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+        "text": "RMM: Reinforced Memory Management for Class-Incremental Learning ",
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+        "text": "Yaoyao Liu1 Bernt Schiele1 Qianru Sun2 ",
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+        "text": "1Max Planck Institute for Informatics, Saarland Informatics Campus 2School of Computing and Information Systems, Singapore Management University {yaoyao.liu, schiele}@mpi-inf.mpg.de qianrusun@smu.edu.sg ",
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+        "text": "Abstract ",
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+        "text": "Class-Incremental Learning (CIL) [40] trains classifiers under a strict memory budget: in each incremental phase, learning is done for new data, most of which is abandoned to free space for the next phase. The preserved data are exemplars used for replaying. However, existing methods use a static and ad hoc strategy for memory allocation, which is often sub-optimal. In this work, we propose a dynamic memory management strategy that is optimized for the incremental phases and different object classes. We call our method reinforced memory management (RMM), leveraging reinforcement learning. RMM training is not naturally compatible with CIL as the past, and future data are strictly non-accessible during the incremental phases. We solve this by training the policy function of RMM on pseudo CIL tasks, e.g., the tasks built on the data of the 0-th phase, and then applying it to target tasks. RMM propagates two levels of actions: Level-1 determines how to split the memory between old and new classes, and Level-2 allocates memory for each specific class. In essence, it is an optimizable and general method for memory management that can be used in any replaying-based CIL method. For evaluation, we plug RMM into two top-performing baselines (LUCIR $^ +$ AANets and POD $+ .$ AANets [30]) and conduct experiments on three benchmarks (CIFAR-100, ImageNet-Subset, and ImageNet-Full). Our results show clear improvements, e.g., boosting POD $+ .$ AANets by $3 . 6 \\%$ , $4 . 4 \\%$ , and $1 . 9 \\%$ in the 25-Phase settings of the above benchmarks, respectively. The code is available at https://class-il.mpi-inf.mpg.de/rmm/. ",
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+        "text": "1 Introduction ",
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+        "text": "Ideally, AI systems should be adaptive to ever-changing environments—where the data are continuously observed by sensors. Their models should be capable of learning new concepts from data while maintaining the ability to recognize previous ones. In practice, the systems often have constrained memory budgets because of which most of the historical data have to be abandoned [20]. However, deep-learning-based AI systems, when continuously updated using new data and limited historical data, often suffer from catastrophic forgetting, as the updates can override knowledge acquired from previous data [33, 34, 39]. ",
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+        "text": "To encourage research on the forgetting problem, Rebuffi et al. [40] defined a standard protocol of class-incremental learning (CIL) for image classification, where the training data of different object classes come in phases. In each phase, the classifier is evaluated on all classes observed so far. As the total memory size is limited [40], CIL systems abandon the majority of the data and only preserve a small number of exemplars, e.g., 20 exemplars per class, which will be used for replaying in subsequent phases. Replaying usually happens for multiple epochs [13, 18, 30, 40], so both the old class exemplars and new class data need to be stored in the limited memory. Existing CIL methods allocate memory between the old and new classes in an arbitrary and static fashion, e.g., 20 per old class vs. 1, 300 per new class for the ImageNet-Full dataset. This causes a serious imbalance between the old and new classes and can exacerbate the problem of catastrophic forgetting. ",
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+            "Figure 1: (a) Existing CIL methods [18, 30, 40] allocate memory between old and new classes in an arbitrary and frozen way, causing the data imbalance between old and new classes and exacerbating the catastrophic forgetting of old knowledge in the learned model. (b) Our proposed method— Reinforced Memory Management (RMM)—is able to learn the optimal and class-specific memory sizes in different incremental phases. Please note we use orange, blue, and green dots to denote the samples observed in the (i-1)-th, $i$ -th, and $( i { + } 1 )$ -th phases, respectively. "
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+        "text": "To address this, we propose to learn an optimal memory management policy for each incremental phase with continuously reinforced model performance and call our method reinforced memory management (RMM). Detailed actions include 1) allocating the memory between the existing (old) and the coming (new) data for each phase, and 2) specifying the memory for each old class according to its recognition difficulty before abandoning any of its data. To this end, we leverage reinforcement learning [26–28, 51, 59] and design a new policy function to contain two sub-functions that propagate two levels of actions in a hierarchical way. Level-1 function determines how to split memory between the old and new data. Its output action is then inputted into the Level-2 function to determine how to allocate memory for each old class. The overall objective of the function is to maximize the cumulative evaluation accuracy across all incremental phases. However, this is not naturally compatible with the standard protocol of CIL [40] where neither past nor future data are accessible for evaluation. To tackle this issue, we propose to pre-train the function on pseudo CIL tasks and then adopt it in the learning process of our target task. In principle, we can build such pseudo tasks using any available categorical data, e.g., the data in the 0-th phase of the target CIL task or the data from another dataset. Even though this is a non-stationary reinforcement learning problem, we can regard the pseudo and target CIL tasks as a sequence of stationary tasks and train the policy function to exploit the dependencies between these consecutive tasks. Such continuous adaptation in non-stationary environments is feasible based on the empirical analysis given in [2]. ",
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+        "text": "Technically, we propose the following method to guarantee the transferability of policy functions between pseudo and target CIL tasks. We take a Level-1 action based on the ratio of the number of new classes to the total number of classes observed so far. A lower (higher) ratio will result in weakening the stability (plasticity) of the classification model. Then, we take a Level-2 action for each individual class conditioned on both the Level-1 action and the training entropy of that class. A higher entropy denotes a more difficult class, leading to more memory allocated to the class. For evaluation, we conduct extensive CIL experiments by plugging RMM into two top-performing methods (LUCIR $+ .$ AANets, POD $^ { + }$ AANets) and testing them on three benchmarks (CIFAR-100, ImageNet-Subset, and ImageNet-Full). Our results show the clear and consistent superiority of RMM, e.g., it boosts the state-of-the-art POD $+$ AANets by $3 . 6 \\%$ , $4 . 4 \\%$ , and $1 . 9 \\%$ in the 25-Phase settings of the above benchmarks, respectively. ",
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+        "text": "Our technical contribution is three-fold. 1) A hierarchical reinforcement learning algorithm called RMM to manage the memory in a way that can be conveniently modified through incremental phases and for different classes. 2) A pseudo task generation strategy that requires only in-domain available data (small-scale) or cross-domain datasets (large-scale), relieving the data incompatibility between reinforcement learning and class-incremental learning. 3) Extensive experiments, visualization, and interpretation for RMM in three CIL benchmarks and using two top models as baselines. ",
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+        "text": "2 Related Work ",
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+        "text": "Incremental Learning [22, 36, 48, 53, 58] continuously updates the model using data coming in a sequence of phases. Similar tasks are also referred to as continual learning [12, 32] and lifelong learning [3, 9]. Recent papers are either task-incremental learning—each phase corresponds to a task (dataset) that contains new data of all seen classes [7, 10, 19, 29, 43, 47, 57], or classincremental learning (CIL)—each phase contains data of a new set of classes, i.e., classes are unseen [4, 6, 18, 25, 31, 37, 40, 41, 49, 52, 55–57]. This paper is concerned with CIL. The key challenge of CIL is the forgetting problem—older classes are forgotten in later phases. Existing methods tackling this can be divided into three categories: memory-based, regularization-based, and network-architecture-based [11, 35]. Memory-based methods preserved a small subset of the old class data (exemplars) to replay the model on them (together with the new class data), in order to relieve the forgetting of the old classes. Some work [18, 40] proposed heuristic strategies to select more representative exemplars from the old class data, and others [31, 47] tried to generate exemplars in optimizable frameworks. None of them changed the allocation of memory for different classes, i.e., all used an arbitrary and static scheme for memory allocation. Regularization-based methods introduce regularization terms in the loss function to consolidate previous knowledge when training the model on new data. The key idea is to enforce predicted label logits [29, 40], features maps [13, 18], or the topology in the feature space [49] of the new model to be close to that of the previous model. Network-architecture-based methods aim to design “incremental network architectures”. Some work [45, 54] gradually extended the network capacity for new data, while others proposed to freeze partial network parameters [1, 30] to preserve the knowledge of the old classes. ",
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+        "text": "Reinforcement Learning defines an agent that needs to decide its actions in an unknown environment by maximizing the expected cumulative reward. It has been widely applied to many optimization problems, e.g., neural architecture search [54, 59] and neural machine translation [38, 46]. Reinforcement learning has also been introduced to solve incremental learning problems. Xu et al. [54] proposed to increase convolution filters once a new task arrives and optimize the increased number by reinforcement learning. Gao et al. [14] proposed an improved version that makes the minimal expansion of the network, reducing memory and computing overheads. Veniat et al. [50] introduced a modular architecture, where each module represents a different atomic skill, and used the REINFORCE algorithm [51] to optimize it. Huang et al. [21] combined reinforcement learning with Net2Net [8] and designed a NAS-based CIL method. In our work, we also use the REINFORCE algorithm [51], but differ in three aspects. First, we are the first to optimize memory allocation for CIL in a reinforced way. Second, we learn the policy functions on generated pseudo CIL tasks, where we can access both past, and future data (for each incremental phase) and thus are able to compute the cross-phase (long-term) rewards. In contrast, the related work [14, 54] could use only current-phase data to estimate a short-term reward. Third, our reinforcement learning has a hierarchical structure that specially fits the nature of the data stream in the CIL settings. ",
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+        "text": "3 Preliminaries ",
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+        "text": "Class-Incremental Learning (CIL) usually assumes $( N { + } 1 )$ learning phases: an initial phase and $N$ incremental phases during which the number of classes gradually increases till the maximum [13, 18, 20, 31]. We assume that total memory $\\mathcal { M }$ is bounded and fixed for all incremental phases [40]. $\\mathcal { M }$ is used to store the exemplars and new coming data as both kinds of data need to be loaded repeatedly during training epochs. In the initial (0-th) phase, data $\\mathcal { D } _ { 0 }$ , containing the training samples of $\\mathcal { C } _ { 0 }$ classes, are used to learn the initial classification model $\\Theta _ { 0 }$ . In the $i$ -th incremental phase, we split $\\mathcal { M }$ into two dynamic partitions: the exemplar memory $\\mathcal { M } _ { \\mathrm { o l d } }$ and new data memory $M _ { \\mathrm { n e w } }$ . We select $\\mathcal { E } _ { t }$ as representative samples of the data seen in the $t$ -th phase, and denote total exemplars $\\mathcal { E } _ { 0 } \\sim \\mathcal { E } _ { i - 1 }$ shortly as $\\mathcal { E } _ { 0 : i - 1 }$ . We save $\\mathcal { E } _ { 0 : i - 1 }$ into $\\mathcal { M } _ { \\mathrm { o l d } }$ and free $M _ { \\mathrm { n e w } }$ . Then, we observe new data that contain $\\mathcal { C } _ { i }$ new classes. We randomly load new data into $M _ { \\mathrm { n e w } }$ until $M _ { \\mathrm { n e w } }$ is full, and all the other new data are discarded. We denote the loaded new data as $\\mathcal { D } _ { i }$ . Then, we initialize $\\Theta _ { i }$ with $\\Theta _ { i - 1 }$ , and train it using $\\mathcal { E } _ { 0 : i - 1 } \\cup \\mathcal { D } _ { i }$ . The resulting model $\\Theta _ { i }$ will be evaluated with a test set containing all classes observed so far. We repeat this training and testing, and report the average accuracy across all phases. ",
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+        "text": "Reinforcement Learning $\\mathbf { \\left( R L \\right) }$ aims to learn an optimal policy function $\\pi$ for an agent interacting in an unknown environment [51, 54, 59]. In the CIL scenario, in each incremental phase, the agent observes the current state $s _ { i }$ from the environment, and then takes an action $a _ { i }$ (how to allocate memory) according to the policy function $\\pi ( a _ { i } | s _ { i } )$ . Subsequently, the environment is updated to a new state $s _ { i + 1 }$ and the reward $r _ { i }$ is calculated to optimize the parameters of $\\pi ( a _ { i } | s _ { i } )$ through cumulative reward back-propagation. Specifically, the learning objective of $\\begin{array} { r } { R _ { i } ^ { ' } = \\sum _ { t = i } ^ { \\infty } \\dot { \\gamma } ^ { t - i } r _ { t } } \\end{array}$ , where $\\gamma \\in [ 0 , 1 )$ i i is a discounting factor that determines the $\\pi ( a _ { i } | s _ { i } )$ is to maximize the expected weights of future rewards. Please note that in our case, theproblem [15, 59], so we remove the discounting factor and $( N { + } 1 )$ $\\textstyle R = \\sum _ { t = 0 } ^ { N } r _ { t }$ task is a finite horizon, which is actually the cumulative validation accuracy of all training CIL tasks. In Section 4, we discuss the proposed RL algorithm for memory allocation and how to generate pseudo tasks for training its policy function. ",
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+        "text": "4 Reinforced Memory Management (RMM) ",
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+        "text": "Our RMM approach learns policy functions that propagate two levels of actions in a hierarchical way, specially designed for CIL. As illustrated in Figure 1 (b), Level-1 determines the memory split between exemplars and new data, and Level-2 allocates the memory for each individual class. We motivate and introduce the formulation of RMM, including the definitions of states, actions, rewards, and hierarchical policy functions in Section 4.1. In Section 4.2, we detail the steps of creating pseudo CIL tasks on which we learn the policy functions. In Section 4.3, we summarize the algorithm. ",
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+        "text": "4.1 Formulation ",
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+        "text": "In the $i$ -th incremental phase CIL, we manage the memory for two kinds of data: exemplars $\\mathcal { E } _ { 0 : i - 1 }$ and new data $\\mathcal { D } _ { i }$ . For the former, we have access to their images and labels so we can allocate a different memory size to a different class, e.g., based on its recognition difficulty. For the latter, we do not have such access before loading the data (otherwise, causing a violation to the CIL protocol), so we are only able to learn a total memory size, i.e., the memory size for all new classes (and then split it evenly for each individual class). Therefore, the memory management in CIL settings is inherently hierarchical: 1) coarse memory allocation between exemplars and new data; and then 2) fine-grained memory allocation among specific classes. To this end, we modify the standard reinforcement learning into a hierarchical structure. ",
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+        "text": "As illustrated in Figure 2 (a), in the $i$ -th incremental phase of CIL (i.e., the environment), the argent receives a state value $s _ { i }$ . Level-1 policy $\\pi _ { \\eta }$ takes $s _ { i }$ as the input to produce an action $a _ { i } ^ { [ 1 ] } \\sim \\pi _ { \\eta } ( s _ { i } )$ $a _ { i } ^ { [ 1 ] }$ determines how to split memory between the exemplars and new data. After that, Level-2 policy $\\pi _ { \\phi }$ takes $s _ { i }$ and $a _ { i } ^ { [ 1 ] }$ as inputs to produce the second action $a _ { i } ^ { [ 2 ] } \\sim \\pi _ { \\phi } ( s _ { i } , a _ { i } ^ { [ 1 ] } )$ that distributes the exemplar memory for each individual class. ",
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+        "text": "States, defined for our CIL settings, should have two properties. 1) Being transferable between CIL tasks, e.g., from a small-scale CIL task including 50 classes (in total) to a large one including 100 classes. The reason is that we need to transfer the policy functions learned from pseudo CIL tasks (defined in Section 4.2) to the target task. The states, the inputs of policy functions, should also be transferable. 2) Being distinct in each incremental phase. This is to enable the state variable to represent a specific forgetting or data imbalance degree at each different learning phase of the CIL model. To fulfill these properties, we formulate the state in the $i$ -th phase as $\\begin{array} { r } { s _ { i } = \\left( \\frac { \\mathcal { C } _ { i } } { \\sum _ { t = 0 } ^ { i - 1 } \\mathcal { C } _ { t } } , \\frac { | \\mathcal { M } _ { \\mathrm { o l d } } | } { | \\mathcal { M } | } \\right) } \\end{array}$ , where $\\mathcal { C } _ { i }$ denotes the number of classes in $\\mathcal { D } _ { i }$ , $\\mathcal { M } _ { \\mathrm { o l d } }$ denotes the memory allocated to exemplars $\\mathcal { E } _ { 0 : i - 1 }$ , and $\\mathcal { M }$ is the total memory. ",
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+        "text": "Level-1 Actions. In the 1-st incremental phase, our Level-1 policy function produces an action to allocate the memory for exemplars ${ \\mathcal { E } } _ { 0 }$ and new data $\\mathcal { D } _ { 1 }$ . We denote this action as $a _ { 1 } ^ { [ 1 ] }$ and assign its value with the ratio of the number of the exemplars $a _ { 1 } ^ { [ 1 ] } \\in ( 0 , 1 )$ . In the $i$ -th phase $( i \\geq 2 )$ ), the definition of $| \\mathcal { E } _ { 0 } |$ $a _ { i } ^ { [ 1 ] }$ to the memory size is different to $a _ { 1 } ^ { [ 1 ] }$ as it is a relative , so we have change over $a _ { i - 1 } ^ { [ 1 ] }$ . Specifically, $a _ { i } ^ { [ 1 ] }$ is the ratio of increased (if its value is positive) or decreased (if negative) memory size of $\\mathcal { M } _ { \\mathrm { o l d } }$ compared to the (i-1)-th phase. Using this definition aims for smooth and continuous memory management. In the formulation, the memory sizes of exemplars $\\mathcal { E } _ { 0 : i - 1 }$ and new data $\\mathcal { D } _ { i }$ are, respectively, ",
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+            "Figure 2: (a) In the $i$ -th phase of the $k$ -th pseudo CIL task, Level-1 policy $\\pi _ { \\eta }$ takes $s _ { i }$ as the input, and produces action $a _ { i } ^ { [ 1 ] }$ . Level-2 policy $\\pi _ { \\phi }$ takes $s _ { i }$ and $a _ { i } ^ { [ 1 ] }$ as the inputs, then produces action $a _ { i } ^ { [ 2 ] }$ . (b) For the $k$ -th pseudo CIL task, we allocate memory for $N$ times (i.e., in $N$ phases) using the policies $\\pi _ { \\eta }$ and $\\pi _ { \\phi }$ , and compute the cumulative reward $R$ . "
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+        "text": "$$\n| \\mathcal { M } _ { \\mathrm { o l d } } | = | \\mathcal { E } _ { 0 : i - 1 } | = \\sum _ { t = 1 } ^ { i } a _ { t } ^ { [ 1 ] } | \\mathcal { M } | , \\quad | \\mathcal { M } _ { \\mathrm { n e w } } | = | \\mathcal { D } _ { i } | = \\left( 1 - \\sum _ { t = 1 } ^ { i } a _ { t } ^ { [ 1 ] } \\right) | \\mathcal { M } | .\n$$",
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+        "text": "We set a constrain $a _ { i } ^ { [ 1 ] } \\in [ - 0 . 1 , 0 . 1 ]$ for $i \\geq 2$ . Otherwise, if $a _ { i } ^ { [ 1 ] }$ is too big, there are not enough exemplars to fill the memory, as most old-class data has been abandoned. If $a _ { i } ^ { [ 1 ] }$ is too small, many exemplars will be permanently deleted in this phase, making it hard or even impossible to adjust $\\mathcal { M } _ { \\mathrm { o l d } }$ back to a high value in the future phases. If $\\textstyle \\sum _ { t = 1 } ^ { i } a _ { t } ^ { [ 1 ] } > 1$ , $M _ { \\mathrm { n e w } }$ will be negative. So, we force Pit=1 a[1]t $\\textstyle \\sum _ { t = 1 } ^ { i } a _ { t } ^ { [ 1 ] } \\leq 1$ by rejection sampling [5], i.e., using $\\pi _ { \\eta }$ to output another action until it is feasible to execute. Note that this situation rarely happens in real training, because when $M _ { \\mathrm { n e w } }$ becomes very low, $\\pi _ { \\eta }$ tends to produce an action to increase it. ",
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+        "text": "Level-2 Actions. Here, we elaborate on how to get class-specific memory allocation. In the $( i - 1 )$ -th phase, we split the classes for $\\mathcal { D } _ { i - 1 }$ into two groups evenly according to training entropy values: classes with higher values (difficult classes) are in one group and the rest in the other group. Therefore, Level-2 action $a _ { i } ^ { [ 2 ] } \\in ( 0 , 1 )$ determines how to split memories between harder and easier classes. During initial experiments, we observed that using two groups already yields improved results and using more groups causes a decrease. ",
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+        "text": "Let $\\mathcal { M } _ { j } ^ { A }$ and $\\mathcal { M } _ { j } ^ { B }$ denote the memory allocated for the high-entropy and low-entropy groups, respectively, in the $j$ -th phase $( j \\le i )$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/84e6a7225e012a5c22100efa9279385763f3b58d8b67af56fa3b6d38c8951314.jpg",
+        "text": "$$\n| \\mathcal { M } _ { j } ^ { A } | = a _ { j + 1 } ^ { [ 2 ] } | \\mathcal { E } _ { j } | = \\frac { a _ { j + 1 } ^ { [ 2 ] } \\mathcal { C } _ { j } } { \\sum _ { t = 1 } ^ { i } \\mathcal { C } _ { t } } | \\mathcal { M } _ { \\mathrm { o l d } } | , | \\mathcal { M } _ { j } ^ { B } | = ( 1 - a _ { j + 1 } ^ { [ 2 ] } ) | \\mathcal { E } _ { j } | = \\frac { ( 1 - a _ { j + 1 } ^ { [ 2 ] } ) \\mathcal { C } _ { j } } { \\sum _ { t = 1 } ^ { i } \\mathcal { C } _ { t } } | \\mathcal { M } _ { \\mathrm { o l d } } | .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Then, we allocate memory evenly to the classes within the group, e.g., if the high-entropy group has 10 classes, each class will have a memory size of $\\frac { 1 } { 1 0 } | \\mathcal { M } _ { j } ^ { A } |$ . ",
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+    {
+        "type": "text",
+        "text": "Rewards. The objective of CIL is that the trained model (in any phase) should be efficient to recognize all classes seen so far. It is intuitive and convenient to use the validation accuracy as the reward in each phase. In the $i$ -th phase, the objective of RMM is to maximize the expected cumulative reward, i.e., $\\textstyle R = \\sum _ { i = 0 } ^ { N } r _ { i }$ , where $r _ { i }$ denotes the validation accuracy in the $i$ -th phase. ",
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+        "type": "text",
+        "text": "4.2 Optimization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the CIL protocol, it is impossible to see past or future data in any incremental phase. It is thus not intuitive how to compute cumulative rewards till the last phase. We propose to solve the issue by generating pseudo CIL tasks (where all data are accessible). ",
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+        "type": "text",
+        "text": "Pseudo CIL Tasks should meet two requirements: 1) their training and validation data are fully accessible for computing cumulative rewards, and 2) they have the same format (e.g., the same number of phases) of the target CIL task. Data Sources: For requirement 1, an intuitive solution is to use $\\mathcal { D } _ { 0 }$ (available in the 0-th phase). Based on the CIL protocol [13, 18, 20, 30], $\\mathcal { D } _ { 0 }$ contains half of the classes of the whole dataset, e.g., 50 classes on CIFAR-100, which supplies enough data to build downsized CIL tasks. When building the tasks, we randomly choose $1 0 \\%$ training samples of each class (from $\\mathcal { D } _ { 0 }$ ) to compose a pseudo validation set (note that we are not allowed to use the original validation set in training). When aiming for larger-scale data in CIL, we can leverage smaller datasets. For example, the pseudo tasks for ImageNet-Subset can be built on the data of CIFAR-100. This is also meaningful to evaluate the transferability of RMM policy functions (discussed in the Ablation Study). Task Generation Protocol is based on requirement 2. If using another dataset, we simply follow its original CIL protocol. If using the data accessed in the 0-th phase (i.e., $\\mathcal { D } _ { 0 }$ ), we can reduce the number of classes (in each phase) by half. For example, for CIFAR-100, we use 50-class $\\mathcal { D } _ { 0 }$ to generate a 5-phase pseudo CIL task as follows: loading 25 classes in the 0-th phase, and after that, five classes per phase. To generate another pseudo task, we simply change the order of classes. ",
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+        "type": "image",
+        "img_path": "images/2676b7321df9fb589fa15a01c9e469e5666e2b81aca84cfb2d226b35f2b252cb.jpg",
+        "image_caption": [
+            "Figure 3: Updating $\\eta$ and $\\phi$ in one epoch. To get stable gradients for $J ( \\eta , \\phi )$ , we create $K$ different pseudo CIL tasks, and run each task for $Z$ times. "
+        ],
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+        "text": "",
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+        "type": "text",
+        "text": "Training. We elaborate the steps of learning Level-1 policy $\\pi _ { \\eta }$ and Level-2 policy $\\pi _ { \\phi }$ in the following. The goal is to optimize the parameters $\\eta$ and $\\phi$ by maximizing the expected cumulative reward $J ( \\eta , \\phi )$ . We denote any pseudo CIL task and its cumulative reward as $\\tau$ and $R$ , respectively, and have, ",
+        "bbox": [
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+        "img_path": "images/56e92a9adacc123b5c869382f4d6cc1397813a9a235d4a290ad811973e5e87a2.jpg",
+        "text": "$$\nJ ( \\eta , \\phi ) = \\mathbb { E } _ { T } \\mathbb { E } _ { \\pi _ { \\eta } , \\pi _ { \\phi } } [ R ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Policy Gradient Estimation. According to the policy gradient theorem [51], we can compute the gradients for $J ( \\eta , \\phi )$ as follows, ",
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+        "type": "equation",
+        "img_path": "images/1769ad66242df41d78c89bcce2e82d9003f5955b35a2809504defd8e101f42f1.jpg",
+        "text": "$$\n\\nabla _ { \\eta , \\phi } J ( \\eta , \\phi ) = \\mathbb { E } _ { T } \\left[ \\sum _ { i = 1 } ^ { N } \\mathbb { E } _ { \\pi _ { \\eta } , \\pi _ { \\phi } } \\big [ \\nabla _ { \\eta , \\phi } \\log ( \\pi _ { \\eta } ( a _ { i } ^ { [ 1 ] } \\vert s _ { i } ) \\pi _ { \\phi } ( a _ { i } ^ { [ 2 ] } \\vert s _ { i } , a _ { i } ^ { [ 1 ] } ) ) R \\big ] \\right] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Following the REINFORCE algorithm [51], we replace the expectations $\\mathbb { E } \\tau [ \\cdot ]$ and $\\mathbb { E } _ { \\pi _ { \\eta } , \\pi _ { \\phi } } [ \\cdot ]$ with sample averages using the Monte Carlo method [16]. Specifically, in each epoch, we create $K$ pseudo tasks and run each task for $Z$ times, as shown in Figure 3. Thus we can derive the empirical approximation of $\\nabla _ { \\eta , \\phi } J ( \\eta , \\phi )$ as, ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/599b0b605f70e3f2c4c7ac76001dc349f5b5ae75e568459f9fd6821adb035525.jpg",
+        "text": "$$\n\\nabla _ { \\eta , \\phi } J ( \\eta , \\phi ) = \\frac { 1 } { Z K } \\sum _ { k = 1 } ^ { K } \\sum _ { z = 1 } ^ { Z } \\sum _ { i = 1 } ^ { N } \\nabla _ { \\eta , \\phi } \\log ( \\pi _ { \\eta } ( a _ { i } ^ { [ 1 ] } | s _ { i } ) \\pi _ { \\phi } ( a _ { i } ^ { [ 2 ] } | s _ { i } , a _ { i } ^ { [ 1 ] } ) ) ( R _ { z } ^ { k } - b ) ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $R _ { z } ^ { k }$ denotes the $z$ -th reward for the $k$ -th pseudo task $\\mathcal { T } _ { k }$ , and $b$ denotes the baseline function— the moving average of previous rewards. Using this baseline function is a common trick in RL to reduce the variance of estimated policy gradients [23, 42, 59]. ",
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+    {
+        "type": "text",
+        "text": "Updating Parameters. We update $\\eta$ and $\\phi$ in each epoch according to the gradient ascent rule [54, 59]: ",
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+    {
+        "type": "equation",
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+        "text": "$$\n\\eta : = \\eta + \\beta _ { 1 } \\nabla _ { \\eta } J ( \\eta , \\phi ) , \\phi : = \\phi + \\beta _ { 2 } \\nabla _ { \\phi } J ( \\eta , \\phi ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\beta _ { 1 }$ and $\\beta _ { 2 }$ are the learning rates. We iterate this update for $m$ epochs in total. ",
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+    {
+        "type": "text",
+        "text": "4.3 Algorithm ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 summarizes the overall training steps of the proposed RMM. There are four loops in the algorithm: 1) we train the RMM agent for $m$ epochs; 2) we create $K$ pseudo CIL tasks in each epoch; 3) we run each pseudo CIL task for $Z$ times; and 4) there are $N { + 1 }$ learning phases each time. Specifically, Line 3 initializes the parameters of policy functions. Line 6 creates the $k$ -th pseudo CIL task. Line 8 initializes the classification model. Lines 10-16 allocate the memory according to the actions produced by RMM policy. Line 17 loads new data. Lines 18-19 train the classification model and compute the accuracy. Line 20 estimates the $z$ -th cumulative reward. Lines 21-22 compute the gradients and update policy functions. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the proposed RMM method on three CIL benchmarks: CIFAR-100 [24], ImageNet-Subset [40], and ImageNet-Full [44], and use two top performing methods LUCIR $+$ AANets and POD $+$ AANets [30] as baselines. Below we introduce the datasets ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1: Input: Data $\\mathcal { D }$ for generating pseudo CIL tasks.   \n2: Output: Policy functions $\\pi _ { \\eta } , \\pi _ { \\phi }$ .   \n3: Initialize $\\eta$ and $\\phi$ ;   \n4: for $m$ epochs do   \n5: for $k$ in $1 , . . . , K$ do   \n6: Create a new pseudo task $\\mathcal { T } _ { k }$ using $\\mathcal { D }$ ;   \n7: for z in 1, ..., Z do   \n8: Initialize classification model $\\Theta _ { 0 }$ ;   \n9: for $_ i$ in 0, ..., N do   \n10: if $i \\geq 1$ do   \n11: Observe $s _ { i }$ and produce $a _ { i } ^ { [ 1 ] } \\sim \\pi _ { \\eta } ( s _ { i } )$ ;   \n12: Allocate $\\mathcal { M } _ { \\mathrm { o l d } }$ and $\\mathcal { M } _ { \\mathrm { n e w } }$ using Eq. 1;   \n13: Produce $a _ { i } ^ { [ 2 ] } \\sim \\pi _ { \\phi } ( a _ { i } ^ { [ 1 ] } , s _ { i } )$ ;   \n14: Allocate $\\{ \\mathcal { M } _ { j } ^ { A } \\} _ { j = 0 } ^ { i }$ and $\\{ \\mathcal { M } _ { j } ^ { B } \\} _ { j = 0 } ^ { i }$ using Eq. 2;   \n15: Update $\\mathcal { E } _ { 0 : i - 1 }$ using herding [40];   \n16: Save $\\mathcal { E } _ { 0 : i - 1 }$ in $\\mathcal { M } _ { \\mathrm { o l d } }$ and free $\\mathcal { M } _ { \\mathrm { n e w } }$ ;   \n17: Observe new data and load $\\mathcal { D } _ { i }$ into $\\mathcal { M } _ { \\mathrm { n e w } }$ randomly;   \n18: Initialize $\\Theta _ { i }$ with $\\Theta _ { i - 1 }$ and train it using $\\mathcal { E } _ { 0 : i - 1 } \\cup \\mathcal { D } _ { i }$ ;   \n19: Compute validation accuracy $r _ { i }$ ;   \n20: Compute $\\begin{array} { r } { R _ { z } ^ { k } = \\sum _ { i = 0 } ^ { N } r _ { i } } \\end{array}$ and update $b$ ;   \n21: Compute $\\nabla _ { \\eta , \\phi } J ( \\eta , \\phi )$ using Eq. 5;   \n22: Update $\\eta$ and $\\phi$ using Eq. 6. ",
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+    {
+        "type": "text",
+        "text": "and implementation details (Section 5.1), followed by the experimental results and analyses (Section 5.2). ",
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+    {
+        "type": "text",
+        "text": "5.1 Datasets and Implementation Details ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets. We use three benchmarks based on two datasets, CIFAR-100 [24] and ImageNet [44], following common settings [13, 18, 40, 30]. CIFAR-100 [24] contains 60, 000 samples of $3 2 \\times 3 2$ color images from 100 classes. There are 500 training and 100 test samples for each class. ImageNet (ILSVRC 2012) [44] contains around 1.3 million samples of $2 2 4 \\times 2 2 4$ color images from 1, 000 classes. There are about 1, 300 training and 50 test samples for each class. ImageNet has two CIL settings: ImageNet-Subset is based on a subset of 100 classes; and ImageNet-Full uses the full set of 1, 000 classes. The 100-class data for the ImageNet-Subset are sampled from ImageNet. For the experiments on PODNet [13] and POD-AANets [30], we use the same class orders and hyperparameters as [13]. For the experiments on LUCIR [18] and LUCIR-AANets [30], we use the same class orders and hyperparameters as [18]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Benchmarks. We follow the benchmark protocol used in [13, 18, 30, 31]. Given a dataset, the initial (the 0-th phase) model is trained on the data of half of the classes. Then, it learns the remaining classes evenly in the subsequent $N$ phases. Assume there is an initial phase and $N$ incremental phases in the CIL system. The total number of incremental phases $N$ is set to be 5, 10 or 25 (for each the setting is called “ $N$ -phase” setting). At the end of each individual phase, the learned model in each phase is evaluated on the test set containing all seen classes. In the tables, we report average accuracy over all phases and the last-phase accuracy, where the latter indicates the degree of forgetting. ",
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+    {
+        "type": "text",
+        "text": "Network Architectures. Following [18, 30, 40, 52], we use a 32-layer ResNet [40] for CIFAR-100 and an 18-layer ResNet [17] for ImageNet. Please note that it is standard to use a shallower ResNet for ImageNet. The 32-layer ResNet consists of an initial convolution layer and three residual blocks (in a single branch). Each block has ten convolution layers with $3 \\times 3$ kernels. The number of filters starts from 16 and is doubled every next block. After these three blocks, there is an average-pooling layer to compress the output feature maps to a feature embedding. The 18-layer ResNet follows the standard settings in [17]. We deploy AANets using the same parameters as its original paper [30]. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/e9b3867b36df741ca6b91bce2d0077969d9f176c67c0f74ed3ab827093051e48.jpg",
+        "table_caption": [
+            "Table 1: Average accuracies $( \\% )$ across all phases using two state-of-the-art methods (LUCIR $+$ AANets and POD $+ .$ AANets [30]) $w /$ and $w / o$ our RMM plugged in. The upper block is for recent CIL methods. For fair comparison, we re-implement these methods using our strict memory budget (see “Memory Budget” in Section 5.1) based on the public code. The results of using another common budget setting and the detailed numbers (confidence intervals and last-phase accuracies) are provided in the supplementary materials. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">CIFAR-100</td><td colspan=\"3\">ImageNet-Subset</td><td colspan=\"3\">ImageNet-Full</td></tr><tr><td>N=5</td><td>10</td><td>25</td><td>5</td><td>10</td><td>25</td><td>5</td><td>10</td><td>25</td></tr><tr><td>LwF[29]</td><td>56.79</td><td>53.05</td><td>50.44</td><td>58.83</td><td>53.60</td><td>50.16</td><td>52.00</td><td>47.87</td><td>47.49</td></tr><tr><td>iCaRL [40]</td><td>60.48</td><td>56.04</td><td>52.07</td><td>67.33</td><td>62.42</td><td>57.04</td><td>50.57</td><td>48.27</td><td>49.44</td></tr><tr><td>LUCIR [18]</td><td>63.34</td><td>62.47</td><td>59.69</td><td>71.21</td><td>68.21</td><td>64.15</td><td>65.16</td><td>62.34</td><td>57.37</td></tr><tr><td>Mnemonics [31]</td><td>64.59</td><td>62.59</td><td>61.02</td><td>72.60</td><td>71.66</td><td>70.52</td><td>65.40</td><td>64.02</td><td>62.05</td></tr><tr><td>PODNet [13]</td><td>64.60</td><td>63.13</td><td>61.96</td><td>76.45</td><td>74.66</td><td>70.15</td><td>66.80</td><td>64.89</td><td>60.28</td></tr><tr><td>LUCIR-AANets [30]</td><td>66.88</td><td>65.53</td><td>63.92</td><td>72.80</td><td>69.71</td><td>68.07</td><td>65.31</td><td>62.99</td><td>61.21</td></tr><tr><td>w/ RMM (ours)</td><td>68.42</td><td>67.17</td><td>64.56</td><td>73.58</td><td>72.83</td><td>72.30</td><td>65.81</td><td>64.10</td><td>62.23</td></tr><tr><td>POD-AANets [30]</td><td>66.61</td><td>64.61</td><td>62.63</td><td>77.36</td><td>75.83</td><td>72.18</td><td>67.97</td><td>65.03</td><td>62.03</td></tr><tr><td>w/ RMM (ours)</td><td>68.86</td><td>67.61</td><td>66.21</td><td>79.52</td><td>78.47</td><td>76.54</td><td>69.21</td><td> 67.45</td><td>63.93</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "For policy functions $\\pi _ { \\eta }$ and $\\pi _ { \\phi }$ , we use two-layer FC networks. All actions are discretized at 0.1 intervals to reduce the search space and get a tolerable training overhead. ",
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+    {
+        "type": "text",
+        "text": "Hyperparameters and Configuration. The training of the classification model $\\Theta$ exactly follows the uniform setting in [13, 18, 30, 31]. On CIFAR-100 (ImageNet-Subset/Full), we train it for 160 (90) epochs in each phase, and divide the learning rate by 10 after 80 (30) and then after 120 (60) epochs. Then, we fine-tune the model for 20 epochs using only exemplars (including the preserved exemplars of the new data to be used in future phases). We use an SGD optimizer and an ADAM optimizer for the classification model and policy functions, respectively. More details are given in the supplementary. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Memory Budget. There are two popular settings about memory budget in related work. One uses a bounded memory budget with a fixed capacity for all phases [18, 31, 40]. Another one allows the memory budget to grow along with phases [18, 20, 49]. The first one is more strict and thus used as the major setting in our paper (note that the results and analyses using the second setting are given in the supplementary materials). In every benchmark, the total budget of memory depends on the phase number $N$ . For example, on CIFAR-100, the total memory budget is set as 7, 000 samples when $N { = } 5$ (7, 000 samples $= 1 0$ classes/phase $\\times 5 0 0$ samples/class $+ ~ 2$ , 000 samples). Please note that 2, 000 is a bounded memory budget allocated since the 0-th phase for saving exemplars. More clarifications about memory budget are given in the supplementary. For fair comparison, we re-implement related methods and report the results in Table 1 if their original results (in the respective papers) were obtained in a different setting of memory budget. ",
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+    {
+        "type": "text",
+        "text": "5.2 Results and Analyses ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Table 1 presents the results of two state-of-the-art methods (LUCIR $+$ AANets and POD $+$ AANets [30]) w/ and $w / o$ our RMM plugged in, and some recent CIL work [13, 18, 29, 31, 40]. Table 2 shows the ablation study in 6 settings. Figure 4 plots the changes of the average number of exemplars per old/new class for the incremental phases. ",
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+        "text": "Comparing to the State-of-the-Art. From Table 1, we make the following observations. 1) Our RMM consistently improves the two top baselines LUCIR $^ +$ AANets and POD $^ +$ AANets [30] in all settings. E.g., LUCIR-AANets $w /$ RMM and POD-AANets w/ RMM respectively get $2 . 7 \\%$ and $3 . 1 \\%$ average improvements on the ImageNet-Subset. 2) Our POD-AANets $w /$ RMM achieves the best performances. Interestingly, we find that our RMM can boost performance more when the number of phases is larger. For example, when $N { = } 2 5$ , RMM improves POD-AANets by $3 . 6 \\%$ and $4 . 4 \\%$ on CIFAR-100 and ImageNet-Subset, respectively. These two numbers are $2 . 3 \\%$ and $2 . 1 \\%$ when $N { = } 5$ ",
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+        "table_body": "<table><tr><td rowspan=\"3\">Ablation Setting</td><td colspan=\"4\">CIFAR-100</td><td colspan=\"4\">ImagNet-Subset</td></tr><tr><td>N=5</td><td></td><td>10</td><td>25</td><td></td><td>5</td><td>10</td><td>25</td></tr><tr><td>Avg</td><td>Last</td><td>Avg Last</td><td>Avg</td><td>Last</td><td>Avg Last</td><td>Avg Last</td><td>Avg</td><td>Last</td></tr><tr><td>1 BaseRow</td><td>66.61</td><td>57.81</td><td>64.61 55.70</td><td>62.63</td><td>52.53</td><td>77.36 70.02</td><td>75.83</td><td>68.97</td><td>72.18 63.89</td></tr><tr><td>2 One-level RL</td><td>67.92 58.61</td><td></td><td>66.94 58.31</td><td></td><td>65.95 56.44</td><td>78.50 72.00</td><td></td><td>78.15 71.00</td><td>75.47 67.47</td></tr><tr><td>3 Two-level RL (Used) 68.86 59.00</td><td></td><td></td><td>67.61 59.03</td><td></td><td>66.21 56.50</td><td>79.52 73.80</td><td></td><td>78.47 71.40</td><td>76.54 68.84</td></tr><tr><td>margin</td><td>+2.3</td><td>+1.2</td><td>+3 +3.3</td><td>+3.6</td><td>+4</td><td>+2.1</td><td>+3.8 +2.6</td><td>+2.4</td><td>+4.4 +5</td></tr><tr><td>4 Two-level RL (T.P.) margin</td><td>68.62 59.40</td><td></td><td>67.22 58.20</td><td></td><td>65.82 56.20</td><td>78.81 72.42</td><td></td><td>77.68 70.77</td><td>75.29 68.81</td></tr><tr><td>5 UpperBound RL</td><td>+2</td><td>+1.6</td><td>+2.6+2.5</td><td></td><td>+3.2 +3.7</td><td>+1.5+2.4</td><td></td><td>+1.9 +1.8</td><td>+3.1 +4.9</td></tr><tr><td>6 Cross Val Fixed</td><td>70.00 61.12</td><td></td><td>68.36 60.00</td><td></td><td>66.56 56.74 65.73 55.51</td><td>80.01 74.31 77.96 70.31</td><td></td><td>78.95 71.97</td><td>76.99 69.14</td></tr><tr><td></td><td>67.50 58.48</td><td></td><td>66.69 57.19</td><td></td><td></td><td></td><td></td><td>76.70 69.08</td><td>74.18 66.10</td></tr></table>",
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+        "type": "text",
+        "text": "Table 2: The evaluation results in the ablation study $( \\% )$ . “T.P.” denotes our results using the Policy functions Transferred from another dataset. “Avg”, “Last”, and “Used” denote the average accuracy over all phases, the last-phase accuracy, and the results used as ours in Table 1, respectively. BaseRow is from the sota method POD-AANets [30]. Row 2 is for learning Level-1 policy. Row 3 is for learning Level-1 and Level-2 policies in a hierarchical way. Row 4 is for using Transferred Policies (from the other dataset in the table), when RL is costly or impossible on target CIL tasks. The bottom lines are two oracles: training the RL model on the target CIL task (Row 5) and using cross-validation to find the best fixed memory allocation between old and new classes (Row 6). ",
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+        "type": "text",
+        "text": "This indicates that the superiority of our RMM is more obvious in challenging settings (where the forgetting problem is more serious due to the more frequent model re-training through phases). ",
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+        "text": "Ablation Settings. Table 2 shows the results of our ablation study. Row 1 is for the baseline method POD-AANets [30]. Row 2 is for learning only Level-1 policy $\\pi _ { \\eta }$ (where each class gets an even split of the memory). Row 3 is for learning both Level-1 policy $\\pi _ { \\eta }$ and Level-2 policy $\\pi _ { \\phi }$ in our proposed hierarchical method, and its results are used in Table 1 as “ours”. Row 4 is for using Policy functions Transferred from another dataset (T.P.), which means on the target CIL dataset there is no training of RMM. Here, for CIFAR-100, we use the policy functions learned on ImageNet-Subset, and vice versa. On the last two rows, we show two oracle settings. Row 5 is the upper bound that assumes all past and future data are accessible during training RMM on the target CIL dataset. Row 6 is for using cross-validation (i.e., all past, future, and validation data are accessible) to find the best fixed memory split between old and new class data, e.g., $\\textstyle \\mathrm { \\frac { o l d } { n e w } } = 0 . 7$ is chosen and then used in all phases. The details of chosen split rates are given in the supplementary materials. ",
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+        "type": "text",
+        "text": "Ablation Results. Hierarchical: In Table 2, when comparing Row 2 to Row 1, it is clear that leveraging reinforcement learning yields better results as it can derive adaptive memory allocation between old and new data. Using class-specific memory management further increases the model performance (i.e., comparing Row 3 to Row 2), even though we divide the classes into only two groups. T.P. (Transferred Policy functions): Comparing Row 4 to Row 3, we can see that using transferred policy functions (trained on another dataset) achieves comparable performance, and Row 4 does not require any reinforcement learning on the target CIL dataset. Oracle: Comparing Row 3 to Row 5, we see that learning RMM on pseudo CIL tasks is comparable to the upper bound case where all training and validate data are accessible, given the fact that the latter needs higher computational overhead and violates the standard CIL protocol. Row 6 results are consistently lower than ours in Row 3, although cross-validation has access to all past, future, and validation data. ",
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+    {
+        "type": "text",
+        "text": "Allocated Memory. Figure 4 shows the change of the average number of samples per class in three ablative settings. Solid and dashed lines represent old and new classes, respectively. From the plots, we have two observations. 1) Learning RMM on the pseudo or target CIL tasks (green and orange lines), we can obtain similar memory management results (i.e., actions). This means the learned policy is transferrable in non-stationary continuous environments. This matches the conclusion of continuous adaptation in [2]. 2) Using our RMM method achieved more balanced memory sizes between exemplars and new data. For example, in the 1-st phase of the 5-phase setting, “UpperBound ",
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+        "type": "image",
+        "img_path": "images/70170b079fa509f7fe9e447e26b1993b1d5aef6ac3af3e5c866ecabdd6dd20c7.jpg",
+        "image_caption": [
+            "Figure 4: The memory allocated for “Old” and “New” across different phases on CIFAR-100. The second and fourth plots are enlarged versions of the first and third plots, respectively. Solid and phase (N=5)dashed lines denote old and new classes, respectively. The baseline is POD-AANets [30]. “Two-level RL” and “UpperBound RL” correspond to Row 3 and Row 5 in Table 2, respectively. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "RL” and “Two-level RL” allocate around 100 samples for both exemplars and new data. While the baseline setting has 40 and 500 samples for them, respectively. It thus addresses the data imbalance problem for CIL in a learnable way. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We propose the reinforced memory management (RMM) method specially for tackling CIL tasks. The hierarchical reinforcement learning (RL) framework (two levels) in RMM is capable of making more adaptive memory allocation actions than using standard RL (one level). Using the generated pseudo tasks in RMM solves the issue of data incompatibility between CIL and RL. Corresponding experimental results show that the policy trained on these pseudo tasks can be directly applied to target tasks without any computational overhead. Our overall method of RMM is generic, and its trained policy (with or without using an in-domain dataset) can be easily incorporated into exemplar replaying-based CIL methods to boost performance. ",
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+    {
+        "type": "text",
+        "text": "Limitations and Societal Impact ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We analyse the limitations and potential negative societal impact in the following three aspects. ",
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+    },
+    {
+        "type": "text",
+        "text": "• Complexity. Training RMM takes an additional time cost. According to Algorithm 1, the cost is $O ( m K Z )$ times higher than the time used for the target CIL task. However, the training of RMM policy is offline and can use a different dataset (see Table 2) — RMM pre-learns a robust policy from synthesized pseudo tasks and can be directly applied for memory management in real CIL tasks. The overhead of applying this policy is very little, e.g., $0 . 6 3 \\%$ and $1 . 1 2 \\%$ of the total training time respectively on CIFAR-100 and ImageNet (Subset and Full), taking $\\mathrm { P O D + }$ AANets as the baseline. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Technical assumptions. We build the framework of RMM based on a series of technical assumptions, which might not directly hold for all real-world continual-learning applications. When applying our method to mission-critical problems, particular care is required when modeling the system. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• Privacy issues. Keeping the old class exemplars has the issue of data privacy. This calls for future research that explicitly forgets or mitigates the identifiable feature of the data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This research was supported by $\\mathbf { A } { ^ { * } \\mathbf { S } } \\mathbf { T } \\mathbf { A } \\mathbf { R }$ under its AME YIRG Grant (Project No. A20E6c0101), Alibaba Innovative Research (AIR) programme, and Max Planck Institute for Informatics. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+    },
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parse/train/H1W1UN9gg/H1W1UN9gg.md

@@ -0,0 +1,547 @@
+# DEEP INFORMATION PROPAGATION
+
+Samuel S. Schoenholz∗ Google Brain
+
+Justin Gilmer∗ Google Brain
+
+Surya Ganguli Stanford University
+
+Jascha Sohl-Dickstein Google Brain
+
+# ABSTRACT
+
+We study the behavior of untrained neural networks whose weights and biases are randomly distributed using mean field theory. We show the existence of depth scales that naturally limit the maximum depth of signal propagation through these random networks. Our main practical result is to show that random networks may be trained precisely when information can travel through them. Thus, the depth scales that we identify provide bounds on how deep a network may be trained for a specific choice of hyperparameters. As a corollary to this, we argue that in networks at the edge of chaos, one of these depth scales diverges. Thus arbitrarily deep networks may be trained only sufficiently close to criticality. We show that the presence of dropout destroys the order-to-chaos critical point and therefore strongly limits the maximum trainable depth for random networks. Finally, we develop a mean field theory for backpropagation and we show that the ordered and chaotic phases correspond to regions of vanishing and exploding gradient respectively.
