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+Enhancing ResNet Image Classification Performance by using
+Parameterized Hypercomplex Multiplication
+Nazmul Shahadat, Anthony S. Maida
+University of Louisiana at Lafayette
+Lafayette LA 70504, USA
+nazmul.ruet@gmail.com, maida@louisiana.edu
+Abstract
+Recently, many deep networks have introduced hy-
+percomplex and related calculations into their architec-
+tures. In regard to convolutional networks for classifica-
+tion, these enhancements have been applied to the con-
+volution operations in the frontend to enhance accuracy
+and/or reduce the parameter requirements while main-
+taining accuracy. Although these enhancements have
+been applied to the convolutional frontend, it has not
+been studied whether adding hypercomplex calculations
+improves performance when applied to the densely con-
+nected backend. This paper studies ResNet architectures
+and incorporates parameterized hypercomplex multipli-
+cation (PHM) into the backend of residual, quaternion,
+and vectormap convolutional neural networks to assess
+the effect. We show that PHM does improve classifica-
+tion accuracy performance on several image datasets,
+including small, low-resolution CIFAR 10/100 and large
+high-resolution ImageNet and ASL, and can achieve
+state-of-the-art accuracy for hypercomplex networks.
+1. Introduction
+Convolutional neural networks (CNNs) have been
+widely used, with great success, in visual classification
+tasks [3, 12] because of their good inductive priors and
+intuitive design.
+Most deep learning building blocks in CNNs use
+real-valued operations.
+However, recent studies have
+explored the complex/hypercomplex space and showed
+that hypercomplex valued networks can perform bet-
+ter than their real-valued counterparts due to the weight
+sharing mechanism embedded in the hypercomplex mul-
+tiplication [7,18]. This weight sharing differs from that
+found in the real-valued convolution operation. Specif-
+ically, quaternion convolutions share weights across in-
+put channels enabling them to discover cross-channel in-
+put relationships that support more accurate prediction
+(a) Validation accuracy comparison for CIFAR-10 data.
+(b) Validation accuracy comparison for CIFAR-100 data.
+Figure 1. Top-1 validation accuracy comparison among orig-
+inal ResNets [7], original quaternion networks [7], original
+vectormap networks [7], our proposed QPHM and VPHM net-
+works for CIFAR benchmarks
+and generalization. The effectiveness of quaternion net-
+works is shown in [7,16,19,23,31].
+The weight-sharing properties of the Hamiltonian
+product allow the discovery of cross-channel relation-
+ships. This is a new plausible inductive bias, namely,
+that there are data correlations across convolutional in-
+put channels that enhance discovery of effective cross-
+channel features. Practitioners have applied these cal-
+culations in the convolution stages of CNNs but not
+to the dense backend where real-valued operations are
+still used. The present paper puts weight-sharing cal-
+culations in the dense backend to further improve CNN
+arXiv:2301.04623v1  [cs.CV]  11 Jan 2023
+
+96
+Top-1 Validation Accuracy
+95.5
+95
+94.5
+HH
+94
+93.5
+93
+ResNet Original Quaternion
+Vectormap
+QPHM
+VPHM
+Original
+Original
+ResNet-18
+ResNet-34
+ResNet-5082
+80
+Top-1 Validation Accuracy
+78
+76
+74
+72
+70
+68
+66
+ResNet Original
+Quaternion
+Vectormap
+QPHM
+VPHM
+Original
+Original
+ResNet-18
+ResNet-34
+ResNet-50performance. To exploit this new type of weight shar-
+ing, we use a parameterized hypercomplex multiplica-
+tion (PHM) [30] layer as a building block. This block
+replaces the real-valued FC layers with hypercomplex
+FC layers. We test the hypothesis using two types of
+hypercomplex CNNs, namely quaternion [8] CNNs and
+vectormap [7] CNNs.
+Our contributions are:
+• Showing the effectiveness of using hypercomplex
+networks in the densely connected backend of a
+CNN.
+• Introducing quaternion networks with PHM based
+dense layer (QPHM) to bring hypercomplex deep
+learning properties to the entire model.
+• Introducing vectormap networks with a PHM based
+dense layer (VPHM) to remove hypercomplex di-
+mensionality constraints from the frontend and
+backend.
+The effectiveness of employing PHM based FC lay-
+ers with hypercomplex networks is seen in Figures 1a
+and 1b.
+We also show that these new models ob-
+tain SOTA results for hypercomplex networks in CIFAR
+benchmarks. Our experiments also show SOTA results
+for American Sign Language (ASL) data. Moreover, our
+models use fewer parameters, FLOPS, and latency com-
+pared to the base model proposed by [7,23] for classifi-
+cation.
+2. Background and Related Work
+2.1. Quaternion Convolution
+Quaternions are four dimensional vectors of the form
+Q = r + ix + jy + kz ; r, x, y, z ∈ R
+(1)
+where, r, x, y, and z are real values and i, j, and k are
+the imaginary values which satisfy i2 = j2 = k2 =
+ijk = −1. Quaternion convolution is defined by con-
+volving a quaternion filter matrix with a quaternion vec-
+tor (or feature map). Let, QF = R + iX + jY + kZ
+be a quaternion filter matrix with R, X, Y, and Z be-
+ing real-valued kernels and QV = r + ix + jy + kz
+be a quaternion input vector with r, x, y, and z being
+real-valued vectors. Quaternion convolution is defined
+below [8].
+QF ⊛ QV = (R ∗ r − X ∗ x − Y ∗ y − Z ∗ z)
++i(R ∗ x + X ∗ r + Y ∗ z − Z ∗ y)
++j(R ∗ y − X ∗ z + Y ∗ r + Z ∗ x)
++k(R ∗ z + X ∗ y − Y ∗ x + Z ∗ r)
+(2)
+There are 16 real-valued convolutions but only four ker-
+nels which are reused.
+This is how the weight shar-
+ing occurs. [18] first described the weight sharing in the
+Hamilton product.
+2.2. Vectormap Convolution
+[7] noted that the Hamilton product and quaternion
+convolution, when used in deep networks, did not re-
+quire the entire Quaternion algebra. They called these
+vectormap convolutions.
+The weight sharing ratio is
+1
+N where N is the dimension of the vectormap, Dvm.
+Let V 3
+in = [v1, v2, v3] be an RGB input vector and
+W 3 = [w1, w2, w3] a weight vector with N = 3. We
+use a permutation τ on inputs so each input vector is
+multiplied by each weight vector element:
+τ(vi) =
+�
+v3
+i = 1
+vi−1
+i > 1
+(3)
+After applying circularly right shifted permutation to
+V 3
+in, a new vector V 3 is formed. The permutation of
+weight τ(W 3) can be found like equation 3. Hence, the
+output vector Vout is:
+V 3
+out = [W 3 · V 3
+in, τ(W 3) · V 3
+in, τ 2(W 3) · V 3
+in]
+(4)
+Here, “·” denotes dot product. The outputs V 3
+out come
+from the linear combination of the elements of V 3
+in and
+W 3. Let the weight filter matrix for a vectormap be
+VF = [A, B, C] and the input vector after linear com-
+bination be Vh = [x, y, z], the vectormap convolution
+between VF , and Vh for Dvm = 3 is:
+�
+�
+R(VF ∗ Vh)
+I (VF ∗ Vh)
+J (VF ∗ Vh)
+�
+� = L ⊙
+�
+�
+A
+B
+C
+C
+A
+B
+B
+C
+A
+�
+� ∗
+�
+�
+x
+y
+z
+�
+�
+(5)
+where, L is a learnable matrix defined as a matrix L ∈
+RDvm×Dvm which is initialized using:
+lij =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+i = 1
+1
+i = j
+1
+j = Cali where Cali = (i + (i − 1)) &
+Cali = Cali − Dvm if Cali > Dvm
+−1
+else.
+(6)
+By choosing Dvm and assigning a new constant matrix
+L ∈ RDvm×Dvm matching Dvm, any dimensional hy-
+percomplex convolution can be used. Vectormap weight
+initialization uses a similar mechanism to complex [25]
+and quaternion [8] weight initialization. Our weight ini-
+tialization follows [7].
+
+2.3. PHM Fully Connected Layer
+The above methods apply to convolutional layers but
+not to fully connected (FC) layers. [30] proposed pa-
+rameterized hypercomplex multiplication (PHM) for FC
+layers. Like vectormaps, PHM can have any dimension.
+If the dimension is four, it is like the Hamilton prod-
+uct. The success of the Hamiltonian product is shown
+in [7,8,16,19,28,31]. Our work uses two different PHM
+dimensions: four for quaternion networks, and five for
+vectormap networks.
+A
+fully
+connected
+layer
+is
+defined
+[30]
+as
+y = FC(x) = Wx + b, where W ∈ Rk×d and
+b ∈ Rk are weights and bias, d and k are input and
+output dimensions, and x ∈ Rd, y ∈ Rk. PHM uses
+the following hypercomplex transform to map input
+x ∈ Rd into output y ∈ Rk as y = PHM (x) = Hx + b,
+where H ∈ Rk×d is the sum of Kronecker products.
+Like Dvm, let the dimension of the PHM module
+be Dphm = N.
+The PHM operation requires that
+both d and k are divisible by N.
+H is the sum
+of Kronecker products of the parameter matrices
+Ai ∈ RN×N and Si ∈ Rk/N×d/N, where i = 1 . . . N:
+H = �N
+i=1 Ai ⊗ Si. Parameter reduction comes from
+reusing matrices A and S in the PHM layer. The ⊗ is
+the Kronecker product. H is multiplied with the input
+in the dense layer. The four dimensional PHM layer is
+explained in [30]. We also use five dimensions which
+is explained here. The learnable parameters for N = 5
+are Pr, Pw, Px, Py, and Pz where P ∈ R1×1. For Ai
+we use the hypercomplex matrix (5 dimensions) which
+is generated in a similar way of vectormap convolution
+(Equations 5 and 6). H is calculated using two learnable
+parameter matrices (Ai, and Si) for N = 5 as follows:
+H =
+�
+�����
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+�
+�����
+�
+��
+�
+A1
+⊗
+�Pr
+�
+����
+S1
++
+�
+�����
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+-1
+0
+0
+0
+0
+0
+-1
+-1
+0
+0
+0
+0
+�
+�����
+�
+��
+�
+A2
+⊗
+�Pw
+�
+� �� �
+S2
++
+�
+�����
+0
+0
+1
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+0
+0
+0
+0
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+0
+0
+0
+1
+-1
+0
+0
+0
+0
+0
+-1
+0
+0
+0
+�
+�����
+�
+��
+�
+A3
+⊗
+�Px
+�
+����
+S3
++
+�
+�����
+0
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+1
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+0
+0
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+0
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+0
+0
+0
+0
+0
+1
+0
+0
+0
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+-1
+0
+0
+�
+�����
+�
+��
+�
+A4
+⊗
+�Py
+�
+����
+S4
++
+�
+�����
+0
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+-1
+0
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+-1
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+�
+�����
+�
+��
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+A5
+⊗
+�Pz
+�
+����
+S5
+=
+�
+�����
+Pr
+Pw
+Px
+Py
+Pz
+−Pz
+Pr
+Pw
+−Px
+−Py
+−Py
+−Pz
+Pr
+−Pw
+Px
+−Px
+−Py
+−Pz
+Pr
+−Pw
+−Pw
+−Px
+−Py
+Pz
+Pr
+�
+�����
+(7)
+Equation 7 for N = 5 expresses the Hamiltonian prod-
+uct of hypercomplex layer. It preserves all PHM layer
+properties.
+3. Proposed Models: QPHM and VPHM
+We propose a new fully hypercomplex model in lieu
+of hypercomplex CNNs that use a real-valued backend
+dense layer. That is, we replace the dense layer with a
+PHM layer to enjoy the benefits of hypercomplex weight
+sharing.
+We chose two base hypercomplex models for the con-
+volutional frontend, the quaternion network and vec-
+tormap network [7, 8] which were using real-valued
+backend layers. To match dimensions with frontend net-
+works, we used a PHM layer at four dimensions with the
+quaternion network and a PHM layer at five dimensions
+with the three dimensional vectormap network. In some
+cases, we also needed to use a PHM layer at five dimen-
+sions with quaternion networks. But we couldn’t use a
+three dimensional PHM layer as the output classes must
+be divisible by the dimensions in the PHM operation.
+Figure 2 shows our proposed PHM based FC layer
+with quaternion convolutional neural networks (QC-
+NNs). At the end of QCNNs (end of layer 4 in Figure
+2 (top)), the output feature maps are flattened. This flat-
+tened layer is normally the input to a fully connected
+layer, but in our proposed method this layer is the input
+layer for the PHM based FC layer. This is represented
+as Pin. The parameterized weight H performs parame-
+terized multiplication to find the hyper-complex output
+Pout. The type of PHM layer depends on the dimensions
+needed. For quaternion networks, we used dimensions
+four and five according to the number of classes in the
+datasets. The figures in Figure 2 (bottom) are expanded
+4D PHM and 5D PHM layer of a single dense layer con-
+nection (red marked in Figure 2 (top)).
+Pin = Prin + Pwin + Pxin + Pyin + Pzin
+(8)
+For the PHM layer with five dimensions, each PHM
+layer accepts five channels of input like Prin, Pwin,
+
+Figure 2. Full hypercomplex network where quaternion convolutional neural networks (QCNNs) are used in the front and PHM
+based fully-connected layers are applied in the back-end. 5-dimensional PHM is explained in Equation 7. Equations 8 and 9
+describe input and output for a 5D PHM layer. 4D PHM is similar.
+Layer
+Output
+size
+Quaternion
+ResNet
+Vectormap
+ResNet
+QPHM
+VPHM
+Stem
+32x32
+3x3Q, 112, std=1
+3x3V, 90, std=1
+3x3Q, 112, std=1
+3x3V, 90, std=1
+Bottleneck
+group 1
+32x32
+�
+�
+1x1Q, 112
+3x3Q, 112
+1x1Q, 448
+�
+� ×3
+�
+�
+1x1V, 90
+3x3V, 90
+1x1V, 360
+�
+� ×3
+�
+�
+1x1QP, 112
+3x3QP, 112
+1x1QP, 448
+�
+� ×3
+�
+�
+1x1VP, 90
+3x3VP, 90
+1x1VP, 390
+�
+� ×3
+Bottleneck
+group 2
+16x16
+�
+�
+1x1Q, 224
+3x3Q, 224
+1x1Q, 896
+�
+� ×4
+�
+�
+1x1V, 180
+3x3V, 180
+1x1V, 720
+�
+� ×4
+�
+�
+1x1QP, 224
+3x3QP, 224
+1x1QP, 896
+�
+� ×4
+�
+�
+1x1VP, 180
+3x3VP, 180
+1x1VP, 720
+�
+� ×4
+Bottleneck
+group 3
+8x8
+�
+�
+1x1Q, 448
+3x3Q, 448
+1x1Q, 1792
+�
+� ×6
+�
+�
+1x1V, 360
+3x3V, 360
+1x1V, 1440
+�
+� ×6
+�
+�
+1x1QP, 448
+3x3QP, 448
+1x1QP, 1792
+�
+� ×6
+�
+�
+1x1VP, 360
+3x3VP, 360
+1x1VP, 1440
+�
+� ×6
+Bottleneck
+group 4
+4x4
+�
+�
+1x1Q, 896
+3x3Q, 896
+1x1Q, 3584
+�
+� ×3
+�
+�
+1x1V, 720
+3x3V, 720
+1x1V, 2880
+�
+� ×3
+�
+�
+1x1QP, 896
+3x3QP, 896
+1x1QP, 3584
+�
+� ×3
+�
+�
+1x1VP, 720
+3x3VP, 720
+1x1VP, 2880
+�
+� ×3
+Pooling
+Layer
+1x1x100
+global average-pool, 100 outputs
+Output
+1x1x100
+fully connected Layer, softmax
+QPHM Layer
+VPHM Layer
+Table 1. The 50-layer architectures tested on CIFAR-100: quaternion ResNet [7, 8], vectormap ResNet [7], our proposed QPHM,
+and VPHM. Input is a 32x32x3 color image. The number of stacked bottleneck modules is specified by multipliers. “Q”, “V”,
+“QP”, “VP”, and “std” denote quaternion convolution, vectormap convolution, QPHM (quaternion network with PHM layer),
+VPHM (vectormap network with PHM layer), and stride respectively. Integers (e.g., 90, 112) denote number of output channels.
+Pxin, Pyin, and Pzin (Equation 8) and produces five
+channels of output like Prout, Pwout, Pxout, Pyout,
+and Pzout which are merged or stacked together to Pout
+as,
+Pout = Prout +Pwout +Pxout +Pyout +Pzout (9)
+Hence,
+the
+representational
+feature
+maps
+persist
+throughout the classification network.
+Similarly, this
+
+Parameterized
+STAGE 1
+STAGE 2
+STAGE 3
+STAGE 4
+Weight
+Pin
+Input
+H
+Horse
+Cat
+224x224x4
+Classification
+Stem Layer
+PHM
+based FC Layer
+QCNN
+64 Filters
+2nd Layer
+3rd Layer
+4rth Layer
+1st Layer
+Filter size 7
+QCNN
+QCNN
+QCNN
+Flatten
+QCNN
+128 Filters
+256 Filters
+ 512 Filters
+Layer
+With stride 2
+64 Filters
+max-pooling
+Stride 2
+Stride 2
+Stride 2
+Stride 1
+5-dimensional PHM layer (VPHM)
+4-dimensional PHM layer (QPHM)
+PPHM (both 4D, and 5D) dense layer is applied in
+the backend of original ResNet [10] which we named
+RPHM (ResNet-with-PHM).
+4. Experiment
+The purpose of the experiments reported herein was
+to test whether replacing the real-valued backend of
+a CNN model with a PHM backend improved clas-
+sification performance. The architectures tested were
+real-valued, quaternion-valued [8, 19], and vectormap
+ResNet [7], either with or without the PHM backend.
+We refer to the quaternion ResNet model with the PHM
+backend as QPHM. Similarly, VPHM, RPHM denote
+the vectormap ResNet, and real-valued ResNet models
+with the PHM backend.
+Our experiments were conducted on the following
+datasets: CIFAR-10/100 [14], the ImageNet300k dataset
+[23] and the American Sign Language Hand Gesture
+color image recognition dataset [5].
+The first two
+datasets have less training samples with small image
+resolutions and the other datasets use a large number
+of training samples with higher resolution images. We
+used these datasets to check our proposed models for
+small and large training samples as well as for small and
+high resolution images. The experiments were run on a
+workstation with an Intel(R) Core(TM) i9-9820X CPU
+@ 3.30GHz, 128 GB memory, and NVIDIA Titan RTX
+GPU (24GB).
+4.1. CIFAR Classification
+In addition to testing the PHM with real-valued,
+quaternion-valued, and vectormap ResNet, we tested the
+network models with three depths: 18, 34, and 50 layers.
+4.1.1
+Method
+We tested all of the above mentioned architectures with
+and without the PHM backend on both the CIFAR-10
+and CIFAR-100 datasets. These datasets were composed
+of 32x32 pixel RGB images falling into either ten classes
+or 100 classes, respectively. Both datasets have 50,000
+training, and 10,000 test examples.
+The models were trained using the same components
+as the real-valued networks, the original quaternion net-
+work, and the original vectormap network using the
+same datasets. All models in Table 2 were trained us-
+ing the same hyperparameters. Our QPHM and VPHM
+design is similar to the quaternion [7,8], and vectormap
+networks [7], respectively. The residual architectures
+differ in the number of output channels than the origi-
+nal hypercomplex networks and the proposed networks
+due to keeping the number of trainable parameters about
+the same. The number of output channels for the resid-
+ual networks is the same as [7] and [19]. Table 1 shows
+the 50-layer architectures tested for CIFAR-100 dataset.
+One goal is to see if the representations generated by
+the PHM based dense layer instead of the real-valued
+dense layer outperforms the quaternion, vectormap, and
+residual baselines reported in [7].
+We also analyzed
+different residual architectures to assess the effect of
+depth on our proposed models. For preprocessing, we
+followed [7]. We used stochastic gradient descent op-
+timization with 0.9 Nesterov momentum.
+The learn-
+ing rate was initially set to 0.1 with warm-up learning
+for the first 10 epochs. For smooth learning, we chose
+cosine learning from epochs 11 to 120. However, we
+were getting about same performance for linear learn-
+ing. All models were trained for 120 epochs and batch
+size was set to 100. This experiment used batch normal-
+ization and 0.0001 weight decay. The implementation
+is on github at-https://github.com/nazmul729/QPHM-
+VPHM.git.
+4.1.2
+Results
+The main results appear in Figure 1 and 3, and in Ta-
+ble 2. Figure 1 gives the overall pattern of results. Fig-
+ure 1a shows results for CIFAR-10. It shows top-1 vali-
+dation accuracy for the five models: real-valued ResNet,
+quaternion-valued ResNet, vectormap ResNet, QPHM,
+and VPHM. Also, results are shown for 18, 34, and 50
+layers. We chose top-1 performance out of three. Fig-
+ure 1b shows the same consistent pattern of results for
+the CIFAR-100 dataset. The magnitude of improvement
+is higher for CIFIR-100 than for CIFAR-10. The results
+are also shown in tabular form in Table 2, along with
+counts of trainable parameters, flops, and latency. It can
+be seen in Table 2 that modifying the backend to have
+a PHM layers has little effect on the parameter count,
+flops, and latency as the input image resolutions, and
+the number of output classes are low.
+The proposed QPHM model attains better top-1 val-
+idation accuracy than the original ResNet, quaternion,
+and vectormap networks for both datasets. The QPHM
+also produces better performance compared to the pro-
+posed VPHM, and RPHM models. Moreover, we com-
+pare our best performance which is obtained by the
+QPHM model, to the deep or shallow complex or hyper-
+complex networks and notice that the QPHM is achieved
+SOTA performance (shown in Table 3) for the CIFAR-
+10 and -100 datasets.
+Table 3 compares different complex or hypercom-
+plex networks top-1 validation accuracy with our best
+result. Our comparison was not limited to [7] and com-
+plex space. The QPHM also gains highest top-1 vali-
+dation accuracy than the relevant CNN models for both
+datasets (shown in Table 4). Tables 2, 3, and 4, show that
+
+Model Name
+Param Count
+FLOPS
+Latency
+Validation Accuracy
+CIFAR-10
+CIFAR-100
+ResNet18 [10]
+11.1M
+0.56G
+0.22ms
+94.08
+72.19
+RPHM18
+11.1M
+0.55G
+0.21ms
+94.74
+77.83
+Quat18 [8]
+8.5M
+0.26G
+0.36ms
+94.08
+71.23
+Vect18 [7]
+7.3M
+0.21G
+0.29ms
+93.95
+72.82
+QPHM18
+8.5M
+0.25G
+0.35ms
+95.03
+77.88
+VPHM18
+7.3M
+0.20G
+0.27ms
+94.97
+77.80
+ResNet34 [10]
+21.2M
+1.16G
+0.29ms
+94.27
+72.19
+RPHM34
+21.1M
+1.15G
+0.28ms
+94.98
+77.80
+Quat34 [8]
+16.3M
+0.438G
+0.57ms
+94.27
+72.76
+Vect34 [7]
+14.04M
+0.35G
+0.45ms
+94.45
+74.12
+QPHM34
+16.3M
+0.432G
+0.54ms
+95.40
+78.51
+VPHM34
+14.03M
+0.34G
+0.44ms
+95.41
+77.23
+ResNet50 [10]
+23.5M
+1.30G
+0.478ms
+93.90
+72.60
+RPHM50
+20.6M
+1.29G
+0.468ms
+95.59
+79.21
+Quat50 [8]
+18.08M
+1.45G
+0.97ms
+93.90
+72.68
+Vect50 [7]
+15.5M
+1.19G
+0.77ms
+94.28
+74.84
+QPHM50
+18.07M
+1.44G
+0.96ms
+95.59
+80.25
+VPHM50
+15.5M
+1.15G
+0.76ms
+95.48
+78.91
+Table 2. Image classification performance on the CIFAR benchmarks for 18, 34 and 50-layer architectures. Here, Quat, Vect,
+QPHM, and VPHM, define the quaternion ResNet, vectormap ResNet, quaternion networks with PHM FC layer, and vectormap
+networks with PHM FC layer, respectively.
+(a) Validation loss versus training.
+(b) Validation accuracy versus training.
+Figure 3. Validation loss and accuracy of 50 layer ResNet [7], quaternion [7], vectormap [7], QPHM, VPHM for CIFAR-100.
+our QPHM model achieves best performance for CIFAR
+10 and 100 datasets with fewer parameters, flops, and la-
+tency.
+4.2. ImageNet Classification
+4.2.1
+Method
+These experiments are performed on a 300k subset of
+the ImageNet dataset which we call ImageNet300k [23].
+[23] explains how the full dataset was sampled. The
+models compared are: standard ResNets [23], quater-
+nion convolutional ResNets [23], and our proposed
+QPHM. We ran 26, 35, and 50-layers architectures using
+“[1, 2, 4, 1]”, “[2, 3, 4, 2]” and “[3, 4, 6, 3]” bottleneck
+block multipliers. Training (all models in Table 5) used
+the same optimizer and hyperparameters as CIFAR clas-
+sification method.
+4.2.2
+Experimental Results
+Table 5 shows the results on the ImageNet300k dataset.
+This result shows that our model takes three millions
+fewer trainable parameters and yields almost 5% higher
+validation performance for the same architectures. Pa-
+
+4.5
+ResNet
+4
+Quaternion
+Vectormap
+3.5
+QPHM
+3
+VPHM
+Loss
+2.5
+2
+1.5
+1
+0.5
+0
+8
+345438
+5
+0
+9
+118
+127
+6
+145
+154
+8
+172
+3
+Epochs90
+80
+70
+60
+Accuracy
+ResNet
+50
+Quaternion
+40
+.Vectormap
+30
+QPHM
+20
+VPHM
+10
+0
+1
+9
+8
+3
+4
+5
+43
+8
+100
+109
+8
+127
+136
+145
+154
+163
+172
+EpochsModel Architecture
+Validation Accuracy
+CIFAR-10
+CIFAR-100
+DCNs [2]
+38.90
+42.6
+DCN [25]
+94.53
+73.37
+QCNN [16]
+77.48
+47.46
+Quat [31]
+77.78
+-
+QCNN [28]
+83.09
+-
+QCNN* [28]
+84.15
+-
+Quaternion18 [7]
+94.80
+71.23
+Quaternion34 [7]
+94.27
+72.76
+Quaternion50 [7]
+93.90
+72.68
+Octonion [27]
+94.65
+75.40
+Vectormap18 [7]
+93.95
+72.82
+Vectormap34 [7]
+94.45
+74.12
+Vectormap50 [7]
+94.28
+74.84
+QPHM-50
+95.59
+80.25
+VPHM-50
+95.48
+78.91
+Table 3.
+Top-1 validation accuracy for hypercomplex net-
+works. DCN stands for deep complex convolutional network.
+* variant used quaternion batch normalization. Quaternion and
+vectormap networks are the base networks [7]
+rameter reduction is not depicted in Table 3 for low res-
+olution CIFAR benchmark images as they have saved
+parameters in thousands. It is also clear that deeper net-
+works perform better than shallow networks.
+4.3. ASL Classification
+4.3.1
+Method
+To compare the proposed QPHM model with other
+networks,
+we
+evaluated
+it
+on
+the
+ASL
+Alpha-
+bet dataset [26] publicly available on Kaggle at
+https://www.kaggle.com/grassknoted/asl-alphabet. This
+dataset has 87,000 hand-gesture images for 29 sign
+classes where each class has about 3,000 images. And,
+the image size is 200 × 200 × 3.
+It has 26 finger spelling alphabet classes for the En-
+glish alphabetic letters and three special characters. Due
+to the divisibility restriction in PHM, we cannot use 29
+classes as 29 is prime. Like all other baseline leave-one-
+out and half-half methods [5,13,15,20,24], we exclude
+one class (letter B) from the training and validation sets.
+We use the same hyperparameters as CIFAR classifica-
+tion method.
+4.3.2
+Experimental Results
+Due to the divisibility limitation, it is not possible to
+evaluate the ASL data using VPHM model as we choose
+PHM with five dimensions for VPHM model. So we
+only tested the QPHM (PHM with four dimensions)
+model to compare with other networks on the ASL
+Model Architecture
+Validation Accuracy
+CIF10
+CIF100
+Convolutional Networks
+ResNet18 [9]
+90.27
+63.41
+ResNet34 [9]
+90.51
+64.52
+ResNet50 [9]
+90.60
+61.68
+ResNet110 [9]
+95.08
+76.63
+ResNet1001 [11]
+95.08
+77.29
+MobileNet [9]
+91.02
+67.44
+Cout [4]
+95.28
+77.54
+Wide Residual Networks
+WRN-28-10
+96.00
+80.75
+WRN-28-10-dropout
+96.11
+81.15
+Our Method
+QPHM50
+95.59
+80.25
+VPHM50
+95.48
+78.91
+QPHM-18-2 (ours)
+96.24
+81.45
+QPHM-50-2 (ours)
+96.63
+82.00
+Table 4. Top-1 validation accuracy comparison among deep
+networks.
+CIF10 and CIF100 stand for CIFAR10 and CI-
+FAR100. Cout is the ResNet-18+cutout. WRN-28-10 [29],
+QPHM-18-2, and QPHM-50-2 stand for wide ResNet 28, 18,
+and 50-layers with the output channel widening factor 10, 2,
+and 2, respectively.
+dataset. Table 6 provides a comparison of top-1 val-
+idation accuracy of our proposed QPHM model with
+other networks in ASL data. Our proposed architecture
+performs state-of-the-art accuracy in this ASL dataset.
+Hence, the representation feature maps in the dense
+layer are very effective for this dataset.
+5. Conclusions
+We replaced the dense backend of existing hypercom-
+plex CNNs for image classification with PHM modules
+to create weight sharing in this layer. This novel de-
+sign improved classification accuracy, reduced parame-
+ter counts, flops, and latency compared to the baseline
+networks. The results support our hypothesis that the
+PHM operation in the densely connected back end pro-
+vides better representations as well as improves accu-
+racy with fewer parameters. These results also high-
+lighted the importance of the calculations in the back-
+end.
+The QPHM and VPHM outperformed the other
+works mentioned in “Experiment” section.
+The pro-
+posed QPHM achieved higher validation accuracy (top-
+1) for all network architectures than the proposed
+VPHM.
+
+Architecture
+Params
+FLOPS
+Latency
+Training
+Accuracy
+Validation
+Accuracy
+ResNet26
+13.6M
+1.72G
+0.75ms
+57.0
+45.48
+Quat ResNet26
+15.1M
+1.30G
+1.71ms
+64.1
+50.09
+QPHM26
+11.4M
+1.18G
+1.7ms
+65.3
+52.23
+ResNet35
+18.5M
+3.57G
+1.02ms
+63.8
+48.99
+Quat ResNet35
+20.5M
+4.59G
+3.15ms
+70.9
+48.11
+QPHM35
+17.5M
+4.10G
+3.15ms
+75.3
+51.84
+ResNet50
+25.5M
+4.01G
+1.46ms
+65.8
+50.92
+Quat ResNet50
+27.6M
+5.82G
+4.21ms
+73.4
+49.69
+QPHM50
+24.5M
+5.32G
+4.16ms
+78.8
+54.38
+Table 5. Classification performance on the ImageNet300k dataset for different ResNet architectures. Top-1 training and validation
+accuracies.
+Architecture
+Top-1 Validation Accuracy
+CNNs
+82%
+HOG-LBP-SVM
+98.36%
+HT with CNN
+96.71%
+RF-JA with l-o-o
+70%
+RF-JA with h-h
+90%
+GF-RF l-o-o
+49%
+GF-RF h-h
+75%
+ESF-MLRF l-o-o
+57%
+ESF-MLRF h-h
+87%
+RF-JP l-o-o
+43%
+RF-JP h-h
+59%
+Faster RCNN
+89.72%
+RCNNA
+94.87%
+DBN
+79%
+CMVA and IF l-o-o
+92.7%
+CMVA and IF h-h
+99.9%
+CNN with ASL
+97.82%
+QPHM
+100.0
+Table 6.
+Top-1 validation accuracy comparison with other
+works on ASL dataset. Here, l-o-o, h-h, HT with CNN [6,21],
+CMVA [24], RF-JA [5], GF-RF [20], ESF-MLRF [15], RF-
+JP [13], RCNN [26], RCNNA [26], DBN [22], and HOG-LBP-
+SVM [17] mean Leave one out, half-half, HYBRID TRANS-
+FORM, CNNs [1] with multiview augmentation and IF Infer-
+ence Fusion, Random Forest with Joint Angles, Gabor Filter-
+based features with Random Forest, Ensemble of Shape Func-
+tion with Multi-Layer Random Forest, Random Forest with
+Joint Positions, Recurrent convolutional neural networks, Re-
+current convolutional neural networks with attention, Deep be-
+lief network, and Histogram of Oriented Gradients (HOG) and
+Local Binary Pattern (LBP) with support vector machine, re-
+spectively.
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+

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+EXACT HYDRODYNAMIC MANIFOLDS FOR THE LINEARIZED
+THREE-DIMENSIONAL BOLTZMANN BGK EQUATION
+FLORIAN KOGELBAUER AND ILYA KARLIN
+Abstract. We perform a complete spectral analysis of the linear three-dimensional
+Boltzmann BGK operator resulting in an explicit transcendental equation for the eigen-
+values. Using the theory of finite-rank perturbations, we prove that there exists a critical
+wave number kcrit which limits the number of hydrodynamic modes in the frequency
+space. This implies that there are only finitely many isolated eigenvalues above the es-
+sential spectrum, thus showing the existence of a finite-dimensional, well-separated linear
+hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results
+can serve as a benchmark for validating approximate theories of hydrodynamic closures
+and moment methods.
+1. Introduction
+The derivation of hydrodynamic equations from kinetic theory is a fundamental, yet not
+completely resolved, problem in thermodynamics and fluids, dating back at least to part
+(b) of Hilbert’s sixth problem [26]. Given the Boltzmann equation or an approximation of
+it, can the the basic equations of fluid dynamics (Euler, Navier–Stokes) be derived directly
+from the dynamics of the distribution function?
+One classical approach is to seek a series expansion in terms of a small parameter, such
+as the relaxation time τ or the Knudsen number ε [39]. One widely used expansion is the
+Chapman–Enskog series [12], where it is assumed that the collision term scales with ε−1,
+thus indicating a (singular) Taylor expansion in ε. Indeed, the zeroth order PDE obtained
+this way gives the Euler equation, while the first order PDE reproduces the Navier–Stokes
+equation. On the linear level, the Navier–Stokes equation is globally dissipative and decay
+of entropy on the kinetic level translates to decay of energy on the fluid level.
+For higher-order expansions, however, we are in trouble. In [4], it was first shown that
+an expansion in terms of Knudsen number can lead to nonphysical properties of the hy-
+drodynamic models: At order two (Burnett equation [12]), the dispersion relation shows
+a change of sign, thus leading to modes which grow in energy (Bobylev instability). In
+particular, the Burnett hydrodynamics are not hyperbolic and there exists no H-theorem
+for them [6].
+1
+arXiv:2301.03069v1  [math-ph]  8 Jan 2023
+
+2
+FLORIAN KOGELBAUER AND ILYA KARLIN
+From a mathematical point of view, of course, there is no guarantee that the expansion
+of a non-local operator in frequency space, i.e., an approximation in terms of local (dif-
+ferential) operators, gives a good approximation for the long-time dynamics of the overall
+system. Among the first to suggest a non-local closure relation was probably Rosenau
+[34]. In a series of works (see, e.g., [19, 18, 21] and references therein), Karlin and Gorban
+derived explicit non-local closures by essentially summing the Chapman–Enskog series for
+all orders. Furthermore, we note that the Chapman–Enskog expansion mixes linear and
+nonlinear terms for the full Boltzmann equation since it only considers powers of ε, while
+the existence (and approximation) of a hydrodynamic manifold can be performed indepen-
+dently of the Knudsen number, for which it only enters as a parameter.
+Spectral properties of linearized kinetic equations are of basic interest in thermodynam-
+ics and have been performed by numerous authors. Already Hilbert himself was concerned
+with the spectral properties of linear integral operators derived from the Boltzmann equa-
+tion [25].
+Carleman [8] proved that the essential spectrum remains the same under a
+compact perturbation (Weyl’s theorem) in the hard sphere case and was able to estimate
+the spectral gap. This result was generalized to a broader class of collision kernels by Grad
+[23] and to soft potentials in [7].
+For spatially uniform Maxwell molecules, a complete spectral description was derived
+in [5] (together with exact special solutions and normal form calculations for the full,
+non-linear problem), see also [11]. Famously, in [15], some fundamental properties of the
+spectrum of a comparably broad class of kinetic operators was derived. In particular, the
+existence of eigenvalue branches and asymptotic expansion of the (small) eigenvalues for
+vanishing wave number was derived. We stress, however, that no analysis for large wave
+numbers or close to the essential spectrum was performed in [15].
+Let us also comment on the relation to Hilbert’s sixth problem. Along these lines, several
+result on the converges to Navier–Stokes (and Euler) equations have been obtained. Al-
+ready Grad [24] was interested in this question. In [15], it is also shown that the semi-group
+generated by the linearized Euler equation converges - for fixed time - to the semi-group
+generated by the linearized Boltzmann equation (and similarly, for the linear Navier–Stokes
+semi-group). In [35], convergence of scaled solutions to the Navier–Stokes equation along
+the lines of [2] was proved. We also mention the results related to convergence rates to the
+equilibrium (hypercoercivity) of the variants of the BGK equation [40, 14]. For an excellent
+review on the mathematical perspective of Hilbert’s sixth problem, we refer to [36].
+In this work, we perform a complete spectral analysis for the Bhatnagar–Gross–Krook
+(BGK) equation [3] linearized around a global Maxwellian.
+The BGK model - despite
+being a comparatively simple approximation to the full Boltzmann equation - shares im-
+portant features such as decay of entropy and the conservation laws of mass, momentum
+and energy [3]. Global existence and estimates of the solution were proved in [32, 33] for
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+3
+the full, non-linear BGK system.
+The single relaxation time τ in the BGK equation will serve as the analog of the Knudsen
+number and fundamental parameter in our analysis. Previous work on the full spectrum
+of kinetic models together with a hydrodynamic interpretation has been performed in [28]
+for the three-dimensional Grad system and in [29] for the linear BGK equation with mass
+density only. A similar independent analysis for the one-dimensional linear BGK with one
+fluid moment was performed in [10, 9] in the context of grossly determined solutions (in
+the sense of [39]), where convergence to the slow manifold is also proven explicitly. While
+the results obtained in [10, 9] are proved for the real line (for which the corresponding
+eigen-distributions are derived), we will focus on the torus TL, for which we expect a dis-
+crete set of eigenvalues.
+Indeed, we will give a complete and (up to the solution of a transcendental equa-
+tion) explicit description of the spectrum of the BGK equation linearized around a global
+Maxwellian. We will show the existence of finitely many discrete eigenvalues above the
+essential spectrum as well as the existence of a critical wave number for each family of
+modes. More precisely, we prove the following:
+Theorem 1.1. The spectrum of the non-dimensional linearized BGK operator L with re-
+laxation time τ around a global Maxwellian is given by
+σ(L) =
+�
+−1
+τ + iR
+�
+∪
+�
+N∈Modes
+�
+|k|<kcrit,N
+{λN(τ|k|)},
+(1.1)
+where Modes = {shear, diff, ac, ac∗} corresponding to the shear mode, the diffusion mode
+and the pair of complex conjugate acoustic modes. The essential spectrum is given by the
+line ℜλ = − 1
+τ , while the discrete spectrum consists of a finite number of discrete, isolated
+eigenvalues. Along with each family of modes, there exists a critical wave number kcrit,N,
+limiting the range of wave numbers for which λN exists.
+Our proof is based on the theory of finite-rank perturbations (see, e.g., [42]), together
+with some properties of the plasma dispersion function, collected in the Appendix for the
+sake of completeness. Furthermore, we give a hydrodynamic interpretation of the results
+by considering the dynamics on the (slow) hydrodynamic manifold (linear combination of
+eigenspaces).
+The paper is structured as follows: In Section 2, we introduce some notation and give
+some basic definitions. In Section 3, we formulate the fundamental equations and perform
+- for completeness - the linearization around a global Maxwellian as well as the non-
+dimensionalization explicitly. Section 4 is devoted to the spectral analysis of the linear part,
+including the derivation of a spectral function describing the discrete spectrum completely.
+We also give a proof of the finiteness of the hydrodynamic spectrum together with a
+description of the modes (shear, diffusion, acoustic) in frequency space. Finally, in Section
+
+4
+FLORIAN KOGELBAUER AND ILYA KARLIN
+(5), we write down the hydrodynamic manifold as a linear combination of eigenvectors and
+derive a closed system for the linear hydrodynamic variables.
+2. Notation and Basic Definitions
+Let H denote a Hilbert space and let T : H → H be a linear operator with domain of
+definition D(H). We denote the spectrum of T as σ(T) and its resolvent set as ρ(T).
+The spectral analysis of the main operator L of the paper (to be defined later) will be
+carried out on the Hilbert space
+Hx,v = L2
+x(T3) × L2
+v(R3, e− |v|2
+2 ),
+(2.1)
+together with the inner product
+⟨f, g⟩x,v =
+�
+T3
+�
+R3 f(x, v)g∗(x, v) e− |v|2
+2 dvdx,
+(2.2)
+where the star denotes complex conjugation.
+Because of the unitary properties of the
+Fourier transform, we can slice the space H for each wave number k and analyze the
+operator Lk (restriction of L to the wave number k) on the Hilbert space
+Hv = L2
+v(R3, e−|v|2),
+(2.3)
+together with the inner product
+⟨f, g⟩v =
+�
+R3 f(v)g∗(v)e− |v|2
+2 dv.
+(2.4)
+For further calculations, let us collect some formulas for definite Gaussian integrals:
+�
+R
+e−Av2 dv =
+� π
+A,
+�
+R3 e−A|v|2 dv =
+� π
+A
+�− 3
+2 ,
+�
+R3|v|2e−A|v|2 dv = 3
+2A
+� π
+A
+� 3
+2 ,
+�
+R
+v2e−Av2 dv =
+√π
+2 A− 3
+2 ,
+(2.5)
+for any A > 0. More generally, for any n ∈ N, we have the useful formula
+�
+R
+v2ne−Av2 dv =
+� π
+A
+(2n − 1)! !
+(2A)n
+.
+(2.6)
+3. Preliminaries and Formulation of the Problem
+We will be concerned with the three-dimensional BGK kinetic equation
+∂f
+∂t + v · ∇xf = −1
+τ QBGK,
+(3.1)
+for the scalar distribution function f : T3
+L ×R3 ×[0, ∞) → R+, f = f(x, v, t) and the BGK
+collision operator
+QBGK =
+�
+f(x, v, t) − feq(n[f], u[f], T[f]; v)
+�
+.
+(3.2)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+5
+Here, T3
+L denotes the three-dimensional torus of length L, the parameter τ > 0 is the
+relaxation time, the equilibrium distribution is given by the standard Gaussian
+feq(n, u, T; v) = n
+�2πkBT
+m
+�− 3
+2
+e−
+m
+2kBT |u−v|2
+,
+(3.3)
+for the molecular mass m and the Boltzmann constant kB, while the five scalar hydrody-
+namic variables are given by the number density,
+n[f](x, t) =
+�
+R3 f(x, v, t) dv,
+(3.4)
+the velocity,
+u[f](x, t) =
+1
+n[f](x, t)
+�
+R3 vf(x, v, t) dv,
+(3.5)
+and the temperature, which is defined implicitly through conservation of energy as
+3
+2
+kB
+m T[f](x, t)n[f](x, t) + n[f](x, t)|u[f](x, t)|2
+2
+=
+�
+R3
+|v|2
+2 f(x, v, t) dv.
+(3.6)
+The physical units are given as [kB] = m2kgs−2K−1 and [kBT] = m2kgs−2 respectively.
+We introduce the moments of the distribution function f as
+M(n)(x, t) =
+�
+R3 f(x, v, t) v⊗ndv,
+(3.7)
+where v⊗0 = 1, v⊗1 = v and
+v⊗n = v ⊗ ... ⊗ v
+�
+��
+�
+n−times
+,
+(3.8)
+for n ≥ 2 is the n-th tensor power. The moment defined in (3.7) is an n-th order symmetric
+tensor, depending on space and time.
+The first three moments relate to the hydrodynamic variables through
+M(0) = n,
+M(1) = nu,
+traceM(2) = n
+�
+|u|2+3kBT
+m
+�
+.
+(3.9)
+Conversely, we can express the hydrodynamic variables in terms of the moments as
+n = M0,
+u = M1
+M0
+,
+kB
+m T = 1
+3
+�traceM2
+M0
+− |M1|2
+M2
+0
+�
+.
+(3.10)
+
+6
+FLORIAN KOGELBAUER AND ILYA KARLIN
+We can reformulate equation (3.1) as an infinite system of coupled momentum equations
+as
+�
+1 + τ ∂
+∂t
+�
+M(n) = −τ∇ · M(n+1) + M(n)
+eq ,
+(3.11)
+for n ≥ 0, where
+M(n)
+eq =
+�
+R3 feq(n[f], u[f], T[f]; v)v⊗n dv.
