Datasets:

zai-org
/

RPC-Bench

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+# CRYSTAL DIFFUSION VARIATIONAL AUTOENCODER FOR PERIODIC MATERIAL GENERATION
+Tian Xie∗, Xiang Fu∗, Octavian-Eugen Ganea∗, Regina Barzilay, Tommi Jaakkola
+Computer Science and Artificial Intelligence Laboratory
+Massachusetts Institute of Technology
+Cambridge, MA 02139, USA
+{txie,xiangfu,oct,regina,tommi}@csail.mit.edu
+# ABSTRACT
+Generating the periodic structure of stable materials is a long-standing challenge for the material design community. This task is difficult because stable materials only exist in a low-dimensional subspace of all possible periodic arrangements of atoms: 1) the coordinates must lie in the local energy minimum defined by quantum mechanics, and 2) global stability also requires the structure to follow the complex, yet specific bonding preferences between different atom types. Existing methods fail to incorporate these factors and often lack proper invariances. We propose a Crystal Diffusion Variational Autoencoder (CDVAE) that captures the physical inductive bias of material stability. By learning from the data distribution of stable materials, the decoder generates materials in a diffusion process that moves atomic coordinates towards a lower energy state and updates atom types to satisfy bonding preferences between neighbors. Our model also explicitly encodes interactions across periodic boundaries and respects permutation, translation, rotation, and periodic invariances. We significantly outperform past methods in three tasks: 1) reconstructing the input structure, 2) generating valid, diverse, and realistic materials, and 3) generating materials that optimize a specific property. We also provide several standard datasets and evaluation metrics for the broader machine learning community.
+# 1 INTRODUCTION
+Solid state materials, represented by the periodic arrangement of atoms in the 3D space, are the foundation of many key technologies including solar cells, batteries, and catalysis (Butler et al., 2018). Despite the rapid progress of molecular generative models and their significant impact on drug discovery, the problem of material generation has many unique challenges. Compared with small molecules, materials have more complex periodic 3D structures and cannot be adequately represented by a simple graph like molecular graphs (Figure 1). In addition, materials can be made up of more than 100 elements in the periodic table, while molecules are generally only made up of a small subset of atoms such as carbon, oxygen, and hydrogen. Finally, the data for training ML models for material design is limited. There are only ${ \sim } 2 0 0 \mathrm { k }$ experimentally known inorganic materials, collected by the ICSD (Belsky et al., 2002), in contrast to close to a billion molecules in ZINC (Irwin & Shoichet, 2005).
+The key challenge of this task is in generating stable materials. Such materials only exist in a lowdimensional subspace of all possible periodic arrangements of atoms: 1) the atom coordinates must lie in the local energy minimum defined by quantum mechanics (QM); 2) global stability also requires the structure to follow the complex, yet specific
+![](images/f9a16b0c92de6c778ad170e8c78bce773bc7a6fa978abde181eb0a9ecf697006.jpg)
+Figure 1: The periodic structure of diamond. The left shows the infinite periodic structure, the middle shows a unit cell representing the periodic structure, and the right shows a multi-graph (Xie & Grossman, 2018) representation.
+bonding preferences between different atom types (section 3.2). The issue of stability is unique to material generation because valency checkers assessing molecular stability are not applicable to materials. Moreover, we also have to encode the interactions crossing periodic boundaries (Figure 1, middle), and satisfy permutation, translation, rotation, and periodic invariances (section 3.1). Our goal is to learn representations that can learn features of stable materials from data, while adhering to the above invariance properties.
+We address these challenges by learning a variational autoencoder (VAE) (Kingma & Welling, 2014) to generate stable 3D materials directly from a latent representation without intermediates like graphs. The key insight is to exploit the fact that all materials in the data distribution are stable, therefore if noise is added to the ground truth structure, denoising it back to its original structure will likely increase stability. We capture this insight by designing a noise conditional score network (NCSN) (Song & Ermon, 2019) as our decoder: 1) the decoder outputs gradients that drive the atom coordinates to the energy local minimum; 2) it also updates atom types based on the neighbors to capture the specific local bonding preferences (e.g., Si-O is preferred over Si-Si and O-O in $\mathrm { S i O } _ { 2 }$ ). During generation, materials are generated using Langevin dynamics that gradually deforms an initial random structure to a stable structure. To capture the necessary invariances and encode the interactions crossing periodic boundaries, we use SE(3) equivariant graph neural networks adapted with periodicity (PGNNs) for both the encoder and decoder of our VAE.
+Our theoretical analysis further reveals an intriguing connection between the gradient field learned by our decoder and an harmonic force field. De facto, the decoder utilizes the latter to estimate the forces on atoms when their coordinates deviate from the equilibrium positions. Consequently, this formulation provides an important physical inductive bias for generating stable materials.
+In this work, we propose Crystal Diffusion Variational AutoEncoder (CDVAE) to generate stable materials by learning from the data distribution of known materials. Our main contributions include:
+• We curate 3 standard datasets from QM simulations and create a set of physically meaningful tasks and metrics for the problem of material generation.
+• We incorporate stability as an inductive bias by designing a noise conditional score network as the decoder of our VAE, which allows us to generate significantly more realistic materials.
+• We encode permutation, translation, rotation, and periodic invariances, as well as interactions crossing periodic boundaries with SE(3) equivariant GNNs adapted with periodicity.
+• Empirically, our model significantly outperforms past methods in tasks including reconstructing an input structure, generating valid, diverse, and realistic materials, and generating materials that optimize specific properties.
+# 2 RELATED WORK
+Material graph representation learning. Graph neural networks have made major impacts in material property prediction. They were first applied to the representation learning of periodic materials by Xie & Grossman (2018) and later enhanced by many studies including Schutt et al. ¨ (2018); Chen et al. (2019). The Open Catalyst Project (OCP) provides a platform for comparing different architectures by predicting energies and forces from the periodic structure of catalytic surfaces (Chanussot et al., 2021). Our encoder and decoder PGNNs directly use GNN architectures developed for the OCP (Klicpera et al., 2020b; 2021; Shuaibi et al., 2021; Godwin et al., 2021), which are also closely related to SE(3) equivariant networks (Thomas et al., 2018; Fuchs et al., 2020).
+Quantum mechanical search of stable materials. Predicting the structure of unknown materials requires very expensive random search and QM simulations, and is considered a grand challenge in materials discovery (Oganov et al., 2019). State-of-the-art methods include random sampling (Pickard & Needs, 2011), evolutionary algorithms (Wang et al., 2012; Glass et al., 2006), substituting elements in known materials (Hautier et al., 2011), etc., but they generally have low success rates and require extensive computation even on relatively small problems.
+Material generative models. Past material generative models mainly focus on two different approaches, and neither incorporate stability as an inductive bias. The first approach treats materials as 3D voxel images, but the process of decoding images back to atom types and coordinates often results in low validity, and the models are not rotationally invariant (Hoffmann et al., 2019; Noh et al., 2019; Court et al., 2020; Long et al., 2021). The second directly encodes atom coordinates, types, and lattices as vectors (Ren et al., 2020; Kim et al., 2020; Zhao et al., 2021), but the models are generally not invariant to any Euclidean transformations. Another related method is to train a force field from QM forces and then apply the learned force field to generate stable materials by minimizing energy (Deringer et al., 2018; Chen & Ong, 2022). This method is conceptually similar to our decoder, but it requires additional force data which is expensive to obtain. Remotely related works include generating contact maps from chemical compositions (Hu et al., 2021; Yang et al., 2021) and building generative models only for chemical compositions (Sawada et al., 2019; Pathak et al., 2020; Dan et al., 2020).
+Molecular conformer generation and protein folding . Our decoder that generates the 3D atomic structures via a diffusion process is closely related to the diffusion models used for molecular conformer generation (Shi et al., 2021; Xu et al., 2021b). The key difference is that our model does not rely on intermediate representations like molecular graphs. G-SchNet (Gebauer et al., 2019) is more closely related to our method because it directly generates 3D molecules atom-by-atom without relying on a graph. Another closely related work is E-NFs (Satorras et al., 2021) that use a flow model to generate 3D molecules. In addition, score-based and energy-based models have also been used for molecular graph generation (Liu et al., 2021) and protein folding (Wu et al., 2021). Flow models have also been used for molecular graph generation (Shi et al., 2020; Luo et al., 2021). However, these generative models do not incorporate periodicity , which makes them unsuitable for materials.
+# 3 PRELIMINARIES
+# 3.1 PERIODIC STRUCTURE OF MATERIALS
+Any material structure can be represented as the periodic arrangement of atoms in the 3D space. As illustrated in Figure 1, we can always find a repeating unit, i.e. a unit cell, to describe the infinite periodic structure of a material. A unit cell that includes $N$ atoms can be fully described by 3 lists: 1) atom types $\pmb { A } = ( a _ { 0 } , . . . , a _ { N } ) \in \mathbb { A } ^ { N }$ , where A denotes the set of all chemical elements; 2) atom coordinates $\pmb { X } = ( \pmb { x } _ { 0 } , . . . , \pmb { x } _ { N } ) \in \mathbb { R } ^ { N \times 3 }$ ; and 3) periodic lattice $\pmb { L } = ( l _ { 1 } , l _ { 2 } , l _ { 3 } ) \in \mathbb { R } ^ { 3 \times 3 }$ . The periodic lattice defines the periodic translation symmetry of the material. Given $\pmb { M } = ( A , X , \pmb { L } )$ , the infinite periodic structure can be represented as,
+$$
+\begin{array} { r } { \{ ( a _ { i } ^ { \prime } , \pmb { x } _ { i } ^ { \prime } ) | a _ { i } ^ { \prime } = a _ { i } , \pmb { x } _ { i } ^ { \prime } = \pmb { x } _ { i } + k _ { 1 } l _ { 1 } + k _ { 2 } l _ { 2 } + k _ { 3 } l _ { 3 } , k _ { 1 } , k _ { 2 } , k _ { 3 } \in \mathbb { Z } \} , } \end{array}
+$$
+where $\boldsymbol { k } _ { 1 } , \boldsymbol { k } _ { 2 } , \boldsymbol { k } _ { 3 }$ are any integers that translate the unit cell using $\pmb { L }$ to tile the entire 3D space.
+The chemical composition of a material denotes the ratio of different elements that the material is composed of. Given the atom types of a material with $N$ atoms $\pmb { A } \in \mathbb { A } ^ { N }$ , the composition can be represented as $\boldsymbol { c } \in \mathbb { R } ^ { | \mathbb { A } | }$ , where $c _ { i } > 0$ denotes the percentage of atom type $i$ and $\textstyle \sum _ { i } { c _ { i } } = 1$ . For example, the composition of diamond in Figure 1 has $c _ { 6 } = 1$ and $c _ { i } = 0$ for $i \neq 6$ because 6 is the atomic number of carbon.
+Invariances for materials. The structure of a material does not change under several invariances. 1) Permutation invariance. Exchanging the indices of any pair of atoms will not change the material. 2) Translation invariance. Translating the atom coordinates $\boldsymbol { X }$ by an arbitrary vector will not change the material. 3) Rotation invariance. Rotating $\boldsymbol { X }$ and $\pmb { L }$ together by an arbitrary rotation matrix will not change the material. 4) Periodic invariance. There are infinite different ways of choosing unit cells with different shapes and sizes, e.g., obtaining a bigger unit cell as an integer multiplier of a smaller unit cell using integer translations. The material will again not change given different choices of unit cells.
+Multi-graph representation for materials. Materials can be represented as a directed multi-graph $\mathcal { G } = \{ \bar { \mathcal { V } } , \bar { \mathcal { E } } \}$ to encode the periodic structures following (Wells et al., 1977; O’Keeffe & Hyde, 1980; Xie & Grossman, 2018), where $\mathcal { V } = \{ v _ { 1 } , . . . , v _ { N } \}$ is the set of nodes representing atoms and ${ \mathcal { E } } =$ $\{ e _ { i j , ( k _ { 1 } , k _ { 2 } , k _ { 3 } ) } | i , j \in \{ 1 , . . . , N \} , k _ { 1 } , k _ { 2 } , k _ { 3 } \in \mathbb { Z } \}$ is the set of edges representing bonds. $e _ { i j , ( k _ { 1 } , k _ { 2 } , k _ { 3 } ) }$ denotes a directed edge from node $i$ at the original unit cell to node $j$ at the cell translated by $k _ { 1 } l _ { 1 } + k _ { 2 } l _ { 2 } + k _ { 3 } l _ { 3 }$ (in Figure 1 right, $( k _ { 1 } , k _ { 2 } , k _ { 3 } )$ are labeled on top of edges). For materials, there is no unique way to define edges (bonds) and the edges are often computed using $\mathbf { k }$ -nearest neighbor (KNN) approaches under periodicity or more advanced methods such as CrystalNN (Pan et al., 2021). Given this directed multi-graph, message-passing neural networks and SE(3)-equivariant networks can be used for the representation learning of materials.
+![](images/76a1088a1b2eb4c9f18fb457bb21e8b7032f46cf2c7456e531b05a39e4c96707.jpg)
+Figure 2: Overview of the proposed CDVAE approach.
+# 3.2 PROBLEM DEFINITION AND ITS PHYSICAL ORIGIN
+Our goal is to generate novel, stable materials $M = ( \pmb { A } , \pmb { X } , \pmb { L } ) \in \mathbb { A } ^ { N } \times \mathbb { R } ^ { N \times 3 } \times \mathbb { R } ^ { 3 \times 3 }$ . The space of stable materials is a subspace in $\mathbb { A } ^ { N } \times \mathbb { R } ^ { N \times 3 } \times \mathbb { R } ^ { 3 \times 3 }$ that satisfies the following constraints. 1) The materials lie in the local minimum of the energy landscape defined by quantum mechanics, with respect to the atom coordinates and lattice, i.e. ${ \partial \bar { E } / \partial X = \mathbf { \bar { 0 } } }$ and $\partial E / \partial \pmb { L } = \mathbf { 0 }$ . 2) The material is globally stable and thus cannot decompose into nearby phases. Global stability is strongly related to bonding preferences between neighboring atoms. For example, in $\mathrm { S i O } _ { 2 }$ , each Si is surrounded by $^ \textrm { \scriptsize 4 O }$ and each O is surrounded by $2 \ S \mathrm { i }$ . This configuration is caused by the stronger bonding preferences between Si-O than Si-Si and O-O.
+Generally, finding novel, stable materials requires very expensive random search and quantum mechanical simulations. To bypass this challenge, we aim to learn a generative model $p ( { \bar { M } } )$ from the empirical distribution of experimentally observed stable materials. A successful generative model will be able to generate novel materials that satisfy the above constraints, which can then be verified using quantum mechanical simulations.
+# 3.3 DIFFUSION MODELS
+Diffusion models are a new class of generative models that have recently shown great success in generating high-quality images (Dhariwal & Nichol, 2021), point clouds (Cai et al., 2020; Luo & Hu, 2021), and molecular conformations (Shi et al., 2021). There are several different types of diffusion models including diffusion probabilistic models (Sohl-Dickstein et al., 2015), noiseconditioned score networks (NCSN) (Song & Ermon, 2019), and denoising diffusion probabilistic models (DDPM) (Ho et al., 2020). We follow ideas from the NCSN (Song & Ermon, 2019) and learn a score network ${ \pmb s } _ { \pmb \theta } ( { \pmb x } )$ to approximate the gradient of a probability density $\nabla _ { \pmb { x } } p ( \pmb { x } )$ at different noise levels. Let $\{ \sigma _ { i } \} _ { i = 1 } ^ { L }$ be a sequence of positive scalars that satisfies $\sigma _ { 1 } / \sigma _ { 2 } = . . . = \sigma _ { L - 1 } / \sigma _ { L } > 1 .$ We define the data distribution perturbed by Gaussian noise $\sigma$ as $\begin{array} { r } { q _ { \sigma } ( \pmb { x } ) = \int p _ { \mathrm { d a t a } } ( \pmb { t } ) \mathcal { N } ( \pmb { x } | \pmb { t } , \sigma ^ { 2 } I ) \mathrm { d } \pmb { t } } \end{array}$ . The goal of NCSN is to learn a score network to jointly estimate the scores of all perturbed data distributions, i.e. $\forall \sigma \in \{ \sigma _ { i } \} _ { i = 1 } ^ { L } : s _ { \theta } ( \boldsymbol { x } , \sigma ) \approx \forall _ { \boldsymbol { x } } q _ { \sigma } ^ { \cdot } ( \boldsymbol { x } )$ . During generation, NCSN uses an annealed Langevin dynamics algorithm to produce samples following the gradient estimated by the score network with a gradually reduced noise level.
+# 4 PROPOSED METHOD
+Our approach generates new materials via a two-step process: 1) We sample a $_ z$ from the latent space and use it to predict 3 aggregated properties of a material: composition $( c )$ , lattice $( L )$ , and number of atoms $( N )$ , which are then used to randomly initialize a material structure $\tilde { M } = ( \tilde { A } , \tilde { X } , L )$ . 2) We perform Langevin dynamics to simultaneously denoise $\tilde { X }$ and $\tilde { A }$ conditioned on $_ z$ to improve both the local and global stability of $\tilde { M }$ and generate the final structure of the new material.
+To train our model, we optimize 3 networks concurrently using stable materials $M = ( A , X , L )$ sampled from the data distribution. 1) A periodic GNN encoder $\mathrm { P G N N } _ { \mathrm { E N C } } ( M )$ that encodes $M$ into a latent representation $_ z$ . 2) A property predictor $\mathrm { M L P _ { A G G } } ( z )$ that predicts the $c , L$ , and $N$ of $M$ from $_ z$ . 3) A periodic GNN decoder $\mathrm { P G N N } _ { \mathrm { D E C } } ( \tilde { M } | z )$ that denoises both $\tilde { X }$ and $\tilde { A }$ conditioned on $_ { z }$ . For 3), the noisy structure $\tilde { M } = ( \tilde { A } , \tilde { X } , L )$ is obtained by adding different levels of noise to $\boldsymbol { X }$ and $\pmb { A }$ . The noise schedules are defined by the predicted aggregated properties, with the motivation of simplifying the task for our decoder from denoising an arbitrary random structure from over ${ \sim } 1 0 0$ elements to a constrained random structure from predicted properties. We train all three networks together by minimizing a combined loss including the aggregated property loss $\mathcal { L } _ { \mathrm { { A G G } } }$ , decoder denoising loss $\mathcal { L } _ { \mathrm { D E C } }$ , and a KL divergence loss ${ \mathcal { L } } _ { \mathrm { K L } }$ for the VAE.
+To capture the interactions across periodic boundaries, we employ a multi-graph representation (section 3.1) for both $M$ and $\tilde { M }$ . We also use SE(3) equivariant GNNs adapted with periodicity as both the encoder and the decoder to ensure the permutation, translation, rotation, and periodic invariances of our model. The CDVAE is summarized in Figure 2 and we explain the individual components of our method below. The implementation details can be found in Appendix B.
+Periodic material encoder. $\mathrm { P G N N } _ { \mathrm { E N C } } ( M )$ encodes a material $M$ as a latent representation $z \in$ $\mathbb { R } ^ { D }$ following the reparameterization trick in VAE (Kingma & Welling, 2014). We use the multigraph representation (refer to section 3.1) to encode $M$ , and $\mathrm { P G N N } _ { \mathrm { E N C } }$ can be parameterized with an SE(3) invariant graph neural network.
+Prediction of aggregated properties. $\mathrm { M L P _ { A G G } } ( z )$ predicts 3 aggregated properties of the encoded material from its latent representation $_ z$ . It is parameterized by 3 separate multilayer perceptrons (MLPs). 1) Composition $\bar { \boldsymbol { c } } \in \mathbb { R } ^ { | \mathbb { A } | }$ is predicted by minimizing the cross entropy between the ground truth composition and predicted composition, i.e. $- \textstyle \sum _ { i } p _ { i } { \bar { \log } } c _ { i }$ . 2) Lattice $\bar { \boldsymbol { L } } \in \mathbb { R } ^ { 3 \times 3 }$ is reduced to 6 unique, rotation invariant parameters with the Niggli algorithm (Grosse-Kunstleve et al., 2004), i.e., the lengths of the 3 lattice vectors, the angles between them, and the values are predicted with an MLP after being normalized to the same scale (Appendix B.1) with an $L _ { 2 }$ loss. 3) Number of atoms $N \in \{ 1 , 2 , \bar { \ldots } \}$ is predicted with a softmax classification loss from the set of possible number of atoms. $\mathcal { L } _ { \mathrm { { A G G } } }$ is a weighted sum of the above 3 losses.
+Conditional score matching decoder. $\mathrm { P G N N } _ { \mathrm { D E C } } ( \tilde { M } | z )$ is a PGNN that inputs a noisy material $\tilde { M }$ with type noises $\sigma _ { A }$ , coordinate noises $\sigma _ { x }$ , as well as a latent $_ { z }$ , and outputs 1) a score $s _ { X } ( \tilde { M } | z ; \sigma _ { A } , \sigma _ { X } ) \in \mathbb { R } ^ { N \times 3 }$ to denoise the coordinate for each atom towards its ground truth value, and 2) a probability distribution of the true atom types $p _ { A } ( \tilde { M } | z ; \sigma _ { A } , \sigma _ { X } ) \in \mathbb { R } ^ { N \times | \mathbb { A } | }$ . We use a SE(3) graph network to ensure the equivariance of $\pmb { s x }$ with respect to the rotation of $\tilde { M }$ . To obtain the noisy structures $\tilde { M }$ , we sample $\sigma _ { A }$ and $\sigma _ { x }$ from two geometric sequences of the same length: $\{ \sigma _ { A , j } \} _ { j = 1 } ^ { \check { L } }$ , $\{ \sigma _ { { \pmb X } , j } \} _ { j = 1 } ^ { L }$ , and add the noises with the following methods. For type noises, we use the type distribution defined by the predicted composition $\begin{array} { r } { \tilde { A } \sim ( \frac { 1 } { 1 + \sigma _ { A } } p _ { A } + \frac { \sigma _ { A } } { 1 + \sigma _ { A } } p _ { c } ) } \end{array}$ , where $p _ { A , i j } = 1$ if atom $i$ $^ c$ has the true atom type to linearly perturb true type distribution $j$ and $p _ { A , i j } = 0$ for all other $j \mathrm { s }$ , and $\scriptstyle { p _ { c } }$ is the predicted composition. For coordinate noises, we add Gaussian noises to the true coordinates $\tilde { X } \sim \mathsf { \bar { N } } ( X , \sigma _ { X } ^ { 2 } I )$ .
+$\mathrm { P G N N } _ { \mathrm { D E C } }$ is parameterized by a SE(3) equivariant PGNN that inputs a multi-graph representation (section 3.1) of the noisy material structure and the latent representation. The node embedding for node $i$ is obtained by the concatenation of the element embedding of $\tilde { a } _ { i }$ and the latent representation $_ z$ , followed by a MLP, $\begin{array} { r } { \pmb { h } _ { i } ^ { 0 } = \mathrm { M L P } ( \pmb { e } _ { \mathrm { a } } ( \tilde { a } _ { i } ) \parallel \pmb { z } ) } \end{array}$ , where $\parallel$ denotes concatenation of two vectors and $e _ { \mathrm { a } }$ is a learned embedding for elements. After $K$ message-passing layers, $\mathrm { P G N N _ { D E C } }$ outputs a vector per node that is equivariant to the rotation of $\tilde { M }$ . These vectors are used to predict the scores, and we follow Song & Ermon (2019); Shi et al. (2021) to parameterize the score network with noise scaling: $s _ { X } ( \tilde { M } | z ; \sigma _ { A } , \sigma _ { X } ) = s _ { X } ( \tilde { M } | z ) / \sigma _ { X }$ . The node representations $h _ { i } ^ { K }$ are used to predict the distribution of true atom types, and the type predictor is the same at all noise levels: $p _ { A } \dot { ( M | z ; \sigma _ { A } , \sigma _ { X } ) } = p _ { A } ( \tilde { M } | z )$ , $p _ { A } ( \tilde { M } | z ) _ { i } = \mathrm { s o f t m a x } ( \mathrm { M L P } ( h _ { i } ^ { K } ) )$ .
+Periodicity influences denoising target. Due to periodicity, a specific atom $i$ may move out of the unit cell defined by $\pmb { L }$ when the noise is sufficiently large. This leads to two different ways to define the scores for node $i$ . 1) Ignore periodicity and define the target score as $\pmb { x } _ { i } - \tilde { \pmb { x } } _ { i }$ ; or 2) Define the target score as the shortest possible displacement between $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ and $\tilde { \mathbf { x } } _ { i }$ considering periodicity, i.e. $\begin{array} { r } { d _ { \operatorname* { m i n } } ( \pmb { x } _ { i } , \tilde { \pmb { x } } _ { i } ) = \operatorname* { m i n } _ { k _ { 1 } , k _ { 2 } , k _ { 3 } } \bar { ( } \pmb { x } _ { i } - \tilde { \pmb { x } } _ { i } + \bar { k } _ { 1 } l _ { 1 } + k _ { 2 } l _ { 2 } + k _ { 3 } l _ { 3 } ) } \end{array}$ . We choose 2) because the scores are the same given two different $\tilde { X }$ that are periodically equivalent, which is mathematically grounded for periodic structures, and empirically results in much more stable training.
+The training loss for the decoder $\mathcal { L } _ { \mathrm { D E C } }$ can be written as,
+$$
+\frac { 1 } { 2 L } \sum _ { j = 1 } ^ { L } \left[ \mathbb { E } _ { q _ { \mathrm { d a t a } ( M ) } } \mathbb { E } _ { q _ { \sigma _ { A , j } , \sigma _ { X , j } } ( \tilde { M } | M ) } \left( \left\| s x ( \tilde { M } | z ) - \frac { d _ { \operatorname* { m i n } } ( X , \tilde { X } ) } { \sigma _ { X , j } } \right\| _ { 2 } ^ { 2 } + \frac { \lambda _ { \mathrm { a } } } { \sigma _ { A , j } } \mathcal { L } _ { \mathrm { a } } ( p _ { A } ( \tilde { M } | z ) , p _ { A } ) \right) \right] ,
+$$
+where $\lambda _ { \mathrm { a } }$ denotes a coefficient for balancing the coordinate and type losses, $\mathcal { L } _ { \mathrm { a } }$ denotes the cross entropy loss over atom types, $_ { p _ { A } }$ denotes the true atom type distribution. Note that to simplify the equation, we follow the loss coefficients in Song & Ermon (2019) for different $\sigma _ { x , j }$ and $\sigma _ { A , j }$ and factor them into Equation 2.
+Material generation with Langevin dynamics. After training the model, we can generate the periodic structure of material given a latent representation $_ z$ . First, we use $_ z$ to predict the aggregated properties: 1) composition $c , \ 2 )$ lattice $\pmb { L }$ , and 3) the number of atoms $N$ . Then, we randomly initialize an initial periodic structure $( A _ { 0 } , X _ { 0 } , L )$ with the aggregated properties and perform an annealed Langevin dynamics (Song & Ermon, 2019) using the decoder, simultaneously updating the atom types and coordinates. During the coordinate update, we map the coordinates back to the unit cell at each step if atoms move out of the cell. The algorithm is summarized in Algorithm 1.
+# Algorithm 1 Material Generation via Annealed Langevin Dynamics
+1: Input: latent representation $_ { z }$ , type and coordinate noise
+levels $\{ \sigma _ { A } \} , ~ \{ \bar { \sigma } _ { X } \}$ , step size $\epsilon$ , number of sampling
+steps $T$
+2: Predict aggregated properties $c , L , N$ from $_ { z }$ .
+3: Uniformly initialize $X _ { 0 }$ within the unit cell by $\pmb { L }$ .
+4: Randomly initialize $\pmb { A } _ { 0 }$ with $^ c$ .
+5: for 6: $j  1$ $L$
+8: $\begin{array} { r l } & { \mathbf { \Phi } _ { \alpha j } ^ { \circ } \gets \epsilon \cdot \boldsymbol { \sigma } _ { X , j } ^ { 2 } / \boldsymbol { \sigma } _ { X , L } ^ { 2 } } \\ & { \mathbf { f } \mathbf { \Phi } \mathbf { f } \gets 1 \mathrm { t o } \mathcal { T } \mathbf { d } \mathbf { 0 } } \\ & { \qquad \mathbf { \Phi } _ { X , t } ^ { \circ } \gets \mathbf { s } _ { X } ( A _ { t - 1 } , X _ { t - 1 } , L \vert z ; \boldsymbol { \sigma } _ { A , j } , \boldsymbol { \sigma } _ { X , j } ) } \\ & { \qquad p _ { A , t } \gets p _ { A } \big ( A _ { t - 1 } , X _ { t - 1 } , L \vert z ; \boldsymbol { \sigma } _ { A , j } , \boldsymbol { \sigma } _ { X , j } \big ) } \\ & { \qquad \mathrm { D r a w } \ X _ { t } ^ { \epsilon } \sim \mathcal { N } \big ( 0 , I \big ) } \\ & { \qquad X _ { t } ^ { \epsilon } \gets X _ { t - 1 } + \alpha _ { j } \mathbf { s } _ { X , t } + \sqrt { 2 \alpha _ { i } } X _ { t } ^ { \epsilon } } \\ & { \qquad X _ { t } \gets \mathrm { b a c k . t o . c e l l } ( X _ { t } ^ { \prime } , L ) } \\ & { \qquad A _ { t } = \mathrm { a r g m a x } p _ { A , t } } \\ & { \qquad X _ { 0 } \gets X _ { T , A _ { 0 } } \gets A _ { T } } \end{array}$
+9:
+10:
+11:
+12:
+13:
+14:
+Connection between the gradient field and a harmonic force field. The gradient field $s _ { X } ( { \tilde { M } } | z )$ is used to update atom coordinates in Langevin dynamics via the force term, $\alpha _ { j } s _ { X , t } .$ . In Appendix A, we show that $\alpha _ { j } { s } _ { X , t }$ is mathematically equivalent $\mathrm { t o } ^ { 2 }$ a harmonic force field ${ \cal F } ( \tilde { \cal X } ) = - k ( \tilde { \cal X } -$ $\boldsymbol { X }$ ) when the noises are small, where $\boldsymbol { X }$ is the equilibrium position of the atoms and $k$ is a force constant. Harmonic force field, i.e. spring-like force field, is a simple yet general physical model that approximates the forces on atoms when they are close to their equilibrium locations. This indicates that our learned gradient field utilizes the harmonic approximation to approximate QM forces without any explicit force data and generates stable materials with this physically motivated inductive bias.
+# 5 EXPERIMENTS
+We evaluate multiple aspects of material generation that are related to real-world material discovery process. Past studies in this field used very different tasks and metrics, making it difficult to compare different methods. Building upon past studies (Court et al., 2020; Ren et al., 2020), we create a set of standard tasks, datasets, and metrics to evaluate and compare models for material generation. Experiment details can be found in Appendix D.
+Tasks. We focus on 3 tasks for material generation. 1) Reconstruction evaluates the ability of the model to reconstruct the original material from its latent representation z. 2) Generation evaluates the validity, property statistics, and diversity of material structures generated by the model. 3) Property optimization evaluates the model’s ability to generate materials that are optimized for a specific property.
+Datasets. We curated 3 datasets representing different types of material distributions. 1) Perov5 (Castelli et al., 2012a;b) includes 18928 perovskite materials that share the same structure but differ in composition. There are 56 elements and all materials have 5 atoms in the unit cell. 2) Carbon-24 (Pickard, 2020) includes 10153 materials that are all made up of carbon atoms but differ in structures. There is 1 element and the materials have $6 \textsuperscript { - } 2 4$ atoms in the unit cells. 3) MP-20 (Jain et al., 2013) includes 45231 materials that differ in both structure and composition. There are
+![](images/578fc93f934b4eaf42ba8a5e6f4ba0d08297c97c2bd4b63b3ba7427bffdf402c.jpg)
+Figure 3: Reconstructed structures of randomly selected materials in the test set. Note our model reconstructs rotated (translated) version of the original material due to the SE(3) invariance.
+Table 1: Reconstruction performance.
+<table><tr><td rowspan="2">Method</td><td colspan="4">Match rate(%)个</td><td colspan="2">RMSE↓</td></tr><tr><td>Perov-5</td><td>Carbon-24</td><td>MP-20</td><td>Perov-5</td><td>Carbon-24</td><td>MP-20</td></tr><tr><td>FTCP</td><td>99.34</td><td>62.28</td><td>69.89</td><td>0.0259</td><td>0.2563</td><td>0.1593</td></tr><tr><td>Cond-DFC-VAE</td><td>51.65</td><td>1</td><td>1</td><td>0.0217</td><td>1</td><td>一</td></tr><tr><td>CDVAE</td><td>97.52</td><td>55.22</td><td>45.43</td><td>0.0156</td><td>0.1251</td><td>0.0356</td></tr></table>
+89 elements and the materials have 1 - 20 atoms in the unit cells. We use a 60-20-20 random split for all of our experiments. Details regarding dataset curation can be found at Appendix C.
+Stability of materials in datasets. Structures in all 3 datasets are obtained from QM simulations and all structures are at local energy minima. Most materials in Perov-5 and Carbon-24 are hypothetical, i.e. they may not have global stability (section 3.2) and likely cannot be synthesized. MP-20 is a realistic dataset that includes most experimentally known inorganic materials with at most 20 atoms in the unit cell, most of which are globally stable. A model achieving good performance in MP-20 has the potential to generate novel materials that can be experimentally synthesized.
+Baselines. We compare CDVAE with the following 4 baselines, which include the latest coordinatebased, voxel-based, and 3D molecule generation methods. FTCP (Ren et al., 2020) is a crystal representation that concatenates real-space properties (atom positions, atom types, etc.) and Fouriertransformed momentum-space properties (diffraction pattern). A 1D CNN-VAE is trained over this representation for crystal generation. Cond-DFC-VAE (Court et al., 2020) encodes and generates crystals with 3D density maps, while employing several modifications over the previous Voxel-VAE (Hoffmann et al., 2019) method. However, the effectiveness is only demonstrated for cubic systems, limiting its usage to the Perov-5 dataset. G-SchNet (Gebauer et al., 2019) is an auto-regressive model that generates 3D molecules by performing atom-by-atom completion using SchNet (Schutt ¨ et al., 2018). Since G-SchNet is unaware of periodicity and cannot generate the lattice $\pmb { L }$ . We adapt G-SchNet to our material generation tasks by constructing the smallest oriented bounding box with PCA such that the introduced periodicity does not cause structural invalidity. P-G-SchNet is our modified G-SchNet that incorporates periodicity. During training, the SchNet encoder inputs the partial periodic structure to predict next atoms. During generation, we first randomly sample a lattice $\pmb { L }$ from training data and autoregressively generate the periodic structure.
+# 5.1 MATERIAL RECONSTRUCTION
+Setup. The first task is to reconstruct the material from its latent representation. We evaluate reconstruction performance by matching the generated structure and the input structure for all materials in the test set. We use StructureMatcher from pymatgen (Ong et al., 2013), which finds the best match between two structures considering all invariances of materials. The match rate is the percentage of materials satisfying the criteria $s \ t \circ 1 \mathrm { = } 0 \ . \ 5$ , angle tol ${ \ o } = 1 0$ , $1 \ t { \bigcirc } 1 = 0 \cdot 3$ . The RMSE is averaged over all matched materials. Because the inter-atomic distances can vary significantly for different materials, the RMSE is normalized by $\sqrt [ 3 ] { V / N }$ , roughly the average atom radius per material. Note G-SchNet is not a VAE so we do not evaluate its reconstruction performance.
+Results. The reconstructed structures are shown in Figure 3 and the metrics are in Table 1. Since our model is SE(3) invariant, the generated structures may be a translated (or rotated) version of the ground truth structure. Our model has a lower RMSE than all other models, indicating its stronger capability to reconstruct the original stable structures. FTCP has a higher match rate than our model.