+
+# 1 INTRODUCTION
+
+Deep neural network architectures have become ubiquitous in machine learning. The success of deep networks is due to the fact that they are highly expressive (Montufar et al., 2014) while simultaneously being relatively easy to optimize (Choromanska et al., 2015; Goodfellow et al., 2014) with strong generalization properties (Recht et al., 2015). Consequently, developments in machine learning often accompany improvements in our ability to train increasingly deep networks. Despite this, designing novel network architectures is frequently equal parts art and science. This is, in part, because a general theory for neural networks that might inform design decisions has lagged behind the feverish pace of design.
+
+A pair of recent papers (Poole et al., 2016; Raghu et al., 2016) demonstrated that random neural networks are exponentially expressive in their depth. Central to their approach was the consideration of networks after random initialization, whose weights and biases were i.i.d. Gaussian distributed. In particular the paper by Poole et al. (2016) developed a “mean field” formalism for treating wide, untrained, neural networks. They showed that these mean field networks exhibit an order-to-chaos transition as a function of the weight and bias variances. Notably the mean field formalism is not closely tied to a specific choice of activation function or loss.
+
+In this paper, we demonstrate the existence of several characteristic “depth” scales that emerge naturally and control signal propagation in these random networks. We then show that one of these depth scales, $\xi _ { c }$ , diverges at the boundary between order and chaos. This result is insensitive to many architectural decisions (such as choice of activation function) and will generically be true at any order-to-chaos transition. We then extend these results to include dropout and we show that even small amounts of dropout destroys the order-to-chaos critical point and consequently removes the divergence in $\xi _ { c }$ . Together these results bound the depth to which signal may propagate through random neural networks.
+
+We then develop a corresponding mean field model for gradients and we show that a duality exists between the forward propagation of signals and the backpropagation of gradients. The ordered and chaotic phases that Poole et al. (2016) identified correspond to regions of vanishing and exploding gradients, respectively. We demonstrate the validity of this mean field theory by computing gradients of random networks on MNIST. This provides a formal explanation of the ‘vanishing gradients’ phenomenon that has long been observed in neural networks (Bengio et al., 1993). We continue to show that the covariance between two gradients is controlled by the same depth scale that limits correlated signal propagation in the forward direction.
+
+Finally, we hypothesize that a necessary condition for a random neural network to be trainable is that information should be able to pass through it. Thus, the depth-scales identified here bound the set of hyperparameters that will lead to successful training. To test this ansatz we train ensembles of deep, fully connected, feed-forward neural networks of varying depth on MNIST and CIFAR10, with and without dropout. Our results confirm that neural networks are trainable precisely when their depth is not much larger than $\xi _ { c }$ . This result is dataset independent and is, therefore, a universal function of network architecture.
+
+A corollary of these result is that asymptotically deep neural networks should be trainable provided they are initialized sufficiently close to the order-to-chaos transition. The notion of “edge of chaos” initialization has been explored previously. Such investigations have been both direct as in Bertschinger et al. (2005); Glorot & Bengio (2010) or indirect, through initialization schemes that favor deep signal propagation such as batch normalization (Ioffe & Szegedy, 2015), orthogonal matrix initialization (Saxe et al., 2014), random walk initialization (Sussillo & Abbott, 2014), composition kernels (Daniely et al., 2016), or residual network architectures (He et al., 2015). The novelty of the work presented here is two-fold. First, our framework predicts the depth at which networks may be trained even far from the order-to-chaos transition. While a skeptic might ask when it would be profitable to initialize a network far from criticality, we respond by noting that there are architectures (such as neural networks with dropout) where no critical point exists and so this more general framework is needed. Second, our work provides a formal, as opposed to intuitive, explanation for why very deep networks can only be trained near the edge of chaos.
+
+# 2 BACKGROUND
+
+We begin by recapitulating the mean-field formalism developed in Poole et al. (2016). Consider a fully-connected, untrained, feed-forward, neural network of depth $L$ with layer width $N _ { l }$ and some nonlinearity $\phi : \mathbb { R }  \mathbb { R }$ . Since this is an untrained neural network we suppose that its weights and biases are respectively i.i.d. as $W _ { i j } ^ { l } \sim N ( 0 , \sigma _ { w } ^ { 2 } / N _ { l } )$ and $b _ { i } ^ { l } \sim N ( 0 , \sigma _ { b } ^ { 2 } )$ . Notationally we set $z _ { i } ^ { l }$ to be the pre-activations of the lth layer and $y _ { i } ^ { l + 1 }$ to be the activations of that layer. Finally, we take the input to the network to be $y _ { i } ^ { 0 } = x _ { i }$ . The propagation of a signal through the network is described by the pair of equations,
+
+$$
+z _ { i } ^ { l } = \sum _ { j } W _ { i j } ^ { l } y _ { j } ^ { l } + b _ { i } ^ { l } ~ y _ { i } ^ { l + 1 } = \phi ( z _ { i } ^ { l } ) .
+$$
+
+Since the weights and biases are randomly distributed, these equations define a probability distribution on the activations and pre-activations over an ensemble of untrained neural networks. The “mean-field” approximation is then to replace $z _ { i } ^ { l }$ by a Gaussian whose first two moments match those of $z _ { i } ^ { l }$ . For the remainder of the paper we will take the mean field approximation as given.
+
+Consider first the evolution of a single input, $x _ { i ; a }$ , as it evolves through the network (as quantified by $y _ { i ; a } ^ { l }$ and $z _ { i ; a } ^ { l } )$ . Since the weights and biases are independent with zero mean, the first two moments of the pre-activations in the same layer will be,
+
+$$
+\mathbb { E } [ z _ { i ; a } ^ { l } ] = 0 \qquad \mathbb { E } [ z _ { i ; a } ^ { l } z _ { j ; a } ^ { l } ] = q _ { a a } ^ { l } \delta _ { i j }
+$$
+
+where $\delta _ { i j }$ is the Kronecker delta. Here $q _ { a a } ^ { l }$ is the variance of the pre-activations in the lth layer due to an input $x _ { i ; a }$ and it is described by the recursion relation,
+
+$$
+q _ { a a } ^ { l } = \sigma _ { w } ^ { 2 } \int { \mathcal { D } } z \phi ^ { 2 } \left( { \sqrt { q _ { a a } ^ { l - 1 } } } z \right) + \sigma _ { b } ^ { 2 }
+$$
+
+where $\textstyle \int { \mathcal { D } } z = { \frac { 1 } { \sqrt { 2 \pi } } } \int d z e ^ { - { \frac { 1 } { 2 } } z ^ { 2 } }$ is the measure for a standard Gaussian distribution. Together these equations completely describe the evolution of a single input through a mean field neural network. For any choice of $\sigma _ { w } ^ { 2 }$ and $\sigma _ { b } ^ { 2 }$ with bounded $\phi$ , eq. 3 has a fixed point at $q ^ { * } = \operatorname* { l i m } _ { l \to \infty } q _ { a a } ^ { l }$ .
+
+The propagation of a pair of signals, $x _ { i ; a } ^ { 0 }$ and $x _ { i ; b } ^ { 0 }$ , through this network can be understood similarly. Here the mean pre-activations are trivially the same as in the single-input case. The independence
+
+of the weights and biases implies that the covariance between different pre-activations in the same layer will be given by, $\mathbb { E } [ z _ { i ; a } ^ { l } z _ { j ; b } ^ { l } ] = q _ { a b } ^ { l } \delta _ { i j }$ . The covariance, $q _ { a b } ^ { l }$ , will be given by the recurrence relation,
+
+$$
+q _ { a b } ^ { l } = \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( u _ { 1 } ) \phi ( u _ { 2 } ) + \sigma _ { b } ^ { 2 }
+$$
+
+where $u _ { 1 } = \sqrt { q _ { a a } ^ { l - 1 } } z _ { 1 }$ and $u _ { 2 } = \sqrt { q _ { b b } ^ { l - 1 } } \left( c _ { a b } ^ { l - 1 } z _ { 1 } + \sqrt { 1 - ( c _ { a b } ^ { l - 1 } ) ^ { 2 } } z _ { 2 } \right)$ , with $c _ { a b } ^ { l } = q _ { a b } ^ { l } / \sqrt { q _ { a a } ^ { l } q _ { b b } ^ { l } }$ are Gaussian approximations to the pre-activations in the preceding layer with the correct covariance matrix. Moreover $c _ { a b } ^ { l }$ is the correlation between the two inputs after $l$ layers.
+
+![](images/47c9c73f7128fa37a106f7cf6b0db68ce60087436f087eae791f03bc88601d0a.jpg)  
+Figure 1: Mean field criticality. (a) The mean field phase diagram showing the boundary between ordered and chaotic phases as a function of $\sigma _ { w } ^ { 2 }$ and $\sigma _ { b } ^ { 2 }$ . (b) The residual $| \boldsymbol { q } ^ { * } - \boldsymbol { q } _ { a a } ^ { l } |$ as a function of depth on a log-scale with $\sigma _ { b } ^ { 2 } = 0 . 0 5$ and $\sigma _ { w } ^ { 2 }$ from 0.01 (red) to 1.7 (purple). Clear exponential behavior is observed. (c) The residual $| c ^ { * } - c _ { a b } ^ { l } |$ as a function of depth on a log-scale. Again, the exponential behavior is clear. The same color scheme is used here as in (b).
+
+Examining eq. 4 it is clear that $c ^ { * } = 1$ is a fixed point of the recurrence relation. To determine whether or not the $c ^ { * } = 1$ is an attractive fixed point the quantity,
+
+$$
+\chi _ { 1 } = \frac { \partial c _ { a b } ^ { l } } { \partial c _ { a b } ^ { l - 1 } } = \sigma _ { w } ^ { 2 } \int \mathcal { D } z \left[ \phi ^ { \prime } \left( \sqrt { q ^ { * } } z \right) \right] ^ { 2 }
+$$
+
+is introduced. Poole et al. (2016) note that the $c ^ { * } = 1$ fixed point is stable if $\chi _ { 1 } < 1$ and is unstable otherwise. Thus, $\chi _ { 1 } = 1$ represents a critical line separating an ordered phase (in which $c ^ { * } = 1$ and all inputs end up asymptotically correlated) and a chaotic phase (in which $c ^ { * } < 1$ and all inputs end up asymptotically decorrelated). For the case of $\phi = \operatorname { t a n h }$ , the phase diagram in fig. 1 (a) is observed.
+
+# 3 ASYMPTOTIC EXPANSIONS AND DEPTH SCALES
+
+Our first contribution is to demonstrate the existence of two depth-scales that arise naturally within the framework of mean field neural networks. Motivating the existence of these depth-scales, we iterate eq. 3 and 4 until convergence for many values of $\sigma _ { w } ^ { 2 }$ between 0.1 and 3.0 and with $\sigma _ { b } ^ { 2 } = 0 . 0 5$ starting with $q _ { a a } ^ { 0 } = q _ { b b } ^ { 0 } = 0 . \mathbf { \bar { 8 } }$ and $c _ { a b } ^ { 0 } = 0 . 6$ . We see, in fig. 1 (b) and (c), that the manner in which both $q _ { a a } ^ { l }$ approaches $q ^ { * }$ and $c _ { a b } ^ { l }$ approaches $c ^ { * }$ is exponential over many orders of magnitude. We therefore anticipate that asymptotically $| q _ { a a } ^ { l } - q ^ { * } | \sim e ^ { - l / \xi _ { q } }$ and $| c _ { a b } ^ { l } - c ^ { * } | \sim e ^ { - l / \xi _ { c } }$ for sufficiently large $l$ . Here, $\xi _ { q }$ and $\xi _ { c }$ define depth-scales over which information may propagate about the magnitude of a single input and the correlation between two inputs respectively.
+
+We will presently prove that $q _ { a a } ^ { l }$ and $c _ { a b } ^ { l }$ are asymptotically exponential. In both cases we will use the same fundamental strategy wherein we expand one of the recurrence relations (either eq. 3 or eq. 4) about its fixed point to get an approximate “asymptotic” recurrence relation. We find that this asymptotic recurrence relation in turn implies exponential decay towards the fixed point over a depth-scale, $\xi _ { x }$ .
+
+We first analyze eq. 3 and identify a depth-scale at which information about a single input may propagate. Let $q _ { a a } ^ { l } = q ^ { * } + \epsilon ^ { l }$ . By construction so long as $\begin{array} { r } { \operatorname* { l i m } _ { l \to \infty } q _ { a a } ^ { l } = q ^ { * } } \end{array}$ exists it follows that $\epsilon ^ { l }  0$ as $l  \infty$ . Eq. 3 may be expanded to lowest order in $\epsilon ^ { l }$ to arrive at an asymptotic recurrence relation (see Appendix 7.1),
+
+$$
+\epsilon ^ { l + 1 } = \epsilon ^ { l } \left[ \chi _ { 1 } + \sigma _ { w } ^ { 2 } \int { \mathcal { D } } z \phi ^ { \prime \prime } \left( { \sqrt { q ^ { * } } } z \right) \phi \left( { \sqrt { q ^ { * } } } z \right) \right] + { \mathcal { O } } \left( ( \epsilon ^ { l } ) ^ { 2 } \right) .
+$$
+
+Notably, the term multiplying $\epsilon ^ { l }$ is a constant. It follows that for large $l$ the asymptotic recurrence relation has an exponential solution, $\epsilon ^ { l } \sim e ^ { - l / \xi _ { q } }$ , with $\xi _ { q }$ given by
+
+$$
+\xi _ { q } ^ { - 1 } = - \log \left[ \chi _ { 1 } + \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { \prime \prime } \left( \sqrt { q ^ { * } } z \right) \phi \left( \sqrt { q ^ { * } } z \right) \right] .
+$$
+
+This establishes $\xi _ { q }$ as a depth scale that controls how deep information from a single input may penetrate into a random neural network.
+
+Next, we consider eq. 4. Using a similar argument (detailed in Appendix 7.2) we can expand about $c _ { a b } ^ { l } = c ^ { * } + \epsilon ^ { l }$ to find an asymptotic recurrence relation,
+
+$$
+\epsilon ^ { l + 1 } = \epsilon ^ { l } \left[ \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ^ { * } ) \phi ^ { \prime } ( u _ { 2 } ^ { * } ) \right] + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) .
+$$
+
+Here $u _ { 1 } ^ { * } = \sqrt { q ^ { * } } z _ { 1 }$ and $u _ { 2 } ^ { * } = \sqrt { q ^ { * } } ( c ^ { * } z _ { 1 } + \sqrt { 1 - ( c ^ { * } ) ^ { 2 } } z _ { 2 } )$ . Thus, once again, we expect that for large $l$ this recurrence will have an exponential solution, $\epsilon ^ { l } \sim e ^ { - l / \xi _ { c } }$ , with $\xi _ { c }$ given by
+
+$$
+\xi _ { c } ^ { - 1 } = - \log \left[ \sigma _ { w } ^ { 2 } \int { \mathcal D } z _ { 1 } { \mathcal D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ^ { * } ) \phi ^ { \prime } ( u _ { 2 } ^ { * } ) \right] .
+$$
+
+In the ordered phase $c ^ { * } = 1$ and so $\xi _ { c } ^ { - 1 } = - \log \chi _ { 1 }$ . Since the transition between order and chaos occurs when $\chi _ { 1 } = 1$ it follows that $\xi _ { c }$ diverges at any order-to-chaos transition so long as $q ^ { * }$ and $c ^ { * }$ exist.
+
+![](images/bd78147f340ab26b1a1f2acb0fc17928f8eb64a6dd13105e1b40508753885a3b.jpg)  
+Figure 2: Depth scales. (a) The iterative correlation map showing $c _ { a b } ^ { l + 1 }$ as a function of $c _ { a b } ^ { l }$ for three different values of $\sigma _ { w } ^ { 2 }$ . Green inset lines show the linearization of the iterative map about the critical point, $e ^ { - 1 / \xi _ { c } }$ . The three curves show networks far in the ordered regime (red), at the edge of chaos (purple), and deep in the chaotic regime (blue). (b) The depth scale for information propagated in a single input, $\xi _ { q }$ as a function of $\sigma _ { w } ^ { 2 }$ for $\sigma _ { b } ^ { 2 } = 0 . 0 1$ (black) to $\sigma _ { b } ^ { 2 } = 0 . 3$ (green). Dashed lines show theoretical predictions while solid lines show measurements. (c) The depth scale for correlations between inputs, $\xi _ { c }$ for the same values of $\sigma _ { b } ^ { 2 }$ . Again dashed lines are the theoretical predictions while solid lines show measurements. Here a clear divergence is observed at the order-to-chaos transition.
+
+These results can be investigated intuitively by plotting $c _ { a b } ^ { l + 1 }$ vs $c _ { a b } ^ { l }$ in fig. 2 (a). In the ordered phase there is only a single fixed point, $c _ { a b } ^ { l } = 1$ . In the chaotic regime we see that a second fixed point develops and the $c _ { a b } ^ { l } = 1$ point becomes unstable. We see that the linearization about the fixed points becomes significantly closer to the trivial map near the order-to-chaos transition.
+
+To test these claims we measure $\xi _ { q }$ and $\xi _ { c }$ directly by iterating the recurrence relations for $q _ { a a } ^ { l }$ and $c _ { a b } ^ { l }$ as before with $q _ { a a } ^ { 0 } = q _ { b b } ^ { 0 } = 0 . \bar { 8 }$ and $c _ { a b } ^ { 0 } = 0 . 6$ . In this case we consider values of $\sigma _ { w } ^ { 2 }$ between
+
+0.1 and 3.0 and $\sigma _ { b } ^ { 2 }$ between 0.01 and 0.3. For each hyperparameter settings we fit the resulting residuals, $| q _ { a a } ^ { l } - q ^ { * } |$ and $| c _ { a b } ^ { l } - c ^ { * } |$ , to exponential functions and infer the depth-scale. We then compare this measured depth-scale to that predicted by the asymptotic expansion. The result of this measurement is shown in fig. 2. In general we see that the agreement is quite good. As expected we see that $\xi _ { c }$ diverges at the critical point.
+
+As observed in Poole et al. (2016) we see that the depth scale for the propagation of information in a single input, $\xi _ { q }$ , is consistently finite and significantly shorter than $\xi _ { c }$ . To understand why this is the case consider eq. 6 and note that for tanh nonlinearities the second term is always negative. Thus, even as $\chi _ { 1 }$ approaches 1 we expect $\begin{array} { r } { \chi _ { 1 } + \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { \prime \prime } ( \sqrt { q ^ { * } } z ) \phi ( \sqrt { q ^ { * } } z ) } \end{array}$ to be substantially smaller than 1.
+
+# 3.1 DROPOUT
+
+The mean field formalism can be extended to include dropout. The main contribution here will be to argue that even infinitesimal amounts of dropout destroys the mean field critical point, and therefore limits the trainable network depth. In the presence of dropout the propagation equation, eq. 1, becomes,
+
+$$
+z _ { i } ^ { l } = \frac { 1 } { \rho } \sum _ { j } W _ { i j } ^ { l } p _ { j } ^ { l } y _ { j } ^ { l } + b _ { i } ^ { l }
+$$
+
+where $p _ { j } \sim \mathrm { B e r n o u l l i } ( \rho )$ and $\rho$ is the dropout rate. As is typically the case we have re-scaled the sum by $\rho ^ { - 1 }$ so that the mean of the pre-activation is invariant with respect to our choice of dropout rate.
+
+Following a similar procedure to the original mean field calculation consider the fate of two inputs, x0i;a and $\mathbf { \bar { \Phi } } _ { x _ { i ; b } } ^ { 0 }$ , as they are propagated through such a random network. We take the dropout masks to be chosen independently for the two inputs mimicking the manner in which dropout is employed in practice. With dropout the diagonal term in the covariance matrix will be (see Appendix 7.3),
+
+$$
+\bar { q } _ { a a } ^ { l } = \frac { \sigma _ { w } ^ { 2 } } { \rho } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { \bar { q } _ { a a } ^ { l - 1 } } z \right) + \sigma _ { b } ^ { 2 } .
+$$
+
+The variance of a single input with dropout will therefore propagate in an identical fashion to the vanilla case with a re-scaling $\sigma _ { w } ^ { 2 } \to \sigma _ { w } ^ { 2 } / \rho$ . Intuitively, this result implies that, for the case of a single input, the presence of dropout simply increases the effective variance of the weights.
+
+Computing the off-diagonal term of the covariance matrix similarly (see Appendix 7.4),
+
+$$
+\bar { q } _ { a b } ^ { l } = \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( \bar { u } _ { 1 } ) \phi ( \bar { u } _ { 2 } ) + \sigma _ { b } ^ { 2 }
+$$
+
+with $\bar { u } _ { 1 } , \bar { u } _ { 2 }$ , and $\bar { c } _ { a b } ^ { l }$ defined by analogy to the mean field equations without dropout. Here, unlike in the case of a single input, the recurrence relation is identical to the recurrence relation without dropout. To see that $\bar { c } ^ { * } = 1$ is no longer a fixed point of these dynamics consider what happens to eq. 12 when we input $\bar { c } ^ { l } = 1$ . For simplicity, we leverage the short range of $\xi _ { q }$ to replace $\bar { q } _ { a a } ^ { l } = \bar { q } _ { b b } ^ { l } = \bar { q } ^ { * }$ . We find (see Appendix 7.5),
+
+$$
+\bar { c } _ { a b } ^ { l + 1 } = 1 - \frac { 1 - \rho } { \rho \bar { q } ^ { * } } \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { \bar { q } ^ { * } } z \right) .
+$$
+
+The s erm is positive for any $\rho < 1$ . This implies that if $\bar { c } _ { a b } ^ { l } = 1$ for any $l$ then $\bar { c } _ { a b } ^ { l + 1 } < 1$ . $c ^ { * } = 1$ is not a fixed point of eq. 12 for any . Since eq. 12 is identical in form to eq. 4 it follows that the depth scale for signal propagation with dropout will likewise be given by eq. 9 with the substitutions $q ^ { * } \to \bar { q } ^ { * }$ and $c ^ { * } \to \bar { c } ^ { * }$ computed using eq. 11 and eq. 12 respectively. Importantly, since there is no longer a sharp critical point with dropout we do not expect a diverging depth scale.
+
+As in networks without dropout we plot, in fig. 3 (a), the iterative map $\bar { c } _ { a b } ^ { l + 1 }$ as a function of $\bar { c } _ { a b } ^ { l }$ Most significantly, we see that the $\bar { c } _ { a b } ^ { l } = 1$ is no longer a fixed point of the dynamics. Instead, as the dropout rate increases $\bar { c } _ { a b } ^ { l }$ gets mapped to decreasing values and the fixed point monotonically decreases.
+
+![](images/7b7daa50fafe45af9ec15a964570b97fa167a83b974c86d18fbc8dbe598eeab1.jpg)  
+Figure 3: Dropout destroys the critical point, and limits the depth to which information can propagate in a deep network. (a) The iterative correlation map showing $\bar { c } _ { a b } ^ { l + 1 }$ as a function of $\bar { c } _ { a b } ^ { l }$ for three different values of the dropout rate $\rho$ for networks tuned close to their critical point. Green inset lines show the linearization of the iterative map about the critical point, $e ^ { - 1 / \xi _ { c } }$ . (b) The asymptotic value of the correlation map, $c ^ { * }$ , as a function of $\sigma _ { w } ^ { 2 }$ for different values of dropout from $\rho = 1$ (black) to $\rho = 0 . 8$ (blue). We see that for all values of dropout except for $\rho = 1$ , $c ^ { * }$ does not show a sharp transition between an ordered phase and a chaotic phase. (c) The correlation depth scale $\xi _ { c }$ as a function of $\sigma _ { w } ^ { 2 }$ for the same values of dropout as in (b). We see here that for all values of $\rho$ except for $\rho = 1$ there is no divergence in $\xi _ { c }$ .
+
+To test these results we plot in fig. 3 (b) the asymptotic correlation, $c ^ { * }$ , as a function of $\sigma _ { w } ^ { 2 }$ for different values of dropout from $\rho = 0 . 8$ to $\rho = 1 . 0$ . As expected, we see that for all $\rho < 1$ there is no sharp transition between $c ^ { * } = 1$ and $c ^ { * } < 1$ . Moreover as the dropout rate increases the correlation $c ^ { * }$ monotonically decreases. Intuitively this makes sense. Identical inputs passed through two different dropout masks will become increasingly dissimilar as the dropout rate increases. In fig. 3 (c) we show the depth scale, $\xi _ { c }$ , as a function of $\sigma _ { w } ^ { 2 }$ for the same range of dropout probabilities. We find that, as predicted, the depth of signal propagation with dropout is drastically reduced and, importantly, there is no longer a divergence in $\xi _ { c }$ . Increasing the dropout rate continues to decrease the correlation depth for constant $\sigma _ { w } ^ { 2 }$ .
+
+# 4 GRADIENT BACKPROPAGATION
+
+There is a duality between the forward propagation of signals and the backpropagation of gradients. To elucidate this connection consider the backpropagation equations given a loss $E$ ,
+
+$$
+\frac { \partial E } { \partial W _ { i j } ^ { l } } = \delta ^ { l } _ { i } \phi ( z _ { j } ^ { l - 1 } ) \qquad \quad \delta _ { i } ^ { l } = \phi ^ { \prime } ( z _ { i } ^ { l } ) \sum _ { j } \delta _ { j } ^ { l + 1 } W _ { j i } ^ { l + 1 }
+$$
+
+with the identification $\delta _ { i } ^ { l } = \partial E / \partial z _ { i } ^ { l }$ . Within mean field theory, it is clear that the scale of fluctuations of the gradient of weights in a layer will be proportional to $\mathbb { E } [ ( \delta _ { i } ^ { l } ) ^ { 2 } ]$ (see appendix 7.6). In contrast to the pre-activations in forward propagation (eq. 1), the $\delta _ { i } ^ { l }$ will typically not be Gaussian distributed even in the large layer width limit.
+
+Nonetheless, we can work out a recurrence relation for the variance of the error, $\tilde { q } _ { a a } ^ { \ l } = \mathbb { E } [ ( \delta _ { i } ^ { l } ) ^ { 2 } ]$ , leveraging the Gaussian ansatz on the pre-activations. In order to do this, however, we must first make an additional approximation that the weights used during forward propagation are drawn independently from the weights used in backpropagation. This approximation is similar in spirit to the vanilla mean field approximation and is reminiscent of work on feedback alignment (Lillicrap et al., 2014). With this in mind we arrive at the recurrence (see appendix 7.7),
+
+$$
+\tilde { q } _ { a a } ^ { l } = \tilde { q } _ { a a } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l } } \chi _ { 1 } .
+$$
+
+The presence of $\chi _ { 1 }$ in the above equation should perhaps not be surprising. In Poole et al. (2016) they show that $\chi _ { 1 }$ is intimately related to the tangent space of a given layer in mean field neural networks. We note that the backpropagation recurrence features an explicit dependence on the ratio of widths of adjacent layers of the network, $N _ { l + 1 } / N _ { l }$ . Here we will consider exclusively constant width networks where this factor is unity. For a discussion of the case of unequal layer widths see Glorot & Bengio (2010).
+
+Since $\chi _ { 1 }$ depends only on the asymptotic $q ^ { * }$ it follows that for constant width networks we expect eq. 15 to again have an exponential solution with,
+
+$$
+\begin{array} { r } { \tilde { q } _ { a a } ^ { l } = \tilde { q } _ { a a } ^ { L } e ^ { - ( L - l ) / \xi } \nabla \qquad \xi _ { \nabla } ^ { - 1 } = - \log \chi _ { 1 } . } \end{array}
+$$
+
+Note that here $\xi _ { \nabla } ^ { - 1 } = - \log \chi _ { 1 }$ both above and below the transition. It follows that $\xi _ { \nabla }$ can be both positive and negative. We conclude that there should be three distinct regimes for the gradients.
+
+1. In the ordered phase, $\chi _ { 1 } < 1$ and so $\xi _ { \nabla } > 0$ . We therefore expect gradients to vanish over a depth $| \xi _ { \nabla } |$ .   
+2. At criticality, $\chi _ { 1 }  1$ and so $\xi _ { \nabla }  \infty$ . Here gradients should be stable regardless of depth.   
+3. In the chaotic phase, $\chi _ { 1 } > 1$ and so $\xi _ { \nabla } < 0$ . It follows that in this regime gradients should explode over a depth $| \xi _ { \nabla } |$ .
+
+Intuitively these three regimes make sense. To see this, recall that perturbations to a weight in layer $l$ can alternatively be viewed as perturbations to the pre-activations in the same layer. In the ordered phase both the perturbed signal and the unperturbed signal will be asymptotically mapped to the same point and the derivative will be small. In the chaotic phase the perturbed and unperturbed signals will become asymptotically decorrelated and the gradient will be large.
+
+![](images/0c1889c4433799e51847a7b432b3e40bab9f9e8dce978c3fdd054c3c4618c1c1.jpg)  
+Figure 4: Gradient backpropagation behaves similarly to signal forward propagation. (a) The 2- norm, $| | \nabla _ { W _ { a b } ^ { l } } E | | _ { 2 } ^ { 2 }$ as a function of layer, $l$ , for a 240 layer random network with a cross-entropy loss on MNIST. Different values of $\sigma _ { w } ^ { 2 }$ from 1.0 (blue) to 4.0 (red) are shown. Clear exponential vanishing $/$ explosion is observed over many orders of magnitude. (b) The depth scale for gradients predicted by theory (dashed line) compared with measurements from experiment (red dots). Similarity between theory and experiment is clear. Deviations near the critical point are primarily due to finite size effects.
+
+To investigate these predictions we construct deep random networks of depth $L = 2 4 0$ and layerwidth $N _ { l } = 3 0 0$ . We then consider the cross-entropy loss of these networks on MNIST. In fig. 4 (a) we plot the layer-by-layer 2-norm of the gradient, $| | \nabla _ { W _ { a b } ^ { l } } E | | _ { 2 } ^ { 2 }$ , as a function of layer, $l$ , for different values of $\sigma _ { w } ^ { 2 }$ . We see that $| | \nabla _ { W _ { a b } ^ { l } } E | | _ { 2 } ^ { 2 }$ behaves exponentially over many orders of magnitude. Moreover, we see that the gradient vanishes in the ordered phase and explodes in the chaotic phase. We test the quantitative predictions of eq. 16 in fig. 4 (b) where we compare $| \xi _ { \nabla } |$ as predicted from theory with the measured depth-scale constructed from exponential fits to the gradient data. Here we see good quantitative agreement between the theoretical predictions from mean field random networks and experimentally realized networks. Together these results suggest that the approximations on the backpropagation equations were representative of deep, wide, random networks.
+
+Finally, we show that the depth scale for correlated signal propagation likewise controls the depth at which information stored in the covariance between gradients can survive. The existence of consistent gradients across similar samples from a training set ought to be especially important for determining whether or not a given neural network architecture can be trained. To establish this depth-scale first note (see Appendix 7.8) that the covariance between gradients of two different inputs, $x _ { i ; 1 }$ and $x _ { i ; 2 }$ , will be proportional to $( \nabla _ { W _ { i j } ^ { l } } E _ { a } ) \cdot ( \nabla _ { W _ { i j } ^ { l } } E _ { b } ) \sim \mathbb { E } [ \breve { \delta } _ { i ; a } ^ { l } \delta _ { i ; b } ^ { l } ] = \tilde { q } _ { a b } ^ { l }$ where $E _ { a }$ is the loss evaluated on $x _ { i ; a }$ and $\delta _ { i ; a } = \partial E _ { a } / \partial z _ { i ; a } ^ { l }$ are appropriately defined errors.
+
+It can be shown (see Appendix 7.9) that $\tilde { q } _ { a b } ^ { \ l }$ features the recurrence relation,
+
+$$
+\tilde { q } _ { a b } ^ { l } = \tilde { q } _ { a b } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l + 2 } } \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ) \phi ^ { \prime } ( u _ { 2 } )
+$$
+
+where $u _ { 1 }$ and $u _ { 2 }$ are defined similarly as for the forward pass. Expanding asymptotically it is clear that to zeroth order in $\epsilon ^ { l }$ , $\tilde { q } _ { a b } ^ { l }$ will have an exponential solution with $\tilde { q } _ { a b } ^ { l } = \tilde { q } _ { a b } ^ { \perp } e ^ { - } ( L - l ) / \xi _ { c }$ with $\xi _ { c }$ as defined in the forward pass.
+
+# 5 EXPERIMENTAL RESULTS
+
+Taken together, the results of this paper lead us to the following hypothesis: a necessary condition for a random network to be trained is that information about the inputs should be able to propagate forward through the network, and information about the gradients should be able to propagate backwards through the network. The preceding analysis shows that networks will have this property precisely when the network depth, $L$ , is not much larger than the depth-scale $\xi _ { c }$ . This criterion is data independent and therefore offers a “universal” constraint on the hyperparameters that depends on network architecture alone. We now explore this relationship between depth of signal propagation and network trainability empirically.
+
+![](images/df16bc2cc2d397471682d32ab781ab579c37ad37f5e96bd35bbd717233b2c136.jpg)  
+Figure 5: Mean field depth scales control trainable hyperparameters. The training accuracy for neural networks as a function of their depth and initial weight variance, $\sigma _ { w } ^ { 2 }$ from a high accuracy (red) to low accuracy (black). In (a) we plot the training accuracy after 200 training steps on MNIST using SGD. Here overlayed in grey dashed lines are different multiples of the depth scale for correlated signal propagation, $n \xi _ { c }$ . We plot the accuracy in (b) after 2000 training steps on CIFAR10 using SGD, in (c) after 14000 training steps on MNIST using SGD, and in (d) after 300 training steps on MNIST using RMSPROP. Here we overlay in white dashed lines $6 \xi _ { c }$ .
+
+To investigate this prediction, we consider random networks of depth $1 0 \leq L \leq 3 0 0$ and $1 \leq \sigma _ { w } ^ { 2 } \leq$ 4 with $\sigma _ { b } ^ { 2 } = 0 . 0 5$ . We train these networks using Stochastic Gradient Descent (SGD) and RMSProp on MNIST and CIFAR10. We use a learning rate of $1 0 ^ { - 3 }$ for SGD when $L \lesssim 2 0 0 , 1 0 ^ { - 4 }$ for larger $L$ , and $1 0 ^ { - 5 }$ for RMSProp. These learning rates were selected by grid search between $1 0 ^ { - 6 }$ and $1 0 ^ { - 2 }$ in exponentially spaced steps of size 10. We note that the depth dependence of learning rate was explored in detail in Saxe et al. (2014). In fig. 5 (a)-(d) we color in red the training accuracy that neural networks achieved as a function of $\sigma _ { w } ^ { 2 }$ and $L$ for different datasets, training time, and choice of minimizer (see Appendix 7.10 for more comparisons). In all cases the neural networks over-fit the data to give a training accuracy of $1 0 0 \%$ and test accuracies of $9 8 \%$ on MNIST and $5 5 \%$ on CIFAR10. We emphasize that the purpose of this study is to demonstrate trainability as opposed to optimizing test accuracy.
+
+We now make the connection between the depth scale, $\xi _ { c }$ , and the maximum trainable depth more precise. Given the arguments in the preceding sections we note that if $L = n \xi _ { c }$ then signal through the network will be attenuated by a factor of $e ^ { n }$ . To understand how much signal can be lost while still allowing for training, we overlay in fig. 5 (a) curves corresponding to $n \xi _ { c }$ from $n = 1$ to 6. We find that networks appear to be trainable when $L \lesssim 6 \xi _ { c }$ . It would be interesting to understand why this is the case.
+
+Motivated by this argument in fig. 5 (b)-(d) in white, dashed, overlay we plot twice the predicted depth scale, $6 \xi _ { c }$ . There is clearly a relationship between the depth of correlated signal propagation and whether or not these networks are trainable. Networks closer to their critical point appear to train more quickly than those further away. Moreover, this relationship has no obvious dependence on dataset, duration of training, or minimizer. We therefore conclude that these bounds on trainable hyperparameters are universal. This in turn implies that to train increasingly deep networks, one must generically be ever closer to criticality.
+
+![](images/b7905c2406fbcbe7bc4ca1cf4ce23925a987a9a05c53736f44721f2c9b723116.jpg)  
+Figure 6: The effect of dropout on trainability. The same scheme as in fig. 5 but with dropout rates of (a) $\rho = 0 . 9 9$ , (b) $\rho = 0 . 9 8$ , and (c) $\rho = 0 . 9 4$ . Even for modest amounts of dropout we see an upper bound on the maximum trainable depth for neural networks. We continue to see good agreement between the prediction of our theory and our experimental training accuracy.
+
+Next we consider the effect of dropout. As we showed earlier, even infinitesimal amounts of dropout disrupt the order-to-chaos phase transition and cause the depth scale to become finite. However, since the effect of a single dropout mask is to simply re-scale the weight variance by $\sigma _ { w } ^ { 2 } \to \sigma _ { w } ^ { 2 } / \rho$ , the gradient magnitude will be stable near criticality, while the input and gradient correlations will not be. This therefore offers a unique opportunity to test whether the relevant depth-scale is $| 1 / \log \chi _ { 1 } |$ or $\xi _ { c }$ .
+
+In fig. 6 we repeat the same experimental setup as above on MNIST with dropout rates $\rho =$ 0.99, 0.98, and 0.94. We observe, first and foremost, that even extremely modest amounts of dropout limit the maximum trainable depth to about $L = 1 0 0$ . We additionally notice that the depth-scale, $\xi _ { c }$ , predicts the trainable region accurately for varying amounts of dropout.
+
+# 6 DISCUSSION
+
+In this paper we have elucidated the existence of several depth-scales that control signal propagation in random neural networks. Furthermore, we have shown that the degree to which a neural network can be trained depends crucially on its ability to propagate information about inputs and gradients through its full depth. At the transition between order and chaos, information stored in the correlation between inputs can propagate infinitely far through these random networks. This in turn implies that extremely deep neural networks may be trained sufficiently close to criticality. However, our contribution goes beyond advocating for hyperparameter selection that brings random networks to be nearly critical. Instead, we offer a general purpose framework that predicts, at the level of mean field theory, which hyperparameters should allow a network to be trained. This is especially relevant when analyzing schemes like dropout where there is no critical point and which therefore imply an upper bound on trainable network depth.
+
+An alternative perspective as to why information stored in the covariance between inputs is crucial for training can be understood by appealing to the correspondence between infinitely wide Bayesian neural networks and Gaussian Processes (Neal, 2012). In particular the covariance, $\dot { q } _ { a b } ^ { l }$ , is intimately related to the kernel of the induced Gaussian Process. It follows that cases in which signal stored in the covariance between inputs may propagate through the network correspond precisely to situations in which the associated Gaussian Process is well defined.
+
+Our work suggests that it may be fruitful to investigate pre-training schemes that attempt to perturb the weights of a neural network to favor information flow through the network. In principle this could be accomplished through a layer-by-layer local criterion for information flow or by selecting the mean and variance in schemes like batch normalization to maximize the covariance depth-scale.
+
+These results suggest that theoretical work on random neural networks can be used to inform practical architectural decisions. However, there is still much work to be done. For instance, the framework developed here does not apply to unbounded activations, such as rectified linear units, where it can be shown that there are phases in which eq. 3 does not have a fixed point. Additionally, the analysis here applies directly only to fully connected feed-forward networks, and will need to be extended to architectures with structured weight matrices such as convolutional networks.
+
+We close by noting that in physics it has long been known that, through renormalization, the behavior of systems near critical points can control their behavior even far from the idealized critical case. We therefore make the somewhat bold hypothesis that a broad class of neural network topologies will be controlled by the fully-connected mean field critical point.
+
+# ACKNOWLEDGMENTS
+
+We thank Ben Poole, Jeffrey Pennington, Maithra Raghu, and George Dahl for useful discussions. We are additionally grateful to RocketAI for introducing us to Temporally Recurrent Online Learning and two-dimensional time.
+
+# REFERENCES
+
+Y Bengio, Paolo Frasconi, and P Simard. The problem of learning long-term dependencies in recurrent networks. In Neural Networks, 1993., IEEE International Conference on, pp. 1183– 1188. IEEE, 1993.
+
+Nils Bertschinger, Thomas Natschlager, and Robert A. Legenstein. At the edge of chaos: Real-time ¨ computations and self-organized criticality in recurrent neural networks. In L. K. Saul, Y. Weiss, and L. Bottou (eds.), Advances in Neural Information Processing Systems 17, pp. 145–152. MIT Press, 2005.
+
+Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun. The ´ loss surfaces of multilayer networks. In AISTATS, 2015.
+
+A. Daniely, R. Frostig, and Y. Singer. Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity. arXiv:1602.05897, 2016.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Aistats, volume 9, pp. 249–256, 2010.
+
+Ian J Goodfellow, Oriol Vinyals, and Andrew M Saxe. Qualitatively characterizing neural network optimization problems. arXiv:1412.6544, 2014.
+
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+
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+
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+
+Guido F Montufar, Razvan Pascanu, Kyunghyun Cho, and Yoshua Bengio. On the number of linear regions of deep neural networks. In Z. Ghahramani, M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 27, pp. 2924–2932. Curran Associates, Inc., 2014.
+
+Radford M Neal. Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012.
+
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+
+M. Raghu, B. Poole, J. Kleinberg, S. Ganguli, and J. Sohl-Dickstein. On the expressive power of deep neural networks. arXiv:1606.05336, June 2016.
+
+Benjamin Recht, Moritz Hardt, and Yoram Singer. Train faster, generalize better: Stability of stochastic gradient descent. arXiv:1509.01240, 2015.
+
+A. M. Saxe, J. L. McClelland, and S. Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. International Conference on Learning Representations, 2014.
+
+David Sussillo and LF Abbott. Random walks: Training very deep nonlinear feed-forward networks with smart initialization. CoRR, vol. abs/1412.6558, 2014.
+
+# 7 APPENDIX
+
+Here we present derivations of results from throughout the paper.
+
+# 7.1 SINGLE INPUT DEPTH-SCALE
+
+# Result:
+
+Consider the recurrence relation for the variance of a single input,
+
+$$
+q _ { a a } ^ { l } = \sigma _ { w } ^ { 2 } \int { \mathcal { D } } z \phi ^ { 2 } \left( { \sqrt { q _ { a a } ^ { l - 1 } } } z \right) + \sigma _ { b } ^ { 2 }
+$$
+
+and a fixed point of the dynamics, $q ^ { * }$ . $q _ { a a } ^ { l }$ can be expanded about the fixed point to yield the asymptotic recurrence relation,
+
+$$
+\epsilon ^ { l + 1 } = \epsilon ^ { l } \left[ \chi _ { 1 } + \sigma _ { w } ^ { 2 } \int { \mathcal { D } } z \phi ^ { \prime \prime } \left( { \sqrt { q ^ { * } } } z \right) \phi \left( { \sqrt { q ^ { * } } } z \right) \right] + { \mathcal { O } } \left( ( \epsilon ^ { l } ) ^ { 2 } \right) .
