diff --git a/parse/train/BkgRe1SFDS/BkgRe1SFDS.md b/parse/train/BkgRe1SFDS/BkgRe1SFDS.md new file mode 100644 index 0000000000000000000000000000000000000000..6b956cbd877ccf69806f1393190874fe2cdd058a --- /dev/null +++ b/parse/train/BkgRe1SFDS/BkgRe1SFDS.md @@ -0,0 +1,440 @@ +# LEARNING WORLD GRAPH DECOMPOSITIONS TO ACCELERATE REINFORCEMENT LEARNING + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Efficiently learning to solve tasks in complex environments is a key challenge for reinforcement learning (RL) agents. We propose to decompose a complex environment using a task-agnostic world graphs, an abstraction that accelerates learning by enabling agents to focus exploration on a subspace of the environment. The nodes of a world graph are important waypoint states and edges represent feasible traversals between them. Our framework has two learning phases: 1) identifying world graph nodes and edges by training a binary recurrent variational autoencoder (VAE) on trajectory data and 2) a hierarchical RL framework that leverages structural and connectivity knowledge from the learned world graph to bias exploration towards task-relevant waypoints and regions. We thoroughly evaluate our approach on a suite of challenging maze tasks and show that using world graphs significantly accelerates RL, achieving higher reward and faster learning. + +# 1 INTRODUCTION + +Many real-world applications, e.g., self-driving cars and in-home robotics, require an autonomous agent to execute different tasks within a single environment that features, e.g. high-dimensional state space, complex world dynamics or structured layouts. In these settings, model-free reinforcement learning (RL) agents often struggle to learn efficiently, requiring a large amount of experience collections to converge to optimal behaviors. Intuitively, an agent could learn more efficiently by focusing its exploration in task-relevant regions, if it has knowledge of the high-level structure of the environment. + +We propose a method to 1) learn and 2) use an environment decomposition in the form of a world graph, a task-agnostic abstraction. World graph nodes are waypoint states, a set of salient states that can summarize agent trajectories and provide meaningful starting points for efficient exploration (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018; Ghosh et al., 2018). The directed and weighted world graph edges characterize feasible traversals among the waypoints. To leverage the world graph, we model hierarchical RL (HRL) agents where a high-level policy chooses a waypoint state as a goal to guide exploration towards task-relevant regions, and a low-level policy strives to reach the chosen goals. + +Our framework consists of two phases. In the task-agnostic phase, we obtain world graphs by training a recurrent variational auto-encoder (VAE) (Chung et al., 2015; Gregor et al., 2015; Kingma & Welling, 2013) with binary latent variables (Nalisnick & Smyth, 2016) over trajectories collected using a random walk policy (Ha & Schmidhuber, 2018) and a curiosity-driven goal-conditioned policy (Ghosh et al., 2018; Nair et al., 2018). World graph nodes are states that are most frequently selected by the binary latent variables, while edges are inferred from empirical transition statistics between neighboring waypoints. In the task-specific phase, taking advantage of the learned world graph for structured exploration, we efficiently train an HRL model (Taylor & Stone, 2009). + +In summary, our main contributions are: + +• A task-agnostic unsupervised approach to learn world graphs, using a recurrent VAE with binary latent variables and a curiosity-driven goal-conditioned policy. • An HRL scheme for the task-specific phase that features multi-goal selection (Wide-thenNarrow) and navigation via world graph traversal. + +![](images/87e8c713c76547993153ec99d35d40aae11b5121b86240dad4d5ac65a6d03b12.jpg) +Figure 1: Top Left: overall pipeline of our 2-phase framework. Top Right (world graph discovery): a subgraph exemplifies traversal between waypoint states (in blue), see Section 3 for more details. Bottom (Hierarhical $R L$ ): an example rollout from our proposed HRL policy with Wide-then-Narrow Manager instructions and world graph traversals, solving a challenging Door-Key task, see Section 4 for more details. + +• Empirical evaluations on multiple tasks in complex 2D grid worlds to validate that our framework produces descriptive world graphs and significantly improves both sample efficiency and final performance on these tasks over baselines, especially thanks to transfer learning from the unsupervised phase and world graph traversal. + +# 2 RELATED WORK + +An understanding of the environment and its dynamics is essential for effective planning and control in model-based RL. For example, a robotics agent often locates or navigates by interpreting a map (Lowry et al., 2015; Thrun, 1998; Angeli et al., 2008). Our exploration strategy draws inspiration from active localization, where robots are actively guided to investigate unfamiliar regions (Fox et al., 1998; Li et al., 2016). Besides mapping, recent works (Azar et al., 2019; Ha & Schmidhuber, 2018; Guo et al., 2018) learn to represent the world with generative latent states (Tian & Gong, 2017; Haarnoja et al., 2018; Racanière et al., 2017). If the latent dynamics are also extrapolated, the latent states can assist planning (Mnih et al., 2016a; Hafner et al., 2018) or model-based RL (Gregor & Besse, 2018; Kaiser et al., 2019). + +While also aiming to model the world, we approach this as abstracting both the structure and dynamics of the environment in a graph representation, where nodes are states from the environment and edges encode actionable efficient transitions between nodes. Existing works (Metzen, 2013; Mannor et al., 2004; Eysenbach et al., 2019; Entezari et al., 2010) have shown benefits of such graph abstractions but typically select nodes only subject to a good coverage the observed state space. Instead, we identify a parsimonious subset of states that can summarize trajectories and provide more useful intermediate landmarks, i.e. waypoints, for navigating complex environments. + +Our method for estimating waypoint states can be viewed as performing automatic (sub)goal discovery. Subgoal and subpolicy learning are two major approaches to identify a set of temporally-extended actions, “skills”, that allow agents to efficiently learn to solve complex tasks. Subpolicy learning identifies policies useful to solve RL tasks, such as option-based methods (Daniel et al., 2016; Bacon et al., 2017) and subtask segmentations (Pertsch et al., 2019; Kipf et al., 2018). Subgoal learning, on the other hand, identifies “important states” to reach ( ¸Sim¸sek et al., 2005). + +Previous works consider various definitions of “important” states: frequently visited states during successful task completions (Digney, 1998; McGovern & Barto, 2001), states introducing the most novel information (Goyal et al., 2019), bottleneck states connecting densely-populated regions (Chen et al., 2007; ¸Sim¸sek et al., 2005), or environment-specific heuristics (Ecoffet et al., 2019). Our work draws intuition from unsupervised temporal segmentation (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018) and imitation learning (Abbeel & $\mathrm { N g }$ , 2004; Hussein et al., 2017). We define “important” states (waypoints) as the most critical states in recovering action sequences generated by some agent, which indicates that these states contain the richest information about the executed policy (Azar et al., 2019). + +![](images/0b8b33e4d41c1c614e19d9ac24f4cdb5614a9211348b71ee40716b3df882e83c.jpg) +Figure 2: Our recurrent latent model with differentiable binary latent units to identify waypoint states. A prior network (left) learns the state-conditioned prior in Beta distribution, $p _ { \psi } ( z _ { t } | s _ { t } ) { = } \mathrm { B e t a } ( \alpha _ { t } , \beta _ { t } )$ . An inference encoder learns an approximate posterior in HardKuma distribution inferred from the state-action sequence input, $q _ { \phi } ( z _ { t } | \mathbf { a } , z ) { = } \mathrm { H a r d } \hat { \mathrm { K } } \mathrm { u m a } ( \tilde { \alpha _ { t } } , \mathbf { \hat { 1 } } )$ . A generation network $p _ { \theta }$ reconstructs $\textbf { \em a }$ from $\{ s _ { t } | z _ { t } = 1 \}$ . + +# 3 LEARNING WORLD GRAPHS + +We propose a method for learning a world graph $\mathcal { G } _ { w }$ , a task-agnostic abstraction of an environment that captures its high-level structure and dynamics. In this work, the primary use of world graphs is to accelerate reinforcement learning of downstream tasks. The nodes of $\mathcal { G } _ { w }$ , denoted by a set of waypoints states $s _ { p } \in \mathcal { V } _ { p }$ , are generically “important” for accomplishing tasks within the environment, and therefore useful as starting points for exploration. Our method identifies such waypoint states from interactions with the environment. In addition, we embed feasible transitions between nearby waypoint states as the edges of $\mathcal { G } _ { w }$ . + +In this work, we define important states in the context of learning $\mathcal { G } _ { w }$ (see Section 2 for alternative definitions). That is, we wish to discover a small set of states that, when used as world graph nodes, concisely summarize the structure and dynamics of the environment. Below, we describe 1) how to collect state-action trajectories and an unsupervised learning objective to identify world graph nodes, and 2) how the graph’s edges (i.e., how to transition between nodes) are formed from trajectories. + +# 3.1 WAYPOINT STATE IDENTIFICATION + +The structure and dynamics of an environment are implicit in the state-action trajectories observed during exploration. To identify world graph nodes from such data, we train a recurrent variational autoencoder (VAE) that, given a sequence of state-action pairs, identifies a subset of the states in the sequence from which the full action sequence can be reconstructed (Figure 2). In particular, the VAE infers binary latent variables that controls whether each state in the sequence is used by the generative decoder, i.e., whether a state is “important” or not. + +Binary Latent VAE The VAE consists of an inference, a generative and a prior network. These are structured as follows: the input to the inference network $q _ { \phi }$ is a trajectory of state-action pairs observed from the environment ${ \tau } = \{ ( s _ { t } , a _ { t } ) \} _ { t = 0 } ^ { T }$ , with ${ \pmb s } = \{ { \boldsymbol s } _ { t } \} _ { t = 0 } ^ { T }$ and $\pmb { a } { = } \{ a _ { t } \} _ { t = 0 } ^ { T }$ denoting the state and action sequences respectively. The output of the inference network is the approximated posterior over a sequence ${ z } = \{ z _ { t } \} _ { t = 0 } ^ { \bar { T } }$ of binary latent variables, denoted as $\varphi _ { \phi } ( \boldsymbol { z } | \boldsymbol { a } , \boldsymbol { s } )$ . The generative network $p _ { \theta }$ computes a distribution over the full action sequence $\textbf { \em a }$ using the masked state sequence, where $s _ { t }$ is masked if $z _ { t } { = } 0$ (we fix $z _ { 0 } { = } z _ { T } { = } 1$ during training), denoted as $p _ { \theta } ( { \pmb a } | { \pmb s } , z )$ . + +Finally, a state-conditioned $p _ { \psi } ( z _ { t } | s _ { t } )$ given by the prior network $p _ { \psi }$ for each $s _ { t }$ encodes the empirical average probability that state $s _ { t }$ is activated for reconstruction. This choice encourages inference to select within a consistent subset of states for use in action reconstruction. In particular, the waypoint states $\nu _ { p }$ are chosen as the states with the largest prior means and during training, once every few iterations, $\nu _ { p }$ is updated based on the current prior network. + +
Algorithm 1: Identifying waypoint states Vp and learning a goal-conditioned policy g Result: Waypoint states Vp and a goal-conditioned policy π g
Initialize network parameters for the recurrent variational inference model V Initialize network parameters for the goal-conditioned policy g Initialize Vp with the initial position of the agent,i.e.Vp = {so = (1,1)}
while VAE reconstruction error has not converged do
forn←1toNdo Sample random waypoint sp ∈ Vp
Navigate agent to sp and perform T-step rollout using a randow walk policy: T𝑛 ←{(s0= Sp,ao),.,(sT,ar)}
gn←ST Navigate agent to Sp and perform T-step rollout using Tg with goal gn:
Tπ ←{(s= Sp,ao),.,(sT,ar)}at~πg(-st,9n) Re-label πg rewards with action reconstruction error as curiosity bonus:
rπ←{1st+1=n-λ·p(at|s,z)}=0
end
Perform policy gradient update of πg using T" and rπ
Update V using T and T
Update Vp as set of states with largest prior mean αs αs+βs
+ +Objective Formally, we optimize the VAE using the following evidence lower bound (ELBO): + +$$ +\mathrm { E L B O } = \mathbb { E } _ { q _ { \phi } ( z | a , s ) } \left[ \log p _ { \theta } ( a | s , z ) \right] - D _ { \mathrm { K L } } \left( q _ { \phi } ( z | a , s ) | p _ { \psi } ( z | s ) \right) . +$$ + +To ensure differentiablity, we apply a continuous relaxation over the discrete $z _ { t }$ . We use the Beta distribution $p _ { \psi } ( z _ { t } ) = \mathrm { B e t a } ( \alpha _ { t } , \beta _ { t } )$ for the prior and the Hard Kumaraswamy distribution $q _ { \psi } ( z _ { t } | { a } , { z } ) = \mathrm { H a r d K u m a } ( \tilde { \alpha } _ { t } , \tilde { \beta } _ { t } )$ for the approximate posterior, which resembles the Beta distribution but is outside the exponential family (Bastings et al., 2019). This choice allows us to sample 0s and 1s without sacrificing differentiability, accomplished via the stretch-and-rectify procedure (Bastings et al., 2019; Louizos et al., 2017) and the reparametrization trick (Kingma & Welling, 2013). Lastly, to prevent the trivial solution of using all states for reconstruction, we use a secondary objective $\mathcal { L } _ { 0 }$ to regularize the $L _ { 0 }$ norm of $_ z$ at a targeted value $\mu _ { 0 }$ (Louizos et al., 2017; Bastings et al., 2019), the desired number of selected states out of $T$ steps, e.g. for when $T = 2 5$ , we set $\mu _ { 0 } = 5$ , meaning ideally 5 out of 25 states are activated for action reconstruction. Another term $\mathcal { L } _ { T }$ to encourage temporal separation between selected states by targeting the number of $0 / 1$ switches among $_ z$ at $2 \mu _ { 0 }$ : + +$$ +\mathcal { L } _ { 0 } = \Big | \Big | \mathbb { E } _ { q _ { \phi } ( z | s , a ) } [ \| z \| _ { 0 } ] - \mu _ { 0 } \Big | \Big | ^ { 2 } , \quad \mathcal { L } _ { T } = \Bigg | \Bigg | \mathbb { E } _ { q _ { \phi } ( z | s , a ) } \left[ \sum _ { t = 0 } ^ { T } \mathbb { 1 } [ z _ { t } \neq z _ { t + 1 } ] \right] - 2 \mu _ { 0 } \Bigg | \Bigg | ^ { 2 } . +$$ + +See Appendix A for details on training the VAE with binary $z _ { t }$ , including integration of the Hard Kumaraswamy distribution and how to regularize the statistics of $_ z$ . + +# 3.2 EXPLORATION FOR WORLD GRAPH DISCOVERY + +Naturally, the latent structure learned by the VAE depends on the trajectories used to train it. Hence, collecting a rich set of trajectories is crucial. Here, we propose a strategy to bootstrap a useful set of trajectories by alternately exploring the environment based on the current iteration’s $\nu _ { p }$ and updating the VAE and $\nu _ { p }$ , repeating this cycle until the action reconstruction accuracy plateaus (Algorithm 1). + +During exploration, we use action replay to navigate the agent to a state drawn from the current iteration’s $\nu _ { p }$ . Although resetting via action replay assumes our underneath environment to be deterministic, in cases where this resetting strategy is infeasible, it may be modified so long as to allow the exploration starting points to expand as the agent discovers more of its environment. For each such starting point, we collect two rollouts. In the first rollout, we perform a random walk to explore the nearby region. In the second rollout, we perform actions using a goal-conditioned policy $\pi _ { g }$ (GCP), setting the final state reached by the random walk as the goal. Both rollouts are used for trianing the VAE and the latter is also used for training $\pi _ { g }$ . + +![](images/0c4516ced121df0c03431d831806e1817ee32a043d24758d20a0950c4f69de1d.jpg) +Figure 3: Left: a standard Feudal Network. Right: using Wide-then-Narrow goals. The Manager first outputs a waypoint state as the wide goal $g ^ { w }$ , then attends to a closer-up area around $g ^ { w }$ to narrow down the final goal $g ^ { n }$ . + +GCP provides a venue to integrate intrinsic motivation, such as curiosity (Burda et al., 2018; Achiam & Sastry, 2017; Pathak et al., 2017; Azar et al., 2019) to generate more diverse rollouts. Specifically, we use the action reconstruction error of the VAE as an intrinsic reward signal when training $\pi _ { g }$ . This choice of curioisty also prevents the VAE from collapsing to the simple behaviors of a vanilla $\pi _ { g }$ . + +# 3.3 EDGE FORMATION + +The final stage is to construct the edges of $\mathcal { G } _ { w }$ , which should ideally capture the environment dynamics, i.e. how to transition between waypoint states. Once VAE training is complete and $\nu _ { p }$ is fixed, we collect random walk rollouts from each of the waypoints $s _ { p } \in \mathcal { V } _ { p }$ to estimate the underlying adjacency matrix (Biggs, 1993). More precisely, we claim a directed edge $s _ { p } \to s _ { q }$ if there exists a random walk trajectory from $s _ { p }$ to $s _ { q }$ that does not intersect a third waypoint. We also consider paths taken by $\pi _ { g }$ (starting at $s _ { p }$ and setting $s _ { q }$ as the goal) and keep the shortest observed path from $s _ { p }$ to $s _ { q }$ as a world graph edge transition. We use the action sequence length of the edge transition between adjacent waypoints as the weight of the edge. As shown experimentally, a key benefit of our approach is the ability to plan over $\mathcal { G } _ { w }$ . To navigate from one waypoint to another, we can use dynamic programming (Sutton, 1998; Feng et al., 2004) to output the optimal traversal of the graph. + +# 4 ACCELERATING REINFORCEMENT LEARNING WITH WORLD GRAPHS + +World graphs present a high-level, task-agnostic abstraction of the environment through waypoints and feasible transition routes between them. A key example of world graph applications for taskspecific RL is structured exploration: instead of exploring the entire environment, RL agents can use world graphs to quickly identify task-relevant regions and bias low-level exploration to these regions. Our framework to leverage world graphs for structured exploration consists of two parts: + +1. Hierarchical RL wherein the high-level policy selects subgoals from $\nu _ { p }$ . +2. Traversals using world graph edges. + +# 4.1 HIERARCHICAL RL OVER WORLD GRAPHS + +Formally, an RL agent learning to solve a task is formulated as a Markov Decision Process: at time $t$ , the agent is in a state $s _ { t }$ , executes an action $a _ { t }$ via a policy $\pi ( a _ { t } | s _ { t } )$ and receives a rewards $r _ { t }$ . The agent’s goal is to maximize its cumulative expected return $\begin{array} { r } { R = \mathbb { E } _ { ( s _ { t } , a _ { t } ) \sim \pi , p , p _ { 0 } } \left[ \sum _ { t \geq 0 } \gamma ^ { t } r _ { t } \right] } \end{array}$ , where $p ( s _ { t + 1 } | s _ { t } , a _ { t } ) , p _ { 0 } ( s _ { 0 } )$ are the transition and initial state distributions. + +To incorporate world graphs with RL, we use a hierarchical approach based on the Feudal Network (FN) (Dayan $\&$ Hinton, 1993; Vezhnevets et al., 2017), depicted in Figure 3. A standard FN + +# Task Description + +Task MultiGoal + +# Environment Characteristics + +Balls are located randomly, dense reward. + +# MultiGoal-Sparse + +Collect randomly spawned balls, each ball gives $+ 1$ reward. To end an episode, the agent has to exit at a designated point. Agents receive a single reward $r \leq 1$ proportional to the number of balls collected upon exiting. + +MultiGoalStochastic Door-Key + +Balls are located randomly, sparse reward. + +Spawn lava blocks at random locations each time step that immediately terminates the episode if stepped on. Agent has to pick up a key to open a door (reward $+ 1 \AA$ and reach the exit point on the other side (reward $+ 1$ ). + +Stochastic environment. Multiple objects: lava and balls are randomly located, dense reward. +Walls, door and key are located randomly. Agents have additional actions: pick and toggle. + +Table 1: An overview of tasks used to evaluate the benefit of using world graphs. Visualizations can be found in Appendix D. + +decomposes the policy of the agent into two separate policies that receive distinct streams of reward: a high-level policy (“Manager”) learns to propose subgoals; a low-level policy (“Worker”) receives subgoals from the Manager as inputs and is rewarded for taking actions in the environment that reach the subgoals. The Manager receives the environment reward defined by the task and therefore must learn to emit subgoals that lead to task completion. The Manager and Worker do not share weights and operate at different temporal resolutions: the Manager only outputs a new subgoal if either the Worker reaches the chosen one or a subgoal horizon $c$ is exceeded. + +For all our experiments, policies are trained using advantage actor-critic (A2C), an on-policy RL algorithm (Wu & Tian, 2016; Pane et al., 2016; Mnih et al., 2016b). To ease optimization, the feature extraction layers of the Manager and Worker that encode $s _ { t }$ are initialized with the corresponding layers from $\pi _ { g }$ , the GCP learned during world graph discovery phase. More details are in Appendix B. + +# 4.2 WIDE-THEN-NARROW GOALS AND WORLD GRAPHS + +To incorporate the world graph, we introduce a Manager policy that factorizes subgoal selection as follows: a wide policy $\pi ^ { w } ( g _ { t } ^ { w } | s _ { t } )$ selects a waypoint state as the wide goal $g ^ { w } \in \mathcal { V } _ { p }$ , and a narrow policy $\pi ^ { n } ( g _ { t } ^ { n } | s _ { t } , g _ { t } ^ { w } )$ selects a state within a local neighborhood of $g _ { t } ^ { w }$ , i.e. its $\epsilon$ -net (Mahadevan $\&$ Maggioni, 2007), as the narrow goal $g ^ { n } \in \{ s : \mathcal { D } ( s , \bar { g } _ { t } ^ { w } ) \leq \epsilon \}$ . The Worker policy $\pi ^ { \mathrm { w o r k e r } } ( a _ { t } | s _ { t } , g _ { t } ^ { n } , g _ { t } ^ { \bar { w } } )$ chooses the action taken by the agent given the current state and the wide and narrow goals from the Manager. A visual illustration is in Figure 4 and training details in Appendix C.2. + +# 4.3 WORLD GRAPH TRAVERSAL + +The wide-then-narrow subgoal format simplifies the search space for the Manager policy. Using waypoints as wide goals also makes it possible to leverage the edges of the world graph for planning and executing the planned traversals. This process breaks down as follows: + +1. When to Traverse: When the agent encounters a waypoint state $s _ { t } \in \mathcal V _ { p }$ , a “traversal” is initiated if $s _ { t }$ has a feasible connection in $\mathcal { G } _ { w }$ to the active wide goal $g _ { t } ^ { w }$ . 2. Planning: Upon triggering a traversal, the optimal traversal route from the initiating state to $g _ { t } ^ { w }$ is estimated from the $\mathcal { G } _ { w }$ edge weights using classic dynamic programming planning (Sutton, 1998; Feng et al., 2004). This yields a sequence of intermediate waypoint states. 3. Execution: Execution of graph traversals depends on the nature of the environment. If deterministic, the agent simply follows the action sequences given by the edges of the traversal. Otherwise, the agent uses the pretrained $\mathbf { G C P } \pi _ { g }$ to sequentially reach each of the intermediate waypoint states along the traversal (we fine-tune $\pi _ { g }$ in parallel where applicable). If the agent fails to reach the next waypoint state within a certain time limit, it stops its current pursuit and a new $( g ^ { w } , g ^ { n } )$ pair is received from the Manager. + +World graph traversal allows the Manager to assign task-relevant wide goals $g ^ { w }$ that can be far away from the agent yet still reachable, which consequentially accelerates learning by focusing exploration around the task-relevant region near $g ^ { w }$ . + +# 5 EXPERIMENTAL VALIDATION + +We now assess each component of our framework on a set of challenging 2D grid worlds. Our ablation studies demonstrate the following benefits of our framework: + +Table 2: On a variety of tasks and environment setups, we evaluate RL models trained with GCP $\pi _ { g }$ initialization, with $\mathcal { G } _ { w }$ world graph travresal, and with both. All models on the right are equipped with WN. Left are baselines for additional comparison. We report final rewards for MultiGoal tasks and success rates for Door-Key are reported. If no result reported, the agent failed to solve the task. + +
TaskSizeA2CFN+πg initOurs
+πg-init+ 9w-traversal+πg-init+Gw-traversal
MultiGoalSmall2.04±0.052.93±0.745.25±0.133.92±0.225.05±0.03
Medium==5.15±0.112.56±0.093.00±0.90
Larger==-2.18±0.122.72±0.59
MultiGoal-SparseSmall==0.39±0.090.24±0.040.42±0.07
Medium0.20±0.040.25±0.03
Larger0.16±0.220.26±0.11
MultiGoal-StochasticSmall1.38±1.201.93±0.163.06±0.312.92±0.45
Medium=2.99±0.122.42±0.242.64±0.14
Larger===0.60±0.12
Door-KeySmall=0.99±0.000.37±0.150.92±0.02
Medium0.56±0.020.76±0.06
Larger=0.26±0.19
+ +Table 3: Comparing learned $\nu _ { p }$ versus random $\mathcal { V } _ { \mathrm { r a n d } }$ as wide subgoals on large mazes, all trained with $\pi _ { g }$ initialization and graph traversal. $\nu _ { p }$ generally is superior in terms of performance and consistency. We report final rewards for MultiGoal tasks and success rates for Door-Key are reported. + +
Waypoint typeMultiGoalMultiGoal-SparseMultiGoal-StochasticDoor-Key
Learned2.72±0.590.26±0.110.60±0.120.26±0.19
Random2.30±0.490.19±0.110.41±0.250.27±0.40