+(3.12)
+The special property of the BGK hierarchy is that the first three moment equations reduce
+to
+∂
+∂tM(0) = −∇ · M(1),
+∂
+∂tM(1) = −∇ · M(2),
+∂
+∂ttraceM(2) = −trace(∇ · M(3)).
+(3.13)
+In particular, the first three moment equations in terms of the hydrodynamic variables
+read
+∂
+∂tn = −∇ · (nu),
+∂
+∂t(nu) = −∇ ·
+�
+R3 v ⊗ vf dv,
+∂
+∂t
+��
+R3
+m|v|2
+2
+f dv
+�
+= −∇ ·
+�
+R3
+|v|2
+2 vf dv.
+(3.14)
+The collision operator QBGK shares some key properties with the collision operator of
+the full Boltzmann equation. Namely, we have that
+�
+R3 QBGK(v)
+�
+�
+1
+v
+|v|2
+�
+� dv = 0,
+(3.15)
+as well as the negativity condition
+⟨QBGKf, f⟩x,v ≤ 0,
+(3.16)
+for all f ∈ Hx,v for which the above expression is defined.
+We will be interested in the linearized dynamics of (3.1) around a global Maxwellian
+φ(v) = n0
+�
+2πkBT0
+m
+�− 3
+2
+e− m|v|2
+2kBT0 ,
+(3.17)
+for the equilibrium density n0 and the equilibrium temperature T0. Setting
+f �→ φ + εf,
+(3.18)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+7
+implies that
+M0 �→ n0 + εM0,
+M1 �→ εM1,
+M2 �→ n0
+kBT0
+m Id3×3 + εM2,
+(3.19)
+and consequently
+n �→ n0 + εM0,
+u �→
+εM1
+n0 + εM0
+,
+T �→ m
+3kB
+�
+3n0 kBT0
+m
++ εtraceM2
+n0 + εM0
+− ε2
+|M1|2
+(n0 + εM0)2
+�
+.
+(3.20)
+Using
+∂n
+∂ε
+����
+ε=0
+= M0,
+∂u
+∂ε
+����
+ε=0
+= M1
+n0
+,
+∂T
+∂ε
+����
+ε=0
+= m
+3kB
+traceM2 − 3T0 kB
+m M0
+n0
+,
+(3.21)
+we can readily calculate:
+∂
+∂ε
+����
+ε=0
+feq[φ + εf] = ∂
+∂ε
+����
+ε=0
+n
+�2πkBT
+m
+�− 3
+2
+e−
+m
+2kBT |u−v|2
+= n0
+�2πkB
+m
+�− 3
+2
+e−
+m
+2kBT0 |v|2
+�
+M0
+n0
++ m
+3kB
+traceM2 − 3T0 kB
+m M0
+n0
+�
+−3
+2
+�
+T
+− 5
+2
+0
++ T
+− 3
+2
+0
+�
+−
+m
+2kBT0
+� �
+−2M1
+n0
+· v
+�
++T
+− 3
+2
+0
+m
+3kB
+traceM2 − 3T0 kB
+m M0
+n0
+|v|2
+�
+− m
+2kB
+�
+(−T −2
+0 )
+�
+,
+(3.22)
+which, after regrouping and cancellations, becomes
+∂
+∂ε
+����
+ε=0
+feq[φ + εf] =
+�2πkBT0
+m
+�− 3
+2
+e−
+m
+2kBT0 |v|2
+�
+M0 −
+m
+kBT0
+traceM2 − 3kBT0
+m M0
+2
++
+�
+− m
+kBT0
+�
+M1 · v + traceM2 − 3 T0kB
+m M0
+6
+|v|2
+� m
+T0kB
+�2�
+.
+(3.23)
+
+8
+FLORIAN KOGELBAUER AND ILYA KARLIN
+Defining the thermal velocity as
+vthermal =
+�
+kB
+m T0,
+(3.24)
+and re-scaling according to
+v �→ vthermalv,
+(3.25)
+implies that
+Mn �→
+�kB
+m T0
+� 3+n
+2
+Mn,
+(3.26)
+which allows us to simplify
+∂
+∂ε
+����
+ε=0
+feq[φ+εf] = (2π)−3/2e− |v|2
+2
+�
+M0 − traceM2 − 3M0
+2
++ M1 · v + traceM2 − 3M0
+6
+|v|2
+�
+.
+(3.27)
+Similarly, we re-scale
+x �→ Lx,
+(3.28)
+which implies that x ∈ T3 henceforth. Defining the thermal time
+tthermal = L
+� m
+kBT0
+,
+(3.29)
+we can re-scale and non-dimensionalize
+t �→ ttthermal,
+τ �→ τtthermal,
+(3.30)
+which leads to the linearized, non-dimensional BGK equation
+∂f
+∂t = −v · ∇xf − 1
+τ f + 1
+τ (2π)−3/2e
+−|v|2
+2
+��5
+2 − |v|2
+2
+�
+M0 + M1 · v + 1
+6(|v|2−3)traceM2
+�
+.
+(3.31)
+Equation (3.31) will be the starting point for further analysis. For later reference, we also
+define the mean free path as
+lmfp = τvthermal.
+(3.32)
+Let us remark that, by equation (3.21), the linearized macro-variables (nlin, ulin, Tlin) are
+related to the moments (M0, M1, traceM2) via the matrix transform
+�
+�
+nlin
+ulin
+Tlin
+�
+� = v3
+thermal
+n0
+�
+�
+n0
+01×3
+0
+03×1
+vthermalI3×3
+03×1
+−T0
+01×3
+T0
+3
+�
+�
+�
+�
+M0
+M1
+traceM2
+�
+� .
+(3.33)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+9
+4. Spectral Analysis of the linearized BGK operator
+In this section, we will carry out a complete spectral analysis of the right-hand side of
+(3.31). This will allow us to draw conclusions on the decay properties of hydrodynamic
+variables, the existence of a critical wave number and the hydrodynamic closure. After
+reformulating the problem in frequency space, we will use the resolvent calculus to formulate
+a condition for the discrete spectrum (Subsection 4.1). Then, we will use properties of the
+plasma dispersion function (see Appendix) to define a spectral function Γτ|k|, whose zeros
+coincide with the discrete, isolated eigenvalues (Subsection 4.2). Then, in Subsection 4.3,
+using Rouch´e’s Theorem, we prove the existence of a critical wave number kcrit such that
+Γτ|k| has no zeros (i.e., there exists no eigenvalues) for |k|> kcrit. Finally, in Subsection
+4.4, we take a closer look at the branches of eigenvalues (modes) and their corresponding
+critical wave numbers.
+4.1. The discrete spectrum of a finite-rank perturbation. To ease notation, we
+define five distinguished vectors associated with the hydrodynamic moments as
+e0(v) = (2π)− 3
+4 ,
+e1(v) = (2π)− 3
+4 v1,
+e2(v) = (2π)− 3
+4 v2,
+e3(v) = (2π)− 3
+4 v3,
+e4(v) = (2π)− 3
+4 |v|2−3
+√
+6
+,
+(4.1)
+which satisfy the orthonormality condition,
+⟨ei, ej⟩v = δij,
+for
+0 ≤ i, j ≤ 4,
+(4.2)
+where δij is the Kronecker’s delta. To ease notation, we denote the projection onto the
+span of {ej}0≤j≤4 as
+P5f =
+4
+�
+j=0
+⟨f, ej⟩vej,
+(4.3)
+for any f ∈ Hv. The linearized dynamics then takes the form
+∂f
+∂t = Lf,
+(4.4)
+for the linear operator
+L = −v · ∇x − 1
+τ + 1
+τ P5.
+(4.5)
+Remark 4.1. Let us recall that any function f ∈ Hv admits a unique expansion as a
+multi-dimensional Hermite series:
+f(v) =
+∞
+�
+n=0
+fn : Hn(v),
+(4.6)
+
+10
+FLORIAN KOGELBAUER AND ILYA KARLIN
+where
+Hn = (−1)ne
+|v|2
+2 ∇ne
+−|v|2
+2
+,
+(4.7)
+and fn is an n-tensor. Since the five basis vectors (4.1) appear in the expansion (4.6) via
+an orthogonal splitting, we have that
+⟨P5f, (1 − P5)f⟩v = 0,
+(4.8)
+for any f ∈ Hv. Hermite expansions were famously used by Grad in his seminal paper [22]
+to establish finite-moment closures.
+From
+⟨Lf, f⟩x,v = ⟨−v · ∇xf − 1
+τ f + 1
+τ P5f, f⟩x,v
+=
+�
+T3
+�
+R3(−v · ∇xf − 1
+τ f + 1
+τ P5f)fe− |v|2
+2 dxdv
+=
+�
+T3
+�
+R3 −1
+τ [(1 − P5)f](P5f + (1 − P5)f)e− |v|2
+2 dxdv
+= −1
+τ ∥(1 − P5)f∥2
+x,v,
+(4.9)
+where we have assumed that f is sufficiently regular to justify the application of the diver-
+gence theorem in x in order to remove the gradient term as well as (4.8), it follows that
+the operator L is dissipative and that
+ℜσ(L) ≤ 0.
+(4.10)
+On the other hand, from (4.9) and from ∥1 − P5∥op= 1, since 1 − P5 is a projection as well,
+it follows that
+⟨Lf, f⟩x,v ≥ −1
+τ ∥f∥2
+x,v.
+(4.11)
+This shows that any solution to (4.4) has to converge to zero, i.e., the global Maxwellian
+is a stable equilibrium up to the conserved quantities from the center mode. On the other
+hand, we infer that the overall convergence rate to equilibrium can be at most − 1
+τ , which
+immediately implies that there cannot be any eigenvalues below the essential spectrum (see
+also the next section).
+Let us proceed with the spectral analysis by switching to frequency space. Since x ∈ T3,
+we can decompose f in a Fourier series as
+f(x, v) =
+∞
+�
+|k|=0
+ˆf(k, v)eix·k,
+(4.12)
+for the Fourier coefficients
+ˆf(k, v) =
+1
+(2π)3
+�
+R3 f(x, v)e−ix·k dx.
+(4.13)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+11
+In frequency space, the operator (4.5) is conjugated to the linear operator
+ˆLk = −iv · kf − 1
+τ f + 1
+τ P5f,
+(4.14)
+which implies that
+σ(L) =
+�
+k∈Z3
+σ( ˆLk).
+(4.15)
+Defining
+fj = ⟨ej, f⟩v,
+(4.16)
+we can define the following relations between the moments and the coefficients (4.16):
+5 − |v|2
+2(2π)
+3
+2
+M0 = 5 − |v|2
+2(2π)
+3
+4
+f0 = f0e0 −
+√
+6
+2 f0e4,
+1
+(2π)
+3
+2
+v · M1 = f1e1 + f2e2 + f3e3,
+|v|2−3
+6(2π)
+3
+2
+traceM2 = e4
+1
+√
+6(2π)
+3
+4
+�
+R
+f|v|2 dv = e4
+1
+√
+6(2π)
+3
+4
+��
+R
+f(|v|2−3) dv + 3M0
+�
+= f2e4 + 3
+√
+6f0e4.
+(4.17)
+For compactness, we bundle these five basis polynomials into a single vector
+e = (e0, e1, e2, e3, e4).
+(4.18)
+First, let us take a look at the spectrum of ˆL0. For k = 0, we see that ˆL collapses to a
+diagonal operator with five dimensional kernel spanned by {ej}0≤j≤4:
+ˆL0ej = −1
+τ (ej − P5ej) = 0,
+0 ≤ j ≤ 4.
+(4.19)
+On the other hand, the operator ˆL0 acts just like − 1
+τ on the orthogonal complement of
+span{ej}0≤j≤4. This shows that
+σ( ˆL0) =
+�
+−1
+τ , 0
+�
+,
+(4.20)
+where the eigenspace associated to zero has dimension five, while the eigenspace associated
+to − 1
+τ has co-dimension five.
+Now, let us analyse ˆLk for k ̸= 0. To ease notation in the following argument, we define
+the operator
+Skf = v · kf,
+(4.21)
+
+12
+FLORIAN KOGELBAUER AND ILYA KARLIN
+for any k ̸= 0, which gives
+σ( ˆLk) = −1
+τ − σ
+�
+iSk − 1
+τ P5
+�
+= −1
+τ − 1
+τ σ (iτSk − P5) .
+(4.22)
+Because the resolvent of Sk is just given by multiplication with (v · k − z)−1, we see
+immediately that σ(Sk) = R, see also [38]. We define the Green’s function matrices as
+GT (z, n, m) = ⟨en, (iτSk − P5 − z)−1em⟩v,
+GS(z, n, m) = ⟨en, (iτSk − z)−1em⟩v,
+(4.23)
+for 0 ≤ n, m ≤ 4 and set GS(z) = {GS(z, n, m)}0≤n≤4, GT (z) = {GT (z, n, m)}0≤n≤4.
+By the second resolvent identity,
+R(z; A) − R(z; B) = R(z; A)(B − A)R(z; B),
+(4.24)
+for any operators A, B and z ∈ ρ(A) ∩ ρ(B), we have for A = iτSk and B = iτSk − P5 that
+(iτSk − P5 − z)−1 = (iτSk − z)−1 + (iτSk − z)−1P5(iτSk − P5 − z)−1.
+(4.25)
+Applying equation (4.25) to em for 0 ≤ m ≤ 4 and rearranging gives
+(iτSk − P5 − z)−1em = (iτSk − z)−1em + (iτSk − z)−1P5(iτSk − P5 − z)−1em
+= (iτSk − z)−1em + (iτSk − z)−1
+4
+�
+j=0
+⟨(iτSk − P5 − z)−1em, ej⟩vej
+= (iτSk − z)−1em +
+4
+�
+j=0
+G∗
+T (z, j, m)(iτSk − z)−1ej,
+(4.26)
+for z ∈ C \ iR. Thus, the resolvent of iτSk − P5 − z includes the resolvent of iτSk as well
+as information from the matrix {GT (z, n, m)}0≤n,m≤4 as coefficients.
+Taking an inner product of (4.26) with en gives
+GT (z, n, m) = GS(z, n, m) +
+4
+�
+j=0
+GT (z, j, m)⟨en, (iτSk − z)−1ej⟩v
+= GS(z, n, m) +
+4
+�
+j=0
+GT (z, j, m)GS(z, n, j)
+(4.27)
+for 0 ≤ n, m ≤ 4 and z ∈ C \ iR, where in the last step, we have used the symmetry of the
+Green’s function matrix. System (4.27) defines twenty-five equations for the coefficients
+GT (z, n, m), which can be re-written more compactly as
+GT = GS + GSGT ,
+(4.28)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+13
+or, equivalently,
+(Id − GS)GT = GS.
+(4.29)
+Equation (4.29) can be interpreted as a special case of Krein’s resolvent identity [30]. This
+shows that we can solve for the entries of GT unless det(Id − GS) = 0, or, to phrase it
+differently, we have that for each wave number k, the discrete spectrum of (iτSk) − P5 can
+be used to infer that
+σdisc( ˆLk) = −1
+τ − 1
+τ
+�
+�
+�z ∈ C : det
+�
+�
+�
+R3 e(v) ⊗ e(v)
+e− |v|2
+2
+iτk · v − z dv − Id
+�
+� = 0
+�
+�
+� .
+(4.30)
+An eigenvalue λ of the operator ˆLk is related to the zero z in (4.30) via
+z = −τλ − 1.
+(4.31)
+In particular, the finite-rank perturbation P5 can only add discrete eigenvalues to the spec-
+trum and we have that σess(iτSk − P5) = σess(iτSk) = iR.
+4.2. Reformulation in terms of the spectral function. We proceed with the spectral
+analysis of (4.5) by rewriting the determinant expression in (4.30). To this end, we note
+that any wave vector k can be written as
+k = Qk(|k|, 0, 0)T ,
+(4.32)
+for some rotation matrix Qk. Defining w = QT
+kv, we have that
+k · v = Qk(|k|, 0, 0)T · v = (|k|, 0, 0) · w = |k|w1,
+(4.33)
+while the vector of basis functions e transforms according to
+e(v) = (2π)− 3
+4
+�
+1, v, |v|2−3
+√
+6
+�
+= (2π)− 3
+4
+�
+1, Qkw, |w|2−3
+√
+6
+�
+=
+�
+�
+1
+0
+0
+0
+Qk
+0
+0
+0
+1
+�
+� e(w).
+(4.34)
+This, together with dv = dw from the orthogonality of Qk, implies that
+det
+�
+�
+�
+R3 e(v) ⊗ e(v)
+e− |v|2
+2
+iτk · v − z dv − Id
+�
+�
+= det
+�
+�
+�
+R3
+�
+�
+1
+0
+0
+0
+Qk
+0
+0
+0
+1
+�
+� e(w) ⊗
+�
+�
+�
+�
+1
+0
+0
+0
+Qk
+0
+0
+0
+1
+�
+� e(w)
+�
+�
+e− |w|2
+2
+iτ|k|w1 − z dw − Id
+�
+�
+= det
+�
+�
+�
+R3 e(w) ⊗ e(w)
+e− |w|2
+2
+iτ|k|w1 − z dw − Id
+�
+� ,
+(4.35)
+
+14
+FLORIAN KOGELBAUER AND ILYA KARLIN
+where we have used the orthogonality of Qk.
+We proceed:
+det
+�
+�
+�
+R3 e(w) ⊗ e(w)
+e− |w|2
+2
+iτ|k|w1 − z dw − Id
+�
+� =
+= det
+�
+���������
+(2π)− 3
+2
+�
+R3
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+w1
+w2
+w3
+|w|2−3
+√
+6
+w1
+w2
+1
+w1w2
+w1w3
+w1
+|w|2−3
+√
+6
+w2
+w1w2
+w2
+2
+w2w3
+w2
+|w|2−3
+√
+6
+w3
+w1w3
+w3w2
+w2
+3
+w3
+|w|2−3
+√
+6
+|w|2−3
+√
+6
+w1
+|w|2−3
+√
+6
+w2
+|w|2−3
+√
+6
+w3
+|w|2−3
+√
+6
+(|w|2−3)2
+6
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+e− |w|2
+2
+iτ|k|w1 − z dw − Id
+�
+���������
+,
+(4.36)
+Integrating out the variables w2 and w3 with the help of (2.5), it follows that
+det
+�
+�
+�
+R3 e(w) ⊗ e(w)
+e− |w|2
+2
+iτ|k|w1 − z dw − Id
+�
+� =
+= det
+�
+�������
+(2π)− 3
+2
+�
+R
+�
+�
+�
+�
+�
+�
+�
+�
+2π
+2πw1
+0
+0
+2π w2
+1−1
+√
+6
+2πw1
+2πw2
+1
+0
+0
+2πw1
+w2
+1−1
+√
+6
+0
+0
+2π
+0
+0
+0
+0
+0
+2π
+0
+2π w2
+1−1
+√
+6
+2πw1
+w2
+1−1
+√
+6
+0
+0
+2π w4
+1−2w2
+1+5
+6
+�
+�
+�
+�
+�
+�
+�
+�
+e−
+w2
+1
+2
+iτ|k|w1 − z dw1 − Id
+�
+�������
+= det
+�
+���
+1
+√
+2π
+�
+R
+�
+�
+�
+�
+1
+w
+w2−1
+√
+6
+w
+w2
+w w2−1
+√
+6
+w2−1
+√
+6
+w w2−1
+√
+6
+w4−2w2+5
+6
+�
+�
+�
+�
+e− w2
+2
+iτ|k|w − z dw − Id
+�
+���
+�
+�
+1
+√
+2π
+�
+R
+e− w2
+2
+iτ|k|w − z − 1
+�
+�
+2
+,
+(4.37)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+15
+where we have used the linearity of the integral and properties of the determinant of block
+matrices. Also, we have used that
+�
+R2(w2
+1 + w2
+2 + w2
+3 − 3)2e−
+w2
+2
+2 −
+w2
+3
+2 dw2dw3
+=
+�
+R2(w4
+1 + w4
+2 + w4
+3 + 9 − 6w2
+1 − 6w2
+2 − 6w2
+3 + 2w2
+1w2
+2 + 2w2
+2w2
+3 + 2w2
+1w2
+3)e−
+w2
+2
+2 −
+w2
+3
+2 dw2dw3
+= 2π
+�
+w4
+1 + 3 + 3 + 9 − 6w2
+1 − 6 − 6 + 2w2
+1 + 2 + 2w2
+1
+�
+= 2π
+�
+w4
+1 − 2w2
+1 + 5
+�
+.
+(4.38)
+For the following calculation, let us define the function
+Z(z) =
+1
+√
+2π
+�
+R
+e− v2
+2
+v − z dv,
+(4.39)
+for z ∈ C \ R. From (4.9), it suffices to consider Z for ℑz > 0. The symmetry property
+Z(z∗) = Z∗(z), however, allows us to extend the function to the whole complex plane (with
+a discontinuity at the real line) once an expression for a half-plane is known.
+Remark 4.2. Integral expressions of the form (4.39) appear frequently in thermodynamics
+and plasma physics [17], where the function (4.39) is called plasma dispersion function
+[13] accordingly. Some properties of Z - including a more explicit expression in terms of
+complex error functions - are collected in the Appendix.
+Using the recurrence relation (A.9), we calculate the first few derivatives of Z in terms
+of polynomials and Z itself:
+dZ
+dz = −1 − zZ,
+d2Z
+dz2 = z + (z2 − 1)Z,
+d3Z
+dz3 = 2 − z2 + (3z − z2)Z,
+d4Z
+dz4 = −5z + z3 + (z4 − 6z2 + 3)Z.
+(4.40)
+Using the identity
+1
+√
+2π
+�
+R
+Hk(v) e− v2
+2
+v − z dv =
+1
+√
+2π
+�
+R
+��
+− d
+dv
+�k
+e− v2
+2
+�
+dv
+v − z = (−1)kk!
+√
+2π
+�
+R
+e− v2
+2
+dv
+(v − z)k+1
+= (−1)k
+√
+2π
+dk
+dzk
+�
+R
+e− v2
+2
+dv
+v − z = (−1)k dkZ
+dzk ,
+(4.41)
+
+16
+FLORIAN KOGELBAUER AND ILYA KARLIN
+together with (4.40) allows us to further simplify the determinant expression in (4.36).
+Indeed, expanding the polynomial matrix in (4.36) in Hermite basis and using (4.41), we
+deduce that
+1
+√
+2π
+�
+R
+�
+�
+�
+�
+1
+w
+w2−1
+√
+6
+w
+w2
+w w2−1
+√
+6
+w2−1
+√
+6
+w w2−1
+√
+6
+w4−2w2+5
+6
+�
+�
+�
+�
+e− w2
+2
+w − ζ dw
+=
+1
+√
+2π
+�
+R
+�
+�
+�
+�
+H0(w)
+H1(w)
+H2(w)
+√
+6
+H1(w)
+H2(w) + H0(w)
+H3(w)+2H1(w)
+√
+6
+H2(w)
+√
+6
+H3(w)+2H1(w)
+√
+6
+H4(w)+4H2(w)+6
+6
+�
+�
+�
+�
+e− w2
+2
+w − ζ dw
+=
+�
+�
+�
+�
+Z
+−Z′
+Z′′
+√
+6
+−Z′
+Z′′ + Z
+− Z′′′+2Z′
+√
+6
+Z′′
+√
+6
+− Z′′′+Z′
+√
+6
+Z(4)+4Z′′+6H0
+6
+�
+�
+�
+�
+=
+�
+�
+�
+�
+Z
+1 + ζZ
+ζ+(ζ2−1)Z
+√
+6
+1 + ζZ
+ζ + ζ2Z
+ζ2+(ζ3−ζ)Z
+√
+6
+ζ+(ζ2−1)Z
+√
+6
+ζ2+(ζ3−ζ)Z
+√
+6
+ζ3−ζ+(ζ4−2ζ2+5)Z
+6
+�
+�
+�
+� .
+(4.42)
+To ease notation, we define the function
+Γτ|k|(ζ) := det
+�
+�
+�
+�
+Z(ζ) − iτ|k|
+1 + ζZ(ζ)
+ζ+(ζ2−1)Z(ζ)
+√
+6
+1 + ζZ(ζ)
+ζ + ζ2Z(ζ) − iτ|k|
+ζ2+(ζ3−ζ)Z(ζ)
+√
+6
+ζ+(ζ2−1)Z(ζ)
+√
+6
+ζ2+(ζ3−ζ)Z(ζ)
+√
+6
+ζ3−ζ+(ζ4−2ζ2+5)Z(ζ)
+6
+− iτ|k|
+�
+�
+�
+�
+= 1
+6
+�
+ζ + 6i|k|3τ 3 − ζ(ζ2 + 5)|k|2τ 2 + 2i(ζ2 + 3)|k|τ
++Z(ζ)(ζ2 − (ζ4 + 4ζ2 + 11)|k|2τ 2 + 2iζ3|k|τ − 5) − 4iZ2(ζ)((ζ2 + 1)|k|τ − iζ)
+�
+,
+(4.43)
+which allows us to conclude that
+det
+�
+�
+�
+R3 e(w) ⊗ e(w)
+e− |v|2
+2
+iτk · v − z dv − Id
+�
+� =
+1
+(i|k|τ)5 (Z(ζ) − iτ|k|)2Γτ|k|(ζ)
+����
+ζ=
+z
+i|k|τ
+,
+(4.44)
+by the scaling properties of the determinant function. Consequently, from (4.30) and (4.31)
+we deduce that
+σdisc( ˆLk) =
+�
+λ ∈ C : Γτ|k|
+�−τλ − 1
+i|k|τ
+�
+= 0
+�
+∪
+�
+λ ∈ C : Z
+�−τλ − 1
+i|k|τ
+�
+= iτ|k|
+�
+. (4.45)
+Typical spectra (4.45) for different wave numbers are shown in Figures 4.1 - 4.3.
+The
+explicit transcendental equation (4.45) determining the discrete spectrum is the first main
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+17
+-π
+-π/2
+0
+π/2
+π
+0.1
+1.
+10.
+100.
+(a) |k|= 1
+-π
+-π/2
+0
+π/2
+π
+0.1
+1.
+10.
+100.
+(b) |k|=
+√
+2
+Figure 4.1. Argument plot of the spectral function (4.44) for τ = 0.5 and
+different values of |k|. The zeros of the function (4.44) in the complex plane
+define eigenvalues of the linearized BGK operator. These are points, were
+a small, counter-clockwise loop runs through the whole rainbow according
+to multiplicity.
+result of our paper. It will allow us to draw further conclusions about the discrete (hydro-
+dynamic) spectrum.
+4.3. Existence of a Critical Wave Number and Finiteness of the Hydrodynamic
+Spectrum. Next, let us prove that there exists a critical wave number kcrit, such that
+σdisc( ˆLk) = ∅,
+for |k|> kcrit.
+(4.46)
+Proof. First, let us recall that any discrete eigenvalue λ of ˆLk (and hence of L) satisfies
+− 1
+τ < ℜλ ≤ 0,
+(4.47)
+by (4.9), which we will assume henceforth (of course, it would in fact follow from a slightly
+more detailed analysis of the following). Since λ and ζ are related by
+λ = −i|k|τζ + 1
+τ
+,
+(4.48)
+this implies that ℜλ = |k|ℑζ − 1
+τ and consequently
+0 < ℑζ ≤
+1
+τ|k|.
+(4.49)
+
+18
+FLORIAN KOGELBAUER AND ILYA KARLIN
+-π
+-π/2
+0
+π/2
+π
+0.1
+1.
+10.
+100.
+(a) |k|=
+√
+3
+-π
+-π/2
+0
+π/2
+π
+0.1
+1.
+10.
+100.
+(b) |k|=
+√
+6
+Figure 4.2. Argument plot of the spectral function (4.44) for τ = 0.5 and
+different values of |k|. The zeros of the function (4.44) in the complex plane
+define eigenvalues of the linearized BGK operator. These are points, were
+a small, counter-clockwise loop runs through the whole rainbow according
+to multiplicity. As we approach the critical wave number, the zeros move
+closer and closer to the essential spectrum (ℜλ = − 1
+τ )
+.
+Our strategy is to apply Rouch´e’s theorem to the function Γτ|k| by splitting it into a
+dominant part plus an (asymptotically) small part.
+To this end, we can focus on the
+family of rectangles Ra = {−a, a, a + i 1
+τ|k|, −a + i 1
+τ|k|} for a > 0. First, let us consider the
+asymptotics of Γτ|k| in ζ for fixed τ|k|.
+Since we are focused on the upper half-plane, we can consider Z+ defined in (A.10) as an
+analytic continuation together with its limit on the real line. In particular, we see from
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+19
+-π
+-π/2
+0
+π/2
+π
+0.1
+1.
+10.
+100.
+(a) |k|=
+√
+8
+-π
+-π/2
+0
+π/2
+π
+0.1
+1.
+10.
+100.
+(b) |k|= 3
+Figure 4.3. Argument plot of the spectral function (4.44) for τ = 0.5 and
+different values of |k|. The zeros of the function (4.44) in the complex plane
+define eigenvalues of the linearized BGK operator. These are points, were
+a small, counter-clockwise loop runs through the whole rainbow according
+to multiplicity. Since the wave number is above kcrit, there exist, indeed,
+no zeros.
+the asymptotics (A.15) that
+Γτ|k|(z) ∼ 1
+6
+�
+ζ + 6i|k|3τ 3 − ζ(ζ2 + 5)|k|2τ 2 + 2i(ζ2 + 3)|k|τ
++Z(ζ)(ζ2 − (ζ4 + 4ζ2 + 11)|k|2τ 2 + 2iζ3|k|τ − 5) − 4iZ2(ζ)((ζ2 + 1)|k|τ − iζ)
+�
+∼ 1
+6
+�
+ζ + 6i|k|3τ 3 − ζ(ζ2 + 5)|k|2τ 2 + 2i(ζ2 + 3)|k|τ
+−
+∞
+�
+n=0
+(2n − 1)! !
+ζ2n+1
+(ζ2 − (ζ4 + 4ζ2 + 11)|k|2τ 2 + 2iζ3|k|τ − 5)
+−4i
+�
+−
+∞
+�
+n=0
+(2n − 1)! !
+ζ2n+1
+�2
+((ζ2 + 1)|k|τ − iζ)
+�
+� ,
+(4.50)
+
+20
+FLORIAN KOGELBAUER AND ILYA KARLIN
+which, after rearranging and regrouping higher-order terms in ζ−1, gives
+Γτ|k|(z) ∼ 1
+6
+�
+ζ + 6i|k|3τ 3 − ζ(ζ2 + 5)|k|2τ 2 + 2i(ζ2 + 3)|k|τ
+− (ζ−1 + ζ−3)(ζ2 − (ζ4 + 4ζ2 + 11)|k|2τ 2 + 2iζ3|k|τ − 5) + O(|ζ|−1)
+−4iζ−2((ζ2 + 1)|k|τ − iζ)
+�
++ O(|ζ|−2)
+∼ 1
+6
+�
+ζ + 6i|k|3τ 3 − |k|2τ 2ζ3 − 5|k|2τ 2ζ + 2i|k|τζ2 + 6i|k|τ
+− ζ + |k|2τ 2ζ3 + 4|k|2τ 2ζ + 11|k|2τ 2ζ−1 − 2i|k|τζ2 − 5ζ−1
+− ζ−1 + |k|2τ 2ζ + 4|k|2τ 2ζ−1 + 11|k|2τ 2ζ−2 − 2i|k|τ − 5ζ−3
+−4i|k|τ − 4i|k|τζ−2 − 4ζ−1 + O(|ζ|−1)
+�
+∼ i(|k|τ)3 + O(|ζ|−1),
+(4.51)
+for |arg(ζ)|≤ π
+2 − δ,
+ζ → ∞, for any real number 0 < δ ≤ π
+2 .
+Remark 4.3. It is a quite remarkable property of the spectral function Γτ|k| that all the
+polynomial terms (up to order four) cancel exactly with the negative-power terms in the
+asymptotic expansion (A.15) to give a constant asymptotic value in the limit. This is due
+to a subtle fine-tuning of the numerical coefficients of the polynomials. This property also
+guarantees the existence of a critical wave number (and hence implies that there are only
+finitely many discrete eigenvalues above the essential spectrum). At the outset, it is by no
+means clear that the spectrum should exhibit this cancellation property. Indeed, numerical
+investigations actually leave this question unanswered [27].
+Let us start with estimating Γτ|k| − i(|k|τ)3 on the real line. Because x �→ |Γτ|k|(x) −
+i(|k|τ)3| is an even function for x ∈ R, we can focus on x > 0. Since Γτ|k|(x) → i(|k|τ)3
+as x → ∞, we know that x �→ |Γτ|k|(x) − i(|k|τ)3| is bounded on the real line. Since
+Γτ|k|(x) − i(|k|τ)3 only contains powers of |k| up to order two, we know that there exists
+a k1 > 0 such that
+|Γτ|k|(x) − i(|k|τ)3|< (|k|τ)3,
+(4.52)
+for all x ∈ R and all |k|> k1.
+By the same token, we conclude that x �→ |Γτ|k|(x +
+i
+|k|τ ) − i(|k|τ)3|, is bounded for x ∈ R
+since (4.50) holds in cone containing the real axis. Therefore, since again Γτ|k|(x +
+i
+|k|τ ) −
+i(|k|τ)3 is bounded for x ∈ R, there exists a k2 > 0 such that
+����Γτ|k|
+�
+x +
+i
+|k|τ
+�
+− i(|k|τ)3
+���� < (|k|τ)3,
+(4.53)
+for all x ∈ R and all |k|> k2.
+Clearly, an estimate of the form (4.53) for all x ∈ R,
+0 ≤ y ≤
+1
+τ|k| and |k|> k3 holds true by compactness and the decay properties of Γτ|k|.
+This shows that, for |k| large enough, we can bound the function Γτ|k| − i(|k|τ)3 on the
+rectangle Ra for any a > 0 by the modulus of i(|k|τ)3, which has no zeros in the strip at
+all (in particular, not in the strip 0 ≤ ℑζ ≤
+1
+τ|k|). For |k| large enough, Rouch´e’s theorem
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+21
+(a) On the real line
+(b) For ℑζ =
+1
+τ|k|
+Figure 4.4. The function ζ �→ |Γτ|k|(ζ) − i(|k|τ)3| on the real line and on
+the line ℑζ =
+1
+τ|k| for τ = 0.5 and |k|= 1 (solid lines) compared to (|k|τ)3
+(dashed lines).
+(a) On the real line
+(b) For ℑζ =
+1
+τ|k|
+Figure 4.5. The function ζ �→ |Γτ|k|(ζ) − i(|k|τ)3| on the real line and on
+the line ℑζ =
+1
+τ|k| for|k|= 4 (solid lines) compared to (|k|τ)3 (dashed lines).
+then implies that Γτ|k| cannot have any zeros for 0 ≤ ℑζ ≤
+1
+τ|k| either.
+This proves the claim.
+□
+Now, let us prove that
+Γ2(λ) :=
+1
+(i|k|τ)3 Γτ|k|
+�−τλ − 1
+i|k|τ
+�
+(4.54)
+has exactly three zeros (one real, two complex conjugate, which we will prove later) for |k|
+small enough.
+
+TiK
+6
+5
+4 F
+3 E
+2
+4
+6
+8
+10[riki(x)-ik33
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+2
+4
+6
+8
+100.14
+0.12
+0.10
+0.08
+2
+6
+8
+10[riki(x)-ik33
+5
+46
+3 E
+2
+4
+6
+8
+1022
+FLORIAN KOGELBAUER AND ILYA KARLIN
+Proof. To this end, we again use the asymptotic expansion (A.15) up to order three for the
+limit |k|→ 0 together with expansion similar to those derived in (4.50) and (4.51):
+Γ2(λ) ∼
+1
+6(i|k|τ)3
+�
+ζ + 6i|k|3τ 3 − ζ(ζ2 + 5)|k|2τ 2 + 2i(ζ2 + 3)|k|τ
++ Z(ζ)(ζ2 − (ζ4 + 4ζ2 + 11)|k|2τ 2 + 2iζ3|k|τ − 5)
+−4iZ2(ζ)((ζ2 + 1)|k|τ − iζ)
+� ���
+ζ= −τλ−1
+i|k|τ
+∼
+1
+6(i|k|τ)3
+�
+ζ + 6i|k|3τ 3 − ζ(ζ2 + 5)|k|2τ 2 + 2i(ζ2 + 3)|k|τ
++ (−ζ−1 − ζ−3 − 3ζ−5 + O(|ζ|−7))(ζ2 − (ζ4 + 4ζ2 + 11)|k|2τ 2 + 2iζ3|k|τ − 5)
+−4i(−ζ−1 − ζ−3 − 3ζ−5 + O(|ζ|−7))2((ζ2 + 1)|k|τ − iζ)
+� ���
+ζ= −τλ−1
+i|k|τ
+,
+(4.55)
+which, after plugging in the transformation (4.48), gives
+Γ2(λ) ∼
+1
+6(i|k|τ)3
+�
+O(|ζ|−3)(|k|τ)2 + 6i(|k|τ)3 + (|k|τ)2 �
+18ζ−1 + 23ζ−3 + 33ζ−5�
+−2i|k|τ
+�
+9ζ−2 + 18ζ−4 + 26ζ−6 + 30ζ−8 + 18ζ−10�
+−
+�
+6ζ−3 + 13ζ−5 + 24ζ−7 + 36
+�� ���
+ζ= −τλ−1
+i|k|τ
+∼
+1
+6(i|k|τ)3
+�
+6i(|k|τ)3 + 18i(|k|τ)3(−τλ − 1)−1 − 18i(|k|τ|)(i|k|τ)2(−τλ − 1)−2
+−6(i|k|τ)3(−τλ − 1)−3 + O(|k|4)
+�
+∼ −
+λ3
+(λτ + 1)3 + O(|k|),
+(4.56)
+i.e., in the limit |k|→ 0, the spectral function (4.43) has a triple zero at λ = 0. The cubic
+scaling in |k| in front of the above expression cancels exactly with the terms inside the
+bracket, leaving only the term λ3 in the limit |k|→ 0. This is consistent with the spectrum
+of ˆL0 containing zero as an isolated eigenvalue, see (4.20). By continuity of the spectrum,
+this implies that the there will emanate exactly three discrete eigenvalues as zeros of the
+spectral function Γτ|k|.
+□
+4.4. Hydrodynamic Modes and their Corresponding Critical Wave Numbers.
+Now, let us take a closer look at the eigenvalues. From (4.43), it follows immediately that
+there exists a sequence of real eigenvalue of algebraic multiplicity two which we call shear
+mode and denote as |k|�→ λshear(|k|).
+A closer look at (4.43) reveals that the function Γτ|k| maps imaginary numbers to imaginary
+numbers (since also Z|iR⊆ iR by (A.6)). As a consequence, Γ2(λ) maps real numbers to
+real numbers. This shows that, together with the above considerations, that, for each wave
+number small enough, there exists exactly one real zero and two complex conjugated zeros.
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+23
+Consequently, apart from the shear mode, there exists a sequence of pairs of complex
+conjugated eigenvalues which we call acoustic modes and denote as |k|�→ λac(|k|) and
+|k|�→ λ∗
+ac(|k|). Figure 4.6 shows the distribution of acoustic modes for a given relaxation
+time and varying wave number.
+Furthermore, there exists another simple, real eigenvalue called diffusion mode which we
+denote as |k|�→ λdiff(|k|). Each mode has its own critical wave number. In conclusion, the
+spectrum is given by
+σ( ˆLk) =
+�
+−1
+τ + iR
+�
+∪ {λshear(|k|), λdiff(|k|), λac(|k|), λ∗
+ac(|k|)},
+(4.57)
+for |k| smaller than the respective critical wave number.
+Remark 4.4. We note that the eigenvalues (and hence the spectrum) depends on wave
+number only through τ|k|. This implies that, while the eigenvectors depend on the full
+wave vector k, the form of the spectrum only depends on the dimensionless parameter τ|k|
+and the existence of the hydrodynamic manifold (as a linear combination of eigenvectors)
+is independent of the relaxation time. If the relaxation time decreases, the critical wave
+number of each mode is increased, thus allowing for more eigenvalues in each family of
+modes. Consequently, decreasing the relaxation time increases the (finite) dimension of
+the hydrodynamic manifold.
+In the limit τ → 0, the eigenvalues accumulate at the essential spectrum and we cannot
+separate a hydrodynamic manifold any longer, since the corresponding spectral projection
+does not exist (no closed contour can be defined that encircles the set of discrete eigenvalues,
+while not intersecting the essential spectrum).
+To finish the spectral analysis, let us derive some information about the critical wave
+number of the four hydrodynamic modes. Since |Z|≤ � π
+2 with equality exactly at zero
+(continuously extended from both sides), we immediately conclude that
+kcrit(λshear) =
+�π
+2
+1
+τ ≈ 1.253311
+τ .
+(4.58)
+from equation (4.45). This is consistent with the result obtained in [29] (equation (2.53)
+in [29]).
+Since the diffusion mode is real, and wanders from zero to − 1
+τ as |k| increases, we can
+recover the critical wave number by taking the limit λ → − 1
+τ (on the branch Z+) in (4.43).
+Since limζ→0,ℑζ>0 Z(ζ) = i�π
+2 , see (A.14), we obtain the critical wave number kcrit(λdiff)
+as a zero of the cubic polynomial
+6(kτ)3 − 11
+�π
+2 (kτ)2 + (6 + 2π) kτ − 5
+�π
+2 = 0.
+(4.59)
+The only real solution is approximately given by
+kcrit(λdiff) ≈ 1.356031
+τ .
+(4.60)
+
+24
+FLORIAN KOGELBAUER AND ILYA KARLIN
+Figure 4.6. The acoustic modes for τ = 0.001 and wave numbers up to
+the critical wave number together withe the vertical line ℜλ = − 1
+τ
+Now, let us turn to the acoustic mode. We know that at the critical wave number, the two
+complex conjugated acoustic modes will merge into the essential spectrum. This happens
+when ℜλ = − 1
+τ . So, let us assume that λ = − 1
+τ − i|k|x, which amount to setting ζ = x in
+(4.43). We obtain two equations (real and imaginary part of Γτ|k|(x)):
+1
+12e−x2 �
+erfi
+� x
+√
+2
+� �√
+2π(τ|k|)2e
+x2
+2 �
+x4 + 4x2 + 11
+�
+− 8πτ|k|
+�
+x2 + 1
+�
+−
+√
+2πe
+x2
+2 �
+x2 − 5
+��
+−4πxerfi
+� x
+√
+2
+�2
+− 2x
+�
+ex2 �
+(τ|k|)2 �
+x2 + 5
+�
+− 1
+�
++
+√
+2πτ|k|e
+x2
+2 x2 − 2π
+��
+= 0,
+1
+12e−x2
+�
+−4πτ|k|
+�
+x2 + 1
+�
+erfi
+� x
+√
+2
+�2
++ erfi
+� x
+√
+2
+� �
+8πx − 2
+√
+2πτ|k|e
+x2
+2 x3
+�
++ 4τ|k|ex2 �
+3τ|k|2+x2 + 3
+�
++
+√
+2πe
+x2
+2 �
+−
+�
+(τ|k|)2 �
+x4 + 4x2 + 11
+��
++ x2 − 5
+�
++ 4πτ|k|
+�
+x2 + 1
+��
+= 0,
+(4.61)
+for x ∈ R. The zero sets of equations (4.61) are shown in Figure 4.7.
+Solving system (4.61) numerically gives the following approximation for the critical wave
+number of the acoustic mode:
+kcrit(λac) = kcrit(λ∗
+ac) ≈ 1.311761
+τ .
+(4.62)
+Remark 4.5. The critical wave numbers obtained before depend inversely on the (non-
+dimensional) relaxation parameter. Transforming back to physical units, we see that the
+
+Im(>)
+1500
+1000
+500
+Re(^)
+-1000
+800
+600
+400
+-200
+500
+-1000
+-1500EXACT HYDRODYNAMICS FROM LINEAR BGK
+25
+Figure 4.7. The zero sets of equations (4.61).
+The intersection of the
+solid line (ℜΓτ|k|(x)) with the dashed line (ℑΓτ|k|(x)) gives the critical wave
+numbers for the acoustic modes (and the diffusion mode on the real line as
+well).
+critical wave number is numerically proportional to the inverse mean-free path (3.32).
+Indeed, we obtain that
+kcrit ∼
+�
+kBT0
+m
+1
+τphys
+∼
+1
+lmfp
+.
+(4.63)
+5. Linear Hydrodynamic Manifolds
+In this section, we give a description of the hydrodynamic manifolds together with
+their respective dynamics. We define the hydrodynamic manifold through the following
+properties:
+(1) It contains an appropriately scaled, spatially independent stationary distribution
+(e.g. global Maxwellian) as a base solution
+(2) The projection onto the hydrodynamic moments along the manifold provide a clo-
+sure of the hydrodynamic moments (mass-density, velocity and temperature)
+(3) It attracts all trajectories in the space of probability-density functions (which are
+close enough to the base solution) exponentially fast, thus acting as a slow manifold
+(4) It is unique.
+From the explicit analysis in the previous section, we know that the addition of P5 only
+adds finitely many discrete eigenvalues to the spectrum. Let us take a closer look at their
+
+.5
+0
+0.5
+0026
+FLORIAN KOGELBAUER AND ILYA KARLIN
+associated eigenvectors.