+$$
+
+# Derivation:
+
+We begin by first expanding to order $\epsilon ^ { l }$ ,
+
+$$
+\begin{array} { l } { \displaystyle q ^ { s } + \epsilon ^ { l + 1 } = \sigma _ { w } ^ { 2 } \int \mathcal { D } z \left[ \phi \left( \sqrt { q ^ { * } + \epsilon ^ { l } } z \right) \right] ^ { 2 } + \sigma _ { b } ^ { 2 } } \\ { \displaystyle \approx \sigma _ { w } ^ { 2 } \int \mathcal { D } z \left[ \phi \left( \sqrt { q ^ { * } } z + \frac { 1 } { 2 } \frac { \epsilon ^ { l } z } { \sqrt { q ^ { * } } } \right) \right] ^ { 2 } + \sigma _ { b } ^ { 2 } } \\ { \displaystyle \approx \sigma _ { w } ^ { 2 } \int \mathcal { D } z \left[ \phi \left( \sqrt { q ^ { * } } z \right) + \frac { 1 } { 2 } \frac { \epsilon ^ { l } z } { \sqrt { q ^ { * } } } \phi ^ { \prime } \left( \sqrt { q ^ { * } } z \right) \right] ^ { 2 } + \sigma _ { b } ^ { 2 } + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) } \\ { \displaystyle \approx \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { q ^ { * } } z \right) + \sigma _ { b } ^ { 2 } + \epsilon ^ { l } \frac { \sigma _ { w } ^ { 2 } } { \sqrt { q ^ { * } } } \int \mathcal { D } z z \phi \left( \sqrt { q ^ { * } } z \right) \phi ^ { \prime } \left( \sqrt { q ^ { * } } z \right) + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) } \\ { \displaystyle \approx q ^ { * } + \epsilon ^ { l } \frac { \sigma _ { w } ^ { 2 } } { \sqrt { q ^ { * } } } \int \mathcal { D } z z \phi \big ( \sqrt { q ^ { * } } z \big ) \phi ^ { \prime } \left( \sqrt { q ^ { * } } z \right) + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) . } \end{array}
+$$
+
+We therefore arrive at the approximate reccurence relation,
+
+$$
+\epsilon ^ { l + 1 } = \epsilon ^ { l } \frac { \sigma _ { w } ^ { 2 } } { \sqrt { q ^ { * } } } \int \mathcal { D } z z \phi ( \sqrt { q ^ { * } } z ) \phi ^ { \prime } \left( \sqrt { q ^ { * } } z \right) + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) .
+$$
+
+Using the identity, $\begin{array} { r } { \int \mathcal { D } z z f ( z ) = \int \mathcal { D } z f ^ { \prime } ( z ) } \end{array}$ we can rewrite this asymptotic recurrence relation as,
+
+$$
+\begin{array} { c } { { \displaystyle \epsilon ^ { l + 1 } = \epsilon ^ { l } \left[ \sigma _ { w } ^ { 2 } \int \mathcal { D } z \left[ \phi ^ { \prime } \left( \sqrt { q ^ { * } } z \right) \right] ^ { 2 } + \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { \prime \prime } \left( \sqrt { q ^ { * } } z \right) \phi \left( \sqrt { q ^ { * } } z \right) \right] + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) } } \\ { { { } } } \\ { { = \displaystyle \epsilon ^ { l } \left[ \chi _ { 1 } + \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { \prime \prime } \left( \sqrt { q ^ { * } } z \right) \phi \left( \sqrt { q ^ { * } } z \right) \right] + \mathcal { O } ( ( \epsilon ^ { l } ) ^ { 2 } ) } } \end{array}
+$$
+
+as required.
+
+# 7.2 TWO INPUT DEPTH-SCALE
+
+# Result:
+
+Consider the recurrence relation for the co-variance of two input,
+
+$$
+q _ { a b } ^ { l } = \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( u _ { 1 } ) \phi ( u _ { 2 } ) + \sigma _ { b } ^ { 2 } ,
+$$
+
+a correlation between the inputs, $c _ { a b } ^ { l } = q _ { a b } ^ { l } / \sqrt { q _ { a a } ^ { l } q _ { b b } ^ { l } }$ , and a fixed point of the dynamics, $c ^ { * }$ . $c _ { a b } ^ { l }$ can be expanded about the fixed point to yield the asymptotic recurrence relation,
+
+$$
+\epsilon ^ { l + 1 } = \epsilon ^ { l } \left[ \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ) \phi ^ { \prime } ( u _ { 2 } ) \right] + \mathcal { O } \left( ( \epsilon ^ { l } ) ^ { 2 } \right) .
+$$
+
+# Derivation:
+
+Since the relaxation of $q _ { a a } ^ { l }$ and $q _ { b b } ^ { l }$ to $q ^ { * }$ occurs much more quickly than the convergence of $q _ { a b } ^ { l }$ we approximate $q _ { a a } ^ { l } = q _ { b b } ^ { l } = q ^ { * }$ as in Poole et al. (2016). We therefore consider the perturbation $q _ { a b } ^ { l } / q ^ { * } = c _ { a b } ^ { l } = c ^ { * } + { \epsilon } ^ { l }$ . It follows that we may make the approximation,
+
+$$
+\begin{array} { r l } & { u _ { 2 } ^ { l } = \sqrt { q ^ { * } } \left( c _ { a b } ^ { l } z _ { 1 } + \sqrt { 1 - ( c _ { a b } ^ { l } ) ^ { 2 } } z _ { 2 } \right) } \\ & { \quad \approx \sqrt { q ^ { * } } \left( c ^ { * } z _ { 1 } + \sqrt { 1 - ( c ^ { * } ) ^ { 2 } - 2 c ^ { * } \epsilon ^ { l } } z _ { 2 } \right) + \sqrt { q ^ { * } } \epsilon ^ { l } z _ { 1 } + \mathcal { O } ( \epsilon ^ { 2 } ) } \end{array}
+$$
+
+We now consider the case where $c ^ { * } < 1$ and $c ^ { * } = 1$ separately; we will later show that these two results agree with one another. First we consider the case where $c ^ { * } < 1$ in which case we may safely expand the above equation to get,
+
+$$
+u _ { 2 } ^ { l } = \sqrt { q ^ { * } } \left( c ^ { * } z _ { 1 } + \sqrt { 1 - ( c ^ { * } ) ^ { 2 } } z _ { 2 } \right) + \sqrt { q ^ { * } } \epsilon ^ { l } \left( z _ { 1 } - \frac { c ^ { * } } { \sqrt { 1 - ( c ^ { * } ) ^ { 2 } } } z _ { 2 } \right) + \mathcal { O } ( \epsilon ^ { 2 } ) .
+$$
+
+This allows us to in turn approximate the recurrence relation,
+
+$$
+\begin{array} { r l } { \frac { d ^ { 2 } + 1 } { d t } = \frac { \sigma _ { 0 } ^ { 4 } } { \sqrt { \sigma _ { 0 } ^ { 4 } } } } & { \frac { 1 } { \sqrt { \sigma _ { 0 } ^ { 4 } } } } \\ { \times } & { \frac { \sigma _ { 0 } ^ { 4 } } { \sigma _ { 0 } ^ { 4 } } \int \mathcal { D } _ { \Sigma ^ { 1 } } \mathcal { D } _ { \Sigma ^ { 2 } } \phi ( s _ { 1 } ^ { \prime } ) \Bigg [ \delta ( u _ { 2 } ^ { \prime } ) + \sqrt { \mathcal { F } } ^ { c } \epsilon ( \frac { \sigma _ { 1 } } { \sqrt { 1 - ( \epsilon ^ { \prime } ) ^ { 2 } } } ) ^ { 2 } \epsilon ( \sigma _ { 2 } ^ { \prime } ) \Bigg ] + \sigma _ { 0 } ^ { 2 } \delta ( \sigma _ { 1 } ^ { 2 } ) } \\ & { \frac { \sigma _ { 1 } ^ { 2 } } { \sqrt { 1 - ( \epsilon ^ { \prime } ) ^ { 2 } } } } \\ & { = \epsilon ^ { * } - \frac { \sigma _ { 0 } ^ { 2 } } { \sqrt { \sigma _ { 0 } ^ { 4 } } } \epsilon \Bigg [ \mathcal { D } _ { \Sigma ^ { 1 } } \mathcal { D } _ { \Sigma ^ { 3 } } ( z _ { 1 } - \frac { \epsilon ^ { \prime } } { \sqrt { 1 - ( \epsilon ^ { \prime } ) ^ { 2 } } } ) \epsilon ( s _ { 1 } ^ { \prime } ) | \phi ( u _ { 2 } ^ { \prime } ) | ^ { 2 } \delta ( z _ { 1 } ^ { \prime } ) } \\ & { = \epsilon ^ { * } - \frac { \sigma _ { 0 } ^ { 2 } } { \sqrt { \sigma _ { 0 } ^ { 4 } } } \epsilon \Bigg [ \int \mathcal { D } _ { \Sigma ^ { 1 } } \mathcal { D } _ { \Sigma ^ { 2 } \Sigma ^ { 3 } } ( \epsilon ( \frac { \sigma _ { 1 } } { \sqrt { 1 + ( \epsilon ^ { \prime } ) ^ { 2 } } } ) ^ { 2 } ( \sigma _ { 2 } ^ { \prime } ) - \frac { \epsilon ^ { * } } { \sqrt { 1 - ( \epsilon ^ { \prime } ) ^ { 2 } } } \int \mathcal { D } _ { \Sigma ^ { 1 } } \mathcal { D } _ { \Sigma ^ { 2 } \Sigma ^ { 3 } } \phi ( s _ { 1 } ^ { \prime } ) | \phi ( z _ { 2 } ^ { \prime } ) \Bigg ] } \\ &  = \epsilon ^ { * } - \frac   \end{array}
+$$
+
+where $u _ { 1 } ^ { * }$ and $u _ { 2 } ^ { * }$ are appropriately defined asymptotic random variables. This leads to the asymptotic recurrence relation,
+
+$$
+\epsilon ^ { l + 1 } = \sigma _ { w } ^ { 2 } \epsilon ^ { l } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ^ { * } ) \phi ^ { \prime } ( u _ { 2 } ^ { * } )
+$$
+
+as required.
+
+We now consider the case where $c ^ { * } = 1$ and $c _ { a b } ^ { l } = 1 - \epsilon ^ { l }$ . In this case the expansion of $u _ { 2 } ^ { l }$ will become,
+
+$$
+u _ { 2 } ^ { l } = \sqrt { q ^ { * } } z _ { 1 } + \sqrt { 2 q ^ { * } \epsilon ^ { l } } z _ { 2 } - \sqrt { q ^ { * } } \epsilon ^ { l } z _ { 1 } + \mathcal { O } ( \epsilon ^ { 3 / 2 } )
+$$
+
+and so the lowest order correction is of order $\mathcal { O } ( \sqrt { \epsilon } ^ { l } )$ as opposed to $\mathcal { O } ( \epsilon ^ { l } )$ . As usual we now expand the recurrence relation, noting that $u _ { 2 } ^ { * } = u _ { 1 } ^ { * }$ is independent of $z _ { 2 }$ when $c ^ { * } = 1$ to find,
+
+$$
+\begin{array} { r l r } { c _ { \omega \delta } ^ { i + 1 } } & { = \frac { \sigma _ { w } ^ { 2 } } { q ^ { * } } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( u _ { 1 } ^ { * } ) \phi ( u _ { 2 } ^ { l } ) + \sigma _ { b } ^ { 2 } } & { ( 4 ; } \\ & { \approx \frac { \sigma _ { w } ^ { 2 } } { q ^ { * } } \displaystyle \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( u _ { 1 } ^ { * } ) \left[ \phi ( u _ { 2 } ^ { * } ) + \left( \sqrt { 2 q ^ { * } t } z _ { 2 } - \sqrt { q ^ { * } } t z _ { 1 } \right) \phi ^ { \prime } ( u _ { 2 } ^ { * } ) + q ^ { * } t ^ { \frac { l } { 2 } } z _ { 2 } ^ { 2 } \phi ^ { \prime \prime } ( u _ { 2 } ^ { * } ) \right] + \sigma _ { b } ^ { 2 } } & \\ & { } & { ( 4 ; } \\ & { = c ^ { * } + \sigma _ { w } ^ { 2 } \epsilon ^ { l } \displaystyle \int \mathcal { D } z \phi ( \sqrt { q ^ { * } } z ) \left[ \phi ^ { \prime \prime } ( \sqrt { q ^ { * } } z ) - \frac { 1 } { \sqrt { q ^ { * } } } z \phi ^ { \prime } ( \sqrt { q ^ { * } } z ) \right] } & { ( 4 ; } \\ & { } & { = c ^ { * } + \sigma _ { w } ^ { 2 } \epsilon ^ { l } \displaystyle \left[ \int \mathcal { D } z \phi ( \sqrt { q ^ { * } } z ) \phi ^ { \prime \prime } ( \sqrt { q ^ { * } } z ) - \frac { 1 } { \sqrt { q ^ { * } } } \displaystyle \int \mathcal { D } z z \phi ( \sqrt { q ^ { * } } z ) \phi ^ { \prime } ( \sqrt { q ^ { * } } z ) \right] } & { ( 4 ; } \\ & { } & { ( 4 ; } \end{array}
+$$
+
+It follows that the asymptotic recurrence relation in this case will be,
+
+$$
+\epsilon ^ { l + 1 } = - \epsilon ^ { l } \sigma _ { w } ^ { 2 } \int \mathcal { D } z \left[ \phi ^ { \prime } ( \sqrt { q ^ { * } } z ) \right] ^ { 2 } = - \epsilon ^ { l } \chi _ { 1 } .
+$$
+
+where $\chi _ { 1 }$ is the stability condition for the ordered phase. We note that although the approximations were somewhat different the asymptotic recurrence relation for $c ^ { * } < 1$ reduces eq. 47 result for $c ^ { * } = 1$ . We may therefore use 4 for all $c ^ { * }$ .
+
+# 7.3 VARIANCE OF AN INPUT WITH DROPOUT
+
+# Result:
+
+In the presence of dropout with rate $\rho$ , the variance of a single input as it is passed through the network is described by the recurrence relation,
+
+$$
+\bar { q } _ { a a } ^ { l } = \frac { \sigma _ { w } ^ { 2 } } { \rho } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { \bar { q } _ { a a } ^ { l - 1 } } z \right) + \sigma _ { b } ^ { 2 } .
+$$
+
+# Derivation:
+
+Recall that the recurrence relation for the pre-activations is given by,
+
+$$
+z _ { i } ^ { l } = \frac { 1 } { \rho } \sum _ { j } W _ { i j } ^ { l } p _ { j } ^ { l } y _ { j } ^ { l } + b _ { i } ^ { l }
+$$
+
+where $p _ { j } ^ { l } \sim \mathrm { B e r n o u l l i } ( \rho )$ . It follows that the variance will be given by,
+
+$$
+\begin{array} { l }  { \displaystyle { \bar { q } _ { a a } ^ { l } = \mathbb { E } [ ( z _ { i } ^ { l } ) ^ { 2 } ] } } \\\ { { \displaystyle ~ = \frac { 1 } { \rho ^ { 2 } } \sum _ { j } \mathbb { E } [ ( W _ { i j } ^ { l } ) ^ { 2 } ] \mathbb { E } [ ( \rho _ { j } ^ { l } ) ^ { 2 } ] \mathbb { E } [ ( y _ { j } ^ { l } ) ^ { 2 } ] + \mathbb { E } [ ( b _ { i } ^ { l } ) ^ { 2 } ] } } \\ { { \displaystyle ~ = \frac { \sigma _ { w } ^ { 2 } } { \rho } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { \bar { q } _ { a a } ^ { l - 1 } } z \right) + \sigma _ { b } ^ { 2 } . } } \end{array}
+$$
+
+where we have used the fact that $\mathbb { E } [ ( p _ { j } ^ { l } ) ^ { 2 } ] = \rho$ .
+
+# 7.4 COVARIANCE OF TWO INPUTS WITH DROPOUT
+
+# Result:
+
+The co-variance between two signals, $z _ { i ; a } ^ { l }$ and $z _ { i ; b } ^ { l }$ , with separate i.i.d. dropout masks $p _ { i ; a } ^ { l }$ and $p _ { i ; b } ^ { l }$ is given by,
+
+$$
+\bar { q } _ { a b } ^ { l } = \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( \bar { u } _ { 1 } ) \phi ( \bar { u } _ { 2 } ) + \sigma _ { b } ^ { 2 } .
+$$
+
+where, in analogy to eq. 4, $\bar { u } _ { 1 } = \sqrt { \bar { q } _ { a a } ^ { l } } z _ { 1 }$ and $\hat { u } _ { 2 } = \sqrt { \hat { q } _ { b b } ^ { l } } \left( \hat { c } _ { a b } ^ { l } z _ { 1 } + \sqrt { 1 - ( \hat { c } _ { a b } ^ { l } ) ^ { 2 } } z _ { 2 } \right) .$
+
+# Derivation:
+
+Proceeding directly we find that,
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } [ z _ { i ; a } ^ { l } z _ { i ; b } ^ { l } ] = \frac { 1 } { \rho ^ { 2 } } \sum _ { j } \mathbb { E } [ ( W _ { i j } ^ { l } ) ^ { 2 } ] \mathbb { E } [ p _ { j ; a } ^ { l } ] \mathbb { E } [ p _ { j ; b } ^ { l } ] \mathbb { E } [ y _ { j ; a } ^ { l } y _ { j ; b } ^ { l } ] + \mathbb { E } [ b _ { i } ^ { l } ] } } \\ & { } & \\ & { } & { = \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ( \bar { u } _ { 1 } ) \phi ( \bar { u } _ { 2 } ) + \sigma _ { b } ^ { 2 } } \end{array}
+$$
+
+where we have used the fact that $\mathbb { E } [ p _ { i ; a } ^ { l } ] = \mathbb { E } [ p _ { i ; b } ^ { l } ] = \rho$ . We have also used the same substitution for $\mathbb { E } [ y _ { j ; a } ^ { l } y _ { j ; b } ^ { l } ]$ used in the original mean field calculation with the appropriate substitution.
+
+7.5 THE LACK OF A $c ^ { * } = 1$ FIXED POINT WITH DROPOUT
+
+# Result:
+
+$c _ { a b } ^ { l } = 1$ then it follows that,
+
+$$
+\bar { c } _ { a b } ^ { l + 1 } = 1 - \frac { 1 - \rho } { \rho \bar { q } ^ { * } } \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { \bar { q } ^ { * } } z \right)
+$$
+
+subject to the approximation, $q _ { a a } ^ { l } \approx q _ { b b } ^ { l } \approx q ^ { * }$ . This implies that $c _ { a b } ^ { l + 1 } < 1$
+
+# Derivation:
+
+Plugging in $c _ { a b } ^ { l } = 1$ with $q _ { a a } ^ { l } \approx q _ { b b } ^ { l } \approx q ^ { * }$ we find that $\bar { u } _ { 1 } = \bar { u } _ { 2 } = \sqrt { q ^ { * } } z _ { 1 }$ . It follows that,
+
+$$
+\begin{array} { r l } & { c _ { a b } ^ { l + 1 } = \frac { q _ { a b } ^ { l + 1 } } { q ^ { * } } } \\ & { \quad = \frac { 1 } { q ^ { * } } \left[ \sigma _ { w } ^ { 2 } \left( \mathcal { D } z \phi ^ { 2 } \left( \sqrt { q ^ { * } } z \right) + \sigma _ { b } ^ { 2 } \right] \right. } \\ & { \quad \left. = \frac { 1 } { q ^ { * } } \left[ \sigma _ { w } ^ { 2 } ( 1 - \rho ^ { - 1 } + \rho ^ { - 1 } ) \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { q ^ { * } } z \right) + \sigma _ { b } ^ { 2 } \right] \right. } \\ & { \quad = \frac { 1 } { q ^ { * } } \left[ \frac { \sigma _ { w } ^ { 2 } } { \rho } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { q ^ { * } } z \right) + \sigma _ { b } ^ { 2 } \right] + \frac { \sigma _ { w } ^ { 2 } } { q ^ { * } } ( 1 - \rho ^ { - 1 } ) \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { q ^ { * } } z \right) } \\ & { \quad = 1 - \frac { 1 - \rho } { \rho \tilde { q } ^ { * } } \sigma _ { w } ^ { 2 } \int \mathcal { D } z \phi ^ { 2 } \left( \sqrt { q ^ { * } } z \right) } \end{array}
+$$
+
+as required. Here we have integrated out $z _ { 2 }$ since nether $\bar { u } _ { 1 }$ nor $\bar { u } _ { 2 }$ depend on it.
+
+# 7.6 MEAN FIELD GRADIENT SCALING
+
+# Result:
+
+In mean field theory the expected magnitude of the gradient $| | \nabla _ { W _ { i j } ^ { l } } E | | ^ { 2 }$ will be proportional to $\mathbb { E } [ ( \delta _ { i } ^ { l } ) ^ { 2 } ]$ .
+
+# Derivation:
+
+We first note that since the $W _ { i j } ^ { l }$ are i.i.d. it follows that,
+
+$$
+\begin{array} { l } { | | \nabla _ { W _ { i j } ^ { l } } E | | ^ { 2 } = \displaystyle \sum _ { i j } \left( \frac { \partial E } { \partial W _ { i j } ^ { l } } \right) ^ { 2 } } \\ { \approx N _ { l } N _ { l + 1 } \mathbb { E } \left[ \left( \frac { \partial E } { \partial W _ { i j } ^ { l } } \right) ^ { 2 } \right] } \end{array}
+$$
+
+where we have used the fact that the first line is related to the sample expectation over the different realizations of the $W _ { i j } ^ { l }$ to approximate it by the analytic expectation in the second line. In mean field theory since the pre-activations in each layer are assumed to be i.i.d. Gaussian it follows that,
+
+$$
+\mathbb { E } \left[ \left( \frac { \partial E } { \partial W _ { i j } ^ { l } } \right) ^ { 2 } \right] = \mathbb { E } [ ( \delta _ { i } ^ { l } ) ^ { 2 } ] \mathbb { E } [ \phi ^ { 2 } ( z _ { j } ^ { l - 1 } ) ]
+$$
+
+and the result follows.
+
+# 7.7 MEAN FIELD BACKPROPAGATION
+
+# Result:
+
+In mean field theory the recursion relation for the variance of the errors, $\tilde { q } ^ { l } = \mathbb { E } [ ( \delta _ { i } ^ { l } ) ^ { 2 } ]$ is given by,
+
+$$
+\tilde { q } _ { a a } ^ { l } = \tilde { q } _ { a a } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l + 2 } } \chi _ { 1 } ( q _ { a a } ^ { l } ) .
+$$
+
+# Derivation:
+
+Computing the variance directly and using mean field approximation,
+
+$$
+\begin{array} { r l } { \tilde { q } _ { a a } ^ { l } = \mathbb { E } [ ( \delta _ { \xi ; a } ^ { l } ) ^ { 2 } ] = \mathbb { E } [ ( \phi ^ { l } ( z _ { : a } ^ { l } ) ) ^ { 2 } ] \underset { \textstyle \mathcal { j } } { \sum } \mathbb { E } [ ( \delta _ { \xi ; a } ^ { l + 1 } ) ^ { 2 } ] \mathbb { E } [ ( W _ { \xi } ^ { l + 1 } ) ^ { 2 } ] } \\ & { = \mathbb { E } [ ( \phi ^ { l } ( z _ { : a } ^ { l } ) ) ^ { 2 } ] \frac { \sigma _ { w } ^ { 2 } } { N _ { l + 1 } } \underset { \textstyle \mathcal { j } } { \sum } \mathbb { E } [ ( \delta _ { \xi ; a } ^ { l + 1 } ) ^ { 2 } ] } \\ & { = \mathbb { E } [ ( \phi ^ { l } ( z _ { : a } ^ { l } ) ) ^ { 2 } ] \frac { N _ { l + 1 } } { N _ { l + 2 } } \sigma _ { w } ^ { 2 } \widehat { q } _ { a a } ^ { l + 1 } } \\ & { = \sigma _ { w } ^ { 2 } \tilde { q } _ { a a } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l + 2 } } \int \mathcal { D } z \left[ \phi ^ { l } \left( \sqrt { q _ { a a } ^ { l } } z \right) \right] ^ { 2 } } \\ & { \approx \tilde { q } _ { a a } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l + 2 } } \chi _ { 1 } } \end{array}
+$$
+
+as required. In the last step we have made the approximation that $q _ { a a } ^ { l } \approx q ^ { * }$ since the depth scale for the variance is short ranged.
+
+# 7.8 MEAN FIELD GRADIENT COVARIANCE SCALING
+
+# Result:
+
+In mean field theory we expect the covariance between the gradients of two different inputs to scale as,
+
+$$
+\begin{array} { r } { \left( \nabla _ { W _ { i j } ^ { l } } E _ { a } \right) \cdot \left( \nabla _ { W _ { i j } ^ { l } } E _ { b } \right) \sim \mathbb { E } [ \delta _ { i ; a } \delta _ { i ; b } ] . } \end{array}
+$$
+
+# Derivation:
+
+We proceed in a manner analogous to Appendix 7.6. Note that in mean field theory since the weights are i.i.d. it follows that
+
+$$
+\begin{array} { r l } { \displaystyle ( \nabla _ { W _ { i j } ^ { l } } E _ { a } ) \cdot ( \nabla _ { W _ { i j } ^ { l } } E _ { b } ) = \sum _ { i j } \frac { \partial E _ { a } } { \partial W _ { i j } ^ { l } } \frac { \partial E _ { b } } { \partial W _ { i j } ^ { l } } } & { } \\ { \displaystyle \approx N _ { l } N _ { l + 1 } \mathbb { E } \left[ \frac { \partial E _ { a } } { \partial W _ { i j } ^ { l } } \frac { \partial E _ { b } } { \partial W _ { i j } ^ { l } } \right] } \end{array}
+$$
+
+where, as before, the final term is approximating the sample expectation. Since the weights in the forward and backwards passes are chosen independently it follows that we can factor the expectation as,
+
+$$
+\mathbb { E } \left[ \frac { \partial E _ { a } } { \partial W _ { i j } ^ { l } } \frac { \partial E _ { b } } { \partial W _ { i j } ^ { l } } \right] = \mathbb { E } [ \delta _ { i ; a } ^ { l } \delta _ { i ; b } ^ { l } ] \mathbb { E } [ \phi ( \boldsymbol { z } _ { i ; a } ^ { l } ) \phi ( \boldsymbol { z } _ { i ; b } ^ { l } ) ]
+$$
+
+and the result follows.
+
+# 7.9 MEAN FIELD BACKPROPAGATION OF COVARIANCE
+
+# Result:
+
+The covariance between the gradients due to two inputs scales as,
+
+$$
+\tilde { q } _ { a b } ^ { l } = \tilde { q } _ { a b } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l + 2 } } \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ) \phi ^ { \prime } ( u _ { 2 } )
+$$
+
+under backpropagation.
+
+# Derivation
+
+As in the analogous derivation for the variance, we compute directly,
+
+$$
+\begin{array} { l } { { \displaystyle { \tilde { q } } _ { a b } ^ { l } = \mathbb { E } [ \delta _ { i ; a } ^ { l } \delta _ { i ; b } ^ { l } ] = \mathbb { E } [ \phi ^ { \prime } ( z _ { i ; a } ) \phi ^ { \prime } ( z _ { i ; b } ) ] \sum _ { j } \mathbb { E } [ \delta _ { j ; a } ^ { l + 1 } \delta _ { j ; b } ^ { l + 1 } ] \mathbb { E } [ ( W _ { j i } ^ { l + 1 } ) ^ { 2 } ] } } \\ { ~ } \\ { { \displaystyle ~ = \tilde { q } _ { a b } ^ { l + 1 } \frac { N _ { l + 1 } } { N _ { l + 2 } } \sigma _ { w } ^ { 2 } \int \mathcal { D } z _ { 1 } \mathcal { D } z _ { 2 } \phi ^ { \prime } ( u _ { 1 } ) \phi ^ { \prime } ( u _ { 2 } ) } } \end{array}
+$$
+
+as required.
+
+Here we include some more experimental figures that investigate the effects of training time, minimizer, and dataset more closely.
+
+![](images/f92f799cc0a7d6777d5621069f5908cb41a2a8d4fe7047cf7bce009f4b7a4ee4.jpg)  
+Figure 7: Training accuracy on MNIST after (a) 45 (b) 304 (c) 2048 and (d) 13780 steps of SGD with learning rate $\mathrm { \bar { 1 0 } ^ { - 3 } }$ .
+
+![](images/bdd7c6274987a163cf2f194a05dae2df2dcbfccbe12c0ecc418ae36a341384d8.jpg)  
+Figure 8: Training accuracy on MNIST after (a) 45 (b) 304 (c) 2048 and (d) 13780 steps of RMSProp with learning rate $1 0 ^ { - 5 }$ .

parse/train/I55UqU-M11y/I55UqU-M11y.md

@@ -0,0 +1,231 @@
+# Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting
+
+Haixu Wu, Jiehui Xu, Jianmin Wang, Mingsheng Long $( \boxtimes )$ School of Software, BNRist, Tsinghua University, China {whx20,xjh20}@mails.tsinghua.edu.cn, {jimwang,mingsheng}@tsinghua.edu.cn
+
+# Abstract
+
+Extending the forecasting time is a critical demand for real applications, such as extreme weather early warning and long-term energy consumption planning. This paper studies the long-term forecasting problem of time series. Prior Transformerbased models adopt various self-attention mechanisms to discover the long-range dependencies. However, intricate temporal patterns of the long-term future prohibit the model from finding reliable dependencies. Also, Transformers have to adopt the sparse versions of point-wise self-attentions for long series efficiency, resulting in the information utilization bottleneck. Going beyond Transformers, we design Autoformer as a novel decomposition architecture with an Auto-Correlation mechanism. We break with the pre-processing convention of series decomposition and renovate it as a basic inner block of deep models. This design empowers Autoformer with progressive decomposition capacities for complex time series. Further, inspired by the stochastic process theory, we design the Auto-Correlation mechanism based on the series periodicity, which conducts the dependencies discovery and representation aggregation at the sub-series level. Auto-Correlation outperforms self-attention in both efficiency and accuracy. In long-term forecasting, Autoformer yields stateof-the-art accuracy, with a $38 \%$ relative improvement on six benchmarks, covering five practical applications: energy, traffic, economics, weather and disease. Code is available at this repository: https://github.com/thuml/Autoformer.
+
+# 1 Introduction
+
+Time series forecasting has been widely used in energy consumption, traffic and economics planning, weather and disease propagation forecasting. In these real-world applications, one pressing demand is to extend the forecast time into the far future, which is quite meaningful for the long-term planning and early warning. Thus, in this paper, we study the long-term forecasting problem of time series, characterizing itself by the large length of predicted time series. Recent deep forecasting models [41, 17, 20, 28, 23, 29, 19, 35] have achieved great progress, especially the Transformer-based models. Benefiting from the self-attention mechanism, Transformers obtain great advantage in modeling long-term dependencies for sequential data, which enables more powerful big models [7, 11].
+
+However, the forecasting task is extremely challenging under the long-term setting. First, it is unreliable to discover the temporal dependencies directly from the long-term time series because the dependencies can be obscured by entangled temporal patterns. Second, canonical Transformers with self-attention mechanisms are computationally prohibitive for long-term forecasting because of the quadratic complexity of sequence length. Previous Transformer-based forecasting models [41, 17, 20] mainly focus on improving self-attention to a sparse version. While performance is significantly improved, these models still utilize the point-wise representation aggregation. Thus, in the process of efficiency improvement, they will sacrifice the information utilization because of the sparse point-wise connections, resulting in a bottleneck for long-term forecasting of time series.
+
+To reason about the intricate temporal patterns, we try to take the idea of decomposition, which is a standard method in time series analysis [1, 27]. It can be used to process the complex time series and extract more predictable components. However, under the forecasting context, it can only be used as the pre-processing of past series because the future is unknown [15]. This common usage limits the capabilities of decomposition and overlooks the potential future interactions among decomposed components. Thus, we attempt to go beyond pre-processing usage of decomposition and propose a generic architecture to empower the deep forecasting models with immanent capacity of progressive decomposition. Further, decomposition can ravel out the entangled temporal patterns and highlight the inherent properties of time series [15]. Benefiting from this, we try to take advantage of the series periodicity to renovate the point-wise connection in self-attention. We observe that the sub-series at the same phase position among periods often present similar temporal processes. Thus, we try to construct a series-level connection based on the process similarity derived by series periodicity.
+
+Based on the above motivations, we propose an original Autoformer in place of the Transformers for long-term time series forecasting. Autoformer still follows residual and encoder-decoder structure but renovates Transformer into a decomposition forecasting architecture. By embedding our proposed decomposition blocks as the inner operators, Autoformer can progressively separate the long-term trend information from predicted hidden variables. This design allows our model to alternately decompose and refine the intermediate results during the forecasting procedure. Inspired by the stochastic process theory [8, 24], Autoformer introduces an Auto-Correlation mechanism in place of self-attention, which discovers the sub-series similarity based on the series periodicity and aggregates similar sub-series from underlying periods. This series-wise mechanism achieves $\mathcal { O } ( L \log L )$ complexity for length- $L$ series and breaks the information utilization bottleneck by expanding the point-wise representation aggregation to sub-series level. Autoformer achieves the state-of-the-art accuracy on six benchmarks. The contributions are summarized as follows:
+
+• To tackle the intricate temporal patterns of the long-term future, we present Autoformer as a decomposition architecture and design the inner decomposition block to empower the deep forecasting model with immanent progressive decomposition capacity. • We propose an Auto-Correlation mechanism with dependencies discovery and information aggregation at the series level. Our mechanism is beyond previous self-attention family and can simultaneously benefit the computation efficiency and information utilization. • Autoformer achieves a $38 \%$ relative improvement under the long-term setting on six benchmarks, covering five real-world applications: energy, traffic, economics, weather and disease.
+
+# 2 Related Work
+
+# 2.1 Models for Time Series Forecasting
+
+Due to the immense importance of time series forecasting, various models have been well developed. Many time series forecasting methods start from the classic tools [32, 9]. ARIMA [6, 5] tackles the forecasting problem by transforming the non-stationary process to stationary through differencing. The filtering method is also introduced for series forecasting [18, 10]. Besides, recurrent neural networks (RNNs) models are used to model the temporal dependencies for time series [36, 26, 40, 22]. DeepAR [28] combines autoregressive methods and RNNs to model the probabilistic distribution of future series. LSTNet [19] introduces convolutional neural networks (CNNs) with recurrent-skip connections to capture the short-term and long-term temporal patterns. Attention-based RNNs [39, 30, 31] introduce the temporal attention to explore the long-range dependencies for prediction. Also, many works based on temporal convolution networks (TCN) [34, 4, 3, 29] attempt to model the temporal causality with the causal convolution. These deep forecasting models mainly focus on the temporal relation modeling by recurrent connections, temporal attention or causal convolution.
+
+Recently, Transformers [35, 38] based on the self-attention mechanism shows great power in sequential data, such as natural language processing [11, 7], audio processing [14] and even computer vision [12, 21]. However, applying self-attention to long-term time series forecasting is computationally prohibitive because of the quadratic complexity of sequence length $L$ in both memory and time. LogTrans [20] introduces the local convolution to Transformer and proposes the LogSparse attention to select time steps following the exponentially increasing intervals, which reduces the complexity to $\mathcal { O } ( L ( \log L ) ^ { 2 } )$ . Reformer [17] presents the local-sensitive hashing (LSH) attention and reduces the complexity to $\mathcal { O } ( L \log L )$ . Informer [41] extends Transformer with KL-divergence based ProbSparse attention and also achieves $\mathcal { O } ( L \log L )$ complexity. Note that these methods are based on the vanilla Transformer and try to improve the self-attention mechanism to a sparse version, which still follows the point-wise dependency and aggregation. In this paper, our proposed Auto-Correlation mechanism is based on the inherent periodicity of time series and can provide series-wise connections.
+
+# 2.2 Decomposition of Time Series
+
+As a standard method in time series analysis, time series decomposition [1, 27] deconstructs a time series into several components, each representing one of the underlying categories of patterns that are more predictable. It is primarily useful for exploring historical changes over time. For the forecasting tasks, decomposition is always used as the pre-processing of historical series before predicting future series [15, 2], such as Prophet [33] with trend-seasonality decomposition and N-BEATS [23] with basis expansion and DeepGLO [29] with matrix decomposition. However, such pre-processing is limited by the plain decomposition effect of historical series and overlooks the hierarchical interaction between the underlying patterns of series in the long-term future. This paper takes the decomposition idea from a new progressive dimension. Our Autoformer harnesses the decomposition as an inner block of deep models, which can progressively decompose the hidden series throughout the whole forecasting process, including both the past series and the predicted intermediate results.
+
+# 3 Autoformer
+
+The time series forecasting problem is to predict the most probable length- $O$ series in the future given the past length- ${ \mathbf { \nabla } } \cdot { I }$ series, denoting as input-I-predict- $O$ . The long-term forecasting setting is to predict the long-term future, i.e. larger $O$ . As aforementioned, we have highlighted the difficulties of long-term series forecasting: handling intricate temporal patterns and breaking the bottleneck of computation efficiency and information utilization. To tackle these two challenges, we introduce the decomposition as a builtin block to the deep forecasting model and propose Autoformer as a decomposition architecture. Besides, we design the Auto-Correlation mechanism to discover the period-based dependencies and aggregate similar sub-series from underlying periods.
+
+# 3.1 Decomposition Architecture
+
+We renovate Transformer [35] to a deep decomposition architecture (Figure 1), including the inner series decomposition block, Auto-Correlation mechanism, and corresponding Encoder and Decoder.
+
+Series decomposition block To learn with the complex temporal patterns in long-term forecasting context, we take the idea of decomposition [1, 27], which can separate the series into trend-cyclical and seasonal parts. These two parts reflect the long-term progression and the seasonality of the series respectively. However, directly decomposing is unrealizable for future series because the future is just unknown. To tackle this dilemma, we present a series decomposition block as an inner operation of Autoformer (Figure 1), which can extract the long-term stationary trend from predicted intermediate hidden variables progressively. Concretely, we adapt the moving average to smooth out periodic fluctuations and highlight the long-term trends. For length- $L$ input series $\breve { \mathcal { X } } \in \mathbb { R } ^ { L \times d }$ , the process is:
+
+$$
+\begin{array} { r l } & { \mathcal { X } _ { \mathrm { t } } = \mathrm { A v g P o o l } ( \mathrm { P a d d i n g } ( \mathcal { X } ) ) } \\ & { \mathcal { X } _ { \mathrm { s } } = \mathcal { X } - \mathcal { X } _ { \mathrm { t } } , } \end{array}
+$$
+
+where $\boldsymbol { \mathcal { X } } _ { \mathrm { s } } , \boldsymbol { \mathcal { X } } _ { \mathrm { t } } \in \mathbb { R } ^ { L \times d }$ denote the seasonal and the extracted trend-cyclical part respectively. We adopt the $\operatorname { A v g P o o l } ( \cdot )$ for moving average with the padding operation to keep the series length unchanged. We use $\mathcal { X } _ { \mathrm { s } } , \mathcal { X } _ { \mathrm { t } } = \mathrm { S e r i e s D e c o m p } ( \mathcal { X } )$ to summarize above equations, which is a model inner block.
+
+Model inputs The inputs of encoder part are the past $I$ time steps $\mathcal { X } _ { \mathrm { e n } } \in \mathbb { R } ^ { I \times d }$ . As a decomposition architecture (Figure 1), the input of Autoformer decoder contains both the seasonal part $\chi _ { \mathrm { d e s } } \in$ $\mathbb { R } ^ { ( \frac { I } { 2 } + O ) \times d }$ and trend-cyclical part $\chi _ { \mathrm { d e t } } \in \mathbb { R } ^ { ( \frac { I } { 2 } + O ) \times d }$ to be refined. Each initialization consists of two parts: the component decomposed from the latter half of encoder’s input $\mathcal { X } _ { \mathrm { e n } }$ with length $\frac { I } { 2 }$ to provide recent information, placeholders with length $O$ filled by scalars. It’s formulized as follows:
+
+$$
+\begin{array} { r l } & { \mathcal { X } _ { \mathrm { e n s } } , \mathcal { X } _ { \mathrm { e n t } } = \mathrm { S e r i e s D e c o m p } ( \mathcal { X } _ { \mathrm { e n } \frac { I } { 2 } : I } ) } \\ & { \qquad \mathcal { X } _ { \mathrm { d e s } } = \mathrm { C o n c a t } ( \mathcal { X } _ { \mathrm { e n s } } , \mathcal { X } _ { 0 } ) } \\ & { \qquad \mathcal { X } _ { \mathrm { d e t } } = \mathrm { C o n c a t } ( \mathcal { X } _ { \mathrm { e n t } } , \mathcal { X } _ { \mathrm { M e a n } } ) , } \end{array}
+$$
+
+![](images/1ebb1575216e8696f19f85fe903bb441210d41e8973469f64d31b9ac1d64fa43.jpg)  
+Figure 1: Autoformer architecture. The encoder eliminates the long-term trend-cyclical part by series decomposition blocks (blue blocks) and focuses on seasonal patterns modeling. The decoder accumulates the trend part extracted from hidden variables progressively. The past seasonal information from encoder is utilized by the encoder-decoder Auto-Correlation (center green block in decoder).
+
+where $\chi _ { \mathrm { e n s } } , \chi _ { \mathrm { e n t } } \in \mathbb { R } ^ { \frac { I } { 2 } \times d }$ denote the seasonal and trend-cyclical parts of $\mathcal { X } _ { \mathrm { e n } }$ respectively, and $\mathcal { X } _ { 0 } , \mathcal { X } _ { \mathrm { M e a n } } \in \mathbb { R } ^ { O \times d }$ denote the placeholders filled with zero and the mean of $\mathcal { X } _ { \mathrm { e n } }$ respectively.
+
+Encoder As shown in Figure 1, the encoder focuses on the seasonal part modeling. The output of the encoder contains the past seasonal information and will be used as the cross information to help the decoder refine prediction results. Suppose we have $N$ encoder layers. The overall equations for $l$ -th encoder layer are summarized as $\mathcal { X } _ { \mathrm { e n } } ^ { \hat { l } ^ { \mathrm { ~ \tiny ~ \cdot ~ } } } = \mathrm { E n c o d e r } ( \mathcal { X } _ { \mathrm { e n } } ^ { l - 1 } )$ . Details are shown as follows:
+
+$$
+\begin{array} { r l } & { S _ { \mathrm { e n } } ^ { l , 1 } , \ l _ { - } = \mathrm { S e r i e s D e c o m p } \Big ( \mathrm { A u t o - C o r r e l a t i o n } ( \mathcal { X } _ { \mathrm { e n } } ^ { l - 1 } ) + \mathcal { X } _ { \mathrm { e n } } ^ { l - 1 } \Big ) } \\ & { S _ { \mathrm { e n } } ^ { l , 2 } , \ l _ { - } = \mathrm { S e r i e s D e c o m p } \Big ( \mathrm { F e e d F o r w a r d } ( S _ { \mathrm { e n } } ^ { l , 1 } ) + S _ { \mathrm { e n } } ^ { l , 1 } \Big ) , } \end{array}
+$$
+
+where $\underline { { { \bf \Pi } } } ^ { 6 6 } \underline { { { \bf \Pi } } } ^ { 5 9 }$ is the eliminated trend part. $\mathcal { X } _ { \mathrm { e n } } ^ { l } = S _ { \mathrm { e n } } ^ { l , 2 } , l \in \{ 1 , \cdots , N \}$ denotes the output of $l$ -th encoder layer and $\mathcal { X } _ { \mathrm { e n } } ^ { 0 }$ is the embedded $\mathcal { X } _ { \mathrm { e n } }$ . $S _ { \mathrm { e n } } ^ { l , i }$ , $i \in \{ 1 , 2 \}$ represents the seasonal component after the -th series decomposition block in the $l$ -th layer respectively. We will give detailed description of Auto-Correlation $( \cdot )$ in the next section, which can seamlessly replace the self-attention.
+
+Decoder The decoder contains two parts: the accumulation structure for trend-cyclical components and the stacked Auto-Correlation mechanism for seasonal components (Figure 1). Each decoder layer contains the inner Auto-Correlation and encoder-decoder Auto-Correlation, which can refine the prediction and utilize the past seasonal information respectively. Note that the model extracts the potential trend from the intermediate hidden variables during the decoder, allowing Autoformer to progressively refine the trend prediction and eliminate interference information for period-based dependencies discovery in Auto-Correlation. Suppose there are $M$ decoder layers. With the latent variable $\chi _ { \mathrm { e n } } ^ { N }$ from the encoder, the equations of $l$ -th decoder layer can be summarized as $\mathcal { X } _ { \mathrm { d e } } ^ { l } =$ $\mathrm { D e c o d e r } ( \mathcal { X } _ { \mathrm { d e } } ^ { l - 1 } , \mathcal { X } _ { \mathrm { e n } } ^ { N } )$ . The decoder can be formalized as follows:
+
+$$
+\begin{array} { r l } & { S _ { \mathrm { d e } } ^ { l , 1 } , \mathcal { T } _ { \mathrm { d e } } ^ { l , 1 } = \mathrm { S e r i e s D e c o m p } \left( \mathrm { A u t o - C o r r e l a t i o n } ( \mathcal { X } _ { \mathrm { d e } } ^ { l - 1 } ) + \mathcal { X } _ { \mathrm { d e } } ^ { l - 1 } \right) } \\ & { S _ { \mathrm { d e } } ^ { l , 2 } , \mathcal { T } _ { \mathrm { d e } } ^ { l , 2 } = \mathrm { S e r i e s D e c o m p } \left( \mathrm { A u t o - C o r r e l a t i o n } ( S _ { \mathrm { d e } } ^ { l , 1 } , \mathcal { X } _ { \mathrm { e n } } ^ { N } ) + S _ { \mathrm { d e } } ^ { l , 1 } \right) } \\ & { S _ { \mathrm { d e } } ^ { l , 3 } , \mathcal { T } _ { \mathrm { d e } } ^ { l , 3 } = \mathrm { S e r i e s D e c o m p } \left( \mathrm { F e e d F o r w a r d } ( S _ { \mathrm { d e } } ^ { l , 2 } ) + S _ { \mathrm { d e } } ^ { l , 2 } \right) } \\ & { \qquad \mathcal { T } _ { \mathrm { d e } } ^ { l } = \mathcal { T } _ { \mathrm { d e } } ^ { l - 1 } + \mathcal { W } _ { l , 1 } \ast \mathcal { T } _ { \mathrm { d e } } ^ { l , 1 } + \mathcal { W } _ { l , 2 } \ast \mathcal { T } _ { \mathrm { d e } } ^ { l , 2 } + \mathcal { W } _ { l , 3 } \ast \mathcal { T } _ { \mathrm { d e } } ^ { l , 3 } , } \end{array}
+$$
+
+where $\mathcal { X } _ { \mathrm { d e } } ^ { l } = { S } _ { \mathrm { d e } } ^ { l , 3 } , l \in \{ 1 , \cdots , M \}$ denotes the output of $l$ -th decoder layer. $\mathcal { X } _ { \mathrm { d e } } ^ { 0 }$ is embedded from $\mathcal { X } _ { \mathrm { d e s } }$ de de for deep transform and $\mathcal { T } _ { \mathrm { d e } } ^ { 0 } = \mathcal { X } _ { \mathrm { d e t } }$ is for accumulatio . $S _ { \mathrm { d e } } ^ { l , i } , T _ { \mathrm { d e } } ^ { l , i } , i \in \{ 1 , 2 , 3 \}$ represent the $i$ $l$ -th layer respectively. $\mathcal { W } _ { l , i } , i \in \{ 1 , 2 , 3 \}$ represents the projector for the $i$ -th extracted trend $\mathcal { T } _ { \mathrm { d e } } ^ { l , i }$ .