+![](images/46996ad50747cc42c9777944916f911c171bbc32b70cd77b6a0f3f7c42b47d98.jpg)
+Figure 4: Structures sampled from $\mathcal { N } ( 0 , 1 )$ and filtered by the validity test.
+Table 2: Generation performance3.
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Data</td><td colspan="2">Validity (%) 4个</td><td colspan="2">COV(%)↑</td><td colspan="3">Property Statistics ↓</td></tr><tr><td>Struc.</td><td>Comp.</td><td>R.</td><td>P</td><td>p</td><td>E</td><td># elem.</td></tr><tr><td>FTCP5</td><td>Perov-5</td><td>0.24</td><td>54.24</td><td>0.00</td><td>0.00</td><td>10.27</td><td>156.0</td><td>0.6297</td></tr><tr><td rowspan="5">Cond-DFC-VAE</td><td>Carbon-24</td><td>0.08</td><td></td><td>0.00</td><td>0.00</td><td>5.206</td><td>19.05</td><td></td></tr><tr><td>MP-20</td><td>1.55</td><td>48.37</td><td>4.72</td><td>0.09</td><td>23.71</td><td>160.9</td><td>0.7363</td></tr><tr><td>Perov-5</td><td>73.60</td><td>82.95</td><td>73.92</td><td>10.13</td><td>2.268</td><td>4.111</td><td>0.8373</td></tr><tr><td>Perov-5</td><td>99.92</td><td>98.79</td><td>0.18</td><td>0.23</td><td>1.625</td><td>4.746</td><td>0.03684</td></tr><tr><td>Carbon-24</td><td>99.94</td><td></td><td>0.00</td><td>0.00</td><td>0.9427</td><td>1.320</td><td></td></tr><tr><td rowspan="4">P-G-SchNet</td><td>MP-20</td><td>99.65</td><td>75.96</td><td>38.33</td><td>99.57</td><td>3.034</td><td>42.09</td><td>0.6411</td></tr><tr><td>Perov-5</td><td>79.63</td><td>99.13</td><td>0.37</td><td>0.25</td><td>0.2755</td><td>1.388</td><td>0.4552</td></tr><tr><td>Carbon-24</td><td>48.39</td><td></td><td>0.00</td><td>0.00</td><td>1.533</td><td>134.7</td><td></td></tr><tr><td>MP-20</td><td>77.51</td><td>76.40</td><td>41.93</td><td>99.74</td><td>4.04</td><td>2.448</td><td>0.6234</td></tr><tr><td rowspan="3">CDVAE</td><td>Perov-5</td><td>100.0</td><td>98.59</td><td>99.45</td><td>98.46</td><td>0.1258</td><td>0.0264</td><td>0.0628</td></tr><tr><td>Carbon-24</td><td>100.0</td><td></td><td>99.80</td><td>83.08</td><td>0.1407</td><td>0.2850</td><td></td></tr><tr><td>MP-20</td><td>100.0</td><td>86.70</td><td>99.15</td><td>99.49</td><td>0.6875</td><td>0.2778</td><td>1.432</td></tr></table>
+This can be explained by the fact that the same set of local structures can be assembled into different stable materials globally (e.g., two different crystal forms of $Z \mathrm { n } S _ { \mathrm {  { \tau } } }$ ). Our model is SE(3) invariant and only encodes local structures, while FTCP directly encodes the absolute coordinates and types of each atom. In Figure 5, we show that CDVAE can generate different plausible arrangements of atoms by sampling 3 Langevin dynamics with different random seeds from the same $_ z$ . We note that this capability could be an advantage since it generates more diverse structures than simply reconstructing the original ones.
+# 5.2 MATERIAL GENERATION
+Setup. The second task is to generate novel, stable materials that are distributionally similar to the test materials. The only high-fidelity evaluation of stability of generated materials is to perform QM calculations, but it is computationally prohibitive to use QM for computing evaluation metrics. We developed several physically meaningful metrics to evaluate the validity, property statistics, and diversity of generated materials. 1) Validity. Following Court et al. (2020), a structure is valid as long as the shortest distance between any pair of atoms is larger than $0 . 5 \mathring \mathrm { A }$ , which is a relative weak criterion. The composition is valid if the overall charge is neutral as computed by SMACT (Davies et al., 2019). 2) Coverage (COV). Inspired by $\mathrm { X u }$ et al. (2021a); Ganea et al. (2021), we define two coverage metrics, COV-R (Recall) and COV-P (Precision), to measure the similarity between ensembles of generated materials and ground truth materials in test set. Intuitively, COV-R measures the percentage of ground truth materials being correctly predicted, and COV-P measures the percentage of predicted materials having high quality (details in Appendix G). 3) Property statistics. We compute the earth mover’s distance (EMD) between the property distribution of generated materials and test materials. We use density ( $\dot { \rho } { } _ { ; }$ , unit $\mathrm { { g } / \mathrm { { c m } ^ { 3 } } } .$ ), energy predicted by an independent GNN ( $E$ , unit eV/atom), and number of unique elements (# elem.) as our properties. Validity and coverage are computed over 10,000 materials randomly sampled from $\mathcal { N } ( 0 , \bar { 1 } )$ . Property statistics is computed over 1,000 valid materials randomly sampled from those that pass the validity test.
+Table 3: Property optimization performance.
+<table><tr><td rowspan="2">Method</td><td colspan="3">Perov-5</td><td colspan="3">Carbon-24</td><td colspan="3">MP-20</td></tr><tr><td>SR5</td><td>SR10</td><td>SR15</td><td>SR5</td><td>SR10</td><td>SR15</td><td>SR5</td><td>SR10</td><td>SR15</td></tr><tr><td>FTCP</td><td>0.06</td><td>0.11</td><td>0.16</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.02</td><td>0.04</td><td>0.05</td></tr><tr><td>Cond-DFC-VAE</td><td>0.55</td><td>0.64</td><td>0.69</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>一</td></tr><tr><td>CDVAE</td><td>0.52</td><td>0.65</td><td>0.79</td><td>0.0</td><td>0.06</td><td>0.06</td><td>0.78</td><td>0.86</td><td>0.90</td></tr></table>
+Results. The generated structures are shown in Figure 4 and the metrics are in Table 2. Our model achieves a higher validity than FTCP, Cond-DFC-VAE, and P-G-SchNet, while G-SchNet achieves a similar validity as ours. The lower structural validity in P-G-SchNet than G-SchNet is likely due to the difficulty of avoiding atom collisions during the autoregressive generation inside a finite periodic box. On the contrary, our G-SchNet baseline constructs the lattice box after the 3D positions of all atoms are generated, and the construction explicitly avoids introducing invalidity. Furthermore, our model also achieves higher COV-R and COV-P than all other models, except in MP-20 our COV-P is similar to G-SchNet and P-G-SchNet. These results indicate that our model generates both diverse (COV-R) and high quality (COV-P) materials. More detailed results on the choice of thresholds for COV-R and COV-P, as well as additional metrics can be found in Appendix G. Finally, our model also significantly outperforms all other models in the property statistics of density and energy, further confirming the high quality of generated materials. We observe that our method tends to generate more elements in a material than ground truth, which explains the lower performance in the statistics of # of elems. than G-SchNet. We hypothesize this is due to the non-Gaussian statistical structure of ground truth materials (details in Appendix D.3), and using a more complex prior, e.g., a flowmodel-transformed Gaussian (Yang et al., 2019), might resolve this issue.
+# 5.3 PROPERTY OPTIMIZATION
+Setup. The third task is to generate materials that optimize a specific property. Following Jin et al. (2018), we jointly train a property predictor $F$ parameterized by an MLP to predict properties of training materials from latent $_ z$ . To optimize properties, we start with the latent representations of testing materials and apply gradient ascent in the latent space to improve the predicted property $F ( \cdot )$ . After applying 5000 gradient steps with step sizes of $1 \times 1 0 ^ { - 3 }$ , 10 materials are decoded from the latent trajectories every 500 steps. We use an independently trained property predictor to select the best one from the 10 decoded materials. Cond-DFC-VAE is a conditional VAE so we directly condition on the target property, sample 10 materials, and select the best one using the property predictor. For all methods, we generate 100 materials following the protocol above. We use the independent property predictor to predict the properties for evaluation. We report the success rate (SR) as the percentage of materials achieving 5, 10, and 15 percentiles of the target property distribution. Our task is to minimize formation energy per atom for all 3 datasets.
+Results. The performance is shown in Table 3. We significantly outperform FTCP, while having a similar performance as Cond-DFC-VAE in Perov-5 (Cond-DFC-VAE cannot work for Carbon-24 and MP-20). Both G-SchNet and P-G-SchNet are incapable of property optimization 6. We note that all models perform poorly on the Carbon-24 dataset, which might be explained by the complex and diverse 3D structures of carbon.
+# 6 CONCLUSIONS AND OUTLOOK
+We have introduced a Crystal Diffusion Variational Autoencoder (CDVAE) to generate the periodic structure of stable materials and demonstrated that it significantly outperforms past methods on the tasks of reconstruction, generation, and property optimization. We note that the last two tasks are far more important for material design than reconstruction because they can be directly used to generate new materials whose properties can then be verified by QM simulations and experiments. We believe CDVAE opens up exciting opportunities for the inverse design of materials for various important applications. Meanwhile, our model is just a first step towards the grand challenge of material design. We provide our datasets and evaluation metrics to the broader machine learning community to collectively develop better methods for the task of material generation.
+# REPRODUCIBILITY STATEMENT
+We have made the following efforts to ensure reproducibility: 1) We provide our code at https:// github.com/txie-93/cdvae; 2)We provide our data and corresponding train/validation/test splits at https://github.com/txie-93/cdvae/tree/main/data; 3) We provide details on experimental configurations in Appendix D.
+# ACKNOWLEDGMENTS
+We thank Peter Mikhael, Jason Yim, Rachel Wu, Bracha Laufer, Gabriele Corso, Felix Faltings, Bowen Jing, and the rest of the RB and TJ group members for their helpful comments and suggestions. The authors gratefully thank DARPA (HR00111920025), the consortium Machine Learning for Pharmaceutical Discovery and Synthesis (mlpds.mit.edu), and MIT-GIST collaboration for support.
+# REFERENCES
+Alec Belsky, Mariette Hellenbrandt, Vicky Lynn Karen, and Peter Luksch. New developments in the inorganic crystal structure database (icsd): accessibility in support of materials research and design. Acta Crystallographica Section B: Structural Science, 58(3):364–369, 2002. 1, 16
+Keith T Butler, Daniel W Davies, Hugh Cartwright, Olexandr Isayev, and Aron Walsh. Machine learning for molecular and materials science. Nature, 559(7715):547–555, 2018. 1
+Ruojin Cai, Guandao Yang, Hadar Averbuch-Elor, Zekun Hao, Serge Belongie, Noah Snavely, and Bharath Hariharan. Learning gradient fields for shape generation. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part III 16, pp. 364–381. Springer, 2020. 4
+Ivano E Castelli, David D Landis, Kristian S Thygesen, Søren Dahl, Ib Chorkendorff, Thomas F Jaramillo, and Karsten W Jacobsen. New cubic perovskites for one-and two-photon water splitting using the computational materials repository. Energy & Environmental Science, 5(10):9034– 9043, 2012a. 6, 16
+Ivano E Castelli, Thomas Olsen, Soumendu Datta, David D Landis, Søren Dahl, Kristian S Thygesen, and Karsten W Jacobsen. Computational screening of perovskite metal oxides for optimal solar light capture. Energy & Environmental Science, 5(2):5814–5819, 2012b. 6, 16
+Lowik Chanussot, Abhishek Das, Siddharth Goyal, Thibaut Lavril, Muhammed Shuaibi, Morgane Riviere, Kevin Tran, Javier Heras-Domingo, Caleb Ho, Weihua Hu, et al. Open catalyst 2020 (oc20) dataset and community challenges. ACS Catalysis, 11(10):6059–6072, 2021. 2, 15
+Chi Chen and Shyue Ping Ong. A universal graph deep learning interatomic potential for the periodic table. arXiv preprint arXiv:2202.02450, 2022. 3
+Chi Chen, Weike Ye, Yunxing Zuo, Chen Zheng, and Shyue Ping Ong. Graph networks as a universal machine learning framework for molecules and crystals. Chemistry of Materials, 31(9):3564– 3572, 2019. 2
+Callum J Court, Batuhan Yildirim, Apoorv Jain, and Jacqueline M Cole. 3-d inorganic crystal structure generation and property prediction via representation learning. Journal of chemical information and modeling, 60(10):4518–4535, 2020. 2, 6, 7, 8
+Yabo Dan, Yong Zhao, Xiang Li, Shaobo Li, Ming Hu, and Jianjun Hu. Generative adversarial networks (gan) based efficient sampling of chemical composition space for inverse design of inorganic materials. npj Computational Materials, 6(1):1–7, 2020. 3
+Daniel W Davies, Keith T Butler, Adam J Jackson, Jonathan M Skelton, Kazuki Morita, and Aron Walsh. Smact: Semiconducting materials by analogy and chemical theory. Journal of Open Source Software, 4(38):1361, 2019. 8, 16
+Volker L Deringer, Chris J Pickard, and Gabor Cs ´ anyi. Data-driven learning of total and local ´ energies in elemental boron. Physical review letters, 120(15):156001, 2018. 3
+Prafulla Dhariwal and Alex Nichol. Diffusion models beat gans on image synthesis. arXiv preprint arXiv:2105.05233, 2021. 4
+Fabian Fuchs, Daniel E. Worrall, Volker Fischer, and Max Welling. Se(3)-transformers: 3d rototranslation equivariant attention networks. In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020. URL https://proceedings.neurips.cc/ paper/2020/hash/15231a7ce4ba789d13b722cc5c955834-Abstract.html. 2
+Octavian-Eugen Ganea, Lagnajit Pattanaik, Connor W Coley, Regina Barzilay, Klavs F Jensen, William H Green, and Tommi S Jaakkola. Geomol: Torsional geometric generation of molecular 3d conformer ensembles. arXiv preprint arXiv:2106.07802, 2021. 8, 18
+Niklas Gebauer, Michael Gastegger, and Kristof Schutt. Symmetry-adapted generation of 3d point ¨ sets for the targeted discovery of molecules. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alche-Buc, E. Fox, and R. Garnett (eds.), ´ Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. 3, 7
+Niklas WA Gebauer, Michael Gastegger, Stefaan SP Hessmann, Klaus-Robert Muller, and Kristof T ¨ Schutt. Inverse design of 3d molecular structures with conditional generative neural networks.¨ arXiv preprint arXiv:2109.04824, 2021. 9
+Colin W Glass, Artem R Oganov, and Nikolaus Hansen. Uspex—evolutionary crystal structure prediction. Computer physics communications, 175(11-12):713–720, 2006. 2
+Jonathan Godwin, Michael Schaarschmidt, Alexander Gaunt, Alvaro Sanchez-Gonzalez, Yulia Rubanova, Petar Velickovi ˇ c, James Kirkpatrick, and Peter Battaglia. Very deep graph neural ´ networks via noise regularisation. arXiv preprint arXiv:2106.07971, 2021. 2
+Ralf W Grosse-Kunstleve, Nicholas K Sauter, and Paul D Adams. Numerically stable algorithms for the computation of reduced unit cells. Acta Crystallographica Section A: Foundations of Crystallography, 60(1):1–6, 2004. 5, 15
+Geoffroy Hautier, Chris Fischer, Virginie Ehrlacher, Anubhav Jain, and Gerbrand Ceder. Data mined ionic substitutions for the discovery of new compounds. Inorganic chemistry, 50(2):656–663, 2011. 2
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and HsuanTien Lin (eds.), Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020. URL https://proceedings.neurips.cc/paper/2020/hash/ 4c5bcfec8584af0d967f1ab10179ca4b-Abstract.html. 4
+Jordan Hoffmann, Louis Maestrati, Yoshihide Sawada, Jian Tang, Jean Michel Sellier, and Yoshua Bengio. Data-driven approach to encoding and decoding 3-d crystal structures. arXiv preprint arXiv:1909.00949, 2019. 2, 7
+Jianjun Hu, Wenhui Yang, Rongzhi Dong, Yuxin Li, Xiang Li, Shaobo Li, and Edirisuriya MD Siriwardane. Contact map based crystal structure prediction using global optimization. CrystEngComm, 23(8):1765–1776, 2021. 3
+John J Irwin and Brian K Shoichet. Zinc- a free database of commercially available compounds for virtual screening. Journal of chemical information and modeling, 45(1):177–182, 2005. 1
+Anubhav Jain, Shyue Ping Ong, Geoffroy Hautier, Wei Chen, William Davidson Richards, Stephen Dacek, Shreyas Cholia, Dan Gunter, David Skinner, Gerbrand Ceder, et al. Commentary: The materials project: A materials genome approach to accelerating materials innovation. APL materials, 1(1):011002, 2013. 6, 16
+Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Junction tree variational autoencoder for molecular graph generation. In International conference on machine learning, pp. 2323–2332. PMLR, 2018. 9
+Sungwon Kim, Juhwan Noh, Geun Ho Gu, Alan Aspuru-Guzik, and Yousung Jung. Generative adversarial networks for crystal structure prediction. ACS central science, 6(8):1412–1420, 2020. 3
+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. In Yoshua Bengio and Yann LeCun (eds.), 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings, 2014. URL http://arxiv.org/ abs/1312.6114. 2, 5
+Johannes Klicpera, Shankari Giri, Johannes T Margraf, and Stephan Gunnemann. Fast and ¨ uncertainty-aware directional message passing for non-equilibrium molecules. arXiv preprint arXiv:2011.14115, 2020a. 15
+Johannes Klicpera, Janek Groß, and Stephan Gunnemann. Directional message passing for molec- ¨ ular graphs. In 8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020. OpenReview.net, 2020b. URL https://openreview. net/forum?id=B1eWbxStPH. 2, 15
+Johannes Klicpera, Florian Becker, and Stephan Gunnemann. Gemnet: Universal directional graph ¨ neural networks for molecules. arXiv preprint arXiv:2106.08903, 2021. 2, 15
+Zhifeng Kong and Wei Ping. On fast sampling of diffusion probabilistic models. arXiv preprint arXiv:2106.00132, 2021. 17
+Meng Liu, Keqiang Yan, Bora Oztekin, and Shuiwang Ji. Graphebm: Molecular graph generation with energy-based models. arXiv preprint arXiv:2102.00546, 2021. 3
+Teng Long, Nuno M Fortunato, Ingo Opahle, Yixuan Zhang, Ilias Samathrakis, Chen Shen, Oliver Gutfleisch, and Hongbin Zhang. Constrained crystals deep convolutional generative adversarial network for the inverse design of crystal structures. npj Computational Materials, 7(1):1–7, 2021. 2
+Shitong Luo and Wei Hu. Diffusion probabilistic models for 3d point cloud generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2837–2845, 2021. 4
+Youzhi Luo, Keqiang Yan, and Shuiwang Ji. Graphdf: A discrete flow model for molecular graph generation. arXiv preprint arXiv:2102.01189, 2021. 3
+Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In International Conference on Machine Learning, pp. 8162–8171. PMLR, 2021. 17
+Juhwan Noh, Jaehoon Kim, Helge S Stein, Benjamin Sanchez-Lengeling, John M Gregoire, Alan Aspuru-Guzik, and Yousung Jung. Inverse design of solid-state materials via a continuous representation. Matter, 1(5):1370–1384, 2019. 2
+Artem R Oganov, Chris J Pickard, Qiang Zhu, and Richard J Needs. Structure prediction drives materials discovery. Nature Reviews Materials, 4(5):331–348, 2019. 2
+M. O’Keeffe and B. G. Hyde. Plane nets in crystal chemistry. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 295(1417):553–618, 1980. ISSN 00804614. URL http://www.jstor.org/stable/36648. 3
+Shyue Ping Ong, William Davidson Richards, Anubhav Jain, Geoffroy Hautier, Michael Kocher, Shreyas Cholia, Dan Gunter, Vincent L Chevrier, Kristin A Persson, and Gerbrand Ceder. Python materials genomics (pymatgen): A robust, open-source python library for materials analysis. Computational Materials Science, 68:314–319, 2013. 7, 15, 18
+Hillary Pan, Alex M Ganose, Matthew Horton, Muratahan Aykol, Kristin A Persson, Nils ER Zimmermann, and Anubhav Jain. Benchmarking coordination number prediction algorithms on inorganic crystal structures. Inorganic chemistry, 60(3):1590–1603, 2021. 3, 15
+Yashaswi Pathak, Karandeep Singh Juneja, Girish Varma, Masahiro Ehara, and U Deva Priyakumar. Deep learning enabled inorganic material generator. Physical Chemistry Chemical Physics, 22 (46):26935–26943, 2020. 3
+Chris J. Pickard. Airss data for carbon at 10gpa and the $c + \mathrm { n + h + o }$ system at 1gpa, 2020. URL https://archive.materialscloud.org/record/2020.0026/v1. 6
+Chris J Pickard and RJ Needs. High-pressure phases of silane. Physical review letters, 97(4):045504, 2006. 16
+Chris J Pickard and RJ Needs. Ab initio random structure searching. Journal of Physics: Condensed Matter, 23(5):053201, 2011. 2, 16
+Zekun Ren, Juhwan Noh, Siyu Tian, Felipe Oviedo, Guangzong Xing, Qiaohao Liang, Armin Aberle, Yi Liu, Qianxiao Li, Senthilnath Jayavelu, et al. Inverse design of crystals using generalized invertible crystallographic representation. arXiv preprint arXiv:2005.07609, 2020. 3, 6, 7, 16
+Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. arXiv preprint arXiv:2202.00512, 2022. 17
+Victor Garcia Satorras, Emiel Hoogeboom, Fabian B Fuchs, Ingmar Posner, and Max Welling. E (n) equivariant normalizing flows for molecule generation in 3d. arXiv preprint arXiv:2105.09016, 2021. 3
+Yoshihide Sawada, Koji Morikawa, and Mikiya Fujii. Study of deep generative models for inorganic chemical compositions. arXiv preprint arXiv:1910.11499, 2019. 3
+Kristof T Schutt, Huziel E Sauceda, P-J Kindermans, Alexandre Tkatchenko, and K-R M ¨ uller. ¨ Schnet–a deep learning architecture for molecules and materials. The Journal of Chemical Physics, 148(24):241722, 2018. 2, 7
+Chence Shi, Minkai Xu, Zhaocheng Zhu, Weinan Zhang, Ming Zhang, and Jian Tang. Graphaf: a flow-based autoregressive model for molecular graph generation. arXiv preprint arXiv:2001.09382, 2020. 3
+Chence Shi, Shitong Luo, Minkai Xu, and Jian Tang. Learning gradient fields for molecular conformation generation. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event, volume 139 of Proceedings of Machine Learning Research, pp. 9558–9568. PMLR, 2021. URL http://proceedings.mlr.press/v139/shi21b.html. 3, 4, 5, 17
+Muhammed Shuaibi, Adeesh Kolluru, Abhishek Das, Aditya Grover, Anuroop Sriram, Zachary Ulissi, and C Lawrence Zitnick. Rotation invariant graph neural networks using spin convolutions. arXiv preprint arXiv:2106.09575, 2021. 2
+Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International Conference on Machine Learning, pp. 2256–2265. PMLR, 2015. 4
+Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alche-Buc, Emily B. Fox, and Roman Garnett (eds.), ´ Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada, pp. 11895– 11907, 2019. URL https://proceedings.neurips.cc/paper/2019/hash/ 3001ef257407d5a371a96dcd947c7d93-Abstract.html. 2, 4, 5, 6
+Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds. arXiv preprint arXiv:1802.08219, 2018. 2
+Yanchao Wang, Jian Lv, Li Zhu, and Yanming Ma. Calypso: A method for crystal structure prediction. Computer Physics Communications, 183(10):2063–2070, 2012. 2
+Logan Ward, Ankit Agrawal, Alok Choudhary, and Christopher Wolverton. A general-purpose machine learning framework for predicting properties of inorganic materials. npj Computational Materials, 2(1):1–7, 2016. 18
+Alexander Frank Wells et al. Three dimensional nets and polyhedra. Wiley, 1977. 3
+Jiaxiang Wu, Tao Shen, Haidong Lan, Yatao Bian, and Junzhou Huang. Se (3)-equivariant energybased models for end-to-end protein folding. bioRxiv, 2021. 3
+Tian Xie and Jeffrey C Grossman. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Physical review letters, 120(14):145301, 2018. 1, 2, 3
+Minkai Xu, Shitong Luo, Yoshua Bengio, Jian Peng, and Jian Tang. Learning neural generative dynamics for molecular conformation generation. In International Conference on Learning Representations, 2021a. URL https://openreview.net/forum?id=pAbm1qfheGk. 8, 18
+Minkai Xu, Lantao Yu, Yang Song, Chence Shi, Stefano Ermon, and Jian Tang. Geodiff: A geometric diffusion model for molecular conformation generation. In International Conference on Learning Representations, 2021b. 3
+Guandao Yang, Xun Huang, Zekun Hao, Ming-Yu Liu, Serge Belongie, and Bharath Hariharan. Pointflow: 3d point cloud generation with continuous normalizing flows. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4541–4550, 2019. 9
+Wenhui Yang, Edirisuriya M Dilanga Siriwardane, Rongzhi Dong, Yuxin Li, and Jianjun Hu. Crystal structure prediction of materials with high symmetry using differential evolution. arXiv preprint arXiv:2104.09764, 2021. 3
+Yong Zhao, Mohammed Al-Fahdi, Ming Hu, Edirisuriya Siriwardane, Yuqi Song, Alireza Nasiri, and Jianjun Hu. High-throughput discovery of novel cubic crystal materials using deep generative neural networks. arXiv preprint arXiv:2102.01880, 2021. 3
+Nils ER Zimmermann and Anubhav Jain. Local structure order parameters and site fingerprints for quantification of coordination environment and crystal structure similarity. RSC Advances, 10 (10):6063–6081, 2020. 18
+# A PROOF FOR THE CONNECTION TO A HARMONIC FORCE FIELD
+We assume the loss in Equation 2 can be minimized to zero when the noises are small, meaning that
+$$
+s _ { X } ( \tilde { A } , \tilde { X } , L | z ) = \frac { d _ { \operatorname* { m i n } } ( X , \tilde { X } ) } { \sigma _ { X , j } } , \forall j > J ,
+$$
+where $\sigma _ { { \pmb X } , j } \in \{ \sigma _ { { \pmb X } , j } \} _ { j = 1 } ^ { L }$ and any noise smaller than $\sigma _ { x , J }$ is considered as small.
+The force term in the Langevin dynamics $\alpha _ { j } { \pmb s } _ { { \pmb X } , t }$ can then be written as
+$$
+\begin{array} { l } { \displaystyle \alpha _ { j } s _ { X } ( \tilde { A } , \tilde { X } , L | z ; \sigma _ { A , j } , \sigma _ { X , j } ) = \epsilon \cdot \sigma _ { X , j } ^ { 2 } / \sigma _ { X , L } ^ { 2 } \cdot s _ { X } ( \tilde { A } , \tilde { X } , L | z ) / \sigma _ { X , j } } \\ { \displaystyle \quad \quad = \epsilon \cdot \frac { \sigma _ { X , j } ^ { 2 } } { \sigma _ { X , L } ^ { 2 } } \cdot \frac { d _ { \operatorname* { m i n } } ( X , \tilde { X } ) } { \sigma _ { X , j } ^ { 2 } } , \forall j > J } \\ { \displaystyle \quad = - \frac { \epsilon } { \sigma _ { X , L } ^ { 2 } } d _ { \operatorname* { m i n } } ( \tilde { X } , X ) , \forall j > J } \end{array}
+$$
+If we write $\epsilon / \sigma _ { X , L } ^ { 2 } = k$ , then,
+$$
+\alpha _ { j } s _ { X } ( \tilde { A } , \tilde { X } , L | z ; \sigma _ { A , j } , \sigma _ { X , j } ) = - k d _ { \operatorname * { m i n } } ( \tilde { X } , X ) , \forall j > J
+$$
+If the noises are small enough that atoms do not cross the periodic boundaries, then we have ${ \pmb d } _ { \mathrm { m i n } } ( { \pmb X } , \tilde { { \pmb X } } ) = { \pmb X } - \tilde { { \pmb X } }$ . Therefore,
+$$
+\alpha _ { j } s _ { X } ( \tilde { A } , \tilde { X } , { \cal L } | z ; \sigma _ { A , j } , \sigma _ { X , j } ) = - k ( \tilde { X } - X ) , \forall j > J .
+$$
+# B IMPLEMENTATION DETAILS
+# B.1 PREDICTION OF LATTICE PARAMETERS
+There are infinitely many different ways of choosing the lattice for the same material. We compute the Niggli reduced lattice (Grosse-Kunstleve et al., 2004) with pymatgen (Ong et al., 2013), which is a unique lattice for any given material. Since the lattice matrix $\pmb { L }$ is not rotation invariant, we instead predict the 6 lattice parameters, i.e. the lengths of the 3 lattice vectors and the angles between them. We normalize the lengths of lattice vectors with $\sqrt [ 3 ] { N }$ , where $N$ is the number of atoms, to ensure that the lengths for materials of different sizes are at the same scale.
+# B.2 MULTI-GRAPH CONSTRUCTION
+For the encoder, we use CrystalNN (Pan et al., 2021) to determine edges between atoms and build a multi-graph representation. For the decoder, since it inputs a noisy structure generated on the fly, the multi-graph must also be built on the fly for both training and generation, and CrystalNN is too slow for that purpose. We use a KNN algorithm that considers periodicity to build the decoder graph where $K = \bar { 2 0 }$ in all of our experiments.
+# B.3 GNN ARCHITECTURE
+We use DimeNet+ $^ +$ adapted for periodicity (Klicpera et al., 2020a;b) as the encoder, which is SE(3) invariant to the input structure. The decoder needs to output an vector per node that is SE(3) equivariant to the input structure. We use GemNet-dQ (Klicpera et al., 2021) as the decoder. We used implementations from the Open Catalysis Project (OCP) (Chanussot et al., 2021), but we reduced the size of hidden dimensions to 128 for faster training. The encoder has 2.2 million parameters and the decoder has 2.3 million parameters.
+# C DATASET CURATION
+# C.1 PEROV-5
+Perovskite is a class of materials that share a similar structure and have the general chemical formula $\mathrm { A B X } _ { 3 }$ . The ideal perovskites have a cubic structure, where the site A atom sits at a corner position, the site B atom sits at a body centered position and site $\mathrm { X }$ atoms sit at face centered positions. Perovskite materials are known for their wide applications. We curate the Perov-5 dataset from an open database that was originally developed for water splitting (Castelli et al., 2012a;b).
+All 18928 materials in the original database are included. In the database, A, B can be any nonradioactive metal and X can be one or several elements from O, N, S, and F. Note that there can be multiple different X atoms in the same material. All materials in Perov-5 are relaxed using density functional theory (DFT), and their relaxed structure can deviate significantly from the ideal structures. A significant portion of the materials are not thermodynamically stable, i.e., they will decompose to nearby phases and cannot be synthesized.
+# C.2 CARBON-24
+Carbon-24 includes various carbon structures obtained via ab initio random structure searching (AIRSS) (Pickard & Needs, 2006; 2011) performed at $1 0 \mathrm { G P a }$ .
+The original dataset includes 101529 carbon structures, and we selected the $10 \%$ of the carbon structure with the lowest energy per atom to create Carbon-24. All 10153 structures in Carbon-24 are relaxed using DFT. The most stable structure is diamond at $1 0 \mathrm { \ G P a }$ . All remaining structures are thermodynamically unstable but may be kinetically stable. Most of the structures cannot be synthesized.
+# C.3 MP-20
+MP-20 includes almost all experimentally stable materials from the Materials Project (Jain et al., 2013) with unit cells including at most 20 atoms. We only include materials that are originally from ICSD (Belsky et al., 2002) to ensure the experimental stability, and these materials represent the majority of experimentally known materials with at most 20 atoms in unit cells.
+To ensure stability, we only select materials with energy above the hull smaller than 0.08 eV/atom and formation energy smaller than 2 eV/atom, following Ren et al. (2020). Differing from Ren et al. (2020), we do not constrain the number of unique elements per material. All materials in MP-20 are relaxed using DFT. Most materials are thermodynamcially stable and have been synthesized.
+# D EXPERIMENT DETAILS
+# D.1 REASONS FOR THE UNSUITABILITY OF SOME METRICS FOR SPECIFIC DATASETS
+In Table 2, property statistics are computed by comparing the earth mover’s distance between the property distribution of generated materials and ground truth materials. So, they are not meaningful for ground truth data.
+Materials in Perov-5 have the same structure, so it is not meaningful to require higher structure diversity.
+Materials in Carbon-24 have the same composition (carbon), so it is not meaningful to require higher composition diversity. In addition, all models have $\sim 1 0 0 \%$ composition validity, so it is not compared in the table.
+# D.2 COMPOSITION VALIDITY CHECKER
+We modified the charge neutrality checker from SMACT (Davies et al., 2019) because the original checker is not suitable for alloys. The checker is based on a list of possible charges for each element and it checks if the material can be charge neutral by enumerating all possible charge combinations. However, it does not consider that metal alloys can be mixed with almost any combination. As a result, for materials composed of all metal elements, we always assume the composition is valid in our validity checker.
+For the ground truth materials in MP-20, the original checker gives a composition validity of ${ \sim } 5 0 \%$ , which significantly underestimates the validity of MP-20 materials (because most of them are experimentally synthesizable and thus valid). Our checker gives a composition validity of ${ \sim } 9 0 \%$ , which is far more reasonable. We note again that these checkers are all empirical and the only high-fidelity evaluation of material stability requires QM simulations.
+# D.3 NON-GAUSSIAN STATISTICAL STRUCTURE OF MATERIALS
+The material datasets are usually biased towards certain material groups. For example, there are lots of lithium-containing materials in MP-20 because it started with battery research. We also find that our decoder tends to underfit the data distribution with a larger $\beta$ in Equation 9. We believe these observations indicate that the statistical structure of the ground truth materials are far from Gaussian. As a result, sampling from $\mathcal { N } ( 0 , 1 )$ may lead to out-of-distribution materials, which explains why our method tends to generate more elements per material than the ground truth.
+# D.4 HYPERPARAMETERS AND TRAINING DETAILS
+The total loss can be written as,
+$$
+{ \mathcal { L } } = { \mathcal { L } } _ { \mathrm { A G G } } + { \mathcal { L } } _ { \mathrm { D E C } } + { \mathcal { L } } _ { \mathrm { K L } } = \lambda _ { \mathrm { c } } { \mathcal { L } } _ { \mathrm { c } } + \lambda _ { L } { \mathcal { L } } _ { L } + \lambda _ { N } { \mathcal { L } } _ { N } + \lambda _ { X } { \mathcal { L } } _ { X } + \lambda _ { A } { \mathcal { L } } _ { A } + \beta { \mathcal { L } } _ { \mathrm { K L } } .
+$$
+We aim to keep each loss term at a similar scale. For all three datasets, we use $\lambda _ { c } = 1 , \lambda _ { L } =$ $1 0 , \lambda _ { N } = 1 , \lambda _ { X } = 1 0 , \mathcal { L } _ { A } = 1$ .
+We tune $\beta$ between $0 . 0 1 , 0 . 0 3 , 0 . 1$ for all three datasets and select the model with best validation loss. For Perov-5, MP-20, we use $\beta = 0 . 0 1$ , and for Carbon-24, we use $\beta = 0 . 0 3$ .
+For the noise levels in $\{ \sigma _ { A , j } \} _ { j = 1 } ^ { L } , \{ \sigma _ { X , j } \} _ { j = 1 } ^ { L }$ , we follow Shi et al. (2021) and set $L = 5 0$ . For all three datasets, we use $\sigma _ { A , \operatorname* { m a x } } = 5 , \sigma _ { A , \operatorname* { m i n } } = 0 . 0 1 , \sigma _ { X , \operatorname* { m a x } } = 1 0 , \sigma _ { X , \operatorname* { m i n } } = 0 . 0 1$ .
