1</td><td>100</td><td>0.17</td><td>1.82</td><td>4.40</td><td>4.67</td></tr><tr><td>250</td><td>0.15</td><td>0.30</td><td>1.00</td><td>1.98</td></tr><tr><td>500</td><td>0.12</td><td>0.17</td><td>0.77</td><td>0.84</td></tr><tr><td>1000</td><td>0.04</td><td>0.09</td><td>0.34</td><td>0.94</td></tr><tr><td rowspan="4">θ2</td><td>100</td><td>0.22</td><td>1.49</td><td>4.35</td><td>4.90</td></tr><tr><td>250</td><td>0.24</td><td>0.52</td><td>1.80</td><td>2.45</td></tr><tr><td>500</td><td>0.23</td><td>0.47</td><td>0.99</td><td>1.48</td></tr><tr><td>1000</td><td>0.13</td><td>0.28</td><td>0.41</td><td>0.79</td></tr></table>
+
+# Nonlinear Biot's equations
+
+From the nonlinear diffusivity equation section, we have shown that the PINN model can solve forward and inverse problems. We then progress to the multiphysics problem represented by the nonlinear Biot's equations. We take  $\Omega = [0,1]^2$ ,  $\mathbb{T} = [0,1]$ , and choose the exact solution in  $\Omega$  as:
+
+$$
+\boldsymbol {u} (x, y, t) := \left[ \begin{array}{l} u \\ v \end{array} \right] = \left[ \begin{array}{l} \sin (x + y + t) \\ \cos (x + y + t) \end{array} \right], \tag {36}
+$$
+
+for the displacement variable where  $u$  and  $v$  are displacements in x- and y-direction, respectively. Note that as we focus on the 2-Dimensional domain; therefore,  $\pmb{u}(x,y,t)$  is composed of two spatial components. For the pressure variable, we choose
+
+$$
+p (x, y, t) := e ^ {(x + y + t)}. \tag {37}
+$$
+
+Here  $x$ ,  $y$ , and  $t$  represent points in  $\mathbf{x}$ -,  $\mathbf{y}$ -direction, and time domain, respectively. The  $\mathcal{N}[\kappa]$  is chosen as:
+
+$$
+\mathcal {N} [ \kappa ] := \kappa_ {0} e ^ {\varepsilon_ {v}}, \tag {38}
+$$
+
+where  $\kappa_0$  represent initial rock matrix conductivity. Again, we assume  $\kappa_0$  to be a scalar in this case, i.e.,  $\kappa_0 = \kappa_0$ . The  $\varepsilon_v$  is the total volumetric strain defined as:
+
+$$
+\varepsilon_ {v} := \operatorname {t r} (\varepsilon) = \sum_ {i = 1} ^ {2} \varepsilon_ {i i}. \tag {39}
+$$
+
+The choice of  $\mathcal{N}[\kappa]$  function is selected to represent the change in a volumetric strain that affects the porous media conductivity, and it is adapted from [53-55]. All the physical constants are set to 1.0; and subsequently,  $\pmb{f}$  is chosen as:
+
+$$
+\boldsymbol {f} (x, y, t) := \left[ \begin{array}{l} f _ {u} (x, y, t) \\ f _ {v} (x, y, t) \end{array} \right], \tag {40}
+$$
+
+where
+
+$$
+f _ {u} (x, y, t) := - 4. 0 \sin (x + y + t) - 2. 0 \cos (x + y + t) - e ^ {(x + y + t)}, \tag {41}
+$$
+
+and
+
+$$
+f _ {v} (x, y, t) := - 4. 0 \cos (x + y + t) - 2. 0 \sin (x + y + t) - e ^ {(x + y + t)}, \tag {42}
+$$
+
+for the momentum balance equation, Eq (6). The source term of the mass balance equation, Eq (10),  $g$  is chosen as:
+
+$$
+\begin{array}{l} g (x, y, t) := \left(\cos (x + y + t) + \sin (x + y + t) - 1\right) \mathrm {e} ^ {\cos (x + y + t) - \sin (x + y + t) + x + y + t} \\ - \cos (x + y + t) + e ^ {x + y + t} - \sin (x + y + t), \tag {43} \\ \end{array}
+$$
+
+to satisfy the exact solution. Furthermore, the boundary conditions and initial conditions are applied using Eqs (36) and (37). The  $\Pi_{\boldsymbol{u}}(x,y,t)$  and  $\Pi_p(x,y,t)$ , Eqs (15) and (17), here act as the physics-informed function.
+
+We generate the exact solution points, Eqs (36) and (37), based on a rectangular mesh  $(\Omega = [0,1]^2)$  with 99 equidistant intervals in both x- and y-direction, i.e.  $\Delta x = \Delta y$ . Using 49 equidistant temporal intervals, in total, we have 500000 examples. Similar to the diffusivity equation case, we draw  $n$  training examples randomly. Subsequently, we split the remaining examples equally for validation and test sets. Again, assuming we have 500000 solution points for the sake of illustration, we use 100 examples to train the model; we then have 249950 examples for both the validation and the test sets.
+
+To recap, the forward modeling of Biot's system aims to predict the displacement  $(\pmb{u})$  and pressure  $(p)$  by specifying the initial and boundary conditions, collocation points, and as well as the physical parameters  $(\mu_l,\lambda_l,\alpha ,\phi ,c_f,K_s,$  and  $\kappa_0)$ . The inverse modeling, however, aims to estimate the physical parameters from observed values of  $\pmb{u}$  and  $p$  with their corresponding values of  $x,y$ , and  $t$ .
+
+In the case of the nonlinear Biot's equations, the architecture of the neural network corresponding to the top of Fig 3 is presented in Fig 7. We have three input nodes and three output nodes for this case. For the forward modeling, the hyperparameters  $N_{hl}$  and  $N_{n}$  are found using a sensitivity analysis, and as argued in the above section, "Training the PINN," we apply the same hyperparameters for the inverse model. Again, we use L-BFGS; a quasi-Newton, fullbatch gradient-based optimization algorithm to minimize the loss function [49] for the forward model. For the inverse problem, we find that combining ADAM, stochastic gradient descent, and L-BFGS might provide faster convergence when training the neural network. Specifically, we use ADAM for the first 10000 iterations and then continue using L-BFGS until the stop criterion is met. Note that we apply the same stop criterion as described above for the diffusivity case. Since the ADAM is a first-order method compared to L-BFGS, which is a second-order model, ADAM is less computationally demanding. Initially, where the weights of the neural network are far from convergence, we speculate that the less computational effort of ADAM is an advantage. However, as we approach the minimum, L-BFGS is likely to provide a better estimate of the steepest descent. Whether these observations could be made when dealing with other types of partial differential equations is an open question, but the use of a similar combined optimization scheme has been reported in the literature for the case of the Navier-Stokes equations [26].
+
+![](images/fab0562fd188bed3d7c4a24b84abbd8c6efe99bce401f1615a505e03196bf62d.jpg)  
+Figure 7: Neural networks architecture used for nonlinear Biot's equations. This figure corresponds to the top part of Fig 3. There are three inputs,  $x$ ,  $y$ , and  $t$ , and three outputs,  $u$ ,  $v$ , and  $p$ . The number of hidden layers,  $N_{hl}$ , and the number of neurons for each hidden layer,  $N_{n}$ , denote the hyperparameters.
+
+# Verification of the forward model
+
+Again, we apply Eq. (19) and obtain:
+
+$$
+M S E = M S E _ {b} + M S E _ {\Pi_ {u}} + M S E _ {\Pi_ {p}} \tag {44}
+$$
+
+where
+
+$$
+M S E _ {b} = \frac {1}{N _ {b}} \sum_ {i = 1} ^ {N _ {b}} \left(\left| u \left(x _ {b} ^ {i}, y _ {b} ^ {i}, t _ {b} ^ {i}\right) - u ^ {i} \right| ^ {2} + \left| v \left(x _ {b} ^ {i}, y _ {b} ^ {i}, t _ {b} ^ {i}\right) - v ^ {i} \right| ^ {2} \right. \tag {45}
+$$
+
+$$
++ \left| p \left(x _ {b} ^ {i}, y _ {b} ^ {i}, t _ {b} ^ {i}\right) - p ^ {i} \right| ^ {2}),
+$$
+
+$$
+M S E _ {\Pi_ {u}} = \frac {1}{N _ {\Pi_ {u}}} \sum_ {i = 1} ^ {N _ {\Pi_ {u}}} \left| \Pi_ {u} \left(x _ {\Pi_ {u}} ^ {i}, y _ {\Pi_ {u}} ^ {i}, t _ {\Pi_ {u}} ^ {i}\right) \right| ^ {2}, \tag {46}
+$$
+
+and
+
+$$
+M S E _ {\Pi_ {p}} = \frac {1}{N _ {\Pi_ {p}}} \sum_ {i = 1} ^ {N _ {\Pi_ {p}}} \left| \Pi_ {p} \left(x _ {\Pi_ {p}} ^ {i}, y _ {\Pi_ {p}} ^ {i}, t _ {\Pi_ {p}} ^ {i}\right) \right| ^ {2}, \tag {47}
+$$
+
+where  $\left\{x_b^i,y_b^i,t_b^i,u^i,v^i,p^i\right\}_{i = 1}^{N_b}$  refer to the initial and boundary training data.  $\left\{x_{\Pi_u}^i,y_{\Pi_u}^i,t_{\Pi_u}^i\right\}_{i = 1}^{N_{\Pi_u}}$  and  $\left\{x_{\Pi_p}^i,y_{\Pi_p}^i,t_{\Pi_p}^i\right\}_{i = 1}^{N_{\Pi_p}}$  specify the collocation points for  $\Pi_{u}(x,y,t)$  and  $\Pi_p(x,y,t)$ , as defined in Eqs (15) and (17). Similar to the diffusivity equation case, these collocation points are sampled using the Latin Hypercube method [51].  $N_{b}$  denotes the number of initial and boundary training data, and  $N_{\Pi_u}$  and  $N_{\Pi_p}$  are the number of collocation points for  $\Pi_{u}$  and  $\Pi_p$ , respectively. For the sake of simplification, in this investigation, we assume  $N_{\Pi_u} = N_{\Pi_p} = N_{\Pi}$  and
+
+$$
+\left\{x _ {\Pi_ {u}} ^ {i}, y _ {\Pi_ {u}} ^ {i}, t _ {\Pi_ {u}} ^ {i} \right\} _ {i = 1} ^ {N _ {\Pi_ {u}}} = \left\{x _ {\Pi_ {p}} ^ {i}, y _ {\Pi_ {p}} ^ {i}, t _ {\Pi_ {p}} ^ {i} \right\} _ {i = 1} ^ {N _ {\Pi_ {p}}} = \left\{x _ {\Pi} ^ {i}, y _ {\Pi} ^ {i}, t _ {\Pi} ^ {i} \right\} _ {i = 1} ^ {N _ {\Pi}}.
+$$
+
+In Fig 8, we illustrate an example of exact solutions of  $u$ ,  $v$ , and  $p$ , and compare them to the prediction values obtained from our test set using the PINN trained with  $N_{b} = 24$ ,  $N_{\Pi} = 20$ ,  $N_{hl} = 6$ , and  $N_{n} = 5$ . This figure demonstrates that the PINN provides good approximations of the exact solutions. The dependency of the prediction accuracy on  $N_{b}$  and  $N_{\Pi}$  with the  $N_{hl}$  and  $N_{n}$  fixed to four and ten, respectively is illustrated in Table 5. Again, to deal with the stochastic behavior of the neural networks, we calculate an average of the relative  $\mathcal{L}^2$  error over many realizations to obtain a clear pattern; see Panels B and C of Table 5. Similar to the diffusivity equation case, we observe the accuracy of the PINN is improved when  $N_{b}$  and  $N_{\Pi}$  are increased. We also note that the total amount of training examples required to achieve high accuracy, i.e., an average  $\mathcal{L}^2$  error less than  $1.0 \times 10^{-3}$ , is in the order of 100s. Again, we can illustrate the impact of the  $\Pi$  terms by considering the case where we train a neural network to interpolate a feed-forward solution based on a known set of solution points with and without making use of the regularization terms. Without using  $\Pi$  we obtain an average relative  $\mathcal{L}^2$  error of  $1.1 \times 10^{-1}$  based on 96 solution points while the average  $\mathcal{L}^2$  error becomes less than  $1.0 \times 10^{-3}$  when including  $\Pi$ .
+
+In Table 6, we present a sensitivity analysis of  $N_{hl}$  and  $N_{n}$  with a fixed size of the training set;  $N_{b} = 96$  and  $N_{\Pi} = 160$ . Once again, the observed trend becomes more apparent by averaging over many training realizations. We can now identify an extremum in the explored space of the hyperparameters corresponding to have  $N_{hl} = 6$  and  $N_{n} = 20$ ; see Panel C of Table 6. We then apply all of the 27 PINN networks trained with that choice of hyperparameters to the test set. We obtain the average  $\mathcal{L}^2$  error of the test set to be  $9.05 \times 10^{-5}$ , which is comparable to that of the validation set,  $9.08 \times 10^{-5}$ .
+
+In Fig 9, we show the behavior of the different loss terms as function of the training iterations, when using  $N_{hl} = 6$ ,  $N_{n} = 20$ ,  $N_{b} = 96$ , and  $N_{\Pi} = 160$ . Similar to the diffusivity equation, we observe that  $MSE_{b}$  is generally higher than  $MSE_{\Pi_{u}}$  and  $MSE_{\Pi_{p}}$ . Furthermore,  $MSE_{\Pi_{u}}$  and  $MSE_{\Pi_{p}}$  are comparable. Unlike the diffusivity equation case (see Fig 5), we observe minor oscillations of  $MSE$ ,  $MSE_{b}$ ,  $MSE_{\Pi_{u}}$ , and  $MSE_{\Pi_{p}}$  during convergence.
+
+![](images/12144c380ae8989caa95192173434f5e4b03f0f096bd74a60152e60f0e0141fc.jpg)
+
+![](images/6a455be744817413d0c8ec1dd3b3001735c79ac14becc5250dcc7c5a61ddc734.jpg)
+
+![](images/8686a47027f12dfaf1e878215ed24dc7ec67ed5c862db1b245ee310d10d26004.jpg)
+
+![](images/c209eb370f94ae00614b40897057c4efda47411ccac22cdcc5c17eddf9e69c14.jpg)
+
+![](images/454c6d6e0c7a1e6dd350cb1dd413e6d0d7c740fc39763ddb0b5eda2a2c1122f8.jpg)
+
+![](images/f0e19696d359279d87365aa4d08b862c482c0e847eead11935da3cf6651107fe.jpg)
+
+![](images/a48ae9a9ca3d956d421143bd955cc8c827deda42068aa0524aeb6274d069960a.jpg)  
+Figure 8: Biot's equations - forward modeling: exact solutions (shown by surface plot) and 100 PINN predictions per time step using the test set (shown by black points). The PINN was trained using  $N_{b} = 24$ ,  $N_{\Pi} = 20$ ,  $N_{hl} = 6$ , and  $N_{n} = 5$ . The top row illustrates the displacement in the x-direction,  $u$ , at (a)  $t = 0.0$ , (b)  $t = 0.5$ , and (c)  $t = 1.0$ . The middle row illustrates the displacement in the y-direction,  $v$ , at (d)  $t = 0.0$ , (e)  $t = 0.5$ , and (f)  $t = 1.0$ . The bottom row illustrates the pressure,  $p$ , at (g)  $t = 0.0$ , (h)  $t = 0.5$ , and (i)  $t = 1.0$ .
+
+![](images/7adabcfbec8045ea92ccb0d859d569a83cb63dde0ea04785feb9451188156533.jpg)
+
+![](images/5a1cb363cb632ca141948c636262a8557a894988f188fb17f26a6d5ce39e23c1.jpg)
+
+# Inverse model of Biot's equations
+
+We rewrite Eq (15) to the parametrized form as:
+
+$$
+\Pi_ {\boldsymbol {u}} = \nabla \cdot \left[ 2 \theta_ {1} \varepsilon (\boldsymbol {u}) + \theta_ {2} \boldsymbol {u} \boldsymbol {I} \right] - \theta_ {3} \nabla \cdot p \boldsymbol {I} - \boldsymbol {f} \text {i n} \Omega \times \mathbb {T}, \tag {48}
+$$
+
+and Eq (17) as:
+
+$$
+\Pi_ {p} = \theta_ {4} \frac {\partial p}{\partial t} + \theta_ {3} \frac {\partial \nabla \cdot \boldsymbol {u}}{\partial t} - \theta_ {5} \nabla \cdot e ^ {\varepsilon_ {v}} (\nabla p - \rho \mathbf {g}) - g \text {i n} \Omega \times \mathbb {T}, \tag {49}
+$$
+
+where
+
+$$
+\theta_ {1} = \mu_ {l}, \quad \theta_ {2} = \lambda_ {l}, \quad \theta_ {3} = \alpha , \quad \theta_ {4} = \phi c _ {f} + \frac {\alpha - \phi}{K _ {s}}, \text {a n d} \quad \theta_ {5} = \kappa_ {0}. \tag {50}
+$$
+
+The II terms now have five additional unknown parameters,  $\theta_{1},\theta_{2},\theta_{3},\theta_{4}$  , and  $\theta_{5}$  that along with the weights and biases of the neural network are adjusted during the training of the network. Once again we apply Eq. (19) and obtain:
+
+$$
+M S E = M S E _ {t r} + M S E _ {\Pi_ {u}} + + M S E _ {\Pi_ {p}} \tag {51}
+$$
+
+Table 5: Biot's equations - forward modeling: PINN performance as function of training set size. Sum of relative  $\mathcal{L}^2$  errors between the exact and predicted values of  $u$ ,  $v$ , and  $p$  for the validation set. The table shows the dependency on the number of the initial and boundary training data,  $N_b$ , and on the number of collocation points,  $N_{\Pi}$ . The hyperparemeters are fixed to 4 layers with 10 neurons per hidden layer.  
+Panel A: Average over 1 realization  
+
+<table><tr><td>NII
+Nb</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td><td>160</td></tr><tr><td>2</td><td>0.4475</td><td>0.4257</td><td>0.5010</td><td>0.2865</td><td>0.2826</td><td>0.1438</td><td>0.1982</td></tr><tr><td>3</td><td>0.1770</td><td>0.1508</td><td>0.2464</td><td>0.2009</td><td>0.2059</td><td>0.1410</td><td>0.1751</td></tr><tr><td>6</td><td>0.4056</td><td>0.4615</td><td>0.0824</td><td>0.0042</td><td>0.1236</td><td>0.1327</td><td>0.1662</td></tr><tr><td>12</td><td>0.0412</td><td>0.0860</td><td>0.0722</td><td>0.0035</td><td>0.0005</td><td>0.0013</td><td>0.0267</td></tr><tr><td>24</td><td>0.0148</td><td>0.0824</td><td>0.0037</td><td>0.0019</td><td>0.0015</td><td>0.0007</td><td>0.0003</td></tr><tr><td>48</td><td>0.0532</td><td>0.0045</td><td>0.0027</td><td>0.0008</td><td>0.0019</td><td>0.0002</td><td>0.0099</td></tr><tr><td>96</td><td>0.0007</td><td>0.0005</td><td>0.0003</td><td>0.0006</td><td>0.0004</td><td>0.0001</td><td>0.0003</td></tr></table>
+
+Panel B: Average over 3 realizations  
+
+<table><tr><td>NII
+Nb</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td><td>160</td></tr><tr><td>2</td><td>0.4738</td><td>0.5502</td><td>0.5458</td><td>0.1493</td><td>0.1310</td><td>0.2138</td><td>0.1804</td></tr><tr><td>3</td><td>0.3415</td><td>0.1240</td><td>0.3578</td><td>0.3044</td><td>0.1573</td><td>0.1192</td><td>0.1562</td></tr><tr><td>6</td><td>0.5052</td><td>0.1423</td><td>0.0733</td><td>0.1688</td><td>0.2066</td><td>0.1158</td><td>0.0005</td></tr><tr><td>12</td><td>0.0551</td><td>0.0566</td><td>0.0044</td><td>0.0087</td><td>0.0472</td><td>0.0037</td><td>0.1108</td></tr><tr><td>24</td><td>0.0498</td><td>0.0135</td><td>0.0045</td><td>0.0005</td><td>0.0009</td><td>0.0016</td><td>0.0013</td></tr><tr><td>48</td><td>0.0034</td><td>0.0016</td><td>0.0246</td><td>0.0012</td><td>0.0027</td><td>0.0002</td><td>0.0015</td></tr><tr><td>96</td><td>0.0004</td><td>0.0002</td><td>0.0006</td><td>0.0005</td><td>0.0006</td><td>0.0001</td><td>0.0014</td></tr></table>
+
+Panel C: Average over 27 realizations  
+
+<table><tr><td>NII
+Nb</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td><td>160</td></tr><tr><td>2</td><td>0.5320</td><td>0.4828</td><td>0.4473</td><td>0.2458</td><td>0.2253</td><td>0.2660</td><td>0.3100</td></tr><tr><td>3</td><td>0.4660</td><td>0.4211</td><td>0.3503</td><td>0.1781</td><td>0.1644</td><td>0.1661</td><td>0.1870</td></tr><tr><td>6</td><td>0.4031</td><td>0.1765</td><td>0.1380</td><td>0.0583</td><td>0.0871</td><td>0.0852</td><td>0.1340</td></tr><tr><td>12</td><td>0.1086</td><td>0.0641</td><td>0.0471</td><td>0.0059</td><td>0.0169</td><td>0.0164</td><td>0.0195</td></tr><tr><td>24</td><td>0.0525</td><td>0.0220</td><td>0.0101</td><td>0.0142</td><td>0.0052</td><td>0.0016</td><td>0.0012</td></tr><tr><td>48</td><td>0.0061</td><td>0.0030</td><td>0.0013</td><td>0.0013</td><td>0.0012</td><td>0.0003</td><td>0.0009</td></tr><tr><td>96</td><td>0.0022</td><td>0.0005</td><td>0.0008</td><td>0.0006</td><td>0.0004</td><td>0.0007</td><td>0.0005</td></tr></table>
+
+where
+
+$$
+\begin{array}{l} M S E _ {t r} = \frac {1}{N _ {t r}} \sum_ {i = 1} ^ {N _ {t r}} \left(\left| u \left(x _ {t r} ^ {i}, y _ {t r} ^ {i}, t _ {t r} ^ {i}\right) - u ^ {i} \right| ^ {2} + \left| v \left(x _ {t r} ^ {i}, y _ {t r} ^ {i}, t _ {t r} ^ {i}\right) - v ^ {i} \right| ^ {2} \right. \tag {52} \\ + \left| p \left(x _ {t r} ^ {i}, y _ {t r} ^ {i}, t _ {t r} ^ {i}\right) - p ^ {i} \right| ^ {2}), \\ \end{array}
+$$
+
+$$
+M S E _ {\Pi_ {u}} = \frac {1}{N _ {\Pi_ {u}}} \sum_ {i = 1} ^ {N _ {\Pi_ {u}}} \left| \Pi_ {u} \left(x _ {\Pi_ {u}} ^ {i}, y _ {\Pi_ {u}} ^ {i}, t _ {\Pi_ {u}} ^ {i}\right) \right| ^ {2}, \tag {53}
+$$
+
+Table 6: Biot's equations - forward modeling: PINN performance as function of hyperparameters. Sum of relative  $\mathcal{L}^2$  errors between the exact and predicted values of  $u$ ,  $v$ , and  $p$  for the validation set. The table shows the dependency on the different number of hidden layers,  $N_{hl}$ , and different number of neurons per layer,  $N_{n}$ . Here, the total number of training and collocation points is fixed to  $N_{b} = 96$  and  $N_{\Pi} = 160$ , respectively.