+For each wave number k ∈ Z3, the eigenvectors associated to
+λN(τ|k|), N ∈ Modes, where Modes = {diff, shear, ac, ac∗}, satisfy the equation
+− iv · k ˆfeig
+N,j − 1
+τ
+ˆfeig
+N,j + 1
+τ P5 ˆfeig
+N,j = λN(τ|k|) ˆfeig
+N,j,
+(5.1)
+where 1 ≤ j ≤ µN(|k|) denotes the geometric multiplicity of λN(τ|k|). Defining
+αN,j,l(k) := ⟨ ˆfeig
+N,j,k(v, k), el(v)⟩v,
+(5.2)
+we can rewrite (5.1) as
+ˆfeig
+N,j(v, k) =
+1
+iτk · v + 1 + τλN(τ|k|)
+4
+�
+l=0
+αN,j,l(k)el(v).
+(5.3)
+To omit cluttering in the notation, we will suppressed the dependence of αN,j,l on k. Taking
+an inner product with ep(v) in (5.3) gives
+αN,j,p =
+4
+�
+l=0
+αN,j,l
+�
+R3 el(v)ep(v)
+e− |v|2
+2
+iτk · v + 1 + τλN(τ|k|) dv,
+(5.4)
+which is equivalent to the non-invertibilty of the matrix (Id − GS) in equation (4.29) and
+(4.30) for z = −1 − τλN(τ|k|). Indeed, denoting αN,j = (αN,j,0, .., αN,j,4), it follows that
+αN,j ∈ ker((Id − GS))|z=−1−τλN(τ|k|).
+(5.5)
+This defines the eigenvector (5.3) for each wave number k and each mode N completely.
+To obtain the closure relation for the linearized hydrodynamic variables (nlin, ulin, Tlin),
+we define a solution to the linearized dynamics (4.4) as
+fhydro(x, v, t) =
+�
+|k|≤kcrit
+�
+N∈Modes
+µN(k)
+�
+j=1
+ˆfeig
+N,j(v, k)eλN(τ|k|)t+ik·x,
+(5.6)
+where we set
+ˆfeig
+N,j,k = 0,
+if |k|> kcrit(λN),
+(5.7)
+and µN(k).
+Following (3.33), let
+Θ := (2π)
+3
+4 v3
+thermal
+n0
+�
+�
+n0
+01×3
+0
+03×1
+vthermalI3×3
+03×1
+−T0
+01×3
+T0
+3
+�
+� ,
+(5.8)
+denote the matrix that realized the linear coordinate change
+(nlin, ulin, Tlin)T = Θe.
+(5.9)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+27
+On the hydrodynamic manifold defined by (5.7), the variables (nlin, ulin, Tlin) evolve ac-
+cording to an explicit (non-local) system. Indeed, in frequency space, denoting the Fourier
+coefficients of (nlin, ulin, Tlin) as (ˆnlin, ˆulin, ˆTlin) we find that
+(ˆnlin, ˆulin, ˆTlin) = Θdiag(⟨ ˆfhydro, e⟩)e,
+(5.10)
+which, setting αN = �µN
+j=1 αN,j and defining
+α := [αshear, αdiff, αac, αac∗],
+(5.11)
+as well as Λ = diag(λshear, λdiff, λac, λac∗), λ = etΛ, can be written more explicitly as
+(ˆnlin, ˆulin, ˆTlin) = Θdiag(αλ)e.
+(5.12)
+We can invert for e,
+e = (Θdiag(αλ))−1(ˆnlin, ˆulin, ˆTlin),
+(5.13)
+and, finally, taking a time derivative, we arrive at
+∂
+∂t(ˆnlin, ˆulin, ˆTlin) = Θdiag(αΛλ)(Θdiag(αλ))−1(ˆnlin, ˆulin, ˆTlin).
+(5.14)
+This defines a (non-local) closure to the linearized dynamics (3.1).
+Since - up to the
+conserved quantities (3.15) - any solution approaches the slow dynamics given by (5.7)
+exponentially fast in time, the closure (5.14) defines the unique, global, hydrodynamic
+limit of (3.1).
+6. Conclusion and Further Perspectives
+We have given a complete and (up to the solution of a transcendental equation) explicit
+description of the spectrum of the three-dimensional BGK equation linearized around a
+global Maxwellian. Further, we identified (and therefore confirmed) the existence of three
+families of modes (shear, diffusion and acoustic) and we gave a description of critical wave
+numbers. The analysis allowed us to infer that the discrete spectrum consists of a finite
+number of eigenvalues, thus implying that the dispersion relation remains bounded also for
+the acoustic modes.
+Let us give an outlook on some future lines of research in this context. We expect that
+the results obtained in this paper are explicit enough to carry out a comparison of viscous
+dissipation versus capillarity as carried out in [37] for the three-dimensional Grad system.
+Furthermore, the explicit knowledge of the spectral function (4.43) allows us to infer more
+refined approximations to the exact non-local hydrodynamics. This will involve expansions
+not in terms of relaxation time or wave number, but much rather in terms of the variable
+ζ in (4.43). This could also improve present numerical methods [27].
+Finally, the spectral properties of the linear three-dimensional BGK equation will also serve
+as the basis for nonlinear analysis in terms of invariant manifolds. Indeed, the fact that the
+discrete spectrum is well separated from the essential spectrum allows us to define a spectral
+projection for the whole set of eigenvalues, thus giving the first-order approximation (in
+terms of nonlinear deformations) to the hydrodynamic manifolds. In particular, we expect
+
+28
+FLORIAN KOGELBAUER AND ILYA KARLIN
+that the theory of thermodynamic projectors [20] may be helpful in proving the nonlinear
+extension.
+Acknowledgement
+This work was supported by European Research Council (ERC) Advanced Grant no.
+834763-PonD (F.K. and I.K.).
+Data Availability Statement
+All data generated or analysed during this study are included in this published article
+(and its supplementary information files).
+Appendix A. Some Properties of the Plasma Dispersion Function
+In the following, we collect some properties of the integral expression (4.39). In partic-
+ular, to evaluate the integral in (4.39) in terms of error functions, we rely on the identities
+in [1, p.297]. Let
+w(z) = e−z2(1 − erf(−iz)),
+z ∈ C,
+(A.1)
+which satisfies the functional identity
+w(−z) = 2e−z2 − w(z),
+z ∈ C.
+(A.2)
+Function (A.1) is called Faddeeva function and is frequently encountered in problems re-
+lated to kinetic equations [17]. We then have that
+w(z) = i
+π
+�
+R
+e−s2
+z − s ds,
+ℑz > 0,
+(A.3)
+and, by relation (A.2), we have for ℑz < 0:
+i
+π
+�
+R
+e−s2
+z − s ds = − i
+π
+�
+R
+e−s2
+(−z) + s ds
+= − i
+π
+�
+R
+e−s2
+(−z) − s ds
+= −w(−z)
+= e−z2[−1 − erf(−iz)].
+(A.4)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+29
+(a) Argument Plot of Z
+(b) Modulus-Argument Plot of Z
+Figure A.1. Complex plots of the function Z.
+Consequently, we obtain
+�
+R
+1
+s − z e− s2
+2 ds =
+�
+R
+e−s2
+s −
+z
+√
+2
+ds
+= iπ i
+π
+�
+R
+e−s2
+z
+√
+2 − s ds
+=
+�
+�
+�
+iπe− z2
+2
+�
+1 − erf
+�
+−iz
+√
+2
+��
+,
+if ℑz > 0,
+iπe− z2
+2
+�
+−1 − erf
+�
+−iz
+√
+2
+��
+,
+if ℑz < 0,
+(A.5)
+where in the first step, we have re-scaled s �→
+√
+2s in the integral. Written more compactly,
+we arrive at
+Z(z) = i
+�π
+2 e− z2
+2
+�
+sign(ℑz) − erf
+�−iz
+√
+2
+��
+,
+ℑz ̸= 0.
+(A.6)
+An an argument plot together with an modulus-argument plot of Z are shown in Figure
+A.1.
+Clearly, Z is discontinuous across the real line (albeit that Z|R exists in the sense
+of principal values as the Hilbert transform of a real Gaussian [13]). The properties
+|Z(z)|≤
+�π
+2 , for z ∈ C \ R,
+0 < arg Z(z) < π for ℑ(z) > 0,
+−π < arg Z(z) < 0 for ℑ(z) < 0,
+(A.7)
+are easy to show and can be read off from the plots (A.1) directly as well.
+Function (A.6) satisfies an ordinary differential equation (in the sense of complex analytic
+
+4
+100=
+T
+2
+元/2
+10=
+0
+-
+-元/2
+-2
+0.1 =
+-元
+-4
+2
+0
+2
+4Im()
+0
+100
+1.
+元/2
+-5/
+-0
+1.0
+Zo()
+一元2
+0.5
+0.1#
+5
+0
+Re()
+530
+FLORIAN KOGELBAUER AND ILYA KARLIN
+functions) on the upper and on the lower half-plane. Indeed, integrating (4.39) by parts
+gives
+1 =
+1
+√
+2π
+�
+R
+(v − z) e− v2
+2
+v − z dv = −zZ +
+1
+√
+2π
+�
+R
+v e− v2
+2
+v − z dv
+= −zZ −
+1
+√
+2π
+�
+R
+e− v2
+2
+(v − z)2 dv = −zZ − d
+dz Z,
+(A.8)
+which implies that Z satisfies the differential equation
+d
+dz Z = −zZ − 1,
+(A.9)
+for z ∈ C \ R. Formula (A.9) can also be used as a recurrence relation for the higher
+derivatives of Z.
+Since we will be interested in function (A.6) for ℑz positive and negative as global functions,
+we define
+Z+(z) = i
+�π
+2 e− z2
+2
+�
+1 − erf
+�−iz
+√
+2
+��
+,
+Z−(z) = i
+�π
+2 e− z2
+2
+�
+−1 − erf
+�−iz
+√
+2
+��
+,
+(A.10)
+for all z ∈ C. Both functions can be extended to analytic functions on the whole complex
+plane via analytic continuation.
+Recall that the error function has the properties that
+erf(−z) = −erf(z),
+erf(z∗) = erf(z)∗,
+(A.11)
+for all z ∈ C, which implies that for x ∈ R,
+erf(ix) = −erf(−ix) = −erf(ix)∗,
+(A.12)
+i.e, the error function maps imaginary numbers to imaginary numbers. Defining the imag-
+inary error function,
+erfi(z) := −ierf(iz),
+(A.13)
+for z ∈ C, which, by (A.12) satisfies erfi|R⊂ R, it follows that for x ∈ R:
+ℜZ+(x) = −
+�π
+2 e− x2
+2 erfi
+� x
+√
+2
+�
+,
+ℑZ+(x) = −
+�π
+2 e− x2
+2 ,
+(A.14)
+similarly for Z−(x).
+Next, let us prove the following asymptotic expansion of Z+:
+Z+(z) ∼ −
+∞
+�
+n=0
+(2n − 1)! !
+z2n+1
+,
+for |arg(z)|≤ π
+2 − δ,
+z → ∞,
+(A.15)
+
+EXACT HYDRODYNAMICS FROM LINEAR BGK
+31
+for any 0 < δ ≤ π
+2 . The proof will be based on a generalized version of Watson’s Lemma
+[41]. To this end, let us define the Laplace transform
+L[f](z) =
+� ∞
+0
+f(x)e−zx dx,
+z ∈ C,
+(A.16)
+of an integrable function f : [0, ∞) → C.
+Lemma A.1. [Generalized Watson’s Lemma] Assume that (A.16) exists for some z = z0 ∈ C
+and assume that f admits an asymptotic expansion of the form
+f(x) =
+N
+�
+n=0
+anxβn−1 + o(xβN−1),
+x > 0,
+x → 0,
+(A.17)
+where an ∈ C and βn ∈ C with ℜβ0 > 0 and ℜβn > ℜβn−1 for 1 ≤ n ≤ N. Then L[f](z)
+admits an asymptotic expansion of the form
+L[f](z) =
+N
+�
+n=0
+anΓ(βn)z−βn + o(z−βN ),
+v,
+z → ∞,
+(A.18)
+for any real number 0 < δ ≤ π
+2 , where Γ is the standard Gamma function.
+For a proof of the above Lemma, we refer e.g. to [16]. Classically, Lemma (A.1) is
+applied to prove that the imaginary error function admits an asymptotic expansion for
+x ∈ R of the form
+erfi(x) ∼ ex2
+√πx
+∞
+�
+k=0
+(2k − 1)! !
+(2x2)k
+,
+for x > 0,
+x → ∞,
+(A.19)
+see also [31], based on the classical version of Watson’s Lemma, whose assumptions are,
+however, unnecessarily restrictive [43].
+For completeness, we recall the derivation of (A.15) based on Lemma A.1. First, let us
+rewrite erfi as a Laplace transform using the change of variables t = √1 − s with dt =
+ds
+2√1−s
+erfi(z) =
+� 1
+0
+d
+dterfi(tz) dt = 2z
+√π
+� 1
+0
+et2z2 dt = 2z
+√π
+� 1
+0
+ez2(1−s)
+ds
+2√1 − s
+= zez2
+√π
+� 1
+0
+1
+√1 − se−sz2 ds = zez2
+√π
+� ∞
+0
+χ[0,1](s)
+√1 − s e−sz2 ds.
+(A.20)
+From the Taylor expansion of the Binomial function, we know that
+1
+√1 − s =
+∞
+�
+n=0
+�− 1
+2
+n
+�
+(−s)n =
+∞
+�
+n=0
+4−n
+�2n
+n
+�
+sn,
+(A.21)
+
+32
+FLORIAN KOGELBAUER AND ILYA KARLIN
+which allows us to apply Lemma (A.1) with βn = n + 1 and an = 4−n�2n
+n
+�
+, thus leading to
+erfi(z) ∼ zez2
+√π
+∞
+�
+n=0
+4−n
+�2n
+n
+�
+Γ(n + 1)z−2(n+1)
+∼ ez2
+√π
+∞
+�
+n=0
+(2n)!
+4nn! z−2n−1
+∼ ez2
+z√π
+∞
+�
+n=0
+(2n − 1)! !
+(2z)n
+,
+(A.22)
+for z → ∞ and |arg(z)|≤ π
+2 − δ, 0 < δ ≤ π
+2 . This is consistent with formula (A.19) for the
+limit along the real line. Finally, we arrive at the following asymptotic expansion for Z:
+Z+(z) ∼ i
+�π
+2 e− z2
+2 −
+∞
+�
+n=0
+(2n − 1)! !
+z2n+1
+,
+for |arg(z)|≤ π
+2 − δ,
+z → ∞,
+(A.23)
+which is, of course, equivalent to
+Z+(z) ∼ −
+∞
+�
+n=0
+(2n − 1)! !
+z2n+1
+,
+for |arg(z)|≤ π
+2 − δ,
+z → ∞,
+(A.24)
+since |e−z2|2= e−2(x2−y2) → 0 for ℜz = x → ∞.
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+ETH Z¨urich, Department of Mechanical and Process Engineering, Leonhardstrasse 27,
+8092 Z¨urich, Switzerland
+Email address: floriank@ethz.ch
+ETH Z¨urich, Department of Mechanical and Process Engineering, Leonhardstrasse 27,
+8092 Z¨urich, Switzerland
+Email address: ikarlin@ethz.ch
+

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+Solving Constrained Reinforcement Learning through Augmented State and
+Reward Penalties
+Hao Jiang 1 Tien Mai 1 Pradeep Varakantham 1
+Abstract
+Constrained Reinforcement Learning has been
+employed to enforce safety constraints on policy
+through the use of expected cost constraints. The
+key challenge is in handling expected cost accu-
+mulated using the policy and not just in a single
+step. Existing methods have developed innova-
+tive ways of converting this cost constraint over
+entire policy to constraints over local decisions
+(at each time step). While such approaches have
+provided good solutions with regards to objective,
+they can either be overly aggressive or conserva-
+tive with respect to costs. This is owing to use
+of estimates for ”future” or ”backward” costs in
+local cost constraints.
+To that end, we provide an equivalent uncon-
+strained formulation to constrained RL that has
+an augmented state space and reward penalties.
+This intuitive formulation is general and has in-
+teresting theoretical properties. More importantly,
+this provides a new paradigm for solving con-
+strained RL problems effectively. As we show in
+our experimental results, we are able to outper-
+form leading approaches on multiple benchmark
+problems from literature.
+1. Introduction
+There are multiple objectives of interest when handling
+safety depending on the type of domain: (a) ensuring safety
+constraint is never violated; (b) ensuring safety constraint is
+not violated in expectation; (c) ensuring the chance of safety
+constraint violation is small (Value at Risk, VaR) (Lucas &
+Klaassen, 1998); (d) ensuring the expected cost of violation
+is bounded (Conditional Value at Risk, CVaR) (Rockafellar
+et al., 2000; Yang et al., 2021); and others. One of the
+main models in Reinforcement Learning to ensure safety is
+Constrained RL, which employs objective (b) above. Our
+focus in this paper is also on Constrained RL.
+1School of Computing and Information Systems, Singapore
+Management University.
+Preprint
+Constrained RL problems are of relevance in domains that
+can be represented using an underlying Constrained Markov
+Decision Problem (CMDP) (Altman, 1999). The main chal-
+lenge in solving Constrained RL problems is the expected
+cost constraint, which requires averaging over multiple tra-
+jectories from the policy. Such problems have many appli-
+cations including but not limited to: (a) electric self driving
+cars reaching destination at the earliest while minimizing
+the risk of not getting stranded on the road with no charge;
+(b) robots moving through unknown terrains to reach a des-
+tination, while having a threshold on the average risk of
+passing through unsafe areas (e.g., a ditch). Broadly, they
+are also applicable to problems robot motion planning (Ono
+et al., 2015; Moldovan & Abbeel, 2012; Chow et al., 2015a),
+resource allocation (Mastronarde & van der Schaar, 2010;
+Junges et al., 2015), and financial engineering (Abe et al.,
+2010; Di Castro et al., 2012).
+Related Work: Many model free approaches have been pro-
+posed to solve Constrained RL problems. One of the initial
+approaches to be developed for addressing such constraints
+is the Lagrangian method (Chow et al., 2015b). However,
+such an approach does not provide either theoretical or em-
+pirical guarantees in ensuring the constraints are enforced.
+To counter the issue of safety guarantees, next set of ap-
+proaches focused on transforming the cost constraint over
+trajectories into cost constraint over individual decisions in
+many different ways. One such approach imposed surrogate
+constraints (El Chamie et al., 2016; G´abor et al., 1998) on
+individual state and action pairs. Since the surrogate con-
+straints are typically stricter than the original constraint on
+the entire trajectory, they were able to provide theoretical
+guarantees on safety. However, the issue with such type of
+approaches is their conservative nature, which can poten-
+tially hamper the expected reward objective. More recent
+approaches such as CPO (Constrained Policy Optimiza-
+tion) (Achiam et al., 2017), Lyapunov (Chow et al., 2019b),
+BVF (Satija et al., 2020a) have since provided more tighter
+local constraints (over individual decisions) and thereby
+have improved the state of art in guaranteeing safety while
+providing high quality solutions (with regards to expected
+reward). In converting a trajectory based constraint to a
+local constraint, there is an estimation of cost involved for
+the rest of the trajectory. Due to such estimation, trans-
+arXiv:2301.11592v1  [cs.LG]  27 Jan 2023
+
+Solving Constrained RL through Augmented State and Reward Penalties
+formed cost constraints over individual decisions are error
+prone. In problems where the estimation is not close to the
+actual, results with such approaches with regards to cost
+constraint enforcement are poor (as we demonstrate in our
+experimental results).
+Contributions:
+To that end, we focus on an approach that relies on exact
+accumulated costs (and not on estimated costs). In this
+paper, we make four key contributions:
+• We provide a re-formulation of the constrained RL prob-
+lem through augmenting the state space with cost ac-
+cumulated so far and also considering reward penalties
+when cost constraint is violated. This builds on the idea
+of augmented MDPs (Hou et al., 2014) employed to
+solve Risk Sensitive MDPs. The key advantage of this
+reformulation is that by penalizing rewards (as opposed
+to the entire expected value that is done typically using
+Lagrangian methods), we get more fine grained control
+on how to handle the constraints. Also, we can utilize
+existing RL methods with minor modifications.
+• We show theoretically that the reward penalties em-
+ployed in the new formulation are not adhoc and can
+equivalently represent different constraints mentioned
+in the first paragraph of introduction, i.e. risk-neural,
+chance constrained (or VAR) and CVaR constraints.
+• We modify existing RL methods (DQN and SAC) to
+solve the re-formulated RL problem with augmented
+state space and reward penalties. A key advantage for
+the new approaches is the knowledge of exact costs
+incurred so far (available within the state space) and this
+allows for assigning credit for cost constraint violations
+more precisely during learning.
+• Finally, we demonstrate the utility of our approach by
+comparing against leading approaches for constrained
+RL on multiple benchmark problems from literature.
+We empirically demonstrate that our approaches are able
+to outperform leading Constrained RL approaches from
+the literature either with respect to expected value or in
+enforcing the cost constraint or both.
+2. Constrained Markov Decision Process
+A Constrained Markov Decision Process (CMDP) (Altman,
+1999) is defined using tuple ⟨S, A, r, p, d, s0, cmax⟩, where
+S is set of states with initial state as s0, A is set of actions,
+r : S × A → R is reward with respect to each state-action
+pair, p : S × A → P is transition probability of each state.
+d : S → d(S) is the cost function and cmax is the maximum
+allowed cumulative cost. Here, we assume that d(s) ≥ 0
+for all s ∈ S. This assumption is not restrictive as one
+can always add positive amounts to d(s) and cmax to meet
+the assumption. The objective in a risk-neural CMDP is to
+compute a policy, π : S × A → [0, 1], which maximizes
+reward over a finite horizon T while ensuring the cumulative
+cost does not exceed the maximum allowed cumulative cost.
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+E
+� T
+�
+t=0
+d(st)|s0, π
+�
+≤ cmax.
+(RN-CMDP)
+The literature has seen other types of constraints, e.g.,
+chance constraints requiring that Pπ(D(τ) > cmax) ≤ α
+for a risk level α ∈ [0, 1], or CVaR ones of the form
+Eπ[(D(τ) − cmax)+] ≤ β. Handling different types of
+constraints would require different techniques. In the next
+section, we present our approach based on augmented state
+and reward penalties that assembles all the aforementioned
+constraint types into one single framework.
+3. Cost Augmented Formulation for Safe RL
+We first present our extended MDP reformulation and pro-
+vide several theoretical findings that connect our extended
+formula with different variants of CMDP. We first focus
+on the case of single-constrained MDP and show how the
+results can be extended to the multi-constrained setting.
+3.1. Extended MDP Reformulation
+We introduce our approach to track the accumulated cost
+at each time period, which allows us to determine states
+that potentially lead to high-cost trajectories. To this end,
+let us define a new MDP with an extended state space
+�
+�S, A, �r, �p, d, s0, cmax
+�
+where �S = {(s, c)| s ∈ S, c ∈
+R+}. That is, each state s′ of the extended MDP includes
+an original state from S and information about the accumu-
+lated cost. We the define the transition probabilities between
+states in the extended space.
+�
+�
+�
+�
+�
+�p(s′
+t+1, c′
+t+1|(st, ct), at) = p(s′
+t+1|st, at)
+if c′
+t+1 = ct + d(st)
+�p(s′
+t+1, c′
+t+1|(st, ct), at) = 0 otherwise
+and new rewards with penalties
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�r(at|(st, ct)) = r(at|st) if ct + d(st) ≤ cmax
+�r(at|(st, ct)) = r(at|st) − ∆(ct + d(st))/γt
+if ct ≤ cmax and ct + d(st) > cmax
+�r(at|(st, ct)) = r(at|st) − ∆d(st)/γt if ct > cmax
+where ∆ is a positive scalar and ∆d(st) and ∆(ct +
+d(st)) are penalties given to the agent if the accumu-
+lated cost exceeds the upper bound cmax. Under these
+
+Solving Constrained RL through Augmented State and Reward Penalties
+reward penalties, the accumulated reward for each tra-
+jectory τ = {(s0, a0), . . . , (sT , aT )} can be written as
+�R(τ) = �
+t γtr(at|st) if D(τ) ≤ cmax and �R(τ) =
+�
+t γtr(at|st) − ∆D(τ) if D(τ) > cmax, where D(τ) is
+the total cost of trajectory τ, i.e., D(τ) = �
+st∈τ d(st). So,
+in fact, we penalize every trajectory that violates the cost
+constraint.
+We now consider the following extended MDP, which han-
+dles the constraints in a relaxed manner through penalties.
+max
+π
+E
+� T
+�
+t=0
+γt�r(at|(st, ct))
+���(s0, c0), π
+�
+(EMDP)
+where c0 = 0. There are also other ways to penalize the
+rewards, allowing us to establish equivalences between the
+extended MDP to other risk-averse CMDP, which we will
+discuss later in the next section.
+3.2. Theoretical Properties
+To demonstrate the generality and power in representation of
+the reward penalties along with state augmentation in the un-
+constrained MDP (EMDP), we provide theoretical properties
+that map reward penalties to different types of constraints
+(expected cost, VaR, CVaR, Worst-case cost):
+(i) Proposition 3.1 states that if the penalty parameter ∆ =
+0, then (EMDP) becomes the classical unconstrained
+MDP.
+(ii) Theorem 3.2 shows that if ∆ = ∞, then (EMDP) is
+equivalent to a worst-case constrained MDP
+(iii) Theorem 3.5 establishes a lower bound on ∆ from
+which any solution to (EMDP) will satisfy the risk-
+neural constraint in (RN-CMDP).
+(iv) Theorem
+3.6
+connects
+(EMDP)
+with
+chance-
+constrained MDP by providing a lower bound for ∆
+from which any solution to (EMDP) will satisfy a VaR
+constraint P(�
+t d(st) ≤ cmax) ≤ α.
+(v) Theorems 3.6 and 3.8 further strengthen the above re-
+sults by showing that, under some different reward set-
+tings, (EMDP) is equivalent to a chance-constrained (or
+VaR) or equivalent to a CVaR CMDP.
+We now describe our theoretical results in detail. All the
+proofs can be found in the appendix. We first state, in Propo-
+sition 3.1, a quite obvious result saying that if we set the
+penalty parameter ∆ = 0, then the MDP with augmented
+state space becomes the original unconstrained MDP.
+Proposition 3.1. If ∆ = 0, then (EMDP) is equivalent to
+the unconstrained MDP maxπ E
+��T
+t=0 γtr(st, at)|s0, π
+�
+.
+It can be seen that increasing ∆ will set more penalties to
+trajectories whose costs exceed the maximum cost allowed
+cmax, which also implies that (EMDP) would lower the prob-
+abilities of taking these trajectories. So, intuitively, if we
+raise ∆ to infinity, then (EMDP) will give policies that yield
+zero probabilities to violating trajectories. We state this
+result in Theorem 3.2 below.
+Theorem 3.2 (Connection to worst-case CMDP). If we
+set ∆ = ∞, then if π∗ solves (EMDP), it also solves the
+following worst-case constrained MDP problem
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+�
+st∈τ
+d(st) ≤ cmax, ∀τ ∼ π.
+(WC-CMDP)
+As a result, π∗ is feasible to the risk-neural CMDP
+(RN-CMDP).
+The above theorem implies that if we set the penalties to
+be very large (e.g., ∞), then all the trajectories generated
+by the optimal policy π∗ will satisfy the constraint, i.e., the
+accumulated cost will not exceed cmax. Such a conservative
+policy would be useful in critical environments where the
+agent is strictly not allowed to go beyond the maximum
+allowed cost cmax. An example would be a routing problem
+for electrical cars where the remaining energy needs not
+become empty before reaching a charging station or the
+destination. Note that the worst-case CMDP (WC-CMDP)
+would be non-stationary and history-dependent, i.e., there
+would be no stationary and history-independent policies
+being optimal for the worst-case CMDP (WC-CMDP). This
+remark is obviously seen, as at a stage, one needs to consider
+the current accumulated cost to make feasible actions. Thus,
+a policy that ignores the historical states and actions would
+be not optimal (or even not feasible) for the worst-case MDP.
+As a result, this worst-case CMDP can not be presented by
+a standard-constrained MDP formulation.
+Theorem 3.2 also tells us that one can get a feasible solution
+to the risk-neural CMDP (RN-CMDP) by just raising ∆ to
+infinity. In fact, ∆ does not need to be infinite to achieve
+feasibility. Below we establish a lower bound for the penalty
+parameter ∆ such that a solution to (EMDP) is always feasi-
+ble to the risk-neural CMDP (RN-CMDP). Let us define Ψ∗
+as the optimal value of the unconstrained MDP problem
+Ψ∗ = max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+.
+and Ψ be the optimal value of the worst-case CMDP
+(WC-CMDP).
+We define a conditional expectation
+�Eπ [D(τ)| D(τ) ≤ cmax] as the expected cost over trajecto-
+ries whose costs are less than cmax
+�Eπ [D(τ)| D(τ) ≤ cmax] =
+�
+τ| D(τ)≤cmax
+Pπ(τ)D(τ)
+
+Solving Constrained RL through Augmented State and Reward Penalties
+where Pπ(τ) is the probability of τ under policy π. Before
+presenting the bound, we first need two lemmas. Lemma 3.3
+establishes a condition under which a policy π is feasible to
+the risk-neural CMDP.
+Lemma
+3.3.
+Let
+φ∗
+=
+cmax
+−
+maxπ
+�
+�Eπ[D(τ)| D(τ) ≤ cmax]
+�
+. Given any policy π, if
+�Eπ[D(τ)| D(τ) > cmax] ≤ φ∗, then Eπ[D(τ)] ≤ cmax.
+Lemma 3.4 below further provides an upper bound for the
+expected cost of violating trajectories under an optimal pol-
+icy given by the extended MDP reformulation (EMDP).
+Lemma 3.4. Given ∆ > 0, let π∗ be an optimal solution
+to (EMDP). We have
+�Eπ∗ [D(τ)| D(τ) > cmax] ≤ Ψ∗ − Ψ
+∆
+.
+Using Lemmas 3.3 and 3.4, we are ready to state the main
+result in Theorem 3.5 below.
+Theorem 3.5 (Connection to the risk-neural CMDP). For
+any ∆ ≥ Ψ∗−Ψ
+φ∗
+, a solution to (EMDP) is always feasible to
+the risk-neural CMDP (RN-CMDP).
+To prove Lemmas 3.3, 3.4, we leverage the fact that the
+objective of (EMDP) can be written equivalently as
+Eπ
+��
+t
+γtr(st, at)
+�
+− ∆�Eπ [D(τ)| D(τ) > cmax]
+(1)
+which allows us to establish a relation between ∆ and
+�Eπ∗ [D(τ)| D(τ) > cmax], where π∗ is an optimal policy
+of (EMDP). The bounds then come from this relation. We
+refer the reader to the appendix for detailed proofs.
+There is also a lower bound for ∆ from which any solution
+to (EMDP) always satisfies a chance constraint (or VaR). To
+state this result, let us first define the following VaR CMDP,
+for any risk level α ∈ [0, 1].
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+Pπ
+�
+(D(τ) > cmax
+�
+≤ α.
+(VaR-CMDP)
+We have the following theorem showing a connection be-
+tween (EMDP) and the VaR CMDP above.
+Theorem 3.6 (Connection to VaR CMDP). For any ∆ ≥
+(Ψ∗ − Ψ)/(αcmax), a solution to (EMDP) is always feasi-
+ble to (VaR-CMDP).
+We also leverage Eq. 1 to prove the theorem by show-
+ing that when ∆ is sufficiently large, the conditional ex-
+pectation �Eπ∗ [D(τ)| D(τ) > cmax] can be bounded from
+above (π∗ is an optimal policy of (EMDP)).
+We then
+can link this to the chance constraint by noting that
+�Eπ∗ [D(τ)| D(τ) > cmax] ≥ cmaxP(D(τ) > cmax).
+Theorem 3.6 tells us that one can just raise ∆ to a suffi-
+ciently large value to meet a chance constraint of any risk
+level. Here, Theorem 3.6 only guarantees feasibility to
+(VaR-CMDP). Interestingly, if we modify the reward penal-
+ties by making them independent of the costs d(s), than
+an equivalent mapping to (VaR-CMDP) can be obtained.
+Specifically, let us re-define the following reward for the
+extended MDP. That is, we replace the cost d(st) by a con-
+stant. Theorem 3.7 below shows that (EMDP) is actually
+equivalent to a chance-constrained CMDP under the new
+reward setting.
+Theorem 3.7 (VaR equivalence). If we modify the reward
+penalties as
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�r(at|(st, ct)) = r(at|st) if ct + d(st) ≤ cmax
+�r(at|(st, ct)) = r(at|st) − ∆(t + 1)/γt
+if ct ≤ cmax and ct + d(st) > cmax
+�r(at|(st, ct)) = r(at|st) − ∆/γt if ct > cmax
+then if π∗ is an optimal solution to (EMDP), then there is
+α∆ ∈ [0; Ψ∗−Ψ
+∆T ] (α is dependent of ∆) such that π∗ is also
+optimal to (VaR-CMDP). Moreover lim∆→∞ α∆ = 0.
+It can be also seen that Theorem 3.2 is a special case of
+Theorem 3.7 when ∆ = ∞.
+We finally connect (EMDP) with a risk-averse CMDP that
+has a CVaR intuition. The theorem below shows that, by
+slightly changing the reward penalties, (EMDP) actually
+solves a risk-averse CMDP problem.
+Theorem 3.8 (CVaR CMDP equivalence). If we modify the
+reward penalties as
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�r(at|(st, ct)) = r(at|st) if ct + d(st) ≤ cmax
+�r(at|(st, ct)) = r(at|st) − ∆(ct + d(st) − cmax)/γt
+if ct ≤ cmax and ct + d(st) > cmax
+�r(at|(st, ct)) = r(at|st) − ∆d(st)/γt if ct > cmax
+then for any ∆ > 0, there is β∆ ∈
+�
+0; Ψ∗−Ψ
+∆
+�
+(β∆ is de-
+pendent of ∆) such that any optimal solution to the extended
+CMDP (EMDP) is also optimal to the following risk-averse
+CMDP
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+Eτ∼π
+�
+(D(τ) − cmax)+�
+≤ β∆.
+(CVaR-CMDP)
+Moreover, lim∆→∞ β∆ = 0.
+
+Solving Constrained RL through Augmented State and Reward Penalties
+In practice, since ∆ is just a scalar, one can just grad-
+ually increase it from 0 to get a desired policy.
+This
+indicates the generality of the unconstrained exended
+MDP formulation (EMDP). In summary, we show that
+(EMDP) brings risk-neural, worst-case and VaR and CVaR
+CMDPs in (RN-CMDP), (WC-CMDP), (VaR-CMDP) and
+(CVaR-CMDP) under one umbrella.
+3.3. Multi-constrained CMDP
+We now discuss extension to CMDP with multiple cost
+constraints (e.g., limited fuel and bounded risk) and show
+how the above theoretical results can be extended to the
+multi-constrained variants. A multi-constrained risk-neural
+CMDP can be formulated as
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+E
+� T
+�
+t=0
+dk(st)|s0, π
+�
+≤ ck
+max, ∀k ∈ [K]
+(MRN-CMDP)
+where [K] denotes the set {1, . . . , K}. Similar to the single
+constraint case, to include cost functions in the rewards, we
+extend the state space to keep track of the accumulated costs
+as �S = {(s, c1, . . . , cK)| s ∈ S, ck ∈ R, ∀k ∈ [K]} and
+define new transitions probabilities as
+�
+�
+�
+�
+�
+�p(st+1, cK
+t+1|(st, cK
+t ), at) = p(st+1|st, at)
+if ck
+t+1 = ck
+t + dk(st)
+�p(st+1, cK
+t+1|(st, cK
+t ), at) = 0 otherwise
+where cK
+t
+= (c1
+t, . . . , cK) for notational simplicity. The
+new rewards are also updated in such a way that every
+trajectory violating the constraints will be penalized.
+�r(at|(st, cK
+t )) = r(at|st) −
+�
+k∈[K]
+∆kδk(ct),
+where δk(ct), ∀k ∈ [K], are defined as follows.
+δk(ct) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0 if , ck
+t + dk(st) ≤ ck
+max
+(ck
+t + dk(st))/γt if ck
+t ≤ ck
+max,
+ck
+t + dk(st) ≥ ck
+max
+dk(st)/γt if ck
+t > ck
+max.
+Here, we allow penalty parameters ∆k to be different over
+constraints. We formulate the extended unconstrained MDP
+as:
+max
+π
+�
+E
+� T
+�
+t=0
+γt�r(at|(st, cK
+t ))
+���(s0, cK
+0 ), π
+��
+.
+(2)
+Similar to the single-constrained case, the reward penalties
+allow us to write the objective function of the extended
+MDP as
+Eπ
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆k�Eπ
+�
+Dk(τ)| Dk(τ) > ck
+max
+�
+(3)
+where Dk(τ) is the accumulated cost dk(st) on trajectory τ,
+i.e., Dk(τ) = �
+st∈τ dk(st). As a result, when ∆k grows,
+the extended MDP will discount the second term of (3), thus
+yielding policies that satisfy or even solve risk-neural or
+risk-averse CMDP problems. Specifically, the following
+results can be proved:
+• When ∆k = ∞, ∀k ∈ [K], then (2) is equivalent to
+worst-case CMDP (i.e., all the trajectories generated
+by the policy will satisfy all the cost constraints).
+• There are lower bounds for ∆k from which any so-
+lution to (2) will be feasible to risk-neural and VaR
+CMDP with multiple constraints.
+• For any ∆k > 0, under different reward penalty set-
+tings, (2) is equivalent to a multi-constrained CVaR
+CMDP or equivalent to a multi-constrained VaR
+CMDP.
+All the detailed proofs and discussions can be found in the
+appendix.
+4. Safe RL Algorithms
+In this section, we update existing RL methods to effectively
+utilize the extended state space and reward penalties, while
+considering RN-CMDP. Due to the theoretical properties
+in the previous section, just by tweaking ∆, we can also
+handle other Constrained MDPs.
+4.1. Safe DQN
+Deep Q Network (DQN) (Mnih et al., 2015) is an efficient
+method to learn in primarily discrete action Reinforcement
+Learning problems. However, the original DQN does not
+consider safety constraints and cannot be applied to any of
+the CMDP variants.
+The main modifications in the updated algorithm, referred
+to as Safe DQN are with regards to exploiting the extended
+state space and the reward penalties based on constraint
+violations. The pseudo code for the Safe DQN algorithm is
+provided in Algorithm 1.
+The impact of extended state space on the algorithm can be
+observed in almost every line of the algorithm. The penalty
+for violation of constraints When selecting an action (line
+4), Safe DQN not consider the feasibility of the action with
+respect to cost. Instead, like in the original DQN, it is purely
+based on the current Q value. The assumption is that the
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Algorithm 1 DQN with Extended State Space
+Initialization: Relay buffer D with capacity N, action-
+value function Q with weight θ, target action-value function
+ˆQ with weight θ− = θ.
+1: for each episode do
+2:
+Initialize with sequence (s0, c0 = 0).
+3:
+for each time step t do
+4:
+Select a random action at with probability ϵ, oth-
+erwise select at = arg maxa Q((st, ct), a; θ).
+5:
+Execute action at, observe (st+1, ct+1), rt.
+6:
+Store ((st, ct), at, rt, (st+1, ct+1)) in D.
+7:
+Update state-cost pair to (st+1, ct+1).
+8:
+Sample ((sj, cj), aj, rj, (sj+1, cj+1)) from D.
+9:
+if cj > cmax then
+10:
+˜rj = r(sj) − ∆d(sj)/γt
+11:
+else if cj+1 > cmax then
+12:
+˜rj = r(sj) − ∆(ct + d(sj))/γt
+13:
+else
+14:
+˜rj = r(sj)
+15:
+end if
+16:
+{”mask” indicates if the episode terminates}
+17:
+yj = ˜rj + γ ∗ maxa′ ˆQ((sj+1, cj+1), a′; θ−) ∗
+maskj+1.
+18:
+Update θ using l = (yi − Q((sj, cj), aj; θ))2.
+19:
+Every C steps reset ˆQ = Q.
+20:
+end for
+21: end for
+penalties accrued due to violation (in lines 9-12) will be suf-
+ficient to force the agent away from cost infeasible actions.
+Once the new rewards are obtained (based on considering
+reward penalties), the Q network is updated using the mean
+square error loss on line 17.
+4.2. Safe SAC
+Soft Actor-Critic (SAC) (Haarnoja et al., 2018) is an off-
+policy algorithm that learns a stochastic policy for discrete
+and continuous action RL problems. SAC employs policy
+entropy in conjunction with value function to ensure more
+exploration. Q value function in SAC is defined as follows:
+Q(s, a) =E[
+∞
+�
+t=0
+γtr(st, at, st+1)+
+α
+∞
+�
+t=1
+γtH(π(·|st))|s0 = s, a0 = a]
+(4)
+where H(.) denotes the entropy of the action distribution
+for a given state, st). SAC also employs the double Q-
+trick, where we use the minimum of two Q value functions
+(Qi(.), i ∈ 1, 2) as the target, y to avoid overestimation.
+y =r(s, a, s′) + γ min
+i=1,2 Qi(s′, ˜a′) − α log π(˜a′|s′)
+(5)
+where ˜a′ ∼ π(·|s′).
+Our algorithm, referred to as Safe SAC builds on SAC by
+having an extended state space and a new action selection
+strategy that exploits the extended state space. In Safe
+DQN, we primarily rely on violation of constraints, so as to
+learn about the bad trajectories and avoid them. While such
+approach works well for discrete action settings and in an
+off policy setting, it is sample inefficient and can be slow
+for actor-critic settings. In Safe SAC, apart from reward
+penalty, we also focus on learning about feasible actions,
+which are generated through the use of the cost accumulated
+so far (available as part of the state space) and a Q value on
+the future cost.
+Formally, we define the optimization to select safe actions
+(at each decision epoch) in Equation 6 and show safe SAC
+algorithm in Algorithm 2. Extending on the double Q trick
+for reward, we also have double Q for future cost, referred to
+as {Qi
+d}i∈1,2. At each step, the objective is to pick an action
+that will maximize the reward Q value for the extended
+state, action minus the weighted entropy of the action. The
+constraint here is to pick only those actions, which will
+not violate the cost constraint. In the left hand side of the
+constraint, we calculate the overall expected cost from: (a)
+(estimate) of the future cost, from the current state; (b)
+(actual) cost incurred so far; and (c) subtracting the (actual)
+cost incurred at the current step, as it is part of both (a) and
+(b);
+arg max
+a
+min
+i=1,2 Qi((s, c), a) − α log π(a|(s, c))
+s.t. max
+i=1,2 Qi
+D((s, c), a) + c − d((s, c)) ≤ cmax, ∀(s, c)
+(6)
+Algorithm 2 (in appendix) provides the pseudo code for
+Safe SAC.
+5. Experiment
+We empirically compare the performance of our approaches
+on both discrete and continuous environments with respect
+to expected reward and expected cost achieved against lead-
+ing benchmark approaches. For an RL benchmark, we use
+the original DQN (Mnih et al., 2015) and it is referred
+to as unsafe DQN, as it does not account for cost con-
+straints. For leading Constrained RL benchmarks, we use
+BVF (Backward Value Function) (Satija et al., 2020b) and
+Lyapunov (Chow et al., 2019a). We mostly show results
+with respect to expected cost constraint, as there are many
+model free approaches that solve the RN-CMDP problem.
+For one example (Safety Gym), we also provide comparison
+when a CVaR constraint is provided. The performance val-
+ues (expected cost and expected reward) along with standard
+deviation in each experiment are averaged over 5 runs.
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Figure 1. Gridworld environment and reward, cost comparison of different approaches
+Figure 2. Highway environment and reward, cost comparison of different approaches
+5.1. GridWorld: RN-CMDP
+For a discrete state and discrete action environment, we
+consider the stochastic 2D grid world problem introduced
+previously (Leike et al., 2017; Chow et al., 2018; Satija
+et al., 2020b; Jain et al., 2021). The grid on the left of Figure
+1 shows the environment. The agent starts at the bottom
+right corner of the map (green cell) and the objective is to
+move to the goal at the bottom left corner (blue cell). The
+agent can only move in the adjoining cells in the cardinal
+directions. Occasionally agent will execute a random action
+with probability p = 0.05 instead of the one selected by
+the agent. It gets a reward of +100 on reaching the goal,
+and a penalty of -1 at every time step. There are a number
+of pits in the map (red cell) and agent gets a random cost
+ranging from 1 to 1.5 on passing through any pit cell. We
+consider an 8x8 grid and the maximum time horizon is 200
+steps, after which the episode terminates. This modified
+GridWorld environment is challenging because agent can
+travel to destination with a short path with a high cost, but if
+it wishes to travel safely, it needs to explore enough to find
+a safe path which is far from the shortest one. We set the
+expected cost threshold, cmax = 2, meaning agent could
+pass at most one pit. For discrete state environments, we
+use the discrete SAC in (Christodoulou, 2019).
+Figure 1 shows the performance of each method with respect
+to expected reward (score) and expected cost (constraint).
+Here are the key observations:
+• With respect to expected reward, among safe approaches,
+Lyapunov achieves the highest reward. However, it
+violates the expected cost constraint by more than twice
+the cost constraint value.
+• Safe SAC and Safe DQN achieve similar expected re-
+ward values, though Safe SAC reaches there faster. This
+high expected reward value is achieved while satisfying
+the expected cost constraint after 1000 episodes.
+• The other constrained RL approach, BVF achieved the
+lowest value while not being able to satisfy the expected
+cost constraint.