+During the training, we use an initial learning rate of 0.001 and reduce the learning rate by a factor of 0.6 if the validation loss does not improve after 30 epochs. The minimum learning rate is 0.0001.
+During the generation, we use $\epsilon = 0 . 0 0 0 1$ and run Langevin dynamics for 100 steps at each noise level.
+# E VISUALIZATION OF MULTIPLE RECONSTRUCTED STRUCTURES
+![](images/014198feed124d83196e3770da4948847d47fbbe02a6692b9c0aeb6b4b8e0a13.jpg)
+Figure 5: Different reconstructed structures from CDVAE from the same $_ { z }$ , following 3 Langevin dynamics sampling with different random seeds.
+# F SAMPLING SPEED FOR MATERIAL GENERATION
+We summarize the speed for generating 10,000 materials for all models in Table 4. FTCP is significantly faster, but the quality of generated materials is very poor as shown in Table 2. CondDFC-VAE is faster than our method in Perov-5, but has a lower quality than our method and only works for cubic systems. It is also unclear how it will perform on larger materials in Carbon-24 and MP-20, because the compute increases cubicily with the increased size of the density map. GSchNet/P-G-SchNet have a comparable sampling time as our method, but have a lower quality. We also note that we did not optimize sampling speed in current work. It is possible to reduce sampling time by using fewer sampling steps without significantly influencing generation quality. There are also many recent works that aim to speed up the sampling process for diffusion models (Nichol & Dhariwal, 2021; Kong & Ping, 2021; Salimans & Ho, 2022).
+Table 4: Time used for generating 10,000 materials on a single RTX 2080 Ti GPU.
+<table><tr><td></td><td>FTCP</td><td>Cond-DFC-VAE</td><td>G-SchNet</td><td>P-G-SchNet</td><td>CDVAE</td></tr><tr><td>Perov-5</td><td>&lt;1min</td><td>0.5h</td><td>2.0h</td><td>2.0h</td><td>3.1 h</td></tr><tr><td>Carbon-24</td><td>&lt;1min</td><td>1</td><td>6.2 h</td><td>6.3h</td><td>5.3 h</td></tr><tr><td>MP-20</td><td>&lt;1min</td><td>1</td><td>6.3 h</td><td>6.3h</td><td>5.8h</td></tr></table>
+# G COVERAGE METRICS FOR MATERIAL GENERATION
+Inspired by $\mathrm { X u }$ et al. (2021a); Ganea et al. (2021), we define six metrics to compare two ensembles of materials: materials generated by a method $\{ M _ { k } \} _ { k \in [ 1 \ldots K ] }$ , and ground truth materials in test data $\{ M _ { l } ^ { * } \} _ { \in [ 1 \ldots L ] }$ .
+We use the Euclidean distance of the CrystalNN fingerprint (Zimmermann & Jain, 2020) and normalized Magpie fingerprint (Ward et al., 2016) to define the structure distance and composition distance between generated and ground truth materials, respectively. They can be written as $D _ { \mathrm { s t r u c . } } ( M _ { k } , M _ { l } ^ { * } )$ and $\cdot \mathrm { \Delta } D _ { \mathrm { c o m p . } } ( M _ { k } , \bar { M } _ { l } ^ { * } )$ . We further define the thresholds for the structure and composition distance as $\delta _ { \mathrm { s t r u c } }$ . and $\delta _ { \mathrm { c o m p . } }$ ., respectively.
+Following the established classification metrics of Precision and Recall, we define the coverage metrics as:
+$$
+\begin{array} { r l } & { \quad \mathrm { C O V - R } \mathrm { ( R e c a l l ) } = \displaystyle \frac { 1 } { L } | \{ l \in [ 1 . . L ] : \exists k \in [ 1 . . K ] , D _ { \mathrm { s t r u c . } } ( M _ { k } , M _ { l } ^ { * } ) < \delta _ { \mathrm { s t r u c . } } , } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad D _ { \mathrm { c o m p . } } ( M _ { k } , M _ { l } ^ { * } ) < \delta _ { \mathrm { c o m p . } } \} } \\ & { \quad \quad \quad \mathrm { A M S D - R } \mathrm { ( R e c a l l ) } = \displaystyle \frac { 1 } { L } \sum _ { l \in [ 1 . . L ] } \operatorname* { m i n } _ { k \in [ 1 . . K ] } D _ { \mathrm { s t r u c . } } ( M _ { k } , M _ { l } ^ { * } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \mathrm { A M C D - R } \mathrm { ( R e c a l l ) } = \displaystyle \frac { 1 } { L } \sum _ { l \in [ 1 . . L ] } \operatorname* { m i n } _ { k \in [ 1 . . K ] } D _ { \mathrm { c o m p . } } ( M _ { k } , M _ { l } ^ { * } ) , } \end{array}
+$$
+where COV is ”Coverage”, AMSD is ”Average Minimum Structure Distance”, AMCD is ”Average Minimum Composition Distance”, and COV-P (precision), AMSD-P (precision), AMCD-P (precision) are defined as in above equations, but with the generated and ground truth material sets swapped. The recall metrics measure how many ground truth materials are correctly predicted, while the precision metrics measure how many generated materials are of high quality (more discussions can be found in Ganea et al. (2021)).
+We note several points on why we define the metrics in their current forms. 1) COV requires both structure and composition distances to be within the thresholds, because generating materials that are structurally close to one ground truth material and compositionally close to another is not meaningful. As a result, AMSD and AMCD are less useful than COV. 2) We use fingerprint distance, rather than RMSE from StructureMatcher (Ong et al., 2013), because the material space is too large for the models to generate enough materials to exactly match the ground truth materials. StructureMatcher first requires the compositions of two materials to exactly match, which will cause all models to have close-to-zero coverage.
+For Perov-5 and Carbon-24, we choose $\delta _ { \mathrm { s t r u c . } } = 0 . 2 , \delta _ { \mathrm { c o m p . } } = 4$ . For MP-20, we choose $\delta _ { \mathrm { s t r u c . } } =$ $0 . 4 , \delta _ { \mathrm { c o m p . } } = 1 0$ . In Figure 6, Figure 7, Figure 8, we show how both COV-R and COV-P change by varying $\bar { \delta } _ { \mathrm { s t r u c } }$ . and $\delta _ { \mathrm { c o m p } }$ . in all three datasets.
+Table 5: Full coverage metrics for the generation task.
+<table><tr><td>Method</td><td>Data</td><td>COV-R↑</td><td>AMSD-R↓</td><td>AMCD-R↓</td><td>COV-P↑</td><td>AMSD-P↓</td><td>AMCD-P↓</td></tr><tr><td rowspan="3">FTCP</td><td>Perov-5</td><td>0.00</td><td>0.7447</td><td>7.212</td><td>0.00</td><td>0.3582</td><td>3.390</td></tr><tr><td>Carbon-24</td><td>0.00</td><td>1.181</td><td>0.00</td><td>0.00</td><td>0.8822</td><td>24.16</td></tr><tr><td>MP-20</td><td>4.72</td><td>0.6542</td><td>9.271</td><td>0.09</td><td>0.1954</td><td>4.378</td></tr><tr><td>Cond-DFC-VAE</td><td>Perov-5</td><td>73.92</td><td>0.1508</td><td>2.773</td><td>10.13</td><td>0.3162</td><td>4.257</td></tr><tr><td rowspan="4">G-SchNet</td><td>Perov-5</td><td>0.18</td><td>0.5962</td><td>1.006</td><td>0.23</td><td>0.4259</td><td>1.3163</td></tr><tr><td>Carbon-24</td><td>0.00</td><td>0.5887</td><td>0.00</td><td>0.00</td><td>0.5970</td><td>0.00</td></tr><tr><td>MP-20</td><td>38.33</td><td>0.5365</td><td>3.233</td><td>99.57</td><td>0.2026</td><td>3.601</td></tr><tr><td>Perov-5</td><td>0.37</td><td>0.5510</td><td>1.0264</td><td>0.25</td><td>0.3967</td><td>1.316</td></tr><tr><td rowspan="4">CDVAE</td><td>Carbon-24</td><td>0.00</td><td>0.6308</td><td>0.00</td><td>0.00</td><td>0.8166</td><td>0.00</td></tr><tr><td>MP-20</td><td>41.93</td><td>0.5327</td><td>3.274</td><td>99.74</td><td>0.1985</td><td>3.567</td></tr><tr><td>Perov-5</td><td>99.45</td><td>0.0482</td><td>0.6969</td><td>98.46</td><td>0.0593</td><td>1.272</td></tr><tr><td>Carbon-24</td><td>99.80</td><td>0.0489</td><td>0.00</td><td>83.08</td><td>0.1343</td><td>0.00</td></tr><tr><td></td><td>MP-20</td><td>99.15</td><td>0.1549</td><td>3.621</td><td>99.49</td><td>0.1883</td><td>4.014</td></tr></table>
+![](images/df4a9ab8291867bf7d43b05e7e5021163cda44e434b159383cce5c820edc7a42.jpg)
+Figure 6: Change of COV-R and COV-P by varying $\delta _ { \mathrm { s t r u c } }$ . and $\delta _ { \mathrm { c o m p } }$ . for Perov-5. Dashed line denotes the current chosen thresholds.
+![](images/3565eef4857703cd54d2fbd09011c227cb6f03ede855b110cc9ce720c5b6f647.jpg)
+Figure 7: Change of COV-R and COV-P by varying $\delta _ { \mathrm { s t r u c } }$ . and $\delta _ { \mathrm { c o m p } }$ . for Carbon-24. Dashed line denotes the current chosen thresholds.
+![](images/c2b820bf4f97c03bb50f377af925aea32bca53720d95774535f0d070c0d563f2.jpg)
+Figure 8: Change of COV-R and COV-P by varying $\delta _ { \mathrm { s t r u c } }$ . and $\delta _ { \mathrm { c o m p } }$ . for MP-20. Dashed line denotes the current chosen thresholds.

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+# SELF-CONSISTENCY IMPROVES CHAIN OF THOUGHT REASONING IN LANGUAGE MODELS
+Xuezhi Wang†‡, Jason Wei†, Dale Schuurmans†, Quoc Le†, Ed H. Chi†, Sharan Narang†, Aakanksha Chowdhery†, Denny Zhou†§
+†Google Research, Brain Team ‡xuezhiw@google.com, §dennyzhou@google.com
+# ABSTRACT
+Chain-of-thought prompting combined with pre-trained large language models has achieved encouraging results on complex reasoning tasks. In this paper, we propose a new decoding strategy, self-consistency, to replace the naive greedy decoding used in chain-of-thought prompting. It first samples a diverse set of reasoning paths instead of only taking the greedy one, and then selects the most consistent answer by marginalizing out the sampled reasoning paths. Self-consistency leverages the intuition that a complex reasoning problem typically admits multiple different ways of thinking leading to its unique correct answer. Our extensive empirical evaluation shows that self-consistency boosts the performance of chain-of-thought prompting with a striking margin on a range of popular arithmetic and commonsense reasoning benchmarks, including GSM8K $( + 1 7 . 9 \% )$ , SVAMP $( + 1 1 . 0 \% )$ , AQuA $( + 1 2 . 2 \% )$ , StrategyQA $( + 6 . 4 \% )$ and ARC-challenge $( + 3 . 9 \% )$ .
+# 1 INTRODUCTION
+Although language models have demonstrated remarkable success across a range of NLP tasks, their ability to demonstrate reasoning is often seen as a limitation, which cannot be overcome solely by increasing model scale (Rae et al., 2021; BIG-bench collaboration, 2021, inter alia). In an effort to address this shortcoming, Wei et al. (2022) have proposed chain-of-thought prompting, where a language model is prompted to generate a series of short sentences that mimic the reasoning process a person might employ in solving a task. For example, given the question “If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot?”, instead of directly responding with “5”, a language model would be prompted to respond with the entire chain-of-thought: “There are 3 cars in the parking lot already. 2 more arrive. Now there are $3 +$ $2 = 5$ cars. The answer is 5.”. It has been observed that chain-of-thought prompting significantly improves model performance across a variety of multi-step reasoning tasks (Wei et al., 2022).
+In this paper, we introduce a novel decoding strategy called self-consistency to replace the greedy decoding strategy used in chain-of-thought prompting (Wei et al., 2022), that further improves language models’ reasoning performance by a significant margin. Self-consistency leverages the intuition that complex reasoning tasks typically admit multiple reasoning paths that reach a correct answer (Stanovich & West, 2000). The more that deliberate thinking and analysis is required for a problem (Evans, 2010), the greater the diversity of reasoning paths that can recover the answer.
+Figure 1 illustrates the self-consistency method with an example. We first prompt the language model with chain-of-thought prompting, then instead of greedily decoding the optimal reasoning path, we propose a “sample-and-marginalize” decoding procedure: we first sample from the language model’s decoder to generate a diverse set of reasoning paths; each reasoning path might lead to a different final answer, so we determine the optimal answer by marginalizing out the sampled reasoning paths to find the most consistent answer in the final answer set. Such an approach is analogous to the human experience that if multiple different ways of thinking lead to the same answer, one has greater confidence that the final answer is correct. Compared to other decoding methods, self-consistency avoids the repetitiveness and local-optimality that plague greedy decoding, while mitigating the stochasticity of a single sampled generation.
+![](images/e52f9a62e8d5a9a773304db1105dd4959b748fa605157f43d522da76b851f3c2.jpg)
+Figure 1: The self-consistency method contains three steps: (1) prompt a language model using chain-of-thought (CoT) prompting; (2) replace the “greedy decode” in CoT prompting by sampling from the language model’s decoder to generate a diverse set of reasoning paths; and (3) marginalize out the reasoning paths and aggregate by choosing the most consistent answer in the final answer set.
+Self-consistency is far simpler than prior approaches that either train an additional verifier (Cobbe et al., 2021) or train a re-ranker given additional human annotations to improve generation quality (Thoppilan et al., 2022). Instead, self-consistency is entirely unsupervised, works off-the-shelf with pre-trained language models, requires no additional human annotation, and avoids any additional training, auxiliary models or fine-tuning. Self-consistency also differs from a typical ensemble approach where multiple models are trained and the outputs from each model are aggregated, it acts more like a “self-ensemble” that works on top of a single language model.
+We evaluate self-consistency on a wide range of arithmetic and commonsense reasoning tasks over four language models with varying scales: the public UL2-20B (Tay et al., 2022) and GPT-3-175B (Brown et al., 2020), and two densely-activated decoder-only language models: LaMDA-137B (Thoppilan et al., 2022) and PaLM-540B (Chowdhery et al., 2022). On all four language models, self-consistency improves over chain-of-thought prompting by a striking margin across all tasks. In particular, when used with PaLM-540B or GPT-3, self-consistency achieves new state-of-the-art levels of performance across arithmetic reasoning tasks, including GSM8K (Cobbe et al., 2021) $( + 1 7 . 9 \%$ absolute accuracy gains), SVAMP (Patel et al., 2021) $( + 1 1 . 0 \% )$ , AQuA (Ling et al., 2017) $( + 1 2 . 2 \% )$ , and across commonsense reasoning tasks such as StrategyQA (Geva et al., 2021) $( + 6 . 4 \% )$ and ARCchallenge (Clark et al., 2018) $( + 3 . 9 \% )$ . In additional experiments, we show self-consistency can robustly boost performance on NLP tasks where adding a chain-of-thought might hurt performance compared to standard prompting (Ye & Durrett, 2022). We also show self-consistency significantly outperforms sample-and-rank, beam search, ensemble-based approaches, and is robust to sampling strategies and imperfect prompts.
+# 2 SELF-CONSISTENCY OVER DIVERSE REASONING PATHS
+A salient aspect of humanity is that people think differently. It is natural to suppose that in tasks requiring deliberate thinking, there are likely several ways to attack the problem. We propose that such a process can be simulated in language models via sampling from the language model’s decoder. For instance, as shown in Figure 1, a model can generate several plausible responses to a math question that all arrive at the same correct answer (Outputs 1 and 3). Since language models are not perfect reasoners, the model might also produce an incorrect reasoning path or make a mistake in one of the reasoning steps (e.g., in Output 2), but such solutions are less likely to arrive at the same answer. That is, we hypothesize that correct reasoning processes, even if they are diverse, tend to have greater agreement in their final answer than incorrect processes.
+We leverage this intuition by proposing the following self-consistency method. First, a language model is prompted with a set of manually written chain-of-thought exemplars (Wei et al., 2022). Next, we sample a set of candidate outputs from the language model’s decoder, generating a diverse set of candidate reasoning paths. Self-consistency is compatible with most existing sampling algorithms, including temperature sampling (Ackley et al., 1985; Ficler & Goldberg, 2017), top- $k$ sampling (Fan et al., 2018; Holtzman et al., 2018; Radford et al., 2019), and nucleus sampling (Holtzman et al., 2020). Finally, we aggregate the answers by marginalizing out the sampled reasoning paths and choosing the answer that is the most consistent among the generated answers.
+<table><tr><td></td><td>GSM8K</td><td>MultiArith</td><td>AQuA</td><td>SVAMP</td><td>CSQA</td><td>ARC-c</td></tr><tr><td>Greedy decode</td><td>56.5</td><td>94.7</td><td>35.8</td><td>79.0</td><td>79.0</td><td>85.2</td></tr><tr><td>Weighted avg (unnormalized)</td><td>56.3±0.0</td><td>90.5±0.0</td><td>35.8±0.0</td><td>73.0±0.0</td><td>74.8±0.0</td><td>82.3 ±0.0</td></tr><tr><td>Weighted avg (normalized)</td><td>22.1 ± 0.0</td><td>59.7 ± 0.0</td><td>15.7 ± 0.0</td><td>40.5± 0.0</td><td>52.1±0.0</td><td>51.7 ± 0.0</td></tr><tr><td>Weighted sum (unnormalized)</td><td>59.9 ± 0.0</td><td>92.2 ± 0.0</td><td>38.2 ± 0.0</td><td>76.2 ± 0.0</td><td>76.2 ± 0.0</td><td>83.5± 0.0</td></tr><tr><td>Weighted sum (normalized)</td><td>74.1 ± 0.0</td><td>99.3 ± 0.0</td><td>48.0± 0.0</td><td>86.8± 0.0</td><td>80.7± 0.0</td><td>88.7 ±0.0</td></tr><tr><td>Unweighted sum (majority vote)</td><td>）74.4 ±0.1</td><td>99.3 ± 0.0</td><td>48.3 ± 0.5</td><td>86.6 ± 0.1</td><td>80.7 ± 0.1</td><td>88.7 ± 0.1</td></tr></table>
+Table 1: Accuracy comparison of different answer aggregation strategies on PaLM-540B.
+In more detail, assume the generated answers ${ \bf a } _ { i }$ are from a fixed answer set, $\mathbf { a } _ { i } \in \mathbb { A }$ , where $i = 1 , \ldots , m$ indexes the $m$ candidate outputs sampled from the decoder. Given a prompt and a question, self-consistency introduces an additional latent variable $\mathbf { r } _ { i }$ , which is a sequence of tokens representing the reasoning path in the $i$ -th output, then couples the generation of $( \mathbf { r } _ { i } , \mathbf { a } _ { i } )$ where $\mathbf { r } _ { i }  \mathbf { a } _ { i }$ , i.e., generating a reasoning path $\mathbf { r } _ { i }$ is optional and only used to reach the final answer ${ \bf a } _ { i }$ . As an example, consider Output 3 from Figure 1: the first few sentences “She eats 3 for breakfast ... So she has $9 e g g s * \mathbb { S } 2 = \mathbb { S } I \mathbb { S } .$ ” constitutes $\mathbf { r } _ { i }$ , while the answer 18 from the last sentence, “The answer is $\$ 18$ , is parsed as ${ \bf a } _ { i }$ .1 After sampling multiple $\left( \mathbf { r } _ { i } , \mathbf { a } _ { i } \right)$ from the model’s decoder, self-consistency applies a marginalization over $\mathbf { r } _ { i }$ by taking a majority vote over ${ \bf a } _ { i }$ , i.e., arg maxa $\begin{array} { r } { \sum _ { i = 1 } ^ { m } \mathbb { 1 } ( \mathbf { a } _ { i } = a ) } \end{array}$ or as we defined as the most “consistent” answer among the final answer set.
+In Table 1, we show the test accuracy over a set of reasoning tasks by using different answer aggregation strategies. In addition to majority vote, one can also weight each $( \mathbf { r } _ { i } , \mathbf { a } _ { i } )$ by $P ( \mathbf { r } _ { i } , \mathbf { a } _ { i } \mid$ prompt, question) when aggregating the answers. Note to compute $P ( \mathbf { r } _ { i } , \mathbf { a } _ { i } \mid$ prompt, question), we can either take the unnormalized probability of the model generating $( \mathbf { r } _ { i } , \mathbf { a } _ { i } )$ given (prompt, question), or we can normalize the conditional probability by the output length (Brown et al., 2020), i.e.,
+$$
+\begin{array} { r } { P ( \mathbf { r } _ { i } , \mathbf { a } _ { i } \mid \mathrm { p r o m p t } , \mathbf { q u e s t i o n } ) = \exp ^ { \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \log P ( t _ { k } | \mathrm { p r o m p t } , \mathbf { q u e s t i o n } , t _ { 1 } , \dots , t _ { k - 1 } ) } , } \end{array}
+$$
+where $\log { P ( t _ { k } \ | }$ prompt, question, $t _ { 1 } , \ldots , t _ { k - 1 } )$ ) is the log probability of generating the $k$ -th token $t _ { k }$ in $\left( \mathbf { r } _ { i } , \mathbf { a } _ { i } \right)$ conditioned on the previous tokens, and $K$ is the total number of tokens in $( \mathbf { r } _ { i } , \mathbf { a } _ { i } )$ . In Table 1, we show that taking the “unweighted sum”, i.e., taking a majority vote directly over ${ \bf a } _ { i }$ yields a very similar accuracy as aggregating using the “normalized weighted sum”. We took a closer look at the model’s output probabilities and found this is because for each $( \mathbf { r } _ { i } , \mathbf { a } _ { i } )$ , the normalized conditional probabilities $P ( \mathbf { r } _ { i } , \mathbf { a } _ { i } \mid$ prompt, question) are quite close to each other, i.e., the language model regards those generations as “similarly likely”.2 Additionally, when aggregating the answers, the results in Table 1 show that the “normalized” weighted sum (i.e., Equation 1) yields a much higher accuracy compared to its unnormalized counterpart. For completeness, in Table 1 we also report the results by taking a “weighted average”, i.e., each $a$ gets a score of its weighted sum divided by $\Sigma _ { i = 1 } ^ { m } \mathbb { 1 } ( \mathbf { a } _ { i } = \bar { a } )$ , which results in a much worse performance.
+Self-consistency explores an interesting space between open-ended text generation and optimal text generation with a fixed answer. Reasoning tasks typically have fixed answers, which is why researchers have generally considered greedy decoding approaches (Radford et al., 2019; Wei et al., 2022; Chowdhery et al., 2022). However, we have found that even when the desired answer is fixed, introducing diversity in the reasoning processes can be highly beneficial; therefore we leverage sampling, as commonly used for open-ended text generation (Radford et al., 2019; Brown et al., 2020; Thoppilan et al., 2022), to achieve this goal. One should note that self-consistency can be applied only to problems where the final answer is from a fixed answer set, but in principle this approach can be extended to open-text generation problems if a good metric of consistency can be defined between multiple generations, e.g., whether two answers agree or contradict each other.
+# 3 EXPERIMENTS
+We conducted a series of experiments to compare the proposed self-consistency method with existing approaches on a range of reasoning benchmarks. We find that self-consistency robustly improves reasoning accuracy for every language model considered, spanning a wide range of model scales.
+# 3.1 EXPERIMENT SETUP
+Tasks and datasets. We evaluate self-consistency on the following reasoning benchmarks.3 • Arithmetic reasoning. For these tasks, we used the Math Word Problem Repository (KoncelKedziorski et al., 2016), including AddSub (Hosseini et al., 2014), MultiArith (Roy & Roth, 2015), and ASDiv (Miao et al., 2020). We also included AQUA-RAT (Ling et al., 2017), a recently published benchmark of grade-school-math problems (GSM8K; Cobbe et al., 2021), and a challenge dataset over math word problems (SVAMP; Patel et al., 2021). • Commonsense reasoning. For these tasks, we used CommonsenseQA (Talmor et al., 2019), StrategyQA (Geva et al., 2021), and the AI2 Reasoning Challenge (ARC) (Clark et al., 2018). • Symbolic Reasoning. We evaluate two symbolic reasoning tasks: last letter concatenation (e.g., the input is “Elon Musk” and the output should be “nk”), and Coinflip (e.g., a coin is heads-up, after a few flips is the coin still heads-up?) from Wei et al. (2022).
+Language models and prompts. We evaluate self-consistency over four transformer-based language models with varying scales:
+• UL2 (Tay et al., 2022) is an encoder-decoder model trained on a mixture of denoisers with 20- billion parameters. UL2 is completely open-sourced4 and has similar or better performance than GPT-3 on zero-shot SuperGLUE, with only 20B parameters and thus is more compute-friendly; • GPT-3 (Brown et al., 2020) with 175-billion parameters. We use two public engines code-davinci001 and code-davinci-002 from the Codex series (Chen et al., 2021) to aid reproducibility;5 • LaMDA-137B (Thoppilan et al., 2022) is a dense left-to-right, decoder-only language model with 137-billion parameters, pre-trained on a mixture of web documents, dialog data and Wikipedia; • PaLM-540B (Chowdhery et al., 2022) is a dense left-to-right, decoder-only language model with 540-billion parameters, pre-trained on a high quality corpus of 780 billion tokens with filtered webpages, books, Wikipedia, news articles, source code, and social media conversations.
+We perform all experiments in the few-shot setting, without training or fine-tuning the language models. For a fair comparison we use the same prompts as in Wei et al. (2022): for all arithmetic reasoning tasks we use the same set of 8 manually written exemplars; for each commonsense reasoning task, 4-7 exemplars are randomly chosen from the training set with manually composed chain-of-thought prompts.6 Full details on the prompts used are given in Appendix A.3.
+Sampling scheme. To sample diverse reasoning paths, we followed similar settings to those suggested in Radford et al. (2019); Holtzman et al. (2020) for open-text generation. In particular, for UL2-20B and LaMDA-137B we applied temperature sampling with $T = 0 . 5$ and truncated at the top- $k$ $k = 4 0$ ) tokens with the highest probability, for PaLM-540B we applied $T = 0 . 7 , k = 4 0$ , and for GPT-3 we use $T = 0 . 7$ without top- $k$ truncation. We provide an ablation study in Section 3.5 to show that self-consistency is generally robust to sampling strategies and parameters.
+# 3.2 MAIN RESULTS
+We report the results of self-consistency averaged over 10 runs, where we sampled 40 outputs independently from the decoder in each run. The baseline we compare to is chain-of-thought prompting with greedy decoding (Wei et al., 2022), referred to as CoT-prompting, which has been previously used for decoding in large language models (Chowdhery et al., 2022).
+Arithmetic Reasoning The results are shown in Table 2.7 Self-consistency improves the arithmetic reasoning performance over all four language models significantly over chain-of-thought prompting. More surprisingly, the gains become more significant when the language model’s scale increases, e.g., we see $+ 3 \% { - } 6 \%$ absolute accuracy improvement over UL2-20B but $+ 9 \% - 2 3 \%$ for LaMDA137B and GPT-3. For larger models that already achieve high accuracy on most tasks (e.g., GPT-3 and PaLM-540B), self-consistency still contributes significant additional gains with $+ 1 2 \% - 1 8 \%$ absolute accuracy on tasks like AQuA and GSM8K, and $+ 7 \% - 1 1 \%$ on SVAMP and ASDiv. With self-consistency, we achieve new state-of-the-art results on almost all tasks: despite the fact that selfconsistency is unsupervised and task-agnostic, these results compare favorably to existing approaches that require task-specific training, or fine-tuning with thousands of examples (e.g., on GSM8K).
+Table 2: Arithmetic reasoning accuracy by self-consistency compared to chain-of-thought prompting (Wei et al., 2022). The previous SoTA baselines are obtained from: a: Relevance and LCA operation classifier (Roy & Roth, 2015), b: Lan et al. (2021), c: Amini et al. (2019), d: Pi et al. (2022), e: GPT-3 175B finetuned with $7 . 5 \mathrm { k }$ examples (Cobbe et al., 2021), $g$ : GPT-3 175B finetuned plus an additional 175B verifier (Cobbe et al., 2021). The best performance for each task is shown in bold.
+<table><tr><td></td><td>Method</td><td>AddSub</td><td>MultiArith</td><td>ASDiv</td><td>AQuA</td><td>SVAMP</td><td>GSM8K</td></tr><tr><td></td><td>Previous SoTA</td><td>94.9a</td><td>60.5a</td><td>75.36</td><td>37.9c</td><td>57.4d</td><td>35e / 55g</td></tr><tr><td>UL2-20B</td><td>CoT-prompting Self-consistency</td><td>18.2 24.8 (+6.6)</td><td>10.7 15.0 (+4.3)</td><td>16.9 21.5 (+4.6)</td><td>23.6 26.9 (+3.3)</td><td>12.6 19.4 (+6.8)</td><td>4.1 7.3 (+3.2)</td></tr><tr><td>LaMDA-137B</td><td>CoT-prompting Self-consistency</td><td>52.9 63.5 (+10.6)</td><td>51.8 75.7 (+23.9)</td><td>49.0 58.2 (+9.2)</td><td>17.7 26.8 (+9.1)</td><td>38.9 53.3 (+14.4)</td><td>17.1 27.7 (+10.6)</td></tr><tr><td>PaLM-540B</td><td>CoT-prompting Self-consistency</td><td>91.9 93.7 (+1.8)</td><td>94.7 99.3 (+4.6)</td><td>74.0 81.9 (+7.9)</td><td>35.8 48.3 (+12.5)</td><td>79.0 86.6 (+7.6)</td><td>56.5 74.4 (+17.9)</td></tr><tr><td>GPT-3 Code-davinci-001</td><td>CoT-prompting Self-consistency</td><td>57.2 67.8 (+10.6)</td><td>59.5 82.7 (+23.2)</td><td>52.7 61.9 (+9.2)</td><td>18.9 25.6 (+6.7)</td><td>39.8 54.5 (+14.7)</td><td>14.6 23.4 (+8.8)</td></tr><tr><td>GPT-3 Code-davinci-002</td><td>CoT-prompting Self-consistency</td><td>89.4 91.6 (+2.2)</td><td>96.2 100.0 (+3.8)</td><td>80.1 87.8 (+7.6)</td><td>39.8 52.0 (+12.2)</td><td>75.8 86.8 (+11.0)</td><td>60.1 78.0 (+17.9)</td></tr></table>
+<table><tr><td></td><td>Method</td><td>CSQA</td><td>StrategyQA</td><td>ARC-e</td><td>ARC-c</td><td>Letter (4)</td><td>Coinflip (4)</td></tr><tr><td></td><td>Previous SoTA</td><td>91.2a</td><td>73.96</td><td>86.4℃</td><td>75.0℃</td><td>N/A</td><td>N/A</td></tr><tr><td>UL2-20B</td><td>CoT-prompting Self-consistency</td><td>51.4 55.7 (+4.3)</td><td>53.3 54.9 (+1.6)</td><td>61.6 69.8 (+8.2)</td><td>42.9 49.5 (+6.8)</td><td>0.0 0.0 (+0.0)</td><td>50.4 50.5 (+0.1)</td></tr><tr><td>LaMDA-137B</td><td>CoT-prompting Self-consistency</td><td>57.9 63.1 (+5.2)</td><td>65.4 67.8 (+2.4)</td><td>75.3 79.3 (+4.0)</td><td>55.1 59.8 (+4.7)</td><td>8.2 8.2 (+0.0)</td><td>72.4 73.5 (+1.1)</td></tr><tr><td>PaLM-540B</td><td>CoT-prompting Self-consistency</td><td>79.0 80.7 (+1.7)</td><td>75.3 81.6 (+6.3)</td><td>95.3 96.4 (+1.1)</td><td>85.2 88.7 (+3.5)</td><td>65.8 70.8 (+5.0)</td><td>88.2 91.2 (+3.0)</td></tr><tr><td>GPT-3 Code-davinci-001</td><td>CoT-prompting Self-consistency</td><td>46.6 54.9 (+8.3)</td><td>56.7 61.7 (+5.0)</td><td>63.1 72.1 (+9.0)</td><td>43.1</td><td>7.8 10.0 (+2.2)</td><td>71.4 75.9 (+4.5)</td></tr><tr><td>GPT-3</td><td>CoT-prompting</td><td>79.0</td><td>73.4</td><td>94.0</td><td>53.7 (+10.6) 83.6</td><td>70.4</td><td>99.0</td></tr></table>
+Table 3: Commonsense and symbolic reasoning accuracy by self-consistency compared to chainof-thought prompting (Wei et al., 2022). The previous SoTA baselines are obtained from: $a$ : DeBERTaV3-large $^ +$ KEAR ( $\mathrm { { X u } }$ et al., 2021b), $b$ : Chowdhery et al. (2022), c: UnifiedQA-FT (Khashabi et al., 2020). The best performance for each task is shown in bold.
+Commonsense and Symbolic Reasoning Table 3 shows the results on commonsense and symbolic reasoning tasks. Similarly, self-consistency yields large gains across all four language models, and obtained SoTA results on 5 out of 6 tasks. For symbolic reasoning, we test the out-of-distribution (OOD) setting where the input prompt contains examples of 2-letters or 2-flips but we test examples of 4-letters and 4-flips (this setting is more challenging as PaLM-540B or GPT-3 can already achieve perfect in-distribution accuracy). In this challenging OOD setting, the gain of self-consistency is still quite significant compared to CoT-prompting with sufficient model sizes.
+To show the effect of the number of sampled reasoning paths, we plot the accuracy (mean and standard deviation over 10 runs) with respect to varying numbers of sampled paths (1, 5, 10, 20, 40) in Figure 2. The results show that sampling a higher number (e.g., 40) of reasoning paths leads to a consistently better performance, further emphasizing the importance of introducing diversity in the reasoning paths. In Table 4, we show self-consistency yields a richer set of reasoning paths compared to greedy decoding with a few example questions from two tasks.
+![](images/91c0b57fce63bbd74bd5c37ac54affcd3ffead95ee070efede9bc60786bb0af7.jpg)
+Figure 2: Self-consistency (blue) significantly improves accuracy over CoT-prompting with greedy decoding (orange) across arithmetic and commonsense reasoning tasks, over LaMDA-137B. Sampling a higher number of diverse reasoning paths consistently improves reasoning accuracy.
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>GSM8K                                                       StrategyQA</td></tr><tr><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>Henry made two stops during his 6O-mile bike trip.Hefirst stopped after 20miles.His second stop was15 milesbefore the end of the trip.How many milesdid he travel between his first and second stops?</td><td rowspan=1 colspan=1>Yes or no:IsAlbany,Georgia the most populousUSAlbany?</td></tr><tr><td rowspan=1 colspan=1>GreedyDecode</td><td rowspan=1 colspan=1>He traveled 6O miles in total.He stopped after 20 miles,so he traveled 60 -20= 40 miles between the first and second stops. The answer is 40.</td><td rowspan=1 colspan=1>Albany, Georgia is the most populous US Al-bany. Thus,the answer is yes.</td></tr><tr><td rowspan=1 colspan=1>SampledPath 1</td><td rowspan=1 colspan=1>Henry travelled a total of 6O miles.His first stop was 2O miles in,and hissecond stop was 15 miles before the end. So between his first and second stopshe travelled 60 -20-15=25 miles. The answer is 25.</td><td rowspan=1 colspan=1>The most populous US Albany is Albany, NewYork．Thus,Albany,Georgia is not the mostpopulous US Albany. So the answer is no.</td></tr><tr><td rowspan=1 colspan=1>SampledPath 2</td><td rowspan=1 colspan=1>He made two stops during a 6O-mile trip. The first was 2O miles into the trip.The second was 15 miles before the end of the trip. This means the second stopWas 60 -15=45 miles into the trip. Since he made the stops in order,the second stop must have been 45 -20 = 25 miles after the first stop.The answer is 25.</td><td rowspan=1 colspan=1>Albany，Georgia has a population of about 88,000. Albany, New York has a population ofabout 95,Ooo. Thus,Albany, Georgia is not themost populous US Albany. So the answer is no.</td></tr></table>
+Table 4: Examples where self-consistency helps repair the errors over greedy decode, on PaLM-540B. Two sampled reasoning paths that are consistent with the ground truth are shown.