+
+Panel A: Average over 1 realization  
+
+<table><tr><td>Nn
+Nhl</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td></tr><tr><td>2</td><td>0.56506</td><td>0.01617</td><td>0.00782</td><td>0.00045</td><td>0.00021</td><td>0.00115</td></tr><tr><td>4</td><td>0.12080</td><td>0.00316</td><td>0.00013</td><td>0.00019</td><td>0.00020</td><td>0.00016</td></tr><tr><td>6</td><td>0.45949</td><td>0.02069</td><td>0.00052</td><td>0.00060</td><td>0.00010</td><td>0.00020</td></tr><tr><td>8</td><td>0.14333</td><td>0.00971</td><td>0.93757</td><td>0.00048</td><td>0.00034</td><td>0.00015</td></tr><tr><td>16</td><td>0.14052</td><td>0.14110</td><td>0.00041</td><td>0.00020</td><td>0.00020</td><td>0.00019</td></tr><tr><td>32</td><td>0.60485</td><td>0.03188</td><td>0.00866</td><td>0.00972</td><td>0.00028</td><td>0.00039</td></tr></table>
+
+Panel B: Average over 3 realizations  
+
+<table><tr><td>Nn
+Nhl</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td></tr><tr><td>2</td><td>0.30761</td><td>0.01074</td><td>0.00346</td><td>0.00019</td><td>0.00026</td><td>0.00020</td></tr><tr><td>4</td><td>0.13684</td><td>0.01101</td><td>0.00032</td><td>0.00006</td><td>0.00007</td><td>0.00008</td></tr><tr><td>6</td><td>0.14338</td><td>0.04415</td><td>0.00018</td><td>0.00012</td><td>0.00009</td><td>0.00007</td></tr><tr><td>8</td><td>0.47702</td><td>0.01634</td><td>0.00020</td><td>0.00010</td><td>0.00008</td><td>0.00007</td></tr><tr><td>16</td><td>0.33020</td><td>0.13525</td><td>0.00081</td><td>0.00041</td><td>0.00013</td><td>0.00013</td></tr><tr><td>32</td><td>0.46854</td><td>0.61381</td><td>0.38437</td><td>0.07503</td><td>0.00087</td><td>0.00010</td></tr></table>
+
+Panel C: Average over 27 realizations  
+
+<table><tr><td>Nn
+Nhl</td><td>2</td><td>5</td><td>10</td><td>20</td><td>40</td><td>80</td></tr><tr><td>2</td><td>0.24203</td><td>0.02952</td><td>0.00180</td><td>0.00062</td><td>0.00021</td><td>0.00023</td></tr><tr><td>4</td><td>0.39596</td><td>0.06005</td><td>0.00368</td><td>0.00039</td><td>0.00010</td><td>0.00014</td></tr><tr><td>6</td><td>0.32610</td><td>0.07829</td><td>0.00018</td><td>0.00009</td><td>0.00010</td><td>0.00014</td></tr><tr><td>8</td><td>0.42070</td><td>0.12314</td><td>0.00037</td><td>0.00013</td><td>0.00010</td><td>0.00012</td></tr><tr><td>16</td><td>0.46364</td><td>0.12222</td><td>0.09613</td><td>0.05190</td><td>0.00012</td><td>0.00011</td></tr><tr><td>32</td><td>0.41372</td><td>0.38439</td><td>0.38473</td><td>0.03998</td><td>0.05080</td><td>0.01052</td></tr></table>
+
+and
+
+$$
+M S E _ {\Pi_ {p}} = \frac {1}{N _ {\Pi_ {p}}} \sum_ {i = 1} ^ {N _ {\Pi_ {p}}} \left| \Pi_ {p} \left(x _ {\Pi_ {p}} ^ {i}, y _ {\Pi_ {p}} ^ {i}, t _ {\Pi_ {p}} ^ {i}\right) \right| ^ {2}, \tag {54}
+$$
+
+where  $\{x_{tr}^i, y_{tr}^i, t_{tr}^i, u^i, v^i, p^i\}_{i=1}^{N_{tr}}$  refer to the set of training data. In contrast to the forward model, we can apply the training points as collocation points when calculating the terms given by Eqs (48) and (49). To recap, the  $N_{tr}$  denotes the number of training data. Similar to the forward model, all physical constants are set to one, i.e.,  $\theta_1, \theta_2, \theta_3, \theta_4$ , and  $\theta_5$  are equal to one. Note that these  $\theta$  are constant throughout the domain.
+
+We use the optimal hyperparameters, i.e.,  $N_{hl} = 6$  and  $N_{n} = 20$  from the sensitivity analysis of the forward model (see Panel C of Table 6). In Table 7, we can observe that this choice also yields the least  $\mathcal{L}^2$  error with respect to both  $p$ ,  $u$ , and  $v$  and the lowest percentage error for the unknown physical parameters  $(\theta_1, \theta_2, \theta_3, \theta_4,$  and  $\theta_5)$ . Moreover, we observe that the  $\mathcal{L}^2$  error of the output space and percentage error of the unknown physical parameters are not much different with different combinations of the hyperparameters.
+
+The performance of the PINN model as a function of the number of training examples  $N_{tr}$  is depicted in Fig 10. We can observe that the stochastic variations in the estimated values of  $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$  and  $\theta_{5}$ , in general, are reduced the more training examples we apply. Moreover, the relative  $\mathcal{L}^2$  errors of  $u, v$ , and  $p$  are always less than  $0.01\%$ . Using
+
+![](images/fec0f640dd10f5528acc4e6e46affbf34844f26c485fc5dd5e62fad757a8eb9b.jpg)  
+Figure 9: Biot's equations - forward modeling: mean square training error plot. This model uses  $N_{hl} = 6$ ,  $N_{n} = 20$ ,  $N_{b} = 96$ , and  $N_{\Pi} = 160$ . MSE,  $MSE_{b}$ ,  $MSE_{\Pi_{u}}$ , and  $MSE_{\Pi_{p}}$  are calculated using Eqs. 44, 45, 46, and 47, respectively.
+
+in the order of 1000 examples, the average estimation error of the physical parameters is in the order of  $1\%$ , but as with the diffusivity case, there is a large variation between the trained PINN models. Within the 27 realizations of the trained PINN models with  $N_{tr} = 1000$  the percentage errors of  $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$  and  $\theta_{5}$  varied between 0.05, 0.02, 0.03, 0.04, and 0.03 and 2.51, 6.14, 4.75, 8.12, and 3.60, respectively. Having 1000 training examples in actual cases, i.e., lab experiments or field observations, is realistic. Hence, also for the Biot's equations, we observe the feasibility of the PINN model to handle the inverse problem by estimating the unknown physical parameters using a reasonably sized data set.
+
+Next, we perform a systematic study of the effect of noise in data, which is created utilizing Eq (35). The results are presented in Table 8 for  $\theta_{1}$ ,  $\theta_{2}$ ,  $\theta_{3}$ ,  $\theta_{4}$ , and  $\theta_{5}$ . Note that these results are an average over ten realizations. We can observe that the percentage error of  $\theta_{1}$ ,  $\theta_{2}$ ,  $\theta_{3}$ ,  $\theta_{4}$ , and  $\theta_{5}$  increase along with the noise level  $(\epsilon)$ , but as  $N_{tr}$  is increased, the impact of the noise is reduced.
+
+All calculations were carried out using a XeonE5_2660v3 processor with a single thread. As an example of the Biot's equations, the CPU time for training the neural networks using  $N_{tr} = 10000$  and 15000 with no noise are 128037 seconds and 186154 seconds, respectively. Note that the reported values are obtained from the model trained using the combined ADAM and L-BFGS. Using L-BFGS alone, the CPU times of model with  $N_{tr} = 10000$  and 15000 are 222681 and 294768 seconds, respectively.
+
+# Conclusion
+
+This paper studies the application of physics-informed neural networks (PINN) for solving the nonlinear diffusivity and Biot's equations in the context of forward and inverse modelings. The following conclusions are drawn:
+
+- PINN can be used to solve the forward modeling problem for the nonlinear diffusivity and Biot's equations, at least for the type of geometries considered in this paper. The displacement and pressure variables of our test sets could be predicted with an average  $\mathcal{L}^2$  error of  $9.05 \times 10^{-5} \pm 3.1 \times 10^{-4}$  based on 27 realizations.  
+- For the inverse modeling cases, PINN can predict all of the unknown physical parameters with an average percentage error of around  $1\%$ ; however, the stochastic variations from one PINN implementation to the next is quite large. Using, for instance, 1000 training examples in the Biot's equation case, the percentage error of the estimated physical parameters over 27 PINN models could vary from 0.02 to 8.12. Increasing the number of training examples reduces this problem. Still, our results indicate it would be essential to do an average over PINN models with different random initialization of the weights and biases. This process may lead to the
+
+Table 7: Biot's equations - inverse modeling: PINN performance as function of hyperparameters. Relative  $\mathcal{L}^2$  error of  $p$ ,  $u$ , and  $v$  and percentage error of  $\theta_{1}$ ,  $\theta_{2}$ ,  $\theta_{3}$ ,  $\theta_{4}$ , and  $\theta_{5}$  for different number of hidden layers,  $N_{hl}$ , and different number of neurons per layer,  $N_{n}$ . The  $N_{tr}$  is fixed at 250. Note that we pick the optimal hyperparameters, i.e.,  $N_{hl} = 6$  and  $N_{n} = 20$  from the sensitivity analysis of the forward model. Results shown in this table are an average over 27 realizations.  
+
+<table><tr><td></td><td>\(N_{n}\) \(N_{hl}\)</td><td>2</td><td>5</td><td>10</td></tr><tr><td rowspan="3">p</td><td>4</td><td>0.00006</td><td>0.00004</td><td>0.00008</td></tr><tr><td>6</td><td>0.00009</td><td>0.00004</td><td>0.00006</td></tr><tr><td>8</td><td>0.00008</td><td>0.00006</td><td>0.00005</td></tr><tr><td rowspan="3">u</td><td>4</td><td>0.00019</td><td>0.00016</td><td>0.00047</td></tr><tr><td>6</td><td>0.00051</td><td>0.00013</td><td>0.00036</td></tr><tr><td>8</td><td>0.00027</td><td>0.00036</td><td>0.00034</td></tr><tr><td rowspan="3">v</td><td>4</td><td>0.00042</td><td>0.00045</td><td>0.00098</td></tr><tr><td>6</td><td>0.00114</td><td>0.00038</td><td>0.00074</td></tr><tr><td>8</td><td>0.00060</td><td>0.00079</td><td>0.00071</td></tr><tr><td rowspan="3">θ1</td><td>4</td><td>0.20180</td><td>0.28665</td><td>1.06223</td></tr><tr><td>6</td><td>0.72512</td><td>0.16844</td><td>0.97916</td></tr><tr><td>8</td><td>0.30534</td><td>0.70613</td><td>0.99229</td></tr><tr><td rowspan="3">θ2</td><td>4</td><td>0.65507</td><td>0.66560</td><td>3.56789</td></tr><tr><td>6</td><td>2.23747</td><td>0.23053</td><td>2.80360</td></tr><tr><td>8</td><td>1.21152</td><td>1.47513</td><td>2.45137</td></tr><tr><td rowspan="3">θ3</td><td>4</td><td>0.02968</td><td>0.05016</td><td>0.23851</td></tr><tr><td>6</td><td>0.33292</td><td>0.02935</td><td>0.10100</td></tr><tr><td>8</td><td>0.22593</td><td>0.11796</td><td>0.07371</td></tr><tr><td rowspan="3">θ4</td><td>4</td><td>0.04834</td><td>0.06162</td><td>0.19135</td></tr><tr><td>6</td><td>0.18104</td><td>0.04666</td><td>0.13785</td></tr><tr><td>8</td><td>0.10711</td><td>0.16124</td><td>0.12127</td></tr><tr><td rowspan="3">θ5</td><td>4</td><td>0.19710</td><td>0.21622</td><td>1.10700</td></tr><tr><td>6</td><td>1.24043</td><td>0.17633</td><td>0.60595</td></tr><tr><td>8</td><td>0.81199</td><td>0.50331</td><td>0.39558</td></tr></table>
+
+requirement of more processing power. This challenge might be even higher when applying PINN to more complex geometries and heterogeneous materials.
+
+- For the inverse modeling, PINN is tolerant to a noise level up to  $5\%$  (the estimation error of physical parameters is approximately less than  $15\%$ ). Again, this requires that one does an average over several PINN realizations. As expected, the result improves when the number of training examples is increased.  
+- We have presented arguments on why the hyperparameters selection process for the forward case is likely to be applicable to the inverse case. For the cases considered here, this was confirmed experimentally. However, this should be explored in more detail by investigating the use of PINN for other types of nonlinear partial differential equations.
+
+Finally, in terms of future work, the capability of the physics-informed neural networks should be tested in the case where the input data is incomplete, i.e.,  $\mathbf{u}$  and  $p$  are not available at the same spatial and temporal coordinates. Besides, one could investigate the potential benefits of training networks using mini-batches. Moreover, smarter initialization of the weights and biases (based on transfer learning principles) could potentially be employed to increase the speed and accuracy of the training procedure [56].
+
+# Acknowledgments
+
+The research leading to these results has received funding from the Danish Hydrocarbon Research and Technology Centre under the Advanced Water Flooding program.
+
+# Author Contributions
+
+Conceptualization: Teeratorn Kadeethum, Thomas M Jorgensen, Hamidreza M Nick
+
+![](images/58f80bd7405b1772bd996dfd34a2b891782cbe83c845713bff8fafe01a5ba324.jpg)  
+Figure 10: Biot's equations - inverse modeling: PINN performance as function of training set size. This figure shows the estimated error dependency on the amount of training data,  $N_{tr}$ . (a)  $\theta_1$ ,  $\theta_2$ ,  $\theta_3$ ,  $\theta_4$ , and  $\theta_5$  and (b)  $u$ ,  $v$ , and  $p$ . The error bars show mean and standard derivation (± 1 SD) based on 27 realizations. The reported error values of  $\theta_1$ ,  $\theta_2$ ,  $\theta_3$ ,  $\theta_4$ , and  $\theta_5$  are percentage errors while the relative  $\mathcal{L}^2$  error is shown for  $u$ ,  $v$ , and  $p$ . Note that there is no noise in this investigation.
+
+![](images/332a1f424a396128bbd82bab758d49b3bbf981c4a36306b23f22f7c0cd7d135e.jpg)
+
+Formal Analysis: Teeratorn Kadeethum
+
+Funding Acquisition: Hamidreza M Nick
+
+Software: Teeratorn Kadeethum
+
+Supervision: Thomas M Jorgensen, Hamidreza M Nick
+
+Validation: Thomas M Jorgensen, Hamidreza M Nick
+
+Writing - Original Draft Preparation: Teeratorn Kadeethum, Thomas M Jorgensen
+
+Writing - Review and Editing: Teeratorn Kadeethum, Thomas M Jorgensen, Hamidreza M Nick
+
+# References
+
+[1] HM Nick, A Raoof, F Centler, M Thullner, and P Regnier. Reactive dispersive contaminant transport in coastal aquifers: numerical simulation of a reactive Henry problem. Journal of contaminant hydrology, 145:90-104, 2013.  
+[2] Kevin Bisdom, Giovanni Bertotti, and Hamidreza M Nick. The impact of in-situ stress and outcrop-based fracture geometry on hydraulic aperture and upscaled permeability in fractured reservoirs. Tectonophysics, 690:63-75, 2016.  
+[3] Sanghyun Lee, Mary F Wheeler, and Thomas Wick. Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Computer Methods in Applied Mechanics and Engineering, 305:111-132, 2016.  
+[4] R Juanes, B Jha, BH Hager, JH Shaw, A Plesch, L Astiz, JH Dieterich, and C Frohlich. Were the May 2012 Emilia-Romagna earthquakes induced? A coupled flow-geomechanics modeling assessment. Geophysical Research Letters, 43(13):6891-6897, 2016.  
+[5] V Vinje, J Brucker, ME Rognes, KA Mardal, and V Haughton. Fluid dynamics in syringomyelia cavities: Effects of heart rate, CSF velocity, CSF velocity waveform and craniovertebral decompression. The neuroradiology journal, page 1971400918795482, 2018.
+
+Table 8: Biot's equations - inverse modeling: PINN performance as function of noise. This figure shows the average percentage errors of  $\theta_{1}$ ,  $\theta_{2}$ ,  $\theta_{3}$ ,  $\theta_{4}$ , and  $\theta_{5}$  for different numbers of training data  $N_{tr}$  corrupted by different noise levels (ε). Here, the neural network architecture is kept fixed to 6 layers and 20 neurons per layer. These results are an average over 10 realizations.  
+
+<table><tr><td colspan="2">Noise (ε)</td><td rowspan="2">0%</td><td rowspan="2">1%</td><td rowspan="2">5%</td><td rowspan="2">10%</td></tr><tr><td colspan="2">Ntr</td></tr><tr><td rowspan="5">θ1</td><td>1000</td><td>0.91</td><td>6.40</td><td>7.15</td><td>29.55</td></tr><tr><td>1500</td><td>0.98</td><td>4.64</td><td>5.70</td><td>14.86</td></tr><tr><td>2000</td><td>0.67</td><td>2.17</td><td>5.26</td><td>15.95</td></tr><tr><td>2500</td><td>0.52</td><td>3.87</td><td>5.33</td><td>12.24</td></tr><tr><td>5000</td><td>0.30</td><td>1.39</td><td>3.32</td><td>7.60</td></tr><tr><td rowspan="5">θ2</td><td>1000</td><td>1.48</td><td>10.26</td><td>12.30</td><td>17.50</td></tr><tr><td>1500</td><td>0.84</td><td>6.95</td><td>5.44</td><td>14.53</td></tr><tr><td>2000</td><td>1.37</td><td>4.38</td><td>6.73</td><td>13.48</td></tr><tr><td>2500</td><td>0.80</td><td>3.95</td><td>2.91</td><td>6.31</td></tr><tr><td>5000</td><td>0.51</td><td>1.97</td><td>2.90</td><td>3.14</td></tr><tr><td rowspan="5">θ3</td><td>1000</td><td>1.19</td><td>2.85</td><td>5.06</td><td>8.10</td></tr><tr><td>1500</td><td>0.87</td><td>1.32</td><td>2.52</td><td>4.80</td></tr><tr><td>2000</td><td>0.58</td><td>1.35</td><td>2.49</td><td>4.11</td></tr><tr><td>2500</td><td>0.11</td><td>0.38</td><td>1.54</td><td>4.12</td></tr><tr><td>5000</td><td>0.10</td><td>0.24</td><td>1.72</td><td>2.36</td></tr><tr><td rowspan="5">θ4</td><td>1000</td><td>0.46</td><td>4.70</td><td>8.35</td><td>13.75</td></tr><tr><td>1500</td><td>0.43</td><td>3.01</td><td>5.22</td><td>8.53</td></tr><tr><td>2000</td><td>0.23</td><td>2.99</td><td>4.12</td><td>10.08</td></tr><tr><td>2500</td><td>0.24</td><td>0.64</td><td>5.88</td><td>9.34</td></tr><tr><td>5000</td><td>0.20</td><td>0.45</td><td>1.62</td><td>6.53</td></tr><tr><td rowspan="5">θ5</td><td>1000</td><td>3.46</td><td>9.22</td><td>16.15</td><td>24.52</td></tr><tr><td>1500</td><td>2.70</td><td>5.64</td><td>14.16</td><td>19.75</td></tr><tr><td>2000</td><td>0.44</td><td>3.76</td><td>11.56</td><td>17.55</td></tr><tr><td>2500</td><td>0.66</td><td>2.23</td><td>6.82</td><td>7.09</td></tr><tr><td>5000</td><td>0.24</td><td>1.28</td><td>4.26</td><td>5.91</td></tr></table>
+
+As the noise  $(\epsilon)$  is increased, the error increases as expected.