+• As expected, Unsafe DQN achieved the highest expected
+reward but was unable to satisfy the expected cost con-
+straint.
+5.2. Highway Environment:RN-CMDP
+Inspired by experiment in GPIRL (Levine et al., 2011), we
+test our safe methods in the highway environment (Leurent,
+2018) of Figure 2. The task in highway environment is to
+navigate a car on a four-lane highway with all other vehicles
+
+HAverage Score in each Episode
+100
+80
+60 
+Score
+40
+20
+0
+Unsafe DQN
+Safe DQN
+BVF
+-20
+Safe SAC
+Lyapunov
+-40
+0
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+EpisodeAverage Constraint in each Episode
+Unsafe DQN
+Safe DON
+15.0
+BVF
+Safe SAC
+12.5
+Lyapunov
+10.0
+Constraint
+7.5
+5.0
+2.5
+0.0
+0
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+EpisodeAverage Score in each Episode
+22.5
+20.0
+17.5
+Score
+15.0
+12.5
+Unsafe DQN
+10.0
+Safe DQN
+BVF
+7.5
+Safe SAC
+Lyapunov
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+0
+EpisodeAverage Constraint in each Episode
+10
+8
+Constraint
+6
+4
+Unsafe DQN
+Safe DQN
+BVF
+2
+Safe SAC
+Lyapunov
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+0
+EpisodeSolving Constrained RL through Augmented State and Reward Penalties
+Figure 3. Safety Gym Environment and reward, cost comparison of different approaches
+acting randomly. The goal for the agent is to maximize its
+reward by travelling on the right lane at the highest speed,
+vmax. However, to ensure safety, we set the constraint on
+the time the agent drives faster than a given speed in the
+rightmost lane.
+Figure 2 shows the expected reward and expected cost per-
+formance of our safe methods compared to that of the bench-
+marks. Safe SAC and Safe DQN were able to get high ex-
+pected rewards while satisfying the expected cost constraint.
+We also provide results on a highway merge environment in
+the appendix.
+5.3. Safety Gym Environment: CVaR-CMDP
+In this environment, we intend to compare the performance
+of our safe methods with a CVaR optimizing CMDP method,
+i.e., WCSAC (Yang et al., 2021). We test all the methods on
+the same environment from (Yang et al., 2021) - StaticEnv
+in Safety Gym (Ray et al., 2019). The environment is shown
+in Figure 3. The point agent has two types of actions: one
+is for turning and another is for moving forward/backward.
+The objective is to reach the goal position while trying to
+avoid hazardous areas. The agent gets a reward of r − 0.2
+in each time step, where r is an original reward signal of
+Safety Gym (distance towards goal plus a constant for being
+within range of goal) while -0.2 functions as a time penalty.
+In each step, if the agent is located in the hazardous area, it
+gets a cost of 1. We set cmax = 8, meaning agent could stay
+in hazardous area for at most 8 time steps. For risk level α
+in WCSAC, we set α = 0.9 and use the almost risk-neutral
+WCSAC, which is proven to reach the best performance in
+both reward and cost in experiment.
+We show the results in Figure 3. As can be seen from the
+figure, Safe SAC is able to achieve similar performance
+to that of WCSAC. Safe DQN was unable to handle this
+environment due to large size of state. For BVF, although
+it reaches a good performance in reward, it violates the
+constraint for many episodes before converging.
+6. Conclusion
+In this paper, we have provided a very generic and scalable
+mechanism for handling a wide variety of policy based cost
+constraints (expected cost, worst-case cost, VaR, CVaR) in
+Constrained MDPs. Lagrangian based approaches, which
+penalize with respect to expected cost are unable to as-
+sign credit appropriately for a cost constraint violation, as
+expected cost averages over all trajectories. Instead, we
+propose to penalize with respect to individual reward while
+maintaining a cost augmented state, thereby providing pre-
+cise credit assignment with regards to cost constraint vio-
+lations. We theoretically demonstrate that this simple cost
+augmented state and reward penalized MDP (referred to
+as EMDP) can represent all the aforementioned cost con-
+straints. We then provide safety aware RL approaches, Safe
+DQN and Safe SAC, which are able to outperform leading
+expected cost constrained RL approaches (Lyapunov and
+BVF) while at the same time providing similar performance
+to leading approach for CVaR constrained RL (WCSAC).
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+
+Solving Constrained RL through Augmented State and Reward Penalties
+A. SAC Pseudocode
+Algorithm 2 provides the pseudocode for the Safe SAC algorithm.
+Algorithm 2 SAC with Extended State Space
+1: Initialize: policy network π with weight θ.
+2: Value Function: Q1, Q2 with weights φ1, φ2, target Q value functions Qtarg,1, Qtarg,2 with weights φtarg
+1
+=
+φ1, φtarg
+2
+= φ2.
+3: Cost Function: Q1
+D, Q2
+D with weights θ1,D, θ2,D, target cost functions Qtarg,1
+D
+, Qtarg,2
+D
+with weights θtarg
+1,D
+=
+θ1,D, θtarg
+2,D = φ2,D.
+4: for episode=1,2,...,N do
+5:
+Get initial state-cost pair (s0, c0 = 0); t ← 1
+6:
+while t ≤ T do
+7:
+tstart ← t
+8:
+while t ≤ tstart + n or t == T do
+9:
+Select action at using Equation 6.
+10:
+Execute at, observe (st+1, ct+1) and rt.
+11:
+t ← t + 1
+12:
+end while
+13:
+{Calculate targets for each network:}
+14:
+˜rt ← if ct > cmax then rt − ∆dt/γt elif ct+1 > cmax then rt − ∆(ct + dt)/γt else rt
+15:
+R ← if t == T then 0 else ˜rt + γ mini=1,2 Qtarg,i((st+1, ct+1), ˜a′) − α log πθ(˜a′), ˜a′ ∼ πθ((st+1, ct+1))
+16:
+RD ← if t == T then 0 else maxi=1,2 Qtarg,i
+D
+((st+1, ct+1), at+1; θD)
+17:
+{Update networks}
+18:
+for i ∈ {t − 1, ..., tstart} do
+19:
+R ← ri + αR, RD ← di + αRD
+20:
+for j = 1, 2 do
+21:
+dφj ← dφj + ∂(R − Qj)2/∂φj
+22:
+dθj,D ← dθj,D + ∂(RD − Qj
+D)2/∂θj,D
+23:
+end for
+24:
+if the policy is safe then
+25:
+dθ ← dθ + ∇θ log π(ai)(minj=1,2 Qtarg,j − α log π(ai))
+26:
+else
+27:
+dθ ← dθ − ∇θ log π(ai)RD
+28:
+end if
+29:
+end for
+30:
+{Update target networks}
+31:
+end while
+32: end for
+B. Proofs
+B.1. Proof of Theorem 3.2
+Theorem 3.2.
+If we set ∆ = ∞, then if π∗ solves (EMDP), it also solves the following worst-case constrained MDP
+problem
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+�
+st∈τ
+d(st) ≤ cmax, ∀τ ∼ π.
+As a result, π∗ is feasible to the risk-neutral CMDP (RN-CMDP).
+Proof. We first see that there is a unique mapping between a trajectory τ = {s0, . . . , sT } from the original MDP to a
+
+Solving Constrained RL through Augmented State and Reward Penalties
+trajectory of the extended MDP τ ′ = {(s0, c0), (s1, c1) . . . , (sT , cT )} with c0 = 0 and ct = �t−1
+i=0 d(st). Under the reward
+penalties, we can write the objective of the extended MDP as
+E
+� T
+�
+t=0
+γtr(at|st, ct)|s0, π
+�
+=
+�
+τ ′={(st,ct)}∼π
+Pπ(τ ′)
+��
+t
+γt�r(at|st, ct)
+�
+=
+�
+τ={s0,s1,...}∼π
+D(τ)≤cmax
+Pπ(τ)
+��
+t
+γtr(st, at)
+�
++
+�
+τ={s0,s1,...}∼π
+D(τ)>cmax
+Pπ(τ)
+��
+t
+γtr(st, at) − ∆
+�
+t
+d(st)
+�
+= Eπ
+��
+t
+γtr(st, at)
+�
+− ∆
+�
+τ∼π
+D(τ)>cmax
+Pπ(τ)D(τ)
+(7)
+As a result, we can rewrite the MDP problem (EMDP) as
+max
+π
+�
+�
+�
+�
+�
+Eπ
+��
+t
+γtr(st, at)
+�
+− ∆
+�
+τ∼π
+D(τ)>cmax
+Pπ(τ)D(τ)
+�
+�
+�
+�
+�
+(8)
+So, if we set ∆ = ∞, to maximize the expected reward, we need to seek a policy that assigns zero probabilities for all
+the trajectories τ such that D(τ) > cmax. Let Π be the set of policies satisfying that condition (and assume that Π is not
+empty), i.e., for any policy π ∈ Π and any trajectory τ such that D(τ) > cmax, Pπ(τ) = 0. This implies that when ∆ = ∞,
+(8) is equivalent to
+Eπ∈Π
+��
+t
+γtr(st, at)
+�
+which is also the worst-case CMDP problem.
+B.2. Proof of Lemma 3.3
+Lemma 3.3. Let φ∗ = cmax − maxπ
+�
+�Eπ[D(τ)| D(τ) ≤ cmax]
+�
+. Given any policy π, if �Eπ[D(τ)| D(τ) > cmax] ≤ φ∗,
+then Eπ[D(τ)] ≤ cmax.
+Proof. For a policy π satisfying �Eπ[D(τ)| D(τ) > cmax] ≤ φ∗, we have
+�
+τ| D(τ)>cmax
+Pπ(τ)D(τ) ≤ cmax − max
+π
+�
+�Eπ[D(τ)| D(τ) ≤ cmax]
+�
+,
+which is equivalent to
+�
+τ| D(τ)>cmax
+Pπ(τ)D(τ) + max
+π
+�
+�Eπ[D(τ)| D(τ) ≤ cmax]
+�
+≤ cmax
+implying
+cmax ≥
+�
+τ| D(τ)>cmax
+Pπ(τ)D(τ) +
+�
+τ| D(τ)≤cmax
+Pπ(τ)D(τ) = Eπ[D(τ)]
+which is the desired inequality.
+B.3. Proof of Lemma 3.4
+Lemma 3.4. Given ∆ > 0, let π∗ be an optimal solution to (EMDP). We have
+�Eπ∗ [D(τ)| D(τ) > cmax] ≤ Ψ∗ − Ψ
+∆
+.
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Proof. We first note that, from (8), we can write
+π∗ = argmaxπ
+�
+�
+�
+�
+�
+Eπ
+��
+t
+γtr(st, at)
+�
+− ∆
+�
+τ
+D(τ)≥cmax
+Pπ(τ)D(τ)
+�
+�
+�
+�
+�
+Let π be an optimal policy to the worst-case CMDP (WC-CMDP). Since π is also feasible to the extended MDP (EMDP),
+we have
+Eπ∗
+��
+t
+γtr(st, at)
+�
+− ∆
+�
+τ
+D(τ)≥cmax
+Pπ∗(τ)D(τ) ≥ Eπ
+��
+t
+γtr(st, at)
+�
+= Ψ
+(9)
+Moreover, since Ψ∗ is the optimal value of the original unconstrained problem Ψ∗ = maxπ E
+��T
+t=0 γtr(st, at)|s0, π
+�
+, we
+should have
+Ψ∗ ≥ Eπ∗
+��
+t
+γtr(st, at)
+�
+(10)
+Combining (24) and (10) gives
+Ψ∗ − ∆
+�
+τ
+D(τ)≥cmax
+Pπ∗(τ)D(τ) ≥ Ψ,
+implying
+�
+τ| D(τ)≥cmax
+Pπ∗(τ)D(τ) ≤ Ψ∗ − Ψ
+∆
+,
+which is the desired inequality.
+B.4. Proof of Theorem 3.5
+Theorem 3.5. For any ∆ ≥ Ψ∗−Ψ
+φ∗
+a solution to (EMDP) is always feasible to the risk-neutral CMDP (RN-CMDP).
+Proof. The theorem is a direct result from Lemmas 3.3 and 3.4. That is, by selecting ∆ ≥ Ψ∗−Ψ
+φ∗
+, from Lemm 3.4 we can
+guarantee that
+�
+τ| D(τ)≥cmax
+Pπ∗(τ)D(τ) ≤ Ψ∗ − Ψ
+∆
+≤ φ∗,
+(11)
+where π∗ is an optimal policy to (EMDP). From Lemma 3.4, (11) also implies that π∗ is also feasible to the risk-neutral
+CMDP (RN-CMDP), as desired.
+B.5. Proof of Theorem 3.6
+Theorem 3.6. For any ∆ ≥ (Ψ∗ − Ψ)/(αcmax), a solution to (EMDP) is always feasible to (VaR-CMDP).
+Proof. We use Lemma 3.4 to see that if π∗ is a solution to (EMDP), then it satisfies
+�Eτ∼π∗
+�
+D(τ)| D(τ) > cmax
+�
+≤ Ψ∗ − Ψ
+∆
+.
+(12)
+On the other hand, we have
+�Eτ∼π∗
+�
+D(τ)| D(τ) > cmax
+�
+=
+�
+τ|D(τ)>cmax
+Pπ∗(τ)D(τ)
+> cmax
+�
+τ|D(τ)>cmax
+Pπ∗(τ)
+= cmaxPπ∗(D(τ) > cmax))
+(13)
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Thus, if we select ∆ ≥ (Ψ∗ − Ψ)/(αcmax), we will have the following chain of inequalities.
+α ≥ Ψ∗ − Ψ
+∆cmax
+(a)
+≥
+1
+cmax
+�Eτ∼π∗
+�
+D(τ)| D(τ) > cmax
+�
+(b)
+≥ Pπ∗(D(τ) > cmax).
+where (a) is due to (12) and (b) is due to (13). This implies that π∗ is feasible to the chance-constrained MDP (VaR-CMDP).
+We complete the proof.
+B.6. Proof of Theorem 3.7
+Theorem 3.7. If we define the reward penalties as
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�r(at|(st, ct)) = r(at|st) if ct + d(st) ≤ cmax
+�r(at|(st, ct)) = r(at|st) − ∆(t + 1)/γt
+if ct ≤ cmax and ct + d(st) > cmax
+�r(at|(st, ct)) = r(at|st) − ∆/γt if ct > cmax
+then if π∗ is an optimal solution to (EMDP), then there is α∆ ∈ [0; Ψ∗−Ψ
+∆T ] (α is dependent of ∆) such that π∗ is also optimal
+to (VaR-CMDP). Moreover lim∆→∞ α∆ = 0.
+Proof. Under the reward setting, we can write the objective of (EMDP) as
+Eπ
+� T
+�
+t=0
+γtr(at|st, ct)|s0
+�
+=
+�
+τ ′={(st,ct)}∼π
+Pπ(τ ′)
+��
+t
+γt�r(at|st, ct)
+�
+=
+�
+τ={s0,s1,...}∼π
+D(τ)≤cmax
+Pπ(τ)
+��
+t
+γtr(st, at)
+�
++
+�
+τ={s0,s1,...}∼π
+D(τ)>cmax
+Pπ(τ)
+��
+t
+γtr(st, at) − ∆T
+�
+= Eπ
+��
+t
+γtr(st, at)
+�
+− ∆TPπ(D(τ) > cmax).
+(14)
+We now show that if π∗ is an optimal policy to (EMDP), then it is also optimal for (VaR-CMDP) with where α∆ =
+Pπ∗(D(τ) > cmax). By contradiction, let us assume that it is not the case. Let π be optimal for (VaR-CMDP). We first see
+that π∗ is feasible to (VaR-CMDP), thus
+Eπ∗
+� T
+�
+t=0
+γtr(st, at)
+�
+< Eπ
+� T
+�
+t=0
+γtr(st, at)
+�
+.
+(15)
+Moreover, since π is feasible to (VaR-CMDP), we have:
+Pπ(D(τ) > cmax) ≤ Pπ∗(D(τ) > cmax).
+(16)
+Combine (15) and (16) and (14), it can be seen that π∗ is not an optimal policy to (EMDP), which is contrary to our initial
+assumption. So, π∗ is an optimal policy for the (VaR-CMDP). We now prove that lim∆→∞ α∆ = 0. To this end, we first
+see that if �π is an optimal solution to the worst-case CMDP (WC-CMDP), then P�π(D(τ) > cmax) = 0. Thus, we have the
+
+Solving Constrained RL through Augmented State and Reward Penalties
+following chain of inequalities
+Ψ∗ − ∆Tα∆ ≥ Eπ∗
+��
+t
+γtr(st, at)
+�
+− ∆TPπ∗(D(τ) > cmax)
+≥ E�π
+��
+t
+γtr(st, at)
+�
+− ∆TP�π(D(τ) > cmax)
+= E�π
+��
+t
+γtr(st, at)
+�
+= Ψ
+Thus
+α∆ ≤ Ψ∗ − Ψ
+∆T
+.
+implying lim∆→∞ α∆ = 0.
+B.7. Proof of Theorem 3.8
+Theorem 3.8. If we define the reward penalties as
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�r(at|(st, ct)) = r(at|st) if ct + d(st) ≤ cmax
+�r(at|(st, ct)) = r(at|st) − ∆(ct + d(st) − cmax)/γt
+if ct ≤ cmax and ct + d(st) > cmax
+�r(at|(st, ct)) = r(at|st) − ∆d(st)/γt if ct > cmax
+then for any ∆ > 0, there is β∆ ∈ [0; Ψ∗−Ψ
+∆
+] (β∆ is dependent of ∆) such that any optimal solution to the extended CMDP
+(EMDP) is also optimal to the following risk-averse CMDP
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+Eτ∼π
+�
+(D(τ) − cmax)+�
+≤ β∆.
+(CVaR-CMDP)
+Moreover, lim∆→∞ β∆ = 0.
+Proof. We first see that, under the reward penalties defined above, the objective of (EMDP) becomes
+Eπ
+� T
+�
+t=0
+γtr(at|st, ct)|s0
+�
+=
+�
+τ ′={(st,ct)}∼π
+Pπ(τ ′)
+��
+t
+γt�r(at|st, ct)
+�
+=
+�
+τ={s0,s1,...}∼π
+D(τ)≤cmax
+Pπ(τ)
+��
+t
+γtr(st, at)
+�
++
+�
+τ={s0,s1,...}∼π
+D(τ)>cmax
+Pπ(τ)
+��
+t
+γtr(st, at) − ∆
+��
+t
+d(st) − cmax
+��
+= Eπ
+��
+t
+γtr(st, at)
+�
+− ∆
+�
+τ∼π
+D(τ)>cmax
+Pπ(τ)(D(τ) − cmax)
+= Eπ
+��
+t
+γtr(st, at)
+�
+− ∆Eτ∼π
+�
+(D(τ) − cmax)+�
+(17)
+We now show that if π∗ is an optimal policy to (EMDP), then it is also optimal for (CVaR-CMDP) with where β∆ =
+Eτ∼π∗
+�
+(D(τ) − cmax)+�
+. By contradiction, let us assume that π∗ is not optimal for (CVaR-CMDP). We then let π be
+optimal for (CVaR-CMDP). We first see that π∗ is feasible to (CVaR-CMDP), thus
+Eπ∗
+� T
+�
+t=0
+γtr(st, at)
+�
+< Eπ
+� T
+�
+t=0
+γtr(st, at)
+�
+(18)
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Moreover, since π is feasible to (CVaR-CMDP), we have:
+Eτ∼π
+�
+(D(τ) − cmax)+�
+≤ β∆ = Eτ∼π∗
+�
+(D(τ) − cmax)+�
+(19)
+Combine (18) and (19) we get
+Eπ∗
+� T
+�
+t=0
+γtr(st, at)
+�
+− ∆Eτ∼π∗
+�
+(D(τ) − cmax)+�
+< Eπ
+� T
+�
+t=0
+γtr(st, at)
+�
+− ∆�Eτ∼π
+�
+(D(τ) − cmax)+�
+(20)
+Using (17), (20) implies that �π yields a strictly better objective value to the extended MDP, as compared to π∗, which
+is contrary to the assumption that π∗ is optimal for (EMDP). So, π∗ should be an optimal policy for the (CVaR-CMDP).
+We now prove that lim∆→∞ β∆ = 0. To this end, we first see that if �π is an optimal solution to the worst-case CMDP
+(WC-CMDP), then �Eτ∼�π
+�
+(D(τ) − cmax)+�
+= 0. Thus, we have the following chain of inequalities:
+Ψ∗ − ∆β∆ ≥ Eπ∗
+��
+t
+γtr(st, at)
+�
+− ∆Eτ∼π∗ �
+(D(τ) − cmax)+�
+≥ E�π
+��
+t
+γtr(st, at)
+�
+− ∆Eτ∼�π
+�
+(D(τ) − cmax)+�
+= E�π
+��
+t
+γtr(st, at)
+�
+(21)
+We recall that E�π [�
+t γtr(st, at)] = Ψ (i.e., objective value of the worst-case CMDP), thus,
+β∆ ≤ Ψ∗ − Ψ
+∆
+,
+implying lim∆→∞ β∆ = 0 as desired.
+C. Multi-constrained MDP
+We present, in the following, a series of theoretical results for the multi-constrained MDP discussed in the main body of the
+paper. Similar to the single-constrained case, we will show that
+• If ∆k = ∞ for all k ∈ [K], then (2) is equivalent to a worst-case CMDP.
+• There is a lower bound for each ∆k such that any optimal policy to (2) will always be feasible to a given risk-neutral or
+chance-constrained MDP.
+• By employing different reward penalty settings, (2) is equivalent to a VaR or CVaR CMDP.
+Since all the proofs are similar to those in the single-constrained case, we keep them brief.
+Proposition C.1. If we set ∆k = ∞ for all k ∈ [K], then the extended MDP is equivalent to the following worst-case
+CMDP
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+�
+st∈τ
+dk(st) ≤ ck
+max, ∀τ ∼ π, ∀k ∈ [K]
+(22)
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Proof. Similar to the proof of Theorem 3.2, we write the objective of the extended MDP as
+E
+� T
+�
+t=0
+γtr(at|st, cK
+t )|s0, π
+�
+=
+�
+τ ′={(st,cK
+t )}∼π
+Pπ(τ ′)
+��
+t
+γt�r(at|st, cK
+t )
+�
+=
+�
+τ={s0,s1,...}∼π
+D(τ)≤cmax
+Pπ(τ)
+��
+t
+γtr(st, at)
+�
++
+�
+τ={s0,s1,...}∼π
+D(τ)>cmax
+Pπ(τ)
+�
+��
+t
+γtr(st, at) −
+�
+k∈[K]
+∆k
+�
+t
+dk(st)
+�
+�
+= Eπ
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆k
+�
+τ∼π
+Dk(τ)≥ck
+max
+Pπ(τ)Dk(τ).
+(23)
+So, if ∆k = ∞, then one needs to seek a policy that assigns zero probabilities to all the trajectories that violate the
+constraints, implying that the extended MDP would yield the same optimal policies as the worst-case CMDP (22).
+Proposition C.2. Let π∗ and π be optimal policies to the extended MDP (2) and the worst-case MDP, and φk = ck
+max −
+maxπ �Eπ[Dk(τ)|Dk(τ) ≤ ck
+max], ∀k ∈ [K]. If we choose ∆k such that ∆k > (Ψ∗ − Ψ)/φk, then any optimal policy of
+(2) is feasible to the risk-neutral CMDP with multiple constraints.
+Proof. Since π is also feasible to (2), we have:
+Eπ∗
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆k
+�
+τ
+Dk(τ)>ck
+max
+Pπ∗(τ)Dk(τ) ≥ Eπ
+��
+t
+γtr(st, at)
+�
+= Ψ.
+(24)
+Moreover, since Ψ∗ is an optimal value of the original unconstrained MDP, we have Ψ∗ ≥ Eπ∗ [�
+t γtr(st, at)], leading to
+�
+k∈[K]
+∆k
+�
+τ
+Dk(τ)≥ck
+max
+Pπ∗(τ)Dk(τ) ≤ Ψ∗ − Ψ.
+(25)
+Moreover, from Lemma 3.3, we know that if �Eπ[Dk(τ)| Dk(τ) > ck
+max] ≤ φk, then Eπ[Dk(τ)] < ck
+max, where
+φk = ck
+max − maxπ �Eπ[Dk(τ)|Dk(τ) ≤ cmax]. Therefore, if we select ∆k ≥ (Ψ∗ − Ψ)/φk, then from (25) we see that
+�Eπ∗[Dk(τ)| Dk(τ) > ck
+max] ≤ φk for all k ∈ [K], implying that π∗ satisfies all the constraints, as desired.
+Proposition C.3. Given any αk ∈ (0, 1), k ∈ [K], if we choose ∆k ≥ (Ψ∗ − Ψ)/(αkck
+max), ∀k ∈ [K], then a solution π∗
+to (2) is always feasible to the following VaR (or chance-constrained) MDP.
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+Pπ
+�
+(Dk(τ) > ck
+max
+�
+≤ αk, ∀k ∈ [K]
+(26)
+Proof. From the proof of Proposition C.2 above, we have the following inequalities
+Ψ∗ − Ψ ≥
+�
+k∈[K]
+∆k
+�
+τ
+Dk(τ)>ck
+max
+Pπ∗(τ)Dk(τ)
+≥
+�
+k∈[K]
+∆kck
+maxPπ∗(Dk(τ) > ck
+max)
+So if we choose ∆k ≥ (Ψ∗ − Ψ)/(αkck
+max), ∀k ∈ [K], we will have
+Ψ∗ − Ψ ≥ (Ψ∗ − Ψ)
+(αkckmax)ck
+maxPπ∗(Dk(τ) > ck
+max), ∀k ∈ [K],
+implying Pπ∗(Dk(τ) > ck
+max) ≤ αk, as desired.
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Proposition C.4. If we define the reward penalties as
+�r(at|(st, cK
+t )) = r(at|st) −
+�
+k∈[K]
+∆kδk(ct), ∀st, at, cK
+t ,
+where δk(ct), ∀k ∈ [K], are defined as follows:
+δk(ct) =
+�
+�
+�
+�
+�
+0 if ck
+t + dk(st) ≤ ck
+max
+(T + 1)/γt if ck
+t ≤ ck
+max, ck
+t + dk(st) > ck
+max
+1/γt if ck
+t > ck
+max,
+then if π∗ is an optimal solution to (EMDP), there is α∆
+∆
+∆
+k ∈ [0; Ψ∗−Ψ
+T ∆k ] (αk is dependent of ∆
+∆
+∆)1 such that π∗ is also optimal
+to the following VaR CMDP
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+Pπ
+�
+(D(τ) > ck
+max
+�
+≤ α∆
+∆
+∆
+k , ∀k ∈ [K].
+(27)
+Moreover lim∆k→∞ α∆
+∆
+∆
+k = 0, ∀k ∈ [K].
+Proof. Similar to the proof of Theorem 3.6, we can write the objective of the extended MDP as
+Eπ
+��
+t
+γtr(at|st, cK
+t )
+�
+= Eπ
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆kTPπ(Dk(τ) > ck
+max)
+Then, in a similar way, if we let α∆
+∆
+∆
+k = TPπ∗(Dk(τ) > ck
+max), then π∗ should be an optimal policy to (27). In addition, we
+can bound αk by deriving the following inequalities.
+Ψ∗ −
+�
+k∈[K]
+∆kTα∆
+∆
+∆
+k ≥ Eπ∗
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆kTPπ∗(Dk(τ) > ck
+max)
+≥ E�π
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆kTP�π(Dk(τ) > ck
+max)
+= E�π
+��
+t
+γtr(st, at)
+�
+= Ψ,
+(28)
+where �π and Ψ are optimal policy and optimal value of the worst-case CMDP (22). This implies
+�
+k∈[K]
+∆kTα∆
+∆
+∆
+k ≤ Ψ∗ − Ψ,
+which tells us that α∆
+∆
+∆
+k ≤ Ψ∗−Ψ
+T ∆k , implying that lim∆k→∞ α∆
+∆
+∆
+k = 0.
+Proposition C.5. For any ∆k > 0, k ∈ [K], if we define the reward penalties as
+�r(at|(st, cK
+t )) = r(at|st) −
+�
+k∈[K]
+∆kδk(ct), ∀st, at, cK
+t ,
+where δk(ct), ∀k ∈ [K], are defined as follows:
+δk(ct) =
+�
+�
+�
+�
+�
+0 if ck
+t + dk(st) ≤ ck
+max
+(ck
+t + dk
+t − ck
+max)/γt if ck
+t ≤ ck
+max, ck
+t + dk(st) > ck
+max
+dk
+t /γt if ck
+t > ck
+max,
+1∆
+∆
+∆ denotes the vector (∆1, . . . , ∆K)
+
+Solving Constrained RL through Augmented State and Reward Penalties
+then there are β∆
+∆
+∆
+k ∈ [0; Ψ∗−Ψ
+∆k ] (β∆
+∆
+∆
+k is dependent of ∆
+∆
+∆) such that any optimal solution π∗ to the extended CMDP (2) is also
+optimal to the following multi-constrained CVaR CMDP
+max
+π
+E
+� T
+�
+t=0
+γtr(st, at)|s0, π
+�
+s.t.
+Eτ∼π
+�
+(D(τ) − cmax)+�
+≤ β∆
+∆
+∆
+k , ∀k ∈ [K]
+(29)
+Moreover, lim∆k→∞ β∆
+∆
+∆
+k = 0.
+Proof. Under the reward setting, we first write the objective function of the extended MDP as
+Eπ
+��
+t
+γtr(at|st, cK
+t )
+�
+= Eπ
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆kEπ
+�
+(Dk(τ) − ck
+max)+�
+Following the same derivations as in the proof of Theorem 3.8, we can further show that, by contradiction, π∗ is also optimal
+for the CVaR CMDP (29) with β∆
+∆
+∆
+k = Eπ∗
+�
+(Dk(τ) − ck
+max)+�
+. To prove lim∆k→∞ β∆
+∆
+∆
+k = 0, we derive similar inequalities
+as in the proof of Proposition C.4, as follows:
+Ψ∗ −
+�
+k∈[K]
+∆kβ∆
+∆
+∆
+k ≥ Eπ∗
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆kEπ∗ �
+(Dk(τ) − ck
+max)+�
+≥ E�π
+��
+t
+γtr(st, at)
+�
+−
+�
+k∈[K]
+∆kE�π
+�
+(Dk(τ) − ck
+max)+�
+= E�π
+��
+t
+γtr(st, at)
+�
+= Ψ,
+implying that β∆
+∆
+∆
+k ≤ Ψ∗−Ψ
+∆k , thus lim∆k→∞ β∆
+∆
+∆
+k = 0 as desired.
+D. Experimental Results on Puddle Environment
+D.1. Continuous Puddle Environment: RN-CMDP
+Inspired by (Jain et al., 2021), we test all the methods on the continuous puddle environment. The environment is shown in
+Figure 4. It is a continuous two-dimensional state-space environment in [0, 1]. The agent starts at the bottom left corner of
+the map (0, 0) and the objective is to move to the goal at the upper right corner (1, 1). The agent can move in four directions
+and occasionally agent will execute a random action with probability p = 0.05 instead of the one selected by the agent.
+In each position transition, noise is drawn from the Uniform[−0.025, 0.025] distribution and added to both coordinates.
+When the agent is within 0.1 L1 distance from the goal state, the agent can be seen as reaching the goal and receive a reward
+of 100 while agent gets a time penalty as -0.1 at each time step. There is a square puddle region centering at (0.5, 0.5) with
+0.4 height. In each time step, if agent is located in the puddle area, it gets a cost of 1. Due to the existence of noise, we
+cannot set the threshold cmax too small as it would be hard for agent to reach the goal, so we set cmax = 8, meaning agent
+could stay in puddle area for at most 8 time steps.
+We show the results in Figure 4. As can be seen from the figure, safe SAC could outperform other methods in both reward
+and cost. Although safe DQN can always satisfy the constraint, it always fail to reach the goal to get the maximum reward.
+For BVF, when the backward value function succeeds to estimate the cost, the reward starts to decrease and worse than safe
+SAC.
+E. Experimental Results on Highway Merge Environment
+We also evaluate our safe methods on another highway environment - merge. The environment is shown in Figure 5
+where agent needs to take actions to complete merging with other vehicles. The rewards are similar to those in highway
+
+Solving Constrained RL through Augmented State and Reward Penalties
+Figure 4. Performance in Puddle Environment
+Figure 5. Merge environment and reward, cost comparison of different approaches
+environment. Figure 5 shows a comparison of our safe methods with other benchmarks. Although Safe DQN fails to
+complete the task in merge environment, Safe SAC still outperforms BVF and unsafe DQN with better score and lower cost.
+The reason that safe DQN fails is that the combinations of extended space is too large in merge environment for safe DQN
+to figure it out. That is also why safe DQN converges quite slowly in highway environment. As safe DQN is unable to deal
+with large size of state space, safe SAC outperforms safe DQN in continuous environments.
+F. Hyperparameters
+In case of discrete environment - GridWorld, the size of state space is 8 × 8 with 18 pits. In Highway environment (including
+merge), related parameters and their values are listed below. There is an additional reward in merge environment named
+mergingspeedreward with value of -0.5. It penalties the agent if it drives with speed less than 30 while merging.
+• lanes count: Number of lanes, setting as 4 in both environments.
+• vehicles count: Number of vehicles on lanes, setting as 50 in both environments.
+• controlled vehicles: Number of agents, setting as 1 in both environments.
+• duration: Duration of the game, setting as 40 in both environments.
+• ego spacing: The space of vehicles, setting as 2 in both environments.
+• vehicles density: The density of vehicles on lanes, setting as 1 in both environments.
+• reward speed range: The range where agent can receive high speed reward, setting as [20, 30] in both environ-
+ments.
+• high speed reward: Reward received when driving with speed in reward speed range, setting as 0.4 in highway
+while 0.2 in merge.
+
+Average Score in each Episode
+75
+Unsafe DQN
+Safe DQN
+BVF
+50
+Safe SAC
+Lyapunov
+25
+0
+Score
+-25
+-50
+-75
+-100
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+0
+EpisodeAverage Constraint in each Episode
+Unsafe DQN
+350
+Safe DON
+BVF
+300
+Safe SAC
+Lyapunov
+250
+Constraint
+200
+150
+100
+50
+0
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000
+EpisodeAverage Score in each Episode
+15
+14
+13
+12
+Unsafe DON
+Safe DQN
+Score
+11
+BVF
+Safe SAC
+10
+Lyapunov
+9
+8
+7
+0
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+EpisodeAverage Constraint in each Episode
+Unsafe DQN
+16
+Safe DON
+BVF
+Safe SAC
+14
+Lyapunov
+Constraint
+12
+10
+8
+6
+2000
+4000
+6000
+8000
+10000
+12000
+14000
+0
+EpisodeSolving Constrained RL through Augmented State and Reward Penalties
+• collision reward: Reward received when colliding with a vehicle, setting as -1 in both environments.
+• right lane reward: Reward received when driving on the right-most lane, setting as 0.1 in both environments.
+• lane change reward: Reward received when taking lane change action, setting as 0 in highway while -0.05 in merge.
+In all the methods, we use networks with a hidden layer size of 64,64,64 along with the ReLu activation and use Adam
+optimizer to optimize the networks. We test our methods on GridWorld, Highway, Safety Gym, Puddle, Highway for 15000,
+15000, 1000, 2000, 15000 episodes respectively and update the network every 4 steps.
+

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+Red Emission from Copper-Vacancy Color Centers in Zinc Sulfide Colloidal
+Nanocrystals
+Sarah M. Thompson,1 C¨uneyt S¸ahin,2, 3 Shengsong Yang,4 Michael E. Flatt´e,3, 5
+Christopher B. Murray,4, 6 Lee C. Bassett,7, ∗ and Cherie R. Kagan1, 8, 9, ∗
+1Department of Electrical and Systems Engineering,
+University of Pennsylvania, Philadelphia Pennsylvania 19104, USA
+2UNAM – National Nanotechnology Research Center and Institute of
+Materials Science and Nanotechnology, Bilkent University, Ankara, Turkey
+3Department of Physics and Astronomy, University of Iowa, Iowa City IA, 52242, USA
+4Department of Chemistry, University of Pennsylvania, Philadelphia PA, 19104, USA
+5Department of Applied Physics, Eindhoven University of Technology,
+P. O. Box 513, 5600 MB Eindhoven, The Netherlands
+6Department of Materials Science and Engineering,
+University of Pennsylvania, Philadelphia PA, 19104, USA
+7Department of Electrical and Systems Engineering,
+University of Pennsylvania, Philadelphia PA, 19104, USA
+8Department of Materials Science and Engineering,
+University of Pennsylvania, Philadelphia Pennsylvania 19104, USA
+9Department of Chemistry, University of Pennsylvania, Philadelphia Pennsylvania 19104, USA
+(Dated: January 12, 2023)
+Copper-doped zinc sulfide (ZnS:Cu) exhibits down-conversion luminescence in the UV, visible,
+and IR regions of the electromagnetic spectrum; the visible red, green, and blue emission is referred
+to as R-Cu, G-Cu, and B-Cu, respectively. The sub-bandgap emission arises from optical transitions
+between localized electronic states created by point defects, making ZnS:Cu a prolific phosphor ma-
+terial and an intriguing candidate material for quantum information science, where point defects
+excel as single-photon sources and spin qubits. Colloidal nanocrystals (NCs) of ZnS:Cu are partic-
+ularly interesting as hosts for the creation, isolation, and measurement of quantum defects, since
+their size, composition, and surface chemistry can be precisely tailored for bio-sensing and opto-
+electronic applications. Here, we present a method for synthesizing colloidal ZnS:Cu NCs that emit
+primarily R-Cu, which has been proposed to arise from the CuZn-VS complex, an impurity-vacancy
+point defect structure analogous to well-known quantum defects in other materials that produce
+favorable optical and spin dynamics. First principles calculations confirm the thermodynamic sta-
+bility and electronic structure of CuZn-VS. Temperature- and time-dependent optical properties of
+ZnS:Cu NCs show blueshifting luminescence and a non-monotonic intensity dependence as temper-
+ature is increased from 19 K to 290 K, for which we propose an empirical dynamical model based
+on thermally-activated coupling between two manifolds of states inside the ZnS bandgap. Under-
+standing of R-Cu emission dynamics, combined with a controlled synthesis method for obtaining
+R-Cu centers in colloidal NC hosts, will greatly facilitate the development of CuZn-VS and related
+complexes as quantum point defects in ZnS.
+I.
+INTRODUCTION
+Controlled impurity doping of wide-bandgap semi-
+conductors can be used to introduce color centers,
+which are point defects that activate sub-bandgap,
+optical
+photoluminescence
+(PL).
+Color
+centers
+can
+serve as sources of tunable PL for bio-imaging and
+opto-electronic applications[1, 2], as well as localized,
+optically-addressable, electronic spin states for applica-
+tions in quantum information science[3, 4].
+For all of
+these applications, colloidal nanocrystals (NCs) can pro-
+vide unique advantages over analogous, bulk materials
+because they can be processed using wet-chemical meth-
+ods, and their large surface areas and effects of quantum
+∗ Corresponding
+authors.
+lbassett@seas.upenn.edu
+&
+ka-
+gan@seas.upenn.edu
+confinement allow for highly tunable optical and elec-
+tronic properties[5].
+Impurity-doped ZnS has long been exploited as a UV,
+visible, and NIR luminescent material in its bulk and
+colloidal NC forms, and it has more recently been studied
+as a potential host material for defect-based quantum
+emitters and quantum bits, or defect qubits[6, 7]. Cu-
+doping of ZnS introduces sub-bandgap red, green, and
+blue PL emission bands that are associated with color
+centers known respectively as R-Cu, G-Cu, and B-Cu.
+R-Cu color centers are particularly interesting thanks to
+their peak PL emission in the NIR biological window.
+However, R-Cu remains under-utilized since it is rarely
+observed in colloidal ZnS:Cu NCs, which typically emit
+visible PL dominated by B-Cu and G-Cu[8, 9].
+Past studies have indicated that R-Cu emission in bulk
+ZnS:Cu arises from a defect complex consisting of a sub-
+stitutional copper impurity (CuZn) and a sulfur vacancy
+arXiv:2301.04223v1  [cond-mat.mtrl-sci]  10 Jan 2023
+
+2
+(VS) in a nearest-neighbor CuZn-VS complex[10]. Com-
+pared to transition metal-doped phosphors like ZnS:Mn
+that rely on electric-dipole-forbidden, intra-d-shell transi-
+tions between substitutional MnZn levels to produce visi-
+ble PL, the mixed orbital character and lowered symme-
+try of the CuZn-VS complex are associated with more
+dipole-allowed radiative transitions[11], and therefore
+shorter optical lifetimes, as desired for many applications.
+Moreover, the symmetry of the CuZn-VS complex is de-
+scribed by the C3v point group, which is characteristic of
+well-developed defect qubits [12, 13] and is a key factor in
+producing favorable defect orbital and spin structures[4].
+R-Cu emission has further been associated with electron
+paramagnetic resonance (EPR) spectra which indicate a
+paramagnetic ground state [14]. These characteristics are
+especially compelling in combination with the favorable
+properties of ZnS as a host for defect qubits, which in-
+clude a high natural abundance of spin-free nuclei and
+relatively weak spin-orbit coupling, as well as the ease
+of ZnS colloidal NC synthesis and surface modification
+compared to hosts materials such as diamond [5, 15].
+Here, we report the synthesis and characterization of
+colloidal ZnS:Cu NCs emitting visible PL dominated by
+R-Cu. We study the structural, compositional, and time-
+and temperature-dependent optical properties of NCs
+synthesized with 0, 0.05, 0.075, and 0.1 mol% Cu:Zn.
+The R-Cu emission intensity scales with the copper con-
+centration, and the R-Cu emission band exhibits com-
+plex temperature- and time-dependent properties that
+are generally consistent with observations of R-Cu in
+bulk ZnS:Cu [16]. In particular, the R-Cu emission peak
+blueshifts with increasing temperature from 19 K to 290
+K, and the R-Cu emission intensity increases with in-
+creasing temperature between 150 K and 270 K, a phe-
+nomenon known as negative thermal quenching (NTQ).
+We propose a single mechanism to explain the blueshift
+and the NTQ based on thermally-activated carrier trans-
+fer between two manifolds of radiative states. This mech-
+anism is consistent with time-resolved PL measurements
+showing the presence of two distinct radiative transitions
+in the R-Cu band at low temperature.
+Drawing from
+first-principles calculations, we discuss the role of defect
+species, spatial arrangement, and charge state in produc-
+ing the manifolds of states responsible for measured R-Cu
+characteristics. A detailed understanding of these char-
+acteristics can facilitate the realization of protocols for
+initialization, readout, and control of the defect’s charge
+and spin states for development of a quantum defect ar-
+chitecture.
+II.
+RESULTS AND DISCUSSION
+A.
+Synthesis of ZnS:Cu NCs with R-Cu Emission
+ZnS NCs are synthesized using the single-source pre-
+cursor approach developed by Zhang et al., [17] in which
+zinc diethyldithiocarbamate (Zn(Ddtc)2) is thermally de-
+FIG. 1.
+ZnS:Cu NC synthesis and structure.
+(a)
+Schematic of the synthesis of R-Cu emitting ZnS:Cu NCs,
+where red represents the application of heat.
+Photographs
+show the reaction vessel before and after NC formation. (b)
+Cu:Zn mol% measured by ICP-OES (black circles) as a func-
+tion of the Cu:Zn mol% added to the synthesis pot.
+The
+component weight of the R-Cu peak when the total PL spec-
+trum is decomposed by non-negative matrix factorization (red
+squares and right-hand vertical axis) also scales with the
+Cu:Zn mol% .
+The R2 values for the linear regression fits
+(black and red lines) are 0.917 and 0.957, respectively. (c)
+TEM image and electron diffraction pattern of a sample of
+ZnS:Cu NCs with 0.1 mol% Cu:Zn. (d) Distribution of NC
+diameters for samples of 100 NCs measured from TEM images
+obtained for each Cu:Zn ratio (0-0.1 mol%).
+composed in oleic acid (OA) and oleylamine (OM); see
+Figure 1a. In previously reported syntheses of colloidal
+ZnS:Cu NCs, the absence of R-Cu emission may result
+from unintentional Cl impurities introduced by CuCl2
+precursors, which are known to quench R-Cu in bulk
+ZnS:Cu along with Al, In, and Ga impurities [10, 14, 16].
+To avoid the introduction of Cl impurities, we instead
+add a fixed volume (0.1 mL) of Cu(CH3COO)2·H2O dis-
+solved in ultrapure DI water, with concentrations corre-
+sponding to Cu:Zn molar ratios of 0 %, 0.05 %, 0.075 %,
+and 0.1 %, to the synthesis pot prior to degassing. In the
+case of undoped ZnS NCs, the 0.1 mL addition consists
+of DI water only. Inductively-coupled plasma - optical
+emission spectroscopy (ICP-OES) and PL measurement
+results (Figure 1b) show that varying the concentration of
+the Cu precursor directly influences the final Cu concen-
+tration in the NC samples, as well as the relative intensity
+of the red PL emitted by the NCs. The PL measurement
+results are discussed further in the next subsection.
+A representative TEM image of ZnS:Cu NCs with 0.1
+mol% Cu:Zn (Figure 1c) shows that the samples are com-
+posed of 7.2±1.2 nm diameter particles.
