Add files using upload-large-folder tool

parse/dev/1W8UwXAQubL/1W8UwXAQubL.md

@@ -0,0 +1,235 @@
+# Multi-Agent Reinforcement Learning is A Sequence Modeling Problem
+
+Muning Wen1,2, Jakub Grudzien Kuba3, Runji Lin4, Weinan Zhang1, Ying Wen1, Jun Wang2,5, Yaodong Yang6,7,† 1Shanghai Jiao Tong University, 2Digital Brain Lab, 3University of Oxford, 4Institute of Automation, Chinese Academy of Science, 5University College London, 6Beijing Institute for General AI, 7Institute for AI, Peking University
+
+# Abstract
+
+Large sequence models (SM) such as GPT series and BERT have displayed outstanding performance and generalization capabilities in natural language process, vision and recently reinforcement learning. A natural follow-up question is how to abstract multi-agent decision making also as an sequence modeling problem and benefit from the prosperous development of the SMs. In this paper, we introduce a novel architecture named Multi-Agent Transformer (MAT) that effectively casts cooperative multi-agent reinforcement learning (MARL) into SM problems wherein the objective is to map agents’ observation sequences to agents’ optimal action sequences. Our goal is to build the bridge between MARL and SMs so that the modeling power of modern sequence models can be unleashed for MARL. Central to our MAT is an encoder-decoder architecture which leverages the multi-agent advantage decomposition theorem to transform the joint policy search problem into a sequential decision making process; this renders only linear time complexity for multiagent problems and, most importantly, endows MAT with monotonic performance improvement guarantee. Unlike prior arts such as Decision Transformer fit only precollected offline data, MAT is trained by online trial and error from the environment in an on-policy fashion. To validate MAT, we conduct extensive experiments on StarCraftII, Multi-Agent MuJoCo, Dexterous Hands Manipulation, and Google Research Football benchmarks. Results demonstrate that MAT achieves superior performance and data efficiency compared to strong baselines including MAPPO and HAPPO. Furthermore, we demonstrate that MAT is an excellent few-short learner on unseen tasks regardless of changes in the number of agents. See our project page at https://sites.google.com/view/multi-agent-transformer(1)
+
+# 1 Introduction
+
+Multi-agent reinforcement learning (MARL) [44, 8] is a challenging problem for its difficulty which arises not only from identifying each individual agent’s policy improvement direction, but also from combining agents’ policy updates jointly which should be beneficial for the whole team. Recently, such difficulty in multi-agent learning has been eased owing to the introduction of centralized training for decentralized execution (CTDE) [11, 45], which allows agents to access the global information and opponents’ actions during the training phase. This framework enables successful developments of methods that directly inherit single-agent algorithms. For examples, COMA replaces the policy-gradient (PG) estimate with a multi-agent PG (MAPG) counterpart [11], MADDPG extends deterministic policy gradient into multi-agent settings with a centralized critic [20, 34], QMIX leverages deep Qnetworks for decentralized agents and introduces a centralized mixing network for Q-value decomposition [29, 36, 26]. MAPPO endowing all agents with the same set of parameters and then training by trust-region methods [46]. PR2 [42] and GR2 [43] methods conduct recursive reasoning under the CTDE framework. These methods, however, cannot cover the whole complexity of multi-agent interactions; in fact, some of them are shown to fail in the simplest cooperative task [15]. To resolve this issue, the multi-agent advantage decomposition theorem was proposed [15, Theorem 1] which captures how different agents contribute to the return and provides an intuition behind the emergence of cooperation through a sequential decision making process scheme. Based on it, HATRPO and HAPPO algorithms [15, 17, 16] were derived which, thanks to the decomposition theorem and sequential update scheme, established new state-of-the-art methods for MARL. However, their limitation is that the agents’ policies are unaware of the purpose to develop cooperation and still rely on a carefully handcrafted maximization objective. Ideally, a team of agents should be aware of the jointness of their training by design, thereby following a holistic and effective paradigm—an ideal solution that is yet to be proposed.
+
+In recent years, sequence models (SM) have made a substantial progress in natural language processing (NLP) [27]. For example, GPT series [3] and BERT models [9], built on autoregressive SMs, have demonstrated remarkable performance on a wide range of downstream tasks and achieved strong performance on few-shot generalization tasks. Although SM are mostly used in language tasks due to its natural fitting with the sequential property of languages, the sequential approaches are not confined to NLP only, but is instead a widely applicable general foundation model [2]. For example, in computer vision (CV), one can split an image into sub-images and align them in a sequence as if they were tokens in NLP tasks [9, 10, 12]. Although the idea of solving CV tasks by SM is straightforward, it serves as the foundation to some of the best-performing CV algorithms [38, 41, 39]. Furthermore, sequential methods are starting to spawn powerful multi-modal visual language models such as Flamingo [1], DALL-E [28], and GATO [30] in the recent past.
+
+Coming with effective and expressive network architectures such as Transformer [40], sequence modeling techniques have also attracted tremendous attention from the RL community, which results in a series of successful offline RL developments based on the Transformer architecture [5, 14, 30, 23]. These methods show great potentials in tackling some of the most fundamental RL training problems, such as long-horizon credit assignment and reward sparsity [37, 24, 25]. For example, by training autoregressive models on pre-collected offline data in a purely supervised way, Decision Transformer [5] bypasses the need for computing cumulative rewards through dynamic programming, but rather generates future actions conditioning on the desired returns, past states and actions. Despite their remarkable successes, none of these methods have been designed to model the most difficult (also unique to MARL) aspect of multi-agent systems—the agents’ interactions. In fact, if we were to simply endow all agents with a Transformer policy and train them independently, their joint performance still could not be guaranteed to improve [15, Proposition 1]. Therefore, while a myriad of powerful SMs are available, MARL—an area that would greatly benefit from SM—has not truly taken advantage of their performance benefit. The key research question to ask is then
+
+# How can we model MARL problems by sequence models ?
+
+In this paper, we take several steps to provide an affirmative answer to the above research question. Our goal is to enhance MARL studies with powerful sequential modeling techniques. To fulfill that, we start by proposing a novel MARL training paradigm which establishes the connection between cooperative MARL problems and sequence modeling problems. Central to the new paradigm are the multi-agent advantage decomposition theorem and sequential update scheme, which effectively transform multi-agent joint policy optimization into a sequential policy search process. As a natural outcome of our findings, we introduce Multi-Agent Transformer (MAT), an encoder-decoder architecture that implements generic MARL solutions through SM. Unlike Decision Transformer [5], MAT is trained online based on trials and errors in an on-policy fashion; therefore, it does not require collecting demonstrations upfront. Importantly, the implementation of the multi-agent advantage decomposition theorem ensures MAT to enjoy monotonic performance improvement guarantee during training. MAT establishes a new state-of-the-art baseline model for cooperative MARL tasks. We justify such a claim by evaluating MAT on the benchmarks of StarCraftII, Multi-Agent MuJoCo, Dexterous Hands Manipulation, and Google Research Football; results show that MAT achieves superior performance over strong baselines, such as MAPPO [46], HAPPO [15], QMIX [29] and UPDeT [13]. Finally, we show that MAT possesses great potentials in task generalizations, which holds regardless of the agent number in new tasks.
+
+# 2 Preliminaries
+
+In this section, we first introduce the cooperative MARL problem formulation and the multi-agent advantage decomposition theorem, which serves as the cornerstone of our work. We then review existing MARL methods that relate to MAT, and finally familiarize the reader with the Transformer.
+
+# 2.1 Problem Formulation
+
+Cooperative MARL problems are often modeled by Markov games $\langle \mathcal { N } , \mathcal { O } , \pmb { \mathcal { A } } , R , P , \gamma \rangle$ [19]. ${ \mathcal { N } } =$ $\{ 1 , \ldots , n \}$ is the set of agents, $\begin{array} { r } { \mathcal { O } = \prod _ { i = 1 } ^ { n } \mathcal { O } ^ { i } } \end{array}$ is the product of local observation spaces of the i=1 agents, namely the joint observation space, $\begin{array} { r } { \pmb { \mathcal { A } } = \prod _ { i = 1 } ^ { n } \pmb { \mathcal { A } } ^ { i } } \end{array}$ is the product of the agents’ action spaces, namely the joint action space, $R : \mathcal { O } \times \pmb { A }  [ - \bar { R } _ { \operatorname* { m a x } } , R _ { \operatorname* { m a x } } ]$ is the joint reward function, $P : \mathcal { O } \times \pmb { \mathcal { A } } \times \mathcal { O }  \mathbb { R }$ is the transition probability function, and $\gamma \in [ 0 , 1 )$ is the discount factor. At time step $t \in \mathbb { N }$ , an agent $i \in \mathcal N$ observes an observation $\mathbf { o } _ { t } ^ { i } \in \mathcal { O } ^ { i }$ (2) $\mathbf { \tilde { \omega } } ( o = \left( o ^ { 1 } , \ldots , o ^ { n } \right)$ is a “joint” observation) and takes an action $\mathbf { a } _ { t } ^ { i }$ according to its policy $\pi ^ { i }$ , which is the $i ^ { \mathrm { { t h } } }$ component of the agents’ joint policy $\pi$ . At each time step, all agents take actions simultaneously based on their observation with no sequential dependencies. The transition kernel $P$ and the joint policy induce the (improper) marginal observation distribution $\begin{array} { r } { \rho _ { \pi } ( \cdot ) \triangleq \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } \mathrm { P r } ( \mathbf { o } _ { t } = o | \pi ) } \end{array}$ . At the end of each time step, the whole team receives a joint reward $R ( \mathbf { o } _ { t } , \mathbf { a } _ { t } )$ and observe $\mathbf { o } _ { t + 1 }$ , whose probability distribution is $P ( \cdot | \mathbf { o } _ { t } , \mathbf { a } _ { t } )$ . Following this process infinitely long, the agents earn a discounted cumulative return of $\begin{array} { r } { R ^ { \gamma } \triangleq \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( \mathbf { o } _ { t } , \mathbf { a } _ { t } ) } \end{array}$ .
+
+# 2.2 Multi-Agent Advantage Decomposition Theorem
+
+The agents evaluate the value of actions and observations with $Q _ { \pi } ( o , a )$ and $V _ { \pi } ( o )$ , defined as
+
+$$
+\begin{array} { r } { \begin{array} { c } { Q _ { \pi } ( o , a ) \triangleq \mathbb { E } _ { \mathbf { o } _ { 1 : \infty } \sim P , \mathbf { a } _ { 1 : \infty } \sim \pi } \left[ R ^ { \gamma } | \mathbf { o } _ { 0 } = o , \mathbf { a } _ { 0 } = a \right] , } \\ { V _ { \pi } ( o ) \triangleq \mathbb { E } _ { \mathbf { a } _ { 0 } \sim \pi , \mathbf { o } _ { 1 : \infty } \sim P , \mathbf { a } _ { 1 : \infty } \sim \pi } \left[ R ^ { \gamma } | \mathbf { o } _ { 0 } = o \right] . } \end{array} } \end{array}
+$$
+
+The jointness of the objective causes difficulties associated with the credit assignment problem— having received a shared reward, individual agents are unable to deduce their own contribution to the team’s success or failure [4]. Indeed, applying traditional RL methods which simply employ the above value functions leads to obstacles in training, such as the growing variance of multi-agent policy gradient (MAPG) estimates [17]. Hence, to tackle these, notions of local value functions [21] and counterfactual baselines [11] have been developed. In this paper, we work with the most general notions of this kind—the multi-agent observation-value functions [15]. That is, for arbitrary disjoint, ordered subsets of agents $i _ { 1 : m } = \overline { { { \{ i _ { 1 } , \ldots , i _ { m } \} } } }$ and $j _ { 1 : h } = \{ j _ { 1 } , \dots , j _ { h } \}$ , for $m , h \le n$ , we define the multi-agent observation-value function by
+
+$$
+Q _ { \pi } ( o , \pmb { a } ^ { i _ { 1 : m } } ) \triangleq \mathbb { E } \big [ R ^ { \gamma } | \mathbf { o } _ { 0 } ^ { i _ { 1 : n } } = o , \mathbf { a } _ { 0 } ^ { i _ { 1 : m } } = \pmb { a } ^ { i _ { 1 : m } } \big ] ,
+$$
+
+which recovers the original state-action value function in Equation (1) when $m = n$ , and the original observation-value function when $m = 0$ (i.e., when the set $i _ { 1 : m }$ is empty). Based on Equation (2), we can further measure the contribution of a chosen subset of agents to the joint return and define the multi-agent advantage function by
+
+$$
+A _ { \pi } ^ { i _ { 1 } ; m } \big ( \pmb { o } , \pmb { a } ^ { j _ { 1 } ; h } , \pmb { a } ^ { i _ { 1 : m } } \big ) \triangleq Q _ { \pi } ^ { j _ { 1 } ; h , ~ i _ { 1 : m } } \big ( \pmb { o } , \pmb { a } ^ { j _ { 1 : h } } , \pmb { a } ^ { i _ { 1 : m } } \big ) - Q _ { \pi } ^ { j _ { 1 : h } } \big ( \pmb { o } , \pmb { a } ^ { j _ { 1 : h } } \big ) .
+$$
+
+The above quantity describes how much better/worse than average the joint action $\textbf { \em a }$ will be if agents $i _ { 1 : m }$ take the joint action $\pmb { a } ^ { i _ { 1 : m } }$ , once $j _ { 1 : h }$ have taken $\pmb { a } ^ { j _ { 1 : h } }$ . Again, when $h = 0$ , the advantage compares the value of $\pmb { a } ^ { i _ { 1 : m } }$ to the baseline value function of the whole team. This value-functional representation of agents’ actions enables studying interactions between them, as well to decompose the joint value function signal, thus helping alleviate the severity of the credit assignment problem [29, 35, 22]. The insights of Equation (3) is accomplished by means of the following theorem.
+
+Theorem 1 (Multi-Agent Advantage Decomposition [17]). Let $i _ { 1 : n }$ be a permutation of agents. Then, for any joint observation $\pmb { o } = \pmb { o } \in \mathcal { O }$ and joint action $\pmb { a } = \pmb { a } ^ { i _ { 1 : n } } \in \mathcal { A }$ , the following equation always holds with no further assumption needed,
+
+$$
+A _ { \pi } ^ { i _ { 1 : n } } \left( o , a ^ { i _ { 1 : n } } \right) = \sum _ { m = 1 } ^ { n } A _ { \pi } ^ { i _ { m } } \left( o , a ^ { i _ { 1 : m - 1 } } , a ^ { i _ { m } } \right) .
+$$
+
+Importantly, this theorem provides an intuition guiding the choice of incrementally improving actions. Suppose that agent $i _ { 1 }$ picks an action $a ^ { i _ { 1 } }$ with positive advantage, $A _ { \pi } ^ { i _ { 1 } } ( o , a ^ { i _ { 1 } } ) \dot { > } 0$ . Then, imagine that for all $j = 2 , \dots , n$ , agent $i _ { j }$ knows the joint action $\mathbf { \Delta } a ^ { i _ { 1 : j - 1 } }$ of its predecessors. In this case, it can choose an action $a ^ { i _ { j } }$ for which the advantage $A _ { \pi } ^ { i _ { j } } ( o , \pmb { a } ^ { i _ { 1 : j - 1 } } , a ^ { i _ { j } } )$ is positive. Altogether, the theorem assures that the joint action $\pmb { a } ^ { i _ { 1 : n } }$ has positive advantage. Furthermore, notice that the joint the complexity of this search is additive, Pni=1 |Ai |, in the sizes of the action spaces. If we were to action has been chosen in perform the search directly in the joint action space, we would browse a set of multiplicative size, steps, each of which searched an individual agent’s action space. Hence, $\begin{array} { r } { \left| \mathcal { A } \right| = \prod _ { i = 1 } ^ { n } \left| \mathcal { A } ^ { i } \right| } \end{array}$ . Later, we will build upon this insight to design a SM that optimizes joint policies efficiently, agent by agent, without the necessity of considering the joint action space at once.
+
+# 2.3 Existing Methods in MARL
+
+We now briefly summarize two state-of-the-art MARL algorithms. Both of them build upon Proximal Policy Optimization (PPO) [33]—a RL method famous for its simplicity and its performance stability.
+
+MAPPO [46] is the first, and the most direct, approach for applying PPO in MARL. It equips all agents with one shared set of parameters and use agents’ aggregated trajectories for the shared policy’s update; at iteration $k + 1$ , it optimizes the policy parameter $\theta _ { k + 1 }$ by maximizing the clip objective of
+
+$$
+\sum _ { i = 1 } ^ { n } \mathbb { E } _ { \mathbf { o } \sim \rho _ { \pi _ { \theta _ { k } } } , \mathbf { a } \sim \pi _ { \theta _ { k } } } \left[ \operatorname* { m i n } \left( \frac { \pi _ { \theta } ( \mathbf { a } ^ { i } | \mathbf { o } ) } { \pi _ { \theta _ { k } } ( \mathbf { a } ^ { i } | \mathbf { o } ) } A _ { \pi _ { \theta _ { k } } } ( \mathbf { o } , \mathbf { a } ) , \mathrm { c l i p } \left( \frac { \pi _ { \theta } ( \mathbf { a } ^ { i } | \mathbf { o } ) } { \pi _ { \theta _ { k } } ( \mathbf { a } ^ { i } | \mathbf { o } ) } , 1 \pm \epsilon \right) A _ { \pi _ { \theta _ { k } } } ( \mathbf { o } , \mathbf { a } ) \right) \right] ,
+$$
+
+where the clip operator clips the input value (if necessary) so that it stays within the interval $[ 1 - \epsilon , 1 + \epsilon ]$ . However, enforcing parameter sharing is equivalent to putting a constraint $\theta ^ { i } = \theta ^ { j } , \forall i , j \in \mathcal { N }$ on the joint policy space, which can lead to an exponentially-worse sub-optimal outcome [15]. This motivates a more principled development of heterogeneous-agent trust-region methods, e.g., HAPPO.
+
+HAPPO [15] is currently one of the SOTA algorithm that fully leverages Theorem (1) to implement multi-agent trust-region learning with monotonic improvement guarantee. During an update, the agents choose a permutation $i _ { 1 : n }$ at random, and then following the order in the permutation, every agent $i _ { m }$ picks $\bar { \pi _ { \mathrm { n e w } } ^ { i _ { m } } } = \pi ^ { i _ { m } }$ that maximizes the objective of
+
+$$
+\mathbb { E } _ { \mathbf { o } \sim \rho _ { \pi _ { \mathrm { o d d } } } , \mathbf { a } ^ { i _ { 1 : m } } - 1 \sim \pi _ { \mathrm { n e r } } ^ { i _ { 1 : m } - 1 } , \mathbf { a } ^ { i _ { m } } \sim \pi _ { \mathrm { o d d } } ^ { i _ { m } } } \left[ \operatorname* { m i n } \left( \mathbf { r } ( \pi ^ { i _ { m } } ) A _ { \pi _ { \mathrm { o d d } } } ^ { i _ { 1 : m } } ( \boldsymbol { o } , \mathbf { a } ^ { i _ { 1 : m } } ) , \mathrm { c l i p } ( \mathbf { r } ( \pi ^ { i _ { m } } ) , 1 \pm \epsilon ) A _ { \pi _ { \mathrm { o d d } } } ^ { i _ { 1 : m } } ( \mathbf { o } , \mathbf { a } ^ { i _ { 1 : m } } ) \right) \right] ,
+$$
+
+where $\mathbf { r } ( \pi ^ { i _ { m } } ) = \pi ^ { i _ { m } } ( \mathbf { a } ^ { i _ { m } } | \mathbf { o } ) / \pi _ { \mathrm { o l d } } ^ { i _ { m } } \left( \mathbf { a } ^ { i _ { m } } | \mathbf { o } \right)$ . Note that the expectation is taken over the newly-updated previous agents’ policies, i.e, $\pi _ { \mathrm { n e w } } ^ { i _ { 1 : m - 1 } }$ ; this reflects an intuition that, under Theorem (1), the agent $i _ { m }$ reacts to its preceding agents $i _ { 1 : m - 1 }$ . However, one drawback of HAPPO is that agent’s policies has to follow the sequential update scheme in the permutation, thus it cannot be run in parallel.
+
+# 2.4 The Transformer Model
+
+Transformer [40] was originally designed for machine translation tasks (e.g., input English, output French). It maintains an encoder-decoder structure, where the encoder maps an input sequence of tokens to latent representations and then the decoder generates a sequence of desired outputs in an auto-regressive manner wherein at each step of inference, the Transformer takes all previously generated tokens as the input. One of the most essential component in Transformer is the scaled dot-product attention, which captures the interrelationship of input sequences. The attention function is written as Attention $\begin{array} { r } { \mathrm { \mathrm { \Omega } } _ { ^ { 1 } } ( \mathbf { Q } , \mathbf { K } , \mathbf { V } ) = \operatorname { s o f t m a x } \big ( \frac { \mathbf { Q } \mathbf { K } ^ { T } } { \sqrt { d _ { k } } } \big ) \mathbf { V } } \end{array}$  V, where the Q, K, V corresponds to the vector of queries, keys and values, which can be learned during training, and the $d _ { k }$ represent the dimension of $\mathbf { Q }$ and $\mathbf { K }$ . Self-attentions refer to cases when $\mathbf { Q } , \mathbf { K } , \mathbf { V }$ share the same set of parameters.
+
+![](images/44c48541656c918c52b627f3e9d5d3661d597e84653999bdf5d132f8ba8df7a9.jpg)  
+Figure 1: Conventional multi-agent learning paradigm (left) wherein all agents take actions simultaneously vs. the multi-agent sequential decision paradigm (right) where agents take actions by following a sequential order, each agent accounts for decisions from preceding agents as red arrows suggest.
+
+Inspired by the attention mechanism, UPDeT [13] handles various observation sizes by decoupling each agent’s observations into a sequence of observation-entities, matching them with different actiongroups, and modeling the relationship between the matched observation-entities with a Transformerbased function for better representation learning in MARL problems. Apart from this, based on the sequential property described in the Theorem (1) and the principle behind HAPPO [15], it is intuitive to think about another Transformer-based implementation for multi-agent trust-region learning. By treating a team of agents as a sequence, the Transformer architecture allows us to model teams of agents with variable numbers and types, while avoiding drawbacks of MAPPO/HAPPO. We will describe in more details how a cooperative MARL problem can be solved by a sequence model.
+
+# 3 The Surprising Connection Between MARL and Sequence Models
+
+To establish the connection between MARL and sequence models, Theorem (1) provides a new angle of understanding the MARL problem from a SM perspective. If each agent knows its predecessors’ actions with an arbitrary decision order, the sum of agents’ local advantages $A _ { \pi } ^ { i _ { j } } \left( o , \pmb { a } ^ { i _ { 1 : m - 1 } } , a ^ { i _ { m } } \right)$ will be exactly equal to the joint advantages $A _ { \pi } ^ { i _ { 1 : n } } ( o , \pmb { a } ^ { i _ { 1 : n } } )$ . This orderly decision setting across agents simplifies the update of their joint policy, where maximizing each agent’s own local advantage is equivalent to maximizing the joint advantage. As such, agents do not need to worry about interference from other agents anymore during the policy update; the local advantage functions have already captured the relationship between agents. This property revealed by Theorem (1) inspires us to propose a multi-agent sequential decision paradigm for MARL problems as show in Figure (1), where we assign agents with an arbitrary decision order (one permutation for each iteration); each agent can access its predecessors’ behaviors, based on which it then takes the optimal decision. This sequential paradigm motivates us to leverage a sequential model, e.g., Transformer, to explicitly capture the sequential relationship between agents described in Theorem (1).
+
+Underpinned by Theorem (1), sequence modeling reduces the complexity growth of MARL problems with the number of agents from multiplicative to additive, thus rendering linear complexity. With the help of the Transformer architecture, we can model policies of heterogeneous agents with an unified network but treat each agent discriminatively with different position, and thus ensuring high sample efficiency while avoiding the exponentially-worse outcome that MAPPO is facing. Besides, in order to guarantee the monotonic improvement of joint policies, HAPPO has to update each policy one-by-one during training, by leveraging previous update results of $\pi ^ { i _ { 1 } } , . . . , \pi ^ { i _ { m - 1 } }$ to improve $\pi ^ { i _ { m } }$ , which becomes critical in computational efficiency at large size of agents. By contramechanism of Transformer architectures allows for batching the ground truth actions $a _ { t } ^ { i _ { 0 } } , . . . , a _ { t } ^ { i _ { n - 1 } }$ ain1 onin the buffer to predict $a _ { t } ^ { i _ { 1 } } , . . . , a _ { t } ^ { i _ { n } }$ and update policies simultaneously, which significantly improves the training speed and makes it feasible for large size of agents. Furthermore, in cases that the number and the type of agents are different, SM can incorporates them into an unified solution through its capability on modeling sequences with flexible sequence length, rather than treat different agent numbers as different tasks. To realize the above idea, we introduce a practical architecture named Multi-Agent Transformer in the next section.
+
+![](images/7d48b9cbf47618a759b37dc1e10c48b8ef89fed8192a3d059497153e41e166e1.jpg)  
+Figure 2: The encoder-decoder architecture of MAT. At each time step, the encoder takes in a sequence of agents’ observations and encodes them into a sequence of latent representations, which is then passed into the decoder. The decoder generate each agent’s optimal action in a sequential and auto-regressive manner. The masked attention blocks ensures agents can only access its preceding agents’ actions during training. We list the full pseudocode of MAT in Appendix A and a video that shows the dynamic data flow of MAT in https://sites.google.com/view/multi-agent-transformer.
+
+# 4 The Multi-Agent Transformer
+
+To implement the sequence modeling paradigm for MARL, our solution is Multi-Agent Transformer (MAT). The idea of applying the Transformer architecture comes from the fact that the mapping between the input of agents’ observation sequence $\left( o ^ { i _ { 1 } } , \ldots , o ^ { i _ { n } } \right)$ and the output of agents’ action sequence $( a ^ { i _ { 1 } } , \ldots , a ^ { i _ { n } } )$ are sequence modeling tasks similar to machine translations. As eluded by Theorem (1), the action $a ^ { i _ { m } }$ depends on all previous agents’ decisions $\mathbf { a } ^ { i _ { 1 : m - 1 } }$ . Hence, our MAT in Figure (2) consists of an encoder, which learns representations of the joint observations, and a decoder which outputs actions for each individual agent in an auto-regressive manner.
+
+The encoder, whose parameters we denote by $\phi$ , takes a sequence of observations $\left( o ^ { i _ { 1 } } , \ldots , o ^ { i _ { n } } \right)$ in arbitrary order and passes them through several computational blocks. Each such block consists of a self-attention mechanism and a multi-layer perceptron (MLP), as well as residual connections to prevent gradient vanishing and network degradation with the increase of depth. We denote the output encoding of the observations as $( \hat { o } ^ { i _ { 1 } } , \dots , \hat { o } ^ { i _ { n } } )$ , which encodes not only the information of agents $( i _ { 1 } , \ldots , i _ { n } )$ but also the high-level interrelationships that represent agents’ interactions. In order to learn expressive representations, in the training phase, we make the encoder to approximate the value functions, whose objective is to minimize the empirical Bellman error by
+
+$$
+L _ { \mathrm { E n c o d e r } } ( \phi ) = \frac { 1 } { T n } \sum _ { m = 1 } ^ { n } \sum _ { t = 0 } ^ { T - 1 } \Big [ R ( \mathbf { o } _ { t } , \mathbf { a } _ { t } ) + \gamma V _ { \vec { \phi } } ( \hat { \mathbf { o } } _ { t + 1 } ^ { i _ { m } } ) - V _ { \phi } ( \hat { \mathbf { o } } _ { t } ^ { i _ { m } } ) \Big ] ^ { 2 } ,
+$$
+
+where $\bar { \phi }$ is the target network’s parameter, which is non-differentiable and updated every few epochs.
+
+The decoder, whose parameters we denote by $\theta$ , passes the embedded joint action $\pmb { a } ^ { i _ { 0 : m - 1 } } , m =$ $\{ 1 , \ldots n \}$ (where $a ^ { i _ { 0 } }$ is an arbitrary symbol indicating the start of decoding) to a sequence of decoding blocks. Crucially, every decoding block comes with a masked self-attention mechanism, where the masking makes sure that, for every $i _ { j }$ , attention is computed only between the $i _ { r } ^ { \mathrm { t h } }$ and the $i _ { j } ^ { \mathrm { t h } }$ action heads wherein $r < j$ so that the sequential update scheme can be maintained. This is then followed by a second masked attention function, which computes the attention between the action heads and observation representations. Finally, the block finishes with an MLP and skipping connections. The output to the last decoder block is a sequence of representations of the joint actions, $\{ \hat { \pmb { a } } ^ { i _ { 0 } : i - 1 } \} _ { i = 1 } ^ { m }$ . This is fed to an MLP that outputs the probability distribution of $i _ { m }$ ’s action, namely, the policy $\pi _ { \theta } ^ { i _ { m } } ( \mathbf { a } ^ { i _ { m } } | \hat { \mathbf { o } } ^ { i _ { 1 : n } } , \mathbf { a } ^ { i _ { 1 : m - 1 } } ) $ . To train the decoder, we minimize the following clipping PPO objective of
+
+![](images/05b4d70e3ed71091e35c8894f1dc7246c770b17c0f70d3241565b25f49c9e247.jpg)  
+Figure 3: Demonstrations of the Bi-DexHands and the HalfCheetah environments.
+
+![](images/2226ea430ed36b5f48270f30a6b9760f1439abfb81507b6a7fab4bd1eaad67f7.jpg)  
+Figure 4: Performance comparisons on the Multi-Agent MuJoCo and the Bi-DexHands benchmarks.
+
+$$
+\begin{array} { l } { { \displaystyle { \cal L } _ { \mathrm { D e c o d e r } } ( \theta ) = - \frac { 1 } { T n } \sum _ { m = 1 } ^ { n } \sum _ { t = 0 } ^ { T - 1 } \operatorname* { m i n } \Big ( \mathrm { \bf r } _ { t } ^ { i _ { m } } ( \theta ) \hat { A } _ { t } , \mathrm { c l i p } ( \mathrm { \bf r } _ { t } ^ { i _ { m } } ( \theta ) , 1 \pm \epsilon ) \hat { A } _ { t } \Big ) } , } \\ { { \displaystyle \mathrm { \bf ~ r } _ { t } ^ { i _ { m } } ( \theta ) = \frac { \pi _ { \theta } ^ { i _ { m } } \big ( { \bf a } _ { t } ^ { i _ { m } } \big | \hat { \bf { \bf { \boldsymbol \Phi } } } _ { t } ^ { i _ { 1 : n } } , \hat { \bf { \bf { a } } } _ { t } ^ { i _ { 1 : m - 1 } } \big ) } { \pi _ { \theta _ { \mathrm { o l d } } } ^ { i _ { m } } \big ( { \bf a } _ { t } ^ { i _ { m } } \big | \hat { \bf { \bf { \boldsymbol \Phi } } } _ { t } ^ { i _ { 1 : n } } , \hat { \bf { a } } _ { t } ^ { i _ { 1 : m - 1 } } \big ) } , } } \end{array}
+$$
+
+where $\hat { A } _ { t }$ is an estimate of the joint advantage function. One can apply generalized advantage estimation (GAE) [32] with $\begin{array} { r } { \hat { V } _ { t } = \frac { 1 } { n } \sum _ { m = 1 } ^ { n } V \big ( \hat { \mathbf { o } } _ { t } ^ { i _ { m } } \big ) } \end{array}$ as a robust estimator for the joint value function. Notably, the action generation process is different between the inference and the training stage. In the inference stage, each action is generated auto-regressively, in the sense that $\mathbf { a } ^ { i _ { m } }$ will be inserted back into the decoder again to generate $\mathrm { a } ^ { i _ { m + 1 } }$ (starting with $\mathrm { \mathbf { a } } ^ { i _ { 0 } }$ and ending with $\mathrm { a } ^ { i _ { n - 1 } }$ ). While during the training stage, the output of all actions, $\mathbf { a } ^ { i _ { 1 : n } }$ can be computed in parallel simply because $\mathbf { a } ^ { i _ { 1 : n - 1 } }$ have already been collected and stored in the replay buffer.
+
+The attention mechanism, which lies in the heart of MAT, encodes observations and actions with a weight matrix calculated by multiplying the embedded queries, $( q ^ { i _ { 1 } } , \ldots , q ^ { i _ { n } } )$ , and keys, $( k ^ { i _ { 1 } } , \ldots , k ^ { i _ { n } } )$ , where each of the weight $w \tilde { ( } q ^ { i _ { r } } , k ^ { i _ { j } } ) = \langle q ^ { i _ { r } ^ { \perp } } , k ^ { i _ { j } } \rangle$ . The embedded values $( v ^ { i _ { 1 } } , \ldots , v ^ { i _ { n } } )$ are multiplied with the weight matrix to output representations. While the unmasked attention in the encoder uses a full weight matrix to extract the interrelationship between agents, i.e., $\hat { \mathbf { O } } ^ { i _ { 1 : n } }$ , the masked attentions in the decoder capture $\mathbf { a } ^ { i _ { 1 : m } }$ with triangular matrices where $\bar { w ( q ^ { i _ { r } } , k ^ { i _ { j } } ) } = 0$ for $r \textless j$ (see an visual illustration in Appendix A). With the properly masked attention mechanism, the decoder can safely output the policy $\pi _ { \theta } ^ { i _ { m + 1 } } ( \mathbf { a } ^ { i _ { m + 1 } } | \hat { \mathbf { o } } ^ { i _ { 1 : n } } , \mathbf { a } ^ { i _ { 1 : m } } )$ , which finishes the implementation of Theorem (1).
+
+The monotonic improvement guarantee. An MAT agent $i _ { m }$ optimizes a trust-region objective that is conditioned on new decisions of agents $i _ { 1 : m - 1 }$ by means of conditioning its policy ratio on them (see Equation (5)). As such, it increases the joint return monotonically like if it followed the sequential update scheme of HAPPO [15, Theorem 2]. However, as oppose to that method, the MAT model does not require $i _ { m }$ to wait until its predecessors make their updates, nor it uses their updated action distribution for importance sampling calculations. In fact, as actions of all agents are outputs of MAT, their clipping objectives can be computed in parallel (during training), thus dominating
+
+Table 1: Performance evaluations of win rate and standard deviation on the SMAC benchmark, where UPDeT’s official codebase supports several Marine-based tasks only.   
+
+<table><tr><td>Task</td><td>Difficulty</td><td>MAT</td><td>MAT-Dec</td><td>MAPPO</td><td>HAPPO</td><td>QMIX</td><td>UPDeT</td><td>Steps</td></tr><tr><td>3m</td><td>Easy</td><td>100.0(1.8)</td><td>100.0(1.1)</td><td>100.0(0.4)</td><td>100.0(1.2)</td><td>96.91.3</td><td>100.0(5.2)</td><td>5e5</td></tr><tr><td>8m</td><td>Easy</td><td>100.0(1.1)</td><td>97.5(2.5)</td><td>96.8(2.9)</td><td>97.5(1.1)</td><td>97.71.9</td><td>96.3(9.7)</td><td>1e6</td></tr><tr><td>1c3s5z</td><td>Easy</td><td>100.0(2.4)</td><td>100.0(0.4)</td><td>100.0(2.2)</td><td>97.5(1.8)</td><td>96.9(1.5)</td><td>/</td><td>2e6</td></tr><tr><td>MMM</td><td>Easy</td><td>100.0(2.2)</td><td>98.1(2.1)</td><td>95.6(4.5)</td><td>81.2(22.9)</td><td>91.2(3.2)</td><td>/</td><td>2e6</td></tr><tr><td>2c vs 64zg</td><td>Hard</td><td>100.0(1.3)</td><td>95.9(2.3)</td><td>100.0(2.7)</td><td>90.0(4.8)</td><td>90.3(4.0)</td><td>/</td><td>5e6</td></tr><tr><td>3s vs 5z</td><td>Hard</td><td>100.0(1.7)</td><td>100.0(1.3)</td><td>100.0(2.5)</td><td>91.9(5.3)</td><td>92.3(4.4)</td><td>/</td><td>5e6</td></tr><tr><td>3s5z</td><td>Hard</td><td>100.0(1.9)</td><td>100.0(3.3)</td><td>72.5(26.5)</td><td>90.0(3.5)</td><td>84.3(5.4)</td><td>/</td><td>3e6</td></tr><tr><td>5m vs 6m</td><td>Hard</td><td>90.6(4.4)</td><td>83.1(4.6)</td><td>88.2(6.2)</td><td>73.8(4.4)</td><td>75.8(3.7)</td><td>90.6(6.1)</td><td>1e7</td></tr><tr><td>8m vs 9m</td><td>Hard</td><td>100.0(3.1)</td><td>95.0(4.6)</td><td>93.8(3.5)</td><td>86.2(4.4)</td><td>92.6(4.0)</td><td>/</td><td>5e6</td></tr><tr><td>10m vs 11m</td><td>Hard</td><td>100.0(1.4)</td><td>100.0(2.0)</td><td>96.3(5.8)</td><td>77.5(9.7)</td><td>95.8(6.1)</td><td>/</td><td>5e6</td></tr><tr><td>25m</td><td>Hard</td><td>100.0(1.3)</td><td>86.9(5.6)</td><td>100.0(2.7)</td><td>70.1(8.1)</td><td>90.2(9.8)</td><td>2.8(3.1)</td><td>2e6</td></tr><tr><td>27m vs 30m</td><td>Hard+</td><td>100.0(0.7)</td><td>95.3(2.2)</td><td>93.1(3.2)</td><td>5.6(2.8)</td><td>39.2(8.8)</td><td>/</td><td>1e7</td></tr><tr><td>MMM2</td><td>Hard+</td><td>93.8(2.6)</td><td>91.2(5.3)</td><td>81.8(10.1)</td><td>68.8(13.7)</td><td>88.3(2.4)</td><td>/</td><td>1e7</td></tr><tr><td>6h vs 8z</td><td>Hard+</td><td>98.8(1.3)</td><td>93.8(4.7)</td><td>88.4(5.7)</td><td>0.3(0.4)</td><td>9.7(3.1)</td><td>/</td><td>1e7</td></tr><tr><td>3s5z vs 3s6z</td><td>Hard+</td><td>96.5(1.3)</td><td>85.3(7.5)</td><td>84.3(19.4)</td><td>82.8(21.2)</td><td>68.8(21.2)</td><td>/</td><td>2e7</td></tr></table>
+
+![](images/3d7ba93c472c245a76d1e9b46ca6a00f09693e351a67723eef51e51c2a4db1c4.jpg)  
+Figure 5: Performance comparison on the Google Research Football tasks with 2-4 agents from left to right respectively.
+
+HAPPO on the time complexity. Lastly, to assure that the limiting joint policy is such that none of the agents is incentivized to change its policy (Nash equilibrium), MAT requires permutating the sequential order of updates at every iteration, which is inline with the discovery in HAPPO [15, Theorem 3].
+
+# 5 Experiments and Results
+
+MAT provides a new solution paradigm for cooperative MARL problems. The key insights of MAT are the sequential update scheme, which is inspired by Theorem (1), as well as the encoder-decoder architecture, which provides a highly-efficient implementation for a sequence modeling perspective. Importantly, MAT inherits the monotonic improvement guarantee, and agents’ policies can be learned in parallel during training. We firmly believe MAT will become a game changer for MARL studies.
+
+To evaluate if MAT meets our expectations, we test MAT on the StarCraftII Multi-Agent Challenge (SMAC) benchmark [31] where MAPPO with parameter sharing [46] has shown superior performance, and the Multi-Agent MuJoCo benchmark [7] where HAPPO [15] shows the current state-of-the-art performance. SMAC and MuJoCo environments are common benchmarks in the MARL field. On top of them, we also test MAT on the Bimanual Dexterous Hands Manipulation (Bi-DexHands) [6] which provides a list of challenging bimanual manipulation tasks (see Figure (3)), and the Google Research Football [18] benchmark with a series of cooperation scenarios in football game.
+
+We apply the same hyper-parameters of baseline algorithms from their original paper to ensure their best performance, and adopt the same hyper-parameter tuning process for our methods with details in Appendix B. To ensure fair comparisons to CTDE methods, we also introduce a CTDE-variant of MAT called MAT-Dec, which essentially adopts a fully decentralized actor for each individual agent (rather than using the decoder proposed in MAT) while keeping the encoder fixed. The critic’s loss for MAT-Dec is $\begin{array} { r } { L ( \phi ) = \frac { 1 } { T } \sum _ { t = 0 } ^ { T - 1 } \left[ R ( \mathbf { o } _ { t } , \mathbf { a } _ { t } ) + \gamma \frac { 1 } { n } \sum _ { m = 1 } ^ { n } V _ { \vec { \phi } } ( \hat { \mathbf { o } } _ { t + 1 } ^ { i _ { m } } ) - \frac { 1 } { n } \sum _ { m = 1 } ^ { n } V _ { \phi } ( \hat { \mathbf { o } } _ { t } ^ { i _ { m } } ) \right] ^ { 2 } , } \end{array}$ and we apply the local advantage estimation $A _ { t } \big ( \hat { \mathbf { o } } _ { t } ^ { i _ { m } } , a ^ { i _ { m } } \big )$ to guide the subsequent policy update.
+
+![](images/475361329427df4de80617968b1a2a1d3b504e83678e33306c556403d6d3eefd.jpg)  
+Figure 6: Performance on the HalfCheetah task with different disabled joints shown in Figure (3a).
+
+# 5.1 Performance on Cooperative MARL Benchmarks
+
+According to Table (1) and Figure (4), MAPPO significantly outperforms HAPPO in SMAC with higher sample efficiency. This verified the homogeneity of SMAC agents and the heterogeneity of multi-agent MuJoCo agents, that are also discovered by Kuba et al. [15]. Take the SMAC task $2 5 m$ as an example, all the marines are equivalent and interchangeable so that agents can learn from their teammate’s experience. Sharing parameters in this settings means leveraging 25 times more examples to train each agents comparing with separated network of HAPPO, and thus enjoying higher learning efficiency. On the other hand, with the heterogeneous settings of multi-agent MuJoCo, training a "foot" agent with experience from a "thigh" agent can surely harm its performance since they represent different functions on the Cheetah. However, MAT outperforms MAPPO and HAPPO in almost all tasks in Table (1) and Figure (4), indicating its modeling capability on both homogeneous and heterogeneous-agent tasks. MAT also enjoys the superior performance over MAT-Dec, which emphasize the importance of the decoder architecture in the MAT design. On the Bi-DexHands tasks, MAT outperforms MAPPO and HAPPO methods by a large margin. We save the Google Football results to Figure (5), where the conclusion stays the same.
+
+# 5.2 MAT as Excellent Few-short Learners
+
+Since Transformer-based models often demonstrate strong generalization performance on few-short tasks [3, 9], we believe MAT can possess strong generalization ability on unseen MARL tasks as well. To validate such an assumption, we design zero-shot and few-shot experiments on SMAC and multi-agent MuJoCo tasks. For SMAC tasks, we pre-train agents on eight tasks involving five types of units (3m, 8m vs 9m, 10m vs 11m, 25m, 3s vs 3z, 2s3z, 3s5z, MMM ) with 10M examples in total and then apply them on six separate and much harder tasks (5m vs 6m, 8m, $2 7 m$ vs 30m, 2s vs 1sc, 1c3s5z, MMM2 ) including seven types of units. This setting is designed to evaluate the generalization ability of MAT when training on simple tasks but transferring to more diverse and complex downstream tasks. In terms of multi-agent MuJoCo, we reuse the models trained on the complete HalfCheetah robot as the pre-trained agent and then directly apply it to six new tasks, each with a different leg being disfunctioned (see Figure (3a)). We investigate the generalization capability of pre-trained models on each downstream task with $0 \%$ (zero-shot), $1 \%$ , $5 \%$ , $10 \%$ few-short new examples, respectively. Note that common MARL baselines such as HAPPO assume fixed number of agents during training, thus it cannot directly handle the cases with changing number of agents.
+
+Table 2: Median evaluation win rate and the standard deviation on the SMAC benchmark for pre-trained models with different number of online examples.   
+
+<table><tr><td rowspan="2">Methods #examples</td><td colspan="4">MAT</td><td colspan="4">MAPPO</td><td colspan="4">MAT-from scratch</td></tr><tr><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td></tr><tr><td>5m vs 6m</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>5.8(3.1)</td><td>18.8(7.1)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>4.3(3.8)</td><td>21.9(12.2)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>1.9(1.3)</td><td>3.8(2.1)</td></tr><tr><td>8m</td><td>100(0.0)</td><td>100(1.2)</td><td>100(0.3)</td><td>100(2.1)</td><td>100(0.0)</td><td>100(1.4)</td><td>100(0.3)</td><td>100(1.4)</td><td>0.0(0.0)</td><td>10.6(23.8)</td><td>92.5(3.7)</td><td>100(1.4)</td></tr><tr><td>27m vs 30m</td><td>0.0(0.0)</td><td>6.3(2.4)</td><td>53.8(16.4)</td><td>71.2(8.2)</td><td>9.4(3.6)</td><td>15(5.9)</td><td>26.2(7.8)</td><td>26.8(9.7)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>0.0(0.3)</td><td>0.3(15.6)</td></tr><tr><td>2s vs 1sc</td><td>0.0(0.0)</td><td>15.6(13.8)</td><td>100(9.7)</td><td>100(0.0)</td><td>0.0(0.0)</td><td>43.1(17.6)</td><td>100(1.1)</td><td>100(1.8)</td><td>0.0(0.0)</td><td>19.3(33.3)</td><td>96.3(6.2)</td><td>100(0.3)</td></tr><tr><td>1c3s5z</td><td>3.1(1.8)</td><td>5.6(5.0)</td><td>82.5(5.5)</td><td>100(2.7)</td><td>3.1(1.8)</td><td>4.3(4.9)</td><td>73.8(13.0)</td><td>97.5(2.1)</td><td>0.0(0.0)</td><td>7.5(4.8)</td><td>87.5(3.9)</td><td>100(1.4)</td></tr><tr><td>MMM2</td><td>0.0(3.6)</td><td>0.0(1.8)</td><td>33.8(13.7)</td><td>62.5(12.1)</td><td>0.0(0.0)</td><td>0.0(1.4)</td><td>13.8(7.0)</td><td>36.2(9.6)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>0.0(0.7)</td></tr></table>
+
+Table 3: Average evaluation score and standard deviation on Multi-Agent MuJoCo for pre-trained models with different number of online examples.   
+
+<table><tr><td rowspan="2">Methods #examples</td><td colspan="4">MAT</td><td colspan="4">MAPPO</td><td colspan="4">MAT-from scratch</td></tr><tr><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td></tr><tr><td>back foot</td><td>2100(89)</td><td>2837(95)</td><td>4691(235)</td><td>5646(79)</td><td>2936(301)</td><td>3017(135)</td><td>3221(119)</td><td>3304(129)</td><td>-0.44(0.4)</td><td>-5.18(11)</td><td>670(1098)</td><td>1635(1184)</td></tr><tr><td>back shin</td><td>4005(316)</td><td>4143(230)</td><td>6077(209)</td><td>7176(74)</td><td>2406(32)</td><td>2542(108)</td><td>2796(137)</td><td>2955(127)</td><td>-0.31(0.1)</td><td>-3.95(17)</td><td>743(537)</td><td>1252(1123)</td></tr><tr><td>back thigh</td><td>5361(45)</td><td>5641(150)</td><td>7101(119)</td><td>7460(61)</td><td>3043(79)</td><td>3060(143)</td><td>3217(3)</td><td>3353（71)</td><td>-0.54(0.3)</td><td>-4.87(7.7)</td><td>930(589)</td><td>2067(861)</td></tr><tr><td>fore foot</td><td>1313(512)</td><td>1955(232)</td><td>4856(146)</td><td>6054(172)</td><td>623（44)</td><td>970(185)</td><td>2025(371)</td><td>2480(239)</td><td>-0.37(0.2)</td><td>-2.25(7.9)</td><td>1821(157)</td><td>2877(106)</td></tr><tr><td>fore shin</td><td>2435(13)</td><td>2617(71)</td><td>3851(57)</td><td>4373(83)</td><td>1715(55)</td><td>2457(125)</td><td>3096(59)</td><td>3310(54)</td><td>-0.15(0.06)</td><td>-0.96(6.0)</td><td>1461(101)</td><td>3003(316)</td></tr><tr><td>fore thigh</td><td>5631(321)</td><td>6448(417)</td><td>7952(109)</td><td>8347(81)</td><td>3087(110)</td><td>3171(83)</td><td>3340(52)</td><td>3519(59)</td><td>-0.29(0.3)</td><td>0.82(14)</td><td>1021(177)</td><td>2600(215)</td></tr></table>
+
+We summarize the zero-shot and few-shot results of each algorithm in Table (2) and (3), where the bold number indicates the best performance. We also provide the performance of MAT if it was given the same amount of data but is trained from scratch, the "MAT-from scratch", as the control group to demonstrate the effectiveness of pre-training process. As both tables suggest, bold numbers are mainly located in the area of MAT, which justify MAT’s strong generalisation performance as a few-short learner. Surprisingly, we find that the few-shot MAT with only $10 \%$ data show even higher rewards than its counterpart that is purely trained on HalfCheetah with the same disabled joints (back foot, back shin and back thigh ) and $100 \%$ full amount data, we believe it is because the pre-train process offers initial weights that are not only closer to optimum but also less likely to stuck in bad local optima than random initialization.
+
+# 6 Conclusion
+
+In the past five years, large sequence models have achieved remarkable successes on solving visual language tasks. In this paper, we take the initial effort to build the connection between multi-agent reinforcement learning (MARL) problems and generic sequence models (SM), with the ambition that MARL researchers can hereafter benefit from the prosperous development on the sequence modeling side. Specifically, we contribute by unifying a general solution to cooperative MARL problems into a Transformer like encoder-decoder model. The proposed Multi-Agent Transformer (MAT) leverages the multi-agent advantage decomposition theorem, which essentially transforms the joint policy optimization process into a sequential decision making process that can be simply implemented by an auto-regressive model. We have demonstrated MAT’s strong empirical performance on three challenging benchmarks against current state-of-the-art MARL solutions including MAPPO and HAPPO. Based on the established connection between MARL and SM, in the future, we plan to bring multi-agent learning tasks into large multi-modal SM, chasing for more generally intelligent models as the most recent success of GATO has already demonstrated [30].
+
+# Acknowledgment
+
+The SJTU team is partially supported by “New Generation of AI 2030” Major Project (2018AAA0100900), Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102), Shanghai Sailing Program (21YF1421900), and National Natural Science Foundation of China (62076161, 62106141).
+
+References   
+[1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katie Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. arXiv preprint arXiv:2204.14198, 2022.   
+[2] Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258, 2021.   
+[3] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. NeurIPS, 2020.   
+[4] Yu-Han Chang, Tracey Ho, and Leslie Kaelbling. All learning is local: Multi-agent learning in global reward games. Advances in neural information processing systems, 16, 2003.   
+[5] Lili Chen, Kevin Lu, Aravind Rajeswaran, Kimin Lee, Aditya Grover, Michael Laskin, Pieter Abbeel, Aravind Srinivas, and Igor Mordatch. Decision transformer: Reinforcement learning via sequence modeling. arXiv preprint arXiv:2106.01345, 2021.   
+[6] Yuanpei Chen, Yaodong Yang, Tianhao Wu, Shengjie Wang, Xidong Feng, Jiechuang Jiang, Stephen Marcus McAleer, Hao Dong, Zongqing Lu, and Song-Chun Zhu. Towards human-level bimanual dexterous manipulation with reinforcement learning. arXiv preprint arXiv:2206.08686, 2022.   
+[7] Christian Schröder de Witt, Bei Peng, Pierre-Alexandre Kamienny, Philip H. S. Torr, Wendelin Böhmer, and Shimon Whiteson. Deep multi-agent reinforcement learning for decentralized continuous cooperative control. CoRR, abs/2003.06709, 2020.   
+[8] Xiaotie Deng, Yuhao Li, David Henry Mguni, Jun Wang, and Yaodong Yang. On the complexity of computing markov perfect equilibrium in general-sum stochastic games. arXiv preprint arXiv:2109.01795, 2021.   
+[9] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. In NAACL-HLT (1), 2019.   
+[10] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2020.   
+[11] Jakob Foerster, Gregory Farquhar, Triantafyllos Afouras, Nantas Nardelli, and Shimon Whiteson. Counterfactual multi-agent policy gradients. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   
+[12] Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick. Masked autoencoders are scalable vision learners. arXiv preprint arXiv:2111.06377, 2021.   
+[13] Siyi Hu, Fengda Zhu, Xiaojun Chang, and Xiaodan Liang. Updet: Universal multi-agent reinforcement learning via policy decoupling with transformers. arXiv preprint arXiv:2101.08001, 2021.   
+[14] Michael Janner, Qiyang Li, and Sergey Levine. Offline reinforcement learning as one big sequence modeling problem. Advances in Neural Information Processing Systems, 34, 2021.   
+[15] Jakub Grudzien Kuba, Ruiqing Chen, Munning Wen, Ying Wen, Fanglei Sun, Jun Wang, and Yaodong Yang. Trust region policy optimisation in multi-agent reinforcement learning. ICLR, 2022.   
+[16] Jakub Grudzien Kuba, Xidong Feng, Shiyao Ding, Hao Dong, Jun Wang, and Yaodong Yang. Heterogeneous-agent mirror learning: A continuum of solutions to cooperative marl. arXiv preprint arXiv:2208.01682, 2022.
+
+[17] Jakub Grudzien Kuba, Muning Wen, Linghui Meng, Haifeng Zhang, David Mguni, Jun Wang, Yaodong Yang, et al. Settling the variance of multi-agent policy gradients. Advances in Neural Information Processing Systems, 34:13458–13470, 2021.
+
+[18] Karol Kurach, Anton Raichuk, Piotr Stanczyk, Micha ´ ł Zaj ˛ac, Olivier Bachem, Lasse Espeholt, Carlos Riquelme, Damien Vincent, Marcin Michalski, Olivier Bousquet, et al. Google research football: A novel reinforcement learning environment. arXiv preprint arXiv:1907.11180, 2019.
+
+[19] Michael L Littman. Markov games as a framework for multi-agent reinforcement learning. In Machine learning proceedings 1994, pages 157–163. Elsevier, 1994.
+
+[20] Ryan Lowe, Yi Wu, Aviv Tamar, Jean Harb, Pieter Abbeel, and Igor Mordatch. Multi-agent actorcritic for mixed cooperative-competitive environments. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 6382–6393, 2017.
+
+[21] Xueguang Lyu, Yuchen Xiao, Brett Daley, and Christopher Amato. Contrasting centralized and decentralized critics in multi-agent reinforcement learning. arXiv preprint arXiv:2102.04402, 2021.
+
+[22] Anuj Mahajan, Tabish Rashid, Mikayel Samvelyan, and Shimon Whiteson. Maven: Multi-agent variational exploration. Advances in Neural Information Processing Systems, 32, 2019.
+
+[23] Linghui Meng, Muning Wen, Yaodong Yang, Chenyang Le, Xiyun Li, Weinan Zhang, Ying Wen, Haifeng Zhang, Jun Wang, and Bo Xu. Offline pre-trained multi-agent decision transformer: One big sequence model conquers all starcraftii tasks. arXiv preprint arXiv:2112.02845, 2021.
+
+[24] David Mguni, Jianhong Wang, Taher Jafferjee, Nicolas Perez-Nieves, Wenbin Song, Yaodong Yang, Feifei Tong, Hui Chen, Jiangcheng Zhu, Yali Du, et al. Learning to shape rewards using a game of switching controls. arXiv preprint arXiv:2103.09159, 2021.
+
+[25] David Henry Mguni, Taher Jafferjee, Jianhong Wang, Nicolas Perez-Nieves, Oliver Slumbers, Feifei Tong, Yang Li, Jiangcheng Zhu, Yaodong Yang, and Jun Wang. Ligs: Learnable intrinsicreward generation selection for multi-agent learning. In International Conference on Learning Representations, 2021.
+
+[26] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. nature, 518(7540):529–533, 2015.
+
+[27] Prakash M Nadkarni, Lucila Ohno-Machado, and Wendy W Chapman. Natural language processing: an introduction. Journal of the American Medical Informatics Association, 18(5):544– 551, 2011.
+
+[28] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In International Conference on Machine Learning, pages 8821–8831. PMLR, 2021.
+
+[29] Tabish Rashid, Mikayel Samvelyan, Christian Schroeder, Gregory Farquhar, Jakob Foerster, and Shimon Whiteson. Qmix: Monotonic value function factorisation for deep multi-agent reinforcement learning. In International Conference on Machine Learning, pages 4295–4304. PMLR, 2018.
+
+[30] Scott Reed, Konrad Zolna, Emilio Parisotto, Sergio Gomez Colmenarejo, Alexander Novikov, Gabriel Barth-Maron, Mai Gimenez, Yury Sulsky, Jackie Kay, Jost Tobias Springenberg, Tom Eccles, Jake Bruce, Ali Razavi, Ashley Edwards, Nicolas Heess, Yutian Chen, Raia Hadsell, Oriol Vinyals, Mahyar Bordbar, and Nando de Freitas. A generalist agent. arXiv preprint arXiv:2205.06175, 2022.
+
+[31] Mikayel Samvelyan, Tabish Rashid, Christian Schroeder de Witt, Gregory Farquhar, Nantas Nardelli, Tim GJ Rudner, Chia-Man Hung, Philip HS Torr, Jakob Foerster, and Shimon Whiteson. The starcraft multi-agent challenge. 2019.
+
+[32] John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. Highdimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015.   
+[33] John Schulman, F. Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. ArXiv, abs/1707.06347, 2017.   
+[34] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In International conference on machine learning, pages 387–395. PMLR, 2014.   
+[35] Kyunghwan Son, Daewoo Kim, Wan Ju Kang, David Earl Hostallero, and Yung Yi. Qtran: Learning to factorize with transformation for cooperative multi-agent reinforcement learning. In International Conference on Machine Learning, pages 5887–5896. PMLR, 2019.   
+[36] Peter Sunehag, Guy Lever, Audrunas Gruslys, Wojciech Marian Czarnecki, Vinicius Zambaldi, Max Jaderberg, Marc Lanctot, Nicolas Sonnerat, Joel Z Leibo, Karl Tuyls, et al. Valuedecomposition networks for cooperative multi-agent learning. arXiv preprint arXiv:1706.05296, 2017.   
+[37] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.   
+[38] Hugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and Hervé Jégou. Going deeper with image transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 32–42, 2021.   
+[39] Zhengzhong Tu, Hossein Talebi, Han Zhang, Feng Yang, Peyman Milanfar, Alan Bovik, and Yinxiao Li. Maxvit: Multi-axis vision transformer. arXiv preprint arXiv:2204.01697, 2022.   
+[40] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.   
+[41] Wenxiao Wang, Lu Yao, Long Chen, Binbin Lin, Deng Cai, Xiaofei He, and Wei Liu. Crossformer: A versatile vision transformer hinging on cross-scale attention. arXiv preprint arXiv:2108.00154, 2021.   
+[42] Ying Wen, Yaodong Yang, Rui Luo, Jun Wang, and Wei Pan. Probabilistic recursive reasoning for multi-agent reinforcement learning. In International Conference on Learning Representations, 2018.   
+[43] Ying Wen, Yaodong Yang, and Jun Wang. Modelling bounded rationality in multi-agent interactions by generalized recursive reasoning. In Christian Bessiere, editor, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, pages 414–421. International Joint Conferences on Artificial Intelligence Organization, 7 2020. Main track.   
+[44] Yaodong Yang and Jun Wang. An overview of multi-agent reinforcement learning from game theoretical perspective. arXiv preprint arXiv:2011.00583, 2020.   
+[45] Yaodong Yang, Ying Wen, Jun Wang, Liheng Chen, Kun Shao, David Mguni, and Weinan Zhang. Multi-agent determinantal q-learning. In International Conference on Machine Learning, pages 10757–10766. PMLR, 2020.   
+[46] Chao Yu, A. Velu, Eugene Vinitsky, Yu Wang, A. Bayen, and Yi Wu. The surprising effectiveness of mappo in cooperative, multi-agent games. ArXiv, abs/2103.01955, 2021.