+
+![](images/611e264868072bea6977edb6f802f5f63a12c602deabbfe382df58f997963cca.jpg)  
+Figure 2: Auto-Correlation (left) and Time Delay Aggregation (right). We utilize the Fast Fourier Transform to calculate the autocorrelation $\mathcal { R } ( \tau )$ , which reflects the time-delay similarities. Then the similar sub-processes are rolled to the same index based on selected delay $\tau$ and aggregated by $\mathcal { R } ( \tau )$ .
+
+The final prediction is the sum of the two refined decomposed components, as $\mathcal { W } _ { S } \ast \mathcal { X } _ { \mathrm { d e } } ^ { M } + \mathcal { T } _ { \mathrm { d e } } ^ { M }$ where is to project the deep transformed seasonal component to the target dimension.
+
+# 3.2 Auto-Correlation Mechanism
+
+As shown in Figure 2, we propose the Auto-Correlation mechanism with series-wise connections to expand the information utilization. Auto-Correlation discovers the period-based dependencies by calculating the series autocorrelation and aggregates similar sub-series by time delay aggregation.
+
+Period-based dependencies It is observed that the same phase position among periods naturally provides similar sub-processes. Inspired by the stochastic process theory [8, 24], for a real discretetime process $\{ \mathcal { X } _ { t } \}$ , we can obtain the autocorrelation $\mathcal { R } _ { \mathcal { X } \mathcal { X } } ( \tau )$ by the following equations:
+
+$$
+\mathcal { R } _ { \mathcal { X } \mathcal { X } } ( \tau ) = \operatorname* { l i m } _ { L  \infty } \frac { 1 } { L } \sum _ { t = 1 } ^ { L } \mathcal { X } _ { t } \mathcal { X } _ { t - \tau } .
+$$
+
+$\mathcal { R } _ { \mathcal { X } \mathcal { X } } ( \tau )$ reflects the time-delay similarity between $\{ \mathcal { X } _ { t } \}$ and its $\tau$ lag series $\{ \mathcal { X } _ { t - \tau } \}$ . As shown in Figure 2, we use the autocorrelation $\mathcal { R } ( \tau )$ as the unnormalized confidence of estimated period length $\tau$ . Then, we choose the most possible $k$ period lengths $\tau _ { 1 } , \cdots , \tau _ { k }$ . The period-based dependencies are derived by the above estimated periods and can be weighted by the corresponding autocorrelation.
+
+1Time delay aggregation The period-based dependencies connect the sub-series among estimated 1periods. Thus, we present the time delay aggregation block (Figure 2), which can roll the series based on selected time delay $\tau _ { 1 } , \cdots , \tau _ { k }$ . This operation can align similar sub-series that are at the same phase position of estimated periods, which is different from the point-wise dot-product aggregation in self-attention family. Finally, we aggregate the sub-series by softmax normalized confidences.
+
+For the single head situation and time series $\mathcal { X }$ with length- $L$ , after the projector, we get query $\mathcal { Q }$ , key $\kappa$ and value $\nu$ . Thus, it can replace self-attention seamlessly. The Auto-Correlation mechanism is:
+
+$$
+\begin{array} { r l } & { \qquad \tau _ { 1 } , \cdots , \tau _ { k } = \underset { \tau \in \{ 1 , \cdots , L \} } { \mathrm { a r g } \mathrm { T o p k } } ( \mathcal { R } _ { \mathcal { Q } , \mathcal { K } } ( \tau ) ) } \\ & { \qquad \widehat { \mathcal { R } } _ { \mathcal { Q } , \mathcal { K } } ( \tau _ { 1 } ) , \cdots , \widehat { \mathcal { R } } _ { \mathcal { Q } , \mathcal { K } } ( \tau _ { k } ) = \mathrm { S o f t M a x } ( \mathcal { R } _ { \mathcal { Q } , \mathcal { K } } ( \tau _ { 1 } ) , \cdots , \mathcal { R } _ { \mathcal { Q } , \mathcal { K } } ( \tau _ { k } ) ) } \\ & { \mathrm { A u t o - C o r r e l a t i o n } ( \mathcal { Q } , \mathcal { K } , \mathcal { V } ) = \underset { i = 1 } { \overset { k } { \sum } } \mathrm { R o l l } ( \mathcal { V } , \tau _ { i } ) \widehat { \mathcal { R } } _ { \mathcal { Q } , \mathcal { K } } ( \tau _ { i } ) , } \end{array}
+$$
+
+where ar $\boldsymbol { \mathrm { \xi ^ { 2 } } } \mathrm { T o p k } ( \cdot )$ is to get the arguments of the Topk autocorrelations and let $k = \lfloor c \times \log L \rfloor$ , $c$ is a hyper-parameter. $\mathcal { R } _ { \mathcal { Q } , \kappa }$ is autocorrelation between series $\mathcal { Q }$ and $\kappa$ . $\mathrm { R o l l } ( \mathcal { X } , \tau )$ represents the operation to $\mathcal { X }$ with time delay $\tau$ , during which elements that are shifted beyond the first position are re-introduced at the last position. For the encoder-decoder Auto-Correlation (Figure 1), $\kappa , \nu$ are from the encoder $\chi _ { \mathrm { e n } } ^ { N }$ and will be resized to length- $O$ , $\mathcal { Q }$ is from the previous block of the decoder.
+
+![](images/b3a1dd188e9913684d89e9998ab12a2c0b4d1275bd0f7f003ea51f10541c2ec2.jpg)  
+Figure 3: Auto-Correlation vs. self-attention family. Full Attention [35] (a) adapts the fully connection among all time points. Sparse Attention [17, 41] (b) selects points based on the proposed similarity metrics. LogSparse Attention [20] (c) chooses points following the exponentially increasing intervals. Auto-Correlation (d) focuses on the connections of sub-series among underlying periods.
+
+For the multi-head version used in Autoformer, with hidden variables of $d _ { \mathrm { m o d e l } }$ channels, $h$ heads, the query, key and value for $i$ -th head are $\mathcal { Q } _ { i } , \mathcal { K } _ { i } , \mathcal { V } _ { i } \in \mathbb { R } ^ { L \times \frac { d _ { \mathrm { m o d e l } } } { h } }$ , $i \in \{ 1 , \cdots , h \}$ . The process is:
+
+$$
+\begin{array} { r l } & { \mathrm { M u l t i H e a d } ( \mathcal { Q } , K , \mathcal { V } ) = \mathcal { W } _ { \mathrm { o u t p u t } } * \mathrm { C o n c a t } ( \mathrm { h e a d } _ { 1 } , \cdot \cdot \cdot , \mathrm { h e a d } _ { h } ) } \\ & { \quad \quad \quad \mathrm { w h e r e ~ h e a d } _ { i } = \mathrm { A u t o - C o r r e l a t i o n } ( \mathcal { Q } _ { i } , K _ { i } , \mathcal { V } _ { i } ) . } \end{array}
+$$
+
+Efficient computation For period-based dependencies, these dependencies point to sub-processes at the same phase position of underlying periods and are inherently sparse. Here, we select the most possible delays to avoid picking the opposite phases. Because we aggregate ${ \mathcal { O } } ( \log L )$ series whose length is $L$ , the complexity of Equations 6 and 7 is $\mathcal { O } ( L \log L )$ . For the autocorrelation computation (Equation 5), given time series $\{ \mathcal { X } _ { t } \}$ , $\mathcal { R } _ { \mathcal { X } \mathcal { X } } ( \tau )$ can be calculated by Fast Fourier Transforms (FFT) based on the Wiener–Khinchin theorem [37]:
+
+$$
+\begin{array} { r l } & { \displaystyle \mathcal { S } _ { \mathcal { X } \mathcal { X } } ( f ) = \mathcal { F } \left( \mathcal { X } _ { t } \right) \mathcal { F } ^ { * } \left( \mathcal { X } _ { t } \right) = \int _ { - \infty } ^ { \infty } \mathcal { X } _ { t } e ^ { - i 2 \pi t f } \mathrm { d } t \overline { { \int _ { - \infty } ^ { \infty } \mathcal { X } _ { t } e ^ { - i 2 \pi t f } \mathrm { d } t } } } \\ & { \displaystyle \mathcal { R } _ { \mathcal { X } \mathcal { X } } ( \tau ) = \mathcal { F } ^ { - 1 } \left( S _ { \mathcal { X } \mathcal { X } } ( f ) \right) = \int _ { - \infty } ^ { \infty } S _ { \mathcal { X } \mathcal { X } } ( f ) e ^ { i 2 \pi f \tau } \mathrm { d } f , } \end{array}
+$$
+
+where $\tau \in \{ 1 , \cdots , L \}$ , $\mathcal { F }$ denotes the FFT and ${ \mathcal { F } } ^ { - 1 }$ is its inverse. $^ *$ denotes the conjugate operation and $\mathcal { S } _ { \mathcal { X X } } ( f )$ is in the frequency domain. Note that the series autocorrelation of all lags in $\{ 1 , \cdots , L \}$ can be calculated at once by FFT. Thus, Auto-Correlation achieves the $\mathcal { O } ( L \log L )$ complexity.
+
+Auto-Correlation vs. self-attention family Different from the point-wise self-attention family, Auto-Correlation presents the series-wise connections (Figure 3). Concretely, for the temporal dependencies, we find the dependencies among sub-series based on the periodicity. In contrast, the self-attention family only calculates the relation between scattered points. Though some selfattentions [20, 41] consider the local information, they only utilize this to help point-wise dependencies discovery. For the information aggregation, we adopt the time delay block to aggregate the similar sub-series from underlying periods. In contrast, self-attentions aggregate the selected points by dot-product. Benefiting from the inherent sparsity and sub-series-level representation aggregation, Auto-Correlation can simultaneously benefit the computation efficiency and information utilization.
+
+# 4 Experiments
+
+We extensively evaluate the proposed Autoformer on six real-world benchmarks, covering five mainstream time series forecasting applications: energy, traffic, economics, weather and disease.
+
+Datasets Here is a description of the six experiment datasets: (1) ETT [41] dataset contains the data collected from electricity transformers, including load and oil temperature that are recorded every
+
+Table 1: Multivariate results with different prediction lengths $O \in \{ 9 6 , 1 9 2 , 3 3 6 , 7 2 0 \}$ . We set the input length $I$ as 36 for ILI and 96 for the others. A lower MSE or MAE indicates a better prediction.   
+
+<table><tr><td colspan="2"></td><td colspan="2">Models Autoformer</td><td colspan="2">Informer[41]</td><td colspan="2">LogTrans[20]</td><td colspan="2">Reformer[17]</td><td colspan="2">LSTNet[19]</td><td colspan="2">LSTM[13]</td><td colspan="2">TCN[3]</td></tr><tr><td colspan="2">Metric</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE MAE</td><td></td><td>MSE MAE</td><td></td><td>MSE</td><td>MAE</td></tr><tr><td rowspan="2">T</td><td rowspan="2">96 192 336</td><td rowspan="2">0.255 0.281 0.339</td><td rowspan="2">0.339 0.340</td><td rowspan="2">0.365 0.533</td><td rowspan="2">0.453 0.563</td><td rowspan="2">0.768 0.989</td><td rowspan="2">0.642 0.757</td><td rowspan="2">0.658</td><td rowspan="2">0.619</td><td rowspan="2">3.142 3.154</td><td rowspan="2">1.365 1.369</td><td rowspan="2">2.041 2.249</td><td rowspan="2">1.073 1.112</td><td rowspan="2">3.041 3.072</td><td rowspan="2"></td><td rowspan="2">1.330 1.339</td></tr><tr><td>1.078 0.827 1.549</td></tr><tr><td></td><td>720 96</td><td>0.422 0.201</td><td>0.372 0.419</td><td>1.363 3.379</td><td>0.887 1.388</td><td>3.048</td><td>1.334</td><td>0.872 1.328</td><td>2.631</td><td>0.972 1.242</td><td>3.160 3.171</td><td>1.369 1.368 2.720</td><td>2.568</td><td>1.238 1.287</td><td>3.105 3.135</td><td>1.348 1.354</td></tr><tr><td>erneera</td><td>192 336</td><td>0.222</td><td>0.317 0.334</td><td>0.274 0.296</td><td>0.368 0.386</td><td>0.258 0.266</td><td>0.357 0.368</td><td></td><td>0.312 0.348</td><td>0.402 0.433</td><td>0.680 0.645 0.725</td><td>0.676</td><td>0.375 0.442</td><td>0.437 0.473</td><td>0.985 0.996</td><td>0.813 0.821</td></tr><tr><td></td><td>720</td><td>0.231 0.254</td><td>0.338 0.361</td><td>0.300 0.373</td><td>0.394 0.439</td><td>0.280 0.283</td><td></td><td>0.380 0.376</td><td>0.350 0.340</td><td>0.433 0.420</td><td>0.828 0.957</td><td>0.727 0.811</td><td>0.439 0.980</td><td>0.473 0.814</td><td>1.000 1.438</td><td>0.824 0.784</td></tr><tr><td>uepeg</td><td>96 192</td><td>0.197</td><td>0.323</td><td>0.847</td><td>0.752</td><td>0.968</td><td>0.812</td><td></td><td>1.065</td><td>0.829</td><td>1.551</td><td>1.058 1.453</td><td></td><td>1.049 3.004</td><td></td><td>1.432</td></tr><tr><td></td><td>336</td><td>0.300 0.509</td><td>0.369 0.524</td><td>1.204</td><td>0.895</td><td>1.040</td><td>0.851</td><td></td><td>1.188</td><td>0.906</td><td>1.477</td><td>1.028</td><td>1.846</td><td>1.179</td><td>3.048</td><td>1.444</td></tr><tr><td></td><td>720</td><td>1.447</td><td>0.941</td><td>1.672 2.478</td><td>1.036 1.310</td><td>1.659 1.941</td><td>1.081 1.127</td><td>1.357 1.510</td><td></td><td>0.976 1.016</td><td>1.507 2.285</td><td>1.031 1.243</td><td>2.136 2.984</td><td>1.231</td><td>3.113</td><td>1.459</td></tr><tr><td></td><td>96</td><td>0.613</td><td>0.388</td><td>0.719</td><td>0.391</td><td>0.684</td><td>0.384</td><td>0.732</td><td></td><td>0.423</td><td></td><td></td><td></td><td>1.427</td><td>3.150</td><td>1.458</td></tr><tr><td>[Tjeee</td><td>192</td><td>0.616</td><td>0.382</td><td>0.696</td><td>0.379</td><td>0.685</td><td>0.390</td><td>0.733</td><td>0.420</td><td></td><td>1.107 1.157</td><td>0.685 0.706</td><td>0.843 0.847</td><td>0.453 0.453</td><td>1.438 1.463</td><td>0.784 0.794</td></tr><tr><td></td><td>336 720</td><td>0.622</td><td>0.337</td><td>0.777</td><td>0.420</td><td>0.733</td><td>0.408</td><td>0.742</td><td>0.420</td><td></td><td>1.216</td><td>0.730</td><td>0.853</td><td>0.455</td><td>1.479</td><td>0.799</td></tr><tr><td></td><td></td><td>0.660</td><td>0.408</td><td>0.864</td><td>0.472</td><td>0.717</td><td>0.396</td><td>0.755</td><td></td><td>0.423</td><td>1.481</td><td>0.805</td><td>1.500</td><td>0.805</td><td>1.499</td><td>0.804</td></tr><tr><td>waaeee</td><td>96</td><td>0.266</td><td>0.336</td><td>0.300</td><td>0.384</td><td>0.458</td><td>0.490</td><td>0.689</td><td></td><td>0.596</td><td>0.594</td><td>0.587</td><td>0.369</td><td>0.406</td><td>0.615</td><td>0.589</td></tr><tr><td></td><td>192 336</td><td>0.307</td><td>0.367</td><td>0.598</td><td>0.544</td><td>0.658</td><td>0.589</td><td>0.752</td><td></td><td>0.638</td><td>0.560</td><td>0.565</td><td>0.416</td><td>0.435</td><td>0.629</td><td>0.600</td></tr><tr><td></td><td>720</td><td>0.359</td><td>0.395</td><td>0.578</td><td>0.523</td><td>0.797</td><td>0.652</td><td>0.639</td><td></td><td>0.596</td><td>0.597</td><td>0.587</td><td>0.455</td><td>0.454</td><td>0.639</td><td>0.608</td></tr><tr><td></td><td></td><td>0.419</td><td>0.428</td><td>1.059</td><td>0.741</td><td>0.869</td><td>0.675</td><td>1.130</td><td></td><td>0.792</td><td>0.618</td><td>0.599</td><td>0.535</td><td>0.520</td><td>0.639</td><td>0.610</td></tr><tr><td></td><td>24</td><td>3.483</td><td></td><td>5.764</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>36</td><td></td><td>1.287</td><td></td><td>1.677</td><td>4.480</td><td>1.444</td><td></td><td>4.400</td><td>1.382</td><td>6.026</td><td>1.770</td><td>5.914</td><td>1.734</td><td>6.624</td><td>1.830</td></tr><tr><td>Ⅱ</td><td>48</td><td>3.103</td><td>1.148</td><td>4.755</td><td>1.467</td><td>4.799</td><td>1.467</td><td></td><td>4.783</td><td>1.448</td><td>5.340</td><td>1.668</td><td>6.631</td><td>1.845</td><td>6.858</td><td>1.879</td></tr><tr><td></td><td></td><td>2.669</td><td>1.085</td><td>4.763</td><td>1.469</td><td>4.800</td><td>1.468</td><td></td><td>4.832</td><td>1.465</td><td>6.080</td><td>1.787</td><td>6.736</td><td>1.857</td><td>6.968</td><td></td></tr><tr><td></td><td>60</td><td></td><td>1.125</td><td>5.264</td><td>1.564</td><td>5.278</td><td></td><td>1.560</td><td>4.882</td><td>1.483</td><td>5.548</td><td>1.720 6.870</td><td></td><td>1.879</td><td>7.127</td><td>1.892 1.918</td></tr><tr><td></td><td></td><td>2.770</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></table>
+
+\* ETT means the ETTm2. See supplementary materials for the full benchmark of ETTh1, ETTh2, ETTm1.
+
+15 minutes between July 2016 and July 2018. (2) Electricity1 dataset contains the hourly electricity consumption of 321 customers from 2012 to 2014. (3) Exchange [19] records the daily exchange rates of eight different countries ranging from 1990 to 2016. (4) Traffic2 is a collection of hourly data from California Department of Transportation, which describes the road occupancy rates measured by different sensors on San Francisco Bay area freeways. (5) Weather3 is recorded every 10 minutes for 2020 whole year, which contains 21 meteorological indicators, such as air temperature, humidity, etc. (6) $I L I ^ { 4 }$ includes the weekly recorded influenza-like illness (ILI) patients data from Centers for Disease Control and Prevention of the United States between 2002 and 2021, which describes the ratio of patients seen with ILI and the total number of the patients. We follow standard protocol and split all datasets into training, validation and test set in chronological order by the ratio of 6:2:2 for the ETT dataset and 7:1:2 for the other datasets.
+
+Implementation details Our method is trained with L2 loss, using the ADAM [16] optimizer with an initial learning rate of $1 0 ^ { - 4 }$ . Batch size is set to 32. The training process is early stopped within 10 epochs. All experiments are repeated three times, implemented in PyTorch [25] and conducted on a single NVIDIA TITAN RTX 24GB GPUs. The hyper-parameter $c$ of Auto-Correlation is in the range of 1 to 3 to trade off performance and efficiency. See supplementary materials for standard deviations and sensitivity analysis. Autoformer contains 2 encoder layers and 1 decoder layer.
+
+Baselines We include 10 baseline methods. For the multivariate setting, we select three latest stateof-the-art transformer-based models: Informer [41], Reformer [17], LogTrans [20], two RNN-based models: LSTNet [19], LSTM [13] and CNN-based TCN [3] as baselines. For the univariate setting, we include more competitive baselines: N-BEATS[23], DeepAR [28], Prophet [33] and ARMIA [1].
+
+Table 2: Univariate results with different prediction lengths $O \in \{ 9 6 , 1 9 2 , 3 3 6 , 7 2 0 \}$ on typical datasets. We set the input length $I$ as 96. A lower MSE or MAE indicates a better prediction.   
+
+<table><tr><td colspan="2">Models Autoformer N-BEATS[23] Informer[41] LogTrans[20] Reformer[17] DeepAR[28] Prophet[33] ARIMA[1]</td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">Metric1</td><td>MSE MAE MSE</td><td></td><td>MAE</td><td>MSE MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE MAE</td><td></td><td>MSE MAE MSE MAE</td><td></td></tr><tr><td rowspan="4"></td><td>96</td><td>0.065 0.189</td><td>0.082</td><td>0.219</td><td>0.088 0.225</td><td>0.082</td><td>0.217</td><td>0.131</td><td>0.288</td><td>0.099</td><td>0.237</td><td></td><td>0.287 0.456 0.211 0.362</td></tr><tr><td>192</td><td>0.118 0.256 0.120</td><td></td><td>0.268</td><td>0.132 0.283</td><td>0.133</td><td>0.284</td><td>0.186</td><td>0.354</td><td>0.154</td><td>0.310</td><td>0.312 0.483 0.261 0.406</td><td></td></tr><tr><td>336</td><td>0.1540.305 0.226</td><td>0.370</td><td>0.180</td><td></td><td>0.336 0.201</td><td>0.361</td><td>0.220</td><td>0.381</td><td>0.277</td><td>0.428</td><td>0.331 0.474 0.317 0.448</td><td></td></tr><tr><td>720</td><td>0.182 0.335 0.188</td><td></td><td>0.338 0.300</td><td></td><td>0.435 0.268</td><td>0.407</td><td>0.267</td><td>0.430</td><td></td><td></td><td>0.332 0.468 0.5340.593 0.366 0.487</td><td></td></tr><tr><td>aepeg</td><td>96</td><td>0.241 0.387 0.156</td><td>0.299</td><td>0.591（</td><td></td><td>0.615 0.279</td><td>0.441</td><td>1.327</td><td>0.944</td><td></td><td></td><td></td><td>0.417 0.515 0.828 0.762 0.112 0.245</td></tr><tr><td></td><td>192</td><td>0.273 0.403 0.669</td><td>0.665</td><td>1.183</td><td></td><td>0.912 1.950</td><td>1.048</td><td>1.258</td><td>0.924</td><td></td><td></td><td></td><td>0.813 0.735 0.909 0.974 0.304 0.404</td></tr><tr><td></td><td>336</td><td>0.508 0.539 0.611</td><td>0.605</td><td></td><td>1.367 0.984 2.438</td><td></td><td>1.262</td><td>2.179</td><td>1.296</td><td></td><td></td><td></td><td>1.331 0.962 1.304 0.988 0.736 0.598</td></tr><tr><td></td><td>720</td><td>0.991 0.768 1.111</td><td>0.860</td><td>1.872</td><td></td><td>1.072 2.010</td><td>1.247</td><td>1.280</td><td>0.953</td><td></td><td></td><td>1.894 1.181 3.238 1.566 1.871 0.935</td><td></td></tr></table>
+
+# 4.1 Main Results
+
+To compare performances under different future horizons, we fix the input length and evaluate models with a wide range of prediction lengths: 96, 192, 336, 720. This setting precisely meets the definition of long-term forecasting. Here are results on both the multivariate and univariate settings.
+
+Multivariate results As for the multivariate setting, Autoformer achieves the consistent state-ofthe-art performance in all benchmarks and all prediction length settings (Table 1). Especially, under the input-96-predict-336 setting, compared to previous state-of-the-art results, Autoformer gives $74 \%$ $1 . 3 3 4 {  } 0 . 3 3 9 _ { . }$ ) MSE reduction in ETT, $18 \%$ $0 . 2 8 0 {  } 0 . 2 3 1$ ) in Electricity, $61 \%$ ( $1 . 3 5 7 {  } 0 . 5 0 9 \rangle$ in Exchange, $15 \%$ $( 0 . 7 3 3 {  } 0 . 6 2 2 )$ in Traffic and $21 \%$ $( 0 . 4 5 5 {  } 0 . 3 5 9 )$ ) in Weather. For the input36-predict-60 setting of ILI, Autoformer makes $43 \%$ $4 . 8 8 2 {  } 2 . 7 7 0$ ) MSE reduction. Overall, Autoformer yields a $38 \%$ averaged MSE reduction among above settings. Note that Autoformer still provides remarkable improvements in the Exchange dataset that is without obvious periodicity. See supplementary materials for detailed showcases. Besides, we can also find that the performance of Autoformer changes quite steadily as the prediction length $O$ increases. It means that Autoformer retains better long-term robustness, which is meaningful for real-world practical applications, such as weather early warning and long-term energy consumption planning.
+
+Univariate results We list the univariate results of two typical datasets in Table 2. Under the comparison with extensive baselines, our Autoformer still achieves state-of-the-art performance for the long-term forecasting tasks. In particular, for the input-96-predict-336 setting, our model achieves $14 \%$ $0 . 1 8 0 {  } 0 . 1 4 5$ MSE reduction on the ETT dataset with obvious periodicity. For the Exchange dataset without obvious periodicity, Autoformer surpasses other baselines by $17 \%$ $( 0 . 6 1 1 {  } 0 . 5 0 8 )$ and shows greater long-term forecasting capacity. Also, we find that ARIMA [1] performs best in the input-96-predict-96 setting of the Exchange dataset but fails in the long-term setting. This situation of ARIMA can be benefited from its inherent capacity for non-stationary economic data but is limited by the intricate temporal patterns of real-world series.
+
+# 4.2 Ablation studies
+
+Table 3: Ablation of decomposition in multivariate ETT with MSE metric. Ours adopts our progressive architecture into other models. Sep employs two models to forecast pre-decomposed seasonal and trend-cyclical components separately. Promotion is the MSE reduction compared to Origin.   
+
+<table><tr><td rowspan="2">Input-96</td><td colspan="3">Transformer[35]</td><td colspan="3">Informer[41]</td><td colspan="3">LogTrans[17]</td><td colspan="3">Reformer[20]</td><td colspan="2">Promotion</td></tr><tr><td>Predict-O| Origin</td><td>Sep</td><td>Ours</td><td>Origin</td><td>Sep</td><td>Ours</td><td>Origin</td><td>Sep</td><td>Ours</td><td></td><td>Origin Sep</td><td>Ours</td><td>Sep</td><td>Ours</td></tr><tr><td>96</td><td>0.604</td><td>0.311</td><td>0.204</td><td>0.365</td><td>0.490</td><td>0.354</td><td>0.768</td><td>0.862</td><td>0.231</td><td>0.658</td><td>0.445</td><td>0.218</td><td>0.069</td><td>0.347</td></tr><tr><td>192</td><td></td><td>1.060 0.760（</td><td>0.266</td><td>0.533</td><td>0.658</td><td>0.432</td><td>0.989</td><td>0.533</td><td>0.378</td><td>1.078</td><td></td><td>0.510 0.336</td><td></td><td>0.300 0.562</td></tr><tr><td>336</td><td>1.413</td><td>0.665</td><td>0.375</td><td>1.363</td><td>1.469</td><td>0.481</td><td>1.334</td><td>0.762</td><td>0.362</td><td>1.549</td><td>1.028</td><td>0.366</td><td>0.434</td><td>1.019</td></tr><tr><td>720</td><td>2.672</td><td>3.200</td><td>0.537</td><td>3.379</td><td>2.766</td><td>0.822</td><td>3.048</td><td>2.601</td><td>0.539</td><td>2.631</td><td>2.845</td><td>0.502</td><td></td><td>0.079 2.332</td></tr></table>
+
+Decomposition architecture With our proposed progressive decomposition architecture, other models can gain consistent promotion, especially as the prediction length $O$ increases (Table 3). This verifies that our method can generalize to other models and release the capacity of other dependencies learning mechanisms, alleviate the distraction caused by intricate patterns. Besides, our architecture outperforms the pre-processing, although the latter employs a bigger model and more parameters. Especially, pre-decomposing may even bring negative effect because it neglects the interaction of components during long-term future, such as Transformer [35] predict-720, Informer [41] predict-336.
+
+Auto-Correlation vs. self-attention family As shown in Table 4, our proposed Auto-Correlation achieves the best performance under various input- ${ \mathbf { \nabla } } J$ -predict- $O$ settings, which verifies the effectiveness of series-wise connections comparing to point-wise self-attentions (Figure 3). Furthermore, we can also observe that Auto-Correlation is memory efficiency from the last column of Table 4, which can be used in long sequence forecasting, such as input-336-predict-1440.
+
+Table 4: Comparison of Auto-Correlation and self-attention in the multivariate ETT. We replace the Auto-Correlation in Autoformer with different self-attentions. The “-” indicates the out-of-memory.   
+
+<table><tr><td rowspan="2" colspan="2">Input Length I Prediction Length O</td><td colspan="3">96</td><td colspan="3">192</td><td colspan="3">336</td></tr><tr><td>336</td><td>720</td><td>1440</td><td>336</td><td>720</td><td>1440</td><td>336</td><td>720</td><td>1440</td></tr><tr><td>Auto- Correlation</td><td>MSE MAE</td><td>0.339 0.372</td><td>0.422 0.419</td><td>0.555 0.496</td><td>0.355 0.392</td><td>0.429 0.430</td><td>0.503 0.484</td><td>0.361 0.406</td><td>0.425 0.440</td><td>0.574 0.534</td></tr><tr><td>Full Attention[35]</td><td>MSE MAE</td><td>0.375 0.425</td><td>0.537 0.502</td><td>0.667 0.589</td><td>0.450 0.470</td><td>0.554 0.533</td><td>- -</td><td>0.501 0.485</td><td>0.647 0.491</td><td>1 =</td></tr><tr><td>LogSparse Attention[20]</td><td>MSE MAE</td><td>0.362 0.413</td><td>0.539 0.522</td><td>0.582 0.529</td><td>0.420 0.450</td><td>0.552 0.513</td><td>0.958 0.736</td><td>0.474 0.474</td><td>0.601 0.524</td><td>- =</td></tr><tr><td>LSH Attention[17]</td><td>MSE MAE</td><td>0.366 0.404</td><td>0.502 0.475</td><td>0.663 0.567</td><td>0.407 0.421</td><td>0.636 0.571</td><td>1.069 0.756</td><td>0.442 0.476</td><td>0.615 0.532</td><td>1 -</td></tr><tr><td>ProbSparse Attention[41]</td><td>MSE MAE</td><td>0.481 0.472</td><td>0.822 0.559</td><td>0.715 0.586</td><td>0.404 0.425</td><td>1.148 0.654</td><td>0.732 0.602</td><td>0.417 0.434</td><td>0.631 0.528</td><td>1.133 0.691</td></tr></table>
+
+# 4.3 Model Analysis
+
+Time series decomposition As shown in Figure 4, without our series decomposition block, the forecasting model cannot capture the increasing trend and peaks of the seasonal part. By adding the series decomposition blocks, Autoformer can aggregate and refine the trend-cyclical part from series progressively. This design also facilitates the learning of the seasonal part, especially the peaks and troughs. This verifies the necessity of our proposed progressive decomposition architecture.
+
+![](images/226af42008cf294d2ad17509100d9847b1a2e1adc1ae3b98a6fc44cc7d5195f9.jpg)  
+Figure 4: Visualization of learned seasonal gradually add the decomposition blocks in $\mathcal { X } _ { \mathrm { d e } } ^ { M }$ and trend-cyclical der from left to rig $\mathcal { T } _ { \mathrm { d e } } ^ { M }$ of the last decoder layer. Wehis case is from ETT dataset under input-96-predict-720 setting. For clearness, we add the linear growth to raw data additionally.
+
+Dependencies learning The marked time delay sizes in Figure 5(a) indicate the most likely periods. Our learned periodicity can guide the model to aggregate the sub-series from the same or neighbor phase of periods by $\mathrm { R o l l } ( \mathcal { X } , \tau _ { i } )$ , $i \in \{ 1 , \cdots , 6 \}$ . For the last time step (declining stage), AutoCorrelation fully utilizes all similar sub-series without omissions or errors compared to self-attentions. This verifies that Autoformer can discover the relevant information more sufficiently and precisely.
+
+Complex seasonality modeling As shown in Figure 6, the lags that Autoformer learns from deep representations can indicate the real seasonality of raw series. For example, the learned lags of the daily recorded Exchange dataset present the monthly, quarterly and yearly periods (Figure 6 (b)). For the hourly recorded Traffic dataset (Figure 6 (c)), the learned lags show the intervals as 24-hours and 168-hours, which match the daily and weekly periods of real-world scenarios. These results show that Autoformer can capture the complex seasonalities of real-world series from deep representations and further provide a human-interpretable prediction.
+
+![](images/1f8ff11071d73f33cde0d525eb867796c34772cd0e34d9f2c2266d99a710dd8f.jpg)  
+Figure 5: Visualization of learned dependencies. For clearness, we select the top-6 time delay sizes $\tau _ { 1 } , \cdots , \tau _ { 6 }$ of Auto-Correlation and mark them in raw series (red lines). For self-attentions, top-6 similar points with respect to the last time step (red stars) are also marked by orange points.
+
+![](images/487dd7c7362dd1bad3b32de704db93d0e84eac245cf0c6717299a42cde5d4a6b.jpg)  
+Figure 6: Statistics of learned lags. For each time series in the test set, we count the top 10 lags learned by decoder for the input-96-predict-336 task. Figure (a)-(d) are the density histograms.
+
+Efficiency analysis We compare the running memory and time among Auto-Correlation-based and self-attention-based models (Figure 7) during the training phase. The proposed Autoformer shows $\mathcal { O } ( L \log L )$ complexity in both memory and time and achieves better long-term sequences efficiency.
+
+![](images/9945f941aa3e41c082433f9479d4023d07440ec79228f1bcc87357764c332377.jpg)  
+Figure 7: Efficiency Analysis. For memory, we replace Auto-Correlation with self-attention family in Autoformer and record the memory with input 96. For running time, we run the Auto-Correlation or self-attentions $1 0 ^ { 3 }$ times to get the execution time per step. The output length increases exponentially.
+
+# 5 Conclusions
+
+This paper studies the long-term forecasting problem of time series, which is a pressing demand for real-world applications. However, the intricate temporal patterns prevent the model from learning reliable dependencies. We propose the Autoformer as a decomposition architecture by embedding the series decomposition block as an inner operator, which can progressively aggregate the longterm trend part from intermediate prediction. Besides, we design an efficient Auto-Correlation mechanism to conduct dependencies discovery and information aggregation at the series level, which contrasts clearly from the previous self-attention family. Autoformer can naturally achieve $\mathcal { O } ( L \log L )$ complexity and yield consistent state-of-the-art performance in extensive real-world datasets.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported by the National Natural Science Foundation of China under Grants 62022050 and 62021002, Beijing Nova Program under Grant Z201100006820041, China’s Ministry of Industry and Information Technology, the MOE Innovation Plan and the BNRist Innovation Fund.