+The NC size
+distribution remains consistent across differently-doped
+samples (Figure 1d). Electron diffraction measurements
+(inset, Figure 1c) exhibit peaks at 2Θ values that corre-
+
+0.25
+100
+(a)
+(b)
+Red PL Component
+0.2
+80
+口
+0.15
+KOH
+60
+Cu(C2HgO2)2°H,0
+Inject
+0.1
+40
+DI water
+Zn(Ddtc)2
+Oleylamine
+Oleic Acid
+0.05
+20
+%
+precursor
+nanocrysta
+degasdegas
+decomposition
+formation
+0
+0
+0
+0.05
+0.1
+Cu:Znmoi%DuringSynthesis
+(c)
+22031
+10
+(d)
+111
+NC Diameter (nm)
+4
+0%0.05%0.075%0.1%
+Cu:Znmol%DuringSvnthesis3
+FIG. 2.
+Room-temperature optical properties.
+(a)
+PL spectra under continuous-wave excitation at λex=375 nm,
+normalized to the PL intensity at λem=442 nm from ZnS NCs
+synthesized with 0–0.1 mol% Cu:Zn. (b) Absorption spectra
+(black curves) and PLE spectra (colored curves) monitoring
+the emission intensity at λem=670 nm as a function of λex,
+from ZnS NCs synthesized with 0–0.1 mol% Cu:Zn.
+spond to the ⟨111⟩, ⟨220⟩, and ⟨311⟩ planes of sphalerite
+(zinc blende), according to PDF# 98-000-04053.
+B.
+Room-Temperature Optical Characterization
+PL emission spectra from NC samples containing the
+four different Cu concentrations are shown in Figure 2a.
+The intrinsic, background PL peak with emission wave-
+length, λem, between 430 nm and 600 nm, is present re-
+gardless of Cu concentration. This PL feature is char-
+acteristic of undoped ZnS NCs and is widely accepted
+to arise from radiative transitions between intrinsic de-
+fect states inside the ZnS bandgap created by Zn and S
+vacancies (VZn and VS) and interstitials (Zni and Si)[18–
+20]. Similar PL is also observed from bulk, undoped ZnS,
+with features being attributed to both intrinsic defects
+and unintentional impurities[21].
+Distinct emission with λem centered at 670 nm is ob-
+served in the Cu-doped NCs with a relative intensity that
+increases with the Cu:Zn molar ratio. The PL spectra
+of Figure 2a are decomposed using nonnegative matrix
+factorization (SI Section 1) in order to calculate the rela-
+tive strengths of the intrinsic and dopant-induced compo-
+nents, yielding the concentration dependence plotted in
+Figure 1b.
+Absorption spectra and λem=670 nm PLE
+spectra for all NC materials are shown in Figure 2b.
+From the absorption spectra, we construct Tauc plots (SI
+Section 2) and extract bandgap energies between 3.77 eV
+and 3.79 eV.
+The λem=670 nm PL and PLE spectra in Figure 2 align
+well with those reported for R-Cu in bulk ZnS:Cu, which
+also peaks at λem=670 nm at room temperature and is at-
+tributed to transitions between VS levels and CuZn levels
+inside the ZnS bandgap.[16] The PLE spectra of Figure
+2b show that the λem=670 nm PL is broadly excited by
+wavelengths in the range λex= 330 – 450 nm in all four
+samples, consistent with measurements by Shionoya et
+al. of R-Cu PLE in bulk ZnS:Cu[10]. The polarization
+dependence of the PLE in their report also indicates a
+nearest-neighbor configuration of VS and CuZn with C3v
+point-group symmetry[10]. R-Cu is quenched when bulk
+ZnS:Cu phosphors are fired in atmospheres containing
+high sulfur pressure[10], further supporting the role of
+VS levels in the PL.
+Compared to their bulk counterparts, impurity dop-
+ing of NC materials can be challenging to achieve and to
+confirm.[22] The alignment between the R-Cu PL/PLE
+spectra we measure here and those arising from bulk
+ZnS:Cu is suggestive of successful Cu doping of the
+ZnS:Cu NCs, since there is extensive evidence that R-
+Cu in bulk ZnS:Cu is activated by Cu substitutionally
+occupying Zn sites.
+We additionally carry out studies
+in which we deposit NC thin films and treat them with
+methanol and methanolic Na2S and Zn(CO2CH3)2·2H2O
+solutions known to remove organic ligands and to strip
+surface cations[23], and to enrich the NC surface in S2-
+or Zn2+, respectively,[24, 25] altering the presence or en-
+vironment of Cu cations if they are on the surface. In
+all cases, the surface treatments do not quench or en-
+hance the R-Cu PL from our NCs, again consistent with
+successful Cu doping of their cores (SI Section 3).
+C.
+Variable Temperature Studies Probing the
+Origins of Cu-Induced Sub-Bandgap PL/PLE
+NCs are drop-cast onto sapphire substrates and loaded
+inside an evacuated cryostat for temperature- and time-
+dependent spectroscopic measurements. Figure 3 shows
+PL/PLE maps of ZnS NCs without Cu doping (Cu:Zn at
+0 mol%) and with Cu doping (Cu:Zn 0.1 mol%), mea-
+sured at 19 K and at 290 K. PL from the undoped
+ZnS NCs is dominated by intrinsic PL at all tempera-
+
+1.5
+(a)
+0.1%
+Normalized PL Intensity
+Increasing
+Cuaddition
+0.075%
+0.5
+0.05%
+0%
+0
+400
+500
+600
+700
+800
+006
+Emission Wavelength,入..(nm)
+(b)
+ExcitationEnergy (eV)
+4.5
+4
+3.5
+3
+2.5
+Cu:Zn 0%
+Absorbance (a.u.)
+0.05%
+0.075%
+E
+e
+0.1%
+300
+350
+400
+450
+500
+Excitation Wavelength, ^ox (nm)4
+FIG. 3.
+Temperature-dependent PL/PLE (a) PL spec-
+tra (top) and PL/PLE maps (below) of films of ZnS NCs
+synthesized with 0 mol% (left) and 0.1 mol% Cu:Zn (right),
+measured at 19 K and room temperature. The PL spectra
+extracted from the 19 K PL/PLE maps are photoexcited at
+λex=375 nm and λex=320 nm for the ZnS NC and ZnS:Cu
+NC films, respectively, to show the clearest peak separation
+and spectrally reduce intrinsic PL in the case of ZnS:Cu NCs.
+(b) Energy level diagrams showing key defect states respon-
+sible for sub-bandgap PL emission in the undoped and doped
+NCs. Dashed lines represent shallow surface defect states. Ar-
+rows are used to indicate assigned radiative transitions in the
+doped NCs, which involve different sub-levels of the CuZn 3d
+shell. The t2 level is indicated with a heavier line to suggest
+that it may be further split into e and a sub-levels depending
+on the CuZn impurity site coordination.
+tures.
+The 19 K PL spectrum from the undoped ZnS
+NCs (λex=375 nm) can be described using three Gaus-
+sian peaks, which are labeled α, β, and γ in Figure 3a.
+Peaks α and β dominate the PL for λex > 330 nm (corre-
+sponding to below-bandgap excitation of the ZnS NCs),
+and their peak emission wavelength varies with λex. The
+third peak observable in the undoped film, peak γ, re-
+mains relatively fixed regardless of λex and is the only
+peak observed in this spectral region for λex <330 nm.
+For ZnS:Cu NCs, the 19 K PL spectrum (λex=320 nm)
+TABLE I. Peak positions and widths for I, II, R-Cu, and α,
+β, and γ, from Gaussian fitting of PL data shown in Fig. 3
+Cu:Zn mol% Label λem (nm) Eem (eV) FWHM (eV)
+0.1
+I
+471
+2.61
+0.13
+0.1
+II
+562
+2.19
+0.21
+0.1
+R-Cu
+709
+1.74
+0.27
+0
+α
+440
+2.81
+0.17
+0
+β
+442
+2.65
+1.12
+0
+γ
+560
+2.20
+0.30
+shows R-Cu emission, as well as blue and green emission
+peaks labeled I and II (Figure 3). The three PL peaks are
+observed for λex ranging from 290 – 420 nm. For λex <
+330 nm, to the blue of the ZnS bandgap wavelength, the
+sub-bandgap intrinsic PL is significantly diminished in
+intensity compared to peaks I, II, and R-Cu. The R-Cu
+peak is distinguishable at all temperatures, and the peak
+emission wavelength blueshifts from 709 nm at 19 K to
+670 nm at room temperature. This observation is similar
+to the reported blueshift in bulk ZnS:Cu from 700 nm at 4
+K to 670 nm at room temperature.[10, 16] Peaks I and II,
+with λem= 471 nm and λem= 562 nm, respectively, are
+quenched at room temperature. The λem and FWHM
+values for PL labeled in Figure 3 are listed in Table I.
+Line plots of the spectral data in the PL/PLE maps of
+Figure 3a are included in SI Section 5.
+Figure 3b shows energy level diagrams containing key
+defect levels believed to activate PL in the undoped and
+doped NCs. In undoped ZnS, intrinsic PL is assigned to
+transitions between Zni, VS, VZn, and Si levels, for which
+the relative energy levels shown are agreed upon, but the
+exact energies are not known[18, 26]. PL peaks activated
+by Cu doping the ZnS NCs, which can be spectrally sep-
+arated from intrinsic PL as discussed in this section, are
+assigned to radiative transitions in the diagram. R-Cu
+emission arises from transitions between states primarily
+associated with VS and the CuZn t2 levels[16]. We assign
+peak I to a transition between VS levels and the lower-
+lying CuZn e levels [8]. This assignment is supported by
+our measurement of a 0.87 eV energy difference between
+peak I and R-Cu PL, similar to the reported 0.86 eV en-
+ergy difference between CuZn t2 and e levels[27]. Peak II
+is assigned to transitions between donor levels that are
+shallower than VS, attributed here to surface defects, and
+the CuZn t2 levels[28]. We note that the state labels and
+identifications in Figure 3b are based on an approximate
+picture of isolated VS and CuZn in cubic ZnS, whereas
+the CuZn-VS is characterized by lowered C3v symmetry
+and hybridization between these levels. We discuss this
+point in more detail later in the next section, drawing
+insight from theoretical calculations.
+Figure 4a shows how time-resolved emission spec-
+troscopy can be used to isolate R-Cu PL from the intrin-
+sic background PL, since most of the intrinsic PL occurs
+within 250 ns of excitation while the R-Cu PL is longer
+lived. The room-temperature PL decay of ZnS:Cu NCs,
+excited with a pulsed excitation source at λex= 375 nm
+
+ZnS NCs
+ZnS:Cu NCs
+(a)
+375nmex.
+320nmex.
+R-Cu
+2
+1.5
+a
+Cts
+Cts
+0
+/105
+/105
+410
+10
+5
+(wu)
+370
+330
+3
+Excitation Wavelength.
+19K
+19K
+410
+2.6
+1
+2
+0.8
+370
+1.5
+0.6 
+0.4
+330
+0.5
+0.2 
+290K
+290K
+290
+430
+530
+630
+730
+830
+430
+530
+630
+730
+830
+Emission Wavelength,
+(nm)
+em
+(b)
+CBM
+CBM
+Vs
+Zn;
+Energy
+II R-Cu
+Vzn
+t2
+Cuzn
+S
+e
+VBM
+VBM5
+FIG. 4.
+Isolation of Cu-Activated PL (a) Time-resolved
+emission spectra from ZnS:Cu NC films at 290 K under 375
+nm, 1 kHz pulsed excitation, in which counts from the first
+250 ns following the laser pulse (black) are plotted separately
+from subsequent counts (red), effectively separating the in-
+trinsic background from the R-Cu peak emission.
+(b) PL
+spectra from ZnS:Cu NC (black trace) and undoped ZnS NC
+(grey trace) films, collected at 19 K with continuous wave, 375
+nm excitation. Intensities are normalized at 430 nm. The dif-
+ference spectrum (red curve) is almost identical to the time-
+gated spectrum from ZnS:Cu NC films under 375 nm, 500
+kHz pulsed excitation (purple dashed curve).
+and monitored at λem= 670 nm, is tri-exponential with
+decay time constants (τi) of τ1=1.85µs, τ2=8.72µs, and
+τ3=26.47µs.
+With 95% confidence, we find that these
+τi are consistent among all three Cu-doped samples (SI
+Section 4). At 19 K, time-resolved emission spectroscopy
+separates peaks I and II as well as R-Cu from the intrin-
+sic background PL. Figure 4b shows that time-gating the
+PL from the ZnS:Cu NC samples yields an almost iden-
+tical spectral shape to that of calculating the difference
+between the normalized, CW PL spectra from the doped
+NCs and the undoped NCs. The CW spectra in Figure 4b
+are normalized such that the PL intensities collected at
+430 nm (the shortest emission wavelength in the measure-
+ment) are the same, as PL at this wavelength is expected
+to arise predominantly from intrinsic defects. The obser-
+vation that nearly identical spectra are obtained, either
+by time-gating the doped spectrum or by subtracting the
+undoped CW spectrum, strongly supports that peaks I
+and II arise from Cu doping, along with the R-Cu peak,
+and these peaks coexist with the intrinsic PL in doped
+samples for sub-bandgap excitation.
+D.
+First-principles calculations
+R-Cu PL has been proposed to arise from a nearest-
+neighbor (NN) complex of CuZn and VS defects, rather
+than more distant associations[10]. To confirm the ther-
+modynamic stability of the NN CuZn-VS complex, we
+use density functional theory (DFT) to calculate the
+formation energies, defect levels, and projected density
+of states (PDOS) for ground-state configurations of the
+complex in several charge states, as well as for the next-
+nearest-neighbor (NNN) complex. The results of these
+calculations are shown in Figure 5. We find that the for-
+mation energy of the NN CuZn-VS complex is lower than
+that of the NNN complex. The formation energy calcu-
+lations in Figure 5a indicate that the two stable charge
+states are either negative (−1) or positive (+1), depend-
+ing on the Fermi level, with the neutral (0) configuration
+always lying higher in energy. This is in contrast to the
+calculation for NN CuZn-VS in an unrelaxed ZnS lattice,
+which significantly increases the formation energy of all
+charge states, but particularly the negative and neutral
+configurations.
+Figure 5b shows the defect levels and their projections
+at k = 0 for each charge state of the NN complex. These
+calculations qualitatively agree with the relative arrange-
+ment of levels in Figure 3b, with orange lines indicating
+the positions of two, higher-energy states derived from
+Zn dangling bonds surrounding the VS site, and green
+lines indicating ten, lower-energy states derived from the
+CuZn d-shell. The total density of states for pure ZnS
+and for ZnS containing a neutral CuZn-VS complex are
+included in SI Section 9. In the negatively-charged com-
+plex, all twelve states are occupied, and the VS-derived
+states are strongly mixed with the CuZn-derived states.
+In the neutral complex, the VS-derived states are only
+partially filled and are no longer mixed with the CuZn-
+derived states. In the positively-charged complex, only
+the CuZn-derived states are occupied and the VS-derived
+states are no longer easily isolated; likely because they
+have been pushed far into the conduction band; however,
+this may be an artifact of well-known DFT bandgap er-
+rors (the estimated bandgap in this calculation is 2.14
+eV, compared to the expected value around 3.6 eV), and
+the VS states may still exist within the bandgap.
+For the R-Cu transition depicted in Figure 3b to occur,
+there must be a hole in the higher-energy CuZn states.
+This hole is likely created by photo-ionization of a CuZn
+electron into the conduction band based on the large
+Stokes shift we observe between peak λem and peak λex
+for R-Cu PL. It has also been proposed that this Stokes
+shift is a result of lattice relaxation around the excited
+complex when a CuZn electron is transferred directly to
+a VS state[10, 29]. In this case, the excited VS level lies
+
+(a)
+250ns-250μs
+290 K
+0-250ns
+R-Cu
+Intrinsic PL
+0
+400
+500
+600
+700
+800
+EmissionWavelength,>
+(nm)
+(b)
+2.5
+R-Cu
+19 K
+Normalized PL Intensity
+6
+2
+1.5
+4
+0.5
+500
+600
+700
+800
+900
+Emission Wavelength,
+(nm)6
+FIG. 5.
+First-principles calculations (a) Formation en-
+ergies for nearest- and next-nearest-neighbor (NN and NNN,
+respectively) associations of CuZn and VS in ZnS, as a func-
+tion of the Fermi level (solid curves).
+Charge states with
+respect to the ZnS lattice are indicated as -1, 0, and +1 for
+the negatively charged, neutral, and positively charged com-
+plex, respectively.
+The dashed curve shows the formation
+energy for the NN complex in an unrelaxed ZnS lattice. All
+formation energy calculations are performed under zinc-rich
+sulfur-poor thermodynamical stability conditions. (b) Defect
+levels at k = 0 for three different charge states of the NN
+CuZn-VS complex. Orange lines indicate VS-derived states,
+and green lines indicate CuZn-derived states. Solid lines indi-
+cate the valence band maximum (0 eV) and conduction band
+minimum (2.14 eV). Dotted lines indicate the Fermi level.
+above the conduction band minimum immediately after
+excitation, and may therefore release an electron to the
+conduction band before being lowered into the bandgap
+following lattice relaxation. If the excited complex re-
+sulting from either of the above processes contains an
+electron in a Vs state, R-Cu emission can subsequently
+occur.
+Otherwise, an electron must be re-captured by
+the complex into a Vs state for R-Cu emission to occur,
+leading to a longer emission lifetime. Based on this ob-
+servation, the electron occupations of the defect levels in
+Figure 5b indicate how the charge state prior to excita-
+tion determines the possible emission pathways, which
+define the emission lifetime and the energy of the R-Cu
+PL.
+E.
+R-Cu Emission Dynamics
+Figure 6 shows how the spectral and temporal char-
+acteristics of the R-Cu PL as a function of temperature
+provide detailed insight regarding the emission mecha-
+nisms and the states involved. At temperatures from 19
+K to 290 K, we measure the PL emission spectrum to find
+the peak λem, and we then measure the corresponding PL
+decay curve for that λem. The PL spectra at each tem-
+perature are converted to energy units (see Methods) and
+fit using Gaussian functions to extract the peak energies
+(Eem) and integrated intensities. The emission spectra
+at 19 K, 110 K, 150 K, 190 K, and 290 K are plotted
+as examples in Figure 6a along with the corresponding
+fit results. The spectral data for all measurement tem-
+peratures are shown in the pseudocolor plot of the inset,
+and fitted spectra for all measurement temperatures not
+included in Figure 6a are shown in SI Section 6.
+For
+the PL decay measurements at each temperature (Fig-
+ure 6b), we find that a tri-exponential decay model most
+effectively describes the data compared to fitting with
+one, two, or four exponential terms or a stretched expo-
+nential decay function. The best-fit lifetime components,
+τi for i = 1,2,3 at every temperature are plotted in Figure
+6c, showing three, well-separated decay lifetimes.
+At the lowest measurement temperature of 19 K (Fig-
+ure 6d), we acquire PL decay curves across the R-Cu
+emission band with 2.5 nm resolution, and we fit the data
+to the tri-exponential model with fixed lifetimes based on
+the fit results from Figure 6c. Figure 6d shows the PL
+amplitudes corresponding to the decay components τ1,
+τ2, and τ3 as a function of λem.
+The fast component
+τ1 likely reflects the tail of one or more peaks outside
+the R-Cu emission band, with little spectral dependence.
+However, separating the slow (τ3) and fast (τ2) PL con-
+tributions this way reveals the presence of two distinct
+peaks at 1.73 eV and 1.82 eV. The observation of en-
+ergetically distinct PL peaks with different lifetimes is
+consistent with the co-existence of two separate radia-
+tive transitions.
+Figure 6e shows the integrated PL intensity over the
+R-Cu band and best-fit Eem at every temperature, ex-
+tracted from Gaussian fits to the PL data in Figure 6a.
+As noted previously, Eem blueshifts as the temperature
+increases, and Figure 6e illustrates that the shift oc-
+curs non-linearly, with a marked inflection between 100
+K and 200 K and saturation at both higher and lower
+temperatures. Meanwhile, the R-Cu emission intensity
+varies non-monotonically with temperature; it decreases
+with increasing temperature from 19 K to 190 K, then
+temporarily increases between 190 K and 210 K, be-
+fore decreasing again at higher temperatures. The ini-
+tial decrease in intensity is consistent with quenching
+through thermally-activated non-radiative recombination
+pathways and is typical for defect emission. The tempo-
+rary increase in intensity with increasing temperature is
+referred to as negative thermal quenching (NTQ) and
+is occasionally observed in defect emission; for example,
+
+(a)
+Formation Energy (eV)
+6
+0
+5.5
+5
+4.5
++1
+--NN,unrelaxed
+4
+NNN.relaxed
+NN,relaxed
+3.5
+0
+0.5
+1
+1.5
+2
+Fermi Level (eV)
+(b)
+(Cuzn-Vs)1- (Cuzn-Vs)0 (Cuzn-Vs)1+
+2.14
+(eV)
+Energy (
+cuZn
+07
+FIG. 6.
+R-Cu Emission Dynamics (a) PL spectra measured at temperatures ranging from 19 K-290 K (data points for five
+representative temperatures are shown, with all data plotted in the inset) and Gaussian fits (solid traces). (b) Time-dependent
+PL emission following pulsed excitation at λex=375 nm, measured at the peak PL wavelength for temperatures from 19 K to
+290 K. (c) PL decay lifetimes extracted from a tri-exponential fit to the time-dependent PL curves in (d) at each measurement
+temperature. (d) Amplitudes of each tri-exponential decay component at a single temperature (19 K) as a function of emission
+energy. (e) Integrated emission intensity (black points) and peak energy (colored points) as a function of temperature, extracted
+from the Gaussian fits of (a). Error bars represent 68% confidence intervals from fit results. The solid black curve is a fit to the
+intensity data using the model described in the text. Red and blue shaded regions represent the relative temperature-dependent
+intensities IA(T) and IB(T) from the best-fit model, and the dashed curve is a sum of the two emission energies resolved in (d),
+weighted by their corresponding best-fit emission intensities. (f) Energy level diagram showing two manifolds of states inside
+the ZnS bandgap with coupled relaxation processes, where radiative recombination from A to G results in 1.73 eV emission
+and radiative recombination from B to G results in 1.82 eV emission.
+in the case of the 2.65 eV PL (referred to as the YS1
+peak) from ZnS:I[30]. NTQ has generally been explained
+by thermally-activated carrier transfer from lower- to
+higher- energy emissive defect states. ZnS:Cu NCs have
+been synthesized in water at room temperature with the
+same Cu(CH3COO)2 precursor[8] and then subsequently
+annealed at 450 ◦C, but the resulting red peak (600 nm at
+room temperature) is not resolvable from other emission
+peaks when the temperature is less than 220 K, making
+it impossible to observe a similar NTQ or blueshift.
+In Figure 6f, we propose an empirical model to cap-
+ture both the temperature-dependent blueshift and the
+NTQ of R-Cu emission. Motivated by the time-resolved
+observations of Figure 6d, we include two radiative re-
+combination transitions with emission energies at 1.73 eV
+(A→G) and 1.82 eV (B→G), corresponding to two dis-
+tinct excited-state configurations. These radiative tran-
+sitions compete with thermally-activated, non-radiative
+transitions that generally tend to quench the emission
+at elevated temperatures. However, as carriers are ther-
+mally excited from state A to state B at temperatures
+with thermal energy corresponding to the energy offset
+ETR, the faster B→G transition increasingly becomes the
+dominant radiative recombination pathway, resulting in
+blueshifted PL and temporary NTQ. This mechanism is
+consistent with our observation that inflection points in
+the PL intensity align with the onset and saturation of
+the blueshift in Eem.
+To quantify this model, we derive the following ana-
+lytical expressions for the temperature-dependent PL in-
+tensities, I(A) and I(B), from the radiative transitions
+occurring from excited states A and B, respectively:
+IA(T) = IA(0)
+krA
+krA + knrA + kTR
+,
+(1)
+IB(T) = IB(0)
+krB
+krB + knrB
++ IA(0)
+kTRkrB
+krB + knrB
+.
+(2)
+Here, krA and krB are the radiative recombination rates
+shown in Figure 6f (solid lines), which are independent
+of temperature. The terms knrA and knrB are rates for
+non-radiative relaxation, and kTR is the rate for non-
+radiative transfer between states A and B (dashed lines
+in Figure 6f). These non-radiative rates are temperature-
+dependent with the form kj = Γj exp(−Ej/kBT), where
+Γj is a proportionality constant, Ej is the activation en-
+ergy of the transition, and kB is Boltzmann’s constant.
+See SI Section 7 for a derivation of these expressions.
+
+(b) 104
+(c)
+200
+19K
+50 
+.110K
+8
+19K
+150K
+(srl)
+150
+103
+72
+6
+190K
+290K
+T3
+290K
+Lifetime 
+250
+100
+4
+1.5
+Energy (eV)
+2
+50
+101
+0
+0
+1.4
+1.6
+1.8
+2
+2.2
+0
+0.5
+0
+100
+200
+300
+Emission Energy (eV)
+Time (ms)
+Temperature (K)
+(d)
+(e)
+(f)
+,=1.134 μs
+Component Amplitude
+00
+1.82
+B
+6
+T2=18.57 μs
+Counts/10
+10
+<Energy (eV)
+T3=180.8 μs
+1.8
+"
+TR
+8
+A
+6
+1.78
+KnrA
+KnrB
+Integrated 
+N
+4
+1.76
+Peak I
+KrA
+KrB
+2
+0
+1.74
+0
+1.6
+1.8
+2
+0
+100
+200
+300
+G
+A
+Emission Energy (eV)
+Temperature (K)8
+The sum of Equations (1) and (2) gives the total PL
+intensity as a function of temperature. This expression
+is used as a model to fit the temperature-dependent PL
+intensity data in Figure 6e, from which we extract best-
+fit values for the energies EnrA, EnrB, and ETR.
+The
+best-fit value of ETR is 153±22 meV. Details of the fit-
+ting procedure along with best-fit results for other pa-
+rameters are included in SI Section 7. To confirm the
+validity of this model, the dashed curve in Figure 6e
+represents a weighted sum of the 1.73 eV and 1.82 eV
+PL contributions, with weights determined by the best-
+fit IA(T) and IB(T) values (the intensities are repre-
+sented by shaded regions in Figure 6e).
+We recover
+a temperature-dependent emission energy that closely
+tracks the measured ∼90 meV blueshift of Eem between
+19 K and 290 K.
+The states A and B in our empirical model might be
+associated with different charge states or spatial config-
+urations of CuZn and VS. Based on the defect level cal-
+culations in Figure 5b, the energy levels associated with
+both CuZn and VS shift in different charge states, as do
+the degree of orbital hybridization between CuZn and VS
+states.
+The emission energies and lifetimes associated
+with different charge states should therefore be different.
+(Note, however, that this remains a qualitative obser-
+vation since the ground-state PDOS calculations do not
+fully capture the energies of excited-state configurations,
+which would be required to calculate emission energies.)
+We also consider the possibility that defects outside of
+CuZn-VS complexes are responsible for activating R-Cu
+at low or high temperatures. Possible candidates for de-
+fects creating donor levels in ZnS include Zni or Cui in-
+terstitial defects, as well as lone VS defects. Cui is a shal-
+low donor in ZnS[31] and is therefore unlikely to play a
+role as the donor level in R-Cu emission at any temper-
+ature. At low temperatures, it is possible that the pri-
+mary donors in the R-Cu emission mechanism are lone
+VS defects in the NC core or at the surface, because their
+more distant interaction with CuZn relative to VS in a
+CuZn-VS complex would be consistent with longer-lived
+and lower-energy radiative recombination. The inverse
+situation, in which lone CuZn defects are primary accep-
+tors at low temperatures, would be similar in a model
+considering holes instead of electrons; however, this sit-
+uation is less likely given the high concentration of VS
+defects expected to exist at the NC surface compared to
+the CuZn defects in these samples. Ultimately, carrier
+transfer between two manifolds of states may be advan-
+tageous for potential defect qubit architectures if it is
+found to be spin-dependent, or mitigated if it is found to
+be detrimental[32].
+We also considered other potential explanations for the
+R-Cu blueshift and NTQ. Previous reports of blueshifted
+R-Cu emission upon increasing temperature from bulk
+ZnS:Cu were attributed to changing occupation in the
+vibrational levels of a highly-localized center[16]. How-
+ever, that explanation is not consistent with the sat-
+uration in the blueshift at high temperatures, which
+we clearly observe and which also appears to occur in
+their measurement around 200 K. Characteristic defect
+PL in bulk and nanocrystalline ZnS:Mn also exhibits
+a blueshift upon increasing temperature with magni-
+tudes between 25 meV and 80 meV, which has been
+attributed to crystal field variations due to lattice ex-
+pansion [33, 34].
+The crystal-field explanation would
+also not predict saturating behavior, and accordingly the
+temperature-dependent blueshift of ZnS:Mn defect PL
+does not saturate, in contrast to the R-Cu observations.
+As an additional indication that crystal field effects can-
+not sufficiently explain the measured R-Cu blueshift, we
+calculate only a 16 ± 0.01 meV increase in the energy sep-
+aration between CuZn and VS levels of the neutral com-
+plex upon performing DFT computations with different
+ZnS lattice constants, corresponding to 0 K and 300 K
+based on the ZnS thermal expansion coefficient[35]. It is
+worth noting that in nanocrystalline ZnS:Mn, NTQ has
+also been reported between 50 K and 300 K, with posi-
+tive thermal quenching resuming above 300 K[34]. The
+authors attributed the NTQ to the thermal depopula-
+tion of localized trap states created by lattice defects,
+organic impurities, and surface defects which are all ex-
+pected to be more prevalent in NCs compared to bulk
+materials. None of the above alternative models for the
+R-Cu blueshift and NTQ can explain the clear presence
+of two distinct emission peaks with different radiative
+lifetimes at low temperature, as we observe in Figure 6d,
+nor do they capture the correspondence in Figure 6e be-
+tween the NTQ regime and the blueshift occurring at
+approximately the same temperature.
+CONCLUSION
+We present a synthetic method for obtaining colloidal
+ZnS:Cu NCs that emit primarily R-Cu, with a tailorable
+intensity depending on the Cu concentration.
+Using
+time- and temperature- resolved measurements and first-
+principles calculations, we find the sub-bandgap PL is
+consistent with radiative transitions from two, coupled
+manifolds of states involving CuZn-VS complexes.
+In
+the future, experimental methods unique to colloidal NC
+platforms will clarify the importance of defect type, lo-
+cation, concentration, and charge state on R-Cu PL. For
+example, post-synthesis NC modification (e.g., surface
+treatments that passivate traps and for remote doping,
+cation exchange, or sulfidation) and the growth of core-
+shell heterostructures will controllably alter the environ-
+ments and compositions of defects as well as the Fermi
+level of NCs.
+Spectroelectrochemical measurements of
+colloidal NC dispersions can also reveal the relationship
+between defect PL and the NC Fermi level. These stud-
+ies, combined with ESR and ODMR measurements that
+probe spin transitions, will yield valuable information
+about the R-Cu electronic structure and spin-dependent
+optical properties for potential development of R-Cu cen-
+ters as defect qubits.
+
+9
+Colloidal NC hosts will also facilitate the isolation and
+study of individual R-Cu color centers, compared to bulk
+hosts which have typically been used for the develop-
+ment of color centers as defect qubits. The deposition
+of sparse dispersions of colloidal NCs reduces the bulk
+purity requirement for single-quantum-emitter measure-
+ments by thousands of times[5, 36]. Furthermore, estab-
+lished methods for colloidal NC luminescence enhance-
+ment by integration with resonant photonic cavities or
+plasmonic nanostructures can improve the efficiency of
+quantum emitter measurements by reducing PL lifetimes
+through Purcell enhancement[37, 38].
+Our investigation of R-Cu color centers also moti-
+vates studying other transition-metal-vacancy complexes
+in ZnS as potential defect qubits; for example, transi-
+tion metals with fewer d-shell electrons than Cu, when
+placed in a similar defect and charge configuration, could
+produce a higher ground-state spin and a greater num-
+ber of internal radiative transitions. Such theoretically
+interesting materials systems are readily available for ex-
+perimentation through colloidal NC synthesis methods,
+which are fast and accessible compared to methods that
+exist for bulk crystals and can be atomically precise[39].
+All of the above opportunities will facilitate the develop-
+ment of color centers in ZnS as quantum defects while
+generally motivating colloidal NCs as a hosts for quan-
+tum defect development and engineering.
+III.
+METHODS
+1.
+Synthesis of Colloidal ZnS:Cu NCs
+A 10 mL solution of Cu(CH3COO)2·H2O in DI water
+is prepared with the appropriate molar concentration of
+Cu, i.e., 0.05%, 0.075%, or 0.1% of the molar concen-
+tration of Zn in the reaction. 0.1 mL of this solution is
+then added to a 50 mL three-necked flask containing 20
+mmol OM. The mixture is degassed for 45 min at 120
+◦C before the injection of 20 mmol OA and 0.2 mmol
+Zn(Ddtc)2, followed by 45 min additional degassing. The
+vessel is then heated to 300 ◦C under a nitrogen atmo-
+sphere and maintained at 300 ◦C for 45 min. It is then
+left to cool to 60 ◦C. The cooled contents are mixed with
+excess ethanol, and NCs are collected via centrifugation,
+washed in ethanol, and re-dispersed in hexanes to a con-
+centration of 10 mg/mL.
+2.
+Tools and Instrumentation
+ICP-OES measurements are collected using a SPEC-
+TRO GENESIS ICP-OES spectrometer. To collect TEM
+images, 2 mg/mL NC dispersions in hexanes are drop-
+cast onto carbon-coated copper grids and imaged us-
+ing a JEOL-1400 TEM. TEM images are analyzed us-
+ing Fiji[40]. Absorption spectra are measured using an
+Agilent Cary 5000 spectrometer. PL and PLE spectra
+are measured using an Edinburgh Instruments FLS1000
+spectrometer with a PMT-980 photodetector. For con-
+tinuous PL and PLE measurements, the excitation source
+is a 450W Xe lamp.
+For time-resolved measurements,
+the excitation source is a 375 nm Picoquant LDH-series
+laser diode. For temperature-controlled measurements,
+samples are placed in an evacuated Advanced Research
+Systems DE-202 cryostat. The illustration of a spherical
+ZnS NC in the Table of Contents graphic was generated
+using NanoCrystal[41].
+3.
+Analysis of PL Emission Spectra
+PL spectra are measured as a distribution function of
+wavelength and converted to energy units prior to Gaus-
+sian fitting to extract the positions and widths of individ-
+ual peaks. To properly account for the nonlinear relation-
+ship between wavelength and energy, we scale the spectra
+using the appropriate Jacobian transformation[42]:
+f(E) = f(λ) hc
+E2
+(3)
+The broad spectral range and large peak widths in our
+measurements make this scaling factor critical in our
+analysis. We find that simply converting the peak wave-
+lengths in the as-measured spectra to energy units would
+result in the extraction of dramatically incorrect peak en-
+ergies. This can be seen in the results of Table I, which
+take the scaling factor into account prior to peak extrac-
+tion on an energy scale.
+4.
+Computational Details
+The electronic structures of the bulk ZnS, and defect
+centers such as CuZn and CuZn-VS complexes in ZnS are
+studied using density functional theory (DFT) with the
+Vienna Simulation Package[43] (VASP). VASP employs
+the Perdew-Burke-Ernzerhof (PBE) functional for the ex-
+change and correlation within the augmented plane wave
+(PAW) scheme [44, 45]. For the supercell, we use two
+different sizes: one with 64 atoms and the other with
+212 atoms. Both calculations yield the same results for
+formation energies, electronic band structure, and total
+and projected DOS. We use a total energy cut-off of 300
+eV, and 6x6x6 and 12x12x12 Monkhorst-Pack k-point
+meshes for the density of states calculations in the larger
+and smaller supercells, respectively.
+The formation energies of differently charged config-
+urations are calculated from the well-known defect for-
+mula, [46]
+Eq
+f(ϵF ) = Eq
+tot − Ebulk
+tot
++ Ecorr +
+�
+i
+niµi
+(4)
++ q(EVBM + ϵF + ∆q/b)
+where the first two terms are the total energies of the
+bulk and defected supercell, and the correction term Ecorr
+
+10
+(first order Makov-Payne correction) originates from the
+interaction between periodic charged supercells.
+The
+chemical potentials µi correspond to adding Cu or re-
+moving Zn and S, ϵF is the Fermi level, EVBM is the
+valence band maximum. The final term ∆q/b is the po-
+tential alignment between the valence band edges for the
+bulk and neutral or charged supercells.
+Hybrid DFT calculations are also completed for the
+64 atom supercell using the B3LYP hybrid functional.
+Charge transition levels of the formation energies are not
+affected by the hybrid calculation, but the conduction
+band minimum is pushed from about 2.14 eV to 3.55
+eV, yielding a band gap energy closer to that which is
+observed experimentally (SI Section 8).
+IV.
+FINANCIAL INTEREST STATEMENT
+The authors declare no competing financial interest.
+ACKNOWLEDGMENTS
+This work was supported by the National Science
+Foundation under Awards DMR-2019444 (S.Y., C.B.M.,
+L.C.B., and C.R.K., for synthesis, measurements, and
+analysis), and DMREF awards DMR-1922278 (L.C.B)
+and DMR-1921877 (C.S. and M. E. F.) for theory, first-
+principles calculations, and data analysis.
+S.M.T. ac-
+knowledges support from the National Science Foun-
+dation Graduate Research Fellowship under Grant No.
+DGE-1845298.
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+
+Supporting Information for: Red Emission from
+Copper-Vacancy Color Centers in Zinc Sulfide
+Colloidal Nanocrystals
+Sarah M. Thompson,† C¨uneyt S¸ahin,‡,¶ Shengsong Yang,§ Michael E. Flatt´e,¶,∥
+Christopher B. Murray,§ Lee C. Bassett,∗,# and Cherie R. Kagan∗,†,@,△
+†Department of Electrical and Systems Engineering, University of Pennsylvania,
+Philadelphia Pennsylvania 19104, USA
+‡UNAM – National Nanotechnology Research Center and Institute of Materials Science
+and Nanotechnology, Bilkent University, Ankara, Turkey
+¶Department of Physics and Astronomy, University of Iowa, Iowa City IA, 52242, USA
+§Department of Chemistry, University of Pennsylvania, Philadelphia PA, 19104, USA
+∥Department of Applied Physics, Eindhoven University of Technology, P. O. Box 513, 5600
+MB Eindhoven, The Netherlands
+⊥Department of Materials Science and Engineering, University of Pennsylvania,
+Philadelphia PA, 19104, USA
+#Department of Electrical and Systems Engineering, University of Pennsylvania,
+Philadelphia PA, 19104, USA
+@Department of Materials Science and Engineering, University of Pennsylvania,
+Philadelphia Pennsylvania 19104, USA
+△Department of Chemistry, University of Pennsylvania, Philadelphia Pennsylvania 19104,
+USA
+E-mail: lbassett@seas.upenn.edu; kagan@seas.upenn.edu
+1
+
+1. Nonnegative Matrix Factorization of Room-Temperature
+Emission Spectra
+We use the built-in nnmf function in MATLAB R2021b to decompose room-temperature
+PL emission spectra from differently-doped ZnS:Cu NCs. The input matrix for factorization
+contains data from all materials and the rank of factors is 2. The relative weights of the
+decomposition components in each spectrum are used to plot the relative strength of the red
+spectral component in Figure 1b of the main text.
+Figure 1: NNMF decomposition of room-temperature PL emission spectra under 375 nm
+excitation from ZnS and ZnS:Cu NCs synthesized with 0 mol%, 0.05 mol%, 0.075 mol%, and
+0.1 mol% Cu:Zn. The two spectral components in the factorization are plotted in blue and
+orange, with their sum plotted in yellow and the measured data plotted in purple.
+2
+
+0at% Cu:Zn
+0.05at% Cu:Zn
+0.015
+0.015
+0.01
+0.01
+0
+0
+400
+600
+800
+400
+600
+800
+Emission Wavelength (nm)
+Emission Wavelength (nm)
+0.075at% Cu:Zn
+0.01
+0.1at% Cu:Zn
+0.01
+0.008
+0.008
+0.002
+0.002
+0
+0
+400
+600
+800
+400
+600
+800
+Emission Wavelength (nm)
+Emission Wavelength (nm)2. Measurement and Calculation of NC Bandgap Ener-
+gies
+Table 1:
+Size distribution and bandgap energies for Cu-doped ZnS NCs synthesized with
+four different Cu:Zn molar ratios.
+Cu:Zn mol%
+NC Size (nm)
+Avg. Calculated Bandgap (eV)
+Measured Bandgap (eV)
+0
+7.0 ± 1.3
+3.85
+3.79
+0.05
+7.4 ± 1.2
+3.84
+3.77
+0.075
+7.2 ± 1.1
+3.85
+3.79
+0.1
+7.5 ± 1.2
+3.83
+3.79
+Figure 2: Tauc plots of absorption spectra data for extraction of NC bandgap energies.
+3
+
+Cu:Zn0mol%
+Cu:Zn0.05mo1%
+10
+10
+Data
+Data
+8
+fit, xint=3.7855
+8
+fit, xint=3.7679
+6
+(ahv)
+6
+a
+4
+4
+2
+2
+0
+n
+3.6
+3.8
+4
+3.6
+3.8
+4
+hy/ey
+hv/eV
+Cu:Zn0.075mo1%
+Cu:Zn0.1mol%
+10
+10
+Data
+Data
+8
+fit. xint=3.7896
+8
+fit, xint=3.7883
+6
+(ahv)
+6
+4
+a
+4
+2
+2
+0
+3.6
+3.8
+4
+3.6
+3.8
+4
+hy/ey
+hv/eVThe average NC radius according to TEM image analysis is used to calculate a bandgap
+energy for each sample using Equation 1, in which R is the NC radius, m∗
+e is the effective
+electron mass (0.25m0), m∗
+h is the effective hole mass (0.59m0), ϵr is the relative permittivity
+(8.9), and Eg is the bandgap energy of bulk ZnS (3.68 eV).1
+Eg +
+π2ℏ2
+2m0R2( 1
+m∗
+e
++ 1
+m∗
+h
+) −
+1.8e2
+4πϵrϵ0R
+(1)
+Figure 3: TEM images of ZnS NCs synthesized with 0 mol%, 0.05 mol%, 0.075 mol%, and
+0.1 mol% Cu:Zn.
+4
+
+Oat% Cu:Zn
+0.05at% cu.zn
+100nm
+100nm
+0.075at%Cu:Zm
+0.1at%Cu:Zn
+100.nm
+100nm3. Effects of Surface Treatments on Red PL
+To measure the effects of altering the presence or environment of Cu cations if they are on the
+surface, we deposit NC solids on MPTS-treated Si wafer substrates and treat them according
+to the description in Figure 4. The treatments we use involve soaking NC films in methanol
+and methanolic Na2S and Zn(CO2CH3)2·2H2O solutions.
+Methanol is known to remove
+organic ligands and to strip surface cations,2 and methanolic Na2S and Zn(CO2CH3)2·2H2O
+solutions are known to enrich the NC surface in S2- or Zn2+, respectively3,4
+Figure 4:
+Room-temperature PL spectra under 375 nm excitation from ZnS:Cu NC
+films before and after treatments in methanol solutions containing either Na2S or
+Zn(CH3COO)2.H2O.
+5
+
+2500
+Untreated film
+1, 4
+2000
+2, 2, 4
+2, 3, 4
+1500
+3, 2, 4
+1000
+Flood sample with methanol for 10 seconds,
+500
+thenspinat2500rpmtoremoveexcess
+2500
+2. Flood sample with 10mM Na,S in methanol
+for 10 seconds, then spin at 2500rpm to
+2000
+counts
+removeexcess
+1500
+3. Flood sample with 10mM Zn(CH,COO)2.2H,0
+1000
+inmethanolfor1oseconds,thenspinat
+Inten
+2500rpmtoremoveexcess
+500
+4.Floodwithmethanolandimmediatelyspin
+0
+at 2500rpm for 30 seconds; repeat a total of
+400
+500
+600
+700
+500
+600
+700
+800
+3 times
+EmissionWavelength
+Emission Wavelength4. Room-Temperature Lifetime Measurements for All
+Copper Concentrations
+The room-temperature PL decay of the 670 nm emission peak from each NC dispersion
+(Figure 5) was measured using a PMT-980 photomultiplier (standard for the FLS1000 Pho-
+toluminescence Spectrometer), a 5 nm monochromator collection bandwidth, and 1 kHz,
+375 nm excitation. The parameters used to fit each signal to a tri-exponential decay (2) are
+given in Table 2 with 95% confidence intervals given in Table 3. The lifetimes τ1, τ2, and τ3
+extracted from these fits are consistent across samples, indicating that changing the copper
+concentration in this doping range has not noticeably altered the recombination kinetics for
+the associated red PL emission.
+I(t) = a1(t)e−t/τ1 + a2(t)e−t/τ2 + a3(t)e−t/τ3 + n
+(2)
+Figure 5: Room-temperature, 670 nm PL decay under 1 kHz, pulsed, 375 nm excitation
+(gray) with the full tri-exponential fit plotted in red and the three separate decay components
+plotted in black. Error bars represent the square root of the number of counts.