+ +• It improves sample efficiency and performance over the baseline HRL model. • It benefits tasks varying in envirionment scale, task type, reward structure, and stochasticity. • The identified waypoints provide superior world representations for solving downstream tasks, as compared to graphs using randomly selected states as nodes. + +Implementation details, snippets of the tasks and mazes are in Appendix C-D. + +# 5.1 ABLATION STUDIES ON 2D GRID WORLDS + +For our ablation studies, we construct 2D grid worlds of increasing sizes (small, medium and large) along with challenging tasks with different reward structures, levels of stochasticity and logic (summarized in Table 1). In all tasks, every action taken by the agent receives a negative reward penalty. We follow a rigorous evaluation protocol (Wu et al., 2017; Ostrovski et al., 2017; Henderson et al., 2018): each experiment is repeated with 3 training seeds. 10 additional validation seeds are used to pick the model with the best reward performance. This model is then tested on 100 testing seeds. We report mean reward and standard deviation. + +We ablate each of the following components in our framework and compare against non-hierarchical (A2C) and hierarchical baselines (FN): + +1. initializing the feature extraction layers of the Manager and Worker from $\pi _ { g }$ , +2. applying Wide-then-Narrow Manager (WN) goal instruction, and +3. allowing the Worker to traverse along $\mathcal { G } _ { w }$ . + +Results are shown in Table 2. In sum, each component improves performance over the baselines. + +Wide and narrow goals Using two goal types is a highly effective way to structure the Manager instructions and enables the Worker to differentiate the transition and local task-solving phases. We note that for small MultiGoal, agents do not benefit much from $\mathcal { G } _ { w }$ traversal: it can rely solely on the guidance from WN goals to master both phases. However with increasing maze size, the Worker struggles to master traversals on its own and thus fails solving the tasks. + +World Graph Traversal As conjectured in Section 4.3, the performance gain of our framework can be explained by the larger range and more targeted exploration strategy. In addition, the Worker does not have to learn long distance transitions with the aid of $\mathcal { G } _ { w }$ traversals. Figure 4 confirms that $\mathcal { G } _ { w }$ traversal speeds up convergence and its effect becomes more evident with larger mazes. Note that the graph learning stage only need 2.4K iterations to converge. Even when taking these additional environment interactions into account, $\mathcal { G } _ { w }$ traversal still exhibits superior sample efficiency, not to mention that the graph is shared among all tasks. Moreover, solving Door-Key involves a complex combination of sub-tasks: find and pick up the key, reach and open the door and finally exit. With limited reward feedback, this is particularly difficult to learn. The ability to traverse along $\mathcal { G } _ { w }$ enables longer-horizon planning on top of the waypoints, thanks to which the agents boost the success rate on medium Door-Key from $0 . 5 6 { \pm } 0 . 0 2$ to $0 . 7 5 { \pm } 0 . 0 6 $ . + +![](images/eeccd173a66cc7f083254caca6b3d6fe078ddac17830e5c24ef8e6d8706c4b0b.jpg) +Figure 4: Validation performance during training (mean and standard-deviation of reward, 3 seeds) for MultiGoal. Left: Comparing $\nu _ { p }$ and $\mathcal { V } _ { \mathrm { r a n d } }$ , with or without traversal, all models use WN and $\pi _ { g }$ initialization. We see that 1) traversal speeds up convergence, 2) $\mathcal { V } _ { \mathrm { r a n d } }$ gives higher variance and slightly worse performance than $\nu _ { p }$ . Right: comparing with or without $\pi _ { g }$ initialization on $\nu _ { p }$ , all models use WN. We see that initializing the task-specific phase with the task-agnostic goal-conditioned policy significantly boosts learning. + +Benefits of Learned Waypoints To highlight the benefit of establishing the waypoints learned by the VAE as nodes for $\mathcal { G } _ { w }$ , we compare against results using a $\mathcal { G } _ { w }$ constructed around randomly selected states $( \nu _ { \mathrm { r a n d } } )$ . The edges of the random-node graph are formed in the same way as described in Section 3.3 and its feature extractor is also initialized from $\pi _ { g }$ . Although granting knowledge acquired during the unsupervised phase to $\mathcal { V } _ { \mathrm { r a n d } }$ is unfair to $\nu _ { p }$ , deploying both initialization and traversal while only varying $\mathcal { V } _ { \mathrm { r a n d } }$ and $\nu _ { p }$ isolates the effect from the nodes to the best extent. The comparative results (in Table 3, learning curves for MultiGoal in Figure 4) suggest $\nu _ { p }$ generally outperforms $\mathcal { V } _ { \mathrm { r a n d } }$ . Door-Key is the only task in which the two matches. However, $\mathcal { V } _ { \mathrm { r a n d } }$ exhibits a large variance, implying that certain sets of random states can be suitable for this task, but using learned waypoints gives strong performance more consistently. + +Initialization with GCP Initializing the weights of the Worker and Manager feature extractors from $\pi _ { g }$ (learned during the task-agnostic phase) consistently benefits learning.In fact, we observe that models starting from scratch fail on almost all tasks within the maximal number of training iterations, unless coupled with $\mathcal { G } _ { w }$ traversal, which is still inferior to using $\pi _ { g }$ -initialization. Particularly, for the small MultiGoal-Stochastic environment, there is a high chance that a lava square blocks traversal; therefore, without the environment knowledge from $\pi _ { g }$ transferred by weight initialization, the interference created by the episode-terminating lava prevents the agent from learning the task. + +# 6 CONCLUSION + +We have shown that world graphs are powerful environment abstractions, which, in particular, are capable of accelerating reinforcement learning. Future works may extend their applications to more challenging RL setups, such as real-world multi-task learning and navigation. It is also interesting to generalize the proposed framework to learn dynamic world graphs for evolving environments, and applying world graphs to multi-agent problems, where agents become part of the world graphs of other agents. + +# REFERENCES + +Pieter Abbeel and Andrew $\mathrm { ~ Y ~ N ~ g ~ }$ . Apprenticeship learning via inverse reinforcement learning. 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Training agent for first-person shooter game with actor-critic curriculum learning. 2016. + +# A RECURRENT VAE WITH DIFFERENTIABLE BINARY LATENT VARIABLES + +As illustrated in the main text, the main objective for the recurrent VAE is the following evidence lower bound with derivation: + +$$ +\begin{array} { r l } & { \log p ( a | s ) = \log \int p ( a | s , z ) d z } \\ & { \qquad = \log \int p ( a | s , z ) p ( z | s ) \frac { q ( z | a , s ) } { q ( z | a , s ) } d z } \\ & { \qquad = \log \int p ( a | s , z ) \frac { p ( z | s ) } { q ( z | a , s ) } q ( z | a , s ) d z } \\ & { \qquad \geq \mathbb { E } _ { q ( z | a , s ) } [ \log p ( a | s , z ) - \log \frac { q ( z | a , s ) } { p ( z | s ) } ] } \\ & { \qquad = \mathbb { E } _ { q ( z | a , s ) } [ \log p ( a | s , z ) ] - D _ { \mathrm { K L } } ( q ( z | a , s ) | | p ( z | s ) ) } \end{array} +$$ + +The inference network $q _ { \psi }$ takes in the trajectories of state-action pairs $\tau$ and at each time step approximates the posterior of the corresponding latent variable $z _ { t }$ . The prior network $p _ { \psi }$ takes the state $s _ { t }$ at each time step and outputs the state-conditioned prior $p _ { \psi } ( s _ { t } )$ . We choose Beta as the prior distribution and the Hard Kuma as the approximated posterior to relax the discrete latent variables to continuous surrogates. + +The Kuma distribution $\mathrm { K u m a } ( \alpha , \beta )$ highly resembles the Beta Distribution in shape but does not come from the exponential family. Similar to Beta, the Kuma distribution also ranges from bimodal (when $\alpha \approx \beta )$ to unimodal $( \alpha / \beta \to 0$ or $\alpha / \beta \to \infty )$ ). Also, when $\alpha = 1$ or $\beta = 1$ , $\operatorname { K u m a } ( \alpha , \beta ) = \operatorname { B e t a } ( \alpha , \beta )$ . We observe empirically better performance when we fix $\beta = 1$ for the Kuma approximated posterior. One major advantage of the Kuma distribution is its simple Cumulative Distribution Function (CDF): + +$$ +F _ { \mathrm { K u m a } } ( x , \alpha , \beta ) = ( 1 - ( 1 - x ^ { \alpha } ) ) ^ { \beta } . +$$ + +It is therefore amendable to the reparametrization trick (Kingma & Welling, 2013; Rezende et al., 2014; Maddison et al., 2016) by sampling from uniform distribution $u \sim \mathcal { U } ( 0 , 1 )$ : + +$$ +z = F _ { \mathrm { K u m a } } ^ { - 1 } ( u ; \alpha , \beta ) \sim \mathrm { K u m a } ( \alpha , \beta ) . +$$ + +Lastly, the KL-divergence between the Kuma and Beta distributions can be approximated in closed form (Nalisnick & Smyth, 2016): + +$$ +\begin{array} { l } { \displaystyle { { \cal D } _ { \mathrm { K L } } ( \mathrm { K u m a } ( a , b ) | \mathrm { B e t a } ( \alpha , \beta ) ) = \frac { a - \alpha } { a } \left( - \gamma - \Psi ( b ) - \frac { 1 } { b } \right) } } \\ { \displaystyle { \phantom { \frac { b - a } { b - a } ( \mathrm { K u m a } ( a , b ) + \log \mathrm { B e t a } ( \alpha , \beta ) - \frac { b - 1 } { b } + ( \beta - 1 ) b \sum _ { m = 1 } ^ { \infty } \frac { 1 } { m + a b } \mathrm { B e t a } \left( \frac { m } { a } , b \right) , } } } \end{array} +$$ + +where $\Psi$ is the Digamma function, $\gamma$ the Euler constant, and the approximation uses the first few terms of the Taylor series expansion. We take the first 5 terms here. + +Next, we make the Kuma distribution “hard” by following the steps in Bastings et al. (2019). First stretch the support to $( r = 0 - \epsilon _ { 1 } , l = 1 + \epsilon _ { 2 }$ ), $\epsilon _ { 1 } , \epsilon _ { 2 } > 0$ , and the resulting CDF distribution takes the form: + +$$ +F _ { S } ( z ) = F _ { \mathrm { K u m a } } \left( { \frac { z - l } { r - l } } ; \alpha , \beta \right) . +$$ + +Then, the non-eligible probabilities for 0’s and 1’s are attained by rectifying all samples below 0 to 0 and above 1 to 1, and other value as it is, that is + +$$ +P ( z = 0 ) = F _ { \mathrm { K u m a } } \left( \frac { - l } { r - l } ; \alpha , \beta \right) , \quad P ( z = 1 ) = 1 - F _ { \mathrm { K u m a } } \left( \frac { 1 - l } { r - l } ; \alpha , \beta \right) . +$$ + +Lastly, we impose two additional regularization terms ${ \mathcal { L } } _ { \prime }$ and $\mathcal { L } _ { T }$ on the approximated posteriors. As described in the main text, ${ \mathcal { L } } _ { \prime }$ prevents the model from selecting all states to reconstruct $\{ a _ { t } \} _ { 0 } ^ { T - 1 }$ by restraining the expected $L _ { 0 }$ norm of $z = \left( z _ { 1 } \cdot \cdot \cdot z _ { T - 1 } \right)$ to approximately be at a targeted value $\mu _ { 0 }$ (Louizos et al., 2017; Bastings et al., 2019). In other words, this objective adds the constraint that there should be $\mu _ { 0 }$ of activated $z _ { t } = 1$ given a sequence of length $T$ . The other term $\mathcal { L } _ { T }$ encourages temporally isolated activation of $z _ { t }$ , meaning the number of transition between 0 and 1 among $z _ { t }$ ’s should roughly be $2 \mu _ { 0 }$ . Note that both expectations in Equation 2 have closed forms for HardKuma. + +$$ +\begin{array} { l } { \displaystyle \mathcal { L } _ { 0 } = \left\| \mathbb { E } _ { q ( \boldsymbol { z } | \boldsymbol { s } , \boldsymbol { a } ) } \left[ \left\| \boldsymbol { z } \right\| _ { 0 } \right] - \mu _ { 0 } \right\| ^ { 2 } , \mathrm { w h e r e } } \\ { \displaystyle \mathbb { E } _ { q ( \boldsymbol { z } | \boldsymbol { s } , \boldsymbol { a } ) } \left[ \left\| \boldsymbol { z } \right\| _ { 0 } \right] = \sum _ { t = 1 } ^ { T } \mathbb { E } _ { q ( \boldsymbol { z } _ { t } | \boldsymbol { s } , \boldsymbol { a } ) } \left[ \mathbb { 1 } _ { \boldsymbol { z } _ { t } \neq 0 } \right] } \\ { \displaystyle \qquad = \sum _ { t = 1 } ^ { T } 1 - p \left( \boldsymbol { z } _ { t } = 0 \right) = \sum _ { t = 1 } ^ { T } 1 - F _ { \mathrm { K u m a } } \left( \frac { - l } { r - l } ; \alpha _ { t } , \beta _ { t } \right) , } \end{array} +$$ + +$$ +\mathbb { E } _ { q ( z | s , a ) } [ \sum _ { t = 1 } ^ { T - 1 } \mathbb { 1 } _ { z _ { t } \neq z _ { t + 1 } } ] = \sum _ { t = 1 } ^ { T - 1 } p \left( z _ { t } = 0 \right) \left( 1 - p \left( z _ { t + 1 } = 0 \right) \right) + \left( 1 - p \left( z _ { t } = 0 \right) \right) p \left( z _ { t + 1 } = 0 \right) . +$$ + +Lagrangian Relaxation. The overall optimization objective consists of action sequence reconstruction, KL-divergence between the posterior and prior, $\mathcal { L } _ { 0 }$ and $\mathcal { L } _ { T }$ (Equation 12). We tune the objective weights $\lambda _ { i }$ using Lagrangian relaxation (Higgins et al., 2017; Bastings et al., 2019; Bertsekas, 1999), treating $\lambda _ { i }$ ’s as learnable parameters and performing alternative optimization between $\lambda _ { i }$ ’s and the model parameters. We observe that as long as their initialization is within a reasonable range, $\lambda _ { i }$ ’s converge to a local optimum: + +$$ +\operatorname* { m a x } _ { \{ \lambda _ { 1 } , 2 , 3 \} } \operatorname* { m i n } _ { \substack { \left\{ \theta , \phi , \psi \right\} } } - \mathbb { E } _ { q _ { \psi } ( z | a , s ) } \left[ \log p _ { \theta } ( a | s , z ) \right] + \lambda _ { 1 } D _ { \mathrm { K L } } \left( q _ { \phi } ( z | a , s ) | p _ { \psi } ( z | s ) \right) + \lambda _ { 2 } \mathcal { L } _ { 0 } + \lambda _ { 3 } \mathcal { L } _ { T } . +$$ + +We observe this approach to produce efficient and stable mini-batch training. + +# B GOAL-CONDITIONED POLICY INITIALIZATION FOR HRL + +Optimizing composite neural networks like HRL (Co-Reyes et al., 2018) is sensitive to weight initialization (Mishkin & Matas, 2015; Le et al., 2015), due to its complexity and lack of clear supervision at various levels. Therefore, taking inspiration from prevailing pre-training procedures in computer vision (Russakovsky et al., 2015; Donahue et al., 2014) and NLP (Devlin et al., 2018; Radford et al., 2019), we take advantage of the weights learned by $\pi _ { g }$ during world graph discovery when initializing the Worker and Manager policies for downstream HRL, as $\pi _ { g }$ has already implicitly embodied much environment dynamics information. + +More specifically, we extract the weights of the feature extractor, i.e. the state encoder, and use them as the initial weights for the state encoders of the HRL policies. Our empirical results demonstrate that such weight initialization consistently improves performance and validates the value of skill/knowledge transfer from GCP (Taylor & Stone, 2009; Barreto et al., 2017). + +# C ADDITIONAL IMPLEMENTATION DETAILS + +Model code folder including all architecture details is shared in comment. + +# C.1 HYPERPARAMETERS FOR VAE TRAINING + +Our models are optimized with Adam (Kingma & Ba, 2014) using mini-batches of size 128, thus spawning 128 asynchronous agents to explore. We use an initial learning rate of 0.0001, with $\bar { \epsilon } = 0 . 0 0 \bar { 1 } , \beta _ { 1 } = \bar { 0 } . 9 , \beta _ { 2 } = 0 . 9 9 \bar { 9 }$ ; gradients are clipped to 40 for inference and generation nets. For HardKuma, we set $l = - 0 . 1$ and $r = 1 . 1$ . The maximum sequence length for BiLSTM is 25. The total number of training iterations is 3600 and model usually converges around 2400 iterations. We train the prior, inference, and generation networks end-to-end. + +We initialize $\lambda _ { i }$ ’s (see Lagrangian Relaxation) to be $\lambda _ { 1 } = 0 . 0 1$ (KL-divergence), $\begin{array} { r } { , \lambda _ { 2 } = 0 . 0 6 ( \mathcal { L } _ { 0 } ) . } \end{array}$ $\lambda _ { 3 } = 0 . 0 2 ( \mathcal { L } _ { T } )$ . After each update of the latent model, we update $\lambda _ { i }$ ’s, whose initial learning rate is 0.0005, by maximizing the original objective in a similar way as using Lagrangian Multiplier. At the end of optimization, $\lambda _ { i }$ ’s converge to locally optimal values. For example, with the medium maze, $\lambda _ { 1 } = 0 . 0 6 7$ for the KL-term, $\lambda _ { 2 } = 0 . 0 7 0$ for the $\mathcal { L } _ { 0 }$ and $\lambda _ { 3 } = 0 . 0 5 1$ for the $\mathcal { L } _ { T }$ term. The total number of waypoints $| \nu _ { p } |$ is set to be $2 0 \%$ of the size of the full state space. + +# C.2 TRAINING HRL MODELS + +The procedure of the Manager and the Worker in sending/receiving orders using either traversal paths among $\nu _ { p }$ from replay buffer for deterministic environments or with $\pi _ { g }$ for stochastic ones follows: + +1. The Manager gives a wide-narrow subgoal pair $( g _ { w } , g _ { n } )$ . +2. The agent takes action based on the Worker policy $\pi ^ { \omega }$ conditioned on $( g _ { w } , g _ { n } )$ and reaches a new state $s ^ { \prime }$ . If $s ^ { \prime } \in \mathcal { V } _ { p }$ , $g _ { w }$ has not yet met, and there exists a valid path basing on the edge paths from the world graph $s ^ { \prime } \to g _ { w }$ , agent then either follows replay actions or $\pi _ { g }$ to reach $g _ { w }$ . If $\pi _ { g }$ still does not reach desired destination in a certain steps, then stop the agent wherever it stands; also $\pi _ { g }$ can be finetuned here. +3. The Worker receives positive reward for reaching $g _ { w }$ for the first time. +4. If agent reaches $g _ { n }$ , the Worker also receives positive rewards and terminates this horizon. +5. The Worker receives negative for every action taken except for during traversal; the Manager receives negative reward for every action taken including traversal. +6. When either $g _ { n }$ is reached or the maximum time step for this horizon is met, the Manager renews its subgoal pair. + +The training of the Worker policy $\pi ^ { \omega }$ follows the same A2C algorithm as $\pi _ { g }$ + +The training of the Manager policy $\pi ^ { m }$ also follows a similar procedure but as it operates at a lower temporal resolution, its value function regresses against the $t _ { m }$ -step discounted reward where $t _ { m }$ covers all actions and rewards generated from the Worker. + +When using the Wide-then-Narrow instruction, the policy gradient for the Manager policy $\pi _ { m }$ becomes: + +$\begin{array} { r } { \boldsymbol { \mathbb { E } } _ { ( s _ { t } , a _ { t } ) \sim \pi , p , p _ { 0 } } \left[ A _ { m , t } \nabla \log \left( \pi ^ { \omega } \left( g _ { w , t } | s _ { t } \right) \pi ^ { n } \left( g _ { n , t } | s _ { t } , g _ { w , t } , s _ { w , t } \right) \right) \right] + \nabla \left[ \mathcal { H } \left( \pi ^ { \omega } \right) + \mathcal { H } \left( \pi ^ { n } ( \cdot | g _ { w , t } ) \right) \right] , } \end{array}$ where $A _ { m , t }$ is the Manager’s advantage at time $t$ . Also, for Manager, as the size of the action space scales linearly with $| S |$ , the exact entropy for the $\pi ^ { m }$ can easily become intractable. Essentially there are $O$ $^ { \prime } \left( | \mathcal { V } | \times \left( N ^ { 2 } \right) \right)$ possible actions. To calculate the entropy exactly, all of them has to be summed, making it easily computationally intractable: + +$$ +\mathcal { H } = \sum _ { w \in \mathcal { V } } \sum _ { w _ { n } \in s _ { w } } \pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \pi ^ { \omega } ( w | s _ { t } ) \log { \nabla \pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \pi ^ { \omega } ( w | s _ { t } ) } . +$$ + +Thus in practice we resort to an effective alternative $\mathcal { H } \left( \pi ^ { \omega } \right) + \mathcal { H } \left( \pi ^ { n } ( \cdot | g _ { w , t } ) \right)$ . + +Psuedo-code for Manager training is in Algorithm 2. + +# C.3 HYPERPARAMETERS FOR HRL + +For training the HRL policies, we inherit most hyperparameters from those used when training $\pi _ { g }$ , as the Manager and the Worker both share similar architectures with $\pi _ { g }$ . The hyperparameters used when training $\pi _ { g }$ follow those from Shang et al. (2019). Because the tasks used in HRL experiments are more difficult than the generic goal-reaching task, we set the maximal number of training iterations to 100K abd training is stopped early if model performance reaches a plateau. The rollout steps for each iteration is 60. Hyperparameters specific to HRL are the horizon $c = 2 0$ and the size of the Manager’s local attention range (that is, the neighborhood around $g ^ { w }$ within which $g ^ { n }$ is selected), which are $N = 5$ for small and medium mazes, and $N = 7$ for the large maze. + +Algorithm 2: Training of $\pi ^ { m }$ for HRL models + +
Clear gradients dθ ←O; while t <= tmax or episode not terminated do Simulate under current policy πm,t-1, πω,t-1; if the Worker has met the previous subgoal or exceeded the horizon c thenReset the set of time steps where πm,t omits a new subgoal Sm = {} and tm = 0.;
Sample a new subgoal gm,t from πm,t; end
Zm,t = fLsTM(CNN(sm,t,sv),hm,tm),Vm,t = fu(zm,t),Tt = fp(2m,t) ;
Sm= SmU {tm} and tm =t;
O, if terminal
Vtmax+1, otherwise
for t = tmax,...1 do R←rt+γR;
if t ∈ Sm then
Am,t ←R-Vm,t;
Accumulate gradients from value loss: dθ ← d0 + 入
Accumulate policy gradients with entropy regularization:80
d0 ← d0+ VlogTm,t(gm,t)Am,t + βVH(πm,t);
end
end
+ +# D 2D GRID WORLD VISUALIZATIONS + +![](images/e873103b674e3b139b1dde9c9a9cc1ebfed8b9e0f0f4150e66c98c1ba56795b4.jpg) +Figure 5: Visualization of the 2D grid environments in our experiments, along with the learned waypoints in blue. + +![](images/86812f5f54acb2108da0bc58155e8700f4575fcfaa3ed83f6a9417d89e4ae0b7.jpg) +Figure 6: Visualization of tasks in our experiments. \ No newline at end of file diff --git a/parse/train/BkgRe1SFDS/BkgRe1SFDS_content_list.json b/parse/train/BkgRe1SFDS/BkgRe1SFDS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..95c8ae23d658b7daf1f844838c9f5a59779d5725 --- /dev/null +++ b/parse/train/BkgRe1SFDS/BkgRe1SFDS_content_list.json @@ -0,0 +1,2338 @@ +[ + { + "type": "text", + "text": "LEARNING WORLD GRAPH DECOMPOSITIONS TO ACCELERATE REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 728, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Efficiently learning to solve tasks in complex environments is a key challenge for reinforcement learning (RL) agents. We propose to decompose a complex environment using a task-agnostic world graphs, an abstraction that accelerates learning by enabling agents to focus exploration on a subspace of the environment. The nodes of a world graph are important waypoint states and edges represent feasible traversals between them. Our framework has two learning phases: 1) identifying world graph nodes and edges by training a binary recurrent variational autoencoder (VAE) on trajectory data and 2) a hierarchical RL framework that leverages structural and connectivity knowledge from the learned world graph to bias exploration towards task-relevant waypoints and regions. We thoroughly evaluate our approach on a suite of challenging maze tasks and show that using world graphs significantly accelerates RL, achieving higher reward and faster learning. ", + "bbox": [ + 233, + 263, + 766, + 444 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 468, + 336, + 483 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many real-world applications, e.g., self-driving cars and in-home robotics, require an autonomous agent to execute different tasks within a single environment that features, e.g. high-dimensional state space, complex world dynamics or structured layouts. In these settings, model-free reinforcement learning (RL) agents often struggle to learn efficiently, requiring a large amount of experience collections to converge to optimal behaviors. Intuitively, an agent could learn more efficiently by focusing its exploration in task-relevant regions, if it has knowledge of the high-level structure of the environment. ", + "bbox": [ + 174, + 497, + 825, + 595 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose a method to 1) learn and 2) use an environment decomposition in the form of a world graph, a task-agnostic abstraction. World graph nodes are waypoint states, a set of salient states that can summarize agent trajectories and provide meaningful starting points for efficient exploration (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018; Ghosh et al., 2018). The directed and weighted world graph edges characterize feasible traversals among the waypoints. To leverage the world graph, we model hierarchical RL (HRL) agents where a high-level policy chooses a waypoint state as a goal to guide exploration towards task-relevant regions, and a low-level policy strives to reach the chosen goals. ", + "bbox": [ + 174, + 602, + 825, + 713 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our framework consists of two phases. In the task-agnostic phase, we obtain world graphs by training a recurrent variational auto-encoder (VAE) (Chung et al., 2015; Gregor et al., 2015; Kingma & Welling, 2013) with binary latent variables (Nalisnick & Smyth, 2016) over trajectories collected using a random walk policy (Ha & Schmidhuber, 2018) and a curiosity-driven goal-conditioned policy (Ghosh et al., 2018; Nair et al., 2018). World graph nodes are states that are most frequently selected by the binary latent variables, while edges are inferred from empirical transition statistics between neighboring waypoints. In the task-specific phase, taking advantage of the learned world graph for structured exploration, we efficiently train an HRL model (Taylor & Stone, 2009). ", + "bbox": [ + 174, + 720, + 825, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In summary, our main contributions are: ", + "bbox": [ + 174, + 839, + 