parse/dev/1W8UwXAQubL/1W8UwXAQubL_content_list.json

@@ -0,0 +1,1096 @@
+[
+    {
+        "type": "text",
+        "text": "Multi-Agent Reinforcement Learning is A Sequence Modeling Problem ",
+        "text_level": 1,
+        "bbox": [
+            258,
+            122,
+            740,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Muning Wen1,2, Jakub Grudzien Kuba3, Runji Lin4, Weinan Zhang1, Ying Wen1, Jun Wang2,5, Yaodong Yang6,7,† 1Shanghai Jiao Tong University, 2Digital Brain Lab, 3University of Oxford, 4Institute of Automation, Chinese Academy of Science, 5University College London, 6Beijing Institute for General AI, 7Institute for AI, Peking University ",
+        "bbox": [
+            251,
+            224,
+            746,
+            313
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            348,
+            535,
+            364
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Large sequence models (SM) such as GPT series and BERT have displayed outstanding performance and generalization capabilities in natural language process, vision and recently reinforcement learning. A natural follow-up question is how to abstract multi-agent decision making also as an sequence modeling problem and benefit from the prosperous development of the SMs. In this paper, we introduce a novel architecture named Multi-Agent Transformer (MAT) that effectively casts cooperative multi-agent reinforcement learning (MARL) into SM problems wherein the objective is to map agents’ observation sequences to agents’ optimal action sequences. Our goal is to build the bridge between MARL and SMs so that the modeling power of modern sequence models can be unleashed for MARL. Central to our MAT is an encoder-decoder architecture which leverages the multi-agent advantage decomposition theorem to transform the joint policy search problem into a sequential decision making process; this renders only linear time complexity for multiagent problems and, most importantly, endows MAT with monotonic performance improvement guarantee. Unlike prior arts such as Decision Transformer fit only precollected offline data, MAT is trained by online trial and error from the environment in an on-policy fashion. To validate MAT, we conduct extensive experiments on StarCraftII, Multi-Agent MuJoCo, Dexterous Hands Manipulation, and Google Research Football benchmarks. Results demonstrate that MAT achieves superior performance and data efficiency compared to strong baselines including MAPPO and HAPPO. Furthermore, we demonstrate that MAT is an excellent few-short learner on unseen tasks regardless of changes in the number of agents. See our project page at https://sites.google.com/view/multi-agent-transformer(1) ",
+        "bbox": [
+            232,
+            378,
+            766,
+            696
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
+            310,
+            739
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Multi-agent reinforcement learning (MARL) [44, 8] is a challenging problem for its difficulty which arises not only from identifying each individual agent’s policy improvement direction, but also from combining agents’ policy updates jointly which should be beneficial for the whole team. Recently, such difficulty in multi-agent learning has been eased owing to the introduction of centralized training for decentralized execution (CTDE) [11, 45], which allows agents to access the global information and opponents’ actions during the training phase. This framework enables successful developments of methods that directly inherit single-agent algorithms. For examples, COMA replaces the policy-gradient (PG) estimate with a multi-agent PG (MAPG) counterpart [11], MADDPG extends deterministic policy gradient into multi-agent settings with a centralized critic [20, 34], QMIX leverages deep Qnetworks for decentralized agents and introduces a centralized mixing network for Q-value decomposition [29, 36, 26]. MAPPO endowing all agents with the same set of parameters and then training by trust-region methods [46]. PR2 [42] and GR2 [43] methods conduct recursive reasoning under the CTDE framework. These methods, however, cannot cover the whole complexity of multi-agent interactions; in fact, some of them are shown to fail in the simplest cooperative task [15]. To resolve this issue, the multi-agent advantage decomposition theorem was proposed [15, Theorem 1] which captures how different agents contribute to the return and provides an intuition behind the emergence of cooperation through a sequential decision making process scheme. Based on it, HATRPO and HAPPO algorithms [15, 17, 16] were derived which, thanks to the decomposition theorem and sequential update scheme, established new state-of-the-art methods for MARL. However, their limitation is that the agents’ policies are unaware of the purpose to develop cooperation and still rely on a carefully handcrafted maximization objective. Ideally, a team of agents should be aware of the jointness of their training by design, thereby following a holistic and effective paradigm—an ideal solution that is yet to be proposed. ",
+        "bbox": [
+            174,
+            753,
+            825,
+            864
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            825,
+            299
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In recent years, sequence models (SM) have made a substantial progress in natural language processing (NLP) [27]. For example, GPT series [3] and BERT models [9], built on autoregressive SMs, have demonstrated remarkable performance on a wide range of downstream tasks and achieved strong performance on few-shot generalization tasks. Although SM are mostly used in language tasks due to its natural fitting with the sequential property of languages, the sequential approaches are not confined to NLP only, but is instead a widely applicable general foundation model [2]. For example, in computer vision (CV), one can split an image into sub-images and align them in a sequence as if they were tokens in NLP tasks [9, 10, 12]. Although the idea of solving CV tasks by SM is straightforward, it serves as the foundation to some of the best-performing CV algorithms [38, 41, 39]. Furthermore, sequential methods are starting to spawn powerful multi-modal visual language models such as Flamingo [1], DALL-E [28], and GATO [30] in the recent past. ",
+        "bbox": [
+            174,
+            304,
+            825,
+            457
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Coming with effective and expressive network architectures such as Transformer [40], sequence modeling techniques have also attracted tremendous attention from the RL community, which results in a series of successful offline RL developments based on the Transformer architecture [5, 14, 30, 23]. These methods show great potentials in tackling some of the most fundamental RL training problems, such as long-horizon credit assignment and reward sparsity [37, 24, 25]. For example, by training autoregressive models on pre-collected offline data in a purely supervised way, Decision Transformer [5] bypasses the need for computing cumulative rewards through dynamic programming, but rather generates future actions conditioning on the desired returns, past states and actions. Despite their remarkable successes, none of these methods have been designed to model the most difficult (also unique to MARL) aspect of multi-agent systems—the agents’ interactions. In fact, if we were to simply endow all agents with a Transformer policy and train them independently, their joint performance still could not be guaranteed to improve [15, Proposition 1]. Therefore, while a myriad of powerful SMs are available, MARL—an area that would greatly benefit from SM—has not truly taken advantage of their performance benefit. The key research question to ask is then ",
+        "bbox": [
+            174,
+            463,
+            825,
+            656
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "How can we model MARL problems by sequence models ? ",
+        "text_level": 1,
+        "bbox": [
+            294,
+            660,
+            702,
+            674
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we take several steps to provide an affirmative answer to the above research question. Our goal is to enhance MARL studies with powerful sequential modeling techniques. To fulfill that, we start by proposing a novel MARL training paradigm which establishes the connection between cooperative MARL problems and sequence modeling problems. Central to the new paradigm are the multi-agent advantage decomposition theorem and sequential update scheme, which effectively transform multi-agent joint policy optimization into a sequential policy search process. As a natural outcome of our findings, we introduce Multi-Agent Transformer (MAT), an encoder-decoder architecture that implements generic MARL solutions through SM. Unlike Decision Transformer [5], MAT is trained online based on trials and errors in an on-policy fashion; therefore, it does not require collecting demonstrations upfront. Importantly, the implementation of the multi-agent advantage decomposition theorem ensures MAT to enjoy monotonic performance improvement guarantee during training. MAT establishes a new state-of-the-art baseline model for cooperative MARL tasks. We justify such a claim by evaluating MAT on the benchmarks of StarCraftII, Multi-Agent MuJoCo, Dexterous Hands Manipulation, and Google Research Football; results show that MAT achieves superior performance over strong baselines, such as MAPPO [46], HAPPO [15], QMIX [29] and UPDeT [13]. Finally, we show that MAT possesses great potentials in task generalizations, which holds regardless of the agent number in new tasks. ",
+        "bbox": [
+            174,
+            678,
+            826,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            318,
+            106
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first introduce the cooperative MARL problem formulation and the multi-agent advantage decomposition theorem, which serves as the cornerstone of our work. We then review existing MARL methods that relate to MAT, and finally familiarize the reader with the Transformer. ",
+        "bbox": [
+            174,
+            121,
+            825,
+            164
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Problem Formulation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            183,
+            364,
+            196
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Cooperative MARL problems are often modeled by Markov games $\\langle \\mathcal { N } , \\mathcal { O } , \\pmb { \\mathcal { A } } , R , P , \\gamma \\rangle$ [19]. ${ \\mathcal { N } } =$ $\\{ 1 , \\ldots , n \\}$ is the set of agents, $\\begin{array} { r } { \\mathcal { O } = \\prod _ { i = 1 } ^ { n } \\mathcal { O } ^ { i } } \\end{array}$ is the product of local observation spaces of the i=1 agents, namely the joint observation space, $\\begin{array} { r } { \\pmb { \\mathcal { A } } = \\prod _ { i = 1 } ^ { n } \\pmb { \\mathcal { A } } ^ { i } } \\end{array}$ is the product of the agents’ action spaces, namely the joint action space, $R : \\mathcal { O } \\times \\pmb { A }  [ - \\bar { R } _ { \\operatorname* { m a x } } , R _ { \\operatorname* { m a x } } ]$ is the joint reward function, $P : \\mathcal { O } \\times \\pmb { \\mathcal { A } } \\times \\mathcal { O }  \\mathbb { R }$ is the transition probability function, and $\\gamma \\in [ 0 , 1 )$ is the discount factor. At time step $t \\in \\mathbb { N }$ , an agent $i \\in \\mathcal N$ observes an observation $\\mathbf { o } _ { t } ^ { i } \\in \\mathcal { O } ^ { i }$ (2) $\\mathbf { \\tilde { \\omega } } ( o = \\left( o ^ { 1 } , \\ldots , o ^ { n } \\right)$ is a “joint” observation) and takes an action $\\mathbf { a } _ { t } ^ { i }$ according to its policy $\\pi ^ { i }$ , which is the $i ^ { \\mathrm { { t h } } }$ component of the agents’ joint policy $\\pi$ . At each time step, all agents take actions simultaneously based on their observation with no sequential dependencies. The transition kernel $P$ and the joint policy induce the (improper) marginal observation distribution $\\begin{array} { r } { \\rho _ { \\pi } ( \\cdot ) \\triangleq \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } \\mathrm { P r } ( \\mathbf { o } _ { t } = o | \\pi ) } \\end{array}$ . At the end of each time step, the whole team receives a joint reward $R ( \\mathbf { o } _ { t } , \\mathbf { a } _ { t } )$ and observe $\\mathbf { o } _ { t + 1 }$ , whose probability distribution is $P ( \\cdot | \\mathbf { o } _ { t } , \\mathbf { a } _ { t } )$ . Following this process infinitely long, the agents earn a discounted cumulative return of $\\begin{array} { r } { R ^ { \\gamma } \\triangleq \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R ( \\mathbf { o } _ { t } , \\mathbf { a } _ { t } ) } \\end{array}$ . ",
+        "bbox": [
+            173,
+            208,
+            826,
+            395
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Multi-Agent Advantage Decomposition Theorem ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            552,
+            428
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The agents evaluate the value of actions and observations with $Q _ { \\pi } ( o , a )$ and $V _ { \\pi } ( o )$ , defined as ",
+        "bbox": [
+            171,
+            439,
+            797,
+            455
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9964eee78a33c9de09bf504cd49e55e5e33eea323b1507e6c2b60a831a03bbc7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { c } { Q _ { \\pi } ( o , a ) \\triangleq \\mathbb { E } _ { \\mathbf { o } _ { 1 : \\infty } \\sim P , \\mathbf { a } _ { 1 : \\infty } \\sim \\pi } \\left[ R ^ { \\gamma } | \\mathbf { o } _ { 0 } = o , \\mathbf { a } _ { 0 } = a \\right] , } \\\\ { V _ { \\pi } ( o ) \\triangleq \\mathbb { E } _ { \\mathbf { a } _ { 0 } \\sim \\pi , \\mathbf { o } _ { 1 : \\infty } \\sim P , \\mathbf { a } _ { 1 : \\infty } \\sim \\pi } \\left[ R ^ { \\gamma } | \\mathbf { o } _ { 0 } = o \\right] . } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            463,
+            671,
+            506
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The jointness of the objective causes difficulties associated with the credit assignment problem— having received a shared reward, individual agents are unable to deduce their own contribution to the team’s success or failure [4]. Indeed, applying traditional RL methods which simply employ the above value functions leads to obstacles in training, such as the growing variance of multi-agent policy gradient (MAPG) estimates [17]. Hence, to tackle these, notions of local value functions [21] and counterfactual baselines [11] have been developed. In this paper, we work with the most general notions of this kind—the multi-agent observation-value functions [15]. That is, for arbitrary disjoint, ordered subsets of agents $i _ { 1 : m } = \\overline { { { \\{ i _ { 1 } , \\ldots , i _ { m } \\} } } }$ and $j _ { 1 : h } = \\{ j _ { 1 } , \\dots , j _ { h } \\}$ , for $m , h \\le n$ , we define the multi-agent observation-value function by ",
+        "bbox": [
+            173,
+            518,
+            826,
+            645
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cbee2162709f7520e8b3b5ff15862083e48254cc99e6c84fc83a44c4ba41273a.jpg",
+        "text": "$$\nQ _ { \\pi } ( o , \\pmb { a } ^ { i _ { 1 : m } } ) \\triangleq \\mathbb { E } \\big [ R ^ { \\gamma } | \\mathbf { o } _ { 0 } ^ { i _ { 1 : n } } = o , \\mathbf { a } _ { 0 } ^ { i _ { 1 : m } } = \\pmb { a } ^ { i _ { 1 : m } } \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            652,
+            663,
+            672
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "which recovers the original state-action value function in Equation (1) when $m = n$ , and the original observation-value function when $m = 0$ (i.e., when the set $i _ { 1 : m }$ is empty). Based on Equation (2), we can further measure the contribution of a chosen subset of agents to the joint return and define the multi-agent advantage function by ",
+        "bbox": [
+            174,
+            686,
+            825,
+            743
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/31b762c35770eb20ffd7c83992c3de84208930092dfe44a0cd5c47484b27ce45.jpg",
+        "text": "$$\nA _ { \\pi } ^ { i _ { 1 } ; m } \\big ( \\pmb { o } , \\pmb { a } ^ { j _ { 1 } ; h } , \\pmb { a } ^ { i _ { 1 : m } } \\big ) \\triangleq Q _ { \\pi } ^ { j _ { 1 } ; h , ~ i _ { 1 : m } } \\big ( \\pmb { o } , \\pmb { a } ^ { j _ { 1 : h } } , \\pmb { a } ^ { i _ { 1 : m } } \\big ) - Q _ { \\pi } ^ { j _ { 1 : h } } \\big ( \\pmb { o } , \\pmb { a } ^ { j _ { 1 : h } } \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            752,
+            728,
+            772
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The above quantity describes how much better/worse than average the joint action $\\textbf { \\em a }$ will be if agents $i _ { 1 : m }$ take the joint action $\\pmb { a } ^ { i _ { 1 : m } }$ , once $j _ { 1 : h }$ have taken $\\pmb { a } ^ { j _ { 1 : h } }$ . Again, when $h = 0$ , the advantage compares the value of $\\pmb { a } ^ { i _ { 1 : m } }$ to the baseline value function of the whole team. This value-functional representation of agents’ actions enables studying interactions between them, as well to decompose the joint value function signal, thus helping alleviate the severity of the credit assignment problem [29, 35, 22]. The insights of Equation (3) is accomplished by means of the following theorem. ",
+        "bbox": [
+            173,
+            786,
+            825,
+            871
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Multi-Agent Advantage Decomposition [17]). Let $i _ { 1 : n }$ be a permutation of agents. Then, for any joint observation $\\pmb { o } = \\pmb { o } \\in \\mathcal { O }$ and joint action $\\pmb { a } = \\pmb { a } ^ { i _ { 1 : n } } \\in \\mathcal { A }$ , the following equation always holds with no further assumption needed, ",
+        "bbox": [
+            171,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2c980f82b18fe7e0d9ac6ff84c2e505ff0ba6e765d0b0538a132da773d89fcab.jpg",
+        "text": "$$\nA _ { \\pi } ^ { i _ { 1 : n } } \\left( o , a ^ { i _ { 1 : n } } \\right) = \\sum _ { m = 1 } ^ { n } A _ { \\pi } ^ { i _ { m } } \\left( o , a ^ { i _ { 1 : m - 1 } } , a ^ { i _ { m } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            141,
+            650,
+            183
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Importantly, this theorem provides an intuition guiding the choice of incrementally improving actions. Suppose that agent $i _ { 1 }$ picks an action $a ^ { i _ { 1 } }$ with positive advantage, $A _ { \\pi } ^ { i _ { 1 } } ( o , a ^ { i _ { 1 } } ) \\dot { > } 0$ . Then, imagine that for all $j = 2 , \\dots , n$ , agent $i _ { j }$ knows the joint action $\\mathbf { \\Delta } a ^ { i _ { 1 : j - 1 } }$ of its predecessors. In this case, it can choose an action $a ^ { i _ { j } }$ for which the advantage $A _ { \\pi } ^ { i _ { j } } ( o , \\pmb { a } ^ { i _ { 1 : j - 1 } } , a ^ { i _ { j } } )$ is positive. Altogether, the theorem assures that the joint action $\\pmb { a } ^ { i _ { 1 : n } }$ has positive advantage. Furthermore, notice that the joint the complexity of this search is additive, Pni=1 |Ai |, in the sizes of the action spaces. If we were to action has been chosen in perform the search directly in the joint action space, we would browse a set of multiplicative size, steps, each of which searched an individual agent’s action space. Hence, $\\begin{array} { r } { \\left| \\mathcal { A } \\right| = \\prod _ { i = 1 } ^ { n } \\left| \\mathcal { A } ^ { i } \\right| } \\end{array}$ . Later, we will build upon this insight to design a SM that optimizes joint policies efficiently, agent by agent, without the necessity of considering the joint action space at once. ",
+        "bbox": [
+            173,
+            195,
+            826,
+            339
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 Existing Methods in MARL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            356,
+            405,
+            371
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now briefly summarize two state-of-the-art MARL algorithms. Both of them build upon Proximal Policy Optimization (PPO) [33]—a RL method famous for its simplicity and its performance stability. ",
+        "bbox": [
+            174,
+            382,
+            823,
+            411
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "MAPPO [46] is the first, and the most direct, approach for applying PPO in MARL. It equips all agents with one shared set of parameters and use agents’ aggregated trajectories for the shared policy’s update; at iteration $k + 1$ , it optimizes the policy parameter $\\theta _ { k + 1 }$ by maximizing the clip objective of ",
+        "bbox": [
+            173,
+            416,
+            825,
+            460
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7fc9c805f8a178e3f197aa0e9cd7dcb061f32fe0b92c30cdd118dc8676683d6.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { n } \\mathbb { E } _ { \\mathbf { o } \\sim \\rho _ { \\pi _ { \\theta _ { k } } } , \\mathbf { a } \\sim \\pi _ { \\theta _ { k } } } \\left[ \\operatorname* { m i n } \\left( \\frac { \\pi _ { \\theta } ( \\mathbf { a } ^ { i } | \\mathbf { o } ) } { \\pi _ { \\theta _ { k } } ( \\mathbf { a } ^ { i } | \\mathbf { o } ) } A _ { \\pi _ { \\theta _ { k } } } ( \\mathbf { o } , \\mathbf { a } ) , \\mathrm { c l i p } \\left( \\frac { \\pi _ { \\theta } ( \\mathbf { a } ^ { i } | \\mathbf { o } ) } { \\pi _ { \\theta _ { k } } ( \\mathbf { a } ^ { i } | \\mathbf { o } ) } , 1 \\pm \\epsilon \\right) A _ { \\pi _ { \\theta _ { k } } } ( \\mathbf { o } , \\mathbf { a } ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            469,
+            776,
+            507
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the clip operator clips the input value (if necessary) so that it stays within the interval $[ 1 - \\epsilon , 1 + \\epsilon ]$ . However, enforcing parameter sharing is equivalent to putting a constraint $\\theta ^ { i } = \\theta ^ { j } , \\forall i , j \\in \\mathcal { N }$ on the joint policy space, which can lead to an exponentially-worse sub-optimal outcome [15]. This motivates a more principled development of heterogeneous-agent trust-region methods, e.g., HAPPO. ",
+        "bbox": [
+            173,
+            513,
+            826,
+            570
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "HAPPO [15] is currently one of the SOTA algorithm that fully leverages Theorem (1) to implement multi-agent trust-region learning with monotonic improvement guarantee. During an update, the agents choose a permutation $i _ { 1 : n }$ at random, and then following the order in the permutation, every agent $i _ { m }$ picks $\\bar { \\pi _ { \\mathrm { n e w } } ^ { i _ { m } } } = \\pi ^ { i _ { m } }$ that maximizes the objective of ",
+        "bbox": [
+            173,
+            575,
+            825,
+            632
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a4d6139b01a2a902c7f7aaf57a8ffaa3bc79294fda5a7d53ef9e10e98075833f.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbf { o } \\sim \\rho _ { \\pi _ { \\mathrm { o d d } } } , \\mathbf { a } ^ { i _ { 1 : m } } - 1 \\sim \\pi _ { \\mathrm { n e r } } ^ { i _ { 1 : m } - 1 } , \\mathbf { a } ^ { i _ { m } } \\sim \\pi _ { \\mathrm { o d d } } ^ { i _ { m } } } \\left[ \\operatorname* { m i n } \\left( \\mathbf { r } ( \\pi ^ { i _ { m } } ) A _ { \\pi _ { \\mathrm { o d d } } } ^ { i _ { 1 : m } } ( \\boldsymbol { o } , \\mathbf { a } ^ { i _ { 1 : m } } ) , \\mathrm { c l i p } ( \\mathbf { r } ( \\pi ^ { i _ { m } } ) , 1 \\pm \\epsilon ) A _ { \\pi _ { \\mathrm { o d d } } } ^ { i _ { 1 : m } } ( \\mathbf { o } , \\mathbf { a } ^ { i _ { 1 : m } } ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            642,
+            816,
+            669
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { r } ( \\pi ^ { i _ { m } } ) = \\pi ^ { i _ { m } } ( \\mathbf { a } ^ { i _ { m } } | \\mathbf { o } ) / \\pi _ { \\mathrm { o l d } } ^ { i _ { m } } \\left( \\mathbf { a } ^ { i _ { m } } | \\mathbf { o } \\right)$ . Note that the expectation is taken over the newly-updated previous agents’ policies, i.e, $\\pi _ { \\mathrm { n e w } } ^ { i _ { 1 : m - 1 } }$ ; this reflects an intuition that, under Theorem (1), the agent $i _ { m }$ reacts to its preceding agents $i _ { 1 : m - 1 }$ . However, one drawback of HAPPO is that agent’s policies has to follow the sequential update scheme in the permutation, thus it cannot be run in parallel. ",
+        "bbox": [
+            174,
+            676,
+            825,
+            736
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 The Transformer Model ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            752,
+            382,
+            768
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Transformer [40] was originally designed for machine translation tasks (e.g., input English, output French). It maintains an encoder-decoder structure, where the encoder maps an input sequence of tokens to latent representations and then the decoder generates a sequence of desired outputs in an auto-regressive manner wherein at each step of inference, the Transformer takes all previously generated tokens as the input. One of the most essential component in Transformer is the scaled dot-product attention, which captures the interrelationship of input sequences. The attention function is written as Attention $\\begin{array} { r } { \\mathrm { \\mathrm { \\Omega } } _ { ^ { 1 } } ( \\mathbf { Q } , \\mathbf { K } , \\mathbf { V } ) = \\operatorname { s o f t m a x } \\big ( \\frac { \\mathbf { Q } \\mathbf { K } ^ { T } } { \\sqrt { d _ { k } } } \\big ) \\mathbf { V } } \\end{array}$ \u0000 V, where the Q, K, V corresponds to the vector of queries, keys and values, which can be learned during training, and the $d _ { k }$ represent the dimension of $\\mathbf { Q }$ and $\\mathbf { K }$ . Self-attentions refer to cases when $\\mathbf { Q } , \\mathbf { K } , \\mathbf { V }$ share the same set of parameters. ",
+        "bbox": [
+            173,
+            779,
+            826,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/44c48541656c918c52b627f3e9d5d3661d597e84653999bdf5d132f8ba8df7a9.jpg",
+        "image_caption": [
+            "Figure 1: Conventional multi-agent learning paradigm (left) wherein all agents take actions simultaneously vs. the multi-agent sequential decision paradigm (right) where agents take actions by following a sequential order, each agent accounts for decisions from preceding agents as red arrows suggest. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            214,
+            84,
+            782,
+            280
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Inspired by the attention mechanism, UPDeT [13] handles various observation sizes by decoupling each agent’s observations into a sequence of observation-entities, matching them with different actiongroups, and modeling the relationship between the matched observation-entities with a Transformerbased function for better representation learning in MARL problems. Apart from this, based on the sequential property described in the Theorem (1) and the principle behind HAPPO [15], it is intuitive to think about another Transformer-based implementation for multi-agent trust-region learning. By treating a team of agents as a sequence, the Transformer architecture allows us to model teams of agents with variable numbers and types, while avoiding drawbacks of MAPPO/HAPPO. We will describe in more details how a cooperative MARL problem can be solved by a sequence model. ",
+        "bbox": [
+            173,
+            358,
+            826,
+            484
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 The Surprising Connection Between MARL and Sequence Models ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            505,
+            758,
+            522
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To establish the connection between MARL and sequence models, Theorem (1) provides a new angle of understanding the MARL problem from a SM perspective. If each agent knows its predecessors’ actions with an arbitrary decision order, the sum of agents’ local advantages $A _ { \\pi } ^ { i _ { j } } \\left( o , \\pmb { a } ^ { i _ { 1 : m - 1 } } , a ^ { i _ { m } } \\right)$ will be exactly equal to the joint advantages $A _ { \\pi } ^ { i _ { 1 : n } } ( o , \\pmb { a } ^ { i _ { 1 : n } } )$ . This orderly decision setting across agents simplifies the update of their joint policy, where maximizing each agent’s own local advantage is equivalent to maximizing the joint advantage. As such, agents do not need to worry about interference from other agents anymore during the policy update; the local advantage functions have already captured the relationship between agents. This property revealed by Theorem (1) inspires us to propose a multi-agent sequential decision paradigm for MARL problems as show in Figure (1), where we assign agents with an arbitrary decision order (one permutation for each iteration); each agent can access its predecessors’ behaviors, based on which it then takes the optimal decision. This sequential paradigm motivates us to leverage a sequential model, e.g., Transformer, to explicitly capture the sequential relationship between agents described in Theorem (1). ",
+        "bbox": [
+            173,
+            537,
+            825,
+            720
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Underpinned by Theorem (1), sequence modeling reduces the complexity growth of MARL problems with the number of agents from multiplicative to additive, thus rendering linear complexity. With the help of the Transformer architecture, we can model policies of heterogeneous agents with an unified network but treat each agent discriminatively with different position, and thus ensuring high sample efficiency while avoiding the exponentially-worse outcome that MAPPO is facing. Besides, in order to guarantee the monotonic improvement of joint policies, HAPPO has to update each policy one-by-one during training, by leveraging previous update results of $\\pi ^ { i _ { 1 } } , . . . , \\pi ^ { i _ { m - 1 } }$ to improve $\\pi ^ { i _ { m } }$ , which becomes critical in computational efficiency at large size of agents. By contramechanism of Transformer architectures allows for batching the ground truth actions $a _ { t } ^ { i _ { 0 } } , . . . , a _ { t } ^ { i _ { n - 1 } }$ ain\u00001 onin the buffer to predict $a _ { t } ^ { i _ { 1 } } , . . . , a _ { t } ^ { i _ { n } }$ and update policies simultaneously, which significantly improves the training speed and makes it feasible for large size of agents. Furthermore, in cases that the number and the type of agents are different, SM can incorporates them into an unified solution through its capability on modeling sequences with flexible sequence length, rather than treat different agent numbers as different tasks. To realize the above idea, we introduce a practical architecture named Multi-Agent Transformer in the next section. ",
+        "bbox": [
+            173,
+            727,
+            826,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d48b9cbf47618a759b37dc1e10c48b8ef89fed8192a3d059497153e41e166e1.jpg",
+        "image_caption": [
+            "Figure 2: The encoder-decoder architecture of MAT. At each time step, the encoder takes in a sequence of agents’ observations and encodes them into a sequence of latent representations, which is then passed into the decoder. The decoder generate each agent’s optimal action in a sequential and auto-regressive manner. The masked attention blocks ensures agents can only access its preceding agents’ actions during training. We list the full pseudocode of MAT in Appendix A and a video that shows the dynamic data flow of MAT in https://sites.google.com/view/multi-agent-transformer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            85,
+            816,
+            337
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            468,
+            823,
+            497
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 The Multi-Agent Transformer ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            457,
+            534
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To implement the sequence modeling paradigm for MARL, our solution is Multi-Agent Transformer (MAT). The idea of applying the Transformer architecture comes from the fact that the mapping between the input of agents’ observation sequence $\\left( o ^ { i _ { 1 } } , \\ldots , o ^ { i _ { n } } \\right)$ and the output of agents’ action sequence $( a ^ { i _ { 1 } } , \\ldots , a ^ { i _ { n } } )$ are sequence modeling tasks similar to machine translations. As eluded by Theorem (1), the action $a ^ { i _ { m } }$ depends on all previous agents’ decisions $\\mathbf { a } ^ { i _ { 1 : m - 1 } }$ . Hence, our MAT in Figure (2) consists of an encoder, which learns representations of the joint observations, and a decoder which outputs actions for each individual agent in an auto-regressive manner. ",
+        "bbox": [
+            173,
+            547,
+            825,
+            645
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The encoder, whose parameters we denote by $\\phi$ , takes a sequence of observations $\\left( o ^ { i _ { 1 } } , \\ldots , o ^ { i _ { n } } \\right)$ in arbitrary order and passes them through several computational blocks. Each such block consists of a self-attention mechanism and a multi-layer perceptron (MLP), as well as residual connections to prevent gradient vanishing and network degradation with the increase of depth. We denote the output encoding of the observations as $( \\hat { o } ^ { i _ { 1 } } , \\dots , \\hat { o } ^ { i _ { n } } )$ , which encodes not only the information of agents $( i _ { 1 } , \\ldots , i _ { n } )$ but also the high-level interrelationships that represent agents’ interactions. In order to learn expressive representations, in the training phase, we make the encoder to approximate the value functions, whose objective is to minimize the empirical Bellman error by ",
+        "bbox": [
+            173,
+            651,
+            825,
+            762
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10cbb876477e3e4be218dadaa0af511fa9fb9ea0cf8eb22fc72163d78c440f12.jpg",
+        "text": "$$\nL _ { \\mathrm { E n c o d e r } } ( \\phi ) = \\frac { 1 } { T n } \\sum _ { m = 1 } ^ { n } \\sum _ { t = 0 } ^ { T - 1 } \\Big [ R ( \\mathbf { o } _ { t } , \\mathbf { a } _ { t } ) + \\gamma V _ { \\vec { \\phi } } ( \\hat { \\mathbf { o } } _ { t + 1 } ^ { i _ { m } } ) - V _ { \\phi } ( \\hat { \\mathbf { o } } _ { t } ^ { i _ { m } } ) \\Big ] ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            768,
+            720,
+            814
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\bar { \\phi }$ is the target network’s parameter, which is non-differentiable and updated every few epochs. ",
+        "bbox": [
+            173,
+            820,
+            823,
+            835
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The decoder, whose parameters we denote by $\\theta$ , passes the embedded joint action $\\pmb { a } ^ { i _ { 0 : m - 1 } } , m =$ $\\{ 1 , \\ldots n \\}$ (where $a ^ { i _ { 0 } }$ is an arbitrary symbol indicating the start of decoding) to a sequence of decoding blocks. Crucially, every decoding block comes with a masked self-attention mechanism, where the masking makes sure that, for every $i _ { j }$ , attention is computed only between the $i _ { r } ^ { \\mathrm { t h } }$ and the $i _ { j } ^ { \\mathrm { t h } }$ action heads wherein $r < j$ so that the sequential update scheme can be maintained. This is then followed by a second masked attention function, which computes the attention between the action heads and observation representations. Finally, the block finishes with an MLP and skipping connections. The output to the last decoder block is a sequence of representations of the joint actions, $\\{ \\hat { \\pmb { a } } ^ { i _ { 0 } : i - 1 } \\} _ { i = 1 } ^ { m }$ . This is fed to an MLP that outputs the probability distribution of $i _ { m }$ ’s action, namely, the policy $\\pi _ { \\theta } ^ { i _ { m } } ( \\mathbf { a } ^ { i _ { m } } | \\hat { \\mathbf { o } } ^ { i _ { 1 : n } } , \\mathbf { a } ^ { i _ { 1 : m - 1 } } ) $ . To train the decoder, we minimize the following clipping PPO objective of ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/05b4d70e3ed71091e35c8894f1dc7246c770b17c0f70d3241565b25f49c9e247.jpg",
+        "image_caption": [
+            "Figure 3: Demonstrations of the Bi-DexHands and the HalfCheetah environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            88,
+            818,
+            215
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2226ea430ed36b5f48270f30a6b9760f1439abfb81507b6a7fab4bd1eaad67f7.jpg",
+        "image_caption": [
+            "Figure 4: Performance comparisons on the Multi-Agent MuJoCo and the Bi-DexHands benchmarks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            251,
+            812,
+            347
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            393,
+            825,
+            465
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e632119bca8ed5a9474e468ead387f89863a23ddd42513b3719d9f53e4fa6f9e.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\cal L } _ { \\mathrm { D e c o d e r } } ( \\theta ) = - \\frac { 1 } { T n } \\sum _ { m = 1 } ^ { n } \\sum _ { t = 0 } ^ { T - 1 } \\operatorname* { m i n } \\Big ( \\mathrm { \\bf r } _ { t } ^ { i _ { m } } ( \\theta ) \\hat { A } _ { t } , \\mathrm { c l i p } ( \\mathrm { \\bf r } _ { t } ^ { i _ { m } } ( \\theta ) , 1 \\pm \\epsilon ) \\hat { A } _ { t } \\Big ) } , } \\\\ { { \\displaystyle \\mathrm { \\bf ~ r } _ { t } ^ { i _ { m } } ( \\theta ) = \\frac { \\pi _ { \\theta } ^ { i _ { m } } \\big ( { \\bf a } _ { t } ^ { i _ { m } } \\big | \\hat { \\bf { \\bf { \\boldsymbol \\Phi } } } _ { t } ^ { i _ { 1 : n } } , \\hat { \\bf { \\bf { a } } } _ { t } ^ { i _ { 1 : m - 1 } } \\big ) } { \\pi _ { \\theta _ { \\mathrm { o l d } } } ^ { i _ { m } } \\big ( { \\bf a } _ { t } ^ { i _ { m } } \\big | \\hat { \\bf { \\bf { \\boldsymbol \\Phi } } } _ { t } ^ { i _ { 1 : n } } , \\hat { \\bf { a } } _ { t } ^ { i _ { 1 : m - 1 } } \\big ) } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            473,
+            727,
+            560
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { A } _ { t }$ is an estimate of the joint advantage function. One can apply generalized advantage estimation (GAE) [32] with $\\begin{array} { r } { \\hat { V } _ { t } = \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n } V \\big ( \\hat { \\mathbf { o } } _ { t } ^ { i _ { m } } \\big ) } \\end{array}$ as a robust estimator for the joint value function. Notably, the action generation process is different between the inference and the training stage. In the inference stage, each action is generated auto-regressively, in the sense that $\\mathbf { a } ^ { i _ { m } }$ will be inserted back into the decoder again to generate $\\mathrm { a } ^ { i _ { m + 1 } }$ (starting with $\\mathrm { \\mathbf { a } } ^ { i _ { 0 } }$ and ending with $\\mathrm { a } ^ { i _ { n - 1 } }$ ). While during the training stage, the output of all actions, $\\mathbf { a } ^ { i _ { 1 : n } }$ can be computed in parallel simply because $\\mathbf { a } ^ { i _ { 1 : n - 1 } }$ have already been collected and stored in the replay buffer. ",
+        "bbox": [
+            173,
+            574,
+            825,
+            674
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The attention mechanism, which lies in the heart of MAT, encodes observations and actions with a weight matrix calculated by multiplying the embedded queries, $( q ^ { i _ { 1 } } , \\ldots , q ^ { i _ { n } } )$ , and keys, $( k ^ { i _ { 1 } } , \\ldots , k ^ { i _ { n } } )$ , where each of the weight $w \\tilde { ( } q ^ { i _ { r } } , k ^ { i _ { j } } ) = \\langle q ^ { i _ { r } ^ { \\perp } } , k ^ { i _ { j } } \\rangle$ . The embedded values $( v ^ { i _ { 1 } } , \\ldots , v ^ { i _ { n } } )$ are multiplied with the weight matrix to output representations. While the unmasked attention in the encoder uses a full weight matrix to extract the interrelationship between agents, i.e., $\\hat { \\mathbf { O } } ^ { i _ { 1 : n } }$ , the masked attentions in the decoder capture $\\mathbf { a } ^ { i _ { 1 : m } }$ with triangular matrices where $\\bar { w ( q ^ { i _ { r } } , k ^ { i _ { j } } ) } = 0$ for $r \\textless j$ (see an visual illustration in Appendix A). With the properly masked attention mechanism, the decoder can safely output the policy $\\pi _ { \\theta } ^ { i _ { m + 1 } } ( \\mathbf { a } ^ { i _ { m + 1 } } | \\hat { \\mathbf { o } } ^ { i _ { 1 : n } } , \\mathbf { a } ^ { i _ { 1 : m } } )$ , which finishes the implementation of Theorem (1). ",
+        "bbox": [
+            173,
+            679,
+            825,
+            809
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The monotonic improvement guarantee. An MAT agent $i _ { m }$ optimizes a trust-region objective that is conditioned on new decisions of agents $i _ { 1 : m - 1 }$ by means of conditioning its policy ratio on them (see Equation (5)). As such, it increases the joint return monotonically like if it followed the sequential update scheme of HAPPO [15, Theorem 2]. However, as oppose to that method, the MAT model does not require $i _ { m }$ to wait until its predecessors make their updates, nor it uses their updated action distribution for importance sampling calculations. In fact, as actions of all agents are outputs of MAT, their clipping objectives can be computed in parallel (during training), thus dominating ",
+        "bbox": [
+            174,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/8150f1f8d6b68e2b5124c1aac4187bc3f81a6103243e4614779468e1c1ad2b56.jpg",
+        "table_caption": [
+            "Table 1: Performance evaluations of win rate and standard deviation on the SMAC benchmark, where UPDeT’s official codebase supports several Marine-based tasks only. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>Difficulty</td><td>MAT</td><td>MAT-Dec</td><td>MAPPO</td><td>HAPPO</td><td>QMIX</td><td>UPDeT</td><td>Steps</td></tr><tr><td>3m</td><td>Easy</td><td>100.0(1.8)</td><td>100.0(1.1)</td><td>100.0(0.4)</td><td>100.0(1.2)</td><td>96.91.3</td><td>100.0(5.2)</td><td>5e5</td></tr><tr><td>8m</td><td>Easy</td><td>100.0(1.1)</td><td>97.5(2.5)</td><td>96.8(2.9)</td><td>97.5(1.1)</td><td>97.71.9</td><td>96.3(9.7)</td><td>1e6</td></tr><tr><td>1c3s5z</td><td>Easy</td><td>100.0(2.4)</td><td>100.0(0.4)</td><td>100.0(2.2)</td><td>97.5(1.8)</td><td>96.9(1.5)</td><td>/</td><td>2e6</td></tr><tr><td>MMM</td><td>Easy</td><td>100.0(2.2)</td><td>98.1(2.1)</td><td>95.6(4.5)</td><td>81.2(22.9)</td><td>91.2(3.2)</td><td>/</td><td>2e6</td></tr><tr><td>2c vs 64zg</td><td>Hard</td><td>100.0(1.3)</td><td>95.9(2.3)</td><td>100.0(2.7)</td><td>90.0(4.8)</td><td>90.3(4.0)</td><td>/</td><td>5e6</td></tr><tr><td>3s vs 5z</td><td>Hard</td><td>100.0(1.7)</td><td>100.0(1.3)</td><td>100.0(2.5)</td><td>91.9(5.3)</td><td>92.3(4.4)</td><td>/</td><td>5e6</td></tr><tr><td>3s5z</td><td>Hard</td><td>100.0(1.9)</td><td>100.0(3.3)</td><td>72.5(26.5)</td><td>90.0(3.5)</td><td>84.3(5.4)</td><td>/</td><td>3e6</td></tr><tr><td>5m vs 6m</td><td>Hard</td><td>90.6(4.4)</td><td>83.1(4.6)</td><td>88.2(6.2)</td><td>73.8(4.4)</td><td>75.8(3.7)</td><td>90.6(6.1)</td><td>1e7</td></tr><tr><td>8m vs 9m</td><td>Hard</td><td>100.0(3.1)</td><td>95.0(4.6)</td><td>93.8(3.5)</td><td>86.2(4.4)</td><td>92.6(4.0)</td><td>/</td><td>5e6</td></tr><tr><td>10m vs 11m</td><td>Hard</td><td>100.0(1.4)</td><td>100.0(2.0)</td><td>96.3(5.8)</td><td>77.5(9.7)</td><td>95.8(6.1)</td><td>/</td><td>5e6</td></tr><tr><td>25m</td><td>Hard</td><td>100.0(1.3)</td><td>86.9(5.6)</td><td>100.0(2.7)</td><td>70.1(8.1)</td><td>90.2(9.8)</td><td>2.8(3.1)</td><td>2e6</td></tr><tr><td>27m vs 30m</td><td>Hard+</td><td>100.0(0.7)</td><td>95.3(2.2)</td><td>93.1(3.2)</td><td>5.6(2.8)</td><td>39.2(8.8)</td><td>/</td><td>1e7</td></tr><tr><td>MMM2</td><td>Hard+</td><td>93.8(2.6)</td><td>91.2(5.3)</td><td>81.8(10.1)</td><td>68.8(13.7)</td><td>88.3(2.4)</td><td>/</td><td>1e7</td></tr><tr><td>6h vs 8z</td><td>Hard+</td><td>98.8(1.3)</td><td>93.8(4.7)</td><td>88.4(5.7)</td><td>0.3(0.4)</td><td>9.7(3.1)</td><td>/</td><td>1e7</td></tr><tr><td>3s5z vs 3s6z</td><td>Hard+</td><td>96.5(1.3)</td><td>85.3(7.5)</td><td>84.3(19.4)</td><td>82.8(21.2)</td><td>68.8(21.2)</td><td>/</td><td>2e7</td></tr></table>",
+        "bbox": [
+            181,
+            123,
+            810,
+            343
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3d7ba93c472c245a76d1e9b46ca6a00f09693e351a67723eef51e51c2a4db1c4.jpg",
+        "image_caption": [
+            "Figure 5: Performance comparison on the Google Research Football tasks with 2-4 agents from left to right respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            364,
+            797,
+            486
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "HAPPO on the time complexity. Lastly, to assure that the limiting joint policy is such that none of the agents is incentivized to change its policy (Nash equilibrium), MAT requires permutating the sequential order of updates at every iteration, which is inline with the discovery in HAPPO [15, Theorem 3]. ",
+        "bbox": [
+            174,
+            550,
+            825,
+            606
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments and Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            628,
+            413,
+            646
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "MAT provides a new solution paradigm for cooperative MARL problems. The key insights of MAT are the sequential update scheme, which is inspired by Theorem (1), as well as the encoder-decoder architecture, which provides a highly-efficient implementation for a sequence modeling perspective. Importantly, MAT inherits the monotonic improvement guarantee, and agents’ policies can be learned in parallel during training. We firmly believe MAT will become a game changer for MARL studies. ",
+        "bbox": [
+            174,
+            662,
+            825,
+            732
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To evaluate if MAT meets our expectations, we test MAT on the StarCraftII Multi-Agent Challenge (SMAC) benchmark [31] where MAPPO with parameter sharing [46] has shown superior performance, and the Multi-Agent MuJoCo benchmark [7] where HAPPO [15] shows the current state-of-the-art performance. SMAC and MuJoCo environments are common benchmarks in the MARL field. On top of them, we also test MAT on the Bimanual Dexterous Hands Manipulation (Bi-DexHands) [6] which provides a list of challenging bimanual manipulation tasks (see Figure (3)), and the Google Research Football [18] benchmark with a series of cooperation scenarios in football game. ",
+        "bbox": [
+            174,
+            738,
+            825,
+            835
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We apply the same hyper-parameters of baseline algorithms from their original paper to ensure their best performance, and adopt the same hyper-parameter tuning process for our methods with details in Appendix B. To ensure fair comparisons to CTDE methods, we also introduce a CTDE-variant of MAT called MAT-Dec, which essentially adopts a fully decentralized actor for each individual agent (rather than using the decoder proposed in MAT) while keeping the encoder fixed. The critic’s loss for MAT-Dec is $\\begin{array} { r } { L ( \\phi ) = \\frac { 1 } { T } \\sum _ { t = 0 } ^ { T - 1 } \\left[ R ( \\mathbf { o } _ { t } , \\mathbf { a } _ { t } ) + \\gamma \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n } V _ { \\vec { \\phi } } ( \\hat { \\mathbf { o } } _ { t + 1 } ^ { i _ { m } } ) - \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n } V _ { \\phi } ( \\hat { \\mathbf { o } } _ { t } ^ { i _ { m } } ) \\right] ^ { 2 } , } \\end{array}$ and we apply the local advantage estimation $A _ { t } \\big ( \\hat { \\mathbf { o } } _ { t } ^ { i _ { m } } , a ^ { i _ { m } } \\big )$ to guide the subsequent policy update. ",
+        "bbox": [
+            174,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/475361329427df4de80617968b1a2a1d3b504e83678e33306c556403d6d3eefd.jpg",
+        "image_caption": [
+            "Figure 6: Performance on the HalfCheetah task with different disabled joints shown in Figure (3a). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            90,
+            800,
+            338
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            387,
+            823,
+            424
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.1 Performance on Cooperative MARL Benchmarks ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            445,
+            558,
+            460
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "According to Table (1) and Figure (4), MAPPO significantly outperforms HAPPO in SMAC with higher sample efficiency. This verified the homogeneity of SMAC agents and the heterogeneity of multi-agent MuJoCo agents, that are also discovered by Kuba et al. [15]. Take the SMAC task $2 5 m$ as an example, all the marines are equivalent and interchangeable so that agents can learn from their teammate’s experience. Sharing parameters in this settings means leveraging 25 times more examples to train each agents comparing with separated network of HAPPO, and thus enjoying higher learning efficiency. On the other hand, with the heterogeneous settings of multi-agent MuJoCo, training a \"foot\" agent with experience from a \"thigh\" agent can surely harm its performance since they represent different functions on the Cheetah. However, MAT outperforms MAPPO and HAPPO in almost all tasks in Table (1) and Figure (4), indicating its modeling capability on both homogeneous and heterogeneous-agent tasks. MAT also enjoys the superior performance over MAT-Dec, which emphasize the importance of the decoder architecture in the MAT design. On the Bi-DexHands tasks, MAT outperforms MAPPO and HAPPO methods by a large margin. We save the Google Football results to Figure (5), where the conclusion stays the same. ",
+        "bbox": [
+            174,
+            473,
+            825,
+            667
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 MAT as Excellent Few-short Learners ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            689,
+            477,
+            704
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Since Transformer-based models often demonstrate strong generalization performance on few-short tasks [3, 9], we believe MAT can possess strong generalization ability on unseen MARL tasks as well. To validate such an assumption, we design zero-shot and few-shot experiments on SMAC and multi-agent MuJoCo tasks. For SMAC tasks, we pre-train agents on eight tasks involving five types of units (3m, 8m vs 9m, 10m vs 11m, 25m, 3s vs 3z, 2s3z, 3s5z, MMM ) with 10M examples in total and then apply them on six separate and much harder tasks (5m vs 6m, 8m, $2 7 m$ vs 30m, 2s vs 1sc, 1c3s5z, MMM2 ) including seven types of units. This setting is designed to evaluate the generalization ability of MAT when training on simple tasks but transferring to more diverse and complex downstream tasks. In terms of multi-agent MuJoCo, we reuse the models trained on the complete HalfCheetah robot as the pre-trained agent and then directly apply it to six new tasks, each with a different leg being disfunctioned (see Figure (3a)). We investigate the generalization capability of pre-trained models on each downstream task with $0 \\%$ (zero-shot), $1 \\%$ , $5 \\%$ , $10 \\%$ few-short new examples, respectively. Note that common MARL baselines such as HAPPO assume fixed number of agents during training, thus it cannot directly handle the cases with changing number of agents. ",
+        "bbox": [
+            174,
+            718,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/9acf59cc8d80a3fe2ad901dd6f997627c3ef331c74bf49ad7ab5cab6437fd2cb.jpg",
+        "table_caption": [
+            "Table 2: Median evaluation win rate and the standard deviation on the SMAC benchmark for pre-trained models with different number of online examples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods #examples</td><td colspan=\"4\">MAT</td><td colspan=\"4\">MAPPO</td><td colspan=\"4\">MAT-from scratch</td></tr><tr><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td></tr><tr><td>5m vs 6m</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>5.8(3.1)</td><td>18.8(7.1)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>4.3(3.8)</td><td>21.9(12.2)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>1.9(1.3)</td><td>3.8(2.1)</td></tr><tr><td>8m</td><td>100(0.0)</td><td>100(1.2)</td><td>100(0.3)</td><td>100(2.1)</td><td>100(0.0)</td><td>100(1.4)</td><td>100(0.3)</td><td>100(1.4)</td><td>0.0(0.0)</td><td>10.6(23.8)</td><td>92.5(3.7)</td><td>100(1.4)</td></tr><tr><td>27m vs 30m</td><td>0.0(0.0)</td><td>6.3(2.4)</td><td>53.8(16.4)</td><td>71.2(8.2)</td><td>9.4(3.6)</td><td>15(5.9)</td><td>26.2(7.8)</td><td>26.8(9.7)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>0.0(0.3)</td><td>0.3(15.6)</td></tr><tr><td>2s vs 1sc</td><td>0.0(0.0)</td><td>15.6(13.8)</td><td>100(9.7)</td><td>100(0.0)</td><td>0.0(0.0)</td><td>43.1(17.6)</td><td>100(1.1)</td><td>100(1.8)</td><td>0.0(0.0)</td><td>19.3(33.3)</td><td>96.3(6.2)</td><td>100(0.3)</td></tr><tr><td>1c3s5z</td><td>3.1(1.8)</td><td>5.6(5.0)</td><td>82.5(5.5)</td><td>100(2.7)</td><td>3.1(1.8)</td><td>4.3(4.9)</td><td>73.8(13.0)</td><td>97.5(2.1)</td><td>0.0(0.0)</td><td>7.5(4.8)</td><td>87.5(3.9)</td><td>100(1.4)</td></tr><tr><td>MMM2</td><td>0.0(3.6)</td><td>0.0(1.8)</td><td>33.8(13.7)</td><td>62.5(12.1)</td><td>0.0(0.0)</td><td>0.0(1.4)</td><td>13.8(7.0)</td><td>36.2(9.6)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>0.0(0.0)</td><td>0.0(0.7)</td></tr></table>",
+        "bbox": [
+            176,
+            123,
+            821,
+            212
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/9a9149bf9747d0c0824dc60fde68e9645173b407b9d91f2f785be9998f703848.jpg",
+        "table_caption": [
+            "Table 3: Average evaluation score and standard deviation on Multi-Agent MuJoCo for pre-trained models with different number of online examples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Methods #examples</td><td colspan=\"4\">MAT</td><td colspan=\"4\">MAPPO</td><td colspan=\"4\">MAT-from scratch</td></tr><tr><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td><td>0%</td><td>1%</td><td>5%</td><td>10%</td></tr><tr><td>back foot</td><td>2100(89)</td><td>2837(95)</td><td>4691(235)</td><td>5646(79)</td><td>2936(301)</td><td>3017(135)</td><td>3221(119)</td><td>3304(129)</td><td>-0.44(0.4)</td><td>-5.18(11)</td><td>670(1098)</td><td>1635(1184)</td></tr><tr><td>back shin</td><td>4005(316)</td><td>4143(230)</td><td>6077(209)</td><td>7176(74)</td><td>2406(32)</td><td>2542(108)</td><td>2796(137)</td><td>2955(127)</td><td>-0.31(0.1)</td><td>-3.95(17)</td><td>743(537)</td><td>1252(1123)</td></tr><tr><td>back thigh</td><td>5361(45)</td><td>5641(150)</td><td>7101(119)</td><td>7460(61)</td><td>3043(79)</td><td>3060(143)</td><td>3217(3)</td><td>3353（71)</td><td>-0.54(0.3)</td><td>-4.87(7.7)</td><td>930(589)</td><td>2067(861)</td></tr><tr><td>fore foot</td><td>1313(512)</td><td>1955(232)</td><td>4856(146)</td><td>6054(172)</td><td>623（44)</td><td>970(185)</td><td>2025(371)</td><td>2480(239)</td><td>-0.37(0.2)</td><td>-2.25(7.9)</td><td>1821(157)</td><td>2877(106)</td></tr><tr><td>fore shin</td><td>2435(13)</td><td>2617(71)</td><td>3851(57)</td><td>4373(83)</td><td>1715(55)</td><td>2457(125)</td><td>3096(59)</td><td>3310(54)</td><td>-0.15(0.06)</td><td>-0.96(6.0)</td><td>1461(101)</td><td>3003(316)</td></tr><tr><td>fore thigh</td><td>5631(321)</td><td>6448(417)</td><td>7952(109)</td><td>8347(81)</td><td>3087(110)</td><td>3171(83)</td><td>3340(52)</td><td>3519(59)</td><td>-0.29(0.3)</td><td>0.82(14)</td><td>1021(177)</td><td>2600(215)</td></tr></table>",
+        "bbox": [
+            176,
+            261,
+            821,
+            343
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We summarize the zero-shot and few-shot results of each algorithm in Table (2) and (3), where the bold number indicates the best performance. We also provide the performance of MAT if it was given the same amount of data but is trained from scratch, the \"MAT-from scratch\", as the control group to demonstrate the effectiveness of pre-training process. As both tables suggest, bold numbers are mainly located in the area of MAT, which justify MAT’s strong generalisation performance as a few-short learner. Surprisingly, we find that the few-shot MAT with only $10 \\%$ data show even higher rewards than its counterpart that is purely trained on HalfCheetah with the same disabled joints (back foot, back shin and back thigh ) and $100 \\%$ full amount data, we believe it is because the pre-train process offers initial weights that are not only closer to optimum but also less likely to stuck in bad local optima than random initialization. ",
+        "bbox": [
+            173,
+            369,
+            825,
+            508
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            529,
+            299,
+            545
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the past five years, large sequence models have achieved remarkable successes on solving visual language tasks. In this paper, we take the initial effort to build the connection between multi-agent reinforcement learning (MARL) problems and generic sequence models (SM), with the ambition that MARL researchers can hereafter benefit from the prosperous development on the sequence modeling side. Specifically, we contribute by unifying a general solution to cooperative MARL problems into a Transformer like encoder-decoder model. The proposed Multi-Agent Transformer (MAT) leverages the multi-agent advantage decomposition theorem, which essentially transforms the joint policy optimization process into a sequential decision making process that can be simply implemented by an auto-regressive model. We have demonstrated MAT’s strong empirical performance on three challenging benchmarks against current state-of-the-art MARL solutions including MAPPO and HAPPO. Based on the established connection between MARL and SM, in the future, we plan to bring multi-agent learning tasks into large multi-modal SM, chasing for more generally intelligent models as the most recent success of GATO has already demonstrated [30]. ",
+        "bbox": [
+            174,
+            559,
+            825,
+            739
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            758,
+            321,
+            776
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The SJTU team is partially supported by “New Generation of AI 2030” Major Project (2018AAA0100900), Shanghai Municipal Science and Technology Major Project (2021SHZDZX0102), Shanghai Sailing Program (21YF1421900), and National Natural Science Foundation of China (62076161, 62106141). ",
+        "bbox": [
+            174,
+            790,
+            825,
+            845
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch, Katie Millican, Malcolm Reynolds, et al. Flamingo: a visual language model for few-shot learning. arXiv preprint arXiv:2204.14198, 2022.   \n[2] Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258, 2021.   \n[3] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. NeurIPS, 2020.   \n[4] Yu-Han Chang, Tracey Ho, and Leslie Kaelbling. All learning is local: Multi-agent learning in global reward games. Advances in neural information processing systems, 16, 2003.   \n[5] Lili Chen, Kevin Lu, Aravind Rajeswaran, Kimin Lee, Aditya Grover, Michael Laskin, Pieter Abbeel, Aravind Srinivas, and Igor Mordatch. Decision transformer: Reinforcement learning via sequence modeling. arXiv preprint arXiv:2106.01345, 2021.   \n[6] Yuanpei Chen, Yaodong Yang, Tianhao Wu, Shengjie Wang, Xidong Feng, Jiechuang Jiang, Stephen Marcus McAleer, Hao Dong, Zongqing Lu, and Song-Chun Zhu. Towards human-level bimanual dexterous manipulation with reinforcement learning. arXiv preprint arXiv:2206.08686, 2022.   \n[7] Christian Schröder de Witt, Bei Peng, Pierre-Alexandre Kamienny, Philip H. S. Torr, Wendelin Böhmer, and Shimon Whiteson. Deep multi-agent reinforcement learning for decentralized continuous cooperative control. CoRR, abs/2003.06709, 2020.   \n[8] Xiaotie Deng, Yuhao Li, David Henry Mguni, Jun Wang, and Yaodong Yang. On the complexity of computing markov perfect equilibrium in general-sum stochastic games. arXiv preprint arXiv:2109.01795, 2021.   \n[9] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. In NAACL-HLT (1), 2019.   \n[10] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2020.   \n[11] Jakob Foerster, Gregory Farquhar, Triantafyllos Afouras, Nantas Nardelli, and Shimon Whiteson. Counterfactual multi-agent policy gradients. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   \n[12] Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick. Masked autoencoders are scalable vision learners. arXiv preprint arXiv:2111.06377, 2021.   \n[13] Siyi Hu, Fengda Zhu, Xiaojun Chang, and Xiaodan Liang. Updet: Universal multi-agent reinforcement learning via policy decoupling with transformers. arXiv preprint arXiv:2101.08001, 2021.   \n[14] Michael Janner, Qiyang Li, and Sergey Levine. Offline reinforcement learning as one big sequence modeling problem. Advances in Neural Information Processing Systems, 34, 2021.   \n[15] Jakub Grudzien Kuba, Ruiqing Chen, Munning Wen, Ying Wen, Fanglei Sun, Jun Wang, and Yaodong Yang. Trust region policy optimisation in multi-agent reinforcement learning. ICLR, 2022.   \n[16] Jakub Grudzien Kuba, Xidong Feng, Shiyao Ding, Hao Dong, Jun Wang, and Yaodong Yang. Heterogeneous-agent mirror learning: A continuum of solutions to cooperative marl. arXiv preprint arXiv:2208.01682, 2022. ",
+        "bbox": [
+            171,
+            82,
+            828,
+            917
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[17] Jakub Grudzien Kuba, Muning Wen, Linghui Meng, Haifeng Zhang, David Mguni, Jun Wang, Yaodong Yang, et al. Settling the variance of multi-agent policy gradients. Advances in Neural Information Processing Systems, 34:13458–13470, 2021. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[18] Karol Kurach, Anton Raichuk, Piotr Stanczyk, Micha ´ ł Zaj ˛ac, Olivier Bachem, Lasse Espeholt, Carlos Riquelme, Damien Vincent, Marcin Michalski, Olivier Bousquet, et al. Google research football: A novel reinforcement learning environment. arXiv preprint arXiv:1907.11180, 2019. ",
+        "bbox": [
+            173,
+            143,
+            823,
+            188
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[19] Michael L Littman. Markov games as a framework for multi-agent reinforcement learning. In Machine learning proceedings 1994, pages 157–163. Elsevier, 1994. ",
+        "bbox": [
+            171,
+            198,
+            825,
+            227
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[20] Ryan Lowe, Yi Wu, Aviv Tamar, Jean Harb, Pieter Abbeel, and Igor Mordatch. Multi-agent actorcritic for mixed cooperative-competitive environments. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 6382–6393, 2017. ",
+        "bbox": [
+            173,
+            238,
+            825,
+            281
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[21] Xueguang Lyu, Yuchen Xiao, Brett Daley, and Christopher Amato. Contrasting centralized and decentralized critics in multi-agent reinforcement learning. arXiv preprint arXiv:2102.04402, 2021. ",
+        "bbox": [
+            173,
+            291,
+            825,
+            333
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[22] Anuj Mahajan, Tabish Rashid, Mikayel Samvelyan, and Shimon Whiteson. Maven: Multi-agent variational exploration. Advances in Neural Information Processing Systems, 32, 2019. ",
+        "bbox": [
+            173,
+            344,
+            823,
+            375
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[23] Linghui Meng, Muning Wen, Yaodong Yang, Chenyang Le, Xiyun Li, Weinan Zhang, Ying Wen, Haifeng Zhang, Jun Wang, and Bo Xu. Offline pre-trained multi-agent decision transformer: One big sequence model conquers all starcraftii tasks. arXiv preprint arXiv:2112.02845, 2021. ",
+        "bbox": [
+            174,
+            385,
+            825,
+            428
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[24] David Mguni, Jianhong Wang, Taher Jafferjee, Nicolas Perez-Nieves, Wenbin Song, Yaodong Yang, Feifei Tong, Hui Chen, Jiangcheng Zhu, Yali Du, et al. Learning to shape rewards using a game of switching controls. arXiv preprint arXiv:2103.09159, 2021. ",
+        "bbox": [
+            173,
+            439,
+            825,
+            481
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[25] David Henry Mguni, Taher Jafferjee, Jianhong Wang, Nicolas Perez-Nieves, Oliver Slumbers, Feifei Tong, Yang Li, Jiangcheng Zhu, Yaodong Yang, and Jun Wang. Ligs: Learnable intrinsicreward generation selection for multi-agent learning. In International Conference on Learning Representations, 2021. ",
+        "bbox": [
+            174,
+            492,
+            826,
+            547
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[26] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. nature, 518(7540):529–533, 2015. ",
+        "bbox": [
+            173,
+            559,
+            823,
+            602
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[27] Prakash M Nadkarni, Lucila Ohno-Machado, and Wendy W Chapman. Natural language processing: an introduction. Journal of the American Medical Informatics Association, 18(5):544– 551, 2011. ",
+        "bbox": [
+            173,
+            613,
+            826,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[28] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever. Zero-shot text-to-image generation. In International Conference on Machine Learning, pages 8821–8831. PMLR, 2021. ",
+        "bbox": [
+            171,
+            666,
+            823,
+            709
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[29] Tabish Rashid, Mikayel Samvelyan, Christian Schroeder, Gregory Farquhar, Jakob Foerster, and Shimon Whiteson. Qmix: Monotonic value function factorisation for deep multi-agent reinforcement learning. In International Conference on Machine Learning, pages 4295–4304. PMLR, 2018. ",
+        "bbox": [
+            173,
+            720,
+            825,
+            776
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[30] Scott Reed, Konrad Zolna, Emilio Parisotto, Sergio Gomez Colmenarejo, Alexander Novikov, Gabriel Barth-Maron, Mai Gimenez, Yury Sulsky, Jackie Kay, Jost Tobias Springenberg, Tom Eccles, Jake Bruce, Ali Razavi, Ashley Edwards, Nicolas Heess, Yutian Chen, Raia Hadsell, Oriol Vinyals, Mahyar Bordbar, and Nando de Freitas. A generalist agent. arXiv preprint arXiv:2205.06175, 2022. ",
+        "bbox": [
+            174,
+            787,
+            826,
+            857
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[31] Mikayel Samvelyan, Tabish Rashid, Christian Schroeder de Witt, Gregory Farquhar, Nantas Nardelli, Tim GJ Rudner, Chia-Man Hung, Philip HS Torr, Jakob Foerster, and Shimon Whiteson. The starcraft multi-agent challenge. 2019. ",
+        "bbox": [
+            174,
+            869,
+            826,
+            911
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[32] John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. Highdimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015.   \n[33] John Schulman, F. Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. ArXiv, abs/1707.06347, 2017.   \n[34] David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In International conference on machine learning, pages 387–395. PMLR, 2014.   \n[35] Kyunghwan Son, Daewoo Kim, Wan Ju Kang, David Earl Hostallero, and Yung Yi. Qtran: Learning to factorize with transformation for cooperative multi-agent reinforcement learning. In International Conference on Machine Learning, pages 5887–5896. PMLR, 2019.   \n[36] Peter Sunehag, Guy Lever, Audrunas Gruslys, Wojciech Marian Czarnecki, Vinicius Zambaldi, Max Jaderberg, Marc Lanctot, Nicolas Sonnerat, Joel Z Leibo, Karl Tuyls, et al. Valuedecomposition networks for cooperative multi-agent learning. arXiv preprint arXiv:1706.05296, 2017.   \n[37] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.   \n[38] Hugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and Hervé Jégou. Going deeper with image transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 32–42, 2021.   \n[39] Zhengzhong Tu, Hossein Talebi, Han Zhang, Feng Yang, Peyman Milanfar, Alan Bovik, and Yinxiao Li. Maxvit: Multi-axis vision transformer. arXiv preprint arXiv:2204.01697, 2022.   \n[40] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.   \n[41] Wenxiao Wang, Lu Yao, Long Chen, Binbin Lin, Deng Cai, Xiaofei He, and Wei Liu. Crossformer: A versatile vision transformer hinging on cross-scale attention. arXiv preprint arXiv:2108.00154, 2021.   \n[42] Ying Wen, Yaodong Yang, Rui Luo, Jun Wang, and Wei Pan. Probabilistic recursive reasoning for multi-agent reinforcement learning. In International Conference on Learning Representations, 2018.   \n[43] Ying Wen, Yaodong Yang, and Jun Wang. Modelling bounded rationality in multi-agent interactions by generalized recursive reasoning. In Christian Bessiere, editor, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20, pages 414–421. International Joint Conferences on Artificial Intelligence Organization, 7 2020. Main track.   \n[44] Yaodong Yang and Jun Wang. An overview of multi-agent reinforcement learning from game theoretical perspective. arXiv preprint arXiv:2011.00583, 2020.   \n[45] Yaodong Yang, Ying Wen, Jun Wang, Liheng Chen, Kun Shao, David Mguni, and Weinan Zhang. Multi-agent determinantal q-learning. In International Conference on Machine Learning, pages 10757–10766. PMLR, 2020.   \n[46] Chao Yu, A. Velu, Eugene Vinitsky, Yu Wang, A. Bayen, and Yi Wu. The surprising effectiveness of mappo in cooperative, multi-agent games. ArXiv, abs/2103.01955, 2021. ",
+        "bbox": [
+            169,
+            79,
+            828,
+            832
+        ],
+        "page_idx": 12
+    }
+]