+
+[6] Maurice Biot. General theory of three-dimensional consolidation. Journal of applied physics, 12(2):155-164, 1941.  
+[7] MA Biot and DG Willis. The elastic coeff cients of the theory of consolidation. J. appl. Mech, 15:594-601, 1957.  
+[8] Karl Terzaghi. Theoretical soil mechanics. Chapman And Hall, Limited.; London, 1951.  
+[9] Herbert F Wang. Theory of linear poroelasticity with applications to geomechanics and hydrogeology. Princeton University Press, 2017.  
+[10] Yder J Masson and SR Pride. Finite-difference modeling of biot's poroelastic equations across all frequencies. Geophysics, 75(2):N33–N41, 2010.  
+[11] Fabian Wenzlau and Tobias M Müller. Finite-difference modeling of wave propagation and diffusion in poroelastic media. Geophysics, 74(4):T55–T66, 2009.  
+[12] Jan Martin Nordbotten. Cell-centered finite volume discretizations for deformable porous media. International journal for numerical methods in engineering, 100(6):399-418, 2014.  
+[13] Irina Sokolova, Muhammad Gusti Bastisya, and Hadi Hajibeygi. Multiscale finite volume method for finite-volume-based simulation of poroelasticity. Journal of Computational Physics, 379:309-324, 2019.  
+[14] Joachim Berdal Haga, Harald Osnes, and Hans Petter Langtangen. On the causes of pressure oscillations in low permeable and low compressible porous media. International Journal for Numerical and Analytical Methods in Geomechanics, 36(12):1507-1522, 2012.  
+[15] Marcio A Murad, Marcio Borges, Jesus A Obregón, and Maicon Correa. A new locally conservative numerical method for two-phase flow in heterogeneous poroelastic media. Computers and Geotechnics, 48:192-207, 2013.  
+[16] Mary Wheeler, Guangri Xue, and Ivan Yotov. Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity. Computational Geosciences, 18(1):57-75, 2014.
+
+[17] Nikolaos Bouklas, Chad M Landis, and Rui Huang. A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels. Journal of the Mechanics and Physics of Solids, 79:21-43, 2015.  
+[18] Nikolaos Bouklas, Chad M Landis, and Rui Huang. Effect of solvent diffusion on crack-tip fields and driving force for fracture of hydrogels. Journal of Applied Mechanics, 82(8), 2015.  
+[19] Jinhyun Choo and Sanghyun Lee. Enriched Galerkin finite elements for coupled poromechanics with local mass conservation. Computer Methods in Applied Mechanics and Engineering, 341:311-332, 2018.  
+[20] Jiangguo Liu, Simon Tavener, and Zhuoran Wang. Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes. SIAM Journal on Scientific Computing, 40(5):B1229-B1252, 2018.  
+[21] R Liu, Mary F Wheeler, Clinton N Dawson, and RH Dean. On a coupled discontinuous/continuous galerkin framework and an adaptive penalty scheme for poroelasticity problems. Computer Methods in Applied Mechanics and Engineering, 198(41-44):3499-3510, 2009.  
+[22] Per Christian Hansen. Discrete inverse problems: insight and algorithms, volume 7. Siam, 2010.  
+[23] Jan S Hesthaven, Gianluigi Rozza, Benjamin Stamm, et al. Certified reduced basis methods for parametrized partial differential equations. Springer, 2016.  
+[24] Jian-Xun Wang, Jin-Long Wu, and Heng Xiao. Physics-informed machine learning approach for reconstructing reynolds stress modeling discrepancies based on dns data. Physical Review Fluids, 2(3):034603, 2017.  
+[25] Maziar Raissi. Deep hidden physics models: Deep learning of nonlinear partial differential equations. The Journal of Machine Learning Research, 19(1):932-955, 2018.  
+[26] Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019.  
+[27] J Nathan Kutz. Deep learning in fluid dynamics. Journal of Fluid Mechanics, 814:1-4, 2017.  
+[28] Zheng Wang, Dunhui Xiao, Fangxin Fang, Rajesh Govindan, Christopher C Pain, and Yike Guo. Model identification of reduced order fluid dynamics systems using deep learning. International Journal for Numerical Methods in Fluids, 86(4):255-268, 2018.  
+[29] Zhen Zhang, Xu-dong Song, Shu-ran Ye, Yi-wei Wang, Chen-guang Huang, Yi-ran An, and Yao-song Chen. Application of deep learning method to reynolds stress models of channel flow based on reduced-order modeling of dns data. Journal of Hydrodynamics, 31(1):58-65, 2019.  
+[30] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097-1105, 2012.  
+[31] Babak Alipanahi, Andrew Delong, Matthew T Weirauch, and Brendan J Frey. Predicting the sequence specificities of dna-and rna-binding proteins by deep learning. Nature biotechnology, 33(8):831, 2015.  
+[32] Dinggang Shen, Guorong Wu, and Heung-Il Suk. Deep learning in medical image analysis. Annual review of biomedical engineering, 19:221–248, 2017.  
+[33] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, 521(7553):436, 2015.  
+[34] Ejaz Ahmed, Michael Jones, and Tim K Marks. An improved deep learning architecture for person re-identification. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3908–3916, 2015.  
+[35] Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Inferring solutions of differential equations using noisy multi-fidelity data. Journal of Computational Physics, 335:736-746, 2017.  
+[36] Heng Xiao, J-L Wu, J-X Wang, Rui Sun, and CJ Roy. Quantifying and reducing model-form uncertainties in reynolds-averaged navier-stokes simulations: A data-driven, physics-informed bayesian approach. Journal of Computational Physics, 324:115-136, 2016.  
+[37] Ricardo Ruiz Baier, Alessio Gizzi, Alessandro Loppini, Christian Cherubini, and Simonetta Filippi. Modelling thermo-electro-mechanical effects in orthotropic cardiac tissue. Communications in Computational Physics, 2019.  
+[38] T. Kadeethum, H.M. Nick, S. Lee, C.N. Richardson, S. Salimzadeh, and F. Ballarin. A Novel Enriched Galerkin Method for Modelling Coupled Flow and Mechanical Deformation in Heterogeneous Porous Media. In 53rd US Rock Mechanics/Geomechanics Symposium, New York, NY, USA, 2019. American Rock Mechanics Association.  
+[39] J. Jaeger, N. Cook, and R. Zimmerman. Fundamentals of Rock Mechanics. Wiley-Blackwell, 4th edition, 2010.  
+[40] Olivier Coussy. Poromechanics. John Wiley & Sons, 2004.
+
+[41] Louis J Durlofsky. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water resources research, 27(5):699-708, 1991.  
+[42] Lars Holden and Oddvar Lia. A tensor estimator for the homogenization of absolute permeability. Transport in porous media, 8(1):37-46, 1992.  
+[43] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. nature, 323(6088):533-536, 1986.  
+[44] Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. science, 313(5786):504-507, 2006.  
+[45] Ian Goodfellow, *Yoshua Bengio*, and Aaron Courville. *Deep learning*. MIT press, 2016.  
+[46] Martin Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaogiang Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.  
+[47] Andreas Griewank et al. On automatic differentiation. Mathematical Programming: recent developments and applications, 6(6):83-107, 1989.  
+[48] George Corliss, Christele Faure, Andreas Griewank, Laurent Hascoet, and Uwe Naumann. Automatic differentiation of algorithms: from simulation to optimization. Springer Science & Business Media, 2002.  
+[49] Dong C Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimization. Mathematical programming, 45(1-3):503-528, 1989.  
+[50] Robert Malouf. A comparison of algorithms for maximum entropy parameter estimation. In proceedings of the 6th conference on Natural language learning-Volume 20, pages 1-7. Association for Computational Linguistics, 2002.  
+[51] M Baudin et al. pydoe: The experimental design package for python, 2015.  
+[52] Igor V Tetko, David J Livingstone, and Alexander I Luik. Neural network studies. 1. comparison of overfitting and overtraining. Journal of chemical information and computer sciences, 35(5):826-833, 1995.  
+[53] Jiancheng Du and R. C K Wong. Application of strain-induced permeability model in a coupled geomechanics-reservoir simulator. Journal of Canadian Petroleum Technology, 46(12):55-61, 2007.  
+[54] Jamal Hussein Abou-Kassem, M Rafiq Islam, and SM Farouq-Ali. Petroleum Reservoir Simulations. Elsevier, 2013.  
+[55] Teeratorn Kadeethum, Hamid Nick, and S Lee. Comparison of two-and three-field formulation discretizations for flow and solid deformation in heterogeneous porous media. In 20th Annual Conference of the International Association for Mathematical Geosciences, 2019.  
+[56] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pages 3320-3328, 2014.

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+# Łukasz Maziarka $^{12}$  Tomasz Danel $^{12}$  Sławomir Mucha $^{2}$  Krzysztof Rataj $^{1}$  Jacek Tabor $^{2}$  Stanisław Jastrzebski $^{3,4}$
+
+# Abstract
+
+Designing a single neural network architecture that performs competitively across a range of molecule property prediction tasks remains largely an open challenge, and its solution may unlock a widespread use of deep learning in the drug discovery industry. To move towards this goal, we propose Molecule Attention Transformer (MAT). Our key innovation is to augment the attention mechanism in Transformer using inter-atomic distances and the molecular graph structure. Experiments show that MAT performs competitively on a diverse set of molecular prediction tasks. Most importantly, with a simple self-supervised pretraining, MAT requires tuning of only a few hyperparameter values to achieve state-of-the-art performance on downstream tasks. Finally, we show that attention weights learned by MAT are interpretable from the chemical point of view.
+
+# 1. Introduction
+
+The task of predicting properties of a molecule lies at the center of applications such as drug discovery or material design. In particular, estimated  $85\%$  drug candidates fail the clinical trials in the United States after a long and costly development process (Wong et al., 2018). Potentially, many of these failures could have been avoided by having correctly predicted a clinically relevant property of a molecule such as its toxicity or bioactivity.
+
+Following the breakthroughs in image (Krizhevsky et al., 2012) and text classification (Vaswani et al., 2017), deep neural networks (DNNs) are expected to revolutionize other fields such as drug discovery or material design (Jr et al.,
+
+2019). However, on many molecular property prediction tasks DNNs are outperformed by shallow models such as support vector machine or random forest (Korotcov et al., 2017; Wu et al., 2018). On the other hand, while DNNs can outperform shallow models on some tasks, they tend to be difficult to train (Ishiguro et al., 2019; Hu et al., 2019), and can require tuning of a large number of hyperparameters. We also observe both issues on our benchmark (see Section 4.2).
+
+Making deep networks easier to train has been the central force behind their widespread use. In particular, one of the most important breakthroughs in deep learning was the development of initialization methods that allowed to train easily deep networks end-to-end (Goodfellow et al., 2016). In a similar spirit, our aim is to develop a deep model that is simple to use out-of-the-box, and achieves strong performance on a wide range of tasks in the field of molecule property prediction.
+
+In this paper we propose the Molecule Attention Transformer (MAT). We adapt Transformer (Devlin et al., 2018) to chemical molecules by augmenting the self-attention with inter-atomic distances and molecular graph structure. Figure 1 shows the architecture. We demonstrate that MAT, in contrast to other tested models, achieves strong performance across a wide range of tasks (see Figure 2). Next, we show that self-supervised pre-training further improves performance, while drastically reducing the time needed for hyperparameter tuning (see Table 3). In these experiments we tuned only the learning rate, testing 7 different values. Finally, we find that MAT has interpretable attention weights. We share pretrained weights at https://github.com/gmum/MAT.
+
+# 2. Related work
+
+Molecule property prediction. Predicting properties of a candidate molecule lies at the heart of many fields such as drug discovery and material design. Broadly speaking, there are two main approaches to predicting molecular properties. First, we can use our knowledge of the underlying physics (Lipinski et al., 1997). However, despite recent advances (Schütt et al., 2017), current approaches remain prohibitively costly to accurately predict many properties of
+
+![](images/65e6d9f097c96932cb35445e1a888158906ece969291d97a079e225115218f81.jpg)  
+Figure 1. Molecule Attention Transformer architecture. We largely base our model on the Transformer encoder. In the first layer we embed each atom using one-hot encoding and atomic features. The main innovation is the Molecule Multi-Head Self-Attention layer that augments attention with distance and graph structure of the molecule. We implement this using a weighted (by  $\lambda_d$ ,  $\lambda_g$ , and  $\lambda_a$ ) element-wise sum of the corresponding matrices.
+
+interest such as bioactivity. The second approach is to use existing data to train a predictive model (Haghighatlari & Hachmann, 2019). Here the key issue is the lack of large datasets. Even for the most popular drug targets, such as 5-HT1A (a popular target for depression), only thousands of active compounds are known. Promising direction is using hybrid approaches such as Wallach et al. (2015) or approaches leveraging domain knowledge and underlying physics to impose a strong prior such as Feinberg et al. (2018).
+
+Deep learning for molecule property prediction. Deep learning has become a valuable tool for modeling molecules. During the years, the community has progressed from using handcrafted representations to representing molecules as strings of symbols, and finally to the currently popular approaches based on molecular graphs.
+
+Graph convolutional networks in each subsequent layer gather information from adjacent nodes in the graph. In this way after  $N$  convolution layers each node has information from its  $N$ -edges distant neighbors. Using the graph structure improves performance in a range of molecule modeling tasks (Wu et al., 2018). Some of the most recent works
+
+implement more sophisticated generalization methods for gathering the neighbor data. Velicković et al. (2017); Shang et al. (2018) propose to augment GCNs with an attention mechanism. Li et al. (2018) introduces a model that dynamically learns neighbourhood function in the graph.
+
+In parallel to these advances, using the three-dimensional structure of the molecule is becoming increasingly popular. Perhaps the most closely related models are 3D Graph Convolutional Neural Network (3DGCN), Message Passing Neural Network (MPNN), and Adaptive Graph Convolutional Network (AGCN) (Cho & Choi, 2018; Gilmer et al., 2017; Li et al., 2018). 3DGCN and MPNN integrate graph and distance information in a single model, which enables them to achieve strong performance on tasks such as solubility prediction. In contrast to them, we additionally allow for a flexible neighbourhood based on self-attention.
+
+Transformer, originally developed for natural language processing (Vaswani et al., 2017), has been recently applied to retrosynthesis in Karpov et al. (2019). They represent compounds as sentences using the SMILES notation (Weininger, 1988). In contrast to them, we represent compounds as a list of atoms, and ensure that models understand the structure of
+
+the molecule by augmenting the self-attention mechanism (see Figure 1). Our ablation studies show it is a critical component of the model.
+
+To summarize, methods related to our model have been proposed in the literature. Our contribution is unifying these ideas in a single model based on the state-of-the-art Transformer architecture that preserves strong performance across many chemical tasks.
+
+How easy is it to use deep learning for molecule property prediction? DNNs performance is not always competitive to methods such as support vector machine or random forest. MoleculeNet is a popular benchmark for methods for molecule property prediction (Wu et al., 2018) that demonstrates this phenomenon. Similar results can be found in Withnall et al. (2019). We reproduce a similar issue on our benchmark. This makes using deep learning less applicable to molecule property prediction because in some cases practitioners might actually benefit from using other methods. Another issue is that graph neural networks, which are the most popular class of models for molecule property prediction, can be difficult to train. Ishiguro et al. (2019) show and try to address the problem that graph neural networks tend to underfit the training set. We also reproduce this issue on our benchmark (see also App. C).
+
+There has been a considerable interest in developing easier to use deep models for molecule property prediction. Goh et al. (2017) pretrains a deep network that takes as an input an image of a molecule. Another studies highlight the need to augment feedforward (Mayr et al., 2018) and graph neural networks (Yang et al., 2019) with handcrafted representations of molecules. Hu et al. (2019) proposes pretraining methods for graph neural networks and shows this largely alleviates the problem of underfitting, present in these architectures (Ishiguro et al., 2019). We take inspiration from Hu et al. (2019) and use one of the three pretraining tasks proposed therein.
+
+Concurrently, Wang et al. (2019); Honda et al. (2019) pretrain a vanilla Transformer (Devlin et al., 2018) that takes as input a text representation (SMILES) of a molecule. Honda et al. (2019) shows that decoding based approach improves data efficiency of the model. A similar approach, specialized to the task of drug-target interaction prediction, was concurrently proposed in Shin et al. (2019). In contrast to them, we adapt Transformer to chemical structures, which in our opinion is crucial for achieving strong empirical performance. We also use a domain-specific pretraining based on Wu et al. (2018). We further confirm importance of both approaches by comparing directly with Honda et al. (2019).
+
+Self-attention based models. Arguably, the attention mechanism (Bahdanau et al., 2014) has been one of the
+
+most important breakthroughs in deep learning. This is perhaps best illustrated by the wide-spread use of Transformer architecture in natural language processing (Vaswani et al., 2017; Devlin et al., 2018).
+
+Multiple prior works have augmented self-attention in Transformer using domain-specific knowledge (Chen et al., 2018; Shaw et al., 2018; Bello et al., 2019; Guo et al., 2019). Guo et al. (2019) encourages Transformer to attend to adjacent words in a sentence, and Chen et al. (2018) encourages another attention-based model to focus on pairs of words in a sentence that are connected in an external knowledge base. Our novelty is applying this successive modeling idea to molecule property prediction.
+
+# 3. Molecule Attention Transformer
+
+As the rich literature on deep learning for molecule property prediction suggests, it is necessary for a model to be flexible enough to represent a range of possible relationships between atoms of a compound. Inspired by its flexibility and strong empirical performance, we base our model on the Transformer encoder (Vaswani et al., 2017; Devlin et al., 2018). It is worth noting that natural language processing has inspired important advances in cheminformatics (Segler et al., 2017; Gómez-Bombarelli et al., 2018), which might be due to similarities between the two domains (Jastrzewski et al., 2016).
+
+Transformer. We begin by briefly introducing the Transformer architecture. On a high level, Transformer for classifications has  $N$  attention blocks followed by a pooling and a classification layer. Each attention block is composed of a multi-head self-attention layer, followed by a feed-forward block that includes a residual connection and layer normalization.
+
+The multi-head self-attention is composed of  $H$  heads. Head  $i$  ( $i = 1, \dots, H$ ) takes as input hidden state  $\mathbf{H}$  and computes first  $\mathbf{Q}_i = \mathbf{H}\mathbf{W}_i^Q$ ,  $\mathbf{K}_i = \mathbf{H}\mathbf{W}_i^H$ , and  $\mathbf{V}_i = \mathbf{H}\mathbf{W}_i^V$ . These are used in the attention operation as follows:
+
+$$
+\mathcal {A} ^ {(i)} = \rho \left(\frac {\mathbf {Q} _ {i} \mathbf {K} _ {i} ^ {T}}{\sqrt {d _ {k}}}\right) \mathbf {V} _ {i}, \tag {1}
+$$
+
+Molecule Self-Attention. Using a naive Transformer architecture would require encoding of chemical molecules as sentences. Instead, inspired by Battaglia et al. (2018), we interpret the self-attention as a soft adjacency matrix between the elements of the input sequence. Following this line of thought, it is natural to augment the self-attention using information about the actual structure of the model. This allows us to avoid using linearized (textual) representation of molecule as input (Jastrzebski et al., 2016), which we expect to be a better inductive bias for the model.
+
+More concretely, we propose the Molecule Self-Attention layer, which we describe in Equation 2. We augment the self-attention matrix as follows: let  $\mathbf{A} \in \{0,1\}^{N_{\mathrm{atoms}} \times N_{\mathrm{atoms}}}$  denote the graph adjacency matrix, and  $\mathbf{D} \in \mathbb{R}^{N_{\mathrm{atoms}} \times N_{\mathrm{atoms}}}$  denote the inter-atomic distances. Let  $\lambda_{a}, \lambda_{d}$ , and  $\lambda_{g}$  denote scalars weighting the self-attention, distance, and adjacency matrices. We modify Equation 1 as follows:
+
+$$
+\mathcal {A} ^ {(i)} = \left(\lambda_ {a} \rho \left(\frac {\mathbf {Q} _ {i} \mathbf {K} _ {i} ^ {T}}{\sqrt {d _ {k}}}\right) + \lambda_ {d} g (\mathbf {D}) + \lambda_ {g} \mathbf {A}\right) \mathbf {V} _ {i}, \quad (2)
+$$
+
+see also Figure 1. We denote  $\lambda_{a}$ ,  $\lambda_{d}$ , and  $\lambda_{g}$  jointly as  $\lambda$ . We use as  $g$  either softmax (normalized over the rows), or an element-wise  $g(d) = \exp(-d)$ . Finally, the distance matrix  $\mathbf{D}$  is computed using RDKit package (Landrum, 2016).
+
+Note that while we use only the adjacency and the distance matrices, MAT can be easily extended to include other types of information, e.g. forces between the atoms.
+
+Molecule Attention Transformer. To define the model, we replace all self-attention layers in the original Transformer encoder by our Molecular Self Attention layers. We embed each atom as a 26 dimensional vector following (Coley et al., 2017), shown in Table 1. In the experiments, we treat  $\lambda_{a}$ ,  $\lambda_{d}$ , and  $\lambda_{g}$  as hyperparameters and keep them frozen during training. Figure 1 illustrates the model.