436, + 853 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "• A task-agnostic unsupervised approach to learn world graphs, using a recurrent VAE with binary latent variables and a curiosity-driven goal-conditioned policy. • An HRL scheme for the task-specific phase that features multi-goal selection (Wide-thenNarrow) and navigation via world graph traversal. ", + "bbox": [ + 214, + 864, + 826, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/87e8c713c76547993153ec99d35d40aae11b5121b86240dad4d5ac65a6d03b12.jpg", + "image_caption": [ + "Figure 1: Top Left: overall pipeline of our 2-phase framework. Top Right (world graph discovery): a subgraph exemplifies traversal between waypoint states (in blue), see Section 3 for more details. Bottom (Hierarhical $R L$ ): an example rollout from our proposed HRL policy with Wide-then-Narrow Manager instructions and world graph traversals, solving a challenging Door-Key task, see Section 4 for more details. " + ], + "image_footnote": [], + "bbox": [ + 184, + 106, + 805, + 372 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• Empirical evaluations on multiple tasks in complex 2D grid worlds to validate that our framework produces descriptive world graphs and significantly improves both sample efficiency and final performance on these tasks over baselines, especially thanks to transfer learning from the unsupervised phase and world graph traversal. ", + "bbox": [ + 218, + 443, + 825, + 500 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 523, + 344, + 540 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "An understanding of the environment and its dynamics is essential for effective planning and control in model-based RL. For example, a robotics agent often locates or navigates by interpreting a map (Lowry et al., 2015; Thrun, 1998; Angeli et al., 2008). Our exploration strategy draws inspiration from active localization, where robots are actively guided to investigate unfamiliar regions (Fox et al., 1998; Li et al., 2016). Besides mapping, recent works (Azar et al., 2019; Ha & Schmidhuber, 2018; Guo et al., 2018) learn to represent the world with generative latent states (Tian & Gong, 2017; Haarnoja et al., 2018; Racanière et al., 2017). If the latent dynamics are also extrapolated, the latent states can assist planning (Mnih et al., 2016a; Hafner et al., 2018) or model-based RL (Gregor & Besse, 2018; Kaiser et al., 2019). ", + "bbox": [ + 173, + 554, + 825, + 680 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While also aiming to model the world, we approach this as abstracting both the structure and dynamics of the environment in a graph representation, where nodes are states from the environment and edges encode actionable efficient transitions between nodes. Existing works (Metzen, 2013; Mannor et al., 2004; Eysenbach et al., 2019; Entezari et al., 2010) have shown benefits of such graph abstractions but typically select nodes only subject to a good coverage the observed state space. Instead, we identify a parsimonious subset of states that can summarize trajectories and provide more useful intermediate landmarks, i.e. waypoints, for navigating complex environments. ", + "bbox": [ + 174, + 686, + 825, + 784 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our method for estimating waypoint states can be viewed as performing automatic (sub)goal discovery. Subgoal and subpolicy learning are two major approaches to identify a set of temporally-extended actions, “skills”, that allow agents to efficiently learn to solve complex tasks. Subpolicy learning identifies policies useful to solve RL tasks, such as option-based methods (Daniel et al., 2016; Bacon et al., 2017) and subtask segmentations (Pertsch et al., 2019; Kipf et al., 2018). Subgoal learning, on the other hand, identifies “important states” to reach ( ¸Sim¸sek et al., 2005). ", + "bbox": [ + 174, + 791, + 825, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Previous works consider various definitions of “important” states: frequently visited states during successful task completions (Digney, 1998; McGovern & Barto, 2001), states introducing the most novel information (Goyal et al., 2019), bottleneck states connecting densely-populated regions (Chen et al., 2007; ¸Sim¸sek et al., 2005), or environment-specific heuristics (Ecoffet et al., 2019). Our work draws intuition from unsupervised temporal segmentation (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018) and imitation learning (Abbeel & $\\mathrm { N g }$ , 2004; Hussein et al., 2017). We define “important” states (waypoints) as the most critical states in recovering action sequences generated by some agent, which indicates that these states contain the richest information about the executed policy (Azar et al., 2019). ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/0b8b33e4d41c1c614e19d9ac24f4cdb5614a9211348b71ee40716b3df882e83c.jpg", + "image_caption": [ + "Figure 2: Our recurrent latent model with differentiable binary latent units to identify waypoint states. A prior network (left) learns the state-conditioned prior in Beta distribution, $p _ { \\psi } ( z _ { t } | s _ { t } ) { = } \\mathrm { B e t a } ( \\alpha _ { t } , \\beta _ { t } )$ . An inference encoder learns an approximate posterior in HardKuma distribution inferred from the state-action sequence input, $q _ { \\phi } ( z _ { t } | \\mathbf { a } , z ) { = } \\mathrm { H a r d } \\hat { \\mathrm { K } } \\mathrm { u m a } ( \\tilde { \\alpha _ { t } } , \\mathbf { \\hat { 1 } } )$ . A generation network $p _ { \\theta }$ reconstructs $\\textbf { \\em a }$ from $\\{ s _ { t } | z _ { t } = 1 \\}$ . " + ], + "image_footnote": [], + "bbox": [ + 207, + 98, + 790, + 251 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 323, + 825, + 406 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 LEARNING WORLD GRAPHS ", + "text_level": 1, + "bbox": [ + 176, + 429, + 441, + 445 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We propose a method for learning a world graph $\\mathcal { G } _ { w }$ , a task-agnostic abstraction of an environment that captures its high-level structure and dynamics. In this work, the primary use of world graphs is to accelerate reinforcement learning of downstream tasks. The nodes of $\\mathcal { G } _ { w }$ , denoted by a set of waypoints states $s _ { p } \\in \\mathcal { V } _ { p }$ , are generically “important” for accomplishing tasks within the environment, and therefore useful as starting points for exploration. Our method identifies such waypoint states from interactions with the environment. In addition, we embed feasible transitions between nearby waypoint states as the edges of $\\mathcal { G } _ { w }$ . ", + "bbox": [ + 174, + 463, + 825, + 561 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we define important states in the context of learning $\\mathcal { G } _ { w }$ (see Section 2 for alternative definitions). That is, we wish to discover a small set of states that, when used as world graph nodes, concisely summarize the structure and dynamics of the environment. Below, we describe 1) how to collect state-action trajectories and an unsupervised learning objective to identify world graph nodes, and 2) how the graph’s edges (i.e., how to transition between nodes) are formed from trajectories. ", + "bbox": [ + 174, + 568, + 825, + 637 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 WAYPOINT STATE IDENTIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 657, + 455, + 671 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The structure and dynamics of an environment are implicit in the state-action trajectories observed during exploration. To identify world graph nodes from such data, we train a recurrent variational autoencoder (VAE) that, given a sequence of state-action pairs, identifies a subset of the states in the sequence from which the full action sequence can be reconstructed (Figure 2). In particular, the VAE infers binary latent variables that controls whether each state in the sequence is used by the generative decoder, i.e., whether a state is “important” or not. ", + "bbox": [ + 174, + 684, + 825, + 768 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Binary Latent VAE The VAE consists of an inference, a generative and a prior network. These are structured as follows: the input to the inference network $q _ { \\phi }$ is a trajectory of state-action pairs observed from the environment ${ \\tau } = \\{ ( s _ { t } , a _ { t } ) \\} _ { t = 0 } ^ { T }$ , with ${ \\pmb s } = \\{ { \\boldsymbol s } _ { t } \\} _ { t = 0 } ^ { T }$ and $\\pmb { a } { = } \\{ a _ { t } \\} _ { t = 0 } ^ { T }$ denoting the state and action sequences respectively. The output of the inference network is the approximated posterior over a sequence ${ z } = \\{ z _ { t } \\} _ { t = 0 } ^ { \\bar { T } }$ of binary latent variables, denoted as $\\varphi _ { \\phi } ( \\boldsymbol { z } | \\boldsymbol { a } , \\boldsymbol { s } )$ . The generative network $p _ { \\theta }$ computes a distribution over the full action sequence $\\textbf { \\em a }$ using the masked state sequence, where $s _ { t }$ is masked if $z _ { t } { = } 0$ (we fix $z _ { 0 } { = } z _ { T } { = } 1$ during training), denoted as $p _ { \\theta } ( { \\pmb a } | { \\pmb s } , z )$ . ", + "bbox": [ + 173, + 775, + 825, + 876 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Finally, a state-conditioned $p _ { \\psi } ( z _ { t } | s _ { t } )$ given by the prior network $p _ { \\psi }$ for each $s _ { t }$ encodes the empirical average probability that state $s _ { t }$ is activated for reconstruction. This choice encourages inference to select within a consistent subset of states for use in action reconstruction. In particular, the waypoint states $\\nu _ { p }$ are chosen as the states with the largest prior means and during training, once every few iterations, $\\nu _ { p }$ is updated based on the current prior network. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/cbbc23f0b93d9dfd2b9ce60847869d9b56475c47b923ecf9289dfec065b51989.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm 1: Identifying waypoint states Vp and learning a goal-conditioned policy g Result: Waypoint states Vp and a goal-conditioned policy π g
Initialize network parameters for the recurrent variational inference model V Initialize network parameters for the goal-conditioned policy g Initialize Vp with the initial position of the agent,i.e.Vp = {so = (1,1)}
while VAE reconstruction error has not converged do
forn←1toNdo Sample random waypoint sp ∈ Vp
Navigate agent to sp and perform T-step rollout using a randow walk policy: T𝑛 ←{(s0= Sp,ao),.,(sT,ar)}
gn←ST Navigate agent to Sp and perform T-step rollout using Tg with goal gn:
Tπ ←{(s= Sp,ao),.,(sT,ar)}at~πg(-st,9n) Re-label πg rewards with action reconstruction error as curiosity bonus:
rπ←{1st+1=n-λ·p(at|s,z)}=0
end
Perform policy gradient update of πg using T" and rπ
Update V using T and T
Update Vp as set of states with largest prior mean αs αs+βs
", + "bbox": [ + 166, + 107, + 823, + 401 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 431, + 823, + 460 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Objective Formally, we optimize the VAE using the following evidence lower bound (ELBO): ", + "bbox": [ + 179, + 465, + 797, + 481 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/37e141c676097bc48ad969339d6ce7edac6486adae84caba87eeb47a2f34f212.jpg", + "text": "$$\n\\mathrm { E L B O } = \\mathbb { E } _ { q _ { \\phi } ( z | a , s ) } \\left[ \\log p _ { \\theta } ( a | s , z ) \\right] - D _ { \\mathrm { K L } } \\left( q _ { \\phi } ( z | a , s ) | p _ { \\psi } ( z | s ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 276, + 486, + 720, + 505 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To ensure differentiablity, we apply a continuous relaxation over the discrete $z _ { t }$ . We use the Beta distribution $p _ { \\psi } ( z _ { t } ) = \\mathrm { B e t a } ( \\alpha _ { t } , \\beta _ { t } )$ for the prior and the Hard Kumaraswamy distribution $q _ { \\psi } ( z _ { t } | { a } , { z } ) = \\mathrm { H a r d K u m a } ( \\tilde { \\alpha } _ { t } , \\tilde { \\beta } _ { t } )$ for the approximate posterior, which resembles the Beta distribution but is outside the exponential family (Bastings et al., 2019). This choice allows us to sample 0s and 1s without sacrificing differentiability, accomplished via the stretch-and-rectify procedure (Bastings et al., 2019; Louizos et al., 2017) and the reparametrization trick (Kingma & Welling, 2013). Lastly, to prevent the trivial solution of using all states for reconstruction, we use a secondary objective $\\mathcal { L } _ { 0 }$ to regularize the $L _ { 0 }$ norm of $_ z$ at a targeted value $\\mu _ { 0 }$ (Louizos et al., 2017; Bastings et al., 2019), the desired number of selected states out of $T$ steps, e.g. for when $T = 2 5$ , we set $\\mu _ { 0 } = 5$ , meaning ideally 5 out of 25 states are activated for action reconstruction. Another term $\\mathcal { L } _ { T }$ to encourage temporal separation between selected states by targeting the number of $0 / 1$ switches among $_ z$ at $2 \\mu _ { 0 }$ : ", + "bbox": [ + 173, + 508, + 826, + 679 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/602b63b98973b4ecac88d73465672660262ace527c7c8cec985e50a21475e94d.jpg", + "text": "$$\n\\mathcal { L } _ { 0 } = \\Big | \\Big | \\mathbb { E } _ { q _ { \\phi } ( z | s , a ) } [ \\| z \\| _ { 0 } ] - \\mu _ { 0 } \\Big | \\Big | ^ { 2 } , \\quad \\mathcal { L } _ { T } = \\Bigg | \\Bigg | \\mathbb { E } _ { q _ { \\phi } ( z | s , a ) } \\left[ \\sum _ { t = 0 } ^ { T } \\mathbb { 1 } [ z _ { t } \\neq z _ { t + 1 } ] \\right] - 2 \\mu _ { 0 } \\Bigg | \\Bigg | ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 218, + 685, + 777, + 729 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "See Appendix A for details on training the VAE with binary $z _ { t }$ , including integration of the Hard Kumaraswamy distribution and how to regularize the statistics of $_ z$ . ", + "bbox": [ + 174, + 733, + 823, + 762 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 EXPLORATION FOR WORLD GRAPH DISCOVERY ", + "text_level": 1, + "bbox": [ + 176, + 779, + 544, + 792 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Naturally, the latent structure learned by the VAE depends on the trajectories used to train it. Hence, collecting a rich set of trajectories is crucial. Here, we propose a strategy to bootstrap a useful set of trajectories by alternately exploring the environment based on the current iteration’s $\\nu _ { p }$ and updating the VAE and $\\nu _ { p }$ , repeating this cycle until the action reconstruction accuracy plateaus (Algorithm 1). ", + "bbox": [ + 174, + 804, + 825, + 861 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "During exploration, we use action replay to navigate the agent to a state drawn from the current iteration’s $\\nu _ { p }$ . Although resetting via action replay assumes our underneath environment to be deterministic, in cases where this resetting strategy is infeasible, it may be modified so long as to allow the exploration starting points to expand as the agent discovers more of its environment. For each such starting point, we collect two rollouts. In the first rollout, we perform a random walk to explore the nearby region. In the second rollout, we perform actions using a goal-conditioned policy $\\pi _ { g }$ (GCP), setting the final state reached by the random walk as the goal. Both rollouts are used for trianing the VAE and the latter is also used for training $\\pi _ { g }$ . ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/0c4516ced121df0c03431d831806e1817ee32a043d24758d20a0950c4f69de1d.jpg", + "image_caption": [ + "Figure 3: Left: a standard Feudal Network. Right: using Wide-then-Narrow goals. The Manager first outputs a waypoint state as the wide goal $g ^ { w }$ , then attends to a closer-up area around $g ^ { w }$ to narrow down the final goal $g ^ { n }$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 816, + 242 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 309, + 825, + 364 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "GCP provides a venue to integrate intrinsic motivation, such as curiosity (Burda et al., 2018; Achiam & Sastry, 2017; Pathak et al., 2017; Azar et al., 2019) to generate more diverse rollouts. Specifically, we use the action reconstruction error of the VAE as an intrinsic reward signal when training $\\pi _ { g }$ . This choice of curioisty also prevents the VAE from collapsing to the simple behaviors of a vanilla $\\pi _ { g }$ . ", + "bbox": [ + 174, + 372, + 825, + 429 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 EDGE FORMATION ", + "text_level": 1, + "bbox": [ + 176, + 445, + 343, + 459 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The final stage is to construct the edges of $\\mathcal { G } _ { w }$ , which should ideally capture the environment dynamics, i.e. how to transition between waypoint states. Once VAE training is complete and $\\nu _ { p }$ is fixed, we collect random walk rollouts from each of the waypoints $s _ { p } \\in \\mathcal { V } _ { p }$ to estimate the underlying adjacency matrix (Biggs, 1993). More precisely, we claim a directed edge $s _ { p } \\to s _ { q }$ if there exists a random walk trajectory from $s _ { p }$ to $s _ { q }$ that does not intersect a third waypoint. We also consider paths taken by $\\pi _ { g }$ (starting at $s _ { p }$ and setting $s _ { q }$ as the goal) and keep the shortest observed path from $s _ { p }$ to $s _ { q }$ as a world graph edge transition. We use the action sequence length of the edge transition between adjacent waypoints as the weight of the edge. As shown experimentally, a key benefit of our approach is the ability to plan over $\\mathcal { G } _ { w }$ . To navigate from one waypoint to another, we can use dynamic programming (Sutton, 1998; Feng et al., 2004) to output the optimal traversal of the graph. ", + "bbox": [ + 173, + 470, + 825, + 611 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 ACCELERATING REINFORCEMENT LEARNING WITH WORLD GRAPHS ", + "text_level": 1, + "bbox": [ + 173, + 631, + 779, + 647 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "World graphs present a high-level, task-agnostic abstraction of the environment through waypoints and feasible transition routes between them. A key example of world graph applications for taskspecific RL is structured exploration: instead of exploring the entire environment, RL agents can use world graphs to quickly identify task-relevant regions and bias low-level exploration to these regions. Our framework to leverage world graphs for structured exploration consists of two parts: ", + "bbox": [ + 174, + 662, + 825, + 733 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. Hierarchical RL wherein the high-level policy selects subgoals from $\\nu _ { p }$ . \n2. Traversals using world graph edges. ", + "bbox": [ + 209, + 744, + 705, + 780 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 HIERARCHICAL RL OVER WORLD GRAPHS ", + "text_level": 1, + "bbox": [ + 173, + 796, + 514, + 810 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Formally, an RL agent learning to solve a task is formulated as a Markov Decision Process: at time $t$ , the agent is in a state $s _ { t }$ , executes an action $a _ { t }$ via a policy $\\pi ( a _ { t } | s _ { t } )$ and receives a rewards $r _ { t }$ . The agent’s goal is to maximize its cumulative expected return $\\begin{array} { r } { R = \\mathbb { E } _ { ( s _ { t } , a _ { t } ) \\sim \\pi , p , p _ { 0 } } \\left[ \\sum _ { t \\geq 0 } \\gamma ^ { t } r _ { t } \\right] } \\end{array}$ , where $p ( s _ { t + 1 } | s _ { t } , a _ { t } ) , p _ { 0 } ( s _ { 0 } )$ are the transition and initial state distributions. ", + "bbox": [ + 173, + 821, + 825, + 890 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To incorporate world graphs with RL, we use a hierarchical approach based on the Feudal Network (FN) (Dayan $\\&$ Hinton, 1993; Vezhnevets et al., 2017), depicted in Figure 3. A standard FN ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Task Description ", + "text_level": 1, + "bbox": [ + 282, + 102, + 362, + 111 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Task MultiGoal ", + "bbox": [ + 183, + 102, + 232, + 119 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Environment Characteristics ", + "text_level": 1, + "bbox": [ + 553, + 102, + 691, + 111 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Balls are located randomly, dense reward. ", + "bbox": [ + 557, + 111, + 732, + 119 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "MultiGoal-Sparse ", + "text_level": 1, + "bbox": [ + 183, + 128, + 263, + 138 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Collect randomly spawned balls, each ball gives $+ 1$ reward. To end an episode, the agent has to exit at a designated point. Agents receive a single reward $r \\leq 1$ proportional to the number of balls collected upon exiting. ", + "bbox": [ + 282, + 111, + 544, + 147 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "MultiGoalStochastic Door-Key ", + "bbox": [ + 183, + 147, + 235, + 175 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Balls are located randomly, sparse reward. ", + "bbox": [ + 555, + 128, + 738, + 138 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Spawn lava blocks at random locations each time step that immediately terminates the episode if stepped on. Agent has to pick up a key to open a door (reward $+ 1 \\AA$ and reach the exit point on the other side (reward $+ 1$ ). ", + "bbox": [ + 282, + 148, + 544, + 185 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Stochastic environment. Multiple objects: lava and balls are randomly located, dense reward. \nWalls, door and key are located randomly. Agents have additional actions: pick and toggle. ", + "bbox": [ + 553, + 147, + 815, + 185 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1: An overview of tasks used to evaluate the benefit of using world graphs. Visualizations can be found in Appendix D. ", + "bbox": [ + 176, + 196, + 823, + 224 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "decomposes the policy of the agent into two separate policies that receive distinct streams of reward: a high-level policy (“Manager”) learns to propose subgoals; a low-level policy (“Worker”) receives subgoals from the Manager as inputs and is rewarded for taking actions in the environment that reach the subgoals. The Manager receives the environment reward defined by the task and therefore must learn to emit subgoals that lead to task completion. The Manager and Worker do not share weights and operate at different temporal resolutions: the Manager only outputs a new subgoal if either the Worker reaches the chosen one or a subgoal horizon $c$ is exceeded. ", + "bbox": [ + 174, + 250, + 825, + 347 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For all our experiments, policies are trained using advantage actor-critic (A2C), an on-policy RL algorithm (Wu & Tian, 2016; Pane et al., 2016; Mnih et al., 2016b). To ease optimization, the feature extraction layers of the Manager and Worker that encode $s _ { t }$ are initialized with the corresponding layers from $\\pi _ { g }$ , the GCP learned during world graph discovery phase. More details are in Appendix B. ", + "bbox": [ + 174, + 353, + 825, + 410 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 WIDE-THEN-NARROW GOALS AND WORLD GRAPHS ", + "text_level": 1, + "bbox": [ + 173, + 425, + 578, + 439 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To incorporate the world graph, we introduce a Manager policy that factorizes subgoal selection as follows: a wide policy $\\pi ^ { w } ( g _ { t } ^ { w } | s _ { t } )$ selects a waypoint state as the wide goal $g ^ { w } \\in \\mathcal { V } _ { p }$ , and a narrow policy $\\pi ^ { n } ( g _ { t } ^ { n } | s _ { t } , g _ { t } ^ { w } )$ selects a state within a local neighborhood of $g _ { t } ^ { w }$ , i.e. its $\\epsilon$ -net (Mahadevan $\\&$ Maggioni, 2007), as the narrow goal $g ^ { n } \\in \\{ s : \\mathcal { D } ( s , \\bar { g } _ { t } ^ { w } ) \\leq \\epsilon \\}$ . The Worker policy $\\pi ^ { \\mathrm { w o r k e r } } ( a _ { t } | s _ { t } , g _ { t } ^ { n } , g _ { t } ^ { \\bar { w } } )$ chooses the action taken by the agent given the current state and the wide and narrow goals from the Manager. A visual illustration is in Figure 4 and training details in Appendix C.2. ", + "bbox": [ + 174, + 452, + 825, + 536 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 WORLD GRAPH TRAVERSAL ", + "text_level": 1, + "bbox": [ + 176, + 551, + 411, + 565 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The wide-then-narrow subgoal format simplifies the search space for the Manager policy. Using waypoints as wide goals also makes it possible to leverage the edges of the world graph for planning and executing the planned traversals. This process breaks down as follows: ", + "bbox": [ + 176, + 577, + 823, + 619 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1. When to Traverse: When the agent encounters a waypoint state $s _ { t } \\in \\mathcal V _ { p }$ , a “traversal” is initiated if $s _ { t }$ has a feasible connection in $\\mathcal { G } _ { w }$ to the active wide goal $g _ { t } ^ { w }$ . 2. Planning: Upon triggering a traversal, the optimal traversal route from the initiating state to $g _ { t } ^ { w }$ is estimated from the $\\mathcal { G } _ { w }$ edge weights using classic dynamic programming planning (Sutton, 1998; Feng et al., 2004). This yields a sequence of intermediate waypoint states. 3. Execution: Execution of graph traversals depends on the nature of the environment. If deterministic, the agent simply follows the action sequences given by the edges of the traversal. Otherwise, the agent uses the pretrained $\\mathbf { G C P } \\pi _ { g }$ to sequentially reach each of the intermediate waypoint states along the traversal (we fine-tune $\\pi _ { g }$ in parallel where applicable). If the agent fails to reach the next waypoint state within a certain time limit, it stops its current pursuit and a new $( g ^ { w } , g ^ { n } )$ pair is received from the Manager. ", + "bbox": [ + 210, + 631, + 825, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "World graph traversal allows the Manager to assign task-relevant wide goals $g ^ { w }$ that can be far away from the agent yet still reachable, which consequentially accelerates learning by focusing exploration around the task-relevant region near $g ^ { w }$ . ", + "bbox": [ + 176, + 803, + 823, + 844 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTAL VALIDATION ", + "text_level": 1, + "bbox": [ + 176, + 864, + 447, + 881 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now assess each component of our framework on a set of challenging 2D grid worlds. Our ablation studies demonstrate the following benefits of our framework: ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/cf6404ac297859fc913d08dceb2daf866c13f6d312aaa9b45a16acb570a106cf.jpg", + "table_caption": [ + "Table 2: On a variety of tasks and environment setups, we evaluate RL models trained with GCP $\\pi _ { g }$ initialization, with $\\mathcal { G } _ { w }$ world graph travresal, and with both. All models on the right are equipped with WN. Left are baselines for additional comparison. We report final rewards for MultiGoal tasks and success rates for Door-Key are reported. If no result reported, the agent failed to solve the task. " + ], + "table_footnote": [], + "table_body": "