parse/dev/CEjuyeZj1jz/CEjuyeZj1jz.md

@@ -0,0 +1,616 @@
+# Finite-Time Analysis of Fully Decentralized Single-Timescale Actor-Critic
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Decentralized Actor-Critic (AC) algorithms have been widely utilized for multi  
+2 agent reinforcement learning (MARL） and have achieved remarkable success.   
+3 Apart from its empirical success, the theoretical convergence property of decen  
+4 tralized AC algorithms is largely unexplored. The existing finite-time convergence   
+5 results are derived based on either double-loop update or two-timescale step sizes   
+6 rule,which is not often adopted in real implementation. In this work,we introduce   
+7 a fully decentralized AC algorithm, where actor, critic,and global reward estimator   
+8 are updated in an alternating manner with step sizes being of the same order, namely,   
+9 we adopt the single-timescale update.Theoretically,using linear approximation for   
+10 value and reward estimation, we show that our algorithm has sample complexity of   
+11 $\tilde { \mathcal { O } } ( \epsilon ^ { - 2 } )$ under Markovian sampling, which matches the optimal complexity with   
+12 double-loop implementation (here, $\tilde { \mathcal { O } }$ hides a log term). The sample complexity   
+13 can be improved to $\mathcal { O } ( \epsilon ^ { - 2 } )$ under the i.i.d. sampling scheme. The central to   
+14 establishing our complexity results is the hidden smoothness of the optimal critic   
+15 variable we revealed. We also provide a local action privacy-preserving version   
+16 of our algorithm and its analysis. Finally,we conduct experiments to show the   
+17 superiority of our algorithm over the existing decentralized AC algorithms.
+
+# 181 Introduction
+
+19 Multi-agent reinforcement learning (MARL)[16,30] has been very successful in various models of   
+20 multi-agent systems,such as robotics [14],autonomous driving [37], Go [25],etc. MARL has been   
+21 extensively explored in the past decades; see,e.g.,[18,20,41,26,8,22]. These works either focus   
+22 on the setting where an central controller is available, or assuming a common reward function for all   
+23 agents. Among the many cooperative MARL settings, the work [42] proposes the fully decentralized   
+24 MARL with networked agents. In this setting, each agent maintains a private heterogeneous reward   
+25 function, and agents can only access local/neighboring information through communicating with its   
+26 neighboring agents on the network. Then, the objective of allagents is to jointly maximize the average   
+27 long-term reward through interacting with environment modeled by multi-agent Markov decision   
+28 process (MDP). They proposed the decentralized Actor-Critic (AC) algorithm to solve this MARL   
+29 problem,and showed its impressive performance. However, the theoretical convergence properties   
+30 of such class of decentralized AC algorithms are largely unexplored; see [41] for a comprehensive   
+31 survey.In this work,our goal is to establish the strong finite-time convergence results under this fully   
+32 decentralized MARL setting. We first review some recent progresses on this line of research below.   
+33 Related works and motivations. The first fully decentralized AC algorithm with provable con  
+34 vergence guarantee was proposed by [42], and they achieved asymptotic convergence results under   
+35 two-time scale step sizes, which requires actor's step sizes to diminish in a faster scale than the critic's   
+36 step sizes.The sample complexities of decentralized AC were established recently. In particular, [6]   
+37 and [11] independently propose two communication eficient decentralized AC algorithms with opti  
+38 mal sample complexity of ${ \mathcal { O } } ( \varepsilon ^ { - 2 } \log ( \varepsilon ^ { - 1 } ) )$ under Markovian sampling scheme. Their analysis are   
+39 based on double-loop implementation, where each policy optimization step follows a nearly accurate   
+40 critic optimization step (a.k.a. policy evaluation),i.e., solving the critic optimization subproblem to   
+41 $\varepsilon$ -accuracy. Such a double-loop scheme requires careful tuning of two additional hyper-parameters,   
+42 which are the batch size and innerloop size.In particular,the batch size and inner loop size need to be   
+43 of order $\mathcal { O } ( \varepsilon ^ { - 1 } )$ and ${ \mathcal { O } } ( \log ( \varepsilon ^ { - 1 } ) )$ in order to achieve their sample complexity results, respectively.   
+44 In practice, single-loop algorithmic framework is often utilized, where one updates the actor and   
+45 critic in an alternating manner by performing only one algorithmic iteration for both of the two   
+46 subproblems; see,e.g.， [23,18,15,39]. The work [38] proposes a new decentralized AC algorithm   
+47 based on such a single-loop alternative update. Nevertheless, they have to adopt two-timescale step   
+48 sizes rule to ensure convergence, which requires actor's step sizes to diminish in a faster scale than   
+49 the critic's step sizes.Due to the separation of the step sizes,the critic optimization sub-problem   
+50 is solved exactly when the number of iterations tends to $\infty$ . Such a restriction on the step size will   
+51 slow down the convergence speed of the algorithm. As a consequence,they only obtain sub-optimal   
+52 sample complexity of $\mathcal { O } ( \varepsilon ^ { - \frac { 5 } { 2 } } )$ . In practice, most algorithms are implemented with single-timescale   
+53 step size rule, where the step sizes for actor and critic updates are of the same order. Though there   
+54 are some theoretical achievements for single-timescale update in other areas such as TDC [31] and   
+55 bi-level optimization [4], similar theoretical understanding under AC setting is largely unexplored.   
+56 Indeed, even when reducing to single-agent seting, the convergence property of single-timescale   
+57 AC algorithm is not well established.The works [9,10] establish the finite-time convergence result   
+58 under a special single-timescale implementation, where they atain the sample complexity of $\mathcal { O } ( \varepsilon ^ { - 2 } )$   
+59 However,their analysis is based on an algorithm where the critic optimization step is formulated as a   
+60 least-square temporal diference (LSTD) at each iteration, where they need to sample the transition   
+61 tuples for $\tilde { \mathcal { O } } ( \varepsilon ^ { - 1 } )$ times to form the data matrix in the LSTD problem. Then, they solve the LSTD   
+62 problem in a closed-form fashion, which requires to invert a matrix of large size.Later,[4] obtains the   
+63 same sample complexity using TD(O) update for critic variables under i.i.d. sampling. Nonetheless,   
+64 their analysis highly relies on the assumption that the Jacobian of the stationary distribution is   
+65 Lipschitz continuous,which is not justified in their work.
+
+i6The above observations motivate us to ask the following question:
+
+Can we establish finite-time convergence result for decentralized AC algorithm with single-timescale step sizes rule?l
+
+，Main contributions.By answering this question positively, we have the following contributions:
+
+· We design a fully decentralized AC algorithm, which employs a single-timescale step sizes rule and adopts Markovian sampling scheme. The proposed algorithm allows communication between agents for every $K _ { c }$ iterations with $K _ { c }$ being any integer lies in $[ 1 , { \mathcal { O } } ( \varepsilon ^ { - { \frac { 1 } { 2 } } } ) ]$ ,rather than communicating at each iteration as adopted by previous single-loop decentralized AC algorithms [38, 42].   
+Using linear approximation for value and reward estimation, we establish the finite-time convergence result for such an algorithm under the standard assumptions. In particular, we show that the algorithm has the sample complexity of $\widetilde { \mathcal { O } } ( \varepsilon ^ { - 2 } )$ , which matches the optimal complexity up to a logarithmic term. In addition, we show that the logarithmic term can be removed under the i.i.d. sampling scheme.Note that these convergence results are valid for all the above mentioned choices for $K _ { c }$   
+To preserve the privacy of local actions, we propose a variant of our algorithm which utilizes noisy local rewards for estimating global rewards. We show that such an algorithm will maintain the optimal sample complexity at the expense of communicating at each iteration.
+
+The underlying principle for obtaining the above convergence results is that we reveal the hidden smoothness of the optimal critic variable, so that we can derive an approximate descent on the averaged critic's optimal gap at each iteration. Consequently, we can resort to the classic convergence analysis for alternating optimization algorithms to establish the approximate ascent property of the overall optimization process,which leads to the final sample complexity results.
+
+89 Another technical highlight is the Lyapunov function we construct for measuring the progress of our   
+90 algorithm. Such a construction is motivated by [4], which analyzes bi-level optimization algorithm.   
+91 However, our Lyapunov function is diferent from theirs as it involves the additional optimal gap of   
+92 averaged critic and reward estimator, which is necessry for dealing with the decentralized setting.
+
+3We finish this section by remarking that our convergence results are even new for single agent AC talgorithms under the settng of single-timescale step sizes rule.
+
+# ：2Preliminary
+
+In this section, we introduce the problem formulation and the policy gradient theorem, which serves as the preliminary for the analyzed decentralzed AC algorithm.
+
+Suppose there are multiple agents aiming to independently optimize a common global objective, and each agent can communicate with its neighbors through a network. To model the topology, we define the graph as $\mathcal { G } = ( \mathcal { N } , \mathcal { E } )$ ,where $\mathcal { N }$ is the set of nodes with $| { \mathcal { N } } | = N$ and $\mathcal { E }$ is the set of edges with $| { \mathcal { E } } | = E$ . In the graph, each node represents an agent, and each edge represents a communication link. The interaction between agents follows the networked multi-agent MDP.
+
+# 2.1Markov decision process
+
+104 A networked multi-agent MDP is defined by a tuple $( \mathcal { G } , S , \{ \mathcal { A } ^ { i } \} _ { i \in \mathcal { N } } , \mathcal { P } , \{ r ^ { i } \} _ { i \in [ N ] } , \gamma )$ ： $\mathcal { G }$ denotes the   
+105 communication topology (the graph), $s$ is the finite state space observed by all agents, $\mathcal { A } ^ { i }$ represents   
+106 the finite action space of agent $i$ .Let $\mathcal { A } : = \mathcal { A } ^ { 1 } \times \cdot \cdot \cdot \stackrel { \textstyle \circ } { \times } \mathcal { A } ^ { N }$ denote the joint action space and   
+107 $\mathcal { P } ( s ^ { \prime } | s , a ) : \mathcal { S } \times \bar { \mathcal { A } } \times \mathcal { S } \to [ \bar { 0 } , 1 ]$ denote the transition probability from any state $s \in S$ to any state   
+108 $s ^ { \prime } \in \mathcal { S }$ for any joint action $a \in { \mathcal { A } }$ ： $r ^ { i } : S \times \mathcal { A }  \mathbb { R }$ is the local reward function that determines the   
+109 reward received by agent $i$ given transition $( s , a )$ ， $\gamma \in [ 0 , 1 ]$ is the discount factor.   
+110 For simplicity, we will use $a : = [ a ^ { 1 } , \cdots , a ^ { N } ]$ to denote the joint action, and $\theta : = [ \theta ^ { 1 } , \cdots , \theta ^ { N } ] \in$   
+111 $\mathbb { R } ^ { d _ { \theta } \times N }$ to denote joint parameters of all actors, with $\theta ^ { i } \in \mathbb { R } ^ { d _ { \theta } }$ . Note that different actors may have   
+112 different number of parameters,which is assumed to be the same for our paper without loss of   
+113 generality. The MDP goes as follows: For a given state $s$ ,each agent make its decision $a ^ { i }$ based   
+114 on its policy $a ^ { i } \sim \pi _ { \theta ^ { i } } ( \cdot | s )$ . The state transits to the next state $s ^ { \prime }$ based on the joint action of all the   
+115 agents: $s ^ { \prime } \sim \mathcal { P } ( \cdot | s , a )$ . Then, each agent will receive its own reward $r ^ { i } ( s , a )$ . For the notation brevity,   
+116 we assume that the reward function mapping is deterministic and does not depend on the next state   
+117 without loss of generality. The stationary distribution induced by the policy $\pi _ { \theta }$ and the transition   
+118 kernel is denoted by $\mu _ { \pi _ { \theta } } ( s )$ ：
+
+9 Our objective is to find a set of policies that maximize the accumulated discounted mean reward :0received by agents
+
+$$
+\theta ^ { * } = \underset { \theta } { \arg \operatorname* { m a x } } J ( \theta ) : = { \mathbb { E } } \left[ \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \bar { r } ( s _ { k } , a _ { k } ) \right] .
+$$
+
+121 Here, $k$ represents the time step. $\begin{array} { r } { \bar { r } ( s _ { k } , a _ { k } ) : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } r ^ { i } ( s _ { k } , a _ { k } ) } \end{array}$ is the mean reward among agents   
+122 at time step $k$ .The randomness of the expectation comes from the initial state distribution $\mu _ { 0 } ( s )$ ,the   
+123 transition kernel $\mathcal { P }$ , and the stochastic policy $\pi _ { \theta ^ { i } } ( \cdot | s )$ ：
+
+# 2.2Policy gradient Theorem
+
+25 Under the discounted reward seting,the global state-value function,action-value function, and   
+26advantage function for policy set $\theta$ ,state $s$ , and action $a$ ,are defined as
+
+$$
+\begin{array} { c l } { { \displaystyle V _ { \pi _ { \theta } } ( s ) : = \mathbb { E } \left[ \displaystyle \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \bar { r } ( s _ { k } , a _ { k } ) \vert s _ { 0 } = s \right] } } \\ { { \displaystyle Q _ { \pi _ { \theta } } ( s , a ) : = \mathbb { E } \left[ \displaystyle \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \bar { r } ( s _ { k } , a _ { k } ) \vert s _ { 0 } = s , a _ { 0 } = a \right] } } \\ { { \displaystyle A _ { \pi _ { \theta } } ( s , a ) : = Q _ { \pi _ { \theta } } ( s , a ) - V _ { \pi _ { \theta } } ( s ) . } } \end{array}
+$$
+
+127To maximize the objective function defined in (1),the policy gradient [28] can be computed as follow
+
+$$
+\nabla _ { \theta } J ( \theta ) = \mathbb { E } _ { s \sim d _ { \pi _ { \theta } } , a \sim \pi _ { \theta } } \left[ \frac { 1 } { 1 - \gamma } A _ { \pi _ { \theta } } ( s , a ) \psi _ { \pi _ { \theta } } ( s , a ) \right] ,
+$$
+
+28where $\begin{array} { r } { d _ { \pi _ { \theta } } ( s ) : = ( 1 - \gamma ) \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } \mathbb { P } ( s _ { k } = s ) } \end{array}$ is thediscountedtatevisitationdistributionuder   
+29policy $\pi _ { \theta }$ ,and $\psi _ { \pi _ { \theta } } ( s , a ) : = \nabla \log \pi _ { \theta } ( s , a )$ is the score function.   
+3oFollowing the derivation of [42],the policy gradient for each agent under discounted reward setting   
+31can be expressed as
+
+$$
+\nabla _ { \theta ^ { i } } J ( \theta ) = \mathbb { E } _ { s \sim d _ { \pi _ { \theta } } , a \sim \pi _ { \theta } } \left[ \frac { 1 } { 1 - \gamma } A _ { \pi _ { \theta } } ( s , a ) \psi _ { \pi _ { \theta ^ { i } } } ( s , a ^ { i } ) \right] .
+$$
+
+# 132 3Decentralized single-timescale actor-critic
+
+# Algorithm 1: Decentralized single-timescale AC (reward estimator version)
+
+1: Initialize: Actor parameter $\theta _ { 0 }$ , critic parameter $\omega _ { 0 }$ , reward estimator parameter $\lambda _ { 0 }$ , initial state $s _ { 0 }$   
+2: for $k = 0 , \cdots , \bar { K } - 1$ do   
+3: Option 1: i.i.d. sampling:   
+4: $s _ { k } \sim \mu _ { \theta _ { k } } ( \cdot ) , a _ { k } \sim \pi _ { \theta _ { k } } ( \cdot | s _ { k } ) , s _ { k + 1 } \sim \mathcal { P } ( \cdot | s _ { k } , a _ { k } ) .$   
+5: Option 2: Markovian sampling:   
+6: $a _ { k } \sim \pi _ { \theta _ { k } } ( \cdot | s _ { k } ) , s _ { k + 1 } \sim \mathcal { P } ( \cdot | s _ { k } , a _ { k } ) .$   
+7:   
+8: Periodical consensus: Compute $\tilde { \omega } _ { k } ^ { i }$ and $\tilde { \lambda } _ { k } ^ { i }$ by (4) and (7).   
+9:   
+10: for $i = 0 , \cdots , N$ in parallel do   
+11: Reward estimator update: Update $\lambda _ { k + 1 } ^ { i }$ by (8).   
+12: Critic update: Update $\boldsymbol { \omega } _ { k + 1 } ^ { i }$ by (5).   
+13: Actor update: Update $\theta _ { k + 1 } ^ { i }$ by (6).   
+14: end for   
+15:end for   
+33We introduce the decentralized single-timescale AC algorithm; see Algorithm 1. In the remaining   
+34parts of this section, we will explain the updates in the algorithm in details.   
+135 In fully-decentralized MARL,each agent can only observe its local reward and action, while trying   
+136 to maximize the global reward (mean reward) defined in (1). The decentralized AC algorithm solves   
+137 the problem by performing online updates in an alternative fashion. Specifically,we have $N$ pairs of   
+138 actor and critic.In order to maximize $J ( \theta )$ ,each critic tries to estimate the global state-value function   
+139 $V _ { \pi _ { \theta } } ( s )$ defined in (2),and each actor then updates its policy parameter based on approximated policy   
+140 gradient. We now provide more details about the algorithm.   
+141 Critics'update.We willuse $\omega ^ { i } \in \mathbb { R } ^ { d _ { \omega } }$ to denote the $i _ { t h }$ crics parameter and $\begin{array} { r } { \bar { \omega } : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \omega ^ { i } } \end{array}$ t0   
+142 represent the averaged parameter of critic. The $i _ { t h }$ critic approximates the global value function as   
+143 $\bar { V _ { \pi _ { \theta } } } ( s ) \approx \hat { V } _ { \omega ^ { i } } ( s )$ ：
+
+146 As w sc $\begin{array} { r } { \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \| \boldsymbol { \omega } ^ { i } - \boldsymbol { \bar { \omega } } \| } \end{array}$ approximation error $\| \bar { \boldsymbol { \omega } } - \boldsymbol { \omega } ^ { * } ( \boldsymbol { \theta } ) \|$ ，which measures the approximation quality of averaged critic.
+
+47In order for critics to reach consensus,we perform the following update for all critics
+
+$$
+\tilde { \omega } _ { k } ^ { i } = \left\{ \begin{array} { l l } { { \sum _ { j = 1 } ^ { N } W ^ { i j } \omega _ { k } ^ { j } } } & { { \quad \mathrm { i f } \ k \ \mathrm { m o d } \ K _ { c } = 0 } } \\ { { \omega _ { k } ^ { i } } } & { { \quad \mathrm { o t h e r w i s e } . } } \end{array} \right.
+$$
+
+148 where $W \in \mathbb { R } ^ { n \times n }$ is a weight matrix for communication among agents, whose property will be   
+149specified in Assumption 5; $K _ { c }$ denotes the consensus frequency.
+
+150To reduce the approximation error, we willperform the local TD(O) update [29] as
+
+$$
+\omega _ { k + 1 } ^ { i } = \prod _ { R _ { \omega } } ( \tilde { \omega } _ { k } ^ { i } + \beta _ { k } g _ { c } ^ { i } ( \xi _ { k } , \omega _ { k } ^ { i } ) ) ,
+$$
+
+151 where $\xi : = ( s , a , s ^ { \prime } )$ represents a transition tuple, $g _ { c } ^ { i } ( \xi , \omega ) : = \delta ^ { i } ( \xi , \omega ) \nabla \hat { V } _ { \omega } ( s )$ is the update direction,   
+152 $\delta ^ { i } ( \xi , \omega ) : = r ^ { i } ( s , a ) + \gamma \hat { V } _ { \omega } ( s ^ { \prime } ) - \hat { V } _ { \omega } ( s )$ is the local temporal difference error (TD-error). $\beta _ { k }$ is the   
+153 step size for critic at iteration $k$ $\prod _ { R _ { \omega } }$ projects the parameter into a ball of radius of $R _ { \omega }$ containing   
+154 the optimal solution, which will be explained when discussing Assumption 1 and 2.   
+155Actors'update. We will use stochastic gradient ascent to update the policy's parameter, and the   
+156 stochastic gradient is calculated based on policy gradient theorem in (3). The advantage function   
+157 $A _ { \pi _ { \theta } } ( s , a )$ can be estimated by
+
+$$
+\delta ( \xi , \theta ) : = \bar { r } ( s , a ) + \gamma V ( s ^ { \prime } ) - V ( s ) ,
+$$
+
+158 with $a$ sampled from $\pi _ { \boldsymbol { \theta } } ( \cdot | \boldsymbol { s } )$ . However, to preserve the privacy of each agents,the local reward   
+159 cannot be shared to other agents under the fully decentralized setting. Thus, the averaged reward   
+160 $\bar { r } ( s _ { k } , a _ { k } )$ is not directly attainable. Consequently, we need a strategy to approximate the averaged   
+161 reward. In this paper, we will adopt the strategy proposed in [42]. In particular, each agent $i$ will have   
+162 a local reward estimator with parameter $\lambda ^ { i } \in \mathbb { R } ^ { d _ { \lambda } }$ , which estimates the global averaged reward as   
+163 $\bar { r } ( s _ { k } , a _ { k } ) \approx \hat { r } _ { \lambda ^ { i } } ( s _ { k } , a _ { k } )$
+
+164Thus,the update of the $i _ { t h }$ actor is given by
+
+$$
+\theta _ { k + 1 } ^ { i } = \theta _ { k } ^ { i } + \alpha _ { k } \hat { \delta } ( \xi _ { k } , \omega _ { k + 1 } ^ { i } , \lambda _ { k + 1 } ^ { i } ) \psi _ { \pi _ { \theta _ { k } ^ { i } } } ( s _ { k } , a _ { k } ^ { i } ) ,
+$$
+
+where $\hat { \delta } ( \xi , \omega , \lambda ) : = \hat { r } _ { \lambda } ( s , a ) + \gamma \hat { V } _ { \omega } ( s ^ { \prime } ) - \hat { V } _ { \omega } ( s )$ is the approximated advantage function. $\alpha _ { k }$ is the step size for actor's update at iteration $k$ ：
+
+Reward estimators’update. Similar to critic, each reward estimator's approximation error can be decomposed into consensus error and the approximation error.
+
+s9For each local reward estimator, we perform the consensus step to minimize the consensus error as
+
+$$
+\tilde { \lambda } _ { k } ^ { i } = \left\{ \sum _ { j = 1 } ^ { N } W ^ { i j } \lambda _ { k } ^ { j } \qquad \mathrm { i f } k \ \mathrm { m o d } \ K _ { c } = 0 \right.
+$$
+
+170To reduce the approximation error, we perform a local update of stochastic gradient descent.
+
+$$
+\lambda _ { k + 1 } ^ { i } = \prod _ { R _ { \lambda } } ( \tilde { \lambda } _ { k } ^ { i } + \eta _ { k } g _ { r } ^ { i } ( \xi _ { k } , \lambda _ { k } ^ { i } ) ) ,
+$$
+
+171 where $g _ { r } ^ { i } ( \xi , \lambda ) : = ( r ^ { i } ( s , a ) - { \hat { r } } _ { \lambda } ( s , a ) ) \nabla { \hat { r } } _ { \lambda } ( s , a )$ is the update direction. $\eta _ { k }$ is the step size for   
+172 reward estimator at iteration $k$ . Note the calculation of $g _ { r } ^ { i } ( \xi , \bar { \lambda } )$ does not require the knowledge of $s ^ { \prime }$   
+173 we use $\xi$ in (8) just for notation brevity. Similar to critics update, $\prod _ { R _ { \lambda } }$ projects the parameter into a   
+174 ball of radius of $R _ { \lambda }$ containing the optimal solution.   
+175In our Algorithm 1, we will use the same order for $\alpha _ { k }$ ， $\beta _ { k }$ ,and $\eta _ { k }$ and hence, our algorithm is in   
+176single-timescale.   
+177 Linear approximation for analysis. In our analysis, we will use linear approximation for both critic   
+178 and reward estimator variables, i.e. $\hat { V } _ { \omega } ( s ) : = \phi ( s ) ^ { T } \omega ; \hat { r } _ { \lambda } ( s , a ) : = \varphi ( s , a ) ^ { T } \lambda ;$ where $\phi ( s ) : S \to $   
+179 $\mathbb { R } ^ { d _ { \omega } }$ and $\varphi ( s , a ) : \mathcal { S } \times \mathcal { A }  \mathbb { R } ^ { d _ { \lambda } }$ are two feature mappings, whose property willbe specified in the   
+180 discussion of Assumption 1.   
+181 Algorithm for preserving the local action. Note that in Algorithm 1, the reward estimators need   
+182 the knowledge of joint actions in order to estimate the global rewards. To preserve the privacy of   
+183 local actions, we further propose a variant of Algorithm 1, which estimates the global rewards by   
+184 communicating noisy local rewards; see [6] for the original idea. However, to maintain the optimal   
+185 sample complexity, such an approach requires $\mathcal { O } ( \log ( \varepsilon ^ { - 1 } ) )$ communication rounds for each iteration.   
+186 We postpone the detailed design and analysis of such an algorithm scheme into Appendix B.   
+187 Remarks on sampling scheme. The unbiased update for critic and actor variables requires sampling   
+188 from $\mu _ { \pi _ { \theta } }$ and $d _ { \pi _ { \theta } }$ ,respectively. However, in practical implementations,states are usually collected   
+189 fromanonline trajectory(Markoviansampling),whosedistributionisgeneralldierentfor $\mu _ { \pi _ { \theta } }$   
+190 and $d _ { \pi _ { \theta } }$ . Such a distribution mismatch willinevitably cause biases during the update of critic and   
+191 actor variables. One has to bound the corresponding error terms when analyzing the algorithm. In   
+192 this work,we will provide the analysis for both sampling schemes.
+
+In this section, we first introduce the technical assumptions used for our analysis,which are standard in the literature.Then, we present the convergence results for both actor and critic variables under i.i.d. sampling and Markovian sampling.
+
+# 4.1 Assumptions
+
+Assumption 1 (bounded rewards and feature vectors). All the local rewards are uniformly bounded, i.e., there exists a positive constants $r _ { \mathrm { m a x } }$ such that $| r ^ { i } ( s , a ) | \leq r _ { \operatorname* { m a x } } ,$ for all feasible $( s , a )$ and $i \in [ N ]$ . The norm of feature vectors are bounded such that for all $s \in S$ ， $a \in { \mathcal { A } }$ $\| \phi ( s ) \| \leq$ $1 , \| \varphi ( s , \overline { { a } } ) \| \leq 1$
+
+？Assumption 1 is standard and commonly adopted; see, e.g.， [3,35,38,24,21]. This assumption can 3be achieved via normalizing the feature vectors.
+
+04Assumption 2 (negative definiteness of $A _ { \theta , \phi }$ and $A _ { \theta , \varphi . }$ ). There exists two positive constants $\lambda _ { \phi } , \lambda _ { \varphi }$   
+55such that for all policy $\theta$ ，the following two matrices are negative definite
+
+$$
+\begin{array} { r l } & { A _ { \theta , \phi } : = \mathbb { E } _ { s \sim \mu _ { \theta } ( s ) } [ \phi ( s ) ( \gamma \phi ( s ^ { \prime } ) ^ { T } - \phi ( s ) ^ { T } ) ] } \\ & { A _ { \theta , \varphi } : = \mathbb { E } _ { s \sim \mu _ { \theta } ( s ) , a \sim \pi _ { \theta } ( \cdot | s ) } [ - \varphi ( s , a ) \varphi ( s , a ) ^ { T } ] , } \end{array}
+$$
+
+06with $\lambda _ { \operatorname* { m a x } } ( A _ { \theta , \phi } ) \leq \lambda _ { \phi } , \lambda _ { \operatorname* { m a x } } ( A _ { \theta , \varphi } ) \leq \lambda _ { \varphi }$ ， where $\lambda _ { \operatorname* { m a x } } ( \cdot )$ represents the largest eigenvalue.
+
+207 Assumption 2 can be achieved when the matrices $\Phi _ { \phi } ~ : = ~ [ \phi ( s _ { 1 } ) , \cdot \cdot \cdot , \phi ( s _ { | S | } ) ]$ and $\Phi _ { \varphi } : = $   
+208 $[ \varphi ( s _ { 1 } , a _ { 1 } ) , \cdot \cdot \cdot , \varphi ( s _ { | S | } , a _ { | A | } ) ]$ have ful row rank, which ensures that the optimal critic and reward   
+209 estimator are unique; see also [24,34]. Together with Assumption 1, we can show that the norm of   
+210 $\omega ^ { \ast } ( \theta )$ and $\lambda ^ { * } ( \theta )$ are bounded by some positive constant, which justifies the projection steps.
+
+Assumption 3 (Lipschitz properties of policy). There exists constants $C _ { \psi } , L _ { \psi } , L _ { \pi }$ such that for ：all $\theta , \theta ^ { \prime } , s \in \mathcal { S }$ and $a \in { \mathcal { A } }$ we have (1). $| \bar { \pi _ { \theta } } ( a | s ) - \pi _ { \theta ^ { \prime } } ( a | s ) | \leq L _ { \pi } \| \theta - \theta ^ { \prime } \|$ (2). $\parallel \psi _ { \theta } ( s , a ) \textrm { -- }$ $\psi _ { \theta ^ { \prime } } ( s , a ) \| \le L _ { \psi } \| \theta - \theta ^ { \prime } \|$ ; (3). $\| \psi _ { \theta } ( s , a ) \| \leq C _ { \psi }$ ：
+
+4Assumption 3 is common for analyzing policy-based algorithms; see, e.g., [33,32,11]. The assump  
+5tion ensures the smoothness of objective function $J ( \theta ) ^ { \top }$ . It holds for a large range of policy classes   
+6such as tabular softmax policy [1], Gaussian policy [7],and Boltzman policy [13].   
+17Assumption 4 (irreducible and aperiodic Markov chain). The Markov chain under $\pi _ { \theta }$ and transition   
+18kernel $\mathcal { P } ( \cdot | s , a )$ is irreducible and aperiodic for any $\theta$ ：   
+219Assumption 4 is a standard assumption, which holds for any uniformly ergodic Markov chains and   
+220 any time-homogeneous Markov chains with finite-state space. It ensures that there exists constants   
+221 $\kappa > 0$ and $\rho \in ( 0 , 1 )$ such that
+
+$$
+\operatorname* { s u p } _ { s \in S } d _ { T V } \big ( \mathbb { P } \big ( s _ { k } \in \cdot | s _ { 0 } = s , \pi _ { \theta } \big ) , \mu _ { \theta } \big ) \leq \kappa \rho ^ { k } , \ \forall k .
+$$
+
+：Assumption 5 (doubly stochastic weight matrix). The communication matrix $W$ is doubly stochastic, 3i.e. each column/row sum up to 1. Moreover, the second largest singular value $\nu$ is smaller than $^ { l }$ ：
+
+Assumption 5 is a common assumption in decentralized optimization and multi-agent reinforcement learning; see, e.g.， [27,5, 6]. It ensures the convergence of consensus error for critic and reward estimator variables.
+
+# 4.2Sample complexity under i.i.d. sampling
+
+Theorem 1 (sample complexity under i.i.d. sampling). Suppose Assumptions 1-5 hold. Consider theupdteofAlgorithm1underi.id.sampling.Let $\begin{array} { r } { \alpha _ { k } \ = \ \frac { \bar { \alpha } } { \sqrt { K } } } \end{array}$ for some positive constant $\bar { \alpha }$
+
+i0 $\begin{array} { r } { \beta _ { k } = \frac { C _ { 9 } } { 2 \lambda _ { \phi } } \alpha _ { k } } \end{array}$ ,and $\begin{array} { r } { \eta _ { k } = \frac { C _ { 1 0 } } { 2 \lambda _ { \varphi } } \alpha _ { k } } \end{array}$ CK≤()heK doeshetbf i1 Then, we have
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \sum _ { i = 1 } ^ { N } \mathbb { E } \left[ \| \omega _ { k } ^ { i } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } \right] \leq \mathcal { O } \left( \frac { 1 } { \sqrt { K } } \right) } \\ & { \displaystyle \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \sum _ { i = 1 } ^ { N } \mathbb { E } \left[ \| \nabla _ { \theta ^ { i } } F ( \theta _ { k } ) \| ^ { 2 } \right] \leq \mathcal { O } \left( \frac { 1 } { \sqrt { K } } \right) + \mathcal { O } ( \varepsilon _ { a p p } + \varepsilon _ { s p } ) , } \end{array}
+$$
+
+233 The proof of Theorem 1 can found in Appendix E.1. It establishes the iteration complexity of   
+234 $\mathcal { O } ( 1 / \sqrt { K } )$ ,or equivalently, sample complexity of $\mathcal { O } ( \varepsilon ^ { - 2 } )$ for Algorithm 1. Note that actors, critics,   
+235 and reward estimators use the step sizes of the same order. The sample complexity matches the   
+236 optimal rate of SGD for general non-convex optimization problem.To explain the errors in(9), let us   
+237 define the approximation error as the following:
+
+$$
+\varepsilon _ { a p p } : = \operatorname* { m a x } _ { \theta , a } \sqrt { \mathbb { E } _ { s \sim \mu _ { \theta } } \left[ | V _ { \pi _ { \theta } } ( s ) - \hat { V } _ { \omega ^ { * } ( \theta ) } ( s ) | ^ { 2 } + | \bar { r } ( s , a ) - \hat { r } _ { \lambda ^ { * } ( \theta ) } ( s , a ) | ^ { 2 } \right] } .
+$$
+
+238 The error $\varepsilon _ { a p p }$ captures the approximation power of critic and reward estimator. Similar terms   
+239 also appear in the literature (see e.g., [35,1, 21]). Such an approximation error becomes zero in   
+240 tabular case. The error $\varepsilon _ { s p }$ is inevitably caused by the mismatch between discounted state visitation   
+241 distribution $d _ { \pi _ { \theta } }$ and stationary distribution $\mu _ { \pi _ { \theta } }$ ; see, e.g., [38,24]. It is defined as
+
+$$
+\varepsilon _ { s p } : = 2 C _ { \theta } ( \log _ { \rho } \kappa ^ { - 1 } + \frac { 1 } { \rho } ) ( 1 - \gamma ) .
+$$
+
+12When $\gamma$ is close to 1, the error becomes small. This is because $d _ { \pi _ { \theta } }$ approaches to $\mu _ { \pi _ { \theta } }$ when $\gamma$ goes to   
+131.In the literature,some works assume that sampling from $d _ { \pi _ { \theta } }$ is permitted, thus eliminate this error;   
+14see,e.g., [4].
+
+# 4.3Sample complexity under markovian sampling
+
+；Theorem 2 (sample complexity under Markovian sampling). Suppose Assumptions 1-5 hold. Consider the update ofAlgorithm $^ { l }$ under Markoviansampling.Let $\begin{array} { r } { \alpha _ { k } = \frac { \bar { \alpha } } { \sqrt { K } } } \end{array}$ for some positive constant $\bar { \alpha }$ $\begin{array} { r } { \beta _ { k } = \frac { C _ { 9 } } { 2 \lambda _ { \phi } } \alpha _ { k } } \end{array}$ and $\begin{array} { r } { \eta _ { k } = \frac { C _ { 1 0 } } { 2 \lambda _ { \varphi } } \alpha _ { k } } \end{array}$ （ $K _ { c } \leq \mathcal { O } ( \alpha _ { k } ^ { - \frac 1 2 } )$ where $K$ is he tl frahe ）we have
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \sum _ { i = 1 } ^ { N } \mathbb { E } \left[ \| \omega _ { k } ^ { i } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } \right] \leq \mathcal { O } \left( \frac { \log ^ { 2 } K } { \sqrt { K } } \right) } \\ & { \displaystyle \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \sum _ { i = 1 } ^ { N } \mathbb { E } \left[ \| \nabla _ { \theta ^ { i } } F ( \theta _ { k } ) \| ^ { 2 } \right] \leq \mathcal { O } \left( \frac { \log ^ { 2 } K } { \sqrt { K } } \right) + \mathcal { O } ( \varepsilon _ { a p p } + \varepsilon _ { s p } ) , } \end{array}
+$$
+
+250where $C _ { 9 } , C _ { 1 0 }$ are positive constants defined in proof.
+
+51 We put the proof of Theorem 2 in Appendix E.2. In Markovian sampling,the updates are biased for   
+52 critics,actors,and reward estimators.The error will decrease as the Markov chain mixes,and the   
+53 logarithmic term is due to the cost for mixing.
+
+Theorem 2 establishes the iteration complexity of $\mathcal { O } ( \log ^ { 2 } K / \sqrt { K } )$ , or equivalently, sample complexity of $\widetilde { \mathcal { O } } ( \varepsilon ^ { - 2 } )$ for Algorithm 1. It matches the state-of-the-art sample complexity of decentralized AC algorithms,which are implemented in double-loop fashion [11, 6].
+
+# 4.4Proof sketch
+
+258 We present the main elements for the proof of Theorem 2, which helps in understanding the difference   
+259 between classical two-timescale/double-loop analysis and our single-timescale analysis. The proof of   
+260 Theorem 1 follows the same framework with simpler sampling scheme.
+
+261 Under Markovian sampling, it is possible to show the following inequality, which characterizes the :62ascent of the objective.
+
+$$
+\begin{array} { r l } & { \mathbb { E } [ J ( \theta _ { k + 1 } ) ] - J ( \theta _ { k } ) \ge \displaystyle \sum _ { i = 1 } ^ { N } \Big [ \frac { \alpha _ { k } } { 2 } \mathbb { E } \| \nabla _ { \theta ^ { i } } J ( \theta _ { k } ) \| ^ { 2 } + \frac { \alpha _ { k } } { 2 } \mathbb { E } \| g _ { a } ^ { i } ( \xi _ { k } , \omega _ { k + 1 } ^ { i } , \lambda _ { k + 1 } ^ { i } ) \| ^ { 2 } } \\ & { \qquad - 8 C _ { \psi } ^ { 2 } \alpha _ { k } \mathbb { E } \| \omega ^ { * } ( \theta _ { k } ) - \omega _ { k + 1 } ^ { i } \| ^ { 2 } - 4 C _ { \psi } ^ { 2 } \alpha _ { k } \mathbb { E } \| \lambda ^ { * } ( \theta _ { k } ) - \lambda _ { k + 1 } ^ { i } \| ^ { 2 } \Big ] } \\ & { \qquad - { \mathcal { O } } ( \log ^ { 2 } ( K ) \alpha _ { k } ^ { 2 } ) - { \mathcal { O } } \big ( ( \varepsilon _ { a p p } + \varepsilon _ { s p } ) \alpha _ { k } \big ) . } \end{array}
+$$
+
+263 To analyze the errors of critic $\lVert \omega ^ { * } ( \theta _ { k } ) - \omega _ { k + 1 } ^ { i } \rVert ^ { 2 }$ and reward estimator $\lVert \lambda ^ { * } ( \theta _ { k } ) - \lambda _ { k + 1 } ^ { i } \rVert ^ { 2 }$ , the two  
+264 timescale analysis requires $\mathcal { O } ( \alpha _ { k } ) < \operatorname* { m i n } \{ \mathcal { O } ( \beta _ { k } ) , \mathcal { O } ( \eta _ { k } ) \}$ in order for these two errors to converge.   
+265 The double-loop approach runs lower-level update for ${ \mathcal { O } } ( \log ( \varepsilon ^ { - 1 } ) )$ times with batch size $\mathcal { O } ( \varepsilon ^ { - 1 } )$   
+266 to drive these errors below $\varepsilon$ and hence, they cannot allow inner loop size and bath size to be $\mathcal { O } ( 1 )$   
+267 simultaneously. To obtain the convergence result for single-timescale update,the idea is to further   
+268 upper bound these two lower-level errors by the quantity $\mathcal { O } ( \alpha _ { k } \mathbb { E } \| g _ { a } ^ { i } ( \xi _ { k } , \omega _ { k + 1 } ^ { i } , \lambda _ { k + 1 } ^ { i } ) \| ^ { 2 } )$ (through a   
+269 series of derivations),and then eliminate these errors by the ascent term $\begin{array} { r l } {  { \frac { \alpha _ { k } } { 2 } \mathbb { E } \bigl \| g _ { a } ^ { i } \bigl ( \xi _ { k } , \omega _ { k + 1 } ^ { i } , \lambda _ { k + 1 } ^ { i } \bigr ) \bigr \| ^ { 2 } } \qquad } & { { } } \end{array}$   
+270 We mainly focus on the analysis of critic's error through the proof sketch. The analysis for reward   
+271estimator's error follows similar procedure.We start by decomposing the error of critic as
+
+$$
+\sum _ { i = 1 } ^ { N } \| \omega _ { k + 1 } ^ { i } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } = \sum _ { i = 1 } ^ { N } ( \| \omega _ { k + 1 } ^ { i } - \bar { \omega } _ { k + 1 } \| ^ { 2 } + \| \bar { \omega } _ { k + 1 } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } ) .
+$$
+
+72The first term represents the consensus error, which can be bounded by the next lemma.
+
+273Lemma 1. Suppose Assumptions l and $5$ hold. Consider the sequence $\left\{ \omega _ { k } ^ { i } \right\}$ generated by Algorithm $^ { l }$ ，   
+274then the following holds
+
+$$
+\| Q \omega _ { k + 1 } \| \leq \nu ^ { \frac { k ^ { \prime } } { K _ { c } } } \| \omega _ { 0 } \| + 4 \sum _ { t = 0 } ^ { k } \nu ^ { \lceil \frac { k ^ { \prime } - 1 - t } { K _ { c } } \rceil } \beta _ { t } \sqrt { N } C _ { \delta } ,
+$$
+
+where $\begin{array} { r } { \omega _ { 0 } : = [ \omega ^ { 1 } , \cdot \cdot \cdot , \omega ^ { N } ] ^ { T } , Q : = I - \frac { 1 } { N } \mathbf { 1 } \mathbf { 1 } ^ { T } , k ^ { \prime } : = \lfloor \frac { k } { K _ { c } } \rfloor * K _ { c } } \end{array}$ . The constant $\nu \in ( 0 , 1 )$ is the second largest singular value of $W$ ：
+
+Based on Lemma 1 and follow the step size rule of Theorem 2,it is possible to show $\| Q \omega _ { k + 1 } \| _ { F } ^ { 2 } =$ $\begin{array} { r } { \sum _ { i = 1 } ^ { N } \| \omega _ { k + 1 } ^ { i } - \bar { \omega } _ { k + 1 } \| ^ { 2 } = \mathcal { O } ( K _ { c } ^ { 2 } \beta _ { k } ^ { 2 } ) } \end{array}$ Let $K _ { c } = \mathcal { O } ( \beta _ { k } ^ { - \frac 1 2 } )$ , we have $\| Q \omega _ { k + 1 } \| _ { F } ^ { 2 } = \mathcal { O } ( \beta _ { k } )$ ，which maintains the optimal rate.
+
+To analyze the second term in (12), we first construct the following Lyapunov function
+
+$$
+\mathbb { V } _ { k } : = - J ( \theta _ { k } ) + \| \bar { \omega } _ { k } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } + \| \bar { \lambda } _ { k } - \lambda ^ { * } ( \theta _ { k } ) \| ^ { 2 } .
+$$
+
+281 Then, it remains to derive an approximate descent property of the term $\lVert \bar { \boldsymbol { \omega } } _ { k } - \boldsymbol { \omega } ^ { * } ( \boldsymbol { \theta } _ { k } ) \rVert ^ { 2 }$ in (13).   
+282 Towards that end, our key step lies in establishing the smoothness of the optimal critic variables   
+283 shown in the next lemma.
+
+i4Lemma 2 (smoothness of optimal critic). Suppose Assumptions 1-3 hold, under the update of 15Algorithm $\boldsymbol { l }$ , there exists a positive constant $L _ { \mu , 1 }$ such that for all $\theta , \theta ^ { \prime }$ , it holds that
+
+$$
+\begin{array} { r } { \| \nabla \omega ^ { * } ( \theta ) - \nabla \omega ^ { * } ( \theta ^ { \prime } ) \| \leq L _ { \mu , 1 } \| \theta - \theta ^ { \prime } \| , } \end{array}
+$$
+
+36where $\nabla \omega ^ { * } ( \theta )$ denotes the Jacobian of $\omega ^ { \ast } ( \theta )$ with respect to $\theta$
+
+This smoothnessproperty isssentialforachievingour $\tilde { \mathcal { O } } ( 1 / \sqrt { K } )$ convergence rate.
+
+To the best of our knowledge,the smoothness of $\omega ^ { \ast } ( \theta )$ has not been justified in the literature.   
+Equipped with Lemma 2,we are able to establish the following lemma.
+
+90Lemma 3 (Error of critic). Under Assumptions 1-5, consider the update of Algorithm 1. Then, it   
+91holds that
+
+$$
+\begin{array} { r l } & { \displaystyle \mathbb { E } [ \| \bar { \omega } _ { k + 1 } - \omega ^ { * } ( \theta _ { k + 1 } ) \| ^ { 2 } ] \le ( 1 + C _ { 9 } \alpha _ { k } ) \| \bar { \omega } _ { k + 1 } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } } \\ & { \qquad + \frac { \alpha _ { k } } { 4 } \displaystyle \sum _ { i = 1 } ^ { N } \| \mathbb { E } [ g _ { a } ^ { i } ( \xi _ { k } , \omega _ { k + 1 } ^ { i } , \lambda _ { k + 1 } ^ { i } ) ] \| ^ { 2 } + { \mathcal O } ( \alpha _ { k } ^ { 2 } ) . } \\ & { \displaystyle \mathbb { E } [ \| \bar { \omega } _ { k + 1 } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } ] \le ( 1 - 2 \lambda _ { \phi } \beta _ { k } ) \| \bar { \omega } _ { k } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } } \\ & { \qquad + C _ { K _ { 1 } } \beta _ { k } \beta _ { k - Z _ { K } } + C _ { K _ { 2 } } \alpha _ { k - Z _ { K } } \beta _ { k } . } \end{array}
+$$
+
+292Here, $Z _ { K } : = \operatorname* { m i n } \{ z \in \mathbb { N } ^ { + } | \kappa \rho ^ { z - 1 } \le \operatorname* { m i n } \{ \alpha _ { k } , \beta _ { k } , \eta _ { k } \} \} ,$ $C _ { 9 }$ ， $\lambda _ { \phi }$ are constants specified in appendix,   
+293and $C _ { K _ { 1 } }$ and $C _ { K _ { 2 } }$ are of order ${ \mathcal { O } } ( \log ( K ) )$ and $\mathcal { O } ( \log ^ { 2 } ( K ) ) d$ respectively.
+
+![](images/51690de4334a89787cd8df9f79b77112e0b40cb6ac372603b25d483fba3a08f5.jpg)  
+Figure 1: Averaged reward versus sample complexity and communication complexity. The vertical axis is the averaged reward over all the agents.
+
+294Plug (15) into (14), we can establish the approximate descent property of $\| \bar { \omega } _ { k } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 }$ in (13):
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } [ \| \bar { \omega } _ { k + 1 } - \omega ^ { * } ( \theta _ { k + 1 } ) \| ^ { 2 } ] \leq ( 1 + C _ { 9 } \alpha _ { k } ) ( 1 - 2 \lambda _ { \phi } \beta _ { k } ) \| \bar { \omega } _ { k } - \omega ^ { * } ( \theta _ { k } ) \| ^ { 2 } } } \\ & { } & { \quad + \frac { \alpha _ { k } } { 4 } \displaystyle \sum _ { i = 1 } ^ { N } \| \mathbb { E } [ g _ { a } ^ { i } ( \xi _ { k } , \omega _ { k + 1 } ^ { i } , \lambda _ { k + 1 } ^ { i } ) ] \| ^ { 2 } } \\ & { } & { \quad + { \mathcal O } ( C _ { K _ { 1 } } \beta _ { k } \beta _ { k - Z _ { K } } + C _ { K _ { 2 } } \alpha _ { k - Z _ { K } } \beta _ { k } ) . } \end{array}
+$$
+
+5Finally, plugging (11), (14),and (16) into (13) gives the ascent of the Lyapunov function, which leads   
+6to our convergence result through steps of standard arguments.
+
+# ，5Numerical results
+
+In this section, our objective is to illustrate the empirical sample complexity and communication complexity of the proposed algorithms. We also implement the algorithm in [6] to serve as a baseline, which employs double-loop algorithmic framework. Our simulation is based on the grounded communication environment proposed in [19]; see Appendix A for detailed set up. Through the discussion, we refer the algorithm in [6] as "DLDAC",the Algorithm 1 as "SDAC-re",the Algorithm 2 as "SDAC-noisy" (see Appendix B). We also provide the result which assumes full reward is available to serve as baseline, which we refer as "SDAC-full". We set $K _ { r } = 5$ for "SDAC-noisy"; $K _ { c } = 1$ for "SDAC-re", "SDAC-noisy", and "SDAC-full". We choose $T _ { c } = 5$ (loop size), $T _ { c } ^ { \prime } = 1$ (critic consensus number every iteration), $T ^ { \prime } = 5$ (reward consensus number every iteration) for "DLDAC".
+
+307 The sample complexity and communication complexity are shown in Figure 1. The results are   
+308 averaged over 10 Monte Carlo runs.As we can see,the proposed two algorithms achieve significantly   
+309 higher reward than "DLDAC" in terms of both sample complexity and communication complexity.   
+310 Moreover,their performances approach the baseline “SDAC-full, where the global reward is assumed   
+311 to be available, indicating that the reward approximation is nearly accurate. Due to space limit, we   
+312 will put additional experiments on the comparison with existing decentralized AC algorithms and the   
+313 ablation study of hyper-parameters to Appendix A.
+
+# 3146Conclusion and future direction
+
+315 In this paper, we studied the convergence of fully decentralized AC algorithm under practical single  
+316 timescale update for the first time. We designed such an algorithm which maintains the optimal   
+317 sample complexity of $\widetilde { \mathcal { O } } ( \varepsilon ^ { - 2 } )$ under less communications. We also proposed a variant to preserve the   
+318 privacy of local actions by communicating noisy rewards. Extensive simulation results demonstrate   
+319 the superiority of our algorithms’ empirical performance over existing decentralized AC algorithms.   
+320 One limitation of our work is that we only study the convergence to stationary point. Thus, we leave   
+321 the research on the avoidance of saddle points and convergence to global optimum as promising   
+322 future directions.
+
+# 3 References
+
+324 [1] A. Agarwal, S. M. Kakade, J. D. Lee, and G. Mahajan. Optimality and approximation with   
+325 policy gradient methods in markov decision processes. In Conference on Learning Theory   
+326 (COLT), pages 64-66,2020.   
+327 [2] J. Baxter and P.L. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial   
+328 Intelligence Research,15:319-350,2001.   
+329 [3] J. Bhandari, D.Russo,and R. Singal. A finite time analysis of temporal difference learning with   
+330 linear function approximation. In Conference on Learning Theory (COLT), pages 1691-1692,   
+331 2018.   
+332 [4] T. Chen, Y. Sun, and W. Yin. Closing the gap: Tighter analysis of alternating stochastic gradient   
+333 methods for bilevel problems. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan,   
+334 editors,Advances in Neural Information Processing Systems,2021.   
+335 [5] Z. Chen, Y.Zhou,and R.Chen. Multi-agent off-policy td learning: Finite-time analysis with near  
+336 optimal sample complexity and communication complexity. arXiv preprint arXiv:2103.13147,   
+337 2021.   
+338 [6] Z. Chen, Y. Zhou, R.-R. Chen, and S. Zou. Sample and communication-effcient decentralized   
+339 actor-critic algorithms with finite-time analysis. In International Conference on Machine   
+340 Learning, pages 3794-3834. PMLR, 2022.   
+341 [7] K.Doya. Reinforcement learning in continuous time and space. Neural Computation,12(1):219-   
+342 245,2000.   
+343 [8]L.Espeholt, H.Soyer,R. Munos, K. Simonyan, V. Mnih,T. Ward, Y.Doron, V.Firoiu,T.Harley,   
+344 I. Dunning, et al. Impala: Scalable distributed deep-rl with importance weighted actor-learner   
+345 architectures. In International Conference on Machine Learning, pages 1407-1416. PMLR,   
+346 2018.   
+347 [9] Z. Fu, Z. Yang, and Z. Wang. Single-timescale actor-critic provably finds globally optimal   
+348 policy. In International Conference on Learning Representations, 2021.   
+349[10] H. Guo,Z.Fu, Z. Yang,and Z. Wang.Decentralized single-timescale actor-critic on zero  
+350 sum two-player stochastic games. In International Conference on Machine Learning, pages   
+351 3899-3909.PMLR,2021.   
+352[11] F. Hairi,J.Liu,and S.Lu. Finite-time convergence and sample complexity of multi-agent actor  
+353 critic reinforcement learning with average reward. In International Conference on Learning   
+354 Representations,2022.   
+355[12] S. M. Kakade. A natural policy gradient. In Proc. Advances in Neural Information Processing   
+356 Systems (NIPS), pages 1531-1538,2002.   
+357 [13] V. R. Konda and V. S. Borkar. Actor-critic-type learning algorithms for Markov decision   
+358 processes. SIAM Journal on Control and Optimization,38(1):94-123,1999.   
+359[14] T.P. Lillicrap,J. J. Hunt, A. Pritzel, N. Heess,T. Erez, Y. Tassa,D. Silver, and D.Wierstra.   
+360 Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.   
+361 [15] Y.Lin, K. Zhang, Z. Yang,Z. Wang,T. Basar, R. Sandhu,and J.Liu. A communication-efficient   
+362 multi-agent actor-critic algorithm for distributed reinforcement learning. In 2019 IEEE 58th   
+363 Conference on Decision and Control (CDC), pages 5562-5567, 2019.   
+364 [16] M. L. Littman. Markov games as a framework for multi-agent reinforcement learning. In   
+365 Machine learning proceedings 1994, pages 157-163. Elsevier, 1994.   
+366[17] Y. Liu, K. Zhang, T. Basar, and W. Yin. An improved analysis of (variance-reduced) policy   
+367 gradient and natural policy gradient methods. Advances in Neural Information Procesing   
+368 Systems,33:7624-7636,2020.   
+369 [18] R. Lowe, Y.I. Wu, A. Tamar,J. Harb, O.Pieter Abbeel, and I. Mordatch. Multi-agent actor-critic   
+370 for mixed cooperative-competitive environments. Advances in neural information processing   
+371 systems,30,2017.   
+372 [19] I. Mordatch and P. Abbeel. Emergence of grounded compositional language in multi-agent   
+373 populations. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   
+374 [20] S. Omidshafiei, J.Pazis, C. Amato, J. P.How,and J. Vian. Deep decentralized multi-task   
+375 multi-agent reinforcement learning under partial observability. In International Conference on   
+376 Machine Learning, pages 2681-2690. PMLR,2017.   
+377 [21] S. Qiu, Z. Yang, J. Ye,and Z. Wang. On the finite-time convergence of actor-critic algorithm.   
+378 In Optimization Foundations for Reinforcement Learning Workshop at Advances in Neural   
+379 Information Processing Systems (NeurIPS), 2019.   
+380 [22] T. Rashid, M. Samvelyan, C.Schroeder, G.Farquhar,J.Foerster,and S.Whiteson. Qmix: Mono  
+381 tonic value function factorisation for deep multi-agent reinforcement learning. In International   
+382 Conference on Machine Learning, pages 4295-4304. PMLR, 2018.   
+383 [23] J. Schulman,F. Wolski,P. Dhariwal, A. Radford,and O. Klimov. Proximal policy optimization   
+384 algorithms. arXiv preprint arXiv:1707.06347, 2017.   
+385 [24] H. Shen,K. Zhang, M. Hong, and T. Chen.Asynchronous advantage actor critic: Non  
+386 asymptotic analysis and linear speedup. ArXiv:20l2.15511, 2020.   
+387[25] D. Silver,J. Schritwieser, K. Simonyan,I. Antonoglou, A. Huang,A. Guez,T. Hubert,L.Baker,   
+388 M.Lai, A. Bolton, et al. Mastering the game of go without human knowledge. nature,   
+389 550(7676):354-359, 2017.   
+390[26] K. Son, D. Kim, W. J. Kang, D. E. Hostallero, and Y. Yi. Qtran: Learning to factorize with   
+391 transformation for cooperative multi-agent reinforcement learning. In International Conference   
+392 on Machine Learning, pages 5887-5896. PMLR, 2019.   
+393 [27] J. Sun, G. Wang, G. B. Giannakis, Q. Yang,and Z. Yang. Finite-sample analysis of decentral  
+394 ized temporal-difference learning with linear function approximation. In Proc. International   
+395 Conference on Artificial Intelligence and Statistics (AISTATS), pages 4485-4495, 2020.   
+396 [28] R. S. Sutton, D. A. McAllester, S. P. Singh,and Y. Mansour. Policy gradient methods for   
+397 reinforcement learning with function approximation. In Proc. Advances in Neural Information   
+398 Processing Systems (NIPS), pages 1057-1063,2000.   
+399 [29] J. N. Tsitsiklis and B. Van Roy. Analysis of temporal-difference learning with function   
+100 approximation. In Advances in neural information processing systems (NIPS), pages 1075-   
+101 1081, 1997.   
+102 [30] O. Vinyals,I. Babuschkin, W. M. Czarnecki,M. Mathieu, A. Dudzik, J. Chung, D. H. Choi,   
+103 R.Powell, T. Ewalds, P. Georgiev, et al. Grandmaster level in starcraft ii using multi-agent   
+104 reinforcement learning. Nature,575(7782):350-354, 2019.   
+105 [31] Y. Wang, S. Zou, and Y. Zhou. Non-asymptotic analysis for two time-scale TDC with general   
+106 smooth function approximation. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and   
+107 J. W. Vaughan,editors,Advances in Neural Information Processing Systems,voume34,ges   
+108 9747-9758. Curran Associates, Inc., 2021.   
+109 [32] Y. F. Wu, W. Zhang,P. Xu, and Q. Gu. A finite-time analysis of two time-scale actor-critic   
+10 methods. Advances in Neural Information Processing Systems, 33:17617-17628,2020.   
+11 [33] P. Xu, F. Gao, and Q. Gu. An improved convergence analysis of stochastic variance-reduced   
+112 policy gradient. In Proc. International Conference on Uncertainty in Artificial Intelligence   
+13 (UAI),2019.   
+I14[34] T. Xu and Y. Liang. Sample complexity bounds for two timescale value-based reinforcement   
+115 learning algorithms. In International Conference on Artificial Intelligence and Statistics, pages   
+116 811-819. PMLR, 2021.   
+417 [35] T. Xu, Z. Wang, and Y. Liang. Improving sample complexity bounds for (natural) actor-critic   
+418 algorithms. In Proc.Advances in Neural Information Processing Systems (NeurIPS),volume 33,   
+419 2020.   
+420 [36] T. Xu, Z. Yang, Z. Wang,and Y. Liang. Doubly robust off-policy actor-critic: Convergence and   
+421 optimality. ArXiv:2102.11866,2021.   
+422 [37] C. Yu, X. Wang, X. Xu, M. Zhang, H. Ge, J. Ren,L. Sun, B. Chen, and G. Tan. Distributed   
+423 multiagent coordinated learning for autonomous driving in highways based on dynamic co  
+424 ordination graphs. IEEE Transactions on Intelligent Transportation Systems, 21(2):735-748,   
+425 2019.   
+426 [38] S. Zeng, T. Chen, A. Garcia, and M. Hong. Learning to coordinate in multi-agent systems: A   
+427 coordinated actor-critic algorithm and finite-time guarantees. arXiv preprint arXiv:2110.05597,   
+428 2021.   
+429 [39] H. Zhang, W. Chen, Z. Huang, M. Li, Y. Yang, W. Zhang, and J. Wang. Bi-level actor-critic   
+430 for multi-agent coordination. In Proceedings of the AAAI Conference on Artificial Intelligence,   
+431 volume 34, pages 7325-7332, 2020.   
+432 [40] K. Zhang, A. Koppel, H. Zhu,and T. Basar. Global convergence of policy gradient methods to   
+433 (almost) locally optimal policies. arXiv preprint arXiv:1906.08383,2019.   
+434 [41] K. Zhang, Z. Yang,and T. Basar. Multi-agent reinforcement learning: A selective overview of   
+435 theories and algorithms. Handbook of Reinforcement Learning and Control, pages 321-384,   
+436 2021.   
+[42] K. Zhang, Z. Yang, H. Liu,T. Zhang,and T. Basar.Fully decentralized multi-agent reinforcement   
+learning with networked agents. In International Conference on Machine Learning, pages   
+5872-5881. PMLR, 2018.
+
+# 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] The limitation is written in an equivalent form as future works in the conclusion section; see Section 6.   
+(c） Did you discuss any potential negative societal impacts of your work? [N/A] We conduct research about the design and analysis of the fundamental actor-critic algorithm, which should not bring any negative societal impact.   
+(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 fullset of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiment..
+
+(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]   
+(b) Did you specify all the training details (e.g. data splits, hyperparameters, how they were chosen)? [Yes]   
+(c） Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(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]
+
+. If you are using existing assets (e.g., code,data, models) or curating/releasing new asets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]
+
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes]   
+(e） Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects..
+
+(a) Did you include the ful 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]