+6
+
+0.05 mo1% Cu:Zn
+0.075 mol% Cu:Zn
+0.1 mol% Cu:Zn
+104
+104
+2103
+102
+102
+101
+101
+101
+0
+50
+100
+150
+200
+250
+0
+50
+100
+150
+200
+250
+0
+50
+100
+150
+200
+250
+Time (μs)
+Time (μs)
+Time (μs)Table 2: Tri-exponential fit parameters for room-temperature PL decay.
+Fit Parameter
+0.05 mol% Cu:Zn
+0.075 mol% Cu:Zn
+0.1 mol% Cu:Zn
+τ1 (µs)
+1.57
+1.83
+1.85
+τ2 (µs)
+8.43
+8.33
+8.72
+τ3 (µs)
+27.65
+26.1
+26.47
+a1 (counts)
+1.07×104
+1.17×104
+1.64×104
+a2(counts)
+8746
+6734
+1.158×104
+a3 (counts)
+1561
+1715
+2278
+n (counts)
+12.95
+7.844
+8.538
+Table 3: 95% confidence bounds of tri-exponential fit parameters for room-temperature PL
+decay.
+Fit Parameter
+0.05 mol% Cu:Zn
+0.075 mol% Cu:Zn
+0.1 mol% Cu:Zn
+τ1 (µs)
+1.44–1.70
+1.67–1.99
+1.72–1.97
+τ2 (µs)
+7.93–8.92
+7.85–8.82
+8.32–9.13
+τ3 (µs)
+26.62–28.68
+25.08–27.12
+25.62–27.33
+a1 (counts)
+1.03×104–1.12×104
+1.11×104–1.23×104
+1.58×104–1.70×104
+a2(counts)
+8245–9247
+6386–7081
+1.108×104–1.209×104
+a3 (counts)
+1401–1720
+1518–1911
+2057–2500
+n (counts)
+12.68–13.22
+7.621–8.066
+8.317–8.759
+7
+
+5. 2D Plots of PL/PLE Data
+Spectral data used to construct the PL/PLE maps in Figure 3 of the main text are shown
+here as 2D plots to provide additional insight.
+Figure 6: PL spectra from undoped ZnS NCs, measured at 19 K and 290 K under excitation
+wavelengths from 290 nm to 420 nm.
+Figure 7: PL spectra from ZnS:Cu NCs, measured at 19 K and 290 K under excitation
+wavelengths from 290 nm to 420 nm.
+8
+
+ZnS NCs, 19 K
+ZnS NCs, 290 K
+10
+3
+Excitation Wavelength
+Excitation Wavelength
+8
+420 nm
+420 nm
+290 nm
+290 nm
+2
+0
+0
+500
+600
+700
+800
+900
+500
+600
+700
+800
+900
+Emission Wavelength (nm)
+Emission Wavelength (nm)ZnS:Cu NCs, 19 K
+ZnS:Cu NCs, 290 K
+8
+10
+Excitation Wavelength
+Excitation Wavelength
+Intensity (counts/10*)
+8
+420 nm
+420 nm
+6
+6
+290 nm
+290 nm
+2
+2
+0
+0
+500
+600
+700
+800
+900
+500
+600
+700
+800
+900
+Emission Wavelength (nm)
+EmissionWavelength(nm)6. Fitting R-Cu Temperature-Dependent Spectra
+PL spectra are fit to two Gaussian peaks at each measurement temperature from 19 K to
+290 K. Signal data are measured in counts per unit wavelength and converted to counts per
+unit energy for Gaussian fitting. For this conversion, the signal intensities are multiplied by
+a Jacobian transformation factor.5 The fits are obtained using a weighted least squares anal-
+ysis, where the weights are the uncertainty at each data point according to the assumption
+that measurement uncertainty is dominated by shot noise. The uncertainty at each point is
+therefore taken to be the Poisson variance, which is simply the number of counts, before any
+correction has been applied to the signal to account for the variable efficiency of the detector
+across wavelengths.
+The dominant peak between 1.73 eV and 1.82 eV at all temperatures corresponds to
+R-Cu emission and the higher-energy peak at approximately 2.2 eV corresponds to peak II
+in the main text. The fit results for all measured spectra are plotted below in Supporting
+Information Figure 8. The R2 values for each fit are given in Supporting Information Figure
+9.
+9
+
+Figure 8: PL spectra measured at 19 K–290 K (blue circles) and corresponding Gaussian
+fits (red lines).
+Figure 9: R2 values for the Gaussian fits in Supplemental Figure 8 as a function of measure-
+ment temperature.
+10
+
+19 K
+10×106
+10
+×106
+30 K
+10 2
+×106
+50 K
+10×106
+70 K
+10×106
+90 K
+data
+data
+data
+data
+data
+8
+ft
+8
+8
+fit
+8
+fit
+8
+fit
+6
+6
+6
+4
+4
+4
+2
+1.4
+1.6
+1.8
+2
+2.2
+2.4
+1.5
+2
+2.5
+1.5
+2
+2.5
+1.5
+2
+2.5
+1.5
+2
+2.5
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+110 K
+×106
+140 K
+150K
+×106
+170 K
+×106
+190 K
+data
+data
+data
+5
+data
+data
+6
+fit
+fit
+6
+fit
+fit
+fit
+4
+Counts
+2
+2
+2.22.4
+2
+2.5
+1.5
+2
+2.5
+1.6
+1.8
+2.2
+0
+1.4
+1.6
+1.8
+2
+1.5
+1.4
+2
+2.4
+1.5
+2
+2.5
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+6×106
+210 K
+230 K
+5×106
+250 K
+5106
+270 K
+×106
+290 K
+6×106
+data
+data
+data
+4
+data
+data
+fit
+4
+fit
+fit
+Counts
+Counts
+2
+2
+1.5
+2.5
+1.5
+2
+2.5
+1.5
+2
+2.5
+1.5
+2
+2.5
+1.4
+2.22.4
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)
+Emission Energy (eV)0.998
+O
+R 0.996
+0.994
+0
+100
+200
+300
+Temperature (K)7. Derivation of NTQ Equation and Best-Fit Results
+Figure 10: Model electronic structure which is proposed to produce the measured R-Cu
+emission dynamics. Transitions occur between a ground state, G, and two excited states, A
+and B. Solid arrows indicate radiative transitions, and dashed arrows indicate nonradiative
+transitions. Labels accompanying each arrow are transition rates. Thermal carrier transfer
+from A to B occurs with a rate kTR and activation energy ETR and is key in producing the
+measured NTQ.
+First, we define the time derivatives of nA(T) and nB(T), which represent the electron
+populations of states A and B as functions of time (t) and temperature (T), using equations
+3 and 4:
+( ∂
+∂t)nA(t, T) = GA(t, T) − nA(t, T)(krA + knrA) − nA(t, T)kTR
+(3)
+( ∂
+∂t)nB(t, T) = GB(t, T) − nB(t, T)(krB + knrB) + nA(t, T)kTR
+(4)
+where krA and krB are the radiative rates of recombination, knrA and knrB are non-radiative
+rates of recombination, and kTR is the non-radiative electron transfer rate between states A
+and B. The non-radiative rates are temperature dependent and given by:
+knri = Γnrie−Enri/kBT
+i = A, B, TR
+Solving for nA(T) and under steady-state conditions gives Equation 5:
+11
+
+B
+....
+一
+A
+....
+-
+-
+-
+KrA
+W
+-
+GnA(T) =
+GA(T)
+krA + knrA + kTR
+(5)
+We make the approximation that generation rates GA(T) and GB(T) are independent of
+temperature. Considering that the PL intensity from radiative A→G transitions, IA(T), is
+equal to the radiative rate krA times the excited state population nA(T), we obtain Equation
+6, where GA(0) = IA(0):
+IA(T) = IA(0)
+krA
+krA + knrA + kTR
+(6)
+When we write out the full, temperature-dependent forms of the non-radiative rates and
+re-arrange the result using the constants CA and CTR defined below, we obtain Equation 7:
+CA = ΓnrA/krA
+CTR = ΓTR/krA
+IA(T) =
+IA(0)
+1 + CAe−EnrA/kBT + CTRe−ET R/kBT
+(7)
+Solving for nB(T) under steady state conditions gives Equation 8, which is dependent
+upon nA(T):
+nB(T) = GB(T) + nA(T)kTR
+krB + knrB
+(8)
+12
+
+Again, multiplying nB(T) by the radiative rate krB to get the PL intensity IB(T) and
+making the approximation that GB(T) is constant (such that IB(0) = GB(0)), we write
+Equation 9:
+IB(T) = IB(0)krB + nA(T)kTRkrB
+krB + knrB
+(9)
+Expanding equation 9 gives us the following exact expression for nB(T):
+IB(T) = IB(0)
+krB
+krB + knrB
++ (
+kTRkrB
+krB + knrB
+)(
+IA(0)
+krA + knrA + kTR
+)
+(10)
+We again use the proportional relationship between the PL intensity from radiative B→G
+transitions and the population nB(T) to write an expression for IB(T). When we write out
+the full, temperature-dependent forms of the non-radiative rates and re-arrange the result
+using the constants CA and CTR as well as CB defined below, we obtain Equation 11:
+CB = ΓnrB/krB
+IB(T) =
+IB(0)
+1 + CBe−EnrB/kBT +
+IA(0)CTRe−ET R/kBT
+(1 + CBe−EnrB/kBT)(1 + CAe−EnrA/kBT + CTRe−ET R/kBT) (11)
+We now use Equations 7 and 11 to fit measured I(T) data, which correspond to the
+integral of the Gaussian fit for the R-Cu peak at every temperature, such that I(T) =
+IA(T) + IB(T). We first use an approximate version of Equation 11 to fit the measured I(T)
+data, then take the resulting parameters as seed values for the fit to the exact equation. To
+write the approximate version of Equation 11, we expand the product in the denominator of
+Equation 11:
+1 + CAe−EnrA/kBT + CBe−EnrB/kBT + CTRe−ET R/kBT + CBCAe−(EnrB+EnrA)/kBT +
+CBCTRe−(EnrB+ET R)/kBT
+13
+
+This expanded product contains two terms with effective activation energies EnrB +EnrA
+and EnrB + ETR. Assuming these effective activation energies are large compared to the
+measurement temperatures in our experiment, we neglect these terms in the approximate
+expression for IB(T).
+The total PL intensity is I(0) = IA(0) + IB(0). We define a proportionality factor WA
+such that WA = IA(0)/I(0) and 1 − WA = IB(0)/I(0). The result is Equation 12
+IB(T) ≈
+(1 − WA)I(0)
+1 + CBe−EnrB/kBT +
+WAI(0)CTRe−ET R/kBT
+1 + CAe−EnrA/kBT + CBe−EnrB/kBT + CTRe−ET R/kBT
+(12)
+Equations 13 and 14 are the exact equations for IA(T) and IB(T) used to fit measured
+I(T) data, in terms of the fit parameters listed below.
+IA(T) =
+WAI(0)
+1 + CAe−EnrA/kBT + CTRe−ET R/kBT
+(13)
+IB(T) =
+(1 − WA)I(0)
+1 + CBe−EnrB/kBT +
+WAI(0)CTRe−ET R/kBT
+(1 + CBe−EnrB/kBT)(1 + CAe−EnrA/kBT + CTRe−ET R/kBT) (14)
+14
+
+We use the measurement results in Figure 6d to fix the value of Wa for fitting and find
+that the energy parameters are reasonable well constrained while the coefficients CA, CB, and
+CTR are virtually unconstrained. The values of the parameters that best describe measured
+I(T) data are given below, with 68% confidence intervals in parentheses:
+WA = 0.9285
+I(0) = 1.074 × 109 counts (1.067, 1.08)
+EnrA = 105.8 meV (68.1, 143.5)
+EnrB = 214.3 meV (166.4, 262.1)
+ETR = 152.7 meV (131.8, 173.6)
+CA = 1372
+CB = 5301
+CTR = 1.47 × 104
+15
+
+8. Hybrid DFT Calculations for Formation Energies
+Figure 11: Formation energies for negatively charged, neutral, and positively charged (-1,
+0, and +1 charges with respect to the ZnS lattice as indicated on plots) nearest- and next-
+nearest-neighbor (NN and NNN, respectively) associations of CuZn and VS in ZnS, as a
+function of the Fermi level. Solid lines indicate calculations performed for a relaxed lattice.
+Dashed lines indicate calculations performed for an unrelaxed lattice. Black and grey lines
+indicate DFT calculations, and the red line indicates hybrid DFT calculations. Details of
+the DFT settings are in the Methods section of the main text.
+16
+
+-NN,unrelaxed
+NNN,relaxed
+1
+0
+NN,relaxed
+6
++1
+NN,relaxed,hybrid calculation
++1
+1
++1
+-1
++1
+0
+1
+2
+3
+FermiLevel(eV)9. Total Density of States of Pure and Defected ZnS
+-4
+-2
+0
+2
+4
+6
+-40
+-20
+0
+20
+40
+Energy (eV)
+DOS (electrons/eV)
+-4
+-2
+0
+2
+4
+6
+-40
+-20
+0
+20
+40
+Energy (eV)
+DOS (electrons/eV)
+Vs
+Vs
+Cu d
+(a)
+(b)
+Figure 12: The total DOS of (a) pure ZnS (b) ZnS with neutral CuZn-VS complex. The CuZn
+d-levels are closer the the valence band maximum and VS states are split into two distinct
+energies due to the nonzero total magnetic moment in the system introduced by the CuZn
+impurity. Here zero of the energy is chosen as the top of the valence band of pure ZnS.
+17
+
+References
+1. Uzar, N.; Arikan, C. M. Synthesis and investigation of optical properties of ZnS nanos-
+tructures. Bull. Mater. Sci. 2011, 34, 287–292.
+2. Goodwin, E. D.; Diroll, B. T.; Oh, S. J.; Paik, T.; Murray, C. B.; Kagan, C. R. Effects of
+Post-Synthesis Processing on CdSe Nanocrystals and Their Solids: Correlation between
+Surface Chemistry and Optoelectronic Properties. Journal of Physical Chemistry C 2014,
+118, 27097–27105.
+3. Oh, S. J. S.; Berry, N. E. N.; Choi, J.-H.; Gaulding, E. A.; Lin, H.; Paik, T.; Diroll, B.
+B. T.; Muramoto, S.; Murray, C. B. C.; Kagan, C. C. R. Designing high-performance
+PbS and PbSe nanocrystal electronic devices through stepwise, post-synthesis, colloidal
+atomic layer deposition. Nano letters 2014, 14, 1559–1566.
+4. Kim, D. K.; Fafarman, A. T.; Diroll, B. T.; Chan, S. H.; Gordon, T. R.; Murray, C. B.;
+Kagan, C. R. Solution-Based Stoichiometric Control over Charge Transport in Nanocrys-
+talline CdSe Devices. ACS nano 2013, 7, 8760–70.
+5. Mooney, J.; Kambhampati, P. Get the Basics Right: Jacobian Conversion of Wavelength
+and Energy Scales for Quantitative Analysis of Emission Spectra. The Journal of Physical
+Chemistry Letters 2013, 4, 3316–3318.
+18
+

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+Model-based Offline Reinforcement Learning with Local Misspecification
+Kefan Dong*, Yannis Flet-Berliac*, Allen Nie*, Emma Brunskill
+Stanford University
+{kefandong,yfletberliac,anie,ebrun}@stanford.edu
+Abstract
+We present a model-based offline reinforcement learning pol-
+icy performance lower bound that explicitly captures dynam-
+ics model misspecification and distribution mismatch and we
+propose an empirical algorithm for optimal offline policy se-
+lection. Theoretically, we prove a novel safe policy improve-
+ment theorem by establishing pessimism approximations to
+the value function. Our key insight is to jointly consider se-
+lecting over dynamics models and policies: as long as a dy-
+namics model can accurately represent the dynamics of the
+state-action pairs visited by a given policy, it is possible to
+approximate the value of that particular policy. We analyze
+our lower bound in the LQR setting and also show compet-
+itive performance to previous lower bounds on policy selec-
+tion across a set of D4RL tasks.
+Introduction
+Offline reinforcement learning (RL) could leverage histor-
+ical decisions made and their outcomes to improve data-
+driven decision-making in areas like marketing (Thomas
+et al. 2017), robotics (Quillen et al. 2018; Yu et al.
+2020, 2021; Swazinna, Udluft, and Runkler 2020; Singh
+et al. 2020), recommendation systems (Swaminathan and
+Joachims 2015), etc. Offline RL is particularly useful when
+it is possible to deploy context-specific decision policies, but
+it is costly or infeasible to do online reinforcement learning.
+Prior work on offline RL for large state and/or action
+spaces has primarily focused on one of two extreme settings.
+One line of work makes minimal assumptions on the under-
+lying stochastic process, requiring only no confounding, and
+leverages importance-sampling estimators of potential poli-
+cies (e.g., Thomas, Theocharous, and Ghavamzadeh (2015);
+Thomas et al. (2019)). Unfortunately, such estimators have a
+variance that scales exponentially with the horizon (Liu et al.
+2018b) and are often ill-suited to long horizon problems1.
+An alternative, which is the majority of work in offline
+RL, is to make a number of assumptions on the domain,
+*These authors contributed equally.
+Copyright © 2022, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1Marginalized importance sampling (MIS) methods (Liu et al.
+2018a; Xie, Ma, and Wang 2019; Yin and Wang 2020; Liu, Bacon,
+and Brunskill 2020) help address this but rely on the system being
+Markov in the underlying state space
+behavior data generation process and the expressiveness of
+the function classes employed. The work in this space typi-
+cally assumes the domain satisfies the Markov assumption,
+which has been recently shown in the off-policy evaluation
+setting to enable provably more efficient policy value esti-
+mation (Kallus and Uehara 2020). Historically, most work
+(e.g., Munos (2003); Farahmand, Munos, and Szepesv´ari
+(2010); Xie and Jiang (2020); Chen and Jiang (2019)) as-
+sumes the batch data set has coverage on any state-action
+pairs that could be visited under any possible policy. More
+recent work relaxes this strong requirement using a pes-
+simism under uncertainty approach that is model-based (Yu
+et al. 2020, 2021; Kidambi et al. 2020), model-free (Liu et al.
+2020) or uses policy search (Curi, Berkenkamp, and Krause
+2020; van Hasselt, Hessel, and Aslanides 2019). Such work
+still relies on realizability/lack of misspecification assump-
+tions. For model-free approaches, a common assumption is
+that the value function class can represent all policies. Liu
+et al. (2020) assume that the value function class is closed
+under (modified) Bellman backups. A recent exception is
+Xie and Jiang (2020), which only requires the optimal Q-
+function to be representable by the value function class.
+However, their sample complexity scales non-optimally (Xie
+and Jiang 2020, Theorem 2), and they also make strong
+assumptions on the data coverage – essentially the dataset
+must visit all states with sufficient probability. Model-based
+approaches such as Malik et al. (2019); Yu et al. (2020) as-
+sume the dynamics class has no misspecification.
+These two lines of work hint at possibilities in the mid-
+dle: can we leverage the sample-efficient benefits of Markov
+structure and allow for minimal assumptions on the data-
+gathering process and potential model misspecification?
+This can be viewed as one step towards more best-in-class
+results for offline RL. Such results are relatively rare in RL,
+which tends to focus on obtaining optimal or near-optimal
+policies for the underlying domain. Yet in many important
+applications, it may be much more practical to hope to iden-
+tify a strong policy within a particular policy class.
+Our insight is that the algorithm may be able to lever-
+age misspecified models and still leverage the Markov as-
+sumption for increased data efficiency. In particular, we take
+a model-based offline RL approach to leverage dynamics
+models that can accurately fit the space of state-action pairs
+visited under a particular policy (local small misspecifica-
+
+tion), rather than being a good model of the entire possi-
+ble state-action space (global small misspecification). Our
+work is most closely related to the recently proposed Min-
+imax Model Learning (MML) algorithm (Voloshin, Jiang,
+and Yue 2021): MML optimizes for the model that mini-
+mizes a value-aware error which upper bounds the differ-
+ence of policy value in learned and real models. If the con-
+sidered model class includes the true model, this can work
+very well, but when the models are misspecified, this can be-
+come overly conservative since it optimizes with respect to
+a worst-case potential state-action distribution shift.
+The key feature of our algorithm is to jointly optimize pol-
+icy and dynamics. Prior model-based offline RL algorithms
+typically estimate dynamics first, and then optimize a policy
+w.r.t. the learned dynamics (Yu et al. 2020, 2021; Voloshin,
+Jiang, and Yue 2021). But when the dynamics model class is
+misspecified, there may not exist a unique “good dynamics”
+that can approximate the value of every policy. As a result,
+the learned policy may have a good estimated value under
+the learned dynamics, but a poor performance in the real en-
+vironment, or the learned policy may be overly conservative
+due to the misestimated dynamics.
+Our paper makes the following contributions. First, we
+provide a finite sample bound that assumes a Markov model,
+leverages the pessimism principle to work with many data-
+gathering distributions, accounts for estimation error in the
+behavior policy and, most importantly, directly accounts
+for dynamics and value function model misspecification
+(see Lemma 3). We prove the misspecification error of our
+method is much tighter than other approaches because we
+only look at the models’ ability to represent visited state-
+action pairs for a particular policy. In that sense, we say
+our algorithm relies on small local model dynamics mis-
+specification. In Theorem 6, we show that when the dynam-
+ics model class does not satisfy realizability, decoupling the
+learning of policy and dynamics is suboptimal. This moti-
+vates our algorithm which jointly optimizes the policy and
+model dynamics across a finite set. Because of the tighter
+pessimistic estimation, we can prove a novel safe policy im-
+provement theorem (see Theorem 4) for offline policy opti-
+mization (OPO). While our primary contribution is theoreti-
+cal, our proposed method for policy selection improves over
+the state-of-the-art MML Voloshin, Jiang, and Yue (2021) in
+a simple linear Gaussian setting, and has solid performance
+on policy selection on a set of D4RL benchmarks.
+Related Works
+There is an extensive and growing body of research on of-
+fline RL and we focus here on methods that also assume a
+Markov domain. Many papers focus on model-free meth-
+ods (e.g., Fujimoto et al. (2018); Kumar et al. (2019, 2020)).
+Nachum et al. (2019) and their follow-ups (Zhang et al.
+2019; Zhang, Liu, and Whiteson 2020) learn a distribution
+correction term, on top of which they perform evaluation or
+policy optimization tasks. Uehara, Huang, and Jiang (2020);
+Jiang and Huang (2020) study the duality between learn-
+ing Q-functions and learning importance weights. Liu et al.
+(2020) explicitly consider the distribution shift in offline RL
+and propose conservative Bellman equations.
+Another line of research uses model-based methods (Ki-
+dambi et al. 2020; Yu et al. 2020, 2021; Matsushima et al.
+2020; Swazinna, Udluft, and Runkler 2020; Fu and Levine
+2021; Farahmand, Barreto, and Nikovski 2017). Gelada
+et al. (2019); Delgrange, Nowe, and P´erez (2022); Voloshin,
+Jiang, and Yue (2021) learn the dynamics using different
+loss functions. Yu et al. (2020) build an uncertainty quan-
+tification on top of the learned dynamics and select a policy
+that optimizes the lower confidence bound. (Argenson and
+Dulac-Arnold 2020; Zhan, Zhu, and Xu 2021) focus on pol-
+icy optimization instead of model learning.
+In Table 1, we compare our error bounds with existing
+results. Our statistical error (introduced by finite dataset) is
+comparable with VAML (Farahmand, Barreto, and Nikovski
+2017), MBS-PI (Liu et al. 2020) and MML (Voloshin, Jiang,
+and Yue 2021). In addition, we consider misspecification er-
+rors and safe policy improvement (SPI).
+Algorithm
+Statistical Error
+Misspecification
+SPI
+VAML
+�
+O
+�
+p
+√n
+�
+2
+(global)
+
+MBS-PI
+�
+O
+�
+Vmaxζ
+(1−γ)2√n
+�
+(global)
+
+MML
+Rn3
+(global)
+
+Ours
+�
+O
+�
+Vmax
+1−γ
+�
+ζ
+n
+�
+(local)
+
+Table 1: Comparison of error bounds with prior works.
+Problem Setup
+A Markov Decision Process (MDP) is defined by a tuple
+⟨T, r, S, A, γ⟩ . S and A denote the state and action spaces.
+T : S × A → ∆(S) is the transition and r : S × A → R+
+is the reward. γ ∈ [0, 1) is the discount factor. For a policy
+π : S → ∆(A), the value function is defined as
+V π
+T (s) = Es0=s,at∼π(st),st+1∼T (st,at)[�∞
+t=0 γtr(st, at)].
+Let Rmax ≜ maxs,a r(s, a) be the maximal reward and
+Vmax ≜ Rmax/(1 − γ). Without loss of generality, we as-
+sume that the initial state is fixed as s0. We use η(T, π) ≜
+V π
+T (s0) to denote the expected value of policy π. Let
+ρπ
+T (s, a) ≜ (1 − γ) �∞
+t=0 γt Prπ
+T (st = s, at = a | s0)
+be the normalized state-action distribution when we execute
+policy π in a domain with dynamics model T. For simplicity
+in this paper we assume the reward function is known.
+An
+offline
+RL
+algorithm
+takes
+a
+dataset
+D
+=
+{(si, ai, s′
+i)}n
+i=1 as input, where n is the size of the dataset.
+Each (si, ai, s′
+i) tuple is drawn independently from a behav-
+ior distribution µ. We assume that µ is consistent with the
+MDP in the sense that µ(· | s, a) = T(s, a) for all (s, a).
+For simplicity, we use ˆE to denote the empirical distribu-
+tion over the dataset D. In this paper, we assume that the
+2VAML only considers linear function approximation and p is
+the dimension of the feature vector.
+3The Rademacher complexity. For the finite hypothesis, the
+best-known upper bound is in the same order of ours.
+
+algorithm has access to an estimated behavior distribution ˆµ
+such that TV(µ, ˆµ) is small. This estimation can be easily
+obtained using a separate dataset (e.g., Liu et al. (2020)).
+The algorithm can access three (finite) function classes
+G, T , Π. G is a class of value functions, T a class of dy-
+namics and Π a class of policies. We assume that g(s, a) ∈
+[0, Vmax] for all g ∈ G. We use T ⋆ to denote the ground-
+truth dynamics. Note that T ⋆ may not be in T . Our goal is
+to return a policy π ∈ Π that maximizes η(T ⋆, π).
+Main Results
+A standard model-based RL algorithm learns the dynamics
+models first, and then uses the learned dynamics to estimate
+the value of a policy, or optimize it. In this approach, it is
+crucial to link the estimation error of the dynamics to the
+estimation error of the value. Therefore, as a starting point,
+we invoke the simulation lemma.
+Lemma 1 (Simulation Lemma (Yu et al. 2020; Kakade and
+Langford 2002)). Consider two MDPs with dynamics T, T ⋆,
+and the same reward function. Then,
+η(T, π) − η(T ⋆, π) =
+γ
+1 − γ E(s,a)∼ρπ
+T [
+Es′∼T (s,a)[V π
+T ⋆(s′)] − Es′∼T ⋆(s,a)[V π
+T ⋆(s′)]
+�
+.
+(1)
+For a fixed ground-truth dynamics T ⋆, we define
+Gπ
+T (s, a) = Es′∼T (s,a)[V π
+T ⋆(s′)] − Es′∼T ⋆(s,a)[V π
+T ⋆(s′)].
+The simulation lemma states that the dynamics will
+provide an accurate estimate of the policy value if
+Es′∼T (s,a)[V π
+T ⋆(s′)] matches Es′∼T ⋆(s,a)[V π
+T ⋆(s′)]. In other
+words, to obtain a good estimate of a policy value, it is suf-
+ficient to minimize the model error Gπ
+T (s, a).
+Since the value function V π
+T ⋆ is unknown, Yu et al. (2020)
+upper bound the model error by introducing a class of test
+functions G : S → R. When V π
+T ⋆ ∈ G, we have
+|Gπ
+T (s,a)|≤supg∈G
+��Es′∼T (s,a)g(s′)−Es′∼T ⋆(s,a)g(s′)]
+��.
+In an offline dataset D, typically we can only observe one
+sample from T ⋆(s, a) per state-action pair. Hence the al-
+gorithm cannot compute this upper bound exactly. In ad-
+dition, the distribution of the dataset D is also different
+from the one required by the simulation lemma ρπ
+T . To ad-
+dress these issues, we explicitly introduce a density ratio
+w : S × A → R+. For a test function g ∈ G and a dynam-
+ics model T, let f g
+T (s, a) ≜ Es′∼T (s,a)[g(s′)]. Recall that ˆE
+denotes the empirical expectation over dataset D. Then our
+model loss is defined as
+ℓw(T, g) = |ˆE[w(s, a)(f g
+T (s, a) − g(s′))]|.
+(2)
+Distribution mismatch. We aim to upper bound policy eval-
+uation error by the loss function even if there are state ac-
+tion pairs with small probability mass under behavior dis-
+tribution µ (i.e., the offline dataset does not have a perfect
+coverage). Following Liu et al. (2020), we treat the un-
+known state-action pairs pessimistically. Let ζ be a fixed
+cutoff threshold. Recall that ˆµ is an estimation of the behav-
+ior distribution. For a policy π and dynamics T, we define
+wπ,T (s, a) ≜ I
+�
+ρπ
+T (s,a)
+ˆµ(s,a) ≤ ζ
+�
+ρπ
+T (s,a)
+ˆµ(s,a) as the truncated den-
+sity ratio. For a fixed policy π, when w = wπ,T ,
+���E(s,a)∼ρπ
+T
+�
+Gπ
+T (s, a)
+����
+≤
+����E(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) ≤ ζ
+�
+Gπ
+T (s, a)
+�����
++
+���E(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+�
+Gπ
+T (s, a)
+����
+≤ |E(s,a)∼ˆµ
+�
+w(s, a)Gπ
+T (s, a)
+�
+|
++ Vmax
+���E(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+�����
+≤ |E(s,a)∼µ
+�
+w(s, a)Gπ
+T (s, a)
+�
+| + ζVmaxTV (ˆµ, µ)
++ Vmax
+������E(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+������.
+Hence, ignoring statistical error due to finite dataset, we can
+upper bound the estimation error |η(T ⋆, π) − η(T, π)| by
+γ
+1 − γ
+�
+sup
+g∈G
+���ℓwπ,T (g, T)
+��� + ζVmaxTV (ˆµ, µ)
++ VmaxE(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+���
+.
+(3)
+Intuitively, the first term measures the error caused by im-
+perfect dynamics T, the second term captures the estimation
+error of the behavior distribution, and the last term comes
+from truncating the density ratios.
+Pessimistic Policy Optimization with Model
+Misspecification
+In this section, we explicitly consider misspecifications of
+the function classes used for representing the value func-
+tion and dynamics models (G and T , respectively). Most
+prior theoretical work on model-based RL make strong as-
+sumptions on the realizability of the dynamics model class.
+For example, in the offline setting, Voloshin, Jiang, and Yue
+(2021) focus on exact realizability of the dynamics model
+(that is, T ⋆ ∈ T ). In the online setting, Jin et al. (2020) pro-
+vide bounds where there is a linear regret term due to global
+model misspecification. Their bounds require a T ∈ T such
+that TV (T(s, a), T ⋆(s, a)) ≤ ϵ for all (s, a), even if the
+state-action pair (s, a) is only visited under some poorly per-
+forming policies. We now show that offline RL tasks can
+need much weaker realizability assumptions on the dynam-
+ics model class.
+Our key observation is that for a given dynamics T and
+policy π, computing the density ratio wπ,T is statistically
+efficient. Note that to compute wπ,T we do not need any
+samples from the true dynamics: instead, we only need to be
+able to estimate the state-action density under a dynamics
+model T for policy π. This allows us to explicitly utilize the
+density ratio to get a relaxed realizability assumption.
+Definition 2. The local value function error for a particular
+
+dynamics model T and policy π is defined as
+ϵV (T, π) ≜ inf
+g∈G |E(s,a)∼µ[wπ,T (s, a)(Es′∼T (s,a)[(g − V π
+T ⋆)(s′)]
++ Es′∼T ⋆(s,a)[(g − V π
+T ⋆)(s′)])]|.
+The term ϵV measures the local misspecification of the
+value function class – that is, the error between the true
+value of the policy V π
+T ⋆ and the best fitting value function
+in the class G – only on the state-action pairs that policy π
+visits under a particular potential dynamics model T. In con-
+trast, previous results (Jin et al. 2020; Nachum et al. 2019;
+Voloshin, Jiang, and Yue 2021) take the global maximum
+error over all (reachable) (s, a), which can be much larger
+than the local misspecification error ϵV (T, π).
+With this local misspecification error, we can establish a
+pessimistic estimation of the true reward. Let E be a high
+probability event under which the loss function ℓwπ,T (T, g)
+is close to its expectation (randomness comes from the
+dataset D). In the Appendix, we define this event formally
+and prove that Pr(E) ≥ 1 − δ. The following lemma gives
+a lower bound on the true reward. Proofs, when omitted, are
+in the Appendix.
+Lemma 3. Let ι = log(2|G||T ||Π|/δ). For any dynamics
+model T and policy π, we define
+lb(T, π) = η(T, π) −
+1
+1 − γ
+�
+sup
+g∈G
+ℓwπ,T (g, T)
++ VmaxE(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+���
+.
+(4)
+Then, under the event E, we have
+η(T ⋆, π) ≥ lb(T, π) −
+1
+1 − γ
+�
+ϵV (T, π)
+− 2Vmax
+�
+ζι/n − ζVmaxTV (ˆµ, µ)
+�
+.
+(5)
+We use this to define our offline policy selection Alg. 1.
+Algorithm 1: Model-based Offline RL with Local
+Misspecification Error
+Require: estimated behavior distribution ˆµ,
+truncation threshold ζ.
+for π ∈ Π, T ∈ T do
+Compute wπ,T (s, a) = I
+�
+ρπ
+T (s,a)
+ˆµ(s,a) ≤ ζ
+�
+ρπ
+T (s,a)
+ˆµ(s,a) .
+Compute lb(T, π) by Eq. (4).
+end
+π ← argmaxπ∈Π maxT ∈T lb(T, π).
+In contrast to existing offline model-based algorithms (Yu
+et al. 2020; Voloshin, Jiang, and Yue 2021), our algorithm
+optimizes the dynamics and policy jointly. For a given dy-
+namics model, policy pair, our Alg. 1 computes the trun-
+cated density ratio wπ,T which does not require collecting
+new samples and then uses this to compute a lower bound
+lb(T, π) (Eq. (4)). Finally, it outputs a policy that maximizes
+the lower bound. We will shortly show this joint optimiza-
+tion can lead to better offline learning.
+Parameter ζ controls the truncation of the stationary im-
+portance weights. Increasing ζ decreases the last term in the
+lower bound objective lb(T, π), but it may also increase the
+variance given the finite dataset size. Note that by setting
+ζ = log(n) and letting n → ∞ (i.e., with infinite data), the
+last term in Eq. (4) and the statistical error converge to zero.
+Safe Policy Improvement
+We now derive a novel safe policy improvement result, up
+to the error terms given below. Intuitively this guarantees
+that the policy returned by Alg. 1 will improve over the be-
+havior policy when possible, which is an attractive property
+in many applied settings. Note that recent work (Voloshin,
+Jiang, and Yue 2021; Yu et al. 2020) on model-based of-
+fline RL does not provide this guarantee when the dynamics
+model class is misspecified. For a fixed policy π, define
+ϵρ(π) ≜ infT ∈T E(s,a)∼ρπ
+T ⋆ [TV (T(s, a), T ⋆(s, a))], (6)
+ϵµ(π) ≜ E(s,a)∼ρπ
+T ⋆
+�
+I
+�ρπ
+T ⋆(s, a)
+ˆµ(s, a)
+> ζ/2
+��
+.
+(7)
+The term ϵρ measures the local misspecification error of the
+dynamics model class in being able to represent the dynam-
+ics for state-action pairs encountered for policy π. ϵµ rep-
+resents that overlap of the dataset for an alternate policy π:
+such a quantity is common in much of offline RL. In the fol-
+lowing theorem, we prove that the true value of the policy
+computed by Alg. 1 is lower bounded by that of the optimal
+policy in the function class with some error terms.
+Theorem 4. Consider a fixed parameter ζ. Let ˆπ be the pol-
+icy computed by Alg. 1 and ˆT = argmaxT lb(T, ˆπ). Let
+ι = log(2|G||T ||Π|/δ). Then, with probability at least 1−δ,
+we have
+η(T ⋆, ˆπ) ≥ sup
+π
+�
+η(T ⋆, π) − 6Vmaxϵρ(π)
+(1 − γ)2
+− Vmaxϵµ(π)
+1 − γ
+�
+− ϵV ( ˆT, ˆπ)
+1 − γ
+− 4Vmax
+1 − γ
+�
+ζι
+n − 2ζVmaxTV (ˆµ, µ)
+1 − γ
+.
+(8)
+To prove Theorem 4, we prove the tightness of lb(T, π) —
+the lower bound maxT lb(T, π) is at least as high as the true
+value of the policy with some errors. Consequently, maxi-
+mizing the lower bound also maximizes the true value of the
+policy. Formally speaking, we have the following Lemma.
+Lemma 5. For any policy π ∈ Π, under the event E we have
+max
+T ∈T lb(T, π) ≥ η(T ⋆, π) − 6Vmaxϵρ(π)/(1 − γ)2
+−
+1
+1 − γ
+�
+Vmaxϵµ(π) − 2Vmax
+�
+ζι/n − ζVmaxTV (ˆµ, µ)
+�
+.
+In the sequel, we present a proof sketch for Lemma 5.
+In this proof sketch, we hide 1/(1 − γ) factors in the big-
+O notation. For a fixed policy π, let ˆT be the minimizer of
+Eq. (6). We prove Lemma 5 by analyzing the terms in the
+definition of lb( ˆT, π) (Eq. (4)) separately.
+i. Following the definition of Eq. (6), we can show that
+∥ρπ
+ˆT − ρπ
+T ⋆∥1
+≤
+O(ϵρ(π)). Consequently we get
+η( ˆT, π) ≥ η(T ⋆, π) − O(ϵρ(π)).
+
+ii. Recall that 0 ≤ g(s, a) ≤ Vmax for all g
+∈ G.
+Then for any (s, a) we have supg∈G |Es′∼ ˆT (s,a)g(s′) −
+Es′∼T ⋆(s,a)g(s′)]| ≤ VmaxTV( ˆT(s, a), T ⋆(s, a)). Com-
+bining the definition of ℓw(g, T), Eq. (6) and statistical
+error we get supg∈G ℓwπ,T (g, T) ≤ �
+O(ϵρ(π) + 1/√n +
+VmaxTV (ˆµ, µ)) under event E.
+iii. For the last term regarding distribution mismatch, we
+combine Eq. (7) and Lemma 8. We can upper bound this
+term by O(ϵρ(π) + ϵµ(π)).
+iv. The final term arises due to the potential estimation error
+in the behavior policy distribution.
+Theorem 4 follows directly from combining Lemma 3 and
+Lemma 5. Note that Theorem 4 accounts for estimation er-
+ror in the behavior policy, misspecification in the dynamics
+model class, and misspecification in the value function class,
+the latter two in a more local, tighter form than prior work.
+Illustrative Example
+To build intuition of where our approach may yield benefits,
+we provide an illustrative example where Alg. 1 has better
+performance than existing approaches: an MDP whose state
+space is partitioned into several parts. The model class is re-
+stricted so that every model can only be accurate on one part
+of the state space. When each deterministic policy only vis-
+its one part of the state space, the local misspecification error
+is small — for each policy, there exists a dynamics model
+in the set which can accurately estimate the distribution of
+states and actions visited under that policy. In contrast, if the
+dynamics are learned to fit the whole state space, the estima-
+tion error will be large.
+More precisely, for a fixed parameter d, consider a MDP
+where S = {s0, · · · , sd} ∪ {sg, sb}. There are d actions
+denoted by a1, · · · , ad. The true dynamics are deterministic
+and given by
+T ⋆(s0, ai) = si,
+T ⋆(si, aj) =
+�sg,
+if I [i = j] ,
+sb,
+if I [i ̸= j] ,
+(9)
+T ⋆(sg, ai) = sg,
+T ⋆(sb, ai) = sb, ∀i ∈ [d].
+(10)
+And the reward is r(s, ai) = I [s = sg] , ∀i ∈ [d].
+The transition function class T is parameterized by θ ∈
+Rd. For a fixed θ, the transition for states s1, . . . , sd is
+Tθ(si, aj) =
+�sg,
+w.p. 1
+2
+�
+1 + e⊤
+j θ
+�
+,
+sb,
+w.p. 1
+2
+�
+1 − e⊤
+j θ
+�
+,
+(11)
+where ej is the j-th standard basis of Rd. The transitions
+for states s0, sg, sb is identical to the true dynamics T ⋆.
+But the transition model Tθ in the function class must use
+the same parameter θ to approximate the dynamics in states
+s1, · · · , sd, which makes it misspecified.
+Decoupling learning the dynamics model and policy is
+suboptimal. Most prior algorithms first learn a dynamics
+model and then do planning with that model. However, note
+here that the optimal action induced by MDP planning given
+a particular Tθ is suboptimal (assuming a uniformly random
+tie-breaking). This is because, for any given θ, that dynam-
+ics model will estimate the dynamics of states s1, · · · , sd
+as being identical, with identical resulting value functions.
+Note this is suboptimality will occur in this example even if
+the dataset is large and covers the state–action pairs visited
+by any possible policy (ϵµ(π) = 0), the value function class
+is tabular and can represent any value function ϵV = 0, the
+behavior policy is known or the resulting estimation error is
+small (TV (ˆµ, µ) = 0, and ζ = 0). In such a case, Theo-
+rem 4 guarantees that with high probability, our algorithm
+will learn the optimal policy because there exist couplings
+of the dynamics models and optimal policies such that the
+local misspecification error ϵρ = 0. This demonstrates that
+prior algorithms (including MML (Voloshin, Jiang, and Yue
+2021)) that decouple the learning of dynamics and policy
+can be suboptimal. We now state this more formally:
+Theorem 6. Consider any (possibly stochastic) algorithm
+that outputs an estimated dynamics Tθ ∈ T . Let πθ be the
+greedy policy w.r.t. Tθ (with ties breaking uniformly at ran-
+dom). Then
+max
+π
+η(T ⋆, π) − η(T ⋆, πθ) ≥ (A − 1)γ2
+A(1 − γ) .
+(12)
+As a side point, we also show that the off-policy estima-
+tion error in Voloshin, Jiang, and Yue (2021) is large when
+the dynamics model class is misspecified in Proposition 7.
+We defer this result to the Appendix.
+Experiments
+While our primary contribution is theoretical, we now inves-
+tigate how our method can be used for offline model-based
+policy selection with dynamics model misspecification. We
+first empirically evaluate our method on Linear-Quadratic
+Regulator (LQR), a commonly used environment in optimal
+control theory (Bertsekas et al. 2000), in order to assess: Can
+Algorithm 1 return the optimal policy when we have both
+model and distribution mismatch? We also evaluate our ap-
+proach using D4RL (Fu et al. 2020), a standard offline RL
+benchmark for continuous control tasks. Here we consider:
+Given policies and dynamics pairs obtained using state-of-
+the-art offline model-based RL methods with ensemble dy-
+namics, does Alg. 1 allow picking the best policy, outper-
+forming previous methods?
+Linear-Quadratic Regulator (LQR)
+LQR is defined by a linear transition dynamics st+1 =
+Ast + Bat + η, where st ∈ Rn and at ∈ Rm are state and
+action at time step t, respectively. η ∼ N(0, σ2I) is ran-
+dom noise. LQR has a quadratic reward function R(s, a) =
+−(sT Qs + aT Ra) with Q ∈ Rn×n and R ∈ Rm×m be-
+ing positive semi-definite matrices, Q, R ⪰ 0. The op-
+timal controller to maximize the sum of future rewards
+�H
+t=1 −(sT
+t Qst+aT
+t Rat) until the end of horizon H has the
+form at = −Kst (K ∈ Rm×n) (Bertsekas et al. 2000). The
+value function is also a quadratic function, V (s) = sT Us+q
+for some constant q and positive semi-definite matrix U ⪰ 0.
+In the experiment, the state space is [−1, 1].
+Misspecified transition classes. Consider a 1D version of
+LQR with A(x) = (1 + x/10), B(x) = (−0.5 − x/10),
+
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+K
+-12
+-9
+-6
+-3
+0
+Return
+Returns of different policies under true environment
+Ours
+MML
+1
+2
+3
+4
+5
+Rank
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Negative of lower bound
+(0.00,-0.25)
+(0.00,0.00)
+(0.00,0.25)
+(0.20,0.25)
+(0.20,-0.25)
+Ranking imposed by Eq 6 on policy-model pair
+(T, )
+Model Loss+Distribution Shift
+0.1
+0.2
+0.3
+0.4
+MBLB
+MML
+MOPO
+D4RL IQM
+Normalized Score
+Figure 1: Left: Visualization of true policy value η(T ⋆, π). Our algorithm picks the optimal policy, whereas MML picks a
+suboptimal policy. Middle: Visualization of negative lower bounds lb(T, π) for different policies and models (indexed by the
+values of (v, u)). Right: We show the interquartile mean (IQM) scores of two model-based lower bounds (MML and MBLB)
+and a recent model-based policy learning algorithm (MOPO) on D4RL.