+
+Pretraining. We experiment with one of the two node-level pretraining tasks proposed in Hu et al. (2019), which involves predicting the masked input nodes. Consistently with Hu et al. (2019), we found it stabilizes learning (see Figure 6) and reduces the need for an extensive hyperparameter search (see Table 3). Given that MAT already achieves good performance using this simple pretraining task, we leave for future work exploring the other tasks proposed in Hu et al. (2019).
+
+Other details. Inspired by Li et al. (2017); Clark et al. (2019), we add an artificial dummy node to the molecule. The dummy node is not connected by an edge to any other atom and the distance to any of them is set to  $10^{6}$ . Our motivation is to allow the model to skip searching for a molecular pattern if none is to find by putting higher attention on that distant node, which is similar to how BERT uses the separation token (Devlin et al., 2018; Clark et al., 2019). We confirm this intuition in Section 4.4 and Section 4.5.
+
+Finally, the distance matrices are calculated from 3D conformers calculated using UFFOPTIMIZEMOLECULE function from the RDKit package (Landrum, 2016), and the default parameters (MAXITERS=200, VDWTHRESH=10.0, CONFID=-1, UNUSEDINTERFRAGINTERACTIONS=True). For each compound we use one pre-computed conformation.
+
+We experimented with sampling more conformations for each compound, but did not observe a consistent boost in performance, however it is possible that using more sophisticated algorithms for compound 3D structure minimization could improve the results. We leave this for future work.
+
+Table 1. Featurization used to embed atoms in MAT.  
+
+<table><tr><td>INDICES</td><td>DESCRIPTION</td></tr><tr><td>0-11</td><td>ATOMIC IDENTITY AS A ONE-HOT VECTOR OF B, N, C, O, F, P, S, CL, BR, I, DUMMY, OTHER</td></tr><tr><td>12-17</td><td>NUMBER OF HEAVY NEIGHBORS AS ONE-HOT VECTOR OF 0, 1, 2, 3, 4, 5</td></tr><tr><td>18-22</td><td>NUMBER OF HYDROGEN ATOMS AS ONE-HOT VECTOR OF 0, 1, 2, 3, 4</td></tr><tr><td>23</td><td>FORMAL CHARGE</td></tr><tr><td>24</td><td>IS IN A RING</td></tr><tr><td>25</td><td>IS Aromatic</td></tr></table>
+
+# 4. Experiments
+
+We begin by comparing MAT to other popular models in the literature on a wide range of tasks. We find that with simple pretraining MAT outperforms other methods, while using a small budget for hyperparameter tuning.
+
+In the rest of this section we try to develop understanding of what makes MAT work well. In particular, we find that individual heads in the multi-headed self-attention layers learn chemically interpretable functions.
+
+# 4.1. Experimental settings
+
+Comparing different models for molecule property prediction is challenging. Despite considerable efforts, the community still lacks a standardized way to compare different models. In our work, we use a similar setting to MoleculeNet (Wu et al., 2018).
+
+Evaluation. Following recommendations of Wu et al. (2018) and the experimental setup of Podlewska & Kafel (2018), we use random split for FreeSolv, ESOL, and MetStab. For all the other datasets we use scaffold split, which assigns compounds that share the same molecular scaffolding to different subsets of the data (Bemis & Murcko, 1996). In regression tasks, the property value was standardized. Test performance is based on the model which gave best results in the validation setting. Each training was repeated 6 times, on different train/validation/test splits. All the other experimental details are reported in the Supplement.
+
+Datasets. We run experiments on a wide range of datasets that represent typical tasks encountered in molecule mod
+
+eling. Below, we include a short description of these tasks, and a more detailed description is moved to App. A.
+
+- FreeSolv, ESOL. Regression tasks used in Wu et al. (2018) for predicting water solubility in terms of the hydration free energy (FreeSolv) and log solubility in mols per litre (ESOL). The datasets have 642 and 1128 molecules, respectively.  
+- Blood-brain barrier permeability (BBBP). Binary classification task used in Wu et al. (2018) for predicting the ability of a molecule to penetrate the blood-brain barrier. The dataset has 2039 molecules.  
+- Estrogen Alpha, Estrogen Beta. The tasks are to predict whether a compound is active towards a given target (Estrogen- $\alpha$ , Estrogen- $\beta$ ) based on experimental data from the ChEMBL database (Gaulton et al., 2011). The datasets have 2398, and 1961 molecules, respectively.  
+- MetStab $_{\text{high}}$ , MetStab $_{\text{low}}$ . Binary classification tasks based on data from Podlewska & Kafel (2018) to predict whether a compound has high (over 2.32 h half-time) or low (lower than 0.6 h half-time) metabolic stability. Both datasets contain the same 2127 molecules.
+
+# 4.2. Molecule Attention Transformer
+
+Models. Similarly to Wu et al. (2018), we test a comprehensive set of baselines that span both shallow and deep models. We compare MAT to the following baselines: GCN (Duvenaud et al., 2015), Random Forest (RF) and Support Vector Machine with RBF kernel (SVM). We also test the following recently proposed models: Edge Attention-based Multi-Relational Graph Convolutional Network (EAGCN) (Shang et al., 2018), and Weave (Kearnes et al., 2016).
+
+Hyperparameter tuning. For each method we extensively tune their hyperparameters using random search (Bergstra & Bengio, 2012). To ensure fair comparison, each model is given the same budget for hyperparameter search. We run two sets of experiments with budget of 150 and 500 evaluations. We include hyperparameter ranges in App. B.
+
+Results. We evaluate models by their average rank according to the test set performance on the 7 datasets. Figure 2 reports ranks of all methods for the two considered hyperparameter budgets (150 and 500). Additionally, we report in Table 2 detailed scores on all datasets. We make three main observations.
+
+First, graph neural networks (GCN, Weave, EAGCN) on average do not outperform the other models. The best graph
+
+model achieves average rank 3.28 compared to 3.14 by RF. On the whole, performance of the deep models improves with larger hyperparameter search budget. This further corroborates the original motivation of our study. Indeed, using common deep learning methods for molecule property prediction is challenging in practice. It requires a large computational budget, and might still result in poor performance.
+
+Second, MAT outperforms the other tested methods in terms of the average rank. MAT achieves average rank of 2.71 and 2.42 for 150 and 500 budgets, compared to 3.14 of RF, which is the second best performing model. This shows that architecture MAT is flexible enough and has the correct inductive bias to perform well on a wide range of tasks.
+
+Examining performance of MAT across individual datasets, we observe that RF and SVM perform better on Estrogen-  $\beta$ , MetStab low, and MetStab high. Both RF and SVM use extended-connectivity fingerprint (Rogers & Hahn, 2010) as input representation, which encodes substructures in the molecule as features. Metabolic stability of a compound depends on existence of particular moieties, which are recognized by enzymes. Therefore a simple structure-based fingerprints perform well in such a setting. Wang et al. (2019); Mayr et al. (2018) show that using fingerprint as input representation improves performance of deep networks on related datasets. These two observations suggest that MAT could benefit from using fingerprints. Instead, we avoid using handcrafted representations, and investigate pretraining as an alternative in the next section. Though fingerprint-based models show excellent performance in all presented tasks, there are datasets on which they fail to match the performance of graph approaches. We observed this also on an energy prediction task (see the extension of our benchmark in App. C).
+
+# 4.3. Pretrained Molecule Attention Transformer
+
+Self-supervised pretraining has revolutionized natural language processing (Devlin et al., 2018) and has improved performance in molecule property prediction (Hu et al., 2019). We apply here node-level self-supervised pretraining from Hu et al. (2019) to MAT. The task is to predict features of masked out nodes. We refer the reader to App. D for more details.
+
+Models. We compare MAT to the two following baselines. First, we apply the same pretraining to EAGCN, which we will refer to as "Pretrained EAGCN". Second, we compare to a concurrent work by Honda et al. (2019). They pretrain a vanilla Transformer by decoding textual representation (SMILES) of molecules. We will refer to their method as "SMILES Transformer".
+
+![](images/61f6eb4bb83284b601ae4d6db51999fba2760aa4d5f0504d818adb1a395643bf.jpg)  
+(a) Hyperparameter search budget of 500 combinations.
+
+![](images/d1cd6f4628e3e9bff050960c05663b5ca6b40450f448e5992ae623e39fb013f7.jpg)  
+(b) Hyperparameter search budget of 150 combinations.  
+Figure 2. The average rank across the 7 datasets in the benchmark. For each model we test 500 (left) or 150 (right) hyperparameter combinations. We split the data using random or scaffold split (according to the dataset description) 6 times into train/validation/test folds and use the mean metrics across the test folds to obtain the ranklists of models. Interestingly, shallow models (RF and SVM) outperform graph models (GCN, EAGCN and Weave).
+
+Table 2. Test performances in the benchmark. For each model we test 500 (top) and 150 (bottom) hyperparameter combinations. On ESOL and FreeSolv we report RMSE (lower is better). The other tasks are evaluated using ROC AUC (higher is better). Experiments are repeated 6 times.  
+(a) Hyperparameter search budget of 500 combinations.  
+
+<table><tr><td></td><td>BBBP</td><td>ESOL</td><td>FREESOLV</td><td>ESTROGEN-α</td><td>ESTROGEN-β</td><td>\( METSTAB_{LOW} \)</td><td>\( METSTAB_{HIGH} \)</td></tr><tr><td>SVM</td><td>.707 ± .000</td><td>.478 ± .054</td><td>.461 ± .077</td><td>.973 ± .000</td><td>.778 ± .000</td><td>.893 ± .030</td><td>.890 ± .029</td></tr><tr><td>RF</td><td>.725 ± .006</td><td>.534 ± .073</td><td>.523 ± .097</td><td>.977 ± .001</td><td>.797 ± .007</td><td>.885 ± .029</td><td>.888 ± .030</td></tr><tr><td>GCN</td><td>.712 ± .010</td><td>.357 ± .032</td><td>.271 ± .048</td><td>.975 ± .003</td><td>.730 ± .006</td><td>.881 ± .031</td><td>.875 ± .036</td></tr><tr><td>WEAVE</td><td>.701 ± .016</td><td>.311 ± .023</td><td>.311 ± .072</td><td>.974 ± .003</td><td>.769 ± .023</td><td>.863 ± .028</td><td>.882 ± .043</td></tr><tr><td>EAGCN</td><td>.680 ± .014</td><td>.316 ± .024</td><td>.345 ± .051</td><td>.961 ± .011</td><td>.781 ± .012</td><td>.883 ± .024</td><td>.868 ± .034</td></tr><tr><td>MAT (OURS)</td><td>.728 ± .008</td><td>.285 ± .022</td><td>.263 ± .046</td><td>.979 ± .003</td><td>.765 ± .007</td><td>.862 ± .038</td><td>.888 ± .027</td></tr></table>
+
+(b) Hyperparameter search budget of 150 combinations.  
+
+<table><tr><td></td><td>BBBP</td><td>ESOL</td><td>FREESOLV</td><td>ESTROGEN-α</td><td>ESTROGEN-β</td><td>\( METSTAB_{LOW} \)</td><td>\( METSTAB_{HIGH} \)</td></tr><tr><td>SVM</td><td>.723 ± .000</td><td>.479 ± .055</td><td>.461 ± .077</td><td>.973 ± .000</td><td>.772 ± .000</td><td>.893 ± .030</td><td>.890 ± .029</td></tr><tr><td>RF</td><td>.721 ± .003</td><td>.534 ± .073</td><td>.524 ± .098</td><td>.977 ± .001</td><td>.791 ± .012</td><td>.892 ± .026</td><td>.888 ± .030</td></tr><tr><td>GCN</td><td>.695 ± .013</td><td>.369 ± .032</td><td>.299 ± .068</td><td>.975 ± .003</td><td>.730 ± .006</td><td>.884 ± .033</td><td>.875 ± .036</td></tr><tr><td>WEAVE</td><td>.702 ± .009</td><td>.298 ± .025</td><td>.298 ± .049</td><td>.974 ± .003</td><td>.769 ± .023</td><td>.863 ± .028</td><td>.885 ± .042</td></tr><tr><td>EAGCN</td><td>.680 ± .014</td><td>.322 ± .052</td><td>.337 ± .042</td><td>.961 ± .011</td><td>.781 ± .012</td><td>.859 ± .024</td><td>.844 ± .037</td></tr><tr><td>MAT (OURS)</td><td>.727 ± .006</td><td>.290 ± .019</td><td>.289 ± .047</td><td>.979 ± .003</td><td>.765 ± .007</td><td>.861 ± .029</td><td>.844 ± .052</td></tr></table>
+
+Hyperparameters. For all methods that use pretraining we reduce the hyperparameter grid to a minimum. We tune only the learning rate in  $\{1e - 3,5e - 4,1e - 4,5e - 5,1e - 5,5e - 6,1e - 6\}$ . We set the other hyperparameters to reasonable defaults based on results from Section 4.2. For MAT and EAGCN, we follow (Devlin et al., 2018) and use the largest model that still fits the GPU memory. For SMILES Transformer we use pretrained weights provided by Honda et al. (2019).
+
+Results. As in previous section, we compare the models based on their average rank on our benchmark. Figure 3 and Table 3 summarize the results.
+
+We observe that Pretrained MAT achieves average rank of 1.57 and outperforms MAT (average rank of 2.14). Importantly, for Pretrained MAT we only tuned the learning rate by evaluating 7 different values. This is in stark contrast to the 500 hyperparameter combinations tested for MAT and EAGCN. To visualize this, in Figure 4 we plot the average test performance of all models as a function of the number of tested hyperparameter combinations. We also note that
+
+![](images/43c9783876a6ff79df5cd42b562f555fcbade1e9037526174f0c607ff4723a68.jpg)  
+Figure 3. The average ranks across the 7 datasets in the benchmark. Pretrained MAT outperforms the other methods, despite a drastically smaller number of tested hyperparameters (7) compared to MAT and EAGCN (500).
+
+Pretrained MAT is more competitive on the three datasets mentioned in the previous section.
+
+We also find that Pretrained MAT outperforms the other two pretrained methods. Pretraining degrades the performance of EAGCN (average rank of 4.0), and SMILES Transformer achieves the worst average rank (average rank of 4.29). This suggests that both the architecture, and the choice of the pretraining task are important for the overall performance of the model.
+
+# 4.4. Ablation studies
+
+To better understand what contributes to the performance of MAT, we run a series of ablation studies on three representative datasets from our benchmark. We leave understanding how these choices interact with pretraining for future work.
+
+For experiments in this section we generated additional splits for ESOL, FreeSolv and BBBP datasets (different than in Section 4.2). For each configuration we select the best hyperparameters settings using random search under a budget of 100 evaluations. Experiments are repeated 3 times.
+
+Dummy node is not so dummy. MAT uses a dummy node that is disconnected from other atoms in the graph (Li et al., 2017). Our intuition is that such functionality can be useful to automatically adapt capacity on small datasets. By attending to the dummy node, the model can effectively choose to avoid changing the internal representation in a given layer. To examine this architectural choice, in Table 4 we compare MAT to a variant that does not include the dummy node. Results show that dummy node improves performance of the model.
+
+Knowing molecular graph and distances between atoms improves performance. Our key architectural innovation is integrating the molecule graph and inter-atomic distances with the self-attention layer in Transformer, as shown in Figure 1. To probe the importance of each of these sources of information, we removed them individually during training. Results in Table 5 suggest that keeping all sources of information results in the most stable performance across the three tasks, which is our primary goal. We also show that MAT can effectively use distance information in a toy task involving 3-dimensional distances between functional groups (see App.F).
+
+Using a more complex featurization does not improve performance. Many models for predicting molecule properties use additional edge features (Coley et al., 2017; Shang et al., 2018; Gilmer et al., 2017). In Table 6 we show that adding additional edge features does not improve MAT performance. This is certainly possible that a more comprehensive set of edge features or a better method to integrate them would improve performance, which we leave for future work. Procedure of using edge features is described in detail in App. E.
+
+# 4.5. Analysis.
+
+To understand MAT better, we investigate attention weights of the model, and the effect of pretraining on the learning dynamics.
+
+What is MAT looking at? In natural language processing, it has been shown that heads in Transformer seem to implement interpretable functions (Htut et al., 2019; Clark et al., 2019). Similarly, we investigate here the chemical function implemented by self-attention heads in MAT. We show patterns found in the model that was pretrained with the atom masking strategy (Hu et al., 2019), and then we verify our findings on a set of molecules extracted from the BBBP testing dataset.
+
+Based on a manual inspection of attention matrices of MAT, we find two broad patterns: (1) many attention heads are almost fully focused on the dummy node, (2) many attention heads focus only on a few atoms. This seems consistent with observations about Transformer in Clark et al. (2019). We also notice that initial self-attention layers learn simple and easily interpretable chemical patterns, while subsequent layers capture more complex arrangements of atoms. In Figure 5 we exemplify attention patterns on a random molecule from the BBBP dataset.
+
+To quantify the above findings, we select six heads from the first layer that fit the second category and seem to implement six patterns: (i) focuses on 2-neighboured aromatic carbons (not substituted); (ii) focuses on sulfurs; (iii) focuses on non-
+
+Table 3. Test set performances of methods that use pretraining. Experiments are repeated 6 times. SMILES refers to SMILES Transformer from Honda et al. (2019).  
+
+<table><tr><td></td><td>BBBP</td><td>ESOL</td><td>FREESOLV</td><td>ESTROGEN-α</td><td>ESTROGEN-β</td><td>METSTABLOW</td><td>METSTABHIGH</td></tr><tr><td>MAT</td><td>.737 ± .009</td><td>.278 ± .020</td><td>.265 ± .042</td><td>.998 ± .000</td><td>.773 ± .012</td><td>.862 ± .025</td><td>.884 ± .030</td></tr><tr><td>EAGCN</td><td>.687 ± .023</td><td>.323 ± .031</td><td>1.244 ± .341</td><td>.994 ± .002</td><td>.770 ± .010</td><td>.861 ± .029</td><td>.839 ± .038</td></tr><tr><td>SMILES</td><td>.717 ± .008</td><td>.356 ± .017</td><td>.393 ± .032</td><td>.953 ± .002</td><td>.757 ± .002</td><td>.860 ± .038</td><td>.881 ± .036</td></tr></table>
+
+![](images/aa58e51cbc83957b112f0e46ba1fdda3281f0f4f5e821d6d62e1ba79686ffd31.jpg)  
+(a) Regression tasks.
+
+![](images/b6d84d38085b9952cf17e7445f111e445cccb7d54c644ee2f3cee21051df4e7b.jpg)  
+(b) Classification tasks.  
+Figure 4. Test performance of all models as a function of the number of tested hyperparameter combinations (on a logarithmic scale). Figures show the aggregated mean RMSE for regression tasks (left) and the aggregated mean ROC AUC for classification tasks (right). Pretrained MAT requires tuning an order of magnitude less hyperparameters, and performs competitively on both sets of tasks.
+
+Table 4. Test performance of MAT model variant without the dummy node (- DUMMY) compared to performance of the original MAT.  
+
+<table><tr><td></td><td>BBBP</td><td>ESOL</td><td>FREESOLV</td></tr><tr><td>MAT</td><td>.723 ± .008</td><td>.286 ± .006</td><td>.250 ± .007</td></tr><tr><td>- DUMMY</td><td>.714 ± .010</td><td>.317 ± .014</td><td>.249 ± .014</td></tr></table>
+
+Table 5. Test performance of MAT with different sources of information removed (equivalent to setting the corresponding  $\lambda$  to zero).  
+
+<table><tr><td></td><td>BBBP</td><td>ESOL</td><td>FREESOLV</td></tr><tr><td>MAT</td><td>.723 ± .008</td><td>.286 ± .006</td><td>.250 ± .007</td></tr><tr><td>- GRAPH</td><td>.716 ± .009</td><td>.316 ± .036</td><td>.276 ± .034</td></tr><tr><td>- DISTANCE</td><td>.729 ± .013</td><td>.281 ± .001</td><td>.281 ± .013</td></tr><tr><td>- ATTENTION</td><td>.692 ± .001</td><td>.306 ± .026</td><td>.329 ± .014</td></tr></table>
+
+Table 6. Test performance of MAT using additional edge features (+ EDGES F.), compared to vanilla MAT.  
+
+<table><tr><td></td><td>BBBP</td><td>ESOL</td><td>FREESOLV</td></tr><tr><td>MAT</td><td>.723 ± .008</td><td>.286 ± .006</td><td>.250 ± .007</td></tr><tr><td>+ EDGES F.</td><td>.683 ± .008</td><td>.314 ± .014</td><td>.358 ± .023</td></tr></table>
+
+ring nitrogens; (iv) focuses on oxygen in carbonyl groups; (v) focuses on 3-neighboured aromatic atoms (positions of aromatic ring substitutions) and on sulfur for different atoms; (vi) focuses on nitrogens in aromatic rings. We found that on the BBBP testing dataset the atoms corresponding to these definitions (queried with SMARTS expressions) have indeed higher attention weights assigned to them than other atoms. For each head, we calculated attention weights for all atoms in all molecules and compared those matching our hypothesis against the other atoms. Their distributions differ significantly ( $p < 0.001$  in Kruskal-Wallis test) for all the patterns. The statistics and experimental details are summarized in App. G.