+# 3.3 SELF-CONSISTENCY HELPS WHEN CHAIN-OF-THOUGHT HURTS PERFORMANCE
+Ye & Durrett (2022) show that sometimes chain-of-thought prompting could hurt performance compared to standard prompting in few-shot in-context learning. Here we perform a study using self-consistency to see if it can help fill in the gap, over a set of common NLP tasks, including (1) Closed-Book Question Answering: BoolQ (Clark et al., 2019), HotpotQA (Yang et al., 2018), and (2) Natural Language Inference: e-SNLI (Camburu et al., 2018), ANLI (Nie et al., 2020) and RTE (Dagan et al., 2005; Bar-Haim et al., 2006; Giampiccolo et al., 2007; Bentivogli et al., 2009).
+The results over PaLM-540B are shown in Table 5. For some tasks (e.g., ANLI-R1, e-SNLI, RTE), adding chain-of-thought does hurt performance compared to standard prompting (Brown et al., 2020), but self-consistency is able to robustly boost the performance and outperform standard prompting, making it a reliable way to add rationales in few-shot in-context learning for common NLP tasks.
+<table><tr><td></td><td>ANLIR1/R2/R3</td><td>e-SNLI</td><td>RTE</td><td>BoolQ</td><td>HotpotQA (EM/F1)</td></tr><tr><td>Standard-prompting (no-rationale)</td><td>69.1 / 55.8 / 55.8</td><td>85.8</td><td>84.8</td><td>71.3</td><td>27.1/36.8</td></tr><tr><td>CoT-prompting (Wei et al.,2022)</td><td>68.8 / 58.9 / 60.6</td><td>81.0</td><td>79.1</td><td>74.2</td><td>28.9/39.8</td></tr><tr><td>Self-consistency</td><td>78.5 / 64.5 / 63.4</td><td>88.4</td><td>86.3</td><td>78.4</td><td>33.8 / 44.6</td></tr></table>
+Table 5: Compare Standard/CoT prompting with self-consistency on common NLP tasks.
+# 3.4 COMPARE TO OTHER EXISTING APPROACHES
+We conduct a set of additional studies and show that self-consistency significantly outperforms existing methods including sample-and-rank, beam search, and ensemble-based approaches.
+Comparison to Sample-and-Rank One commonly used approach to improve generation quality is sample-and-rank, where multiple sequences are sampled from the decoder and then ranked according to each sequence’s log probability (Adiwardana et al., 2020). We compare self-consistency with sample-and-rank on GPT-3 code-davinci-001, by sampling the same number of sequences from the decoder as self-consistency and taking the final answer from the top-ranked sequence. The results are shown in Figure 3. While sample-and-rank does improve the accuracy with additionally sampled sequences and ranking, the gain is much smaller compared to self-consistency.
+![](images/211c58f5fc739ddbb34ca25bc8a91e11996e73243a0ad766cce0146bc55d347b.jpg)
+Figure 3: Self-consistency significantly outperforms sample-and-rank with the same # of samples.
+Comparison to Beam Search In Table 6, we compare self-consistency with beam search decoding on the UL2-20B model. For a fair comparison we report the accuracy under the same number of beams and reasoning paths. On both tasks self-consistency outperforms beam search significantly. Note self-consistency can also adopt beam search to decode each reasoning path (results are shown as “Self-consistency using beam search”), but its performance is worse compared to self-consistency with sampling. The reason is that beam search yields a lower diversity in the outputs (Li & Jurafsky, 2016), while in self-consistency the diversity of the reasoning paths is the key to a better performance.
+<table><tr><td></td><td>Beam size / Self-consistency paths</td><td>1</td><td>5</td><td>10</td><td>20</td><td>40</td></tr><tr><td rowspan="2">AQuA</td><td>Beam search decoding (top beam)</td><td>23.6 23.6</td><td>19.3 19.8 ± 0.3</td><td>16.1</td><td>15.0</td><td>10.2</td></tr><tr><td>Self-consistency using beam search Self-consistency using sampling</td><td>19.7 ± 2.5</td><td>24.9 ± 2.6</td><td>21.2 ±0.7 25.3 ± 1.8</td><td>24.6 ± 0.4 26.7 ± 1.0</td><td>24.2 ±0.5 26.9 ± 0.5</td></tr><tr><td rowspan="2">MultiArith</td><td></td><td>10.7</td><td>12.0</td><td>11.3</td><td></td><td></td></tr><tr><td>Beam search decoding (top beam)</td><td></td><td></td><td></td><td>11.0</td><td>10.5</td></tr><tr><td rowspan="2"></td><td>Self-consistency using beam search</td><td>10.7</td><td>11.8 ± 0.0</td><td>11.4 ± 0.1</td><td>12.3 ± 0.1</td><td>10.8 ±0.1</td></tr><tr><td>Self-consistency using sampling</td><td>9.5 ± 1.2</td><td>11.3 ± 1.2</td><td>12.3 ± 0.8</td><td>13.7 ± 0.9</td><td>14.7 ± 0.3</td></tr></table>
+Table 6: Compare self-consistency with beam search decoding on the UL2-20B model.
+Comparison to Ensemble-based Approaches We further compare self-consistency to ensemblebased methods for few-shot learning. In particular, we consider ensembling by: (1) prompt order permutation: we randomly permute the exemplars in the prompt 40 times to mitigate model’s sensitivity to prompt order (Zhao et al., 2021; Lu et al., 2021); and (2) multiple sets of prompts (Gao et al., 2021): we manually write 3 different sets of prompts. We took majority vote of the answers from greedy decoding in both approaches as an ensemble. Table 7 shows that compared to self-consistency, existing ensemble-based approaches achieve a much smaller gain.8 In addition, note that self-consistency is different from a typical model-ensemble approach, where multiple models are trained and their outputs are aggregated. Self-consistency acts more like a “self-ensemble” on top of a single language model. We additionally show the results of ensembling multiple models in Appendix A.1.3 where the model-ensembles perform much worse compared to self-consistency.
+<table><tr><td></td><td>GSM8K</td><td>MultiArith</td><td>SVAMP</td><td>ARC-e</td><td>ARC-c</td></tr><tr><td>CoT (Wei et al., 2022)</td><td>17.1</td><td>51.8</td><td>38.9</td><td>75.3</td><td>55.1</td></tr><tr><td>Ensemble (3 sets of prompts)</td><td>18.6 ± 0.5</td><td>57.1 ± 0.7</td><td>42.1 ± 0.6</td><td>76.6 ± 0.1</td><td>57.0± 0.2</td></tr><tr><td>Ensemble (40 prompt permutations)</td><td>19.2 ± 0.1</td><td>60.9 ± 0.2</td><td>42.7 ± 0.1</td><td>76.9 ± 0.1</td><td>57.0 ±0.1</td></tr><tr><td>Self-Consistency (4O sampled paths)</td><td>27.7 ± 0.2</td><td>75.7 ± 0.3</td><td>53.3 ± 0.2</td><td>79.3 ± 0.3</td><td>59.8 ± 0.2</td></tr></table>
+Table 7: Self-consistency outperforms prompt-order and multi-prompt ensembles on LaMDA-137B.
+8Self-consistency is compatible with both ensemble approaches and we show the results in Appendix A.1.4.
+# 3.5 ADDITIONAL STUDIES
+We conducted a number of additional experiments to analyze different aspects of the self-consistency method, including its robustness to sampling strategies and parameters, and how it works with imperfect prompts and non-natural-language reasoning paths.
+Self-Consistency is Robust to Sampling Strategies and Scaling We show self-consistency is robust to sampling strategies and parameters, by varying $T$ in temperature sampling (Ackley et al., 1985; Ficler & Goldberg, 2017), $k$ in top- $k$ sampling (Fan et al., 2018; Holtzman et al., 2018; Radford et al., 2019), and $p$ in nucleus sampling (Holtzman et al., 2020), over PaLM-540B in Figure 4 (left). Figure 4 (right) shows that self-consistency robustly improves performance across all scales for the LaMDA-137B model series. The gain is relatively lower for smaller models due to certain abilities (e.g., arithmetic) only emerge when the model reaches a sufficient scale (Brown et al., 2020).
+![](images/87a328209d36c386b2c5751a4a40e895999983a1e5c87663caac19ecc466a090.jpg)
+Figure 4: GSM8K accuracy. (Left) Self-consistency is robust to various sampling strategies and parameters. (Right) Self-consistency improves performance across language model scales.
+Self-Consistency Improves Robustness to Imperfect Prompts For few-shot learning with manually constructed prompts, human annotators sometimes make minor mistakes when creating the prompts. We further study if self-consistency can help improve a language model’s robustness to imperfect prompts.9 We show the results in Table 8: while imperfect prompts decrease accuracy with greedy decoding $1 7 . 1  1 4 . 9$ ), self-consistency can fill in the gaps and robustly improve the results.
+Additionally, we found that the consistency (in terms of $\%$ of decodes agreeing with the final aggregated answer) is highly correlated with accuracy (Figure 5, over GSM8K). This suggests that one can use self-consistency to provide an uncertainty estimate of the model in its generated solutions. In other words, one can use low consistency as an indicator that the model has low confidence; i.e., self-consistency confers some ability for the model to “know when it doesn’t know”.
+<table><tr><td rowspan="3">LaMDA-137B</td><td>Prompt with correct chain-of-thought</td><td>17.1</td></tr><tr><td>Prompt with imperfect chain-of-thought + Self-consistency (40 paths)</td><td>14.9 23.4</td></tr><tr><td>Prompt with equations + Self-consistency (40 paths)</td><td>5.0 6.5</td></tr><tr><td rowspan="2">PaLM-540B</td><td>Zero-shot CoT (Kojima et al., 2022)</td><td>43.0</td></tr><tr><td>+ Self-consistency (40 paths)</td><td>69.2</td></tr></table>
+Table 8: Self-consistency works under imperfect prompts, equation prompts and zero-shot chain-of-thought for GSM8K.
+![](images/904d0744036049f41de9ff1b79bfb3e5c595bb15c1aa6fad815946070f5097c4.jpg)
+Figure 5: The consistency is correlated with model’s accuracy.
+Self-Consistency Works for Non-Natural-Language Reasoning Paths and Zero-shot CoT We also tested the generality of the self-consistency concept to alternative forms of intermediate reasoning like equations (e.g., from “There are 3 cars in the parking lot already. 2 more arrive. Now there are $3 + 2 = 5$ cars.” to $" 3 + 2 = 5 "$ ). The results are shown in Table 8 (“Prompt with equations”): self-consistency still improves accuracy by generating intermediate equations; however, compared to generating natural language reasoning paths, the gain is smaller since the equations are much shorter and less opportunity remains for generating diversity in the decoding process. In addition, we tested self-consistency with zero-shot chain-of-thought (Kojima et al., 2022) and show that self-consistency works for zero-shot CoT as well and improves the results significantly $( + 2 6 . 2 \% )$ in Table 8.
+# 4 RELATED WORK
+Reasoning in language models. Language models are known to struggle in Type 2 tasks, such as arithmetic, logical and commonsense reasoning (Evans, 2010). Previous work has primarily focused on specialized approaches for improving reasoning (Andor et al., 2019; Ran et al., 2019; Geva et al., 2020; Pi˛ekos et al., 2021). Compared to prior work, self-consistency is applicable to a wide range of reasoning tasks without any additional supervision or fine-tuning, while still substantially improving the performance of the chain-of-thought prompting approach proposed in Wei et al. (2022).
+Sampling and re-ranking in language models. Multiple decoding strategies for language models have been proposed in the literature, e.g., temperature sampling (Ackley et al., 1985; Ficler & Goldberg, 2017), top- $k$ sampling (Fan et al., 2018; Holtzman et al., 2018; Radford et al., 2019), nucleus sampling (Holtzman et al., 2020), minimum Bayes risk decoding (Eikema & Aziz, 2020; Shi et al., 2022), and typical decoding (Meister et al., 2022). Other work has sought to explicitly promote diversity in the decoding process (Batra et al., 2012; Li et al., 2016; Vijayakumar et al., 2018).
+Re-ranking is another common approach to improve generation quality in language models (Adiwardana et al., 2020; Shen et al., 2021). Thoppilan et al. (2022) collect additional human annotations to train a re-ranker for response filtering. Cobbe et al. (2021) train a “verifier” to re-rank generated solutions, which substantially improves the solve rate on math tasks compared to just fine-tuning the language model. Elazar et al. (2021) improve the consistency of factual knowledge extraction by extending pre-training with an additional consistency loss. All these methods require either training an additional re-ranker or collecting additional human annotation, while self-consistency requires no additional training, fine-tuning, nor extra data collection.
+Extract reasoning paths. Some previous work has considered task-specific approaches for identifying reasoning paths, such as constructing semantic graphs (Xu et al., 2021a), learning an RNN to retrieve reasoning paths over the Wikipedia graph (Asai et al., 2020), fine-tuning with human annotated reasoning paths on math problems (Cobbe et al., 2021), or training an extractor with heuristic-based pseudo reasoning paths (Chen et al., 2019). More recently, the importance of diversity in the reasoning processes has been noticed, but only leveraged via task-specific training, either through an additional QA model over extracted reasoning paths (Chen et al., 2019), or by the introduction of latent variables in a commonsense knowledge graph (Yu et al., 2022). Compared to these approaches, self-consistency is far simpler and requires no additional training. The approach we propose simply couples the generation of reasoning paths and a final answer by sampling from the decoder, using aggregation to recover the most consistent answer without additional modules.
+Consistency in language models. Some prior work has shown that language models can suffer from inconsistency in conversation (Adiwardana et al., 2020), explanation generation (Camburu et al., 2020), and factual knowledge extraction (Elazar et al., 2021). Welleck et al. (2020) use “consistency” to refer to generating an infinite-length sequence in recurrent language models. Nye et al. (2021) improve the logical consistency of samples from a System 1 model by adding a System 2-inspired logical reasoning module. In this paper we focus on a slightly different notion of “consistency”, i.e., utilizing answer consistency among diverse reasoning paths to improve accuracy.
+# 5 CONCLUSION AND DISCUSSION
+We introduced a simple yet effective method called self-consistency, and observed that it significantly improves accuracy in a range of arithmetic and commonsense reasoning tasks, across four large language models with varying scales. Beyond accuracy gains, self-consistency is also useful for collecting rationales when performing reasoning tasks with language models, and for providing uncertainty estimates and improved calibration of language model outputs.
+One limitation of self-consistency is that it incurs more computation cost. In practice people can try a small number of paths (e.g., 5 or 10) as a starting point to realize most of the gains while not incurring too much cost, as in most cases the performance saturates quickly (Figure 2). As part of future work, one could use self-consistency to generate better supervised data to fine-tune the model, such that the model can give more accurate predictions in a single inference run after fine-tuning. In addition, we observed that language models can sometimes generate incorrect or nonsensical reasoning paths (e.g., the StrategyQA example in Table 4, the two population numbers are not exactly correct), and further work is needed to better ground models’ rationale generations.
+# REPRODUCIBILITY STATEMENT
+In experiments, we included four different language models with varying scales. Two of them are public models: UL2 is a completely open-sourced model with model checkpoints available at https:// github.com/google-research/google-research/tree/master/ul2; GPT-3 is also a public model with public API available at https://openai.com/api/. For GPT-3, we have included two public engines (“code-davinci-001” and “code-davinci-002”) to further aid reproducibility, as Codex is currently free so anyone can reproduce the results. In addition, as our results make use of LaMDA-137B and PaLM-540B that are not publicly available, we provide the exact input prompts for all tasks in Appendix A.3 (and note that we do not perform any finetuning and only apply prompting to off-the-shelf language models).
+# ETHICS STATEMENT
+As we stated in the discussion, language models can sometimes generate nonsensical or non-factual reasoning paths, so one should use language models’ outputs with extra caution. We deal with reasoning tasks mostly and the generated rationales are only used for inspecting how a model reaches its answer. One could potentially use the generated rationales to further check why the model makes certain mistakes or whether the model contains any biases when performing a certain task. For language model in real-world use, further work is needed to better ground models’ predictions and improve model’s factuality and safety, to ensure the models do not cause harms to users.
+# REFERENCES
+David H. Ackley, Geoffrey E. Hinton, and Terrence J. Sejnowski. A learning algorithm for boltzmann machines. Cognitive Science, 9(1):147–169, 1985. ISSN 0364-0213. URL https://www. sciencedirect.com/science/article/pii/S0364021385800124.
+Daniel Adiwardana, Minh-Thang Luong, David R. So, Jamie Hall, Noah Fiedel, Romal Thoppilan, Zi Yang, Apoorv Kulshreshtha, Gaurav Nemade, Yifeng Lu, and Quoc V. Le. Towards a human-like open-domain chatbot, 2020.
+Aida Amini, Saadia Gabriel, Shanchuan Lin, Rik Koncel-Kedziorski, Yejin Choi, and Hannaneh Hajishirzi. MathQA: Towards interpretable math word problem solving with operation-based formalisms. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 2357–2367. Association for Computational Linguistics, June 2019. URL https://aclanthology.org/N19-1245.
+Daniel Andor, Luheng He, Kenton Lee, and Emily Pitler. Giving BERT a calculator: Finding operations and arguments with reading comprehension. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), 2019. URL https://aclanthology. org/D19-1609.
+Akari Asai, Kazuma Hashimoto, Hannaneh Hajishirzi, Richard Socher, and Caiming Xiong. Learning to retrieve reasoning paths over wikipedia graph for question answering. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum? id $=$ SJgVHkrYDH.
+Roy Bar-Haim, Ido Dagan, Bill Dolan, Lisa Ferro, Danilo Giampiccolo, Bernardo Magnini, and Idan Szpektor. The second pascal recognising textual entailment challenge. In Proceedings of the second PASCAL challenges workshop on recognising textual entailment, 2006.
+Dhruv Batra, Payman Yadollahpour, Abner Guzman-Rivera, and Gregory Shakhnarovich. Diverse m-best solutions in markov random fields. In Proceedings of the 12th European Conference on Computer Vision - Volume Part V, ECCV’12, pp. 1–16, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642337147. URL https://doi.org/10.1007/978-3-642-33715-4_1.
+Luisa Bentivogli, Peter Clark, Ido Dagan, and Danilo Giampiccolo. The fifth pascal recognizing textual entailment challenge. In TAC, 2009.
+BIG-bench collaboration. Beyond the imitation game: Measuring and extrapolating the capabilities of language models. In preparation, 2021. URL https://github.com/google/ BIG-bench/.
+Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In Advances in Neural Information Processing Systems, 2020. URL https://proceedings.neurips.cc/paper/2020/ file/1457c0d6bfcb4967418bfb8ac142f64a-Paper.pdf.
+Oana-Maria Camburu, Tim Rocktäschel, Thomas Lukasiewicz, and Phil Blunsom. esnli: Natural language inference with natural language explanations. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 9539–9549. Curran Associates, Inc., 2018. URL http://papers.nips.cc/paper/ 8163-e-snli-natural-language-inference-with-natural-language-explanations. pdf.
+Oana-Maria Camburu, Brendan Shillingford, Pasquale Minervini, Thomas Lukasiewicz, and Phil Blunsom. Make up your mind! adversarial generation of inconsistent natural language explanations. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 4157–4165, Online, July 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020. acl-main.382. URL https://aclanthology.org/2020.acl-main.382.
+Jifan Chen, Shih-Ting Lin, and Greg Durrett. Multi-hop question answering via reasoning chains. CoRR, abs/1910.02610, 2019. URL http://arxiv.org/abs/1910.02610.
+Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, et al. Evaluating large language models trained on code. arXiv preprint arXiv:2107.03374, 2021.
+Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, Parker Schuh, Kensen Shi, Sasha Tsvyashchenko, Joshua Maynez, Abhishek Rao, Parker Barnes, Yi Tay, Noam Shazeer, Vinodkumar Prabhakaran, Emily Reif, Nan Du, Ben Hutchinson, Reiner Pope, James Bradbury, Jacob Austin, Michael Isard, Guy Gur-Ari, Pengcheng Yin, Toju Duke, Anselm Levskaya, Sanjay Ghemawat, Sunipa Dev, Henryk Michalewski, Xavier Garcia, Vedant Misra, Kevin Robinson, Liam Fedus, Denny Zhou, Daphne Ippolito, David Luan, Hyeontaek Lim, Barret Zoph, Alexander Spiridonov, Ryan Sepassi, David Dohan, Shivani Agrawal, Mark Omernick, Andrew M. Dai, Thanumalayan Sankaranarayana Pillai, Marie Pellat, Aitor Lewkowycz, Erica Moreira, Rewon Child, Oleksandr Polozov, Katherine Lee, Zongwei Zhou, Xuezhi Wang, Brennan Saeta, Mark Diaz, Orhan Firat, Michele Catasta, Jason Wei, Kathy Meier-Hellstern, Douglas Eck, Jeff Dean, Slav Petrov, and Noah Fiedel. Palm: Scaling language modeling with pathways, 2022. URL https://arxiv.org/abs/2204.02311.
+Christopher Clark, Kenton Lee, Ming-Wei Chang, Tom Kwiatkowski, Michael Collins, and Kristina Toutanova. Boolq: Exploring the surprising difficulty of natural yes/no questions. In NAACL, 2019.
+Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and Oyvind Tafjord. Think you have solved question answering? try arc, the ai2 reasoning challenge. ArXiv, abs/1803.05457, 2018.
+Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. Training verifiers to solve math word problems, 2021.
+Ido Dagan, Oren Glickman, and Bernardo Magnini. The pascal recognising textual entailment challenge. In Machine Learning Challenges Workshop, pp. 177–190. Springer, 2005.
+Bryan Eikema and Wilker Aziz. Is MAP decoding all you need? the inadequacy of the mode in neural machine translation. In Proceedings of the 28th International Conference on Computational Linguistics, pp. 4506–4520, Barcelona, Spain (Online), December 2020. International Committee on Computational Linguistics. URL https://aclanthology.org/2020.coling-main. 398.
+Yanai Elazar, Nora Kassner, Shauli Ravfogel, Abhilasha Ravichander, Eduard Hovy, Hinrich Schütze, and Yoav Goldberg. Measuring and improving consistency in pretrained language models. Transactions of the Association for Computational Linguistics, 9:1012–1031, 2021. doi: 10.1162/tacl_a_00410. URL https://aclanthology.org/2021.tacl-1.60.
+Jonathan St BT Evans. Intuition and reasoning: A dual-process perspective. Psychological Inquiry, 21(4):313–326, 2010.
+Angela Fan, Mike Lewis, and Yann Dauphin. Hierarchical neural story generation. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 889–898, Melbourne, Australia, July 2018. Association for Computational Linguistics. doi: 10.18653/v1/P18-1082. URL https://aclanthology.org/P18-1082.
+Jessica Ficler and Yoav Goldberg. Controlling linguistic style aspects in neural language generation. In Proceedings of the Workshop on Stylistic Variation, pp. 94–104, Copenhagen, Denmark, September 2017. Association for Computational Linguistics. doi: 10.18653/v1/W17-4912. URL https: //aclanthology.org/W17-4912.
+Tianyu Gao, Adam Fisch, and Danqi Chen. Making pre-trained language models better few-shot learners. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pp. 3816–3830, Online, August 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.acl-long.295. URL https://aclanthology.org/2021.acl-long. 295.
+Mor Geva, Ankit Gupta, and Jonathan Berant. Injecting numerical reasoning skills into language models. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, 2020. doi: 10.18653/v1/2020.acl-main.89. URL https://aclanthology.org/2020. acl-main.89.
+Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, Dan Roth, and Jonathan Berant. Did aristotle use a laptop? A question answering benchmark with implicit reasoning strategies. Transactions of the Association for Computational Linguistics, 2021. URL https://aclanthology.org/ 2021.tacl-1.21.
+Danilo Giampiccolo, Bernardo Magnini, Ido Dagan, and Bill Dolan. The third pascal recognizing textual entailment challenge. In Proceedings of the ACL-PASCAL workshop on textual entailment and paraphrasing, pp. 1–9. Association for Computational Linguistics, 2007.
+Ari Holtzman, Jan Buys, Maxwell Forbes, Antoine Bosselut, David Golub, and Yejin Choi. Learning to write with cooperative discriminators. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 1638–1649, Melbourne, Australia, July 2018. Association for Computational Linguistics. doi: 10.18653/v1/P18-1152. URL https://aclanthology.org/P18-1152.
+Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. The curious case of neural text degeneration. In International Conference on Learning Representations, 2020. URL https: //openreview.net/forum?id=rygGQyrFvH.
+Mohammad Javad Hosseini, Hannaneh Hajishirzi, Oren Etzioni, and Nate Kushman. Learning to solve arithmetic word problems with verb categorization. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2014. doi: 10.3115/v1/D14-1058. URL https://aclanthology.org/D14-1058.
+Daniel Khashabi, Sewon Min, Tushar Khot, Ashish Sabharwal, Oyvind Tafjord, Peter Clark, and Hannaneh Hajishirzi. UNIFIEDQA: Crossing format boundaries with a single QA system. In Findings of the Association for Computational Linguistics: EMNLP 2020, pp. 1896–1907, Online, November 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020.findings-emnlp.171. URL https://aclanthology.org/2020.findings-emnlp.171.
+Takeshi Kojima, Shixiang Shane Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa. Large language models are zero-shot reasoners. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. URL https://openreview.net/forum?id $_ { \cdot } =$ e2TBb5y0yFf.
+Rik Koncel-Kedziorski, Subhro Roy, Aida Amini, Nate Kushman, and Hannaneh Hajishirzi. MAWPS: A math word problem repository. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, 2016. doi: 10.18653/v1/N16-1136. URL https://aclanthology.org/N16-1136.
+Yihuai Lan, Lei Wang, Qiyuan Zhang, Yunshi Lan, Bing Tian Dai, Yan Wang, Dongxiang Zhang, and Ee-Peng Lim. MWPToolkit: An open-source framework for deep learning-based math word problem solvers. arXiv preprint arXiv:2109.00799, 2021. URL https://arxiv.org/abs/ 2109.00799.
+Jiwei Li and Dan Jurafsky. Mutual information and diverse decoding improve neural machine translation, 2016. URL https://arxiv.org/abs/1601.00372.
+Jiwei Li, Will Monroe, and Dan Jurafsky. A simple, fast diverse decoding algorithm for neural generation. CoRR, abs/1611.08562, 2016. URL http://arxiv.org/abs/1611.08562.
+Wang Ling, Dani Yogatama, Chris Dyer, and Phil Blunsom. Program induction by rationale generation: Learning to solve and explain algebraic word problems. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), 2017. doi: 10.18653/v1/P17-1015. URL https://aclanthology.org/P17-1015.
+Yao Lu, Max Bartolo, Alastair Moore, Sebastian Riedel, and Pontus Stenetorp. Fantastically ordered prompts and where to find them: Overcoming few-shot prompt order sensitivity. ArXiv, abs/2104.08786, 2021.
+Clara Meister, Tiago Pimentel, Gian Wiher, and Ryan Cotterell. Typical decoding for natural language generation. arXiv preprint arXiv:2202.00666, 2022.
+Shen Yun Miao, Chao Chun Liang, and Keh Yih Su. A diverse corpus for evaluating and developing English math word problem solvers. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, 2020. URL https://aclanthology.org/2020. acl-main.92.
+Yixin Nie, Adina Williams, Emily Dinan, Mohit Bansal, Jason Weston, and Douwe Kiela. Adversarial NLI: A new benchmark for natural language understanding. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. Association for Computational Linguistics, 2020.
+Maxwell Nye, Michael Henry Tessler, Joshua B. Tenenbaum, and Brenden M. Lake. Improving coherence and consistency in neural sequence models with dual-system, neuro-symbolic reasoning. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan (eds.), Advances in Neural Information Processing Systems, 2021. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ uyKk_avJ-p4.
+Arkil Patel, Satwik Bhattamishra, and Navin Goyal. Are NLP models really able to solve simple math word problems? In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 2080–2094, Online, June 2021. Association for Computational Linguistics. doi: 10.18653/v1/2021.naacl-main. 168. URL https://aclanthology.org/2021.naacl-main.168.
+Xinyu Pi, Qian Liu, Bei Chen, Morteza Ziyadi, Zeqi Lin, Yan Gao, Qiang Fu, Jian-Guang Lou, and Weizhu Chen. Reasoning like program executors, 2022.
+Piotr Pi˛ekos, Mateusz Malinowski, and Henryk Michalewski. Measuring and improving BERT’s mathematical abilities by predicting the order of reasoning. In Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers), 2021. doi: 10.18653/v1/2021.acl-short.49. URL https://aclanthology.org/2021.acl-short.49.
+Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. 2019.
+Jack W Rae, Sebastian Borgeaud, Trevor Cai, Katie Millican, Jordan Hoffmann, Francis Song, John Aslanides, Sarah Henderson, Roman Ring, Susannah Young, et al. Scaling language models: Methods, analysis & insights from training gopher. arXiv preprint arXiv:2112.11446, 2021.
+Qiu Ran, Yankai Lin, Peng Li, Jie Zhou, and Zhiyuan Liu. NumNet: Machine reading comprehension with numerical reasoning. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), 2019. doi: 10.18653/v1/D19-1251. URL https://aclanthology. org/D19-1251.
+Subhro Roy and Dan Roth. Solving general arithmetic word problems. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, 2015. doi: 10.18653/v1/ D15-1202. URL https://aclanthology.org/D15-1202.
+Jianhao Shen, Yichun Yin, Lin Li, Lifeng Shang, Xin Jiang, Ming Zhang, and Qun Liu. Generate & rank: A multi-task framework for math word problems. In Findings of the Association for Computational Linguistics: EMNLP 2021, pp. 2269–2279, Punta Cana, Dominican Republic, November 2021. Association for Computational Linguistics. URL https://aclanthology. org/2021.findings-emnlp.195.
+Freda Shi, Daniel Fried, Marjan Ghazvininejad, Luke Zettlemoyer, and Sida I. Wang. Natural language to code translation with execution. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, pp. 3533–3546, Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguistics. URL https://aclanthology. org/2022.emnlp-main.231.
+Keith E Stanovich and Richard F West. Individual differences in reasoning: Implications for the rationality debate? Behavioral and brain sciences, 23(5):645–665, 2000. URL https: //pubmed.ncbi.nlm.nih.gov/11301544/.
+Alon Talmor, Jonathan Herzig, Nicholas Lourie, and Jonathan Berant. CommonsenseQA: A question answering challenge targeting commonsense knowledge. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), 2019. URL https://aclanthology. org/N19-1421.
+Yi Tay, Mostafa Dehghani, Vinh Q. Tran, Xavier Garcia, Jason Wei, Xuezhi Wang, Hyung Won Chung, Dara Bahri, Tal Schuster, Steven Zheng, Denny Zhou, Neil Houlsby, and Donald Metzler. Unifying language learning paradigms, 2022. URL https://arxiv.org/abs/2205.05131.
+Romal Thoppilan, Daniel De Freitas, Jamie Hall, Noam Shazeer, Apoorv Kulshreshtha, Heng-Tze Cheng, Alicia Jin, Taylor Bos, Leslie Baker, Yu Du, et al. Lamda: Language models for dialog applications. arXiv preprint arXiv:2201.08239, 2022. URL https://arxiv.org/abs/ 2201.08239.
+Ashwin Vijayakumar, Michael Cogswell, Ramprasaath Selvaraju, Qing Sun, Stefan Lee, David Crandall, and Dhruv Batra. Diverse beam search for improved description of complex scenes. Proceedings of the AAAI Conference on Artificial Intelligence, 32, Apr. 2018. URL https: //ojs.aaai.org/index.php/AAAI/article/view/12340.
+Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed Chi, Quoc Le, and Denny Zhou. Chain of thought prompting elicits reasoning in large language models. Conference on Neural Information Processing Systems (NeurIPS), 2022. URL https://arxiv. org/pdf/2201.11903.
+Sean Welleck, Ilia Kulikov, Jaedeok Kim, Richard Yuanzhe Pang, and Kyunghyun Cho. Consistency of a recurrent language model with respect to incomplete decoding. In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pp. 5553–5568, Online, November 2020. Association for Computational Linguistics. doi: 10.18653/v1/2020. emnlp-main.448. URL https://aclanthology.org/2020.emnlp-main.448.
+Weiwen Xu, Yang Deng, Huihui Zhang, Deng Cai, and Wai Lam. Exploiting reasoning chains for multi-hop science question answering. In Findings of the Association for Computational Linguistics: EMNLP 2021, pp. 1143–1156, Punta Cana, Dominican Republic, November 2021a. Association for Computational Linguistics. URL https://aclanthology.org/2021. findings-emnlp.99.
+Yichong Xu, Chenguang Zhu, Shuohang Wang, Siqi Sun, Hao Cheng, Xiaodong Liu, Jianfeng Gao, Pengcheng He, Michael Zeng, and Xuedong Huang. Human parity on commonsenseqa: Augmenting self-attention with external attention, 2021b. URL https://arxiv.org/abs/ 2112.03254.
+Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Bengio, William Cohen, Ruslan Salakhutdinov, and Christopher D. Manning. HotpotQA: A dataset for diverse, explainable multi-hop question answering. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 2369–2380, Brussels, Belgium, October-November 2018. Association for Computational Linguistics. doi: 10.18653/v1/D18-1259. URL https://aclanthology.org/D18-1259.
+Xi Ye and Greg Durrett. The unreliability of explanations in few-shot prompting for textual reasoning. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho (eds.), Advances in Neural Information Processing Systems, 2022. URL https://openreview.net/forum? id=Bct2f8fRd8S.
+Wenhao Yu, Chenguang Zhu, Lianhui Qin, Zhihan Zhang, Tong Zhao, and Meng Jiang. Diversifying content generation for commonsense reasoning with mixture of knowledge graph experts. In Findings of Annual Meeting of the Association for Computational Linguistics (ACL), 2022.
+Zihao Zhao, Eric Wallace, Shi Feng, Dan Klein, and Sameer Singh. Calibrate before use: Improving few-shot performance of language models. In Marina Meila and Tong Zhang (eds.), Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research. PMLR, 2021. URL https://proceedings.mlr.press/ v139/zhao21c.html.
+# A APPENDIX
+A.1 ADDITIONAL EXPERIMENT RESULTS
+# A.1.1 ROBUSTNESS TO SAMPLING STRATEGIES AND PARAMETERS
+In Figure 6 we ablate the results with respect to different sampling strategies and parameters by varying $T$ in temperature sampling and $k$ in Top- $k$ sampling, on LaMDA-137B. We show that self-consistency is robust to various sampling strategies and parameters.
+![](images/5682763857dd59665317b37d4e2ff1237798960f4e4fb66b5db1538fb257bd34.jpg)
+Figure 6: GSM8K accuracy over LaMDA-137B. Self-consistency works under various sampling strategies and sampling parameters.
+In Figure 7 and Figure 8, we show the results of self-consistency compared with greedy decoding a single path over LaMDA-137B and PaLM-540B, respectively. Self-consistency improves over greedy decode by a quite significant margin on both models, on top of high accuracy already achieved by scaling up model sizes.