+Q = 1, R = 1 and noise η ∼ N(0, 0.05). Our true dy-
+namics is given by x∗ = 6, and the corresponding optimal
+policy has K = −1.1. Function classes used by Alg. 1 are
+finite and computed as follows: (i) the value function class
+G contains the value functions of 1D LQR with parameters
+x ∈ {2, 4, 10} and K ∈ {−1.1, −0.9, −0.7}; (ii) the transi-
+tion class T is misspecified. We use the following transition
+class Tu ∈ T parametrized by u,
+Tu =
+�st+1 = A(x∗)st − B(x∗)at,
+st ∈ [u, u + 1],
+st+1 = st,
+otherwise,
+with u ∈ {−0.75, −0.5, −0.25, 0, 0.25}. In other words,
+the capacity of the transition class is limited – each func-
+tion can only model the true dynamics of a part of the
+states; (iii) the policy class is given by πv parameterized
+by v, and πv(s) = −1.1(s − v) + N(0, 0.01) with v ∈
+{−0.6, −0.4, −0.2, 0, 0.2, 0.4, 0.6}. Intuitively, πv tries to
+push the state toward s = v.
+Since the state and action spaces are one dimensional, we
+can compute the density ratio wπ,T efficiently by discretiza-
+tion. The implementation details are deferred to Appendix.
+Baseline. We compare our algorithm to minimizing MML
+loss as described in the OPO algorithm of Voloshin, Jiang,
+and Yue (2021, Algorithm 2). MML strictly outperformed
+VAML (Farahmand, Barreto, and Nikovski 2017) as shown
+in the experiments of (Voloshin, Jiang, and Yue 2021);
+hence, we only compare to MML in our experiments.
+Results. Figure 1 (Left) shows the return of different poli-
+cies under the true environment. Our method picks the op-
+timal policy for the true model, whereas MML picks the
+wrong policy. In Figure 1 (Middle), we also visualize dif-
+ferent terms in the definition of lb(T, π) (Eq. (5)). Note that
+the model loss for different policy is different (model loss for
+(v, u) = (0, 0) is significantly larger than (0.0.−0.25), even
+if the dynamics are the same). This is because the model loss
+is evaluated with a different density ratio.
+This highlights the main benefit of our method over the
+baseline. Since the model class is misspecified, maximizing
+over the weight function w in the MML loss results in an
+unrealistically large loss value for some models. However,
+if the chosen policy does not visit the part of the state space
+with a large error, there is no need to incur a high penalty.
+D4RL
+D4RL (Fu et al. 2020) is an offline RL standardized bench-
+mark designed and commonly used to evaluate the progress
+of offline RL algorithms. This benchmark is standard for
+evaluating offline policy learning algorithms. Here, we use
+a state-of-the-art policy learning algorithm MOPO (Yu et al.
+2020) to propose a set of policy-transition model tuples –
+for N policy hyperparameters and K transition models, we
+can get M × K tuples: {(π1, T1), (π1, T2), ..., (πN, TK)}.
+The MOPO algorithm learns an ensemble of transition mod-
+els and randomly chooses one to sample trajectories during
+each episode of training. Instead, we choose one transition
+model to generate trajectories for the policy throughout the
+entire training. In our experiment, we choose M = 1 and
+K = 5, and train each tuple for 5 random seeds on Hopper
+and HalfCheetah tasks (see Appendix). We then compute the
+model-based lower bound for each (πi, Tj), and select the
+optimal policy that has the highest lower bound. We learn the
+dynamics using 300k iterations and we train each policy us-
+ing 100k gradient iterations steps with SAC (Haarnoja et al.
+2018) as the policy gradient algorithm, imitating MOPO (Yu
+et al. 2020) policy gradient update.
+MML.
+Voloshin, Jiang, and Yue (2021) recommended
+two practical implementations for computing MML lower
+bounds. The implementation parametrizes w(s, a)V (s′)
+jointly via a new function h(s, a, s′). We refer readers to
+Prop 3.5 from Voloshin, Jiang, and Yue (2021) for a detailed
+explanation. We describe how we parametrize this function
+as follows:
+• Linear: Voloshin, Jiang, and Yue (2021) showed that if
+T, V, µ are all from the linear function classes, then a
+model T that minimizes MML loss is both unique and
+identifiable. This provides a linear parametrization of
+h(s, a, s′) = ψ(s, a, s′)T θ, where ψ is a basis function.
+We choose ψ to be either a squared basis function or a
+polynomial basis function with degree 2.
+• Kernel: Using a radial basis function (RBF) over S ×
+
+Dataset Type
+Env
+MOPO
+MML
+(Squared)
+MML
+(Polynomial)
+MML
+(RKHS)
+MBLB
+(Linear)
+MBLB
+(Quad)
+medium
+hopper
+175.4
+(95.3)
+379.4
+(466.4)
+375.6
+(459.5)
+375.0
+(459.9)
+591.7
+(523.1)
+808.5
+(502.7)
+med-expert
+hopper
+183.8
+(94.4)
+160.9
+(131.5)
+116.5
+(148.4)
+61.4
+(35.0)
+261.1
+(157.9)
+242.5
+(134.0)
+expert
+hopper
+80.4
+(63.4)
+93.8
+(87.9)
+61.6
+(61.9)
+70.0
+(56.2)
+118.2
+(61.6)
+121.0
+(72.5)
+medium
+halfcheetah
+599.8
+(668.4)
+1967.6
+(1707.5)
+2625.1
+(937.2)
+3858.2
+(1231.1)
+3290.4
+(1753.1)
+2484.2
+(1526.8)
+med-expert
+halfcheetah
+-486.6
+(48.1)
+-188.5
+(137.2)
+-77.0
+(252.5)
+-343.2
+(225.2)
+207.4
+(509.5)
+192.8
+(432.0)
+Table 2: We report the mean and (standard deviation) of selected policy’s simulator environment performance across 5 random
+seeds. MML and MBLB are used as model-selection procedures where they select the best policy for each seed. Our method is
+choosing the most near-optimal policy across the datasets.
+A × S and computing K((s, a, s′), (˜s, ˜a, ˜s′)), Voloshin,
+Jiang, and Yue (2021) showed that there exists a closed-
+form solution to compute the maxima of the MML loss
+(RKHS). Here, there is no need for any gradient update,
+we only sample s′ from T.
+MBLB (Ours).
+For a continuous control task, we compute
+our model-based lower bound (MBLB) as follows:
+Compute η(T, π). Although it is reasonable to directly use a
+value function V π
+T trained during policy learning to compute
+η(T, π), Paine et al. (2020); Kumar et al. (2021) points out
+how this value function often severely over-estimates the ac-
+tual discounted return. Therefore, we estimate the expected
+value of policy π using the generalized advantage estima-
+tor (GAE) (Schulman et al. 2016). For a sequence of tran-
+sitions {st, at, r(st, at), st+1}t∈[0,N], it is defined as: At =
+�t+N
+t′=t (γλ)t′−t(r(st′, at′) + γVφ(st′+1) − Vφ(st′)), with λ
+a fixed hyperparameter and Vφ the value function estimator
+at the previous optimization iteration. Then, to estimate the
+value function, we solve the non-linear regression problem
+minimizeφ
+�t+N
+t′=t (Vφ(st′)− ˆVt′)2 where ˆVt = At+Vφ(st′).
+We also provide a comparison to using the standard TD-1
+Fitted Q Evaluation (FQE) (Le, Voloshin, and Yue 2019) in-
+stead in Table A1 in the Appendix. We find that using GAE
+provides better policy evaluation estimations.
+Behavior density modeling. We use a state-of-the-art nor-
+malizing flow probability model to estimate the density of
+state-action pairs (Papamakarios et al. 2021). For ρπ
+T , we
+sample 10,000 trajectories from T, π, and estimate the cor-
+responding density; for the behavior distribution µ, we use
+the given dataset D. We empirically decide the number of
+training epochs that will give the model the best fit.
+Compute supg∈G |ℓwπ,T (g, T)|. We parametrize g either as
+a linear function of state: g(s) = mT s, or a quadratic func-
+tion of the state: g(s) = sT Ms + b. We use gradient ascent
+on ℓwπ,T (g, T) to maximize this objective.
+Results. We report the results in Table 2. There is gen-
+eral overlap across seeds for the performance between vari-
+ous methods, but our approach has the best average perfor-
+mance or is within the standard deviation of the best. We also
+show that for different choices of how we parameterize the
+w(s, a)V (s′) distribution (MML) and how we choose the
+family of g test function (MBLB), we are selecting differ-
+ent final policies. However, overall, MBLB can pick better-
+performing final policies with two different parametrizations
+while MML is choosing lower-performing policies with its
+three parametrizations. We find that our approach of select-
+ing among the set of policies computed from each of the
+models used by MOPO consistently outperforms the policy
+produced by MOPO in the considered tasks.
+To summarize these results, we report the interquartile
+mean (IQM) scores of each method in Figure 1 (Right). IQM
+is an outlier robust metric proposed by Agarwal et al. (2021)
+to compare deep RL algorithms. We create the plot by sam-
+pling with replacement over all runs on all datasets 50000
+times. Though there is significant overlap, our method gen-
+erally outperforms policies learned from MOPO.
+Conclusion
+There are many directions for future work. The current
+lb(T, π) implementation with density ratio wπ,T (s, a) is not
+differentiable: an interesting question is to make this differ-
+entiable so that we can directly optimize a policy. Another
+interesting question would be to construct estimators for the
+local misspecification errors ϵρ, ϵµ and ϵV , which could be
+used to refine the model class to optimize performance.
+To conclude, this paper studies model-based offline rein-
+forcement learning with local model misspecification errors,
+and proves a novel safe policy improvement theorem. Our
+theoretical analysis shows the benefit of this tighter analy-
+sis and approach. We illustrate the advantage of our method
+over prior work in a small linear quadratic example and
+also demonstrate that it is competitive or has stronger per-
+formance than recent model-based offline RL methods on
+policy selection in a set of D4RL tasks.
+
+Acknowledgment
+Research reported in this paper was sponsored in part by
+NSF grant #2112926, the DEVCOM Army Research Lab-
+oratory under Cooperative Agreement W911NF-17-2-0196
+(ARL IoBT CRA) and a Stanford Hoffman-Yee grant. The
+views and conclusions contained in this document are those
+of the authors and should not be interpreted as representing
+the official policies, either expressed or implied, of the Army
+Research Laboratory or the U.S.Government. The U.S. Gov-
+ernment is authorized to reproduce and distribute reprints for
+Government purposes notwithstanding any copyright nota-
+tion herein.
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+Missing Proofs
+High Probability Events
+In this section, we introduce concentration inequalities and define the high probability events.
+Define the following quantities
+L(π, g, T) = E(s,a,s′)∼µ
+�
+wπ,T (s, a)(Ex∼T (s,a)[g(x)] − Ex∼T ⋆(s,a)[g(x)])
+�
+,
+(13)
+l(π, g, T) = E(s,a,s′)∼D[wπ,T (s, a)(f g
+T (s, a) − g(s′))].
+(14)
+Recall that ι = log(2|G||T ||Π|/δ). Consider the event
+E =
+�
+|L(π, g, T) − l(π, g, T)| ≤ 2Vmax
+�
+ζι
+n ,
+∀π ∈ Π, g ∈ G, T ∈ T
+�
+.
+(15)
+In the following, we show that
+Pr (E) ≥ 1 − δ.
+(16)
+Recall that D = {(si, ai, s′
+i)}n
+i=1 where (si, ai, s′
+i) ∼ µ are i.i.d. samples from distribution µ. For fixed π ∈ Π, g ∈ G, T ∈ T ,
+we have E[ˆl(π, g, T)] = l(π, g, T). Meanwhile, note that
+|wπ,T (s, a)(f g
+T (s, a) − g(s′))| ≤ ζVmax,
+(17)
+E(s,a,s′)∼µ[wπ,T (s, a)2(f g
+T (s, a) − g(s′))2]
+(18)
+≤ E(s,a,s′)∼ρπ
+T [wπ,T (s, a)(f g
+T (s, a) − g(s′))2] ≤ V 2
+maxζ.
+(19)
+By Bernstein inequality, with probability at least 1 − δ/(|G||T ||Π|),
+|L(π, g, T) − l(π, g, T)| ≤
+�
+2V 2
+maxζ log(2|G||T ||Π|/δ)
+n
++ ζVmax
+3n
+log(2|G||T ||Π|/δ)
+(20)
+Recall that ι = log(2|G||T ||Π|/δ). When n ≥ ζ we have
+|L(π, g, T) − l(π, g, T)| ≤ 2Vmax
+�
+ζι
+n .
+(21)
+Note that when n < ζ, E trivially holds. As a result, applying union bound we prove Eq. (16).
+Proof of Lemma 3
+Proof. In the following, we consider a fixed policy π and dynamics T ∈ T . We use w to denote wπ,T when the context is clear.
+By basic algebra we get
+���E(s,a)∼ρπ
+T [Gπ
+T (s, a)]
+���
+(22)
+≤
+����E(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) ≤ ζ
+�
+Gπ
+T (s, a)
+����� + E(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+�
+|Gπ
+T (s, a)|
+�
+(23)
+≤
+��E(s,a)∼ˆµ[w(s, a)Gπ
+T (s, a)]
+�� + VmaxE(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+��
+.
+(24)
+Note that
+E(s,a)∼ˆµ[w(s, a)Gπ
+T (s, a)] =
+�
+s,a
+ˆµ(s, a)w(s, a)Gπ
+T (s, a)
+(25)
+=
+�
+s,a
+(ˆµ(s, a) − µ(s, a) + µ(s, a))w(s, a)Gπ
+T (s, a)
+(26)
+=
+�
+s,a
+µ(s, a)w(s, a)Gπ
+T (s, a) +
+�
+s,a
+(ˆµ(s, a) − µ(s, a))w(s, a)Gπ
+T (s, a)
+(27)
+≤ E(s,a)∼µ[w(s, a)Gπ
+T (s, a)] +
+�
+s,a
+|ˆµ(s, a) − µ(s, a)|ζVmax
+(28)
+≤ E(s,a)∼µ[w(s, a)Gπ
+T (s, a)] + ζVmaxTV (ˆµ, µ) .
+(29)
+
+Continuing Eq. (24) we get
+���E(s,a)∼ρπ
+T [Gπ
+T (s, a)]
+���
+(30)
+≤
+��E(s,a)∼µ[w(s, a)Gπ
+T (s, a)]
+�� + VmaxE(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+��
++ ζVmaxTV (ˆµ, µ) .
+(31)
+Consequently, in the following we prove
+��E(s,a)∼µ[w(s, a)Gπ
+T (s, a)]
+�� ≤ sup
+g∈G
+ℓw(g, T) + ϵV (T, π) + 2Vmax
+�
+ζι
+n .
+Let Lw(g, T) =
+��E(s,a,s′)∼µ
+�
+w(s, a)(Ex∼T (s,a)[g(x)] − Ex∼T ⋆(s,a)[g(x)])
+��� be the population error. Recall that under the high
+probability event E in Eq. (15), for any g ∈ G and T ∈ T
+|Lw(g, T) − ℓw(g, T)| ≤ 2Vmax
+�
+ζι
+n .
+(32)
+Now by the definition of Gπ
+T (s, a), for any g ∈ G we have
+��E(s,a)∼µ[w(s, a)Gπ
+T (s, a)]
+��
+(33)
+=
+��E(s,a)∼µ
+�
+w(s, a)
+�
+Es′∼T (s,a)[V π
+T ⋆(s′)] − Es′∼T ⋆(s,a)[V π
+T ⋆(s′)]
+����
+(34)
+≤
+��E(s,a)∼µ
+�
+w(s, a)
+�
+Es′∼T (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)]
+����
+(35)
++
+��E(s,a)∼µ
+�
+w(s, a)
+�
+Es′∼T (s,a)[g(s′) − V π
+T ⋆(s′)] + Es′∼T ⋆(s,a)[g(s′) − V π
+T ⋆(s′)]
+����.
+(36)
+Define
+ˆg = argmin
+g∈G
+��E(s,a)∼µ
+�
+w(s, a)
+�
+Es′∼T (s,a)[g(s′) − V π
+T ⋆(s′)] + Es′∼T ⋆(s,a)[g(s′) − V π
+T ⋆(s′)]
+����.
+Since g is arbitrarily, continuing Eq. (36) and recalling Definition 2 we get
+��E(s,a)∼µ[w(s, a)Gπ
+T (s, a)]
+��
+(37)
+≤
+��E(s,a)∼µ
+�
+w(s, a)
+�
+Es′∼T (s,a)[ˆg(s′)] − Es′∼T ⋆(s,a)[ˆg(s′)]
+���� + ϵV (T, π)
+(38)
+≤ sup
+g∈G
+��E(s,a)∼µ
+�
+w(s, a)
+�
+Es′∼T (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)]
+���� + ϵV (T, π).
+(39)
+Combining Eq. (39) and Eq. (32) we get,
+��E(s,a)∼µ[w(s, a)Gπ
+T (s, a)]
+�� ≤ sup
+g∈G
+Lw(g, T) + ϵV (T, π)
+(40)
+≤ sup
+g∈G
+ℓw(g, T) + ϵV (T, π) + 2Vmax
+�
+ζι
+n .
+(41)
+Now plugging in Eq. (31) we get,
+���E(s,a)∼ρπ
+T [Gπ
+T (s, a)]
+���
+≤ sup
+g∈G
+ℓw(g, T) + ϵV (T, π) + 2Vmax
+�
+ζι
+n + VmaxE(s,a)∼ρπ
+T
+�
+I
+�ρπ
+T (s, a)
+ˆµ(s, a) > ζ
+��
++ ζVmaxTV (ˆµ, µ) .
+Finally, combining with simulation lemma (Lemma 1) we finish the proof.
+Proof of Lemma 5
+Proof of Lemma 5. Consider a fixed π ∈ Π. When the context is clear, we use ϵρ and ϵµ to denote ϵρ(π) and ϵµ(π) respectively.
+Consider the dynamics
+ˆT = argmin
+T ∈T
+E(s,a)∼ρπ
+T ⋆ [TV (T(s, a), T ⋆(s, a))].
+(42)
+By the definition of ϵρ we get
+E(s,a)∼ρπ
+T ⋆
+�
+TV
+�
+ˆT(s, a), T ⋆(s, a)
+��
+≤ ϵρ.
+
+Applying Lemma 9 we get
+��ρπ
+ˆT − ρπ
+T ⋆
+��
+1 ≤
+ϵρ
+(1 − γ).
+(43)
+The rest of the proof is organized in the following way. We bound the three terms in RHS of Eq. (4) respectively as follows
+η( ˆT, π) ≥ η(T ⋆, π) − Vmax
+1 − γ ϵρ,
+(44)
+sup
+g∈G
+ℓw(g, ˆT) ≤ 2Vmaxϵρ
+1 − γ
++ 2Vmax
+�
+ζι
+n + ζVmaxTV (ˆµ, µ) ,
+(45)
+E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�
+ρπ
+ˆT (s, a)
+ˆµ(s, a) > ζ
+��
+≤ ϵµ +
+3ϵρ
+(1 − γ).
+(46)
+Then we combine these inequalities together to prove Lemma 5.
+Step 1: Proving Eq. (44). Note that for every T and π, η(T, π) =
+1
+1−γ ⟨ρπ
+T , r⟩ where r is the reward function. Then we have
+η(T ⋆, π) − η( ˆT, π) =
+1
+1 − γ
+�
+ρπ
+T ⋆ − ρπ
+ˆT , r
+�
+≤
+1
+1 − γ
+��ρπ
+T ⋆ − ρπ
+ˆT
+��
+1 ∥r∥∞ .
+(47)
+Combining with Eq. (43) we get Eq. (44).
+Step 2: Proving Eq. (45). For any fixed function g ∈ G. Let w = wπ, ˆT be a shorthand. Define
+Lw(g, T) =
+��E(s,a,s′)∼µ[w(s, a)(f g
+T (s, a) − g(s′))]
+��
+to be the population error. Then we have
+Lw(g, ˆT)
+=
+���E(s,a)∼µ
+���
+w(s, a)
+�
+Es′∼ ˆT (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)]
+�����
+≤
+���E(s,a)∼ˆµ
+�
+w(s, a)
+�
+Es′∼ ˆT (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)]
+����� + ζVmaxTV (ˆµ, µ)
+=
+�����E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�
+ρπ
+ˆT (s, a)
+ˆµ(s, a) ≤ ζ
+� �
+Es′∼ ˆT (s,a)[g(s′)] − Es′∼T ⋆(s,a)[g(s′)]
+������� + ζVmaxTV (ˆµ, µ)
+≤ VmaxE(s,a)∼ρπ
+ˆ
+T
+�
+I
+�
+ρπ
+ˆT (s, a)
+ˆµ(s, a) ≤ ζ
+�
+TV
+�
+ˆT(s, a), T ⋆(s, a)
+��
++ ζVmaxTV (ˆµ, µ)
+≤ VmaxE(s,a)∼ρπ
+T ⋆
+�
+TV
+�
+ˆT(s, a), T ⋆(s, a)
+��
++ Vmaxϵρ
+1 − γ + ζVmaxTV (ˆµ, µ)
+(By Eq. (43))
+≤ Vmax
+�
+ϵρ +
+ϵρ
+1 − γ
+�
++ ζVmaxTV (ˆµ, µ) ≤ 2Vmaxϵρ
+1 − γ
++ ζVmaxTV (ˆµ, µ) .
+Under event E we have
+ℓw(g, ˆT) ≤ Lw(g, ˆT) + 2Vmax
+�
+ζι
+n .
+(48)
+Because g is arbitrary, we get Eq. (45).
+Step 3: Proving Eq. (46). Note that
+E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�ρˆπ
+T (s, a)
+ˆµ(s, a) > ζ
+��
+(49)
+= E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�
+ρπ
+ˆT (s, a)
+ρπ
+T ⋆(s, a)
+ρπ
+T ⋆(s, a)
+ˆµ(s, a)
+> ζ
+��
+(50)
+≤ E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�
+ρπ
+ˆT (s, a)
+ρπ
+T ⋆(s, a) > 2
+��
++ E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�ρπ
+T ⋆(s, a)
+ˆµ(s, a)
+> ζ/2
+��
+.
+(51)
+
+With the help of Lemma 8, we can upper bound the first term of Eq. (51) by the total variation between ρπ
+ˆT and ρπ
+T ⋆. Combining
+Lemma 8 and Eq. (43) we get
+E(s,a)∼ρπ
+ˆ
+T
+�
+I
+� ρˆπ
+T (s, a)
+ρπ
+T ⋆(s, a) > 2
+��
+≤
+2ϵρ
+1 − γ .
+(52)
+On the other hand, by combining Eq. (43) and the definition of ϵµ we get
+E(s,a)∼ρπ
+ˆ
+T
+�
+I
+�ρπ
+T ⋆(s, a)
+ˆµ(s, a)
+> ζ/2
+��
+≤ E(s,a)∼ρπ
+T ⋆
+�
+I
+�ρπ
+T ⋆(s, a)
+ˆµ(s, a)
+> ζ/2
+��
++
+ϵρ
+1 − γ ≤ ϵµ +
+ϵρ
+1 − γ .
+Consequently, we get Eq. (46).
+Now we stitch Eq. (43), Eq. (44) and Eq. (45) together. Combining with the definition of lb( ˆT, π) in Eq. (4), we have
+lb( ˆT, π) = η( ˆT, π) −
+1
+1 − γ
+�
+sup
+g∈G
+���ℓwπ,T (g, ˆT)
+��� + VmaxE(s,a)∼ρπ
+T
+�
+I
+�
+ρπ
+ˆT (s, a)
+ˆµ(s, a) > ζ
+��
++ 2ζVmaxTV (ˆµ, µ)
+�
+≥ η(T ⋆, π) − Vmaxϵρ
+1 − γ − 2Vmaxϵρ
+(1 − γ)2 − 2Vmax
+1 − γ
+�
+ζι
+n − Vmax
+1 − γ
+� 3ϵρ
+1 − γ + ϵµ
+�
+− 2ζVmaxTV (ˆµ, µ)
+1 − γ
+≥ η(T ⋆, π) − 6Vmaxϵρ
+(1 − γ)2 − Vmaxϵµ
+1 − γ − 2Vmax
+1 − γ
+�
+ζι
+n − 2ζVmaxTV (ˆµ, µ)
+1 − γ
+.
+Note that ˆT ∈ T , we have
+max
+T ∈T lb(T, π) ≥ lb( ˆT, π),
+(53)
+which finishes the proof.
+Proof of Theorem 4
+Proof of Theorem 4. Let ˆT, ˆπ ← argmaxT ∈T ,π∈Π lb(T, π) be the dynamics and policy that maximizes the lower bound. Note
+that ˆπ is the output of Algorithm 1.
+Now under the event E, by Lemma 5, for any policy π we have
+max
+T ∈T lb(T, π) ≥ η(T ⋆, π) − 6Vmaxϵρ(π)
+(1 − γ)2
+− Vmaxϵµ(π)
+1 − γ
+− 2Vmax
+1 − γ
+�
+ζι
+n − 2ζVmaxTV (ˆµ, µ)
+1 − γ
+.
+(54)
+On the other hand, under the event E, by Lemma 3 we get
+η(T ⋆, π) ≥ lb( ˆT, ˆπ) − ϵV ( ˆT, ˆπ)
+1 − γ
+− 2Vmax
+1 − γ
+�
+ζι
+n .
+(55)
+By the optimality of ˆT, ˆπ, we have lb( ˆT, ˆπ) ≥ supT ∈T lb(T, π) for any π. As a result, combining with Eq. (54) and Eq. (55)
+we get
+η(T ⋆, ˆπ) ≥ lb( ˆT, ˆπ) − ϵV ( ˆT, ˆπ)
+1 − γ
+− 2Vmax
+1 − γ
+�
+ζι
+n
+(56)
+≥ sup
+π∈Π
+sup
+T ∈T
+lb(T, π) − ϵV ( ˆT, ˆπ)
+1 − γ
+− 2Vmax
+1 − γ
+�
+ζι
+n
+(57)
+≥ sup
+π
+�
+η(T ⋆, π) − 6Vmaxϵρ(π)
+(1 − γ)2
+− Vmaxϵµ(π)
+1 − γ
+�
+− ϵV ( ˆT, ˆπ)
+1 − γ
+− 4Vmax
+1 − γ
+�
+ζι
+n − 2ζVmaxTV (ˆµ, µ)
+1 − γ
+.
+(58)
+Proof of Theorem 6
+Proof of Theorem 6. Note that for any fixed θ ∈ Rd, the transition function for state s1, · · · , sd are identical. As a result,
+Qπ
+Tθ(si, aj) = Qπ
+Tθ(si′, aj), ∀i, i′ ∈ [d] for any policy π. Recall that πθ is the optimal policy of Tθ (with ties breaking uniformly
+at random). Therefore, πθ(s0) = 1/A and πθ(si) = πθ(si′), ∀i, i′ ∈ [d].
+By the definition of the ground-truth dynamics T ⋆ in Eqs. (9)-(10), we have Qπθ
+T ⋆(si, aj) = I [i = j]
+γ
+1−γ . Therefore,
+η(T ⋆, πθ) = γ
+A
+d
+�
+i=1
+Qπθ
+T ⋆(si, πθ(si)) ≤ γ
+A max
+a
+d
+�
+i=1
+Qπθ
+T ⋆(si, a) ≤
+γ2
+A(1 − γ).
+(59)
+
+Since maxπ η(T ⋆, π) =
+γ2
+1−γ , we have
+max
+π
+η(T ⋆, π) − η(T ⋆, πθ) ≥ (A − 1)γ2
+A(1 − γ) .
+OPE Error of MML
+In this section, we show that the off-policy estimation error in Voloshin, Jiang, and Yue (2021) can be large when the dynamics
+model class is misspecified in Proposition 7.
+The MML algorithm requires an density ratio class W : S × A → R+ and prove that when wπ,T ∈ W and V π
+T ⋆ ∈ G,
+|η(T, π) − η(T ⋆, π)| ≤ γ min
+T ∈T
+max
+w∈W,g∈G |ℓw(g, T)|.
+(60)
+Unfortunately, this is suboptimal since the error may not converge to zero even given infinite data:
+Proposition 7. Consider the set the dynamics class T = {Tθ : θ ∈ Sd−1, θi ≥ 0, ∀i ∈ [d]}. Let Π = {πx : x ∈ [d]} where
+πx(si) = ax for 0 ≤ i ≤ d and πx(sg) = πx(sb) = a1. Let W be the density ratio class induced by π running on {T ⋆} ∪ T .
+Even with G = {V πx
+T ⋆ : x ∈ [d]} and infinite number of data, we have
+min
+T ∈T
+max
+w∈W,g∈G |ℓw(g, T)| ≥
+γ
+8(1 − γ).
+(61)
+In contrast, the error terms in Theorem 4 converge to 0 when ζ > poly(d, 1/(1 − γ)) and n → ∞ in the same setting.
+Proof of Proposition 7. Recall that we set the dynamics class T = {Tθ : θ ∈ Sd−1}. Let Π = {πx : x ∈ [d]} where
+πx(si) = ax for 0 ≤ i ≤ d and πx(sg) = πx(sb) = a1. Let W be the density ratio induced by π. For any x ∈ [d], we can
+compute
+ρπx
+T ⋆(s0, ai) = (1 − γ)I [i = x] ,
+ρπx
+T ⋆(si, aj) = γ(1 − γ)I [i = x, j = x] ,
+(62)
+ρπx
+T ⋆(sg, aj) = γ2(1 − γ)I [j = 1] ,
+ρπx
+T ⋆(sb, aj) = 0.
+(63)
+Let µ be uniform distribution over 3d + d2 state action pairs. Then we can define W = {wx : x ∈ [d]} where wx(s, a) ≜
+1
+1−γ
+ρπx
+T ⋆(s,a)
+µ(s,a) .
+Now for any fixed θ ∈ Sd−1, θ ≥ 0, consider
+max
+w∈W,g∈G |ℓw(g, Tθ)|.
+(64)
+Let x = argmini θi. We claim that
+ℓwx(V πx
+T ⋆ , Tθ) ≥
+γ
+8(1 − γ).
+Indeed, with infinite data we have
+ℓwx(V πx
+T ⋆ , Tθ) =
+��E(s,a)∼µ
+�
+wx(s, a)
+�
+Es′∼T (s,a)[V πx
+T ⋆ (s′)] − Es′∼T ⋆(s,a)[V πx
+T ⋆ (s′)]
+����
+=
+1
+1 − γ
+���E(s,a)∼ρπx
+T ⋆
+��
+Es′∼T (s,a)[V πx
+T ⋆ (s′)] − Es′∼T ⋆(s,a)[V πx
+T ⋆ (s′)]
+�����.
+Recall that Tθ = T ⋆ for states s0, sg, sb. As a result, we continue the equation by
+1
+1 − γ
+���E(s,a)∼ρπx
+T ⋆
+��
+Es′∼T (s,a)[V πx
+T ⋆ (s′)] − Es′∼T ⋆(s,a)[V πx
+T ⋆ (s′)]
+�����
+= γ
+��Es′∼T (sx,ax)[V πx
+T ⋆ (s′)] − Es′∼T ⋆(sx,ax)[V πx
+T ⋆ (s′)]
+��
+(by the definition of ρ)
+= γ
+����
+1
+2(1 + θx)V πx
+T ⋆ (sg) + 1
+2(1 − θx)V πx
+T ⋆ (sb) − V πx
+T ⋆ (sg)
+����
+(by the definition of Tθ)
+= γ
+2 (1 − θx)(V πx
+T ⋆ (sg) − V πx
+T ⋆ (sb)).
+By basic algebra, V πx
+T ⋆ (sg) = (1 − γ)−1 and V πx
+T ⋆ (sb) = 0. As a result, we get
+ℓwx(V πx
+T ⋆ , Tθ) ≥
+γ
+2(1 − γ)(1 − θx).
+(65)
+Recall that x = argmini θi. Since θ ∈ Sd−1 and θi ≥ 0, ∀i, we have 1 = �d
+i=1 θ2
+i ≥ dθ2
+x. As a result, when d > 2 we have
+θx ≤ 1/
+√
+2. Therefore
+ℓwx(V πx
+T ⋆ , Tθ) ≥
+γ
+2(1 − γ)(1 − θx) ≥
+γ
+8(1 − γ).
+(66)
+
+Helper Lemmas
+In this section, we present several helper lemmas used in Appendix .
+Lemma 8. For two distribution p, q over x ∈ X, if we have ∥p − q∥1 ≤ ϵ, then for any ζ > 1,
+Ex∼p
+�
+I
+�p(x)
+q(x) > ζ
+��
+≤
+ζ
+ζ − 1ϵ.
+Proof. Define E(x) = I
+�
+p(x)
+q(x) > ζ
+�
+. Note that under event E(x) we have
+p(x) > q(x)ζ =⇒ p(x) − q(x) > q(x)(ζ − 1).
+(67)
+As a result,
+ϵ ≥ ∥p − q∥1 ≥
+�
+|p(x) − q(x)|E(x) dx
+(68)
+≥
+�
+(ζ − 1)q(x)E(x) dx = Ex∼q[E(x)](ζ − 1)
+(69)
+≥ (Ex∼p[E(x)] − ϵ)(ζ − 1).
+(70)
+By algebraic manipulation we get Ex∼p[E(x)] ≤
+ζ
+ζ−1ϵ.
+Lemma 9. Consider a fixed policy π and two dynamics model T, ¯T. Suppose
+E(s,a)∼ρπ
+T
+�
+TV
+�
+T(s, a), ¯T(s, a)
+��
+≤ ϵ,
+we get
+��ρπ
+T − ρπ
+¯T
+��
+1 ≤
+1
+1 − γ ϵ.
+(71)
+Proof. First of all let G, ¯G be the transition kernel from S × A to S × A induced by T, π and ¯T, π respectively. Then for any
+distribution ρ ∈ ∆(S × A) we have
+��Gρ − ¯Gρ
+��
+1 ≤ E(s,a)∼ρ
+�
+TV
+� ¯T(s, a), T(s, a)
+��
+.
+(72)
+Let ρh (or ¯ρh) be the state-action distribution on step h under dynamics T (or ¯T). Then we have
+ρh − ¯ρh =
+�
+Gh − ¯Gh�
+ρ0 =
+h−1
+�
+h′=0
+¯Gh−h′−1�
+G − ¯G
+�
+Gh′ρ0.
+(73)
+As a result,
+∥ρh − ¯ρh∥1 ≤
+h−1
+�
+h′=0
+��� ¯Gh−h′−1�
+G − ¯G
+�
+Gh′ρ0
+���
+1
+(74)
+≤
+h−1
+�
+h′=0
+���
+�
+G − ¯G
+�
+Gh′ρ0
+���
+1 ≤
+h−1
+�
+h′=0
+E(s,a)∼ρh′
+�
+TV
+� ¯T(s, a), T(s, a)
+��
+.
+(75)
+It follows that
+��ρπ
+T − ρπ
+¯T
+��
+1 ≤ (1 − γ)
+∞
+�
+h=0
+γh ∥ρh − ¯ρh∥1
+(76)
+≤(1 − γ)
+∞
+�
+h=0
+γh
+h−1
+�
+h′=0
+E(s,a)∼ρh′
+�
+TV
+� ¯T(s, a), T(s, a)
+��
+(77)
+≤(1 − γ)
+∞
+�
+h=0
+γh
+1 − γ E(s,a)∼ρh
+�
+TV
+� ¯T(s, a), T(s, a)
+��
+(78)
+=
+∞
+�
+h=0
+γhE(s,a)∼ρh
+�
+TV
+� ¯T(s, a), T(s, a)
+��
+(79)
+=
+1
+1 − γ E(s,a)∼ρπ
+T
+�
+TV
+� ¯T(s, a), T(s, a)
+��
+.
+(80)
+
+LQR Experimental Details
+Data generation
+The
+of���ine
+dataset
+is
+generated
+by
+running
+several
+πv
+under
+the
+true
+dynamics
+with
+v
+∈
+{−1, −0.75, −0.5, −0.25, 0, 0.25, 0.5, 0.75} and added noise N(0, 0.5) to the policy. As a result, the behavior dataset
+covers most of the state-action space. The dataset contains 2000 trajectories with length 20 from each policy.
+Implementation
+We compute the density ratio by approximating the behavior distribution µ and the state-action distribution ρπ
+T respectively. By
+discretizing the state-action space into 10 × 10 bins uniformly, the distribution µ(s, a) is approximated by the frequency of the
+corresponding bin. For ρπ
+T , we first collect 2000 trajectories of policy π under T and compute the distribution similarly. Because
+all the function classes are finite, we enumerate over the function classes to compute lb(T, π) for every pair of dynamics and
+policy.
+Hyperparameters
+In the experiments, we use the following hyperparameters.
+• Cutoff threshold in Line 3 of Alg. 1: ζ = 50.
+• Random seeds for three runs: 1, 2, 3.
+• State noise: η ∼ N(0, 0.05).
+• Policy noise: N(0, 0.01).
+• Discount factor: γ = 0.9.
+• Mean of initial state: 0.5.
+• Noise added to initial state: 0.2.
+• Number of trajectories per policy: 2000.
+We do not require parameter tuning for optimization procedures. We tried cutoff threshold with ζ ∈ {10, 20, 50} and number
+of trajectories in {20, 500, 2000}. Smaller cutoff leads to an over-pessimistic lower bound, and fewer trajectories introduce
+variance to the final result.
+Computing resources
+These experiments run on a machine with 2 CPUs, 4GB RAM, and Ubuntu 20.04. We don’t require GPU resources. We use
+Python 3.9.5 and numpy 1.20.2.
+D4RL Experimental Details
+Tasks
+Hopper. The Hopper task is to make a hopper with three joints and four body parts hop forward as fast as possible. The state
+space is 11-dimension, the action is a 3-dimensional continuous space.
+HalfCheetah. The HalfCheetah task is to make a 2D robot with 7 rigid links, including 2 legs and a torso run forward as fast
+as possible. The state space is 17-dimension, the action is a 6-dimensional continuous space.
+Model Choice and Hyperparameters
+For all the dynamics, each model is parametrized as a 4-layer feedforward neural network with 200 hidden units. For the
+SAC (Haarnoja et al. 2018) updates (serving as the policy gradient updates subroutine), the function approximations used for
+the policy and value function are 2-layer feedforward neural networks with 256 hidden units.
+The hyperparameter choices for behavior density modeling are based on the training progress of the normalizing flow model.
+We pre-select a few (less than 10) combinations of hyperparameters and pick the set that gives us the lowest training loss.
+Usually, this is not the best practice. However, the small number of combinations (non-exhaustive search) and small model size
+reduced our concern for training set overfitting.
+MOPO (Yu et al. 2020):
+• Batch size: 100.
+• Rollout horizon: 5.
+• Lambda: 1.
+MBLB:
+• Random seeds for five runs: 1, 2, 3, 4, 5.
+
+• Number of trajectories to sample: 100.
+• Rollout horizon: 5.
+• Batch size: 32.
+• Cutoff threshold in Line 3 of Alg. 1: ζ = 5.
+• Discount factor γ: 0.99.
+• GAE λ: 0.95.
+• g function latent size: 8.
+MML:
+• Random seeds for five runs: 1, 2, 3, 4, 5.
+• Batch size: 32.
+• Basis function class: square, polynomial
+• Ratio-Value function parametrization: linear, reproducing kernel hilbert space (RKHS)
+For MML, we first need to make a decision on how to parametrize h(s, a, s′). If we choose a linear parametrization such as
+h(s, a, s′) = ψ(s, a, s′)T θ, we need to decide what ψ is. There are two obvious choices: ψ(x) = [x, x2, 1] (square basis func-
+tion), or a polynomial basis function with degree 2: given x = [x1, x2, ..., xd], ψ(x) = [x2
+1, x1x2, x1x3, ..., x2
+2, x2x3, ..., x2
+d],
+which can be efficiently computed as the upper triangular entries of xxT . If we choose the ratio-value function parametrization
+to be RKHS, then we use radial basis function (RBF) as K((s, a, s′), (˜s, ˜a, ˜s′)).
+Computing resources
+These experiments run on a machine with 4 CPUs, 10GB RAM, and Ubuntu 20.04. We don’t require GPU resources. We use
+Python 3.9.5 and numpy 1.20.2.
+Algorithms
+We describe the MML and MBLB algorithms in this section. Algorithm 2 describes how we compute MBLB. Note that we
+compute three components of lower bound explicitly. Algorithm 3 describes how we compute MML with linear parametrization.
+Algorithm 4 describes how we compute MML with RKHS parametrization.
+Algorithm 2: MBLB: Model-based Lower Bound
+Input: offline RL data D; set of dynamics, policy pairs
+[(π1, T1), ..., (πK, TK)], Vmax, γ, ζ.
+Output: optimal policy π∗
+ˆµ(·, ·) = trainFlow (D)
+scores = []
+for i ← 1...K do
+Qπi = trainFQE (Sample (D, Ti, πi), πi)
+ρTi
+πi(·, ·) = trainFlow (Sample (D, Ti, πi))
+η = E(s,a)∼D[Qπi(s, πi(s))]
+Initialize (θ)
+L = 0; ∆ = 0
+for (s, a, s′) ∈ D do
+w = max(min(
+ρ
+Ti
+πi(s,a)
+ˆµ(s,a) , ζ), 0)
+ℓ = −|w · (Ex∼Ti(s)[gθ(x)] − gθ(s′))|
+θ = θ + ∇θℓ
+∆ = ∆ − Vmax · I
+�
+ρ
+Ti
+πi(s,a)
+ˆµ(s,a) > ζ
+�
+L = L + ℓ
+end
+score =
+1
+|D|(η +
+1
+1−γ (∆ + L))
+scores ← score
+end
+i = argmax(scores)
+return πi
+
+Algorithm 3: MML-Linear: Minimax Model Learning Bound
+Input: offline RL data D; set of dynamics, policy pairs
+[(π1, T1), ..., (πK, TK)].
+Output: optimal policy π∗
+scores = []
+for i ← 1...K do
+Initialize (θ)
+L = 0
+for (s, a, s′) ∈ D do
+ℓ = −(Ex∼Ti(s)[ψ(s, a, x)T θ] − ψ(s, a, s′)T θ)
+θ = θ + ∇θℓ
+L = L + ℓ
+end
+score =
+L
+|D|
+scores ← score
+end
+i = argmax(scores)
+return πi
+Algorithm 4: MML-RKHS: Minimax Model Learning Bound
+Input: offline RL data D; set of dynamics, policy pairs
+[(π1, T1), ..., (πK, TK)], kernel K.
+Output: optimal policy π∗
+scores = []
+for i ← 1...K do
+L = 0
+for (s, a, s′), (˜s, ˜a, ˜s′) ∈ D do
+ℓ1 = Ex∼T (s),˜x∼T (˜s)[K((s, a, x), (˜s, ˜a, ˜x))]
+ℓ2 = −2Ex∼T (s)[K((s, a, x), (˜s, ˜a, ˜s′))]
+ℓ3 = K((s, a, s′), (˜s, ˜a, ˜s′))
+L = L + ℓ1 + ℓ2 + ℓ3
+end
+score =
+L
+|D|
+scores ← score
+end
+i = argmax(scores)
+return πi
+D4RL Additional Experiments
+Ablation Study
+We conduct an ablation study in Table A1 where we evaluate the final performance of the policies selected using either FQE
+with TD-1 estimation or FQE with GAE estimation. We observe that using GAE for offline policy selection allows for picking
+better policies on average.
+MBLB with RKHS
+In this section, we derive the closed-form solution to supg∈G ℓw(g, T) when the test function g belongs to a reproducing kernel
+Hilbert space (RKHS), and empirically evaluate the MBLB method with RKHS parameterization.
+Let K : S ×S → R be a symmetric and positive definite kernel and HK its corresponding RKHS with inner product ⟨·, ·⟩HK.
+Then we have the following lemma.
+Lemma 10. When G = {g ∈ HK : ⟨g, g⟩HK ≤ 1}, we have
+sup
+g∈G
+ℓw(g, T)2 = Es,a,s′∼D,x∼T (s,a)E˜s,˜a,˜s′∼D,˜x∼T (˜s,˜a) [w(s, a)w(˜s, ˜a)(K(x, ˜x) + K(s′, ˜s′) − K(x, ˜s′) − K(˜x, s′)]
+(81)
+Proof. Let Kx ≜ K(x, ·) ∈ HK. By the reproducing property, we have ⟨Kx, Ky⟩HK = K(x, y) and ⟨Kx, g⟩HK = g(x). As a
+
+Dataset Type
+Environment
+FQE
+(TD-1)
+FQE
+(GAE)
+medium
+hopper
+507.8
+(549.6)
+533.5
+(532.6)
+med-expert
+hopper
+149.3
+(146.2)
+261.1
+(157.9)
+expert
+hopper
+39.0
+(34.6)
+120.7
+(78.7)
+medium
+halfcheetah
+1802.5
+(1011.9)
+2117.4
+(1215.6)
+med-expert
+halfcheetah
+302.1
+(605.2)
+394.9
+(632.0)
+Table A1: We report the mean and (standard deviation) of the selected policy’s environment performance across 3 random seeds
+using different variants of FQE.