+
+Effect of pretraining. Wu et al. (2018) observed that using pretraining stabilizes and speeds up training of graph convolutional models. We observe a similar effect in our case. Figure 6 reports training error of MAT and Pretrained MAT on the ESOL (left), and the FreeSolv (right) datasets. We use the learning rate that achieved the best generalization on each dataset in Sec. 4.3. The experiments are repeated 6 times. On both datasets, Pretrained MAT converges faster and has a lower variance of training error across repetitions. Mean standard deviation of training error for Pretrained MAT (MAT) is 0.027 (0.057) and 0.040 (0.076) for ESOL
+
+![](images/90f7f84e42dab8dbe6872eea3b56339eb70fb5a2aa502968f37ba3ea43289d5e.jpg)  
+Figure 5. The heatmaps show selected self-attention weights from the first layer of MAT, on a random molecule from the BBBP dataset (center). The atoms, which these heads focus on, are marked with the same color as the corresponding matrix. The interpretation of the presented patterns is described in the text.
+
+and FreeSolv, respectively.
+
+![](images/a20326c199962e3361f567ebc320d21eef8431a1b9215fc8a59a0a71c265d654.jpg)  
+(a) ESOL  
+Figure 6. Training of MAT with (blue) and without (orange) pretraining, on ESOL (left) and FreeSolv (right). Pretraining stabilizes training (smaller variance of the training error) and improves convergence speed.
+
+![](images/fa4dea71c7d254c4e41deecd772edd449d688f2ab17bc98e81ba293787dd3715.jpg)  
+(b) FreeSolv
+
+# 5. Conclusions.
+
+In this work we propose Molecule Attention Transformer as a versatile architecture for molecular property prediction. In contrast to other tested models, MAT performs well across a wide range of molecule property prediction tasks. Moreover, inclusion of self-supervised pretraining further improves its performance, and drastically reduces the need for tuning of hyperparameters.
+
+We hope that our work will widen adoption of deep learning in applications involving molecular property prediction,
+
+as well as inspire new modeling approaches. One particularly promising avenue for future work is exploring better pretraining tasks for MAT.
+
+# References
+
+Bahdanau, D., Cho, K., and Bengio, Y. Neural machine translation by jointly learning to align and translate, 2014.  
+Battaglia, P. W., Hamrick, J. B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V. F., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., Faulkner, R., Gülçehre, C., Song, F., Ballard, A. J., Gilmer, J., Dahl, G. E., Vaswani, A., Allen, K., Nash, C., Langston, V., Dyer, C., Heess, N., Wierstra, D., Kohli, P., Botvinick, M., Vinyals, O., Li, Y., and Pascanu, R. Relational inductive biases, deep learning, and graph networks. CoRR, abs/1806.01261, 2018.  
+Bello, I., Zoph, B., Vaswani, A., Shlens, J., and Le, Q. V. Attention augmented convolutional networks. CoRR, abs/1904.09925, 2019.  
+Bemis, G. W. and Murcko, M. A. The properties of known drugs. 1. molecular frameworks. Journal of medicinal chemistry, 39(15):2887-2893, 1996.  
+Bergstra, J. and Bengio, Y. Random search for hyperparameter optimization. *J. Mach. Learn. Res.*, 13(1): 281-305, February 2012. ISSN 1532-4435.  
+Chen, G., Chen, P., Hsieh, C.-Y., Lee, C.-K., Liao, B., Liao, R., Liu, W., Qiu, J., Sun, Q., Tang, J., et al. Alchemy: A quantum chemistry dataset for benchmarking ai models. arXiv preprint arXiv:1906.09427, 2019.  
+Chen, Q., Zhu, X., Ling, Z.-H., Inkpen, D., and Wei, S. Neural natural language inference models enhanced with external knowledge. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 2406-2417, Melbourne, Australia, July 2018. Association for Computational Linguistics.  
+Cho, H. and Choi, I. S. Three-dimensionally embedded graph convolutional network (3DGCN) for molecule interpretation. CoRR, abs/1811.09794, 2018.  
+Clark, K., Khandelwal, U., Levy, O., and Manning, C. D. What does bert look at? an analysis of bert's attention. arXiv preprint arXiv:1906.04341, 2019.  
+Coley, C. W., Barzilay, R., Green, W. H., Jaakkola, T. S., and Jensen, K. F. Convolutional embedding of attributed molecular graphs for physical property prediction. Journal of Chemical Information and Modeling, 57(8):1757-1772, 2017. doi: 10.1021/acs.jcim.6b00601. PMID: 28696688.
+
+Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K. Bert: Pre-training of deep bidirectional transformers for language understanding, 2018. cite arxiv:1810.04805Comment: 13 pages.  
+Duvenaud, D. K., Maclaurin, D., Iparraguirre, J., Bombarell, R., Hirzel, T., Aspuru-Guzik, A., and Adams, R. P. Convolutional networks on graphs for learning molecular fingerprints. In Cortes, C., Lawrence, N. D., Lee, D. D., Sugiyama, M., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 28, pp. 2224-2232. Curran Associates, Inc., 2015.  
+Feinberg, E. N., Sur, D., Wu, Z., Husic, B. E., Mai, H., Li, Y., Sun, S., Yang, J., Ramsundar, B., and Pande, V. S. Potentialnet for molecular property prediction. ACS Central Science, 4(11):1520-1530, 2018. doi: 10.1021/acscentsci.8b00507.  
+Gaulton, A., Bellis, L. J., Bento, A. P., Chambers, J., Davies, M., Hersey, A., Light, Y., McGlinchey, S., Michalovich, D., Al-Lazikani, B., and Overington, J. P. ChEMBL: a large-scale bioactivity database for drug discovery. *Nucleic Acids Research*, 40(D1):D1100–D1107, 09 2011. ISSN 0305-1048. doi: 10.1093/nar/gkr777.  
+Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. Neural message passing for quantum chemistry. In Precup, D. and Teh, Y. W. (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 1263-1272, International Convention Centre, Sydney, Australia, 06-11 Aug 2017. PMLR.  
+Goh, G., Siegel, C., Vishnu, A., Hodas, N., and Baker, N. Chemception: A deep neural network with minimal chemistry knowledge matches the performance of expert-developed qsar/qspr models. arXiv preprint arXiv:1706.06689, 06 2017.  
+Gómez-Bombarelli, R., Wei, J. N., Duvenaud, D., Hernández-Lobato, J., Sánchez-Lengeling, B., Sheberla, D., Aguilera-Iparraguirre, J., Hirzel, T. D., Adams, R. P., and Aspuru-Guzik, A. Automatic chemical design using a data-driven continuous representation of molecules. ACS Central Science, 4(2):268-276, 02 2018. doi: 10.1021/acscentsci.7b00572.  
+Goodfellow, I., Bengio, Y., and Courville, A. Deep Learning. MIT Press, 2016.  
+Guo, M., Zhang, Y., and Liu, T. Gaussian transformer: a lightweight approach for natural language inference. In AAAI 2019, 2019.  
+Haghhighatlari, M. and Hachmann, J. Advances of machine learning in molecular modeling and simulation. Current Opinion in Chemical Engineering, 23:51 - 57, 2019.
+
+ISSN 2211-3398. doi: https://doi.org/10.1016/j.coche.2019.02.009. Frontiers of Chemical Engineering: Molecular Modeling.  
+Honda, S., Shi, S., and Ueda, H. R. Smiles transformer: Pre-trained molecular fingerprint for low data drug discovery, 2019.  
+Htut, P. M., Phang, J., Bordia, S., and Bowman, S. R. Do attention heads in bert track syntactic dependencies?, 2019.  
+Hu, W., Liu, B., Gomes, J., Zitnik, M., Liang, P., Pande, V. S., and Leskovec, J. Pre-training graph neural networks. CoRR, abs/1905.12265, 2019.  
+Ishiguro, K., Maeda, S., and Koyama, M. Graph warp module: an auxiliary module for boosting the power of graph neural networks. CoRR, abs/1902.01020, 2019.  
+Jastrzebski, S., Lesniak, D., and Czarnecki, W. M. Learning to smile (s). arXiv preprint arXiv:1602.06289, 2016.  
+Jr, J. F. R., Florea, L., de Oliveira, M. C. F., Diamond, D., and Jr, O. N. O. A survey on big data and machine learning for chemistry, 2019.  
+Karpov, P., Godin, G., and Tetko, I. V. A transformer model for retrosynthesis. In International Conference on Artificial Neural Networks, pp. 817-830. Springer, 2019.  
+Kearnes, S., McCloskey, K., Berndl, M., Pande, V., and Riley, P. Molecular graph convolutions: Moving beyond fingerprints. Journal of Computer-Aided Molecular Design, 30, 03 2016. doi: 10.1007/s10822-016-9938-8.  
+Kim, S., Chen, J., Cheng, T., Gindulyte, A., He, J., He, S., Li, Q., Shoemaker, B. A., Thiessen, P. A., Yu, B., et al. Pubchem 2019 update: improved access to chemical data. *Nucleic acids research*, 47(D1):D1102–D1109, 2018.  
+Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Korotcov, A., Tkachenko, V., Russo, D. P., and Ekins, S. Comparison of deep learning with multiple machine learning methods and metrics using diverse drug discovery data sets. Molecular Pharmaceutics, 14(12):4462-4475, 12 2017. doi: 10.1021/acs.molpharmaceut.7b00578.  
+Krizhevsky, A., Sutskever, I., and Hinton, G. E. Imagenet classification with deep convolutional neural networks. In Pereira, F., Burges, C. J. C., Bottou, L., and Weinberger, K. Q. (eds.), Advances in Neural Information Processing Systems 25, pp. 1097-1105. Curran Associates, Inc., 2012.  
+Landrum, G. Rdkit: Open-source cheminformatics software. 2016.
+
+Li, J., Cai, D., and He, X. Learning graph-level representation for drug discovery. arXiv preprint arXiv:1709.03741, 2017.  
+Li, R., Wang, S., Zhu, F., and Huang, J. Adaptive graph convolutional neural networks. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.  
+Lipinski, C. A., Lombardo, F., Dominy, B. W., and Feeney, P. J. Experimental and computational approaches to estimate solubility and permeability in drug discovery and development settings. Advanced Drug Delivery Reviews, 23(1):3 - 25, 1997. ISSN 0169-409X. doi: https://doi.org/10.1016/S0169-409X(96)00423-1. In Vitro Models for Selection of Development Candidates.  
+Mayr, A., Klambauer, G., Unterthiner, T., Steijaert, M., Wegner, J. K., Ceulemans, H., Clevert, D.-A., and Hochreiter, S. Large-scale comparison of machine learning methods for drug target prediction on chembl. Chem. Sci., 9: 5441-5451, 2018. doi: 10.1039/C8SC00148K.  
+Podlewska, S. and Kafel, R. Metstabon—online platform for metabolic stability predictions. International journal of molecular sciences, 19(4):1040, 2018.  
+Ramsundar, B., Eastman, P., Walters, P., Pande, V., Leswing, K., and Wu, Z. Deep Learning for the Life Sciences. O'Reilly Media, 2019.  
+Rogers, D. and Hahn, M. Extended-connectivity fingerprints. Journal of chemical information and modeling, 50 (5):742-754, 2010.  
+Schütt, K. T., Arbabzadah, F., Chmiela, S., Müller, K. R., and Tkatchenko, A. Quantum-chemical insights from deep tensor neural networks. Nature Communications, 8: 13890, Jan 2017. doi: 10.1038/ncomms13890.  
+Segler, M., Kogej, T., Tyrchan, C., and Waller, M. Generating focused molecule libraries for drug discovery with recurrent neural networks. ACS Central Science, 4, 01 2017. doi: 10.1021/acscentsci.7b00512.  
+Shang, C., Liu, Q., Chen, K.-S., Sun, J., Lu, J., Yi, J., and Bi, J. Edge Attention-based Multi-Relational Graph Convolutional Networks. arXiv e-prints, art. arXiv:1802.04944, Feb 2018.  
+Shaw, P., Uszkoreit, J., and Vaswani, A. Self-attention with relative position representations. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), pp. 464-468, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-2074.
+
+Shin, B., Park, S., Kang, K., and Ho, J. C. Self-attention based molecule representation for predicting drug-target interaction. In Doshi-Velez, F., Fackler, J., Jung, K., Kale, D., Ranganath, R., Wallace, B., and Wiens, J. (eds.), Proceedings of the 4th Machine Learning for Healthcare Conference, volume 106 of Proceedings of Machine Learning Research, pp. 230–248, Ann Arbor, Michigan, 09–10 Aug 2019. PMLR.  
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. CoRR, abs/1706.03762, 2017.  
+Veličković, P., Cucurull, G., Casanova, A., Romero, A., Liò, P., and Bengio, Y. Graph Attention Networks. arXiv e-prints, art. arXiv:1710.10903, Oct 2017.  
+Wallach, I., Dzamba, M., and Heifets, A. Atomnet: A deep convolutional neural network for bioactivity prediction in structure-based drug discovery. *ArXiv*, abs/1510.02855, 2015.  
+Wang, S., Guo, Y., Wang, Y., Sun, H., and Huang, J. Smilesbert: Large scale unsupervised pre-training for molecular property prediction. In Proceedings of the 10th ACM International Conference on Bioinformatics, Computational Biology and Health Informatics, BCB '19, pp. 429-436, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450366663. doi: 10.1145/3307339.3342186.  
+Weininger, D. Smiles, a chemical language and information system. 1. introduction to methodology and encoding rules. Journal of chemical information and computer sciences, 28(1):31-36, 1988.  
+Withnall, M., Lindelöf, E., Engkvist, O., and Chen, H. Building attention and edge convolution neural networks for bioactivity and physical-chemical property prediction, Sep 2019.  
+Wong, C. H., Siah, K. W., and Lo, A. W. Estimation of clinical trial success rates and related parameters. *Bio/statistics*, 20(2):273-286, 01 2018. ISSN 1465-4644. doi: 10.1093/biostatistics/kxx069.  
+Wu, Z., Ramsundar, B., Feinberg, E. N., Gomes, J., Geniesse, C., Pappu, A. S., Leswing, K., and Pande, V. Moleculenet: a benchmark for molecular machine learning. Chem. Sci., 9:513-530, 2018. doi: 10.1039/C7SC02664A.  
+Yang, K., Swanson, K., Jin, W., Coley, C., Eiden, P., Gao, H., Guzman-Perez, A., Hopper, T., Kelley, B., Mathea, M., et al. Analyzing learned molecular representations for property prediction. Journal of chemical information and modeling, 59(8):3370-3388, 2019.
+
+# A. Dataset details.
+
+We include below a more detailed description of the datasets used in our benchmark.
+
+- FreeSolv, ESOL. Regression tasks. Popular tasks for predicting water solubility in terms of the hydration free energy (FreeSolv) and logS (ESOL). Solubility of molecules is an important property that influences the bioavailability of drugs.  
+- Blood-brain barrier permeability (BBBP). Binary classification task. The blood-brain barrier (BBB) separates the central nervous system from the bloodstream. Predicting BBB penetration is especially relevant in drug design when the goal for the molecule is either to reach the central nervous system or the contrary – not to affect the brain.  
+- MetStab $_{\text{high}}$ , MetStab $_{\text{low}}$ . Binary classification tasks. The metabolic stability of a compound is a measure of the half-life time of the compound within an organism. The compounds for this task were taken from (Podlewska & Kafel, 2018), where compounds were divided into three sets: high, medium, and low stability. In this paper we concatenated these sets in order to build two classification tasks: MetStab $_{\text{high}}$  (discriminating high against others) and MetStab $_{\text{low}}$  (discriminating low against others).  
+- Estrogen Alpha, Estrogen Beta. Binary classification tasks. Often in drug discovery, it is important that a molecule is not potent towards a given target. Modulating of the estrogen receptors changes the genomic expression throughout the body, which in turn may lead to the development of cancer. For these tasks, the compounds with known activities towards the receptors were extracted from ChEMBL (Gaulton et al., 2011) database and divided into active and inactive sets based on their reported inhibition constant (Ki), being  $< 100$  nM and  $>1000$  nM, respectively.
+
+# B. Other experimental details
+
+In this section we include details for hyperparameters and training settings used in Section 4.2.
+
+Molecule Attention Trainsformer. Table 7 shows hyperparameter ranges used in experiments for MAT. A short description of these hyperparameters is listed below:
+
+- MODEL DIM - size of embedded atom features,  
+- LAYERS NUMBER - number of encoder module repeats (N in Figure 1),
+
+- ATTENTION HEADS NUMBER - number of molecule self-attention heads,  
+- PFFs NUMBER - number of dense layers in the position-wise feed forward block ( $\mathbf{K}$  in Figure 1),  
+-  $\lambda_{att}$  - self-attention weight  $\lambda_{att}$ ,  
+-  $\lambda_{dist} -$  distance weight  $\lambda_{d}$  
+- DISTANCE MATRIX KERNEL - function  $g$  used to transform the distance matrix  $\mathbf{D}$  
+- MODEL DROPOUT - dropout applied after the embedding layer, position-wise feed forward layers, and residual layers (before sum operation),  
+- WEIGHT DECAY - optimizer weight decay,  
+- LEARNING RATE - (see Equation 3)  
+- EPOCHS NUMBER - number of epochs for which the model is trained  
+- BATCH SIZE - batch size used during the training of the model  
+- WARMUP FACTOR - fraction of epochs after which we end with increasing the learning rate linearly and begin with decreasing it proportionally to the inverse square root of the step number. (see Equation 3)
+
+Table 7. Molecule Attention Transformer hyperparameters ranges  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>BATCH SIZE</td><td>8, 16, 32, 64, 128</td></tr><tr><td>LEARNING RATE</td><td>.01, .005, .001, .0005, .0001</td></tr><tr><td>EPOCHS</td><td>30, 100</td></tr><tr><td>MODEL DIM</td><td>32, 64, 128, 256, 512, 1024</td></tr><tr><td>LAYERS NUMBER</td><td>1, 2, 4, 6, 8</td></tr><tr><td>ATTENTION HEADS NUMBER</td><td>1, 2, 4, 8, 16</td></tr><tr><td>PFFS NUMBER</td><td>1</td></tr><tr><td>λatt</td><td>0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1</td></tr><tr><td>λdistance</td><td>0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1</td></tr><tr><td>DISTANCE MATRIX KERNEL</td><td>‘SOFTMAX’, ‘EXP’</td></tr><tr><td>MODEL DROPOUT</td><td>.0, .1, .2</td></tr><tr><td>WEIGHT DECAY</td><td>.0, .00001, .0001, .001, .01</td></tr><tr><td>WARMUP FACTOR</td><td>.0, .1, .2, .3, .4, .5</td></tr></table>
+
+As suggested in (Vaswani et al., 2017), for optimization of MAT we used Adam optimizer (Kingma & Ba, 2014), with learning rate scheduler given by the following formula:
+
+$$
+\begin{array}{l} \text {S t e p} _ {L R} = \text {o p t i m i z e r} \quad f a c t o r \cdot \text {m o d e l} \quad \dim^ {- 0. 5}. \\ \cdot \min  \left(\text {s t e p} n u m ^ {- 0. 5}, \text {s t e p} n u m \cdot \text {w a r m u p s t e p s} ^ {- 0. 5}\right). \tag {3} \\ \end{array}
+$$
+
+Where optimizer factor is given by  $100 \cdot$  LEARNING RATE and warmup steps is given by WARMUP FACTOR  $\cdot$  total train steps number.
+
+After  $N$  layers embedding of the molecule is calculated by taking the mean of returned by the network vector representations of all atoms (Global pooling in Figure 1). Then it is passed to the single linear layer, which returns the prediction.
+
+SVM, RF, GCN, Weave. In our experiments, we used DeepChem (Ramsundar et al., 2019) implementation of baseline algorithms (SVM, RF, GCN, Weave). We used the same hyperparameters for tuning as were used in DeepChem, having regard to their proposed default values (we list them in Tables 8 - 11).
+
+RF and SVM work on the vector representation of molecule given by the Extended-connectivity fingerprints (Rogers & Hahn, 2010). ECFP vectors were calculated using class CIRCULARFINGERPRINT from the DeepChem package, with default parameters (RADIUS=2, SIZE=2048).