parse/dev/CEjuyeZj1jz/CEjuyeZj1jz_content_list.json

@@ -0,0 +1,1608 @@
+[
+    {
+        "type": "text",
+        "text": "Finite-Time Analysis of Fully Decentralized Single-Timescale Actor-Critic ",
+        "text_level": 1,
+        "bbox": [
+            235,
+            122,
+            764,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            580,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            333
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Decentralized Actor-Critic (AC) algorithms have been widely utilized for multi  \n2 agent reinforcement learning (MARL） and have achieved remarkable success.   \n3 Apart from its empirical success, the theoretical convergence property of decen  \n4 tralized AC algorithms is largely unexplored. The existing finite-time convergence   \n5 results are derived based on either double-loop update or two-timescale step sizes   \n6 rule,which is not often adopted in real implementation. In this work,we introduce   \n7 a fully decentralized AC algorithm, where actor, critic,and global reward estimator   \n8 are updated in an alternating manner with step sizes being of the same order, namely,   \n9 we adopt the single-timescale update.Theoretically,using linear approximation for   \n10 value and reward estimation, we show that our algorithm has sample complexity of   \n11 $\\tilde { \\mathcal { O } } ( \\epsilon ^ { - 2 } )$ under Markovian sampling, which matches the optimal complexity with   \n12 double-loop implementation (here, $\\tilde { \\mathcal { O } }$ hides a log term). The sample complexity   \n13 can be improved to $\\mathcal { O } ( \\epsilon ^ { - 2 } )$ under the i.i.d. sampling scheme. The central to   \n14 establishing our complexity results is the hidden smoothness of the optimal critic   \n15 variable we revealed. We also provide a local action privacy-preserving version   \n16 of our algorithm and its analysis. Finally,we conduct experiments to show the   \n17 superiority of our algorithm over the existing decentralized AC algorithms. ",
+        "bbox": [
+            145,
+            345,
+            767,
+            594
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "181 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            613,
+            312,
+            631
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "19 Multi-agent reinforcement learning (MARL)[16,30] has been very successful in various models of   \n20 multi-agent systems,such as robotics [14],autonomous driving [37], Go [25],etc. MARL has been   \n21 extensively explored in the past decades; see,e.g.,[18,20,41,26,8,22]. These works either focus   \n22 on the setting where an central controller is available, or assuming a common reward function for all   \n23 agents. Among the many cooperative MARL settings, the work [42] proposes the fully decentralized   \n24 MARL with networked agents. In this setting, each agent maintains a private heterogeneous reward   \n25 function, and agents can only access local/neighboring information through communicating with its   \n26 neighboring agents on the network. Then, the objective of allagents is to jointly maximize the average   \n27 long-term reward through interacting with environment modeled by multi-agent Markov decision   \n28 process (MDP). They proposed the decentralized Actor-Critic (AC) algorithm to solve this MARL   \n29 problem,and showed its impressive performance. However, the theoretical convergence properties   \n30 of such class of decentralized AC algorithms are largely unexplored; see [41] for a comprehensive   \n31 survey.In this work,our goal is to establish the strong finite-time convergence results under this fully   \n32 decentralized MARL setting. We first review some recent progresses on this line of research below.   \n33 Related works and motivations. The first fully decentralized AC algorithm with provable con  \n34 vergence guarantee was proposed by [42], and they achieved asymptotic convergence results under   \n35 two-time scale step sizes, which requires actor's step sizes to diminish in a faster scale than the critic's   \n36 step sizes.The sample complexities of decentralized AC were established recently. In particular, [6]   \n37 and [11] independently propose two communication eficient decentralized AC algorithms with opti  \n38 mal sample complexity of ${ \\mathcal { O } } ( \\varepsilon ^ { - 2 } \\log ( \\varepsilon ^ { - 1 } ) )$ under Markovian sampling scheme. Their analysis are   \n39 based on double-loop implementation, where each policy optimization step follows a nearly accurate   \n40 critic optimization step (a.k.a. policy evaluation),i.e., solving the critic optimization subproblem to   \n41 $\\varepsilon$ -accuracy. Such a double-loop scheme requires careful tuning of two additional hyper-parameters,   \n42 which are the batch size and innerloop size.In particular,the batch size and inner loop size need to be   \n43 of order $\\mathcal { O } ( \\varepsilon ^ { - 1 } )$ and ${ \\mathcal { O } } ( \\log ( \\varepsilon ^ { - 1 } ) )$ in order to achieve their sample complexity results, respectively.   \n44 In practice, single-loop algorithmic framework is often utilized, where one updates the actor and   \n45 critic in an alternating manner by performing only one algorithmic iteration for both of the two   \n46 subproblems; see,e.g.， [23,18,15,39]. The work [38] proposes a new decentralized AC algorithm   \n47 based on such a single-loop alternative update. Nevertheless, they have to adopt two-timescale step   \n48 sizes rule to ensure convergence, which requires actor's step sizes to diminish in a faster scale than   \n49 the critic's step sizes.Due to the separation of the step sizes,the critic optimization sub-problem   \n50 is solved exactly when the number of iterations tends to $\\infty$ . Such a restriction on the step size will   \n51 slow down the convergence speed of the algorithm. As a consequence,they only obtain sub-optimal   \n52 sample complexity of $\\mathcal { O } ( \\varepsilon ^ { - \\frac { 5 } { 2 } } )$ . In practice, most algorithms are implemented with single-timescale   \n53 step size rule, where the step sizes for actor and critic updates are of the same order. Though there   \n54 are some theoretical achievements for single-timescale update in other areas such as TDC [31] and   \n55 bi-level optimization [4], similar theoretical understanding under AC setting is largely unexplored.   \n56 Indeed, even when reducing to single-agent seting, the convergence property of single-timescale   \n57 AC algorithm is not well established.The works [9,10] establish the finite-time convergence result   \n58 under a special single-timescale implementation, where they atain the sample complexity of $\\mathcal { O } ( \\varepsilon ^ { - 2 } )$   \n59 However,their analysis is based on an algorithm where the critic optimization step is formulated as a   \n60 least-square temporal diference (LSTD) at each iteration, where they need to sample the transition   \n61 tuples for $\\tilde { \\mathcal { O } } ( \\varepsilon ^ { - 1 } )$ times to form the data matrix in the LSTD problem. Then, they solve the LSTD   \n62 problem in a closed-form fashion, which requires to invert a matrix of large size.Later,[4] obtains the   \n63 same sample complexity using TD(O) update for critic variables under i.i.d. sampling. Nonetheless,   \n64 their analysis highly relies on the assumption that the Jacobian of the stationary distribution is   \n65 Lipschitz continuous,which is not justified in their work. ",
+        "bbox": [
+            145,
+            645,
+            826,
+            839
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            844,
+            825,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            90,
+            825,
+            356
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            362,
+            826,
+            502
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "i6The above observations motivate us to ask the following question: ",
+        "bbox": [
+            151,
+            508,
+            604,
+            523
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Can we establish finite-time convergence result for decentralized AC algorithm with single-timescale step sizes rule?l ",
+        "bbox": [
+            158,
+            529,
+            823,
+            558
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "，Main contributions.By answering this question positively, we have the following contributions: ",
+        "bbox": [
+            156,
+            563,
+            810,
+            579
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "· We design a fully decentralized AC algorithm, which employs a single-timescale step sizes rule and adopts Markovian sampling scheme. The proposed algorithm allows communication between agents for every $K _ { c }$ iterations with $K _ { c }$ being any integer lies in $[ 1 , { \\mathcal { O } } ( \\varepsilon ^ { - { \\frac { 1 } { 2 } } } ) ]$ ,rather than communicating at each iteration as adopted by previous single-loop decentralized AC algorithms [38, 42].   \nUsing linear approximation for value and reward estimation, we establish the finite-time convergence result for such an algorithm under the standard assumptions. In particular, we show that the algorithm has the sample complexity of $\\widetilde { \\mathcal { O } } ( \\varepsilon ^ { - 2 } )$ , which matches the optimal complexity up to a logarithmic term. In addition, we show that the logarithmic term can be removed under the i.i.d. sampling scheme.Note that these convergence results are valid for all the above mentioned choices for $K _ { c }$   \nTo preserve the privacy of local actions, we propose a variant of our algorithm which utilizes noisy local rewards for estimating global rewards. We show that such an algorithm will maintain the optimal sample complexity at the expense of communicating at each iteration. ",
+        "bbox": [
+            215,
+            584,
+            825,
+            791
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The underlying principle for obtaining the above convergence results is that we reveal the hidden smoothness of the optimal critic variable, so that we can derive an approximate descent on the averaged critic's optimal gap at each iteration. Consequently, we can resort to the classic convergence analysis for alternating optimization algorithms to establish the approximate ascent property of the overall optimization process,which leads to the final sample complexity results. ",
+        "bbox": [
+            166,
+            797,
+            825,
+            867
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "89 Another technical highlight is the Lyapunov function we construct for measuring the progress of our   \n90 algorithm. Such a construction is motivated by [4], which analyzes bi-level optimization algorithm.   \n91 However, our Lyapunov function is diferent from theirs as it involves the additional optimal gap of   \n92 averaged critic and reward estimator, which is necessry for dealing with the decentralized setting. ",
+        "bbox": [
+            147,
+            90,
+            826,
+            147
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3We finish this section by remarking that our convergence results are even new for single agent AC talgorithms under the settng of single-timescale step sizes rule. ",
+        "bbox": [
+            156,
+            154,
+            823,
+            183
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "：2Preliminary",
+        "text_level": 1,
+        "bbox": [
+            158,
+            200,
+            305,
+            218
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce the problem formulation and the policy gradient theorem, which serves as the preliminary for the analyzed decentralzed AC algorithm. ",
+        "bbox": [
+            163,
+            231,
+            823,
+            261
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Suppose there are multiple agents aiming to independently optimize a common global objective, and each agent can communicate with its neighbors through a network. To model the topology, we define the graph as $\\mathcal { G } = ( \\mathcal { N } , \\mathcal { E } )$ ,where $\\mathcal { N }$ is the set of nodes with $| { \\mathcal { N } } | = N$ and $\\mathcal { E }$ is the set of edges with $| { \\mathcal { E } } | = E$ . In the graph, each node represents an agent, and each edge represents a communication link. The interaction between agents follows the networked multi-agent MDP. ",
+        "bbox": [
+            173,
+            265,
+            825,
+            337
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1Markov decision process ",
+        "text_level": 1,
+        "bbox": [
+            161,
+            352,
+            383,
+            366
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "104 A networked multi-agent MDP is defined by a tuple $( \\mathcal { G } , S , \\{ \\mathcal { A } ^ { i } \\} _ { i \\in \\mathcal { N } } , \\mathcal { P } , \\{ r ^ { i } \\} _ { i \\in [ N ] } , \\gamma )$ ： $\\mathcal { G }$ denotes the   \n105 communication topology (the graph), $s$ is the finite state space observed by all agents, $\\mathcal { A } ^ { i }$ represents   \n106 the finite action space of agent $i$ .Let $\\mathcal { A } : = \\mathcal { A } ^ { 1 } \\times \\cdot \\cdot \\cdot \\stackrel { \\textstyle \\circ } { \\times } \\mathcal { A } ^ { N }$ denote the joint action space and   \n107 $\\mathcal { P } ( s ^ { \\prime } | s , a ) : \\mathcal { S } \\times \\bar { \\mathcal { A } } \\times \\mathcal { S } \\to [ \\bar { 0 } , 1 ]$ denote the transition probability from any state $s \\in S$ to any state   \n108 $s ^ { \\prime } \\in \\mathcal { S }$ for any joint action $a \\in { \\mathcal { A } }$ ： $r ^ { i } : S \\times \\mathcal { A }  \\mathbb { R }$ is the local reward function that determines the   \n109 reward received by agent $i$ given transition $( s , a )$ ， $\\gamma \\in [ 0 , 1 ]$ is the discount factor.   \n110 For simplicity, we will use $a : = [ a ^ { 1 } , \\cdots , a ^ { N } ]$ to denote the joint action, and $\\theta : = [ \\theta ^ { 1 } , \\cdots , \\theta ^ { N } ] \\in$   \n111 $\\mathbb { R } ^ { d _ { \\theta } \\times N }$ to denote joint parameters of all actors, with $\\theta ^ { i } \\in \\mathbb { R } ^ { d _ { \\theta } }$ . Note that different actors may have   \n112 different number of parameters,which is assumed to be the same for our paper without loss of   \n113 generality. The MDP goes as follows: For a given state $s$ ,each agent make its decision $a ^ { i }$ based   \n114 on its policy $a ^ { i } \\sim \\pi _ { \\theta ^ { i } } ( \\cdot | s )$ . The state transits to the next state $s ^ { \\prime }$ based on the joint action of all the   \n115 agents: $s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot | s , a )$ . Then, each agent will receive its own reward $r ^ { i } ( s , a )$ . For the notation brevity,   \n116 we assume that the reward function mapping is deterministic and does not depend on the next state   \n117 without loss of generality. The stationary distribution induced by the policy $\\pi _ { \\theta }$ and the transition   \n118 kernel is denoted by $\\mu _ { \\pi _ { \\theta } } ( s )$ ： ",
+        "bbox": [
+            140,
+            376,
+            825,
+            464
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            467,
+            825,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "9 Our objective is to find a set of policies that maximize the accumulated discounted mean reward :0received by agents ",
+        "bbox": [
+            151,
+            601,
+            825,
+            630
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a57ed5cc7c9fb93570921368e8ce0c1b57746264812e7c7d38f2463cc1960d1.jpg",
+        "text": "$$\n\\theta ^ { * } = \\underset { \\theta } { \\arg \\operatorname* { m a x } } J ( \\theta ) : = { \\mathbb { E } } \\left[ \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } \\bar { r } ( s _ { k } , a _ { k } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            635,
+            648,
+            679
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "121 Here, $k$ represents the time step. $\\begin{array} { r } { \\bar { r } ( s _ { k } , a _ { k } ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } r ^ { i } ( s _ { k } , a _ { k } ) } \\end{array}$ is the mean reward among agents   \n122 at time step $k$ .The randomness of the expectation comes from the initial state distribution $\\mu _ { 0 } ( s )$ ,the   \n123 transition kernel $\\mathcal { P }$ , and the stochastic policy $\\pi _ { \\theta ^ { i } } ( \\cdot | s )$ ： ",
+        "bbox": [
+            142,
+            685,
+            825,
+            731
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2Policy gradient Theorem ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            744,
+            385,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "25 Under the discounted reward seting,the global state-value function,action-value function, and   \n26advantage function for policy set $\\theta$ ,state $s$ , and action $a$ ,are defined as ",
+        "bbox": [
+            148,
+            770,
+            828,
+            799
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f96bfa53476484060390d862d284a352d87ba9b063f4163070d173319b84328a.jpg",
+        "text": "$$\n\\begin{array} { c l } { { \\displaystyle V _ { \\pi _ { \\theta } } ( s ) : = \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } \\bar { r } ( s _ { k } , a _ { k } ) \\vert s _ { 0 } = s \\right] } } \\\\ { { \\displaystyle Q _ { \\pi _ { \\theta } } ( s , a ) : = \\mathbb { E } \\left[ \\displaystyle \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } \\bar { r } ( s _ { k } , a _ { k } ) \\vert s _ { 0 } = s , a _ { 0 } = a \\right] } } \\\\ { { \\displaystyle A _ { \\pi _ { \\theta } } ( s , a ) : = Q _ { \\pi _ { \\theta } } ( s , a ) - V _ { \\pi _ { \\theta } } ( s ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            804,
+            666,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "127To maximize the objective function defined in (1),the policy gradient [28] can be computed as follow ",
+        "bbox": [
+            145,
+            90,
+            825,
+            106
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d65a256814f9d06dcb77a9163440bea3bd179613f8d84903f4daf7bf6520825a.jpg",
+        "text": "$$\n\\nabla _ { \\theta } J ( \\theta ) = \\mathbb { E } _ { s \\sim d _ { \\pi _ { \\theta } } , a \\sim \\pi _ { \\theta } } \\left[ \\frac { 1 } { 1 - \\gamma } A _ { \\pi _ { \\theta } } ( s , a ) \\psi _ { \\pi _ { \\theta } } ( s , a ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            108,
+            674,
+            142
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "28where $\\begin{array} { r } { d _ { \\pi _ { \\theta } } ( s ) : = ( 1 - \\gamma ) \\sum _ { k = 0 } ^ { \\infty } \\gamma ^ { k } \\mathbb { P } ( s _ { k } = s ) } \\end{array}$ is thediscountedtatevisitationdistributionuder   \n29policy $\\pi _ { \\theta }$ ,and $\\psi _ { \\pi _ { \\theta } } ( s , a ) : = \\nabla \\log \\pi _ { \\theta } ( s , a )$ is the score function.   \n3oFollowing the derivation of [42],the policy gradient for each agent under discounted reward setting   \n31can be expressed as ",
+        "bbox": [
+            147,
+            143,
+            825,
+            174
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            179,
+            823,
+            208
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d08883530bded5e0135b1e0a46e783d80c973d9c221815cf2f8239098fde1a8b.jpg",
+        "text": "$$\n\\nabla _ { \\theta ^ { i } } J ( \\theta ) = \\mathbb { E } _ { s \\sim d _ { \\pi _ { \\theta } } , a \\sim \\pi _ { \\theta } } \\left[ \\frac { 1 } { 1 - \\gamma } A _ { \\pi _ { \\theta } } ( s , a ) \\psi _ { \\pi _ { \\theta ^ { i } } } ( s , a ^ { i } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            209,
+            686,
+            244
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "132 3Decentralized single-timescale actor-critic ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            257,
+            555,
+            276
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: Decentralized single-timescale AC (reward estimator version) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            300,
+            673,
+            315
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize: Actor parameter $\\theta _ { 0 }$ , critic parameter $\\omega _ { 0 }$ , reward estimator parameter $\\lambda _ { 0 }$ , initial state $s _ { 0 }$   \n2: for $k = 0 , \\cdots , \\bar { K } - 1$ do   \n3: Option 1: i.i.d. sampling:   \n4: $s _ { k } \\sim \\mu _ { \\theta _ { k } } ( \\cdot ) , a _ { k } \\sim \\pi _ { \\theta _ { k } } ( \\cdot | s _ { k } ) , s _ { k + 1 } \\sim \\mathcal { P } ( \\cdot | s _ { k } , a _ { k } ) .$   \n5: Option 2: Markovian sampling:   \n6: $a _ { k } \\sim \\pi _ { \\theta _ { k } } ( \\cdot | s _ { k } ) , s _ { k + 1 } \\sim \\mathcal { P } ( \\cdot | s _ { k } , a _ { k } ) .$   \n7:   \n8: Periodical consensus: Compute $\\tilde { \\omega } _ { k } ^ { i }$ and $\\tilde { \\lambda } _ { k } ^ { i }$ by (4) and (7).   \n9:   \n10: for $i = 0 , \\cdots , N$ in parallel do   \n11: Reward estimator update: Update $\\lambda _ { k + 1 } ^ { i }$ by (8).   \n12: Critic update: Update $\\boldsymbol { \\omega } _ { k + 1 } ^ { i }$ by (5).   \n13: Actor update: Update $\\theta _ { k + 1 } ^ { i }$ by (6).   \n14: end for   \n15:end for   \n33We introduce the decentralized single-timescale AC algorithm; see Algorithm 1. In the remaining   \n34parts of this section, we will explain the updates in the algorithm in details.   \n135 In fully-decentralized MARL,each agent can only observe its local reward and action, while trying   \n136 to maximize the global reward (mean reward) defined in (1). The decentralized AC algorithm solves   \n137 the problem by performing online updates in an alternative fashion. Specifically,we have $N$ pairs of   \n138 actor and critic.In order to maximize $J ( \\theta )$ ,each critic tries to estimate the global state-value function   \n139 $V _ { \\pi _ { \\theta } } ( s )$ defined in (2),and each actor then updates its policy parameter based on approximated policy   \n140 gradient. We now provide more details about the algorithm.   \n141 Critics'update.We willuse $\\omega ^ { i } \\in \\mathbb { R } ^ { d _ { \\omega } }$ to denote the $i _ { t h }$ crics parameter and $\\begin{array} { r } { \\bar { \\omega } : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\omega ^ { i } } \\end{array}$ t0   \n142 represent the averaged parameter of critic. The $i _ { t h }$ critic approximates the global value function as   \n143 $\\bar { V _ { \\pi _ { \\theta } } } ( s ) \\approx \\hat { V } _ { \\omega ^ { i } } ( s )$ ： ",
+        "bbox": [
+            174,
+            321,
+            784,
+            518
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            535,
+            825,
+            564
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            569,
+            825,
+            654
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            660,
+            825,
+            707
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "146 As w sc $\\begin{array} { r } { \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\| \\boldsymbol { \\omega } ^ { i } - \\boldsymbol { \\bar { \\omega } } \\| } \\end{array}$ approximation error $\\| \\bar { \\boldsymbol { \\omega } } - \\boldsymbol { \\omega } ^ { * } ( \\boldsymbol { \\theta } ) \\|$ ，which measures the approximation quality of averaged critic. ",
+        "bbox": [
+            142,
+            712,
+            826,
+            758
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "47In order for critics to reach consensus,we perform the following update for all critics ",
+        "bbox": [
+            150,
+            762,
+            732,
+            779
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c598bf8c5edab326c5b5f77eabf65e0560422616152b275d36cb4c82dcd670d3.jpg",
+        "text": "$$\n\\tilde { \\omega } _ { k } ^ { i } = \\left\\{ \\begin{array} { l l } { { \\sum _ { j = 1 } ^ { N } W ^ { i j } \\omega _ { k } ^ { j } } } & { { \\quad \\mathrm { i f } \\ k \\ \\mathrm { m o d } \\ K _ { c } = 0 } } \\\\ { { \\omega _ { k } ^ { i } } } & { { \\quad \\mathrm { o t h e r w i s e } . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            780,
+            647,
+            821
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "148 where $W \\in \\mathbb { R } ^ { n \\times n }$ is a weight matrix for communication among agents, whose property will be   \n149specified in Assumption 5; $K _ { c }$ denotes the consensus frequency. ",
+        "bbox": [
+            140,
+            823,
+            830,
+            853
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "150To reduce the approximation error, we willperform the local TD(O) update [29] as ",
+        "bbox": [
+            140,
+            858,
+            715,
+            873
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7df98e1e2c9a46c215cdce8f3351206e727f08821e231258f164bdaf3e8f60d3.jpg",
+        "text": "$$\n\\omega _ { k + 1 } ^ { i } = \\prod _ { R _ { \\omega } } ( \\tilde { \\omega } _ { k } ^ { i } + \\beta _ { k } g _ { c } ^ { i } ( \\xi _ { k } , \\omega _ { k } ^ { i } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            875,
+            609,
+            910
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "151 where $\\xi : = ( s , a , s ^ { \\prime } )$ represents a transition tuple, $g _ { c } ^ { i } ( \\xi , \\omega ) : = \\delta ^ { i } ( \\xi , \\omega ) \\nabla \\hat { V } _ { \\omega } ( s )$ is the update direction,   \n152 $\\delta ^ { i } ( \\xi , \\omega ) : = r ^ { i } ( s , a ) + \\gamma \\hat { V } _ { \\omega } ( s ^ { \\prime } ) - \\hat { V } _ { \\omega } ( s )$ is the local temporal difference error (TD-error). $\\beta _ { k }$ is the   \n153 step size for critic at iteration $k$ $\\prod _ { R _ { \\omega } }$ projects the parameter into a ball of radius of $R _ { \\omega }$ containing   \n154 the optimal solution, which will be explained when discussing Assumption 1 and 2.   \n155Actors'update. We will use stochastic gradient ascent to update the policy's parameter, and the   \n156 stochastic gradient is calculated based on policy gradient theorem in (3). The advantage function   \n157 $A _ { \\pi _ { \\theta } } ( s , a )$ can be estimated by ",
+        "bbox": [
+            140,
+            89,
+            825,
+            150
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            155,
+            825,
+            198
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/050afe0fecef61900890702f119d64dca283abb87e6a3271ae198534ad6e1e0e.jpg",
+        "text": "$$\n\\delta ( \\xi , \\theta ) : = \\bar { r } ( s , a ) + \\gamma V ( s ^ { \\prime } ) - V ( s ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            204,
+            619,
+            222
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "158 with $a$ sampled from $\\pi _ { \\boldsymbol { \\theta } } ( \\cdot | \\boldsymbol { s } )$ . However, to preserve the privacy of each agents,the local reward   \n159 cannot be shared to other agents under the fully decentralized setting. Thus, the averaged reward   \n160 $\\bar { r } ( s _ { k } , a _ { k } )$ is not directly attainable. Consequently, we need a strategy to approximate the averaged   \n161 reward. In this paper, we will adopt the strategy proposed in [42]. In particular, each agent $i$ will have   \n162 a local reward estimator with parameter $\\lambda ^ { i } \\in \\mathbb { R } ^ { d _ { \\lambda } }$ , which estimates the global averaged reward as   \n163 $\\bar { r } ( s _ { k } , a _ { k } ) \\approx \\hat { r } _ { \\lambda ^ { i } } ( s _ { k } , a _ { k } )$ ",
+        "bbox": [
+            142,
+            228,
+            825,
+            314
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "164Thus,the update of the $i _ { t h }$ actor is given by ",
+        "bbox": [
+            147,
+            318,
+            460,
+            333
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/80c4e4d02510ee0a582f3b7accf35f492b9bb597b949e7acb3865efabea7fda4.jpg",
+        "text": "$$\n\\theta _ { k + 1 } ^ { i } = \\theta _ { k } ^ { i } + \\alpha _ { k } \\hat { \\delta } ( \\xi _ { k } , \\omega _ { k + 1 } ^ { i } , \\lambda _ { k + 1 } ^ { i } ) \\psi _ { \\pi _ { \\theta _ { k } ^ { i } } } ( s _ { k } , a _ { k } ^ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            339,
+            658,
+            364
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\delta } ( \\xi , \\omega , \\lambda ) : = \\hat { r } _ { \\lambda } ( s , a ) + \\gamma \\hat { V } _ { \\omega } ( s ^ { \\prime } ) - \\hat { V } _ { \\omega } ( s )$ is the approximated advantage function. $\\alpha _ { k }$ is the step size for actor's update at iteration $k$ ： ",
+        "bbox": [
+            163,
+            371,
+            823,
+            402
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Reward estimators’update. Similar to critic, each reward estimator's approximation error can be decomposed into consensus error and the approximation error. ",
+        "bbox": [
+            161,
+            409,
+            820,
+            438
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "s9For each local reward estimator, we perform the consensus step to minimize the consensus error as ",
+        "bbox": [
+            150,
+            443,
+            820,
+            458
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f539efa72736e16b125f439a9161fed467e853ddd5ce935999b5d6ef76917eb3.jpg",
+        "text": "$$\n\\tilde { \\lambda } _ { k } ^ { i } = \\left\\{ \\sum _ { j = 1 } ^ { N } W ^ { i j } \\lambda _ { k } ^ { j } \\qquad \\mathrm { i f } k \\ \\mathrm { m o d } \\ K _ { c } = 0 \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            463,
+            647,
+            507
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "170To reduce the approximation error, we perform a local update of stochastic gradient descent. ",
+        "bbox": [
+            140,
+            512,
+            774,
+            527
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0fee6db238a581cd1a22a22b0521163942db3a2c409d4fabd97c1567dbc30699.jpg",
+        "text": "$$\n\\lambda _ { k + 1 } ^ { i } = \\prod _ { R _ { \\lambda } } ( \\tilde { \\lambda } _ { k } ^ { i } + \\eta _ { k } g _ { r } ^ { i } ( \\xi _ { k } , \\lambda _ { k } ^ { i } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            534,
+            607,
+            568
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "171 where $g _ { r } ^ { i } ( \\xi , \\lambda ) : = ( r ^ { i } ( s , a ) - { \\hat { r } } _ { \\lambda } ( s , a ) ) \\nabla { \\hat { r } } _ { \\lambda } ( s , a )$ is the update direction. $\\eta _ { k }$ is the step size for   \n172 reward estimator at iteration $k$ . Note the calculation of $g _ { r } ^ { i } ( \\xi , \\bar { \\lambda } )$ does not require the knowledge of $s ^ { \\prime }$   \n173 we use $\\xi$ in (8) just for notation brevity. Similar to critics update, $\\prod _ { R _ { \\lambda } }$ projects the parameter into a   \n174 ball of radius of $R _ { \\lambda }$ containing the optimal solution.   \n175In our Algorithm 1, we will use the same order for $\\alpha _ { k }$ ， $\\beta _ { k }$ ,and $\\eta _ { k }$ and hence, our algorithm is in   \n176single-timescale.   \n177 Linear approximation for analysis. In our analysis, we will use linear approximation for both critic   \n178 and reward estimator variables, i.e. $\\hat { V } _ { \\omega } ( s ) : = \\phi ( s ) ^ { T } \\omega ; \\hat { r } _ { \\lambda } ( s , a ) : = \\varphi ( s , a ) ^ { T } \\lambda ;$ where $\\phi ( s ) : S \\to $   \n179 $\\mathbb { R } ^ { d _ { \\omega } }$ and $\\varphi ( s , a ) : \\mathcal { S } \\times \\mathcal { A }  \\mathbb { R } ^ { d _ { \\lambda } }$ are two feature mappings, whose property willbe specified in the   \n180 discussion of Assumption 1.   \n181 Algorithm for preserving the local action. Note that in Algorithm 1, the reward estimators need   \n182 the knowledge of joint actions in order to estimate the global rewards. To preserve the privacy of   \n183 local actions, we further propose a variant of Algorithm 1, which estimates the global rewards by   \n184 communicating noisy local rewards; see [6] for the original idea. However, to maintain the optimal   \n185 sample complexity, such an approach requires $\\mathcal { O } ( \\log ( \\varepsilon ^ { - 1 } ) )$ communication rounds for each iteration.   \n186 We postpone the detailed design and analysis of such an algorithm scheme into Appendix B.   \n187 Remarks on sampling scheme. The unbiased update for critic and actor variables requires sampling   \n188 from $\\mu _ { \\pi _ { \\theta } }$ and $d _ { \\pi _ { \\theta } }$ ,respectively. However, in practical implementations,states are usually collected   \n189 fromanonline trajectory(Markoviansampling),whosedistributionisgeneralldierentfor $\\mu _ { \\pi _ { \\theta } }$   \n190 and $d _ { \\pi _ { \\theta } }$ . Such a distribution mismatch willinevitably cause biases during the update of critic and   \n191 actor variables. One has to bound the corresponding error terms when analyzing the algorithm. In   \n192 this work,we will provide the analysis for both sampling schemes. ",
+        "bbox": [
+            140,
+            575,
+            825,
+            632
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            638,
+            825,
+            667
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            672,
+            825,
+            733
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            738,
+            825,
+            823
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            828,
+            825,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first introduce the technical assumptions used for our analysis,which are standard in the literature.Then, we present the convergence results for both actor and critic variables under i.i.d. sampling and Markovian sampling. ",
+        "bbox": [
+            173,
+            119,
+            825,
+            161
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 Assumptions ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            166,
+            302,
+            181
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 (bounded rewards and feature vectors). All the local rewards are uniformly bounded, i.e., there exists a positive constants $r _ { \\mathrm { m a x } }$ such that $| r ^ { i } ( s , a ) | \\leq r _ { \\operatorname* { m a x } } ,$ for all feasible $( s , a )$ and $i \\in [ N ]$ . The norm of feature vectors are bounded such that for all $s \\in S$ ， $a \\in { \\mathcal { A } }$ $\\| \\phi ( s ) \\| \\leq$ $1 , \\| \\varphi ( s , \\overline { { a } } ) \\| \\leq 1$ ",
+        "bbox": [
+            171,
+            190,
+            825,
+            248
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "？Assumption 1 is standard and commonly adopted; see, e.g.， [3,35,38,24,21]. This assumption can 3be achieved via normalizing the feature vectors. ",
+        "bbox": [
+            156,
+            255,
+            823,
+            284
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "04Assumption 2 (negative definiteness of $A _ { \\theta , \\phi }$ and $A _ { \\theta , \\varphi . }$ ). There exists two positive constants $\\lambda _ { \\phi } , \\lambda _ { \\varphi }$   \n55such that for all policy $\\theta$ ，the following two matrices are negative definite ",
+        "bbox": [
+            150,
+            286,
+            823,
+            314
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/febd38e063cf1efe83595f9e7ed70661379bdaf44cd045cea6cccc933aaff52e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { A _ { \\theta , \\phi } : = \\mathbb { E } _ { s \\sim \\mu _ { \\theta } ( s ) } [ \\phi ( s ) ( \\gamma \\phi ( s ^ { \\prime } ) ^ { T } - \\phi ( s ) ^ { T } ) ] } \\\\ & { A _ { \\theta , \\varphi } : = \\mathbb { E } _ { s \\sim \\mu _ { \\theta } ( s ) , a \\sim \\pi _ { \\theta } ( \\cdot | s ) } [ - \\varphi ( s , a ) \\varphi ( s , a ) ^ { T } ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            314,
+            651,
+            356
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "06with $\\lambda _ { \\operatorname* { m a x } } ( A _ { \\theta , \\phi } ) \\leq \\lambda _ { \\phi } , \\lambda _ { \\operatorname* { m a x } } ( A _ { \\theta , \\varphi } ) \\leq \\lambda _ { \\varphi }$ ， where $\\lambda _ { \\operatorname* { m a x } } ( \\cdot )$ represents the largest eigenvalue. ",
+        "bbox": [
+            148,
+            356,
+            776,
+            372
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "207 Assumption 2 can be achieved when the matrices $\\Phi _ { \\phi } ~ : = ~ [ \\phi ( s _ { 1 } ) , \\cdot \\cdot \\cdot , \\phi ( s _ { | S | } ) ]$ and $\\Phi _ { \\varphi } : = $   \n208 $[ \\varphi ( s _ { 1 } , a _ { 1 } ) , \\cdot \\cdot \\cdot , \\varphi ( s _ { | S | } , a _ { | A | } ) ]$ have ful row rank, which ensures that the optimal critic and reward   \n209 estimator are unique; see also [24,34]. Together with Assumption 1, we can show that the norm of   \n210 $\\omega ^ { \\ast } ( \\theta )$ and $\\lambda ^ { * } ( \\theta )$ are bounded by some positive constant, which justifies the projection steps. ",
+        "bbox": [
+            140,
+            378,
+            826,
+            438
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3 (Lipschitz properties of policy). There exists constants $C _ { \\psi } , L _ { \\psi } , L _ { \\pi }$ such that for ：all $\\theta , \\theta ^ { \\prime } , s \\in \\mathcal { S }$ and $a \\in { \\mathcal { A } }$ we have (1). $| \\bar { \\pi _ { \\theta } } ( a | s ) - \\pi _ { \\theta ^ { \\prime } } ( a | s ) | \\leq L _ { \\pi } \\| \\theta - \\theta ^ { \\prime } \\|$ (2). $\\parallel \\psi _ { \\theta } ( s , a ) \\textrm { -- }$ $\\psi _ { \\theta ^ { \\prime } } ( s , a ) \\| \\le L _ { \\psi } \\| \\theta - \\theta ^ { \\prime } \\|$ ; (3). $\\| \\psi _ { \\theta } ( s , a ) \\| \\leq C _ { \\psi }$ ： ",
+        "bbox": [
+            156,
+            439,
+            828,
+            483
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4Assumption 3 is common for analyzing policy-based algorithms; see, e.g., [33,32,11]. The assump  \n5tion ensures the smoothness of objective function $J ( \\theta ) ^ { \\top }$ . It holds for a large range of policy classes   \n6such as tabular softmax policy [1], Gaussian policy [7],and Boltzman policy [13].   \n17Assumption 4 (irreducible and aperiodic Markov chain). The Markov chain under $\\pi _ { \\theta }$ and transition   \n18kernel $\\mathcal { P } ( \\cdot | s , a )$ is irreducible and aperiodic for any $\\theta$ ：   \n219Assumption 4 is a standard assumption, which holds for any uniformly ergodic Markov chains and   \n220 any time-homogeneous Markov chains with finite-state space. It ensures that there exists constants   \n221 $\\kappa > 0$ and $\\rho \\in ( 0 , 1 )$ such that ",
+        "bbox": [
+            153,
+            489,
+            828,
+            532
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            150,
+            534,
+            826,
+            563
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            570,
+            826,
+            613
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f4caf3bdbda363064efc5175d6d0bd141d5fcec2fa062996782800d440e1b7a5.jpg",
+        "text": "$$\n\\operatorname* { s u p } _ { s \\in S } d _ { T V } \\big ( \\mathbb { P } \\big ( s _ { k } \\in \\cdot | s _ { 0 } = s , \\pi _ { \\theta } \\big ) , \\mu _ { \\theta } \\big ) \\leq \\kappa \\rho ^ { k } , \\ \\forall k .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            613,
+            656,
+            641
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "：Assumption 5 (doubly stochastic weight matrix). The communication matrix $W$ is doubly stochastic, 3i.e. each column/row sum up to 1. Moreover, the second largest singular value $\\nu$ is smaller than $^ { l }$ ： ",
+        "bbox": [
+            156,
+            642,
+            825,
+            671
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 5 is a common assumption in decentralized optimization and multi-agent reinforcement learning; see, e.g.， [27,5, 6]. It ensures the convergence of consensus error for critic and reward estimator variables. ",
+        "bbox": [
+            169,
+            678,
+            825,
+            722
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2Sample complexity under i.i.d. sampling ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            728,
+            495,
+            744
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (sample complexity under i.i.d. sampling). Suppose Assumptions 1-5 hold. Consider theupdteofAlgorithm1underi.id.sampling.Let $\\begin{array} { r } { \\alpha _ { k } \\ = \\ \\frac { \\bar { \\alpha } } { \\sqrt { K } } } \\end{array}$ for some positive constant $\\bar { \\alpha }$ ",
+        "bbox": [
+            168,
+            753,
+            825,
+            786
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "i0 $\\begin{array} { r } { \\beta _ { k } = \\frac { C _ { 9 } } { 2 \\lambda _ { \\phi } } \\alpha _ { k } } \\end{array}$ ,and $\\begin{array} { r } { \\eta _ { k } = \\frac { C _ { 1 0 } } { 2 \\lambda _ { \\varphi } } \\alpha _ { k } } \\end{array}$ CK≤()heK doeshetbf i1 Then, we have ",
+        "bbox": [
+            151,
+            787,
+            825,
+            821
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64b23754249a99edfe92b9114220a0d1920a0ecc4d3c081cc61ebb1066208617.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\sum _ { i = 1 } ^ { N } \\mathbb { E } \\left[ \\| \\omega _ { k } ^ { i } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } \\right] \\leq \\mathcal { O } \\left( \\frac { 1 } { \\sqrt { K } } \\right) } \\\\ & { \\displaystyle \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\sum _ { i = 1 } ^ { N } \\mathbb { E } \\left[ \\| \\nabla _ { \\theta ^ { i } } F ( \\theta _ { k } ) \\| ^ { 2 } \\right] \\leq \\mathcal { O } \\left( \\frac { 1 } { \\sqrt { K } } \\right) + \\mathcal { O } ( \\varepsilon _ { a p p } + \\varepsilon _ { s p } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            821,
+            702,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "233 The proof of Theorem 1 can found in Appendix E.1. It establishes the iteration complexity of   \n234 $\\mathcal { O } ( 1 / \\sqrt { K } )$ ,or equivalently, sample complexity of $\\mathcal { O } ( \\varepsilon ^ { - 2 } )$ for Algorithm 1. Note that actors, critics,   \n235 and reward estimators use the step sizes of the same order. The sample complexity matches the   \n236 optimal rate of SGD for general non-convex optimization problem.To explain the errors in(9), let us   \n237 define the approximation error as the following: ",
+        "bbox": [
+            138,
+            114,
+            826,
+            189
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d9707e90a0074fc416133a505459d489498ff56f898a9e2bcd4f590e0c20da4.jpg",
+        "text": "$$\n\\varepsilon _ { a p p } : = \\operatorname* { m a x } _ { \\theta , a } \\sqrt { \\mathbb { E } _ { s \\sim \\mu _ { \\theta } } \\left[ | V _ { \\pi _ { \\theta } } ( s ) - \\hat { V } _ { \\omega ^ { * } ( \\theta ) } ( s ) | ^ { 2 } + | \\bar { r } ( s , a ) - \\hat { r } _ { \\lambda ^ { * } ( \\theta ) } ( s , a ) | ^ { 2 } \\right] } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            194,
+            745,
+            229
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "238 The error $\\varepsilon _ { a p p }$ captures the approximation power of critic and reward estimator. Similar terms   \n239 also appear in the literature (see e.g., [35,1, 21]). Such an approximation error becomes zero in   \n240 tabular case. The error $\\varepsilon _ { s p }$ is inevitably caused by the mismatch between discounted state visitation   \n241 distribution $d _ { \\pi _ { \\theta } }$ and stationary distribution $\\mu _ { \\pi _ { \\theta } }$ ; see, e.g., [38,24]. It is defined as ",
+        "bbox": [
+            140,
+            233,
+            825,
+            291
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4aca707c109aebb50742790ccac450a3497baa4d6c1d804059a57c39e261cd9c.jpg",
+        "text": "$$\n\\varepsilon _ { s p } : = 2 C _ { \\theta } ( \\log _ { \\rho } \\kappa ^ { - 1 } + \\frac { 1 } { \\rho } ) ( 1 - \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            296,
+            617,
+            328
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "12When $\\gamma$ is close to 1, the error becomes small. This is because $d _ { \\pi _ { \\theta } }$ approaches to $\\mu _ { \\pi _ { \\theta } }$ when $\\gamma$ goes to   \n131.In the literature,some works assume that sampling from $d _ { \\pi _ { \\theta } }$ is permitted, thus eliminate this error;   \n14see,e.g., [4]. ",
+        "bbox": [
+            151,
+            333,
+            826,
+            376
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3Sample complexity under markovian sampling ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            391,
+            539,
+            406
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "；Theorem 2 (sample complexity under Markovian sampling). Suppose Assumptions 1-5 hold. Consider the update ofAlgorithm $^ { l }$ under Markoviansampling.Let $\\begin{array} { r } { \\alpha _ { k } = \\frac { \\bar { \\alpha } } { \\sqrt { K } } } \\end{array}$ for some positive constant $\\bar { \\alpha }$ $\\begin{array} { r } { \\beta _ { k } = \\frac { C _ { 9 } } { 2 \\lambda _ { \\phi } } \\alpha _ { k } } \\end{array}$ and $\\begin{array} { r } { \\eta _ { k } = \\frac { C _ { 1 0 } } { 2 \\lambda _ { \\varphi } } \\alpha _ { k } } \\end{array}$ （ $K _ { c } \\leq \\mathcal { O } ( \\alpha _ { k } ^ { - \\frac 1 2 } )$ where $K$ is he tl frahe ）we have ",
+        "bbox": [
+            156,
+            416,
+            826,
+            484
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c80e82dfcb93775d73f2f824889f96ba920fc04a245faeaa4625232d868e778e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\sum _ { i = 1 } ^ { N } \\mathbb { E } \\left[ \\| \\omega _ { k } ^ { i } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } \\right] \\leq \\mathcal { O } \\left( \\frac { \\log ^ { 2 } K } { \\sqrt { K } } \\right) } \\\\ & { \\displaystyle \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\sum _ { i = 1 } ^ { N } \\mathbb { E } \\left[ \\| \\nabla _ { \\theta ^ { i } } F ( \\theta _ { k } ) \\| ^ { 2 } \\right] \\leq \\mathcal { O } \\left( \\frac { \\log ^ { 2 } K } { \\sqrt { K } } \\right) + \\mathcal { O } ( \\varepsilon _ { a p p } + \\varepsilon _ { s p } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            488,
+            714,
+            577
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "250where $C _ { 9 } , C _ { 1 0 }$ are positive constants defined in proof. ",
+        "bbox": [
+            140,
+            580,
+            527,
+            597
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "51 We put the proof of Theorem 2 in Appendix E.2. In Markovian sampling,the updates are biased for   \n52 critics,actors,and reward estimators.The error will decrease as the Markov chain mixes,and the   \n53 logarithmic term is due to the cost for mixing. ",
+        "bbox": [
+            148,
+            606,
+            826,
+            648
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 establishes the iteration complexity of $\\mathcal { O } ( \\log ^ { 2 } K / \\sqrt { K } )$ , or equivalently, sample complexity of $\\widetilde { \\mathcal { O } } ( \\varepsilon ^ { - 2 } )$ for Algorithm 1. It matches the state-of-the-art sample complexity of decentralized AC algorithms,which are implemented in double-loop fashion [11, 6]. ",
+        "bbox": [
+            165,
+            655,
+            825,
+            702
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4Proof sketch ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            717,
+            300,
+            732
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "258 We present the main elements for the proof of Theorem 2, which helps in understanding the difference   \n259 between classical two-timescale/double-loop analysis and our single-timescale analysis. The proof of   \n260 Theorem 1 follows the same framework with simpler sampling scheme. ",
+        "bbox": [
+            142,
+            742,
+            825,
+            785
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "261 Under Markovian sampling, it is possible to show the following inequality, which characterizes the :62ascent of the objective. ",
+        "bbox": [
+            147,
+            790,
+            825,
+            820
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1f7596b1cccefa71303b4672bdf56e3d558929113030290c0e5a164889713bf.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ J ( \\theta _ { k + 1 } ) ] - J ( \\theta _ { k } ) \\ge \\displaystyle \\sum _ { i = 1 } ^ { N } \\Big [ \\frac { \\alpha _ { k } } { 2 } \\mathbb { E } \\| \\nabla _ { \\theta ^ { i } } J ( \\theta _ { k } ) \\| ^ { 2 } + \\frac { \\alpha _ { k } } { 2 } \\mathbb { E } \\| g _ { a } ^ { i } ( \\xi _ { k } , \\omega _ { k + 1 } ^ { i } , \\lambda _ { k + 1 } ^ { i } ) \\| ^ { 2 } } \\\\ & { \\qquad - 8 C _ { \\psi } ^ { 2 } \\alpha _ { k } \\mathbb { E } \\| \\omega ^ { * } ( \\theta _ { k } ) - \\omega _ { k + 1 } ^ { i } \\| ^ { 2 } - 4 C _ { \\psi } ^ { 2 } \\alpha _ { k } \\mathbb { E } \\| \\lambda ^ { * } ( \\theta _ { k } ) - \\lambda _ { k + 1 } ^ { i } \\| ^ { 2 } \\Big ] } \\\\ & { \\qquad - { \\mathcal { O } } ( \\log ^ { 2 } ( K ) \\alpha _ { k } ^ { 2 } ) - { \\mathcal { O } } \\big ( ( \\varepsilon _ { a p p } + \\varepsilon _ { s p } ) \\alpha _ { k } \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            825,
+            784,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "263 To analyze the errors of critic $\\lVert \\omega ^ { * } ( \\theta _ { k } ) - \\omega _ { k + 1 } ^ { i } \\rVert ^ { 2 }$ and reward estimator $\\lVert \\lambda ^ { * } ( \\theta _ { k } ) - \\lambda _ { k + 1 } ^ { i } \\rVert ^ { 2 }$ , the two  \n264 timescale analysis requires $\\mathcal { O } ( \\alpha _ { k } ) < \\operatorname* { m i n } \\{ \\mathcal { O } ( \\beta _ { k } ) , \\mathcal { O } ( \\eta _ { k } ) \\}$ in order for these two errors to converge.   \n265 The double-loop approach runs lower-level update for ${ \\mathcal { O } } ( \\log ( \\varepsilon ^ { - 1 } ) )$ times with batch size $\\mathcal { O } ( \\varepsilon ^ { - 1 } )$   \n266 to drive these errors below $\\varepsilon$ and hence, they cannot allow inner loop size and bath size to be $\\mathcal { O } ( 1 )$   \n267 simultaneously. To obtain the convergence result for single-timescale update,the idea is to further   \n268 upper bound these two lower-level errors by the quantity $\\mathcal { O } ( \\alpha _ { k } \\mathbb { E } \\| g _ { a } ^ { i } ( \\xi _ { k } , \\omega _ { k + 1 } ^ { i } , \\lambda _ { k + 1 } ^ { i } ) \\| ^ { 2 } )$ (through a   \n269 series of derivations),and then eliminate these errors by the ascent term $\\begin{array} { r l } {  { \\frac { \\alpha _ { k } } { 2 } \\mathbb { E } \\bigl \\| g _ { a } ^ { i } \\bigl ( \\xi _ { k } , \\omega _ { k + 1 } ^ { i } , \\lambda _ { k + 1 } ^ { i } \\bigr ) \\bigr \\| ^ { 2 } } \\qquad } & { { } } \\end{array}$   \n270 We mainly focus on the analysis of critic's error through the proof sketch. The analysis for reward   \n271estimator's error follows similar procedure.We start by decomposing the error of critic as ",
+        "bbox": [
+            140,
+            89,
+            826,
+            194
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            198,
+            828,
+            228
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bafcbe78f6245c75f58e098c39af226fd6e002a1f24c6896c051b39c3bc43829.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { N } \\| \\omega _ { k + 1 } ^ { i } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } = \\sum _ { i = 1 } ^ { N } ( \\| \\omega _ { k + 1 } ^ { i } - \\bar { \\omega } _ { k + 1 } \\| ^ { 2 } + \\| \\bar { \\omega } _ { k + 1 } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            233,
+            736,
+            276
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "72The first term represents the consensus error, which can be bounded by the next lemma. ",
+        "bbox": [
+            150,
+            289,
+            746,
+            304
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "273Lemma 1. Suppose Assumptions l and $5$ hold. Consider the sequence $\\left\\{ \\omega _ { k } ^ { i } \\right\\}$ generated by Algorithm $^ { l }$ ，   \n274then the following holds ",
+        "bbox": [
+            142,
+            308,
+            828,
+            338
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcc878cad72d73bdf0089a693a34bc3381bffac0c167c88f6b0b07be1e9193dc.jpg",
+        "text": "$$\n\\| Q \\omega _ { k + 1 } \\| \\leq \\nu ^ { \\frac { k ^ { \\prime } } { K _ { c } } } \\| \\omega _ { 0 } \\| + 4 \\sum _ { t = 0 } ^ { k } \\nu ^ { \\lceil \\frac { k ^ { \\prime } - 1 - t } { K _ { c } } \\rceil } \\beta _ { t } \\sqrt { N } C _ { \\delta } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            343,
+            669,
+            387
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\omega _ { 0 } : = [ \\omega ^ { 1 } , \\cdot \\cdot \\cdot , \\omega ^ { N } ] ^ { T } , Q : = I - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ { T } , k ^ { \\prime } : = \\lfloor \\frac { k } { K _ { c } } \\rfloor * K _ { c } } \\end{array}$ . The constant $\\nu \\in ( 0 , 1 )$ is the second largest singular value of $W$ ： ",
+        "bbox": [
+            165,
+            392,
+            820,
+            426
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Based on Lemma 1 and follow the step size rule of Theorem 2,it is possible to show $\\| Q \\omega _ { k + 1 } \\| _ { F } ^ { 2 } =$ $\\begin{array} { r } { \\sum _ { i = 1 } ^ { N } \\| \\omega _ { k + 1 } ^ { i } - \\bar { \\omega } _ { k + 1 } \\| ^ { 2 } = \\mathcal { O } ( K _ { c } ^ { 2 } \\beta _ { k } ^ { 2 } ) } \\end{array}$ Let $K _ { c } = \\mathcal { O } ( \\beta _ { k } ^ { - \\frac 1 2 } )$ , we have $\\| Q \\omega _ { k + 1 } \\| _ { F } ^ { 2 } = \\mathcal { O } ( \\beta _ { k } )$ ，which maintains the optimal rate. ",
+        "bbox": [
+            169,
+            435,
+            826,
+            486
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To analyze the second term in (12), we first construct the following Lyapunov function ",
+        "bbox": [
+            168,
+            489,
+            745,
+            506
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/610f3d8974dd34ffa23094cb4d2d38bccf0067588d74e7dc31506768ecda5a65.jpg",
+        "text": "$$\n\\mathbb { V } _ { k } : = - J ( \\theta _ { k } ) + \\| \\bar { \\omega } _ { k } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } + \\| \\bar { \\lambda } _ { k } - \\lambda ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            511,
+            679,
+            530
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "281 Then, it remains to derive an approximate descent property of the term $\\lVert \\bar { \\boldsymbol { \\omega } } _ { k } - \\boldsymbol { \\omega } ^ { * } ( \\boldsymbol { \\theta } _ { k } ) \\rVert ^ { 2 }$ in (13).   \n282 Towards that end, our key step lies in establishing the smoothness of the optimal critic variables   \n283 shown in the next lemma. ",
+        "bbox": [
+            140,
+            544,
+            826,
+            587
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "i4Lemma 2 (smoothness of optimal critic). Suppose Assumptions 1-3 hold, under the update of 15Algorithm $\\boldsymbol { l }$ , there exists a positive constant $L _ { \\mu , 1 }$ such that for all $\\theta , \\theta ^ { \\prime }$ , it holds that ",
+        "bbox": [
+            151,
+            590,
+            825,
+            621
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7ffb63ac40a9efe4533d63bede6d5960dd6334388dd83363078d235665b8088c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| \\nabla \\omega ^ { * } ( \\theta ) - \\nabla \\omega ^ { * } ( \\theta ^ { \\prime } ) \\| \\leq L _ { \\mu , 1 } \\| \\theta - \\theta ^ { \\prime } \\| , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            626,
+            629,
+            643
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "36where $\\nabla \\omega ^ { * } ( \\theta )$ denotes the Jacobian of $\\omega ^ { \\ast } ( \\theta )$ with respect to $\\theta$ ",
+        "bbox": [
+            150,
+            648,
+            589,
+            666
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This smoothnessproperty isssentialforachievingour $\\tilde { \\mathcal { O } } ( 1 / \\sqrt { K } )$ convergence rate. ",
+        "bbox": [
+            169,
+            676,
+            728,
+            694
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge,the smoothness of $\\omega ^ { \\ast } ( \\theta )$ has not been justified in the literature.   \nEquipped with Lemma 2,we are able to establish the following lemma. ",
+        "bbox": [
+            165,
+            698,
+            826,
+            728
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "90Lemma 3 (Error of critic). Under Assumptions 1-5, consider the update of Algorithm 1. Then, it   \n91holds that ",
+        "bbox": [
+            150,
+            731,
+            825,
+            760
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d41d4b68b08b6ee5898a7839f9243bf48b8c9223f7c6a98961769e0df5a4cdee.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathbb { E } [ \\| \\bar { \\omega } _ { k + 1 } - \\omega ^ { * } ( \\theta _ { k + 1 } ) \\| ^ { 2 } ] \\le ( 1 + C _ { 9 } \\alpha _ { k } ) \\| \\bar { \\omega } _ { k + 1 } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } } \\\\ & { \\qquad + \\frac { \\alpha _ { k } } { 4 } \\displaystyle \\sum _ { i = 1 } ^ { N } \\| \\mathbb { E } [ g _ { a } ^ { i } ( \\xi _ { k } , \\omega _ { k + 1 } ^ { i } , \\lambda _ { k + 1 } ^ { i } ) ] \\| ^ { 2 } + { \\mathcal O } ( \\alpha _ { k } ^ { 2 } ) . } \\\\ & { \\displaystyle \\mathbb { E } [ \\| \\bar { \\omega } _ { k + 1 } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } ] \\le ( 1 - 2 \\lambda _ { \\phi } \\beta _ { k } ) \\| \\bar { \\omega } _ { k } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } } \\\\ & { \\qquad + C _ { K _ { 1 } } \\beta _ { k } \\beta _ { k - Z _ { K } } + C _ { K _ { 2 } } \\alpha _ { k - Z _ { K } } \\beta _ { k } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            763,
+            745,
+            868
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "292Here, $Z _ { K } : = \\operatorname* { m i n } \\{ z \\in \\mathbb { N } ^ { + } | \\kappa \\rho ^ { z - 1 } \\le \\operatorname* { m i n } \\{ \\alpha _ { k } , \\beta _ { k } , \\eta _ { k } \\} \\} ,$ $C _ { 9 }$ ， $\\lambda _ { \\phi }$ are constants specified in appendix,   \n293and $C _ { K _ { 1 } }$ and $C _ { K _ { 2 } }$ are of order ${ \\mathcal { O } } ( \\log ( K ) )$ and $\\mathcal { O } ( \\log ^ { 2 } ( K ) ) d$ respectively. ",
+        "bbox": [
+            137,
+            880,
+            830,
+            914
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/51690de4334a89787cd8df9f79b77112e0b40cb6ac372603b25d483fba3a08f5.jpg",
+        "image_caption": [
+            "Figure 1: Averaged reward versus sample complexity and communication complexity. The vertical axis is the averaged reward over all the agents. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            108,
+            797,
+            270
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "294Plug (15) into (14), we can establish the approximate descent property of $\\| \\bar { \\omega } _ { k } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 }$ in (13): ",
+        "bbox": [
+            140,
+            332,
+            815,
+            348
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5aa66d81d53d48fd30a1131032bf0f8325b2113adfb2d414728dae438b927946.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\mathbb { E } [ \\| \\bar { \\omega } _ { k + 1 } - \\omega ^ { * } ( \\theta _ { k + 1 } ) \\| ^ { 2 } ] \\leq ( 1 + C _ { 9 } \\alpha _ { k } ) ( 1 - 2 \\lambda _ { \\phi } \\beta _ { k } ) \\| \\bar { \\omega } _ { k } - \\omega ^ { * } ( \\theta _ { k } ) \\| ^ { 2 } } } \\\\ & { } & { \\quad + \\frac { \\alpha _ { k } } { 4 } \\displaystyle \\sum _ { i = 1 } ^ { N } \\| \\mathbb { E } [ g _ { a } ^ { i } ( \\xi _ { k } , \\omega _ { k + 1 } ^ { i } , \\lambda _ { k + 1 } ^ { i } ) ] \\| ^ { 2 } } \\\\ & { } & { \\quad + { \\mathcal O } ( C _ { K _ { 1 } } \\beta _ { k } \\beta _ { k - Z _ { K } } + C _ { K _ { 2 } } \\alpha _ { k - Z _ { K } } \\beta _ { k } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            352,
+            728,
+            438
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5Finally, plugging (11), (14),and (16) into (13) gives the ascent of the Lyapunov function, which leads   \n6to our convergence result through steps of standard arguments. ",
+        "bbox": [
+            151,
+            440,
+            825,
+            469
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "，5Numerical results",
+        "text_level": 1,
+        "bbox": [
+            156,
+            488,
+            354,
+            506
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, our objective is to illustrate the empirical sample complexity and communication complexity of the proposed algorithms. We also implement the algorithm in [6] to serve as a baseline, which employs double-loop algorithmic framework. Our simulation is based on the grounded communication environment proposed in [19]; see Appendix A for detailed set up. Through the discussion, we refer the algorithm in [6] as \"DLDAC\",the Algorithm 1 as \"SDAC-re\",the Algorithm 2 as \"SDAC-noisy\" (see Appendix B). We also provide the result which assumes full reward is available to serve as baseline, which we refer as \"SDAC-full\". We set $K _ { r } = 5$ for \"SDAC-noisy\"; $K _ { c } = 1$ for \"SDAC-re\", \"SDAC-noisy\", and \"SDAC-full\". We choose $T _ { c } = 5$ (loop size), $T _ { c } ^ { \\prime } = 1$ (critic consensus number every iteration), $T ^ { \\prime } = 5$ (reward consensus number every iteration) for \"DLDAC\". ",
+        "bbox": [
+            169,
+            518,
+            826,
+            645
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "307 The sample complexity and communication complexity are shown in Figure 1. The results are   \n308 averaged over 10 Monte Carlo runs.As we can see,the proposed two algorithms achieve significantly   \n309 higher reward than \"DLDAC\" in terms of both sample complexity and communication complexity.   \n310 Moreover,their performances approach the baseline “SDAC-full, where the global reward is assumed   \n311 to be available, indicating that the reward approximation is nearly accurate. Due to space limit, we   \n312 will put additional experiments on the comparison with existing decentralized AC algorithms and the   \n313 ablation study of hyper-parameters to Appendix A. ",
+        "bbox": [
+            140,
+            650,
+            825,
+            748
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3146Conclusion and future direction ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            767,
+            472,
+            785
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "315 In this paper, we studied the convergence of fully decentralized AC algorithm under practical single  \n316 timescale update for the first time. We designed such an algorithm which maintains the optimal   \n317 sample complexity of $\\widetilde { \\mathcal { O } } ( \\varepsilon ^ { - 2 } )$ under less communications. We also proposed a variant to preserve the   \n318 privacy of local actions by communicating noisy rewards. Extensive simulation results demonstrate   \n319 the superiority of our algorithms’ empirical performance over existing decentralized AC algorithms.   \n320 One limitation of our work is that we only study the convergence to stationary point. Thus, we leave   \n321 the research on the avoidance of saddle points and convergence to global optimum as promising   \n322 future directions. ",
+        "bbox": [
+            142,
+            797,
+            826,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3 References ",
+        "text_level": 1,
+        "bbox": [
+            153,
+            89,
+            267,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "324 [1] A. Agarwal, S. M. Kakade, J. D. Lee, and G. Mahajan. Optimality and approximation with   \n325 policy gradient methods in markov decision processes. In Conference on Learning Theory   \n326 (COLT), pages 64-66,2020.   \n327 [2] J. Baxter and P.L. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial   \n328 Intelligence Research,15:319-350,2001.   \n329 [3] J. Bhandari, D.Russo,and R. Singal. A finite time analysis of temporal difference learning with   \n330 linear function approximation. In Conference on Learning Theory (COLT), pages 1691-1692,   \n331 2018.   \n332 [4] T. Chen, Y. Sun, and W. Yin. Closing the gap: Tighter analysis of alternating stochastic gradient   \n333 methods for bilevel problems. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan,   \n334 editors,Advances in Neural Information Processing Systems,2021.   \n335 [5] Z. Chen, Y.Zhou,and R.Chen. Multi-agent off-policy td learning: Finite-time analysis with near  \n336 optimal sample complexity and communication complexity. arXiv preprint arXiv:2103.13147,   \n337 2021.   \n338 [6] Z. Chen, Y. Zhou, R.-R. Chen, and S. Zou. Sample and communication-effcient decentralized   \n339 actor-critic algorithms with finite-time analysis. In International Conference on Machine   \n340 Learning, pages 3794-3834. PMLR, 2022.   \n341 [7] K.Doya. Reinforcement learning in continuous time and space. Neural Computation,12(1):219-   \n342 245,2000.   \n343 [8]L.Espeholt, H.Soyer,R. Munos, K. Simonyan, V. Mnih,T. Ward, Y.Doron, V.Firoiu,T.Harley,   \n344 I. Dunning, et al. Impala: Scalable distributed deep-rl with importance weighted actor-learner   \n345 architectures. In International Conference on Machine Learning, pages 1407-1416. PMLR,   \n346 2018.   \n347 [9] Z. Fu, Z. Yang, and Z. Wang. Single-timescale actor-critic provably finds globally optimal   \n348 policy. In International Conference on Learning Representations, 2021.   \n349[10] H. Guo,Z.Fu, Z. Yang,and Z. Wang.Decentralized single-timescale actor-critic on zero  \n350 sum two-player stochastic games. In International Conference on Machine Learning, pages   \n351 3899-3909.PMLR,2021.   \n352[11] F. Hairi,J.Liu,and S.Lu. Finite-time convergence and sample complexity of multi-agent actor  \n353 critic reinforcement learning with average reward. In International Conference on Learning   \n354 Representations,2022.   \n355[12] S. M. Kakade. A natural policy gradient. In Proc. Advances in Neural Information Processing   \n356 Systems (NIPS), pages 1531-1538,2002.   \n357 [13] V. R. Konda and V. S. Borkar. Actor-critic-type learning algorithms for Markov decision   \n358 processes. SIAM Journal on Control and Optimization,38(1):94-123,1999.   \n359[14] T.P. Lillicrap,J. J. Hunt, A. Pritzel, N. Heess,T. Erez, Y. Tassa,D. Silver, and D.Wierstra.   \n360 Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.   \n361 [15] Y.Lin, K. Zhang, Z. Yang,Z. Wang,T. Basar, R. Sandhu,and J.Liu. A communication-efficient   \n362 multi-agent actor-critic algorithm for distributed reinforcement learning. In 2019 IEEE 58th   \n363 Conference on Decision and Control (CDC), pages 5562-5567, 2019.   \n364 [16] M. L. Littman. Markov games as a framework for multi-agent reinforcement learning. In   \n365 Machine learning proceedings 1994, pages 157-163. Elsevier, 1994.   \n366[17] Y. Liu, K. Zhang, T. Basar, and W. Yin. An improved analysis of (variance-reduced) policy   \n367 gradient and natural policy gradient methods. Advances in Neural Information Procesing   \n368 Systems,33:7624-7636,2020.   \n369 [18] R. Lowe, Y.I. Wu, A. Tamar,J. Harb, O.Pieter Abbeel, and I. Mordatch. Multi-agent actor-critic   \n370 for mixed cooperative-competitive environments. Advances in neural information processing   \n371 systems,30,2017.   \n372 [19] I. Mordatch and P. Abbeel. Emergence of grounded compositional language in multi-agent   \n373 populations. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.   \n374 [20] S. Omidshafiei, J.Pazis, C. Amato, J. P.How,and J. Vian. Deep decentralized multi-task   \n375 multi-agent reinforcement learning under partial observability. In International Conference on   \n376 Machine Learning, pages 2681-2690. PMLR,2017.   \n377 [21] S. Qiu, Z. Yang, J. Ye,and Z. Wang. On the finite-time convergence of actor-critic algorithm.   \n378 In Optimization Foundations for Reinforcement Learning Workshop at Advances in Neural   \n379 Information Processing Systems (NeurIPS), 2019.   \n380 [22] T. Rashid, M. Samvelyan, C.Schroeder, G.Farquhar,J.Foerster,and S.Whiteson. Qmix: Mono  \n381 tonic value function factorisation for deep multi-agent reinforcement learning. In International   \n382 Conference on Machine Learning, pages 4295-4304. PMLR, 2018.   \n383 [23] J. Schulman,F. Wolski,P. Dhariwal, A. Radford,and O. Klimov. Proximal policy optimization   \n384 algorithms. arXiv preprint arXiv:1707.06347, 2017.   \n385 [24] H. Shen,K. Zhang, M. Hong, and T. Chen.Asynchronous advantage actor critic: Non  \n386 asymptotic analysis and linear speedup. ArXiv:20l2.15511, 2020.   \n387[25] D. Silver,J. Schritwieser, K. Simonyan,I. Antonoglou, A. Huang,A. Guez,T. Hubert,L.Baker,   \n388 M.Lai, A. Bolton, et al. Mastering the game of go without human knowledge. nature,   \n389 550(7676):354-359, 2017.   \n390[26] K. Son, D. Kim, W. J. Kang, D. E. Hostallero, and Y. Yi. Qtran: Learning to factorize with   \n391 transformation for cooperative multi-agent reinforcement learning. In International Conference   \n392 on Machine Learning, pages 5887-5896. PMLR, 2019.   \n393 [27] J. Sun, G. Wang, G. B. Giannakis, Q. Yang,and Z. Yang. Finite-sample analysis of decentral  \n394 ized temporal-difference learning with linear function approximation. In Proc. International   \n395 Conference on Artificial Intelligence and Statistics (AISTATS), pages 4485-4495, 2020.   \n396 [28] R. S. Sutton, D. A. McAllester, S. P. Singh,and Y. Mansour. Policy gradient methods for   \n397 reinforcement learning with function approximation. In Proc. Advances in Neural Information   \n398 Processing Systems (NIPS), pages 1057-1063,2000.   \n399 [29] J. N. Tsitsiklis and B. Van Roy. Analysis of temporal-difference learning with function   \n100 approximation. In Advances in neural information processing systems (NIPS), pages 1075-   \n101 1081, 1997.   \n102 [30] O. Vinyals,I. Babuschkin, W. M. Czarnecki,M. Mathieu, A. Dudzik, J. Chung, D. H. Choi,   \n103 R.Powell, T. Ewalds, P. Georgiev, et al. Grandmaster level in starcraft ii using multi-agent   \n104 reinforcement learning. Nature,575(7782):350-354, 2019.   \n105 [31] Y. Wang, S. Zou, and Y. Zhou. Non-asymptotic analysis for two time-scale TDC with general   \n106 smooth function approximation. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and   \n107 J. W. Vaughan,editors,Advances in Neural Information Processing Systems,voume34,ges   \n108 9747-9758. Curran Associates, Inc., 2021.   \n109 [32] Y. F. Wu, W. Zhang,P. Xu, and Q. Gu. A finite-time analysis of two time-scale actor-critic   \n10 methods. Advances in Neural Information Processing Systems, 33:17617-17628,2020.   \n11 [33] P. Xu, F. Gao, and Q. Gu. An improved convergence analysis of stochastic variance-reduced   \n112 policy gradient. In Proc. International Conference on Uncertainty in Artificial Intelligence   \n13 (UAI),2019.   \nI14[34] T. Xu and Y. Liang. Sample complexity bounds for two timescale value-based reinforcement   \n115 learning algorithms. In International Conference on Artificial Intelligence and Statistics, pages   \n116 811-819. PMLR, 2021.   \n417 [35] T. Xu, Z. Wang, and Y. Liang. Improving sample complexity bounds for (natural) actor-critic   \n418 algorithms. In Proc.Advances in Neural Information Processing Systems (NeurIPS),volume 33,   \n419 2020.   \n420 [36] T. Xu, Z. Yang, Z. Wang,and Y. Liang. Doubly robust off-policy actor-critic: Convergence and   \n421 optimality. ArXiv:2102.11866,2021.   \n422 [37] C. Yu, X. Wang, X. Xu, M. Zhang, H. Ge, J. Ren,L. Sun, B. Chen, and G. Tan. Distributed   \n423 multiagent coordinated learning for autonomous driving in highways based on dynamic co  \n424 ordination graphs. IEEE Transactions on Intelligent Transportation Systems, 21(2):735-748,   \n425 2019.   \n426 [38] S. Zeng, T. Chen, A. Garcia, and M. Hong. Learning to coordinate in multi-agent systems: A   \n427 coordinated actor-critic algorithm and finite-time guarantees. arXiv preprint arXiv:2110.05597,   \n428 2021.   \n429 [39] H. Zhang, W. Chen, Z. Huang, M. Li, Y. Yang, W. Zhang, and J. Wang. Bi-level actor-critic   \n430 for multi-agent coordination. In Proceedings of the AAAI Conference on Artificial Intelligence,   \n431 volume 34, pages 7325-7332, 2020.   \n432 [40] K. Zhang, A. Koppel, H. Zhu,and T. Basar. Global convergence of policy gradient methods to   \n433 (almost) locally optimal policies. arXiv preprint arXiv:1906.08383,2019.   \n434 [41] K. Zhang, Z. Yang,and T. Basar. Multi-agent reinforcement learning: A selective overview of   \n435 theories and algorithms. Handbook of Reinforcement Learning and Control, pages 321-384,   \n436 2021.   \n[42] K. Zhang, Z. Yang, H. Liu,T. Zhang,and T. Basar.Fully decentralized multi-agent reinforcement   \nlearning with networked agents. In International Conference on Machine Learning, pages   \n5872-5881. PMLR, 2018. ",
+        "bbox": [
+            143,
+            99,
+            828,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            84,
+            830,
+            919
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            93,
+            828,
+            481
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            502,
+            254,
+            518
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors.. ",
+        "bbox": [
+            214,
+            530,
+            339,
+            544
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work?[Yes] The limitation is written in an equivalent form as future works in the conclusion section; see Section 6.   \n(c） Did you discuss any potential negative societal impacts of your work? [N/A] We conduct research about the design and analysis of the fundamental actor-critic algorithm, which should not bring any negative societal impact.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?[Yes] ",
+        "bbox": [
+            238,
+            547,
+            825,
+            680
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            684,
+            493,
+            699
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the fullset of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] ",
+        "bbox": [
+            236,
+            703,
+            733,
+            734
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiment.. ",
+        "bbox": [
+            212,
+            738,
+            393,
+            752
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(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]   \n(b) Did you specify all the training details (e.g. data splits, hyperparameters, how they were chosen)? [Yes]   \n(c） Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(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] ",
+        "bbox": [
+            238,
+            756,
+            826,
+            875
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": ". If you are using existing assets (e.g., code,data, models) or curating/releasing new asets... ",
+        "bbox": [
+            220,
+            878,
+            823,
+            893
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] ",
+        "bbox": [
+            235,
+            897,
+            694,
+            911
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(b) Did you mention the license of the assets? [Yes]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   \n(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes]   \n(e） Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            202,
+            90,
+            825,
+            196
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects.. ",
+        "bbox": [
+            214,
+            200,
+            705,
+            215
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the ful text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b） Did you describe any potential participant risks,with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c） Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            236,
+            219,
+            825,
+            309
+        ],
+        "page_idx": 12
+    }
+]