+![](images/dd06aeb61120fc868024f45c264d037f8b03f44182e787809b7cfde355dece76.jpg)
+Figure 7: Self-consistency (blue) significantly improves accuracy across various arithmetic and commonsense reasoning tasks, over LaMDA-137B. Sampling a higher number of diverse reasoning paths consistently improves reasoning accuracy.
+We further show additional sampled reasoning paths from the LaMDA-137B model in Table 12, and sampled reasoning paths from the PaLM-540B model in Table 13. We see that the diversity in the additionally sampled reasoning paths indeed helps the model arrive at a more correct final answer after aggregation.
+# A.1.2 ROBUSTNESS TO DIFFERENT SETS OF PROMPTS
+In Table 9, we further show that self-consistency is quite robust to different sets of input prompts. We manually wrote 3 different sets of chain-of-thought as prompts to the model. Across all sets of prompts, self-consistency yields consistent gains over the original CoT approach.
+# A.1.3 COMPARED TO MODEL ENSEMBLES
+Additionally, we provide results of directly ensembling the outputs from multiple language models. The results are shown in Table 10, by greedily decoding sequences from 3 language models and taking the majority vote (averaged over 10 runs). Note this is a typical ensemble approach (averaging over the predictions over multiple models) and it achieves a performance significantly worse than self-consistency (self-consistency over PaLM-540B gets an accuracy of $7 4 . 4 \%$ ), as lower-capacity models drag down the performance of higher-capacity models. In addition, this approach is limited in two ways: 1) It requires multiple models for an ensemble which might not always be available, while self-consistency only requires one single model to “self-ensemble”; 2) If one of the models is much weaker, it can actually hurt the final performance.
+![](images/528b99ea5220667091bc895ecec5856d21dd707b103c238b2499d6362bf59214.jpg)
+Figure 8: Self-consistency (blue) significantly improves accuracy across various arithmetic and commonsense reasoning tasks, over PaLM-540B. Sampling a higher number of diverse reasoning paths consistently helps reasoning accuracy.
+<table><tr><td></td><td>Prompt set 1 (used in the main text)丨Prompt set 2丨Prompt set 3</td><td></td><td></td></tr><tr><td>CoT (Wei et al., 2022)</td><td>56.5</td><td>54.6</td><td>54.0</td></tr><tr><td>Self-consistency</td><td>74.4 (+17.9)</td><td>72.1 (+17.5)</td><td>70.4 (+16.4)</td></tr></table>
+Table 9: GSM8K accuracy over PaLM-540B. The results show robustness of self-consistency with respect to different prompts in the input.
+<table><tr><td></td><td>Method</td><td>GSM8K accuracy</td></tr><tr><td>Single model</td><td>PaLM-540B,greedy/self-consistency</td><td>56.5 / 74.4</td></tr><tr><td rowspan="4">Ensemble of models</td><td>LaMDA-137B+PaLM-540B</td><td>36.9 ± 0.5</td></tr><tr><td>PaLM-540B + GPT-3 (code-davinci-001,175B)</td><td>36.6 ± 0.4</td></tr><tr><td>LaMDA-137B + GPT-3 (code-davinci-001,175B)</td><td>16.0 ± 0.8</td></tr><tr><td>LaMDA-137B +PaLM-540B + GPT-3 (code-davinci-001,175B)</td><td>33.3 ± 0.7</td></tr></table>
+Table 10: Comparison of GSM8K accuracy over multiple-model ensembles.
+# .1.4 COMBINING SELF-CONSISTENCY WITH OTHER ENSEMBLING STRATEGIE
+Self-consistency is completely compatible with other ensemble strategies, although the gains achieved by self-consistency are significantly higher than other ensemble strategies (and can “override” the performance gains achieved by other ensemble strategies). We further performed experiments and include the results in Table 11 (for a fair comparison, we use 40 sets of prompts, or 40 prompt permutations to compare with self-consistency with 40 paths, all experiments are based on PaLM540B).
+<table><tr><td></td><td>GSM8K accuracy</td></tr><tr><td>Different sets of prompts (x40)</td><td>58.9</td></tr><tr><td>Prompt-permutation (x40)</td><td>59.6</td></tr><tr><td>Self-consistency (x40)</td><td>74.4</td></tr><tr><td>Self-consistency +different sets of prompts (x40)</td><td>75.4</td></tr><tr><td>Self-consistency + prompt-permutation (x40)</td><td>73.8</td></tr></table>
+Table 11: Combining self-consistency with other ensembling strategies.
+# A.2 DETAILS ON RESOURCES AND INFERENCE
+For all four language models we perform prompting-based inference only. For UL2 we use TPU v3 $2 \mathrm { x } 2$ configuration, 4 chips, 8 cores). For GPT-3 models the experiments are done though the public API.10 For LaMDA-137B we use TPU v3 (8x8 configuration, 64 chips, 128 cores). For PaLM-540B we use TPU v4 (4x4x12 configuration, 192 chips, 384 cores). Most inference jobs take 1 to 4 hours (over about 1,000 examples) for each task on UL2 and LaMDA-137B, and about 2 to 12 hours on PaLM-540B. Some tasks (e.g., commonsense reasoning) take longer but do not exceed 2 days for each task.
+For GPT-3 models, we use $1 2 8 \mathrm { m a x }$ tokens for all methods, without frequency penalty or presence penalty. For all models, we take the generated outputs until the start of the next $\ " \mathrm { Q } : \ " $ to parse the final answers, consistent with our prompting format.
+Table 12: Additional examples where self-consistency helps repair the errors over greedy decode on LaMDA-137B. Two sampled reasoning paths that are consistent with the ground truth are shown.
+<table><tr><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>Greedy Decode</td><td rowspan=1 colspan=1>Sampled Path 1</td><td rowspan=1 colspan=1>Sampled Path 2</td></tr><tr><td rowspan=1 colspan=1>[MultiArith] Sam had 10 socks.If he threw away3 old onesthatdidn&#x27;t fitandbought 36newones,how many socks would hehave?</td><td rowspan=1 colspan=1>Sam had 1O socks.He threwaway 3 old ones that didn&#x27;t fit.He bought 36 new ones.Thatmeans he has 10+36-3=41socks. The answer is 41.</td><td rowspan=1 colspan=1>Sam had 10 socks.Hethrew away 3old ones.Sonow he has 7 left. Then hebought 36 new ones．Sonow he has 7 + 36 = 43socks. The answer is 43.</td><td rowspan=1 colspan=1>He had 1O socks.He threw away3 old ones that didn&#x27;t ft.Hebought 36 new ones.So in totalhe has 10-3+36=43 socks.Theanswer is 43.</td></tr><tr><td rowspan=1 colspan=1>[SVAMP] In a school there are697 girls and the rest are boys.If thereare 228 moregirls thanboys.How many boys are therein that school?</td><td rowspan=1 colspan=1>There are 697 girls and the restare boys.That means there are697+ 228 = 925 girls. Therest are boys.So the number ofboys is 925-697 = 228. Theanswer is 228.</td><td rowspan=1 colspan=1>The number of boys is thedifference between 697and 228.This is 697 - 228= 469.The answer is 469.</td><td rowspan=1 colspan=1>In the beginning there are 697girls and the rest are boys.So,thenumber of boys must be 697 - 228= 469 boys. The answer is 469.</td></tr><tr><td rowspan=1 colspan=1>[AQuA] There arelO oranges ina basket.Find the no.of ways inwhich2 oranges are chosen fromthe basket?(a)45 (b) 90 (c)120(d) 150 (e) 180</td><td rowspan=1 colspan=1>There are 1O oranges in a bas-ket.So there are 10 waysin which 2 oranges are chosenfrom the basket.</td><td rowspan=1 colspan=1>There are 1O oranges in abasket.We need to choose2 oranges from the basket.So the number of ways are10C2.So the answer is (a).</td><td rowspan=1 colspan=1>There are 1O oranges in the bas-ket.If you choose 2 oranges,youhave 10C2=45 ways of choosing2 oranges. So the answer is (a).</td></tr><tr><td rowspan=1 colspan=1>[CommonsenseQA] The manlaidon thesoft mossand lookedup at the trees,where was theman?(a) niagra falls (b) forest(c) waterfall (d) ground (e) tree</td><td rowspan=1 colspan=1>The answer must be somethingthat is soft and mossy. Of theabove choices,only ground issoft and mossy. So the answeris (d).</td><td rowspan=1 colspan=1>Theanswermustbesomewhere where mossis found.Of the abovechoices,only forests areknown for having moss.So the answer is (b).</td><td rowspan=1 colspan=1>The answer must be a place wherethe man is laying down and look-ing up at trees.Of the abovechoices,only forest isaplacewhere people lay down and lookat trees. So the answer is (b).</td></tr><tr><td rowspan=1 colspan=1>[ARC] Which of thesewill mostlikely harm a habitat?(a) plant-ing trees. (b) water pollution. (c)rainfall. (d) sunlight.</td><td rowspan=1 colspan=1>Planting trees will not harm ahabitat. So the answer is (a).</td><td rowspan=1 colspan=1>The answer is (b)，waterpollution,because it willmost likely harm the habi-tat. The other answers arenot likely to harm the habi-tat.</td><td rowspan=1 colspan=1>The answeris (b), water pollution.Treesare good fora habitat, rain-fall is necessary for a habitat,andsunlight is necessary fora habitat.So the answer is (b).</td></tr><tr><td rowspan=1 colspan=1>[ARC-challenge] Which pro-cessin thecarbon cycle takes thelongest to complete? (a) emis-sion of waste.(b) respiration inanimals.(c) photosynthesis inplants.(d) formation of fossil fu-els.</td><td rowspan=1 colspan=1>The process of photosynthesisin plants takes the longest tocomplete. So the answer is (c).</td><td rowspan=1 colspan=1>The formation of fossil fu- els takes the longest timeto complete.So the an-swer is (d).</td><td rowspan=1 colspan=1>The process that takes the longesttime to complete is the forma-tion of fossil fuels,which hap-pens over millions of years. Sothe answer is (d).</td></tr></table>
+Table 13: Additional examples where self-consistency helps repair the errors over greedy decode on PaLM-540B. Two sampled reasoning paths that are consistent with the ground truth are shown.
+<table><tr><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>Greedy Decode</td><td rowspan=1 colspan=1>Sampled Path 1</td><td rowspan=1 colspan=1>Sampled Path 2</td></tr><tr><td rowspan=1 colspan=1>[GSM8K] Kyle bought lastyear&#x27;s best-selling book for$19.50. This is with a 25% dis-count from the original price.What was the original price ofthebook?</td><td rowspan=1 colspan=1>The book was sold for $19.50.This is 25% less than the origi-nal price.This means the originalprice was 100%． 25% of100%is 25%．So the original pricewas 100%- 25%= 75%.Theoriginal price was 75% of $19.50.75% of $19.50 is 75% * $19.50 =$14.625. The answer is $14.625.</td><td rowspan=1 colspan=1>The original price of the bookis $19.50.Since it was dis-counted by 25%,the originalprice was 19.50 / 0.75 = 26.The answer is 26.</td><td rowspan=1 colspan=1>Kyle bought a book that is soldfor $19.50.With a 25% discount,the original price must be higher.If we call the original price X,then the discounted price is 0.75x.Since this is $19.50,we can write0.75x = 19.50． Solving this,wefind that the original price is X =19.50 /0.75 = $26. The answer is$26.</td></tr><tr><td rowspan=1 colspan=1>[AQuA] The capacity of atank of dimensions (8 m x 6m x 2.5 m) is (a) 120 litres (b)1200 litres (c) 12000 litres (d)120000 litres (e) None of these</td><td rowspan=1 colspan=1>The volume of the tank is 8 m *6 m *2.5 m=120 m.So thecapacity of the tank is 120 m.Sothe answer is (c).</td><td rowspan=1 colspan=1>The volume of the tank is 8m* 6m* 2.5m=120m.This isequal to 120000 litres. So theanswer is (d)</td><td rowspan=1 colspan=1>The volume of the tank is 8 m *6m * 2.5 m=120 m3.One cu-bic meter contains 1Ooo litres,sothe tank contains 120 m3 *1000liters/m3 = 120,000 liters.So theanswer is (d).</td></tr><tr><td rowspan=1 colspan=1>[StrategyQA] Yesorno:Can the Great Depression betreated with Prozac?</td><td rowspan=1 colspan=1>The Great Depression was aneconomic depression．Prozac isa drug used to treat depression.Thus, the Great Depression couldbe treated with Prozac. So the an-swer is yes.</td><td rowspan=1 colspan=1>Prozac is an anti-depressantmedication. The Great Depres-sion is not a disease.Thus,Prozac cannot treat the GreatDepression. So the answer isno.</td><td rowspan=1 colspan=1>The Great Depression is a time pe-riod.Prozac is used to treat de-pression (the mental illness). Thus,the Great Depression can not betreated with Prozac.So the answeris no.</td></tr><tr><td rowspan=1 colspan=1>[ARC-easy] Which source ofenergy is nonrenewable? (a)coal.(b) sunlight.(c) water.(d) wood.</td><td rowspan=1 colspan=1>Coal,sunlight,and water areall renewable sources of energy.Wood is a nonrenewable source ofenergy.So the answer is (d).</td><td rowspan=1 colspan=1>A nonrenewable source of en-ergy is one in which the sourceis not replenished in a shortamount of time.Thus, the onlyoption that is nonrenewable iscoal. So the answer is (a).</td><td rowspan=1 colspan=1>Coal is a fossil fuel and is formedfrom dead organisms from the past.Coal cannot be produced in a shortperiod of time,so it is considereda nonrenewable source of energy.So the answer is (a).</td></tr><tr><td rowspan=1 colspan=1>[ARC-challenge] Studentsstudiedthe revolutionofEartharound the Sun.Duringwhichmonth willFlorida have themost sunlight energy avail-able?(a)March.(b)June.(c)September.(d) December.</td><td rowspan=1 colspan=1>The Earth is closest to the Sunin December, hence Florida willhave the most sunlight energyavailable in December. So the an-swer is (d).</td><td rowspan=1 colspan=1>Earth is closest to the Sun onJune 21,the summer solstice.So the answer is (b).</td><td rowspan=1 colspan=1>Florida is closer to the sun duringJune,so it will have the most sun-light energy during that month. Sothe answer is (b).</td></tr></table>
+# A.3 FULL SETS OF PROMPTS
+We list the full details of the prompts used for two newly-introduced datasets, AQUA-RAT (Ling et al., 2017) and AI2 Reasoning Challenge (ARC) (Clark et al., 2018), where we manually composed the example chain-of-thought in this paper, in Table 14 and Table 15, respectively.
+Table 14: Few-shot exemplars for AQUA-RAT.
+<table><tr><td>Q: John found that the average of 15 numbers is 40.If 10 is added to each number then the mean of the numbers is? Answer Choices: (a) 50 (b) 45 (c) 65 (d) 78 (e) 64</td></tr><tr><td>A: If 10 is added to each number, then the mean of the numbers also increases by 10.So the new mean would be 5O. The answer is (a).</td></tr><tr><td>Q: If a/b=3/4 and 8a + 5b=22,then find the value of a. Answer Choices: (a) 1/2 (b)3/2 (c)5/2 (d) 4/2 (e) 7/2</td></tr><tr><td>A: If a/b=3/4,then b=4a/3.So 8a + 5(4a/3)=22.This simplifies to 8a +20a/3 =22,which means 44a/3=22. So a is equal to 3/2. The answer is (b).</td></tr><tr><td>Q: A person is traveling at 2O km/hr and reached his destinyin 2.5 hr then find the distance?Answer Choices:</td></tr><tr><td>(a) 53 km (b) 55 km (c) 52 km (d) 60 km (e) 50 km A: The distance that the person traveled would have been 20 km/hr *2.5 hrs = 50 km. The answer is (e).</td></tr><tr><td>Q: How many keystrokes are neded to type the numbers from 1 to 50o? Answer Choices: (a)1156 (b)1392 (c) 1480 (d) 1562 (e) 1788</td></tr><tr><td>A: There are 9one-digit numbers from1 to 9.There are 90 two-digit numbers from 10 to 99.There are 401 three-digit numbers from 100 to 500.9 + 90(2) + 401(3)=1392.The answer is (b).</td></tr></table>
+Q: George wants to warm his hands quickly by rubbing them. Which skin surface will produce the most heat? (a) dry palms. (b) wet palms. (c) palms covered with oil. (d) palms covered with lotion.
+A: Dry surfaces will more likely cause more friction via rubbing than other smoother surfaces, hence dry palms will produce the most heat. The answer is (a).
+Q: Which factor will most likely cause a person to develop a fever? (a) a leg muscle relaxing after exercise. (b) a bacterial population in the bloodstream. (c) several viral particles on the skin. (d) carbohydrates being digested in the stomach.
+A: Option (b), bacterial population is the most likely cause for a person developing fever. The answer is (b). Q: Which change in the state of water particles causes the particles to become arranged in a fixed position? (a) boiling. (b) melting. (c) freezing. (d) evaporating.
+A: When water is freezed, the particles are arranged in a fixed position; the particles are still moving for all other options. The answer is (c).
+Q: When a switch is used in an electrical circuit, the switch can (a) cause the charge to build. (b) increase and decrease the voltage. (c) cause the current to change direction. (d) stop and start the flow of current. A: The function of a switch is to start and stop the flow of a current. The answer is (d).
+Table 15: Few-shot exemplars for ARC easy/challenge.
+Table 16: Few-shot exemplars for HotpotQA (closed-book setting).
+<table><tr><td>Q: Which magazine was started first Arthur&#x27;s Magazine or First for Women?</td></tr><tr><td>A: Arthur&#x27;s Magazine started in 1844. First for Women started in 1989.So Arthur&#x27;s Magazine was started first. The answer is Arthur&#x27;s Magazine.</td></tr><tr><td>Q: The Oberoi family is part of a hotel company that has a head office in what city?</td></tr><tr><td>A: The Oberoi family is part of the hotel company called The Oberoi Group.The Oberoi Group has its head office in Delhi.The answeris Delhi.</td></tr><tr><td>Q: What nationality was James Henry Miller&#x27;s wife?</td></tr><tr><td>A: James Henry Miller&#x27;s wife is June Miller. June Miller is an American.The answer is American.</td></tr><tr><td>Q: The Dutch-Belgian television series that &quot;House of Anubis&quot; was based on first aired in what year?</td></tr><tr><td>A: &quot;House of Anubis&quot;is basedon the Dutch-Belgian television series Het Huis Anubis.Het Huis Anubis is first</td></tr></table>
+As additional information, we also list the exact set of prompts used for all arithmetic reasoning tasks in Table 17, since there are multiple sets of prompts introduced in Wei et al. (2022). The prompts for CommonsenseQA and StrategyQA are the same as used in Wei et al. (2022).
+We provide the exact prompts used for common NLP tasks in the following tables as well, including NLI (Table 18, Table 19, Table 20) and Closed-Book Question-Answering tasks (Table 16, Table 21).
+Q: There are 15 trees in the grove. Grove workers will plant trees in the grove today. After they are done, there will be 21 trees. How many trees did the grove workers plant today?
+A: We start with 15 trees. Later we have 21 trees. The difference must be the number of trees they planted. So, they must have planted $2 1 - 1 5 = 6$ trees. The answer is 6.
+Q: If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in the parking lot? A: There are 3 cars in the parking lot already. 2 more arrive. Now there are $3 + 2 = 5$ cars. The answer is 5. Q: Leah had 32 chocolates and her sister had 42. If they ate 35, how many pieces do they have left in total? A: Leah had 32 chocolates and Leah’s sister had 42. That means there were originally $3 2 + 4 2 = 7 4$ chocolates. 35 have been eaten. So in total they still have $7 4 - 3 5 = 3 9$ chocolates. The answer is 39. Q: Jason had 20 lollipops. He gave Denny some lollipops. Now Jason has 12 lollipops. How many lollipops did Jason give to Denny?
+A: Jason had 20 lollipops. Since he only has 12 now, he must have given the rest to Denny. The number of lollipops he has given to Denny must have been $2 0 - 1 2 = 8 $ lollipops. The answer is 8.
+Q: Shawn has five toys. For Christmas, he got two toys each from his mom and dad. How many toys does he have now?
+A: He has 5 toys. He got 2 from mom, so after that he has $5 + 2 = 7$ toys. Then he got 2 more from dad, so in total he has $7 + 2 = 9$ toys. The answer is 9.
+Q: There were nine computers in the server room. Five more computers were installed each day, from monday to thursday. How many computers are now in the server room?
+A: There are 4 days from monday to thursday. 5 computers were added each day. That means in total $4 ^ { * } 5 =$ 20 computers were added. There were 9 computers in the beginning, so now there are $9 + 2 0 = 2 9$ computers. The answer is 29.
+Q: Michael had 58 golf balls. On tuesday, he lost 23 golf balls. On wednesday, he lost 2 more. How many golf balls did he have at the end of wednesday?
+A: Michael initially had 58 balls. He lost 23 on Tuesday, so after that he has $5 8 - 2 3 = 3 5$ balls. On Wednesday he lost 2 more so now he has $3 5 - 2 = 3 3$ balls. The answer is 33.
+Q: Olivia has $\$ 23$ . She bought five bagels for $\$ 3$ each. How much money does she have left?
+A: She bought 5 bagels for $\$ 3$ each. This means she spent $5 * \ S 3 = \ S 1 5$ on the bagels. She had $\$ 23$ in beginning, so now she has $\$ 23-\$ 15=58$ . The answer is 8.
+Premise:
+"Conceptually cream skimming has two basic dimensions - product and geography."
+Based on this premise, can we conclude the hypothesis "Product and geography are what make cream skimming work." is true?
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: Based on "cream skimming has two basic dimensions" we can’t infer that these two dimensions are what make cream skimming work. The answer is it is not possible to tell.
+Premise:
+"One of our member will carry out your instructions minutely."
+Based on this premise, can we conclude the hypothesis "A member of my team will execute your orders with
+immense precision." is true?
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: "one of" means the same as "a member of", "carry out" means the same as "execute", and "minutely" means the same as "immense precision". The answer is yes.
+Premise:
+"Fun for adults and children."
+Based on this premise, can we conclude the hypothesis "Fun for only children." is true?
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: "adults and children" contradicts "only children". The answer is no.
+Premise:
+"He turned and smiled at Vrenna."
+Based on this premise, can we conclude the hypothesis "He smiled at Vrenna who was walking slowly behind
+him with her mother." is true?
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: the premise does not say anything about "Vrenna was walking". The answer is it is not possible to tell.
+Premise:
+"well you see that on television also"
+Based on this premise, can we conclude the hypothesis "You can see that on television, as well." is true?
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: "also" and "as well" mean the same thing. The answer is yes.
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: "Vrenna and I both" contradicts "neither Vrenna nor myself". The answer is no.
+Premise:
+"A person on a horse jumps over a broken down airplane."
+Based on this premise, can we conclude the hypothesis "A person is training his horse for a competition." is true? OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: The person is not necessarily training his horse. The answer is it is not possible to tell. Premise:
+"A person on a horse jumps over a broken down airplane."
+Based on this premise, can we conclude the hypothesis "A person is at a diner, ordering an omelette." is true? OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: One jumping horse cannot be in a diner ordering food. The answer is no. Premise:
+"A person on a horse jumps over a broken down airplane."
+Based on this premise, can we conclude the hypothesis "A person is outdoors, on a horse." is true? OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: A broken down airplane is outdoors. The answer is yes. Premise:
+"Children smiling and waving at camera."
+Based on this premise, can we conclude the hypothesis "They are smiling at their parents." is true?
+OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: Just because they are smiling and waving at a camera does not imply their parents or anyone is anyone behind it. The answer is it is not possible to tell.
+Premise:
+"Children smiling and waving at camera."
+Based on this premise, can we conclude the hypothesis "The kids are frowning." is true? OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A: One cannot be smiling and frowning at the same time. The answer is no.
+Premise:
+"Children smiling and waving at camera."
+Based on this premise, can we conclude the hypothesis "There are children present." is true? OPTIONS:
+- yes
+- no
+- it is not possible to tell
+A:The children must be present to see them smiling and waving. The answer is yes.
+Premise:
+"No Weapons of Mass Destruction Found in Iraq Yet."
+Based on this premise, can we conclude the hypothesis "Weapons of Mass Destruction Found in Iraq." is true?
+A: "No Weapons of Mass Destruction Found" contradicts "Weapons of Mass Destruction Found". The answer is no.
+Premise:
+"A place of sorrow, after Pope John Paul II died, became a place of celebration, as Roman Catholic faithful gathered in downtown Chicago to mark the installation of new Pope Benedict XVI."
+Based on this premise, can we conclude the hypothesis "Pope Benedict XVI is the new leader of the Roman Catholic Church." is true?’
+A: "installation of new Pope Benedict XVI." means "Pope Benedict XVI is the new leader". The answer is yes.
+Premise:
+"A man is due in court later charged with the murder 26 years ago of a teenager whose case was the first to be featured on BBC One’s Crimewatch. Colette Aram, 16, was walking to her boyfriend’s house in Keyworth, Nottinghamshire, on 30 October 1983 when she disappeared. Her body was later found in a field close to her home. Paul Stewart Hutchinson, 50, has been charged with murder and is due before Nottingham magistrates later."
+Based on this premise, can we conclude the hypothesis "Paul Stewart Hutchinson is accused of having stabbed a girl." is true?
+A: The premise does not say Paul Stewart Hutchinson "stabbed" this girl. The answer is no.
+Premise:
+"Herceptin was already approved to treat the sickest breast cancer patients, and the company said, Monday, it will discuss with federal regulators the possibility of prescribing the drug for more breast cancer patients." Based on this premise, can we conclude the hypothesis "Herceptin can be used to treat breast cancer." is true?
+A: "Herceptin was approved to treat breast cancer" implies that "Herceptin can be used to treat breast cancer".
+The answer is yes.
+Q: does system of a down have 2 singers?
+A: System of a Down currently consists of Serj Tankian, Daron Malakian, Shavo Odadjian and John Dolmayan.
+Serj and Daron do vocals, so the band does have two singers. The answer is yes.
+Q: do iran and afghanistan speak the same language?
+A: Iran and Afghanistan both speak the Indo-European language Persian. The answer is yes.
+Q: is a cello and a bass the same thing?
+A: The cello is played sitting down with the instrument between the knees, whereas the double bass is played standing or sitting on a stool. The answer is no.
+Q: can you use oyster card at epsom station?
+A: Epsom railway station serves the town of Epsom in Surrey and is not in the London Oyster card zone. The answer is no.

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+# Langevin Quasi-Monte Carlo
+Sifan Liu Department of Statistics Stanford University Stanford, CA 94305 sfliu@stanford.edu
+# Abstract
+Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful algorithms for sampling from complex high-dimensional distributions. To sample from a distribution with density $\pi \overset { \cdot } { ( } \theta ) \propto \overset { \cdot } { \exp ( - U ( \theta ) ) }$ , LMC iteratively generates the next sample by taking a step in the gradient direction $\nabla U$ with added Gaussian perturbations. Expectations w.r.t. the target distribution $\pi$ are estimated by averaging over LMC samples. In ordinary Monte Carlo, it is well known that the estimation error can be substantially reduced by replacing independent random samples by quasi-random samples like low-discrepancy sequences. In this work, we show that the estimation error of LMC can also be reduced by using quasirandom samples. Specifically, we propose to use completely uniformly distributed (CUD) sequences with certain low-discrepancy property to generate the Gaussian perturbations. Under smoothness and convexity conditions, we prove that LMC with a low-discrepancy CUD sequence achieves smaller error than standard LMC. The theoretical analysis is supported by compelling numerical experiments, which demonstrate the effectiveness of our approach.
+# 1 Introduction
+Sampling from probability distributions is a crucial task in both statistics and machine learning. However, when the target distribution does not permit exact sampling, researchers often rely on Markov chain Monte Carlo (MCMC) methods. These techniques simulate a Markov chain that converges to the target distribution as its stationary distribution. Recently, MCMC samplers based on discretizing the continuous-time Langevin diffusion have become popular, due to its ease of implementation and ability to handle stochastic gradients (Welling and Teh, 2011).
+The primary focus of this work is on the quality of samples generated by Langevin Monte Carlo (LMC) algorithms in terms of estimating the expectation $\mathbb { E } _ { \theta \sim \pi } \left[ f ( \theta ) \right]$ for some integrand $f$ by sample averages. In the context of Bayesian inference, the target distribution $\pi$ is typically the posterior distribution, and computing the posterior expectation, posterior variance, or confidence intervals are of great interest. In the context of post-selection inference, the target distribution $\pi$ is the probability distribution conditioned on the selection event, and computing the selection-adjusted p-value is the main task. LMC has been widely used in this problem as well (Markovic and Taylor, 2016; Shi et al., 2022). In all these situations, the accuracy of the sample average estimator is critical and affects the downstream data analysis.
+In traditional Monte Carlo sampling, it is well known that using quasi-Monte Carlo (QMC) samples, instead of independent and identically distributed (i.i.d.) random samples, can lead to significant error reduction. So it is natural to ask whether we can apply QMC techniques to improve Langevin Monte Carlo sampling as well. In this work, we introduce the Langevin quasi-Monte Carlo (LQMC) algorithm, which replaces the i.i.d. random inputs in the LMC algorithm with quasi-random numbers.
+![](images/bea1dbc0489080cb7836fa89b3f143e12237f65da9d81a18d220019f9c986b24.jpg)
+Figure 1: Scatter plots of 251 points generated from Mersenne Twister 19937 (left) and 251 points generated from a linear congruential generator (LCG) of period 251. Points from an entire period of a pseudo-random number generator (right) fill the unit square more evenly than the same number of points from a PRNG with a larger period (left).
+These quasi-random numbers are carefully designed to sample from the target distribution more evenly and more balanced, leading to improved estimation accuracy.
+Not all quasi-Monte Carlo point sets are suitable for simulating Markov chains. Suppose the Markov chain is driven by a sequence of uniform random vectors in the unit cube. A sufficient condition for the sequence is known as completely uniformly distributed (CUD). In our implementation of the driving sequence, we use an entire period of a pseudo-random number generator (PRNG). While modern computer simulations often use PRNGs with a large period, such as Mersenne Twister with a period of $2 ^ { 1 9 9 3 7 } - 1$ , our approach runs through the entire period of a PRNG with a relatively small period in the LMC algorithm. The advantage of using an entire period of a PRNG is that the points are more evenly distributed, which is more desirable for numerical integration. We illustrate the balancing property of an entire PRNG in Figure 1.
+The main contributions of this paper are threefold. First, we propose a novel technique of using quasi-random numbers in Langevin-type algorithms, which can be applied to a wide range of such algorithms by substituting i.i.d. random numbers with a sequence of quasi-random numbers. The quasi-random numbers are constructed similarly as usual PRNGs, therefore no extra computational complexity is required. Second, we evaluate the performance of the proposed LQMC algorithm in a variety of numerical experiments, demonstrating that it can significantly reduce the mean squared error (MSE) of traditional LMC by a factor ranging from 2 to 500, depending on the problem. Finally, we provide theoretical analysis showing that LQMC can reduce the Monte Carlo part of the error from $O ( n ^ { - 1 / 2 } )$ to $O ( n ^ { - 1 + \delta } )$ for any $\delta > 0$ in situations where the Markov chain is strongly contracting and the integrand function $f$ is sufficiently regular. This error reduction is consistent with the usual improvement achieved by using quasi-Monte Carlo in place of plain Monte Carlo.
+The rest of the paper is organized as follows. In Section 2, we provide some background on LMC and QMC, followed by a review of related work. Section 3 describes the LQMC algorithm and its implementation details. In Section 4, we present theoretical guarantees for the proposed method. Finally, in Section 5, we provide empirical results to evaluate the performance of LQMC and compare it with the standard LMC algorithm.
+# 2 Backgrounds
+This section provides some background on Langevin Monte Carlo and quasi-Monte Carlo.
+# 2.1 Langevin Monte Carlo
+Suppose we want to sample from the target distribution $\pi ( \theta ) \propto \exp ( - U ( \theta ) )$ where $\theta \in \mathbb { R } ^ { d }$ and $U$ is known as the potential function. LMC algorithms are based on Euler-Maruyama discretization of the Langevin diffusion $\theta ( t )$ , which satisfies the stochastic differential equation
+$$
+\mathrm { d } \theta ( t ) = - \nabla U ( \theta ( t ) ) \mathrm { d } t + \sqrt { 2 } \mathrm { d } W _ { t } ,
+$$
+where $\{ W _ { t } \} _ { t \ge 0 }$ is a $d$ -dimensional standard Brownian motion. Under mild technical conditions, the Langevin diffusion $\theta ( t )$ has $\pi$ as its unique invariant distribution (Roberts and Tweedie, 1996). With a discretization step size $h$ , LMC updates the sample $\theta _ { k }$ by
+$$
+\theta _ { k + 1 }  \theta _ { k } - h \nabla U ( \theta _ { k } ) + \sqrt { 2 h } \xi _ { k + 1 }
+$$
+where $\xi _ { k } \overset { i i d } { \sim } \mathcal { N } ( 0 , I _ { d } )$
+In many applications, we are interested in computing the expectation $\mu : = \mathbb { E } _ { \theta \sim \pi } \left[ f ( \theta ) \right]$ over $\pi$ for some $\pi$ -integrable function $f$ . The LMC estimator of $\mu$ is the sample average
+$$
+{ \hat { \mu } } _ { n } = { \frac { 1 } { n } } \sum _ { k = 1 } ^ { n } f ( \theta _ { k } ) ,
+$$
+where $n$ is the number of iterations.
+Teh et al. (2016) provide an asymptotic bias-variance decomposition of the MSE of the weighted average $\frac { \sum _ { k = 1 } ^ { n } h _ { k } f ( \theta _ { k } ) } { \sum _ { k = 1 } ^ { n } h _ { k } }$ and show that the optimal step size scales as $h _ { k } \asymp k ^ { - 1 / 3 }$ , leading to an MSE of order $O ( n ^ { - 2 / 3 } )$ . Here $h _ { k }$ is the step size used at the $k$ -th iteration. Vollmer et al. (2016) generalize this result to the non-asymptotic setting with a constant step size $h$ . They show that the MSE is of order $\begin{array} { r } { O ( h ^ { 2 } + \frac { 1 } { n h } ) } \end{array}$ , where $\bar { h } ^ { 2 }$ corresponds to the squared bias and $\scriptstyle { \frac { 1 } { n h } }$ corresponds to the variance.
+# 2.2 Quasi-Monte Carlo
+QMC is an alternative to Monte Carlo for numerical integration and is well-known for having much higher accuracy than Monte Carlo. QMC is primarily designed to numerically evaluate the integral $\begin{array} { r } { \mu = \int _ { [ 0 , 1 ] ^ { d } } f ( \dot { \mathbf u } ) \mathrm d \mathbf u } \end{array}$ . It estimates $\mu$ by taking points $\bar { \mathbf { u } } _ { i } \in [ \bar { 0 } , 1 ] ^ { d }$ and let the estimator be
+$$
+{ \hat { \mu } } = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } f ( \mathbf { u } _ { i } ) .
+$$
+Unlike Monte Carlo which takes $\mathbf { u } _ { i }$ to be identically independently distributed (i.i.d.), QMC constructs the point set $\{ { \mathbf { u } } _ { i } \} _ { i = 1 } ^ { n }$ that aims to minimize the star discrepancy
+$$
+D _ { n } ^ { * } = D _ { n } ^ { * } ( \mathbf { u } _ { 1 } , \ldots , \mathbf { u } _ { n } ) = \operatorname* { s u p } _ { \mathbf { a } \in [ 0 , 1 ] ^ { d } } \bigg \lvert \frac { 1 } { n } \sum _ { i = 1 } ^ { n } 1 \{ \mathbf { u } _ { i } \in [ \mathbf { 0 } , \mathbf { a } ) \} - \prod _ { j = 1 } ^ { d } a _ { j } \bigg \rvert .