+
+Table 8. SVM hyperparameter ranges  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td rowspan="2">C</td><td>.25, .4375, .625, .8125, 1., 1.1875, 1.375, 1.5625, 1.75, 1.9375, 2.125, 2.3125, 2.5, 2.6875, 2.875, 3.0625, 3.25, 3.4375, 3.625, 3.8125, 4.</td></tr><tr><td rowspan="2">.0125, .021875, .03125, .040625, .05, .059375, .06875, .078125, .0875, .096875, .10625, .115625, .125, .134375, .14375, .153125, .1625, .171875, .18125, .190625, .2</td></tr><tr><td>GAMMA</td></tr></table>
+
+Table 9. RF hyperparameter ranges  
+
+<table><tr><td colspan="2">PARAMETERS</td></tr><tr><td>N ESTIMATORS</td><td>125, 218, 312, 406, 500, 593, 687, 781, 875, 968, 1062, 1156, 1250, 1343, 1437, 1531, 1625, 1718, 1812, 1906, 2000</td></tr></table>
+
+Table 10. GCN hyperparameter ranges  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>BATCH SIZE</td><td>64, 128, 256</td></tr><tr><td>LEARNING RATE</td><td>0.002, 0.001, 0.0005</td></tr><tr><td>N FILTERS</td><td>64, 128, 192, 256</td></tr><tr><td>N FULLY CONNECTED NODES</td><td>128, 256, 512</td></tr></table>
+
+EAGCN Table 12 shows hyperparameter ranges used in experiments for EAGCN. For EAGCN with weighted struc
+
+Table 11. Weave hyperparameter ranges  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>BATCH SIZE</td><td>16, 32, 64, 128</td></tr><tr><td>NB EPOCH</td><td>20, 40, 60, 80, 100</td></tr><tr><td>LEARNING RATE</td><td>0.002, 0.001, 0.00075, 0.0005</td></tr><tr><td>N GRAPH FEAT</td><td>32, 64, 96, 128, 256</td></tr><tr><td>N PAIR FEAT</td><td>14</td></tr></table>
+
+ture number of convolutional features  $n\_ sgc = n\_ sgc\_ 1 + n\_ sgc\_ 2 + n\_ sgc\_ 3 + n\_ sgc\_ 4 + n\_ sgc\_ 5.$
+
+Table 12. EAGCN hyperparameter ranges  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>BATCH SIZE</td><td>16, 32, 64, 128, 256, 512</td></tr><tr><td>EAGCN STRUCTURE</td><td>&#x27;CONCATE&#x27;, &#x27;WEIGHTED&#x27;</td></tr><tr><td>NUM EPOCHS</td><td>30, 100</td></tr><tr><td>LEARNING RATE</td><td>.01, .005, .001, .0005, .0001</td></tr><tr><td>DROPOUT</td><td>.0, .1, .3</td></tr><tr><td>WEIGHT DECAY</td><td>.0, .001, .01, .0001</td></tr><tr><td>N CONV LAYERS</td><td>1, 2, 4, 6</td></tr><tr><td>N DENSE LAYERS</td><td>1, 2, 3, 4</td></tr><tr><td>N SGC 1</td><td>30, 60</td></tr><tr><td>N SGC 2</td><td>5, 10, 15, 20, 30</td></tr><tr><td>N SGC 3</td><td>5, 10, 15, 20, 30</td></tr><tr><td>N SGC 4</td><td>5, 10, 15, 20, 30</td></tr><tr><td>N SGC 5</td><td>5, 10, 15, 20, 30</td></tr><tr><td>DENSE DIM</td><td>16, 32, 64, 128</td></tr></table>
+
+# C. Additional results for Sec. 4.2
+
+Predicting internal energy We run an additional experiment on a regression task related to quantum mechanics. From the Alchemy dataset (Chen et al., 2019), which is a dataset of 12 quantum properties calculated for  $200\mathrm{K}$  molecules, we have chosen internal energy at  $298.15\mathrm{K}$  to further test the performance of our model. We hypothesize that our molecule self-attention should perform particularly well in tasks involving atom level interactions such as energy prediction.
+
+Table 13 presents mean absolute errors of three methods: one classical method (RF), one graph method (GCN), and our pretrained MAT. We use original train-valid/test splits of the dataset. For RF and GCN we run a random search with budget of 500 hyperparameter sets. For pretrained MAT, we tune only the learning rate, that is selected from  $\{1e - 3,5e - 4,1e - 4,5e - 5,1e - 5,5e - 6,1e - 6\}$ .
+
+MAT achieves a slightly lower error than GCN. As can be expected, both graph methods can learn internal energy function correctly because of the locality preserved in the graph structure. The classical method based on fingerprints gives MAE that is almost two orders of magnitude higher than MAE of the other methods in the comparison.
+
+Table 13. Test results for internal energy prediction reported as MAE. All methods were tuned with a random search with budget of 500 hyperparameter combinations.  
+
+<table><tr><td></td><td>U (INTERNAL ENERGY)</td></tr><tr><td>RF</td><td>.380</td></tr><tr><td>GCN</td><td>.006</td></tr><tr><td>MAT</td><td>.004</td></tr></table>
+
+![](images/8de4bf30b00c174983ec590d4a0d5df4016e25653ae24460d4ef2e9af0da8a31.jpg)  
+Figure 7. Training loss of MAT and GCN as a function of the number of layers (left) and model dimensionality (right).
+
+![](images/1ce07000f1a11e485fb43e80268f0be1ab641e4897cd1373ae7ee42dd0680162.jpg)
+
+Training error for graph-based neural networks Ishiguro et al. (2019) show that graph neural networks suffer from underfitting of the training set and their performance does not scale well with the complexity of the network. We reproduce their experiments and confirm that this problem is indeed present for both GCN and MAT. According to Figure 7, the training loss of GCN and MAT flattens at some point and stops decreasing even if we keep increasing the number of layers and model dimensionality. Despite this issue, for almost all settings, MAT achieves lower training error than GCN.
+
+# D. Additional details for Sec. 4.3
+
+Task description. As a node-level pretraining task we chose masking from (Hu et al., 2019) which is a version of BERT masked language model adapted to graph structured data. The idea is that predicting masked nodes based on their neighbourhood will encourage model to capture domain specific relationships between atoms.
+
+For each molecular graph we randomly replace  $15\%$  of input nodes (atom attributes) with special mask token. After forward pass we apply linear model to corresponding node embeddings to predict masked node attributes. In case of EAGCN we additionally mask attributes of edges connected to masked nodes to prevent model from learning simple value copying.
+
+Pretraining setting. Training dataset consisted of  $2\mathrm{m}$  n molecules sampled from the ZINC15 database. Models
+
+were trained for 8 epochs with learning rate set to 0.001 and batch size 256. MAT was optimized with Noam optimizer (described in App. B), whereas for EAGCN we used Adam (Kingma & Ba, 2014). In both cases procedure minimized binary cross entropy loss.
+
+Fine-tuning setting. All our pretrained models are finetuned on the target tasks for 100 epochs, with batch size equal to 32 and learning rate selected from the set of  $\{1e - 3,5e - 4,1e - 4,5e - 5,1e - 5,5e - 6,1e - 6\}$ .
+
+In Estrogen Alpha experiments we excluded three molecules (with the highest number of atoms) from the dataset, due to the memory issues.
+
+Table 14. Pretrained MAT hyperparameters  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>MODEL DIM</td><td>1024</td></tr><tr><td>LAYERS NUMBER</td><td>8</td></tr><tr><td>ATTENTION HEADS NUMBER</td><td>16</td></tr><tr><td>PFFS NUMBER</td><td>1</td></tr><tr><td>λatt</td><td>.33</td></tr><tr><td>λdistance</td><td>.33</td></tr><tr><td>DISTANCE MATRIX KERNEL</td><td>&#x27;EXP&#x27;</td></tr><tr><td>MODEL DROPOUT</td><td>.0</td></tr><tr><td>WEIGHT DECAY</td><td>.0</td></tr></table>
+
+Table 15. Pretrained EAGCN hyperparameters  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>EAGCN STRUCTURE</td><td>&#x27;WEIGHTED&#x27;</td></tr><tr><td>DROPOUT</td><td>.0</td></tr><tr><td>WEIGHT DECAY</td><td>.0</td></tr><tr><td>N CONV LAYERS</td><td>8</td></tr><tr><td>N DENSE LAYERS</td><td>1</td></tr><tr><td>N SGC</td><td>1080</td></tr></table>
+
+SMILES Transformer. We used pretrained weights of SMILES-Transformers conducted by Honda et al. (2019). In this setting, according to the authors, we used MLP with 1 hidden layer, with 100 hidden units, that works on the 1024-dimensional molecule embedding returned by the pretrained transformer. We trained this MLP on the target tasks for 100 epochs, with batch size equal to 32 and learning rate selected from the set of  $\{1e - 3,5e - 4,1e - 4,5e - 5,1e - 5,5e - 6,1e - 6\}$ .
+
+# E. Additional results for Sec. 4.4
+
+Edge features. Every bond in the molecule was embedded by the vector of edge features (we used features similar to described in (Shang et al., 2018)). Every edge feature was then passed through linear layer, followed by ReLU activation, which returned one single value for every
+
+single edge (if there is no edge between atoms, we pass zero vector through the layer). This results in the matrix  $\mathbf{E} \in \mathbb{R}^{N_{\mathrm{atoms}} \times N_{\mathrm{atoms}}}$  which was then used in Molecule Self-Attention layer, instead of the adjacency matrix.
+
+Table 16. Edge Features used for experiments form Table 6  
+
+<table><tr><td>ATTRIBUTE</td><td>DESCRIPTION</td></tr><tr><td>BOND ORDER</td><td>VALUES FROM SET {1, 1.5, 2, 3}</td></tr><tr><td>AROMATICITY</td><td>IS Aromatic</td></tr><tr><td>CONJUGATION</td><td>IS CONJUGATED</td></tr><tr><td>RING STATUS</td><td>IS IN A RING</td></tr></table>
+
+# F. Toy task
+
+Task description. The essential feature of Molecule Attention Transformer is that it augments the self-attention module using molecule structure. Here we investigate MAT on a task heavily reliant on distances between atoms; we are primarily interested in how the performance of MAT depends on  $\lambda_{a}$ ,  $\lambda_{d}$ ,  $\lambda_{g}$  that are used to weight the adjacency and the distance matrices in Equation 2.
+
+Naturally, many properties of molecules depend on their geometry. For instance, steric effect happens when a spatial proximity of a given group, blocks reaction from happening, due to an overlap in electronic groups. However, this type of reasoning can be difficult to learn based only on the graph information, as it does not always reflect the geometry well. Furthermore, focusing on distance information might require selecting low values for either  $\lambda_{g}$  or  $\lambda_{a}$  (see Figure 1).
+
+To illustrate this, we designed a toy task to predict whether or not two substructures are closer to each other in space than a predefined threshold; see also Figure 8a. We expect that MAT will work significantly better than a vanilla graph convolutional network if  $\lambda_{d}$  is tuned well.
+
+Experimental setting. We construct the dataset by sampling 2677 molecules from PubChem (Kim et al., 2018), and use  $20\AA$  threshold between  $-\mathrm{NH}_2$  fragment and tert-butyl group to determine the binary label. The threshold was selected so that positive and negative examples are well balanced.
+
+Results. First, we plot Molecule Attention Transformer performance as a function of  $\lambda_{d}$  in Figure 8b for three settings of  $\lambda$ :  $\lambda_{a} = 0$  (blue),  $\lambda_{a} = \lambda_{g}$  (orange), and  $\lambda_{g} = 0$  (green). In all cases we find that using distance information improves the performance significantly. Additionally, we found that GCN achieves 0.93 AUC on this task, compared to 0.98 by MAT with  $\lambda_{d} = 1.0$ . These results both motivate tuning  $\lambda$ , and show that MAT can efficiently use distance information if it is important for the task at hand.
+
+![](images/703f992bc7172029fcd9712bc38c8b90e862083f3ba2e9073dec53bec92aa0a0.jpg)  
+(a) The toy task is to predict whether two substructures  $\left(-\mathrm{NH}_{2}\right.$  fragment and tert-butyl group) co-occur within given distance.
+
+![](images/9ef996bb3009e6273f6df2b191addee41ed644c8bba55529c2f2eed25767b08e.jpg)  
+(b) Molecule Attention Transformer performance on the toy task as a function of  $\lambda_{d}$ , for different settings of  $\lambda_{g}$  and  $\lambda_{a}$ .  
+Figure 8. MAT can efficiently use the inter-atomic distances to solve the toy task (see left). Additionally, the performance is heavily dependent on  $\lambda_{d}$ , which motivates tuning  $\lambda$  in the main experiments (see right).
+
+Further details. The molecules in the toy task dataset were downloaded from PubChem. The SMARTS query used to find the compounds was (C([C;H3])([C;H3])([C;H3]).[NX3H2]). All molecules were then filtered so that only those with exactly one tert-butyl group and one  $-\mathrm{NH}_2$  fragment were left. For each of them, five conformers were created with RDKit implementation of the Universal Force Field (UFF).
+
+The task is a binary classification of the distance between two molecule fragments. If the euclidean distance between  $-\mathrm{NH}_2$  fragment and tert-butyl group is greater than a given threshold, the label is 1 (0 otherwise). As the distance we mean the distance between the closest heavy atoms in these two fragments across calculated conformers. We used  $20\AA$  as the threshold as it leads to a balanced dataset. There are 2677 compounds total from which 1140 are in a positive class. The dataset was randomly split into training, validation, and test datasets.
+
+In experiments the hyperparameters that yielded promising results on our datasets were used (listed in Table 17). The values of  $\lambda$  parameters were tuned, and their scores are shown in Figure 8b. All three  $\lambda$  parameters  $(\lambda_d, \lambda_g, \lambda_a)$  sum to 1 in all experiments.
+
+To compare our results with a standard graph convolutional neural network, we run a grid search over hyperparameters shown in Table 18. The hyperparameters for which the best validation AUC score was reached are emboldened, and their test AUC score is  $0.925 \pm 0.006$ .
+
+Table 17. MAT hyperparameters used.  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>BATCH SIZE</td><td>16</td></tr><tr><td>LEARNING RATE</td><td>0.0005</td></tr><tr><td>EPOCHS</td><td>100</td></tr><tr><td>MODEL DIM</td><td>64</td></tr><tr><td>MODEL N</td><td>4</td></tr><tr><td>MODEL H</td><td>8</td></tr><tr><td>MODEL N DENSE</td><td>2</td></tr><tr><td>MODEL DENSE OUTPUT NONLINEARITY</td><td>’TANH’</td></tr><tr><td>DISTANCE MATRIX KERNEL</td><td>’SOFTMAX’</td></tr><tr><td>MODEL DROPOUT</td><td>0.0</td></tr><tr><td>WEIGHT DECAY</td><td>0.001</td></tr><tr><td>OPTIMIZER</td><td>’ADAM_ANNEAL’</td></tr><tr><td>AGGREGATION TYPE</td><td>’MEAN’</td></tr></table>
+
+Table 18. Hyperparameters used for tuning GCN.  
+
+<table><tr><td></td><td>PARAMETERS</td></tr><tr><td>BATCH SIZE</td><td>16, 32, 64</td></tr><tr><td>LEARNING RATE</td><td>0.0005</td></tr><tr><td>EPOCHS</td><td>20, 40, 60, 80, 100</td></tr><tr><td>N FILTERS</td><td>64, 128</td></tr><tr><td>N FULLY CONNECTED NODES</td><td>128, 256</td></tr></table>
+
+# G. Interpretability analysis
+
+Table 19. Statistics of the six attention head patterns described in the text. Each head function is defined by a SMARTS that selects atoms with high attention weights. For each atom in the dataset we calculated mean weight assigned to them by the corresponding attention head (average column value of the attention matrix). Calculated means and standard deviations show the difference between attention weights of matching atoms  $(\mu^{+},\sigma^{+})$  against the other atoms  $(\mu^{-},\sigma^{-})$ .
+
+<table><tr><td>HEAD</td><td>I</td><td>II</td><td>III</td><td>IV</td><td>V</td><td>VI</td></tr><tr><td>SMARTS</td><td>[c;D2]</td><td>[S,S]</td><td>[N;R0]</td><td>O=*</td><td>[A;D3]</td><td>N</td></tr><tr><td>μ+</td><td>.136</td><td>.330</td><td>.061</td><td>.095</td><td>.043</td><td>.228</td></tr><tr><td>σ+</td><td>.080</td><td>.280</td><td>.074</td><td>.120</td><td>.032</td><td>.171</td></tr><tr><td>μ-</td><td>.008</td><td>.001</td><td>.002</td><td>.006</td><td>.006</td><td>.005</td></tr><tr><td>σ-</td><td>.032</td><td>.003</td><td>.016</td><td>.034</td><td>.014</td><td>.009</td></tr></table>
+
+We found several patterns in the self-attention heads by looking at the first layer of MAT. These patterns correspond to chemical structures that can be found in molecules. For
+
+each such pattern found in a qualitative manner, we tested quantitatively if our hypotheses are true about what these particular attention heads represent.
+
+For each pattern found in one of the attention heads, we construct a SMARTS expression describing atoms that belong to our hypothetical molecular structures. Then, all atoms matching the pattern are extracted from the BBBP dataset, and their mean attention weights (average column value of the attention matrix) are compared against atoms that do not match the pattern. Table 19 shows the distributions of attention weights for matching and not matching atoms. Atoms which match the SMARTS expression have significantly higher attention weights  $(\mu^{+} > \mu^{-})$ .