parse/dev/YQA28p7qNz/YQA28p7qNz.md

@@ -0,0 +1,233 @@
+# 3D-LLM: Injecting the 3D World into Large Language Models
+
+Yining Hong University of California, Los Angeles
+
+Haoyu Zhen Shanghai Jiao Tong University
+
+Peihao Chen South China University of Technology
+
+Shuhong Zheng University of Illinois Urbana-Champaign
+
+Yilun Du Massachusetts Institute of Technology
+
+Zhenfang Chen MIT-IBM Watson AI Lab
+
+Chuang Gan UMass Amherst and MIT-IBM Watson AI Lab
+
+# Abstract
+
+Large language models (LLMs) and Vision-Language Models (VLMs) have been proven to excel at multiple tasks, such as commonsense reasoning. Powerful as these models can be, they are not grounded in the 3D physical world, which involves richer concepts such as spatial relationships, affordances, physics, layout, and so on. In this work, we propose to inject the 3D world into large language models and introduce a whole new family of 3D-LLMs. Specifically, 3D-LLMs can take 3D point clouds and their features as input and perform a diverse set of 3D-related tasks, including captioning, dense captioning, 3D question answering, task decomposition, 3D grounding, 3D-assisted dialog, navigation, and so on. Using three types of prompting mechanisms that we design, we are able to collect over 1M 3D-language data covering these tasks. To efficiently train 3D-LLMs, we first utilize a 3D feature extractor that obtains 3D features from rendered multi-view images. Then, we use 2D VLMs as our backbones to train our 3D-LLMs. By introducing a 3D localization mechanism, 3D-LLMs can better capture 3D spatial information. Experiments on held-out evaluation dataset, ScanQA, SQA3D and 3DMV-VQA, outperform state-of-the-art baselines. In particular, experiments on ScanQA show that our model outperforms state-of-the-art baselines by a large margin (e.g., the BLEU-1 score surpasses state-of-the-art score by $9 \%$ ). Furthermore, experiments on our held-in datasets for 3D captioning, task composition, and 3D-assisted dialogue show that our model outperforms 2D VLMs. Qualitative examples also show that our model could perform more tasks beyond the scope of existing LLMs and VLMs. Project Page: : https://vis-www.cs.umass.edu/3dllm/.
+
+# 1 Introduction
+
+In the past several years, we have witnessed a surge of large language models (LLMs) (e.g., GPT4 [33]) that excel at multiple tasks, such as communication and commonsense reasoning. Recent works have explored aligning images and videos with LLM for a new generation of multi-modal LLMs (e.g., Flamingo [15], BLIP-2 [29]) that equip LLMs with the ability to understand and reason about 2D images. However, as powerful as the models can be in communication and reasoning, they are not grounded in the real 3D physical world, which involves richer concepts such as spatial relationships, affordances, physics and interaction so on. Therefore, such LLMs pale in comparison with the robots depicted in sci-fi movies - the assistants that could understand the 3D environments, as well as perform reasoning and planning based on the 3D understandings.
+
+![](images/3a27a47ae0b5f4af284acd63d3e7627dd933189016401b7dee6c599162ed8e4d.jpg)  
+Figure 1: Examples from our generated 3D-language data, which covers multiple 3D-related tasks.
+
+To this end, we propose to inject the 3D world into large language models, and introduce a whole new family of 3D-LLMs that could take 3D representations (i.e., 3D point clouds with their features) as input, and perform a series of 3D-related tasks. By taking the 3D representations of scenes as input, LLMs are blessed with twofold advantages: (1) long-term memories about the entire scene can be stored in the holistic 3D representations, instead of episodic partial-view observations. (2) 3D properties such as affordances and spatial relationships can be reasoned from 3D representations, far beyond the scope of language-based or 2D image-based LLMs.
+
+One major challenge of training the proposed 3D-LLMs lies in data acquisition. Unlike the vast amount of paired 2D-images-and-text data on the Internet, the scarcity of 3D data hinders the development of 3D-based foundation models. 3D data paired with language descriptions are even harder to obtain. To address this, we propose a set of unique data generation pipelines that could generate large-scale 3D data paired with language. Specifically, we make use of ChatGPT [33] and devise three efficient prompting procedures for communication between 3D data and language. In this way, we are able to obtain approximately one million 3D-language data covering a diverse set of tasks, including but not limited to 3D captioning, dense captioning, 3D question answering, 3D task decomposition, 3D grounding, 3D-assisted dialog, navigation and so on, as shown in Figure 1.
+
+The next challenge resides in how to obtain meaningful 3D features that could align with language features for 3D-LLMs. One way is to train 3D encoders from scratch using a similar contrastivelearning paradigm for the alignment between 2D images and language (e.g., CLIP [36]). However, this paradigm consumes tremendous data, time, and GPU resources. From another perspective, there are numerous recent works that build 3D features from 2D multi-view images (e.g., concept fusion [24], 3D-CLR [20]). Inspired by this, we also utilize a 3D feature extractor that constructs 3D features from the 2D pretrained features of rendered multi-view images. Recently, there are also quite a few visual-language models (e.g., BLIP-2 [29], Flamingo [15]) utilizing the 2D pretrained CLIP features for training their VLMs. Since our extracted 3D features are mapped to the same feature space as 2D pretrained features, we can seamlessly use 2D VLMs as our backbones and input the 3D features for the efficient training of 3D-LLMs.
+
+One crucial aspect of 3D-LLMs, different from vanilla LLMs and 2D VLMs, is that 3D-LLMs are expected to have an underlying 3D spatial sense of information. Thus, we develop a 3D localization mechanism that bridges the gap between language and spatial locations. Specifically, we append 3D position embeddings to the extracted 3D features to better encode spatial information. In addition, we append a series of location tokens to the 3D-LLMs, and localization can be trained via outputting location tokens given the language descriptions of specific objects in the scenes. In this way, 3D-LLMs could better capture 3D spatial information.
+
+To sum up, our paper has the following contributions:
+
+• We introduce a new family of 3D-based Large Language models (3D-LLMs) that can take 3D points with features and language prompts as input, and perform a variety of 3D-related tasks. We focus on tasks beyond the scope of vanilla LLMs or 2D-LLMs, such as tasks about holistic scene understanding, 3D spatial relationships, affordances and 3D planning.   
+• We devise novel data collection pipelines that could generate large-scale 3D-language data. Based on the pipelines, we collect a dataset that has over 1M 3D-language data that cover a diverse set of 3D-related tasks, including but not limited to 3D captioning, dense captioning, 3D question answering, task decomposition, 3D grounding, 3D-assisted dialog, navigation, and so on. • We use a 3D feature extractor that extracts meaningful 3D features from rendered multi-view images. We utilize 2D pretrained VLMs as our backbones for efficient training. We introduce a 3D localization mechanism for training the 3D-LLMs to better capture 3D spatial information. • Experiments on held-out evaluation dataset, ScanQA, SQA3D and 3DMV-VQA, outperform stateof-the-art baselines. In particular, 3D LLMs outperform baselines by a large margin on ScanQA (e.g., $9 \%$ for BLEU-1 and $10 \%$ for CIDER). Experiments on held-in datasets for 3D captioning, task composition, and 3D-assisted dialogue show that our model outperforms 2D VLMs. Qualitative studies further demonstrate that our model is able to handle a diverse set of tasks.   
+• We release our 3D-LLMs, the 3D-language dataset, and language-aligned 3D features of the dataset for future research development 1.
+
+# 2 Related Works
+
+Large Language Models. Our work is closely related to large language models [4, 14, 37, 10, 34] (LLMs) like GPT-3 [4] and PaLM [10], which are able to handle different language tasks with a single model and show strong generalization abilities. These models are typically trained on massive textual data with self-supervised training targets like predicting the next tokens [4, 37] or reconstructing the masked tokens [14, 38]. To better align these LLMs’ predictions to human instructions, improve the models’ generalization abilities on unseen tasks, a series of instruction tuning methods [35, 42] and datasets [11, 13] have been proposed. In this work, we aim to inject the 3D world into large language models, understanding rich 3D concepts such as spatial relations, affordances, and physics.
+
+Vision-Language Pre-trained Models. Our work is also related to vision-language pre-trained models that connect images and natural language [30, 31, 18, 36, 25]. Some research [36, 25] learn to train models from scratch with massive image-language pairs and apply them to downstream tasks like visual question answering [19, 47], captioning [7], and referring expression comprehension [46] with finetuning. Other researchers have connected pre-trained vision models and pre-trained LLMs
+
+# Box-Demonstration-Instruction based Prompting
+
+# Bounding-Box Context:
+
+Room 1: table: [0.4, 0.1, 0.1], chair: [0.4, 0.3, 0.4]… Room2: desk: [0.5, 0.2, 0.3] Instruction (Shortened Version): You are an AI visual assistant / conversation generator / question generator in the room. You need to generate a high-level task and its low-level actions/ 4-10 round conversations between a human and a robot assistant / question-answer pairs about the scene.
+
+# Demonstration: For Room 1:
+
+<Task Decomposition> High-Level Task: make up. Low-Level Actions: 1. go to the cabinet 2. take out cosmetics from the cabinet 3. bring the cosmetics to the table with mirror 4. open the lamp 5. make up <3D-Assisted Dialog> Human: Bring me a toothbrush. Robot: Sure! Where is it?. Human: Maybe in the sink. Robot: I have arrived mat the sink and find no toothbrush. I found an closed bathroom cabinet. Is it possible to contain what you need?... <QA> Question: Is there a table between the chair and the tv? Answer: Yes. Generate similar responses for Room 2.
+
+# Response : For Room 2:
+
+<Task Decomposition> High-Level Task: Get ready for work. Low-Level Actions: 1. go to the toilet 2. Wash hands in the sink cabine. 3. Gog to the mirror 4. Brush teeth and wash face 5. Turn on the lamp 6. Leave the room through the door <3D-Assisted Dialog> Human: Turn on the light. Robot: Sure! Which light?. Human: The wall lamp. Robot: I have turned it on. Is it bright enough? Human: No. Are there other lights? Robot: I see a desk lamp.
+
+![](images/2d8067fe9ace609f15e8f9852ad7489400bc9d0782906893d5f97d8c4ae5d1b0.jpg)
+
+![](images/303393e5d97603b76d23b93708f9b4766ccf92f118e128a0eedb3e68fd4b6013.jpg)  
+Figure 2: 3D-language data generation pipelines.
+
+with additional learnable neural modules like perceiver [2] and QFormers [30], leveraging perception abilities in pre-trained vision models, and reasoning and generalization capacities in LLMs. Inspired by these previous works, we plan to build an AI assistant that could understand the 3D world and perform corresponding 3D reasoning and planning. This is not trivial and we need to overcome obstacles like how to handle the problem of data sparsity, how to align the 3D world with 2D images, and how to capture 3D spatial information.
+
+3D & Language. Another line of research that is similar to ours is 3D and language [5, 45, 8, 20, 1, 16, 22, 45, 3]. ScanQA [45] requires a model to answer questions related to the 3D world; ScanRefer [5] asks a model to localize a region that the text expression refer to; 3D captioning [8] tests models’ abilities to generate captions describing the 3D scenes. However, these 3D tasks and their corresponding models are usually task-specific and could only handle cases within the same distribution of the training sets without generalization. Different from them, we aim to build a 3D model that could handle different tasks at the same time and enable new abilities like 3D-assistant dialog and task decomposition.
+
+# 3 3D-Language Data Generation
+
+The community has witnessed the proliferation of multi-modal data thanks to easy access to a tremendous amount of 2D image and text pairs on the internet. However, when it comes to 3D-related data, obtaining multimodal resource is not easy, due to not only the scarcity of 3D assets, but also the difficulty of providing language data for 3D assets. There are some existing datasets that contain 3D-language data (e.g., ScanQA [45], ScanRefer [5]). However, they are limited with regard to both quantity and diversity, restricted to only one task per dataset. How to generate a 3D-language dataset that can be utilized for all kinds of 3D-related tasks is well worth delving into.
+
+Inspired by the recent success of large language models like GPT [33], we propose to leverage such models for 3D-language data collection. Specifically, as shown in Figure 2, we have three ways to prompt a text-only GPT for generating data. 1) boxes-demonstration-instruction based prompting. We input the axis-aligned bounding boxes (AABB) of both the rooms and the objects in the 3D scenes, providing information about the semantics and spatial locations of the scene. We then provide specific instructions to the GPT model to generate diverse data. We give 0-3 few-shot demonstration examples of the GPT model showing what kind of data it is instructed to generate. 2) ChatCaptioner based prompting. We utilize techniques similar to [48], in which ChatGPT is prompted to ask a series of informative questions about an image and BLIP-2 [29] answers the questions. In order to collect 3D-related data, we first sample several images from different views of a 3D scene. These images are fed into ChatGPT and BLIP-2 to get the caption of each image. We then leverage ChatGPT to summarize all these captions, which contain information about different regions, to form a global 3D description of the entire scene. 3) Revision based prompting. It can be used to transfer one type of 3D data to another.
+
+![](images/f4888587452207b2b2eccbf38248389e715b2a2c61af78659527a15e3d47ef08.jpg)  
+Figure 3: Overview of our 3D-LLM framework. The first two columns show our 3D feature extractor. We first render a few multi-view images from the 3D scene, extract 2D dense features, and then construct 3D features from these multi-view images using three kinds of methods. And then, the 3D features and input language prompts are input to the 3D-LLMs to generate responses.
+
+Given the prompting pipelines, GPT is able to generate various types of 3D-language data as summarized in Figure 1. More data generation details and prompt designs are shown in the Appendix.
+
+We mainly establish our 3D-language dataset upon several 3D assets:
+
+• Objaverse is a universe of 800K 3D objects. However, since the language descriptions were extracted from online sources and not examined by humans, most objects have very noisy descriptions (e.g., with urls) or no descriptions. We utilize ChatCaptioner based prompting to generate high-quality 3D-related descriptions for the scenes and reivison-based prompting to generate questions. • Scannet [12] is a richly-annotated dataset of approximately 1k 3D indoor scenes. It provides semantics and bounding boxes of the objects in the scenes.   
+• Habitat-Matterport (HM3D) [39] is a dataset of 3D environments of embodied AI. HM3DSem [44] further adds semantic annotations and bounding boxes for more than 200 scenes of HM3D. We use the pre-segmented rooms of HM3D in 3D-CLR [20].
+
+# 4 3D-LLM
+
+# 4.1 Overview
+
+In this section, we introduce how we train our 3D-LLMs. We argue that it’s hard to train 3D-LLMs from scratch, since our collected 3D-language dataset is still not the size of billion-scale imagelanguage dataset used to train 2D VLMs. Furthermore, for 3D scenes, there are no available pretrained encoders like those for 2D images (e.g., CLIP ViT encoders). Thus, retraining 3D-language models from scratch is data-inefficient and resource-heavy. Recently, researchers have proposed to extract 3D features from 2D multi-view images [24, 20]. Using these alignment methods, we could use pretrained image encoders to extract image features, and then map the features to the 3D data. Since the pretrained image features serve as inputs to 2D VLMs, the mapped 3d features of the same feature space can also be seamlessly fed into the pretrained 2D VLMs, which we use as our backbones to train 3D-LLMs. We also propose a 3D localization mechanism to boost the model’s ability to capture 3D spatial information. Figure 3 shows our framework.
+
+# 4.2 3D Feature Extractor
+
+The first step of training 3D-LLMs is to build meaningful 3D features that could be aligned with language features. For 2D images, there exist feature extractors like CLIP, which learn visual models from language supervision. The models are pretrained using billion-scale internet data of imagelanguage pairs. It’s hard to pre-train such feature learners from scratch, since there are no 3D-language assets comparable to internet-scale image-language pairs in terms of quantity and diversity.
+
+On the contrary, numerous methods have been proposed to extract 3D features from 2D multi-view images [24, 20, 17, 21]. Inspired by these works, we extract features for 3D points by rendering the 3D scenes in several different views, and construct 3D features from rendered image features.
+
+We first extract pixel-aligned dense features for rendered images following [24]. Then, we utilize three methods to construct 3D features from rendered image features. These methods are designed for different types of 3D data.
+
+• Direct Reconstruction. We directly reconstruct point cloud from rgbd images rendered from the 3D data using ground-truth camera matrixes. The features are directly mapped to the reconstructed 3D points. This method is suitable for rendered rgbd data with perfect camera poses and intrinsics. • Feature Fusion. Similar to [24], we fuse 2D features into 3D maps using gradslam [27]. Different from dense mapping methods, the features are fused in addition to depths and colors. This method is suitable for 3D data with noisy depth map renderings, or noisy camera poses and intrinsics. • Neural Field. We utilize [20], which constructs 3D compact representation using neural voxel field [40]. Specifically, each voxel in the field has a feature in addition to density and color. Then we align 3D features in the rays and 2D features in the pixels using MSE loss. This method is for 3D data with RGB renderings but no depth data, and noisy camera poses and intrinsics.
+
+In this way, we are able to obtain the $< N$ , $\mathcal { D } _ { v } >$ -dim 3D features of each 3D scene, where $N$ is the number of points in the point cloud, and $\mathcal { D } _ { v }$ is the feature dimension.
+
+# 4.3 Training 3D-LLMs
+
+# 4.3.1 2D VLMs as backbones
+
+In addition to the feature extractor, training 3D-LLMs from scratch is also non-trivial. In fact, according to [29, 15], the training of 2D VLMs only begins to show "signs of life" after consuming half a billion images. They usually use frozen and pre-trained image encoders such as CLIP to extract features for 2D images. Considering that with 3D feature extractor, the 3D features can be mapped into the same feature space as 2D images, it’s reasonable to use these 2D VLMs as our backbones.
+
+The perceiver architecture proposed by [23] leverages an asymmetric attention mechanism to iteratively distill inputs into a tight latent bottleneck, allowing it to handle very large inputs of arbitrary input sizes, thus can tackle different modalities. This architecture is utilized in VLMs like Flamingo [15]. BLIP-2 [29] also utilizes a similar structure called QFormer. The 2D image features, output from frozen image encoders, are flattened and sent to the perceiver to generate a fixed-sized input. Given that our 3D features are in the same feature space as the 2D features by the 3D feature extractor, and that perceiver is able to handle inputs of arbitrary input sizes of the same feature dimension, point cloud features with arbitrary sizes could also be fed into the perceiver. Therefore, we use the 3D feature extractor to extract the 3D features in the same feature space as the features of the frozen image encoders. Then, we use pretrained 2D VLMs as our backbones, input the aligned 3D features to train 3D-LLMs with our collected 3D-language dataset.
+
+# 4.3.2 3D Localization Mechanism
+
+Notice that since 3D features are reconstructed via 2D pretrained feature extractor that has been aligned with language (e.g., CLIP [36] and EVA-CLIP [41]), localization can be performed by directly calculating the similarity between 3D features and language features. However, Apart from building 3D features, which can be aligned with language semantics, it’s also essential that the model itself could capture 3D spatial information. To this end, we propose a 3D localization mechanism that boosts 3D LLMs’ abilities to absorb spatial information. It consists of two parts:
+
+Augmenting 3D features with position embeddings Besides the 3D features aggregated from 2D multi-view features, we also add position embeddings to the features. Supposing the feature dim is $\mathcal { D } _ { v }$ , we generate sin/cos position embeddings of the three dimensions, each has an embedding size $\mathcal { D } _ { v } / 3$ . We concatenate the embeddings of all three dimensions, and add them to the 3D features with a weight.
+
+Augmenting LLM vocabularies with location tokens In order to align 3D spatial locations with LLMs, we propose to embed 3D locations in the vocabularies, following [6] and [43]. To be specific, the region to be grounded can be denoted as a sequence of discrete tokens representing the bounding box in the form of AABB. The continuous corner coordinates of the bounding boxes are uniformly discretized to voxel integers as location tokens $\langle x _ { m i n } , y _ { m i n } , z _ { m i n } , x _ { m a x } , y _ { m a x } , z _ { m a x } \rangle$ . After adding these additional location tokens, we unfreeze the weights for these tokens in the input and output embeddings of language models.
+
+# 5 Experiments
+
+We first introduce the architecture, and training and evaluation protocols. In Sec 5.1, we analyze the held-out experiments on ScanQA [3], SQA3D [32], and 3DMV-VQA [20] Dataset. Sec 5.2 covers more analysis on held-in evaluation and qualitative examples.
+
+Architecture We experiment on three backbone 2D VLMs for 3D-LLMs: Flamingo 9B, BLIP-2 Vit-g $\mathrm { \Omega } _ { \mathrm { { ; } } } \mathrm { 0 p t } 2 . 7 \mathrm { B }$ , BLIP-2 Vit- $\mathrm { g }$ FlanT5-XL. For BLIP-2, during pre-training the 3D-LLMs, we initialize the model from BLIP-2 checkpoints released in LAVIS library [28], and finetune the parameters for the QFormer. 3D features are 1408-dim features, same as EVA_CLIP [41] hidden feature dim used by BLIP-2. We keep most parts of the LLMs (i.e., Opt and FlanT5) frozen, except the weights for the newly-added location tokens in the input and the output embeddings. For Flamingo, we initialize the model from the Flamingo9B checkpoint released in OpenFlamingo repository [2]. We finetune the parameters for perceiver, gated cross attention layers, and the weights for additional location tokens in the input and output embeddings. 3D features are 1024-dim features, same as CLIP hidden feature dim used by Flamingo. For generating class-agnostic (generic) object masks for the 2D pixel-aligned dense feature extraction, we follow [24] and use the Mask2Former (M2F) [9] or the segment anything (SAM) [26].
+
+Training & Evaluation Datasets & Protocols We split our datasets into two genres, held-in datasets and held-out datasets. Specifically, our 3D-language data generation pipeline generates the held-in datasets of multiple tasks. We utilize training sets of held-in datasets for pre-training foundation 3D-LLMs, and their validation sets can be applied for held-in evaluation. During pre-training, we mix the held-in datasets of all tasks. The models are trained with the standard language modeling loss to output responses. Held-out datasets, on the other hand, are not used in training the foundation 3D-LLMs. We use three held-out 3D question answering datasets for held-out evaluation: ScanQA, SQA3D and 3DMV-VQA.
+
+# 5.1 Held-Out Evaluation
+
+# 5.1.1 Experiments on ScanQA
+
+We finetune our pretrained 3D-LLMs on the ScanQA dataset and compare with baseline models.
+
+Baselines & Evaluation Metrics We include representative baseline models on the benchmark. ScanQA is the state-of-the-art method on the benchmark that uses VoteNet to obtain object proposals, and then fuse them with language embeddings. ScanRefer+MCAN is a baseline that identifies the referred object and the MCAN model is applied to the image surrounding the localized object. VoteNet+MCAN detects objects in a 3D space, extracts their features, and uses them in a standard VQA model. Notably, these baseline models all extract explicit object representations from a pretrained localization module. In addition to these baselines, we also design several LLM-based baselines. LLaVA is a visual instruction tuning that connects a vision encoder and LLM for generalpurpose visual and language understanding. We use its pretrained model and do zero-shot evaluation on our dataset. We use a single random image as input. We use LLaVA 13B model. ULIP encoders $^ +$ LLMs use existing pre-trained 3D encoders with LLMs, for comparison between 3D pre-trained encoders, and 2D encoders for feature encoding. Single Image $^ +$ Pretrained VLMs use our 2D VLM backbones (i.e., flamingo and BLIP-2), replace the 3D inputs of 3D-LLMs with single image features to train the models, and then finetune on ScanQA dataset. Multi-View Image $^ +$ Pretrained VLMs use our 2D VLM backbones, replace the 3D inputs of 3D-LLMs with concatenated features of multi-view images to train the models, and then finetune on ScanQA dataset. We report BLEU, ROUGE-L, METEOR, CIDEr for robust answer matching. We also use exact match (EM) metric.
+
+Table 1: Experimental results on ScanQA validation set. \* Means the models use explicit object representations. B-1, B-2, B-3, B-4 denote BLEU-1, BLEU-2, BLEU-3, BLEU-4 respectively. M2F denotes mask2former, SAM denotes Segment Anything.   
+
+<table><tr><td></td><td>B-1</td><td>B-2</td><td>B-3</td><td>B-4</td><td>METEOR</td><td>ROUHE-L</td><td>CIDER</td><td>EM</td></tr><tr><td>VoteNet+MCAN*</td><td>28.0</td><td>16.7</td><td>10.8</td><td>6.2</td><td>11.4</td><td>29.8</td><td>54.7</td><td>17.3</td></tr><tr><td>ScanRefer+MCAN*</td><td>26.9</td><td>16.6</td><td>11.6</td><td>7.9</td><td>11.5</td><td>30</td><td>55.4</td><td>18.6</td></tr><tr><td>ScanQA*</td><td>30.2</td><td>20.4</td><td>15.1</td><td>10.1</td><td>13.1</td><td>33.3</td><td>64.9</td><td>21.0</td></tr><tr><td>LLaVA(zero-shot)</td><td>7.1</td><td>2.6</td><td>0.9</td><td>0.3</td><td>10.5</td><td>12.3</td><td>5.7</td><td>0.0</td></tr><tr><td>ULIPPointMLP+flant5</td><td>18.4</td><td>7.2</td><td>2.7</td><td>1.4</td><td>7.4</td><td>18.1</td><td>26.9</td><td>7.5</td></tr><tr><td>ULIPPointMLP+opt</td><td>19.1</td><td>7.3</td><td>2.7</td><td>1.9</td><td>7.4</td><td>18.2</td><td>28.0</td><td>8.4</td></tr><tr><td>ULIPPointBERT+flant5</td><td>29.2</td><td>17.9</td><td>10.3</td><td>6.1</td><td>11.6</td><td>28.1</td><td>50.9</td><td>14.5</td></tr><tr><td>ULIPPointBERT+opt</td><td>28.8</td><td>16.9</td><td>9.7</td><td>5.9</td><td>11.3</td><td>27.9</td><td>50.5</td><td>13.8</td></tr><tr><td>flamingo-SingleImage</td><td>23.8</td><td>14.5</td><td>9.2</td><td>8.5</td><td>10.7</td><td>29.6</td><td>52</td><td>16.9</td></tr><tr><td>flamingo-MultiView</td><td>25.6</td><td>15.2</td><td>9.2</td><td>8.4</td><td>11.3</td><td>31.1</td><td>55</td><td>18.8</td></tr><tr><td>BLIP2-flant5-SingleImage</td><td>28.6</td><td>15.1</td><td>9.0</td><td>5.1</td><td>10.6</td><td>25.8</td><td>42.6</td><td>13.3</td></tr><tr><td>BLIP2-flant5-MultiView</td><td>29.7</td><td>16.2</td><td>9.8</td><td>5.9</td><td>11.3</td><td>26.6</td><td>45.7</td><td>13.6</td></tr><tr><td>3D-LLM(M2F, flamingo)</td><td>30.3</td><td>17.8</td><td>12.0</td><td>7.2</td><td>12.2</td><td>32.3</td><td>59.2</td><td>20.4</td></tr><tr><td>3D-LLM (M2F,BLIP2-opt)</td><td>35.9</td><td>22.5</td><td>16.0</td><td>9.4</td><td>13.8</td><td>34.0</td><td>63.8</td><td>19.3</td></tr><tr><td>3D-LLM (SAM, BLIP2-opt)</td><td>35.0</td><td>21.7</td><td>15.5</td><td>9.5</td><td>14.0</td><td>34.5</td><td>67.1</td><td>19.8</td></tr><tr><td>3D-LLM (M2F, BLIP2-flant5)</td><td>39.3</td><td>25.2</td><td>18.4</td><td>12.0</td><td>14.5</td><td>35.7</td><td>69.4</td><td>20.5</td></tr><tr><td>3D-LLM(SAM, BLIP2-flant5)</td><td>37.5</td><td>24.1</td><td>17.6</td><td>12.9</td><td>15.1</td><td>37.5</td><td>74.5</td><td>21.2</td></tr></table>
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Format</td><td colspan="6">test set</td><td rowspan="2">Avg.</td></tr><tr><td>What</td><td>Is</td><td>How</td><td>Can</td><td>Which</td><td>Others</td></tr><tr><td>Blind test</td><td>SQ→A</td><td>26.75</td><td>63.34</td><td>43.44</td><td>69.53</td><td>37.89</td><td>43.41</td><td>43.65</td></tr><tr><td>ScanQA(w/o stxt)</td><td>VQ→A</td><td>28.58</td><td>65.03</td><td>47.31</td><td>66.27</td><td>43.87</td><td>42.88</td><td>45.27</td></tr><tr><td>ScanQA</td><td>VSQ→A</td><td>31.64</td><td>63.80</td><td>46.02</td><td>69.53</td><td>43.87</td><td>45.34</td><td>46.58</td></tr><tr><td>ScanQA+aux task</td><td>VSQ→AL</td><td>33.48</td><td>66.10</td><td>42.37</td><td>69.53</td><td>43.02</td><td>46.40</td><td>47.20</td></tr><tr><td>MCAN</td><td>VSQ→A</td><td>28.86</td><td>59.66</td><td>44.09</td><td>68.34</td><td>40.74</td><td>40.46</td><td>43.42</td></tr><tr><td>ClipBERT</td><td>VSQ→A</td><td>30.24</td><td>60.12</td><td>38.71</td><td>63.31</td><td>42.45</td><td>42.71</td><td>43.31</td></tr><tr><td>Unified QA</td><td>VSQ →A</td><td>33.01</td><td>50.43</td><td>31.91</td><td>56.51</td><td>45.17</td><td>41.11</td><td>41.00</td></tr><tr><td>Unified QA</td><td>VSQ→A</td><td>27.58</td><td>47.99</td><td>34.05</td><td>59.47</td><td>40.91</td><td>39.77</td><td>38.71</td></tr><tr><td>GPT-3</td><td>VSQ→A</td><td>39.67</td><td>45.99</td><td>40.47</td><td>45.56</td><td>36.08</td><td>38.42</td><td>41.00</td></tr><tr><td>GPT-3</td><td>VSQ→A</td><td>28.90</td><td>46.42</td><td>28.05</td><td>40.24</td><td>30.11</td><td>36.07</td><td>34.57</td></tr><tr><td>3D-LLM</td><td>VSQ→A</td><td>37.05</td><td>65.18</td><td>45.81</td><td>67.46</td><td>51.00</td><td>49.82</td><td>49.79</td></tr></table>
+
+Table 2: Experimental Results on SQA3D test set. In the Format column, “V" means the 3D visual inputs, “S" means the situation inputs, “Q" and “A" denote questions and answers respectively. Here we use 3D-LLM (SAM, BLIP2-flant5).
+
+Result Analysis We report our results on ScanQA validation set in Table 1. We observe a significant increase in the evaluation metrics. For example, for BLEU-1, our model outperforms the state-ofthe-art ScanQA model by ${ \sim } 9 \%$ for validation set. For CIDER, we report a ${ \sim } 1 0 \%$ gain compared to ScanQA, and much higher than other 3D-based baselines. These results show that by injecting 3D into LLMs, the models can generate answers that are much more similar to the ground-truth answers. Furthermore, 3D-based baselines use object detectors like VoteNet to segment the objects, and then send per-object features into their models, while our inputs are holistic 3D features without explicit object representations. This shows that our model could perform visual reasoning about objects and their relationships even without explicit object representations. We then examine whether 2D VLMs have the same ability. We find that by taking single-view images or multi-view images as inputs, the performances drop much compared to 3D-LLMs. Specifically, multi-view images also contain information about the whole scene. However, they have much lower performances compared to 3D-LLMs, probably because features of multi-view images are disorganized, thus losing 3D-related information.
+
+# 5.1.2 Experiments on SQA3D
+
+SQA3D [32] requires the tested agent to first understand its situation (position, orientation, etc.) in the 3D scene as described by text, then reason about its surrounding environment and answer a question under that situation. We finetune our pretrained 3D-LLMs on the SQA3D dataset and compare with baseline models. We include all baseline models introduced by the original paper. Specifically, ScanQA+aux task achieves the SOTA performance by adding two auxilliary tasks: prediction the
+
+position and rotation of the agent situation. Table 2 shows the results. We can see that our 3D-LLM outperforms all baseline models a lot, even without training with auxiliary tasks and losses.   
+
+<table><tr><td>Methods</td><td>Concept</td><td>Counting</td><td>Relation</td><td>Comparison</td><td>Overall</td></tr><tr><td>NS-VQA*</td><td>59.8</td><td>21.5</td><td>33.4</td><td>61.6</td><td>38.0</td></tr><tr><td>3D-Feature+LSTM</td><td>61.2</td><td>22.4</td><td>49.9</td><td>61.3</td><td>48.2</td></tr><tr><td>3D-CLR*</td><td>66.1</td><td>41.3</td><td>57.6</td><td>72.3</td><td>57.7</td></tr><tr><td>famingo-SingleImage</td><td>58.7</td><td>18.5</td><td>38.4</td><td>60.1</td><td>40.3</td></tr><tr><td>flamingo-MultiView</td><td>60.0</td><td>18.3</td><td>40.2</td><td>61.4</td><td>41.6</td></tr><tr><td>BLIP-SingleImage</td><td>58.0</td><td>20.4</td><td>42.3</td><td>62.3</td><td>43.1</td></tr><tr><td>BLIP-MultiView</td><td>61.9</td><td>21.1</td><td>48.0</td><td>62.3</td><td>47.1</td></tr><tr><td>3D-LLM (M2F, flamingo)</td><td>68.9</td><td>32.4</td><td>61.6</td><td>68.3</td><td>58.6</td></tr><tr><td>3D-LLM (M2F, BLIP2-opt)</td><td>63.4</td><td>30.7</td><td>57.6</td><td>65.2</td><td>54.9</td></tr><tr><td>3D-LLM (SAM,BLIP2-opt)</td><td>73.4</td><td>24.5</td><td>63.2</td><td>77.6</td><td>61.5</td></tr><tr><td>3D-LLM(M2F,BLIP2-flanT5)</td><td>68.1</td><td>31.4</td><td>55.1</td><td>69.7</td><td>54.6</td></tr><tr><td>3D-LLM(SAM,BLIP2-flanT5)</td><td>76.3</td><td>30.2</td><td>64.3</td><td>80.2</td><td>64.0</td></tr></table>
+
+Table 3: Experimental results on 3DMV-VQA dataset. \* denotes using explicit object representations and neuro-symbolic reasoning.
+
+# 5.1.3 Experiments on 3DMV-VQA
+
+We finetune our pretrained 3D-LLMs on the 3DMV-VQA dataset and compare with baseline models. We include all baseline models introduced by the original paper. Specifically, 3D-CLR [20] is the SOTA achieves the SOTA performance via neuro-symbolic reasoning based on 3D features.
+
+Result Analysis Table 3 shows the performances on 3DMV-VQA. We can see that 3D-LLMs outperform state-of-the-art baseline model in the question types of concept and relation, and also in the overall performance. Our model also outperforms 3D-Feature+LSTM, demonstrating the power of LLMs over vanilla language models with similar 3D features as inputs. Overall, 3D-based methods outshine 2D-based versions of the methods. Our 3D-LLMs outperform their corresponding 2D VLMs with image input, further demonstrating the importance of 3D representations for 3D-LLMs.
+
+<table><tr><td>Tasks</td><td>Models</td><td colspan="6">BLEU-1 BLEU-2 BLEU-3 BLEU-4 METEOR ROUGH-L</td></tr><tr><td rowspan="8">3D Captioning</td><td>flamingo-SingleImage</td><td>29.0</td><td>17.9</td><td>12.5</td><td>12.1</td><td>12.4</td><td>28.2</td></tr><tr><td>flamingo-MultiView</td><td>29.5</td><td>18.6</td><td>13.7</td><td>12.4</td><td>14.0</td><td>29.0</td></tr><tr><td>BLIP2-flant5-SingleImage</td><td>30.3</td><td>18.3</td><td>14.5</td><td>12.0</td><td>13.1</td><td>30.9</td></tr><tr><td>BLIP2-flant5-MultiView</td><td>34.4</td><td>23.9</td><td>18.0</td><td>14.1</td><td>17.5</td><td>35.7</td></tr><tr><td>3D-LLM (flamingo)</td><td>36.1</td><td>24.5</td><td>18.7</td><td>15.6</td><td>17.6</td><td>35.8</td></tr><tr><td>3D-LLM (BLIP2-opt)</td><td>35.7</td><td>26.7</td><td>20.3</td><td>15.9</td><td>18.7</td><td>40.1</td></tr><tr><td>3D-LLM (BLIP2-t5)</td><td>39.8</td><td>31.0</td><td>24.7</td><td>20.1</td><td>17.7</td><td>42.6</td></tr><tr><td>3D-LLM(SAM,BLIP2-t5) 4</td><td>44.5</td><td>38.6</td><td>29.5</td><td>24.2</td><td>22.1</td><td>45.4</td></tr><tr><td rowspan="8">3D-assisted Dialog</td><td>flant5</td><td>27.4</td><td>16.5</td><td>11.1</td><td>8.7</td><td>9.5</td><td>27.5</td></tr><tr><td>flamingo-SingleImage</td><td>29.4</td><td>18.7</td><td>11.3</td><td>9.4</td><td>10.0</td><td>26.8</td></tr><tr><td>flamingo-MultiView</td><td>30.6</td><td>21.3</td><td>11.9</td><td>9.1</td><td>10.4</td><td>27.9</td></tr><tr><td>BLIP2-flant5-SingleImage</td><td>28.4</td><td>17.3</td><td>10.6</td><td>9.1</td><td>10.2</td><td>27.4</td></tr><tr><td>BLIP2-flant5-MultiView</td><td>32.4</td><td>20.9</td><td>12.1</td><td>9.5</td><td>11.0</td><td>29.5</td></tr><tr><td>3D-LLM (flamingo)</td><td>35.0</td><td>22.8</td><td>15.4</td><td>10.6</td><td>16.0</td><td>34.2</td></tr><tr><td>3D-LLM (BLIP2-opt)</td><td>39.6</td><td>27.5</td><td>20.5</td><td>16.2</td><td>18.4</td><td>38.6</td></tr><tr><td>3D-LLM (BLIP2-flant5)</td><td>39.0</td><td>27.8</td><td>21.2</td><td>16.6</td><td>18.9</td><td>39.3</td></tr><tr><td rowspan="8"></td><td>3D-LLM(SAM,BLIP2-t5)</td><td>40.5</td><td>29.4</td><td>23.9</td><td>21.4</td><td>19.6</td><td>40.8</td></tr><tr><td>flant5</td><td>25.5</td><td>21.1</td><td>16.7</td><td>6.0</td><td>13.9</td><td>28.4</td></tr><tr><td>flamingo-SingleImage</td><td>31.4</td><td>23.0</td><td>18.8</td><td>7.1</td><td>15.6</td><td>30.6</td></tr><tr><td>flamingo-MultiView</td><td>33.1</td><td>24.7</td><td>21.4</td><td>7.3</td><td>16.1</td><td>33.2</td></tr><tr><td>BLIP2-flant5-SingleImage</td><td>32.2</td><td>25.3</td><td>18.2</td><td>6.9</td><td>15.0</td><td>31.0</td></tr><tr><td>Task Decomposition BLIP2-flant5-MultiView</td><td>33.1</td><td>27.0</td><td>20.6</td><td>6.9</td><td>15.5</td><td>34.0</td></tr><tr><td>3D-LLM (flamingo)</td><td>32.9</td><td>25.6</td><td>20.2</td><td>6.4</td><td>16.0</td><td>33.5</td></tr><tr><td>3D-LLM (BLIP2-opt)</td><td>34.1</td><td>27.7</td><td>20.8</td><td>7.6</td><td>16.5</td><td>35.4</td></tr><tr><td></td><td>3D-LLM (BLIP2-flant5) 3D-LLM(SAM,BLIP2-t5) 31.6</td><td>33.9</td><td>28.1 22.3</td><td>20.7 17.2</td><td>7.4 8.8</td><td>15.9 14.0</td><td>37.8 38.3</td></tr></table>
+
+Table 4: Experimental Results on Held-In Datasets. 3D-LLMs outperform 2D VLMs.
+
+# 5.2 More Extensive Evaluation
+
+Held-In Evaluation We carry out experiments on held-in datasets of three tasks: 3D captioning, 3D-assited dialog and task decomposition. The baselines include 2D VLMs as for the held-out evaluation. We add one language-only baseline: FlanT5, which examines LLMs’ ability to complete these tasks without any visual input. To evaluate the quality of responses, we include BLEU, ROUGEL, METEOR, CIDEr as our metrics. We report the held-in evaluation performances in Table 4. From the table, we could see that 3D-LLMs could generate high-quality responses, outperforming both 2D VLMs and language-only LLMs.
+
+Qualitative Examples In Figure 4, we show qualitative examples of 3D-LLM’s predictions. We can see that our 3D-LLM is able to perform a variety of tasks.
+
+![](images/218d0c0b87c7b86eda74bbdf574c22b8ebdf7aee0bcdaf46c083e9a46088f3bf.jpg)  
+Figure 4: Qualitative examples of 3D-LLM’s prediction.
+
+# 6 Conclusion
+
+In this paper, we propose a new family of 3D-LLMs that can take 3D representations as inputs and generate responses. We introduce a series of 3D-language data generation pipelines to generate a dataset of 1M 3D-language pairs to train our 3D-LLMs. Our 3D-LLMs leverage 2D pretrained VLMs as backbones and a novel 3D localization mechanism. Experiments show that our 3D-LLMs outperform state-of-the-art baseline models on ScanQA datasets, and could perform a diverse set of 3D-related tasks. A limitation is that the 3D feature extractor relies on 2D multi-view images, and thus all 3D scenes need to be rendered so that they can be trained in 3D-LLMs, which introduces an additional rendering process.
+
+# 7 Acknowledgements
+
+This work was supported by the MIT-IBM Watson AI Lab, DARPA MCS, DSO grant DSOCO21072, and gift funding from MERL, Cisco, Sony, and Amazon. We would also like to thank the computation support from AiMOS, a server cluster for the IBM Research AI Hardware Center.
+
+# References
+
+[1] P. Achlioptas, A. Abdelreheem, F. Xia, M. Elhoseiny, and L. J. Guibas. ReferIt3D: Neural listeners for fine-grained 3D object identification in real-world scenes. In ECCV, 2020.   
+[2] A. Awadalla, I. Gao, J. Gardner, J. Hessel, Y. Hanafy, W. Zhu, K. Marathe, Y. Bitton, S. Gadre, J. Jitsev, S. Kornblith, P. W. Koh, G. Ilharco, M. Wortsman, and L. Schmidt. Openflamingo, Mar. 2023.   
+[3] D. Azuma, T. Miyanishi, S. Kurita, and M. Kawanabe. ScanQA: 3D question answering for spatial scene understanding. 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 19107–19117, 2022.   
+[4] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, pages 1877–1901, 2020.   
+[5] D. Z. Chen, A. X. Chang, and M. Nießner. ScanRefer: 3D object localization in RGB-D scans using natural language. 16th European Conference on Computer Vision (ECCV), 2020.   
+[6] T. Chen, S. Saxena, L. Li, D. J. Fleet, and G. E. Hinton. Pix2seq: A language modeling framework for object detection. ArXiv, abs/2109.10852, 2021.   
+[7] X. Chen, H. Fang, T.-Y. Lin, R. Vedantam, S. Gupta, P. Dollár, and C. L. Zitnick. Microsoft coco captions: Data collection and evaluation server. arXiv preprint arXiv:1504.00325, 2015.   
+[8] Z. Chen, A. Gholami, M. Nießner, and A. X. Chang. Scan2cap: Context-aware dense captioning in rgb-d scans. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3193–3203, 2021.   
+[9] B. Cheng, A. Choudhuri, I. Misra, A. Kirillov, R. Girdhar, and A. G. Schwing. Mask2former for video instance segmentation. ArXiv, abs/2112.10764, 2021.   
+[10] A. Chowdhery, S. Narang, J. Devlin, M. Bosma, G. Mishra, A. Roberts, P. Barham, H. W. Chung, C. Sutton, S. Gehrmann, et al. Palm: Scaling language modeling with pathways. arXiv preprint arXiv:2204.02311, 2022.   
+[11] H. W. Chung, L. Hou, S. Longpre, B. Zoph, Y. Tay, W. Fedus, E. Li, X. Wang, M. Dehghani, S. Brahma, et al. Scaling instruction-finetuned language models. arXiv preprint arXiv:2210.11416, 2022.   
+[12] A. Dai, A. X. Chang, M. Savva, M. Halber, T. A. Funkhouser, and M. Nießner. ScanNet: Richly-annotated 3D reconstructions of indoor scenes. 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2432–2443, 2017.   
+[13] Databricks. Free dolly: Introducing the world’s first truly open instruction-tuned llm, 2023.   
+[14] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.   
+[15] J.-B. A. et al. Flamingo: a visual language model for few-shot learning. 2022.   
+[16] M. Feng, Z. Li, Q. Li, L. Zhang, X. Zhang, G. Zhu, H. Zhang, Y. Wang, and A. S. Mian. Free-form description guided 3d visual graph network for object grounding in point cloud. 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 3702–3711, 2021.   
+[17] S. Y. Gadre, M. Wortsman, G. Ilharco, L. Schmidt, and S. Song. CLIP on wheels: Zero-shot object navigation as object localization and exploration. ArXiv, abs/2203.10421, 2022.   
+[18] T. Gong, C. Lyu, S. Zhang, Y. Wang, M. Zheng, Q. Zhao, K. Liu, W. Zhang, P. Luo, and K. Chen. MultiModal-GPT: A vision and language model for dialogue with humans. arXiv preprint arXiv:2305.04790, 2023.   
+[19] Y. Goyal, T. Khot, D. Summers-Stay, D. Batra, and D. Parikh. Making the V in VQA matter: Elevating the role of image understanding in Visual Question Answering. In Conference on Computer Vision and Pattern Recognition (CVPR), 2017.   
+[20] Y. Hong, C. Lin, Y. Du, Z. Chen, J. B. Tenenbaum, and C. Gan. 3D concept learning and reasoning from multi-view images, 2023.   
+[21] C. Huang, O. Mees, A. Zeng, and W. Burgard. Visual language maps for robot navigation, 2023.   
+[22] P.-H. Huang, H.-H. Lee, H.-T. Chen, and T.-L. Liu. Text-guided graph neural networks for referring 3D instance segmentation. In AAAI, 2021.   
+[23] A. Jaegle, F. Gimeno, A. Brock, A. Zisserman, O. Vinyals, and J. Carreira. Perceiver: General perception with iterative attention. In International Conference on Machine Learning, 2021.   
+[24] K. M. Jatavallabhula, A. Kuwajerwala, Q. Gu, M. Omama, T. Chen, S. Li, G. Iyer, S. Saryazdi, N. Keetha, A. Tewari, J. B. Tenenbaum, C. M. de Melo, M. Krishna, L. Paull, F. Shkurti, and A. Torralba. Conceptfusion: Open-set multimodal 3D mapping, 2023.   
+[25] C. Jia, Y. Yang, Y. Xia, Y.-T. Chen, Z. Parekh, H. Pham, Q. Le, Y.-H. Sung, Z. Li, and T. Duerig. Scaling up visual and vision-language representation learning with noisy text supervision. In International Conference on Machine Learning, pages 4904–4916. PMLR, 2021.   
+[26] A. Kirillov, E. Mintun, N. Ravi, H. Mao, C. Rolland, L. Gustafson, T. Xiao, S. Whitehead, A. C. Berg, W.-Y. Lo, P. Dollár, and R. B. Girshick. Segment anything. ArXiv, abs/2304.02643, 2023.   
+[27] J. Krishna Murthy, S. Saryazdi, G. Iyer, and L. Paull. gradslam: Dense slam meets automatic differentiation. arXiv, 2020.   
+[28] D. Li, J. Li, H. Le, G. Wang, S. Savarese, and S. C. H. Hoi. LAVIS: A library for language-vision intelligence, 2022.   
+[29] J. Li, D. Li, S. Savarese, and S. Hoi. BLIP-2: Bootstrapping language-image pre-training with frozen image encoders and large language models, 2023.   
+[30] J. Li, D. Li, S. Savarese, and S. Hoi. BLIP-2: Bootstrapping language-image pre-training with frozen image encoders and large language models. arXiv preprint arXiv:2301.12597, 2023.   
+[31] H. Liu, C. Li, Q. Wu, and Y. J. Lee. Visual instruction tuning. arXiv preprint arXiv:2304.08485, 2023.   
+[32] X. Ma, S. Yong, Z. Zheng, Q. Li, Y. Liang, S.-C. Zhu, and S. Huang. Sqa3d: Situated question answering in 3d scenes, 2023.   
+[33] OpenAI. GPT-4 technical report, 2023.   
+[34] OpenAI. GPT-4 technical report. ArXiv, abs/2303.08774, 2023.   
+[35] L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, et al. Training language models to follow instructions with human feedback. Advances in Neural Information Processing Systems, 35:27730–27744, 2022.   
+[36] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pages 8748–8763. PMLR, 2021.   
+[37] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.   
+[38] C. Raffel, N. Shazeer, A. Roberts, K. Lee, S. Narang, M. Matena, Y. Zhou, W. Li, and P. J. Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. The Journal of Machine Learning Research, 21(1):5485–5551, 2020.   
+[39] S. K. Ramakrishnan, A. Gokaslan, E. Wijmans, O. Maksymets, A. Clegg, J. M. Turner, E. Undersander, W. Galuba, A. Westbury, A. X. Chang, M. Savva, Y. Zhao, and D. Batra. Habitat-matterport 3D dataset (HM3D): 1000 large-scale 3D environments for embodied AI. In Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2021.   
+[40] C. Sun, M. Sun, and H.-T. Chen. Direct voxel grid optimization: Super-fast convergence for radiance fields reconstruction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 5459–5469, June 2022.   
+[41] Q. Sun, Y. Fang, L. Wu, X. Wang, and Y. Cao. Eva-clip: Improved training techniques for clip at scale, 2023.   
+[42] Z. Sun, Y. Shen, Q. Zhou, H. Zhang, Z. Chen, D. Cox, Y. Yang, and C. Gan. Principle-driven self-alignment of language models from scratch with minimal human supervision. arXiv e-prints, pages arXiv–2305, 2023.   
+[43] P. Wang, A. Yang, R. Men, J. Lin, S. Bai, Z. Li, J. Ma, C. Zhou, J. Zhou, and H. Yang. Unifying architectures, tasks, and modalities through a simple sequence-to-sequence learning framework. In International Conference on Machine Learning, 2022.   
+[44] K. Yadav, R. Ramrakhya, S. K. Ramakrishnan, T. Gervet, J. Turner, A. Gokaslan, N. Maestre, A. X. Chang, D. Batra, M. Savva, et al. Habitat-matterport 3D semantics dataset. arXiv preprint arXiv:2210.05633, 2022.   
+[45] S. Ye, D. Chen, S. Han, and J. Liao. 3D question answering. IEEE transactions on visualization and computer graphics, PP, 2021.   
+[46] L. Yu, P. Poirson, S. Yang, A. C. Berg, and T. L. Berg. Modeling context in referring expressions. In Computer Vision–ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part II 14, pages 69–85. Springer, 2016.   
+[47] P. Zhang, Y. Goyal, D. Summers-Stay, D. Batra, and D. Parikh. Yin and Yang: Balancing and answering binary visual questions. In Conference on Computer Vision and Pattern Recognition (CVPR), 2016.   
+[48] D. Zhu, J. Chen, K. Haydarov, X. Shen, W. Zhang, and M. Elhoseiny. Chatgpt asks, blip-2 answers: Automatic questioning towards enriched visual descriptions, 2023.