+
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parse/train/I55UqU-M11y/I55UqU-M11y_content_list.json

@@ -0,0 +1,1041 @@
+[
+    {
+        "type": "text",
+        "text": "Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Haixu Wu, Jiehui Xu, Jianmin Wang, Mingsheng Long $( \\boxtimes )$ School of Software, BNRist, Tsinghua University, China {whx20,xjh20}@mails.tsinghua.edu.cn, {jimwang,mingsheng}@tsinghua.edu.cn ",
+        "bbox": [
+            191,
+            224,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            303,
+            535,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Extending the forecasting time is a critical demand for real applications, such as extreme weather early warning and long-term energy consumption planning. This paper studies the long-term forecasting problem of time series. Prior Transformerbased models adopt various self-attention mechanisms to discover the long-range dependencies. However, intricate temporal patterns of the long-term future prohibit the model from finding reliable dependencies. Also, Transformers have to adopt the sparse versions of point-wise self-attentions for long series efficiency, resulting in the information utilization bottleneck. Going beyond Transformers, we design Autoformer as a novel decomposition architecture with an Auto-Correlation mechanism. We break with the pre-processing convention of series decomposition and renovate it as a basic inner block of deep models. This design empowers Autoformer with progressive decomposition capacities for complex time series. Further, inspired by the stochastic process theory, we design the Auto-Correlation mechanism based on the series periodicity, which conducts the dependencies discovery and representation aggregation at the sub-series level. Auto-Correlation outperforms self-attention in both efficiency and accuracy. In long-term forecasting, Autoformer yields stateof-the-art accuracy, with a $38 \\%$ relative improvement on six benchmarks, covering five practical applications: energy, traffic, economics, weather and disease. Code is available at this repository: https://github.com/thuml/Autoformer. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            626,
+            310,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Time series forecasting has been widely used in energy consumption, traffic and economics planning, weather and disease propagation forecasting. In these real-world applications, one pressing demand is to extend the forecast time into the far future, which is quite meaningful for the long-term planning and early warning. Thus, in this paper, we study the long-term forecasting problem of time series, characterizing itself by the large length of predicted time series. Recent deep forecasting models [41, 17, 20, 28, 23, 29, 19, 35] have achieved great progress, especially the Transformer-based models. Benefiting from the self-attention mechanism, Transformers obtain great advantage in modeling long-term dependencies for sequential data, which enables more powerful big models [7, 11]. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, the forecasting task is extremely challenging under the long-term setting. First, it is unreliable to discover the temporal dependencies directly from the long-term time series because the dependencies can be obscured by entangled temporal patterns. Second, canonical Transformers with self-attention mechanisms are computationally prohibitive for long-term forecasting because of the quadratic complexity of sequence length. Previous Transformer-based forecasting models [41, 17, 20] mainly focus on improving self-attention to a sparse version. While performance is significantly improved, these models still utilize the point-wise representation aggregation. Thus, in the process of efficiency improvement, they will sacrifice the information utilization because of the sparse point-wise connections, resulting in a bottleneck for long-term forecasting of time series. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To reason about the intricate temporal patterns, we try to take the idea of decomposition, which is a standard method in time series analysis [1, 27]. It can be used to process the complex time series and extract more predictable components. However, under the forecasting context, it can only be used as the pre-processing of past series because the future is unknown [15]. This common usage limits the capabilities of decomposition and overlooks the potential future interactions among decomposed components. Thus, we attempt to go beyond pre-processing usage of decomposition and propose a generic architecture to empower the deep forecasting models with immanent capacity of progressive decomposition. Further, decomposition can ravel out the entangled temporal patterns and highlight the inherent properties of time series [15]. Benefiting from this, we try to take advantage of the series periodicity to renovate the point-wise connection in self-attention. We observe that the sub-series at the same phase position among periods often present similar temporal processes. Thus, we try to construct a series-level connection based on the process similarity derived by series periodicity. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on the above motivations, we propose an original Autoformer in place of the Transformers for long-term time series forecasting. Autoformer still follows residual and encoder-decoder structure but renovates Transformer into a decomposition forecasting architecture. By embedding our proposed decomposition blocks as the inner operators, Autoformer can progressively separate the long-term trend information from predicted hidden variables. This design allows our model to alternately decompose and refine the intermediate results during the forecasting procedure. Inspired by the stochastic process theory [8, 24], Autoformer introduces an Auto-Correlation mechanism in place of self-attention, which discovers the sub-series similarity based on the series periodicity and aggregates similar sub-series from underlying periods. This series-wise mechanism achieves $\\mathcal { O } ( L \\log L )$ complexity for length- $L$ series and breaks the information utilization bottleneck by expanding the point-wise representation aggregation to sub-series level. Autoformer achieves the state-of-the-art accuracy on six benchmarks. The contributions are summarized as follows: ",
+        "bbox": [
+            174,
+            263,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• To tackle the intricate temporal patterns of the long-term future, we present Autoformer as a decomposition architecture and design the inner decomposition block to empower the deep forecasting model with immanent progressive decomposition capacity. • We propose an Auto-Correlation mechanism with dependencies discovery and information aggregation at the series level. Our mechanism is beyond previous self-attention family and can simultaneously benefit the computation efficiency and information utilization. • Autoformer achieves a $38 \\%$ relative improvement under the long-term setting on six benchmarks, covering five real-world applications: energy, traffic, economics, weather and disease. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Models for Time Series Forecasting ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            603,
+            459,
+            618
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Due to the immense importance of time series forecasting, various models have been well developed. Many time series forecasting methods start from the classic tools [32, 9]. ARIMA [6, 5] tackles the forecasting problem by transforming the non-stationary process to stationary through differencing. The filtering method is also introduced for series forecasting [18, 10]. Besides, recurrent neural networks (RNNs) models are used to model the temporal dependencies for time series [36, 26, 40, 22]. DeepAR [28] combines autoregressive methods and RNNs to model the probabilistic distribution of future series. LSTNet [19] introduces convolutional neural networks (CNNs) with recurrent-skip connections to capture the short-term and long-term temporal patterns. Attention-based RNNs [39, 30, 31] introduce the temporal attention to explore the long-range dependencies for prediction. Also, many works based on temporal convolution networks (TCN) [34, 4, 3, 29] attempt to model the temporal causality with the causal convolution. These deep forecasting models mainly focus on the temporal relation modeling by recurrent connections, temporal attention or causal convolution. ",
+        "bbox": [
+            173,
+            628,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recently, Transformers [35, 38] based on the self-attention mechanism shows great power in sequential data, such as natural language processing [11, 7], audio processing [14] and even computer vision [12, 21]. However, applying self-attention to long-term time series forecasting is computationally prohibitive because of the quadratic complexity of sequence length $L$ in both memory and time. LogTrans [20] introduces the local convolution to Transformer and proposes the LogSparse attention to select time steps following the exponentially increasing intervals, which reduces the complexity to $\\mathcal { O } ( L ( \\log L ) ^ { 2 } )$ . Reformer [17] presents the local-sensitive hashing (LSH) attention and reduces the complexity to $\\mathcal { O } ( L \\log L )$ . Informer [41] extends Transformer with KL-divergence based ProbSparse attention and also achieves $\\mathcal { O } ( L \\log L )$ complexity. Note that these methods are based on the vanilla Transformer and try to improve the self-attention mechanism to a sparse version, which still follows the point-wise dependency and aggregation. In this paper, our proposed Auto-Correlation mechanism is based on the inherent periodicity of time series and can provide series-wise connections. ",
+        "bbox": [
+            174,
+            800,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Decomposition of Time Series ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            162,
+            418,
+            178
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As a standard method in time series analysis, time series decomposition [1, 27] deconstructs a time series into several components, each representing one of the underlying categories of patterns that are more predictable. It is primarily useful for exploring historical changes over time. For the forecasting tasks, decomposition is always used as the pre-processing of historical series before predicting future series [15, 2], such as Prophet [33] with trend-seasonality decomposition and N-BEATS [23] with basis expansion and DeepGLO [29] with matrix decomposition. However, such pre-processing is limited by the plain decomposition effect of historical series and overlooks the hierarchical interaction between the underlying patterns of series in the long-term future. This paper takes the decomposition idea from a new progressive dimension. Our Autoformer harnesses the decomposition as an inner block of deep models, which can progressively decompose the hidden series throughout the whole forecasting process, including both the past series and the predicted intermediate results. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Autoformer ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            358,
+            303,
+            376
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The time series forecasting problem is to predict the most probable length- $O$ series in the future given the past length- ${ \\mathbf { \\nabla } } \\cdot { I }$ series, denoting as input-I-predict- $O$ . The long-term forecasting setting is to predict the long-term future, i.e. larger $O$ . As aforementioned, we have highlighted the difficulties of long-term series forecasting: handling intricate temporal patterns and breaking the bottleneck of computation efficiency and information utilization. To tackle these two challenges, we introduce the decomposition as a builtin block to the deep forecasting model and propose Autoformer as a decomposition architecture. Besides, we design the Auto-Correlation mechanism to discover the period-based dependencies and aggregate similar sub-series from underlying periods. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Decomposition Architecture ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            408,
+            531
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We renovate Transformer [35] to a deep decomposition architecture (Figure 1), including the inner series decomposition block, Auto-Correlation mechanism, and corresponding Encoder and Decoder. ",
+        "bbox": [
+            174,
+            541,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Series decomposition block To learn with the complex temporal patterns in long-term forecasting context, we take the idea of decomposition [1, 27], which can separate the series into trend-cyclical and seasonal parts. These two parts reflect the long-term progression and the seasonality of the series respectively. However, directly decomposing is unrealizable for future series because the future is just unknown. To tackle this dilemma, we present a series decomposition block as an inner operation of Autoformer (Figure 1), which can extract the long-term stationary trend from predicted intermediate hidden variables progressively. Concretely, we adapt the moving average to smooth out periodic fluctuations and highlight the long-term trends. For length- $L$ input series $\\breve { \\mathcal { X } } \\in \\mathbb { R } ^ { L \\times d }$ , the process is: ",
+        "bbox": [
+            173,
+            577,
+            825,
+            689
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/282df0854e95928f14b62ed4e853f369b7025077bb7d2e1cc6605fe5b7efcf53.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { X } _ { \\mathrm { t } } = \\mathrm { A v g P o o l } ( \\mathrm { P a d d i n g } ( \\mathcal { X } ) ) } \\\\ & { \\mathcal { X } _ { \\mathrm { s } } = \\mathcal { X } - \\mathcal { X } _ { \\mathrm { t } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            691,
+            599,
+            728
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { \\mathcal { X } } _ { \\mathrm { s } } , \\boldsymbol { \\mathcal { X } } _ { \\mathrm { t } } \\in \\mathbb { R } ^ { L \\times d }$ denote the seasonal and the extracted trend-cyclical part respectively. We adopt the $\\operatorname { A v g P o o l } ( \\cdot )$ for moving average with the padding operation to keep the series length unchanged. We use $\\mathcal { X } _ { \\mathrm { s } } , \\mathcal { X } _ { \\mathrm { t } } = \\mathrm { S e r i e s D e c o m p } ( \\mathcal { X } )$ to summarize above equations, which is a model inner block. ",
+        "bbox": [
+            174,
+            732,
+            825,
+            775
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Model inputs The inputs of encoder part are the past $I$ time steps $\\mathcal { X } _ { \\mathrm { e n } } \\in \\mathbb { R } ^ { I \\times d }$ . As a decomposition architecture (Figure 1), the input of Autoformer decoder contains both the seasonal part $\\chi _ { \\mathrm { d e s } } \\in$ $\\mathbb { R } ^ { ( \\frac { I } { 2 } + O ) \\times d }$ and trend-cyclical part $\\chi _ { \\mathrm { d e t } } \\in \\mathbb { R } ^ { ( \\frac { I } { 2 } + O ) \\times d }$ to be refined. Each initialization consists of two parts: the component decomposed from the latter half of encoder’s input $\\mathcal { X } _ { \\mathrm { e n } }$ with length $\\frac { I } { 2 }$ to provide recent information, placeholders with length $O$ filled by scalars. It’s formulized as follows: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ca468ee556859c90f1fc9a7b942f1134137418316e28b4815e04a5396ca9b82.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { X } _ { \\mathrm { e n s } } , \\mathcal { X } _ { \\mathrm { e n t } } = \\mathrm { S e r i e s D e c o m p } ( \\mathcal { X } _ { \\mathrm { e n } \\frac { I } { 2 } : I } ) } \\\\ & { \\qquad \\mathcal { X } _ { \\mathrm { d e s } } = \\mathrm { C o n c a t } ( \\mathcal { X } _ { \\mathrm { e n s } } , \\mathcal { X } _ { 0 } ) } \\\\ & { \\qquad \\mathcal { X } _ { \\mathrm { d e t } } = \\mathrm { C o n c a t } ( \\mathcal { X } _ { \\mathrm { e n t } } , \\mathcal { X } _ { \\mathrm { M e a n } } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            858,
+            624,
+            916
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/1ebb1575216e8696f19f85fe903bb441210d41e8973469f64d31b9ac1d64fa43.jpg",
+        "image_caption": [
+            "Figure 1: Autoformer architecture. The encoder eliminates the long-term trend-cyclical part by series decomposition blocks (blue blocks) and focuses on seasonal patterns modeling. The decoder accumulates the trend part extracted from hidden variables progressively. The past seasonal information from encoder is utilized by the encoder-decoder Auto-Correlation (center green block in decoder). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            88,
+            825,
+            286
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\chi _ { \\mathrm { e n s } } , \\chi _ { \\mathrm { e n t } } \\in \\mathbb { R } ^ { \\frac { I } { 2 } \\times d }$ denote the seasonal and trend-cyclical parts of $\\mathcal { X } _ { \\mathrm { e n } }$ respectively, and $\\mathcal { X } _ { 0 } , \\mathcal { X } _ { \\mathrm { M e a n } } \\in \\mathbb { R } ^ { O \\times d }$ denote the placeholders filled with zero and the mean of $\\mathcal { X } _ { \\mathrm { e n } }$ respectively. ",
+        "bbox": [
+            174,
+            368,
+            823,
+            400
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Encoder As shown in Figure 1, the encoder focuses on the seasonal part modeling. The output of the encoder contains the past seasonal information and will be used as the cross information to help the decoder refine prediction results. Suppose we have $N$ encoder layers. The overall equations for $l$ -th encoder layer are summarized as $\\mathcal { X } _ { \\mathrm { e n } } ^ { \\hat { l } ^ { \\mathrm { ~ \\tiny ~ \\cdot ~ } } } = \\mathrm { E n c o d e r } ( \\mathcal { X } _ { \\mathrm { e n } } ^ { l - 1 } )$ . Details are shown as follows: ",
+        "bbox": [
+            173,
+            412,
+            826,
+            469
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf5b96dae4c206d1e884f885670db030d85b3cb046d302cbd50a4dac521f6449.jpg",
+        "text": "$$\n\\begin{array} { r l } & { S _ { \\mathrm { e n } } ^ { l , 1 } , \\ l _ { - } = \\mathrm { S e r i e s D e c o m p } \\Big ( \\mathrm { A u t o - C o r r e l a t i o n } ( \\mathcal { X } _ { \\mathrm { e n } } ^ { l - 1 } ) + \\mathcal { X } _ { \\mathrm { e n } } ^ { l - 1 } \\Big ) } \\\\ & { S _ { \\mathrm { e n } } ^ { l , 2 } , \\ l _ { - } = \\mathrm { S e r i e s D e c o m p } \\Big ( \\mathrm { F e e d F o r w a r d } ( S _ { \\mathrm { e n } } ^ { l , 1 } ) + S _ { \\mathrm { e n } } ^ { l , 1 } \\Big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            474,
+            704,
+            529
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\underline { { { \\bf \\Pi } } } ^ { 6 6 } \\underline { { { \\bf \\Pi } } } ^ { 5 9 }$ is the eliminated trend part. $\\mathcal { X } _ { \\mathrm { e n } } ^ { l } = S _ { \\mathrm { e n } } ^ { l , 2 } , l \\in \\{ 1 , \\cdots , N \\}$ denotes the output of $l$ -th encoder layer and $\\mathcal { X } _ { \\mathrm { e n } } ^ { 0 }$ is the embedded $\\mathcal { X } _ { \\mathrm { e n } }$ . $S _ { \\mathrm { e n } } ^ { l , i }$ , $i \\in \\{ 1 , 2 \\}$ represents the seasonal component after the -th series decomposition block in the $l$ -th layer respectively. We will give detailed description of Auto-Correlation $( \\cdot )$ in the next section, which can seamlessly replace the self-attention. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Decoder The decoder contains two parts: the accumulation structure for trend-cyclical components and the stacked Auto-Correlation mechanism for seasonal components (Figure 1). Each decoder layer contains the inner Auto-Correlation and encoder-decoder Auto-Correlation, which can refine the prediction and utilize the past seasonal information respectively. Note that the model extracts the potential trend from the intermediate hidden variables during the decoder, allowing Autoformer to progressively refine the trend prediction and eliminate interference information for period-based dependencies discovery in Auto-Correlation. Suppose there are $M$ decoder layers. With the latent variable $\\chi _ { \\mathrm { e n } } ^ { N }$ from the encoder, the equations of $l$ -th decoder layer can be summarized as $\\mathcal { X } _ { \\mathrm { d e } } ^ { l } =$ $\\mathrm { D e c o d e r } ( \\mathcal { X } _ { \\mathrm { d e } } ^ { l - 1 } , \\mathcal { X } _ { \\mathrm { e n } } ^ { N } )$ . The decoder can be formalized as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61949f61e35543e7ee1153307307222fd6b2e0d60dadeae1cc3fce06995e170c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { S _ { \\mathrm { d e } } ^ { l , 1 } , \\mathcal { T } _ { \\mathrm { d e } } ^ { l , 1 } = \\mathrm { S e r i e s D e c o m p } \\left( \\mathrm { A u t o - C o r r e l a t i o n } ( \\mathcal { X } _ { \\mathrm { d e } } ^ { l - 1 } ) + \\mathcal { X } _ { \\mathrm { d e } } ^ { l - 1 } \\right) } \\\\ & { S _ { \\mathrm { d e } } ^ { l , 2 } , \\mathcal { T } _ { \\mathrm { d e } } ^ { l , 2 } = \\mathrm { S e r i e s D e c o m p } \\left( \\mathrm { A u t o - C o r r e l a t i o n } ( S _ { \\mathrm { d e } } ^ { l , 1 } , \\mathcal { X } _ { \\mathrm { e n } } ^ { N } ) + S _ { \\mathrm { d e } } ^ { l , 1 } \\right) } \\\\ & { S _ { \\mathrm { d e } } ^ { l , 3 } , \\mathcal { T } _ { \\mathrm { d e } } ^ { l , 3 } = \\mathrm { S e r i e s D e c o m p } \\left( \\mathrm { F e e d F o r w a r d } ( S _ { \\mathrm { d e } } ^ { l , 2 } ) + S _ { \\mathrm { d e } } ^ { l , 2 } \\right) } \\\\ & { \\qquad \\mathcal { T } _ { \\mathrm { d e } } ^ { l } = \\mathcal { T } _ { \\mathrm { d e } } ^ { l - 1 } + \\mathcal { W } _ { l , 1 } \\ast \\mathcal { T } _ { \\mathrm { d e } } ^ { l , 1 } + \\mathcal { W } _ { l , 2 } \\ast \\mathcal { T } _ { \\mathrm { d e } } ^ { l , 2 } + \\mathcal { W } _ { l , 3 } \\ast \\mathcal { T } _ { \\mathrm { d e } } ^ { l , 3 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            738,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { X } _ { \\mathrm { d e } } ^ { l } = { S } _ { \\mathrm { d e } } ^ { l , 3 } , l \\in \\{ 1 , \\cdots , M \\}$ denotes the output of $l$ -th decoder layer. $\\mathcal { X } _ { \\mathrm { d e } } ^ { 0 }$ is embedded from $\\mathcal { X } _ { \\mathrm { d e s } }$ de de for deep transform and $\\mathcal { T } _ { \\mathrm { d e } } ^ { 0 } = \\mathcal { X } _ { \\mathrm { d e t } }$ is for accumulatio . $S _ { \\mathrm { d e } } ^ { l , i } , T _ { \\mathrm { d e } } ^ { l , i } , i \\in \\{ 1 , 2 , 3 \\}$ represent the $i$ $l$ -th layer respectively. $\\mathcal { W } _ { l , i } , i \\in \\{ 1 , 2 , 3 \\}$ represents the projector for the $i$ -th extracted trend $\\mathcal { T } _ { \\mathrm { d e } } ^ { l , i }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/611e264868072bea6977edb6f802f5f63a12c602deabbfe382df58f997963cca.jpg",
+        "image_caption": [
+            "Figure 2: Auto-Correlation (left) and Time Delay Aggregation (right). We utilize the Fast Fourier Transform to calculate the autocorrelation $\\mathcal { R } ( \\tau )$ , which reflects the time-delay similarities. Then the similar sub-processes are rolled to the same index based on selected delay $\\tau$ and aggregated by $\\mathcal { R } ( \\tau )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The final prediction is the sum of the two refined decomposed components, as $\\mathcal { W } _ { S } \\ast \\mathcal { X } _ { \\mathrm { d e } } ^ { M } + \\mathcal { T } _ { \\mathrm { d e } } ^ { M }$ where is to project the deep transformed seasonal component to the target dimension. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Auto-Correlation Mechanism ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 2, we propose the Auto-Correlation mechanism with series-wise connections to expand the information utilization. Auto-Correlation discovers the period-based dependencies by calculating the series autocorrelation and aggregates similar sub-series by time delay aggregation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Period-based dependencies It is observed that the same phase position among periods naturally provides similar sub-processes. Inspired by the stochastic process theory [8, 24], for a real discretetime process $\\{ \\mathcal { X } _ { t } \\}$ , we can obtain the autocorrelation $\\mathcal { R } _ { \\mathcal { X } \\mathcal { X } } ( \\tau )$ by the following equations: ",
+        "bbox": [
+            173,
+            457,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fb64cef2f5829fd289fb477f5c815edeb42f5543fa2e2009089f66659843dcdb.jpg",
+        "text": "$$\n\\mathcal { R } _ { \\mathcal { X } \\mathcal { X } } ( \\tau ) = \\operatorname* { l i m } _ { L  \\infty } \\frac { 1 } { L } \\sum _ { t = 1 } ^ { L } \\mathcal { X } _ { t } \\mathcal { X } _ { t - \\tau } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\mathcal { R } _ { \\mathcal { X } \\mathcal { X } } ( \\tau )$ reflects the time-delay similarity between $\\{ \\mathcal { X } _ { t } \\}$ and its $\\tau$ lag series $\\{ \\mathcal { X } _ { t - \\tau } \\}$ . As shown in Figure 2, we use the autocorrelation $\\mathcal { R } ( \\tau )$ as the unnormalized confidence of estimated period length $\\tau$ . Then, we choose the most possible $k$ period lengths $\\tau _ { 1 } , \\cdots , \\tau _ { k }$ . The period-based dependencies are derived by the above estimated periods and can be weighted by the corresponding autocorrelation. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1Time delay aggregation The period-based dependencies connect the sub-series among estimated 1periods. Thus, we present the time delay aggregation block (Figure 2), which can roll the series based on selected time delay $\\tau _ { 1 } , \\cdots , \\tau _ { k }$ . This operation can align similar sub-series that are at the same phase position of estimated periods, which is different from the point-wise dot-product aggregation in self-attention family. Finally, we aggregate the sub-series by softmax normalized confidences. ",
+        "bbox": [
+            173,
+            627,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the single head situation and time series $\\mathcal { X }$ with length- $L$ , after the projector, we get query $\\mathcal { Q }$ , key $\\kappa$ and value $\\nu$ . Thus, it can replace self-attention seamlessly. The Auto-Correlation mechanism is: ",
+        "bbox": [
+            171,
+            704,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e1702fd112f4259c360baac62f3388d05edd202271aca6d8571fd7c14922488.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\qquad \\tau _ { 1 } , \\cdots , \\tau _ { k } = \\underset { \\tau \\in \\{ 1 , \\cdots , L \\} } { \\mathrm { a r g } \\mathrm { T o p k } } ( \\mathcal { R } _ { \\mathcal { Q } , \\mathcal { K } } ( \\tau ) ) } \\\\ & { \\qquad \\widehat { \\mathcal { R } } _ { \\mathcal { Q } , \\mathcal { K } } ( \\tau _ { 1 } ) , \\cdots , \\widehat { \\mathcal { R } } _ { \\mathcal { Q } , \\mathcal { K } } ( \\tau _ { k } ) = \\mathrm { S o f t M a x } ( \\mathcal { R } _ { \\mathcal { Q } , \\mathcal { K } } ( \\tau _ { 1 } ) , \\cdots , \\mathcal { R } _ { \\mathcal { Q } , \\mathcal { K } } ( \\tau _ { k } ) ) } \\\\ & { \\mathrm { A u t o - C o r r e l a t i o n } ( \\mathcal { Q } , \\mathcal { K } , \\mathcal { V } ) = \\underset { i = 1 } { \\overset { k } { \\sum } } \\mathrm { R o l l } ( \\mathcal { V } , \\tau _ { i } ) \\widehat { \\mathcal { R } } _ { \\mathcal { Q } , \\mathcal { K } } ( \\tau _ { i } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            738,
+            730,
+            835
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where ar $\\boldsymbol { \\mathrm { \\xi ^ { 2 } } } \\mathrm { T o p k } ( \\cdot )$ is to get the arguments of the Topk autocorrelations and let $k = \\lfloor c \\times \\log L \\rfloor$ , $c$ is a hyper-parameter. $\\mathcal { R } _ { \\mathcal { Q } , \\kappa }$ is autocorrelation between series $\\mathcal { Q }$ and $\\kappa$ . $\\mathrm { R o l l } ( \\mathcal { X } , \\tau )$ represents the operation to $\\mathcal { X }$ with time delay $\\tau$ , during which elements that are shifted beyond the first position are re-introduced at the last position. For the encoder-decoder Auto-Correlation (Figure 1), $\\kappa , \\nu$ are from the encoder $\\chi _ { \\mathrm { e n } } ^ { N }$ and will be resized to length- $O$ , $\\mathcal { Q }$ is from the previous block of the decoder. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/b3a1dd188e9913684d89e9998ab12a2c0b4d1275bd0f7f003ea51f10541c2ec2.jpg",
+        "image_caption": [
+            "Figure 3: Auto-Correlation vs. self-attention family. Full Attention [35] (a) adapts the fully connection among all time points. Sparse Attention [17, 41] (b) selects points based on the proposed similarity metrics. LogSparse Attention [20] (c) chooses points following the exponentially increasing intervals. Auto-Correlation (d) focuses on the connections of sub-series among underlying periods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            88,
+            823,
+            292
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the multi-head version used in Autoformer, with hidden variables of $d _ { \\mathrm { m o d e l } }$ channels, $h$ heads, the query, key and value for $i$ -th head are $\\mathcal { Q } _ { i } , \\mathcal { K } _ { i } , \\mathcal { V } _ { i } \\in \\mathbb { R } ^ { L \\times \\frac { d _ { \\mathrm { m o d e l } } } { h } }$ , $i \\in \\{ 1 , \\cdots , h \\}$ . The process is: ",
+        "bbox": [
+            171,
+            364,
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+            398
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/62c004a35e71d95d2adf122bec64c3548f70fa05424bd3536c4b4e3582ae4569.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { M u l t i H e a d } ( \\mathcal { Q } , K , \\mathcal { V } ) = \\mathcal { W } _ { \\mathrm { o u t p u t } } * \\mathrm { C o n c a t } ( \\mathrm { h e a d } _ { 1 } , \\cdot \\cdot \\cdot , \\mathrm { h e a d } _ { h } ) } \\\\ & { \\quad \\quad \\quad \\mathrm { w h e r e ~ h e a d } _ { i } = \\mathrm { A u t o - C o r r e l a t i o n } ( \\mathcal { Q } _ { i } , K _ { i } , \\mathcal { V } _ { i } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            285,
+            401,
+            714,
+            438
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Efficient computation For period-based dependencies, these dependencies point to sub-processes at the same phase position of underlying periods and are inherently sparse. Here, we select the most possible delays to avoid picking the opposite phases. Because we aggregate ${ \\mathcal { O } } ( \\log L )$ series whose length is $L$ , the complexity of Equations 6 and 7 is $\\mathcal { O } ( L \\log L )$ . For the autocorrelation computation (Equation 5), given time series $\\{ \\mathcal { X } _ { t } \\}$ , $\\mathcal { R } _ { \\mathcal { X } \\mathcal { X } } ( \\tau )$ can be calculated by Fast Fourier Transforms (FFT) based on the Wiener–Khinchin theorem [37]: ",
+        "bbox": [
+            173,
+            445,
+            826,
+            530
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2d3d564e0485f3fffbb3432f06a689f28d47f204dc0cc229606fec48019eb783.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathcal { S } _ { \\mathcal { X } \\mathcal { X } } ( f ) = \\mathcal { F } \\left( \\mathcal { X } _ { t } \\right) \\mathcal { F } ^ { * } \\left( \\mathcal { X } _ { t } \\right) = \\int _ { - \\infty } ^ { \\infty } \\mathcal { X } _ { t } e ^ { - i 2 \\pi t f } \\mathrm { d } t \\overline { { \\int _ { - \\infty } ^ { \\infty } \\mathcal { X } _ { t } e ^ { - i 2 \\pi t f } \\mathrm { d } t } } } \\\\ & { \\displaystyle \\mathcal { R } _ { \\mathcal { X } \\mathcal { X } } ( \\tau ) = \\mathcal { F } ^ { - 1 } \\left( S _ { \\mathcal { X } \\mathcal { X } } ( f ) \\right) = \\int _ { - \\infty } ^ { \\infty } S _ { \\mathcal { X } \\mathcal { X } } ( f ) e ^ { i 2 \\pi f \\tau } \\mathrm { d } f , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            534,
+            722,
+            606
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\tau \\in \\{ 1 , \\cdots , L \\}$ , $\\mathcal { F }$ denotes the FFT and ${ \\mathcal { F } } ^ { - 1 }$ is its inverse. $^ *$ denotes the conjugate operation and $\\mathcal { S } _ { \\mathcal { X X } } ( f )$ is in the frequency domain. Note that the series autocorrelation of all lags in $\\{ 1 , \\cdots , L \\}$ can be calculated at once by FFT. Thus, Auto-Correlation achieves the $\\mathcal { O } ( L \\log L )$ complexity. ",
+        "bbox": [
+            176,
+            609,
+            820,
+            652
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Auto-Correlation vs. self-attention family Different from the point-wise self-attention family, Auto-Correlation presents the series-wise connections (Figure 3). Concretely, for the temporal dependencies, we find the dependencies among sub-series based on the periodicity. In contrast, the self-attention family only calculates the relation between scattered points. Though some selfattentions [20, 41] consider the local information, they only utilize this to help point-wise dependencies discovery. For the information aggregation, we adopt the time delay block to aggregate the similar sub-series from underlying periods. In contrast, self-attentions aggregate the selected points by dot-product. Benefiting from the inherent sparsity and sub-series-level representation aggregation, Auto-Correlation can simultaneously benefit the computation efficiency and information utilization. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            809,
+            312,
+            827
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We extensively evaluate the proposed Autoformer on six real-world benchmarks, covering five mainstream time series forecasting applications: energy, traffic, economics, weather and disease. ",
+        "bbox": [
+            173,
+            840,
+            823,
+            869
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets Here is a description of the six experiment datasets: (1) ETT [41] dataset contains the data collected from electricity transformers, including load and oil temperature that are recorded every ",
+        "bbox": [
+            173,
+            882,
+            821,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/52266c9dbd8161d0a8bdaa7f10b0a3b498678a2fbad400c2e8c8e72446796d73.jpg",
+        "table_caption": [
+            "Table 1: Multivariate results with different prediction lengths $O \\in \\{ 9 6 , 1 9 2 , 3 3 6 , 7 2 0 \\}$ . We set the input length $I$ as 36 for ILI and 96 for the others. A lower MSE or MAE indicates a better prediction. "
+        ],
+        "table_footnote": [
+            "\\* ETT means the ETTm2. See supplementary materials for the full benchmark of ETTh1, ETTh2, ETTm1. "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"2\">Models Autoformer</td><td colspan=\"2\">Informer[41]</td><td colspan=\"2\">LogTrans[20]</td><td colspan=\"2\">Reformer[17]</td><td colspan=\"2\">LSTNet[19]</td><td colspan=\"2\">LSTM[13]</td><td colspan=\"2\">TCN[3]</td></tr><tr><td colspan=\"2\">Metric</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE MAE</td><td></td><td>MSE MAE</td><td></td><td>MSE</td><td>MAE</td></tr><tr><td rowspan=\"2\">T</td><td rowspan=\"2\">96 192 336</td><td rowspan=\"2\">0.255 0.281 0.339</td><td rowspan=\"2\">0.339 0.340</td><td rowspan=\"2\">0.365 0.533</td><td rowspan=\"2\">0.453 0.563</td><td rowspan=\"2\">0.768 0.989</td><td rowspan=\"2\">0.642 0.757</td><td rowspan=\"2\">0.658</td><td rowspan=\"2\">0.619</td><td rowspan=\"2\">3.142 3.154</td><td rowspan=\"2\">1.365 1.369</td><td rowspan=\"2\">2.041 2.249</td><td rowspan=\"2\">1.073 1.112</td><td rowspan=\"2\">3.041 3.072</td><td rowspan=\"2\"></td><td rowspan=\"2\">1.330 1.339</td></tr><tr><td>1.078 0.827 1.549</td></tr><tr><td></td><td>720 96</td><td>0.422 0.201</td><td>0.372 0.419</td><td>1.363 3.379</td><td>0.887 1.388</td><td>3.048</td><td>1.334</td><td>0.872 1.328</td><td>2.631</td><td>0.972 1.242</td><td>3.160 3.171</td><td>1.369 1.368 2.720</td><td>2.568</td><td>1.238 1.287</td><td>3.105 3.135</td><td>1.348 1.354</td></tr><tr><td>erneera</td><td>192 336</td><td>0.222</td><td>0.317 0.334</td><td>0.274 0.296</td><td>0.368 0.386</td><td>0.258 0.266</td><td>0.357 0.368</td><td></td><td>0.312 0.348</td><td>0.402 0.433</td><td>0.680 0.645 0.725</td><td>0.676</td><td>0.375 0.442</td><td>0.437 0.473</td><td>0.985 0.996</td><td>0.813 0.821</td></tr><tr><td></td><td>720</td><td>0.231 0.254</td><td>0.338 0.361</td><td>0.300 0.373</td><td>0.394 0.439</td><td>0.280 0.283</td><td></td><td>0.380 0.376</td><td>0.350 0.340</td><td>0.433 0.420</td><td>0.828 0.957</td><td>0.727 0.811</td><td>0.439 0.980</td><td>0.473 0.814</td><td>1.000 1.438</td><td>0.824 0.784</td></tr><tr><td>uepeg</td><td>96 192</td><td>0.197</td><td>0.323</td><td>0.847</td><td>0.752</td><td>0.968</td><td>0.812</td><td></td><td>1.065</td><td>0.829</td><td>1.551</td><td>1.058 1.453</td><td></td><td>1.049 3.004</td><td></td><td>1.432</td></tr><tr><td></td><td>336</td><td>0.300 0.509</td><td>0.369 0.524</td><td>1.204</td><td>0.895</td><td>1.040</td><td>0.851</td><td></td><td>1.188</td><td>0.906</td><td>1.477</td><td>1.028</td><td>1.846</td><td>1.179</td><td>3.048</td><td>1.444</td></tr><tr><td></td><td>720</td><td>1.447</td><td>0.941</td><td>1.672 2.478</td><td>1.036 1.310</td><td>1.659 1.941</td><td>1.081 1.127</td><td>1.357 1.510</td><td></td><td>0.976 1.016</td><td>1.507 2.285</td><td>1.031 1.243</td><td>2.136 2.984</td><td>1.231</td><td>3.113</td><td>1.459</td></tr><tr><td></td><td>96</td><td>0.613</td><td>0.388</td><td>0.719</td><td>0.391</td><td>0.684</td><td>0.384</td><td>0.732</td><td></td><td>0.423</td><td></td><td></td><td></td><td>1.427</td><td>3.150</td><td>1.458</td></tr><tr><td>[Tjeee</td><td>192</td><td>0.616</td><td>0.382</td><td>0.696</td><td>0.379</td><td>0.685</td><td>0.390</td><td>0.733</td><td>0.420</td><td></td><td>1.107 1.157</td><td>0.685 0.706</td><td>0.843 0.847</td><td>0.453 0.453</td><td>1.438 1.463</td><td>0.784 0.794</td></tr><tr><td></td><td>336 720</td><td>0.622</td><td>0.337</td><td>0.777</td><td>0.420</td><td>0.733</td><td>0.408</td><td>0.742</td><td>0.420</td><td></td><td>1.216</td><td>0.730</td><td>0.853</td><td>0.455</td><td>1.479</td><td>0.799</td></tr><tr><td></td><td></td><td>0.660</td><td>0.408</td><td>0.864</td><td>0.472</td><td>0.717</td><td>0.396</td><td>0.755</td><td></td><td>0.423</td><td>1.481</td><td>0.805</td><td>1.500</td><td>0.805</td><td>1.499</td><td>0.804</td></tr><tr><td>waaeee</td><td>96</td><td>0.266</td><td>0.336</td><td>0.300</td><td>0.384</td><td>0.458</td><td>0.490</td><td>0.689</td><td></td><td>0.596</td><td>0.594</td><td>0.587</td><td>0.369</td><td>0.406</td><td>0.615</td><td>0.589</td></tr><tr><td></td><td>192 336</td><td>0.307</td><td>0.367</td><td>0.598</td><td>0.544</td><td>0.658</td><td>0.589</td><td>0.752</td><td></td><td>0.638</td><td>0.560</td><td>0.565</td><td>0.416</td><td>0.435</td><td>0.629</td><td>0.600</td></tr><tr><td></td><td>720</td><td>0.359</td><td>0.395</td><td>0.578</td><td>0.523</td><td>0.797</td><td>0.652</td><td>0.639</td><td></td><td>0.596</td><td>0.597</td><td>0.587</td><td>0.455</td><td>0.454</td><td>0.639</td><td>0.608</td></tr><tr><td></td><td></td><td>0.419</td><td>0.428</td><td>1.059</td><td>0.741</td><td>0.869</td><td>0.675</td><td>1.130</td><td></td><td>0.792</td><td>0.618</td><td>0.599</td><td>0.535</td><td>0.520</td><td>0.639</td><td>0.610</td></tr><tr><td></td><td>24</td><td>3.483</td><td></td><td>5.764</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>36</td><td></td><td>1.287</td><td></td><td>1.677</td><td>4.480</td><td>1.444</td><td></td><td>4.400</td><td>1.382</td><td>6.026</td><td>1.770</td><td>5.914</td><td>1.734</td><td>6.624</td><td>1.830</td></tr><tr><td>Ⅱ</td><td>48</td><td>3.103</td><td>1.148</td><td>4.755</td><td>1.467</td><td>4.799</td><td>1.467</td><td></td><td>4.783</td><td>1.448</td><td>5.340</td><td>1.668</td><td>6.631</td><td>1.845</td><td>6.858</td><td>1.879</td></tr><tr><td></td><td></td><td>2.669</td><td>1.085</td><td>4.763</td><td>1.469</td><td>4.800</td><td>1.468</td><td></td><td>4.832</td><td>1.465</td><td>6.080</td><td>1.787</td><td>6.736</td><td>1.857</td><td>6.968</td><td></td></tr><tr><td></td><td>60</td><td></td><td>1.125</td><td>5.264</td><td>1.564</td><td>5.278</td><td></td><td>1.560</td><td>4.882</td><td>1.483</td><td>5.548</td><td>1.720 6.870</td><td></td><td>1.879</td><td>7.127</td><td>1.892 1.918</td></tr><tr><td></td><td></td><td>2.770</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "15 minutes between July 2016 and July 2018. (2) Electricity1 dataset contains the hourly electricity consumption of 321 customers from 2012 to 2014. (3) Exchange [19] records the daily exchange rates of eight different countries ranging from 1990 to 2016. (4) Traffic2 is a collection of hourly data from California Department of Transportation, which describes the road occupancy rates measured by different sensors on San Francisco Bay area freeways. (5) Weather3 is recorded every 10 minutes for 2020 whole year, which contains 21 meteorological indicators, such as air temperature, humidity, etc. (6) $I L I ^ { 4 }$ includes the weekly recorded influenza-like illness (ILI) patients data from Centers for Disease Control and Prevention of the United States between 2002 and 2021, which describes the ratio of patients seen with ILI and the total number of the patients. We follow standard protocol and split all datasets into training, validation and test set in chronological order by the ratio of 6:2:2 for the ETT dataset and 7:1:2 for the other datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementation details Our method is trained with L2 loss, using the ADAM [16] optimizer with an initial learning rate of $1 0 ^ { - 4 }$ . Batch size is set to 32. The training process is early stopped within 10 epochs. All experiments are repeated three times, implemented in PyTorch [25] and conducted on a single NVIDIA TITAN RTX 24GB GPUs. The hyper-parameter $c$ of Auto-Correlation is in the range of 1 to 3 to trade off performance and efficiency. See supplementary materials for standard deviations and sensitivity analysis. Autoformer contains 2 encoder layers and 1 decoder layer. ",
+        "bbox": [
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+            694,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines We include 10 baseline methods. For the multivariate setting, we select three latest stateof-the-art transformer-based models: Informer [41], Reformer [17], LogTrans [20], two RNN-based models: LSTNet [19], LSTM [13] and CNN-based TCN [3] as baselines. For the univariate setting, we include more competitive baselines: N-BEATS[23], DeepAR [28], Prophet [33] and ARMIA [1]. ",
+        "bbox": [
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+            787,
+            825,
+            843
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/0b627e1d44ceb110476fbbd79f418eefb3c5de77241146900a5c22035547c876.jpg",
+        "table_caption": [
+            "Table 2: Univariate results with different prediction lengths $O \\in \\{ 9 6 , 1 9 2 , 3 3 6 , 7 2 0 \\}$ on typical datasets. We set the input length $I$ as 96. A lower MSE or MAE indicates a better prediction. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Models Autoformer N-BEATS[23] Informer[41] LogTrans[20] Reformer[17] DeepAR[28] Prophet[33] ARIMA[1]</td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\"></td></tr><tr><td colspan=\"2\">Metric1</td><td>MSE MAE MSE</td><td></td><td>MAE</td><td>MSE MAE</td><td>MSE</td><td>MAE</td><td>MSE</td><td>MAE</td><td>MSE MAE</td><td></td><td>MSE MAE MSE MAE</td><td></td></tr><tr><td rowspan=\"4\"></td><td>96</td><td>0.065 0.189</td><td>0.082</td><td>0.219</td><td>0.088 0.225</td><td>0.082</td><td>0.217</td><td>0.131</td><td>0.288</td><td>0.099</td><td>0.237</td><td></td><td>0.287 0.456 0.211 0.362</td></tr><tr><td>192</td><td>0.118 0.256 0.120</td><td></td><td>0.268</td><td>0.132 0.283</td><td>0.133</td><td>0.284</td><td>0.186</td><td>0.354</td><td>0.154</td><td>0.310</td><td>0.312 0.483 0.261 0.406</td><td></td></tr><tr><td>336</td><td>0.1540.305 0.226</td><td>0.370</td><td>0.180</td><td></td><td>0.336 0.201</td><td>0.361</td><td>0.220</td><td>0.381</td><td>0.277</td><td>0.428</td><td>0.331 0.474 0.317 0.448</td><td></td></tr><tr><td>720</td><td>0.182 0.335 0.188</td><td></td><td>0.338 0.300</td><td></td><td>0.435 0.268</td><td>0.407</td><td>0.267</td><td>0.430</td><td></td><td></td><td>0.332 0.468 0.5340.593 0.366 0.487</td><td></td></tr><tr><td>aepeg</td><td>96</td><td>0.241 0.387 0.156</td><td>0.299</td><td>0.591（</td><td></td><td>0.615 0.279</td><td>0.441</td><td>1.327</td><td>0.944</td><td></td><td></td><td></td><td>0.417 0.515 0.828 0.762 0.112 0.245</td></tr><tr><td></td><td>192</td><td>0.273 0.403 0.669</td><td>0.665</td><td>1.183</td><td></td><td>0.912 1.950</td><td>1.048</td><td>1.258</td><td>0.924</td><td></td><td></td><td></td><td>0.813 0.735 0.909 0.974 0.304 0.404</td></tr><tr><td></td><td>336</td><td>0.508 0.539 0.611</td><td>0.605</td><td></td><td>1.367 0.984 2.438</td><td></td><td>1.262</td><td>2.179</td><td>1.296</td><td></td><td></td><td></td><td>1.331 0.962 1.304 0.988 0.736 0.598</td></tr><tr><td></td><td>720</td><td>0.991 0.768 1.111</td><td>0.860</td><td>1.872</td><td></td><td>1.072 2.010</td><td>1.247</td><td>1.280</td><td>0.953</td><td></td><td></td><td>1.894 1.181 3.238 1.566 1.871 0.935</td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 Main Results ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            299,
+            305,
+            314
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To compare performances under different future horizons, we fix the input length and evaluate models with a wide range of prediction lengths: 96, 192, 336, 720. This setting precisely meets the definition of long-term forecasting. Here are results on both the multivariate and univariate settings. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Multivariate results As for the multivariate setting, Autoformer achieves the consistent state-ofthe-art performance in all benchmarks and all prediction length settings (Table 1). Especially, under the input-96-predict-336 setting, compared to previous state-of-the-art results, Autoformer gives $74 \\%$ $1 . 3 3 4 {  } 0 . 3 3 9 _ { . }$ ) MSE reduction in ETT, $18 \\%$ $0 . 2 8 0 {  } 0 . 2 3 1$ ) in Electricity, $61 \\%$ ( $1 . 3 5 7 {  } 0 . 5 0 9 \\rangle$ in Exchange, $15 \\%$ $( 0 . 7 3 3 {  } 0 . 6 2 2 )$ in Traffic and $21 \\%$ $( 0 . 4 5 5 {  } 0 . 3 5 9 )$ ) in Weather. For the input36-predict-60 setting of ILI, Autoformer makes $43 \\%$ $4 . 8 8 2 {  } 2 . 7 7 0$ ) MSE reduction. Overall, Autoformer yields a $38 \\%$ averaged MSE reduction among above settings. Note that Autoformer still provides remarkable improvements in the Exchange dataset that is without obvious periodicity. See supplementary materials for detailed showcases. Besides, we can also find that the performance of Autoformer changes quite steadily as the prediction length $O$ increases. It means that Autoformer retains better long-term robustness, which is meaningful for real-world practical applications, such as weather early warning and long-term energy consumption planning. ",
+        "bbox": [
+            173,
+            375,
+            826,
+            542
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Univariate results We list the univariate results of two typical datasets in Table 2. Under the comparison with extensive baselines, our Autoformer still achieves state-of-the-art performance for the long-term forecasting tasks. In particular, for the input-96-predict-336 setting, our model achieves $14 \\%$ $0 . 1 8 0 {  } 0 . 1 4 5$ MSE reduction on the ETT dataset with obvious periodicity. For the Exchange dataset without obvious periodicity, Autoformer surpasses other baselines by $17 \\%$ $( 0 . 6 1 1 {  } 0 . 5 0 8 )$ and shows greater long-term forecasting capacity. Also, we find that ARIMA [1] performs best in the input-96-predict-96 setting of the Exchange dataset but fails in the long-term setting. This situation of ARIMA can be benefited from its inherent capacity for non-stationary economic data but is limited by the intricate temporal patterns of real-world series. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 Ablation studies ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            693,
+            326,
+            707
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f3267c9ffb0709b04d20d44b69cea4a5d6f80e9e1ad7cf53d50e148f69ec1a8b.jpg",
+        "table_caption": [
+            "Table 3: Ablation of decomposition in multivariate ETT with MSE metric. Ours adopts our progressive architecture into other models. Sep employs two models to forecast pre-decomposed seasonal and trend-cyclical components separately. Promotion is the MSE reduction compared to Origin. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Input-96</td><td colspan=\"3\">Transformer[35]</td><td colspan=\"3\">Informer[41]</td><td colspan=\"3\">LogTrans[17]</td><td colspan=\"3\">Reformer[20]</td><td colspan=\"2\">Promotion</td></tr><tr><td>Predict-O| Origin</td><td>Sep</td><td>Ours</td><td>Origin</td><td>Sep</td><td>Ours</td><td>Origin</td><td>Sep</td><td>Ours</td><td></td><td>Origin Sep</td><td>Ours</td><td>Sep</td><td>Ours</td></tr><tr><td>96</td><td>0.604</td><td>0.311</td><td>0.204</td><td>0.365</td><td>0.490</td><td>0.354</td><td>0.768</td><td>0.862</td><td>0.231</td><td>0.658</td><td>0.445</td><td>0.218</td><td>0.069</td><td>0.347</td></tr><tr><td>192</td><td></td><td>1.060 0.760（</td><td>0.266</td><td>0.533</td><td>0.658</td><td>0.432</td><td>0.989</td><td>0.533</td><td>0.378</td><td>1.078</td><td></td><td>0.510 0.336</td><td></td><td>0.300 0.562</td></tr><tr><td>336</td><td>1.413</td><td>0.665</td><td>0.375</td><td>1.363</td><td>1.469</td><td>0.481</td><td>1.334</td><td>0.762</td><td>0.362</td><td>1.549</td><td>1.028</td><td>0.366</td><td>0.434</td><td>1.019</td></tr><tr><td>720</td><td>2.672</td><td>3.200</td><td>0.537</td><td>3.379</td><td>2.766</td><td>0.822</td><td>3.048</td><td>2.601</td><td>0.539</td><td>2.631</td><td>2.845</td><td>0.502</td><td></td><td>0.079 2.332</td></tr></table>",
+        "bbox": [
+            173,
+            771,
+            825,
+            868
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Decomposition architecture With our proposed progressive decomposition architecture, other models can gain consistent promotion, especially as the prediction length $O$ increases (Table 3). This verifies that our method can generalize to other models and release the capacity of other dependencies learning mechanisms, alleviate the distraction caused by intricate patterns. Besides, our architecture outperforms the pre-processing, although the latter employs a bigger model and more parameters. Especially, pre-decomposing may even bring negative effect because it neglects the interaction of components during long-term future, such as Transformer [35] predict-720, Informer [41] predict-336. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            825,
+            161
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Auto-Correlation vs. self-attention family As shown in Table 4, our proposed Auto-Correlation achieves the best performance under various input- ${ \\mathbf { \\nabla } } J$ -predict- $O$ settings, which verifies the effectiveness of series-wise connections comparing to point-wise self-attentions (Figure 3). Furthermore, we can also observe that Auto-Correlation is memory efficiency from the last column of Table 4, which can be used in long sequence forecasting, such as input-336-predict-1440. ",
+        "bbox": [
+            174,
+            174,
+            825,
+            244
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a1c7a7b0f3818ccf9690fa4d7b20f84d23baf3a2cad4a8205686e2fe885f2ed.jpg",
+        "table_caption": [
+            "Table 4: Comparison of Auto-Correlation and self-attention in the multivariate ETT. We replace the Auto-Correlation in Autoformer with different self-attentions. The “-” indicates the out-of-memory. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\">Input Length I Prediction Length O</td><td colspan=\"3\">96</td><td colspan=\"3\">192</td><td colspan=\"3\">336</td></tr><tr><td>336</td><td>720</td><td>1440</td><td>336</td><td>720</td><td>1440</td><td>336</td><td>720</td><td>1440</td></tr><tr><td>Auto- Correlation</td><td>MSE MAE</td><td>0.339 0.372</td><td>0.422 0.419</td><td>0.555 0.496</td><td>0.355 0.392</td><td>0.429 0.430</td><td>0.503 0.484</td><td>0.361 0.406</td><td>0.425 0.440</td><td>0.574 0.534</td></tr><tr><td>Full Attention[35]</td><td>MSE MAE</td><td>0.375 0.425</td><td>0.537 0.502</td><td>0.667 0.589</td><td>0.450 0.470</td><td>0.554 0.533</td><td>- -</td><td>0.501 0.485</td><td>0.647 0.491</td><td>1 =</td></tr><tr><td>LogSparse Attention[20]</td><td>MSE MAE</td><td>0.362 0.413</td><td>0.539 0.522</td><td>0.582 0.529</td><td>0.420 0.450</td><td>0.552 0.513</td><td>0.958 0.736</td><td>0.474 0.474</td><td>0.601 0.524</td><td>- =</td></tr><tr><td>LSH Attention[17]</td><td>MSE MAE</td><td>0.366 0.404</td><td>0.502 0.475</td><td>0.663 0.567</td><td>0.407 0.421</td><td>0.636 0.571</td><td>1.069 0.756</td><td>0.442 0.476</td><td>0.615 0.532</td><td>1 -</td></tr><tr><td>ProbSparse Attention[41]</td><td>MSE MAE</td><td>0.481 0.472</td><td>0.822 0.559</td><td>0.715 0.586</td><td>0.404 0.425</td><td>1.148 0.654</td><td>0.732 0.602</td><td>0.417 0.434</td><td>0.631 0.528</td><td>1.133 0.691</td></tr></table>",
+        "bbox": [
+            173,
+            290,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 Model Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            506,
+            320,
+            521
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Time series decomposition As shown in Figure 4, without our series decomposition block, the forecasting model cannot capture the increasing trend and peaks of the seasonal part. By adding the series decomposition blocks, Autoformer can aggregate and refine the trend-cyclical part from series progressively. This design also facilitates the learning of the seasonal part, especially the peaks and troughs. This verifies the necessity of our proposed progressive decomposition architecture. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/226af42008cf294d2ad17509100d9847b1a2e1adc1ae3b98a6fc44cc7d5195f9.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of learned seasonal gradually add the decomposition blocks in $\\mathcal { X } _ { \\mathrm { d e } } ^ { M }$ and trend-cyclical der from left to rig $\\mathcal { T } _ { \\mathrm { d e } } ^ { M }$ of the last decoder layer. Wehis case is from ETT dataset under input-96-predict-720 setting. For clearness, we add the linear growth to raw data additionally. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            611,
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+            733
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dependencies learning The marked time delay sizes in Figure 5(a) indicate the most likely periods. Our learned periodicity can guide the model to aggregate the sub-series from the same or neighbor phase of periods by $\\mathrm { R o l l } ( \\mathcal { X } , \\tau _ { i } )$ , $i \\in \\{ 1 , \\cdots , 6 \\}$ . For the last time step (declining stage), AutoCorrelation fully utilizes all similar sub-series without omissions or errors compared to self-attentions. This verifies that Autoformer can discover the relevant information more sufficiently and precisely. ",
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Complex seasonality modeling As shown in Figure 6, the lags that Autoformer learns from deep representations can indicate the real seasonality of raw series. For example, the learned lags of the daily recorded Exchange dataset present the monthly, quarterly and yearly periods (Figure 6 (b)). For the hourly recorded Traffic dataset (Figure 6 (c)), the learned lags show the intervals as 24-hours and 168-hours, which match the daily and weekly periods of real-world scenarios. These results show that Autoformer can capture the complex seasonalities of real-world series from deep representations and further provide a human-interpretable prediction. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "image",
+        "img_path": "images/1f8ff11071d73f33cde0d525eb867796c34772cd0e34d9f2c2266d99a710dd8f.jpg",
+        "image_caption": [
+            "Figure 5: Visualization of learned dependencies. For clearness, we select the top-6 time delay sizes $\\tau _ { 1 } , \\cdots , \\tau _ { 6 }$ of Auto-Correlation and mark them in raw series (red lines). For self-attentions, top-6 similar points with respect to the last time step (red stars) are also marked by orange points. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/487dd7c7362dd1bad3b32de704db93d0e84eac245cf0c6717299a42cde5d4a6b.jpg",
+        "image_caption": [
+            "Figure 6: Statistics of learned lags. For each time series in the test set, we count the top 10 lags learned by decoder for the input-96-predict-336 task. Figure (a)-(d) are the density histograms. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            261,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Efficiency analysis We compare the running memory and time among Auto-Correlation-based and self-attention-based models (Figure 7) during the training phase. The proposed Autoformer shows $\\mathcal { O } ( L \\log L )$ complexity in both memory and time and achieves better long-term sequences efficiency. ",
+        "bbox": [
+            174,
+            503,
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+            546
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/9945f941aa3e41c082433f9479d4023d07440ec79228f1bcc87357764c332377.jpg",
+        "image_caption": [
+            "Figure 7: Efficiency Analysis. For memory, we replace Auto-Correlation with self-attention family in Autoformer and record the memory with input 96. For running time, we run the Auto-Correlation or self-attentions $1 0 ^ { 3 }$ times to get the execution time per step. The output length increases exponentially. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            768,
+            307,
+            786
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper studies the long-term forecasting problem of time series, which is a pressing demand for real-world applications. However, the intricate temporal patterns prevent the model from learning reliable dependencies. We propose the Autoformer as a decomposition architecture by embedding the series decomposition block as an inner operator, which can progressively aggregate the longterm trend part from intermediate prediction. Besides, we design an efficient Auto-Correlation mechanism to conduct dependencies discovery and information aggregation at the series level, which contrasts clearly from the previous self-attention family. Autoformer can naturally achieve $\\mathcal { O } ( L \\log L )$ complexity and yield consistent state-of-the-art performance in extensive real-world datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            553,
+            107
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This work was supported by the National Natural Science Foundation of China under Grants 62022050 and 62021002, Beijing Nova Program under Grant Z201100006820041, China’s Ministry of Industry and Information Technology, the MOE Innovation Plan and the BNRist Innovation Fund. ",
+        "bbox": [
+            174,
+            119,
+            826,
+            162
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            180,
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+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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+        "bbox": [
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+]

parse/train/S1Dh8Tg0-/S1Dh8Tg0-.md

@@ -0,0 +1,318 @@
+# FIX YOUR CLASSIFIER: THE MARGINAL VALUE OFTRAINING THE LAST WEIGHT LAYER
+
+Elad Hoffer, Itay Hubara, Daniel Soudry   
+Department of Electrical Engineering   
+Technion   
+Haifa, 320003, Israel   
+elad.hoffer, itay.hubara, daniel.soudry@gmail.com
+
+# ABSTRACT
+
+Neural networks are commonly used as models for classification for a wide variety of tasks. Typically, a learned affine transformation is placed at the end of such models, yielding a per-class value used for classification. This classifier can have a vast number of parameters, which grows linearly with the number of possible classes, thus requiring increasingly more resources.