TaskSizeA2CFN+πg initOurs
+πg-init+ 9w-traversal+πg-init+Gw-traversal
MultiGoalSmall2.04±0.052.93±0.745.25±0.133.92±0.225.05±0.03
Medium==5.15±0.112.56±0.093.00±0.90
Larger==-2.18±0.122.72±0.59
MultiGoal-SparseSmall==0.39±0.090.24±0.040.42±0.07
Medium0.20±0.040.25±0.03
Larger0.16±0.220.26±0.11
MultiGoal-StochasticSmall1.38±1.201.93±0.163.06±0.312.92±0.45
Medium=2.99±0.122.42±0.242.64±0.14
Larger===0.60±0.12
Door-KeySmall=0.99±0.000.37±0.150.92±0.02
Medium0.56±0.020.76±0.06
Larger=0.26±0.19
", + "bbox": [ + 176, + 98, + 821, + 253 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/847bc395d5c3394e94cf251a43c7da6c325d114220ea1be19dced25f131ba821.jpg", + "table_caption": [ + "Table 3: Comparing learned $\\nu _ { p }$ versus random $\\mathcal { V } _ { \\mathrm { r a n d } }$ as wide subgoals on large mazes, all trained with $\\pi _ { g }$ initialization and graph traversal. $\\nu _ { p }$ generally is superior in terms of performance and consistency. We report final rewards for MultiGoal tasks and success rates for Door-Key are reported. " + ], + "table_footnote": [], + "table_body": "
Waypoint typeMultiGoalMultiGoal-SparseMultiGoal-StochasticDoor-Key
Learned2.72±0.590.26±0.110.60±0.120.26±0.19
Random2.30±0.490.19±0.110.41±0.250.27±0.40
", + "bbox": [ + 272, + 332, + 725, + 364 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "• It improves sample efficiency and performance over the baseline HRL model. • It benefits tasks varying in envirionment scale, task type, reward structure, and stochasticity. • The identified waypoints provide superior world representations for solving downstream tasks, as compared to graphs using randomly selected states as nodes. ", + "bbox": [ + 217, + 440, + 825, + 506 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implementation details, snippets of the tasks and mazes are in Appendix C-D. ", + "bbox": [ + 176, + 517, + 681, + 532 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 ABLATION STUDIES ON 2D GRID WORLDS ", + "text_level": 1, + "bbox": [ + 174, + 547, + 513, + 563 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For our ablation studies, we construct 2D grid worlds of increasing sizes (small, medium and large) along with challenging tasks with different reward structures, levels of stochasticity and logic (summarized in Table 1). In all tasks, every action taken by the agent receives a negative reward penalty. We follow a rigorous evaluation protocol (Wu et al., 2017; Ostrovski et al., 2017; Henderson et al., 2018): each experiment is repeated with 3 training seeds. 10 additional validation seeds are used to pick the model with the best reward performance. This model is then tested on 100 testing seeds. We report mean reward and standard deviation. ", + "bbox": [ + 173, + 573, + 825, + 671 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We ablate each of the following components in our framework and compare against non-hierarchical (A2C) and hierarchical baselines (FN): ", + "bbox": [ + 173, + 678, + 823, + 707 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. initializing the feature extraction layers of the Manager and Worker from $\\pi _ { g }$ , \n2. applying Wide-then-Narrow Manager (WN) goal instruction, and \n3. allowing the Worker to traverse along $\\mathcal { G } _ { w }$ . ", + "bbox": [ + 212, + 718, + 736, + 771 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results are shown in Table 2. In sum, each component improves performance over the baselines. ", + "bbox": [ + 173, + 781, + 805, + 796 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Wide and narrow goals Using two goal types is a highly effective way to structure the Manager instructions and enables the Worker to differentiate the transition and local task-solving phases. We note that for small MultiGoal, agents do not benefit much from $\\mathcal { G } _ { w }$ traversal: it can rely solely on the guidance from WN goals to master both phases. However with increasing maze size, the Worker struggles to master traversals on its own and thus fails solving the tasks. ", + "bbox": [ + 174, + 810, + 825, + 881 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "World Graph Traversal As conjectured in Section 4.3, the performance gain of our framework can be explained by the larger range and more targeted exploration strategy. In addition, the Worker does not have to learn long distance transitions with the aid of $\\mathcal { G } _ { w }$ traversals. Figure 4 confirms that $\\mathcal { G } _ { w }$ traversal speeds up convergence and its effect becomes more evident with larger mazes. Note that the graph learning stage only need 2.4K iterations to converge. Even when taking these additional environment interactions into account, $\\mathcal { G } _ { w }$ traversal still exhibits superior sample efficiency, not to mention that the graph is shared among all tasks. Moreover, solving Door-Key involves a complex combination of sub-tasks: find and pick up the key, reach and open the door and finally exit. With limited reward feedback, this is particularly difficult to learn. The ability to traverse along $\\mathcal { G } _ { w }$ enables longer-horizon planning on top of the waypoints, thanks to which the agents boost the success rate on medium Door-Key from $0 . 5 6 { \\pm } 0 . 0 2$ to $0 . 7 5 { \\pm } 0 . 0 6 $ . ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/eeccd173a66cc7f083254caca6b3d6fe078ddac17830e5c24ef8e6d8706c4b0b.jpg", + "image_caption": [ + "Figure 4: Validation performance during training (mean and standard-deviation of reward, 3 seeds) for MultiGoal. Left: Comparing $\\nu _ { p }$ and $\\mathcal { V } _ { \\mathrm { r a n d } }$ , with or without traversal, all models use WN and $\\pi _ { g }$ initialization. We see that 1) traversal speeds up convergence, 2) $\\mathcal { V } _ { \\mathrm { r a n d } }$ gives higher variance and slightly worse performance than $\\nu _ { p }$ . Right: comparing with or without $\\pi _ { g }$ initialization on $\\nu _ { p }$ , all models use WN. We see that initializing the task-specific phase with the task-agnostic goal-conditioned policy significantly boosts learning. " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 821, + 251 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 361, + 825, + 487 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Benefits of Learned Waypoints To highlight the benefit of establishing the waypoints learned by the VAE as nodes for $\\mathcal { G } _ { w }$ , we compare against results using a $\\mathcal { G } _ { w }$ constructed around randomly selected states $( \\nu _ { \\mathrm { r a n d } } )$ . The edges of the random-node graph are formed in the same way as described in Section 3.3 and its feature extractor is also initialized from $\\pi _ { g }$ . Although granting knowledge acquired during the unsupervised phase to $\\mathcal { V } _ { \\mathrm { r a n d } }$ is unfair to $\\nu _ { p }$ , deploying both initialization and traversal while only varying $\\mathcal { V } _ { \\mathrm { r a n d } }$ and $\\nu _ { p }$ isolates the effect from the nodes to the best extent. The comparative results (in Table 3, learning curves for MultiGoal in Figure 4) suggest $\\nu _ { p }$ generally outperforms $\\mathcal { V } _ { \\mathrm { r a n d } }$ . Door-Key is the only task in which the two matches. However, $\\mathcal { V } _ { \\mathrm { r a n d } }$ exhibits a large variance, implying that certain sets of random states can be suitable for this task, but using learned waypoints gives strong performance more consistently. ", + "bbox": [ + 174, + 511, + 825, + 651 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Initialization with GCP Initializing the weights of the Worker and Manager feature extractors from $\\pi _ { g }$ (learned during the task-agnostic phase) consistently benefits learning.In fact, we observe that models starting from scratch fail on almost all tasks within the maximal number of training iterations, unless coupled with $\\mathcal { G } _ { w }$ traversal, which is still inferior to using $\\pi _ { g }$ -initialization. Particularly, for the small MultiGoal-Stochastic environment, there is a high chance that a lava square blocks traversal; therefore, without the environment knowledge from $\\pi _ { g }$ transferred by weight initialization, the interference created by the episode-terminating lava prevents the agent from learning the task. ", + "bbox": [ + 174, + 675, + 825, + 773 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 803, + 318, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We have shown that world graphs are powerful environment abstractions, which, in particular, are capable of accelerating reinforcement learning. Future works may extend their applications to more challenging RL setups, such as real-world multi-task learning and navigation. It is also interesting to generalize the proposed framework to learn dynamic world graphs for evolving environments, and applying world graphs to multi-agent problems, where agents become part of the world graphs of other agents. 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", + "bbox": [ + 169, + 664, + 825, + 693 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A RECURRENT VAE WITH DIFFERENTIABLE BINARY LATENT VARIABLES", + "text_level": 1, + "bbox": [ + 169, + 101, + 805, + 119 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "As illustrated in the main text, the main objective for the recurrent VAE is the following evidence lower bound with derivation: ", + "bbox": [ + 173, + 132, + 820, + 161 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/74a121592da2d503de3dc3d7827353b7b30c2e1c28cc4cac718fdf83190bd04f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\log p ( a | s ) = \\log \\int p ( a | s , z ) d z } \\\\ & { \\qquad = \\log \\int p ( a | s , z ) p ( z | s ) \\frac { q ( z | a , s ) } { q ( z | a , s ) } d z } \\\\ & { \\qquad = \\log \\int p ( a | s , z ) \\frac { p ( z | s ) } { q ( z | a , s ) } q ( z | a , s ) d z } \\\\ & { \\qquad \\geq \\mathbb { E } _ { q ( z | a , s ) } [ \\log p ( a | s , z ) - \\log \\frac { q ( z | a , s ) } { p ( z | s ) } ] } \\\\ & { \\qquad = \\mathbb { E } _ { q ( z | a , s ) } [ \\log p ( a | s , z ) ] - D _ { \\mathrm { K L } } ( q ( z | a , s ) | | p ( z | s ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 165, + 704, + 323 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The inference network $q _ { \\psi }$ takes in the trajectories of state-action pairs $\\tau$ and at each time step approximates the posterior of the corresponding latent variable $z _ { t }$ . The prior network $p _ { \\psi }$ takes the state $s _ { t }$ at each time step and outputs the state-conditioned prior $p _ { \\psi } ( s _ { t } )$ . We choose Beta as the prior distribution and the Hard Kuma as the approximated posterior to relax the discrete latent variables to continuous surrogates. ", + "bbox": [ + 173, + 325, + 826, + 396 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The Kuma distribution $\\mathrm { K u m a } ( \\alpha , \\beta )$ highly resembles the Beta Distribution in shape but does not come from the exponential family. Similar to Beta, the Kuma distribution also ranges from bimodal (when $\\alpha \\approx \\beta )$ to unimodal $( \\alpha / \\beta \\to 0$ or $\\alpha / \\beta \\to \\infty )$ ). Also, when $\\alpha = 1$ or $\\beta = 1$ , $\\operatorname { K u m a } ( \\alpha , \\beta ) = \\operatorname { B e t a } ( \\alpha , \\beta )$ . We observe empirically better performance when we fix $\\beta = 1$ for the Kuma approximated posterior. One major advantage of the Kuma distribution is its simple Cumulative Distribution Function (CDF): ", + "bbox": [ + 173, + 401, + 826, + 487 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1dc9fcfd1b674b818890fa11824ac8c456e5655cba6f91429501c5b1f3f88f5c.jpg", + "text": "$$\nF _ { \\mathrm { K u m a } } ( x , \\alpha , \\beta ) = ( 1 - ( 1 - x ^ { \\alpha } ) ) ^ { \\beta } .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 491, + 620, + 511 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "It is therefore amendable to the reparametrization trick (Kingma & Welling, 2013; Rezende et al., 2014; Maddison et al., 2016) by sampling from uniform distribution $u \\sim \\mathcal { U } ( 0 , 1 )$ : ", + "bbox": [ + 173, + 516, + 826, + 545 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/711a6644a08fe094230354b7b31abc259d6f5b7390a9b99aaef14f2692d8d138.jpg", + "text": "$$\nz = F _ { \\mathrm { K u m a } } ^ { - 1 } ( u ; \\alpha , \\beta ) \\sim \\mathrm { K u m a } ( \\alpha , \\beta ) .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 550, + 622, + 570 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lastly, the KL-divergence between the Kuma and Beta distributions can be approximated in closed form (Nalisnick & Smyth, 2016): ", + "bbox": [ + 171, + 575, + 825, + 604 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/fd54f638e2b6e99559ad144aef4fae1cf3cc7c17eb72c14ffd6565080319944e.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { { \\cal D } _ { \\mathrm { K L } } ( \\mathrm { K u m a } ( a , b ) | \\mathrm { B e t a } ( \\alpha , \\beta ) ) = \\frac { a - \\alpha } { a } \\left( - \\gamma - \\Psi ( b ) - \\frac { 1 } { b } \\right) } } \\\\ { \\displaystyle { \\phantom { \\frac { b - a } { b - a } ( \\mathrm { K u m a } ( a , b ) + \\log \\mathrm { B e t a } ( \\alpha , \\beta ) - \\frac { b - 1 } { b } + ( \\beta - 1 ) b \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m + a b } \\mathrm { B e t a } \\left( \\frac { m } { a } , b \\right) , } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 223, + 609, + 772, + 688 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "where $\\Psi$ is the Digamma function, $\\gamma$ the Euler constant, and the approximation uses the first few terms of the Taylor series expansion. We take the first 5 terms here. ", + "bbox": [ + 171, + 691, + 823, + 720 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Next, we make the Kuma distribution “hard” by following the steps in Bastings et al. (2019). First stretch the support to $( r = 0 - \\epsilon _ { 1 } , l = 1 + \\epsilon _ { 2 }$ ), $\\epsilon _ { 1 } , \\epsilon _ { 2 } > 0$ , and the resulting CDF distribution takes the form: ", + "bbox": [ + 176, + 727, + 823, + 768 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/f498e055bf839a3b4e68963b2175603bfe1bb354a3a065bb4dfd8246322b0e21.jpg", + "text": "$$\nF _ { S } ( z ) = F _ { \\mathrm { K u m a } } \\left( { \\frac { z - l } { r - l } } ; \\alpha , \\beta \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 772, + 607, + 808 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Then, the non-eligible probabilities for 0’s and 1’s are attained by rectifying all samples below 0 to 0 and above 1 to 1, and other value as it is, that is ", + "bbox": [ + 171, + 813, + 826, + 842 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/192632507efa1b88897b77a696970eb5a96eaf5ccbf0d7d10e691c5285e775e8.jpg", + "text": "$$\nP ( z = 0 ) = F _ { \\mathrm { K u m a } } \\left( \\frac { - l } { r - l } ; \\alpha , \\beta \\right) , \\quad P ( z = 1 ) = 1 - F _ { \\mathrm { K u m a } } \\left( \\frac { 1 - l } { r - l } ; \\alpha , \\beta \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 847, + 761, + 881 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Lastly, we impose two additional regularization terms ${ \\mathcal { L } } _ { \\prime }$ and $\\mathcal { L } _ { T }$ on the approximated posteriors. As described in the main text, ${ \\mathcal { L } } _ { \\prime }$ prevents the model from selecting all states to reconstruct $\\{ a _ { t } \\} _ { 0 } ^ { T - 1 }$ by restraining the expected $L _ { 0 }$ norm of $z = \\left( z _ { 1 } \\cdot \\cdot \\cdot z _ { T - 1 } \\right)$ to approximately be at a targeted value $\\mu _ { 0 }$ (Louizos et al., 2017; Bastings et al., 2019). In other words, this objective adds the constraint that there should be $\\mu _ { 0 }$ of activated $z _ { t } = 1$ given a sequence of length $T$ . The other term $\\mathcal { L } _ { T }$ encourages temporally isolated activation of $z _ { t }$ , meaning the number of transition between 0 and 1 among $z _ { t }$ ’s should roughly be $2 \\mu _ { 0 }$ . Note that both expectations in Equation 2 have closed forms for HardKuma. ", + "bbox": [ + 173, + 893, + 826, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/05375c9ba77cfcf9223c575f1bb6085fd0a88dec24f1342685259b651bd523bc.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { L } _ { 0 } = \\left\\| \\mathbb { E } _ { q ( \\boldsymbol { z } | \\boldsymbol { s } , \\boldsymbol { a } ) } \\left[ \\left\\| \\boldsymbol { z } \\right\\| _ { 0 } \\right] - \\mu _ { 0 } \\right\\| ^ { 2 } , \\mathrm { w h e r e } } \\\\ { \\displaystyle \\mathbb { E } _ { q ( \\boldsymbol { z } | \\boldsymbol { s } , \\boldsymbol { a } ) } \\left[ \\left\\| \\boldsymbol { z } \\right\\| _ { 0 } \\right] = \\sum _ { t = 1 } ^ { T } \\mathbb { E } _ { q ( \\boldsymbol { z } _ { t } | \\boldsymbol { s } , \\boldsymbol { a } ) } \\left[ \\mathbb { 1 } _ { \\boldsymbol { z } _ { t } \\neq 0 } \\right] } \\\\ { \\displaystyle \\qquad = \\sum _ { t = 1 } ^ { T } 1 - p \\left( \\boldsymbol { z } _ { t } = 0 \\right) = \\sum _ { t = 1 } ^ { T } 1 - F _ { \\mathrm { K u m a } } \\left( \\frac { - l } { r - l } ; \\alpha _ { t } , \\beta _ { t } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 179, + 727, + 290 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c14baa5bbf179469a3b3c656cae666e14f93fa77dc76184a1900553fb7dcd0c1.jpg", + "text": "$$\n\\mathbb { E } _ { q ( z | s , a ) } [ \\sum _ { t = 1 } ^ { T - 1 } \\mathbb { 1 } _ { z _ { t } \\neq z _ { t + 1 } } ] = \\sum _ { t = 1 } ^ { T - 1 } p \\left( z _ { t } = 0 \\right) \\left( 1 - p \\left( z _ { t + 1 } = 0 \\right) \\right) + \\left( 1 - p \\left( z _ { t } = 0 \\right) \\right) p \\left( z _ { t + 1 } = 0 \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 189, + 337, + 808, + 378 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Lagrangian Relaxation. The overall optimization objective consists of action sequence reconstruction, KL-divergence between the posterior and prior, $\\mathcal { L } _ { 0 }$ and $\\mathcal { L } _ { T }$ (Equation 12). We tune the objective weights $\\lambda _ { i }$ using Lagrangian relaxation (Higgins et al., 2017; Bastings et al., 2019; Bertsekas, 1999), treating $\\lambda _ { i }$ ’s as learnable parameters and performing alternative optimization between $\\lambda _ { i }$ ’s and the model parameters. We observe that as long as their initialization is within a reasonable range, $\\lambda _ { i }$ ’s converge to a local optimum: ", + "bbox": [ + 174, + 406, + 825, + 491 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/36b2ff3e96953adb6e0be1a1825d306d29a4210bf18425ebfaffb56135e650d9.jpg", + "text": "$$\n\\operatorname* { m a x } _ { \\{ \\lambda _ { 1 } , 2 , 3 \\} } \\operatorname* { m i n } _ { \\substack { \\left\\{ \\theta , \\phi , \\psi \\right\\} } } - \\mathbb { E } _ { q _ { \\psi } ( z | a , s ) } \\left[ \\log p _ { \\theta } ( a | s , z ) \\right] + \\lambda _ { 1 } D _ { \\mathrm { K L } } \\left( q _ { \\phi } ( z | a , s ) | p _ { \\psi } ( z | s ) \\right) + \\lambda _ { 2 } \\mathcal { L } _ { 0 } + \\lambda _ { 3 } \\mathcal { L } _ { T } .\n$$", + "text_format": "latex", + "bbox": [ + 187, + 497, + 785, + 523 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We observe this approach to produce efficient and stable mini-batch training. ", + "bbox": [ + 176, + 530, + 676, + 545 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B GOAL-CONDITIONED POLICY INITIALIZATION FOR HRL", + "text_level": 1, + "bbox": [ + 176, + 564, + 678, + 583 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Optimizing composite neural networks like HRL (Co-Reyes et al., 2018) is sensitive to weight initialization (Mishkin & Matas, 2015; Le et al., 2015), due to its complexity and lack of clear supervision at various levels. Therefore, taking inspiration from prevailing pre-training procedures in computer vision (Russakovsky et al., 2015; Donahue et al., 2014) and NLP (Devlin et al., 2018; Radford et al., 2019), we take advantage of the weights learned by $\\pi _ { g }$ during world graph discovery when initializing the Worker and Manager policies for downstream HRL, as $\\pi _ { g }$ has already implicitly embodied much environment dynamics information. ", + "bbox": [ + 173, + 595, + 825, + 695 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "More specifically, we extract the weights of the feature extractor, i.e. the state encoder, and use them as the initial weights for the state encoders of the HRL policies. Our empirical results demonstrate that such weight initialization consistently improves performance and validates the value of skill/knowledge transfer from GCP (Taylor & Stone, 2009; Barreto et al., 2017). ", + "bbox": [ + 174, + 702, + 825, + 757 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C ADDITIONAL IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 779, + 553, + 795 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Model code folder including all architecture details is shared in comment. ", + "bbox": [ + 176, + 809, + 656, + 824 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.1 HYPERPARAMETERS FOR VAE TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 840, + 503, + 856 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Our models are optimized with Adam (Kingma & Ba, 2014) using mini-batches of size 128, thus spawning 128 asynchronous agents to explore. We use an initial learning rate of 0.0001, with $\\bar { \\epsilon } = 0 . 0 0 \\bar { 1 } , \\beta _ { 1 } = \\bar { 0 } . 9 , \\beta _ { 2 } = 0 . 9 9 \\bar { 9 }$ ; gradients are clipped to 40 for inference and generation nets. For HardKuma, we set $l = - 0 . 1$ and $r = 1 . 1$ . The maximum sequence length for BiLSTM is 25. The total number of training iterations is 3600 and model usually converges around 2400 iterations. We train the prior, inference, and generation networks end-to-end. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We initialize $\\lambda _ { i }$ ’s (see Lagrangian Relaxation) to be $\\lambda _ { 1 } = 0 . 0 1$ (KL-divergence), $\\begin{array} { r } { , \\lambda _ { 2 } = 0 . 0 6 ( \\mathcal { L } _ { 0 } ) . } \\end{array}$ $\\lambda _ { 3 } = 0 . 0 2 ( \\mathcal { L } _ { T } )$ . After each update of the latent model, we update $\\lambda _ { i }$ ’s, whose initial learning rate is 0.0005, by maximizing the original objective in a similar way as using Lagrangian Multiplier. At the end of optimization, $\\lambda _ { i }$ ’s converge to locally optimal values. For example, with the medium maze, $\\lambda _ { 1 } = 0 . 0 6 7$ for the KL-term, $\\lambda _ { 2 } = 0 . 0 7 0$ for the $\\mathcal { L } _ { 0 }$ and $\\lambda _ { 3 } = 0 . 0 5 1$ for the $\\mathcal { L } _ { T }$ term. The total number of waypoints $| \\nu _ { p } |$ is set to be $2 0 \\%$ of the size of the full state space. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C.2 TRAINING HRL MODELS ", + "text_level": 1, + "bbox": [ + 174, + 239, + 390, + 253 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The procedure of the Manager and the Worker in sending/receiving orders using either traversal paths among $\\nu _ { p }$ from replay buffer for deterministic environments or with $\\pi _ { g }$ for stochastic ones follows: ", + "bbox": [ + 174, + 265, + 825, + 295 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1. The Manager gives a wide-narrow subgoal pair $( g _ { w } , g _ { n } )$ . \n2. The agent takes action based on the Worker policy $\\pi ^ { \\omega }$ conditioned on $( g _ { w } , g _ { n } )$ and reaches a new state $s ^ { \\prime }$ . If $s ^ { \\prime } \\in \\mathcal { V } _ { p }$ , $g _ { w }$ has not yet met, and there exists a valid path basing on the edge paths from the world graph $s ^ { \\prime } \\to g _ { w }$ , agent then either follows replay actions or $\\pi _ { g }$ to reach $g _ { w }$ . If $\\pi _ { g }$ still does not reach desired destination in a certain steps, then stop the agent wherever it stands; also $\\pi _ { g }$ can be finetuned here. \n3. The Worker receives positive reward for reaching $g _ { w }$ for the first time. \n4. If agent reaches $g _ { n }$ , the Worker also receives positive rewards and terminates this horizon. \n5. The Worker receives negative for every action taken except for during traversal; the Manager receives negative reward for every action taken including traversal. \n6. When either $g _ { n }$ is reached or the maximum time step for this horizon is met, the Manager renews its subgoal pair. ", + "bbox": [ + 210, + 304, + 825, + 494 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The training of the Worker policy $\\pi ^ { \\omega }$ follows the same A2C algorithm as $\\pi _ { g }$ ", + "bbox": [ + 176, + 506, + 678, + 521 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The training of the Manager policy $\\pi ^ { m }$ also follows a similar procedure but as it operates at a lower temporal resolution, its value function regresses against the $t _ { m }$ -step discounted reward where $t _ { m }$ covers all actions and rewards generated from the Worker. ", + "bbox": [ + 176, + 526, + 823, + 570 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "When using the Wide-then-Narrow instruction, the policy gradient for the Manager policy $\\pi _ { m }$ becomes: ", + "bbox": [ + 173, + 575, + 821, + 604 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\boldsymbol { \\mathbb { E } } _ { ( s _ { t } , a _ { t } ) \\sim \\pi , p , p _ { 0 } } \\left[ A _ { m , t } \\nabla \\log \\left( \\pi ^ { \\omega } \\left( g _ { w , t } | s _ { t } \\right) \\pi ^ { n } \\left( g _ { n , t } | s _ { t } , g _ { w , t } , s _ { w , t } \\right) \\right) \\right] + \\nabla \\left[ \\mathcal { H } \\left( \\pi ^ { \\omega } \\right) + \\mathcal { H } \\left( \\pi ^ { n } ( \\cdot | g _ { w , t } ) \\right) \\right] , } \\end{array}$ where $A _ { m , t }$ is the Manager’s advantage at time $t$ . Also, for Manager, as the size of the action space scales linearly with $| S |$ , the exact entropy for the $\\pi ^ { m }$ can easily become intractable. Essentially there are $O$ $^ { \\prime } \\left( | \\mathcal { V } | \\times \\left( N ^ { 2 } \\right) \\right)$ possible actions. To calculate the entropy exactly, all of them has to be summed, making it easily computationally intractable: ", + "bbox": [ + 174, + 609, + 825, + 689 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f2f8eb2b755e7a0bca9642c720d6a05acc8e754043512c8b836b2075a363f0e3.jpg", + "text": "$$\n\\mathcal { H } = \\sum _ { w \\in \\mathcal { V } } \\sum _ { w _ { n } \\in s _ { w } } \\pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \\pi ^ { \\omega } ( w | s _ { t } ) \\log { \\nabla \\pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \\pi ^ { \\omega } ( w | s _ { t } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 261, + 694, + 735, + 729 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Thus in practice we resort to an effective alternative $\\mathcal { H } \\left( \\pi ^ { \\omega } \\right) + \\mathcal { H } \\left( \\pi ^ { n } ( \\cdot | g _ { w , t } ) \\right)$ . ", + "bbox": [ + 176, + 733, + 687, + 750 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Psuedo-code for Manager training is in Algorithm 2. ", + "bbox": [ + 173, + 755, + 519, + 770 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C.3 HYPERPARAMETERS FOR HRL ", + "text_level": 1, + "bbox": [ + 176, + 786, + 429, + 801 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For training the HRL policies, we inherit most hyperparameters from those used when training $\\pi _ { g }$ , as the Manager and the Worker both share similar architectures with $\\pi _ { g }$ . The hyperparameters used when training $\\pi _ { g }$ follow those from Shang et al. (2019). Because the tasks used in HRL experiments are more difficult than the generic goal-reaching task, we set the maximal number of training iterations to 100K abd training is stopped early if model performance reaches a plateau. The rollout steps for each iteration is 60. Hyperparameters specific to HRL are the horizon $c = 2 0$ and the size of the Manager’s local attention range (that is, the neighborhood around $g ^ { w }$ within which $g ^ { n }$ is selected), which are $N = 5$ for small and medium mazes, and $N = 7$ for the large maze. ", + "bbox": [ + 173, + 811, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/69965e8760f4ff4df4061f6a7d51654966da8e25dc1350b945fd2a8ae267845c.jpg", + "table_caption": [ + "Algorithm 2: Training of $\\pi ^ { m }$ for HRL models " + ], + "table_footnote": [], + "table_body": "
Clear gradients dθ ←O; while t <= tmax or episode not terminated do Simulate under current policy πm,t-1, πω,t-1; if the Worker has met the previous subgoal or exceeded the horizon c thenReset the set of time steps where πm,t omits a new subgoal Sm = {} and tm = 0.;
Sample a new subgoal gm,t from πm,t; end
Zm,t = fLsTM(CNN(sm,t,sv),hm,tm),Vm,t = fu(zm,t),Tt = fp(2m,t) ;
Sm= SmU {tm} and tm =t;
O, if terminal
Vtmax+1, otherwise
for t = tmax,...1 do R←rt+γR;
if t ∈ Sm then
Am,t ←R-Vm,t;
Accumulate gradients from value loss: dθ ← d0 + 入
Accumulate policy gradients with entropy regularization:80
d0 ← d0+ VlogTm,t(gm,t)Am,t + βVH(πm,t);
end
end
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" + ], + "image_footnote": [], + "bbox": [ + 207, + 726, + 792, + 814 + ], + "page_idx": 15 + } +] \ No newline at end of file diff --git a/parse/train/BkgRe1SFDS/BkgRe1SFDS_middle.json b/parse/train/BkgRe1SFDS/BkgRe1SFDS_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..f66df972d40a1e111a47c0ade8a9fb4688c954f8 --- /dev/null +++ b/parse/train/BkgRe1SFDS/BkgRe1SFDS_middle.json @@ -0,0 +1,45689 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 446, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 447, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 447, + 97 + ], + "score": 1.0, + "content": "LEARNING WORLD GRAPH DECOMPOSITIONS", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 99, + 448, + 117 + ], + "spans": [ + { + "bbox": [ + 104, + 99, + 448, + 117 + ], + "score": 1.0, + "content": "TO ACCELERATE REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 209, + 469, + 352 + ], + "lines": [ + { + "bbox": [ + 141, + 208, + 469, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 208, + 469, + 223 + ], + "score": 1.0, + "content": "Efficiently learning to solve tasks in complex environments is a key challenge", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 233 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 233 + ], + "score": 1.0, + "content": "for reinforcement learning (RL) agents. 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In these settings, model-free reinforcement", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "learning (RL) agents often struggle to learn efficiently, requiring a large amount of experience", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 505, + 452 + ], + "score": 1.0, + "content": "collections to converge to optimal behaviors. Intuitively, an agent could learn more efficiently by", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "focusing its exploration in task-relevant regions, if it has knowledge of the high-level structure of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 462, + 161, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 161, + 472 + ], + "score": 1.0, + "content": "environment.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 394, + 506, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 490 + ], + "score": 1.0, + "content": "We propose a method to 1) learn and 2) use an environment decomposition in the form of a world", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "graph, a task-agnostic abstraction. World graph nodes are waypoint states, a set of salient states", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "score": 1.0, + "content": "that can summarize agent trajectories and provide meaningful starting points for efficient explo-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "ration (Chatzigiorgaki & Skodras, 2009; Jayaraman et al., 2018; Ghosh et al., 2018). The directed and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "weighted world graph edges characterize feasible traversals among the waypoints. To leverage the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "world graph, we model hierarchical RL (HRL) agents where a high-level policy chooses a waypoint", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "state as a goal to guide exploration towards task-relevant regions, and a low-level policy strives to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 201, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 201, + 567 + ], + "score": 1.0, + "content": "reach the chosen goals.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 478, + 506, + 567 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 571, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "score": 1.0, + "content": "Our framework consists of two phases. In the task-agnostic phase, we obtain world graphs by training", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "a recurrent variational auto-encoder (VAE) (Chung et al., 2015; Gregor et al., 2015; Kingma &", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 605 + ], + "score": 1.0, + "content": "Welling, 2013) with binary latent variables (Nalisnick & Smyth, 2016) over trajectories collected", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "using a random walk policy (Ha & Schmidhuber, 2018) and a curiosity-driven goal-conditioned", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "policy (Ghosh et al., 2018; Nair et al., 2018). 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For example, a robotics agent often locates or navigates by interpreting a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "map (Lowry et al., 2015; Thrun, 1998; Angeli et al., 2008). Our exploration strategy draws inspiration", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 485 + ], + "score": 1.0, + "content": "from active localization, where robots are actively guided to investigate unfamiliar regions (Fox", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "et al., 1998; Li et al., 2016). Besides mapping, recent works (Azar et al., 2019; Ha & Schmidhuber,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "2018; Guo et al., 2018) learn to represent the world with generative latent states (Tian & Gong, 2017;", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 506, + 518 + ], + "score": 1.0, + "content": "Haarnoja et al., 2018; Racanière et al., 2017). If the latent dynamics are also extrapolated, the latent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "states can assist planning (Mnih et al., 2016a; Hafner et al., 2018) or model-based RL (Gregor &", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 528, + 241, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 241, + 540 + ], + "score": 1.0, + "content": "Besse, 2018; Kaiser et al., 2019).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "While also aiming to model the world, we approach this as abstracting both the structure and dynamics", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "of the environment in a graph representation, where nodes are states from the environment and edges", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "encode actionable efficient transitions between nodes. Existing works (Metzen, 2013; Mannor et al.,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "2004; Eysenbach et al., 2019; Entezari et al., 2010) have shown benefits of such graph abstractions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "but typically select nodes only subject to a good coverage the observed state space. 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Our", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "work draws intuition from unsupervised temporal segmentation (Chatzigiorgaki & Skodras, 2009;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 278, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 334, + 291 + ], + "score": 1.0, + "content": "Jayaraman et al., 2018) and imitation learning (Abbeel &", + "type": "text" + }, + { + "bbox": [ + 334, + 279, + 348, + 290 + ], + "score": 0.38, + "content": "\\mathrm { N g }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 278, + 505, + 291 + ], + "score": 1.0, + "content": ", 2004; Hussein et al., 2017). We define", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "“important” states (waypoints) as the most critical states in recovering action sequences generated", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "by some agent, which indicates that these states contain the richest information about the executed", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 311, + 211, + 323 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 211, + 323 + ], + "score": 1.0, + "content": "policy (Azar et al., 2019).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 340, + 270, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 272, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 272, + 355 + ], + "score": 1.0, + "content": "3 LEARNING WORLD GRAPHS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 303, + 381 + ], + "score": 1.0, + "content": "We propose a method for learning a world graph", + "type": "text" + }, + { + "bbox": [ + 303, + 368, + 317, + 379 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 366, + 505, + 381 + ], + "score": 1.0, + "content": ", a task-agnostic abstraction of an environment", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "that captures its high-level structure and dynamics. In this work, the primary use of world graphs", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 408, + 402 + ], + "score": 1.0, + "content": "is to accelerate reinforcement learning of downstream tasks. The nodes of", + "type": "text" + }, + { + "bbox": [ + 409, + 390, + 422, + 401 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 389, + 506, + 402 + ], + "score": 1.0, + "content": ", denoted by a set of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 400, + 507, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 172, + 414 + ], + "score": 1.0, + "content": "waypoints states", + "type": "text" + }, + { + "bbox": [ + 172, + 401, + 205, + 413 + ], + "score": 0.92, + "content": "s _ { p } \\in \\mathcal { V } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 400, + 507, + 414 + ], + "score": 1.0, + "content": ", are generically “important” for accomplishing tasks within the environment,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 410, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 425 + ], + "score": 1.0, + "content": "and therefore useful as starting points for exploration. Our method identifies such waypoint states", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "from interactions with the environment. In addition, we embed feasible transitions between nearby", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 434, + 249, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 232, + 446 + ], + "score": 1.0, + "content": "waypoint states as the edges of", + "type": "text" + }, + { + "bbox": [ + 232, + 434, + 245, + 444 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 434, + 249, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 372, + 463 + ], + "score": 1.0, + "content": "In this work, we define important states in the context of learning", + "type": "text" + }, + { + "bbox": [ + 372, + 451, + 386, + 461 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "(see Section 2 for alternative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "definitions). That is, we wish to discover a small set of states that, when used as world graph nodes,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "concisely summarize the structure and dynamics of the environment. Below, we describe 1) how to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 483, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 507, + 496 + ], + "score": 1.0, + "content": "collect state-action trajectories and an unsupervised learning objective to identify world graph nodes,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 493, + 497, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 497, + 507 + ], + "score": 1.0, + "content": "and 2) how the graph’s edges (i.e., how to transition between nodes) are formed from trajectories.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 279, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 281, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 281, + 533 + ], + "score": 1.0, + "content": "3.1 WAYPOINT STATE IDENTIFICATION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "The structure and dynamics of an environment are implicit in the state-action trajectories observed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "during exploration. To identify world graph nodes from such data, we train a recurrent variational", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "autoencoder (VAE) that, given a sequence of state-action pairs, identifies a subset of the states in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "the sequence from which the full action sequence can be reconstructed (Figure 2). In particular, the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "VAE infers binary latent variables that controls whether each state in the sequence is used by the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 598, + 354, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 354, + 610 + ], + "score": 1.0, + "content": "generative decoder, i.e., whether a state is “important” or not.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "Binary Latent VAE The VAE consists of an inference, a generative and a prior network. 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A generation network", + "type": "text" + }, + { + "bbox": [ + 311, + 238, + 321, + 246 + ], + "score": 0.81, + "content": "p _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 235, + 367, + 248 + ], + "score": 1.0, + "content": "reconstructs", + "type": "text" + }, + { + "bbox": [ + 368, + 238, + 375, + 245 + ], + "score": 0.75, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 235, + 396, + 248 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 396, + 236, + 435, + 247 + ], + "score": 0.92, + "content": "\\{ s _ { t } | z _ { t } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 235, + 439, + 248 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 322 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 105, + 256, + 506, + 323 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 340, + 270, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 272, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 272, + 355 + ], + "score": 1.0, + "content": "3 LEARNING WORLD GRAPHS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 303, + 381 + ], + "score": 1.0, + "content": "We propose a method for learning a world graph", + "type": "text" + }, + { + "bbox": [ + 303, + 368, + 317, + 379 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 366, + 505, + 381 + ], + "score": 1.0, + "content": ", a task-agnostic abstraction of an environment", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "that captures its high-level structure and dynamics. In this work, the primary use of world graphs", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 389, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 408, + 402 + ], + "score": 1.0, + "content": "is to accelerate reinforcement learning of downstream tasks. The nodes of", + "type": "text" + }, + { + "bbox": [ + 409, + 390, + 422, + 401 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 389, + 506, + 402 + ], + "score": 1.0, + "content": ", denoted by a set of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 400, + 507, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 172, + 414 + ], + "score": 1.0, + "content": "waypoints states", + "type": "text" + }, + { + "bbox": [ + 172, + 401, + 205, + 413 + ], + "score": 0.92, + "content": "s _ { p } \\in \\mathcal { V } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 400, + 507, + 414 + ], + "score": 1.0, + "content": ", are generically “important” for accomplishing tasks within the environment,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 410, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 425 + ], + "score": 1.0, + "content": "and therefore useful as starting points for exploration. Our method identifies such waypoint states", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "from interactions with the environment. In addition, we embed feasible transitions between nearby", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 434, + 249, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 232, + 446 + ], + "score": 1.0, + "content": "waypoint states as the edges of", + "type": "text" + }, + { + "bbox": [ + 232, + 434, + 245, + 444 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 434, + 249, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 366, + 507, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 450, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 372, + 463 + ], + "score": 1.0, + "content": "In this work, we define important states in the context of learning", + "type": "text" + }, + { + "bbox": [ + 372, + 451, + 386, + 461 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "(see Section 2 for alternative", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "definitions). That is, we wish to discover a small set of states that, when used as world graph nodes,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "concisely summarize the structure and dynamics of the environment. Below, we describe 1) how to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 483, + 507, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 507, + 496 + ], + "score": 1.0, + "content": "collect state-action trajectories and an unsupervised learning objective to identify world graph nodes,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 493, + 497, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 497, + 507 + ], + "score": 1.0, + "content": "and 2) how the graph’s edges (i.e., how to transition between nodes) are formed from trajectories.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 450, + 507, + 507 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 279, + 532 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 281, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 281, + 533 + ], + "score": 1.0, + "content": "3.1 WAYPOINT STATE IDENTIFICATION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 542, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "The structure and dynamics of an environment are implicit in the state-action trajectories observed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 567 + ], + "score": 1.0, + "content": "during exploration. To identify world graph nodes from such data, we train a recurrent variational", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "autoencoder (VAE) that, given a sequence of state-action pairs, identifies a subset of the states in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 588 + ], + "score": 1.0, + "content": "the sequence from which the full action sequence can be reconstructed (Figure 2). In particular, the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "VAE infers binary latent variables that controls whether each state in the sequence is used by the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 598, + 354, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 354, + 610 + ], + "score": 1.0, + "content": "generative decoder, i.e., whether a state is “important” or not.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 543, + 506, + 610 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 627 + ], + "score": 1.0, + "content": "Binary Latent VAE The VAE consists of an inference, a generative and a prior network. These", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 352, + 639 + ], + "score": 1.0, + "content": "are structured as follows: the input to the inference network", + "type": "text" + }, + { + "bbox": [ + 352, + 627, + 363, + 638 + ], + "score": 0.84, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "is a trajectory of state-action pairs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 636, + 507, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 231, + 653 + ], + "score": 1.0, + "content": "observed from the environment", + "type": "text" + }, + { + "bbox": [ + 231, + 637, + 298, + 650 + ], + "score": 0.92, + "content": "{ \\tau } = \\{ ( s _ { t } , a _ { t } ) \\} _ { t = 0 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 636, + 321, + 653 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 322, + 637, + 367, + 650 + ], + "score": 0.92, + "content": "{ \\pmb s } = \\{ { \\boldsymbol s } _ { t } \\} _ { t = 0 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 636, + 385, + 653 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 637, + 433, + 650 + ], + "score": 0.92, + "content": "\\pmb { a } { = } \\{ a _ { t } \\} _ { t = 0 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 636, + 507, + 653 + ], + "score": 1.0, + "content": "denoting the state", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "and action sequences respectively. 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Algorithm 1: Identifying waypoint states Vp and learning a goal-conditioned policy g Result: Waypoint states Vp and a goal-conditioned policy π g
Initialize network parameters for the recurrent variational inference model V Initialize network parameters for the goal-conditioned policy g Initialize Vp with the initial position of the agent,i.e.Vp = {so = (1,1)}
while VAE reconstruction error has not converged do
forn←1toNdo Sample random waypoint sp ∈ Vp
Navigate agent to sp and perform T-step rollout using a randow walk policy: T𝑛 ←{(s0= Sp,ao),.,(sT,ar)}
gn←ST Navigate agent to Sp and perform T-step rollout using Tg with goal gn:
Tπ ←{(s= Sp,ao),.,(sT,ar)}at~πg(-st,9n) Re-label πg rewards with action reconstruction error as curiosity bonus:
rπ←{1st+1=n-λ·p(at|s,z)}=0
end
Perform policy gradient update of πg using T" and rπ
Update V using T and T
Update Vp as set of states with largest prior mean αs αs+βs