+$$
+The star discrepancy measures the uniformity of the point sets by comparing the fraction of points inside $[ \mathbf { 0 } , \mathbf { a } )$ and the volume $\textstyle \prod _ { j = 1 } ^ { d } a _ { j }$ , taking supreme over all the rectangles inside $[ 0 , 1 ] ^ { d }$ anchored at 0. QMC can generate points with $D _ { n } ^ { * } = O ( n ^ { - 1 } ( \log n ) ^ { d - 1 } )$ , thus QMC is also known as low-discrepancy sequence. Commonly used QMC points include Sobol’ sequence (Sobol’, 1967), Niederreiter’s sequence (Niederreiter, 1987), Halton’s sequence, and lattice rules. For a comprehensive survey, we refer to the monograph Dick and Pillichshammer (2010). If the integrand $f$ has bounded variation in the sense of Hardy and Krause $\| f \| _ { \mathrm { H K } }$ , then the Koksma-Hlawka inequality (see e.g. Dick and Pillichshammer (2010)) bounds the integration error by
+$$
+| \hat { \mu } - \mu | \leq D _ { n } ^ { * } \cdot \| f \| _ { \mathrm { H K } } \leq O ( n ^ { - 1 } ( \log n ) ^ { d - 1 } ) .
+$$
+While the Koksma-Hlawka inequality shows that QMC is asymptotically better than usual Monte Carlo, it doesn’t provide a practical way to estimate the error. Moreover, integrands might have infinite Hardy-Krause variation.
+One can apply randomization techniques to QMC to address both problems. Common randomization techniques include random shifts (Cranley and Patterson, 1976) and scrambling (Owen, 1995). For RQMC samples $\mathbf { u } _ { 1 } , \ldots , \mathbf { u } _ { n }$ , each $\mathbf { u } _ { i } \sim \dot { \mathrm { U n i f } } ( [ 0 , 1 ] ^ { d } )$ individually but they have the low-discrepancy property collectively with probability 1. One can estimate the error by multiple independent random replicates. For sufficiently smooth $f$ , the scrambled Sobol’ sequence has variance $O ( \tilde { ( } n ^ { - 3 } ( \log n ) ^ { d - 1 } )$ (Owen, 1997a,b).
+# 2.3 Related work
+The first attempt to apply quasi-random numbers to simulate stochastic differential equations was made by Hofmann and Mathé (1997). They showed that if a numerical scheme is weakly convergent with i.i.d. samples, then using completely uniformly distributed (CUD) sequences also leads to consistent estimation. They also demonstrated that certain low-discrepancy sequences are not suitable for simulating SDEs. There have also been some efforts to apply QMC to MCMC. Owen and Tribble (2005) proposed to apply CUD sequences to a Metropolis algorithm and showed that the method is consistent in problems with finite state spaces. Chen et al. (2011) generalized the consistency result to continuous state spaces under the assumption that the Markov chain is a contraction. More recently, Dick et al. (2016); Dick and Rudolf (2014) proved that there exists constructions of the driving sequence $\{ { \bf u } _ { k } \} _ { k \ge 1 }$ such that the discrepancy between the empirical distribution of MCMC samples and the target distribution is bounded by $O ( n ^ { - 1 / 2 } ( \log n ) ^ { 1 / 2 } )$ , the same rate achieved by random inputs. Another line of applying QMC to Markov chains is known as array-RQMC proposed by L’Ecuyer et al. (2008). Array-RQMC runs in parallel multiple Markov chains, and each iteration involves a complicated reordering of the states so that the low-discrepancy among the chains is maintained. Empirically, it achieves significantly smaller estimation error than usual MCMC, but theoretical guarantees remain a challenging open problem.
+There has been a growing interest in using QMC techniques in various machine learning tasks, such as variational inference (Buchholz et al., 2018; Liu and Owen, 2021), policy learning and evaluation (Arnold et al., 2022), reinforcement learning with evolution strategies (Choromanski et al., 2019; Rowland et al., 2018), compression of large datasets (Dick and Feischl, 2021), example selection in stochastic gradient descent (SGD) (Lu et al., 2021), and deep learning for solving partial differential equations (Longo et al., 2021).
+Numerous efforts have been devoted to improving LMC and stochastic gradient Langevin dynamics (SGLD). To overcome the instability of Euler-Maruyama discretization, various numerical schemes have been proposed, including higher-order integrators (Chen et al., 2015), underdamped LMC (Cheng et al., 2018), and stochastic Runge-Kutta diffusion (Li et al., 2019). For SGLD, variance reduction techniques such as SAGA and SVGR (Dubey et al., 2016) and control variates (Baker et al., 2019) have been proposed. LMC also provides a useful perspective for optimization, as demonstrated by the analyses in Chen et al. (2016); Dalalyan (2017); Raginsky et al. (2017); Xu et al. (2018); Erdogdu et al. (2018). Our contribution is orthogonal to all the aforementioned work, as our algorithm only modifies the random numbers used in the algorithm. Therefore, our method can be combined with other algorithms without interference.
+# 3 QMC for LMC
+In the LMC algorithm, we can think of the Markov chain as being driven by a sequence of uniform variables $\mathbf { u } _ { k }$ in the unit cube $[ 0 , 1 ] ^ { d }$ . For instance, the Gaussian perturbation can be represented as $\xi _ { k } = \Phi ^ { - 1 } ( { \mathbf { u } } _ { k } )$ , where $\Phi ^ { - 1 }$ denotes the inverse Gaussian CDF applied element-wise to $\mathbf { u } _ { k }$ . If a stochastic gradient is employed, the randomness associated with the stochastic gradient can also be expressed as uniform variables. Therefore, we can write the transition of the Markov chain as $\theta _ { k + 1 } \bar { = } \psi ( \theta _ { k } , { \bf u } _ { k + 1 } )$ . In typical computer experiments, $\mathbf { u } _ { k }$ are not really i.i.d. but are deterministic pseudo-random numbers. In this section, we will describe an alternative method of generating the pseudo-random numbers $\mathbf { u } _ { k }$ , which are carefully constructed and can lead to more accurate sample averages.
+The idea here is to use point sets that are more evenly distributed such as QMC points, which can lead to significant improvement in the usual Monte Carlo estimation. However, caution is required when using QMC points to simulate an SDE like (1). This is because the correlation between successive QMC samples may introduce undesired behavior in the Markov chain, as demonstrated in (Tribble, 2007, Section 3.2). To avoid the dependence among successive values, we require that the blocks of points $( v _ { i } , v _ { i + 1 } , \ldots , v _ { i + d - 1 } )$ for any lag $d$ are uniformly distributed. This notion of uniformity is formally known as completely uniformly distributed (CUD, Korobov (1948)), which we define next.
+We say an infinite sequence $\{ \mathbf { u } _ { i } \} _ { i = 1 } ^ { \infty } \subseteq [ 0 , 1 ] ^ { d }$ is uniformly distributed on $[ 0 , 1 ] ^ { d }$ if the star discrepancy $D ^ { * } \big ( \{ { \bf \dot { u } } _ { i } \} _ { i = 1 } ^ { n } \big )$ goes to 0 as $n \to \infty$ , where the star discrepancy is defined in Equation 3.
+Definition 3.1 (Completely uniformly distributed sequence (CUD)). An infinite sequence $\{ v _ { i } \} _ { i = 0 } ^ { \infty } \subset [ 0 , 1 ]$ is called completely uniformly distributed, if for all positive integer $d _ { \mathrm { { z } } }$ the sequence $\{ ( v _ { k } , \ldots , v _ { k + d - 1 } ) \} _ { k = 0 } ^ { \infty } \subseteq \mathbb { R } ^ { d }$ is uniformly distributed on $[ 0 , 1 ] ^ { d }$ . $A$ triangular array $\begin{array} { r c l } { \mathbf { v } _ { n } } & { = } & { \left( v _ { n , 1 } , \ldots , v _ { n , N _ { n } } \right) } \end{array}$ is called array-CUD, if for all positive integer $d _ { \mathrm { { z } } }$ , $D ^ { * } ( ( v _ { n , 1 } , \dots , v _ { n , d } ) , ( v _ { n , 2 } , \dots , v _ { n , d + 1 } ) , \dots , ( v _ { n , N _ { n } - d + 1 } , \dots , v _ { n , N _ { n } } ) ) \to 0$ as $n \to \infty$ , $N _ { n }  \infty$ .
+In other words, the subsequent $d$ -tuples in a CUD sequence are uniformly distributed in the $d .$ - dimensional unit cube for any positive dimension $d$ . Now we are ready to present the main algorithm.
+# 3.1 LQMC algorithm
+Let $\{ v _ { i } \} _ { i = 0 } ^ { \infty }$ be a CUD sequence. Let $\mathbf { u } _ { k } = ( v _ { k d } , \ldots , v _ { ( k + 1 ) d - 1 } ) \in \mathbb { R } ^ { d }$ be the $k$ -th non-overlapping $d$ -tuple from the sequence $k \geq 0 ,$ . A CUD sequence is often constructed deterministically. They can further be randomized using the Cranley-Patterson (i.e. random shift) rotation (Cranley and Patterson, 1976)
+$$
+\mathbf { u } _ { k }  \mathbf { u } _ { k } + \Delta \mod 1 ,
+$$
+where $\Delta \sim \mathrm { U n i f } ( [ 0 , 1 ] ^ { d } )$ . The Cranley-Patterson rotation randomly shifts each dimension of $\mathbf { u } _ { k }$ by a uniform random number separately. Then each $\mathbf { u } _ { k }$ is uniformly distributed on $[ 0 , 1 ] ^ { d }$ . If we apply the inverse Gaussian CDF to each coordinate of $\mathbf { u } _ { k }$ , then $\Phi ^ { - 1 } ( \bar { \mathbf { u } } _ { k } ) \sim \mathcal { N } ( 0 , I _ { d } )$ . In the Langevin-type algorithms, we will let ${ \xi _ { k } = \Phi ^ { - 1 } ( { \mathbf { u } } _ { k } ) }$ and use $\xi _ { k }$ as the Gaussian perturbation in the $k$ -th iteration. Specifically, each iteration takes the form
+$$
+\theta _ { k + 1 } = \theta _ { k } - h \nabla U ( \theta _ { k } ) + \sqrt { 2 h } \cdot \Phi ^ { - 1 } ( \mathbf { u } _ { k + 1 } ) , \quad k \geq 0 .
+$$
+Thus the transition map is $\psi ( \theta , { \mathbf { u } } ) = \theta - h \nabla U ( \theta ) + \sqrt { 2 h } \Phi ^ { - 1 } ( { \mathbf { u } } )$ . In practice, we can only run finite many iterations. In the following, we will describe how to construct a finite CUD sequence and feed it into the LMC algorithm.
+# 3.2 Construction of CUD sequences
+A finite CUD (array-CUD) sequence is often implemented by using an entire period of a pseudo random number generator with a small period (Tribble, 2007). There exist other constructions of CUD sequences. For further details, interested readers can refer to Levin (1999). We propose to use the linear-feedback shift register (LFSR) provided in Chen (2011), because it has demonstrated good performance and the computational effort required is comparable to other commonly used PRNGs.
+The binary Galois LFSR (Tausworthe generator, Tausworthe (1965)) of order $m$ updates the states $b _ { i } \in \{ 0 , 1 \}$ recursively by
+$$
+b _ { i } = \sum _ { j = 0 } ^ { m - 1 } a _ { j } b _ { i - m + j } \mod 2 , \quad i \geq m
+$$
+with initial states $b _ { 0 } , b _ { 1 } , \dotsc , b _ { m - 1 }$ pre-specified. The $m$ -tuple $( b _ { i } , b _ { i + 1 } , \ldots , b _ { i + m - 1 } ) \in { \bf G F } ( 2 ) ^ { m }$ can only take $2 ^ { m }$ different values. If there is an $m$ -tuple that is all zero, then all $b _ { i }$ ’s in this sequence must be zero. So the period of the sequence $\{ b _ { i } \} _ { i \ge 0 }$ is at most $n = 2 ^ { m } - 1$ . Moreover, the period is exactly equal to $2 ^ { m } - 1$ if and only if the characteristic polynomial
+$$
+x ^ { m } + a _ { m - 1 } x ^ { m - 1 } + \ldots + a _ { 1 } x + a _ { 0 }
+$$
+is a primitive polynomial over $\mathrm { G P } ( 2 )$ (Niederreiter, 1992, Lemma 9.1). Given the states $\{ b _ { i } \} _ { i \ge 0 }$ and an offset $s > 0$ such that $\operatorname* { g c d } ( s , 2 ^ { \acute { m } } - 1 ) = 1$ , $v _ { i }$ is computed with
+$$
+v _ { i } = \sum _ { j = 0 } ^ { m - 1 } b _ { s i + j } 2 ^ { - j - 1 } , \quad i = 0 , 1 , \ldots , 2 ^ { m } - 2 .
+$$
+That is, for each $i$ , we take the $m$ -tuple $( b _ { s i + j } ) _ { 0 \leq j < m }$ and interpret it as the binary expansion of $v _ { i }$ For the next step, we jump $s$ bits ahead in the sequence $\{ b _ { i } \} _ { i \ge 0 }$ and use the $m$ -tuple starting from $b _ { s ( i + 1 ) }$ . Chen (2011) provided a table of the LFSR generators for $1 0 \leq m \leq 3 2$ . They searched the offsets so that the LFSR has good equi-distributed properties. Our experiments use the LFSR generators listed there.
+Given the sequence $\{ v _ { i } \} _ { i = 0 } ^ { n - 1 }$ of length $n$ , we repeat it $d$ times and arrange $v _ { i }$ ’s in the following $n \times d$ matrix
+$$
+\left( \begin{array} { c c c c } { { v _ { 0 } } } & { { v _ { 1 } } } & { { \cdot \cdot \cdot } } & { { v _ { d - 1 } } } \\ { { v _ { d } } } & { { v _ { d + 1 } } } & { { \cdot \cdot \cdot } } & { { v _ { 2 d - 1 } } } \\ { { \vdots } } & { { \vdots } } & { { \ddots } } & { { \vdots } } \\ { { v _ { ( n - 1 ) d } } } & { { v _ { ( n - 1 ) d + 1 } } } & { { \cdot \cdot \cdot } } & { { v _ { n d - 1 } } } \end{array} \right) .
+$$
+We run the LMC algorithm $n = 2 ^ { m } - 1$ iterations. The $k$ -th uniform vector $\mathbf { u } _ { k }$ is the $k$ -th row of the above matrix. The procedure is summarized in Algorithm 1.
+# Algorithm 1 Langevin quasi-Monte Carlo (LQMC)
+Input: Number of iterations $n = 2 ^ { m } - 1$ such that $\operatorname* { g c d } ( 2 ^ { m } - 1 , d ) = 1$ , step size $h$ , initial value $\theta _ { 0 }$ Generate an LFSR sequence $\{ v _ { i } \} _ { i \ge 0 }$ of period $2 ^ { m } - 1$ . Let $\mathbf u _ { k } = ( v _ { ( k - 1 ) d } , \dots , v _ { k d - 1 } ) \in [ 0 , 1 ] ^ { d }$ , for $1 \leq k \leq n$ . Apply Cranley-Patterson rotation (random shift) to $\mathbf { u } _ { k }$ ’s. for $k  1 , \ldots , n$ do $\theta _ { k } \gets \theta _ { k - 1 } - h \nabla U ( \theta _ { k - 1 } ) + \sqrt { 2 h } \Phi ^ { - 1 } ( \mathbf { u } _ { k } )$ end for
+Output: $\theta _ { 1 } , \ldots , \theta _ { n }$
+If $\operatorname* { g c d } ( n , d ) = 1$ , then each column of the matrix (5) contains no repeated values. This means that among the $n = 2 ^ { m } - 1$ iterations of the LQMC algorithm, each dimension uses one value in each sub-interval ( k2m , k+12m ] at most once (0 ≤ k ≤ 2m − 1). This perfect one-dimensional stratification is one of the reasons why CUD may achieve smaller estimation error than pseudo-random numbers. If $\operatorname* { g c d } ( n , d ) > 1$ , then we take $d ^ { \prime }$ to be the smallest integer greater than $d$ and co-prime with $n$ . We then create the matrix in (5) similarly but with $d ^ { \prime }$ columns. In the LQMC algorithm, we take $\mathbf { u } _ { k }$ to be the $k$ -th row of the matrix but only use the first $d$ coordinates.
+Algorithm 1 may seem to be restricted by having a fixed number of iterations, $n = 2 ^ { m } - 1$ . However, in practice, the LQMC algorithm can be started with an initial value of $m$ . If the chain does not converge after $2 ^ { m } - 1$ iterations, one can continue the chain with another freshly generated LFSR, possibly with a larger period. This allows for flexibility in adjusting the number of iterations based on the convergence of the chain. Additionally, if a burn-in period is required, one can first run the algorithm with an LFSR of a small period to serve as the burn-in stage and then continue with a larger LFSR. Furthermore, running multiple chains with independent random shifts is embarrassingly parallel. We present the algorithm in the form of the basic LMC algorithm with accurate gradient and constant learning rate. However, as we noted previously, other Langevin-type algorithms can also utilize the CUD sequence directly by substituting the pseudo-random numbers with the LFSR sequence.
+# 4 Theoretical guarantee
+Here we study the estimation error $\textstyle | n ^ { - 1 } \sum _ { k = 1 } ^ { n } f ( \theta _ { k } ) - \pi ( f ) |$ of LQMC for some test function $f$ that is 1-Lipschitz and bounded. As the first attempt to prove the convergence rate of using QMC in LMC, we impose the relatively strong conditions of smoothness and convexity.
+Assumption 1. The potential function $U$ is $L$ -smooth
+$$
+\| \nabla U ( \theta ) - \nabla U ( \theta ^ { \prime } ) \| _ { 2 } \le L \| \theta - \theta ^ { \prime } \| _ { 2 } , \quad \forall \theta , \theta ^ { \prime } ,
+$$
+and $M$ -strongly convex
+$$
+U ( \theta ^ { \prime } ) \geq U ( \theta ) + \nabla U ( \theta ) ^ { \top } ( \theta ^ { \prime } - \theta ) + \frac { M } { 2 } \| \theta ^ { \prime } - \theta \| _ { 2 } ^ { 2 } , \quad \forall \theta , \theta ^ { \prime } .
+$$
+We will also assume a constant step size $h$ . While LMC with vanishing step sizes converges weakly to the target distribution, in practice a constant step size is often used (Vollmer et al., 2016; Brosse et al., 2018). With a constant step size, we can derive a non-asymptotic error bound for LQMC.
+Assumption 1 implies that if the step size $\begin{array} { r } { h \le \frac { 2 } { L + M } } \end{array}$ , then the transition map $\psi$ is a strong contraction with parameter $\rho = 1 - h M$ , i.e.
+$$
+\| \psi ( \boldsymbol { \theta } , { \mathbf u } ) - \psi ( \boldsymbol { \theta } ^ { \prime } , { \mathbf u } ) \| _ { 2 } = \| \boldsymbol { \theta } - \boldsymbol { \theta } ^ { \prime } - h ( \nabla U ( \boldsymbol { \theta } ) - \nabla U ( \boldsymbol { \theta } ^ { \prime } ) ) \| _ { 2 } \le \rho \| \boldsymbol { \theta } - \boldsymbol { \theta } ^ { \prime } \| _ { 2 } .
+$$
+See e.g. Lemma 2 of Dalalyan and Karagulyan (2019). The strong contraction implies that if we start two chains from $\theta$ and $\theta ^ { \prime }$ , and use the same random numbers at every step, then the two chains will merge exponentially fast. In other words, the state $\theta _ { k }$ largely depends on the most recent iterations and quickly forgets about the past history. Formally, let $\mathbf w _ { k } ^ { ( \ell ) } = ( \bar { \mathbf u } _ { k } , \dots , \mathbf u _ { k - \ell + 1 } )$ denote the random numbers used in the most recent $\ell$ steps. Define the $\ell \cdot$ -step transition as
+$$
+\theta _ { k } = \psi _ { \ell } ( \theta _ { k - \ell } , \mathbf { w } _ { k } ^ { ( \ell ) } )
+$$
+and let $\bar { f } _ { \ell } ( \mathbf w _ { k } ^ { ( \ell ) } )$ denote the value of $f ( \theta _ { k } )$ marginalized over $\theta _ { k - \ell } \sim \pi$ , i.e.
+$$
+\bar { f } _ { \ell } ( \mathbf { w } _ { k } ^ { ( \ell ) } ) = \int f \circ \psi _ { \ell } ( x , \mathbf { w } _ { k } ^ { ( \ell ) } ) \pi ( \mathrm { d } x ) .
+$$
+Thus $\bar { f } _ { \ell } ( \mathbf { w } _ { k } ^ { ( \ell ) } )$ only depends on the most recent $\ell$ iterations. Due to the strong contraction, $\big | \bar { f } _ { \ell } \big ( \mathbf { w } _ { k } ^ { ( \ell ) } \big ) -$ $f ( \theta _ { k } ) |$ decays exponentially fast with $\ell$ . So for large $\ell$ , the estimation error of $\scriptstyle n ^ { - 1 } \sum _ { k = 1 } ^ { n } f ( \theta _ { k } )$ is close to the error of $\begin{array} { r } { \frac { 1 } { n - \ell } \sum _ { k = \ell + 1 } ^ { n } \bar { f } _ { \ell } ( \mathbf { w } _ { k } ^ { ( \ell ) } ) } \end{array}$ . The latter can be viewed as a $d { \boldsymbol { \ell } }$ -dimensional numerical integration scheme based on the point set $\{ \mathbf { w } _ { k } ^ { ( \ell ) } \} _ { k = \ell + 1 } ^ { n }$ . By leveraging the discrepancy bound of the LFSR sequence and assuming that $\bar { f } _ { \ell }$ has bounded variation in the sense of Hardy and Krause, we can derive an error bound using the Koksma-Hlawka inequality (4). Now we state the main error bound and leave the detailed proof in the Appendix A.
+Theorem 4.1 (Error bound of LQMC). Let Assumption 1 hold. Define the step size h ≤ 2L+M , $\rho = 1 - h M$ , $\ell = \lceil ( 1 / 2 ) \log _ { \rho } h \rceil$ . Let $\theta _ { 1 } , \ldots , \theta _ { n }$ be the output of Algorithm 1 which runs $n$ iterations with step size $\begin{array} { r } { h \le \frac { 2 } { L + M } } \end{array}$ . Assume the LFSR sequence $\{ v _ { i } \} _ { i \ge 0 }$ in use has period $n = 2 ^ { m } - 1$ , offset $s$ , and $g c d ( m , n ) = \operatorname* { g c d } ( d \ell , n ) = 1$ . If $\bar { f } _ { \ell }$ has bounded variation in the sense of Hardy and Krause, then as $n \to \infty$ we have
+$$
+\left| \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f ( \theta _ { k } ) - \pi ( f ) \right| \le C _ { 1 } n ^ { - 1 + \delta } + C _ { 2 } h ^ { 1 / 2 } , \quad \forall \delta > 0 .
+$$
+Here $\delta$ hides poly-logarithmic factors $( \log n ) ^ { d }$ , $C _ { 1 }$ depends on $d , \ell$ and $\| \bar { f } _ { \ell } \| _ { H K }$ , and $\begin{array} { r } { C _ { 2 } = \frac { 3 \sqrt { 2 } } { 2 } \frac { L } { M } d + } \end{array}$ $\operatorname* { m a x } _ { 0 \leq k \leq n } \| \theta _ { k } \| + \mathbb { E } _ { \pi } \left[ \| \theta \| \right]$
+The upper bound consists of two terms. The first term represents the numerical integration error, which arises from the discrepancy of the point set used in the integration scheme. By utilizing low-discrepancy CUD sequences, we can reduce this numerical integration error (the first term) from the standard rate of $O ( n ^ { - 1 / 2 } )$ to a faster rate of $O ( n ^ { - 1 + \delta } )$ for any $\delta > 0$ . However, it is important to note that when using a constant step size $h$ in LMC, the bias term (second term) does not vanish. This bias term includes not only the discretization error of the Langevin diffusion, but also the difference between $f ( \theta _ { k } )$ and its truncated version $\bar { f } _ { \ell } ( \mathbf w _ { k } ^ { ( \ell ) } )$ . Consequently, the bias term in our analysis is larger than the bias term in Vollmer et al. (2016), which employs different techniques and assumptions based on the Poisson equation.
+The theorem’s assumption of finite Hardy-Krause variation is a common requirement in error bounds for QMC methods, and it can be challenging to verify in practice. Basu and Owen (2016) provide sufficient conditions in order for $f \circ \psi _ { \ell }$ to have finite HK variation, requiring the $\ell \cdot$ -step transition $\psi _ { \ell }$ to be sufficiently smooth. In the next section, we aim to assess the practical performance of the proposed LQMC algorithm through numerical experiments.
+# 5 Numerical experiments
+To comprehensively evaluate the performance of the algorithm, we will consider both convex and non-convex potentials, both low-dimensional and high-dimensional state spaces, both accurate and stochastic gradients, both smooth and discontinuous integrands, as well as different learning rate schedules. Additional numerical results can be found in the Appendix B.
+![](images/711d95f81584702ce987bb3b555e889589fcac7e1ae0f7fa9aa7d395db72c8fa.jpg)
+Figure 2: Bayesian logistic regression with accurate gradients (top) and stochastic gradients (bottom).
+# 5.1 Bayesian logistic regression
+We first consider the Bayesian logistic model
+$$
+\begin{array} { r } { y _ { i } \mid x _ { i } \sim \mathrm { B e r n o u l l i } ( ( 1 + \exp ( - x _ { i } ^ { \mathsf { T } } \beta ) ) ^ { - 1 } ) , \quad 1 \le i \le N , } \\ { \beta \sim \mathcal { N } ( 0 , I _ { d } ) . \qquad } \end{array}
+$$
+We take $N = 2 0$ , $d = 1 0$ . The features $x _ { i }$ are generated from $\mathcal { N } ( 0 , \Sigma )$ with $\Sigma _ { i j } = 2 ^ { - | i - j | }$ . The coefficients $\beta$ and the data $y _ { i }$ ’s are generated from the same model. We consider the test functions $f ( x ) = x _ { j } , x _ { j } ^ { 2 } , \mathbf { 1 } _ { \{ x _ { j } > 0 \} }$ for $j = 1 , \ldots , d$ . The step size $h$ is fixed to 0.001.
+We compute the MSE of the estimator based on usual LMC and the proposed LQMC with CUD sequences and report the MSE averaged over all coordinates and 20 random replicates. We do not have a closed form for the expectations $\mathbb { E } \left[ f \right]$ , so the ground truth is estimated using a high-accuracy estimator proposed in He et al. (2023) using scrambled Sobol’ sequence with a very large sample size.
+In Figure 2 (top panel), we present a log-log plot of the MSE against the number of iterations. Across all three test functions, we observe that LQMC reduces the MSE by a factor ranging from 4 to 8. As the number of iterations increases, the curve corresponding to LQMC reaches a plateau. This behavior can be attributed to the discretization error inherent in the unadjusted LMC, which cannot be further reduced by increasing the number of iterations.
+In the bottom panel of Figure 2, we increase the number of observations to $N = 1 0 0$ and incorporate stochastic gradient estimation in the Langevin algorithm. Specifically, at each iteration, we estimate the gradient using a random subset of 10 observations. The results demonstrate that LQMC still provides a big improvement when $n$ is smaller than $2 ^ { 1 4 }$ . However, as $n$ surpasses $2 ^ { 1 4 }$ , we observe that the LQMC curve flattens again. It is worth noting that the improvement achieved by LQMC in this scenario is less pronounced compared to the previous example, primarily due to the presence of noise in the gradient estimates.
+# 5.2 Bayesian linear regression
+Now we try a higher-dimensional example with Bayesian linear regression. The model is defined as
+$$
+\begin{array} { r l } & { \boldsymbol { y } _ { i } \sim \mathcal { N } ( x _ { i } ^ { \intercal } \beta , \sigma ^ { 2 } = 4 ^ { - 1 } ) , \quad 1 \leq i \leq N , } \\ & { \beta \sim \mathcal { N } ( 0 , I ) . } \end{array}
+$$
+We take $d = 1 0 0$ and $N = 2 0$ . We generate $x _ { i } \in \mathbb { R } ^ { d }$ similarly as in the logistic regression example. The test functions and step size are also unchanged. The posterior distribution of $\beta$ has the closed form $\begin{array} { r } { \mathcal { N } \left( ( \frac { X ^ { \top } X } { \sigma ^ { 2 } } + I ) ^ { - 1 } \frac { X ^ { \widehat { \mathsf { T } } } Y } { \sigma ^ { 2 } } , ( \frac { X ^ { \top } X } { \sigma ^ { 2 } } + I ) ^ { - 1 } \right) } \end{array}$ . The results are shown in Figure 3. We see that even at 100 dimension, LQMC still brings a substantial improvement over LMC in terms of MSE. In particular, for the integrand $f ( x ) = x _ { j }$ , LQMC achieves a reduction in MSE of approximately 500-fold compared to LMC.
+![](images/bbfcb350aef010496d700422f157e1fb65400bdfea5ca9ef459b33cbb7646348.jpg)
+Figure 3: Bayesian linear regression in 100 dimensions.
+![](images/8d6b00ad90b084168d4d46cfd574ad385bcee892df89ec67464f8e036835017e.jpg)
+Figure 4: Crossed random effect.
+# 5.3 A hierarchical Bayesian model
+We consider a hierarchical Bayesian model known as the crossed random effect model
+$$
+\begin{array} { r l } & { Y _ { i j } \sim { \mathcal N } ( \mu + a _ { i } + b _ { j } , 1 ) , \quad 1 \leq i \leq I , \ 1 \leq j \leq J , } \\ & { \mu \sim { \mathcal N } ( 0 , 1 ) , \ a _ { i } \stackrel { i i d } { \sim } { \mathcal N } ( 0 , \sigma _ { a } ^ { 2 } ) , \ b _ { j } \stackrel { i i d } { \sim } { \mathcal N } ( 0 , \sigma _ { b } ^ { 2 } ) , } \\ & { \log ( \sigma _ { a } ^ { 2 } ) , \ \log ( \sigma _ { b } ^ { 2 } ) \stackrel { i i d } { \sim } { \mathcal N } ( 0 , 1 ) . } \end{array}
+$$
+The goal is to sample from the posterior distribution of $( \mu , \mathbf { a } , \mathbf { b } , \log ( \sigma _ { a } ^ { 2 } ) , \log ( \sigma _ { b } ^ { 2 } ) )$ , which has dimension $d = I + J + 3$ . We take $I = 3$ , $J = 5$ . We will consider the test functions $f ( x ) = x _ { j }$ $( 1 \leq j \leq d )$ . The ground truth of $\mathbb { E } \left[ f ( x ) \right]$ is estimated by Langevin dynamics with Metropolis adjustments (MALA) using a large sample size.
+We will compare the performance of the LQMC algorithm using three different step sizes: a constant step size of $1 0 ^ { - 4 }$ , a constant step size of $1 0 ^ { - 2 }$ , and decreasing step sizes with $h _ { k } \bar { = } c _ { 0 } ( c _ { 1 } + k ) ^ { - 1 / 3 }$ . The choice of $c _ { 0 }$ and $c _ { 1 }$ ensures that the step size decreases from $\mathrm { 1 0 ^ { - 2 } }$ to $1 0 ^ { - 4 }$ throughout the entire algorithm. The use of the exponent $- 1 / 3$ in the decreasing step sizes is recommended in Teh et al. (2016). The results of these comparisons are presented in Figure 4.
+In the small step size case (left panel), we observe that the errors of LMC and LQMC are initially comparable for small values of $n$ . This is because the algorithm converges slowly, and thus the error is dominated by the bias. However, as $n$ increases, the improvement of LQMC becomes evident. In the large step size case (middle panel), the MSE of LQMC is consistently smaller than that of LMC even for small values of $n$ . This is because the algorithm converges faster to the target distribution with a larger step size $h$ . Therefore, the improvement of LQMC is more pronounced. Interestingly, in this particular example, using decreasing step sizes yields similar accuracy to using a constant step size of $1 0 ^ { - 4 }$ . It is worth noting that the MSE of LMC does not decrease at a rate of $n ^ { - 2 / 3 }$ as in Teh et al. (2016). This is because the line in the plot does not represent the accuracy against the iteration $k$ within a single training process. Instead, it reflects the accuracy achieved after completing all $n$ iterations of the algorithm, considering different values of $n$ .
+![](images/044ec0893d8ac106377d731757837a9257b8c1e1fc339530b893fe2cdfbbd550.jpg)
+Figure 5: Double well potential.
+# 5.4 Nonconvex potential
+Finally we investigate a double-well potential function $\begin{array} { r } { U ( x ) = \frac { 1 } { 4 } x ^ { 2 } - \frac { 1 } { 2 } \log ( 1 + x ^ { 2 } ) } \end{array}$ from Pagès and Panloup (2018). We know $\mathbb { E } \left[ x \right] = 0$ and $\mathbb { i } \left[ \mathbf { 1 } _ { \{ \underline { { x } } \geq 0 \} } \right] = 0 . 5$ . The second moment $\mathbb { E } \left[ x ^ { 2 } \right]$ is computed by Gaussian quadrature. See the results in Figure 5. Since the potential has two separate local minimums, it takes longer for the Langevin algorithm to explore the space sufficiently and converge to the target distribution. Once converged, the improvement of LQMC over LMC is still significant.
+# Acknowledgments and Disclosure of Funding
+The author thanks Prof. Art Owen for helpful conversations. This work was partially funded by the NSF grant DMS-2152780 and the Stanford Data Science Scholars program.
+# References
+Arnold, S. M., L’Ecuyer, P., Chen, L., Chen, Y.-F., and Sha, F. (2022). Policy learning and evaluation with randomized quasi-Monte Carlo. In International Conference on Artificial Intelligence and Statistics, pages 1041–1061. PMLR.
+Baker, J., Fearnhead, P., Fox, E. B., and Nemeth, C. (2019). Control variates for stochastic gradient MCMC. Statistics and Computing, 29:599–615.
+Basu, K. and Owen, A. B. (2016). Transformations and Hardy–Krause variation. SIAM Journal on Numerical Analysis, 54(3):1946–1966.
+Brosse, N., Durmus, A., and Moulines, E. (2018). The promises and pitfalls of stochastic gradient Langevin dynamics. Advances in Neural Information Processing Systems, 31.
+Buchholz, A., Wenzel, F., and Mandt, S. (2018). Quasi-Monte Carlo variational inference. In International Conference on Machine Learning, pages 668–677. PMLR.
+Chen, C., Carlson, D., Gan, Z., Li, C., and Carin, L. (2016). Bridging the gap between stochastic gradient MCMC and stochastic optimization. In Artificial Intelligence and Statistics, pages 1051–1060. PMLR.
+Chen, C., Ding, N., and Carin, L. (2015). On the convergence of stochastic gradient MCMC algorithms with high-order integrators. Advances in neural information processing systems, 28.
+Chen, S. (2011). Consistency and convergence rate of Markov chain quasi Monte Carlo with examples. PhD thesis, Stanford University.
+Chen, S., Dick, J., and Owen, A. (2011). Consistency of Markov chain quasi-Monte Carlo on continuous state spaces. The Annals of Statistics, 39(2):673–701.
+Cheng, X., Chatterji, N. S., Bartlett, P. L., and Jordan, M. I. (2018). Underdamped Langevin MCMC: A non-asymptotic analysis. In Conference on learning theory, pages 300–323. PMLR.
+Choromanski, K., Pacchiano, A., Parker-Holder, J., and Tang, Y. (2019). Structured Monte Carlo sampling for nonisotropic distributions via determinantal point processes. arXiv preprint arXiv:1905.12667.