+
+In this work we argue that this classifier can be fixed, up to a global scale constant, with little or no loss of accuracy for most tasks, allowing memory and computational benefits. Moreover, we show that by initializing the classifier with a Hadamard matrix we can speed up inference as well. We discuss the implications for current understanding of neural network models.
+
+# 1 INTRODUCTION
+
+Deep neural network have become a widely used model for machine learning, achieving state-ofthe-art results on many tasks. The most common task these models are used for is to perform classification, as in the case of convolutional neural networks (CNNs) used to classify images to a semantic category. CNN models are currently considered the standard for visual tasks, allowing far better accuracy than preceding approaches (Krizhevsky et al., 2012; He et al., 2016; Szegedy et al., 2015).
+
+Training NN models and using them for inference requires large amounts of memory and computational resources, thus, extensive amount of research has been done lately to reduce the size of networks. Han et al. (2015) used weight sharing and specification, Micikevicius et al. (2017) used mixed precision to reduce the size of the neural networks by half. Tai et al. (2015) and Jaderberg et al. (2014) used low rank approximations to speed up NNs.
+
+Hubara et al. (2016b), Li et al. (2016) and Zhou et al. (2016), used a more aggressive approach, in which weights, activations and gradients were quantized to further reduce computation during training. Although aggressive quantization benefits from smaller model size, the extreme compression rate comes with a loss of accuracy.
+
+Past work noted the fact that predefined (Park & Sandberg, 1991) and random (Huang et al., 2006) projections can be used together with a learned affine transformation to achieve competitive results on several tasks. In this study suggest the reversed proposal - that common NN models used can learn useful representation even without modifying the final output layer, which often holds a large number of parameters that grows linearly with number of classes.
+
+# 1.1 CLASSIFIERS IN CONVOLUTIONAL NEURAL NETWORKS
+
+Convolutional neural networks (CNNs) are commonly used to solve a variety of spatial and temporal tasks. CNNs are usually composed of a stack of convolutional parameterized layers, spatial pooling layers and fully connected layers, separated by non-linear activation functions. Earlier architectures of CNNs (LeCun et al., 1998; Krizhevsky et al., 2012) used a set of fully-connected layers at later stage of the network, presumably to allow classification based on global features of an image. The final classifier can also be replaced with a convolutional layer with output feature maps matching the number of classes, as demonstrated by Springenberg et al. (2014).
+
+Despite the enormous number of trainable parameters these layers added to the model, they are known to have a rather marginal impact on the final performance of the network (Zeiler & Fergus, 2014) and are easily compressed and reduced after a model was trained by simple means such as matrix decomposition and sparsification (Han et al., 2015). Further more, modern architecture choices are characterized with the removal of most of the fully connected layers (Lin et al., 2013; Szegedy et al., 2015; He et al., 2016), which was found to lead to better generalization and overall accuracy, together with a huge decrease in the number of trainable parameters.
+
+Additionally, numerous works showed that CNNs can be trained in a metric learning regime (Bromley et al., 1994; Schroff et al., 2015; Hoffer & Ailon, 2015), where no explicit classification layer was introduced and the objective regarded only distance measures between intermediate representations. Hardt & Ma (2017) suggested an all-convolutional network variant, where they kept the original initialization of the classification layer fixed with no negative impact on performance on the Cifar10 dataset. All of these properties provide evidence that fully-connected layers are in fact redundant and play a small role in learning and generalization.
+
+Despite the apparent minor role they play, fully-connected layers are still commonly used as classification layers, transforming from the dimension of network features $N$ to the number of required class categories $C$ . Therefore, each classification model must hold $N \cdot C$ number of trainable parameters that grows in a linear manner with the number of classes. This property still holds when the fully-connected layer is replaced with a convolutional classifier as shown by Springenberg et al. (2014).
+
+In this work we claim that for common use-cases of convolutional network, the parameters used for the final classification transform are completely redundant, and can be replaced with a predetermined linear transform. As we will show for the first time, this property holds even in largescale models and classification tasks, such as recent architectures trained on the ImageNet benchmark (Deng et al., 2009).
+
+The use of a fixed transform can, in many cases, allow a huge decrease in model parameters, and a possible computational benefit. We suggest that existing models can, with no other modification, devoid their classifier weights, which can help the deployment of those models in devices with low computation ability and smaller memory capacity. Moreover, as we keep the classifier fixed, less parameters need to be updated, reducing the communication cost for models deployed in distributed systems. The use of a fixed transform which does not depend on the number classes can allow models to scale to a large number of possible outputs, without a linear cost in the number of parameters. We also suggest that these finding might shed light on the importance of the preceding non-linear layers to learning and generalization.
+
+# 2 USING A FIXED CLASSIFIER
+
+# 2.1 FULLY-CONNECTED CLASSIFIERS
+
+We focus our attention on the final representation obtained by the network (the last hidden layer), before the classifier. We denote these representation as $x = F ( z ; \theta )$ where $F$ is assumed to be a deep neural network with input $z$ and parameters $\theta$ , e.g., a convolutional network, trained by backpropagation.
+
+In common NN models, this representation is followed by an additional affine transformation
+
+$$
+\boldsymbol { y } = \boldsymbol { W } ^ { T } \boldsymbol { x } + \boldsymbol { b }
+$$
+
+where $W$ and $b$ are also trained by back-propagation.
+
+For input $x$ of $N$ length, and $C$ different possible outputs, $W$ is required to be a matrix of $N \times$ $C$ . Training is done using cross-entropy loss, by feeding the network outputs through a softmax activation
+
+$$
+v _ { i } = \frac { e ^ { y _ { i } } } { \sum _ { j } ^ { C } e ^ { y _ { j } } } , i \in \{ 1 , \ldots , C \}
+$$
+
+and reducing the expected negative log likelihood with respect to ground-truth target $t \in \{ 1 , \ldots , C \}$ , by minimizing
+
+$$
+\mathcal { L } ( \boldsymbol { x } , t ) = - \log \boldsymbol { v _ { t } } = - \boldsymbol { w _ { t } } \cdot \boldsymbol { x } - b _ { t } + \log \left( \sum _ { j } ^ { C } e ^ { \boldsymbol { w _ { j } } \cdot \boldsymbol { x } + b _ { j } } \right)
+$$
+
+where $w _ { i }$ is the $i$ -th column of $W$ .
+
+# 2.2 CHOOSING THE PROJECTION MATRIX
+
+To evaluate our conjecture regarding the importance of the final classification transformation, we replaced the trainable parameter matrix $W$ with a fixed orthonormal projection $Q \in \mathbb { R } ^ { N \times C }$ , such that $\forall i \neq j : q _ { i } \cdot q _ { j } = 0$ and $\| q _ { i } \| _ { 2 } = 1$ , where $q _ { i }$ is the ith column of $Q$ . This can be ensured by a simple random sampling and singular-value decomposition
+
+As the rows of classifier weight matrix are fixed with an equally valued $L _ { 2 }$ norm, we find it beneficial to also restrict the representation of $x$ by normalizing it to reside on the $n$ -dimensional sphere
+
+$$
+{ \hat { x } } = { \frac { x } { \| x \| _ { 2 } } }
+$$
+
+This allows faster training and convergence, as the network does not need to account for changes in the scale of its weights.
+
+We now face the problem that $q _ { i } \cdot { \hat { x } }$ is bounded between $- 1$ and 1. This causes convergence issues, as the softmax function is scale sensitive, and the network is affected by the inability to re-scale its input. This is similar to the phenomenon described by Vaswani et al. (2017) with respect to softmax function used for attention mechanisms. In the same spirit, we can amend this issue with a fixed scale $T$ applied to softmax inputs $\begin{array} { r } { f ( y ) = \mathrm { s o f t m a x } ( \frac { 1 } { T } y ) } \end{array}$ , also known as a softmax temperature. However, this introduces an additional hyper-parameter which may differ between networks and datasets. Instead, we suggest to introduce a single scalar parameter $\alpha$ to learn the softmax scale, effectively functioning as an inverse of the softmax temperature $\textstyle { \frac { 1 } { T } }$ .
+
+Using normalized weights and an additional scale coefficient is similar in spirit to weightnormalization (Salimans $\&$ Kingma, 2016), with the difference that we use a single scale for all entries in the weight matrix, in contrast to a scale for each row that Salimans & Kingma (2016) uses.
+
+We keep the additional vector of bias parameters $b \in \mathbb { R } ^ { C }$ , and train using the same negative-loglikelihood criterion. More explicitly, our classifier output is now
+
+$$
+v _ { i } = \frac { e ^ { \alpha q _ { i } \cdot \hat { x } + b _ { i } } } { \sum _ { j } ^ { C } e ^ { \alpha q _ { j } \cdot \hat { x } + b _ { j } } } , ~ i \in \{ 1 , \dots , C \}
+$$
+
+and we minimize the loss:
+
+$$
+\mathcal { L } ( x , t ) = - \alpha q _ { t } \cdot \frac { x } { \| x \| _ { 2 } } + b _ { t } + \log \left( \sum _ { i = 1 } ^ { C } \exp \left( \alpha q _ { i } \cdot \frac { x } { \| x \| _ { 2 } } + b _ { i } \right) \right)
+$$
+
+where we recall $x$ is the final representation obtained by the network for a specific sample, and $t \in \{ 1 , \ldots , C \}$ is the ground-truth label for that sample.
+
+Observing the behavior of the $\alpha$ parameter over time revealed a logarithmic growth depicted in graph 1. Interestingly, this is the same behavior exhibited by the norm of a learned classifier, first described by Hoffer et al. (2017) and linked to the generalization of the network. This was recently explained by the under-review work of Soudry et al. (2018) as convergence to a max margin classifier. We suggest that using a single parameter will enable a simpler examination and possible further exploration of this phenomenon and its implications.
+
+We note that as $- 1 \leq q _ { i } \cdot \hat { x } \leq 1$ , we also found it possible to train the network with a simple cosine angle loss:
+
+$$
+\begin{array} { r } { \mathcal { L } ( \hat { x } , t ) = \left\{ \begin{array} { l l } { q _ { i } \cdot \hat { x } - 1 , } & { \mathrm { i f ~ } i = t , } \\ { q _ { i } \cdot \hat { x } + 1 , } & { \mathrm { o t h e r w i s e . } } \end{array} \right. } \end{array}
+$$
+
+allowing to discard the softmax function and its scale altogether, but resulting in a slight decrease in final validation accuracy compared to original models.
+
+![](images/01404488485590a6b8450c7b405ac6733926c95f6bdd8fcc86d34e0f75348a55.jpg)  
+Figure 1: The softmax scale coefficient $\alpha$ was observed to follow a logarithmic growth over the course of training.
+
+# 2.3 USING A FIXED HADMARD MATRIX
+
+We further suggest the use of a Hadamard matrix (Hedayat et al., 1978) as the final classification transform. Hadamard matrix $H$ is an $n \times n$ matrix, where all of its entries are either $+ 1$ or $- 1$ . Further more, $H$ is orthogonal, such that $H H ^ { T } = n I _ { n }$ where $I _ { n }$ is the identity matrix.
+
+We can use a truncated Hadamard matrix $\hat { H } \in \{ - 1 , 1 \} ^ { C \times N }$ where all $C$ rows are orthogonal as our final classification layer such that
+
+$$
+\boldsymbol { y } = \hat { H } \hat { \boldsymbol { x } } + \boldsymbol { b }
+$$
+
+This usage allows two main benefits:
+
+• A deterministic, low-memory and easily generated matrix that can be used to classify. • Removal of the need to perform a full matrix-matrix multiplication - as multiplying by a Hadamard matrix can be done by simple sign manipulation and addition.
+
+We note that $n$ must be a multiple of 4, but it can be easily truncated to fit normally defined networks.
+
+We also note the similarity of using a Hadamard matrix as a final classifier to methods of weight binarization such as the one suggested by Courbariaux et al. (2015). As the classifier weights are fixed to need only 1-bit precision, it is now possible to focus our attention on the features preceding it.
+
+# 3 EXPERIMENTAL RESULTS
+
+Table 1: Validation accuracy results on learned vs. fixed classifier   
+
+<table><tr><td>Network</td><td>Dataset</td><td>Learned</td><td>Fixed</td><td>#Params</td><td>% Fixed params</td></tr><tr><td>Resnet56 (He et al., 2016)</td><td>Cifar10</td><td>93.03%</td><td>93.14%</td><td>855,770</td><td>0.07%</td></tr><tr><td>DenseNet(k=12)(Huang et al.,2017)</td><td>Cifar100</td><td>77.73%</td><td>77.67%</td><td>800,032</td><td>4.2%</td></tr><tr><td>Resnet50 (He et al.,2016)</td><td>ImageNet</td><td>75.3%</td><td>75.3%</td><td>25,557,032</td><td>8.01%</td></tr><tr><td>DenseNeti69(Huang et al.,2017)</td><td>ImageNet</td><td>76.2%</td><td>76%</td><td>14,149,480</td><td>11.76%</td></tr><tr><td>ShuffleNet(Zhang et al.,2017b)</td><td>ImageNet</td><td>65.9%</td><td>65.4%</td><td>1,826,555</td><td>52.56%</td></tr></table>
+
+# 3.1 CIFAR10/100
+
+We used the well known Cifar10 and Cifar100 datasets by Krizhevsky (2009) as an initial test-bed to explore the idea of a fixed classifier. Cifar10 is an image classification benchmark dataset containing 50, 000 training images and 10, 000 test images. The images are in color and contain $3 2 \times 3 2$ pixels. There are 10 possible classes of various animals and vehicles. Cifar100 holds the same number of images of same size, but contains 100 different classes.
+
+![](images/b790e6c1ea18c50f28412ff764663fcde6fa2026c98f0476f2de5e25afbd9420.jpg)  
+Figure 2: Comparing training and validation error of fixed and learned classifier (ResNet56, Cifar10)
+
+![](images/7a16b5745f40f0ce6805d78f0a07c622f8323a8bec49b7537382ee3993d030f1.jpg)  
+Figure 3: Comparing fixed vs. trained variable scale $\alpha$ (ResNet56, Cifar10)
+
+We trained a residual network of He et al. (2016) on the Cifar10 dataset. We used a network of depth 56 and the same hyper-parameters used in the original work. We compared two variants: the original model with a learned classifier, and our version, where a fixed transformation is used. The results shown in figure 2 demonstrate that although the training error is considerably lower for the network with learned classifier, both models achieve the same classification accuracy on the validation set. Our conjecture is that with our new fixed parameterization, the network can no longer increase the norm of a given sample’s representation - thus learning its label requires more effort. As this may happen for specific seen samples - it affects only training error.
+
+We also compared using a fixed scale variable $\alpha$ at different values vs. a learned parameter. Results for $\alpha = \{ 0 . 1 , 1 , 1 0 \}$ are depicted in figure 3 for both training and validation error. As can be seen, similar validation accuracy can be obtained using a fixed scale value (in this case $\alpha = 1$ or 10 will suffice) at the expense of another hyper-parameter to seek. In all our experiments we opted to train this parameter instead. In all experiments the $\alpha$ scale parameter was regularized with the same weight decay coefficient used on original classifier.
+
+We then followed to train a model on the Cifar100 dataset. We used the DenseNet-BC model of Huang et al. (2017) with depth of 100 layers and $k = 1 2$ . We continued to train according to the original regime and setting described for this network and dataset. Naturally, the higher number of classes caused the number of parameters to grow and encompass about $4 \%$ of the whole model. Validation accuracy for the fixed-classifier model remained equally good as the original model, and we continued to observe the same training curve.
+
+# 3.2 IMAGENET
+
+In order to validate our results on a more challenging dataset, we used the Imagenet dataset introduced by Deng et al. (2009). The Imagenet dataset spans over 1000 visual classes, and over 1.2 million samples. CNNs used to classify Imagenet such as Krizhevsky et al. (2012), He et al. (2016),
+
+Szegedy et al. (2016) usually have a hidden representation leading to the final classifier of at least 1024 dimensions. This architectural choice, together with the large number of classes, causes the size of classifier to exceed millions of parameters and taking a sizable share from the entire model size.
+
+We evaluated our fixed classifier method on Imagenet using Resnet50 by He et al. (2016) with the same training regime and hyper-parameters. By using a fixed classifier, approximately 2-million parameters were removed from the model, accounting for about $8 \%$ of the model parameters. Following the same procedure, we trained a Densenet169 model (Huang et al., 2017) for which a fixed classifier reduced about $1 2 \%$ of the parameters. Similarly to results on Cifar10 dataset, we observed the same convergence speed and approximately the same final accuracy on both the validation and training sets.
+
+Furthermore, we were interested in evaluating more challenging models where the classifier parameters constitutes the majority amount. For this reason we chose the Shufflenet architecture (Zhang et al., 2017b), which was designed to be used in low memory and limited computing platforms. The Shufflenet network contains about 1.8 million parameters, out of which 0.96 million are part of the final classifier. Fixing the classifier resulted with a model with only 0.86 million parameters. This model was trained and found, again, to converge to similar validation accuracy as the original.
+
+Interestingly, this method allowed Imagenet training in an under-specified regime, where there are more training samples than number of parameters. This is an unconventional regime for modern deep networks, which are usually over-specified to have many more parameters than training samples (Zhang et al., 2017a). Moreover, many recent theoretical results related to neural network training (Soudry & Hoffer, 2017; Xie et al., 2016; Safran & Shamir, 2016; Soltanolkotabi et al., 2017; Soudry & Carmon, 2016) and even generalization (Gunasekar et al., 2017; Advani & Saxe, 2017; Wilson et al., 2017) usually assume over-specification.
+
+Table 1 summarizes our fixed-classifier results on convolutional networks, comparing to originally reported results. We offer our drop-in replacement for learned classifier that can be used to train models with fixed classifiers and replicate our results1.
+
+# 3.3 LANGUAGE MODELING
+
+As language modeling requires classification of all possible tokens available in the task vocabulary, we were interested to see if a fixed classifier can be used, possible saving a very large number of trainable parameters (vocabulary size can have tens or even hundreds of thousands of different words). Recent works have already found empirically that using the same weights for both word embedding and classifier can yield equal or better results than using a separate pair of weights (Inan et al., 2016; Press & Wolf, 2017; Vaswani et al., 2017). This is compliant with our findings that the linear classifier is largely redundant. To examine further reduction in the number of parameters, we removed both classifier and embedding weights and replaced them with a fixed transform.
+
+We trained a language model on the WikiText2 dataset described in Merity et al. (2016), using the same setting in Merity et al. (2017). We used a recurrent model with 2-layers of LSTM (Hochreiter & Schmidhuber, 1997) and embedding $^ +$ hidden size of 512. As the vocabulary of WikiText2 holds about $3 3 K$ different words, the expected number of parameters in embedding and classifier is about 34-million. This number makes for about $8 9 \%$ from the $3 8 M$ parameters used for the whole model.
+
+We found that using a random orthogonal transform yielded poor results compared to learned embedding. We suspect that, in oppose to image classification benchmarks, the embedding layer in language models holds information of the words similarities and relations, thus requiring a fine initialization. To test our intuition, we opted to use pre-trained embeddings using word2vec algorithm by Mikolov et al. (2013) or PMI factorization as suggested by Levy & Goldberg (2014). We find that using fixed word2vec embeddings, we achieve much better results. Specifically, we use $8 9 \%$ less parameters than the fully learned model, and obtain only somewhat worse perplexity.
+
+We argue that this implies a required structure in word embedding that stems from semantic relatedness between words and the natural imbalance between classes. However, we suggest that with a much more cost effective ways to train word embeddings (e.g., Mikolov et al. (2013)), we can narrow the gap and avoid their cost when training bigger models.
+
+Table 2: Validation perplexity results   
+
+<table><tr><td>Network</td><td>Dataset</td><td>Learned</td><td>Fixed</td><td>#Params</td><td>% Fixed params</td></tr><tr><td>2-layer LSTM (h=512)</td><td>WikiText-2</td><td>74.1</td><td>81.2</td><td>38,312,446</td><td>88.94%</td></tr></table>
+
+# 4 DISCUSSION
+
+# 4.1 IMPLICATIONS TO FUTURE DNN MODELS AND USE CASES
+
+In the last couple of years a we observe a rapid growth in the number of classes benchmark datasets contain, for example: Cifar100 (Krizhevsky, 2009), ImageNet1K, ImageNet22k (Deng et al., 2009) and language modeling (Merity et al., 2016). Therefore the computational demands of the final classifier will increase as well and should be considered no less than the architecture chosen. We use the work by Sun et al. (2017) as our use case, which introduced JFT-300M - an internal Google dataset with over 18K different classes. Using a Resnet50 (He et al., 2016), with a 2048 sized representation, this led to a model with over 36M parameters. This means that over $6 0 \%$ of the model parameters reside in the final classification layer.
+
+Sun et al. (2017) further describes the difficulty in distributing this amount of parameters between the training servers, and the need to split them between 50 sub-layers. We also note the fact that the training procedure needs to account for synchronization after each parameter update - which must incur a non-trivial overhead.
+
+Our work can help considerably in this kind of scenario - where using a fixed classifier removes the need to do any gradient synchronization for the final layer. Furthermore, using a Hadamard matrix, we can remove the need to save the transformation altogether, and make it more efficient, allowing considerable memory and computational savings.
+
+# 4.2 POSSIBLE CAVEATS
+
+We argue that our method works due to the ability of preceding layers in the network to learn separable representations that are easily classified even when the classifier itself is fixed. This property can be affected when the ratio between learned features and number of classes is small – that is, when $C > N$ . We’ve been experimenting with such cases, for example Imagenet classification $C = 1 0 0 0 ^ { \circ } ,$ ) using mobilenet-0.5 (Howard et al., 2017) where $N = 5 1 2$ , or reduced version of ResNet (He et al., 2016) where $N = 2 5 6$ . In both scenarios, our method converged similarly to a fully learned classifier reaching the same final validation accuracy. This is strengthening our finding, showing that even in cases in which $C > N$ , fixed classifier can provide equally good results.
+
+Another possible issue may appear when the possible classes are highly correlated. As a fixed orthogonal classifier does not account for this kind of correlation, it may prove hard for the network to learn in this case. This may suggest another reason for the difficulties we experienced in training a language model using an orthogonal fixed classifier, as word classes tend to have highly correlated instances.
+
+# 4.3 FUTURE WORK
+
+Understanding that linear classifiers used in NN models are largely redundant allows us to consider new approaches in training and understanding these models.
+
+Recent works (Neyshabur et al., 2017; Bartlett et al., 2017) suggested a connection between generalization capabilities of models and various norm-related quantities of their weights. Such results might be potentially simplified in our model, since we have a single scalar variable (i.e., scale), which seems to be the only relevant parameter in the model (since we normalize the last hidden layer, and fix the last weight layer).
+
+The use of fixed classifiers might be further simplified in Binarized Neural Networks (Hubara et al., 2016a), where the activations and weights are restricted to $\pm 1$ during propagations. In this case the norm of the last hidden layer is constant for all samples (equal to the square root of the hidden layer width). This constant can be absorbed into the scale constant $\alpha$ , and there is no need in a per-sample normalization as in eq. 1.
+
+We also plan to further explore more efficient ways to learn word embedding, where similar redundancy in classifier weights may suggest simpler forms of token representations - such as low-rank or sparse versions, allowing similar benefits to the fixed transformations we suggested.
+
+# 5 CONCLUSION
+
+In this work we suggested removing the parameters from the classification layer used in deep neural networks. We showed empirical results suggesting that keeping the classifier fixed cause little or no decline in classification performance for common balanced datasets such as Cifar and Imagenet, while allowing a noticeable reduction in trainable parameters. We argue that fixing the last layer can reduce the computational complexity for training as well as the communication cost in distributed learning. Furthermore, using a Hadamard matrix as classifier might lead to some computational benefits when properly implemented, and save memory otherwise spent on large amount of transformation coefficients. As datasets tend to become more complex by time (e.g., Cifar100, ImageNet1K, ImageNet22k, JFT-300M, and language modeling) we believe that resource hungry affine transformation should remain fixed during training, at least partially.
+
+We also found that new efficient methods to create pre-defined word embeddings should be explored, as they require huge amount of parameters that can possibly be avoided when learning a new task. Based on these findings, we recommend future research to focus on representations learned by the non-linear part of neural networks - up to the final classifier, as it seems to be highly redundant.
+
+# ACKNOWLEDGMENTS
+
+The research leading to these results has received funding from the Taub Foundation, and the European Research Council under European Unions Horizon 2020 Program, ERC Grant agreement no. 682203 SpeedInfTradeoff.
+
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+        "text": "FIX YOUR CLASSIFIER: THE MARGINAL VALUE OFTRAINING THE LAST WEIGHT LAYER",
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+        "text": "Elad Hoffer, Itay Hubara, Daniel Soudry   \nDepartment of Electrical Engineering   \nTechnion   \nHaifa, 320003, Israel   \nelad.hoffer, itay.hubara, daniel.soudry@gmail.com ",
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+        "text": "ABSTRACT ",
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+        "text": "Neural networks are commonly used as models for classification for a wide variety of tasks. Typically, a learned affine transformation is placed at the end of such models, yielding a per-class value used for classification. This classifier can have a vast number of parameters, which grows linearly with the number of possible classes, thus requiring increasingly more resources. ",
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+        "text": "In this work we argue that this classifier can be fixed, up to a global scale constant, with little or no loss of accuracy for most tasks, allowing memory and computational benefits. Moreover, we show that by initializing the classifier with a Hadamard matrix we can speed up inference as well. We discuss the implications for current understanding of neural network models. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
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+        "text": "Deep neural network have become a widely used model for machine learning, achieving state-ofthe-art results on many tasks. The most common task these models are used for is to perform classification, as in the case of convolutional neural networks (CNNs) used to classify images to a semantic category. CNN models are currently considered the standard for visual tasks, allowing far better accuracy than preceding approaches (Krizhevsky et al., 2012; He et al., 2016; Szegedy et al., 2015). ",
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+        "text": "Training NN models and using them for inference requires large amounts of memory and computational resources, thus, extensive amount of research has been done lately to reduce the size of networks. Han et al. (2015) used weight sharing and specification, Micikevicius et al. (2017) used mixed precision to reduce the size of the neural networks by half. Tai et al. (2015) and Jaderberg et al. (2014) used low rank approximations to speed up NNs. ",
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+    {
+        "type": "text",
+        "text": "Hubara et al. (2016b), Li et al. (2016) and Zhou et al. (2016), used a more aggressive approach, in which weights, activations and gradients were quantized to further reduce computation during training. Although aggressive quantization benefits from smaller model size, the extreme compression rate comes with a loss of accuracy. ",
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+        "text": "Past work noted the fact that predefined (Park & Sandberg, 1991) and random (Huang et al., 2006) projections can be used together with a learned affine transformation to achieve competitive results on several tasks. In this study suggest the reversed proposal - that common NN models used can learn useful representation even without modifying the final output layer, which often holds a large number of parameters that grows linearly with number of classes. ",
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+        "text": "Convolutional neural networks (CNNs) are commonly used to solve a variety of spatial and temporal tasks. CNNs are usually composed of a stack of convolutional parameterized layers, spatial pooling layers and fully connected layers, separated by non-linear activation functions. Earlier architectures of CNNs (LeCun et al., 1998; Krizhevsky et al., 2012) used a set of fully-connected layers at later stage of the network, presumably to allow classification based on global features of an image. The final classifier can also be replaced with a convolutional layer with output feature maps matching the number of classes, as demonstrated by Springenberg et al. (2014). ",
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+        "text": "Despite the enormous number of trainable parameters these layers added to the model, they are known to have a rather marginal impact on the final performance of the network (Zeiler & Fergus, 2014) and are easily compressed and reduced after a model was trained by simple means such as matrix decomposition and sparsification (Han et al., 2015). Further more, modern architecture choices are characterized with the removal of most of the fully connected layers (Lin et al., 2013; Szegedy et al., 2015; He et al., 2016), which was found to lead to better generalization and overall accuracy, together with a huge decrease in the number of trainable parameters. ",
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+    {
+        "type": "text",
+        "text": "Additionally, numerous works showed that CNNs can be trained in a metric learning regime (Bromley et al., 1994; Schroff et al., 2015; Hoffer & Ailon, 2015), where no explicit classification layer was introduced and the objective regarded only distance measures between intermediate representations. Hardt & Ma (2017) suggested an all-convolutional network variant, where they kept the original initialization of the classification layer fixed with no negative impact on performance on the Cifar10 dataset. All of these properties provide evidence that fully-connected layers are in fact redundant and play a small role in learning and generalization. ",
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+    {
+        "type": "text",
+        "text": "Despite the apparent minor role they play, fully-connected layers are still commonly used as classification layers, transforming from the dimension of network features $N$ to the number of required class categories $C$ . Therefore, each classification model must hold $N \\cdot C$ number of trainable parameters that grows in a linear manner with the number of classes. This property still holds when the fully-connected layer is replaced with a convolutional classifier as shown by Springenberg et al. (2014). ",
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+    {
+        "type": "text",
+        "text": "In this work we claim that for common use-cases of convolutional network, the parameters used for the final classification transform are completely redundant, and can be replaced with a predetermined linear transform. As we will show for the first time, this property holds even in largescale models and classification tasks, such as recent architectures trained on the ImageNet benchmark (Deng et al., 2009). ",
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+    {
+        "type": "text",
+        "text": "The use of a fixed transform can, in many cases, allow a huge decrease in model parameters, and a possible computational benefit. We suggest that existing models can, with no other modification, devoid their classifier weights, which can help the deployment of those models in devices with low computation ability and smaller memory capacity. Moreover, as we keep the classifier fixed, less parameters need to be updated, reducing the communication cost for models deployed in distributed systems. The use of a fixed transform which does not depend on the number classes can allow models to scale to a large number of possible outputs, without a linear cost in the number of parameters. We also suggest that these finding might shed light on the importance of the preceding non-linear layers to learning and generalization. ",
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+    {
+        "type": "text",
+        "text": "2 USING A FIXED CLASSIFIER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 FULLY-CONNECTED CLASSIFIERS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We focus our attention on the final representation obtained by the network (the last hidden layer), before the classifier. We denote these representation as $x = F ( z ; \\theta )$ where $F$ is assumed to be a deep neural network with input $z$ and parameters $\\theta$ , e.g., a convolutional network, trained by backpropagation. ",
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+    {
+        "type": "text",
+        "text": "In common NN models, this representation is followed by an additional affine transformation ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/fde10dac5420c3aceecb01820957ab7dbf1fbce49978b2e2a5ea54c5bb9bf8c0.jpg",
+        "text": "$$\n\\boldsymbol { y } = \\boldsymbol { W } ^ { T } \\boldsymbol { x } + \\boldsymbol { b }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $W$ and $b$ are also trained by back-propagation. ",
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+    {
+        "type": "text",
+        "text": "For input $x$ of $N$ length, and $C$ different possible outputs, $W$ is required to be a matrix of $N \\times$ $C$ . Training is done using cross-entropy loss, by feeding the network outputs through a softmax activation ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25c2ab06a67ab444591ab0502e196f5a5aa9d9fcb90b42512d066e7930086ccc.jpg",
+        "text": "$$\nv _ { i } = \\frac { e ^ { y _ { i } } } { \\sum _ { j } ^ { C } e ^ { y _ { j } } } , i \\in \\{ 1 , \\ldots , C \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "and reducing the expected negative log likelihood with respect to ground-truth target $t \\in \\{ 1 , \\ldots , C \\}$ , by minimizing ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b3d14507ef55d95c686fa5ef8ab6b3015a2a093c918b2cbf7797030d80b6899.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { x } , t ) = - \\log \\boldsymbol { v _ { t } } = - \\boldsymbol { w _ { t } } \\cdot \\boldsymbol { x } - b _ { t } + \\log \\left( \\sum _ { j } ^ { C } e ^ { \\boldsymbol { w _ { j } } \\cdot \\boldsymbol { x } + b _ { j } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i }$ is the $i$ -th column of $W$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 CHOOSING THE PROJECTION MATRIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To evaluate our conjecture regarding the importance of the final classification transformation, we replaced the trainable parameter matrix $W$ with a fixed orthonormal projection $Q \\in \\mathbb { R } ^ { N \\times C }$ , such that $\\forall i \\neq j : q _ { i } \\cdot q _ { j } = 0$ and $\\| q _ { i } \\| _ { 2 } = 1$ , where $q _ { i }$ is the ith column of $Q$ . This can be ensured by a simple random sampling and singular-value decomposition ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As the rows of classifier weight matrix are fixed with an equally valued $L _ { 2 }$ norm, we find it beneficial to also restrict the representation of $x$ by normalizing it to reside on the $n$ -dimensional sphere ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e103c5611e81caaa54a0ade79607e528fa17c016724bb5f6d1435fc20ad5a230.jpg",
+        "text": "$$\n{ \\hat { x } } = { \\frac { x } { \\| x \\| _ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This allows faster training and convergence, as the network does not need to account for changes in the scale of its weights. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now face the problem that $q _ { i } \\cdot { \\hat { x } }$ is bounded between $- 1$ and 1. This causes convergence issues, as the softmax function is scale sensitive, and the network is affected by the inability to re-scale its input. This is similar to the phenomenon described by Vaswani et al. (2017) with respect to softmax function used for attention mechanisms. In the same spirit, we can amend this issue with a fixed scale $T$ applied to softmax inputs $\\begin{array} { r } { f ( y ) = \\mathrm { s o f t m a x } ( \\frac { 1 } { T } y ) } \\end{array}$ , also known as a softmax temperature. However, this introduces an additional hyper-parameter which may differ between networks and datasets. Instead, we suggest to introduce a single scalar parameter $\\alpha$ to learn the softmax scale, effectively functioning as an inverse of the softmax temperature $\\textstyle { \\frac { 1 } { T } }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Using normalized weights and an additional scale coefficient is similar in spirit to weightnormalization (Salimans $\\&$ Kingma, 2016), with the difference that we use a single scale for all entries in the weight matrix, in contrast to a scale for each row that Salimans & Kingma (2016) uses. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We keep the additional vector of bias parameters $b \\in \\mathbb { R } ^ { C }$ , and train using the same negative-loglikelihood criterion. More explicitly, our classifier output is now ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2d0fc3e0e32139c343616479290b01b28566acd8510ce605b72ae3e450d250bb.jpg",
+        "text": "$$\nv _ { i } = \\frac { e ^ { \\alpha q _ { i } \\cdot \\hat { x } + b _ { i } } } { \\sum _ { j } ^ { C } e ^ { \\alpha q _ { j } \\cdot \\hat { x } + b _ { j } } } , ~ i \\in \\{ 1 , \\dots , C \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and we minimize the loss: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c80281ba57874c0b4f61a5fd1f30bc5cd5a86deb91d51a7179b7537b88deca4.jpg",
+        "text": "$$\n\\mathcal { L } ( x , t ) = - \\alpha q _ { t } \\cdot \\frac { x } { \\| x \\| _ { 2 } } + b _ { t } + \\log \\left( \\sum _ { i = 1 } ^ { C } \\exp \\left( \\alpha q _ { i } \\cdot \\frac { x } { \\| x \\| _ { 2 } } + b _ { i } \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we recall $x$ is the final representation obtained by the network for a specific sample, and $t \\in \\{ 1 , \\ldots , C \\}$ is the ground-truth label for that sample. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Observing the behavior of the $\\alpha$ parameter over time revealed a logarithmic growth depicted in graph 1. Interestingly, this is the same behavior exhibited by the norm of a learned classifier, first described by Hoffer et al. (2017) and linked to the generalization of the network. This was recently explained by the under-review work of Soudry et al. (2018) as convergence to a max margin classifier. We suggest that using a single parameter will enable a simpler examination and possible further exploration of this phenomenon and its implications. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We note that as $- 1 \\leq q _ { i } \\cdot \\hat { x } \\leq 1$ , we also found it possible to train the network with a simple cosine angle loss: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a0290bbd1b2624d3cd3e25a46ea80b9d96a15c71ea3d42966530e775ea19d37.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( \\hat { x } , t ) = \\left\\{ \\begin{array} { l l } { q _ { i } \\cdot \\hat { x } - 1 , } & { \\mathrm { i f ~ } i = t , } \\\\ { q _ { i } \\cdot \\hat { x } + 1 , } & { \\mathrm { o t h e r w i s e . } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "allowing to discard the softmax function and its scale altogether, but resulting in a slight decrease in final validation accuracy compared to original models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/01404488485590a6b8450c7b405ac6733926c95f6bdd8fcc86d34e0f75348a55.jpg",
+        "image_caption": [
+            "Figure 1: The softmax scale coefficient $\\alpha$ was observed to follow a logarithmic growth over the course of training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 USING A FIXED HADMARD MATRIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We further suggest the use of a Hadamard matrix (Hedayat et al., 1978) as the final classification transform. Hadamard matrix $H$ is an $n \\times n$ matrix, where all of its entries are either $+ 1$ or $- 1$ . Further more, $H$ is orthogonal, such that $H H ^ { T } = n I _ { n }$ where $I _ { n }$ is the identity matrix. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can use a truncated Hadamard matrix $\\hat { H } \\in \\{ - 1 , 1 \\} ^ { C \\times N }$ where all $C$ rows are orthogonal as our final classification layer such that ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d8dbd7fc2d50d61c1820fbb4b56791b2fd1e62a774a7f43dfd922900b49ec79.jpg",
+        "text": "$$\n\\boldsymbol { y } = \\hat { H } \\hat { \\boldsymbol { x } } + \\boldsymbol { b }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This usage allows two main benefits: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• A deterministic, low-memory and easily generated matrix that can be used to classify. • Removal of the need to perform a full matrix-matrix multiplication - as multiplying by a Hadamard matrix can be done by simple sign manipulation and addition. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We note that $n$ must be a multiple of 4, but it can be easily truncated to fit normally defined networks. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We also note the similarity of using a Hadamard matrix as a final classifier to methods of weight binarization such as the one suggested by Courbariaux et al. (2015). As the classifier weights are fixed to need only 1-bit precision, it is now possible to focus our attention on the features preceding it. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/4b3a0814417dd85a8aa427f724a61b8dcd7271a337bf9c656aaa0975d4a69682.jpg",
+        "table_caption": [
+            "Table 1: Validation accuracy results on learned vs. fixed classifier "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Dataset</td><td>Learned</td><td>Fixed</td><td>#Params</td><td>% Fixed params</td></tr><tr><td>Resnet56 (He et al., 2016)</td><td>Cifar10</td><td>93.03%</td><td>93.14%</td><td>855,770</td><td>0.07%</td></tr><tr><td>DenseNet(k=12)(Huang et al.,2017)</td><td>Cifar100</td><td>77.73%</td><td>77.67%</td><td>800,032</td><td>4.2%</td></tr><tr><td>Resnet50 (He et al.,2016)</td><td>ImageNet</td><td>75.3%</td><td>75.3%</td><td>25,557,032</td><td>8.01%</td></tr><tr><td>DenseNeti69(Huang et al.,2017)</td><td>ImageNet</td><td>76.2%</td><td>76%</td><td>14,149,480</td><td>11.76%</td></tr><tr><td>ShuffleNet(Zhang et al.,2017b)</td><td>ImageNet</td><td>65.9%</td><td>65.4%</td><td>1,826,555</td><td>52.56%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.1 CIFAR10/100 ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We used the well known Cifar10 and Cifar100 datasets by Krizhevsky (2009) as an initial test-bed to explore the idea of a fixed classifier. Cifar10 is an image classification benchmark dataset containing 50, 000 training images and 10, 000 test images. The images are in color and contain $3 2 \\times 3 2$ pixels. There are 10 possible classes of various animals and vehicles. Cifar100 holds the same number of images of same size, but contains 100 different classes. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b790e6c1ea18c50f28412ff764663fcde6fa2026c98f0476f2de5e25afbd9420.jpg",
+        "image_caption": [
+            "Figure 2: Comparing training and validation error of fixed and learned classifier (ResNet56, Cifar10) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a16b5745f40f0ce6805d78f0a07c622f8323a8bec49b7537382ee3993d030f1.jpg",
+        "image_caption": [
+            "Figure 3: Comparing fixed vs. trained variable scale $\\alpha$ (ResNet56, Cifar10) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We trained a residual network of He et al. (2016) on the Cifar10 dataset. We used a network of depth 56 and the same hyper-parameters used in the original work. We compared two variants: the original model with a learned classifier, and our version, where a fixed transformation is used. The results shown in figure 2 demonstrate that although the training error is considerably lower for the network with learned classifier, both models achieve the same classification accuracy on the validation set. Our conjecture is that with our new fixed parameterization, the network can no longer increase the norm of a given sample’s representation - thus learning its label requires more effort. As this may happen for specific seen samples - it affects only training error. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We also compared using a fixed scale variable $\\alpha$ at different values vs. a learned parameter. Results for $\\alpha = \\{ 0 . 1 , 1 , 1 0 \\}$ are depicted in figure 3 for both training and validation error. As can be seen, similar validation accuracy can be obtained using a fixed scale value (in this case $\\alpha = 1$ or 10 will suffice) at the expense of another hyper-parameter to seek. In all our experiments we opted to train this parameter instead. In all experiments the $\\alpha$ scale parameter was regularized with the same weight decay coefficient used on original classifier. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We then followed to train a model on the Cifar100 dataset. We used the DenseNet-BC model of Huang et al. (2017) with depth of 100 layers and $k = 1 2$ . We continued to train according to the original regime and setting described for this network and dataset. Naturally, the higher number of classes caused the number of parameters to grow and encompass about $4 \\%$ of the whole model. Validation accuracy for the fixed-classifier model remained equally good as the original model, and we continued to observe the same training curve. ",
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+        "type": "text",
+        "text": "3.2 IMAGENET ",
+        "text_level": 1,
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+        "text": "In order to validate our results on a more challenging dataset, we used the Imagenet dataset introduced by Deng et al. (2009). The Imagenet dataset spans over 1000 visual classes, and over 1.2 million samples. CNNs used to classify Imagenet such as Krizhevsky et al. (2012), He et al. (2016), ",
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+        "type": "text",
+        "text": "Szegedy et al. (2016) usually have a hidden representation leading to the final classifier of at least 1024 dimensions. This architectural choice, together with the large number of classes, causes the size of classifier to exceed millions of parameters and taking a sizable share from the entire model size. ",
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+        "type": "text",
+        "text": "We evaluated our fixed classifier method on Imagenet using Resnet50 by He et al. (2016) with the same training regime and hyper-parameters. By using a fixed classifier, approximately 2-million parameters were removed from the model, accounting for about $8 \\%$ of the model parameters. Following the same procedure, we trained a Densenet169 model (Huang et al., 2017) for which a fixed classifier reduced about $1 2 \\%$ of the parameters. Similarly to results on Cifar10 dataset, we observed the same convergence speed and approximately the same final accuracy on both the validation and training sets. ",
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+        "type": "text",
+        "text": "Furthermore, we were interested in evaluating more challenging models where the classifier parameters constitutes the majority amount. For this reason we chose the Shufflenet architecture (Zhang et al., 2017b), which was designed to be used in low memory and limited computing platforms. The Shufflenet network contains about 1.8 million parameters, out of which 0.96 million are part of the final classifier. Fixing the classifier resulted with a model with only 0.86 million parameters. This model was trained and found, again, to converge to similar validation accuracy as the original. ",
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+        "type": "text",
+        "text": "Interestingly, this method allowed Imagenet training in an under-specified regime, where there are more training samples than number of parameters. This is an unconventional regime for modern deep networks, which are usually over-specified to have many more parameters than training samples (Zhang et al., 2017a). Moreover, many recent theoretical results related to neural network training (Soudry & Hoffer, 2017; Xie et al., 2016; Safran & Shamir, 2016; Soltanolkotabi et al., 2017; Soudry & Carmon, 2016) and even generalization (Gunasekar et al., 2017; Advani & Saxe, 2017; Wilson et al., 2017) usually assume over-specification. ",
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+        "type": "text",
+        "text": "Table 1 summarizes our fixed-classifier results on convolutional networks, comparing to originally reported results. We offer our drop-in replacement for learned classifier that can be used to train models with fixed classifiers and replicate our results1. ",
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+    {
+        "type": "text",
+        "text": "3.3 LANGUAGE MODELING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As language modeling requires classification of all possible tokens available in the task vocabulary, we were interested to see if a fixed classifier can be used, possible saving a very large number of trainable parameters (vocabulary size can have tens or even hundreds of thousands of different words). Recent works have already found empirically that using the same weights for both word embedding and classifier can yield equal or better results than using a separate pair of weights (Inan et al., 2016; Press & Wolf, 2017; Vaswani et al., 2017). This is compliant with our findings that the linear classifier is largely redundant. To examine further reduction in the number of parameters, we removed both classifier and embedding weights and replaced them with a fixed transform. ",
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+        "type": "text",
+        "text": "We trained a language model on the WikiText2 dataset described in Merity et al. (2016), using the same setting in Merity et al. (2017). We used a recurrent model with 2-layers of LSTM (Hochreiter & Schmidhuber, 1997) and embedding $^ +$ hidden size of 512. As the vocabulary of WikiText2 holds about $3 3 K$ different words, the expected number of parameters in embedding and classifier is about 34-million. This number makes for about $8 9 \\%$ from the $3 8 M$ parameters used for the whole model. ",
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+        "type": "text",
+        "text": "We found that using a random orthogonal transform yielded poor results compared to learned embedding. We suspect that, in oppose to image classification benchmarks, the embedding layer in language models holds information of the words similarities and relations, thus requiring a fine initialization. To test our intuition, we opted to use pre-trained embeddings using word2vec algorithm by Mikolov et al. (2013) or PMI factorization as suggested by Levy & Goldberg (2014). We find that using fixed word2vec embeddings, we achieve much better results. Specifically, we use $8 9 \\%$ less parameters than the fully learned model, and obtain only somewhat worse perplexity. ",
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+        "type": "text",
+        "text": "We argue that this implies a required structure in word embedding that stems from semantic relatedness between words and the natural imbalance between classes. However, we suggest that with a much more cost effective ways to train word embeddings (e.g., Mikolov et al. (2013)), we can narrow the gap and avoid their cost when training bigger models. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "table",
+        "img_path": "images/5afa0bfa88c22165bd2e08524e5d4125ba7283298189734a8f7776811cb801d3.jpg",
+        "table_caption": [
+            "Table 2: Validation perplexity results "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Dataset</td><td>Learned</td><td>Fixed</td><td>#Params</td><td>% Fixed params</td></tr><tr><td>2-layer LSTM (h=512)</td><td>WikiText-2</td><td>74.1</td><td>81.2</td><td>38,312,446</td><td>88.94%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 IMPLICATIONS TO FUTURE DNN MODELS AND USE CASES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the last couple of years a we observe a rapid growth in the number of classes benchmark datasets contain, for example: Cifar100 (Krizhevsky, 2009), ImageNet1K, ImageNet22k (Deng et al., 2009) and language modeling (Merity et al., 2016). Therefore the computational demands of the final classifier will increase as well and should be considered no less than the architecture chosen. We use the work by Sun et al. (2017) as our use case, which introduced JFT-300M - an internal Google dataset with over 18K different classes. Using a Resnet50 (He et al., 2016), with a 2048 sized representation, this led to a model with over 36M parameters. This means that over $6 0 \\%$ of the model parameters reside in the final classification layer. ",
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+    {
+        "type": "text",
+        "text": "Sun et al. (2017) further describes the difficulty in distributing this amount of parameters between the training servers, and the need to split them between 50 sub-layers. We also note the fact that the training procedure needs to account for synchronization after each parameter update - which must incur a non-trivial overhead. ",
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+    {
+        "type": "text",
+        "text": "Our work can help considerably in this kind of scenario - where using a fixed classifier removes the need to do any gradient synchronization for the final layer. Furthermore, using a Hadamard matrix, we can remove the need to save the transformation altogether, and make it more efficient, allowing considerable memory and computational savings. ",
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+    {
+        "type": "text",
+        "text": "4.2 POSSIBLE CAVEATS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We argue that our method works due to the ability of preceding layers in the network to learn separable representations that are easily classified even when the classifier itself is fixed. This property can be affected when the ratio between learned features and number of classes is small – that is, when $C > N$ . We’ve been experimenting with such cases, for example Imagenet classification $C = 1 0 0 0 ^ { \\circ } ,$ ) using mobilenet-0.5 (Howard et al., 2017) where $N = 5 1 2$ , or reduced version of ResNet (He et al., 2016) where $N = 2 5 6$ . In both scenarios, our method converged similarly to a fully learned classifier reaching the same final validation accuracy. This is strengthening our finding, showing that even in cases in which $C > N$ , fixed classifier can provide equally good results. ",
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+    {
+        "type": "text",
+        "text": "Another possible issue may appear when the possible classes are highly correlated. As a fixed orthogonal classifier does not account for this kind of correlation, it may prove hard for the network to learn in this case. This may suggest another reason for the difficulties we experienced in training a language model using an orthogonal fixed classifier, as word classes tend to have highly correlated instances. ",
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+    {
+        "type": "text",
+        "text": "4.3 FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Understanding that linear classifiers used in NN models are largely redundant allows us to consider new approaches in training and understanding these models. ",
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+    },
+    {
+        "type": "text",
+        "text": "Recent works (Neyshabur et al., 2017; Bartlett et al., 2017) suggested a connection between generalization capabilities of models and various norm-related quantities of their weights. Such results might be potentially simplified in our model, since we have a single scalar variable (i.e., scale), which seems to be the only relevant parameter in the model (since we normalize the last hidden layer, and fix the last weight layer). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The use of fixed classifiers might be further simplified in Binarized Neural Networks (Hubara et al., 2016a), where the activations and weights are restricted to $\\pm 1$ during propagations. In this case the norm of the last hidden layer is constant for all samples (equal to the square root of the hidden layer width). This constant can be absorbed into the scale constant $\\alpha$ , and there is no need in a per-sample normalization as in eq. 1. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also plan to further explore more efficient ways to learn word embedding, where similar redundancy in classifier weights may suggest simpler forms of token representations - such as low-rank or sparse versions, allowing similar benefits to the fixed transformations we suggested. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work we suggested removing the parameters from the classification layer used in deep neural networks. We showed empirical results suggesting that keeping the classifier fixed cause little or no decline in classification performance for common balanced datasets such as Cifar and Imagenet, while allowing a noticeable reduction in trainable parameters. We argue that fixing the last layer can reduce the computational complexity for training as well as the communication cost in distributed learning. Furthermore, using a Hadamard matrix as classifier might lead to some computational benefits when properly implemented, and save memory otherwise spent on large amount of transformation coefficients. As datasets tend to become more complex by time (e.g., Cifar100, ImageNet1K, ImageNet22k, JFT-300M, and language modeling) we believe that resource hungry affine transformation should remain fixed during training, at least partially. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We also found that new efficient methods to create pre-defined word embeddings should be explored, as they require huge amount of parameters that can possibly be avoided when learning a new task. Based on these findings, we recommend future research to focus on representations learned by the non-linear part of neural networks - up to the final classifier, as it seems to be highly redundant. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The research leading to these results has received funding from the Taub Foundation, and the European Research Council under European Unions Horizon 2020 Program, ERC Grant agreement no. 682203 SpeedInfTradeoff. ",
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parse/train/rkgpv2VFvr/rkgpv2VFvr.md

@@ -0,0 +1,498 @@
+# SHARING KNOWLEDGE IN MULTI-TASK DEEP REINFORCEMENT LEARNING
+
+Carlo D’Eramo & Davide Tateo   
+Department of Computer Science   
+TU Darmstadt, IAS   
+Hochschulstraße 10, 64289, Darmstadt, Germany   
+{carlo.deramo,davide.tateo}@tu-darmstadt.de   
+Andrea Bonarini & Marcello Restelli   
+Politecnico di Milano, DEIB   
+Piazza Leonardo da Vinci 32, 20133, Milano   
+{andrea.bonarini,marcello.restelli}@polimi.it
+
+# Jan Peters
+
+TU Darmstadt, IAS Hochschulstraße 10, 64289, Darmstadt, Germany Max Planck Institute for Intelligent Systems Max-Planck-Ring 4, 72076, Tübingen, Germany jan.peters@tu-darmstadt.de
+
+# ABSTRACT
+
+We study the benefit of sharing representations among tasks to enable the effective use of deep neural networks in Multi-Task Reinforcement Learning. We leverage the assumption that learning from different tasks, sharing common properties, is helpful to generalize the knowledge of them resulting in a more effective feature extraction compared to learning a single task. Intuitively, the resulting set of features offers performance benefits when used by Reinforcement Learning algorithms. We prove this by providing theoretical guarantees that highlight the conditions for which is convenient to share representations among tasks, extending the wellknown finite-time bounds of Approximate Value-Iteration to the multi-task setting. In addition, we complement our analysis by proposing multi-task extensions of three Reinforcement Learning algorithms that we empirically evaluate on widely used Reinforcement Learning benchmarks showing significant improvements over the single-task counterparts in terms of sample efficiency and performance.