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+    {
+        "type": "text",
+        "text": "Multilogue-Net: A Context Aware RNN for Multi-modal Emotion Detection and Sentiment Analysis in Conversation",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Aman Shenoy*",
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+    {
+        "type": "text",
+        "text": "Birla Inst. of Technology and Science, Pilani",
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+    {
+        "type": "text",
+        "text": "Pilani, RA, India",
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+        "text": "f2016393@pilani(bits-pilani.ac.in",
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+    {
+        "type": "text",
+        "text": "Ashish Sardana",
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+    {
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+        "text": "NVIDIA Graphics",
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+    {
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+        "text": "Bengaluru, KA, India",
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+    {
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+        "text": "asardana@nvidia.com",
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+    {
+        "type": "text",
+        "text": "Abstract",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Sentiment Analysis and Emotion Detection in conversation is key in several real-world applications, with an increase in modalities available aiding a better understanding of the underlying emotions. Multi-modal Emotion Detection and Sentiment Analysis can be particularly useful, as applications will be able to use specific subsets of available modalities, as per the available data. Current systems dealing with Multi-modal functionality fail to leverage and capture - the context of the conversation through all modalities, the dependency between the listener(s) and speaker emotional states, and the relevance and relationship between the available modalities. In this paper, we propose an end to end RNN architecture that attempts to take into account all the mentioned drawbacks. Our proposed model, at the time of writing, out-performs the state of the art on a benchmark dataset on a variety of accuracy and regression metrics.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-modal Emotion Detection and Sentiment Analysis in conversation is gathering a lot of attention recently considering its potential use cases owing to the rapid growth of online social media platforms such as YouTube, Facebook, Instagram, Twitter etc. (Chen et al., 2017, Poria et al., 2016, Poria et al., 2017, Zadeh et al., 2016b, Zadeh et al., 2017), especially knowing that information obtained from any combination of more than one of the available modalities (e.g. text, audio, video) can be used to produce meaningful results.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The current state of the art systems on multimodal emotion detection and sentiment analysis do not treat the modalities in accordance to the information they are capable of holding (e.g. textual information is significantly more likely to hold",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "contextual information then audio or video features are), lack an adequate fusion mechanism, and fail to effectively capture the context of a conversation in a multi-modal setting. In addition to the lack of proper usage of the available modalities, models also fail to effectively capture the flow of a conversation, the separation between speaker and listener states, and the emotional effect a speakers utterance has on the listener (s) in dyadic conversations.",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our proposed model Multilogue-Net, attempts to embed basic domain knowledge and takes insight from Poria et al. (2019), assuming that the sentiment or emotion governing a particular utterance predominantly depends on 4 factors interlocutor state, interlocutor intent, the preceding and future emotions, and the context of the conversation. Interlocutor intent amongst the mentioned is particularly difficult to model due to its dependency of prior knowledge about the speaker, but modelling the other 3 separately, yet in an interrelated manner was theorized to produce meaningful results if managed to be captured effectively. The key intention was to attempt to simulate the setting in which an utterance is said, and use the actual utterance at that point to be able to gain better insights regarding emotion and sentiment of that utterance. The model uses information from all modalities learning multiple state vectors (representing interlocutor state) for a given utterance, followed by a pairwise attention mechanism inspired by Ghosal et al. (2018), attempting to better capture the relationship between all pairs of the available modalities.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The model uses two gated recurrent units (GRU) (Chung et al., 2014) for each modality for modelling interlocutor state and emotion. Along with these GRU's, the model also uses an interconnected context network, consisting of the same number of GRU's as the number of available modalities, to model a different learned context representation for each modality. The incoming utterance",
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+    {
+        "type": "aside_text",
+        "text": "arXiv:2002.08267v3 [cs.CL] 22 Jun 2020",
+        "bbox": [
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+    },
+    {
+        "type": "page_footnote",
+        "text": "* The following work was pursued when author was an intern at NVIDIA Graphics, Bengaluru",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "representations and the historical GRU outputs are used at every timestamp to be able to arrive at a prediction for that timestamp.",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The model produces  $m$  different representations at every timestamp (Where  $m$  is the number of modalities), where each representation is the emotional state at that timestamp as conveyed by each of the modalities. These  $m$  representations are used by the fusion mechanism to incorporate information from each of the  $m$  representations to be able to arrive at the final prediction for that timestamp. We understand that the usage of the pairwise attention mechanism, along with the Emotion GRU are what make the model flexible across tasks.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The usage of only the text representation as input to the context GRUs has been observed to be key to the results, as the context of the conversation would be better captured by textual information then it would have with audio or video information. We believe that Multilogue-net performs better than the current state of the art (Ghosal et al., 2018) on multi-modal datasets because of better context representation leveraging all available modalities.<sup>1</sup>",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The remaining sections of the paper are arranged as follows: Section 2 discusses related work; Section 3 discusses the model in detail; Section 4 provides experimental results, dataset details, and analysis; Section 5 contains our ablation studies and its implications; and finally Section 6 speaks on potential future work, and concludes our paper.",
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+    {
+        "type": "text",
+        "text": "2 Related Work",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Multi-modal Emotion recognition and Sentiment Analysis has always attracted attention in multiple fields such as natural language processing, psychology, cognitive science, and so on (Picard, 2010). Previous works have been done studying factors of variation that have a more direct correlation with emotion, such as Ekman et al. (1992), who found correlation between emotion and facial cues, and a lot of studies extensively focus on emotions and their relationship with one another such as Plutchiks wheel of emotions, which defines eight primary emotion types, each of which has a multitude of emotions as sub-types.",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Early work done to leverage multi-modal information for emotion recognition includes works such as Datcu and Rothkrantz (2012), who fused",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "acoustic information with visual cues for emotion recognition and Eyben et al. (2010), who used contextual information for emotion recognition in multi-modal settings. More recently, deep recurrent neural networks have been used to be able make the best of the learned representations of the modalities available to be able to give very effective and accurate emotion and sentiment predictions. Poria et al. (2017) successfully used RNN-based deep networks for multi-modal emotion recognition, which was followed by multiple other works (Chen et al., 2017; Zadeh et al., 2018a; Zadeh et al., 2018c) giving results far better than what was seen before. Recent works also include works such as Hazarika et al. (2018), who used memory networks for emotion recognition in dyadic conversations, where two distinct memory networks enabled interspeaker interaction.",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Some works such as DialogueRNN (Majumder et al., 2018), though focused on emotion recognition and sentiment analysis using a single modality (text), works very well in a multi-modal setting by just replacing the text representation with a concatenated vector of all the modality representations. DialogueRNN effectively leveraged the separation between the speakers by maintaining two independent gated recurrent units to keep track of the interlocutor states, also effectively capturing context in the conversation, yielding state-of-the-art performance on uni-modal data. Even though DialogueRNN was able to give reasonably good results on multi-modal data, the lack of an adequate fusion mechanism and the lack of focus on a multi-modal representation held its multi-modal performance back.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Apart from the kind of works shown before, where a methodology or a model was proposed, works such as Poria et al. (2019) spoke extensively about the research challenges and advancements in emotion detection in conversation and gave a comprehensive overview of the problem. Most recently Ghosal et al. (2018) introduced the idea of learning the relationship between pairs of all available modalities using pairwise attention, in a multimodal setting, where similar attributes learned by multiple modalities are emphasized and differences between the modality representations are diminished. Pairwise attention proved to be incredibly effective yielding state-of-the-art performance on multi-modal data with just simple representations for each modality.",
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+    {
+        "type": "page_footnote",
+        "text": "<sup>1</sup>A basic model and training implementation of Multilogue-Net can be found at https://github.com/amanshenoy/multilogue-net.",
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+    {
+        "type": "text",
+        "text": "3 Proposed Methodology",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Problem Formulation",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Let there be a  $P$  number of participants  $p_1, p_2, \\ldots, p_P$  in the conversation. The problem is defined such that for every utterance  $u_1, u_2, \\ldots, u_N$  uttered by any participant(s), a sentiment score is allotted along with a predicted emotion label (one of happy, sad, angry, surprise, disgust, and fear). Each utterance corresponds to a particular participant of the conversation, allowing this formulation of the problem to also capture the average sentiment of a participant in the conversation. Predictions over utterances also avoid problems such as classification during long moments of silence when predictions are made for a fixed time interval, and is also mostly common practice.",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For every utterance  $u_{t}(p)$ , where  $p$  is the party who uttered the utterance, there exist three independent representations,  $t_{t} \\in \\mathbb{R}^{D_{t}}$ ,  $a_{t} \\in \\mathbb{R}^{D_{a}}$ , and  $v_{t} \\in \\mathbb{R}^{D_{v}}$ , and are obtained using the feature extractors further explained in section 4.2.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This gives us our overall formulation of the problem, which is to be able to learn a function which would take as input three independent representations of a particular utterance, information regarding the previous emotional state of the participant, and a representation of the current context of the conversation - to be able to map to an output prediction of a sentiment score and emotion label.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Details regarding how these representations are updated and how the output is generated using these inputs are described in detail below.",
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+    {
+        "type": "text",
+        "text": "3.2 Model Details",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Modelling was done under the underlying assumption that the sentiment or emotion of an utterance predominantly depends on four factors as mentioned before:",
+        "bbox": [
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+    },
+    {
+        "type": "list",
+        "sub_type": "text",
+        "list_items": [
+            "- Interlocutor State",
+            "- Interlocutor Intent",
+            "- Context of the conversation until that point",
+            "- Previous interlocutor states and emotions of a particular participant in the conversation"
+        ],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The proposed model attempts to model three out of the mentioned four explicitly, and assume that interlocutor intent will be modelled implicitly during model training. Interlocutor state is modelled using a state GRU (will be referred to as  $sGRU$ ),",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A context GRU is used to keep track of the context of the conversation ( $cGRU$ ), and an emotion GRU ( $eGRU$ ) is used to keep track of the emotional state of that particular participant. Finally, a pairwise attention mechanism, which uses the emotion representation of all modalities at a particular timestamp is used to leverage the important modalities and relevant combination of the modalities for emotion or sentiment prediction at that timestamp.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c34e7778da243f7711db919a17bfb0023082fe04b2a090a52ddce68eb90026d4.jpg",
+        "image_caption": [
+            "Figure 1: Description of all the state updates at times-tamp  $t$  for a single participant  $p_1$"
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Every utterance has three independent feature representations (text, audio, and video features),  $t_t \\in \\mathbb{R}^{D_t}$ ,  $a_t \\in \\mathbb{R}^{D_a}$ , and  $v_t \\in \\mathbb{R}^{D_v}$ . Each of these feature representations are treated and operated on independently until the pairwise attention mechanism. The model consists of two GRUs (state GRU, and emotion GRU) for every modality and participant, and a context GRU for each modality common to all participants in the conversation (If  $p$  is the number of participants and  $m$  is the number of modalities, the model would have a total of  $2mp + m$  GRUs). The inputs at the current timestamp and the previous state, context, and emotion representations are operated on to be able to arrive at the prediction at that timestamp. Figure 1 describes the updates at a particular timestamp and the role of each GRU is further explained below.",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 Context GRU (cGRU)",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Context GRU  $(cGRU)$  for each modality aims to capture the context of the conversation by jointly encoding the utterance representation of that modality (at timestamp  $t$  in the given diagram)  $(t_{t} \\in \\mathbb{R}^{D_{t}}$ ,  $a_{t} \\in \\mathbb{R}^{D_{a}}$ , or  $v_{t} \\in \\mathbb{R}^{D_{v}}$ ) and the previous times- tamp speaker state GRU output of that modality. This accounts for inter-speaker and inter-utterance dependencies to produce an effective context rep",
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f6ec70f61cf8fe3a8ad36a9aa5898bf8d51a12b676025ee92651ffb28fab3f2.jpg",
+        "image_caption": [
+            "Figure 2: State updates and final prediction output in a conversation between two participants  $p_1$  and  $p_2$ , where the updates of each participant at a timestamp is as given in figure 1"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            147,
+            74,
+            850,
+            338
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "resentation. The current utterance  $t_t$ ,  $a_t$ , or  $v_t$ , changes the state of that speaker from  $(s_t^t, s_t^a, s_t^v)$  to  $(s_{t+1}^t, s_{t+1}^a, s_{t+1}^v)$ . To capture this change in context we use GRU cell  $cGRU$  having output size  $D_c$ , using  $t_t$ ,  $a_t$ , or  $v_t$  and  $(s_t^t, s_t^a, s_t^v)$  as:",
+        "bbox": [
+            114,
+            417,
+            489,
+            498
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nc _ {t + 1} ^ {t} = c G R U \\left(c _ {t} ^ {t}, \\left(t _ {t} \\oplus s _ {t} ^ {t}\\right)\\right) \\tag {1}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            514,
+            487,
+            533
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nc _ {t + 1} ^ {a} = c G R U \\left(c _ {t} ^ {a}, \\left(a _ {t} \\oplus s _ {t} ^ {a}\\right)\\right) \\tag {2}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            561,
+            487,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nc _ {t + 1} ^ {v} = c G R U \\left(c _ {t} ^ {v}, \\left(v _ {t} \\oplus s _ {t} ^ {v}\\right)\\right) \\tag {3}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            607,
+            487,
+            626
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Where  $D_{c}$  is the size of the context vectors  $c_{t + 1}^{t}$ ,  $c_{t + 1}^{a}$ , and  $c_{t + 1}^{v}$ .  $D_{t}$ ,  $D_{a}$ , and  $D_{v}$  are the sizes of utterance representations of text, audio, and video respectively.  $\\oplus$  represents the concatenation operation,  $D_{s}$  is the size of all the state vectors  $s_{t + 1}^{t}$ ,  $s_{t + 1}^{a}$ , and  $s_{t + 1}^{v}$ ; and all GRU weight and biases shapes are such that they produce the expected shape of outputs taking the given shape of inputs.",
+        "bbox": [
+            114,
+            636,
+            489,
+            765
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 State GRU (sGRU)",
+        "text_level": 1,
+        "bbox": [
+            115,
+            776,
+            329,
+            791
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The network keeps track of the participants involved in a conversation by employing a  $p * m$  number of (sGRU)'s, where  $p$  is the number participants in the conversation and  $m$  is the number of available modalities. The sGRU associated with a participant outputs fixed size vectors which serve as an encoding to represent the interlocutor state,",
+        "bbox": [
+            114,
+            797,
+            490,
+            909
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and are directly used for both emotion and sentiment prediction, and updating the context vectors.",
+        "bbox": [
+            509,
+            417,
+            884,
+            449
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "All the state vectors are initialized to null at the first timestamp. For a timestamp  $t$ , the state vector of participant  $p$  and modality  $m \\in \\{t, a, v\\}$  is updated using the input feature representation of that modality and simple attention over all the context vectors until that timestamp. The simple attention mechanism over all the context vectors is described by the following equations:",
+        "bbox": [
+            509,
+            450,
+            884,
+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\alpha = \\operatorname {s o f t m a x} \\left(m _ {t} ^ {T} W _ {\\alpha} \\left[ c _ {1} ^ {m}, c _ {2} ^ {m}, \\dots , c _ {t} ^ {m} \\right]\\right) \\tag {4}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            537,
+            607,
+            882,
+            626
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\na t t _ {t} = \\alpha \\left[ c _ {1} ^ {m}, c _ {2} ^ {m}, \\dots , c _ {t} ^ {m} \\right] ^ {T} \\tag {5}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            598,
+            655,
+            882,
+            674
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Where  $m_t^T \\in \\{t_t^T, a_t^T, v_t^T\\}$ ,  $W_\\alpha \\in \\mathbb{R}^{D_{t,a,v} \\times D_c}$ ,  $\\alpha^T \\in \\mathbb{R}^{(t-1)}$ , and  $att_t \\in \\mathbb{R}^{D_c}$ . In equation 4, we calculate attention scores over all previous context representations of all previous utterances, highlighting the relative importance of all the previous context vectors to  $m_t$ . A softmax layer is applied to amplify this relative importance, and finally equation 5 the final output of attention over context  $att_t$  is calculated by pooling the previous context vectors with  $\\alpha$ .",
+        "bbox": [
+            509,
+            681,
+            885,
+            841
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We then employ  $sGRU^{t,a,v}$  to update  $s_t^{t,a,v}$  to  $s_{t+1}^{t,a,v}$  on the basis of incoming utterance representations for each modality  $m_t^T \\in \\{t_t^T, a_t^T, v_t^T\\}$  and the context representations  $att_t^T, att_t^a$ , and  $att_v^v$  using",
+        "bbox": [
+            509,
+            844,
+            884,
+            910
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "GRU cells  $sGRU_{t}^{t}$ ,  $sGRU_{t}^{a}$ , and  $sGRU_{t}^{v}$ , each of output size  $D_{s}$ .",
+        "bbox": [
+            115,
+            74,
+            489,
+            105
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {t + 1} ^ {t} = s G R U \\left(s _ {t} ^ {t}, \\left(t _ {t} \\oplus a t t _ {t + 1} ^ {t}\\right)\\right) \\tag {6}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            124,
+            487,
+            143
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {t + 1} ^ {a} = s G R U \\left(s _ {t} ^ {a}, \\left(a _ {t} \\oplus a t t _ {t + 1} ^ {a}\\right)\\right) \\tag {7}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            174,
+            170,
+            487,
+            187
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ns _ {t + 1} ^ {v} = s G R U \\left(s _ {t} ^ {v}, \\left(v _ {t} \\oplus a t t _ {t + 1} ^ {v}\\right)\\right) \\tag {8}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            174,
+            215,
+            487,
+            233
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where  $D_{s}$  is the size of all the state vectors  $s_{t + 1}^{t}, s_{t + 1}^{a},$  and  $s_{t + 1}^{v}.D_{t}, D_{a}, D_{v}$  are the sizes of utterance representations of text, audio, and video respectively.  $\\oplus$  represents concatenation operation, and all GRU weights shapes are such that they produce the expected shape of outputs taking the given shape of inputs.",
+        "bbox": [
+            115,
+            242,
+            489,
+            354
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The intended purpose of using this as the input to  $sGRU^{t,a,v}$  is to model the dependency of the speaker state on the context of the conversation as understood by the utterances until that point, along with the utterance representation at that point. The output of the  $sGRU$  for modality  $m$  and times-tamp  $t$  serves as an encoding of the speaker state as conveyed by modality  $m$ , at time  $t$ .",
+        "bbox": [
+            115,
+            356,
+            490,
+            483
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.3 Emotion GRU (eGRU)",
+        "text_level": 1,
+        "bbox": [
+            115,
+            495,
+            356,
+            510
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The emotion GRU serves as the decoder for the encoding produced by the state GRU. The emotion GRU uses the previous timestamp  $eGRU$  output, and the encoding provided by  $sGRU$  to produce an emotion or sentiment representation which is further used by the pairwise attention mechanism to be able to produce the relevant output for prediction. At timestamp  $(t + 1)$  the emotion vectors are updated as:",
+        "bbox": [
+            114,
+            516,
+            489,
+            659
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ne _ {t + 1} ^ {t} = e G R U \\left(e _ {t} ^ {t}, s _ {t + 1} ^ {t}\\right) \\tag {9}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            208,
+            678,
+            487,
+            697
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ne _ {t + 1} ^ {a} = e G R U \\left(e _ {t} ^ {a}, s _ {t + 1} ^ {a}\\right) \\tag {10}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            208,
+            724,
+            487,
+            741
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\ne _ {t + 1} ^ {v} = e G R U \\left(e _ {t} ^ {v}, s _ {t + 1} ^ {v}\\right) \\tag {11}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            208,
+            769,
+            487,
+            787
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where  $D_{e}$  is the size of all the emotion vectors  $e_{t + 1}^{t}, e_{t + 1}^{a}$ , and  $e_{t + 1}^{v}.D_{t}, D_{a},$  and  $D_{v}$  are the sizes of utterance representations of text, audio, and video respectively.  $D_{e}$  is the size of the state vectors  $s_{t + 1}^{t}, s_{t + 1}^{a}$ , and  $s_{t + 1}^{v}$ ; and all GRU weights shapes are such that they produce the expected shape of outputs taking the given shape of inputs.",
+        "bbox": [
+            115,
+            796,
+            489,
+            909
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The emotion GRU acts as a decoder to the encoding produced by the associated state GRU, producing a vector which can be used for both sentiment and emotion prediction.",
+        "bbox": [
+            509,
+            74,
+            885,
+            139
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.4 Pairwise Attention Mechanism",
+        "text_level": 1,
+        "bbox": [
+            510,
+            147,
+            811,
+            161
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The emotion GRU for each timestamp will produce an  $m$  number of vectors (where  $m$  is the number of modalities available). Pairwise attention is then used over these  $m$  vectors to produce the final prediction output. In particular pairwise attention is calculated over the following pairs in our case  $(e^v, e^t), (e^t, e^a)$ , and  $(e^a, e^v)$ . Pairwise attention for pair  $(e^v, e^t)$  would be calculated as follows:",
+        "bbox": [
+            509,
+            166,
+            885,
+            294
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8bf6f89254a51886bd7d32722236e03897cbf1eb8d20fcfff18178be7bf37ae.jpg",
+        "image_caption": [
+            "Figure 3: Pairwise attention mechanism used as the fusion mechanism followed by the final prediction layer"
+        ],
+        "image_footnote": [],
+        "bbox": [
+            542,
+            313,
+            857,
+            546
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nB _ {1} = e ^ {v}. \\left(e ^ {t}\\right) ^ {T}, B _ {2} = e ^ {t}. \\left(e ^ {v}\\right) ^ {T} \\tag {12}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            583,
+            618,
+            882,
+            637
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nN _ {1} = \\operatorname {s o f t m a x} \\left(B _ {1}\\right), N _ {2} = \\operatorname {s o f t m a x} \\left(B _ {2}\\right) \\tag {13}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            521,
+            670,
+            882,
+            687
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nO _ {1} = N _ {1}. e ^ {t}, O _ {2} = N _ {2}. e ^ {v} \\tag {14}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            600,
+            712,
+            882,
+            728
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nA _ {1} = O _ {1} \\odot e ^ {v}, A _ {2} = O _ {2} \\odot e ^ {t} \\tag {15}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            584,
+            750,
+            882,
+            766
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {p a i r w i s e} \\left(e ^ {v}, e ^ {t}\\right) = A _ {1} \\oplus A _ {2} \\tag {16}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            586,
+            789,
+            882,
+            806
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where  $B_{1}, B_{2} \\in \\mathbb{R}^{D_{e} \\times D_{e}}$ ;  $N_{1}, N_{2} \\in \\mathbb{R}^{D_{e} \\times D_{e}}$ ;  $A_{1}, A_{2} \\in \\mathbb{R}^{D_{e} \\times D_{e}}$ ; and pairwise  $(e^{v}, e^{t}) \\in \\mathbb{R}^{D_{e} \\times 2D_{e}}$ ;  $\\odot$  represents element-wise product; and  $\\oplus$  represents concatenation.",
+        "bbox": [
+            510,
+            810,
+            884,
+            876
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A complete analysis on the pairwise attention mechanism has been done by Ghosal et al. (2018),",
+        "bbox": [
+            509,
+            877,
+            884,
+            909