parse/dev/eLgK35G3A5d/eLgK35G3A5d.md

@@ -0,0 +1,596 @@
+# ANNEALED FISHER IMPLICIT SAMPLER
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Sampling from an un-normalized target distribution is an important problem in many scientific fields. An implicit sampler uses a parametric transform $x = G _ { \theta } ( z )$ to push forward an easy-to-sample latent code $z$ to obtain a sample $x$ . Such samplers are favored for fast inference speed and flexible architecture. Thus it is appealing to train an implicit sampler for sampling from the un-normalized target. In this paper, we propose a novel approach to training an implicit sampler by minimizing the Fisher Divergence between sampler and target distribution. We find that the trained sampler works well for relatively simple targets but may fail for more complicated multi-modal targets. To improve the training for multi-modal targets, we propose another adaptive training approach that trains the sampler to gradually learn a sequence of annealed distributions. We construct the annealed distribution path to bridge a simple distribution and the complicated target. With the annealed approach, the sampler is capable of handling challenging multi-modal targets. In addition, we also introduce a few MCMC correction steps after the sampler to better spread the samples. We call our proposed sampler the Annealed Fisher Implicit Sampler (AFIS). We test AFIS on several sampling benchmarks. The experiments show that our AFIS outperforms baseline methods in many aspects. We also show in theory that the added MC correction steps get faster mixing by using the learned sampler as MCMC’s initialization.
+
+# 1 INTRODUCTION
+
+Sampling from an un-normalized distribution is an important problem in many scientific fields such as Bayesian statistics (Green, 1995), biology (Schutte et al., 1999), physics simulations (Olsson, ¨ 1995), machine learning (Andrieu et al., 2003), and so on. Typically, the problem is formulated as: given a known differentiable un-normalized target potential function $\log p ( x )$ , one wants to sample from the target distribution. Due to the success of deep neural networks, there is increasing popularity to train a deep generative model to learn to sample(Hu et al., 2018; Wu et al., 2020; Matthews et al., 2022; Corenflos et al., 2021). Such learned models which can approximately sample from target distribution are called samplers.
+
+Training a neural network (i.e., a parameterized transform) $x = G _ { \theta } ( z )$ to push forward an easyto-sample latent code $z \sim p _ { Z } ( z )$ to obtain a sample is an appealing approach. Such approaches are favored for fast sampling because they only need a single-time forward pass of neural network transform. Let $G _ { \theta } ( . )$ denote the parametric transform and $q ( x )$ the un-normalized target distribution with unknown normalizing constant $\begin{array} { r } { Z = \int q ( x ) d x } \end{array}$ . Let $p _ { \theta } ( x )$ denote the sampler-induced distribution. Some previous work takes a normalizing flow model as sampler, and then minimizes the KL divergence between sampler-induced and target distributions regardless of normalizing constant: $\mathcal { D } _ { K L } \big ( \bar { p } _ { \theta } , q \big ) = \mathbb { E } _ { x \sim p _ { \theta } } \big [ \log ^ { \big ( } p _ { \theta } ( x ) - \log q ( x ) \dot { + } \log Z \big ]$ . Note that $Z$ is parameter-free and can be ignored during training. However, minimizing KL divergence relies on explicit log-likelihood of sampler-induced distribution, which can not be computed in a general transform. Such transform with no explicit likelihood is referred to as an implicit sampler.
+
+In this paper, we will focus on implicit samplers. Note that the annoying normalizing constant vanishes when considering the score function of a distribution, $s ( x ) = \nabla _ { x } \log p ( x )$ . Thus, we can take the score-based divergence to constructively get rid of the unknown normalizing constant for implicit samplers. Fisher divergence (FD), which is a popular score-based probability divergence, and its variants have obtained much success in recent years, especially in training deep generative models such as energy-based models (Kingma & Cun, 2010; Martens et al., 2012; Song et al., 2019), score based diffusion models (Song et al., 2020; Kingma et al., 2021; Vahdat et al., 2021; Song & Ermon, 2019; Ho et al., 2020), etc. Assume $p ( x ) , q ( x )$ are two probability densities. The Fisher Divergence between $p$ and $q$ is defined as
+
+$$
+\mathcal { D } _ { F D } ( \boldsymbol { p } , \boldsymbol { q } ) = \frac { 1 } { 2 } \mathbb { E } _ { \boldsymbol { x } \sim \boldsymbol { p } ( \boldsymbol { x } ) } \lVert \nabla _ { \boldsymbol { x } } \log \boldsymbol { p } ( \boldsymbol { x } ) - \nabla _ { \boldsymbol { x } } \log \boldsymbol { q } ( \boldsymbol { x } ) \rVert _ { 2 } ^ { 2 } .
+$$
+
+It is always no less than 0 and equals to 0 if and only if $p ( x ) = q ( x )$ a.s. under probability measure $p$ . Fisher Divergence is suitable for measuring the dissimilarity between sampler and un-normalized target distribution. So as to be used for training the implicit sampler.
+
+In this paper, we firstly propose a novel approach to learning a sampler by minimizing the Fisher Divergence between sampler and un-normalized target distributions. We call such a sampler the Fisher Implicit Sampler. We then show that the proposed sampler is capable of handling relatively simple target distribution, but would fail for more challenging multi-modal targets.
+
+To remedy this issue and unlock the full potential of the Fisher Implicit Sampler, we additionally propose a novel adaptive training approach that trains the implicit sampler gradually using a sequence of annealed distributions instead of the target distribution. We anneal the target distribution to bridge the hard-to-sample target and an easy-to-sample prior. More precisely, we extend the target distribution $q ( x )$ to a sequence of annealed distributions $\{ q _ { k } ( x ) \} _ { k }$ for $k = 0 , \ldots , K$ , where $q _ { K } ( x )$ is the target density and $q _ { 0 } ( x )$ is an easy-to-sample prior distribution, typically a normal distribution. The design of such an annealed path gradually reduces the learning difficulty for the sampler.
+
+![](images/32d4bfa5ca48c8afb01d954f630d2d649fa4b3bd9bbec99db65bf5bfebd67829.jpg)  
+Figure 1: Illustration of proposed Annealed Fisher Implicit Sampler.
+
+Moreover, we find that a few steps of MC correction after the sampler help the samples spread better with little cost, as also used in some previous work (Wu et al., 2020; Arbel et al., 2021; Matthews et al., 2022). Combining all together, we call our proposed sampler the Annealed Fisher Implicit Sampler (AFIS), as illustrated in Figure 1. We validate our AFIS on sampling benchmarks, showing improvements over baseline approaches.
+
+The main contributions of our work are summarized as follows:
+
+• We propose a novel loss function to minimize the Fisher Divergence. We show that minimizing the proposed loss is equivalent to minimizing the Fisher Divergence between sampler and target distribution. Note that our objective is largely different from other ones in previous work.   
+• We provide an insightful understanding of the difficulty in learning multi-modal targets by minimizing Fisher Divergence. We facilitate the annealing technique on training samplers based on our understanding.   
+• We bring in a novel annealing technique and MC correction steps with our sampler, leading to improved sampling performance with little additional cost.
+
+# 2 BACKGROUND
+
+# 2.1 TRAIN IMPLICIT SAMPLERS WITH SCORE-BASED DIVERGENCE
+
+The learning-to-sample problem arises in many application fields of machine learning. Assume we only have access to an un-normalized target distribution $q ( x )$ (or its logarithm $\log q ( x ) )$ , and the goal is to approximately sample from the target. In recent years, training a neural networkbased transform to approximately sample from target distribution is an appealing method. Such a transform is called a neural sampler. Let $G _ { \theta }$ denote a neural network which transforms a relatively simple latent code $z \sim p _ { 0 } ( z )$ to a sample $x = G _ { \theta } ( z )$ . Here, $p _ { Z } ( z )$ is an easy-to-sample latent distribution, usually the standard Normal distribution. A general neural sampler does not have an explicit expression of the log-likelihood function, which we name them implicit samplers. Because of the un-normalized target distribution and unavailable log-likelihood, training implicit samplers by minimizing KL or related divergence always fails. An alternative way is to consider score-based divergence.
+
+The Stein Neural Sampler of Hu et al. (2018) is trained by minimizing Stein’s Discrepancy between sampler and target distributions. The Stein Discrepancy (SD) (Gorham & Mackey, 2015) is defined as
+
+$$
+\mathcal { D } _ { S D } ( p , q ) = \operatorname* { s u p } _ { \mathbf { f } \in \mathcal { F } } \biggl \{ \mathbb { E } _ { x \sim p } \langle \nabla _ { x } \log q ( x ) , \mathbf { f } ( x ) \rangle + \langle \nabla _ { x } , \mathbf { f } ( x ) \rangle \biggr \} ,
+$$
+
+The calculation of Stein’s discrepancy relies on solving a maximization problem w.r.t. test function f . When the function class $\mathcal { F }$ is carefully chosen, the optimal f may have an explicit solution or easier formulation. For instance, $\mathrm { H u }$ et al. (2018) found that if $\mathcal { F }$ is taken to be $\mathcal { F } \overset { \cdot } { = } \{ \mathbf { f } : \mathbb { E } _ { p } \| \mathbf { f } \| _ { 2 } ^ { 2 } \leq$ $\delta \}$ , the SD is equivalent to a regularized representation
+
+$$
+\mathcal { D } _ { S D } ( p , q ) = \operatorname* { m a x } _ { f } \biggl \{ \mathbb { E } _ { x \sim p } \langle \nabla _ { x } \log q ( x ) , \mathbf { f } ( x ) \rangle + \langle \nabla _ { x } , \mathbf { f } ( x ) \rangle - \lambda \bigl [ \mathbf { f } ^ { T } \mathbf { f } \bigr ] \biggr \} .
+$$
+
+They used two neural networks: $G _ { \theta }$ to parametrize an implicit sampler and $\mathbf { f } _ { \eta }$ to parametrize the test function. Let $p _ { \theta } ( x )$ denote the implicit sampler distribution induced by $x = G _ { \theta } ( z )$ with $z \sim p _ { Z } ( z )$ . Stein Neural Sampler solves a minimax problem on parameter pair $( \theta , \eta )$ to obtain a sampler that minimizes the SD between sampler and target by
+
+$$
+\operatorname* { m i n } _ { \theta } \operatorname* { m a x } _ { \eta } L ( \theta , \eta ) = \operatorname* { m i n } _ { \theta } \operatorname* { m a x } _ { \eta } \Biggl \{ \mathbb { E } _ { x \sim p _ { \theta } } \langle \nabla _ { x } \log q ( x ) , \mathbf { f } _ { \eta } ( x ) \rangle + \langle \nabla _ { x } , \mathbf { f } _ { \eta } ( x ) \rangle - \lambda \big [ \mathbf { f } _ { \eta } ^ { T } \mathbf { f } _ { \eta } \big ] \Biggr \} .
+$$
+
+Here the notion $x \sim p _ { \theta }$ means $x = G _ { \theta } ( z )$ with $z \sim p _ { Z } ( z )$ . They called the above SD the Fisher Stein Discrepancy and the corresponding sampler FSD Neural Sampler.
+
+The Stein Neural Sampler opens the door to training implicit samplers by minimizing score-based Divergence. In fact, the FSD Neural Sampler calculates a surrogate of Fisher Divergence. The FSD’s test function f provides an approximation of Fisher Divergence. However, as we show in Section 3.1, their calculation of Fisher Divergence only provides partial gradient updates of the sampler’s parameters, thus leading to training failure even for simple target.
+
+# 2.2 SCORE FUNCTION ESTIMATION
+
+Since the implicit sampler does not have an explicit log-likelihood function or score function, training it with score-based divergence requires inevitably estimating the score function (or equivalent component). Score matching (Hyvarinen & Dayan, 2005) and its variants provided powerful ap- ¨ proaches to estimating score function through samples. Assume one only has available samples $x \sim p$ , and wants to use a parametric approximated distribution $q _ { \phi } ( x )$ to approximate $p$ . Such an approximation can be made by minimizing the Fisher Divergence between $p$ and $q _ { \phi }$ . We can rewrite the Fisher Divergence as
+
+$$
+\begin{array} { r } { \mathcal { D } _ { F D } ( p , q _ { \phi } ) = \mathbb { E } _ { x \sim p } \bigg \{ \| \nabla _ { x } \log p ( x ) \| _ { 2 } ^ { 2 } + \| \nabla _ { x } \log q _ { \phi } ( x ) \| _ { 2 } ^ { 2 } - 2 \langle \nabla _ { x } \log p ( x ) , \nabla _ { x } \log q _ { \phi } ( x ) \rangle \bigg \} . } \end{array}
+$$
+
+Under certain conditions, the equality $\begin{array} { r } { { \mathbb { E } } _ { x \sim p } \langle \nabla _ { x } \log p ( x ) , \nabla _ { x } \log q _ { \phi } ( x ) \rangle \ = \ - { \mathbb { E } } _ { x \sim p } \Delta \log q _ { \phi } ( x ) } \end{array}$ ∆ log qϕ(x) = Pi ∂2∂x2 l holds (usually referred to as Stein’s Identity(Stein, 1981; Gorham denotes the Laplacian operator applied on $\&$ Mackey, 2017)) . Here $\log q _ { \phi } ( x )$ . Combining this equality and noting that the first term of FD $\mathbb { E } _ { x \sim p } \| \nabla _ { x } \log p ( x ) \| _ { 2 } ^ { 2 }$ does not rely on parameter $\phi$ , we have that minimizing $\mathcal { D } _ { F D } ( p , q _ { \phi } )$ is equivalent to minimizing the following objective
+
+$$
+\mathcal { L } ( \phi ) = \mathbb { E } _ { x \sim p } \bigg \{ \| \nabla _ { x } \log q _ { \phi } ( x ) \| _ { 2 } ^ { 2 } + 2 \Delta \log q _ { \phi } ( x ) \bigg \} .
+$$
+
+This objective can be estimated only through samples from $p$ , thus is tractable when $q _ { \phi }$ is welldefined. More specifically, one only needs to define a score network $s _ { \phi } ( x ) \colon \mathbb { R } ^ { D } \to \mathbb { R } ^ { D }$ instead of a density to estimate the score function of $p$ in some cases. This technique was proposed in Hyvarinen ¨ & Dayan (2005) named after Score Matching. Other variants of score matching were also studied (Song et al., 2019; Vincent, 2011; Pang et al., 2020; Meng et al., 2020; Lu et al., 2022; Bao et al., 2020). Score Matching related techniques have been widely used in training energy-based models and score-based diffusion models in recent years. In this paper, we use score matching related techniques to estimate the score function of the sampler’s distribution.
+
+# 3 ANNEALED FISHER IMPLICIT SAMPLER
+
+3.1 MINIMIZING THE FISHER DIVERGENCE: S2D LOSS
+
+Let $G _ { \theta } ( . ) \colon \mathbb { R } ^ { D _ { Z } } \to \mathbb { R } ^ { D _ { X } }$ be an implicit sampler (i.e., a neural transform), $p _ { Z }$ latent distribution, $p _ { \theta }$ sampler induced distribution $x = \overset { } { G } _ { \theta } ( z )$ , and $q ( x )$ un-normalized target. Our goal is to pull close the FD between $p _ { \theta }$ and $q$ in order to train the sampler. Recall the definition of Fisher Divergence between $p _ { \theta } , q$ is
+
+$$
+\mathcal { D } _ { F D } ( p _ { \theta } , q ) = \mathbb { E } _ { x \sim p _ { \theta } } \| \nabla _ { x } \log p _ { \theta } ( x ) - \nabla _ { x } \log q ( x ) \| _ { 2 } ^ { 2 } .
+$$
+
+For our learning-to-sample setting, the target score function $\nabla _ { x } \log { q ( x ) }$ is known. A direct solution seems work if one uses an additional score network $s _ { \phi } ( . ) \colon \mathbb { R } ^ { D _ { X } } \to \bar { \mathbb { R } } ^ { D _ { X } }$ to approximate sampler’s score function. Samples from implicit sampler is cheap to obtain, so estimating sampler’s score function is not hard with score matching related techniques. We call this step the Score Estimation Step. With a good approximated $s _ { \phi } ( x )$ of sampler’s score function, one may wish to minimize the approximated Fisher Divergence to update the sampler
+
+$$
+\theta ^ { * } = \arg \operatorname* { m i n } _ { \theta } \mathbb { E } _ { x = G _ { \theta } ( z ) , z \sim p _ { Z } ( z ) } \| s _ { \phi } ( x ) - \nabla _ { x } \log q ( x ) \| _ { 2 } ^ { 2 } .
+$$
+
+We call this step the Score Difference Minimization Step. By alternating the above two steps, one may wish the Fisher divergence will be minimized, thus the training of sampler is done. We name the resulting approach the Direct Method. Interestingly, the Direct Method coincides with FSD Neural Sampler as we state in Proposition 1. We put detailed proof in Appendix A due to limited pages.
+
+Proposition 1. Estimating the sampler’s score function $s _ { \phi } ( . )$ with score matching is equivalent to maximizing the Fisher Stein Discrepancy objective to obtain $F S D$ ’s optimal test function. More specially, the optimal score estimation $s ^ { * }$ and FSD optimal test function $\mathbf { f } ^ { * }$ satisfy
+
+$$
+\mathbf { f } ^ { * } ( x ) = \frac { 1 } { 2 \lambda } \big [ \nabla _ { x } \log q ( x ) - s ^ { * } ( x ) \big ] .
+$$
+
+Moreover, the Direct method is equivalent to FSD when training implicit Sampler.
+
+Although the direct method seems reasonable, it fails as we show in the experiment on a simple Banana target in Figure 2. We find that even if sampler’s score function is estimated perfectly at each iteration, the direct method still gives only partial parameter gradient for minimizing the Fisher Divergence. We start by analyzing Fisher Divergence’s gradient w.r.t. sampler’s parameter. The Fisher Divergence is
+
+![](images/e6f8f22c49293e1885c936dea95da974a163888ecc213c5d1a45e861b79a89d5.jpg)  
+Figure 2: Direct method fails for simple Banana distribution while S2D loss succeeds.
+
+One wants to adjust $\theta$ to minimize $\mathcal { L } _ { F D } ( \boldsymbol { \theta } )$ . The $\theta$ gradient of the above objective writes
+
+$$
+\frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } } \| s _ { d } ( x ) - s _ { \theta } ( x ) \| ^ { 2 } = \mathbb { E } _ { p _ { \theta } } \| s _ { d } ( x ) - s _ { \theta } ( x ) \| ^ { 2 } \frac { \partial } { \partial \theta } \log p _ { \theta } ( x ) + \mathbb { E } _ { p _ { \theta } } 2 \big ( s _ { \theta } ( x ) - s _ { d } ( x ) \big ) ^ { T } \frac { \partial } { \partial \theta } s _ { \theta } ( x ) .
+$$
+
+The first gradient term coincides with the direct approach if we asynchronously estimate the sampler’s score function perfectly. More precisely, with perfect score estimation $\boldsymbol { s } _ { \phi } ( \boldsymbol { \hat { x } } ) = \nabla _ { \boldsymbol { x } } \log p _ { \theta } ( \boldsymbol { x } )$ we have
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { \partial } { \partial \theta } \mathbb { E } _ { x \sim p _ { \theta } } \| \nabla _ { x } \log q ( x ) - s _ { \phi } ( x ) \| _ { 2 } ^ { 2 } } \\ & { \displaystyle = \frac { \partial } { \partial \theta } \int \| \nabla _ { x } \log q ( x ) - s _ { \phi } ( x ) \| _ { 2 } ^ { 2 } p _ { \theta } ( x ) d x = \int \| \nabla _ { x } \log q ( x ) - s _ { \phi } ( x ) \| _ { 2 } ^ { 2 } \frac { \partial } { \partial \theta } p _ { \theta } ( x ) d x } \\ & { \displaystyle = \int \| \nabla _ { x } \log q ( x ) - s _ { \phi } ( x ) \| _ { 2 } ^ { 2 } p _ { \theta } ( x ) \frac { \partial } { \partial \theta } \log p _ { \theta } ( x ) d x = \mathbb { E } _ { x \sim p _ { \theta } } \| \nabla _ { x } \log q ( x ) - s _ { \phi } ( x ) \| _ { 2 } ^ { 2 } \frac { \partial } { \partial \theta } \log p _ { \theta } ( x ) . } \end{array}
+$$
+
+The above equation reveals that the direct method only takes partial gradient to minimize the FD between sampler and target. In many cases, this partial gradient leads to training failure as we observe in Figure 2. In (Hu et al., 2018), FSD Neural Sampler used Kernelized Stein Discrepancy trained implicit sampler as initialization before training with FSD. However, such initialization limits the usage of FSD because the optimization might start from a local minima which is close to KSD’s local minima and can potentially be mislead the sampler.
+
+In order to minimize the Fisher Divergence correctly, we propose a novel training objective called Score Square Difference loss (S2D) which accounts for the full parameter gradient to minimize the Fisher Divergence. The S2D loss is defined as the difference of target and sampler’s square score norm, where the sampler’s score function is estimated asynchronously with a score network $s _ { \phi } ( x )$ . More precisely, our S2D loss is defined as
+
+$$
+\begin{array} { r } { \mathcal { L } _ { S 2 D } ( \theta ) : = \mathbb { E } _ { x \sim p _ { \theta } } \bigg \{ \| \nabla _ { x } \log q ( x ) \| _ { 2 } ^ { 2 } - \| s _ { \phi } ( x ) \| _ { 2 } ^ { 2 } \bigg \} , } \end{array}
+$$
+
+where $s _ { \phi } ( . )$ is the estimated score function of sampler distribution. The score function is usually estimated by score matching related techniques. The notation $x \sim p _ { \theta }$ means $x = G _ { \theta } ( z ) , z \sim p _ { Z } ( z )$ . The following proposition 2 shows that, if the sampler score function is estimated perfectly, the parameter gradient of S2D loss is the same as the gradient of Fisher Divergence.
+
+Proposition 2. Assume $\boldsymbol { s } _ { \phi } ( \boldsymbol { x } ) = \nabla _ { \boldsymbol { x } } \log p _ { \theta } ( \boldsymbol { x } )$ . Then the following equality holds:
+
+$$
+\frac { \partial } { \partial \theta } \mathcal { L } _ { S 2 D } ( \theta ) = \frac { \partial } { \partial \theta } \mathcal { L } _ { F D } ( \theta ) .
+$$
+
+We give the detailed proof in Appendix B. This proposition says that, if we alternate between score estimation of sampler’s score function, and minimization of the S2D loss, we are actually minimizing the Fisher Divergence between sampler and target. The S2D loss is a surrogate of Fisher Divergence which can provide the same parameter gradient as Fisher Divergence. So minimizing the S2D loss gives the same results as minimizing the intractable Fisher Divergence. Figure 3 gives an illustration of the relation between S2D loss and Fisher Divergence. The black curve stands for the intractable Fisher Divergence. Green curve represents the S2D loss. The S2D loss shares the same gradient parameter as Fisher Divergence. We refer to a sampler trained with such approach the Fisher Implicit Sampler (FIS). We give an algorithm for FIS in Algorithm 1. We take standard score matching as an illustration of score estimation step, but other score estimation techniques such as denoising score matching and sliced score matching also works.
+
+# Algorithm 1: Fisher Implicit Sampler training
+
+Input: un-normalized target $\log q ( x )$ , latent distribution $p _ { Z } ( z )$ , implicit sampler $G _ { \theta }$ , score network $s _ { \phi }$ , mini-batch size $\mathbf { B }$ , max iteration $\mathbf { M }$ .   
+Randomly initialize $( \theta ^ { ( 0 ) } , \phi ^ { ( 0 ) } )$ .   
+for $t$ in $\it 0 . M$ do # update score network parameter Get mini-batch $x _ { i } = \bar { G _ { \theta ^ { ( t ) } } } ( z _ { i } ) , z _ { i } \sim p _ { Z } ( z ) , i = 1 , . . , B$ . Calculate score matching objective: $\begin{array} { r } { \mathcal { L } _ { S M } ( \boldsymbol { \phi } ) = \frac { 1 } { B } \sum _ { i = 1 } ^ { B } \biggl [ \| s _ { \boldsymbol { \phi } } ( x _ { i } ) \| _ { 2 } ^ { 2 } + 2 \langle \nabla _ { \boldsymbol { x } } , s _ { \boldsymbol { \phi } } ( x _ { i } ) \rangle \biggr ] . } \end{array}$ Minimize ${ \mathcal { L } } _ { S M } ( \phi )$ to get $\phi ^ { ( t + 1 ) }$ . # update sampler parameter Get mini-batch latent code $z _ { i } \sim p _ { Z } ( z ) , i = 1 , . . . , B$ . Use re-parametrization trick to calculate S2D loss for sampler $\begin{array} { r } { \dot { \mathcal { L } } _ { S 2 D } ( \dot { \theta } ) = \frac { 1 } { B } \sum _ { i = 1 } ^ { B } \left[ \| \nabla _ { x } \log q ( G _ { \theta } ( z _ { i } ) ) \| _ { 2 } ^ { 2 } - \| s _ { \phi ^ { ( t + 1 ) } } ( \dot { G _ { \theta } } ( z _ { i } ) ) \| _ { 2 } ^ { 2 } \right] . } \end{array}$ Minimize $\mathcal { L } _ { S 2 D } ( \theta )$ to get $\theta ^ { ( t + 1 ) }$ .
+
+end return $( \theta , \phi )$ .
+
+![](images/481ef58caba92912744ef498168af2d95e91be31068139220c80a41c2e5841ff.jpg)  
+Figure 3: S2D loss and Fisher Divergence. The S2D loss shares the same parameter gradient as Fisher Divergence if sampler’s score is estimated perfectly asynchronously. Thus minimizing the S2D loss to update the sampler is equivalent to minimizing the Fisher Divergence between sampler and target.   
+Figure 2 shows that our proposed FIT (S2D loss) can successfully train an implicit sampler from scratch to sample from the famous banana shape distribution. While the Direct method fails to train the correct sampler.
+
+Although FIT is capable of handling benchmark targets, we find that FIT fails on more challenging multi-modal targets with very separated modes. To remedy the multi-modal failure issues and fully unlock the potential of the S2D loss, we propose to combine the annealing techniques with FIT for multi-modal targets. The idea of annealing is widely used in sampling and stochastic optimization literature (Neal, 2001; Salimans et al., 2015; Chen et al., 2016; Doucet et al., 2001; Van Laarhoven & Aarts, 1987). The technique constructs a distribution bridge between a relatively simple prior distribution and a complicated target. The learning (or other operations such as sampling or optimization) are gradually operated on each middle distribution from prior to the tar
+
+get. Typically, the annealing technique can lower the barrier of operation of the target by dispersing the difficulty to all middle distributions.
+
+# 3.2 ANNEALED FISHER IMPLICIT TRAINING
+
+By executing FIT steps repeatedly, the sampler is trained to minimize the Fisher divergence between $p _ { \theta }$ and target $q$ . However, directly minimizing the Fisher divergence is problematic in practice. If the sampler’s distribution is too dissimilar to the target, the Fisher divergence could be hard to estimate accurately as mentioned in (Wenliang & Kanagawa, 2020). The Fisher divergence can be small under any tolerance even if two distribution are largely different in terms of KL divergence. More precisely, the Fisher Divergence is likely inaccurate if two distributions are too dissimilar. Due to this issue, the sampler might not be able to estimated the Fisher divergence accurately, making the training fail. In fact, above issue occurs a lot in real applications. Sampler is often initialized to concentrate around the origin, while the target distribution rarely concentrates around the origin.
+
+To remedy the inaccurate Score Estimation issue, we need to guide the sampler to start from learning a relatively simple target, and then the more challenging one. Based on such intuition, we introduce a gradual relaxation of target distribution. More precisely, we construct a sequence of annealed distributions $\{ q _ { k } \} , k \in \{ 0 , . . , K \}$ which gradually transform a relatively simple distribution $q _ { 0 }$ to target distribution $q _ { K } = q$ . Typically, $q _ { 0 }$ is chosen as $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ for simplicity. We let the sampler gradually learn to sample from each $q _ { k }$ with $k$ increasing from $k = 0$ to $k = K$ . Since when $k$ is small $q _ { k }$ is simpler than $q _ { K }$ , the estimation of Fisher divergence is easier. Thus the sampler can learn to approximate $q _ { k }$ . When one gradually turns $k$ to $k = K$ , the sampler will gradually learn to sample from our final target $q _ { K } = q$ .
+
+Such easy-to-hard technique is commonly known as annealing techniques. Marinari & Parisi (1992); Geyer & Thompson (1995) proved faster mixing time with temperature annealed target. Wenzel et al. (2020) utilized anneal path to connect model and posterior in Bayesian inference regime. Mandt et al. (2016); Huang et al. (2018); Fu et al. (2019) annealed the KL regularization in variational inference. D’Angelo & Fortuin (2021) proposed to anneal the target when running Stein Variational Gradient Descent algorithm for better mixing speed. Perhaps the most similar annealing approach to ours is Neal (2001); Wu et al. (2020) which construct a geometric distribution path $p _ { k } ( x )$ between a Gaussian prior and target density. We utilize a similar anneal path as in Wu et al. (2020).
+
+In this paper, we anneal the target distribution $q$ with a geometric interpolation starting with a standard Gaussian distribution as prior
+
+$$
+\begin{array} { r } { \log q _ { k } ( x ) = \lambda _ { k } \log q _ { K } ( x ) + ( 1 - \lambda _ { k } ) \log q _ { 0 } ( x ) } \end{array}
+$$
+
+with $q _ { 0 } = \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ and $0 \leq \lambda _ { k } \leq 1$ a pre-defined annealing schedule function with $\lambda _ { 0 } = 0 , \lambda _ { K } = 1$ The score function is then linearly interpolated with
+
+$$
+\begin{array} { r } { \nabla _ { x } \log q _ { k } ( x ) = \lambda _ { k } \nabla _ { x } \log q _ { K } ( x ) + ( 1 - \lambda _ { k } ) \nabla _ { x } \log q _ { 0 } ( x ) , } \end{array}
+$$
+
+where $\nabla _ { x } \log q _ { K } ( x ) = \nabla _ { x } \log q ( x )$ is the target score function and $\nabla _ { x } \log { q _ { 0 } ( x ) }$ the prior score. For standard Normal prior, we have $\nabla _ { x } \log q _ { 0 } ( x ) = - x$ . We name our FIS sampler combined with annealing technique the Annealed Fisher Implicit Training. Because of the pages limitation, we put the full AFIS algorithm in Appendix F. By annealing the target distribution to a sequence of easier-to-learn targets, we divide the difficulty of sampler to learn one final distribution to learn sequentially from less difficult targets. Thus the sampler will not be bothered by inaccurate Fisher divergence estimation and training failure. Figure 1 gives a brief summary of how AFIS works. The Annealed Fisher Implicit Sampler is trained along annealed distributions progressively.
+
+# 3.3 MONTE CARLO CORRECTION
+
+Deterministic sampler suffers from mode-connection issue. The issue says that a deterministic transform can not fully disconnect two modes as studied in Wu et al. (2020). Such issue limit the use of a pure deterministic sampler. Recent works show that combining stochastic corrections with deterministic transforms could improve the sampling performance (Wu et al., 2020; Song et al., 2020; Song & Ermon, 2019). MCMC(Hastings, 1970; Roberts & Rosenthal, 1998; Xifara et al., 2014; Neal, 2011) is a commonly used stochastic transform family. By running MCMC, one can approximated sample from some un-normalized target distribution. Thus a few steps MCMC is a nice way to serve as stochastic corrections.
+
+In particular, after training the sampler, we take the generated samples $x = G _ { \theta } ( z ) , z \sim p _ { 0 } ( z )$ as initialization and run several MCMC as correction steps to spread samples for better diversity. Both energy-based and score-based MCMC can be used. We take the Langevin MC as an illustration and put more details of MC corrections in Appendix C. Note that our method is not limited to these MC corrections.
+
+Langevin Dynamic Correction A set of particles is assumed to reach $q ( x )$ as a stationary distribution if it is driven by a Langevin Dynamic with local updates
+
+$$
+d X _ { t } = \nabla _ { X _ { t } } \log q ( X _ { t } ) / 2 + d W _ { t } ,
+$$
+
+where $W _ { t }$ is standard Brownian motion. The discrete scheme of Langevin Correction is given by
+
+$$
+X ^ { ( t + 1 ) } = X ^ { ( t ) } + \frac { \epsilon } { 2 } \nabla \log q ( X ^ { ( t ) } ) + \sqrt { \epsilon } Z ^ { ( t ) } ,
+$$
+
+where $Z ^ { ( t ) } \sim \mathcal { N } ( 0 ; I )$ . The Fokker-Planck equation tells that under certain conditions, $q ( x )$ is the only stationary distribution of above diffusion dynamic. About 20 updates of steps is sufficient to have good enough correction effects in practice.
+
+The combination of deterministic sampler and stochastic correction in fact gives faster mixing for MCMC. The deterministic sampler sample particles coarsely near target’s high density modes. After that, the MCMC helps the particle spread better around each modes. In particular, we show that Langevin mixing time can be controlled by Fisher divergence between sampler distribution and target. Taking advantage of flexible neural network architecture, AFIS can be trained to match target score at any precision. The Theorem 1 shows that Langevin Correction’s mixing time can be reduced by well trained sampler.
+
+Theorem 1. Assume the target potential $\log q ( x )$ is smooth and satisfies
+
+$$
+\operatorname* { l i m } _ { \| x \| _ { 2 } \to + \infty } \left( \frac { \| \nabla \log q ( x ) \| _ { 2 } ^ { 2 } } { 2 } - \Delta \log q ( x ) \right) = + \infty .
+$$
+
+![](images/053e63e030b10b771291c8d877939da98dad199b289915ff7c81c70690b12f64.jpg)  
+Figure 4: Sample comparison on Double Well targets. (a) real samples; (b) samples from trained AFIS with 5 steps of HMC correction; (c) samples from trained AFIS; (d) samples from trained FIS without annealing; (e) samples from trained FSD-NS. All samplers and score networks use the same architecture.
+
+Assume generated distribution $p$ induced by AFIS $x = G ( z )$ is trained to match Fisher divergence under $\delta$ precision $\mathcal { D } _ { F } ( p , q ) \leq \delta$ . Then there exists a positive constant $\lambda$ and a dimension-free positive constant $C$ which only depend on target distribution $q ( x )$ , such that under Langevin diffusion with initial distribution $p _ { 0 } = p$ ,
+
+$$
+d X _ { t } = \nabla \log q ( x ) / 2 d t + d W _ { t } ,
+$$
+
+the diffusion time
+
+$$
+T ^ { * } = \operatorname* { m a x } \biggl \{ 0 , \frac { 1 } { 2 \lambda } \bigl [ C + \log ( \frac { \delta } { \epsilon } ) \bigr ] \biggr \}
+$$
+
+is enough to control the KL divergence between corrected distribution $p _ { T }$ and target $q$ under tolerance ϵ.
+
+In practice, the AFIS can be trained to achieve any precision to match target under Fisher Divergence. The above theorem says, the better AFIS is trained, the shorter time for MC correction is needed to achieve same tolerance in terms of KL divergence. We provide the detailed proof in Appendix D.
+
+# 3.4 COMBINING ALL: THE ANNEALED FISHER IMPLICIT SAMPLER
+
+Combining the S2D loss, the annealed technique, and MC corrections, we obtain our final sampler: the Annealed Fisher Implicit Sampler (AFIS) with MC corrections. Figure 4 shows a comparison of trained sampler’s samples on Double Well distribution. Double Well is a usually used bi-variate testing target with two separated modes. The figure shows that the AFIS with a few steps of MC correction gives the best samples. The AFIS with no MCMC correction can not fully separate two disjoint modes. The FIS (without annealing) fails to learn the two modes. The FSD-NS (or the Direct Method) also fails for training. To be concluded, the experiments show that S2D loss, annealed technique, and MC correction all contribute to successful learning.
+
+# 4 EXPERIMENTS
+
+# 4.1 AFIS FOR SYNTHETIC TARGET
+
+For sanity check, we apply AFIS on some toy target distributions as used in Hu et al. (2018); Rezende & Mohamed (2015). The anneal path $p _ { \lambda } ( x ) \propto \bar { \exp } ( \lambda \log p _ { t a r g e t } ( x ) + ( 1 - \lambda ) \log p _ { p r i o r } ( x ) )$ starts from a Normal distribution when $\lambda = 0$ and ends with the target when $\lambda = 1$ . Let $M$ be the number of max iterations, and $t$ be the current training iteration. We set $\lambda _ { i }$ to grow linearly from 0 to 1 when $i < 9 M / 1 0$ . We train the sampler with real target $\log q ( x )$ for rest $M / 1 0$ iterations. The annealed path reduces the bar of learning to sample, resulting relatively accurate updating direction for the current sampler. The sampler is guided along the annealed path towards the target. We defer the detailed experiment settings and more results to Appendix E.1.
+
+Specifically, we visualize the sample results on three distributions with hard-to-sample characteristics such as multi-modality and periodicity, as shown in Figure 5. It shows that samples from our AFIS+MC method perfectly match all target distributions. For quantitative comparison, we calculate the Maximum Mean Discrepancy between the pure HMC samples and all samplers’ samples. The FSD-NS does not converge when training, so we omit the result of FSD-NS in comparison. Since the task focuses on training implicit samplers, we do not compare other explicit samplers. Table 1 summarizes the results of the MMD evaluation of all samplers. In all datasets, our AFIS consistently performs better than FIS. With additional MC correction steps, we always get lower MMD compared to the pure AFIS method.
+
+![](images/30b421d000ec025d09c14ecb35dcf9bb99faf884be5da29499f2988ae3559759.jpg)  
+Figure 5: Target and AFIS+MC samples.
+
+Table 1: MMD (with rbf kernel) evaluation for synthetic targets. Additional 10 Langevin MC correction steps are used in AFIS+MC sampler. The lower the metric, the better the sampler.   
+
+<table><tr><td>Target</td><td>banana</td><td>double well</td><td>t1</td><td>t2</td><td>t3</td></tr><tr><td>FIS(ours)</td><td>1.12e-2±1.07e-3</td><td>3.51e-1±3.06e-3</td><td>4.54e-2±4.10e-3</td><td>7.48e-2±1.81e-3</td><td>5.18e-2±2.68e-3</td></tr><tr><td>AFIS(ours)</td><td>7.07e-4±1.72e-4</td><td>1.07e-2±1.37e-3</td><td>3.31e-3±1.13e-3</td><td>4.64e-2±2.53e-3</td><td>2.65e-2±1.91e-3</td></tr><tr><td>AFIS+MC(ours)</td><td>2.45e-4±1.20e-4</td><td>5.99e-3±1.33e-3</td><td>2.15e-3±8.37e-4</td><td>3.61e-2±2.67e-3</td><td>2.20e-2±1.97e-3</td></tr></table>
+
+# 4.2 BAYESIAN REGRESSION
+
+We also test our Implicit Sampler on Bayesian regression tasks as in Song et al. (2017). HMC is a good baseline for such tasks, as pointed out in Neklyudov et al. (2020); Neklyudov & Welling (2022). The inference of the Bayesian logistic regression model aims to sample from the posterior distribution. We compare FIS (no anneal), AFIS, and $\mathbf { A F I S + M C }$ on Australian, German, and Heart datasets. To evaluate samples’ quality, we run HMC as a baseline to obtain approximated samples from target distributions and calculate Maximum Mean Discrepancy between samples from implicit samplers and HMC baseline. Table 2 shows the results of the Bayesian inference experiments. Other than FSD-NS, which always fails during training, our generators can generate high-quality samples. Moreover, annealed technique and MC correction steps further improve sample quality. Experimental details can be found in Appendix E.2.
+
+Table 2: MMD (with rbf kernel) evaluation for posterior sampling. Additional 10 Langevin MC correction steps are used in AFIS+MC sampler. The lower the metric, the better the sampler.   
+
+<table><tr><td>Posterior</td><td>Australian</td><td>German</td><td>Heart</td></tr><tr><td>FIS(ours)</td><td>7.99e-3±2.81e-4</td><td>1.91e-4±6.48e-6</td><td>9.84e-5±1.08e-5</td></tr><tr><td>AFIS(ours)</td><td>6.30e-3±2.50e-4</td><td>2.42e-6±4.02e-7</td><td>3.66e-5±1.08e-5</td></tr><tr><td>AFIS+MC(ours)</td><td>2.16e-3±1.08e-4</td><td>2.46e-6±3.97e-7</td><td>3.64e-5±1.07e-5</td></tr></table>
+
+# 5 CONCLUSION
+
+We have presented a novel approach for training an implicit sampler to sample from un-normalized density. Our approach minimizes the Fisher Divergence with the aid of an asynchronous score network. We show theoretically that our method can accurately minimize the Fisher Divergence for the implicit sampler, which is the first one as far as we know. Besides, our approach uses both the annealing technique and stochastic corrections for improved sampling performance. We also prove the faster mixing for MC correction. We test our approach on commonly used synthetic target generation and Bayesian regression benchmarks and observe ideal performance.
+
+# ETHICS STATEMENT
+
+Our work proposes an approach to train an implicit sampler by minimizing Fisher Divergence between sampler and target distribution. Since the research is a fundamental methodology in machine learning, the negative consequences of the methodology seem not obvious.
+
+# REPRODUCIBILITY STATEMENT
+
+We provide details of our approach and sampler in Appendix. We provide complete proofs of all theoretical results also in Appendix. We also propose the python code for implementation. We state that our research is reproducible.
+
+# REFERENCES
+
+Christophe Andrieu, Nando De Freitas, Arnaud Doucet, and Michael I Jordan. An introduction to mcmc for machine learning. Machine learning, 50(1):5–43, 2003.
+
+Michael Arbel, Alex Matthews, and Arnaud Doucet. Annealed flow transport monte carlo. In International Conference on Machine Learning, pp. 318–330. PMLR, 2021.
+
+Fan Bao, Chongxuan Li, Kun Xu, Hang Su, Jun Zhu, and Bo Zhang. Bi-level score matching for learning energy-based latent variable models. Advances in Neural Information Processing Systems, 33:18110–18122, 2020.
+
+MS Bartlett. Approximate confidence intervals. Biometrika, 40(1/2):12–19, 1953.
+
+Changyou Chen, David Carlson, Zhe Gan, Chunyuan Li, and Lawrence Carin. Bridging the gap between stochastic gradient mcmc and stochastic optimization. In Artificial Intelligence and Statistics, pp. 1051–1060. PMLR, 2016.
+
+Adrien Corenflos, James Thornton, George Deligiannidis, and Arnaud Doucet. Differentiable particle filtering via entropy-regularized optimal transport. In International Conference on Machine Learning, pp. 2100–2111. PMLR, 2021.
+
+Francesco D’Angelo and Vincent Fortuin. Annealed stein variational gradient descent. In Third Symposium on Advances in Approximate Bayesian Inference, 2021.
+
+Arnaud Doucet, Nando de Freitas, and Neil Gordon. An introduction to sequential monte carlo methods. In Sequential Monte Carlo methods in practice, pp. 3–14. Springer, 2001.
+
+Hao Fu, Chunyuan Li, Xiaodong Liu, Jianfeng Gao, Asli C¸ elikyilmaz, and Lawrence Carin. Cyclical annealing schedule: A simple approach to mitigating kl vanishing. In NAACL-HLT (1), 2019.
+
+Charles J Geyer and Elizabeth A Thompson. Annealing markov chain monte carlo with applications to ancestral inference. Journal of the American Statistical Association, 90(431):909–920, 1995.
+
+Jackson Gorham and Lester Mackey. Measuring sample quality with stein’s method. Advances in Neural Information Processing Systems, 28, 2015.
+
+Jackson Gorham and Lester Mackey. Measuring sample quality with kernels. In International Conference on Machine Learning, pp. 1292–1301. PMLR, 2017.
+
+Peter J Green. Reversible jump markov chain monte carlo computation and bayesian model determination. Biometrika, 82(4):711–732, 1995.
+
+Leonard Gross. Logarithmic sobolev inequalities. American Journal of Mathematics, 97(4):1061– 1083, 1975.
+
+WK Hastings. Monte carlo sampling methods using markov chains and their applications. Biometrika, 57(1):97–109, 1970.
+
+Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840–6851, 2020.
+
+Tianyang Hu, Zixiang Chen, Hanxi Sun, Jincheng Bai, Mao Ye, and Guang Cheng. Stein neural sampler. arXiv preprint arXiv:1810.03545, 2018.
+
+Chin-Wei Huang, Shawn Tan, Alexandre Lacoste, and Aaron C Courville. Improving explorability in variational inference with annealed variational objectives. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
+
+Aapo Hyvarinen and Peter Dayan. Estimation of non-normalized statistical models by score match-¨ ing. Journal of Machine Learning Research, 6(4), 2005.
+
+Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. Advances in neural information processing systems, 34:21696–21707, 2021.
+
+Durk P Kingma and Yann Cun. Regularized estimation of image statistics by score matching. Advances in neural information processing systems, 23, 2010.
+
+Cheng Lu, Kaiwen Zheng, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Maximum likelihood training for score-based diffusion odes by high order denoising score matching. In International Conference on Machine Learning, pp. 14429–14460. PMLR, 2022.
+
+Stephan Mandt, James McInerney, Farhan Abrol, Rajesh Ranganath, and David Blei. Variational tempering. In Arthur Gretton and Christian C. Robert (eds.), Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine Learning Research, pp. 704–712, Cadiz, Spain, 09–11 May 2016. PMLR.
+
+Enzo Marinari and Giorgio Parisi. Simulated tempering: a new monte carlo scheme. EPL (Europhysics Letters), 19(6):451, 1992.
+
+James Martens, Ilya Sutskever, and Kevin Swersky. Estimating the hessian by back-propagating curvature. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pp. 963–970, 2012.
+
+Agdg Matthews, M. Arbel, D. J. Rezende, and A. Doucet. Continual repeated annealed flow transport monte carlo. 2022.
+
+Chenlin Meng, Lantao Yu, Yang Song, Jiaming Song, and Stefano Ermon. Autoregressive score matching. Advances in Neural Information Processing Systems, 33:6673–6683, 2020.
+
+Radford M Neal. Annealed importance sampling. Statistics and computing, 11(2):125–139, 2001.
+
+Radford M Neal. Mcmc using hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo, pp. 139–188. Chapman and Hall/CRC, 2011.
+
+Kirill Neklyudov and Max Welling. Orbital mcmc. In International Conference on Artificial Intelligence and Statistics, pp. 5790–5814. PMLR, 2022.
+
+Kirill Neklyudov, Max Welling, Evgenii Egorov, and Dmitry Vetrov. Involutive mcmc: a unifying framework. In International Conference on Machine Learning, pp. 7273–7282. PMLR, 2020.
+
+Peter Olsson. Two phase transitions in the fully frustrated xy model. Physical review letters, 75(14): 2758, 1995.
+
+Tianyu Pang, Kun Xu, Chongxuan Li, Yang Song, Stefano Ermon, and Jun Zhu. Efficient learning of generative models via finite-difference score matching. Advances in Neural Information Processing Systems, 33:19175–19188, 2020.
+
+Grigorios A Pavliotis. Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations, volume 60. Springer, 2014.
+
+Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In International conference on machine learning, pp. 1530–1538. PMLR, 2015.
+
+Gareth O Roberts and Jeffrey S Rosenthal. Optimal scaling of discrete approximations to langevin diffusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(1): 255–268, 1998.
+
+Tim Salimans, Diederik Kingma, and Max Welling. Markov chain monte carlo and variational inference: Bridging the gap. In International conference on machine learning, pp. 1218–1226. PMLR, 2015.
+
+Ch Schutte, Alexander Fischer, Wilhelm Huisinga, and Peter Deuflhard. A direct approach to con- ¨ formational dynamics based on hybrid monte carlo. Journal of Computational Physics, 151(1): 146–168, 1999.
+
+Jiaming Song, Shengjia Zhao, and Stefano Ermon. A-nice-mc: Adversarial training for mcmc. Advances in Neural Information Processing Systems, 30, 2017.
+
+Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. Advances in Neural Information Processing Systems, 32, 2019.
+
+Yang Song, Sahaj Garg, Jiaxin Shi, and Stefano Ermon. Sliced score matching: A scalable approach to density and score estimation. In Proceedings of the Thirty-Fifth Conference on Uncertainty in Artificial Intelligence, UAI 2019, Tel Aviv, Israel, July 22-25, 2019, pp. 204, 2019.
+
+Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020.
+
+Charles M Stein. Estimation of the mean of a multivariate normal distribution. The annals of Statistics, pp. 1135–1151, 1981.
+
+Arash Vahdat, Karsten Kreis, and Jan Kautz. Score-based generative modeling in latent space. Advances in Neural Information Processing Systems, 34:11287–11302, 2021.
+
+Peter JM Van Laarhoven and Emile HL Aarts. Simulated annealing. In Simulated annealing: Theory and applications, pp. 7–15. Springer, 1987.
+
+Pascal Vincent. A connection between score matching and denoising autoencoders. Neural computation, 23(7):1661–1674, 2011.
+
+Li K Wenliang and Heishiro Kanagawa. Blindness of score-based methods to isolated components and mixing proportions. arXiv e-prints, pp. arXiv–2008, 2020.
+
+Florian Wenzel, Kevin Roth, Bastiaan Veeling, Jakub Swiatkowski, Linh Tran, Stephan Mandt, Jasper Snoek, Tim Salimans, Rodolphe Jenatton, and Sebastian Nowozin. How good is the Bayes posterior in deep neural networks really? In Hal Daume III and Aarti Singh (eds.), ´ Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pp. 10248–10259. PMLR, 13–18 Jul 2020.
+
+Hao Wu, Jonas Kohler, and Frank No ¨ e. Stochastic normalizing flows. ´ Advances in Neural Information Processing Systems, 33:5933–5944, 2020.
+
+Tatiana Xifara, Chris Sherlock, Samuel Livingstone, Simon Byrne, and Mark Girolami. Langevin diffusions and the metropolis-adjusted langevin algorithm. Statistics & Probability Letters, 91: 14–19, 2014.
+
+Takuya Yamano. Skewed jensen—fisher divergence and its bounds. Foundations, 1(2):256–264, 2021.
+
+# A PROOF OF PROPOSITION 1
+
+We provide the proof of Proposition 1 here.
+
+Proof. With fixed $p$ and known target $q$ , the optimal test function $\mathbf { f } ^ { * }$ has representation
+
+$$
+\mathbf { f } ^ { * } = \arg \operatorname* { m i n } _ { \mathbf { f } } \mathcal { L } ( \mathbf { f } )
+$$
+
+Where functional $\mathcal { L } ( \mathbf { f } )$ has integral representation
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } ( f ) = \mathbb { E } _ { x \sim p } \bigg \{ \langle \nabla _ { x } \log q ( x ) , \mathbf { f } ( x ) \rangle + \langle \nabla _ { x } , \mathbf { f } ( x ) \rangle - \lambda [ \mathbf { f } ^ { T } ( x ) \mathbf { f } ( x ) ] \bigg \} } \\ { \displaystyle \qquad = \int p ( x ) \langle \nabla _ { x } \log q ( x ) , \mathbf { f } ( x ) \rangle + p ( x ) \langle \nabla _ { x } , \mathbf { f } ( x ) \rangle - \lambda p ( x ) [ \mathbf { f } ^ { T } ( x ) \mathbf { f } ( x ) ] d x } \\ { \displaystyle \qquad = \int l ( x , \mathbf { f } , \nabla \mathbf { f } ) d x . } \end{array}
+$$
+
+Here $\begin{array} { r } { l ( x , \mathbf { f } , \nabla \mathbf { f } ) = \int p ( x ) \langle \nabla _ { x } \log q ( x ) , \mathbf { f } ( x ) \rangle + p ( x ) \langle \nabla _ { x } , \mathbf { f } ( x ) \rangle - \lambda p ( x ) [ \mathbf { f } ^ { T } ( x ) \mathbf { f } ( x ) ] . } \end{array}$ . By EulerLagrange equation, the optimal function f satisfies
+
+$$
+\frac { \partial l } { \partial { \bf f } } - \frac { d } { d x } ( \frac { \partial l } { \partial { \bf f ^ { \prime } } } ) + \frac { \partial ^ { 2 } } { \partial x ^ { 2 } } ( \frac { \partial l } { \partial { \bf f ^ { \prime \prime } } } ) = 0 .
+$$
+
+By calculation, we have
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial l } { \partial { \bf f } } ( { \boldsymbol x } ) = p ( { \boldsymbol x } ) \nabla \log q ( { \boldsymbol x } ) - 2 \lambda p ( { \boldsymbol x } ) { \bf f } ( { \boldsymbol x } ) } \\ { \displaystyle \frac { d } { d x } ( \frac { \partial l } { \partial { \bf f ^ { \prime } } } ) ( { \boldsymbol x } ) = \nabla _ { { \boldsymbol x } } p ( { \boldsymbol x } ) } \\ { \displaystyle \frac { \partial l } { \partial { \bf f ^ { \prime \prime } } } ( { \boldsymbol x } ) = 0 . } \end{array}
+$$
+
+So the optimal $\mathbf { f } ^ { * }$ satisfies the Euler-Lagrange equation as
+
+$$
+p ( x ) \nabla _ { x } \log q ( x ) - 2 \lambda p ( x ) \mathbf { f } ( x ) - \nabla _ { x } p ( x ) = 0 .
+$$
+
+Divide the both side with $p ( x )$ and note that $\nabla _ { x } p ( x ) / p ( x ) = \nabla _ { x } \log p ( x )$ , the equation turns to
+
+$$
+\mathbf { f } ^ { * } ( x ) = \frac { 1 } { 2 \lambda } \big [ \nabla _ { x } \log q ( x ) - \nabla _ { x } \log p ( x ) \big ] .
+$$
+
+Next consider optimal $s ^ { * }$ . The $s ^ { * }$ is obtained by minimizing the Score Matching objective, which is equivalent to minimizing the Fisher divergence between $p$ and $s$ induced family, thus the optimal $\boldsymbol { s } ^ { * } ( \bar { \boldsymbol { x } } ) = \nabla _ { \boldsymbol { x } } \log \boldsymbol { p } ( \boldsymbol { x } )$ . Substitute $\nabla _ { x } \log p ( \bar { x ) }$ with $s ^ { * }$ into $f ^ { * }$ formula, we have
+
+$$
+\mathbf { f } ^ { * } ( x ) = \frac { 1 } { 2 \lambda } \big [ \nabla _ { x } \log q ( x ) - s ^ { * } ( x ) \big ] .
+$$
+
+# B PROOF OF PROPOSITION 2
+
+In this section, we prove that the S2D loss and Fisher Divergence shares exactly the same parameter gradient.
+
+Proof. Let $p _ { \theta }$ denote sampler’s distribution. $s _ { \theta }$ denote the true but unknown sampler’s score function. $q$ denotes the known un-normalized target. For rest of the proof, the notion $\| x \|$ represents the $L ^ { 2 }$ norm of a vector in $D _ { X }$ dimensional Euclidean space $x ~ \in \mathbb { R } ^ { D _ { X } }$ . Recall that the Fisher Divergence is defined as
+
+$$
+\begin{array} { r } { \mathcal { L } _ { F D } ( \theta ) = \mathbb { E } _ { x \sim p _ { \theta } } \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Thus the sampler parameter gradient of Fisher Divergence writes
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } } \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } = \frac { \partial } { \partial \theta } \int \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| _ { 2 } ^ { 2 } p _ { \theta } ( x ) d x } \\ { = \displaystyle \int \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| _ { 2 } ^ { 2 } \frac { \partial } { \partial \theta } p _ { \theta } ( x ) d x + \displaystyle \int p _ { \theta } ( x ) \frac { \partial } { \partial \theta } \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| _ { 2 } ^ { 2 } d x } \\ { = \mathbb { E } _ { p _ { \theta } } \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } \frac { \partial } { \partial \theta } \log p _ { \theta } ( x ) + \mathbb { E } _ { p _ { \theta } } 2 ( s _ { \theta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \frac { \partial } { \partial \theta } s _ { \theta } ( x ) } \\ { = ( 1 ) + ( 2 ) . } \end{array}
+$$
+
+The first term can be estimated with
+
+$$
+\begin{array} { l } { \displaystyle ( 1 ) = \int \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } \frac { \partial } { \partial \theta } p _ { \theta } ( x ) d x } \\ { \displaystyle = \frac { \partial } { \partial \theta } \int \mathbf { s g } \big [ \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } \big ] p _ { \theta } ( x ) } \\ { \displaystyle = \frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } } \mathbf { s g } \bigg [ \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } \bigg ] . } \end{array}
+$$
+
+Here the operator sg denotes stop gradient operator with respect to parameter $\theta$ . $\mathbf { s g } [ f _ { \theta } ]$ stop the parameter dependence of $\theta$ for function $f$ , meaning that one can only evaluate $f _ { \theta } ( x )$ point-wise but can not obtain the $\theta$ gradient of $f _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ . Here we stop the gradient of function $\| \nabla \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 }$ , so we can use another score network $s _ { \phi }$ to approximate $s _ { \theta }$ point-wise, regardless of the $\theta$ parameter dependence. Next we consider the second term. The second term turns to
+
+$$
+\begin{array} { r l } & { 2 ) = \mathbb { E } _ { p \rho ^ { 2 } } ( s _ { \vartheta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \frac { \partial } { \partial \theta } s _ { \vartheta } ( x ) } \\ & { = \mathbb { E } _ { p _ { \vartheta } ^ { 2 } } ( s _ { \vartheta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \frac { \partial } { \partial \theta } \nabla _ { x } \log p _ { \vartheta } ( x ) } \\ & { = 2 \displaystyle \int p _ { \vartheta } ( x ) ( s _ { \vartheta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \frac { \partial } { \partial \theta } \frac { \partial } { \partial x } \log p _ { \vartheta } ( x ) d x } \\ & { = 2 \displaystyle \int p _ { \vartheta } ( x ) ( s _ { \vartheta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \frac { \partial } { \partial \theta } \left[ \frac { 1 } { p _ { \vartheta } ( x ) } \frac { \partial p _ { \vartheta } ( x ) } { \partial x } \right] d x } \\ & { = 2 \displaystyle \int ( s _ { \vartheta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \left[ \frac { \partial } { \partial \theta } \frac { \partial } { \partial x } p _ { \vartheta } ( x ) \right] d x - 2 \displaystyle \int p _ { \vartheta } ( x ) ( s _ { \vartheta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \left[ \frac { \partial \log p _ { \vartheta } ( x ) } { \partial x } \frac { \partial } { \partial x } \right. } \\ & { \left. - { \langle \Phi \rangle _ { * } } _ { * } ( x ) \right] ( \vartheta ) } \end{array}
+$$
+
+Looking at (3), we have
+
+$$
+\begin{array} { r l } & { ( 3 ) = 2 \displaystyle \int ( s _ { \theta } ( x ) - \nabla _ { x } \log q ( x ) ) ^ { T } \Big [ \frac { \partial } { \partial \theta } \frac { \partial } { \partial x ^ { p _ { \theta } ( x ) } } \Big ] d x } \\ & { = 2 \displaystyle \int \frac { \partial } { \partial \theta } \Big \{ \bf s g \Big [ ( \delta _ { \theta } ( x ) - \nabla _ { x } \log q ( x ) ) \Big ] ^ { T } \frac { \partial } { \partial x ^ { p _ { \theta } ( x ) } } \Big \} d x } \\ & { = 2 \displaystyle \frac { \partial } { \partial \theta } \int \frac { \partial } { \partial c } p _ { \theta } ( x + \epsilon \nu ) d x , \quad \boldsymbol { v } = \bf s g \Big [ ( \delta _ { \theta } ( x ) - \nabla _ { x } \log q ( x ) ) \Big ] , \boldsymbol { \epsilon } = 0 } \\ & { = 2 \displaystyle \frac { \partial } { \partial \theta } \frac { \partial } { \partial \epsilon } \int p _ { \theta } ( x + \epsilon \nu ) d x } \\ & { = 2 \displaystyle \frac { \partial } { \partial \theta } \frac { \partial } { \partial \epsilon } 1 } \\ & { = 0 . } \end{array}
+$$
+
+Above equality holds because of $\textstyle \int p _ { \theta } ( x + \epsilon v ) d x = 1$ holds for all $v , \theta , \epsilon$ . If we view $\epsilon$ as a shift strength parameter, the above equality recovers the first order Bartlett identity (Bartlett, 1953).
+
+Next we turns to term (4). Note that
+
+$$
+\begin{array} { r l } & { ( 4 ) _ { \perp } - - 2 \int m ( | x | ^ { \prime } ( s _ { \perp } \langle x \rangle - \nabla _ { x } \cdot | \mathbf { g } _ { \mathbf { Q } } | \langle x \rangle ) ^ { 2 } | \begin{array} { l } { \mathrm { R e } _ { \perp } \langle x \rangle  x  \mathrm { R e } _ { \perp } \langle x \rangle } \\ { \mathrm { a s } } \end{array} | \mathrm { d } x \equiv \mathrm { i } ( \mathrm { R e } _ { \perp } \langle x \rangle ) } \\ & { = \ \int \int m ( \frac { 1 } { s _ { \perp } }  x   x   x   x  ) ^ { 2 } \mathrm { R e } _ { \perp }  x   x   x   x   x  } \\ & { = - 2 \int ( | x _ { \perp } \langle x \rangle - \nabla _ { x } | \mathbf { g } _ { \mathbf { Q } } \langle x \rangle ) ^ { 2 } \mathrm { R e } _ { \perp }  x   x   x   x  } \\ & { \qquad \mathrm { a s } } \\ & { - 2 \frac { B } { \omega } \int \mathrm { d } s \{ | \omega _ { \perp } \langle x \rangle - \nabla _ { x } | \mathbf { g } _ { \mathbf { Q } } \langle x \rangle \} ^ { 2 } \frac { \partial } { \partial x } \mathrm { d } x \mathrm { d } s  x   x   x  } \\ & { = - \frac { \partial } { \partial t } \int \mathrm { d } s \{  x \langle x \rangle - \nabla _ { x } \cdot | \mathbf { g } _ { \mathbf { Q } } \langle x \rangle   x   x  \mathrm { d } x \overline { { \mathbf { Q } } _ { \mathbf { Q } } \langle x \rangle } \} | \mathbf { g } _ { \mathbf { Q } }  x   x  } \\ &  = - \frac { \partial } { \partial t } \int \mathrm { d } s \{  x   x  - \nabla _ { x } \cdot | \mathbf { g } _ { \mathbf { Q } } \langle x   x   x  \mathrm { d } x \overline { { \mathbf { Q } _ { \mathbf { Q } } \langle x \rangle } } | \mathbf { g } _ { \mathbf { Q } }  x  \} \\ &  - \frac { \partial } { \partial t } \overline { { w } } _  \end{array}
+$$
+
+Combining all above, we calculate the parameter derivative as
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } } \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } } \\ & { = ( 1 ) + ( 2 ) = ( 1 ) + ( 3 ) + ( 4 ) } \\ & { \displaystyle = \frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } } \mathbf { s g } \bigg [ \| \nabla _ { x } \log q ( x ) - s _ { \theta } ( x ) \| ^ { 2 } \bigg ] + 0 - 2 \frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } ( x ) } \bigg \{ \mathbf { s g } \bigg [ \big ( s _ { \theta } ( x ) - \nabla _ { x } \log q ( x ) \big ) \bigg ] ^ { T } \mathbf { s g } \bigg [ s _ { \theta } ( x ) \bigg ] \bigg \} } \\ & { = \displaystyle \frac { \partial } { \partial \theta } \mathbb { E } _ { p _ { \theta } } \bigg \{ \mathbf { s g } \bigg [ \| \nabla _ { x } \log q ( x ) \| ^ { 2 } \bigg ] - \mathbf { s g } \bigg [ \| s _ { \theta } ( x ) \| ^ { 2 } \bigg ] \bigg \} . } \end{array}
+$$
+
+Thus the equivalent loss function
+
+$$
+\mathcal { L } _ { S 2 D } ( \theta ) = \mathbb { E } _ { p _ { \theta } } \bigg \{ \mathbf { s g } \bigg [ \| \nabla _ { x } \log q ( x ) \| ^ { 2 } \bigg ] - \mathbf { s g } \bigg [ \| s _ { \theta } ( x ) \| ^ { 2 } \bigg ] \bigg \} .
+$$
+
+Share the same parameter gradients as the Fisher divergence which is intractable. Since we only need the $x$ gradient of sampler score function $s _ { \theta }$ (because the stop gradient operator), so we can estimate $s _ { \theta } ( x )$ through another score network $s _ { \phi } ( x )$ with samples consistently obtained from sampler. With above objective function, we could minimize the Fisher divergence between $p _ { \theta }$ and $q$ . □
+
+# C INTRODUCTION TO METROPOLIS-HASTINGS AND HAMILTONIAN CORRECTION
+
+Assume the target distribution is $p ( x )$ , the MH MCMC requires a proposal distribution $p ( \tilde { x } | x )$ to propose candidate samples $\tilde { x } \sim q ( \tilde { x } | x )$ . The Markov chain then accept the candidate sample with probability r = min{ p(˜x)q(x|x˜)p(x)q(˜x|x) , . Under some conditions, the chain will eventually reach $p ( x )$ as stationary distribution. The proposal distribution can be symmetric or non-symmetric. Conditional gaussian $q ( \tilde { x } | x ) = \mathcal { N } ( x ; \sigma ^ { 2 } )$ is a usual choice. Proposals based on score function $q ( \tilde { x } | x ) = \mathcal { N } ( x +$ $\begin{array} { r } { \frac { \epsilon } { 2 } \nabla _ { x } \log p ( x ) , \sigma ^ { 2 } ) } \end{array}$ is also popular (Xifara et al., 2014). If one consider an auxiliary state space of $( x , v )$ and execute the proposal in such space, the MC schedule is called Hamiltonian Monte Carlo. The Hamiltonian Monte Carlo execute a Monte Carlo dynamic in auxiliary space. With current sample $X ^ { ( t ) }$ . The HMC sample a momentum vector from an auxiliary distribution $V ^ { ( t ) } \sim$ $\exp ( - v ^ { T } M ^ { - 1 } v / 2 )$ . The joint sample $( X ^ { ( t ) } , V ^ { ( t ) } )$ updated by running a Hamiltonian Dynamics in joint space via
+
+$$
+{ \frac { d X _ { t } } { d t } } = { \frac { \partial H } { \partial V } } , { \frac { d V _ { t } } { d t } } = - { \frac { \partial H } { \partial X } } .
+$$
+
+Here $\begin{array} { r } { H ( x , v ) = - \log p ( x ) + \frac { 1 } { 2 } v ^ { T } M ^ { - 1 } v } \end{array}$ is the Hamiltonian of such mechanical system. HMC has many advantage that it mixes well for high-dimensional targets, and travels in joints space thus not easy to be trapped in local minima. Leap frog integrator is usually a practical choice for numerical updates (Neal, 2011). To make Markov Chain detail balanced, additional Metropolis correction is also needed for a Hamiltonian proposal. In short words, HMC iteratively accepts new position and momentum pair $( \tilde { x } , \tilde { v } )$ with rate $\operatorname* { m i n } 1$ , $\frac { H ( \tilde { x } , \tilde { v } ) } { H ( x , v ) }$ where $( \tilde { x } , \tilde { v } ) = L e a p F r o g ( x , v )$ as approximated Hamiltonian proposal.
+
+# D PROOF OF THEOREM 1
+
+We give the proof of Theorem 1 here. To begin with, we give a lemma to bound KL divergence with Fisher divergence as shown in Yamano (2021)
+
+Lemma 2. For fixed $q$ , there exists a dimension-free positive constant c such that for every distribution $p$ which is both integral and log-integral with respect to $q$ , and $p$ has same support as $q$ , we have
+
+$$
+\mathcal { D } _ { K L } \leq \frac { c } { 2 } \mathcal { D } _ { F } ( p , q ) .
+$$
+
+proof of lemma. For every 1st order smooth function $f$ , assume both $| f | ^ { 2 }$ and $\| \nabla f \| _ { 2 } ^ { 2 }$ are integrable with respect to $q$ , the log-Sobolev’s inequality (Gross, 1975) shows that there exist a dimension-free positive constant $c$ , such that
+
+$$
+\int | f | ^ { 2 } \log | f | q ( x ) d x \leq c \int \| \nabla f \| ^ { 2 } q ( x ) d x + \| f \| _ { 2 } ^ { 2 } \log \| f \| _ { 2 } ^ { 2 } .
+$$
+
+Here $\begin{array} { r } { \| f \| _ { 2 } ^ { 2 } = \int | f | ^ { 2 } q ( x ) d x } \end{array}$ . Replace $f = { \sqrt { p / q } }$ , we have
+
+$$
+L H S = { \frac { 1 } { 2 } } \int ( p / q ) \log ( p / q ) q = { \frac { 1 } { 2 } } \mathbb { E } _ { p } \log ( p / q ) = { \mathcal { D } } _ { K L } ( p , q ) .
+$$
+
+So we have
+
+$$
+\nabla { \frac { \sqrt { p } } { \sqrt { q } } } = { \frac { 1 } { 2 } } \left[ { \frac { { \frac { \nabla p } { \sqrt { p } } } { \sqrt { q } } - { \frac { \nabla q } { \sqrt { q } } } { \sqrt { p } } } { q } } \right] = { \frac { 1 } { 2 } } \left[ { \sqrt { { \frac { p } { q } } } } { \frac { \nabla p } { p } } - { \sqrt { { \frac { p } { q } } } } { \frac { \nabla q } { q } } \right] = { \frac { 1 } { 2 } } { \sqrt { { \frac { p } { q } } } } \left[ \nabla \log p - \nabla \log q \right] .
+$$
+
+Thus the first term in RHS is
+
+$$
+\begin{array} { l } { { \displaystyle c \int \| \nabla f \| ^ { 2 } q ( x ) d x } = { \displaystyle c \int \| \nabla \sqrt { \frac { p } { q } } \| ^ { 2 } q ( x ) p = \frac { c } { 2 } \int \| \nabla \log p - \nabla \log q \| ^ { 2 } p } } \\ { ~ } \\ { { \displaystyle ~ = \mathbb { E } _ { p } \| \nabla \log p - \nabla \log q \| ^ { 2 } = \mathcal { D } _ { F } ( p , q ) } . } \end{array}
+$$
+
+Note that $\begin{array} { r } { \| f \| _ { 2 } ^ { 2 } = \int | f | ^ { 2 } q ( x ) d x = \int ( p / q ) q = \int p = 1 } \end{array}$ . We combine both sides to conclude
+
+$$
+\frac { c } { 2 } { \cal D } _ { K L } ( p , q ) \leq \frac { 1 } { 2 } { \cal D } _ { F } ( p , q ) + 0 .
+$$
+
+So we have
+
+$$
+\mathcal { D } _ { K L } ( p , q ) \leq \frac { c } { 2 } \mathcal { D } _ { F } ( p , q ) ,
+$$
+
+where $c$ be another positive constant.
+
+The above lemma shows that KL divergence is upper bounded with Fisher divergence, which we are using to train the sampler. With above lemma, we can calculate mixing time for Langevin correction in proof below
+
+Proof. Assume target satisfies
+
+$$
+\operatorname* { l i m } _ { \| x \| _ { 2 } \to + \infty } ( \frac { | \nabla \log q ( x ) | _ { 2 } ^ { 2 } } { 2 } - \Delta \log q ( x ) ) = + \infty .
+$$
+
+then their exits a constant $\lambda > 0$ , such that Poincare inequality holds for each $f \in C ^ { 1 } ( \mathbb { R } ^ { d } ) \cap L ^ { 2 } ( q )$ with $\mathbb { E } _ { q } f = 0$ Theorem 4.3 in Pavliotis (2014)
+
+$$
+\lambda \| f \| _ { L ^ { 2 } ( q ) } ^ { 2 } \leq \| \nabla f \| _ { L ^ { 2 } ( q ) } ^ { 2 } .
+$$
+
+Let $p _ { 0 }$ denotes the ASS distribution, which is trained to be bounded with $\mathcal { D } _ { F } ( p _ { 0 } , q ) ~ \leq ~ \delta$ . By lemma, the KL between initial distribution $p _ { 0 }$ and target $q$ is bounded by Fisher divergence with a dimension-free constant $c$
+
+$$
+\mathcal { D } _ { K L } ( p _ { 0 } , q ) \leq \frac { c } { 2 } \mathcal { D } _ { F } ( p _ { 0 } , q ) \leq \delta \leq + \infty
+$$
+
+With Poincare’s inequality holds, the $\mathrm { K L }$ along Langevin diffusion $d X _ { t } = \nabla \log q ( X _ { t } ) / 2 + d W _ { t }$ decays exponentially fast as in Theorem 4.6 in Pavliotis (2014)
+
+$$
+\begin{array} { r l } & { \mathcal { D } _ { K L } ( p _ { t } , q ) \leq \exp ( - 2 \lambda t ) \mathcal { D } _ { K L } ( p _ { 0 } , q ) } \\ & { \quad \quad \quad \quad \quad \leq \exp ( - 2 \lambda t ) \frac { c } { 2 } \mathcal { D } _ { F } ( p _ { 0 } , q ) } \\ & { \quad \quad \quad \quad \leq \exp ( - 2 \lambda t ) \frac { c } { 2 } \delta . } \end{array}
+$$
+
+Thus if we want $\mathcal { D } _ { K L } ( p _ { t } , q )$ to be controlled under tolerance $\epsilon$ , we only need diffused time $t$ to satisfies
+
+$$
+t \geq \frac { 1 } { 2 \lambda } \bigg [ \log ( \frac { c } { 2 } ) + \log ( \frac { \delta } { \epsilon } ) \bigg ] = \frac { 1 } { 2 \lambda } \bigg [ C + \log ( \frac { \delta } { \epsilon } ) \bigg ] .
+$$
+
+where we place $\begin{array} { r } { C = \log \left( \frac { c } { 2 } \right) } \end{array}$ to be another constant. The diffusion time must be positive, thus we take
+
+$$
+T ^ { * } = \operatorname* { m a x } \{ 0 , \frac { 1 } { 2 \lambda } \left[ C + \log ( \frac { \delta } { \epsilon } ) \right] \} ,
+$$
+
+and finish the proof.
+
+# E EXPERIMENTAL DETAILS AND MORE RESULTS
+
+# E.1 SYNTHETIC TARGET
+
+For toy 2-dimensional data experiments, we use a 3-layer MLP neural network with 200 hidden units in each layer as the sampler. The activation of the sampler is chosen as LeakyReLU non-linearity with a 0.2 coefficient. The score network is a 3-layer MLP with 200 hidden units in each layer. The activation of the score network is GELU non-linearity.
+
+When reporting the numbers in Tab 1, we compute MMD metrics based on a total of 2000 samples. We run 20 independent experiments for each target and algorithm to calculate the mean and standard deviation.
+
+Figure 6 visualizes the model capabilities of FIS, AFIS, and AFIS $^ +$ MC samplers for matching three 2-dimensional target energy functions.
+
+![](images/644b974a7c57ab2510a6539ca82d2bf200da13524e551ec0fda5ba7e7186d09f.jpg)  
+Figure 6: Comparison between samples generated by FIS, AFIS and AFIS $+ \mathbf { M } \mathbf { C }$ on three 2D energy functions.
+
+# E.2 BAYESIAN REGRESSION
+
+For high-dimensional data experiments, we also use 3-layer MLP neural networks as the sampler and score network, respectively. The activation of the sampler is chosen as LeakyReLU non-linearity with a 0.2 coefficient. The activation of the score network is GELU non-linearity. For Australian and Heart distributions, we use 400 hidden units in each layer and 600 hidden units for German distribution.
+
+When reporting the numbers in Tab 2, we compute the MMD metric based on a total of 2000 samples. We run 20 independent experiments for each target and algorithm to calculate the mean and standard deviation. For the basic settings of Bayesian Regression problems, readers could refer to Song et al. (2017) for more details.
+
+# F FULL AFIS ALGORITHM
+
+This section gives the full Annealed Fisher Implicit Sampler training algorithm.
+
+# Algorithm 2: Annealed Fisher Implicit Sampler training algorithm
+
+<table><tr><td>gorimnZ.AmcaiedTTsnerhnpnenSanpilertranngaigorim log qprior(x) ; latent distribution pz(z), implicit sampler Gθ, score network s𝜙, mini-batch size B,max iteration M. Randomly initialize (0(o),(0)). forkin1:Kdo</td></tr><tr><td># anneal the target set logqk(x) = λk log q(x)+ (1-λk)logqprior(x) for tin 1:M do</td></tr><tr><td>#update score network parameter Get mini-batch from sampler xi = Gθ(t)(zi), zi ~ pz(z),i=1,.,B. Calculate score matching objective</td></tr><tr><td>B 1 s(x)2+2(Vx,s(xi))]. Lsm(Φ)= B ?</td></tr><tr><td>i=1 Minimize Lsm(Φ) to get (t+1).</td></tr><tr><td>#update samplerparameter Get mini-batch latent code zi ~ pz(z),i = 1,...,B. Use re-parametrization trick to calculate S2D loss for sampler</td></tr><tr><td>B Ls2D(0) = Vxlogqk(Gθ(zi)-(t+1)(G(zi))2</td></tr></table>