+
+# 1 INTRODUCTION
+
+Multi-Task Learning (MTL) ambitiously aims to learn multiple tasks jointly instead of learning them separately, leveraging the assumption that the considered tasks have common properties which can be exploited by Machine Learning (ML) models to generalize the learning of each of them. For instance, the features extracted in the hidden layers of a neural network trained on multiple tasks have the advantage of being a general representation of structures common to each other. This translates into an effective way of learning multiple tasks at the same time, but it can also improve the learning of each individual task compared to learning them separately (Caruana, 1997). Furthermore, the learned representation can be used to perform Transfer Learning (TL), i.e. using it as a preliminary knowledge to learn a new similar task resulting in a more effective and faster learning than learning the new task from scratch (Baxter, 2000; Thrun & Pratt, 2012).
+
+The same benefits of extraction and exploitation of common features among the tasks achieved in MTL, can be obtained in Multi-Task Reinforcement Learning (MTRL) when training a single agent on multiple Reinforcement Learning (RL) problems with common structures (Taylor & Stone, 2009; Lazaric, 2012). In particular, in MTRL an agent can be trained on multiple tasks in the same domain, e.g. riding a bicycle or cycling while going towards a goal, or on different but similar domains, e.g. balancing a pendulum or balancing a double pendulum1. Considering recent advances in Deep Reinforcement Learning (DRL) and the resulting increase in the complexity of experimental benchmarks, the use of Deep Learning (DL) models, e.g. deep neural networks, has become a popular and effective way to extract common features among tasks in MTRL algorithms (Rusu et al., 2015; Liu et al., 2016; Higgins et al., 2017). However, despite the high representational capacity of DL models, the extraction of good features remains challenging. For instance, the performance of the learning process can degrade when unrelated tasks are used together (Caruana, 1997; Baxter, 2000); another detrimental issue may occur when the training of a single model is not balanced properly among multiple tasks (Hessel et al., 2018).
+
+Recent developments in MTRL achieve significant results in feature extraction by means of algorithms specifically developed to address these issues. While some of these works rely on a single deep neural network to model the multi-task agent (Liu et al., 2016; Yang et al., 2017; Hessel et al., 2018; Wulfmeier et al., 2019), others use multiple deep neural networks, e.g. one for each task and another for the multi-task agent (Rusu et al., 2015; Parisotto et al., 2015; Higgins et al., 2017; Teh et al., 2017). Intuitively, achieving good results in MTRL with a single deep neural network is more desirable than using many of them, since the training time is likely much less and the whole architecture is easier to implement. In this paper we study the benefits of shared representations among tasks. We theoretically motivate the intuitive effectiveness of our method, deriving theoretical guarantees that exploit the theoretical framework provided by Maurer et al. (2016), in which the authors present upper bounds on the quality of learning in MTL when extracting features for multiple tasks in a single shared representation. The significancy of this result is that the cost of learning the shared representation decreases with a factor $\mathcal { O } ( { } ^ { 1 / \sqrt { T } } )$ , where $T$ is the number of tasks for many function approximator hypothesis classes. The main contribution of this work is twofold.
+
+1. We derive upper confidence bounds for Approximate Value-Iteration (AVI) and Approximate Policy-Iteration $( \mathsf { A P I } ) ^ { 2 }$ (Farahmand, 2011) in the MTRL setting, and we extend the approximation error bounds in Maurer et al. (2016) to the case of multiple tasks with different dimensionalities. Then, we show how to combine these results resulting in, to the best of our knowledge, the first proposed extension of the finite-time bounds of AVI/API to MTRL. Despite being an extension of previous works, we derive these results to justify our approach showing how the error propagation in AVI/API can theoretically benefit from learning multiple tasks jointly.
+
+2. We leverage these results proposing a neural network architecture, for which these bounds hold with minor assumptions, that allow us to learn multiple tasks with a single regressor extracting a common representation. We show an empirical evidence of the consequence of our bounds by means of a variant of Fitted $Q$ -Iteration (FQI) (Ernst et al., 2005), based on our shared network and for which our bounds apply, that we call Multi Fitted $Q$ -Iteration (MFQI). Then, we perform an empirical evaluation in challenging RL problems proposing multitask variants of the Deep $Q$ -Network (DQN) (Mnih et al., 2015) and Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2015) algorithms. These algorithms are practical implementations of the more general AVI/API framework, designed to solve complex problems. In this case, the bounds apply to these algorithms only with some assumptions, e.g. stationary sampling distribution. The outcome of the empirical analysis joins the theoretical results, showing significant performance improvements compared to the singletask version of the algorithms in various RL problems, including several MuJoCo (Todorov et al., 2012) domains.
+
+# 2 PRELIMINARIES
+
+Let $B ( \mathcal { X } )$ be the space of bounded measurable functions w.r.t. the $\sigma$ -algebra $\sigma _ { \mathcal { X } }$ , and similarly $B ( \mathcal { X } , L )$ be the same bounded by $L < \infty$ .
+
+A Markov Decision Process (MDP) is defined as a 5-tuple $\mathcal { M } = < { \mathcal { S } } , { \mathcal { A } } , { \mathcal { P } } , { \mathcal { R } } , \gamma >$ , where $s$ is the state space, $\mathcal { A }$ is the action space, ${ \mathcal { P } } : { \mathcal { S } } \times { \mathcal { A } }  { \mathcal { S } }$ is the transition distribution where $\mathcal { P } ( s ^ { \prime } | s , a )$
+
+is the probability of reaching state $s ^ { \prime }$ when performing action $a$ in state $s$ , $\mathcal { R } : \mathcal { S } \times \mathcal { A } \times \mathcal { S } $ $\mathbb { R }$ is the reward function, and $\gamma \in \mathsf { \Gamma } ( 0 , 1 ]$ is the discount factor. A deterministic policy $\pi$ maps, for each state, the action to perform: $\dot { \pi } : \mathcal { S }  \mathcal { A }$ . Given a policy $\pi$ , the value of an action $a$ in a state $s$ represents the expected discounted cumulative reward obtained by performing $a$ in $s$ and following $\pi$ thereafter: $\begin{array} { r } { Q ^ { \pi } ( s , a ) \triangleq \mathbb { E } [ \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } r _ { i + k + 1 } \vert s _ { i } = s , a _ { i } = a , \pi ] } \end{array}$ , where $r _ { i + 1 }$ is the reward obtained after the $i$ -th transition. The expected discounted cumulative reward is maximized by following the optimal policy $\pi ^ { * }$ which is the one that determines the optimal action values, i.e., the ones that satisfy the Bellman optimality equation (Bellman, 1954): $Q ^ { \ast } ( s , a ) \ { \triangleq }$ $\begin{array} { r } { \int _ { \mathcal { S } } \mathcal { P } ( s ^ { \prime } | s , a ) \left[ \mathcal { R } ( s , a , s ^ { \prime } ) + \gamma \operatorname* { m i x } _ { a ^ { \prime } } Q ^ { * } ( s ^ { \prime } , a ^ { \prime } ) \right] \dot { d s ^ { \prime } } . } \end{array}$ . The solution of the Bellman optimality equation is the fixed point of the optimal Bellman operator $T ^ { * } : B ( S \times { \mathcal { A } } ) \to B ( S \times { \mathcal { A } } )$ defined as $\begin{array} { r } { ( \mathcal { T } ^ { * } Q ) ( s , a ) \triangleq \int _ { S } \mathcal { P } ( s ^ { \prime } | s , a ) [ \mathcal { R } ( s , a , s ^ { \prime } ) + \gamma \operatorname* { m a x } _ { a ^ { \prime } } Q ( s ^ { \prime } , a ^ { \prime } ) ] d s ^ { \prime } . } \end{array}$ In the MTRL setting, there are multiple MDPs $\mathcal { M } ^ { ( t ) } = < S ^ { ( t ) } , \mathcal { A } ^ { ( t ) } , \mathcal { P } ^ { ( t ) } , \mathcal { R } ^ { ( t ) } , \gamma ^ { ( t ) } >$ where $t \in \{ 1 , \ldots , T \}$ and $T$ is the number of MDPfunction $\mathcal { M } ^ { ( t ) }$ $\pi _ { t } : { \mathcal { S } } ^ { ( t ) }  A ^ { ( t ) }$ induces an action-valuethis setting, the goal is to $\begin{array} { r } { Q _ { t } ^ { \pi _ { t } } \big ( s ^ { ( t ) } , a ^ { ( t ) } \big ) = \mathbb E [ \sum _ { k = 0 } ^ { \infty } \gamma ^ { k } r _ { i \downarrow + k + 1 } ^ { ( t ) } | s _ { i } = s ^ { ( t ) } , a _ { i } = a ^ { ( t ) } , \pi _ { t } ] . } \end{array}$ maximize the sum of the expected cumulative discounted reward of each task.
+
+In our theoretical analysis of the MTRL problem, the complexity of representation plays a central role. As done in Maurer et al. (2016), we consider the Gaussian complexity, a variant of the well-known Rademacher complexity, to measure the complexity of the representation. Given a set $\bar { \mathbf { X } } \in \mathcal { X } ^ { T n }$ of $n$ input samples for each task $t \in \{ 1 , \ldots , T \}$ , and a class $\mathcal { H }$ composed of $k \in \{ 1 , \ldots , K \}$ functions, the Gaussian complexity of a random set $\mathcal { H } ( \bar { \mathbf { X } } ) = \{ ( h _ { k } ( X _ { t i } ) ) : h \in \mathcal { H } \} \subseteq \mathbb { R } ^ { K T n }$ is defined as follows:
+
+$$
+G ( \mathcal { H } ( \bar { \mathbf { X } } ) ) = \mathbb { E } \left[ \operatorname* { s u p } _ { h \in \mathcal { H } } \sum _ { t k i } \gamma _ { t k i } h _ { k } ( X _ { t i } ) \bigg | X _ { t i } \right] ,
+$$
+
+where $\gamma _ { t k i }$ are independent standard normal variables. We also need to define the following quantity, taken from Maurer (2016): let $\gamma$ be a vector of $m$ random standard normal variables, and $f \in { \mathcal { F } }$ : $Y  \mathbb { R } ^ { m }$ , with $Y \subseteq \mathbb { R } ^ { n }$ , we define
+
+$$
+O ( \mathcal { F } ) = \operatorname* { s u p } _ { \substack { y , y ^ { \prime } \in Y , y \neq y ^ { \prime } } } \mathbb { E } \left[ \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { \langle \gamma , f ( y ) - f ( y ^ { \prime } ) \rangle } { \| y - y ^ { \prime } \| } \right] .
+$$
+
+Equation 2 can be viewed as a Gaussian average of Lipschitz quotients, and appears in the bounds provided in this work. Finally, we define $L ( \mathcal { F } )$ as the upper bound of the Lipschitz constant of all the functions $f$ in the function class $\mathcal { F }$ .
+
+# 3 THEORETICAL ANALYSIS
+
+The following theoretical study starts from the derivation of theoretical guarantees for MTRL in the AVI framework, extending the results of Farahmand (2011) in the MTRL scenario. Then, to bound the approximation error term in the AVI bound, we extend the result described in Maurer (2006) to MTRL. As we discuss, the resulting bounds described in this section clearly show the benefit of sharing representation in MTRL. To the best of our knowledge, this is the first general result for MTRL; previous works have focused on finite MDPs (Brunskill & Li, 2013) or linear models (Lazaric & Restelli, 2011).
+
+# 3.1 MULTI-TASK REPRESENTATION LEARNING
+
+The multi-task representation learning problem consists in learning simultaneously a set of $T$ tasks $\mu _ { t }$ , modeled as probability measures over the space of the possible input-output pairs $( x , y )$ , with $x \in \mathcal { X }$ and $y \in \mathbb { R }$ , being $\mathcal { X }$ the input space. Let $w \in \mathcal { W } : \mathcal { X } \to \mathbb { R } ^ { \bar { J } }$ , $h \in \mathcal { H } : \mathbb { R } ^ { J } \overset { } { \to } \bar { \mathbb { R } } ^ { K }$ and $f \in \mathcal { F } : \mathbb { R } ^ { K }  \mathbb { R }$ be functions chosen from their respective hypothesis classes. The functions in the hypothesis classes must be Lipschitz continuous functions. Let $\bar { \bf Z } = ( { \bf Z } _ { 1 } , \ldots , { \bf Z } _ { T } )$ be the multi-sample over the set of tasks $\pmb { \mu } = ( \mu _ { 1 } , \ldots , \mu _ { T } )$ , where $\mathbf { Z } _ { t } \ = \ \left( Z _ { t 1 } , \ldots , Z _ { t n } \right) \ \sim \ \mu _ { t } ^ { n }$ and $Z _ { t i } = ( X _ { t i } ^ { - } , Y _ { t i } ) \sim \mu _ { t }$ . We can formalize our regression problem as the following minimization
+
+problem:
+
+$$
+\operatorname* { m i n } \left\{ \frac { 1 } { n T } \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { N } \ell ( f _ { t } ( h ( w _ { t } ( X _ { t i } ) ) ) , Y _ { t i } ) : \mathbf { f } \in \mathcal { F } ^ { T } , h \in \mathcal { H } , \mathbf { w } \in \mathcal { W } ^ { T } \right\} ,
+$$
+
+where we use $\mathbf { f } = ( f _ { 1 } , \ldots , f _ { T } )$ , $\mathbf { w } = ( w _ { 1 } , \hdots , w _ { T } )$ , and define the minimizers of Equation (3) as $\hat { \mathbf { w } }$ , $\hat { h }$ , and $\hat { \bf f }$ . We assume that the loss function $\ell : \mathbb { R } \times \mathbb { R } \to [ 0 , 1 ]$ is 1-Lipschitz in the first argument for every value of the second argument. While this assumption may seem restrictive, the result obtained can be easily scaled to the general case. To use the principal result of this section, for a generic loss function $\ell ^ { \prime }$ , it is possible to use $\ell ( \cdot ) = \ell ^ { \prime } ( \cdot ) / \epsilon _ { \mathrm { m a x } }$ , where $\epsilon _ { \mathrm { m a x } }$ is the maximum value of $\ell ^ { \prime }$ . The expected loss over the tasks, given w, $h$ and f is the task-averaged risk:
+
+$$
+\varepsilon _ { \mathrm { a v g } } ( \mathbf { w } , h , \mathbf { f } ) = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } \left[ \ell ( f _ { t } ( h ( w _ { t } ( X ) ) ) , Y ) \right]
+$$
+
+The minimum task-averaged risk, given the set of tasks $\pmb { \mu }$ and the hypothesis classes $\nu , \varkappa$ and $\mathcal { F }$ is $\varepsilon _ { \mathrm { a v g } } ^ { * }$ , and the corresponding minimizers are $\mathbf { w } ^ { * }$ , $h ^ { * }$ and $\mathbf { f } ^ { * }$ .
+
+# 3.2 MULTI-TASK APPROXIMATE VALUE ITERATION BOUND
+
+We start by considering the bound for the AVI framework which applies for the single-task scenario. Theorem 1. (Theorem 3.4 of Farahmand (2011)) Let K be a positive integer, and Qmax ≤ Rmax1−γ . Then for any sequence $( Q _ { k } ) _ { k = 0 } ^ { K } \subset B ( S \times \mathcal { A } , Q _ { m a x } )$ and the corresponding sequence $( \varepsilon _ { k } ) _ { k = 0 } ^ { K - 1 }$ , where $\varepsilon _ { k } = \lVert Q _ { k + 1 } - T ^ { * } Q _ { k } \rVert _ { \nu } ^ { 2 }$ , we have:
+
+$$
+\Vert Q ^ { * } - Q ^ { \pi _ { K } } \Vert _ { 1 , \rho } \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \left[ \operatorname* { i n f } _ { r \in [ 0 , 1 ] } C _ { { \cal V } I , \rho , \nu } ^ { \frac { 1 } { 2 } } ( K ; r ) \mathcal { E } ^ { \frac { 1 } { 2 } } ( \varepsilon _ { 0 } , . . . , \varepsilon _ { K - 1 } ; r ) + \frac { 2 } { 1 - \gamma } \gamma ^ { K } R _ { m a x } \right] ,
+$$
+
+where
+
+$$
+\begin{array} { l } { { \displaystyle C _ { W , \rho , \nu } ( K ; r ) = \left( \frac { 1 - \gamma } { 2 } \right) ^ { 2 } \operatorname* { s u p } _ { \pi _ { 1 } ^ { \prime } , \ldots , \pi _ { K } ^ { \prime } } \sum _ { k = 0 } ^ { K - 1 } a _ { k } ^ { 2 ( 1 - r ) } \left[ \sum _ { m \ge 0 } \gamma ^ { m } \Big ( c _ { W _ { 1 } , \rho , \nu } ( m , K - k ; \pi _ { K } ^ { \prime } ) \right. } } \\ { { \displaystyle \left. + c _ { W _ { 2 } , \rho , \nu } ( m + 1 ; \pi _ { k + 1 } ^ { \prime } , \ldots , \pi _ { K } ^ { \prime } ) \Big ) \right] ^ { 2 } , } } \end{array}
+$$
+
+with E (ε0, . . . , εK−1; r) = PK−1k=0 , the two coefficients $c _ { V I _ { 1 } , \rho , \nu } , c _ { V I _ { 2 } , \rho , \nu } ,$ , the distributions $\rho$ and $\nu$ , and the series $\alpha _ { k }$ are defined as in Farahmand (2011).
+
+In the multi-task scenario, let the average approximation error across tasks be:
+
+$$
+\varepsilon _ { \mathrm { a v g } , k } \bigl ( \hat { \mathbf { w } } _ { k } , \hat { h } _ { k } , \hat { \mathbf { f } } _ { k } \bigr ) = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \lVert Q _ { t , k + 1 } - T _ { t } ^ { * } Q _ { t , k } \rVert _ { \nu } ^ { 2 } ,
+$$
+
+where $Q _ { t , k + 1 } = \hat { f } _ { t , k } \circ \hat { h } _ { k } \circ \hat { w } _ { t , k }$ , and $\tau _ { t } ^ { * }$ is the optimal Bellman operator of task $t$
+
+In the following, we extend the AVI bound of Theorem 1 to the multi-task scenario, by computing the average loss across tasks and pushing inside the average using Jensen’s inequality.
+
+Theorem 2. Let K be a positive integer, and $\begin{array} { r } { Q _ { m a x } \leq \frac { R _ { m a x } } { 1 - \gamma } } \end{array}$ . Then for any sequence $( Q _ { k } ) _ { k = 0 } ^ { K } \subset B ( S \times$ $A , Q _ { m a x } )$ and the corresponding sequence $\left( \varepsilon _ { a \nu g , k } \right) _ { k = 0 } ^ { K - 1 }$ , where $\varepsilon _ { a \nu g , k } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \lVert Q _ { t , k + 1 } - \mathcal { T } _ { t } ^ { * } Q _ { t , k } \rVert _ { \nu } ^ { 2 } ,$ we have:
+
+$$
+\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \lVert Q _ { t } ^ { * } - Q _ { t } ^ { \pi _ { K } } \rVert _ { 1 , \rho } \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \left[ \operatorname* { i n f } _ { r \in [ 0 , 1 ] } C _ { M } ^ { \frac { 1 } { 2 } } ( K ; r ) \mathcal { E } _ { a v g } ^ { \frac { 1 } { 2 } } ( \varepsilon _ { a v g , 0 } , \dots , \varepsilon _ { a v g , K - 1 } ; r ) + \frac { 2 \gamma ^ { K } R _ { m a x , a v g } } { 1 - \gamma } \right] .
+$$
+
+$$
+\begin{array} { r l } & { v i t h \ E _ { a v g } = \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { a v g , k } , \gamma = \underset { t \in \{ 1 , \ldots , T \} } { \operatorname* { m a x } } \gamma _ { t } , C _ { V I } ^ { \frac { 1 } { 2 } } ( K ; r ) = \underset { t \in \{ 1 , \ldots , T \} } { \operatorname* { m a x } } C _ { V J , \rho , \nu } ^ { \frac { 1 } { 2 } } ( K ; t , r ) , R _ { m a x , a v g } = } \\ & { \frac { 1 } { \bar { r } } \sum _ { t = 1 } ^ { T } R _ { m a x , t } a n d \alpha _ { k } = \left\{ \begin{array} { l l } { \frac { ( 1 - \gamma ) \gamma ^ { K - k - 1 } } { 1 - \gamma ^ { K + 1 } } } & { 0 \leq k < K , } \\ { \frac { ( 1 - \gamma ) \gamma ^ { K } } { 1 - \gamma ^ { K + 1 } } } & { k = K } \end{array} \right. . } \end{array}
+$$
+
+Remarks Theorem 2 retains most of the properties of Theorem 3.4 of Farahmand (2011), except that the regression error in the bound is now task-averaged. Interestingly, the second term of the sum in Equation (8) depends on the average maximum reward for each task. In order to obtain this result we use an overly pessimistic bound on $\gamma$ and the concentrability coefficients, however this approximation is not too loose if the MDPs are sufficiently similar.
+
+# 3.3 MULTI-TASK APPROXIMATION ERROR BOUND
+
+We bound the task-averaged approximation error $\varepsilon _ { \mathrm { a v g } }$ at each AVI iteration $k$ involved in (8) following a derivation similar to the one proposed by Maurer et al. (2016), obtaining:
+
+Theorem 3. Let $\pmb { \mu }$ , W , $\mathcal { H }$ and $\mathcal { F }$ be defined as above and assume $0 \in \mathcal H$ and $f ( 0 ) = 0 , \forall f \in { \mathcal { F } }$ . Then for $\delta > 0$ with probability at least $1 - \delta$ in the draw of $\begin{array} { r } { \bar { \mathbf Z } \sim \prod _ { t = 1 } ^ { T } \mu _ { t } ^ { n } } \end{array}$ we have that
+
+$$
+\begin{array} { r l } & { \varepsilon _ { a v g } \big ( \widehat { \mathbf { w } } , \widehat { h } , \widehat { \mathbf { f } } \big ) \leq L ( \mathcal { F } ) \left( c _ { 1 } \frac { L ( \mathcal { H } ) \operatorname* { s u p } _ { l \in \{ 1 , \ldots , T \} } G ( \mathcal { W } ( \mathbf { X } _ { l } ) ) } { n } + c _ { 2 } \frac { \operatorname* { s u p } _ { \mathbf { w } } \| \mathbf { w } ( \bar { \mathbf { X } } ) \| O ( \mathcal { H } ) } { n T } \right. } \\ & { \qquad \left. + c _ { 3 } \frac { \operatorname* { m i n } _ { p \in P } G ( \mathcal { H } ( p ) ) } { n T } \right) + c _ { 4 } \frac { \operatorname* { s u p } _ { h , \mathbf { w } } \| h ( \mathbf { w } ( \bar { \mathbf { X } } ) ) \| O ( \mathcal { F } ) } { n \sqrt { T } } + \sqrt { \frac { 8 \ln ( \frac { 3 } { \delta } ) } { n T } } + \varepsilon _ { a v g } ^ { * } . } \end{array}
+$$
+
+Remarks The assumptions $0 \in \mathcal H$ and $f ( 0 ) = 0$ for all $f \in { \mathcal { F } }$ are not essential for the proof and are only needed to simplify the result. For reasonable function classes, the Gaussian complexity√ $G ( \mathcal { W } ( \dot { \mathbf { X } } _ { l } ) )$ is $\mathcal { O } ( \sqrt { n } )$ . If $\dot { \operatorname { s u p } } _ { \mathbf { w } } \lVert \mathbf { w } ( \bar { \mathbf { X } } ) \rVert$ and $\mathrm { s u p } _ { h , \mathbf { w } } \| h ( \mathbf { w } ( { \bar { \mathbf { X } } } ) ) \|$ can be uniformly bounded, then they are $\mathcal { O } ( \sqrt { n T } )$ . For some function classes, the Gaussian average of Lipschitz quotients $O ( \cdot )$ can be bounded independently from the number of samples. Given these assumptions, the first and the fourth term of the right hand side of Equation (9), which represent respectively the cost of learning the meta-state space w and the task-specific f mappings, are both ${ \mathcal { O } } ( { } ^ { 1 } / { \sqrt { n } } )$ . The second term represents the cost of learning the multi-task representation $h$ and is $\mathcal { O } ( { } ^ { 1 / \sqrt { n T } } )$ , thus vanishing in the multi-task limit $T \to \infty$ . The third term can be removed if $\forall h \in { \mathcal { H } }$ , $\exists p _ { 0 } \in P : h ( p ) = { \bar { 0 } }$ ; even when this assumption does not hold, this term can be ignored for many classes of interest, e.g. neural networks, as it can be arbitrarily small.
+
+The last term to be bounded in (9) is the minimum average approximation error $\varepsilon _ { \mathrm { a v g } } ^ { * }$ at each AVI iteration . Recalling that the task-averaged approximation error is defined as in (7), applying Theorem 5.3 by Farahmand (2011) we obtain:
+
+Lemma 4. Let $Q _ { t , k } ^ { * } , \forall t \in \{ 1 , \ldots , T \}$ be the minimizers of $\mathbf { \psi } _ { \nu g , k } ^ { \mathrm { ~ c ~ } } , \check { t } _ { k } = \arg \operatorname* { m a x } _ { t \in \{ 1 , \dots , T \} } \| Q _ { t , k + 1 } ^ { * } -$ $\begin{array} { r } { \mathcal { T } _ { t } ^ { * } Q _ { t , k } \| _ { \nu } ^ { 2 } , } \end{array}$ , and $b _ { k , i } = \lVert Q _ { \check { t } _ { k } , i + 1 } - \mathcal { T } _ { \check { t } } ^ { * } Q _ { \check { t } _ { k } , i } \rVert _ { \nu }$ , then:
+
+$$
+\varepsilon _ { a v g , k } ^ { * } \leq \left( \| Q _ { \tilde { t } _ { k } , k + 1 } ^ { * } - ( T _ { \tilde { t } } ^ { * } ) ^ { k + 1 } Q _ { \tilde { t } _ { k } , 0 } \| _ { \nu } + \sum _ { i = 0 } ^ { k - 1 } ( \gamma _ { \tilde { t } _ { k } } C _ { A E } ( \nu ; \tilde { t } _ { k } , P ) ) ^ { i + 1 } b _ { k , k - 1 - i } \right) ^ { 2 } ,
+$$
+
+with $C _ { A E }$ defined as in Farahmand (2011).
+
+Final remarks The bound for MTRL is derived by composing the results in Theorems 2 and 3, and Lemma 4. The results above highlight the advantage of learning a shared representation. The bound in Theorem 2 shows that a small approximation error is critical to improve the convergence towards the optimal action-value function, and the bound in Theorem 3 shows that the cost of learning the shared representation at each AVI iteration is mitigated by using multiple tasks. This is particularly beneficial when the feature representation is complex, e.g. deep neural networks.
+
+# 3.4 DISCUSSION
+
+As stated in the remarks of Equation (9), the benefit of MTRL is evinced by the second component of the bound, i.e. the cost of learning $h$ , which vanishes with the increase of the number of tasks. Obviously, adding more tasks require the shared representation to be large enough to include all of them, undesirably causing the term $\mathrm { s u p } _ { h , \mathbf { w } } \| h ( \mathbf { w } ( \mathbf { \bar { X } } ) ) \|$ in the fourth component of the bound to increase. This introduces a tradeoff between the number of features and number of tasks; however, for a reasonable number of tasks the number of features used in the single-task case is enough to handle them, as we show in some experiments in Section 5. Notably, since the AVI/API framework provided by Farahmand (2011) provides an easy way to include the approximation error of a generic function approximator, it is easy to show the benefit in MTRL of the bound in Equation (9). Despite being just multi-task extensions of previous works, our results are the first one to theoretically show the benefit of sharing representation in MTRL. Moreover, they serve as a significant theoretical motivation, besides to the intuitive ones, of the practical algorithms that we describe in the following sections.
+
+![](images/7374a975de557af8e91619c393dab8803cd8e60b7ba849d8b4177c03e09398fc.jpg)  
+Figure 1: (a) The architecture of the neural network we propose to learn $T$ tasks simultaneously. The $w _ { t }$ block maps each input $x _ { t }$ from task $\mu _ { t }$ to a shared set of layers $h$ which extracts a common representation of the tasks. Eventually, the shared representation is specialized in block $f _ { t }$ and the output $y _ { t }$ of the network is computed. Note that each block can be composed of arbitrarily many layers. (b) Results of FQI and MFQI averaged over 4 tasks in Car-On-Hill, showing $\| Q ^ { * } - Q ^ { \pi \kappa } \|$ on the left, and the discounted cumulative reward on the right. (c) Results of MFQI showing $\lVert Q ^ { * } - Q ^ { \pi _ { K } } \rVert$ for increasing number of tasks. Both results in (b) and (c) are averaged over 100 experiments, and show the $9 5 \%$ confidence intervals.
+
+# 4 SHARING REPRESENTATIONS
+
+We want to empirically evaluate the benefit of our theoretical study in the problem of jointly learning $T$ different tasks $\mu _ { t }$ , introducing a neural network architecture for which our bounds hold. Following our theoretical framework, the network we propose extracts representations $w _ { t }$ from inputs $x _ { t }$ for each task $\mu _ { t }$ , mapping them to common features in a set of shared layers $h$ , specializing the learning of each task in respective separated layers $f _ { t }$ , and finally computing the output $y _ { t } = ( \bar { f _ { t } } \circ h \circ w _ { t } ) ( x _ { t } \bar  ) =$ $f _ { t } ( h ( w _ { t } ( x _ { t } ) ) )$ (Figure 1(a)). The idea behind this architecture is not new in the literature. For instance, similar ideas have already been used in DQN variants to improve exploration on the same task via bootstrapping (Osband et al., 2016) and to perform MTRL (Liu et al., 2016).
+
+The intuitive and desirable property of this architecture is the exploitation of the regularization effect introduced by the shared representation of the jointly learned tasks. Indeed, unlike learning a single task that may end up in overfitting, forcing the model to compute a shared representation of the tasks helps the regression process to extract more general features, with a consequent reduction in the variance of the learned function. This intuitive justification for our approach, joins the theoretical benefit proven in Section 3. Note that our architecture can be used in any MTRL problem involving a regression process; indeed, it can be easily used in value-based methods as a $Q$ -function regressor, or in policy search as a policy regressor. In both cases, the targets are learned for each task $\mu _ { t }$ in its respective output block $f _ { t }$ . Remarkably, as we show in the experimental Section 5, it is straightforward to extend RL algorithms to their multi-task variants only through the use of the proposed network architecture, without major changes to the algorithms themselves.
+
+# 5 EXPERIMENTAL RESULTS
+
+To empirically evince the effect described by our bounds, we propose an extension of FQI (Ernst et al., 2005; Riedmiller, 2005), that we call MFQI, for which our AVI bounds apply. Then, to empirically evaluate our approach in challenging RL problems, we introduce multi-task variants of two well-known DRL algorithms: DQN (Mnih et al., 2015) and DDPG (Lillicrap et al., 2015), which we call Multi Deep $Q$ -Network (MDQN) and Multi Deep Deterministic Policy Gradient (MDDPG) respectively. Note that for these methodologies, our AVI and API bounds hold only with the simplifying assumption that the samples are i.i.d.; nevertheless they are useful to show the benefit of our method also in complex scenarios, e.g. MuJoCo (Todorov et al., 2012). We remark that in these experiments we are only interested in showing the benefit of learning multiple tasks with a shared representation w.r.t. learning a single task; therefore, we only compare our methods with the single task counterparts, ignoring other works on MTRL in literature. Experiments have been developed using the MushroomRL library (D’Eramo et al., 2020), and run on an NVIDIA $\textsuperscript { \textregistered }$ DGX StationTM and Intel $\textsuperscript { \textregistered }$ AI DevCloud. Refer to Appendix B for all the details and our motivations about the experimental settings.
+
+![](images/0484b9ead8bc7c15505dd7ea802286fd29b45b9ea1a114762d40a879ee85ea95.jpg)  
+Figure 2: Discounted cumulative reward averaged over 100 experiments of DQN and MDQN for each task and for transfer learning in the Acrobot problem. An epoch consists of 1, 000 steps, after which the greedy policy is evaluated for 2, 000 steps. The $9 5 \%$ confidence intervals are shown.
+
+# 5.1 MULTI FITTED $Q$ -ITERATION
+
+As a first empirical evaluation, we consider FQI, as an example of an AVI algorithm, to show the effect described by our theoretical AVI bounds in experiments. We consider the Car-On-Hill problem as described in Ernst et al. (2005), and select four different tasks from it changing the mass of the car and the value of the actions (details in Appendix B). Then, we run separate instances of FQI with a single task network for each task respectively, and one of MFQI considering all the tasks simultaneously. Figure 1(b) shows the $L _ { 1 }$ -norm of the difference between $Q ^ { * }$ and $Q ^ { \pi _ { K } }$ averaged over all the tasks. It is clear how MFQI is able to get much closer to the optimal $Q$ -function, thus giving an empirical evidence of the AVI bounds in Theorem 2. For completeness, we also show the advantage of MFQI w.r.t. FQI in performance. Then, in Figure 1(c) we provide an empirical evidence of the benefit of increasing the number of tasks in MFQI in terms of both quality and stability.
+
+# 5.2 MULTI DEEP $Q$ -NETWORK
+
+As in Liu et al. (2016), our MDQN uses separate replay memories for each task and the batch used in each training step is built picking the same number of samples from each replay memory. Furthermore, a step of the algorithm consists of exactly one step in each task. These are the only minor changes to the vanilla DQN algorithm we introduce, while all other aspects, such as the use of the target network, are not modified. Thus, the time complexity of MDQN is considerably lower than vanilla DQN thanks to the learning of $T$ tasks with a single model, but at the cost of a higher memory complexity for the collection of samples for each task. We consider five problems with similar state spaces, sparse rewards and discrete actions: Cart-Pole, Acrobot, Mountain-Car, Car-On-Hill, and Inverted-Pendulum. The implementation of the first three problems is the one provided by the OpenAI Gym library Brockman et al. (2016), while Car-On-Hill is described in Ernst et al. (2005) and Inverted-Pendulum in Lagoudakis & Parr (2003).
+
+Figure 2(a) shows the performance of MDQN w.r.t. to vanilla DQN that uses a single-task network structured as the multi-task one in the case with $T = 1$ . The first three plots from the left show good performance of MDQN, which is both higher and more stable than DQN. In Car-On-Hill, MDQN is slightly slower than DQN to reach the best performance, but eventually manages to be more stable. Finally, the Inverted-Pendulum experiment is clearly too easy to solve for both approaches, but it is still useful for the shared feature extraction in MDQN. The described results provide important hints about the better quality of the features extracted by MDQN w.r.t. DQN. To further demonstrate this, we evaluate the performance of DQN on Acrobot, arguably the hardest of the five problems, using a single-task network with the shared parameters in $h$ initialized with the weights of a multi-task network trained with MDQN on the other four problems. Arbitrarily, the pre-trained weights can be adjusted during the learning of the new task or can be kept fixed and only the remaining randomly initialized parameters in w and f are trained. From Figure 2(b), the advantages of initializing the weights are clear. In particular, we compare the performance of DQN without initialization w.r.t. DQN with initialization in three settings: in Unfreeze-0 the initialized weights are adjusted, in NoUnfreeze they are kept fixed, and in Unfreeze-10 they are kept fixed until epoch 10 after which they start to be optimized. Interestingly, keeping the shared weights fixed shows a significant performance improvement in the earliest epochs, but ceases to improve soon. On the other hand, the adjustment of weights from the earliest epochs shows improvements only compared to the uninitialized network in the intermediate stages of learning. The best results are achieved by starting to adjust the shared weights after epoch 10, which is approximately the point at which the improvement given by the fixed initialization starts to lessen.
+
+![](images/fa652fc9d8ebaac6972a4ca0cd646bb486d20bebea3e01890c8f3892eebfb6b9.jpg)  
+Figure 3: Discounted cumulative reward averaged over 40 experiments of DDPG and MDDPG for each task and for transfer learning in the Inverted-Double-Pendulum and Hopper problems. An epoch consists of 10, 000 steps, after which the greedy policy is evaluated for 5, 000 steps. The $9 5 \%$ confidence intervals are shown.
+
+# 5.3 MULTI DEEP DETERMINISTIC POLICY GRADIENT
+
+In order to show how the flexibility of our approach easily allows to perform MTRL in policy search algorithms, we propose MDDPG as a multi-task variant of DDPG. As an actor-critic method, DDPG requires an actor network and a critic network. Intuitively, to obtain MDDPG both the actor and critic networks should be built following our proposed structure. We perform separate experiments on two sets of MuJoCo Todorov et al. (2012) problems with similar continuous state and action spaces: the first set includes Inverted-Pendulum, Inverted-Double-Pendulum, and Inverted-Pendulum-Swingup as implemented in the pybullet library, whereas the second set includes Hopper-Stand, Walker-Walk, and Half-Cheetah-Run as implemented in the DeepMind Control SuiteTassa et al. (2018). Figure 3(a) shows a relevant improvement of MDDPG w.r.t. DDPG in the pendulum tasks. Indeed, while in Inverted-Pendulum, which is the easiest problem among the three, the performance of MDDPG is only slightly better than DDPG, the difference in the other two problems is significant. The advantage of MDDPG is confirmed in Figure 3(c) where it performs better than DDPG in Hopper and equally good in the other two tasks. Again, we perform a TL evaluation of DDPG in the problems where it suffers the most, by initializing the shared weights of a single-task network with the ones of a multi-task network trained with MDDPG on the other problems. Figures 3(b) and 3(d) show evident advantages of pre-training the shared weights and a significant difference between keeping them fixed or not.
+
+# 6 RELATED WORKS
+
+Our work is inspired from both theoretical and empirical studies in MTL and MTRL literature. In particular, the theoretical analysis we provide follows previous results about the theoretical properties of multi-task algorithms. For instance, Cavallanti et al. (2010) and Maurer (2006) prove the theoretical advantages of MTL based on linear approximation. More in detail, Maurer (2006) derives bounds on MTL when a linear approximator is used to extract a shared representation among tasks. Then, Maurer et al. (2016), which we considered in this work, describes similar results that extend to the use of non-linear approximators. Similar studies have been conducted in the context of MTRL. Among the others, Lazaric & Restelli (2011) and Brunskill & Li (2013) give theoretical proofs of the advantage of learning from multiple MDPs and introduces new algorithms to empirically support their claims, as done in this work.
+
+Generally, contributions in MTRL assume that properties of different tasks, e.g. dynamics and reward function, are generated from a common generative model. About this, interesting analyses consider Bayesian approaches; for instance Wilson et al. (2007) assumes that the tasks are generated from a hierarchical Bayesian model, and likewise Lazaric & Ghavamzadeh (2010) considers the case when the value functions are generated from a common prior distribution. Similar considerations, which however does not use a Bayesian approach, are implicitly made in Taylor et al. (2007), Lazaric et al. (2008), and also in this work.
+
+In recent years, the advantages of MTRL have been empirically evinced also in DRL, especially exploiting the powerful representational capacity of deep neural networks. For instance, Parisotto et al. (2015) and Rusu et al. (2015) propose to derive a multi-task policy from the policies learned by DQN experts trained separately on different tasks. Rusu et al. (2015) compares to a therein introduced variant of DQN, which is very similar to our MDQN and the one in Liu et al. (2016), showing how their method overcomes it in the Atari benchmark Bellemare et al. (2013). Further developments, extend the analysis to policy search (Yang et al., 2017; Teh et al., 2017), and to multi-goal RL (Schaul et al., 2015; Andrychowicz et al., 2017). Finally, Hessel et al. (2018) addresses the problem of balancing the learning of multiple tasks with a single deep neural network proposing a method that uniformly adapts the impact of each task on the training updates of the agent.