+result,
+sup
+g∈G
+ℓw(g, T)2 =
+sup
+g:⟨g,g⟩HK ≤1
+Es,a,s′∼D,x∼T (s,a)[w(s, a)(⟨Kx, g⟩HK − ⟨Ks′, g⟩HK)]2
+(82)
+=
+sup
+g:⟨g,g⟩HK ≤1
+�
+Es,a,s′∼D,x∼T (s,a)[w(s, a)(Kx − Ks′)], g
+�2
+HK
+(83)
+= ∥Es,a,s′∼D,x∼T (s,a)[w(s, a)(Kx − Ks′)]∥2
+HK
+(Cauchy-Schwarz)
+=
+�
+Es,a,s′∼D,x∼T (s,a)[w(s, a)(Kx − Ks′)], E˜s,˜a,˜s′∼D,˜x∼T (˜s,˜a)[w(˜s, ˜a)(K˜x − K˜s′)]
+�
+HK
+(84)
+= Es,a,s′∼D,x∼T (s,a)E˜s,˜a,˜s′∼D,˜x∼T (˜s,˜a)[⟨w(s, a)(Kx − Ks′), w(˜s, ˜a)(K˜x − K˜s′)⟩HK]
+(85)
+= Es,a,s′∼D,x∼T (s,a)E˜s,˜a,˜s′∼D,˜x∼T (˜s,˜a)[w(s, a)w(˜s, ˜a)(K(x, ˜x) + K(s′, ˜s′) − K(x, ˜s′) − K(˜x, s′)].
+(86)
+Table A2 presents the performance of the MBLB algorithm with RKHS parameterization. On most of the environments,
+MBLB-RKHS performs better than/comparable with MML-RKHS. However, MBLB-Quad consistently outperforms MBLB-
+RKHS on all the environments. We suspect that MBLB-RKHS could outperform MBLB-Quad with different choices of kernels
+because the quadratic parameterization can be seen as a special case of RKHS parameterization (with quadratic kernels).
+Dataset Type
+Env
+MOPO
+MML
+(Squared)
+MML
+(Polynomial)
+MML
+(RKHS)
+MBLB
+(Linear)
+MBLB
+(Quad)
+MBLB
+(RKHS)
+medium
+hopper
+175.4
+(95.3)
+379.4
+(466.4)
+375.6
+(459.5)
+375.0
+(459.9)
+591.7
+(523.1)
+808.5
+(502.7)
+317.8
+(476.4)
+med-expert
+hopper
+183.8
+(94.4)
+160.9
+(131.5)
+116.5
+(148.4)
+61.4
+(35.0)
+261.1
+(157.9)
+242.5
+(134.0)
+208.1
+(144.3)
+expert
+hopper
+80.4
+(63.4)
+93.8
+(87.9)
+61.6
+(61.9)
+70.0
+(56.2)
+118.2
+(61.6)
+121.0
+(72.5)
+120.9
+(61.8)
+medium
+halfcheetah
+599.8
+(668.4)
+1967.6
+(1707.5)
+2625.1
+(937.2)
+3858.2
+(1231.1)
+3290.4
+(1753.1)
+2484.2
+(1526.8)
+2229.7
+(1949.8)
+med-expert
+halfcheetah
+-486.6
+(48.1)
+-188.5
+(137.2)
+-77.0
+(252.5)
+-343.2
+(225.2)
+207.4
+(509.5)
+192.8
+(432.0)
+-2.1
+(690.6)
+Table A2: We report the mean and (standard deviation) of selected policy’s simulator environment performance across 5 random
+seeds. MML and MBLB are used as model-selection procedures where they select the best policy for each seed. Our method is
+choosing the most near-optimal policy across the datasets.
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Normalized Score (τ)
+0.00
+0.25
+0.50
+0.75
+1.00
+Fraction of runs with score > τ
+MBLB
+MML
+MOPO
+Figure A1: Performance profile between three methods.
+

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+Semidefinite Relaxations for Robust Multiview Triangulation
+Linus H¨arenstam-Nielsen1, Niclas Zeller2, Daniel Cremers1
+1Technical University of Munich, 2Karlsruhe University of Applied Sciences
+linus.nielsen@tum.de, niclas.zeller@h-ka.de, cremers@tum.de
+Abstract
+We propose the first convex relaxation for multiview tri-
+angulation that is robust to both noise and outliers. To this
+end, we extend existing semidefinite relaxation approaches
+to loss functions that include a truncated least squares cost
+to account for outliers. We propose two formulations, one
+based on epipolar constraints and one based on the frac-
+tional reprojection equations. The first is lower dimensional
+and remains tight under moderate noise and outlier levels,
+while the second is higher dimensional and therefore slower
+but remains tight even under extreme noise and outlier lev-
+els. We demonstrate through extensive experiments that the
+proposed approach allows us to compute provably optimal
+reconstructions and that empirically the relaxations remain
+tight even under significant noise and a large percentage of
+outliers.
+1. Introduction
+Triangulation refers to the problem of recovering the 3D
+location of a set of points from their observed 2D locations
+in two or more images under known camera transforma-
+tions.
+Since the 2D projections are typically noisy (due
+to lens distortions or inaccurate feature point localization),
+the optimal solution is often phrased as a non-convex opti-
+mization problem. While solutions are mostly computed us-
+ing faster but sub-optimal local optimization methods, there
+have been some efforts to compute globally optimal triangu-
+lations [1, 4]. While these works show that one can obtain
+globally optimal solutions for triangulation problems with
+noisy input, their practical value remains limited as they are
+not well adapted to the challenges of real-world data where
+even a single outlier can deteriorate the result.
+Despite their often slower runtime, globally optimal
+methods offer several advantages: Firstly, in safety-critical
+systems it may be required to complement the computed so-
+lution with some guarantee that it really is the best solution
+or at least within a bound of the optimal solution. Secondly,
+in many offline applications runtime is actually not critical
+and then one may want to trade off better accuracy for extra
+(a) 22 views, no outliers
+(b) 22 views, 19 outliers
+Figure 1.
+Example of a triangulated point from the Reichstag
+dataset. Blue point: ground truth. Red point: non-robust global
+optimum found by the relaxation found by [1] (Eq. (T)). Green
+point: robust global optimum found by our proposed relaxation in
+Eq. (RT).
+runtime. Thirdly, globally optimal solutions of real-world
+problems can serve as ground truth for assessing the perfor-
+mance of local optimization methods.
+In this work, we revisit the problem of computing prov-
+ably optimal triangulations in the presense of outliers. To
+this end, we develop two possible convex relaxations for
+the truncated least squares cost function so as to combine
+the robustness with the capacity to compute globally opti-
+mal solutions. Our main contributions can be summarized
+as follows:
+• We extend the convex triangulation methods from [1]
+and [4] with a truncated least squares cost function and
+derive the corresponding convex relaxations.
+• We show that the relaxations are always tight in the
+noise-free and outlier-free case by explicitly construct-
+ing the globally optimal Lagrange multipliers (which
+1
+arXiv:2301.11431v1  [cs.CV]  26 Jan 2023
+
+furthermore satisfy the corank 1, restricted slater and
+non-branch point criteria required for local stability
+with respect to noise).
+• We validate empirically that both relaxations remain
+tight even under large amounts of noise and high out-
+lier ratios.
+To the best of our knowledge, this is the first example of
+a successful semidefinite relaxation of a robust estimation
+problem with reprojection errors.
+2. Related work
+Triangulation is a core subroutine for structure from mo-
+tion and therefore has been studied extensively. For two
+views there are many solution variants, including comput-
+ing the roots of a degree 6 polynomial [9] for the repro-
+jection error, and [15] for the angular error. Multiview tri-
+angulation is typically performed using non-optimal meth-
+ods such as local search or the linear method from [9].
+Robust triangulation is typically tackled using RANSAC
+[13, 16, 20] where a 2-view solver is used repeatedly for
+randomly sampled pairs of views until an inlier set can be
+established.
+Semidefinite relaxations have been used to obtain certi-
+fiably optimal algorithms for many computer vision prob-
+lems. Examples include semidefinite relaxations for parti-
+tioning, grouping and restoration [14], for minimizing re-
+projection error [12], for multiview triangulation [1, 4], for
+essential matrix estimation [28], for hand-eye calibration
+[7, 22, 23], for robust point cloud registration [24, 26, 27],
+and for 3D shape from 2D landmarks [25].
+The work
+[26] also considers outlier-robust estimation applied to rota-
+tion averaging, mesh registration, absolute pose registration
+and category-level object pose+shape estimation. Solving
+semidefinite relaxations is typically slow and memory in-
+tensive, stemming from the fact that the number of vari-
+ables is the square of the number of variables in the orig-
+inal problem. However there has been recent interest in
+developing solvers that can scale to larger problems. In-
+cluding [6] which uses a reformulation in terms of eigen-
+value optimization based on [10] which can take advantage
+of GPUs, and [26] which uses efficient non-global solvers
+for speeding up the convergence of the global solver.
+In a limited number of cases, semidefinite relaxations
+can be shown to always solve the original problem when
+excluding degenerate configurations.
+Including the dual
+quaternion formulation of hand-eye calibration [7] and 2-
+view triangulation using epipolar constraints [1]. In both
+cases the problem has two quadratic constraints one of
+which equals zero. Another case is the rotation alignment
+problem which has a closed form solution in terms of an
+eigenvalue decomposition (quaternion formulation) or sin-
+gular value decomposition (rotation matrix formulation).
+Outlier-robust estimation is inapproximable in general
+[2], so one typically has to rely on empirical experiments
+to validate how stable the relaxation is. Though it is some
+times possible to find sets of special cases where the relax-
+ation can be shown to be always tight (or non-tight), as in
+the recent work [19] for robust rotation alignment of point
+clouds.
+3. Notation and preliminaries
+For t, s ∈ R3 we write [t]× for the 3×3 skew-symmetric
+matrix such that t × s = [t]×s. Sk is the set of k × k
+real symmetric matrices. (a; b) denotes the vertical con-
+catenation of vectors a and b and for a collection of vec-
+tors a1, . . . , an the subscript-free version denotes the cor-
+responding stacked vector a = (a1; . . . ; an).
+We use a
+bar to denote the homogeneous version of a vector, that is
+¯a := (a; 1). When dimensionality is understood we define
+ei to be the ith unit vector and Ei = eieT
+i . For a vector of
+monomials m = (m1; . . . ; md) we define em
+mi as the unit
+vector whose only non-zero entry corresponds to the index
+of mi in m, meaning em
+mi = ei ∈ Rd. For a vector x ∈ Rk
+we define:
+Mx :=
+� I
+−x
+−xT
+∥x∥2
+�
+∈ Sk+1
+(1)
+such that for y ∈ Rk we have ¯yT Mx¯y = ∥x − y∥2. The
+operator ⊗ denotes the Kronecker product, and ⊕ denotes
+the tensor sum. For example, for 2 × 2 matrices A and B:
+A ⊕ B =
+�A
+0
+0
+B
+�
+, A ⊗ B =
+�a11B
+a12B
+a21B
+a22B
+�
+.
+(2)
+3.1. Semidefinite relaxations
+As a general strategy, we aim to solve the triangu-
+lation problem by relaxing a Quadratically Constrained
+Quadratic Program, which has the following form:
+min
+z∈Rd
+zT Mz
+s.t.
+zT Ez = 1
+zT Aiz = 0,
+i = 1, . . . , k.
+(3)
+This is a very general formulation with applications in com-
+puter vision but it is NP-hard to solve in most cases, so an
+imperfect method is typically necessary. One such strat-
+egy is to lift the problem from Rd to Sd by introducing a
+new variable Z = zzT and using the fact that zT Mz =
+tr(MzzT ) = tr(MZ) to arrive at:
+min
+Z∈Sd
+tr(MZ)
+s.t.
+tr(EZ) = 1
+tr(AiZ) = 0,
+i = 1, . . . , k
+Z ≽ 0.
+(4)
+2
+
+Eq. (4) is a relaxation of Eq. (3) since if z satisfies the con-
+straints of Eq. (3) we always have that Z = zzT satisfies
+the constraints of Eq. (4) with the same objective value.
+However, the converse is unfortunately not always true. In
+particular, if ˆZ is optimal for Eq. (4) we can obtain a cor-
+responding solution ˆz for Eq. (3) with the same objective
+value if and only if ˆZ is rank one. In this case we have
+ˆZ = ˆzˆzT and we then say that the relaxation is tight.
+The main advantage of working with the relaxation
+Eq. (4) as opposed to the original problem Eq. (3) is that the
+relaxation is a convex optimization problem, in particular
+a semidefinite program, for which a variety of polynomial-
+time solvers are available, including [3, 17]. We can verify
+whether a potential solution to Eq. (3) is optimal by comput-
+ing the corresponding Lagrange multipliers, as summarized
+in the following fact:
+Fact 1. If ˆz ∈ Rd satisfies the constraints of Eq. (3) (primal
+feasibility) and there are Lagrange multipliers ˆλ ∈ R, ˆξ ∈
+Rk and a corresponding multiplier matrix S(ˆλ, ˆξ) = M +
+�k
+i=1 ˆξiAi − ˆλE satisfying:
+i) Dual feasibility: S(ˆλ, ˆξ) ≽ 0
+ii) Complementarity: S(ˆλ, ˆξ)ˆz = 0
+then the relaxation Eq. (4) is tight and ˆz is optimal for
+Eq. (3).
+If the relaxation is not tight we can at best expect an
+optimal ˆZ to generate an approximation of the optimal ˆz.
+Therefore, a key metric to consider when applying a re-
+laxation is the percentage of encountered problem cases
+in which it remains tight. Fortunately, [5] shows that the
+relaxation is well behaved for problems that are close in
+parameter-space to solutions where the multiplier matrix
+has corank 11, which we will show later occurs in the noise-
+free case. We restate the main result in loose terms here:
+Fact 2. If we, in addition to the conditions in Fact 1, have
+that S(λ, µ) is corank 1 and ACQ (which is a smoothness
+condition, see [5] Definition 3.1) holds, then the relaxation
+Eq. (4) is locally stable, meaning that it will remain tight
+also for perturbed objective functions M + ε ˜
+M for small
+enough ε.
+The practical usefulness of Fact 2 comes from the con-
+sideration that it’s often possible to show that the relaxation
+is tight and the stability conditions hold for noise-free mea-
+surements. This means that there is some surrounding re-
+gion of noisy measurements for which the relaxation is tight
+as well. There is also a version of Fact 2 which covers per-
+turbations to the constraints, however, we will not make use
+of it here.
+1corank(A) = n - rank(A) for an n × n matrix A.
+4. Relaxations for multiview triangulation
+Given n views of a point X from cameras located at Pi =
+(Ri, ti) ∈ SE(3) with intrinsic matrices Ki ∈ R3×3, and
+with, possibly noisy, observations denoted as ˜xi ∈ R2, the
+n-view triangulation problem with reprojection error is de-
+fined as:
+min
+X∈R3
+n
+�
+i=1
+∥˜xi − π(Ki, Pi, X)∥2
+(5)
+where π(Ki, Pi, X) is the reprojection of the point X ∈ R3
+to camera i. This is a nonconvex problem but it is not yet in
+QCQP form as in Eq. (3) since π(Ki, Pi, X) is not quadratic
+in X. In the next section we will recap two ways in which
+it can be converted to a QCQP, from which we can generate
+the corresponding semidefinite relaxations.
+4.1. Triangulation with epipolar constraints
+As described in [1] we can reformulate Eq. (5) as a poly-
+nomial optmization problem of degree 2 by reparametrizing
+X in terms of it’s n reprojections xi subject to the epipolar
+constraints:
+min
+xi∈R2
+n
+�
+i=1
+∥xi − ˜xi∥2
+s.t.
+¯xT
+i Fij ¯xj = 0
+i, j = 1, . . . , n
+i ̸= j
+(6)
+where Fij = K−T
+i
+[tij]×RijK−1
+j
+is the fundamental matrix
+corresponding to the relative transformation between poses
+i and j. Since the estimated reprojections xi all satisfy the
+epipolar constraints, the solution of Eq. (5) can be recovered
+exactly from Eq. (6) using the linear method from [9].
+Using the parametrization z = (x; 1) = ¯x the semidefi-
+nite relaxation of Eq. (6) is:
+min
+Z∈S2n+1
+tr(M˜xZ)
+s.t.
+tr(En+1Z) = 1
+tr( ¯FijZ) = 0,
+i = 1, . . . , k
+Z ≽ 0
+(T)
+where ¯Fij
+∈
+S2n+1 is defined such that ¯xT ¯Fij ¯x
+=
+¯xT
+i Fij ¯xj. It was shown already in both [1] and [5] that
+Eq. (T) is a locally stable relaxation for noise-free measure-
+ments, whenever the views are not co-planar. In particu-
+lar, since noise-free observations ˜x by definition satisfy the
+constraints of the original problem Eq. (T), the solution is
+obtained by setting z = (˜x; 1), and since M˜x is positive
+semidefinite and corank 1, the conditions of Fact 1 are sat-
+isfied by setting all Lagrange multipliers to zero, such that
+ˆS = M˜x.
+3
+
+(a) 3 views
+(b) 5 views
+(c) 7 views
+(d) 3 views, 1 outlier
+(e) 5 views, 2 outliers
+(f) 7 views, 4 outliers
+Figure 2. Examples of simulated triangulation problems from Sec. 5.1 with σ = 50px for various number of views and outliers. Blue
+point: ground truth, Red point: non-robust global optimum found by the relaxation found by [4] (Eq. (TF)). Green point: robust global
+optimum found by our proposed relaxation in Eq. (RTF). With no outliers the robust and non-robust methods give the same result.
+4.2. Triangulation with fraction constraints
+As an alternative to Eq. (6) we can also solve Eq. (5) by
+explicitly parametrizing the 3D point X in homogeneous
+coordinates and multiplying out the fractional equations:
+min
+¯
+X∈R4,xi∈R2
+n
+�
+i=1
+∥xi − ˜xi∥2
+s.t.
+¯XT ¯X = 1
+xk
+i bT
+i ¯X − aT
+ik ¯X = 0
+i = 1, . . . , n
+k = 1, 2
+(7)
+Where ai1, ai2 and bi are given by the rows of the corre-
+sponding camera matrix Ki
+�
+RT
+i
+−RT
+i ti
+�
+. A naive ap-
+proach to relaxing Eq. (7) would be to use the parametriza-
+tion z = (x; ¯X), but unfortunately, as shown in [4], this
+leads to a problem whose optimal value is always zero. To
+circumvent this issue, [4] proposes parametrizing the prob-
+lem in terms of all possible products between the elements
+of x and X. They also show through experiments that while
+the resulting relaxation has more parameters and constraints
+than Eq. (T), it is also tight in a significantly wider range of
+cases, leading to a tradeoff between reliability and compu-
+tation time.
+4
+
+We will use a similar relaxation, though we will skip
+the initial change of variables to get a slightly different
+but equivalent formulation which can be extended to the
+robust case more conveniently. We start by setting z =
+(x ⊗ ¯X; ¯X) = ¯x ⊗ ¯X and then we multiply each repro-
+jection constraint in Eq. (7) with zj to get 8n + 4 quadratic
+constraints:
+(xk
+i bT
+i ¯X − aT
+ik ¯X)zj = zT (e¯x
+xk
+i ⊗ bi − e¯x
+1 ⊗ aik)eT
+j z
+= 0
+(8)
+Note that in Eq. (8) we have made use of the unit vector no-
+tation from Sec. 3, meaning in particular e¯x
+xk
+i = e2i+k and
+e¯x
+1 = e2n+1. We also need to indoduce constraints to pre-
+serve the fact that z comes from a (2n + 1) × 4 kronecker
+product. When Z = zzT is rank one, it turns out that this
+condition is equivalent to Z being composed of 2n+1 sym-
+metric 4×4 blocks, see [4] for more details. We will denote
+this constraint as Z ∈ kron(2n + 1, 4). The relaxation of
+can now be written as2:
+min
+Z∈S8n+4
++
+tr(Z(M˜x ⊗ I4))
+s.t.
+tr(Z(08n×8n ⊕ I4)) = 1
+Z ∈ kron(2n + 1, 4)
+tr(Z(e¯x
+xk
+i ⊗ bi − e¯x
+1 ⊗ aik)eT
+j ) = 0
+i = 1, . . . n,
+k = 1, 2
+j = 1, . . . , 8n + 4.
+(TF)
+We have now introduced two relaxations for the multiview
+triangulation problem. In the next two sections we will ex-
+tend each to the robust case.
+4.3. Robust triangulation with epipolar constraints
+Now that we have introduced the two main relaxations
+of Eq. (6) we move to the the main contribution of this pa-
+per, which is to introduce the corresponding truncated least
+squares (TLS) extensions. Similarly to [26] we will use the
+fact that the TLS cost function can be written as a minimiza-
+tion problem by introducing a binary decision variable for
+each residual
+ρi(r2
+i ) = min(r2
+i , ci) =
+min
+θi∈{0,1} θir2
+i + (1 − θi)ci
+(9)
+where ci > 0 is the square of the inlier threshold. Meaning
+that the TLS extension of Eq. (6) can be written as:
+min
+xi∈R2,θi∈R
+n
+�
+i=1
+�
+θi∥xi − ˜xi∥2 + (1 − θi)ci
+�
+s.t.
+¯xT
+i Fij ¯xj = 0,
+θ2
+i − θi = 0.
+i, j = 1, . . . , n
+i ̸= j.
+(10)
+2The cost functions in Eq. (7) and Eq. (TF) are equivalent, since ( ¯
+X ⊗
+¯x)T (M˜x ⊗ I4)( ¯
+X ⊗ ¯x) = (¯xT M˜x¯x) ¯
+XT ¯
+X.
+However this cost function is a 3rd degree polynomial in
+the variables as it contains terms like θi∥xi∥2, so we can’t
+apply the relaxation directly. But we can obtain a 2nd order
+formulation by noting that θ2
+i = θi implies θi∥xi − ˜xi∥2 =
+∥θixi − θi˜xi∥2 and making the substitution yi = θixi:
+min
+yi∈R2,θi∈R
+n
+�
+i=1
+�
+∥yi − θi˜xi∥2 + (1 − θi)ci
+�
+s.t.
+(yi; θi)T Fij(yj; θj) = 0
+θ2
+i − θi = 0
+θiyi = yi
+i, j = 1, . . . , n,
+i ̸= j.
+(11)
+The last set of constraints θiyi = yi is redundant but we’ve
+found that it is necessary for the relaxation to remain tight
+in the presence of noise. We can recover the solution to
+Eq. (10) from Eq. (11) by triangulating the estimated inliers
+and setting each xi to be the reprojection of the resulting
+point onto view i.
+Using the parametrization z = (y; θ; 1) the semidefinite
+relaxation of Eq. (11) is:
+min
+Z∈S3n+1
++
+tr(M c
+˜xZ)
+s.t.
+tr( ¯FijZ) = 0
+Zθi,θi − Z1,θi = 0
+Zθi,yi − Z1,yi = 0
+tr(E3n+1Z) = 1
+i, j = 1, . . . , n
+i ̸= j
+(RT)
+where M c
+˜x is the robust extension of M˜x, defined as:
+M c
+˜x =
+�
+�
+I
+−B(˜x)
+0
+−B(˜x)T
+diag(∥˜xi∥2)
+−c
+0
+−cT
+�n
+i=0 ci
+�
+� ,
+B(˜x) =
+�
+�
+�
+�
+�
+˜x1
+0
+. . .
+0
+0
+˜x2
+. . .
+0
+...
+...
+...
+0
+0
+0
+0
+˜xn
+�
+�
+�
+�
+� .
+(12)
+and Zmi,mj is the entry of Z corresponding to the index of
+the monomials mi and mj in z. As shown in [2] solving
+Eq. (11) in the presence of outliers is NP hard even in the
+noise-free case. However, in the noise-free and outlier-free
+case we can show that the relaxation is tight with a corank 1
+multiplier matrix, meaning that the relaxation is also locally
+stable with respect to noise, assuming ACQ holds:
+Theorem 1. Assuming ACQ holds, the relaxation Eq. (RT)
+is tight locally stable for noise-free and outlier-free mea-
+surements ˜xi, i = 1, . . . , n.
+5
+
+Proof. Partiton the lagrange multipliers as ξ = (ϕ; µ; η),
+where ϕij ∈ R µi ∈ R2 and η ∈ R corresponds to the
+constraints (yi; θi)T Fij(yj; θj) = 0, θiyi = yi and θ2
+i = θi
+respectively. Then we have:
+S(λ, ϕ, µ, η) =
+F(ϕ) +
+�
+�
+I
+−B(˜xi − µi)
+−µ
+∗
+diag(∥˜xi∥2 + 2ηi)
+− 1
+2c − η
+∗
+∗
+�n
+i=1 ci − λ
+�
+� . (13)
+Where F(ϕ) = �
+ij ϕij ¯Fij. Now let ˆλ = ˆϕij = ˆµi = 0
+and ˆηi = 1
+2ci to get:
+ˆS = S(ˆλ, ˆϕ, ˆµ, ˆη) = S(0, 0, 0, 1
+2c) =
+�
+�
+I
+−B(˜xi)
+0
+∗
+diag(∥˜xi∥2 + ci)
+−c
+∗
+∗
+�n
+i=1 ci
+�
+� .
+(14)
+This way, with ˆz = (˜x; 1n; 1) we have ˆSˆz = 0. And fur-
+thermore, for arbitrary x, θ, α:
+(x; θ; α)T ˆS(x; θ; α) =
+=
+n
+�
+i=0
+�
+∥xi∥2 − 2θi˜xi + θ2
+i (∥˜xi∥2+ci) − 2ciθiα + ciα2
+�
+=
+n
+�
+i=0
+�
+∥xi − θi˜xi∥2 + ci(α − θi)2
+�
+≥ 0
+so ˆS is positive semidefinite.
+So the relaxation is tight
+by Fact 1.
+And since the only nonzero solution to
+(x; θ; α)T ˆS(x; θ; α) = 0 up to scale is (x; θ; α) = ˆz we
+have that ˆS is corank 1. So, assuming ACQ holds, the re-
+laxation is locally stable by Fact 2.
+In the following section we will furthermore introduce a
+higher order relaxation which can handle higher noise and
+outlier levels.
+4.4. Robust triangulation with fraction constraints
+Since the fractional constraints in Eq. (TF) are more sta-
+ble with respect to noise than the epipolar constraints in
+Eq. (T), we might also expect that extending Eq. (TF) to
+handle outliers will result in a relaxation which is more sta-
+ble than Eq. (RT). In this section we will show how the
+robust extension can be formulated, and as we will see in
+Sec. 5 it is indeed significantly more stable with respect to
+both noise and outliers.
+In order to extend Eq. (TF) to handle outliers we will
+proceed in a similar manner as in the case with epipolar
+constraints. Starting by writing the cost function in terms of
+Problem
+Relaxation
+Robust
+Constraints
+Variables
+Eq. (6)
+Eq. (T)
+
+1
+2n2 − 1
+2n + 1
+2n + 1
+Eq. (11)
+Eq. (RT)
+
+1
+2n2 + 2.5n + 1
+3n + 1
+Eq. (7)
+Eq. (TF)
+
+28n2 + 14n + 1
+8n + 4
+Eq. (15)
+Eq. (RTF)
+
+51n2 + 65n + 1
+12n + 4
+Table 1. Summary of relaxations for the triangulation problem and
+its robust extension.
+the 2nd order variables yi = θixi:
+min
+¯
+X∈R4,xi∈R2
+n
+�
+i=1
+�
+∥yi − θi˜xi∥2 + (1 − θi)ci
+�
+s.t.
+¯XT ¯X = 1
+yk
+i bT
+i ¯X − aT
+ik ¯X = 0
+θ2
+i − θi = 0
+θiyi = yi
+i = 1, . . . , n
+k = 1, 2.
+(15)
+For convenience we will denote the vertical concatenation
+of y and θ as (y; θ) = yθ. For the relaxation we will then
+use the parametrization z = (yθ⊗ ¯X; ¯X) = ¯yθ⊗ ¯X and gen-
+erate redundant constraints from θ2
+i − θi = 0 and θiyi = yi
+by multiplying each equation by ¯Xs ¯Xt for s, t = 1, . . . , 4.
+Resulting in the following relaxation:
+min
+Z∈S12n+4
++
+tr(Z((M c
+˜x ⊗ I4) ⊕ 04×4))
+s.t.
+tr(Z(012n×12n ⊕ I4)) = 1
+Z ∈ kron(3n + 1, 4)
+tr(Z(e¯yθ
+yk
+i ⊗ bi − e¯yθ
+θi ⊗ aik)eT
+j ) = 0
+Z ¯
+Xsθi, ¯
+Xtθi − Z ¯
+Xs, ¯
+Xtθi = 0
+Z ¯
+Xsθi, ¯
+Xtyi − Z ¯
+Xs, ¯
+Xtyi = 0
+i = 1, . . . n,
+k = 1, 2
+s, t = 1, . . . , 4
+j = 1, . . . , 12n + 4.
+(RTF)
+Similarly to the epipolar case we are able to show that the
+relaxation is tight in the noise and outlier-free case by ex-
+plicitly constructing the globally optimal Lagrange multi-
+pliers. Using the same approach we are also able to show
+part of the criteria required for local stability with respect to
+noise in the outlier-free case. See Appendix A for details.
+With this we have 4 relaxations for the triangulation
+problem corresponding to the non-robust and robust case
+with the epipolar and the fractional parametrization. We
+summarize the relaxations and their number of variables and
+constraints in Tab. 1.
+6
+
+0
+20
+40
+60
+80
+100
+noise level ( )
+20
+40
+60
+80
+100
+% tight relaxations
+3 views
+0 outliers
+1 outliers
+2 outliers
+3 outliers
+4 outliers
+5 outliers
+robust epipolar (RT)
+robust fractional (RTF)
+0
+20
+40
+60
+80
+100
+noise level ( )
+20
+40
+60
+80
+100
+% tight relaxations
+5 views
+0
+20
+40
+60
+80
+100
+noise level ( )
+20
+40
+60
+80
+100
+% tight relaxations
+7 views
+0
+20
+40
+60
+80
+100
+noise level ( )
+10
+2
+10
+1
+100
+101
+average error
+3 views
+0
+20
+40
+60
+80
+100
+noise level ( )
+10
+2
+10
+1
+100
+101
+average error
+5 views
+0
+20
+40
+60
+80
+100
+noise level ( )
+10
+2
+10
+1
+100
+101
+average error
+7 views
+Figure 3. Average number of tight relaxation (top) and estimation error (bottom) for 3, 5 and 7 views for the robust epipolar relaxation
+Eq. (RT) and the robust fractional relaxation Eq. (RTF). We generate experiments for various noise levels and number of outliers as
+described in Sec. 5.1.
+4.5. Rounding in the non-tight case
+For non-tight cases the optimal ˆZ will have rank of at
+least 2, which means we can’t recover the optimal solution
+ˆz for the original problem Eq. (3). However we can still
+construct an approximate solution through a rounding pro-
+cedure. We start by setting ˆz to be the eigenvector corre-
+sponding to the minimal eigenvalue, normalized such that
+ˆzT Eˆz = 1 as in Eq. (3). We then apply a different pro-
+cedure for each problem depending on the constraints. For
+Eq. (T) we triangulate the resulting ˆxi (which in this case
+will generally not satisfy the epipolar constraints) using the
+linear method from [9] after rounding the smallest singu-
+lar value of the data matrix to 0. For Eq. (RT) we do the
+same except that we first determine the inlier parameters ˆθi
+by rounding the corresponding entries of ˆz to 0 or 1. For
+Eq. (TF) and Eq. (RTF) we compute the best-fitting tensor
+product decomposition of ˆz using a singular value decom-
+position, as described in [21] and use the same method as in
+the epipolar case for determining the inlier parameters.
+5. Experiments
+We
+implement
+all
+relaxations
+using
+CVXPY
+[8]
+with
+the
+solver
+MOSEK
+[3]
+using
+the
+setting
+0
+20
+40
+60
+80
+100
+noise level ( )
+0
+20
+40
+60
+80
+100
+% tight relaxations
+25 views
+0 outliers
+10 outliers
+20 outliers
+25 outliers
+0
+20
+40
+60
+80
+100
+noise level ( )
+0
+20
+40
+60
+80
+100
+% tight relaxations
+30 views
+0
+20
+40
+60
+80
+100
+noise level ( )
+10
+2
+10
+1
+100
+average error
+25 views
+0
+20
+40
+60
+80
+100
+noise level ( )
+10
+2
+10
+1
+100
+average error
+30 views
+Figure 4. Average number of tight relaxation (top) and estimation
+error(bottom) for 25 and 30 views using Eq. (RT).
+MSK DPAR INTPNT CO TOL REL GAP
+=
+10−14 for
+the simulated experiments and 10−10 for the Reichstag
+7
+
+5
+10
+15
+20
+25
+30
+number of views
+10
+1
+100
+101
+solver time (s)
+epipolar (T)
+fractional (TF)
+robust epipolar (RT)
+robust fractional (RTF)
+Figure 5. Average computation time for each solver, averaged over
+all noise levels and number of outliers.
+experiments, all other parameters are left on their defaults.
+We find that working in units of pixels results in poorly
+conditioned solutions, leading to ˆz not satisfying the
+constraints to high accuracy even in cases where the
+problem is known to be tight. To avoid this issue we use
+the change of variables xi →
+1
+W xi and adjust the intrinsics
+accordingly. Since the scaling is the same for each point
+the optimal solution remains unchanged, but we get much
+closer to rank one solutions in practice due to the improved
+numerical stability.
+5.1. Simulated experiments
+We simulate triangulation problems as initially proposed
+in [18] by placing n cameras on a sphere of radius 2 and
+sample a point to be triangulated from the unit cube, see
+Fig. 2 for some examples. The same setup was also used for
+experiments in [1, 4]. For the reprojection model we simu-
+late a pinhole camera with dimensions with width W =
+2108 and height H = 1162, focal length f = 1012.0027
+and principal point p = (1054, 581). We simulate noisy
+observations by adding Gaussian noise with standard devi-
+ation σ to the ground truth image coordinates. When gener-
+ating an outlier we select a view at random and replace the
+measurement with a random point in the image.
+We run the experiment for each method at various dif-
+ferent noise levels and number of outliers. For each noise
+level we run Eq. (RT) 375 times and Eq. (RTF) 60 times
+for n = 3, 5 and 7 views and in each case add up to n − 2
+outliers. The percentage of tight relaxations and the estima-
+tion error can be seen in Fig. 3. We also run Eq. (RT) 30
+times each for n = 25 and 30 with 0, 10, 20 and 25 out-
+liers, the results of which can be seen in Fig. 4. We don’t
+run Eq. (RTF) for these cases since we run into memory
+limitations with MOSEK.
+From Fig. 3 we can see that in general the fractional
+relaxation in Eq. (15) is significantly more stable than the
+epipolar relaxation Eq. (RT). In fact, across all experiments
+the fractional relaxation is tight in 99.8% of cases. How-
+ever, we can also note that the epipolar relaxation remains
+viable for lower noise levels, for instance in the case with
+n = 7 the relaxations perform similarly in terms of aver-
+age estimation error up until σ ≈ 60px and 3 outliers, after
+which the percentage of tight relaxations drop drastically.
+As can be seen from the average solver timings in Fig. 5
+0
+1
+2
+3
+4
+5
+number of outliers
+75
+80
+85
+90
+95
+100
+% tight relaxations
+3, 5, 7 views
+3 views
+5 views
+7 views
+25 views
+30 views
+robust epipolar (RT)
+robust fractional (RTF)
+0
+5
+10
+15
+20
+25
+number of outliers
+0
+20
+40
+60
+80
+100
+% tight relaxations
+25, 30 views
+0
+1
+2
+3
+4
+5
+number of outliers
+10
+2
+10
+1
+100
+101
+error (m)
+3, 5, 7 views
+0
+5
+10
+15
+20
+25
+number of outliers
+10
+2
+10
+1
+100
+101
+error (m)
+25, 30 views
+Figure 6. Average number of tight relaxation (top) and estimation
+error(bottom) for Eq. (RT) and Eq. (RTF) on the Reichstag dataset
+as descriped in section Sec. 5.2.
+the fractional relaxations is also over one order of magni-
+tude slower than the epipolar relaxation, meaning that it
+might be preferable to use Eq. (RT) in cases where the qual-
+ity of observations is known to be high.
+5.2. Reichstag dataset
+We also validate our relaxations on the Reichstag dataset
+from [11]. The dataset consits of 75 views of roughly 18k
+3D points. We use the ground truth correspondences es-
+timated by structure from motion as detailed in [11] and
+generate each triangulation problem by selecting n views
+which all observe a common point. We then add up to n−2
+outliers by replacing the ground truth observations with a
+randomly selected keypoints in the same image. See Fig. 1
+for an example point with n = 22 views and 19 outliers.
+For n = 3, 5 and 7 views we run Eq. (RT) 375 times and
+Eq. (RTF) 60 times for each possible number of outliers.
+And similarly we run Eq. (RT) 120 times for n = 25 and 30
+views. The results are summarized in Fig. 6.
+Similarly to the simulated experiments we can note that
+the percentage of tight relaxations (and mean error) de-
+creases (and increases) steadily as more outliers are added,
+with a sharp drop when the number of inliers gets close
+to 2, with the fractional method outperforming the epipo-
+lar method.
+6. Conclusion
+We proposed a global optimization framework robust
+multiview triangulation. To this end we derive semidefi-
+nite relaxations for triangulation losses that incorporate a
+truncated quadratic cost making them robust to both noise
+and outliers. On synthetic and real data we confirm that
+provably optimal triangulations can be computed and relax-
+ations remain empirically tight despite significant amounts
+of noise and outliers.
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+
+A. Local stability of fractional method
+In this section we will prove two of the criteria required
+for local stability for the robust fractional method Eq. (RTF)
+for noise-free and outlier-free measurements. Local stabil-
+ity for the non-robust case was shown already in [4] but we
+will provide an alternate proof here in our notation, since
+it will lead into the extension to the robust case. For this
+we will need the stronger version of Fact 2, which we will
+restate here loosely (see [5] Theorem 4.5 for more details).
+Using the definition A(ξ) = �k
+i=1 ξiAi:
+Fact 3. If we, in addition to the conditions in Fact 1, have
+that:
+(i) (ACQ) ACQ holds
+(ii) (smoothness) The the constraint set is smooth with re-
+spect to pertubations to the constraints
+(iii) (non-branch point) The nullspace of the multiplier ma-
+trix and the tangent space of the constrant-set at the
+optimum don’t intersect nontrivially: ker( ˆS) ∩ Tˆz =
+{0}
+(iv) (restricted slater) There exists ξ′, λ′ such that A(ξ′)−
+λ′E is positive definite on the subspace of vectors z⊥
+for which ˆSz⊥ = 0 and ˆzT z⊥ ̸= 0. In other words the
+part of the nullspace of ˆS which is orthogonal to the
+solution ˆz.
+The
+tangent
+space
+in
+(iii)
+is
+given
+by
+Tˆz
+=
+ker(ˆzT A1; . . . ; ˆzT Ak; ˆzT E).
+A.1. Non-robust version
+We will show (iii-iv) for a version of Eq. (TF) with some-
+what less constraints, noting that if we show (iii-iv) for the
+problem with less constraints we can then add in the remain-
+ing constraints back in and set the corresponding multipliers
+to zero to show that (iii-iv) holds for the original problem as
+well. Note however that since we don’t show (i-ii) the full
+proof is incomplete and is left for future work.
+Theorem 2. Assuming (i-ii) holds, the fractional relax-
+ation Eq. (TF) is tight and locally stable for noise-free and
+outlier-free measurements ˜xi, i = 1, . . . , n.
+Proof. We start by partitioning the Lagrange multipliers as
+ξ = (ϕ; α). Where ϕ = (ϕ1; . . . ; ϕ2n), and each ϕi ∈ R4
+contains the multipliers corresponding to ith reprojection
+constraint multiplied by the entries of ¯X (recall that there
+are two reprojection constraints per observation). Note that
+in the original formulation we also multiply by all the en-
+tries of x ⊗ ¯X as well, but as we will see these are not
+necessary for the proof to hold. And α corresponds to the
+kronecker product constraints.
+Since the observations ˜x are noise free we can denote the
+corresponding unique3 3D point in homogeneous coordi-
+nates as ˆX ∈ R4, normalized such that ∥ ˆX∥ = 1. It will be
+convenient to introduce the reparametrization u = ˜x which
+is the same as the observation vector, except partitioned
+such that u = (u1; . . . ; u2n), ui ∈ R, i.e. u2i+k = ˜xik
+for i = 1, . . . , n, k = 1, 2. The primal optimum is then ob-
+tained at ˆz = ¯u ⊗ ˆX, which is verified by setting ˆξ = ˆλ = 0
+to get ˆSˆz = (M˜x ⊗ I4)(¯u ⊗ ˆX) = (M˜x¯u) ⊗ ˆX = 0.
+We then note that, due to the properties of the kronecker
+product4 and that M˜x is positive semidefintie with corank 1,
+we have that M˜x ⊗ I4 is positive semidefinite with corank
+4. So the conditions of Fact 1 are satisfied.
+Since the nullspace ker( ˆS) is 4-dimensional and contains
+the four orthogonal vectors ˆz = ¯u ⊗ ˆX and ˆzl = ¯u ⊗ ˆXl
+where ˆXT ˆXl = 0, ˆXT
+l ˆXk = 0 for k ̸= l = 1, 2, 3 we can
+parametrize z⊥ from (iv) as z⊥ = ¯u⊗ ˆX⊥ where ˆXT
+⊥ ˆX = 0.
+For (iii) we need to show that the vectors that span
+ker( ˆS) are not in Tˆz, i.e. for any z ∈ ker( ˆS) either that
+ˆzT Aiz ̸= 0 for some constraint i, or that ˆzT Ez ̸= 0. This is
+the case since ˆzT Eˆz = 1 ̸= 0 and, letting Kijst be the kro-
+necker constraint matrix corresponding to index st of block
+ij, ˆzT Kijstzl = uiuj( ˆXs ˆXlt − ˆXt ˆXls) is nonzero for at
+least some index ijst unless u = 0 or ˆX and ˆXl are paral-
+lel, which is not the case by construction.
+To show (iv), we set α′ = λ′ = 0 and ϕ′
+i = uibi − ai,
+and verify that with z⊥ as above:
+zT
+⊥A(ϕ′, 0)z⊥ =
+2n
+�
+i=1
+ˆXT
+⊥ϕ′
+i(uibi − ai) ˆX⊥
+=
+2n
+�
+i=1
+((uibi − ai)T ˆX⊥)2 > 0
+(16)
+where the final strict inequality follows from the fact that
+each term is strictly positive as (uibi − ai)T ˆX = 0 by the
+original constraints and ˆX⊥ is orthogonal to ˆX.
+We note that, while not all constraints used in Eq. (TF)
+are required for (iii-iv) to hold, we have found some cases
+where adding the additional constraints results in a tighter
+relaxation in the presence of noise, so we used the full set
+of constraints in our experiments.
+A.2. Robust version
+We now move on to the robust fractional method
+Theorem 3. Assuming (i-ii) holds, the fractional relax-
+ation Eq. (RTF) is tight and locally stable for noise-free and
+outlier-free measurements ˜xi, i = 1, . . . , n.
+3assuming the observations are not degenerate, e.g. not all on a line.
+4For matrices A ∈ Sn, B ∈ Sm with eigenvalues αi, βj the eigen-
+values of the kronecker product A ⊗ B are given by the products of the
+eigenvalues αiβj for i = 1, . . . , n, j = 1, . . . , m.
+10
+
+Proof. Partition
+the
+Lagrange
+multipliers
+as
+ξ
+=
+(ϕ; µ; η; α), where as in Theorem 2 ϕ corresponds to the
+reprojection constraints and α corresponds to the kronecker
+constraints. We let µ ∈ R32n correspond to the constraints
+¯Xs ¯Xt(yikθi − yik) = 0 for s, t = 1, 2, 3, 4, k = 1, 2
+and i = 1, . . . , n.
+And finally we similarly have that
+η ∈ R16n = (η1; . . . ; ηn), ηi ∈ R16 corresponds to the
+constraints ¯Xs ¯Xt(θ2
+i − θi) = 0. For each view i we collect
+the corresponding subset of η into a 4×4 matrix Hi defined
+such that ¯XT Hi ¯X = �4
+s,t=1 ηist ¯Xs ¯Xt.
+To verify the global optimum we start by setting ˆz =
+¯uθ ⊗ ˆX where uθ = (˜x; 1n). We then note that the con-
+straint matrices for for the ηi-constraints can be written as a
+kronecker product to get:
+S(0, 0, η, 0) = M c
+˜x ⊗ I4 +
+n
+�
+i=1
+Ti ⊗ Hi
+(17)
+where each Ti ∈ S3n+1 is defined such that ¯yT
+θ Ti¯yθ = θ2
+i −
+θi for arbitrary yθ as in Sec. 4.4. We then set ˆη such that
+ˆHi = ciI4 and ˆϕ = ˆµ = ˆα = ˆλ = 0 to get:
+ˆS = S(0, 0, ˆη, 0) = (M c
+˜x +
+n
+�
+i=1
+ciTi) ⊗ I4.
+(18)
+Now, by the same argument as in Theorem 1 the matrix
+M c
+˜x + �n
+i=1 ciTi is positive semidefinite with corank 1, so
+ˆS is positive semidefinite with corank 4. Meaning that the
+conditions of Fact 1 are satisfied. (iii) also follows using
+the same argument based on the kronecker constraints as in
+Theorem 2.
+Finally, for (iv) we note that ker( ˆS) is spanned by ˆz and
+ˆzl = ¯uθ ⊗ ˆXl, l = 1, 2, 3, so by setting µ′ = η′ = α′ =
+λ′ = 0 and ϕ′
+i = uibi − ai restricted slater for ˆS follows in
+the same way as in Eq. (16).
+11
+