+Cranley, R. and Patterson, T. N. (1976). Randomization of number theoretic methods for multiple integration. SIAM Journal on Numerical Analysis, 13(6):904–914.
+Dalalyan, A. (2017). Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent. In Conference on Learning Theory, pages 678–689. PMLR.
+Dalalyan, A. S. and Karagulyan, A. (2019). User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient. Stochastic Processes and their Applications, 129(12):5278–5311.
+Dick, J. and Feischl, M. (2021). A quasi-monte carlo data compression algorithm for machine learning. Journal of Complexity, 67:101587.
+Dick, J. and Pillichshammer, F. (2010). Digital Sequences, Discrepancy and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge.
+Dick, J. and Rudolf, D. (2014). Discrepancy estimates for variance bounding Markov chain quasiMonte Carlo. Electron. J. Probab, 19(105):1–24.
+Dick, J., Rudolf, D., and Zhu, H. (2016). Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo. Annals of Applied Probability, 26(5):3178–3205.
+Dubey, K. A., J Reddi, S., Williamson, S. A., Poczos, B., Smola, A. J., and Xing, E. P. (2016). Variance reduction in stochastic gradient Langevin dynamics. Advances in neural information processing systems, 29.
+Erdogdu, M. A., Mackey, L., and Shamir, O. (2018). Global non-convex optimization with discretized diffusions. Advances in Neural Information Processing Systems, 31.
+He, Z., Zheng, Z., and Wang, X. (2023). On the error rate of importance sampling with randomized quasi-Monte Carlo. SIAM Journal on Numerical Analysis, 61(2):515–538.
+Hofmann, N. and Mathé, P. (1997). On quasi-Monte Carlo simulation of stochastic differential equations. Mathematics of computation, 66(218):573–589.
+Korobov, N. (1948). On functions with uniformly distributed fractional parts. In Dokl. Akad. Nauk SSSR, volume 62, pages 21–22.
+L’Ecuyer, P., Lécot, C., and Tuffin, B. (2008). A randomized quasi-Monte Carlo simulation method for Markov chains. Operations Research, 56(4):958–975.
+Levin, M. B. (1999). Discrepancy estimates of completely uniformly distributed and pseudorandom number sequences. International Mathematics Research Notices, 1999(22):1231–1251.
+Li, X., Wu, Y., Mackey, L., and Erdogdu, M. A. (2019). Stochastic Runge-Kutta accelerates Langevin Monte Carlo and beyond. Advances in neural information processing systems, 32.
+Liu, S. and Owen, A. B. (2021). Quasi-Monte Carlo quasi-Mewton in variational Bayes. The Journal of Machine Learning Research, 22(1):11043–11065.
+Longo, M., Mishra, S., Rusch, T. K., and Schwab, C. (2021). Higher-order quasi-monte carlo training of deep neural networks. SIAM Journal on Scientific Computing, 43(6):A3938–A3966.
+Lu, Y., Meng, S. Y., and De Sa, C. (2021). A general analysis of example-selection for stochastic gradient descent. In International Conference on Learning Representations.
+Markovic, J. and Taylor, J. (2016). Bootstrap inference after using multiple queries for model selection. arXiv preprint arXiv:1612.07811.
+Niederreiter, H. (1987). Point sets and sequences with small discrepancy. Monatshefte für Mathematik, 104:273–337.
+Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. SIAM.
+Owen, A. B. (1995). Randomly permuted $( t , m , s )$ -nets and $( t , s )$ -sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, pages 299–317, New York. Springer-Verlag.
+Owen, A. B. (1997a). Monte Carlo variance of scrambled net quadrature. SIAM Journal of Numerical Analysis, 34(5):1884–1910.
+Owen, A. B. (1997b). Scrambled net variance for integrals of smooth functions. Annals of Statistics, 25(4):1541–1562.
+Owen, A. B. and Tribble, S. D. (2005). A quasi-Monte Carlo Metropolis algorithm. Proceedings of the National Academy of Sciences, 102(25):8844–8849.
+Pagès, G. and Panloup, F. (2018). Weighted multilevel langevin simulation of invariant measures. Annals of Applied Probability, 28(6):3358–3417.
+Raginsky, M., Rakhlin, A., and Telgarsky, M. (2017). Non-convex learning via stochastic gradient Langevin dynamics: a nonasymptotic analysis. In Conference on Learning Theory, pages 1674– 1703. PMLR.
+Roberts, G. O. and Tweedie, R. L. (1996). Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli, pages 341–363.
+Rowland, M., Choromanski, K. M., Chalus, F., Pacchiano, A., Sarlos, T., Turner, R. E., and Weller, A. (2018). Geometrically coupled monte carlo sampling. Advances in Neural Information Processing Systems, 31.
+Shi, J., Liu, C., and Mackey, L. (2022). Sampling with mirrored stein operators. In International Conference on Learning Representations.
+Sobol’, I. M. (1967). On the distribution of points in a cube and the approximate evaluation of integrals. Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 7(4):784–802.
+Tausworthe, R. C. (1965). Random numbers generated by linear recurrence modulo two. Mathematics of Computation, 19(90):201–209.
+Teh, Y. W., Thiery, A. H., and Vollmer, S. J. (2016). Consistency and fluctuations for stochastic gradient Langevin dynamics. Journal of Machine Learning Research, 17.
+Tribble, S. D. (2007). Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences. PhD thesis, Citeseer.
+Vollmer, S. J., Zygalakis, K. C., and Teh, Y. W. (2016). Exploration of the (non-) asymptotic bias and variance of stochastic gradient Langevin dynamics. The Journal of Machine Learning Research, 17(1):5504–5548.
+Welling, M. and Teh, Y. W. (2011). Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-11), pages 681–688.
+Xu, P., Chen, J., Zou, D., and Gu, Q. (2018). Global convergence of Langevin dynamics based algorithms for nonconvex optimization. Advances in Neural Information Processing Systems, 31.
+# A Proofs
+# A.1 Proof of Theorem 4.1
+We start by decomposing the error $\textstyle { \frac { 1 } { n } } \sum _ { k = 1 } ^ { n } f ( \theta _ { k } ) - \pi ( f ) |$ into three parts
+$$
+\begin{array} { l } { \displaystyle \sum _ { = 1 } ^ { n } f ( \theta _ { k } ) - \pi ( f ) \bigg | \le \bigg | \frac { 1 } { n } \sum _ { k = \ell + 1 } ^ { n } f ( \theta _ { k } ) - \frac { 1 } { n } \sum _ { k = \ell + 1 } ^ { n } \bar { f } _ { \ell } ( { \mathbf w } _ { k } ^ { ( \ell ) } ) \bigg | + \bigg | \frac { 1 } { n } \sum _ { k = \ell + 1 } ^ { n } \bar { f } _ { \ell } ( { \mathbf w } _ { k } ^ { ( \ell ) } ) - \pi ( f ) \bigg | + \frac { \ell } { n } 2 \| f \| _ { \infty } } \\ { = ( I ) + ( I I ) + \frac { 2 \ell } { n } \| f \| _ { \infty } . } \end{array}
+$$
+We first upper bound $( I )$ .
+Lemma 1 (Upper bound of $( I )$ ; adapted from Lemma 6.1.4 of Chen (2011)). If the transition map $\psi$ is a contraction with parameter $\rho$ and if $f$ is $^ { l }$ -Lipschitz, then
+$$
+\lvert \bar { f } _ { \ell } ( \mathbf w _ { k } ^ { ( \ell ) } ) - f ( \theta _ { k } ) \rvert \le \left( \operatorname* { m a x } _ { 0 \le i \le n } \Vert \theta _ { i } \Vert + \mathbb { E } _ { \pi } \left[ \Vert \theta \Vert \right] \right) \rho ^ { \ell } .
+$$
+Proof of Lemma $^ { l }$ . Note that
+$$
+\begin{array} { r l } { | \bar { f } _ { \ell } ( \mathbf w _ { k } ^ { ( \ell ) } ) - f ( \theta _ { k } ) | \le \displaystyle \int | f ( \psi _ { \ell } ( \theta , \mathbf w _ { k } ^ { ( \ell ) } ) ) - f ( \psi _ { \ell } ( \theta _ { k - \ell } , \mathbf w _ { k } ^ { ( \ell ) } ) ) | \pi ( \mathrm d \theta ) } & { } \\ { \displaystyle } & { \le \displaystyle \int \| ( \psi _ { \ell } ( \theta , \mathbf w _ { k } ^ { ( \ell ) } ) ) - ( \psi _ { \ell } ( \theta _ { k - \ell } , \mathbf w _ { k } ^ { ( \ell ) } ) ) \| \pi ( \mathrm d \theta ) } \\ { \displaystyle } & { \le \rho \displaystyle \int \| ( \psi _ { \ell - 1 } ( \theta , \mathbf w _ { k - 1 } ^ { ( \ell - 1 ) } ) ) - ( \psi _ { \ell - 1 } ( \theta _ { k - \ell } , \mathbf w _ { k - 1 } ^ { ( \ell - 1 ) } ) ) \| \pi ( \mathrm d \theta ) } \\ { \displaystyle } & { \le \rho ^ { \ell } \displaystyle \int \| \theta - \theta _ { k - \ell } \| \pi ( \mathrm d \theta ) } \\ { \displaystyle } & { \le \rho ^ { \ell } ( \operatorname* { m a x } _ { 0 \le i \le n } \| \theta _ { i } \| + \mathbb { E } _ { \pi } \left[ \| \theta \| \right] ) . } \end{array}
+$$
+To bound $( I I )$ , note that $\begin{array} { r } { \frac { 1 } { n - \ell } \sum _ { k = \ell + 1 } ^ { n } \bar { f } _ { \ell } ( \mathbf { w } _ { k } ^ { ( \ell ) } ) } \end{array}$ is estimating
+$$
+\mathbb { E } \left[ \bar { f } _ { \ell } ( \mathbf { w } ^ { ( \ell ) } ) \right] = \int \psi _ { \ell } ( \boldsymbol { \theta } , \mathbf { w } ^ { ( \ell ) } ) \pi ( \mathrm { d } \boldsymbol { \theta } ) \mathrm { d } \mathbf { w } ^ { ( \ell ) } = : \pi P _ { \ell } ( \boldsymbol { f } ) .
+$$
+Here, $\pi P _ { \ell }$ denote the distribution of the $\ell$ -step state $\theta _ { \ell }$ starting from $\theta _ { 0 } \sim \pi$ . So we have the further decomposition
+$$
+\begin{array} { r l } & { ( I I ) \le \displaystyle \left| \frac { 1 } { n - \ell } \sum _ { k = \ell + 1 } ^ { n } \bar { f } _ { \ell } ( \mathbf w _ { k } ^ { ( \ell ) } ) - \pi ( f ) \right| + \frac { \ell } { n - \ell } \| f \| _ { \infty } } \\ & { \quad \le | \pi ( f ) - \pi P _ { \ell } ( f ) | + \displaystyle \left| \frac { 1 } { n - \ell } \sum _ { k = \ell + 1 } ^ { n } \bar { f } _ { \ell } ( \mathbf w _ { k } ^ { ( \ell ) } ) - \pi P _ { \ell } ( f ) \right| + \frac { \ell } { n - \ell } \| f \| _ { \infty } } \\ & { \quad \le ( I I ) ^ { \prime } + ( I I ) ^ { \prime \prime } + \displaystyle \frac { \ell } { n - \ell } \| f \| _ { \infty } . } \end{array}
+$$
+The first term $( I I ) ^ { \prime }$ is due to the discretization in time. The second term $( I I ) ^ { \prime \prime }$ is the numerical integration error.
+To bound $( I I ) ^ { \prime }$ , we use the following result.
+Lemma 2 (Upper bound on discretization error $( I I ) ^ { \prime } $ ). Under Assumption $I$ , we have for $f$ 1- Lipschitz,
+$$
+| \pi ( f ) - \pi P _ { \ell } ( f ) | \leq \frac { 3 \sqrt { 2 } } { 2 } \frac { L } { M } h ^ { 1 / 2 } d .
+$$
+Proof of Lemma 2. We let $\theta ( t )$ be the continuous-time Langevin diffusion with $\theta ( 0 ) = \theta _ { 0 } \sim \pi$ , $W _ { t _ { k + 1 } } - W _ { t _ { k } } = \sqrt { h } \xi _ { k + 1 }$ , where $\xi _ { k + 1 } \overset { i i d } { \sim } \mathcal { N } ( 0 , I _ { d } )$ , $t _ { k } = k h$ . So we have
+$$
+\theta ( t _ { k + 1 } ) = \theta ( t _ { k } ) - \int _ { t _ { k } } ^ { t _ { k + 1 } } \nabla U ( \theta ( s ) ) \mathrm { d } s + \sqrt { 2 h } \xi _ { k + 1 }
+$$
+and
+$$
+\theta _ { k + 1 } = \theta _ { k } - h \nabla U ( \theta _ { k } ) + \sqrt { 2 h } \xi _ { k + 1 } .
+$$
+Combing the previous two equations gives
+$$
+\theta ( t _ { k + 1 } ) - \theta _ { k + 1 } = \theta ( t _ { k } ) - \theta _ { k } - h [ \nabla U ( \theta ( t _ { k } ) ) - \nabla U ( \theta _ { k } ) ] - \int _ { t _ { k } } ^ { t _ { k + 1 } } \nabla U ( \theta ( s ) ) - \nabla U ( \theta ( t _ { k } ) ) \mathrm { d } s .
+$$
+Let $\Delta _ { k } = \theta ( t _ { k } ) - \theta _ { k }$ . The last display reads
+$$
+\Delta _ { k + 1 } = \Delta _ { k } - h [ \nabla U ( \theta _ { k } + \Delta _ { k } ) - \nabla U ( \theta _ { k } ) ] - \int _ { t _ { k } } ^ { t _ { k + 1 } } \nabla U ( \theta ( s ) ) - \nabla U ( \theta ( t _ { k } ) ) \mathrm { d } s .
+$$
+By the contracting property (6) in the main paper,
+$$
+\begin{array} { r } { \| \Delta _ { k } - h [ \nabla U ( \theta _ { k } + \Delta _ { k } ) - \nabla U ( \theta _ { k } ) ] \| \le \rho \| \Delta _ { k } \| . } \end{array}
+$$
+Taking expectation and use $L$ -smoothness of $U$ , we have
+$$
+\mathbb { E } \left[ \Vert \Delta _ { k + 1 } \Vert \right] \leq \rho \mathbb { E } \left[ \Vert \Delta _ { k } \Vert \right] + L \int _ { t _ { k } } ^ { t _ { k + 1 } } \mathbb { E } \left[ \Vert \theta ( s ) - \theta ( t _ { k } ) \Vert \right] \mathrm { d } s .
+$$
+By Lemma 3 of Dalalyan and Karagulyan (2019), E√ $\left[ \lVert \nabla U ( \theta ) \rVert _ { 2 } ^ { 2 } \right] \leq L d$ . So we have $\mathbb { E } \left[ \lVert \nabla U ( \theta ) \rVert \right] \leq$ $\sqrt { d \mathbb { E } \left[ \| \nabla U ( \theta ) \| _ { 2 } ^ { 2 } \right] } \leq \sqrt { L } d$ . Because $\theta ( t )$ is a stationary process,
+$$
+\begin{array} { r l } { \displaystyle \int _ { t _ { k } } ^ { t _ { k + 1 } } \mathbb { E } \left[ \| \theta ( s ) - \theta ( t _ { k } ) \| \right] \mathrm { d } s = } & { \displaystyle \int _ { 0 } ^ { h } \mathbb { E } \left[ \| \theta ( t ) - \theta ( 0 ) \| \right] \mathrm { d } t } \\ & { \quad \quad \quad = \displaystyle \int _ { 0 } ^ { h } \mathbb { E } \left[ \| - \int _ { 0 } ^ { t } \nabla U ( \theta ( s ) ) \mathrm { d } s + \sqrt { 2 } W _ { t } \| \right] \mathrm { d } t } \\ & { \quad \quad \le \displaystyle \int _ { 0 } ^ { h } \int _ { 0 } ^ { t } \mathbb { E } \left[ \| \nabla U ( \theta ( s ) ) \| \right] \mathrm { d } s \mathrm { d } t + \displaystyle \int _ { 0 } ^ { h } \sqrt { 2 } \mathbb { E } \left[ \| W _ { t } \| \right] \mathrm { d } t } \\ & { \quad \quad = \displaystyle \frac { h ^ { 2 } } { 2 } \sqrt { L } d + \displaystyle \int _ { 0 } ^ { h } \sqrt { 2 t } \mathbb { E } \left[ \| \xi _ { 1 } \| \right] \mathrm { d } t . } \end{array}
+$$
+Note that
+$$
+\mathbb { E } \left[ \Vert \xi _ { 1 } \Vert \right] = { \sqrt { 2 } } { \frac { \Gamma ( d / 2 + 1 / 2 ) } { \Gamma ( d / 2 ) } } \leq { \sqrt { 2 } } ( { \frac { d + 1 } { 2 } } ) ^ { 1 / 2 } = { \sqrt { d + 1 } } .
+$$
+Thus,
+$$
+\begin{array} { r l r } {  { \int _ { t _ { k } } ^ { t _ { k + 1 } } \mathbb { E } [ \| \theta ( s ) - \theta ( t _ { k } ) \| ] \mathrm { d } s \le \frac { 1 } { 2 } L ^ { 1 / 2 } h ^ { 2 } d + \frac { 3 \sqrt { 2 } } { 2 } h ^ { 3 / 2 } d ^ { 1 / 2 } } } \\ & { } & { \le \frac { \sqrt { 2 } } { 2 } h ^ { 3 / 2 } d + \frac { 3 \sqrt { 2 } } { 2 } h ^ { 3 / 2 } d ^ { 1 / 2 } } \\ & { } & { \le \frac { 3 \sqrt { 2 } } { 2 } h ^ { 3 / 2 } d . } \end{array}
+$$
+Denote $\begin{array} { r } { r = \frac { 3 \sqrt { 2 } } { 2 } L h ^ { 3 / 2 } d . } \end{array}$ . So
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } \left[ \| \Delta _ { k + 1 } \| \right] \leq \rho \mathbb { E } \left[ \| \Delta _ { k } \| \right] + r \leq \rho ^ { k + 1 } \mathbb { E } \left[ \| \Delta _ { 0 } \| \right] + \displaystyle \sum _ { i = 0 } ^ { k } \rho ^ { i } r } \\ { \leq \displaystyle \frac { r } { 1 - \rho } = \frac { 3 \sqrt { 2 } } { 2 } \frac { L } { M } h ^ { 1 / 2 } d } \end{array}
+$$
+Therefore, for any $k \geq 1$ ,
+$$
+\begin{array} { r l } & { | \pi ( f ) - \pi P _ { k } ( f ) | = \left| \mathbb { E } \left[ f ( \theta ( t _ { k } ) ) - \mathbb { E } \left[ f ( \theta _ { k } ) \right] \right] \le \mathbb { E } \left[ | f ( \theta ( t _ { k } ) ) - f ( \theta _ { k } ) | \right] \right| } \\ & { \quad \quad \quad \le \mathbb { E } \left[ \left. \Delta _ { k } \right. \right] \le \displaystyle \frac { 3 \sqrt { 2 } } { 2 } \frac { L } { M } h ^ { 1 / 2 } d . } \end{array}
+$$
+If we use a noisy gradient $\hat { \boldsymbol g } ( \boldsymbol \theta _ { k } ) = \nabla U ( \boldsymbol \theta _ { k } ) + \boldsymbol e _ { k }$ where $e _ { k }$ is the noise with mean zero and bounded variance such that $\mathbb { E } ( | | e _ { k } | | _ { 2 } ^ { 2 } ) \leq \sigma ^ { 2 }$ , then an extra term $2 h \sigma$ will appear in Lemma 2. As $\sigma ^ { 2 }$ is usually expected to be proportional to the dimension , this additional term is of the same order as the other term.
+Theorem A.1 (Theorem 9.8 of Niederreiter (1992)). Let $v _ { 0 } , v _ { 1 } , \ldots$ be an LFSR with offset s and period $n = 2 ^ { m } - 1$ which satisfy $g c d ( m , n ) = 1$ . Then the sequence $\{ \mathbf { u } _ { i } \} _ { i = 0 } ^ { n - 1 } \subset [ 0 , 1 ] ^ { s }$ with $\mathbf { u } _ { i } = ( v _ { i } , v _ { i + 1 } , \dots , v _ { i + s - 1 } )$ has, on average, star-discrepancy
+$$
+O ( n ^ { - 1 } ( \log n ) ^ { d + 1 } \log \log n )
+$$
+with an implied constant depending only on $d$ and the average is taken over all primitive polynomials over $G F ( 2 )$ of degree $m$ .
+Proof of Theorem 4.1. The error on the left-hand-side is bounded by
+$$
+( I ) + ( I I ) ^ { \prime } + ( I I ) ^ { \prime \prime } + \frac { 4 \ell } { n } \| f \| _ { \infty } .
+$$
+Lemma 1 shows that $\begin{array} { r } { ( I ) \leq ( \operatorname* { m a x } _ { 0 \leq i < n } \| \theta _ { i } \| + \mathbb { E } _ { \pi } \left[ \| \theta \| \right] ) \rho ^ { \ell } \leq ( \operatorname* { m a x } _ { 0 \leq i \leq n } \| \theta _ { i } \| + \mathbb { E } _ { \pi } \left[ \| \theta \| \right] ) h ^ { 1 / 2 } } \end{array}$ since $\ell = \lceil ( 1 / 2 ) \log _ { \rho } h \rceil$ . Lemma 2 shows that $\begin{array} { r } { ( I I ) ^ { \prime } \leq \frac { 3 \sqrt { 2 } } { 2 } \frac { L } { M } d h ^ { 1 / 2 } } \end{array}$ . Denote $C _ { 2 } = \operatorname* { m a x } _ { 0 \leq i \leq n } \left\| \theta _ { i } \right\| +$ $\begin{array} { r } { \mathbb { E } _ { \pi } \left[ \| \theta \| \right] + \frac { 3 \sqrt { 2 } } { 2 } \frac { L } { M } d } \end{array}$ . So $( I ) + ( I I ) ^ { \prime } \leq C _ { 2 } h ^ { 1 / 2 }$ .
+By Theorem A.1 and the condition that $\operatorname* { g c d } ( d \ell , n ) = 1$ , the star-discrepancy $D ^ { * } \big ( \{ \bar { w } _ { k } ^ { ( \ell ) } \} _ { k \geq 1 } \big )$ is upper bounded by $O ( n ^ { - 1 } ( \log n ) ^ { d \ell + 1 } \log \log n )$ . Finally, by Koksma-Hlawka inequality, we have $( I I ) ^ { \prime \prime } \le \| \bar { f } _ { \ell } \| _ { \mathrm { H K } } \cdot D ^ { * } ( \{ \bar { w } _ { k } ^ { ( \ell ) } \} _ { k \ge 1 } )$ . Thus, $\begin{array} { r } { ( I I ) ^ { \prime \prime } + \frac { 4 \ell } { n } \| f \| _ { \infty } \leq C _ { 1 } n ^ { - 1 + \delta } } \end{array}$ , where $\delta$ hides the polylogarithmic terms in $\log n$ and $C _ { 1 }$ depends on $d , \ell , \| \bar { f } _ { \ell } \| _ { \mathrm { H K } }$ .
+Therefore, the upper bound becomes
+$$
+( I ) + ( I I ) ^ { \prime } + ( I I ) ^ { \prime \prime } + \frac { 4 \ell } { n } \| f \| _ { \infty } \leq C _ { 1 } n ^ { - 1 + \delta } + C _ { 2 } h ^ { 1 / 2 } .
+$$
+# B Additional numerical results
+The primary contribution of this work is to improve LMC as a Monte Carlo sampling algorithm, not as an optimization algorithm. Therefore, our main focus is on providing a better estimation of $\pi ( f )$ for some function of interest. Downstream tasks relying on such expectations can also benefit from LQMC. For posterior prediction, it is essential to recognize that the prediction error is not solely determined by the sampling method. Even with infinite perfect samples from the posterior, the prediction error can still arise due to model misspecification, noisy data, biased sampling, etc. So the improvement achieved by LQMC might be less pronounced when assessing the prediction error.
+To investigate the performance of LQMC in a posterior prediction setting, we conducted experiments similar to those presented in Dubey et al. (2016) using three UCI datasets. Each dataset was split into a training set $( 7 0 \% )$ , a validation set $( 1 0 \% )$ , and a test set $( 2 0 \% )$ . We performed a tuning process for the constant step size on a grid using the validation set and evaluated the prediction error on the test set. Each iteration computes the stochastic gradient using 32 data points sampled at random. Details of the datasets are in Table 1.
+<table><tr><td>Datasets</td><td>Parkinsons</td><td>Bike</td><td>Protein</td></tr><tr><td>N (number of instances)</td><td>5875</td><td>17379</td><td>45730</td></tr><tr><td>p (number of features)</td><td>21</td><td>12</td><td>9</td></tr></table>
+Table 1: Summary of datasets used for Bayesian posterior prediction.
+The results are presented in Figure 6. The $x$ -axes represent the total number of iterations of Langevin algorithm and the $y$ -axes represent the test error. The error bars represent the variation across 10 random replicates. It is evident that LQMC reduced the test error, although the improvement is not substantial. This aligns with our initial expectation, as the proposed method primarily enhances the accuracy of estimating the posterior mean. However, the test error often consists of other sources of error, thus the improvement achieved by the proposed method in reducing the test error might be limited.
+![](images/f6acd57f6cfbd7761952326b942ddaeef1cc2dc6b22870c352cfb01ab6a1d5e7.jpg)
+Figure 6: Test error versus number of iterations for the three UCI datasets.

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+# SegViT: Semantic Segmentation with Plain Vision Transformers
+Bowen Zhang1∗, Zhi Tian2∗, Quan Tang4, Xiangxiang $\mathbf { C h u ^ { 2 } }$ , Xiaolin Wei2, Chunhua Shen3, Yifan Liu1
+1 The University of Adelaide, Australia 2 Meituan Inc. 3 Zhejiang University, China 4 South China University of Technology, China
+# Abstract
+We explore the capability of plain Vision Transformers (ViTs) for semantic segmentation and propose the SegViT. Previous ViT-based segmentation networks usually learn a pixel-level representation from the output of the ViT. Differently, we make use of the fundamental component—attention mechanism, to generate masks for semantic segmentation. Specifically, we propose the Attention-to-Mask (ATM) module, in which the similarity maps between a set of learnable class tokens and the spatial feature maps are transferred to the segmentation masks. Experiments show that our proposed $\mathrm { S e g V i T }$ using the ATM module outperforms its counterparts using the plain ViT backbone on the ADE20K dataset and achieves new state-of-the-art performance on COCO-Stuff-10K and PASCAL-Context datasets. Furthermore, to reduce the computational cost of the ViT backbone, we propose query-based down-sampling (QD) and query-based up-sampling (QU) to build a Shrunk structure. With the proposed Shrunk structure, the model can save up to $4 0 \%$ computations while maintaining competitive performance.
+# 1 Introduction
+Semantic segmentation is a dense prediction task in computer vision that requires pixel-level classification of an input image. Fully Convolutional Networks (FCN) [1] are widely used in recent state-of-the-art methods. This paradigm includes a deep convolutional neural network as the encoder/backbone and a segmentation-oriented decoder to provide dense predictions. A $1 \times 1$ convolutional layer is usually applied to a representative feature map to obtain the pixel level predictions. To achieve higher performance, previous works [2–4] focus on enriching the context information or fusing multi-scale information. However, the correlations among spatial locations are hard to model explicitly in FCNs due to the limited receptive field.
+Recently, Vision Transformers (ViT) [5], which make use of the spatial attention mechanism are introduced to the field of computer vision. Unlike typical convolution-based backbones, the ViT has a plain and non-hierarchical architecture that keeps the resolution of the feature maps all the way through. The lack of the down-sampling process (excluding tokenizing the image) brings differences to the architecture to do the semantic segmentation task using ViT backbone. Various semantic segmentation methods [6–8] based on ViT backbones have achieved promising performance due to the powerful representation learned from the pre-trained backbones. However, the potential of the attention mechanism is not fully explored.
+Different from previous per-pixel classification paradigm [6–8], we consider learning a meaningful class token and then finding local patches with higher similarity to it. To achieve this goal, we propose the Attention-to-Mask (ATM) module. More specifically, we employ a transformer block that takes the learnable class tokens as queries and transfers the spatial feature maps as keys and values. A dot-product operator calculates the similarity maps between queries and keys. We encourage regions belonging to the same category to generate larger similarity values for the corresponding category (i.e. a specific class token). Fig. 1 visualizes the similarity maps between the features and the ‘Table’ and ‘Chair’ tokens. By simply applying a Sigmoid operation, we can transfer the similarity maps to the masks. Meanwhile, following the design of a typical transformer block, a Softmax operation is also applied to the similarity maps to get the cross attention maps. The ‘Table’ and ‘Chair’ tokens are then updated as in any regular transformer decoders, by a weighted sum of the values with the cross attention maps as the weights. Since the mask is a byproduct of the regular attentive calculations, negligible computation is involved during the operation.
+Building upon this efficient ATM module, we propose a new semantic segmentation paradigm with the plain ViT structure, dubbed $\mathrm { { S e g V i T } }$ . In the paradigm, several ATM modules are employed on different layers, and we get the final segmentation mask by adding the outputs from different layers together. $\mathrm { S e g V i T }$ outperforms its ViT-based counterparts with less computational cost. However, compared with previous encoder-decoder structures that use hierarchical networks as encoders, ViT backbones as encoders are generally heavier. To further reduce the computational cost, we employ a Shrunk structure consisting of query-based down-sampling (QD) and query-based up-sampling (QU). The QD can be inserted into the ViT backbone to reduce the resolution by half and QU is used parallel to the backbone to recover the resolution. The Shrunk structure together with the ATM module as the decoder can reduce up to $4 0 \%$ computations while maintaining competitive performance.
+We summarize our main contributions as follows:
+• We propose an Attention-to-Mask (ATM) decoder module that is effective and efficient for semantic segmentation. For the first time, we utilize the spatial information in attention maps to generate mask predictions for each category, which can work as a new paradigm for semantic segmentation.
+• We managed to apply our ATM decoder module to the plain, non-hierarchical ViT backbones in a cascade manner and designed a structure namely $\mathrm { S e g V i T }$ that achieves mIoU $5 5 . 2 \%$ on the competitive ADE20K dataset which is the best and lightest among methods that use ViT backbones. We also benchmark our method on the PASCAL-Context dataset $( 6 5 . 3 \%$ mIoU) and COCO-Stuff-10K dataset $( 5 0 . 3 \%$ mIoU) and achieve new state-of-the-art performance.
+We further explore the architecture of ViT backbones and work out a Shrunk structure to apply to the backbone to reduce the overall computational cost while still maintaining competitive performance. This alleviates the disadvantage of ViT backbones that are usually more computationally intensive compared to their hierarchical counterparts. Our Shrunk version of $\mathrm { S e g V i T }$ on the ADE20K dataset reaches mIoU $5 5 . 1 \%$ with the computational cost of 373.5 GFLOPs which is about $4 0 \%$ off compared to the original SegViT (637.9 GFLOPs).
+# 2 Related Work
+Semantic segmentation. Semantic segmentation which requires pixel-level classification on an input image is a fundamental task in computer vision. Fully Convolutional Networks (FCN) used to be the dominant approach to this task. Initial per-pixel approaches such as [9, 10] attribute the class label to each pixel based on the per-pixel probability. To enlarge the receptive field, several approaches [11, 12] have proposed dilated convolutions or apply spatial pyramid pooling to capture contextual information of multiple scales. With the introduction of attention mechanisms, [13, 14, 6] replace the feature merge conducted by convolutions and pooling with attention to better capture long-range dependencies.
+Recent works [15, 8, 16] decouple the per-pixel classification process. They reconstruct the structure by using a fixed number of learnable tokens and use them as weights for the transformation to apply on feature maps. Binary matching rather than cross-entropy is used to allow overlaps between feature maps and learnable tokens are used to dynamically generate classification probabilities. This paradigm enables the classification process to be conducted globally and alleviates the burden for the decoder to do per-pixel classification, which as a result, is more precise and the performance is generally better. However, for those methods, the feature map is still calculated in a static manner, usually requiring feature merge modules such as FPN [4].
+![](images/d4684106657fd91e50f1ae4dbfc23811e0a8d9bf529682b3f8d4363fdb795cf5.jpg)
+Figure 1: The overall concept of our Attention-to-Mask decoder. In a typical attentive process, the dot-product is first calculated between queries and keys to measure the similarity (as illustrated on the left). If the similarity map is applied with Softmax operation on the spatial dimension, the output is the typical attention map (multiple heads are summed together). However, if the same similarity map is applied with a per-pixel operation Sigmoid, it produces a mask that indicates the area with certain similarity. Based on the assumption that the tokens within the same category have higher similarity, we can train a token vector to have high similarity within tokens of the specific category and low similarity elsewhere. In the meantime, this process does not violate the attention mechanism. Thus, it can process alongside the original transformer layers.
+Transformers for vision. Attention-based transformer backbones have become powerful alternatives to standard convolution based networks for image classification tasks. The original ViT [5] is a plain, non-hierarchical architecture. Various hierarchical transformers such as [17–21] have been presented afterwards. These methods inherit some designs from convolution based networks such as hierarchical structures, pooling and down-sampling with convolutions. As a result, they can be used as a straightforward replacement for convolutional based networks and applied with previous decoder heads for tasks such as semantic segmentation.
+Plain-backbone decoders. High-resolution feature maps generated by backbones are important for dense prediction tasks such as semantic segmentation. Typical hierarchical transformers use feature merge techniques such as FPN [4] or dilated backbones to generate high-resolution feature maps. However, for plain, non-hierarchical transformer backbones, the resolution remains the same for all layers. SETR [6] proposed a simple strategy to treat transformer outputs in a sequence-to-sequence perspective to solve segmentation tasks. Segmenter [8] joints random initialized class embeddings and the transformer patch embeddings together and applies several self-attention layers to the joint token sequence to obtain updated class embeddings and patch embeddings semantic prediction. In our study, we consider learning a class token and then finding local patches with higher similarities with the help of the attention map, making the inference process more direct and efficient.
+# 3 Method
+# 3.1 Encoder
+Given an input image $I \in \mathbb { R } ^ { H \times W \times 3 }$ , a plain vision transformer backbone reshapes it into a sequence of tokens $\dot { \mathcal { F } } _ { 0 } \in \mathbb { R } ^ { \tilde { L } \times C }$ where ${ \cal L } = { \cal H } \bar { W } / P ^ { 2 }$ , $P$ is the patch size and $C$ is the number of channels. Learnable position embeddings of the same size of $\mathcal { F } _ { 0 }$ are added to capture the positional information. Then, the token sequence $\mathcal { F } _ { 0 }$ is applied with $m$ transformer layers to get the output. We define the output tokens for each layer as $[ \mathcal { F } _ { 1 } , \mathcal { F } _ { 2 } , \ldots , \mathcal { F } _ { m } ] \in \mathbb { R } ^ { L \times C }$ . Typically, a transformer layer consists of a multi-head self-attention block followed by a point-wise multilayer perceptron block with layer norm in between and then a residual connection is added afterward. The transformer layers are stacked repetitively several times. For a plain vision transformer like ViT, there are no other modules involved and for each layer, the number of the tokens is not changed.
+![](images/34cbf2f58eed90953a49e4ee13e4b168073eba542e2899120d61dd93c2b7b2f1.jpg)
+Figure 2: The overall SegViT structure with the ATM module. The Attention-to-Mask (ATM) module inherits the typical transformer decoder structure. It takes in randomly initialized class embeddings as queries and the feature maps from the ViT backbone to generate keys and values. The outputs of the ATM module are used as the input queries for the next layer. The ATM module is carried out sequentially with inputs from different layers of the backbone as keys and values in a cascade manner. A linear transform is then applied to the output of the ATM module to produce the class predictions for each token. The mask for the corresponding class is transferred from the similarities between queries and keys in the ATM module.