parse/dev/gE_vt-w4LhL/gE_vt-w4LhL.md

@@ -0,0 +1,252 @@
+# Squeezeformer: An Efficient Transformer for Automatic Speech Recognition
+
+Sehoon $\mathbf { K } \mathbf { i m } ^ { * 1 }$ , Amir Gholami∗1, Albert Shaw†‡1, Nicholas Lee†1, Karttikeya Mangalam1, Jitendra Malik1, Michael W. Mahoney123, Kurt Keutzer1
+
+1University of California, Berkeley 2ICSI 3LBNL {sehoonkim, amirgh, nicholas_lee, mangalam, malik, mahoneymw, keutzer}@berkeley.edu Albertshaw@google.com
+
+# Abstract
+
+The recently proposed Conformer model has become the de facto backbone model for various downstream speech tasks based on its hybrid attention-convolution architecture that captures both local and global features. However, through a series of systematic studies, we find that the Conformer architecture’s design choices are not optimal. After re-examining the design choices for both the macro and micro-architecture of Conformer, we propose Squeezeformer which consistently outperforms the state-of-the-art ASR models under the same training schemes. In particular, for the macro-architecture, Squeezeformer incorporates (i) the Temporal U-Net structure which reduces the cost of the multi-head attention modules on long sequences, and (ii) a simpler block structure of multi-head attention or convolution modules followed up by feed-forward module instead of the Macaron structure proposed in Conformer. Furthermore, for the micro-architecture, Squeezeformer (i) simplifies the activations in the convolutional block, (ii) removes redundant Layer Normalization operations, and (iii) incorporates an efficient depthwise downsampling layer to efficiently sub-sample the input signal. Squeezeformer achieves state-of-the-art results of $7 . 5 \%$ , $6 . 5 \%$ , and $6 . 0 \%$ word-error-rate (WER) on LibriSpeech test-other without external language models, which are $3 . 1 \%$ , $1 . 4 \%$ , and $0 . 6 \%$ better than Conformer-CTC with the same number of FLOPs. Our code is open-sourced and available online [25].
+
+# 1 Introduction
+
+The increasing success of end-to-end neural network models has been a huge driving force for the drastic advancements in various automatic speech recognition (ASR) tasks. While both convolutional neural networks (CNN) [19, 27, 29, 36, 61] and Transformers [24, 31, 32, 56, 57] have drawn attention as popular backbone architectures for ASR models, each of them has several limitations. Generally, CNN models lack the ability to capture global contexts and Transformers involve prohibitive computing and memory overhead. To overcome these shortcomings, Conformer [16] has recently proposed a novel convolution-augmented Transformer architecture. Due to its ability to synchronously capture global and local features from audio signals, Conformer has become the de facto model not only for ASR tasks, but also for various end-to-end speech processing tasks [17]. Furthermore, it has also achieved state-of-the-art performance in combination with recent developments in self-supervised learning methodologies as well [37, 62]. While the Conformer architecture was introduced as an autoregressive RNN-Transducer (RNN-T) [14] model in its original setting, it has been adopted with less critique to non-autoregressive schemes such as Connectionist Temporal Classification (CTC) [15] as well [38].
+
+![](images/6a966fc8d69f0dfca207c707aab43dea72301ec2184f3705f2f0839da5c1aab3.jpg)  
+Figure 1: (Left) We perform a series of systematic studies on macro and micro architecture to redesign the Conformer architecture towards our Squeezeformer architecture. The bars and the line indicate the WER on LibriSpeech test-other dataset and the FLOPs, respectively. For each design modification, we strictly improve WER until our final Squeezeformer model outperforms Conformer by $1 . 4 0 \%$ WER improvement with the same number of FLOPs. See Tab. 1 for the details. (Right) LibriSpeech test-other WER vs. FLOPs for Squeezeformer and other state-of-the-art ASR models. Conformer- $\mathrm { { C T C ^ { * } } }$ is our own reproduction to the best performance as possible and the others are the reported numbers in their papers [4, 27, 36]. Our architecture scales well to smaller and larger models to constantly outperform other models by a large margin throughout the entire FLOPs range. See Tab. 3 for the details. For both plots, the lower the WER, the better; however, we plotted in reverse for better visualization.
+
+Despite being a key architecture in speech processing tasks, the Conformer architecture has some limitations that can be improved upon. First, Conformer still suffers from the quadratic complexity of the attention mechanism limiting its efficiency on long sequence lengths. This problem is further highlighted by the long sequence lengths of typical audio inputs as also pointed out in [46]. Furthermore, the Conformer architecture is relatively more complicated than Transformer architectures used in other domains such as in natural language processing [7, 44, 51] or computer vision [9, 10, 49]. For instance, the Conformer architecture incorporates multiple different normalization schemes and activation functions, the Macaron structure [34], as well as back-to-back multi-head attention (MHA) and convolution modules. This level of complexity makes it difficult to efficiently deploy the model on dedicated hardware platforms for inference [26, 39, 55]. More importantly, this raises the question of whether such design choices are necessary and optimal for achieving good performance in ASR tasks.
+
+In this paper, we perform a careful and systematic analysis of each of the design choices with the goal of achieving lower word-error-rate (WER) for a given computational budget. We developed a much simpler and more efficient hybrid attention-convolution architecture in both its macro and micro-design that consistently outperforms the state-of-the-art ASR models. In particular, we make the following contributions in our proposed Squeezeformer model:
+
+• We find a high temporal redundancy in the learned feature representations of neighboring speech frames especially deeper in the network, which results in unnecessary computational overhead. To address this, we incorporate the temporal U-Net structure in which a downsampling layer halves the sampling rate at the middle of the network, and a light upsampling layer recovers the temporal resolution at the end for training stability (§ 3.1.1).   
+• We redesign the hybrid attention-convolution architecture based on our observation that the backto-back MHA and convolution modules with the Macaron structure are suboptimal. In particular, we propose a simpler block structure similar to the standard Transformer block [7, 51], where the MHA and convolution modules are each directly followed by a single feed forward module $( \ S \ 3 . 1 . 2 )$ .   
+• We finely examine the micro-architecture of the network and found several modifications that simplify the model overall and greatly improve the accuracy and efficiency. This includes (i) activation unification that replaces GLU activations with Swish (§ 3.2.1), (ii) Layer Normalization
+
+![](images/5bfbfab0b9ed5aa15ba0479dd25d335c2908998b4be733be3df2efb6269735d2.jpg)  
+Figure 2: (Left) The Conformer architecutre and (Right) the Squeezeformer architecture which comprises of the Temporal U-Net structure for downsampling and upsampling of the sampling rate, the standard Transformer-style block structure that only uses Post-Layer Normalization, and the depthwise separable subsampling layer.
+
+simplification by replacing redundant pre-Layer Normalization layers with a scaled post-Layer Normalization which incorporates a learnable scaling for the residual path that can be merged with other layers to be zero-cost during inference $\lbrace \ S 3 . 2 . 2 )$ , and (iii) incorporation of a depthwise separable convolution for the first sub-sampling layer that results in a significant floating point operations (FLOPs) reduction $( \ S \ 3 . 2 . 3 )$ .
+
+• We show that the Squeezeformer architecture scales well with both smaller and larger models and consistently outperforms other state-of-the-art ASR models when trained under the same settings (Tab. 4.2, $\ S 4 . 2$ ). Furthermore, we justify the final model architecture of Squeezeformer with a reverse ablation study for the design choices (Tab. 4, $\ S 4 . 3$ ).
+
+# 2 Related Work
+
+The recent advancements in end-to-end ASR can be broadly categorized into (1) model architecture and (2) training methods.
+
+Model Architecture for End-to-end ASR. The recent end-to-end ASR models are typically composed of an encoder, which takes as input a speech signal (i.e., sequence of speech frames) and extracts high-level acoustic features, and a decoder, which converts the extracted features from the encoder into a sequence of text. The model architecture of the encoder determines the representational power of an ASR model and its ability to extract acoustic features from input signals. Therefore, a strong architecture is critical for overall performance.
+
+One of the popular choices for a backbone model architecture is convolutional neural network (CNN). End-to-end deep CNN models have been first explored in [29, 61], and further improved by introducing depth-wise separable convolution [21, 47, 50] in QuartzNet [27] and the Squeezeand-Excitation module [23] in CitriNet [36] and ContextNet [19]. However, since CNNs often fail to capture global contexts, Transformer [51] models have also been widely adopted in backbone architectures due to their ability to capture long-range dependencies between speech frames [24, 31, 32, 56, 57]. Recently, [16] has proposed a novel model architecture named Conformer, which augments Transformers with convolutions to model both global and local dependencies efficiently. With the Conformer architecture as our starting point, we focus on designing a next-generation model architecture for ASR that is simpler, more accurate, and more efficient.
+
+The hybrid attention-convolution architecture of Conformer has enabled the state-of-the-art results in many speech tasks. However, the quadratic complexity of the attention layer still proves to be cost prohibitive at larger sequence lengths. While different approaches have been proposed to reduce the cost of MHA in ASR [5, 46, 58, 59], their main focus is not changing the overall architecture design, and their optimizations can also be applied to our model, as they are orthogonal to the developments for Squeezeformer. Efficient-Conformer [4] introduces the progressive downsampling scheme and grouped attention to reduce the training and inference costs of Conformer. Our work incorporates a similar progressive downsampling, but also introduces an up-sampling mechanism with skip connections from the earlier layers inspired by the U-Net [45] architecture in computer vision and the U-Time [43] architecture for sleep signal analysis. We find this to be critical for training stability and overall performance. In addition, through systematic experiments, we completely refactor the Conformer block by carefully redesigning both the macro and micro-architectures.
+
+Table 1: Starting from Conformer as the baseline, we redesign the architecture towards Squeezeformer through a series of systematic studies on macro and micro architecture. Note that for each design change, the WER on LibriSpeech test-clean and test-other datasets improves consistently. For comparison, we include the number of parameters and FLOPs for a 30s input in the last two columns.   
+
+<table><tr><td>Model</td><td>Design change</td><td>test-clean test-other|</td><td></td><td>|Params (M)| GFLOPs</td><td></td></tr><tr><td>Conformer-CTC-M</td><td>Baseline</td><td>3.20</td><td>7.90</td><td>27.4</td><td>71.7</td></tr><tr><td></td><td>+ Temporal U-Net ($ 3.1.1)</td><td>2.97</td><td>7.28</td><td>27.5</td><td>57.0</td></tr><tr><td></td><td>+ Transformer-style Block (§ 3.1.2)</td><td>2.93</td><td>7.12</td><td>27.5</td><td>57.0</td></tr><tr><td></td><td>+ Unified activations (§ 3.2.1)</td><td>2.88</td><td>7.09</td><td>28.7</td><td>58.4</td></tr><tr><td></td><td>+ Simplified LayerNorm (§ 3.2.2)</td><td>2.85</td><td>6.89</td><td>28.7</td><td>58.4</td></tr><tr><td>Squeezeformer-SM</td><td>+ DW sep. subsampling (§ 3.2.3)</td><td>2.79</td><td>6.89</td><td>28.2</td><td>42.7</td></tr><tr><td>Squeezeformer-M</td><td>+ Model scale-up ($ 3.2.3)</td><td>2.56</td><td>6.50</td><td>55.6</td><td>72.0</td></tr></table>
+
+Training Methodology for End-to-end ASR. In the past few years, various self-supervised learning methodologies based on contrastive learning [2, 52, 62] or masked prediction [1, 6, 22] have been proposed to push forward the ASR performance. While a model pre-trained with self-supervised tasks generally outperforms when finetuned on a target ASR task, training strategies are not the main focus in this work as they can be applied independently to the underlying architecture.
+
+# 3 Architecture Design
+
+The Conformer architecture has been widely adopted by the speech community and is used as a backbone for different speech tasks. At a macro-level, Conformer incorporates the Macaron structure [34] comprised of four modules per block, as shown in Fig. 2 (Left). These blocks are stacked multiple times to construct the Conformer architecture. In this work, we carefully reexamine the design choices in Conformer, starting first with its macro-architecture, and then its microarchitecture design. We choose Conformer-CTC-M as the baseline model for the case study, and we compare word-error-rate (WER) on LibriSpeech test-other as a performance metric for each architecture. Furthermore, we measure FLOPs on a 30s audio input as a proxy for model efficiency. While we acknowledge that FLOPs may not always be a linear indicator of hardware and runtime efficiency, we choose FLOPs as it is hardware agnostic and is statically computable. However, we do measure the final end-to-end throughput of our changes, ensuring up to $1 . 3 4 \times$ consistent improvement in runtime for different versions of Squeezeformer (Tab. 4.2)
+
+# 3.1 Macro-Architecture Design
+
+We first focus on designing the macro structure of Squeezeformer, i.e., how the blocks and modules are organized in a global scale.
+
+# 3.1.1 Temporal U-Net Architecture
+
+The hybrid attention-convolution structure enables Conformer to capture both global and local interactions. However, the attention operation has a quadratic FLOPs complexity with respect to the input sequence length. We propose to lighten this extra overhead by computing attention over a reduced sequence length. In the Conformer model itself, the input sampling rate is reduced from $1 0 \mathrm { m s }$ to $4 0 \mathrm { m s }$ with a convolutional subsampling block at the base of the network. However, this rate is kept constant throughout the network, with all the attention and convolution operations operating at a constant temporal scale.
+
+To this end, we begin by studying the temporal redundancy in the learned feature representations. In particular, we analyze how the learned feature embeddings per speech frame are differentiated through the Conformer model depth. We randomly sample 100 audio signals from LibriSpeech’s dev-other dataset, and process them through the Conformer blocks, recording their per-block activations. We then measure the average cosine similarity between two neighboring embedding vectors. The results are plotted as the solid lines in Fig. 3. We observe that the embeddings for the speech frames directly next to each other have an average similarity of $9 5 \%$ at the topmost layer, and even those 4 speech frames away from each other have a similarity of more than $80 \%$ . This reveals that there is an increasing temporal redundancy as inputs are processed through the Conformer blocks deeper in the network. We hypothesize that this redundancy in feature embedding vectors causes unnecessary computational overhead and that the sequence length can be reduced deeper in the network without loss in accuracy.
+
+![](images/1fad821fb9ce98ef500fa758eeed93ed6552d0b4fbbc048e48d38242324d8cee.jpg)  
+Figure 3: Cosine similarity between two embedding vectors of neighboring speech frames with varying adjacency distances across the Conformer blocks. The temporal dimension is downsampled after the 7th block and upsampled before the 16th block in the Temporal U-Net structure.
+
+![](images/bec441ea699c28ec9951f7127d981bf98196d5d12ad33a3e58f372913a5b5dcc.jpg)  
+Figure 4: (Left) Back-to-back preLN and postLN at the boundary of the blocks. (Right) The preLN can be replaced with the learned scaling that readjusts the magnitude of the activation that goes into the subsequent module.
+
+As our first macro-architecture improvement step, we change the Conformer model to incorporate subsampling of the embedding vectors after it has been processed by the early blocks of the model. In particular, we keep the sample rate to be $4 0 \mathrm { m s }$ up to the 7th block, and afterwards we subsample to a rate of $8 0 \mathrm { m s }$ per input sequence by using a pooling layer. For the pooling layer we use a depthwise separable convolution with stride 2 and kernel size 3 to merge the redundancies across neighboring embeddings. This decreases the attention complexity by $4 \times$ and also reduces the redundancies of the features. This temporal downsampling shares similarities with computer vision models, which often downsample the input image spatially to save compute and develop hierarchical level features [10, 20, 30, 48], and with the approach of Efficient Conformer [4].
+
+However, the temporal downsampling alone leads to an unstable and diverging training behaviour $( \ S 4 . 3 )$ . One possible reason for this is the lack of enough resolution for the decoder after subsampling the rate to $8 0 \mathrm { m s }$ . The decoder maps an embedding for each speech frame into a single label, e.g., character, and therefore requires sufficient resolution for successful decoding of the full sequence. Inspired from successful architectures for dense prediction in computer vision such as U-Net [45], we incorporate the Temporal U-Net structure to recover the resolution at the end of the network through an upsampling layer as shown in Fig. 2. This upsampling block takes the embedding vectors processed by the $4 0 \mathrm { m s }$ and $8 0 \mathrm { m s }$ sampling rates, and produces an embedding with a rate of 40ms by adding them together via a skip connection. To the best of our knowledge, the closest work to our Temporal U-Net is the approach proposed in [43], in which the U-Net structure is incorporated into a fully-convolutional model to downsample sleep signals.
+
+This change not only reduces the total FLOPs by $20 \%$ compared to Conformer1, but also improves the test-other WER by $0 . 6 2 \%$ from $7 . 9 0 \%$ to $7 . 2 8 \%$ (Tab. 1, 2nd row). Furthermore, analyzing the cosine similarity shows that the Temporal U-Net architecture prevents the neighboring embeddings from becoming too similar to each others at the later blocks, in particular at the final block directly connected to the decoder, as shown in Fig. 3 as the dashed lines.
+
+# 3.1.2 Transformer-Style Block
+
+The Conformer block consists of a sequence of feed-forward (‘F’), multi-head attention (MHA, ‘M’), convolution (‘C’), and another feed-forward module (‘F’). We denote this as the FMCF structure. Note that the convolutional kernel sizes in ASR models are rather large, e.g., 31 in Conformer, which makes its behaviour similar to attention in mixing global information. This is stark contrast to convolutional kernels in computer vision, which often have small $3 \times 3$ kernels and hence benefit greatly from attention’s global processing. As such, placing the convolution and MHA module with a similar functionality back-to-back (i.e., the MC substructure) does not seem prudent. Hence, we consider an MF/CF structure, which is motivated by considering the convolution module as a local MHA module. Furthermore, we drop the Macaron structure [34], as MHA modules followed by feed-forward modules have been more widely adopted in the literature [7, 9, 44, 51]. In a nutshell, we simplify the architecture to be similar to the standard Transformer network and denote the blocks MF and CF substructures, as shown in Fig. 2. This modification further improves the test-other WER by $0 . 1 6 \%$ from $7 . 2 8 \%$ to $7 . 1 2 \%$ and marginally improves the test-clean WER without affecting the FLOPs (Tab. 1, 3rd row).
+
+# 3.2 Micro-Architecture Design
+
+So far we have designed the macro structure of Squeezeformer by incorporating seminal architecture principles from computer vision and natural language processing into Conformer. In this subsection, we now focus on optimizing the micro structure of the individual modules. We show that we can further simplify the module architectures while improving both efficiency and performance.
+
+# 3.2.1 Unified Activations
+
+Conformer uses Swish activation for most of the blocks. However, it switches to a Gated Linear Unit (GLU) for its convolution module. Such a heterogeneous design seems over-complicated and unnecessary. From a practical standpoint, the use of multiple activations complicates hardware deployment, as an efficient implementation requires dedicated logic design, look up tables, or custom approximations [26, 39, 55]. For instance, on low-end edge devices with no dedicated vector processing unit, supporting additional non-linear operations would require additional look up tables or advanced algorithms [12, 13]. To address this, we propose to replace the GLU activation with Swish, unifying the choice of activation function throughout the entire model. We keep the expansion rate for the convolution modules. As shown in the 4th row of Tab. 1, this change does not entail noticeable changes in WER and FLOPs but only simplifies the architecture.
+
+# 3.2.2 Simplified Layer Normalizations
+
+Continuing our micro-architecture improvements, we note that the Conformer model incorporates redundant Layer Normalizations (LayerNorm), as shown in Fig. 4 (Left). This is because the Conformer model contains both a post-LayerNorm (postLN) that applies LayerNorm in between the residual blocks, as well as pre-LayerNorm (preLN) which applies LayerNorm inside the residual connection. While it is hypothesized that preLN stabilizes training and postLN benefits performance [53], these two modules used together lead to redundant back-to-back operations. Aside from the architectural redundancy, LayerNorm can be computationally expensive [26, 55] due to its global reduction operations.
+
+However, we found that naïvely removing the preLN or postLN leads to training instability and convergence failure $( \ S \ 4 . 3 )$ . Investigating the cause of failure, we observe that a typical trained Conformer model has orders of magnitude differences in the norms of the learnable scale variables of the back-to-back preLN and postLN. In particular, we found that the preLN would scale down the input signal by a large value, giving more weight to the skip connection. Therefore, it is important to use a scaling layer when replacing the preLN component to allow the network to control this weight. This idea is also on par with several training stabilization strategies in other domains. For instance, NF-Net [3] proposed adaptive (i.e., learnable) scaling before and after the residual blocks to stabilize training without normalization. Furthermore, DeepNet [53] also recently proposed to add non-trainable rule-based scaling to the skip connections to stabilize preLN in Transformers.
+
+Table 2: Detailed architecture configurations for Conformer-CTC (baseline) and Squeezeformer. For comparison, we include the number of parameters and FLOPs for a 30s input in the last two columns.   
+
+<table><tr><td>Model</td><td>#Layers</td><td>Dimension</td><td>#Heads</td><td>Params (M)</td><td>GFLOPs</td></tr><tr><td>Conformer-CTC-S</td><td>16</td><td>144</td><td>4</td><td>8.7</td><td>26.2</td></tr><tr><td>Squeezeformer-XS</td><td>16</td><td>144</td><td>4</td><td>9.0</td><td>15.8</td></tr><tr><td>Squeezeformer-S</td><td>18</td><td>196</td><td>4</td><td>18.6</td><td>26.3</td></tr><tr><td>Conformer-CTC-M</td><td>16</td><td>256</td><td>4</td><td>27.4</td><td>71.7</td></tr><tr><td>Squeezeformer-SM</td><td>16</td><td>256</td><td>4</td><td>28.2</td><td>42.7</td></tr><tr><td>Squeezeformer-M</td><td>20</td><td>324</td><td>4</td><td>55.6</td><td>72.0</td></tr><tr><td>Conformer-CTC-L</td><td>18</td><td>512</td><td>8</td><td>121.5</td><td>280.6</td></tr><tr><td>Squeezeformer-ML</td><td>18</td><td>512</td><td>8</td><td>125.1</td><td>169.2</td></tr><tr><td>Squeezeformer-L</td><td>22</td><td>640</td><td>8</td><td>236.3</td><td>277.9</td></tr></table>
+
+Inspired by these computer vision advancements, we propose to replace preLN with a learnable scaling layer that scales and shifts the activations, $\mathrm { S c a l i n g } \bar { ( } x ) = \gamma x \bar { + } \beta$ , with learnable scale and bias vectors $\gamma$ and $\beta$ of the size of feature dimension. For homogeneity of architectural design, we then replace the preLN throughout all the modules with the postLN-then-scaling as illustrated in Fig. 2 (Right) and make the entire model postLN-only. Note that the learned scaling parameters can be merged into the weights of the subsequent linear layer, as the architecture illustrated in Fig. 2 (Right), and hence have zero inference cost. With the learned scaling, our model further improves the test-other WER by $0 . 2 0 \%$ from $7 . 0 9 \%$ to $6 . 8 9 \%$ (Tab. 1, 5th row).
+
+# 3.2.3 Depthwise Separable Subsampling
+
+We now shift our focus from the Conformer blocks to the subsampling block. While it is easy to overlook this single module at the beginning of the architecture, we note that it accounts for a significant portion of the overall FLOPs count, up to $28 \%$ for Conformer-CTC-M with a 30-second input. This is because the subsampling layer uses two vanilla convolution operations each of which has a stride 2. To reduce the overhead of this layer, we replace the second convolution operation with a depthwise separable convolution while keeping the kernel size and stride the same. We leave the first convolution operation as is since it is equivalent to a depthwise convolution with the input dimension 1. This saves an additional $22 \%$ of the baseline FLOPs without a test-other WER drop and even a $0 . 0 6 \%$ improvement in test-clean WER (Tab. 1, 6th row). An important point to note here is that generally depthwise separable convolutions are hard to efficiently map to hardware accelerators, in part due its low arithmetic intensity. However, given the large FLOPs reduction, we consistently observe an overall improvement in the total inference throughput of up to $1 . 3 4 \times$ as reported in Tab. 3, as compared to the baseline Conformer models.
+
+We name our final model with all these improvements as Squeezeformer-SM. Compared to ConformerCTC-M, our initial baseline, Squeezeformer-SM improves WER by $1 . 0 1 \%$ from $7 . 9 0 \%$ to $6 . 8 9 \%$ with $40 \%$ less FLOPs. Given the smaller FLOPs of Squeezeformer-SM, we also scale up the model to a similar FLOPs cost as Conformer-CTC-M. In particular, we scale both depth and width of the model together following the practice in [8]. Scaling up the model achieves additional test-other WER gain of $0 . 3 9 \%$ from $6 . 8 9 \%$ to $6 . 5 0 \%$ (Tab. 1, 7th row), and we name this architecture Squeezeformer-M.
+
+# 4 Results
+
+# 4.1 Experiment Setup
+
+Models. Following the procedure described in $\ S 3$ , we construct Squeezeformer variants with different size and FLOPs: we apply the macro and micro-architecture changes in $\ S \ 3 . 1$ and $\ S 3 . 2$ , respectively, to construct Squeezeformer-XS, SM, and ML from Conformer-S, M, and L, retaining the model size. Afterwards, we construct Squeezeformer-S, M, and L by scaling up each model to match the FLOPs of the corresponding Conformer. The detailed architecture configurations are described in Tab. 2.
+
+While there are multiple options available for the decoder such as RNN-Transducer (RNN-T) [14] and Connectionist Temporal Classification (CTC) [15], we use a CTC decoder whose non-autoregressive decoding method benefits training and inference latency [36]. However, the main focus of this work is the model architecture design of the encoder, which can be orthogonal to the decoder type.
+
+Table 3: WER $( \% )$ comparison on LibriSpeech dev and test datasets for Squeezeformer and other state-of-the-art CTC models for ASR including Conformer-CTC, QuartzNet [27], CitriNet [36], Transformer-CTC [31], and Efficient Conformer-CTC [4]. For comparison, we include the number of parameters, FLOPs, and throughput (Thp) on a single NVIDIA Tesla A100 GPU for a 30s input in the last three columns. ∗The performance numbers for Conformer-CTC are based on our own reproduction to the best performance as possible. All the other performance numbers are from the corresponding papers. †With and ‡without the grouped attention.   
+
+<table><tr><td>Model</td><td>dev-clean</td><td>dev-other</td><td>test-clean</td><td>test-other</td><td>Params (M)</td><td>GFLOPs</td><td>Thp (ex/s)</td></tr><tr><td>Conformer-CTC-S* [16]</td><td>4.21</td><td>10.54</td><td>4.06</td><td>10.58</td><td>8.7</td><td>26.2</td><td>613</td></tr><tr><td>QuartzNet 5x5 [27]</td><td>5.39</td><td>15.69</td><td>1</td><td>1</td><td>6.7</td><td>20.2</td><td>1</td></tr><tr><td>Citrinet 256 [36]</td><td>-</td><td>1</td><td>3.78</td><td>9.60</td><td>10.3</td><td>16.8</td><td>1</td></tr><tr><td>Squeezeformer-Xs</td><td> 3.63</td><td>9.30</td><td> 3.74</td><td>9.09</td><td>9.0</td><td>15.8</td><td>763</td></tr><tr><td>Conformer-CTC-M* [16]</td><td>2.94</td><td>7.80</td><td>3.20</td><td>7.90</td><td>27.4</td><td>71.7</td><td>463</td></tr><tr><td>QuartzNet 5x10 [27]</td><td>4.14</td><td>12.33</td><td>1</td><td>1</td><td>12.8</td><td>38.5</td><td>-</td></tr><tr><td>QuartzNet 5x15 [27]</td><td>3.98</td><td>11.58</td><td>3.90</td><td>11.28</td><td>18.9</td><td>55.7</td><td>-</td></tr><tr><td>Citrinet 512 [36]</td><td>1</td><td>1</td><td>3.11</td><td>7.82</td><td>37.0</td><td>63.1</td><td>1</td></tr><tr><td>Eff. Conformer-CTCt [4]</td><td>-</td><td>1</td><td>3.57</td><td>8.99</td><td>13.2</td><td>26.0</td><td>=</td></tr><tr><td>Eff. Conformer-CTC‡ [4]</td><td></td><td>-</td><td>3.58</td><td>8.88</td><td>13.2</td><td>32.5</td><td>=</td></tr><tr><td>Squeezeformer-S</td><td>2.80</td><td>7.49</td><td>3.08</td><td>7.47</td><td>18.6</td><td>26.3</td><td>602</td></tr><tr><td> Squeezeformer-SM</td><td>2.71</td><td>6.98</td><td>2.79</td><td>6.89</td><td>28.2</td><td>42.7</td><td>558</td></tr><tr><td>Conformer-CTC-L* [16]</td><td>2.61</td><td>6.45</td><td>2.80</td><td>6.55</td><td>121.5</td><td>280.6</td><td>200</td></tr><tr><td>Citrinet 1024 [36]</td><td>1</td><td>-</td><td>2.52</td><td>6.22</td><td>143.1</td><td>246.3</td><td>-</td></tr><tr><td> Squeezeformer-M</td><td>2.43</td><td>6.51</td><td>2.56</td><td>6.50</td><td>55.6</td><td>72.0</td><td>431</td></tr><tr><td>Squeezeformer-ML</td><td> 2.34</td><td>6.08</td><td>2.61</td><td>6.05</td><td>125.1</td><td>169.2</td><td>268</td></tr><tr><td>Transformer-CTC [31]</td><td>2.6</td><td>7.0</td><td>2.7</td><td>6.8</td><td>255.2</td><td>621.1</td><td>1</td></tr><tr><td> Squeezeformer-L</td><td> 2.27</td><td> 5.77</td><td>2.47</td><td> 5.97</td><td>236.3</td><td>277.9</td><td>207</td></tr></table>
+
+Another subtlety when evaluating models is the use of external language models (LM). In many prior works [18, 24, 35, 40, 54, 56], decoders are often augmented with external LMs such as pre-trained 4-gram or Transformer, which boost the final WER by re-scoring the outputs in a more lexically accurate manner. However, we compare the results without external LMs to fairly compare the true representation power of the model architectures alone − external LMs can be incorporated as an orthogonal optimization afterward.
+
+Training Details. Because the training recipes and codes for Conformer have not been open-sourced, we train it to reproduce the best performance numbers as possible. We train both Conformer-CTC and Squeezeformer on the LibriSpeech-960hr [41] for 500 epochs on Google’s cloud TPUs v3 with batch size 1024 for the small and medium variants and 2048 for the large variants. We use AdamW [33] optimizer with weight decay 5e-4 for all models. More details for the training and evaluation setup are given in $\ S \operatorname { A . 1 }$ and $\ S \ A . 2$ .
+
+# 4.2 Main Results
+
+In Tab. 3 we compare the WER of Squeezeformer with Conformer-CTC and other state-of-the-art CTC-based ASR models including QuartzNet [27], CitriNet [36], Transformer [31], and Efficient Conformer [4] on the clean and other datasets. Note that the performance numbers for Conformer$\mathrm { C T C } ^ { 2 }$ are based on our own reproduction to the best performance as possible due to the absence of public training recipes or codes. For simplicity, we denote WER as test-clean/test-other without $\%$ throughout the section.
+
+Squeezeformer vs. Conformer. Our smallest model Squeezeformer-XS outperforms ConformerCTC-S by 0.32/1.49 (3.74/9.09 vs. 4.06/10.58) with $1 . 6 6 \times$ FLOPs reduction. Compared with Conformer-CTC-M, Squeezeformer-S achieves 0.12/0.43 WER improvement (3.08/7.47 vs.
+
+Table 4: Ablation studies for the design choices made in Squeezeformer, including Temporal U-Net, LayerNorm, and activation in the convolution module. ∗Without the upsampling layer, the model fails to converge.   
+
+<table><tr><td>Ablation</td><td>| Model</td><td>dev-clean dev-other</td><td></td></tr><tr><td> Ours</td><td>Squeezeformer-M</td><td>2.43</td><td>6.51</td></tr><tr><td>Temporal U-Net (§ 3.1.1),</td><td>No skip connection No upsampling</td><td>2.78 N/A*</td><td>7.38 N/A*</td></tr><tr><td>LayerNorm (§ 3.2.2)</td><td>PostLN only PreLNonly</td><td>5.60 3.02</td><td>14.00 8.27</td></tr><tr><td>Convolution module (§ 3.2.1) |No Swish</td><td></td><td>2.53</td><td>6.73</td></tr></table>
+
+3.20/7.90) with $1 . 4 7 \times$ smaller size and $2 . 7 3 \times$ less FLOPs, and Squeezeformer-SM further improves WER by 0.41/1.01 (2.79/6.89 vs. 3.20/7.90) with a comparable size and $1 . 7 0 \times$ less FLOPs. Compared with Conformer-CTC-L, Squeezeformer-M shows $0 . 2 4 / 0 . 0 5$ WER improvement $( 2 . 5 6 / 6 . 5 0 $ vs. 2.80/6.55) with significant size and FLOPs reductions of $2 . 1 8 \times$ and $3 . 9 0 \times$ , respectively, and Squeezeformer-ML shows 0.19/0.50 WER improvement (2.61/6.05 vs. 2.80/6.55) with a similar size and $1 . 6 6 \times$ less FLOPs. Finally, our largest model Squeezeformer-L improves WER by 0.33/0.58 upon Conformer-CTC-L with the same FLOPs count, achieving the state-of-the-art result of 2.47/5.97.
+
+Squeezeformer vs. Other ASR Models. As can be seen in Tab. 3, our model consistently outperforms QuartzNet, CitriNet, and Transformer with comparable or smaller model sizes and FLOPs counts. A notable result is a comparison against Efficient-Conformer: our model outperforms the efficientlydesigned Efficient Conformer by a large margin of 0.79/1.99 (2.79/6.89 vs. 3.58/8.88) with the same FLOPs count. The overall results are summarized as a plot in Fig. 1 (Right) where Squeezeformer consistently outperforms other models across all FLOPs regimes.
+
+# 4.3 Ablation Studies
+
+In this section, we provide additional ablation studies for the design choices made for individual architecture components using Squeezeformer-M as the base model. See Tab. 4. Unless specified, we use the same hyperparameter settings as in the main experiment.
+
+Temporal U-Net. In the 2nd row of Tab. 4, the model clearly underperforms by 0.35/0.87 without the skip connection from the downsampling layer to the upsampling layer. This shows that the high-resolution information collected in the early layers is critical for successful decoding. The 3rd row in Tab. 4 shows that our model completely fails to converge without the upsampling layer due to training stability, even with several different peak learning rates of $\{ 0 . 5 , 1 . 0 , 1 . 5 \} \mathrm { e } . 3$ .
+
+LayerNorm. In the 4th line of Tab. 4, we show that WER drops significantly by 3.17/7.49 when we apply the PostLN-only scheme without the learned scaling layer. Another alternative design choice is to apply the PreLN-only scheme without the learned scaling, which also results in a noticeable WER degradation of $0 . 5 9 / 1 . 7 6$ as shown in the 5th line of Tab. 4. In both cases, the model fails to converge, so we report the best WER before divergence. The results suggest that the learned scaling layer plays a key role for training stabilization and better WER.
+
+Convolution Module. When ablating the GLU activation in the convolution modules, another possible design choice is to drop it without replacing it with the Swish activation. This, however, results in 0.10/0.22 worse WER as shown in the last line of Tab. 4.
+
+# 5 Conclusions
+
+In this work, we performed a series of systematic ablation studies on the macro and micro architecture of the Conformer architecture, and we proposed a novel hybrid attention-convolution architecture that is simpler and consistently achieves better performance than other models for a wide range of computational budgets. The key novel components of Squeezeformer’s macro-architecture is the incorporation of the Temporal U-Net structure which downsamples audio signals in the second half of the network to reduce the temporal redundancy between adjacent features and save compute, as well as the MF/CF block structure similar to the standard Transformer-style which simplifies the architecture and improves performance. Furthermore, the micro-architecture of Squeezeformer simplifies the activations throughout the model and replaces redundant LayerNorms with the scaled postLN, which is more efficient and leads to better accuracy. We also drastically reduce the subsampling cost at the beginning of the model by incorporating a depthwise separable convolution. We perform extensive testing of the proposed architecture and find that Squeezeformer scales very well across different model sizes and FLOPs regimes, surpassing prior model architectures when trained under the same settings. Our code along with the checkpoints for all of the trained models is open-sourced and available online [25].
+
+# Acknowledgments
+
+The authors acknowledge contributions from Dr. Zhewei Yao, Aniruddha Nrusimha, and Jiachen Lian. We also acknowledge gracious support from Google Cloud, Google TRC team, and specifically Jonathan Caton, Prof. David Patterson, and Dr. Ed Chi. We would also like to acknowledge the Sky Pilot team from UC Berkeley. Prof. Keutzer’s lab is sponsored by Intel corporation, Intel VLAB team, Intel One-API center of excellence, as well as funding through BDD and BAIR. Sehoon Kim would like to acknowledge the support from Korea Foundation for Advanced Studies (KFAS). Amir Gholami was supported through funding from Samsung SAIT. Michael W. Mahoney would also like to acknowledge the UC Berkeley CLTC, ARO, NSF, and ONR. Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred.
+
+# References
+
+[1] Alexei Baevski, Wei-Ning Hsu, Qiantong Xu, Arun Babu, Jiatao Gu, and Michael Auli. Data2vec: A general framework for self-supervised learning in speech, vision and language. arXiv preprint arXiv:2202.03555, 2022.   
+[2] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael Auli. Wav2vec 2.0: A framework for self-supervised learning of speech representations. Advances in Neural Information Processing Systems, 33:12449–12460, 2020.   
+[3] Andy Brock, Soham De, Samuel L Smith, and Karen Simonyan. High-performance large-scale image recognition without normalization. In International Conference on Machine Learning, pages 1059–1071. PMLR, 2021.   
+[4] Maxime Burchi and Valentin Vielzeuf. Efficient Conformer: Progressive downsampling and grouped attention for automatic speech recognition. arXiv preprint arXiv:2109.01163, 2021.   
+[5] Xuankai Chang, Aswin Shanmugam Subramanian, Pengcheng Guo, Shinji Watanabe, Yuya Fujita, and Motoi Omachi. End-to-end ASR with adaptive span self-attention. In INTERSPEECH, pages 3595–3599, 2020.   
+[6] Sanyuan Chen, Chengyi Wang, Zhengyang Chen, Yu Wu, Shujie Liu, Zhuo Chen, Jinyu Li, Naoyuki Kanda, Takuya Yoshioka, Xiong Xiao, et al. WavLM: Large-scale self-supervised pre-training for full stack speech processing. arXiv preprint arXiv:2110.13900, 2021.   
+[7] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional Transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.   
+[8] Piotr Dollár, Mannat Singh, and Ross Girshick. Fast and accurate model scaling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 924–932, 2021.   
+[9] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.   
+[10] Haoqi Fan, Bo Xiong, Karttikeya Mangalam, Yanghao Li, Zhicheng Yan, Jitendra Malik, and Christoph Feichtenhofer. Multiscale Vision Transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6824–6835, 2021.
+
+[11] John S Garofolo. Timit acoustic phonetic continuous speech corpus. Linguistic Data Consortium, 1993, 1993.
+
+[12] Xue Geng, Jie Lin, Bin Zhao, Anmin Kong, Mohamed M Sabry Aly, and Vijay Chandrasekhar. Hardware-aware softmax approximation for deep neural networks. In Asian Conference on Computer Vision, pages 107–122. Springer, 2018.
+
+[13] Xue Geng, Jie Lin, Bin Zhao, Zhe Wang, Mohamed M Sabry Aly, and Vijay Chandrasekhar. Hardware-aware exponential approximation for deep neural network. 2018.
+
+[14] Alex Graves. Sequence transduction with recurrent neural networks. arXiv preprint arXiv:1211.3711, 2012.
+
+[15] Alex Graves, Santiago Fernández, Faustino Gomez, and Jürgen Schmidhuber. Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks. In Proceedings of the 23rd international conference on Machine learning, pages 369–376, 2006.
+
+[16] Anmol Gulati, James Qin, Chung-Cheng Chiu, Niki Parmar, Yu Zhang, Jiahui Yu, Wei Han, Shibo Wang, Zhengdong Zhang, Yonghui Wu, and Ruoming Pang. Conformer: Convolutionaugmented Transformer for speech recognition. arXiv preprint arXiv:2005.08100, 2020.
+
+[17] Pengcheng Guo et al. Recent developments on ESPNet toolkit boosted by Conformer. In ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5874–5878. IEEE, 2021.
+
+[18] Kyu J Han, Ramon Prieto, and Tao Ma. State-of-the-art speech recognition using multi-stream self-attention with dilated 1d convolutions. In 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 54–61. IEEE, 2019.
+
+[19] Wei Han, Zhengdong Zhang, Yu Zhang, Jiahui Yu, Chung-Cheng Chiu, James Qin, Anmol Gulati, Ruoming Pang, and Yonghui Wu. ContextNet: Improving convolutional neural networks for automatic speech recognition with global context. arXiv preprint arXiv:2005.03191, 2020.
+
+[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
+
+[21] Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. MobileNets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017.
+
+[22] Wei-Ning Hsu, Benjamin Bolte, Yao-Hung Hubert Tsai, Kushal Lakhotia, Ruslan Salakhutdinov, and Abdelrahman Mohamed. HuBERT: Self-supervised speech representation learning by masked prediction of hidden units. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 29:3451–3460, 2021.
+
+[23] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-Excitation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132–7141, 2018.
+
+[24] Shigeki Karita et al. A comparative study on Transformer vs RNN in speech applications. In 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 449–456. IEEE, 2019.
+
+[25] Sehoon Kim. https://github.com/kssteven418/squeezeformer.
+
+[26] Sehoon Kim, Amir Gholami, Zhewei Yao, Michael W Mahoney, and Kurt Keutzer. I-BERT: Integer-only bert quantization. In International conference on machine learning, pages 5506– 5518. PMLR, 2021.
+
+[27] Samuel Kriman, Stanislav Beliaev, Boris Ginsburg, Jocelyn Huang, Oleksii Kuchaiev, Vitaly Lavrukhin, Ryan Leary, Jason Li, and Yang Zhang. QuartzNet: Deep automatic speech recognition with 1d time-channel separable convolutions. In ICASSP, pages 6124–6128. IEEE, 2020.
+
+[28] Taku Kudo and John Richardson. Sentencepiece: A simple and language independent subword tokenizer and detokenizer for neural text processing. arXiv preprint arXiv:1808.06226, 2018.
+
+[29] Jason Li, Vitaly Lavrukhin, Boris Ginsburg, Ryan Leary, Oleksii Kuchaiev, Jonathan M Cohen, Huyen Nguyen, and Ravi Teja Gadde. Jasper: An end-to-end convolutional neural acoustic model. arXiv preprint arXiv:1904.03288, 2019.
+
+[30] Yanghao Li, Chao-Yuan Wu, Haoqi Fan, Karttikeya Mangalam, Bo Xiong, Jitendra Malik, and Christoph Feichtenhofer. Improved Multiscale Vision Transformers for classification and detection. arXiv preprint arXiv:2112.01526, 2021.
+
+[31] Tatiana Likhomanenko, Qiantong Xu, Vineel Pratap, Paden Tomasello, Jacob Kahn, Gilad Avidov, Ronan Collobert, and Gabriel Synnaeve. Rethinking evaluation in ASR: Are our models robust enough? arXiv preprint arXiv:2010.11745, 2020.
+
+[32] Chunxi Liu, Frank Zhang, Duc Le, Suyoun Kim, Yatharth Saraf, and Geoffrey Zweig. Improving RNN Transducer based ASR with auxiliary tasks. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 172–179. IEEE, 2021.
+
+[33] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017.
+
+[34] Yiping Lu, Zhuohan Li, Di He, Zhiqing Sun, Bin Dong, Tao Qin, Liwei Wang, and Tie-Yan Liu. Understanding and improving Transformer from a multi-particle dynamic system point of view. arXiv preprint arXiv:1906.02762, 2019.
+
+[35] Christoph Lüscher, Eugen Beck, Kazuki Irie, Markus Kitza, Wilfried Michel, Albert Zeyer, Ralf Schlüter, and Hermann Ney. RWTH ASR Systems for LibriSpeech: Hybrid vs attention–w/o data augmentation. arXiv preprint arXiv:1905.03072, 2019.
+
+[36] Somshubra Majumdar, Jagadeesh Balam, Oleksii Hrinchuk, Vitaly Lavrukhin, Vahid Noroozi, and Boris Ginsburg. Citrinet: Closing the gap between non-autoregressive and autoregressive end-to-end models for automatic speech recognition. arXiv preprint arXiv:2104.01721, 2021.
+
+[37] Edwin G Ng, Chung-Cheng Chiu, Yu Zhang, and William Chan. Pushing the limits of nonautoregressive speech recognition. arXiv preprint arXiv:2104.03416, 2021.
+
+[38] NVDIA Nemo. https://github.com/nvidia/nemo.
+
+[39] NVDLA Primer. http://nvdla.org/primer.html, 2021.
+
+[40] Jing Pan, Joshua Shapiro, Jeremy Wohlwend, Kyu J Han, Tao Lei, and Tao Ma. ASAPPASR: Multistream CNN and self-attentive SRU for SOTA speech recognition. arXiv preprint arXiv:2005.10469, 2020.
+
+[41] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur. Librispeech: an ASR corpus based on public domain audio books. In 2015 IEEE international conference on acoustics, speech and signal processing (ICASSP), pages 5206–5210. IEEE, 2015.
+
+[42] Daniel S Park, William Chan, Yu Zhang, Chung-Cheng Chiu, Barret Zoph, Ekin D Cubuk, and Quoc V Le. Specaugment: A simple data augmentation method for automatic speech recognition. arXiv preprint arXiv:1904.08779, 2019.
+
+[43] Mathias Perslev, Michael Jensen, Sune Darkner, Poul Jørgen Jennum, and Christian Igel. UTime: A fully convolutional network for time series segmentation applied to sleep staging. Advances in Neural Information Processing Systems, 32, 2019.
+
+[44] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. 2018.
+
+[45] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015.
+
+[46] Kyuhong Shim, Jungwook Choi, and Wonyong Sung. Understanding the role of self attention for efficient speech recognition. In International Conference on Learning Representations, 2021.   
+[47] Laurent Sifre and Stéphane Mallat. Rigid-motion scattering for texture classification. arXiv preprint arXiv:1403.1687, 2014.   
+[48] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.   
+[49] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image Transformers & distillation through attention. In International Conference on Machine Learning, pages 10347–10357. PMLR, 2021.   
+[50] Vincent Vanhoucke. Learning visual representations at scale. 2014.   
+[51] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.   
+[52] Chengyi Wang, Yu Wu, Yao Qian, Kenichi Kumatani, Shujie Liu, Furu Wei, Michael Zeng, and Xuedong Huang. Unispeech: Unified speech representation learning with labeled and unlabeled data. In International Conference on Machine Learning, pages 10937–10947. PMLR, 2021.   
+[53] Hongyu Wang, Shuming Ma, Li Dong, Shaohan Huang, Dongdong Zhang, and Furu Wei. DeepNet: Scaling Transformers to 1,000 layers. arXiv preprint arXiv:2203.00555, 2022.   
+[54] Yongqiang Wang et al. Transformer-based acoustic modeling for hybrid speech recognition. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 6874–6878. IEEE, 2020.   
+[55] Joonsang Yu, Junki Park, Seongmin Park, Minsoo Kim, Sihwa Lee, Dong Hyun Lee, and Jungwook Choi. NN-LUT: Neural approximation of non-linear operations for efficient Transformer inference. arXiv preprint arXiv:2112.02191, 2021.   
+[56] Frank Zhang, Yongqiang Wang, Xiaohui Zhang, Chunxi Liu, Yatharth Saraf, and Geoffrey Zweig. Faster, simpler and more accurate hybrid asr systems using wordpieces. arXiv preprint arXiv:2005.09150, 2020.   
+[57] Qian Zhang, Han Lu, Hasim Sak, Anshuman Tripathi, Erik McDermott, Stephen Koo, and Shankar Kumar. Transformer transducer: A streamable speech recognition model with Transformer encoders and RNN-T loss. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7829–7833. IEEE, 2020.   
+[58] Shucong Zhang, Erfan Loweimi, Peter Bell, and Steve Renals. Stochastic attention head removal: A simple and effective method for improving Transformer based ASR models. arXiv preprint arXiv:2011.04004, 2020.   
+[59] Shucong Zhang, Erfan Loweimi, Peter Bell, and Steve Renals. On the usefulness of selfattention for automatic speech recognition with Transformers. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 89–96. IEEE, 2021.   
+[60] Xiaohui Zhang, Frank Zhang, Chunxi Liu, Kjell Schubert, Julian Chan, Pradyot Prakash, Jun Liu, Ching-Feng Yeh, Fuchun Peng, Yatharth Saraf, et al. Benchmarking lf-mmi, ctc and rnn-t criteria for streaming asr. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 46–51. IEEE, 2021.   
+[61] Ying Zhang, Mohammad Pezeshki, Philémon Brakel, Saizheng Zhang, Cesar Laurent Yoshua Bengio, and Aaron Courville. Towards end-to-end speech recognition with deep convolutional neural networks. arXiv preprint arXiv:1701.02720, 2017.   
+[62] Yu Zhang, James Qin, Daniel S Park, Wei Han, Chung-Cheng Chiu, Ruoming Pang, Quoc V Le, and Yonghui Wu. Pushing the limits of semi-supervised learning for automatic speech recognition. arXiv preprint arXiv:2010.10504, 2020.