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We use the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 413, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 179, + 428 + ], + "score": 1.0, + "content": "Beta distribution", + "type": "text" + }, + { + "bbox": [ + 179, + 414, + 276, + 426 + ], + "score": 0.9, + "content": "p _ { \\psi } ( z _ { t } ) = \\mathrm { B e t a } ( \\alpha _ { t } , \\beta _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 413, + 506, + 428 + ], + "score": 1.0, + "content": "for the prior and the Hard Kumaraswamy distribution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 427, + 507, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 249, + 439 + ], + "score": 0.9, + "content": "q _ { \\psi } ( z _ { t } | { a } , { z } ) = \\mathrm { H a r d K u m a } ( \\tilde { \\alpha } _ { t } , \\tilde { \\beta } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 427, + 507, + 441 + ], + "score": 1.0, + "content": "for the approximate posterior, which resembles the Beta dis-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "tribution but is outside the exponential family (Bastings et al., 2019). 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Hence,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 661 + ], + "score": 1.0, + "content": "collecting a rich set of trajectories is crucial. 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For", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 102, + 85, + 504, + 318 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 102, + 85, + 504, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 102, + 85, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 102, + 85, + 504, + 318 + ], + "score": 0.533, + "html": "
Algorithm 1: Identifying waypoint states Vp and learning a goal-conditioned policy g Result: Waypoint states Vp and a goal-conditioned policy π g
Initialize network parameters for the recurrent variational inference model V Initialize network parameters for the goal-conditioned policy g Initialize Vp with the initial position of the agent,i.e.Vp = {so = (1,1)}
while VAE reconstruction error has not converged do
forn←1toNdo Sample random waypoint sp ∈ Vp
Navigate agent to sp and perform T-step rollout using a randow walk policy: T𝑛 ←{(s0= Sp,ao),.,(sT,ar)}
gn←ST Navigate agent to Sp and perform T-step rollout using Tg with goal gn:
Tπ ←{(s= Sp,ao),.,(sT,ar)}at~πg(-st,9n) Re-label πg rewards with action reconstruction error as curiosity bonus:
rπ←{1st+1=n-λ·p(at|s,z)}=0
end
Perform policy gradient update of πg using T" and rπ
Update V using T and T
Update Vp as set of states with largest prior mean αs αs+βs
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This choice allows us to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "sample 0s and 1s without sacrificing differentiability, accomplished via the stretch-and-rectify proce-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "dure (Bastings et al., 2019; Louizos et al., 2017) and the reparametrization trick (Kingma & Welling,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "2013). 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Another term", + "type": "text" + }, + { + "bbox": [ + 489, + 505, + 504, + 515 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { T }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 452, + 528 + ], + "score": 1.0, + "content": "to encourage temporal separation between selected states by targeting the number of", + "type": "text" + }, + { + "bbox": [ + 452, + 515, + 466, + 525 + ], + "score": 0.73, + "content": "0 / 1", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "switches", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 525, + 175, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 136, + 540 + ], + "score": 1.0, + "content": "among", + "type": "text" + }, + { + "bbox": [ + 136, + 528, + 144, + 536 + ], + "score": 0.75, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 525, + 154, + 540 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 154, + 527, + 171, + 538 + ], + "score": 0.89, + "content": "2 \\mu _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 525, + 175, + 540 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 402, + 507, + 540 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 134, + 543, + 476, + 578 + ], + "lines": [ + { + "bbox": [ + 134, + 543, + 476, + 578 + ], + "spans": [ + { + "bbox": [ + 134, + 543, + 476, + 578 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { 0 } = \\Big | \\Big | \\mathbb { E } _ { q _ { \\phi } ( z | s , a ) } [ \\| z \\| _ { 0 } ] - \\mu _ { 0 } \\Big | \\Big | ^ { 2 } , \\quad \\mathcal { L } _ { T } = \\Bigg | \\Bigg | \\mathbb { E } _ { q _ { \\phi } ( z | s , a ) } \\left[ \\sum _ { t = 0 } ^ { T } \\mathbb { 1 } [ z _ { t } \\neq z _ { t + 1 } ] \\right] - 2 \\mu _ { 0 } \\Bigg | \\Bigg | ^ { 2 } .", + "type": "interline_equation", + "image_path": "602b63b98973b4ecac88d73465672660262ace527c7c8cec985e50a21475e94d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 134, + 543, + 476, + 554.6666666666666 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 134, + 554.6666666666666, + 476, + 566.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 134, + 566.3333333333333, + 476, + 577.9999999999999 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 581, + 504, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 354, + 594 + ], + "score": 1.0, + "content": "See Appendix A for details on training the VAE with binary", + "type": "text" + }, + { + "bbox": [ + 354, + 583, + 364, + 592 + ], + "score": 0.83, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 581, + 505, + 594 + ], + "score": 1.0, + "content": ", including integration of the Hard", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 593, + 379, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 593, + 368, + 605 + ], + "score": 1.0, + "content": "Kumaraswamy distribution and how to regularize the statistics of", + "type": "text" + }, + { + "bbox": [ + 369, + 595, + 375, + 603 + ], + "score": 0.78, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 593, + 379, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 581, + 505, + 605 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 617, + 333, + 628 + ], + "lines": [ + { + "bbox": [ + 105, + 617, + 336, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 336, + 630 + ], + "score": 1.0, + "content": "3.2 EXPLORATION FOR WORLD GRAPH DISCOVERY", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "Naturally, the latent structure learned by the VAE depends on the trajectories used to train it. 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Right: using Wide-then-Narrow goals. 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In the first rollout, we perform a random walk to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "score": 1.0, + "content": "explore the nearby region. In the second rollout, we perform actions using a goal-conditioned policy", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 118, + 280 + ], + "score": 0.8, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "(GCP), setting the final state reached by the random walk as the goal. Both rollouts are used for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 276, + 343, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 326, + 292 + ], + "score": 1.0, + "content": "trianing the VAE and the latter is also used for training", + "type": "text" + }, + { + "bbox": [ + 327, + 280, + 338, + 290 + ], + "score": 0.85, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 276, + 343, + 292 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 308 + ], + "score": 1.0, + "content": "GCP provides a venue to integrate intrinsic motivation, such as curiosity (Burda et al., 2018; Achiam", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 306, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 506, + 318 + ], + "score": 1.0, + "content": "& Sastry, 2017; Pathak et al., 2017; Azar et al., 2019) to generate more diverse rollouts. Specifically,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 469, + 329 + ], + "score": 1.0, + "content": "we use the action reconstruction error of the VAE as an intrinsic reward signal when training", + "type": "text" + }, + { + "bbox": [ + 469, + 318, + 481, + 329 + ], + "score": 0.86, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 317, + 505, + 329 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 326, + 497, + 343 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 481, + 343 + ], + "score": 1.0, + "content": "choice of curioisty also prevents the VAE from collapsing to the simple behaviors of a vanilla", + "type": "text" + }, + { + "bbox": [ + 481, + 330, + 492, + 340 + ], + "score": 0.86, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 326, + 497, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 353, + 210, + 364 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 212, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 212, + 366 + ], + "score": 1.0, + "content": "3.3 EDGE FORMATION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 373, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 291, + 387 + ], + "score": 1.0, + "content": "The final stage is to construct the edges of", + "type": "text" + }, + { + "bbox": [ + 291, + 374, + 304, + 385 + ], + "score": 0.9, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 374, + 506, + 387 + ], + "score": 1.0, + "content": ", which should ideally capture the environment", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 482, + 398 + ], + "score": 1.0, + "content": "dynamics, i.e. how to transition between waypoint states. Once VAE training is complete and", + "type": "text" + }, + { + "bbox": [ + 483, + 385, + 495, + 397 + ], + "score": 0.89, + "content": "\\nu _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 394, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 366, + 410 + ], + "score": 1.0, + "content": "fixed, we collect random walk rollouts from each of the waypoints", + "type": "text" + }, + { + "bbox": [ + 367, + 396, + 400, + 408 + ], + "score": 0.93, + "content": "s _ { p } \\in \\mathcal { V } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 394, + 506, + 410 + ], + "score": 1.0, + "content": "to estimate the underlying", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 410, + 420 + ], + "score": 1.0, + "content": "adjacency matrix (Biggs, 1993). More precisely, we claim a directed edge", + "type": "text" + }, + { + "bbox": [ + 411, + 408, + 446, + 419 + ], + "score": 0.9, + "content": "s _ { p } \\to s _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "if there exists", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 234, + 430 + ], + "score": 1.0, + "content": "a random walk trajectory from", + "type": "text" + }, + { + "bbox": [ + 234, + 420, + 244, + 430 + ], + "score": 0.86, + "content": "s _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 417, + 256, + 430 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 256, + 419, + 267, + 430 + ], + "score": 0.87, + "content": "s _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "that does not intersect a third waypoint. We also consider", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 165, + 442 + ], + "score": 1.0, + "content": "paths taken by", + "type": "text" + }, + { + "bbox": [ + 166, + 430, + 177, + 441 + ], + "score": 0.87, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 429, + 223, + 442 + ], + "score": 1.0, + "content": "(starting at", + "type": "text" + }, + { + "bbox": [ + 224, + 431, + 234, + 441 + ], + "score": 0.87, + "content": "s _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 429, + 281, + 442 + ], + "score": 1.0, + "content": "and setting", + "type": "text" + }, + { + "bbox": [ + 281, + 431, + 291, + 441 + ], + "score": 0.85, + "content": "s _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "as the goal) and keep the shortest observed path from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 107, + 442, + 117, + 452 + ], + "score": 0.85, + "content": "s _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 439, + 128, + 453 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 128, + 442, + 139, + 452 + ], + "score": 0.86, + "content": "s _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "as a world graph edge transition. We use the action sequence length of the edge transition", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "between adjacent waypoints as the weight of the edge. As shown experimentally, a key benefit of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 265, + 475 + ], + "score": 1.0, + "content": "our approach is the ability to plan over", + "type": "text" + }, + { + "bbox": [ + 266, + 462, + 279, + 473 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 462, + 506, + 475 + ], + "score": 1.0, + "content": ". To navigate from one waypoint to another, we can use", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "dynamic programming (Sutton, 1998; Feng et al., 2004) to output the optimal traversal of the graph.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 106, + 500, + 477, + 513 + ], + "lines": [ + { + "bbox": [ + 104, + 500, + 478, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 500, + 478, + 514 + ], + "score": 1.0, + "content": "4 ACCELERATING REINFORCEMENT LEARNING WITH WORLD GRAPHS", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "World graphs present a high-level, task-agnostic abstraction of the environment through waypoints", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 536, + 507, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 507, + 549 + ], + "score": 1.0, + "content": "and feasible transition routes between them. A key example of world graph applications for task-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "specific RL is structured exploration: instead of exploring the entire environment, RL agents can use", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 558, + 507, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 507, + 572 + ], + "score": 1.0, + "content": "world graphs to quickly identify task-relevant regions and bias low-level exploration to these regions.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 570, + 461, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 461, + 582 + ], + "score": 1.0, + "content": "Our framework to leverage world graphs for structured exploration consists of two parts:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 128, + 590, + 432, + 618 + ], + "lines": [ + { + "bbox": [ + 128, + 587, + 433, + 605 + ], + "spans": [ + { + "bbox": [ + 128, + 587, + 416, + 605 + ], + "score": 1.0, + "content": "1. 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Right: using Wide-then-Narrow goals. 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Visualizations can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 167, + 208, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 208, + 179 + ], + "score": 1.0, + "content": "be found in Appendix D.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 507, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 211 + ], + "score": 1.0, + "content": "decomposes the policy of the agent into two separate policies that receive distinct streams of reward:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "a high-level policy (“Manager”) learns to propose subgoals; a low-level policy (“Worker”) receives", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "subgoals from the Manager as inputs and is rewarded for taking actions in the environment that reach", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "the subgoals. 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More details are in Appendix B.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 106, + 337, + 354, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 356, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 356, + 350 + ], + "score": 1.0, + "content": "4.2 WIDE-THEN-NARROW GOALS AND WORLD GRAPHS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "To incorporate the world graph, we introduce a Manager policy that factorizes subgoal selection as fol-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 184, + 383 + ], + "score": 1.0, + "content": "lows: a wide policy", + "type": "text" + }, + { + "bbox": [ + 184, + 370, + 228, + 381 + ], + "score": 0.93, + "content": "\\pi ^ { w } ( g _ { t } ^ { w } | s _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 369, + 386, + 383 + ], + "score": 1.0, + "content": "selects a waypoint state as the wide goal", + "type": "text" + }, + { + "bbox": [ + 387, + 370, + 421, + 381 + ], + "score": 0.92, + "content": "g ^ { w } \\in \\mathcal { V } _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 369, + 506, + 383 + ], + "score": 1.0, + "content": ", and a narrow policy", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 379, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 107, + 380, + 164, + 392 + ], + "score": 0.92, + "content": "\\pi ^ { n } ( g _ { t } ^ { n } | s _ { t } , g _ { t } ^ { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 379, + 352, + 394 + ], + "score": 1.0, + "content": "selects a state within a local neighborhood of", + "type": "text" + }, + { + "bbox": [ + 352, + 381, + 364, + 392 + ], + "score": 0.89, + "content": "g _ { t } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 379, + 396, + 394 + ], + "score": 1.0, + "content": ", i.e. its", + "type": "text" + }, + { + "bbox": [ + 397, + 383, + 402, + 390 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 379, + 472, + 394 + ], + "score": 1.0, + "content": "-net (Mahadevan", + "type": "text" + }, + { + "bbox": [ + 472, + 381, + 481, + 390 + ], + "score": 0.36, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 379, + 506, + 394 + ], + "score": 1.0, + "content": "Mag-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 103, + 387, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 103, + 387, + 237, + 407 + ], + "score": 1.0, + "content": "gioni, 2007), as the narrow goal", + "type": "text" + }, + { + "bbox": [ + 237, + 391, + 338, + 403 + ], + "score": 0.92, + "content": "g ^ { n } \\in \\{ s : \\mathcal { D } ( s , \\bar { g } _ { t } ^ { w } ) \\leq \\epsilon \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 387, + 419, + 407 + ], + "score": 1.0, + "content": ". 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A visual illustration is in Figure 4 and training details in Appendix C.2.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 252, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 254, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 254, + 450 + ], + "score": 1.0, + "content": "4.3 WORLD GRAPH TRAVERSAL", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 457, + 504, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "The wide-then-narrow subgoal format simplifies the search space for the Manager policy. 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The Worker policy", + "type": "text" + }, + { + "bbox": [ + 419, + 391, + 505, + 403 + ], + "score": 0.92, + "content": "\\pi ^ { \\mathrm { w o r k e r } } ( a _ { t } | s _ { t } , g _ { t } ^ { n } , g _ { t } ^ { \\bar { w } } )", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "chooses the action taken by the agent given the current state and the wide and narrow goals from the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 414, + 433, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 433, + 425 + ], + "score": 1.0, + "content": "Manager. A visual illustration is in Figure 4 and training details in Appendix C.2.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 103, + 358, + 506, + 425 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 437, + 252, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 254, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 254, + 450 + ], + "score": 1.0, + "content": "4.3 WORLD GRAPH TRAVERSAL", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 457, + 504, + 491 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "The wide-then-narrow subgoal format simplifies the search space for the Manager policy. Using", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "waypoints as wide goals also makes it possible to leverage the edges of the world graph for planning", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 480, + 408, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 408, + 492 + ], + "score": 1.0, + "content": "and executing the planned traversals. This process breaks down as follows:", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 106, + 457, + 505, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 129, + 500, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 129, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 129, + 499, + 406, + 513 + ], + "score": 1.0, + "content": "1. When to Traverse: When the agent encounters a waypoint state", + "type": "text" + }, + { + "bbox": [ + 406, + 500, + 438, + 512 + ], + "score": 0.91, + "content": "s _ { t } \\in \\mathcal V _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 499, + 505, + 513 + ], + "score": 1.0, + "content": ", a “traversal” is", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 139, + 509, + 435, + 526 + ], + "spans": [ + { + "bbox": [ + 139, + 509, + 186, + 526 + ], + "score": 1.0, + "content": "initiated if", + "type": "text" + }, + { + "bbox": [ + 186, + 512, + 196, + 522 + ], + "score": 0.85, + "content": "s _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 509, + 309, + 526 + ], + "score": 1.0, + "content": "has a feasible connection in", + "type": "text" + }, + { + "bbox": [ + 309, + 511, + 323, + 522 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 509, + 416, + 526 + ], + "score": 1.0, + "content": "to the active wide goal", + "type": "text" + }, + { + "bbox": [ + 417, + 512, + 429, + 523 + ], + "score": 0.89, + "content": "g _ { t } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 509, + 435, + 526 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 128, + 524, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 128, + 524, + 505, + 538 + ], + "score": 1.0, + "content": "2. Planning: Upon triggering a traversal, the optimal traversal route from the initiating state", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 140, + 536, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 140, + 536, + 152, + 550 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 153, + 537, + 165, + 548 + ], + "score": 0.89, + "content": "g _ { t } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 536, + 252, + 550 + ], + "score": 1.0, + "content": "is estimated from the", + "type": "text" + }, + { + "bbox": [ + 253, + 537, + 266, + 547 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 536, + 505, + 550 + ], + "score": 1.0, + "content": "edge weights using classic dynamic programming planning", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 141, + 547, + 500, + 560 + ], + "spans": [ + { + "bbox": [ + 141, + 547, + 500, + 560 + ], + "score": 1.0, + "content": "(Sutton, 1998; Feng et al., 2004). This yields a sequence of intermediate waypoint states.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 129, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 129, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "3. Execution: Execution of graph traversals depends on the nature of the environment. If", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 141, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 141, + 572, + 506, + 586 + ], + "score": 1.0, + "content": "deterministic, the agent simply follows the action sequences given by the edges of the", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 141, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 141, + 583, + 349, + 597 + ], + "score": 1.0, + "content": "traversal. 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If the agent fails to reach the next waypoint state within a certain time limit, it", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 141, + 615, + 457, + 630 + ], + "spans": [ + { + "bbox": [ + 141, + 615, + 281, + 630 + ], + "score": 1.0, + "content": "stops its current pursuit and a new", + "type": "text" + }, + { + "bbox": [ + 281, + 616, + 316, + 628 + ], + "score": 0.93, + "content": "( g ^ { w } , g ^ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 615, + 457, + 630 + ], + "score": 1.0, + "content": "pair is received from the Manager.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 51, + "bbox_fs": [ + 128, + 499, + 506, + 630 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 636, + 504, + 669 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 409, + 650 + ], + "score": 1.0, + "content": "World graph traversal allows the Manager to assign task-relevant wide goals", + "type": "text" + }, + { + "bbox": [ + 410, + 637, + 422, + 648 + ], + "score": 0.89, + "content": "g ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 635, + 505, + 650 + ], + "score": 1.0, + "content": "that can be far away", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "from the agent yet still reachable, which consequentially accelerates learning by focusing exploration", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 106, + 658, + 268, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 252, + 671 + ], + "score": 1.0, + "content": "around the task-relevant region near", + "type": "text" + }, + { + "bbox": [ + 252, + 659, + 264, + 669 + ], + "score": 0.89, + "content": "g ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 658, + 268, + 671 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 58, + "bbox_fs": [ + 105, + 635, + 505, + 671 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 274, + 698 + ], + "lines": [ + { + "bbox": [ + 104, + 683, + 276, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 683, + 276, + 700 + ], + "score": 1.0, + "content": "5 EXPERIMENTAL VALIDATION", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 60 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "We now assess each component of our framework on a set of challenging 2D grid worlds. Our", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 721, + 386, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 386, + 732 + ], + "score": 1.0, + "content": "ablation studies demonstrate the following benefits of our framework:", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 61.5, + "bbox_fs": [ + 105, + 708, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 78, + 503, + 201 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 78, + 503, + 201 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 78, + 503, + 201 + ], + "spans": [ + { + "bbox": [ + 108, + 78, + 503, + 201 + ], + "score": 0.981, + "html": "