+# 3.2 Decoder
+Mask-to-Attention (ATM). Cross attention can be described as the mapping between two sequences of tokens. We define two token sequences as $\mathcal { G } \in \mathbb { R } ^ { N \times C }$ with the length $N$ equals to the number of classes and $\mathcal { F } _ { i } \in \mathbb { R } ^ { L \times C }$ . First, linear transformations are applied to each of them to form query (Q), key (K) and values (V), as presented by Eq. (1).
+$$
+Q = \phi _ { q } ( \mathcal { G } ) \in \mathbb { R } ^ { N \times C } , K = \phi _ { k } ( \mathcal { F } _ { i } ) \in \mathbb { R } ^ { L \times C } , V = \phi _ { v } ( \mathcal { F } _ { i } ) \in \mathbb { R } ^ { L \times C } ,
+$$
+The similarity map is calculated between the query and the key. Following the scaled dot-product attention mechanism, the similarity map and attention map are calculated by:
+$$
+\begin{array} { c } { S ( Q , K ) = \displaystyle \frac { Q K ^ { T } } { \sqrt { d _ { k } } } \in \mathbb { R } ^ { N \times L } , } \\ { A t t e n t i o n ( \mathcal { G } , \mathcal { F } _ { i } ) = \displaystyle \mathtt { S o f t m a x } ( S ( Q , K ) ) V \in \mathbb { R } ^ { N \times C } , } \end{array}
+$$
+where $\sqrt { d _ { k } }$ is a scaling factor with $d _ { k }$ equals to the dimension of the keys. The shape of the similarity map $S ( Q , K )$ is determined by the length of the two token sequences $N$ and $L$ . The attention mechanism is then to update $\mathcal { G }$ by a weighted sum of $V$ , where the weight assigned to the summation is the similarity map applied with Softmax along the dimension $L$ .
+Dot-product attention uses the Softmax function to exclusively concentrate the attention on the token that has the most similarity. However, we suppose that the tokens other than ones that yield maximum similarities are also meaningful. Based on this intuition, we design a lightweight module that generates semantic predictions more directly. To be more specific, we assign $\mathcal { G }$ as the class embeddings for the segmentation task and ${ \mathcal { F } } _ { i }$ as the output of layer $i$ of the ViT backbone. We pair a semantic mask to each token in $\mathcal { G }$ to represent the semantic prediction for each class. The calculation for the mask is:
+$$
+M a s k ( \mathcal { G } , \mathcal { F } _ { i } ) = \operatorname { S i g m o i d } ( S ( Q , K ) ) \in \mathbb { R } ^ { N \times L }
+$$
+The shape of the masks is $N \times L$ , which can be further reshaped to $N \times H / P \times W / P$ . The structure of the ATM mechanism is illustrated in the right part in Fig. 2. Masks are the middle output of the cross attention. The final output tokens from the ATM module are used for classification. We apply a linear transformation followed by a Softmax activation to the output class tokens to get class probability predictions. Note that we follow [15] to add a ‘no object’ category $( \varnothing )$ in case the image doesn’t contain certain classes. During inference, the output is produced by the dot-product between the class probability and the mask groups.
+![](images/937a25918d13b9bb305644659197a9d1f00fc926dfb7e1a6dcf30600feacaa80.jpg)
+Figure 3: The structure comparison between $\mathbf { S e g V i T }$ with a single layer and the Shrunk version. (a) illustrates the $\mathrm { S e g V i T }$ structure with ATM module used once with the last layer of the ViT backbone as the input to generate predictions. (b) uses the query-based down-sampling (QD) module to implement a naive way to shrink the resolution of the features of the backbone from $^ { 1 / 1 6 }$ to $^ { 1 / 3 2 }$ and thus reduces the overall computational cost. (c) is the proposed (shrunk) version which applies the additional query-based up-sampling module. The Shrunk version can save up to $40 \%$ of computational cost when using the ViT-Large backbone without much sacrifice to the performance.
+Plain backbones such as ViT does not have multiple stages with features of different scale. Thus, structures such as FPN to merge features with multiple scales are not applicable. However, features other than the last layer contain rich low-level semantic information and are beneficial to the performance. We designed a structure that can make use of the feature maps from different layers of ViT to compact with our ATM decoder namely $\mathrm { S e g V i T }$ . In this study, we also found a way to compact the computational cost for the ViT backbone without sacrificing performance. This proposed Shrunk version of $\mathrm { S e g V i T }$ uses query-based down-sampling (QD) module together with a query-based up-sampling (QU) module to compress the ViT backbone and bring an overall reduction to the computational cost.
+The SegViT structure. As illustrated in Fig. 2, an ATM decoder takes in $N$ tokens as the class embeddings and another sequence of tokens as the base to calculate keys and values for the ATM module to generate masks. The output of the ATM is $N$ updated tokens and $N$ masks corresponding to each class token. We use random initialized learnable tokens as the class embeddings and the output of the last layer of the ViT backbone as the base first. To make use of multi-layer information, the output of the first ATM decoder is then used as the class embeddings for the next ATM decoder with the output of another layer of the ViT backbone as the base. This process is repeated another time so that we can get three groups of tokens and masks. Formally, the loss function of each layer can be formulated as,
+$$
+\mathcal { L } _ { o v e r a l l } = \mathcal { L } _ { c l s } + \mathcal { L } _ { m a s k } = \mathcal { L } _ { c l s } + \lambda _ { f o c a l } \mathcal { L } _ { I o U } + \lambda _ { d i c e } \mathcal { L } _ { d i c e }
+$$
+In each group, the output tokens are supervised by the classification loss $( \mathcal { L } _ { c l s } )$ which is mentioned above and the masks are summed orderly and supervised by the mask loss $( \mathcal { L } _ { m a s k } )$ which is a linear combination of a focal loss [22] and a dice loss [23] multiplied by hyper-parameters $\lambda _ { f o c a l }$ and $\lambda _ { d i c e }$ respectively as in DETR [24]. The loss of all three groups are then summed together. We have further experiments to show that this design is beneficial and efficient.
+The Shrunk structure. Plain transformer backbones such as ViT is known to have larger computational cost than their counterparts with similar performance. We propose a Shrunk structure using query-based down-sampling (QD) and up-sampling (QU). Since the shape of the output of the attention module is determined by the shape of the query, we can apply down-sampling before the query transformation to realize the QD or insert new query tokens during the cross attention to realize the QU. By changing the resolution with the number of query tokens, the spatial size is changed according to the cross attention, providing more flexibility to preserve (recover) important regions. To be more specific, in the QD layer, we use the nearest sampling to reduce the number of the query tokens while keep the size of the key and value tokens. When passing through a transformer layer, the values are weighted and summed by the attention map between query tokens and the key tokens. This is non-linear downsampling that will pay more attention to the important regions. In the QU layer, we employ a transformer decoder structure [25] and initialize new learnable tokens as queries based on the desired output resolution.
+As shown in Fig. 3, we design the $\mathrm { S e g V i T }$ structure with one single layer as the baseline (a). We first try a naive approach (b), which is to apply the QD once at the $^ 1 / 3$ depth of the backbone (e.g., the 8th layer of a backbone with 24 layers) to down-sample the resolution of the layer output from $^ { 1 / 1 6 }$ to $1 / \bar { 3 } 2$ so as to reduce the overall computational cost. The performance drops as expected since the QD process involves information lose.
+To compensate for the information loss in the naive ‘shrunk’ version, we further apply two QU layers in parallel with the backbone. This is our proposed Shrunk version (c). The first QU layer takes in features with $^ { 1 / 1 6 }$ resolution from the low level of the backbone. Its output is then used as the query to make cross attention with the down-sampled features with $^ { 1 / 3 2 }$ resolution from the last layer of the backbone. The shape of the output of this QU structure is of $^ { 1 / 1 6 }$ resolution.
+Directly reducing the number of the query tokens inevitably harms the final performance. However, with our designed QU layer and the ATM module, the Shrunk structure is able to reduce $4 0 \%$ of overall computational cost while still being competitive in performance.
+# 4 Experiments
+# 4.1 Datasets
+ADE20K [26] is a challenging scene parsing dataset which contains 20, 210 images as the training set and 2, 000 images as the validation set with 150 semantic classes.
+COCO-Stuff-10K [27] is a scene parsing benchmark with 9, 000 training images and 1, 000 test images. Even though the dataset contains 182 categories, not all categories exist in the test split. We follow the implementation of mmsegmentation [28] with 171 categories to conduct the experiments.
+PASCAL-Context [29] is a dataset with 4, 996 images in training set and 5, 104 images in the validation set. There are 60 semantic classes in total, including a class representing ‘background’.
+# 4.2 Implementation details
+Transformer backbone. We use the naive ViT [5] as the backbone. In particular, we use its ‘Base’ variation for most ablation studies and provide results on the ‘Large’ variation. Since there can be a huge difference with different pre-trained weights, as suggested by Segmenter [8], we use the weights provided by Augreg [30] following the counterparts [8, 31] for a fair comparison. The weights are obtained by training on ImageNet-21k with strong data augmentation and regularization. For a simple reference, we report that for pre-trained weights provided by ViT [5] and Augreg [30], the mIoU scores using the same training recipe on ADE20K dataset are $5 1 . 7 \%$ and $5 4 . 6 \%$ , respectively. Training settings. We use MMSegmentation [28] and follow the commonly used training settings. During training, we applied data augmentation sequentially via random horizontal flipping, random resize with the ration between 0.5 and 2.0 and random cropping $5 1 2 \times 5 1 2$ for all except that we use $4 8 0 \times 4 8 0$ for PASCAL-Context and $6 4 0 \times 6 4 0$ for ViT-large on ADE20K). The batch size is 16 for all datasets with a total iteration of $1 6 0 k$ , $8 0 k$ and $8 0 k$ for ADE20k, COCO-Stuff-10k and PASCAL-Context respectively. Evaluation metric. We use the mean Intersection over Union (mIoU) as the metric to evaluate the performance. ‘ss’ means single-scale testing and ‘ms’ test time augmentation with multi-scaled (0.5, 0.75, 1.0, 1.25, 1.5, 1.75) inputs. All reported mIoU scores are in a percentage format. All reported computational costs in GFLOPs are measured using the fvcore 2 library.
+# 4.3 Comparisons with the State-of-the-art Methods
+Results on ADE20K. Table 1 reports the comparison with the state-of-the-art methods on ADE20K validation set using ViT backbone. The SegViT uses the ATM module with multi-layer inputs from the original ViT backbone, while the Shrunk is the one that conducts QD to the ViT backbone and saves $4 0 \%$ of the computational cost without sacrificing too much performance. Our method achieves
+Table 1: Experiment results on the ADE20K val. split. ‘ms’ means that mIoU is calculated using multi-scale inference. ‘†’ means the models use the backbone weights pre-trained by AugReg [30]. ‘\*’ represents the model is reproduced under the same settings as the official repo. The GFLOPs is measured at single-scale inference with the given crop size.
+<table><tr><td>Method</td><td>Backbone</td><td>Crop Size</td><td>GFLOPs</td><td>mIoU (ss)</td><td>mIoU (ms)</td></tr><tr><td>UperNet* [32]</td><td>ViT-Base</td><td>512 × 512</td><td>&gt;250</td><td>46.6</td><td>47.5</td></tr><tr><td>DPT*[7]</td><td>ViT-Base</td><td>512 × 512</td><td>219.8</td><td>47.2</td><td>47.9</td></tr><tr><td>SETR-MLA* [6]</td><td>ViT-Base</td><td>512 × 512</td><td>113.5</td><td>48.2</td><td>49.3</td></tr><tr><td>Segmenter* [8]</td><td>ViT-Base</td><td>512 × 512</td><td>129.6</td><td>49.0</td><td>50.0</td></tr><tr><td>StructToken [31]</td><td>ViT-Base</td><td>512 × 512</td><td>&gt;150</td><td>50.9</td><td>51.8</td></tr><tr><td>SegViT (Ours)</td><td>ViT-Base</td><td>512 × 512</td><td>120.9</td><td>51.3</td><td>53.0</td></tr><tr><td>DPT* [7]</td><td>ViT-Large†</td><td>640 × 640</td><td>479.0</td><td>49.2</td><td>49.5</td></tr><tr><td>UperNet* [32]</td><td>ViT-Larget</td><td>640 × 640</td><td>&gt;700</td><td>48.6</td><td>50.0</td></tr><tr><td>SETR-MLA [6]</td><td>ViT-Large</td><td>512 × 512</td><td>368.6</td><td>48.6</td><td>50.3</td></tr><tr><td>MCIBI [33]</td><td>ViT-Large</td><td>512 × 512</td><td>&gt;400</td><td>1</td><td>50.8</td></tr><tr><td>Segmenter [8]</td><td>ViT-Large†</td><td>640 × 640</td><td>671.8</td><td>51.8</td><td>53.6</td></tr><tr><td>StructToken [31]</td><td>ViT-Larget</td><td>640 × 640</td><td>&gt;700</td><td>52.8</td><td>54.2</td></tr><tr><td>SegViT (Shrunk, ours)</td><td>ViT-Large†</td><td>640 × 640</td><td>373.5</td><td>53.9</td><td>55.1</td></tr><tr><td>SegViT (ours)</td><td>ViT-Larget</td><td>640 × 640</td><td>637.9</td><td>54.6</td><td>55.2</td></tr></table>
+$5 5 . 2 \%$ in terms of mIoU with the ViT-Large backbone. It is $1 . 0 \%$ better than the recent StructToken [31] using the same backbone. Besides, our Shrunk version can also achieve a similar performance $5 5 . 1 \%$ with computational cost 373.5 GFLOPs which is much less than the ViT-Large backbone alone (612.3 GFLOPs).
+Results on COCO-Stuff-10K. Table 2 shows the result on the COCO-Stuff-10K dataset. Our method achieves $5 0 . 3 \%$ which is higher than the previous state-to-the-art StrucToken by $1 . 2 \%$ with less computational cost. Our Shrunk version achieves $4 9 . 4 \%$ with 224.8 GFLOPs, which is similar to the computational cost of a dilated ResNet-101 backbone but with much higher performance.
+Table 2: Experiment results on the COCO-Stuff-10K test. split. Following published methods, we report the results with multi-scale inference (denoted by ‘ms’). The GFLOPs is measured at single scale inference with a crop size of $5 1 2 \times 5 1 2$ .
+<table><tr><td>Method</td><td>Backbone</td><td>GFLOPs</td><td>mIoU (ms)</td></tr><tr><td>DANet [34]</td><td>Dilated-ResNet-101</td><td>289.3</td><td>39.7</td></tr><tr><td>MaskFormer[15]</td><td>ResNet-101-fpn</td><td>81.7</td><td>39.8</td></tr><tr><td>EMANet [35]</td><td>Dilated-ResNet-101</td><td>247.4</td><td>39.9</td></tr><tr><td>SpyGR [36]</td><td>ResNet-101-fpn</td><td>v80</td><td>39.9</td></tr><tr><td>OCRNet [3]</td><td>HRNetV2-W48</td><td>167.9</td><td>40.5</td></tr><tr><td>GINet [37]</td><td>JPU-ResNet-101</td><td>&gt;200</td><td>40.6</td></tr><tr><td>RecoNet [38]</td><td>Dilated-ResNet-101</td><td>&gt;200</td><td>41.5</td></tr><tr><td>ISNet [39]</td><td>Dilated-ResNeSt-101</td><td>228.3</td><td>42.1</td></tr><tr><td>MCIBI [33]</td><td>ViT-Large</td><td>&gt;380</td><td>44.9</td></tr><tr><td>StructToken [31]</td><td>ViT-Large</td><td>&gt;400</td><td>49.1</td></tr><tr><td>SegViT (Shrunk, ours)</td><td>ViT-Large</td><td>224.8</td><td>49.4</td></tr><tr><td>SegViT (ours)</td><td>ViT-Large</td><td>383.9</td><td>50.3</td></tr></table>
+Results on PASCAL-Context. Table 3 shows the results on the PASCAL-Context dataset. We follow HRNet [40] to evaluate our method and report the results under 59 classes (without background) and 60 classes (with background). $\mathrm { S e g V i T }$ reaches mIoU $6 5 . 3 \%$ and $5 9 . 3 \%$ respectively for those two metrics that outperform the state-of-the-art methods using the ViT backbones with less computational cost.
+Table 3: Expperiment results on the PASCAL-Context val. split. Following published methods, we report the results with multi-scale inference (denoted by ‘ms’). $\mathrm { m I o U _ { 5 9 } }$ : mIoU averaged over 59 classes (without background). $\mathrm { m I o U _ { 6 0 } }$ : mIoU averaged over 60 classes (59 classes plus background). Both metrics were used in the literature; and we report for the 60 classes. The GFLOPs is measured at single scale inference with a crop size of $4 8 0 \times 4 8 0$ .
+<table><tr><td>Method</td><td>Backbone</td><td>GFLOPs</td><td>mIoU59 (ms)</td><td>mIoU60 (ms)</td></tr><tr><td>RefineNet [41]</td><td>ResNet-152</td><td></td><td>=</td><td>47.3</td></tr><tr><td>UNet++ [42]</td><td>ResNet-101</td><td>=</td><td>47.7</td><td>1</td></tr><tr><td>PSPNet [11]</td><td>Dilated-ResNet-101</td><td>157.0</td><td>47.8</td><td></td></tr><tr><td>Ding et al. [43]</td><td>ResNet-101</td><td>=</td><td>51.6</td><td></td></tr><tr><td>EncNet [44]</td><td>Dilated-ResNet-101</td><td>192.1</td><td>52.6</td><td>=</td></tr><tr><td>HRNet [40]</td><td>HRNetV2-W48</td><td>82.7</td><td>54.0</td><td>48.3</td></tr><tr><td>NRD [45]</td><td>ResNet-101</td><td>42.9</td><td>54.1</td><td>49.0</td></tr><tr><td>GFFNet [46]</td><td>Dilated-ResNet-101</td><td>=</td><td>54.3</td><td>1</td></tr><tr><td>EfficientFCN [47]</td><td>ResNet-101</td><td>52.8</td><td>55.3</td><td></td></tr><tr><td>OCRNet [3]</td><td>HRNetV2-W48</td><td>143.9</td><td>56.2</td><td>=</td></tr><tr><td>SETR-MLA [6]</td><td>ViT-Large</td><td>318.5</td><td>1</td><td>55.8</td></tr><tr><td>Segmenter [8]</td><td>ViT-Large</td><td>346.2</td><td>1</td><td>59.0</td></tr><tr><td>SegViT (Shrunk, ours)</td><td>ViT-Large</td><td>186.9</td><td>63.7</td><td>57.4</td></tr><tr><td>SegViT (ours)</td><td>ViT-Large</td><td>321.6</td><td>65.3</td><td>59.3</td></tr></table>
+# 4.4 Ablation Study
+In this section, we conduct the ablation study to show the effectiveness of our proposed methods.
+Effect of the ATM module. Table 4 shows the effect of the ATM module. We set the SETR-naive as the baseline, which uses two $1 \times 1$ convolutions to get per-pixel classifications directly from the last layer of the ViT-Base transformer output. We can see that by applying the ATM module and supervise with a regular cross-entropy loss, ATM is capable of providing $0 . 5 \%$ of performance boost. However, it is more beneficial to decouple the classification and mask prediction process and use the mask and classification supervision separately ( $3 . 1 \%$ increase).
+Ablation of using different layers as input for SegViT. Table 5 shows the performance boost that multiple layers input can provide. We can see that the performance boost of feature maps from additional lower layers is obvious $( + 1 . 3 \% )$ . We then involved more layers of features and see further performance gains. We empirically choose to use three layers for its best performance.
+Table 4: Comparison between our proposed ATM module with other methods. ‘CE loss’ indicates the cross-entropy loss that is commonly used in semantic segmentation. The experiments are carried out on the ViT-Base backbone using ADE20K dataset.
+<table><tr><td>Decoder</td><td>Loss</td><td>mIoU (ss)</td></tr><tr><td>SETR</td><td>CE loss</td><td>46.5</td></tr><tr><td>ATM</td><td>CE loss</td><td>47.0 (+0.5)</td></tr><tr><td>ATM</td><td>Lmask loss</td><td>49.6 (+3.1)</td></tr></table>
+Table 5: Ablation results of using different layer inputs to the $\mathrm { S e g V i T }$ structure on ADE20K dataset using ViT-Base as the backbone. Involving multi-layer features can bring obvious performance gain.
+<table><tr><td></td><td>Used layers</td><td>mIoU (ss)</td></tr><tr><td>Single</td><td>[12]</td><td>49.6</td></tr><tr><td>Cascade</td><td>[6,12]</td><td>50.9 (+1.3)</td></tr><tr><td>Cascade</td><td>[6,8,12]</td><td>51.3 (+1.7)</td></tr><tr><td>Cascade</td><td>[3,6,9,12]</td><td>51.2 (+1.6)</td></tr></table>
+Ablation for the ATM Decoder. We conduct experiments to show the effectiveness of the proposed ATM decoder
+$\mathbf { S e g V i T }$ on hierarchical backbones. Shown in Table 6, the $\mathrm { S e g V i T }$ structure is also able to apply to hierarchical backbones. We choose the most competitive methods Maskformer [15] and Mask2former [48] for comparison. Results indicate that even though our method is not designed for hierarchical backbones, we can still achieve competitive performance while being efficient in terms of computational cost.
+Table 6: The experiments use the Swin-Tiny [18] backbone and are carried out on the ADE20K dataset. The GFLOPs are measured at single scale inference with a crop size of $5 1 2 \times 5 1 2$ . QD: query-based down-saumping. QU: query-based upsampling.
+<table><tr><td>Method</td><td>mIoU (ss)</td><td>GFLOPs</td></tr><tr><td>Maskformer [15]</td><td>46.7</td><td>57.3</td></tr><tr><td>Mask2former [48]</td><td>47.7</td><td>73.7</td></tr><tr><td>SegViT (Ours)</td><td>47.1</td><td>48.0</td></tr></table>
+Table 7: Ablation of the QD module in terms of the targets and methods to down-sample. The experiments are carried out on the ViT-Large backbone of ADE20K dataset.
+<table><tr><td>Applied to</td><td>Methods</td><td>mIoU (ss)</td></tr><tr><td>Q</td><td>Conv</td><td>44.5</td></tr><tr><td>Q,K,V</td><td>Nearest</td><td>52.6</td></tr><tr><td>Q</td><td>Nearest</td><td>53.9</td></tr></table>
+Ablation for the QD module. The motivation to use QD is to make use of the pre-train weights of the backbone. As in Table 7, if we use a stride 2 convolution with learnable parameters to downsample the query, it will destroy the pre-train weights and dramatically decrease the performance. If the down-sampling is applied to both Q and (K, V), there will be an inevitable loss in information during the down-sampling process which is reflected in the weaker performance. We found that applying $2 \times 2$ nearest down-sampling on query only for the QD module is the better option.
+Ablation of the components in Shrunk structure. Shown in Table 8, we studied the effect of each component (QD and QU) in the Shrunk structure. The results presented matches the structures illustrated in Fig. 3. When QD is applied, the performance decreases by $2 . 7 \%$ from the ‘Single’ ATM head. However, by applying QU, the performance is recovered. QD learns a non-linear downsampling by the attention mechanism between key and query. One query will attend to several keys. QU is used to preserve the resolution and at the same time provide low-level feature information. We can see that by using QD and QU jointly, the performance can be retained and the computational cost is reduced. ATM module can also be used as the decoder to form our Shrunk structure to further boost performance.
+Table 8: Ablation results of Shrunk version on the ADE20K dataset. The GFLOPs are measured at single scale inference with a crop size of $5 1 2 \times 5 1 2$ on ViT-Base backbone.
+<table><tr><td> Structure</td><td>QD</td><td>QU</td><td>Head</td><td>mIoU (ss)</td><td>GFLOPs</td></tr><tr><td>Single</td><td></td><td></td><td>SETR</td><td>46.5</td><td>107.3</td></tr><tr><td>Single</td><td></td><td></td><td>ATM</td><td>49.6 (+3.1)</td><td>115.8</td></tr><tr><td>Naive Shrunk</td><td>&lt;</td><td></td><td>ATM</td><td>46.9 (+0.4)</td><td>74.1</td></tr><tr><td>Shrunk</td><td></td><td>√</td><td>ATM</td><td>50.0 (+3.5)</td><td>97.1</td></tr></table>
+# 5 Conclusion
+We proposed an effective structure using plain ViT transformer backbones termed $\mathrm { S e g V i T }$ for the semantic segmentation task. For the first time, we utilize spatial information in attention maps for semantic segmentation. To implement this idea, we proposed an Attention-to-mask (ATM) module that can derive mask predictions during the attention calculation process. We show on a number of semantic segmentation benchmarks that our method is efficient and achieves state-of-the-art performance. We also proposed a Shrunk structure which is applied to the backbone and capable of reducing $4 0 \%$ of the computational cost while still maintaining competitive performance. We believe both structures can be strong paradigms, especially for semantic segmentation using ViT backbones. Last but not the least, our method still has some limitations. One of the limitations is that the large amount of GPU memory consumed by the global attention mechanism might not be supported by some devices, which might restrict the applicability of our structures.
+Acknowledgments C. Shen’s participation was in part supported by a major grant from Zhejiang Provincial Government. This work was also supported by the start-up funding of the University of Adelaide. [grant number 15130411]. This research was supported by Meituan.
+# References
+IEEE Conf. Comp. Vis. Patt. Recogn., pp. 3431–3440, 2015.
+[2] L.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. Yuille, “Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 40, no. 4, pp. 834–848, 2017.
+[3] Y. Yuan, X. Chen, and J. Wang, “Object-contextual representations for semantic segmentation,” in Proc. Eur. Conf. Comp. Vis., pp. 173–190, Springer, 2020.
+[4] T.-Y. Lin, P. Dollár, R. Girshick, K. He, B. Hariharan, and S. Belongie, “Feature pyramid networks for object detection,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 2117–2125, 2017.
+[5] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, et al., “An image is worth 16x16 words: Transformers for image recognition at scale,” Proc. Int. Conf. Learn. Representations, 2021.
+[6] S. Zheng, J. Lu, H. Zhao, X. Zhu, Z. Luo, Y. Wang, Y. Fu, J. Feng, T. Xiang, P. H. Torr, et al., “Rethinking semantic segmentation from a sequence-to-sequence perspective with transformers,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 6881–6890, 2021.
+[7] R. Ranftl, A. Bochkovskiy, and V. Koltun, “Vision transformers for dense prediction,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 12179–12188, 2021.
+[8] R. Strudel, R. Garcia, I. Laptev, and C. Schmid, “Segmenter: Transformer for semantic segmentation,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 7262–7272, 2021.
+[9] C. Farabet, C. Couprie, L. Najman, and Y. LeCun, “Learning hierarchical features for scene labeling,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 35, no. 8, pp. 1915–1929, 2012.
+[10] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 3431–3440, 2015.
+[11] H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid scene parsing network,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., 2017.
+[12] L.-C. Chen, Y. Zhu, G. Papandreou, F. Schroff, and H. Adam, “Encoder-decoder with atrous separable convolution for semantic image segmentation,” in Proc. Eur. Conf. Comp. Vis., pp. 801–818, 2018.
+[13] J. Fu, J. Liu, J. Jiang, Y. Li, Y. Bao, and H. Lu, “Scene segmentation with dual relation-aware attention network,” IEEE Trans. Neural Netw. & Learn. Syst., vol. 32, no. 6, pp. 2547–2560, 2020.
+[14] C. Yu, J. Wang, C. Gao, G. Yu, C. Shen, and N. Sang, “Context prior for scene segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 12416–12425, 2020.
+[15] B. Cheng, A. Schwing, and A. Kirillov, “Per-pixel classification is not all you need for semantic segmentation,” Proc. Advances in Neural Inf. Process. Syst., vol. 34, 2021.
+[16] W. Zhang, J. Pang, K. Chen, and C. C. Loy, “K-net: Towards unified image segmentation,” Proc. Advances in Neural Inf. Process. Syst., vol. 34, 2021.
+[17] W. Wang, E. Xie, X. Li, D.-P. Fan, K. Song, D. Liang, T. Lu, P. Luo, and L. Shao, “Pyramid vision transformer: A versatile backbone for dense prediction without convolutions,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 568–578, 2021.
+[18] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. Guo, “Swin transformer: Hierarchical vision transformer using shifted windows,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 10012–10022, 2021.
+[19] X. Chu, Z. Tian, Y. Wang, B. Zhang, H. Ren, X. Wei, H. Xia, and C. Shen, “Twins: Revisiting the design of spatial attention in vision transformers,” Proc. Advances in Neural Inf. Process. Syst., vol. 34, 2021.
+[20] E. Xie, W. Wang, Z. Yu, A. Anandkumar, J. M. Alvarez, and P. Luo, “Segformer: Simple and efficient design for semantic segmentation with transformers,” Proc. Advances in Neural Inf. Process. Syst., vol. 34, 2021.
+[21] Y.-H. Wu, Y. Liu, X. Zhan, and M.-M. Cheng, “P2t: Pyramid pooling transformer for scene understanding,” IEEE Trans. Pattern Anal. Mach. Intell., 2022.
+[22] T.-Y. Lin, P. Goyal, R. Girshick, K. He, and P. Dollár, “Focal loss for dense object detection,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 2980–2988, 2017.
+[23] F. Milletari, N. Navab, and S.-A. Ahmadi, “V-net: Fully convolutional neural networks for volumetric medical image segmentation,” in 3DV, pp. 565–571, IEEE, 2016.
+[24] N. Carion, F. Massa, G. Synnaeve, N. Usunier, A. Kirillov, and S. Zagoruyko, “End-to-end object detection with transformers,” in Proc. Eur. Conf. Comp. Vis., pp. 213–229, Springer, 2020.
+[25] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” Proc. Advances in Neural Inf. Process. Syst., vol. 30, 2017.
+[26] B. Zhou, H. Zhao, X. Puig, S. Fidler, A. Barriuso, and A. Torralba, “Scene parsing through ade20k dataset,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 633–641, 2017.
+[27] H. Caesar, J. Uijlings, and V. Ferrari, “Coco-stuff: Thing and stuff classes in context,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 1209–1218, 2018.
+[28] MMSegmentation, “MMSegmentation: OpenMMLab semantic segmentation toolbox and benchmark.” https://github.com/open-mmlab/mmsegmentation, 2020.
+[29] R. Mottaghi, X. Chen, X. Liu, N.-G. Cho, S.-W. Lee, S. Fidler, R. Urtasun, and A. Yuille, “The role of context for object detection and semantic segmentation in the wild,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 891–898, 2014.
+[30] A. Steiner, A. Kolesnikov, X. Zhai, R. Wightman, J. Uszkoreit, and L. Beyer, “How to train your vit? data, augmentation, and regularization in vision transformers,” arXiv: Comp. Res. Repository, 2021.
+[31] F. Lin, Z. Liang, J. He, M. Zheng, S. Tian, and K. Chen, “Structtoken: Rethinking semantic segmentation with structural prior,” arXiv: Comp. Res. Repository, 2022.
+[32] T. Xiao, Y. Liu, B. Zhou, Y. Jiang, and J. Sun, “Unified perceptual parsing for scene understanding,” in Proc. Eur. Conf. Comp. Vis., pp. 418–434, 2018.
+[33] Z. Jin, T. Gong, D. Yu, Q. Chu, J. Wang, C. Wang, and J. Shao, “Mining contextual information beyond image for semantic segmentation,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 7231–7241, 2021.
+[34] J. Fu, J. Liu, H. Tian, Y. Li, Y. Bao, Z. Fang, and H. Lu, “Dual attention network for scene segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 3146–3154, 2019.
+[35] X. Li, Z. Zhong, J. Wu, Y. Yang, Z. Lin, and H. Liu, “Expectation-maximization attention networks for semantic segmentation,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 9167–9176, 2019.
+[36] X. Li, Y. Yang, Q. Zhao, T. Shen, Z. Lin, and H. Liu, “Spatial pyramid based graph reasoning for semantic segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 8950–8959, 2020.
+[37] T. Wu, Y. Lu, Y. Zhu, C. Zhang, M. Wu, Z. Ma, and G. Guo, “Ginet: Graph interaction network for scene parsing,” in Proc. Eur. Conf. Comp. Vis., pp. 34–51, Springer, 2020.
+[38] W. Chen, X. Zhu, R. Sun, J. He, R. Li, X. Shen, and B. Yu, “Tensor low-rank reconstruction for semantic segmentation,” in Proc. Eur. Conf. Comp. Vis., pp. 52–69, Springer, 2020.
+[39] Z. Jin, B. Liu, Q. Chu, and N. Yu, “Isnet: Integrate image-level and semantic-level context for semantic segmentation,” in Proc. IEEE Int. Conf. Comp. Vis., pp. 7189–7198, 2021.
+[40] K. Sun, Y. Zhao, B. Jiang, T. Cheng, B. Xiao, D. Liu, Y. Mu, X. Wang, W. Liu, and J. Wang, “Highresolution representations for labeling pixels and regions,” arXiv: Comp. Res. Repository, 2019.
+[41] G. Lin, A. Milan, C. Shen, and I. Reid, “RefineNet: Multi-path refinement networks for high-resolution semantic segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 1925–1934, 2017.
+[42] Z. Zhou, M. M. R. Siddiquee, N. Tajbakhsh, and J. Liang, “Unet++: A nested U-net architecture for medical image segmentation,” in Proc. Deep Learning in Medical Image Analysis Workshop, pp. 3–11, 2018.
+[43] H. Ding, X. Jiang, B. Shuai, A. Q. Liu, and G. Wang, “Context contrasted feature and gated multi-scale aggregation for scene segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 2393–2402, 2018.
+[44] H. Zhang, K. Dana, J. Shi, Z. Zhang, X. Wang, A. Tyagi, and A. Agrawal, “Context encoding for semantic segmentation,” in Proc. IEEE Conf. Comp. Vis. Patt. Recogn., pp. 7151–7160, 2018.
+[45] B. Zhang, Z. Tian, C. Shen, et al., “Dynamic neural representational decoders for high-resolution semantic segmentation,” Proc. Advances in Neural Inf. Process. Syst., vol. 34, 2021.
+[46] X. Li, H. Zhao, L. Han, Y. Tong, S. Tan, and K. Yang, “Gated fully fusion for semantic segmentation,” in Proc. AAAI Conf. Artificial Intell., vol. 34, pp. 11418–11425, 2020.
+[47] J. Liu, J. He, J. Zhang, J. Ren, and H. Li, “EfficientFCN: Holistically-guided decoding for semantic segmentation,” in Proc. Eur. Conf. Comp. Vis., 2020.
+[48] B. Cheng, I. Misra, A. G. Schwing, A. Kirillov, and R. Girdhar, “Masked-attention mask transformer for universal image segmentation,” 2022.
+# Checklist
+1. For all authors...
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
+(b) Did you describe the limitations of your work? [Yes]
+(c) Did you discuss any potential negative societal impacts of your work? [No] We conduct experiments on a fundamental task of semantic segmentation. This technique may be used for editing fake images to mislead the public if being used by someone who has ulterior motives.
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+2. If you are including theoretical results...
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+3. If you ran experiments...
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We included the code for the main experiments in the supplemental materials. All the code will be released upon acceptance.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See implementation details.
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We fix the random seed and other random operators
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See the implementation details and GFLOPs in tables.
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+(a) If your work uses existing assets, did you cite the creators? [Yes]
+(b) Did you mention the license of the assets? [No] The license can be found in their own homepages.
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our trained model in the supplemental material.
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] All the data are public benchmarks.
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] All the data are public benchmarks.
+5. If you used crowdsourcing or conducted research with human subjects...
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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