parse/dev/gE_vt-w4LhL/gE_vt-w4LhL_content_list.json

@@ -0,0 +1,1300 @@
+[
+    {
+        "type": "text",
+        "text": "Squeezeformer: An Efficient Transformer for Automatic Speech Recognition ",
+        "text_level": 1,
+        "bbox": [
+            225,
+            122,
+            774,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sehoon $\\mathbf { K } \\mathbf { i m } ^ { * 1 }$ , Amir Gholami∗1, Albert Shaw†‡1, Nicholas Lee†1, Karttikeya Mangalam1, Jitendra Malik1, Michael W. Mahoney123, Kurt Keutzer1 ",
+        "bbox": [
+            179,
+            224,
+            820,
+            256
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1University of California, Berkeley 2ICSI 3LBNL {sehoonkim, amirgh, nicholas_lee, mangalam, malik, mahoneymw, keutzer}@berkeley.edu Albertshaw@google.com ",
+        "bbox": [
+            179,
+            260,
+            818,
+            304
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            332,
+            535,
+            348
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The recently proposed Conformer model has become the de facto backbone model for various downstream speech tasks based on its hybrid attention-convolution architecture that captures both local and global features. However, through a series of systematic studies, we find that the Conformer architecture’s design choices are not optimal. After re-examining the design choices for both the macro and micro-architecture of Conformer, we propose Squeezeformer which consistently outperforms the state-of-the-art ASR models under the same training schemes. In particular, for the macro-architecture, Squeezeformer incorporates (i) the Temporal U-Net structure which reduces the cost of the multi-head attention modules on long sequences, and (ii) a simpler block structure of multi-head attention or convolution modules followed up by feed-forward module instead of the Macaron structure proposed in Conformer. Furthermore, for the micro-architecture, Squeezeformer (i) simplifies the activations in the convolutional block, (ii) removes redundant Layer Normalization operations, and (iii) incorporates an efficient depthwise downsampling layer to efficiently sub-sample the input signal. Squeezeformer achieves state-of-the-art results of $7 . 5 \\%$ , $6 . 5 \\%$ , and $6 . 0 \\%$ word-error-rate (WER) on LibriSpeech test-other without external language models, which are $3 . 1 \\%$ , $1 . 4 \\%$ , and $0 . 6 \\%$ better than Conformer-CTC with the same number of FLOPs. Our code is open-sourced and available online [25]. ",
+        "bbox": [
+            233,
+            364,
+            766,
+            627
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            657,
+            316,
+            674
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The increasing success of end-to-end neural network models has been a huge driving force for the drastic advancements in various automatic speech recognition (ASR) tasks. While both convolutional neural networks (CNN) [19, 27, 29, 36, 61] and Transformers [24, 31, 32, 56, 57] have drawn attention as popular backbone architectures for ASR models, each of them has several limitations. Generally, CNN models lack the ability to capture global contexts and Transformers involve prohibitive computing and memory overhead. To overcome these shortcomings, Conformer [16] has recently proposed a novel convolution-augmented Transformer architecture. Due to its ability to synchronously capture global and local features from audio signals, Conformer has become the de facto model not only for ASR tasks, but also for various end-to-end speech processing tasks [17]. Furthermore, it has also achieved state-of-the-art performance in combination with recent developments in self-supervised learning methodologies as well [37, 62]. While the Conformer architecture was introduced as an autoregressive RNN-Transducer (RNN-T) [14] model in its original setting, it has been adopted with less critique to non-autoregressive schemes such as Connectionist Temporal Classification (CTC) [15] as well [38]. ",
+        "bbox": [
+            174,
+            679,
+            825,
+            844
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a966fc8d69f0dfca207c707aab43dea72301ec2184f3705f2f0839da5c1aab3.jpg",
+        "image_caption": [
+            "Figure 1: (Left) We perform a series of systematic studies on macro and micro architecture to redesign the Conformer architecture towards our Squeezeformer architecture. The bars and the line indicate the WER on LibriSpeech test-other dataset and the FLOPs, respectively. For each design modification, we strictly improve WER until our final Squeezeformer model outperforms Conformer by $1 . 4 0 \\%$ WER improvement with the same number of FLOPs. See Tab. 1 for the details. (Right) LibriSpeech test-other WER vs. FLOPs for Squeezeformer and other state-of-the-art ASR models. Conformer- $\\mathrm { { C T C ^ { * } } }$ is our own reproduction to the best performance as possible and the others are the reported numbers in their papers [4, 27, 36]. Our architecture scales well to smaller and larger models to constantly outperform other models by a large margin throughout the entire FLOPs range. See Tab. 3 for the details. For both plots, the lower the WER, the better; however, we plotted in reverse for better visualization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            88,
+            823,
+            268
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            441,
+            823,
+            470
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Despite being a key architecture in speech processing tasks, the Conformer architecture has some limitations that can be improved upon. First, Conformer still suffers from the quadratic complexity of the attention mechanism limiting its efficiency on long sequence lengths. This problem is further highlighted by the long sequence lengths of typical audio inputs as also pointed out in [46]. Furthermore, the Conformer architecture is relatively more complicated than Transformer architectures used in other domains such as in natural language processing [7, 44, 51] or computer vision [9, 10, 49]. For instance, the Conformer architecture incorporates multiple different normalization schemes and activation functions, the Macaron structure [34], as well as back-to-back multi-head attention (MHA) and convolution modules. This level of complexity makes it difficult to efficiently deploy the model on dedicated hardware platforms for inference [26, 39, 55]. More importantly, this raises the question of whether such design choices are necessary and optimal for achieving good performance in ASR tasks. ",
+        "bbox": [
+            173,
+            477,
+            825,
+            642
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we perform a careful and systematic analysis of each of the design choices with the goal of achieving lower word-error-rate (WER) for a given computational budget. We developed a much simpler and more efficient hybrid attention-convolution architecture in both its macro and micro-design that consistently outperforms the state-of-the-art ASR models. In particular, we make the following contributions in our proposed Squeezeformer model: ",
+        "bbox": [
+            173,
+            648,
+            825,
+            718
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We find a high temporal redundancy in the learned feature representations of neighboring speech frames especially deeper in the network, which results in unnecessary computational overhead. To address this, we incorporate the temporal U-Net structure in which a downsampling layer halves the sampling rate at the middle of the network, and a light upsampling layer recovers the temporal resolution at the end for training stability (§ 3.1.1).   \n• We redesign the hybrid attention-convolution architecture based on our observation that the backto-back MHA and convolution modules with the Macaron structure are suboptimal. In particular, we propose a simpler block structure similar to the standard Transformer block [7, 51], where the MHA and convolution modules are each directly followed by a single feed forward module $( \\ S \\ 3 . 1 . 2 )$ .   \n• We finely examine the micro-architecture of the network and found several modifications that simplify the model overall and greatly improve the accuracy and efficiency. This includes (i) activation unification that replaces GLU activations with Swish (§ 3.2.1), (ii) Layer Normalization ",
+        "bbox": [
+            173,
+            722,
+            826,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/5bfbfab0b9ed5aa15ba0479dd25d335c2908998b4be733be3df2efb6269735d2.jpg",
+        "image_caption": [
+            "Figure 2: (Left) The Conformer architecutre and (Right) the Squeezeformer architecture which comprises of the Temporal U-Net structure for downsampling and upsampling of the sampling rate, the standard Transformer-style block structure that only uses Post-Layer Normalization, and the depthwise separable subsampling layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            87,
+            810,
+            297
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "simplification by replacing redundant pre-Layer Normalization layers with a scaled post-Layer Normalization which incorporates a learnable scaling for the residual path that can be merged with other layers to be zero-cost during inference $\\lbrace \\ S 3 . 2 . 2 )$ , and (iii) incorporation of a depthwise separable convolution for the first sub-sampling layer that results in a significant floating point operations (FLOPs) reduction $( \\ S \\ 3 . 2 . 3 )$ . ",
+        "bbox": [
+            186,
+            376,
+            825,
+            445
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• We show that the Squeezeformer architecture scales well with both smaller and larger models and consistently outperforms other state-of-the-art ASR models when trained under the same settings (Tab. 4.2, $\\ S 4 . 2$ ). Furthermore, we justify the final model architecture of Squeezeformer with a reverse ablation study for the design choices (Tab. 4, $\\ S 4 . 3$ ). ",
+        "bbox": [
+            176,
+            450,
+            825,
+            506
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            525,
+            321,
+            542
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The recent advancements in end-to-end ASR can be broadly categorized into (1) model architecture and (2) training methods. ",
+        "bbox": [
+            174,
+            553,
+            823,
+            582
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Model Architecture for End-to-end ASR. The recent end-to-end ASR models are typically composed of an encoder, which takes as input a speech signal (i.e., sequence of speech frames) and extracts high-level acoustic features, and a decoder, which converts the extracted features from the encoder into a sequence of text. The model architecture of the encoder determines the representational power of an ASR model and its ability to extract acoustic features from input signals. Therefore, a strong architecture is critical for overall performance. ",
+        "bbox": [
+            174,
+            593,
+            825,
+            676
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One of the popular choices for a backbone model architecture is convolutional neural network (CNN). End-to-end deep CNN models have been first explored in [29, 61], and further improved by introducing depth-wise separable convolution [21, 47, 50] in QuartzNet [27] and the Squeezeand-Excitation module [23] in CitriNet [36] and ContextNet [19]. However, since CNNs often fail to capture global contexts, Transformer [51] models have also been widely adopted in backbone architectures due to their ability to capture long-range dependencies between speech frames [24, 31, 32, 56, 57]. Recently, [16] has proposed a novel model architecture named Conformer, which augments Transformers with convolutions to model both global and local dependencies efficiently. With the Conformer architecture as our starting point, we focus on designing a next-generation model architecture for ASR that is simpler, more accurate, and more efficient. ",
+        "bbox": [
+            173,
+            683,
+            825,
+            821
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The hybrid attention-convolution architecture of Conformer has enabled the state-of-the-art results in many speech tasks. However, the quadratic complexity of the attention layer still proves to be cost prohibitive at larger sequence lengths. While different approaches have been proposed to reduce the cost of MHA in ASR [5, 46, 58, 59], their main focus is not changing the overall architecture design, and their optimizations can also be applied to our model, as they are orthogonal to the developments for Squeezeformer. Efficient-Conformer [4] introduces the progressive downsampling scheme and grouped attention to reduce the training and inference costs of Conformer. Our work incorporates a similar progressive downsampling, but also introduces an up-sampling mechanism with skip connections from the earlier layers inspired by the U-Net [45] architecture in computer vision and the U-Time [43] architecture for sleep signal analysis. We find this to be critical for training stability and overall performance. In addition, through systematic experiments, we completely refactor the Conformer block by carefully redesigning both the macro and micro-architectures. ",
+        "bbox": [
+            174,
+            828,
+            823,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c43f76d02f321b4ffeb0b8400a07b53f39be51ac7bfeb07e384227f00f4b78f.jpg",
+        "table_caption": [
+            "Table 1: Starting from Conformer as the baseline, we redesign the architecture towards Squeezeformer through a series of systematic studies on macro and micro architecture. Note that for each design change, the WER on LibriSpeech test-clean and test-other datasets improves consistently. For comparison, we include the number of parameters and FLOPs for a 30s input in the last two columns. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Design change</td><td>test-clean test-other|</td><td></td><td>|Params (M)| GFLOPs</td><td></td></tr><tr><td>Conformer-CTC-M</td><td>Baseline</td><td>3.20</td><td>7.90</td><td>27.4</td><td>71.7</td></tr><tr><td></td><td>+ Temporal U-Net ($ 3.1.1)</td><td>2.97</td><td>7.28</td><td>27.5</td><td>57.0</td></tr><tr><td></td><td>+ Transformer-style Block (§ 3.1.2)</td><td>2.93</td><td>7.12</td><td>27.5</td><td>57.0</td></tr><tr><td></td><td>+ Unified activations (§ 3.2.1)</td><td>2.88</td><td>7.09</td><td>28.7</td><td>58.4</td></tr><tr><td></td><td>+ Simplified LayerNorm (§ 3.2.2)</td><td>2.85</td><td>6.89</td><td>28.7</td><td>58.4</td></tr><tr><td>Squeezeformer-SM</td><td>+ DW sep. subsampling (§ 3.2.3)</td><td>2.79</td><td>6.89</td><td>28.2</td><td>42.7</td></tr><tr><td>Squeezeformer-M</td><td>+ Model scale-up ($ 3.2.3)</td><td>2.56</td><td>6.50</td><td>55.6</td><td>72.0</td></tr></table>",
+        "bbox": [
+            194,
+            156,
+            812,
+            279
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            295,
+            825,
+            377
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Training Methodology for End-to-end ASR. In the past few years, various self-supervised learning methodologies based on contrastive learning [2, 52, 62] or masked prediction [1, 6, 22] have been proposed to push forward the ASR performance. While a model pre-trained with self-supervised tasks generally outperforms when finetuned on a target ASR task, training strategies are not the main focus in this work as they can be applied independently to the underlying architecture. ",
+        "bbox": [
+            174,
+            387,
+            825,
+            458
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 Architecture Design ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            477,
+            372,
+            493
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Conformer architecture has been widely adopted by the speech community and is used as a backbone for different speech tasks. At a macro-level, Conformer incorporates the Macaron structure [34] comprised of four modules per block, as shown in Fig. 2 (Left). These blocks are stacked multiple times to construct the Conformer architecture. In this work, we carefully reexamine the design choices in Conformer, starting first with its macro-architecture, and then its microarchitecture design. We choose Conformer-CTC-M as the baseline model for the case study, and we compare word-error-rate (WER) on LibriSpeech test-other as a performance metric for each architecture. Furthermore, we measure FLOPs on a 30s audio input as a proxy for model efficiency. While we acknowledge that FLOPs may not always be a linear indicator of hardware and runtime efficiency, we choose FLOPs as it is hardware agnostic and is statically computable. However, we do measure the final end-to-end throughput of our changes, ensuring up to $1 . 3 4 \\times$ consistent improvement in runtime for different versions of Squeezeformer (Tab. 4.2) ",
+        "bbox": [
+            173,
+            507,
+            825,
+            674
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Macro-Architecture Design ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            690,
+            403,
+            705
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We first focus on designing the macro structure of Squeezeformer, i.e., how the blocks and modules are organized in a global scale. ",
+        "bbox": [
+            174,
+            713,
+            821,
+            741
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 Temporal U-Net Architecture ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            756,
+            429,
+            770
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The hybrid attention-convolution structure enables Conformer to capture both global and local interactions. However, the attention operation has a quadratic FLOPs complexity with respect to the input sequence length. We propose to lighten this extra overhead by computing attention over a reduced sequence length. In the Conformer model itself, the input sampling rate is reduced from $1 0 \\mathrm { m s }$ to $4 0 \\mathrm { m s }$ with a convolutional subsampling block at the base of the network. However, this rate is kept constant throughout the network, with all the attention and convolution operations operating at a constant temporal scale. ",
+        "bbox": [
+            174,
+            779,
+            825,
+            877
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To this end, we begin by studying the temporal redundancy in the learned feature representations. In particular, we analyze how the learned feature embeddings per speech frame are differentiated through the Conformer model depth. We randomly sample 100 audio signals from LibriSpeech’s dev-other dataset, and process them through the Conformer blocks, recording their per-block activations. We then measure the average cosine similarity between two neighboring embedding vectors. The results are plotted as the solid lines in Fig. 3. We observe that the embeddings for the speech frames directly next to each other have an average similarity of $9 5 \\%$ at the topmost layer, and even those 4 speech frames away from each other have a similarity of more than $80 \\%$ . This reveals that there is an increasing temporal redundancy as inputs are processed through the Conformer blocks deeper in the network. We hypothesize that this redundancy in feature embedding vectors causes unnecessary computational overhead and that the sequence length can be reduced deeper in the network without loss in accuracy. ",
+        "bbox": [
+            173,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fad821fb9ce98ef500fa758eeed93ed6552d0b4fbbc048e48d38242324d8cee.jpg",
+        "image_caption": [
+            "Figure 3: Cosine similarity between two embedding vectors of neighboring speech frames with varying adjacency distances across the Conformer blocks. The temporal dimension is downsampled after the 7th block and upsampled before the 16th block in the Temporal U-Net structure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            108,
+            459,
+            262
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/bec441ea699c28ec9951f7127d981bf98196d5d12ad33a3e58f372913a5b5dcc.jpg",
+        "image_caption": [
+            "Figure 4: (Left) Back-to-back preLN and postLN at the boundary of the blocks. (Right) The preLN can be replaced with the learned scaling that readjusts the magnitude of the activation that goes into the subsequent module. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            513,
+            108,
+            805,
+            243
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            382,
+            825,
+            520
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As our first macro-architecture improvement step, we change the Conformer model to incorporate subsampling of the embedding vectors after it has been processed by the early blocks of the model. In particular, we keep the sample rate to be $4 0 \\mathrm { m s }$ up to the 7th block, and afterwards we subsample to a rate of $8 0 \\mathrm { m s }$ per input sequence by using a pooling layer. For the pooling layer we use a depthwise separable convolution with stride 2 and kernel size 3 to merge the redundancies across neighboring embeddings. This decreases the attention complexity by $4 \\times$ and also reduces the redundancies of the features. This temporal downsampling shares similarities with computer vision models, which often downsample the input image spatially to save compute and develop hierarchical level features [10, 20, 30, 48], and with the approach of Efficient Conformer [4]. ",
+        "bbox": [
+            174,
+            522,
+            825,
+            647
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "However, the temporal downsampling alone leads to an unstable and diverging training behaviour $( \\ S 4 . 3 )$ . One possible reason for this is the lack of enough resolution for the decoder after subsampling the rate to $8 0 \\mathrm { m s }$ . The decoder maps an embedding for each speech frame into a single label, e.g., character, and therefore requires sufficient resolution for successful decoding of the full sequence. Inspired from successful architectures for dense prediction in computer vision such as U-Net [45], we incorporate the Temporal U-Net structure to recover the resolution at the end of the network through an upsampling layer as shown in Fig. 2. This upsampling block takes the embedding vectors processed by the $4 0 \\mathrm { m s }$ and $8 0 \\mathrm { m s }$ sampling rates, and produces an embedding with a rate of 40ms by adding them together via a skip connection. To the best of our knowledge, the closest work to our Temporal U-Net is the approach proposed in [43], in which the U-Net structure is incorporated into a fully-convolutional model to downsample sleep signals. ",
+        "bbox": [
+            174,
+            654,
+            825,
+            806
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This change not only reduces the total FLOPs by $20 \\%$ compared to Conformer1, but also improves the test-other WER by $0 . 6 2 \\%$ from $7 . 9 0 \\%$ to $7 . 2 8 \\%$ (Tab. 1, 2nd row). Furthermore, analyzing the cosine similarity shows that the Temporal U-Net architecture prevents the neighboring embeddings from becoming too similar to each others at the later blocks, in particular at the final block directly connected to the decoder, as shown in Fig. 3 as the dashed lines. ",
+        "bbox": [
+            174,
+            813,
+            825,
+            854
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            823,
+            119
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1.2 Transformer-Style Block",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            397,
+            148
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The Conformer block consists of a sequence of feed-forward (‘F’), multi-head attention (MHA, ‘M’), convolution (‘C’), and another feed-forward module (‘F’). We denote this as the FMCF structure. Note that the convolutional kernel sizes in ASR models are rather large, e.g., 31 in Conformer, which makes its behaviour similar to attention in mixing global information. This is stark contrast to convolutional kernels in computer vision, which often have small $3 \\times 3$ kernels and hence benefit greatly from attention’s global processing. As such, placing the convolution and MHA module with a similar functionality back-to-back (i.e., the MC substructure) does not seem prudent. Hence, we consider an MF/CF structure, which is motivated by considering the convolution module as a local MHA module. Furthermore, we drop the Macaron structure [34], as MHA modules followed by feed-forward modules have been more widely adopted in the literature [7, 9, 44, 51]. In a nutshell, we simplify the architecture to be similar to the standard Transformer network and denote the blocks MF and CF substructures, as shown in Fig. 2. This modification further improves the test-other WER by $0 . 1 6 \\%$ from $7 . 2 8 \\%$ to $7 . 1 2 \\%$ and marginally improves the test-clean WER without affecting the FLOPs (Tab. 1, 3rd row). ",
+        "bbox": [
+            173,
+            159,
+            825,
+            352
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Micro-Architecture Design ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            368,
+            400,
+            383
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "So far we have designed the macro structure of Squeezeformer by incorporating seminal architecture principles from computer vision and natural language processing into Conformer. In this subsection, we now focus on optimizing the micro structure of the individual modules. We show that we can further simplify the module architectures while improving both efficiency and performance. ",
+        "bbox": [
+            174,
+            391,
+            825,
+            446
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 Unified Activations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            462,
+            357,
+            477
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Conformer uses Swish activation for most of the blocks. However, it switches to a Gated Linear Unit (GLU) for its convolution module. Such a heterogeneous design seems over-complicated and unnecessary. From a practical standpoint, the use of multiple activations complicates hardware deployment, as an efficient implementation requires dedicated logic design, look up tables, or custom approximations [26, 39, 55]. For instance, on low-end edge devices with no dedicated vector processing unit, supporting additional non-linear operations would require additional look up tables or advanced algorithms [12, 13]. To address this, we propose to replace the GLU activation with Swish, unifying the choice of activation function throughout the entire model. We keep the expansion rate for the convolution modules. As shown in the 4th row of Tab. 1, this change does not entail noticeable changes in WER and FLOPs but only simplifies the architecture. ",
+        "bbox": [
+            174,
+            482,
+            825,
+            621
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 Simplified Layer Normalizations ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            635,
+            450,
+            650
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Continuing our micro-architecture improvements, we note that the Conformer model incorporates redundant Layer Normalizations (LayerNorm), as shown in Fig. 4 (Left). This is because the Conformer model contains both a post-LayerNorm (postLN) that applies LayerNorm in between the residual blocks, as well as pre-LayerNorm (preLN) which applies LayerNorm inside the residual connection. While it is hypothesized that preLN stabilizes training and postLN benefits performance [53], these two modules used together lead to redundant back-to-back operations. Aside from the architectural redundancy, LayerNorm can be computationally expensive [26, 55] due to its global reduction operations. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            766
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "However, we found that naïvely removing the preLN or postLN leads to training instability and convergence failure $( \\ S \\ 4 . 3 )$ . Investigating the cause of failure, we observe that a typical trained Conformer model has orders of magnitude differences in the norms of the learnable scale variables of the back-to-back preLN and postLN. In particular, we found that the preLN would scale down the input signal by a large value, giving more weight to the skip connection. Therefore, it is important to use a scaling layer when replacing the preLN component to allow the network to control this weight. This idea is also on par with several training stabilization strategies in other domains. For instance, NF-Net [3] proposed adaptive (i.e., learnable) scaling before and after the residual blocks to stabilize training without normalization. Furthermore, DeepNet [53] also recently proposed to add non-trainable rule-based scaling to the skip connections to stabilize preLN in Transformers. ",
+        "bbox": [
+            174,
+            772,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/81d263a0ce16a0acdd451ec0998455195722e83d2978a19f5e420c170582794c.jpg",
+        "table_caption": [
+            "Table 2: Detailed architecture configurations for Conformer-CTC (baseline) and Squeezeformer. For comparison, we include the number of parameters and FLOPs for a 30s input in the last two columns. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>#Layers</td><td>Dimension</td><td>#Heads</td><td>Params (M)</td><td>GFLOPs</td></tr><tr><td>Conformer-CTC-S</td><td>16</td><td>144</td><td>4</td><td>8.7</td><td>26.2</td></tr><tr><td>Squeezeformer-XS</td><td>16</td><td>144</td><td>4</td><td>9.0</td><td>15.8</td></tr><tr><td>Squeezeformer-S</td><td>18</td><td>196</td><td>4</td><td>18.6</td><td>26.3</td></tr><tr><td>Conformer-CTC-M</td><td>16</td><td>256</td><td>4</td><td>27.4</td><td>71.7</td></tr><tr><td>Squeezeformer-SM</td><td>16</td><td>256</td><td>4</td><td>28.2</td><td>42.7</td></tr><tr><td>Squeezeformer-M</td><td>20</td><td>324</td><td>4</td><td>55.6</td><td>72.0</td></tr><tr><td>Conformer-CTC-L</td><td>18</td><td>512</td><td>8</td><td>121.5</td><td>280.6</td></tr><tr><td>Squeezeformer-ML</td><td>18</td><td>512</td><td>8</td><td>125.1</td><td>169.2</td></tr><tr><td>Squeezeformer-L</td><td>22</td><td>640</td><td>8</td><td>236.3</td><td>277.9</td></tr></table>",
+        "bbox": [
+            251,
+            128,
+            746,
+            282
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Inspired by these computer vision advancements, we propose to replace preLN with a learnable scaling layer that scales and shifts the activations, $\\mathrm { S c a l i n g } \\bar { ( } x ) = \\gamma x \\bar { + } \\beta$ , with learnable scale and bias vectors $\\gamma$ and $\\beta$ of the size of feature dimension. For homogeneity of architectural design, we then replace the preLN throughout all the modules with the postLN-then-scaling as illustrated in Fig. 2 (Right) and make the entire model postLN-only. Note that the learned scaling parameters can be merged into the weights of the subsequent linear layer, as the architecture illustrated in Fig. 2 (Right), and hence have zero inference cost. With the learned scaling, our model further improves the test-other WER by $0 . 2 0 \\%$ from $7 . 0 9 \\%$ to $6 . 8 9 \\%$ (Tab. 1, 5th row). ",
+        "bbox": [
+            173,
+            305,
+            825,
+            416
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.2.3 Depthwise Separable Subsampling ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            435,
+            465,
+            450
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now shift our focus from the Conformer blocks to the subsampling block. While it is easy to overlook this single module at the beginning of the architecture, we note that it accounts for a significant portion of the overall FLOPs count, up to $28 \\%$ for Conformer-CTC-M with a 30-second input. This is because the subsampling layer uses two vanilla convolution operations each of which has a stride 2. To reduce the overhead of this layer, we replace the second convolution operation with a depthwise separable convolution while keeping the kernel size and stride the same. We leave the first convolution operation as is since it is equivalent to a depthwise convolution with the input dimension 1. This saves an additional $22 \\%$ of the baseline FLOPs without a test-other WER drop and even a $0 . 0 6 \\%$ improvement in test-clean WER (Tab. 1, 6th row). An important point to note here is that generally depthwise separable convolutions are hard to efficiently map to hardware accelerators, in part due its low arithmetic intensity. However, given the large FLOPs reduction, we consistently observe an overall improvement in the total inference throughput of up to $1 . 3 4 \\times$ as reported in Tab. 3, as compared to the baseline Conformer models. ",
+        "bbox": [
+            174,
+            457,
+            825,
+            636
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We name our final model with all these improvements as Squeezeformer-SM. Compared to ConformerCTC-M, our initial baseline, Squeezeformer-SM improves WER by $1 . 0 1 \\%$ from $7 . 9 0 \\%$ to $6 . 8 9 \\%$ with $40 \\%$ less FLOPs. Given the smaller FLOPs of Squeezeformer-SM, we also scale up the model to a similar FLOPs cost as Conformer-CTC-M. In particular, we scale both depth and width of the model together following the practice in [8]. Scaling up the model achieves additional test-other WER gain of $0 . 3 9 \\%$ from $6 . 8 9 \\%$ to $6 . 5 0 \\%$ (Tab. 1, 7th row), and we name this architecture Squeezeformer-M. ",
+        "bbox": [
+            173,
+            643,
+            825,
+            726
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            750,
+            266,
+            767
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Experiment Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            780,
+            339,
+            795
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Models. Following the procedure described in $\\ S 3$ , we construct Squeezeformer variants with different size and FLOPs: we apply the macro and micro-architecture changes in $\\ S \\ 3 . 1$ and $\\ S 3 . 2$ , respectively, to construct Squeezeformer-XS, SM, and ML from Conformer-S, M, and L, retaining the model size. Afterwards, we construct Squeezeformer-S, M, and L by scaling up each model to match the FLOPs of the corresponding Conformer. The detailed architecture configurations are described in Tab. 2. ",
+        "bbox": [
+            174,
+            808,
+            825,
+            877
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "While there are multiple options available for the decoder such as RNN-Transducer (RNN-T) [14] and Connectionist Temporal Classification (CTC) [15], we use a CTC decoder whose non-autoregressive decoding method benefits training and inference latency [36]. However, the main focus of this work is the model architecture design of the encoder, which can be orthogonal to the decoder type. ",
+        "bbox": [
+            174,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/26054a3addbae605ba7954ce17abe7cb5c29201382bf276c72804a2af0c3c238.jpg",
+        "table_caption": [
+            "Table 3: WER $( \\% )$ comparison on LibriSpeech dev and test datasets for Squeezeformer and other state-of-the-art CTC models for ASR including Conformer-CTC, QuartzNet [27], CitriNet [36], Transformer-CTC [31], and Efficient Conformer-CTC [4]. For comparison, we include the number of parameters, FLOPs, and throughput (Thp) on a single NVIDIA Tesla A100 GPU for a 30s input in the last three columns. ∗The performance numbers for Conformer-CTC are based on our own reproduction to the best performance as possible. All the other performance numbers are from the corresponding papers. †With and ‡without the grouped attention. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>dev-clean</td><td>dev-other</td><td>test-clean</td><td>test-other</td><td>Params (M)</td><td>GFLOPs</td><td>Thp (ex/s)</td></tr><tr><td>Conformer-CTC-S* [16]</td><td>4.21</td><td>10.54</td><td>4.06</td><td>10.58</td><td>8.7</td><td>26.2</td><td>613</td></tr><tr><td>QuartzNet 5x5 [27]</td><td>5.39</td><td>15.69</td><td>1</td><td>1</td><td>6.7</td><td>20.2</td><td>1</td></tr><tr><td>Citrinet 256 [36]</td><td>-</td><td>1</td><td>3.78</td><td>9.60</td><td>10.3</td><td>16.8</td><td>1</td></tr><tr><td>Squeezeformer-Xs</td><td> 3.63</td><td>9.30</td><td> 3.74</td><td>9.09</td><td>9.0</td><td>15.8</td><td>763</td></tr><tr><td>Conformer-CTC-M* [16]</td><td>2.94</td><td>7.80</td><td>3.20</td><td>7.90</td><td>27.4</td><td>71.7</td><td>463</td></tr><tr><td>QuartzNet 5x10 [27]</td><td>4.14</td><td>12.33</td><td>1</td><td>1</td><td>12.8</td><td>38.5</td><td>-</td></tr><tr><td>QuartzNet 5x15 [27]</td><td>3.98</td><td>11.58</td><td>3.90</td><td>11.28</td><td>18.9</td><td>55.7</td><td>-</td></tr><tr><td>Citrinet 512 [36]</td><td>1</td><td>1</td><td>3.11</td><td>7.82</td><td>37.0</td><td>63.1</td><td>1</td></tr><tr><td>Eff. Conformer-CTCt [4]</td><td>-</td><td>1</td><td>3.57</td><td>8.99</td><td>13.2</td><td>26.0</td><td>=</td></tr><tr><td>Eff. Conformer-CTC‡ [4]</td><td></td><td>-</td><td>3.58</td><td>8.88</td><td>13.2</td><td>32.5</td><td>=</td></tr><tr><td>Squeezeformer-S</td><td>2.80</td><td>7.49</td><td>3.08</td><td>7.47</td><td>18.6</td><td>26.3</td><td>602</td></tr><tr><td> Squeezeformer-SM</td><td>2.71</td><td>6.98</td><td>2.79</td><td>6.89</td><td>28.2</td><td>42.7</td><td>558</td></tr><tr><td>Conformer-CTC-L* [16]</td><td>2.61</td><td>6.45</td><td>2.80</td><td>6.55</td><td>121.5</td><td>280.6</td><td>200</td></tr><tr><td>Citrinet 1024 [36]</td><td>1</td><td>-</td><td>2.52</td><td>6.22</td><td>143.1</td><td>246.3</td><td>-</td></tr><tr><td> Squeezeformer-M</td><td>2.43</td><td>6.51</td><td>2.56</td><td>6.50</td><td>55.6</td><td>72.0</td><td>431</td></tr><tr><td>Squeezeformer-ML</td><td> 2.34</td><td>6.08</td><td>2.61</td><td>6.05</td><td>125.1</td><td>169.2</td><td>268</td></tr><tr><td>Transformer-CTC [31]</td><td>2.6</td><td>7.0</td><td>2.7</td><td>6.8</td><td>255.2</td><td>621.1</td><td>1</td></tr><tr><td> Squeezeformer-L</td><td> 2.27</td><td> 5.77</td><td>2.47</td><td> 5.97</td><td>236.3</td><td>277.9</td><td>207</td></tr></table>",
+        "bbox": [
+            171,
+            196,
+            828,
+            474
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            493,
+            823,
+            522
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Another subtlety when evaluating models is the use of external language models (LM). In many prior works [18, 24, 35, 40, 54, 56], decoders are often augmented with external LMs such as pre-trained 4-gram or Transformer, which boost the final WER by re-scoring the outputs in a more lexically accurate manner. However, we compare the results without external LMs to fairly compare the true representation power of the model architectures alone − external LMs can be incorporated as an orthogonal optimization afterward. ",
+        "bbox": [
+            174,
+            527,
+            825,
+            611
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Training Details. Because the training recipes and codes for Conformer have not been open-sourced, we train it to reproduce the best performance numbers as possible. We train both Conformer-CTC and Squeezeformer on the LibriSpeech-960hr [41] for 500 epochs on Google’s cloud TPUs v3 with batch size 1024 for the small and medium variants and 2048 for the large variants. We use AdamW [33] optimizer with weight decay 5e-4 for all models. More details for the training and evaluation setup are given in $\\ S \\operatorname { A . 1 }$ and $\\ S \\ A . 2$ . ",
+        "bbox": [
+            174,
+            617,
+            825,
+            700
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 Main Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            718,
+            305,
+            733
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Tab. 3 we compare the WER of Squeezeformer with Conformer-CTC and other state-of-the-art CTC-based ASR models including QuartzNet [27], CitriNet [36], Transformer [31], and Efficient Conformer [4] on the clean and other datasets. Note that the performance numbers for Conformer$\\mathrm { C T C } ^ { 2 }$ are based on our own reproduction to the best performance as possible due to the absence of public training recipes or codes. For simplicity, we denote WER as test-clean/test-other without $\\%$ throughout the section. ",
+        "bbox": [
+            173,
+            741,
+            825,
+            824
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Squeezeformer vs. Conformer. Our smallest model Squeezeformer-XS outperforms ConformerCTC-S by 0.32/1.49 (3.74/9.09 vs. 4.06/10.58) with $1 . 6 6 \\times$ FLOPs reduction. Compared with Conformer-CTC-M, Squeezeformer-S achieves 0.12/0.43 WER improvement (3.08/7.47 vs. ",
+        "bbox": [
+            174,
+            830,
+            825,
+            872
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4654b804d434057458b82b098afb4d7b762a2fdbffbda9585988e14460b8de41.jpg",
+        "table_caption": [
+            "Table 4: Ablation studies for the design choices made in Squeezeformer, including Temporal U-Net, LayerNorm, and activation in the convolution module. ∗Without the upsampling layer, the model fails to converge. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Ablation</td><td>| Model</td><td>dev-clean dev-other</td><td></td></tr><tr><td> Ours</td><td>Squeezeformer-M</td><td>2.43</td><td>6.51</td></tr><tr><td>Temporal U-Net (§ 3.1.1),</td><td>No skip connection No upsampling</td><td>2.78 N/A*</td><td>7.38 N/A*</td></tr><tr><td>LayerNorm (§ 3.2.2)</td><td>PostLN only PreLNonly</td><td>5.60 3.02</td><td>14.00 8.27</td></tr><tr><td>Convolution module (§ 3.2.1) |No Swish</td><td></td><td>2.53</td><td>6.73</td></tr></table>",
+        "bbox": [
+            271,
+            128,
+            730,
+            251
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3.20/7.90) with $1 . 4 7 \\times$ smaller size and $2 . 7 3 \\times$ less FLOPs, and Squeezeformer-SM further improves WER by 0.41/1.01 (2.79/6.89 vs. 3.20/7.90) with a comparable size and $1 . 7 0 \\times$ less FLOPs. Compared with Conformer-CTC-L, Squeezeformer-M shows $0 . 2 4 / 0 . 0 5$ WER improvement $( 2 . 5 6 / 6 . 5 0 $ vs. 2.80/6.55) with significant size and FLOPs reductions of $2 . 1 8 \\times$ and $3 . 9 0 \\times$ , respectively, and Squeezeformer-ML shows 0.19/0.50 WER improvement (2.61/6.05 vs. 2.80/6.55) with a similar size and $1 . 6 6 \\times$ less FLOPs. Finally, our largest model Squeezeformer-L improves WER by 0.33/0.58 upon Conformer-CTC-L with the same FLOPs count, achieving the state-of-the-art result of 2.47/5.97. ",
+        "bbox": [
+            173,
+            266,
+            825,
+            364
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Squeezeformer vs. Other ASR Models. As can be seen in Tab. 3, our model consistently outperforms QuartzNet, CitriNet, and Transformer with comparable or smaller model sizes and FLOPs counts. A notable result is a comparison against Efficient-Conformer: our model outperforms the efficientlydesigned Efficient Conformer by a large margin of 0.79/1.99 (2.79/6.89 vs. 3.58/8.88) with the same FLOPs count. The overall results are summarized as a plot in Fig. 1 (Right) where Squeezeformer consistently outperforms other models across all FLOPs regimes. ",
+        "bbox": [
+            174,
+            369,
+            825,
+            454
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 Ablation Studies ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            470,
+            328,
+            486
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide additional ablation studies for the design choices made for individual architecture components using Squeezeformer-M as the base model. See Tab. 4. Unless specified, we use the same hyperparameter settings as in the main experiment. ",
+        "bbox": [
+            174,
+            496,
+            823,
+            539
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Temporal U-Net. In the 2nd row of Tab. 4, the model clearly underperforms by 0.35/0.87 without the skip connection from the downsampling layer to the upsampling layer. This shows that the high-resolution information collected in the early layers is critical for successful decoding. The 3rd row in Tab. 4 shows that our model completely fails to converge without the upsampling layer due to training stability, even with several different peak learning rates of $\\{ 0 . 5 , 1 . 0 , 1 . 5 \\} \\mathrm { e } . 3$ . ",
+        "bbox": [
+            174,
+            544,
+            825,
+            614
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "LayerNorm. In the 4th line of Tab. 4, we show that WER drops significantly by 3.17/7.49 when we apply the PostLN-only scheme without the learned scaling layer. Another alternative design choice is to apply the PreLN-only scheme without the learned scaling, which also results in a noticeable WER degradation of $0 . 5 9 / 1 . 7 6$ as shown in the 5th line of Tab. 4. In both cases, the model fails to converge, so we report the best WER before divergence. The results suggest that the learned scaling layer plays a key role for training stabilization and better WER. ",
+        "bbox": [
+            174,
+            619,
+            825,
+            704
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Convolution Module. When ablating the GLU activation in the convolution modules, another possible design choice is to drop it without replacing it with the Swish activation. This, however, results in 0.10/0.22 worse WER as shown in the last line of Tab. 4. ",
+        "bbox": [
+            176,
+            709,
+            825,
+            752
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            772,
+            307,
+            789
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we performed a series of systematic ablation studies on the macro and micro architecture of the Conformer architecture, and we proposed a novel hybrid attention-convolution architecture that is simpler and consistently achieves better performance than other models for a wide range of computational budgets. The key novel components of Squeezeformer’s macro-architecture is the incorporation of the Temporal U-Net structure which downsamples audio signals in the second half of the network to reduce the temporal redundancy between adjacent features and save compute, as well as the MF/CF block structure similar to the standard Transformer-style which simplifies the architecture and improves performance. Furthermore, the micro-architecture of Squeezeformer simplifies the activations throughout the model and replaces redundant LayerNorms with the scaled postLN, which is more efficient and leads to better accuracy. We also drastically reduce the subsampling cost at the beginning of the model by incorporating a depthwise separable convolution. We perform extensive testing of the proposed architecture and find that Squeezeformer scales very well across different model sizes and FLOPs regimes, surpassing prior model architectures when trained under the same settings. Our code along with the checkpoints for all of the trained models is open-sourced and available online [25]. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            825,
+            189
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            208,
+            328,
+            224
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The authors acknowledge contributions from Dr. Zhewei Yao, Aniruddha Nrusimha, and Jiachen Lian. We also acknowledge gracious support from Google Cloud, Google TRC team, and specifically Jonathan Caton, Prof. David Patterson, and Dr. Ed Chi. We would also like to acknowledge the Sky Pilot team from UC Berkeley. Prof. Keutzer’s lab is sponsored by Intel corporation, Intel VLAB team, Intel One-API center of excellence, as well as funding through BDD and BAIR. Sehoon Kim would like to acknowledge the support from Korea Foundation for Advanced Studies (KFAS). Amir Gholami was supported through funding from Samsung SAIT. Michael W. Mahoney would also like to acknowledge the UC Berkeley CLTC, ARO, NSF, and ONR. Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred. ",
+        "bbox": [
+            174,
+            238,
+            825,
+            363
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            383,
+            266,
+            400
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] Alexei Baevski, Wei-Ning Hsu, Qiantong Xu, Arun Babu, Jiatao Gu, and Michael Auli. Data2vec: A general framework for self-supervised learning in speech, vision and language. arXiv preprint arXiv:2202.03555, 2022.   \n[2] Alexei Baevski, Yuhao Zhou, Abdelrahman Mohamed, and Michael Auli. Wav2vec 2.0: A framework for self-supervised learning of speech representations. Advances in Neural Information Processing Systems, 33:12449–12460, 2020.   \n[3] Andy Brock, Soham De, Samuel L Smith, and Karen Simonyan. High-performance large-scale image recognition without normalization. In International Conference on Machine Learning, pages 1059–1071. PMLR, 2021.   \n[4] Maxime Burchi and Valentin Vielzeuf. Efficient Conformer: Progressive downsampling and grouped attention for automatic speech recognition. arXiv preprint arXiv:2109.01163, 2021.   \n[5] Xuankai Chang, Aswin Shanmugam Subramanian, Pengcheng Guo, Shinji Watanabe, Yuya Fujita, and Motoi Omachi. End-to-end ASR with adaptive span self-attention. In INTERSPEECH, pages 3595–3599, 2020.   \n[6] Sanyuan Chen, Chengyi Wang, Zhengyang Chen, Yu Wu, Shujie Liu, Zhuo Chen, Jinyu Li, Naoyuki Kanda, Takuya Yoshioka, Xiong Xiao, et al. WavLM: Large-scale self-supervised pre-training for full stack speech processing. arXiv preprint arXiv:2110.13900, 2021.   \n[7] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional Transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.   \n[8] Piotr Dollár, Mannat Singh, and Ross Girshick. Fast and accurate model scaling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 924–932, 2021.   \n[9] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.   \n[10] Haoqi Fan, Bo Xiong, Karttikeya Mangalam, Yanghao Li, Zhicheng Yan, Jitendra Malik, and Christoph Feichtenhofer. Multiscale Vision Transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6824–6835, 2021. ",
+        "bbox": [
+            174,
+            398,
+            828,
+            919
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[11] John S Garofolo. Timit acoustic phonetic continuous speech corpus. Linguistic Data Consortium, 1993, 1993. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[12] Xue Geng, Jie Lin, Bin Zhao, Anmin Kong, Mohamed M Sabry Aly, and Vijay Chandrasekhar. Hardware-aware softmax approximation for deep neural networks. In Asian Conference on Computer Vision, pages 107–122. Springer, 2018. ",
+        "bbox": [
+            174,
+            128,
+            821,
+            172
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[13] Xue Geng, Jie Lin, Bin Zhao, Zhe Wang, Mohamed M Sabry Aly, and Vijay Chandrasekhar. Hardware-aware exponential approximation for deep neural network. 2018. ",
+        "bbox": [
+            173,
+            180,
+            821,
+            210
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[14] Alex Graves. Sequence transduction with recurrent neural networks. arXiv preprint arXiv:1211.3711, 2012. ",
+        "bbox": [
+            173,
+            219,
+            823,
+            248
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[15] Alex Graves, Santiago Fernández, Faustino Gomez, and Jürgen Schmidhuber. Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks. In Proceedings of the 23rd international conference on Machine learning, pages 369–376, 2006. ",
+        "bbox": [
+            173,
+            257,
+            825,
+            301
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[16] Anmol Gulati, James Qin, Chung-Cheng Chiu, Niki Parmar, Yu Zhang, Jiahui Yu, Wei Han, Shibo Wang, Zhengdong Zhang, Yonghui Wu, and Ruoming Pang. Conformer: Convolutionaugmented Transformer for speech recognition. arXiv preprint arXiv:2005.08100, 2020. ",
+        "bbox": [
+            173,
+            309,
+            825,
+            353
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[17] Pengcheng Guo et al. Recent developments on ESPNet toolkit boosted by Conformer. In ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 5874–5878. IEEE, 2021. ",
+        "bbox": [
+            173,
+            361,
+            823,
+            405
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[18] Kyu J Han, Ramon Prieto, and Tao Ma. State-of-the-art speech recognition using multi-stream self-attention with dilated 1d convolutions. In 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 54–61. IEEE, 2019. ",
+        "bbox": [
+            173,
+            414,
+            823,
+            457
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[19] Wei Han, Zhengdong Zhang, Yu Zhang, Jiahui Yu, Chung-Cheng Chiu, James Qin, Anmol Gulati, Ruoming Pang, and Yonghui Wu. ContextNet: Improving convolutional neural networks for automatic speech recognition with global context. arXiv preprint arXiv:2005.03191, 2020. ",
+        "bbox": [
+            173,
+            465,
+            825,
+            510
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016. ",
+        "bbox": [
+            173,
+            517,
+            825,
+            560
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[21] Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. MobileNets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017. ",
+        "bbox": [
+            173,
+            569,
+            823,
+            613
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[22] Wei-Ning Hsu, Benjamin Bolte, Yao-Hung Hubert Tsai, Kushal Lakhotia, Ruslan Salakhutdinov, and Abdelrahman Mohamed. HuBERT: Self-supervised speech representation learning by masked prediction of hidden units. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 29:3451–3460, 2021. ",
+        "bbox": [
+            174,
+            622,
+            823,
+            679
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[23] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-Excitation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132–7141, 2018. ",
+        "bbox": [
+            173,
+            688,
+            823,
+            718
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[24] Shigeki Karita et al. A comparative study on Transformer vs RNN in speech applications. In 2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU), pages 449–456. IEEE, 2019. ",
+        "bbox": [
+            176,
+            727,
+            823,
+            770
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[25] Sehoon Kim. https://github.com/kssteven418/squeezeformer. ",
+        "bbox": [
+            174,
+            779,
+            609,
+            795
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[26] Sehoon Kim, Amir Gholami, Zhewei Yao, Michael W Mahoney, and Kurt Keutzer. I-BERT: Integer-only bert quantization. In International conference on machine learning, pages 5506– 5518. PMLR, 2021. ",
+        "bbox": [
+            174,
+            803,
+            823,
+            845
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[27] Samuel Kriman, Stanislav Beliaev, Boris Ginsburg, Jocelyn Huang, Oleksii Kuchaiev, Vitaly Lavrukhin, Ryan Leary, Jason Li, and Yang Zhang. QuartzNet: Deep automatic speech recognition with 1d time-channel separable convolutions. In ICASSP, pages 6124–6128. IEEE, 2020. ",
+        "bbox": [
+            174,
+            856,
+            826,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[28] Taku Kudo and John Richardson. Sentencepiece: A simple and language independent subword tokenizer and detokenizer for neural text processing. arXiv preprint arXiv:1808.06226, 2018. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[29] Jason Li, Vitaly Lavrukhin, Boris Ginsburg, Ryan Leary, Oleksii Kuchaiev, Jonathan M Cohen, Huyen Nguyen, and Ravi Teja Gadde. Jasper: An end-to-end convolutional neural acoustic model. arXiv preprint arXiv:1904.03288, 2019. ",
+        "bbox": [
+            174,
+            128,
+            823,
+            172
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[30] Yanghao Li, Chao-Yuan Wu, Haoqi Fan, Karttikeya Mangalam, Bo Xiong, Jitendra Malik, and Christoph Feichtenhofer. Improved Multiscale Vision Transformers for classification and detection. arXiv preprint arXiv:2112.01526, 2021. ",
+        "bbox": [
+            173,
+            181,
+            823,
+            224
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[31] Tatiana Likhomanenko, Qiantong Xu, Vineel Pratap, Paden Tomasello, Jacob Kahn, Gilad Avidov, Ronan Collobert, and Gabriel Synnaeve. Rethinking evaluation in ASR: Are our models robust enough? arXiv preprint arXiv:2010.11745, 2020. ",
+        "bbox": [
+            173,
+            233,
+            825,
+            276
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[32] Chunxi Liu, Frank Zhang, Duc Le, Suyoun Kim, Yatharth Saraf, and Geoffrey Zweig. Improving RNN Transducer based ASR with auxiliary tasks. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 172–179. IEEE, 2021. ",
+        "bbox": [
+            173,
+            286,
+            825,
+            329
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[33] Ilya Loshchilov and Frank Hutter. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017. ",
+        "bbox": [
+            173,
+            338,
+            825,
+            367
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[34] Yiping Lu, Zhuohan Li, Di He, Zhiqing Sun, Bin Dong, Tao Qin, Liwei Wang, and Tie-Yan Liu. Understanding and improving Transformer from a multi-particle dynamic system point of view. arXiv preprint arXiv:1906.02762, 2019. ",
+        "bbox": [
+            174,
+            376,
+            823,
+            420
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[35] Christoph Lüscher, Eugen Beck, Kazuki Irie, Markus Kitza, Wilfried Michel, Albert Zeyer, Ralf Schlüter, and Hermann Ney. RWTH ASR Systems for LibriSpeech: Hybrid vs attention–w/o data augmentation. arXiv preprint arXiv:1905.03072, 2019. ",
+        "bbox": [
+            174,
+            429,
+            825,
+            472
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[36] Somshubra Majumdar, Jagadeesh Balam, Oleksii Hrinchuk, Vitaly Lavrukhin, Vahid Noroozi, and Boris Ginsburg. Citrinet: Closing the gap between non-autoregressive and autoregressive end-to-end models for automatic speech recognition. arXiv preprint arXiv:2104.01721, 2021. ",
+        "bbox": [
+            176,
+            481,
+            825,
+            523
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[37] Edwin G Ng, Chung-Cheng Chiu, Yu Zhang, and William Chan. Pushing the limits of nonautoregressive speech recognition. arXiv preprint arXiv:2104.03416, 2021. ",
+        "bbox": [
+            174,
+            534,
+            823,
+            563
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[38] NVDIA Nemo. https://github.com/nvidia/nemo. ",
+        "bbox": [
+            173,
+            571,
+            526,
+            587
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[39] NVDLA Primer. http://nvdla.org/primer.html, 2021. ",
+        "bbox": [
+            174,
+            597,
+            552,
+            613
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[40] Jing Pan, Joshua Shapiro, Jeremy Wohlwend, Kyu J Han, Tao Lei, and Tao Ma. ASAPPASR: Multistream CNN and self-attentive SRU for SOTA speech recognition. arXiv preprint arXiv:2005.10469, 2020. ",
+        "bbox": [
+            173,
+            621,
+            826,
+            664
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[41] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur. Librispeech: an ASR corpus based on public domain audio books. In 2015 IEEE international conference on acoustics, speech and signal processing (ICASSP), pages 5206–5210. IEEE, 2015. ",
+        "bbox": [
+            173,
+            672,
+            823,
+            717
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[42] Daniel S Park, William Chan, Yu Zhang, Chung-Cheng Chiu, Barret Zoph, Ekin D Cubuk, and Quoc V Le. Specaugment: A simple data augmentation method for automatic speech recognition. arXiv preprint arXiv:1904.08779, 2019. ",
+        "bbox": [
+            173,
+            726,
+            825,
+            768
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[43] Mathias Perslev, Michael Jensen, Sune Darkner, Poul Jørgen Jennum, and Christian Igel. UTime: A fully convolutional network for time series segmentation applied to sleep staging. Advances in Neural Information Processing Systems, 32, 2019. ",
+        "bbox": [
+            178,
+            779,
+            825,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[44] Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. 2018. ",
+        "bbox": [
+            169,
+            830,
+            825,
+            859
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[45] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-Net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015. ",
+        "bbox": [
+            174,
+            869,
+            826,
+            912
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[46] Kyuhong Shim, Jungwook Choi, and Wonyong Sung. Understanding the role of self attention for efficient speech recognition. In International Conference on Learning Representations, 2021.   \n[47] Laurent Sifre and Stéphane Mallat. Rigid-motion scattering for texture classification. arXiv preprint arXiv:1403.1687, 2014.   \n[48] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.   \n[49] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image Transformers & distillation through attention. In International Conference on Machine Learning, pages 10347–10357. PMLR, 2021.   \n[50] Vincent Vanhoucke. Learning visual representations at scale. 2014.   \n[51] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.   \n[52] Chengyi Wang, Yu Wu, Yao Qian, Kenichi Kumatani, Shujie Liu, Furu Wei, Michael Zeng, and Xuedong Huang. Unispeech: Unified speech representation learning with labeled and unlabeled data. In International Conference on Machine Learning, pages 10937–10947. PMLR, 2021.   \n[53] Hongyu Wang, Shuming Ma, Li Dong, Shaohan Huang, Dongdong Zhang, and Furu Wei. DeepNet: Scaling Transformers to 1,000 layers. arXiv preprint arXiv:2203.00555, 2022.   \n[54] Yongqiang Wang et al. Transformer-based acoustic modeling for hybrid speech recognition. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 6874–6878. IEEE, 2020.   \n[55] Joonsang Yu, Junki Park, Seongmin Park, Minsoo Kim, Sihwa Lee, Dong Hyun Lee, and Jungwook Choi. NN-LUT: Neural approximation of non-linear operations for efficient Transformer inference. arXiv preprint arXiv:2112.02191, 2021.   \n[56] Frank Zhang, Yongqiang Wang, Xiaohui Zhang, Chunxi Liu, Yatharth Saraf, and Geoffrey Zweig. Faster, simpler and more accurate hybrid asr systems using wordpieces. arXiv preprint arXiv:2005.09150, 2020.   \n[57] Qian Zhang, Han Lu, Hasim Sak, Anshuman Tripathi, Erik McDermott, Stephen Koo, and Shankar Kumar. Transformer transducer: A streamable speech recognition model with Transformer encoders and RNN-T loss. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7829–7833. IEEE, 2020.   \n[58] Shucong Zhang, Erfan Loweimi, Peter Bell, and Steve Renals. Stochastic attention head removal: A simple and effective method for improving Transformer based ASR models. arXiv preprint arXiv:2011.04004, 2020.   \n[59] Shucong Zhang, Erfan Loweimi, Peter Bell, and Steve Renals. On the usefulness of selfattention for automatic speech recognition with Transformers. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 89–96. IEEE, 2021.   \n[60] Xiaohui Zhang, Frank Zhang, Chunxi Liu, Kjell Schubert, Julian Chan, Pradyot Prakash, Jun Liu, Ching-Feng Yeh, Fuchun Peng, Yatharth Saraf, et al. Benchmarking lf-mmi, ctc and rnn-t criteria for streaming asr. In 2021 IEEE Spoken Language Technology Workshop (SLT), pages 46–51. IEEE, 2021.   \n[61] Ying Zhang, Mohammad Pezeshki, Philémon Brakel, Saizheng Zhang, Cesar Laurent Yoshua Bengio, and Aaron Courville. Towards end-to-end speech recognition with deep convolutional neural networks. arXiv preprint arXiv:1701.02720, 2017.   \n[62] Yu Zhang, James Qin, Daniel S Park, Wei Han, Chung-Cheng Chiu, Ruoming Pang, Quoc V Le, and Yonghui Wu. Pushing the limits of semi-supervised learning for automatic speech recognition. arXiv preprint arXiv:2010.10504, 2020. ",
+        "bbox": [
+            171,
+            55,
+            826,
+            911
+        ],
+        "page_idx": 12
+    }
+]