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the role of each one of the intermediate variables has been described. These equations (12, 13, 14, 15, 16) calculate  ${}^{m}C_{2}$  pairwise fusion representations, which are further concatenated to make the final prediction as described below.",
+        "bbox": [
+            114,
+            74,
+            492,
+            156
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.5 Final Predictions",
+        "text_level": 1,
+        "bbox": [
+            115,
+            164,
+            312,
+            179
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The prediction layer varies based on whether a sentiment or emotion prediction is expected. For sentiment prediction first all three pairs of pairwise attention i.e.  $pairwise(e^v, e^t)$ ,  $pairwise(e^a, e^t)$ , and  $pairwise(e^v, e^a)$  at that timestamp are concatenated along with the emotion GRU outputs at that timestamp  $(e_t^t, e_t^a)$ , and  $e_t^v$  and the concatenated layer is passed through a fully connected layer followed by a softmax or tanh layer based on the nature of the expected prediction. For sentiment prediction between -1 and +1 at timestamp t the output layer would equate as follows:",
+        "bbox": [
+            114,
+            183,
+            490,
+            376
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\np w = p w \\left(e ^ {v}, e ^ {t}\\right) \\oplus p w \\left(e ^ {a}, e ^ {t}\\right) \\oplus p w \\left(e ^ {v}, e ^ {a}\\right) \\tag {17}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            124,
+            404,
+            489,
+            423
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nL _ {t} = p w \\oplus e _ {t} ^ {t} \\oplus e _ {t} ^ {a} \\oplus e _ {t} ^ {v} \\tag {18}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            208,
+            451,
+            489,
+            469
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {p r e d} _ {\\text {s e n t i m e n t} (t)} = \\tanh  \\left(W _ {L} L _ {t}\\right) \\tag {19}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            492,
+            487,
+            510
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Where pairwise  $(e^v, e^t)$  has been represented as  $pw(e^v, e^t)$ ; and  $W_L \\in \\mathbb{R}^{9D_e \\times 1}$ .",
+        "bbox": [
+            114,
+            516,
+            487,
+            548
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For emotion prediction we use a fully connected layer along with a final softmax layer to calculate 6 emotion class probabilities from  $L_{t}$ .",
+        "bbox": [
+            114,
+            549,
+            489,
+            596
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nl _ {t} = \\operatorname {R e L U} \\left(W _ {l} L _ {t} + b _ {l}\\right) \\tag {20}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            613,
+            487,
+            630
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\nP _ {t} = \\operatorname {s o f t m a x} \\left(W _ {\\text {s m a x}} l _ {t} + b _ {\\text {s m a x}}\\right) \\tag {21}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            154,
+            653,
+            487,
+            671
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\operatorname {p r e d} _ {\\text {e m o t i o n} (t)} = \\operatorname {a r g m a x} \\left(P _ {t}\\right) \\tag {22}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            695,
+            487,
+            712
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "text": "\n$$\n\\begin{array}{c} \\text {W h e r e} W _ {l} \\in \\mathbb {R} ^ {D _ {l} \\times 9 D _ {e}}; b _ {l} = \\in \\mathbb {R} ^ {D _ {l}}; W _ {s m a x} \\in \\\\ \\mathbb {R} ^ {c \\times D _ {l}}; b _ {s m a x} \\in \\mathbb {R} ^ {c} \\text {a n d} P _ {t} \\in \\mathbb {R} ^ {c} \\end{array}\n$$\n",
+        "text_format": "latex",
+        "bbox": [
+            115,
+            719,
+            487,
+            752
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.6 Training",
+        "text_level": 1,
+        "bbox": [
+            115,
+            760,
+            245,
+            777
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fairly standard practices have been employed for the training of the model. Categorical cross-entropy has been used along with L2-regularization as the loss function during training for emotion prediction, to maximize likelihood over each of the classes.",
+        "bbox": [
+            114,
+            780,
+            489,
+            860
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Mean Square Error (MSE) along with L2 regularization has been employed as loss function during training for sentiment regression. The usage of a",
+        "bbox": [
+            114,
+            860,
+            490,
+            910
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c72d52d61c956c854c4ab49fb53c326e9641644a955fdf755de2943d8756420a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Metric</td><td>A2</td><td>F1</td></tr><tr><td colspan=\"3\">Text + Audio</td></tr><tr><td>BC-LSTM</td><td>79.30</td><td>-</td></tr><tr><td>MMMU-BA</td><td>80.58</td><td>-</td></tr><tr><td>DialogueRNN</td><td>78.81</td><td>79.12</td></tr><tr><td>Multilogue-net</td><td>80.12</td><td>78.84</td></tr><tr><td colspan=\"3\">Video + Audio</td></tr><tr><td>BC-LSTM</td><td>62.10</td><td>-</td></tr><tr><td>MMMU-BA</td><td>65.16</td><td>-</td></tr><tr><td>DialogueRNN</td><td>63.22</td><td>60.14</td></tr><tr><td>Multilogue-net</td><td>69.55</td><td>63.40</td></tr><tr><td colspan=\"3\">Text + Video</td></tr><tr><td>BC-LSTM</td><td>80.20</td><td>-</td></tr><tr><td>MMMU-BA</td><td>81.51</td><td>-</td></tr><tr><td>DialogueRNN</td><td>79.88</td><td>79.10</td></tr><tr><td>Multilogue-net</td><td>80.66</td><td>79.62</td></tr><tr><td colspan=\"3\">Text + Audio + Video</td></tr><tr><td>BC-LSTM</td><td>80.30</td><td>-</td></tr><tr><td>MMMU-BA</td><td>82.31</td><td>-</td></tr><tr><td>DialogueRNN</td><td>79.80</td><td>79.48</td></tr><tr><td>Multilogue-net</td><td>81.19</td><td>80.10</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Multilogue-Net performance on CMU-MOSI in comparison with the current and previous state-of-the-art on the dataset. A2 indicating accuracy with 2 classes, and F1 indicating F1 score.",
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+    {
+        "type": "text",
+        "text": "saturating output layer and a loss function that does not undo the saturation, leads to the model to stop training when it makes extreme predictions (close to -1 or +1) due to very small gradients. Using initialization strategies that start at smaller model weights, mini-batch gradient descent-based Adam (Kingma and Ba, 2014) optimizer, and using L2 regularization is used to avoid this failure mode.",
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments, Datasets, and Results",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 Datasets",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate our model using two benchmark datasets - CMU Multi-modal Opinion-level Sentiment Intensity (CMU-MOSI) (Zadeh et al., 2016a) and the recently published CMU Multi-modal Opinion Sentiment and Emotion Intensity (CMU-MOSEI) dataset (Zadeh et al., 2018b).",
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+    {
+        "type": "text",
+        "text": "4.1.1 CMU-MOSI",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "CMU-MOSI dataset consists of 93 videos spanning over 2199 utterances. Each utterance has a sentiment label associated with it. It has 52, 10 & 31 videos in training, validation & test set accounting for 1151, 296 & 752 utterances. CMU-MOSEI",
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+    },
+    {
+        "type": "text",
+        "text": "has 3229 videos with 22676 utterances from more than 1000 online YouTube speakers. The training, validation & test set consist of 16216, 1835 & 4625 utterances, respectively. Each utterance in CMUMOSI dataset has been annotated as either positive or negative.",
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+    },
+    {
+        "type": "text",
+        "text": "4.1.2 CMU-MOSEI",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In CMU-MOSEI dataset labels are in a continuous range of -3 to +3 and are accompanied by an emotion label being one of six emotions. However, in this work we also project the instances of CMU-MOSEI in a two-class classification setup with values  $\\geq 0$  signifies positive sentiments and values  $< 0$  signify negative sentiments. We have called this A2 accuracy (accuracy with 2 classes). Along with this we have also shown results for continuous range prediction between -3 and +3, and emotion prediction with the 6 emotion labels for each utterance in CMU-MOSEI. We have used A2 as a metric to be consistent with the previous published works on CMU-MOSEI dataset (Ghosal et al., 2018; Zadeh et al., 2018b). CMU-MOSEI has further been used for other comprehensive experiments due to its large sizer and easier feature extraction",
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+    {
+        "type": "text",
+        "text": "4.2 Uni-modal Feature Extraction",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.2.1 CMU-MOSEI",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use the CMU-Multi-modal Data SDK (Zadeh et al., 2018b) for feature extraction. For MOSEI dataset, sentiment label-level features were provided where text features used were GloVe embeddings (Pennington et al., 2014), visual features extracted by Facet (Stckli et al., 2017) & acoustic features by OpenSMILE (Eyben et al., 2010). Thereafter, we compute the average of sentiment label-level features in an utterance to obtain the utterance-level features. For each sentiment label-level feature, the dimension of the feature vector is set to 300 (text), 35 (visual) & 384 (acoustic).",
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+    },
+    {
+        "type": "text",
+        "text": "4.2.2 CMU-MOSI",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In contrast, for MOSI dataset we use utterance level features provided in Poria et al. (2017). These utterance-level features represent the outputs of a convolutional neural network (Karpathy et al., 2014), 3D convolutional neural network (Ji et al., 2010) & openSMILE (Eyben et al., 2010) for text, visual & acoustic modalities, respectively. Dimensions of utterance-level features are 100, 100 & 73 for text, visual & acoustic, respectively.",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bffba207437805566f071975444fffa9e169d9c2afb3c25da1baf10872eb115e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Metric</td><td>A2</td><td>F1</td><td>MAE</td><td>r</td></tr><tr><td colspan=\"5\">T + A</td></tr><tr><td>MMMU-BA</td><td>79.74</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DialogueRNN</td><td>79.80</td><td>78.32</td><td>-</td><td>-</td></tr><tr><td>Multilogue-net</td><td>80.18</td><td>79.88</td><td>-</td><td>-</td></tr><tr><td colspan=\"5\">V + A</td></tr><tr><td>MMMU-BA</td><td>76.66</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DialogueRNN</td><td>73.90</td><td>73.92</td><td>-</td><td>-</td></tr><tr><td>Multilogue-net</td><td>75.16</td><td>74.04</td><td>-</td><td>-</td></tr><tr><td colspan=\"5\">V + T</td></tr><tr><td>MMMU-BA</td><td>79.40</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DialogueRNN</td><td>78.90</td><td>78.12</td><td>-</td><td>-</td></tr><tr><td>Multilogue-net</td><td>80.06</td><td>79.84</td><td>-</td><td>-</td></tr><tr><td colspan=\"5\">T + A + V</td></tr><tr><td>Graph-MFN</td><td>76.90</td><td>77.00</td><td>0.71</td><td>0.54</td></tr><tr><td>MMMU-BA</td><td>79.80</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DialogueRNN</td><td>79.98</td><td>79.82</td><td>0.69</td><td>0.42</td></tr><tr><td>Multilogue-net</td><td>82.10</td><td>80.01</td><td>0.59</td><td>0.50</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Multilogue-Net performance on CMU-MOSEI Sentiment Labels compared to previous state-of-the-art models on regression and accuracy Metrics. All metrics apart from MAE represents higher values for better results, MAE represents lower values for better results.",
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+    {
+        "type": "text",
+        "text": "4.3 Experiments",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate our proposed approach on CMU-MOSI (test-set) on accuracy and F1 score, and CMU-MOSEI (dev-set) on accuracy, F1 score, mean absolute error  $(MAE)$ , Pearson score  $(r)$ , and accuracy's on the emotion labels. Due to the lack of speaker information in CMU-MOSI we were not able to use the CMU-Multi-modal Data SDK for sentiment label extraction, to be able to evaluate our approach on CMU-MOSI on mean absolute error and Pearson score.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results have also been reported for usage of two of the three available modalities. Uni-modal performance has not been reported as the focus of the paper is the effective usage of multi-modal data. In a uni-modal setting the model would not be using the fusion mechanism and the output would be equivalent to having a few dense layers after the emotion GRU to directly output the final prediction. F1 scores have not been mentioned by most previous models being used for comparison, but have been reported for Multilogue-Net for additional comparison to any future models using CMU-MOSI dataset.",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the performance of Multilogue-",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/d13ce9bb85c5b155522f0390ee986ed828ee1c6801212d27137497b9f139c749.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"13\">MOSEI Emotions (Text + Video + Audio)</td></tr><tr><td>Emotion</td><td colspan=\"2\">Anger</td><td colspan=\"2\">Disgust</td><td colspan=\"2\">Fear</td><td colspan=\"2\">Happy</td><td colspan=\"2\">Sad</td><td colspan=\"2\">Surprise</td></tr><tr><td>Metric</td><td>WA</td><td>F1</td><td>WA</td><td>F1</td><td>WA</td><td>F1</td><td>WA</td><td>F1</td><td>WA</td><td>F1</td><td>WA</td><td>F1</td></tr><tr><td>Graph-MFN</td><td>62.6</td><td>72.8</td><td>69.1</td><td>76.6</td><td>62.0</td><td>89.9</td><td>66.3</td><td>66.3</td><td>60.4</td><td>66.9</td><td>53.7</td><td>85.5</td></tr><tr><td>Multilogue-Net</td><td>83.1</td><td>80.9</td><td>90.3</td><td>87.3</td><td>89.7</td><td>87.0</td><td>70.0</td><td>68.4</td><td>76.1</td><td>74.5</td><td>87.4</td><td>84.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Multilogue-Net performance on MOSEI Emotion Labels compared with that of Graph-MFN on weighted accuracy and F1 score. MOSEI Emotion label results were presented by only one model, and comprehensive results have not been published for the same.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Net on CMU-MOSI dataset, comparing to the current state of the art (Ghosal et al., 2018), previous state-of-the-art (Poria et al., 2017), and DialogueRNN (Majumder et al., 2018) (Multi-modal performance of DialogueRNN has not been reported by Majumder et al. (2018), and we have run these experiments additionally for a better comparative study, where concatenating the input representations has been used as a fusion mechanism). Our model consistently outperforms the previous state-of-the-art but performs better only on one of the subsets of the modalities when compared to the current state-of-the-art.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In comparison to MMMU-BA our model also lacks in Multi-modal performance. We theorize that the model performance is lacking because of the low number of training examples (CMU-MOSI consists only of 93 conversations out of which 62 were used for training), in contrast to our model which has a high capacity (Relative to models being compared with). Since Multilogue-Net learns a lot of intermediate representations in order to make a prediction, it would need a larger dataset with more variability to be able to learn meaningful representations. The proposition that performance lacks due to a lack of training examples is backed by the results on CMU-MOSEI (demonstrated in a comparative setting in Table 2 and 3) where the model consistently outperforms the current state-of-the-art on most metrics.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "On CMU-MOSEI, our model seems to perform very consistently on both sentiment and emotion labels. The model outperforms the current state of the art on all but one metric (both classification and accuracy) on sentiment labels in the tri-modal setting. Multilogue-Net also outperforms the current state of the art on the emotion labels by a considerable margin (This is also attributed to the fact that not a lot of models have presented results on these labels).",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Similar observations are made in both datasets,",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where the tri-modal metrics show the best performance, and audio + video show the worst relative performance (suggesting the importance of text in a multi-modal setting). Textual information seems to be the guiding factor for multi-modal performance, with video and audio features simply acting as a push to the uni-modal performance on text.",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We theorize that the performance of Multilogue-Net is majorly attributed to its increased capacity as compared to previous models. Effective usage of this increased capacity, using representations inspired from a basic understanding of conversation, along with a larger dataset for training have been key in achieving the improved results.",
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+    {
+        "type": "text",
+        "text": "5 Ablation Studies and Analysis",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Until now, some architectural considerations, such as the use of  $eGRU$  and the fusion mechanism, have been briefly explained but not empirically justified. This section aims to get empirical evidence regarding the effectiveness of these modules. Since our model completely hinges around the usage of the context and state GRU's, our ablation studies and analysis have focused on the fusion mechanism and emotion GRU ( $eGRU$ ) only.",
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+    {
+        "type": "text",
+        "text": "5.1 Fusion Mechanism",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The effectiveness of the fusion mechanism can be very easily examined by observing the results of the model on both tasks - Sentiment Regression and Emotion Recognition, with and without the fusion mechanism. Table 4 shows these results on CMU-MOSEI modality subsets.",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The bi-modal results in table 4 involve evaluating the pairwise attention module only once (Since there is only one pair available), directly followed by the prediction layer. The tri-modal case on the other hand involves evaluating the pairwise attention module thrice (Once for each pair). In general, the number of times this module will have to be evaluated for  $m$  modalities is  $^m C_2$ , which raises",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/142328d1eaa3c5b14453fe739332440f787cb3609c0805b17c3445195d5ae641.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Fusion Mechanism</td><td>A2</td><td>MAE</td></tr><tr><td colspan=\"3\">Text + Audio</td></tr><tr><td>without</td><td>75.78</td><td>-</td></tr><tr><td>with</td><td>80.18</td><td>-</td></tr><tr><td colspan=\"3\">Video + Audio</td></tr><tr><td>without</td><td>75.66</td><td>-</td></tr><tr><td>with</td><td>75.16</td><td>-</td></tr><tr><td colspan=\"3\">Text + Video</td></tr><tr><td>without</td><td>76.80</td><td>-</td></tr><tr><td>with</td><td>80.06</td><td>-</td></tr><tr><td colspan=\"3\">Text + Audio + Video</td></tr><tr><td>without</td><td>79.80</td><td>0.66</td></tr><tr><td>with</td><td>82.10</td><td>0.59</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "a fair concern regarding the trade-off between the additional computational cost and performance.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We empirically observe that the additional computational cost can be considered negligible in context of the increased performance, largely attributing to the non-parametric nature of the fusion mechanism and the relatively small number of additional parameters in the prediction layer ( $6D_{e}$  for the sentiment regression;  $36D_{e}$  for emotion recognition).",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The fusion mechanism seems to clearly be beneficial in all of the reported cases apart from video + audio, implying that the fusion mechanism is useful only in the cases the text representation is used. This further strengthens our claim that the text representation guides tri-modal performance.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Emotion GRU (eGRU)",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Unlike as done with the fusion mechanism, the effectiveness of the  $eGRU$  cannot be examined by evaluating metrics with and without it. Removing the Emotion GRU would clearly be detrimental to the results, and would not convey the intention of having it.",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The primary intention of having the  $eGRU$  can be considered to be maintaining consistency between tasks. To better understand what this means table 5 quantitatively demonstrates this effect. The model was trained separately for Emotion Detection and Sentiment Regression tasks. After both the models were trained satisfactorily, a particular sample from the test set (test sample 6) was inferred on. We then retrieved the intermediate text repre",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/3e88cd50e3b94836a9e016c6be7e3f2a08c9328c68f1990d22b3fbe2693e4cb7.jpg",
+        "table_caption": [
+            "Table 4: Multilogue-Net performance on CMU-MOSEI with and without the fusion mechanism - for 'without' fusion we have concatenated all the representations and directly passed them to the prediction layer."
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Representation</td><td>Euclidean Distance</td></tr><tr><td colspan=\"2\">Sample 6 with t = 4</td></tr><tr><td>s4t</td><td>4.6 units</td></tr><tr><td>c4t</td><td>6.1 units</td></tr><tr><td>e4t</td><td>26.4 units</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Euclidean Distance between the same representations for Sentiment Regression as compared to Emotion Detection. (Distances have been converted to units for convenience and easier comparison)",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "sentations  $(e_4^t, c_4^t$ , and  $s_4^t$ ; superscript  $t$  indicating text modality) at a particular timestamp  $(t = 4)$  for both models on that sample. The Euclidean Distance between these two sets of representations (one for each task) was evaluated and have been shown in table 5, where we can clearly observe that the euclidean distance between the emotion representations is much larger as compared to the state and context representations.",
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+        "text": "This shows that for both tasks, interlocutor state and context representations are relatively similar to each other, whereas the emotion state representation is more varied and task dependant. This not only allows us to use the same cGRU and sGRU weights across tasks, but would also allow us to train for multiple tasks in parallel using a different eGRU for each task - giving us consistent and accurate predictions across multiple tasks. Analysis of such a network, and whether training for multiple tasks in parallel aids one another, has not been covered in this paper and is left to our future work.",
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+    {
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+        "text": "6 Conclusion",
+        "text_level": 1,
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+        "text": "In this paper, we have presented an RNN architecture for multi-modal sentiment analysis and emotion detection in conversation. In contrast to the current state-of-the-art models, our model focuses on effectively capturing the context of a conversation and treats each modality independently, taking into account the information a particular modality is capable of holding. Our model consistently performs well on benchmark datasets such as CMU-MOSI and CMU-MOSEI in any multi-modal setting.",
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+    {
+        "type": "text",
+        "text": "The model can be further extended to have better feature extractors, and increase both the number of modalities and the number of participants in the conversation. Due to the lack of availability of datasets consisting of these extensions with emotion or sentiment labels, we have left this to our future work.",
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+    {
+        "type": "text",
+        "text": "References",
+        "text_level": 1,
+        "bbox": [
+            117,
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+            215,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Minghai Chen, Sen Wang, Paul Pu Liang, Tadas Baltruaiitis, Amir Zadeh, and Louis-Philippe Morency. 2017. Multimodal sentiment analysis with word-level fusion and reinforcement learning. In Proceedings of the 19th ACM International Conference on Multimodal Interaction, pages 163-171.",
+            "Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Y. Bengio. 2014. Empirical evaluation of gated recurrent neural networks on sequence modeling.",
+            "Dragos Datcu and Lon Rothkrantz. 2012. Semantic audiovisual data fusion for automatic emotion recognition.",
+            "P. Ekman, Edmund Rolls, David Perrett, and H. Ellis. 1992. Facial expressions of emotion: An old controversy and new findings: Discussion. Royal Society of London Philosophical Transactions Series B, 335:69-.",
+            "Florian Eyben, Martin Wllmer, and Bjrn Schuller. 2010. opensmile - the munich versatile and fast opensource audio feature extractor. pages 1459-1462.",
+            "Deepanway Ghosal, Md Shad Akhtar, Dushyant Chauhan, Soujanya Poria, and Pushpak Bhattacharyya Asif Ekbal. 2018. Contextual inter-modal attention for multi-modal sentiment analysis. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 3454-3466.",
+            "Devamanyu Hazarika, Soujanya Poria, Amir Zadeh, Erik Cambria, Louis-Philippe Morency, and Roger Zimmermann. 2018. Conversational memory network for emotion recognition in dyadic dialogue videos. volume 2018, pages 2122-2132.",
+            "Shuiwang Ji, Wei Xu, Ming Yang, and Kai Yu. 2010. 3d convolutional neural networks for human action recognition. volume 35, pages 495-502.",
+            "Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei. 2014. Large-scale video classification with convolutional neural networks. pages 1725-1732.",
+            "Diederik Kingma and Jimmy Ba. 2014. Adam: A method for stochastic optimization. International Conference on Learning Representations.",
+            "Navonil Majumder, Soujanya Poria, Devamanyu Hazarika, Rada Mihalcea, Alexander Gelbukh, and Erik Cambria. 2018. Dialoguernn: An attentive rn for emotion detection in conversations.",
+            "Jeffrey Pennington, Richard Socher, and Christoper Manning. 2014. Glove: Global vectors for word representation. volume 14, pages 1532-1543.",
+            "Rosalind Picard. 2010. Affective computing: From laughter to ieee. IEEE Transactions on Affective Computing, 1:11-17."
+        ],
+        "bbox": [
+            117,
+            97,
+            490,
+            908
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "list",
+        "sub_type": "ref_text",
+        "list_items": [
+            "Soujanya Poria, Erik Cambria, Devamanyu Hazarika, Navonil Majumder, Amir Zadeh, and Louis-Philippe Morency. 2017. Context-dependent sentiment analysis in user-generated videos. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, pages 873-883.",
+            "Soujanya Poria, Iti Chaturvedi, Erik Cambria, and Amir Hussain. 2016. Convolutional mkl based multimodal emotion recognition and sentiment analysis. In IEEE 16th International Conference on Data Mining (ICDM), pages 439-448.",
+            "Soujanya Poria, Navonil Majumder, Rada Mihalcea, and Eduard Hovy. 2019. Emotion recognition in conversation: Research challenges, datasets, and recent advances. arXiv:1905.02947.",
+            "Sabrina Stckli, Michael Schulte-Mecklenbeck, Stefan Borer, and Andrea Samson. 2017. Facial expression analysis with affdex and facet: A validation study. Behavior Research Methods, 50.",
+            "Amir Zadeh, Minghai Chen, Soujanya Poria, Erik Cambria, and Louis-Philippe Morency. 2017. Tensor fusion network for multimodal sentiment analysis. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 1103-1114.",
+            "Amir Zadeh, Paul Liang, Navonil Mazumder, Soujanya Poria, Erik Cambria, and Louis-Philippe Morency. 2018a. Memory fusion network for multi-view sequential learning.",
+            "Amir Zadeh, Paul Liang, Soujanya Poria, Erik Cambria, and Louis-Philippe Morency. 2018b. Multi-modal language analysis in the wild: Cmu-mosei dataset and interpretable dynamic fusion graph. pages 2236-2246.",
+            "Amir Zadeh, Paul Liang, Soujanya Poria, Prateek Vij, Erik Cambria, and Louis-Philippe Morency. 2018c. Multi-attention recurrent network for human communication comprehension. Proceedings of the 2018 AAAI Conference on Artificial Intelligence, 2018.",
+            "Amir Zadeh, Rowan Zellers, Eli Pincus, and Louis-Philippe Morency. 2016a. Mosi: Multimodal corpus of sentiment intensity and subjectivity analysis in online opinion videos.",
+            "Amir Zadeh, Rowan Zellers, Eli Pincus, and Louis-Philippe Morency. 2016b. Multimodal sentiment intensity analysis in videos: Facial gestures and verbal messages. Intelligent Systems, IEEE, pages 82-88."
+        ],
+        "bbox": [
+            512,
+            76,
+            884,
+            797
+        ],
+        "page_idx": 9
+    }
+]