TaskSizeA2CFN+πg initOurs
+πg-init+ 9w-traversal+πg-init+Gw-traversal
MultiGoalSmall2.04±0.052.93±0.745.25±0.133.92±0.225.05±0.03
Medium==5.15±0.112.56±0.093.00±0.90
Larger==-2.18±0.122.72±0.59
MultiGoal-SparseSmall==0.39±0.090.24±0.040.42±0.07
Medium0.20±0.040.25±0.03
Larger0.16±0.220.26±0.11
MultiGoal-StochasticSmall1.38±1.201.93±0.163.06±0.312.92±0.45
Medium=2.99±0.122.42±0.242.64±0.14
Larger===0.60±0.12
Door-KeySmall=0.99±0.000.37±0.150.92±0.02
Medium0.56±0.020.76±0.06
Larger=0.26±0.19
", + "type": "table", + "image_path": "cf6404ac297859fc913d08dceb2daf866c13f6d312aaa9b45a16acb570a106cf.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 78, + 503, + 119.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 119.0, + 503, + 160.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 160.0, + 503, + 201.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 106, + 208, + 505, + 253 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 206, + 504, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 492, + 223 + ], + "score": 1.0, + "content": "Table 2: On a variety of tasks and environment setups, we evaluate RL models trained with GCP", + "type": "text" + }, + { + "bbox": [ + 492, + 210, + 504, + 221 + ], + "score": 0.82, + "content": "\\pi _ { g }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 219, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 183, + 232 + ], + "score": 1.0, + "content": "initialization, with", + "type": "text" + }, + { + "bbox": [ + 184, + 220, + 197, + 230 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 219, + 506, + 232 + ], + "score": 1.0, + "content": "world graph travresal, and with both. All models on the right are equipped", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "with WN. Left are baselines for additional comparison. We report final rewards for MultiGoal tasks", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 241, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 504, + 253 + ], + "score": 1.0, + "content": "and success rates for Door-Key are reported. If no result reported, the agent failed to solve the task.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "table", + "bbox": [ + 167, + 263, + 444, + 289 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 167, + 263, + 444, + 289 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 167, + 263, + 444, + 289 + ], + "spans": [ + { + "bbox": [ + 167, + 263, + 444, + 289 + ], + "score": 0.953, + "html": "
Waypoint typeMultiGoalMultiGoal-SparseMultiGoal-StochasticDoor-Key
Learned2.72±0.590.26±0.110.60±0.120.26±0.19
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TaskSizeA2CFN+πg initOurs
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MultiGoalSmall2.04±0.052.93±0.745.25±0.133.92±0.225.05±0.03
Medium==5.15±0.112.56±0.093.00±0.90
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MultiGoal-StochasticSmall1.38±1.201.93±0.163.06±0.312.92±0.45
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Larger===0.60±0.12
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Medium0.56±0.020.76±0.06
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We report mean reward and standard deviation.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 455, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "We ablate each of the following components in our framework and compare against non-hierarchical", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 548, + 264, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 264, + 561 + ], + "score": 1.0, + "content": "(A2C) and hierarchical baselines (FN):", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 537, + 505, + 561 + ] + }, + { + "type": "index", + "bbox": [ + 130, + 569, + 451, + 611 + ], + "lines": [ + { + "bbox": [ + 128, + 567, + 452, + 584 + ], + "spans": [ + { + "bbox": [ + 128, + 567, + 435, + 584 + ], + "score": 1.0, + "content": "1. initializing the feature extraction layers of the Manager and Worker from", + "type": "text" + }, + { + "bbox": [ + 435, + 572, + 447, + 582 + ], + "score": 0.84, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 567, + 452, + 584 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 128, + 583, + 406, + 596 + ], + "spans": [ + { + "bbox": [ + 128, + 583, + 406, + 596 + ], + "score": 1.0, + "content": "2. applying Wide-then-Narrow Manager (WN) goal instruction, and", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 129, + 598, + 312, + 612 + ], + "spans": [ + { + "bbox": [ + 129, + 598, + 294, + 612 + ], + "score": 1.0, + "content": "3. allowing the Worker to traverse along", + "type": "text" + }, + { + "bbox": [ + 295, + 599, + 308, + 610 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 598, + 312, + 612 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + } + ], + "index": 27, + "bbox_fs": [ + 128, + 567, + 452, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 493, + 631 + ], + "lines": [ + { + "bbox": [ + 106, + 618, + 494, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 494, + 632 + ], + "score": 1.0, + "content": "Results are shown in Table 2. In sum, each component improves performance over the baselines.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 618, + 494, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 642, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "Wide and narrow goals Using two goal types is a highly effective way to structure the Manager", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "instructions and enables the Worker to differentiate the transition and local task-solving phases. We", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 663, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 104, + 663, + 366, + 677 + ], + "score": 1.0, + "content": "note that for small MultiGoal, agents do not benefit much from", + "type": "text" + }, + { + "bbox": [ + 366, + 664, + 380, + 675 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 663, + 505, + 677 + ], + "score": 1.0, + "content": "traversal: it can rely solely on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 506, + 688 + ], + "score": 1.0, + "content": "the guidance from WN goals to master both phases. However with increasing maze size, the Worker", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 686, + 394, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 394, + 699 + ], + "score": 1.0, + "content": "struggles to master traversals on its own and thus fails solving the tasks.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 642, + 506, + 699 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "World Graph Traversal As conjectured in Section 4.3, the performance gain of our framework", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "can be explained by the larger range and more targeted exploration strategy. In addition, the Worker", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 356, + 299 + ], + "score": 1.0, + "content": "does not have to learn long distance transitions with the aid of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 356, + 288, + 369, + 298 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 370, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "traversals. Figure 4 confirms that", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 120, + 309 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 120, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "traversal speeds up convergence and its effect becomes more evident with larger mazes. Note that", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "the graph learning stage only need 2.4K iterations to converge. Even when taking these additional", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 264, + 333 + ], + "score": 1.0, + "content": "environment interactions into account,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 264, + 320, + 278, + 331 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 278, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "traversal still exhibits superior sample efficiency, not to", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "mention that the graph is shared among all tasks. Moreover, solving Door-Key involves a complex", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "combination of sub-tasks: find and pick up the key, reach and open the door and finally exit. With", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 458, + 366 + ], + "score": 1.0, + "content": "limited reward feedback, this is particularly difficult to learn. The ability to traverse along", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 459, + 353, + 472, + 364 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 472, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "enables", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "longer-horizon planning on top of the waypoints, thanks to which the agents boost the success rate on", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 375, + 309, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 205, + 387 + ], + "score": 1.0, + "content": "medium Door-Key from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 205, + 375, + 250, + 386 + ], + "score": 0.84, + "content": "0 . 5 6 { \\pm } 0 . 0 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 250, + 375, + 261, + 387 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 261, + 375, + 305, + 386 + ], + "score": 0.85, + "content": "0 . 7 5 { \\pm } 0 . 0 6 ", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 306, + 375, + 309, + 387 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 16 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 79, + 503, + 199 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 79, + 503, + 199 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 503, + 199 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 503, + 199 + ], + "score": 0.971, + "type": "image", + "image_path": "eeccd173a66cc7f083254caca6b3d6fe078ddac17830e5c24ef8e6d8706c4b0b.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 503, + 119.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 119.0, + 503, + 159.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 159.0, + 503, + 199.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 208, + 505, + 259 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 208, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 506, + 219 + ], + "score": 1.0, + "content": "Figure 4: Validation performance during training (mean and standard-deviation of reward, 3 seeds) for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 217, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 104, + 217, + 212, + 230 + ], + "score": 1.0, + "content": "MultiGoal. Left: Comparing", + "type": "text" + }, + { + "bbox": [ + 212, + 219, + 223, + 229 + ], + "score": 0.89, + "content": "\\nu _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 217, + 239, + 230 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 240, + 219, + 261, + 228 + ], + "score": 0.9, + "content": "\\mathcal { V } _ { \\mathrm { r a n d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 217, + 444, + 230 + ], + "score": 1.0, + "content": ", with or without traversal, all models use WN and", + "type": "text" + }, + { + "bbox": [ + 444, + 219, + 455, + 229 + ], + "score": 0.88, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 217, + 506, + 230 + ], + "score": 1.0, + "content": "initialization.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 227, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 288, + 241 + ], + "score": 1.0, + "content": "We see that 1) traversal speeds up convergence, 2)", + "type": "text" + }, + { + "bbox": [ + 288, + 228, + 310, + 239 + ], + "score": 0.9, + "content": "\\mathcal { V } _ { \\mathrm { r a n d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 227, + 506, + 241 + ], + "score": 1.0, + "content": "gives higher variance and slightly worse performance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 237, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 124, + 252 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 124, + 239, + 135, + 249 + ], + "score": 0.88, + "content": "\\nu _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 237, + 262, + 252 + ], + "score": 1.0, + "content": ". Right: comparing with or without", + "type": "text" + }, + { + "bbox": [ + 262, + 240, + 273, + 249 + ], + "score": 0.87, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 237, + 333, + 252 + ], + "score": 1.0, + "content": "initialization on", + "type": "text" + }, + { + "bbox": [ + 333, + 239, + 344, + 249 + ], + "score": 0.88, + "content": "\\nu _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 237, + 506, + 252 + ], + "score": 1.0, + "content": ", all models use WN. We see that initializing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 248, + 463, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 463, + 261 + ], + "score": 1.0, + "content": "the task-specific phase with the task-agnostic goal-conditioned policy significantly boosts learning.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 356, + 299 + ], + "score": 1.0, + "content": "does not have to learn long distance transitions with the aid of", + "type": "text" + }, + { + "bbox": [ + 356, + 288, + 369, + 298 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "traversals. Figure 4 confirms that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 120, + 309 + ], + "score": 0.88, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "traversal speeds up convergence and its effect becomes more evident with larger mazes. Note that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 505, + 322 + ], + "score": 1.0, + "content": "the graph learning stage only need 2.4K iterations to converge. Even when taking these additional", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 319, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 264, + 333 + ], + "score": 1.0, + "content": "environment interactions into account,", + "type": "text" + }, + { + "bbox": [ + 264, + 320, + 278, + 331 + ], + "score": 0.89, + "content": "\\mathcal { G } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 319, + 506, + 333 + ], + "score": 1.0, + "content": "traversal still exhibits superior sample efficiency, not to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "mention that the graph is shared among all tasks. Moreover, solving Door-Key involves a complex", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "combination of sub-tasks: find and pick up the key, reach and open the door and finally exit. With", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 458, + 366 + ], + "score": 1.0, + "content": "limited reward feedback, this is particularly difficult to learn. 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Training agent for first-person shooter game with actor-critic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 115, + 537, + 227, + 549 + ], + "spans": [ + { + "bbox": [ + 115, + 537, + 227, + 549 + ], + "score": 1.0, + "content": "curriculum learning. 2016.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 107, + 525, + 505, + 549 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 104, + 80, + 493, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 494, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 494, + 95 + ], + "score": 1.0, + "content": "A RECURRENT VAE WITH DIFFERENTIABLE BINARY LATENT VARIABLES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 502, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 504, + 119 + ], + "score": 1.0, + "content": "As illustrated in the main text, the main objective for the recurrent VAE is the following evidence", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 224, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 224, + 129 + ], + "score": 1.0, + "content": "lower bound with derivation:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 131, + 431, + 256 + ], + "lines": [ + { + "bbox": [ + 180, + 131, + 431, + 256 + ], + "spans": [ + { + "bbox": [ + 180, + 131, + 431, + 256 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\log p ( a | s ) = \\log \\int p ( a | s , z ) d z } \\\\ & { \\qquad = \\log \\int p ( a | s , z ) p ( z | s ) \\frac { q ( z | a , s ) } { q ( z | a , s ) } d z } \\\\ & { \\qquad = \\log \\int p ( a | s , z ) \\frac { p ( z | s ) } { q ( z | a , s ) } q ( z | a , s ) d z } \\\\ & { \\qquad \\geq \\mathbb { E } _ { q ( z | a , s ) } [ \\log p ( a | s , z ) - \\log \\frac { q ( z | a , s ) } { p ( z | s ) } ] } \\\\ & { \\qquad = \\mathbb { E } _ { q ( z | a , s ) } [ \\log p ( a | s , z ) ] - D _ { \\mathrm { K L } } ( q ( z | a , s ) | | p ( z | s ) ) } \\end{array}", + "type": "interline_equation", + "image_path": "74a121592da2d503de3dc3d7827353b7b30c2e1c28cc4cac718fdf83190bd04f.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 180, + 131, + 431, + 172.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 180, + 172.66666666666666, + 431, + 214.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 180, + 214.33333333333331, + 431, + 255.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 258, + 506, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 256, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 203, + 272 + ], + "score": 1.0, + "content": "The inference network", + "type": "text" + }, + { + "bbox": [ + 204, + 261, + 215, + 271 + ], + "score": 0.86, + "content": "q _ { \\psi }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 256, + 403, + 272 + ], + "score": 1.0, + "content": "takes in the trajectories of state-action pairs", + "type": "text" + }, + { + "bbox": [ + 404, + 261, + 411, + 268 + ], + "score": 0.78, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 256, + 505, + 272 + ], + "score": 1.0, + "content": "and at each time step", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 269, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 363, + 283 + ], + "score": 1.0, + "content": "approximates the posterior of the corresponding latent variable", + "type": "text" + }, + { + "bbox": [ + 363, + 271, + 372, + 281 + ], + "score": 0.83, + "content": "z _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 269, + 453, + 283 + ], + "score": 1.0, + "content": ". 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Similar to Beta, the Kuma distribution also ranges from bi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 164, + 354 + ], + "score": 1.0, + "content": "modal (when", + "type": "text" + }, + { + "bbox": [ + 165, + 342, + 198, + 353 + ], + "score": 0.89, + "content": "\\alpha \\approx \\beta )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 341, + 257, + 354 + ], + "score": 1.0, + "content": "to unimodal", + "type": "text" + }, + { + "bbox": [ + 257, + 341, + 303, + 353 + ], + "score": 0.9, + "content": "( \\alpha / \\beta \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 341, + 317, + 354 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 317, + 341, + 367, + 353 + ], + "score": 0.91, + "content": "\\alpha / \\beta \\to \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 341, + 428, + 354 + ], + "score": 1.0, + "content": "). 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We observe empirically better performance when we fix", + "type": "text" + }, + { + "bbox": [ + 449, + 353, + 475, + 363 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "Kuma approximated posterior. 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We take the first 5 terms here.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 108, + 576, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "Next, we make the Kuma distribution “hard” by following the steps in Bastings et al. (2019). 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Similar to Beta, the Kuma distribution also ranges from bi-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 164, + 354 + ], + "score": 1.0, + "content": "modal (when", + "type": "text" + }, + { + "bbox": [ + 165, + 342, + 198, + 353 + ], + "score": 0.89, + "content": "\\alpha \\approx \\beta )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 341, + 257, + 354 + ], + "score": 1.0, + "content": "to unimodal", + "type": "text" + }, + { + "bbox": [ + 257, + 341, + 303, + 353 + ], + "score": 0.9, + "content": "( \\alpha / \\beta \\to 0", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 341, + 317, + 354 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 317, + 341, + 367, + 353 + ], + "score": 0.91, + "content": "\\alpha / \\beta \\to \\infty )", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 341, + 428, + 354 + ], + "score": 1.0, + "content": "). Also, when", + "type": "text" + }, + { + "bbox": [ + 428, + 342, + 458, + 352 + ], + "score": 0.9, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 341, + 472, + 354 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 473, + 342, + 502, + 353 + ], + "score": 0.9, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 341, + 506, + 354 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 352, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 217, + 364 + ], + "score": 0.9, + "content": "\\operatorname { K u m a } ( \\alpha , \\beta ) = \\operatorname { B e t a } ( \\alpha , \\beta )", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 352, + 449, + 365 + ], + "score": 1.0, + "content": ". We observe empirically better performance when we fix", + "type": "text" + }, + { + "bbox": [ + 449, + 353, + 475, + 363 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 352, + 506, + 365 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "Kuma approximated posterior. 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We take the first 5 terms here.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 106, + 548, + 505, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 576, + 504, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 589 + ], + "score": 1.0, + "content": "Next, we make the Kuma distribution “hard” by following the steps in Bastings et al. (2019). First", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 195, + 600 + ], + "score": 1.0, + "content": "stretch the support to", + "type": "text" + }, + { + "bbox": [ + 195, + 587, + 290, + 599 + ], + "score": 0.87, + "content": "( r = 0 - \\epsilon _ { 1 } , l = 1 + \\epsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 587, + 296, + 600 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 297, + 588, + 338, + 599 + ], + "score": 0.87, + "content": "\\epsilon _ { 1 } , \\epsilon _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 587, + 505, + 600 + ], + "score": 1.0, + "content": ", and the resulting CDF distribution takes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 597, + 147, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 147, + 611 + ], + "score": 1.0, + "content": "the form:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 575, + 505, + 611 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 612, + 372, + 640 + ], + "lines": [ + { + "bbox": [ + 239, + 612, + 372, + 640 + ], + "spans": [ + { + "bbox": [ + 239, + 612, + 372, + 640 + ], + "score": 0.95, + "content": "F _ { S } ( z ) = F _ { \\mathrm { K u m a } } \\left( { \\frac { z - l } { r - l } } ; \\alpha , \\beta \\right) .", + "type": "interline_equation", + "image_path": "f498e055bf839a3b4e68963b2175603bfe1bb354a3a065bb4dfd8246322b0e21.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 239, + 612, + 372, + 626.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 239, + 626.0, + 372, + 640.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 644, + 506, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "Then, the non-eligible probabilities for 0’s and 1’s are attained by rectifying all samples below 0 to 0", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 655, + 298, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 298, + 667 + ], + "score": 1.0, + "content": "and above 1 to 1, and other value as it is, that is", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 643, + 505, + 667 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 671, + 466, + 698 + ], + "lines": [ + { + "bbox": [ + 144, + 671, + 466, + 698 + ], + "spans": [ + { + "bbox": [ + 144, + 671, + 466, + 698 + ], + "score": 0.91, + "content": "P ( z = 0 ) = F _ { \\mathrm { K u m a } } \\left( \\frac { - l } { r - l } ; \\alpha , \\beta \\right) , \\quad P ( z = 1 ) = 1 - F _ { \\mathrm { K u m a } } \\left( \\frac { 1 - l } { r - l } ; \\alpha , \\beta \\right) .", + "type": "interline_equation", + "image_path": "192632507efa1b88897b77a696970eb5a96eaf5ccbf0d7d10e691c5285e775e8.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 144, + 671, + 466, + 680.0 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 144, + 680.0, + 466, + 689.0 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 144, + 689.0, + 466, + 698.