+
+# 7 CONCLUSION
+
+We have theoretically proved the advantage in RL of using a shared representation to learn multiple tasks w.r.t. learning a single task. We have derived our results extending the AVI/API bounds (Farahmand, 2011) to MTRL, leveraging the upper bounds on the approximation error in MTL provided in Maurer et al. (2016). The results of this analysis show that the error propagation during the AVI/API iterations is reduced according to the number of tasks. Then, we proposed a practical way of exploiting this theoretical benefit which consists in an effective way of extracting shared representations of multiple tasks by means of deep neural networks. To empirically show the advantages of our method, we carried out experiments on challenging RL problems with the introduction of multi-task extensions of FQI, DQN, and DDPG based on the neural network structure we proposed. As desired, the favorable empirical results confirm the theoretical benefit we described.
+
+# ACKNOWLEDGMENTS
+
+This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. #640554 (SKILLS4ROBOTS) and No. #713010 (GOALRobots). This project has also been supported by grants from NVIDIA, the NVIDIA DGX Station, and the Intel
+
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+
+# A PROOFS
+
+A.1 APPROXIMATED VALUE-ITERATION BOUNDS
+
+Proof of Theorem 2. We compute the average expected loss across tasks:
+
+$$
+\begin{array} { r l } & { \frac { 1 } { T } \frac { \gamma } { \Gamma } = \Vert \boldsymbol { Q } _ { \mu } ^ { * } - \boldsymbol { Q } _ { \mu } ^ { * * } \Vert _ { \mathcal { S } } , } \\ & { \qquad \leq \frac { 1 } { T } \frac { \gamma } { \Gamma } \frac { 2 \gamma } { \Gamma + 1 } ( \frac { 2 \gamma } { 1 - \gamma _ { 1 } } ) ^ { 2 } [ \frac { \sin [ \Gamma ] } { \Gamma \Theta ( \mathcal { R } ; \mathcal { S } ) } ( \boldsymbol { \bar { X } } ; \boldsymbol { i } , \gamma ) \boldsymbol { \bar { E } } ^ { \tilde { \Sigma } } ( \varepsilon _ { \mathrm { f r o m } } , \dots , \varepsilon _ { 1 } , \dots , \boldsymbol { i } , \nu ) + \frac { 2 } { 1 - \gamma _ { 1 } } \gamma _ { \mathrm { f r o m } ; \mathcal { S } } ^ { \tilde { \kappa } } ] } \\ & { \qquad \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \frac { 1 } { \gamma } [ \frac { \sin [ \Gamma ] } { \Gamma \Theta ( \mathcal { R } ; \mathcal { S } ) } ( \boldsymbol { \bar { X } } ; \boldsymbol { i } , \gamma ) \boldsymbol { \bar { E } } ^ { \tilde { \Sigma } } ( \varepsilon _ { \mathrm { f r o m } } , \dots , \boldsymbol { \bar { e } } , \kappa _ { 1 } , \dots , \boldsymbol { i } , \gamma ) + \frac { 2 } { 1 - \gamma _ { 1 } } \gamma _ { \mathrm { f r o m } ; \mathcal { S } } ^ { \tilde { \kappa } } ] } \\ & { \qquad \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } [ \frac { \gamma } { T } \frac { 1 } { \Gamma + 1 } ( \frac { \sin [ \Gamma ] } { \Theta ( \mathcal { R } ; \mathcal { S } ) } ( \boldsymbol { \bar { X } } ; \boldsymbol { i } , \gamma ) \boldsymbol { \bar { E } } ^ { \tilde { \Sigma } } ( \varepsilon _ { \mathrm { f r o m } } , \dots , \boldsymbol { \bar { e } } , \kappa _ { 1 } , \dots , \boldsymbol { i } , \gamma ) ) + \frac { 2 } { 1 - \gamma } \gamma ^ { \kappa } R _ { \mathrm { f r o m . 5 } } ] } \\ &  \qquad \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } [ \frac { \gamma } { T } [ \frac { \sin [ \Gamma ] } { \sin [ \Theta ] } ( \boldsymbol { \bar { X } } ; \boldsymbol { \bar { X } } ^ { \tilde { \Sigma } } ( \boldsymbol  \ \end{array}
+$$
+
+with $\gamma = \operatorname* { m a x } _ { t \in \{ 1 , \dots , T \} } \gamma _ { t } , C _ { \mathrm { V I } } ^ { \frac { 1 } { 2 } } ( K ; r ) = \operatorname* { m a x } _ { t \in \{ 1 , \dots , T \} } C _ { \mathrm { V I } , \rho , \nu } ^ { \frac { 1 } { 2 } } ( K ; t , r ) .$ , and $R _ { \mathrm { m a x , a v g } } = 1 / T \sum _ { t = 1 } ^ { T } R _ { \mathrm { m a x } , t }$ . Considering the term 1/T PTt=1 hE 12 (εt,0, . . . , εt,K−1; t, r)i = 1/T PTt=1 PK−1k=0 α2rt,kεt,k 12 let
+
+$$
+\alpha _ { k } = \left\{ \begin{array} { l l } { \frac { ( 1 - \gamma ) \gamma ^ { K - k - 1 } } { 1 - \gamma ^ { K + 1 } } } & { 0 \leq k < K , } \\ { \frac { ( 1 - \gamma ) \gamma ^ { K } } { 1 - \gamma ^ { K + 1 } } } & { k = K } \end{array} \right. ,
+$$
+
+then we bound
+
+$$
+\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \sum _ { k = 0 } ^ { K - 1 } \alpha _ { t , k } ^ { 2 r } \varepsilon _ { t , k } \right) ^ { \frac { 1 } { 2 } } \leq \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { t , k } \right) ^ { \frac { 1 } { 2 } } .
+$$
+
+Using Jensen’s inequality:
+
+$$
+\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { t , k } \right) ^ { \frac { 1 } { 2 } } \leq \left( \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \varepsilon _ { t , k } \right) ^ { \frac { 1 } { 2 } } .
+$$
+
+So, now we can write (11) as
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \lVert Q _ { t } ^ { * } - Q _ { t } ^ { \pi _ { K } } \rVert _ { 1 , \rho } \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \left[ \underset { r \in [ 0 , 1 ] } { \operatorname* { i n f } } C _ { \mathrm { V I } } ^ { \frac { 1 } { 2 } } ( K ; r ) \mathcal { E } _ { \mathrm { a v g } } ^ { \frac { 1 } { 2 } } ( \varepsilon _ { \mathrm { a v g } , 0 } , \dotsc , \varepsilon _ { \mathrm { a v g } , K - 1 } ; r ) \right. } \\ & { \quad \quad \quad \quad \quad \left. + \frac { 2 } { 1 - \gamma } \gamma ^ { K } R _ { \operatorname* { m a x } , \mathrm { a v g } } \right] , } \end{array}
+$$
+
+$\varepsilon _ { \mathrm { a v g } , k } = 1 / T \sum _ { t = 1 } ^ { T } \varepsilon _ { \mathrm { t } , k }$ and $\begin{array} { r } { \mathcal { E } _ { \mathrm { a v g } } \big ( \varepsilon _ { \mathrm { a v g } , 0 } , \dots , \varepsilon _ { \mathrm { a v g } , K - 1 } ; r \big ) = \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { \mathrm { a v g } , k } . } \end{array}$
+
+Proof of Lemma 4. Let us start from the definition of optimal task-averaged risk:
+
+$$
+\varepsilon _ { \mathrm { a v g } , k } ^ { * } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \lVert Q _ { t , k + 1 } ^ { * } - T _ { t } ^ { * } Q _ { t , k } \rVert _ { \nu } ^ { 2 } ,
+$$
+
+where $Q _ { t , k } ^ { * }$ , with $t \in [ 1 , T ]$ , are the minimizers of $\varepsilon _ { \mathrm { a v g } , k }$
+
+Consider the task $\check { t }$ such that
+
+$$
+\check { t } _ { k } = \arg \operatorname* { m a x } _ { t \in \{ 1 , \dots , T \} } \| Q _ { t , k + 1 } ^ { * } - T _ { t } ^ { * } Q _ { t , k } \| _ { \nu } ^ { 2 } ,
+$$
+
+we can write the following inequality:
+
+$$
+\sqrt { \varepsilon _ { \mathrm { a v g } , k } ^ { * } } \leq \| Q _ { \tilde { t } _ { k } , k + 1 } ^ { * } - \mathcal { T } _ { \tilde { t } } ^ { * } Q _ { \tilde { t } _ { k } , k } \| _ { \nu } .
+$$
+
+By the application of Theorem 5.3 by Farahmand (2011) to the right hand side, and defining $\dot { b _ { k , i } } = \lVert \bar { Q _ { \tilde { t } _ { k } , i + 1 } } - T _ { \check { t } } ^ { * } Q _ { \tilde { t } _ { k } , i } \rVert _ { \nu }$ , we obtain:
+
+$$
+\sqrt { \varepsilon _ { \mathrm { a v g } , k } ^ { * } } \leq \| Q _ { \bar { t } _ { k } , k + 1 } ^ { * } - ( \gamma _ { \bar { t } } ^ { * } ) ^ { k + 1 } Q _ { \bar { t } _ { k } , 0 } \| _ { \nu } + \sum _ { i = 0 } ^ { k - 1 } ( \gamma _ { \bar { t } _ { k } } C _ { \mathrm { A E } } ( \nu ; \check { t } _ { k } , P ) ) ^ { i + 1 } b _ { k , k - 1 - i } .
+$$
+
+Squaring both sides yields the result:
+
+$$
+\varepsilon _ { \mathrm { a v g } , k } ^ { * } \leq \left( \| Q _ { \bar { t } _ { k } , k + 1 } ^ { * } - ( T _ { \bar { t } } ^ { * } ) ^ { k + 1 } Q _ { \bar { t } _ { k } , 0 } \| _ { \nu } + \sum _ { i = 0 } ^ { k - 1 } ( \gamma _ { \bar { t } _ { k } } C _ { \mathrm { A E } } ( \nu ; \check { t } _ { k } , P ) ) ^ { i + 1 } b _ { k , k - 1 - i } \right) ^ { 2 } .
+$$
+
+# A.2 APPROXIMATED POLICY-ITERATION BOUNDS
+
+We start by considering the bound for the API framework:
+
+Theorem 5. (Thefor any sequence Letand $K$ be a positive integer, and he corresponding sequen $\begin{array} { r } { Q _ { m a x } \leq \frac { R _ { m a x } } { 1 - \gamma } } \end{array}$ . Then where $( Q _ { k } ) _ { k = 0 } ^ { K - 1 } \subset B ( S \times \mathcal { A } , Q _ { m a x } )$ $( \varepsilon _ { k } ) _ { k = 0 } ^ { K - 1 }$ $\varepsilon _ { k } = \lVert Q _ { k } - Q ^ { \pi _ { k } } \rVert _ { \nu } ^ { 2 }$ , we have:
+
+$$
+\| Q ^ { * } - Q ^ { \pi \kappa } \| _ { 1 , \rho } \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \left[ \operatorname* { i n f } _ { r \in [ 0 , 1 ] } C _ { P I , \rho , \nu } ^ { \frac 1 2 } ( K ; r ) \mathcal { E } ^ { \frac 1 2 } ( \varepsilon _ { 0 } , \dots , \varepsilon _ { K - 1 } ; r ) + \gamma ^ { K - 1 } R _ { m a x } \right] ,
+$$
+
+where
+
+$$
+\begin{array} { r l } { { \displaystyle C _ { P I , \rho , \nu } ( K ; r ) = } } & { { } } \\ { { \displaystyle \left( \frac { 1 - \gamma } { 2 } \right) ^ { 2 } \operatorname* { s u p } _ { \pi _ { 0 } ^ { ' } , \ldots , \pi _ { K } ^ { ' } } \sum _ { k = 0 } ^ { K - 1 } a _ { k } ^ { 2 ( 1 - r ) } } } & { { ~ \left( \sum _ { m \ge 0 } \gamma ^ { m } c _ { P I _ { 1 } , \rho , \nu } ( K - k - 1 , m + 1 ; \pi _ { k + 1 } ^ { \prime } ) + \right. } } \\ { { } } & { { \left. \sum _ { m \ge 1 } \gamma ^ { m } c _ { P I _ { 2 } , \rho , \nu } ( K - k - 1 , m ; \pi _ { k + 1 } ^ { \prime } , \pi _ { k } ^ { \prime } ) + c _ { P I _ { 3 } , \rho , \nu } \right) ^ { 2 } ; } } \end{array}
+$$
+
+with butio $\begin{array} { r } { \mathcal { E } ( \varepsilon _ { 0 } , \dots , \varepsilon _ { K - 1 } ; r ) = \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { k } } \end{array}$ , the three coefficients fined as in Farahmand $C _ { P I _ { 1 } , \rho , \nu } , C _ { P I _ { 2 } , \rho , \nu } , C _ { P I _ { 3 } , \rho , \nu } ,$ , the distri$\rho$ $\nu$ $\alpha _ { k }$
+
+From Theorem 5, by computing the average loss across tasks and pushing inside the average using Jensen’s inequality, we derive the API bounds averaged on multiple tasks.
+
+Theorem 6. Let K be a positive integer, and $\begin{array} { r } { Q _ { m a x } \leq \frac { R _ { m a x } } { 1 - \gamma } } \end{array}$ . Then for any sequence $( Q _ { k } ) _ { k = 0 } ^ { K - 1 } \subset B ( S \times$ $A , Q _ { m a x } )$ and the corresponding sequence $( \varepsilon _ { a \nu g , k } ) _ { k = 0 } ^ { K - 1 }$ , where $\varepsilon _ { a \nu g , k } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \lVert Q _ { t , k } - Q _ { t } ^ { \pi _ { k } } \rVert _ { \nu } ^ { 2 }$ have:
+
+$$
+\begin{array} { r l } & { \frac { 1 } { T } \displaystyle \sum _ { t = 1 } ^ { T } \| Q _ { t } ^ { * } - Q _ { t } ^ { \pi \kappa } \| _ { 1 , \rho } \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \left[ \operatorname* { i n f } _ { r \in [ 0 , 1 ] } C _ { P I } ^ { \frac { 1 } { 2 } } ( K ; r ) \mathcal { E } _ { a \nu g } ^ { \frac { 1 } { 2 } } ( \varepsilon _ { a \nu g , 0 } , \dots , \varepsilon _ { a \nu g , K - 1 } ; r ) \right. } \\ & { \qquad \left. + \gamma ^ { K - 1 } R _ { m a x , a \nu g } \right] , } \end{array}
+$$
+
+$$
+\begin{array} { r l } & { v i t h \ E _ { a v g } = \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { a v g , k } , \gamma = \underset { t \in \{ 1 , \ldots , T \} } { \operatorname* { m a x } } \gamma _ { t } , C _ { P I } ^ { \frac { 1 } { 2 } } ( K ; r ) = \underset { t \in \{ 1 , \ldots , T \} } { \operatorname* { m a x } } C _ { P I , \rho , \nu } ^ { \frac { 1 } { 2 } } ( K ; t , r ) , R _ { m a x , a v g } = } \\ & { \frac { 1 } { \bar { r } } \sum _ { t = 1 } ^ { T } R _ { m a x , t } a n d \alpha _ { k } = \left\{ \begin{array} { l l } { \frac { ( 1 - \gamma ) \gamma ^ { K - k - 1 } } { 1 - \gamma ^ { K + 1 } } } & { 0 \leq k < K , } \\ { \frac { ( 1 - \gamma ) \gamma ^ { K } } { 1 - \gamma ^ { K + 1 } } } & { k = K } \end{array} \right. . } \end{array}
+$$
+
+Proof of Theorem $6$ . The proof is very similar to the one for AVI. We compute the average expected loss across tasks:
+
+$$
+\begin{array} { r l } & { \frac { 1 } { T } \frac { \gamma } { \Gamma } \Vert \mathscr { Q } _ { \xi } ^ { \lambda } - \mathscr { Q } _ { \xi } ^ { \lambda \kappa } \Vert _ { \xi , \theta } , } \\ & { \quad \le \frac { 1 } { T } \frac { \gamma } { \Gamma + 1 } \frac { 2 \gamma _ { \kappa } } { ( 1 - \gamma ) ^ { 2 } } \Bigg [ \frac { \mathrm { i n f ~ } } { \Gamma \Theta _ { 1 , \theta } ^ { \lambda } } C _ { \tilde { \kappa } , \xi \kappa } ^ { \frac { \lambda } { \alpha } } ( \mathcal { K } _ { \xi , \xi } , \gamma ) E ^ { \frac { \lambda } { 2 } } ( \varepsilon _ { \kappa , \eta } , \dots , \varepsilon _ { \kappa - 1 } ; \xi , \boldsymbol { \kappa } ) + \gamma _ { \kappa } ^ { K - 1 } \tilde { H } _ { \mathrm { m a x } } \Bigg ] } \\ & { \quad \le \frac { 2 \gamma _ { \kappa } } { ( 1 - \gamma ) ^ { 2 } } \frac { 1 } { T } \Bigg [ \frac { 1 } { \Gamma \Theta _ { 1 , \theta } ^ { \lambda } } C _ { \tilde { \kappa } , \xi \kappa } ^ { \frac { \lambda } { \alpha } } ( \mathcal { K } _ { \xi , \xi } , \gamma ) E ^ { \frac { \lambda } { 2 } } ( \varepsilon _ { \kappa , \eta } , \dots , \varepsilon _ { \kappa - 1 } ; \xi , \boldsymbol { \kappa } ) - \gamma _ { \kappa } ^ { K } \tilde { H } _ { \mathrm { m a x } } \Bigg ] } \\ & { \quad \le \frac { 2 \gamma _ { \kappa } } { ( 1 - \gamma ) ^ { 2 } } \Bigg [ \frac { 1 } { \gamma } \frac { \gamma } { \Gamma + 1 } \Bigg ( \frac { \mathrm { i n f ~ } } { \Gamma \Theta _ { 1 , \theta } ^ { \lambda } } C _ { \tilde { \kappa } , \xi \kappa } ^ { \frac { \lambda } { \alpha } } ( \mathcal { K } _ { \xi , \xi } , \gamma ) E ^ { \frac { \lambda } { 2 } } ( \varepsilon _ { \kappa , \eta } , \dots , \varepsilon _ { \kappa - 1 } ; \boldsymbol { \kappa } , \boldsymbol { \kappa } ) + \gamma _ { \kappa } ^ { K - 1 } \tilde { H } _ { \mathrm { m a x } } \Bigg ] } \\ &  \quad \le \frac { 2 \gamma _ { \kappa } } { ( 1 - \gamma ) ^ { 2 } } \Bigg [ \frac { 1 } { \gamma } \frac { \gamma } { \Gamma \Theta _ { 1 , \theta } ^ { \lambda } } \Bigg ( \frac { 1 }  \Gamma  \end{array}
+$$
+
+Using Jensen’s inequality as in the AVI scenario, we can write (15) as:
+
+$$
+\begin{array} { r l } & { \frac { 1 } { T } \displaystyle \sum _ { t = 1 } ^ { T } \| Q _ { t } ^ { * } - Q _ { t } ^ { \pi \kappa } \| _ { 1 , \rho } \leq \frac { 2 \gamma } { ( 1 - \gamma ) ^ { 2 } } \left[ \operatorname* { i n f } _ { r \in [ 0 , 1 ] } C _ { \mathrm { P I } } ^ { \frac { 1 } { 2 } } ( K ; r ) \mathcal { E } _ { \mathrm { a v g } } ^ { \frac { 1 } { 2 } } ( \varepsilon _ { \mathrm { a v g } , 0 } , \dots , \varepsilon _ { \mathrm { a v g } , K - 1 } ; r ) \right. } \\ & { \qquad \left. + \gamma ^ { K - 1 } R _ { \mathrm { m a x , a v g } } \right] , } \end{array}
+$$
+
+$\varepsilon _ { \mathrm { a v g } , k } = 1 / T \sum _ { t = 1 } ^ { T } \varepsilon _ { \mathrm { t } , k }$ and $\begin{array} { r } { \mathcal { E } _ { \mathrm { a v g } } \big ( \varepsilon _ { \mathrm { a v g } , 0 } , \dots , \varepsilon _ { \mathrm { a v g } , K - 1 } ; r \big ) = \sum _ { k = 0 } ^ { K - 1 } \alpha _ { k } ^ { 2 r } \varepsilon _ { \mathrm { a v g } , k } . } \end{array}$
+
+# A.3 APPROXIMATION BOUNDS
+
+Proof of Theorem 3. Let $w _ { 1 } ^ { * } , \ldots , w _ { T } ^ { * } .$ , $h ^ { * }$ and $f _ { 1 } ^ { * } , \ldots , f _ { T } ^ { * }$ be the minimizers of $\varepsilon _ { \mathrm { a v g } } ^ { * }$ , then:
+
+$$
+\begin{array} { r l } & { \quad \varepsilon _ { \mathrm { a v g } } ( \hat { \mathbf { w } } , \hat { h } , \hat { \mathbf { f } } ) - \varepsilon _ { \mathrm { a v g } } ^ { * } = \underbrace { \left( \varepsilon _ { \mathrm { a v g } } ( \hat { \mathbf { w } } , \hat { h } , \hat { \mathbf { f } } ) - \frac { 1 } { n T } \sum _ { t } \ell ( \hat { f } _ { t } ( \hat { h } ( \hat { w } _ { t } ( X _ { t i } ) ) ) , Y _ { t i } ) \right) } _ { A } } \\ & { \quad \quad \quad + \underbrace { \left( \frac { 1 } { n T } \sum _ { t i } \ell ( \hat { f } _ { t } ( \hat { h } ( \hat { w } _ { t } ( X _ { t i } ) ) ) , Y _ { t i } ) - \frac { 1 } { n T } \sum _ { t i } \ell ( \int _ { t } ^ { * } ( h ^ { * } ( w _ { t } ^ { * } ( X _ { t i } ) ) ) , Y _ { t i } ) \right) } _ { B } } \\ & { \quad \quad \quad + \underbrace { \left( \frac { 1 } { n T } \sum _ { t i } \ell ( { f } _ { t } ^ { * } ( h ^ { * } ( w _ { t } ^ { * } ( X _ { t i } ) ) ) , Y _ { t i } ) - \varepsilon _ { \mathrm { a v g } } ^ { * } \right) } _ { G } . } \end{array}
+$$
+
+We proceed to bound the three components individually:
+
+• $C$ can be bounded using Hoeffding’s inequality, with probability $1 - \delta / 2$ by $\sqrt { \ln ( 2 / \delta ) \big / ( 2 n T ) }$ , as it contains only $n T$ random variables bounded in the interval $[ 0 , 1 ]$ ;
+
+• $B$ can be bounded by 0, by definition of ˆw, $\hat { h }$ and $\hat { \bf f }$ , as they are the minimizers of Equation (3); • the bounding of $A$ is less straightforward and is described in the following.
+
+We define the following auxiliary function spaces:
+
+$$
+\begin{array} { r l } & { \mathbb { W } ^ { \prime } = \{ x \in \mathcal { X } \to ( w _ { t } ( x _ { t i } ) ) : ( w _ { 1 } , \dots , w _ { T } ) \in \mathbb { W } ^ { T } \} , } \\ & { } \\ & { \mathbb { F } ^ { \prime } = \big \{ y \in \mathbb { R } ^ { K T n } \to ( f _ { t } ( y _ { t i } ) ) : ( f _ { 1 } , \dots , f _ { T } ) \in \mathcal { F } ^ { T } \big \} , } \end{array}
+$$
+
+and the following auxiliary sets:
+
+$$
+\begin{array} { r l } & { S = \big \{ ( \ell ( f _ { t } ( h ( w _ { t } ( X _ { t i } ) ) ) , Y _ { t i } ) ) : f \in \mathcal { F } ^ { x } , h \in \mathcal { H } , w \in \mathcal { W } ^ { x } \big \} \subseteq \mathbb { R } ^ { x \cdot n } , } \\ & { S ^ { \prime } = \mathcal { F } ^ { \prime } ( \mathcal { H } ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) ) = \big \{ ( f _ { t } ( h ( w _ { t } ( X _ { t i } ) ) ) ) : f \in \mathcal { F } ^ { T } , h \in \mathcal { H } , w \in \mathcal { W } ^ { T } \big \} \subseteq \mathbb { R } ^ { T n } , } \\ & { S ^ { \prime \prime } = \mathcal { H } ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) = \big \{ ( h ( w _ { t } ( X _ { t i } ) ) ) : h \in \mathcal { H } , w \in \mathcal { W } ^ { T } \big \} \subseteq \mathbb { R } ^ { K T n } , } \end{array}
+$$
+
+which will be useful in our proof.
+
+Using Theorem 9 by Maurer et al. (2016), we can write:
+
+$$
+\begin{array} { r l r } {  { \varepsilon _ { \mathrm { a v g } } ( \hat { \mathbf { w } } , \hat { h } , \hat { \mathbf { f } } ) - \frac { 1 } { n T } \sum _ { t i } \ell ( \hat { f } _ { t } ( \hat { h } ( \hat { w } _ { t } ( X _ { t i } ) ) ) , Y _ { t i } ) } } \\ & { } & { \leq \operatorname* { s u p } _ { \mathbf { w } \in \mathcal { W } ^ { T } , h \in \mathcal { H } , \mathbf { f } \in \mathcal { F } ^ { T } } ( \varepsilon _ { \mathrm { a v g } } ( \mathbf { w } , h , \mathbf { f } ) - \frac { 1 } { n T } \sum _ { t i } \ell ( f _ { t } ( h ( w _ { t } ( X _ { t i } ) ) ) , Y _ { t i } ) ) } \\ & { } & { \leq \frac { \sqrt { 2 \pi } G ( S ) } { n T } + \sqrt { \frac { 9 \ln ( \frac { 2 } { \delta } ) } { 2 n T } } , \ ~ ( 1 ) } \end{array}
+$$
+
+then by Lipschitz property of the loss function $\ell$ and the contraction lemma Corollary 11 Maurer et al. (2016): ${ \bar { G ( S ) } } \leq { \bar { G } } ( { \bar { S ^ { \prime } } } )$ . By Theorem 12 by Maurer et al. (2016), for universal constants $c _ { 1 } ^ { \prime }$ and $c _ { 2 } ^ { \prime }$ :
+
+$$
+G ( S ^ { \prime } ) \leq c _ { 1 } ^ { \prime } L ( \mathcal { F } ^ { \prime } ) G ( S ^ { \prime \prime } ) + c _ { 2 } ^ { \prime } D ( S ^ { \prime \prime } ) O ( \mathcal { F } ^ { \prime } ) + \operatorname* { m i n } _ { y \in Y } G ( \mathcal { F } ( y ) ) ,
+$$
+
+where $L ( \mathcal { F } ^ { \prime } )$ is the largest value for the Lipschitz constants in the function space ${ \mathcal { F } } ^ { \prime }$ , and $D ( S ^ { \prime \prime } )$ is the Euclidean diameter of the set $S ^ { \prime \prime }$ .
+
+Using Theorem 12 by Maurer et al. (2016) again, for universal constants $c _ { 1 } ^ { \prime \prime }$ and $c _ { 2 } ^ { \prime \prime }$ :
+
+$$
+G ( S ^ { \prime \prime } ) \leq c _ { 1 } ^ { \prime \prime } L ( \mathcal { H } ) G ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) + c _ { 2 } ^ { \prime \prime } D ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) O ( \mathcal { H } ) + \operatorname* { m i n } _ { p \in P } G ( \mathcal { H } ( p ) ) .
+$$
+
+Putting (19) and (20) together:
+
+$$
+\begin{array} { r l } & { G ( S ^ { \prime } ) \leq c _ { 1 } ^ { \prime } L ( \mathcal { F } ^ { \prime } ) \left( c _ { 1 } ^ { \prime \prime } L ( \mathcal { H } ) G ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) + c _ { 2 } ^ { \prime \prime } D ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) O ( \mathcal { H } ) + \underset { p \in P } { \operatorname* { m i n } } G ( \mathcal { H } ( p ) ) \right) } \\ & { \qquad + c _ { 2 } ^ { \prime } D ( S ^ { \prime \prime } ) O ( \mathcal { F } ^ { \prime } ) + \underset { y \in \bar { \mathbf { Y } } } { \operatorname* { m i n } } G ( \mathcal { F } ( y ) ) } \\ & { = c _ { 1 } ^ { \prime } c _ { 1 } ^ { \prime \prime } L ( \mathcal { F } ^ { \prime } ) L ( \mathcal { H } ) G ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) + c _ { 1 } ^ { \prime } c _ { 2 } ^ { \prime \prime } L ( \mathcal { F } ^ { \prime } ) D ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) O ( \mathcal { H } ) + c _ { 1 } ^ { \prime } L ( \mathcal { F } ^ { \prime } ) \underset { p \in P } { \operatorname* { m i n } } G ( \mathcal { H } ( p ) ) } \\ & { \qquad + c _ { 2 } ^ { \prime } D ( S ^ { \prime \prime } ) O ( \mathcal { F } ^ { \prime } ) + \underset { y \in \bar { \mathbf { Y } } } { \operatorname* { m i n } } G ( \mathcal { F } ( y ) ) . } \end{array}
+$$
+
+At this point, we have to bound the individual terms in the right hand side of (21), following the same procedure proposed by Maurer et al. (2016).
+
+Firstly, to bound $L ( \mathcal { F } ^ { \prime } )$ , let $y , y ^ { \prime } \in \mathbb { R } ^ { K T n }$ , where $y = \left( y _ { t i } \right)$ with $y _ { t i } \in \mathbb { R } ^ { K }$ and $y ^ { \prime } = ( y _ { t i } ^ { \prime } )$ with $y _ { t i } ^ { \prime } \in \mathbb { R } ^ { K }$ . We can write the following:
+
+$$
+\begin{array} { r l } { \| f ( y ) - f ( y ^ { \prime } ) \| ^ { 2 } = \displaystyle \sum _ { t i } \left( f _ { t } ( y _ { t i } ) - f _ { t } ( y _ { t i } ^ { \prime } ) \right) ^ { 2 } } & { } \\ { \leq L ( \mathcal F ) ^ { 2 } \displaystyle \sum _ { t i } \| y _ { t i } - y _ { t i } ^ { \prime } \| ^ { 2 } } & { } \\ { = L ( \mathcal F ) ^ { 2 } \| y - y ^ { \prime } \| ^ { 2 } , } \end{array}
+$$
+
+whence $L ( \mathcal { F } ^ { \prime } ) \leq L ( \mathcal { F } )$ .
+
+Then, we bound:
+
+$$
+\begin{array}{c} G ( \mathcal { W } ^ { \prime } ( \bar { \mathbf { X } } ) ) = \mathbb { E } \left[ \operatorname* { s u p } _ { \mathbf { w } \in \mathcal { W } ^ { T } } \sum _ { k t i } \gamma _ { k t i } w _ { t k } ( X _ { t i } ) \Bigg | X _ { t i } \right] \leq \sum _ { t } \operatorname* { s u p } _ { l \in \{ 1 , \ldots , T \} } \mathbb { E } \left[ \operatorname* { s u p } _ { w \in \mathcal { W } } \sum _ { k i } \gamma _ { k l i } w _ { k } ( X _ { l i } ) \Bigg | X _ { l i } \right]  \\ { = T _ { \mathbf { \Phi } _ { l \in \{ 1 , \ldots , T \} } } G ( \mathcal { W } ( \mathbf { X } _ { l } ) ) . } \end{array}
+$$
+
+Then, since it is possible to bound the Euclidean diameter using the norm of the supremum value in the set, we bound $D ( S ^ { \prime \prime } ) \leq 2 \operatorname* { s u p } _ { h , \mathbf { w } } \| h ( \mathbf { w } ( \bar { \mathbf { X } } ) ) \|$ and $D ( \mathcal { W } ^ { \prime } ( \bar { \bar { \mathbf { X } } } ) ) \leq 2 \operatorname* { s u p } _ { \mathbf { w } \in \mathcal { W } ^ { T } } \| \mathbf { \bar { w } } ( \bar { \mathbf { X } } ) \|$ .
+
+Also, we bound $O ( \mathcal { F } ^ { \prime } )$ :
+
+$$
+\begin{array} { r l } { \mathbb { E } \left[ \displaystyle \operatorname* { s u p } _ { j \in \mathcal { E } ^ { * } ( \mathcal { N } , \theta ) } \langle \gamma , g ( y ) - g ( \bar { y } ^ { \prime } ) \rangle \right] = } & { \mathbb { E } \left[ \displaystyle \operatorname* { s u p } _ { j \in \mathcal { E } ^ { * } \times \mathcal { F } } \displaystyle \sum _ { \hat { \epsilon } } \gamma _ { \hat { \epsilon } } \langle f ( y _ { \hat { \epsilon } } ) - f ( \bar { y } _ { \hat { \epsilon } } ^ { \prime } ) \rangle \right] } \\ & { = \displaystyle \sum _ { \hat { \epsilon } } \mathbb { E } \left[ \displaystyle \operatorname* { s u p } _ { j \in \mathcal { E } ^ { * } } \displaystyle \sum _ { \hat { \epsilon } } \gamma _ { \hat { \epsilon } } ( f ( y _ { \hat { \epsilon } \hat { \epsilon } } ) - f ( \bar { y } _ { \hat { \epsilon } } ^ { \prime } ) ) \right] } \\ & { \leq \sqrt { T } \left( \displaystyle \sum _ { \hat { \epsilon } } \mathbb { E } \left[ \displaystyle \operatorname* { s u p } _ { j \in \mathcal { E } ^ { * } } \displaystyle \sum _ { \hat { \epsilon } } \gamma _ { \hat { \epsilon } } ( f ( y _ { \hat { \epsilon } \hat { \epsilon } } ) - f ( \bar { y } _ { \hat { \epsilon } } ^ { \prime } ) ) \right] ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } \\ & { \leq \sqrt { T } \left( \displaystyle \sum _ { \hat { \epsilon } } O ( \mathcal { F } ) ^ { 2 } \sum _ { \hat { \epsilon } } \| y _ { \hat { \epsilon } } - y _ { \hat { \epsilon } } ^ { \prime } \| ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } \\ & { = \sqrt { T } O ( \mathcal { F } ) \| y - y ^ { \prime } \| , } \end{array}
+$$
+
+whence $O ( \mathcal { F } ^ { \prime } ) \leq \sqrt { T } O ( \mathcal { F } )$ .
+
+To minimize the last term, it is possible to choose $y _ { 0 } = 0$ , as $f ( 0 ) = 0 , \forall f \in { \mathcal { F } }$ , resulting in $\begin{array} { r } { \operatorname* { m i n } _ { y \in Y } G ( \mathcal { F } ( y ) ) = G ( \mathcal { F } ( 0 ) ) = \bar { 0 } } \end{array}$ .
+
+Then, substituting in (21), and recalling that $G ( S ) \leq G ( S ^ { \prime } )$ :
+
+$$
+\begin{array} { r l } & { G ( S ) \leq c _ { 1 } ^ { \prime } c _ { 1 } ^ { \prime \prime } L ( \mathcal { F } ) L ( \mathcal { H } ) T \underset { \iota \in \{ 1 , \ldots , T \} } { \operatorname* { s u p } } G ( \mathcal { W } ( \mathbf { X } _ { l } ) ) + 2 c _ { 1 } ^ { \prime } c _ { 2 } ^ { \prime \prime } L ( \mathcal { F } ) \underset { \mathbf { w } \in \mathcal { W } ^ { T } } { \operatorname* { s u p } } \| \mathbf { w } ( \bar { \mathbf { X } } ) \| O ( \mathcal { H } ) } \\ & { \qquad + c _ { 1 } ^ { \prime } L ( \mathcal { F } ) \underset { p \in P } { \operatorname* { m i n } } G ( \mathcal { H } ( p ) ) + 2 c _ { 2 } ^ { \prime } \underset { h , \mathbf { w } } { \operatorname* { s u p } } \| h ( \mathbf { w } ( \bar { \mathbf { X } } ) ) \| \sqrt { T } O ( \mathcal { F } ) . } \end{array}
+$$
+
+Now, the first term $A$ of (17) can be bounded substituting (25) in (18):
+
+$$
+\begin{array} { r l } { c _ { \mathrm { s r g } } ( \boldsymbol { \Phi } , \boldsymbol { \hat { h } } , \boldsymbol { \hat { \mathbf { f } } } ) - \frac { 1 } { n T } \displaystyle \sum _ { t = 1 } ^ { 1 } \ell ( \hat { f } _ { t } ( \hat { h } ( \hat { w } _ { t } ( X _ { t } ) ) ) , Y _ { t i } ) } & { } \\ { \leq \frac { \sqrt { 2 \pi } } { n T } \big ( c _ { 1 } ^ { \prime } c _ { 1 } ^ { \prime \prime } L ( Y ) L ( \mathcal { H } ) T \operatorname* { s u p } _ { \iota \in ( \mathcal { H } \setminus \mathcal { N } _ { t } ) ) } G ( \mathcal { W } ( \mathbf { X } _ { t } ) ) + 2 c _ { 1 } ^ { \prime } c _ { 2 } ^ { \prime \prime } L ( \mathcal { F } ) \operatorname* { s u p } _ { \mathrm { w i t } } \| \mathbf { w } ( \boldsymbol { \bar { X } } ) \| O ( \mathcal { H } ) } \\ { } & { ~ + c _ { 1 } ^ { \prime } L ( \mathcal { F } ) \operatorname* { m i n } _ { \bar { \theta } \in \mathcal { P } } G ( \mathcal { H } ( \boldsymbol { p } ) ) + 2 c _ { 2 } ^ { \prime } \operatorname* { s u p } _ { \bar { \theta } \in \mathcal { W } } \| h ( \mathbf { w } ( \boldsymbol { \bar { X } } ) ) \| \sqrt { T } O ( \mathcal { F } ) \big ) + \sqrt { \frac { 9 \ln ( \frac { 2 } { \delta } ) } { 2 n T } } } \\ { } & { = c _ { 1 } \frac { L ( \mathcal { F } ) L ( \mathcal { H } ) \operatorname* { s u p } _ { \bar { \theta } \in \{ S , \dots , T \} } G ( \mathcal { W } ( \mathbf { X } _ { t } ) ) } { n } + c _ { 2 } \frac { \operatorname* { s u p } _ { \mathrm { w i t } } \| \mathbf { w } ( \bar { \mathbf { X } } ) \| \mathcal { D } ( \mathcal { H } ) } { n T } } \\ { } & { ~ + c _ { 3 } \frac { L ( \mathcal { F } ) \operatorname* { m i n } _ { \bar { \theta } \in \mathcal { P } } G ( \mathcal { H } ( \mathcal { p } ) ) } { n T } + c _ { 4 } \frac { \operatorname* { s u p } _ { \bar { \theta } \in \mathcal { W } } \| \hat { b } ( \mathbf { w } ( \boldsymbol { \bar { X } } ) ) \| Q ( \mathcal { F } ) } { n \mathcal { F } T } + \sqrt { \frac { 9 \ln ( \frac { 2 } { \delta } ) } { 2 n T } } . } \end{array}
+$$
+
+A union bound between $A$ , $B$ and $C$ of (17) completes the proof:
+
+$$
+\begin{array} { r l } & { \varepsilon _ { \mathrm { a v g } } ( \widehat { \mathbf { w } } , \widehat { \mathbf { h } } , \widehat { \mathbf { f } } ) - \varepsilon _ { \mathrm { a v g } } ^ { * } \leq c _ { 1 } \frac { L ( \mathcal { F } ) L ( \mathcal { H } ) \operatorname* { s u p } _ { l \in \{ 1 , \ldots , T \} } G ( \mathcal { W } ( \mathbf { X } _ { l } ) ) } { n } } \\ & { \qquad + c _ { 2 } \frac { \operatorname* { s u p } _ { \mathbf { w } } \left\| \mathbf { w } ( \widehat { \mathbf { X } } ) \right\| L ( \mathcal { F } ) O ( \mathcal { H } ) } { n T } } \\ & { \qquad + c _ { 3 } \frac { L ( \mathcal { F } ) \operatorname* { i n i n } _ { p \in \mathcal { F } } G ( \mathcal { H } ( \mathcal { p } ) ) } { n T } } \\ & { \qquad + c _ { 4 } \frac { \operatorname* { s u p } _ { h , \mathbf { w } } \left\| h ( \mathbf { x } ( \widehat { \mathbf { X } } ) ) \right\| O ( \mathcal { F } ) } { n \sqrt { T } } } \\ & { \qquad + \sqrt { \frac { 8 \ln ( \frac { 3 } { \delta } ) } { n T } } . } \end{array}
+$$
+
+# B ADDITIONAL DETAILS OF EMPIRICAL EVALUATION
+
+# B.1 MULTI FITTED $Q$ -ITERATION
+
+We consider Car-On-Hill problem with discount factor 0.95 and horizon 100. Running Adam optimizer with learning rate 0.001 and using a mean squared loss, we train a neural network composed of 2 shared layers of 30 neurons each, with sigmoidal activation function, as described in Riedmiller (2005). We select 8 tasks for the problem changing the mass of the car $m$ and the value of the discrete actions $a$ (Table 1). Figure 1(b) is computed considering the first four tasks, while Figure 1(c) considers task 1 in the result with 1 task, tasks 1 and 2 for the result with 2 tasks, tasks 1, 2, 3, and 4 for the result with 4 tasks, and all the tasks for the result with 8 tasks. To run FQI and MFQI, for each task we collect transitions running an extra-tree trained following the procedure and setting in Ernst et al. (2005), using an $\epsilon$ -greedy policy with $\epsilon = 0 . 1$ , to obtain a small, but representative dataset. The optimal $Q$ -function for each task is computed by tree-search3 for 100 states uniformly picked from the state space, and the 2 discrete actions, for a total of 200 state-action tuples.
+
+# B.2 MULTI DEEP $Q$ -NETWORK
+
+The five problems we consider for this experiment are: Cart-Pole, Acrobot, Mountain-Car, Car-OnHill, and Inverted-Pendulum4. The discount factors are respectively 0.99, 0.99, 0.99, 0.95, and 0.95. The horizons are respectively $5 0 0 , 1 , 0 0 0 , 1 , 0 0 0 , 1 0 0$ , and 3, 000. The network we use consists of 80 ReLu units for each $w _ { t } , t \in \{ 1 , \ldots , T \}$ block, with $T = 5$ . Then, the shared block $h$ consists of one layer with 80 ReLu units and another one with 80 sigmoid units. Eventually, each $f _ { t }$ has a number of linear units equal to the number of discrete actions $\bar { a _ { i } ^ { ( t ) } , i } \in \{ 1 , \dots , \# \mathcal { A } ^ { ( t ) } \}$ of task $\mu _ { t }$ which outputs the action-value $Q _ { t } ( s , a _ { i } ^ { ( t ) } ) = y _ { t } ( s , a _ { i } ^ { ( t ) } ) = f _ { t } ( h ( w _ { t } ( s ) ) , a _ { i } ^ { ( t ) } ) , \forall s \in S ^ { ( t ) }$ . The use of sigmoid units in the second layer of $h$ is due to our choice to extract meaningful shared features bounded between 0 and 1 to be used as input of the last linear layer, as in most RL approaches. In practice, we have also found that sigmoid units help to reduce task interference in multi-task networks, where instead the linear response of ReLu units cause a problematic increase in the feature values. Furthermore, the use of a bounded feature space reduces the $\mathrm { s u p } _ { h , \mathbf { w } } \| h ( \mathbf { w } ( { \bar { \mathbf { X } } } ) ) \|$ term in the upper bound of Theorem 3, corresponding to the upper bound of the diameter of the feature space, as shown in Appendix A. The initial replay memory size for each task is 100 and the maximum size is 5, 000. We use Huber loss with Adam optimizer using learning rate $1 0 ^ { - 3 }$ and batch size of 100 samples for each task. The target network is updated every 100 steps. The exploration is $\varepsilon$ -greedy with $\varepsilon$ linearly decaying from 1 to 0.01 in the first 5, 000 steps.
+
+Table 1: Different values of the mass of the car and available actions chosen for the Car-On-Hill tasks in the MFQI empirical evaluation.   
+
+<table><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>Mass</td><td rowspan=1 colspan=2>Action set</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{-4.0;4.0}</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{-4.0;4.0}</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{-4.5;4.5}</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>1.2</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>{-4.5;4.5}</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=2>{-4.125;4.125}</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=2>{-4.25;4.25}</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>0.8</td><td rowspan=1 colspan=2>{-4.375;4.375}</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.85</td><td rowspan=1 colspan=2>{-4.0;4.0}</td></tr></table>
+
+# B.3 MULTI DEEP DETERMINISTIC POLICY GRADIENT
+
+The two set of problems we consider for this experiment are: one including Inverted-Pendulum, Inverted-Double-Pendulum, and Inverted-Pendulum-Swingup, and another one including HopperStand, Walker-Walk, and Half-Cheetah- $R u n ^ { 5 }$ . The discount factors are 0.99 and the horizons are $1 , 0 0 0$ for all problems. The actor network is composed of 600 ReLu units for each $w _ { t } , t \in \{ 1 , \ldots , T \}$ block, with $T = 3$ . The shared block $h$ has 500 units with ReLu activation function as for MDQN. Finally, each $f _ { t }$ has a number of tanh units equal to the number of dimensions of the continuous actions $a ^ { ( t ) } \in \mathcal { A } ^ { ( t ) }$ of task $\mu _ { t }$ which outputs the policy $\pi _ { t } ( s ) = y _ { t } ( s ) = f _ { t } ( h ( w _ { t } ( s ) ) ) , \forall s \in S ^ { ( t ) }$ . On the other hand, the critic network consists of the same $w _ { t }$ units of the actor, except for the use of sigmoidal units in the $h$ layer, as in MDQN. In addition to this, the actions $a ^ { ( t ) }$ are given as input to $h$ . Finally, each $f _ { t }$ has a single linear unit $Q _ { t } ( s , a ^ { ( t ) } ) = y _ { t } ( s , a ^ { ( t ) } ) = f _ { t } ( h ( w _ { t } ( s ) , \bar { a } ^ { ( t ) } ) ) , \forall s \in \mathcal { S } ^ { ( t ) }$ . The initial replay memory size for each task is 64 and the maximum size is 50, 000. We use Huber loss to update the critic network and the policy gradient to update the actor network. In both cases the optimization is performed with Adam optimizer and batch size of 64 samples for each task. The learning rate of the actor is $1 0 ^ { - 4 }$ and the learning rate of the critic is $1 0 ^ { - 3 }$ . Moreover, we apply $\ell _ { 2 }$ -penalization to the critic network using a regularization coefficient of 0.01. The target networks are updated with soft-updates using $\tau = 1 0 ^ { \bar { - } 3 }$ . The exploration is performed using the action computed by the actor network adding a noise generated with an Ornstein-Uhlenbeck process with $\theta = 0 . 1 5$ and $\sigma = 0 . 2$ . Note that most of these values are taken from the original DDPG paper Lillicrap et al. (2015), which optimizes them for the single-task scenario.