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 708, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 320, + 721 + ], + "score": 1.0, + "content": "Lastly, we impose two additional regularization terms", + "type": "text" + }, + { + "bbox": [ + 320, + 709, + 330, + 720 + ], + "score": 0.88, + "content": "{ \\mathcal { L } } _ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 708, + 348, + 721 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 348, + 709, + 363, + 720 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 708, + 505, + 721 + ], + "score": 1.0, + "content": "on the approximated posteriors. As", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 102, + 716, + 504, + 739 + ], + "spans": [ + { + "bbox": [ + 102, + 716, + 217, + 739 + ], + "score": 1.0, + "content": "described in the main text,", + "type": "text" + }, + { + "bbox": [ + 217, + 721, + 228, + 731 + ], + "score": 0.85, + "content": "{ \\mathcal { L } } _ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 716, + 467, + 739 + ], + "score": 1.0, + "content": "prevents the model from selecting all states to reconstruct", + "type": "text" + }, + { + "bbox": [ + 468, + 719, + 504, + 733 + ], + "score": 0.93, + "content": "\\{ a _ { t } \\} _ { 0 } ^ { T - 1 }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 219, + 95 + ], + "score": 1.0, + "content": "by restraining the expected", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 219, + 83, + 232, + 93 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 232, + 82, + 268, + 95 + ], + "score": 1.0, + "content": "norm of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 269, + 83, + 342, + 95 + ], + "score": 0.91, + "content": "z = \\left( z _ { 1 } \\cdot \\cdot \\cdot z _ { T - 1 } \\right)", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 343, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "to approximately be at a targeted value", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 95, + 118, + 105 + ], + "score": 0.81, + "content": "\\mu _ { 0 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 119, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "(Louizos et al., 2017; Bastings et al., 2019). 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Note that both expectations in Equation 2 have closed forms for HardKuma.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 142, + 445, + 230 + ], + "lines": [ + { + "bbox": [ + 145, + 142, + 445, + 230 + ], + "spans": [ + { + "bbox": [ + 145, + 142, + 445, + 230 + ], + "score": 0.65, + "content": "\\begin{array} { l } { \\displaystyle \\mathcal { L } _ { 0 } = \\left\\| \\mathbb { E } _ { q ( \\boldsymbol { z } | \\boldsymbol { s } , \\boldsymbol { a } ) } \\left[ \\left\\| \\boldsymbol { z } \\right\\| _ { 0 } \\right] - \\mu _ { 0 } \\right\\| ^ { 2 } , \\mathrm { w h e r e } } \\\\ { \\displaystyle \\mathbb { E } _ { q ( \\boldsymbol { z } | \\boldsymbol { s } , \\boldsymbol { a } ) } \\left[ \\left\\| \\boldsymbol { z } \\right\\| _ { 0 } \\right] = \\sum _ { t = 1 } ^ { T } \\mathbb { E } _ { q ( \\boldsymbol { z } _ { t } | \\boldsymbol { s } , \\boldsymbol { a } ) } \\left[ \\mathbb { 1 } _ { \\boldsymbol { z } _ { t } \\neq 0 } \\right] } \\\\ { \\displaystyle \\qquad = \\sum _ { t = 1 } ^ { T } 1 - p \\left( \\boldsymbol { z } _ { t } = 0 \\right) = \\sum _ { t = 1 } ^ { T } 1 - F _ { \\mathrm { K u m a } } \\left( \\frac { - l } { r - l } ; \\alpha _ { t } , \\beta _ { t } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "05375c9ba77cfcf9223c575f1bb6085fd0a88dec24f1342685259b651bd523bc.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 145, + 142, + 445, + 171.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 145, + 171.33333333333334, + 445, + 200.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 145, + 200.66666666666669, + 445, + 230.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 267, + 495, + 300 + ], + "lines": [ + { + "bbox": [ + 116, + 267, + 495, + 300 + ], + "spans": [ + { + "bbox": [ + 116, + 267, + 495, + 300 + ], + "score": 0.78, + "content": "\\mathbb { E } _ { q ( z | s , a ) } [ \\sum _ { t = 1 } ^ { T - 1 } \\mathbb { 1 } _ { z _ { t } \\neq z _ { t + 1 } } ] = \\sum _ { t = 1 } ^ { T - 1 } p \\left( z _ { t } = 0 \\right) \\left( 1 - p \\left( z _ { t + 1 } = 0 \\right) \\right) + \\left( 1 - p \\left( z _ { t } = 0 \\right) \\right) p \\left( z _ { t + 1 } = 0 \\right) .", + "type": "interline_equation", + "image_path": "c14baa5bbf179469a3b3c656cae666e14f93fa77dc76184a1900553fb7dcd0c1.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 116, + 267, + 495, + 278.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 116, + 278.0, + 495, + 289.0 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 116, + 289.0, + 495, + 300.0 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 322, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 506, + 335 + ], + "score": 1.0, + "content": "Lagrangian Relaxation. 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We tune the objective", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 139, + 357 + ], + "score": 1.0, + "content": "weights", + "type": "text" + }, + { + "bbox": [ + 139, + 345, + 150, + 356 + ], + "score": 0.87, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "using Lagrangian relaxation (Higgins et al., 2017; Bastings et al., 2019; Bertsekas, 1999),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 140, + 368 + ], + "score": 1.0, + "content": "treating", + "type": "text" + }, + { + "bbox": [ + 140, + 357, + 150, + 367 + ], + "score": 0.86, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 356, + 454, + 368 + ], + "score": 1.0, + "content": "’s as learnable parameters and performing alternative optimization between", + "type": "text" + }, + { + "bbox": [ + 455, + 356, + 465, + 367 + ], + "score": 0.88, + "content": "\\lambda _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "’s and the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 367, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 486, + 378 + ], + "score": 1.0, + "content": "model parameters. 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The hyperparameters used when", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 140, + 679 + ], + "score": 1.0, + "content": "training", + "type": "text" + }, + { + "bbox": [ + 140, + 667, + 152, + 678 + ], + "score": 0.86, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "follow those from Shang et al. (2019). 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If agent reaches", + "type": "text" + }, + { + "bbox": [ + 208, + 331, + 219, + 342 + ], + "score": 0.85, + "content": "g _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 330, + 504, + 343 + ], + "score": 1.0, + "content": ", the Worker also receives positive rewards and terminates this horizon.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 128, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "5. 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When either", + "type": "text" + }, + { + "bbox": [ + 194, + 371, + 206, + 381 + ], + "score": 0.85, + "content": "g _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "is reached or the maximum time step for this horizon is met, the Manager", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 381, + 238, + 393 + ], + "spans": [ + { + "bbox": [ + 141, + 381, + 238, + 393 + ], + "score": 1.0, + "content": "renews its subgoal pair.", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + } + ], + "index": 16.5, + "bbox_fs": [ + 128, + 241, + 506, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 415, + 413 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 411, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 242, + 417 + ], + "score": 1.0, + "content": "The training of the Worker policy", + "type": "text" + }, + { + "bbox": [ + 243, + 402, + 255, + 411 + ], + "score": 0.87, + "content": "\\pi ^ { \\omega }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 397, + 399, + 417 + ], + "score": 1.0, + "content": "follows the same A2C algorithm as", + "type": "text" + }, + { + "bbox": [ + 400, + 403, + 411, + 414 + ], + "score": 0.85, + "content": "\\pi _ { g }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 397, + 411, + 417 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 417, + 504, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 248, + 431 + ], + "score": 1.0, + "content": "The training of the Manager policy", + "type": "text" + }, + { + "bbox": [ + 248, + 419, + 263, + 428 + ], + "score": 0.88, + "content": "\\pi ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "also follows a similar procedure but as it operates at a lower", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 504, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 352, + 443 + ], + "score": 1.0, + "content": "temporal resolution, its value function regresses against the", + "type": "text" + }, + { + "bbox": [ + 352, + 430, + 364, + 440 + ], + "score": 0.88, + "content": "t _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 428, + 491, + 443 + ], + "score": 1.0, + "content": "-step discounted reward where", + "type": "text" + }, + { + "bbox": [ + 491, + 430, + 504, + 441 + ], + "score": 0.87, + "content": "t _ { m }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 441, + 339, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 339, + 452 + ], + "score": 1.0, + "content": "covers all actions and rewards generated from the Worker.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 417, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 503, + 479 + ], + "lines": [ + { + "bbox": [ + 104, + 453, + 504, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 489, + 472 + ], + "score": 1.0, + "content": "When using the Wide-then-Narrow instruction, the policy gradient for the Manager policy", + "type": "text" + }, + { + "bbox": [ + 489, + 459, + 504, + 469 + ], + "score": 0.83, + "content": "\\pi _ { m }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 466, + 147, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 147, + 482 + ], + "score": 1.0, + "content": "becomes:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 453, + 504, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 546 + ], + "lines": [ + { + "bbox": [ + 111, + 483, + 502, + 497 + ], + "spans": [ + { + "bbox": [ + 111, + 483, + 502, + 497 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\boldsymbol { \\mathbb { E } } _ { ( s _ { t } , a _ { t } ) \\sim \\pi , p , p _ { 0 } } \\left[ A _ { m , t } \\nabla \\log \\left( \\pi ^ { \\omega } \\left( g _ { w , t } | s _ { t } \\right) \\pi ^ { n } \\left( g _ { n , t } | s _ { t } , g _ { w , t } , s _ { w , t } \\right) \\right) \\right] + \\nabla \\left[ \\mathcal { H } \\left( \\pi ^ { \\omega } \\right) + \\mathcal { H } \\left( \\pi ^ { n } ( \\cdot | g _ { w , t } ) \\right) \\right] , } \\end{array}", + "type": "inline_equation", + "image_path": "9ffc0eea67dc3f57b1ef0f703548515d10800cb2907adbea88eab6289e65c13e.jpg" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 133, + 514 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 502, + 154, + 513 + ], + "score": 0.91, + "content": "A _ { m , t }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 500, + 297, + 514 + ], + "score": 1.0, + "content": "is the Manager’s advantage at time", + "type": "text" + }, + { + "bbox": [ + 297, + 503, + 302, + 511 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 500, + 505, + 514 + ], + "score": 1.0, + "content": ". Also, for Manager, as the size of the action space", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 185, + 525 + ], + "score": 1.0, + "content": "scales linearly with", + "type": "text" + }, + { + "bbox": [ + 185, + 512, + 198, + 524 + ], + "score": 0.89, + "content": "| S |", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 511, + 300, + 525 + ], + "score": 1.0, + "content": ", the exact entropy for the", + "type": "text" + }, + { + "bbox": [ + 300, + 513, + 315, + 522 + ], + "score": 0.88, + "content": "\\pi ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 511, + 505, + 525 + ], + "score": 1.0, + "content": "can easily become intractable. Essentially there", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 507, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 121, + 537 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 121, + 524, + 130, + 534 + ], + "score": 0.44, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 524, + 186, + 537 + ], + "score": 0.86, + "content": "^ { \\prime } \\left( | \\mathcal { V } | \\times \\left( N ^ { 2 } \\right) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 523, + 507, + 537 + ], + "score": 1.0, + "content": "possible actions. To calculate the entropy exactly, all of them has to be summed,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 534, + 288, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 288, + 549 + ], + "score": 1.0, + "content": "making it easily computationally intractable:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 483, + 507, + 549 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 550, + 450, + 578 + ], + "lines": [ + { + "bbox": [ + 160, + 550, + 450, + 578 + ], + "spans": [ + { + "bbox": [ + 160, + 550, + 450, + 578 + ], + "score": 0.93, + "content": "\\mathcal { H } = \\sum _ { w \\in \\mathcal { V } } \\sum _ { w _ { n } \\in s _ { w } } \\pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \\pi ^ { \\omega } ( w | s _ { t } ) \\log { \\nabla \\pi ^ { n } ( w _ { n } | s _ { w } , s _ { t } ) \\pi ^ { \\omega } ( w | s _ { t } ) } .", + "type": "interline_equation", + "image_path": "f2f8eb2b755e7a0bca9642c720d6a05acc8e754043512c8b836b2075a363f0e3.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 160, + 550, + 450, + 578 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 581, + 421, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 421, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 315, + 596 + ], + "score": 1.0, + "content": "Thus in practice we resort to an effective alternative", + "type": "text" + }, + { + "bbox": [ + 315, + 581, + 416, + 594 + ], + "score": 0.89, + "content": "\\mathcal { H } \\left( \\pi ^ { \\omega } \\right) + \\mathcal { H } \\left( \\pi ^ { n } ( \\cdot | g _ { w , t } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 579, + 421, + 596 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 579, + 421, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 598, + 318, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 597, + 318, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 318, + 612 + ], + "score": 1.0, + "content": "Psuedo-code for Manager training is in Algorithm 2.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 597, + 318, + 612 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 263, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 623, + 265, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 265, + 636 + ], + "score": 1.0, + "content": "C.3 HYPERPARAMETERS FOR HRL", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 479, + 658 + ], + "score": 1.0, + "content": "For training the HRL policies, we inherit most hyperparameters from those used when training", + "type": "text" + }, + { + "bbox": [ + 479, + 646, + 491, + 656 + ], + "score": 0.85, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 642, + 506, + 658 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 361, + 667 + ], + "score": 1.0, + "content": "the Manager and the Worker both share similar architectures with", + "type": "text" + }, + { + "bbox": [ + 361, + 656, + 372, + 667 + ], + "score": 0.87, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 654, + 505, + 667 + ], + "score": 1.0, + "content": ". The hyperparameters used when", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 140, + 679 + ], + "score": 1.0, + "content": "training", + "type": "text" + }, + { + "bbox": [ + 140, + 667, + 152, + 678 + ], + "score": 0.86, + "content": "\\pi _ { g }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "follow those from Shang et al. (2019). Because the tasks used in HRL experiments are", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "more difficult than the generic goal-reaching task, we set the maximal number of training iterations", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "to 100K abd training is stopped early if model performance reaches a plateau. The rollout steps for", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 396, + 712 + ], + "score": 1.0, + "content": "each iteration is 60. Hyperparameters specific to HRL are the horizon", + "type": "text" + }, + { + "bbox": [ + 396, + 699, + 426, + 709 + ], + "score": 0.89, + "content": "c = 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "and the size of the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 373, + 723 + ], + "score": 1.0, + "content": "Manager’s local attention range (that is, the neighborhood around", + "type": "text" + }, + { + "bbox": [ + 374, + 710, + 386, + 721 + ], + "score": 0.89, + "content": "g ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 709, + 443, + 723 + ], + "score": 1.0, + "content": "within which", + "type": "text" + }, + { + "bbox": [ + 443, + 710, + 455, + 721 + ], + "score": 0.88, + "content": "g ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "is selected),", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 719, + 421, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 147, + 733 + ], + "score": 1.0, + "content": "which are", + "type": "text" + }, + { + "bbox": [ + 148, + 721, + 176, + 731 + ], + "score": 0.9, + "content": "N = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 719, + 315, + 733 + ], + "score": 1.0, + "content": "for small and medium mazes, and", + "type": "text" + }, + { + "bbox": [ + 315, + 721, + 343, + 731 + ], + "score": 0.9, + "content": "N = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 719, + 421, + 733 + ], + "score": 1.0, + "content": "for the large maze.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 642, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 100, + 495, + 398 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 85, + 294, + 97 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 84, + 295, + 99 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 211, + 99 + ], + "score": 1.0, + "content": "Algorithm 2: Training of", + "type": "text" + }, + { + "bbox": [ + 211, + 86, + 226, + 96 + ], + "score": 0.86, + "content": "\\pi ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 84, + 295, + 99 + ], + "score": 1.0, + "content": "for HRL models", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 100, + 495, + 398 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 100, + 495, + 398 + ], + "spans": [ + { + "bbox": [ + 108, + 100, + 495, + 398 + ], + "score": 0.45, + "html": "
Clear gradients dθ ←O; while t <= tmax or episode not terminated do Simulate under current policy πm,t-1, πω,t-1; if the Worker has met the previous subgoal or exceeded the horizon c thenReset the set of time steps where πm,t omits a new subgoal Sm = {} and tm = 0.;
Sample a new subgoal gm,t from πm,t; end
Zm,t = fLsTM(CNN(sm,t,sv),hm,tm),Vm,t = fu(zm,t),Tt = fp(2m,t) ;
Sm= SmU {tm} and tm =t;
O, if terminal
Vtmax+1, otherwise
for t = tmax,...1 do R←rt+γR;
if t ∈ Sm then
Am,t ←R-Vm,t;
Accumulate gradients from value loss: dθ ← d0 + 入
Accumulate policy gradients with entropy regularization:80
d0 ← d0+ VlogTm,t(gm,t)Am,t + βVH(πm,t);
end
end
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Clear gradients dθ ←O; while t <= tmax or episode not terminated do Simulate under current policy πm,t-1, πω,t-1; if the Worker has met the previous subgoal or exceeded the horizon c thenReset the set of time steps where πm,t omits a new subgoal Sm = {} and tm = 0.;
Sample a new subgoal gm,t from πm,t; end
Zm,t = fLsTM(CNN(sm,t,sv),hm,tm),Vm,t = fu(zm,t),Tt = fp(2m,t) ;
Sm= SmU {tm} and tm =t;
O, if terminal
Vtmax+1, otherwise
for t = tmax,...1 do R←rt+γR;
if t ∈ Sm then
Am,t ←R-Vm,t;
Accumulate gradients from value loss: dθ ← d0 + 入
Accumulate policy gradients with entropy regularization:80
d0 ← d0+ VlogTm,t(gm,t)Am,t + βVH(πm,t);
end
end
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Algorithm 1: Identifying waypoint states Vp and learning a goal-conditioned policy g Result: Waypoint states Vp and a goal-conditioned policy π g
Initialize network parameters for the recurrent variational inference model V Initialize network parameters for the goal-conditioned policy g Initialize Vp with the initial position of the agent,i.e.Vp = {so = (1,1)}
while VAE reconstruction error has not converged do
forn←1toNdo Sample random waypoint sp ∈ Vp
Navigate agent to sp and perform T-step rollout using a randow walk policy: T𝑛 ←{(s0= Sp,ao),.,(sT,ar)}
gn←ST Navigate agent to Sp and perform T-step rollout using Tg with goal gn:
Tπ ←{(s= Sp,ao),.,(sT,ar)}at~πg(-st,9n) Re-label πg rewards with action reconstruction error as curiosity bonus:
rπ←{1st+1=n-λ·p(at|s,z)}=0
end
Perform policy gradient update of πg using T" and rπ
Update V using T and T
Update Vp as set of states with largest prior mean αs αs+βs
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Medium0.20±0.040.25±0.03
Larger0.16±0.220.26±0.11
MultiGoal-StochasticSmall1.38±1.201.93±0.163.06±0.312.92±0.45
Medium=2.99±0.122.42±0.242.64±0.14
Larger===0.60±0.12
Door-KeySmall=0.99±0.000.37±0.150.92±0.02
Medium0.56±0.020.76±0.06
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Clear gradients dθ ←O; while t <= tmax or episode not terminated do Simulate under current policy πm,t-1, πω,t-1; if the Worker has met the previous subgoal or exceeded the horizon c thenReset the set of time steps where πm,t omits a new subgoal Sm = {} and tm = 0.;
Sample a new subgoal gm,t from πm,t; end
Zm,t = fLsTM(CNN(sm,t,sv),hm,tm),Vm,t = fu(zm,t),Tt = fp(2m,t) ;
Sm= SmU {tm} and tm =t;
O, if terminal
Vtmax+1, otherwise
for t = tmax,...1 do R←rt+γR;
if t ∈ Sm then
Am,t ←R-Vm,t;
Accumulate gradients from value loss: dθ ← d0 + 入
Accumulate policy gradients with entropy regularization:80
d0 ← d0+ VlogTm,t(gm,t)Am,t + βVH(πm,t);
end
end
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Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, Andrew Y. Ng ´ + +Computer Science Department, Stanford University zxie,sidaw,danilevy,anie,ang @cs.stanford.edu, jiweil,jurafsky @stanford.edu + +# ABSTRACT + +Data noising is an effective technique for regularizing neural network models. While noising is widely adopted in application domains such as vision and speech, commonly used noising primitives have not been developed for discrete sequencelevel settings such as language modeling. In this paper, we derive a connection between input noising in neural network language models and smoothing in $n$ - gram models. Using this connection, we draw upon ideas from smoothing to develop effective noising schemes. We demonstrate performance gains when applying the proposed schemes to language modeling and machine translation. Finally, we provide empirical analysis validating the relationship between noising and smoothing. + +# 1 INTRODUCTION + +Language models are a crucial component in many domains, such as autocompletion, machine translation, and speech recognition. A key challenge when performing estimation in language modeling is the data sparsity problem: due to large vocabulary sizes and the exponential number of possible contexts, the majority of possible sequences are rarely or never observed, even for very short subsequences. + +In other application domains, data augmentation has been key to improving the performance of neural network models in the face of insufficient data. In computer vision, for example, there exist well-established primitives for synthesizing additional image data, such as by rescaling or applying affine distortions to images (LeCun et al., 1998; Krizhevsky et al., 2012). Similarly, in speech recognition adding a background audio track or applying small shifts along the time dimension has been shown to yield significant gains, especially in noisy settings (Deng et al., 2000; Hannun et al., 2014). However, widely-adopted noising primitives have not yet been developed for neural network language models. + +Classic $n$ -gram models of language cope with rare and unseen sequences by using smoothing methods, such as interpolation or absolute discounting (Chen & Goodman, 1996). Neural network models, however, have no notion of discrete counts, and instead use distributed representations to combat the curse of dimensionality (Bengio et al., 2003). Despite the effectiveness of distributed representations, overfitting due to data sparsity remains an issue. Existing regularization methods, however, are typically applied to weights or hidden units within the network (Srivastava et al., 2014; Le et al., 2015) instead of directly considering the input data. + +In this work, we consider noising primitives as a form of data augmentation for recurrent neural network-based language models. By examining the expected pseudocounts from applying the noising schemes, we draw connections between noising and linear interpolation smoothing. Using this connection, we then derive noising schemes that are analogues of more advanced smoothing methods. We demonstrate the effectiveness of these schemes for regularization through experiments on language modeling and machine translation. Finally, we validate our theoretical claims by examining the empirical effects of noising. + +# 2 RELATED WORK + +Our work can be viewed as a form of data augmentation, for which to the best of our knowledge there exists no widely adopted schemes in language modeling with neural networks. Classical regularization methods such as $L _ { 2 }$ -regularization are typically applied to the model parameters, while dropout is applied to activations which can be along the forward as well as the recurrent directions (Zaremba et al., 2014; Semeniuta et al., 2016; Gal, 2015). Others have introduced methods for recurrent neural networks encouraging the hidden activations to remain stable in norm, or constraining the recurrent weight matrix to have eigenvalues close to one (Krueger & Memisevic, 2015; Arjovsky et al., 2015; Le et al., 2015). These methods, however, all consider weights and hidden units instead of the input data, and are motivated by the vanishing and exploding gradient problem. + +Feature noising has been demonstrated to be effective for structured prediction tasks, and has been interpreted as an explicit regularizer (Wang et al., 2013). Additionally, Wager et al. (2014) show that noising can inject appropriate generative assumptions into discriminative models to reduce their generalization error, but do not consider sequence models (Wager et al., 2016). + +The technique of randomly zero-masking input word embeddings for learning sentence representations has been proposed by Iyyer et al. (2015), Kumar et al. (2015), and Dai & Le (2015), and adopted by others such as Bowman et al. (2015). However, to the best of our knowledge, no analysis has been provided besides reasoning that zeroing embeddings may result in a model ensembling effect similar to that in standard dropout. This analysis is applicable to classification tasks involving sum-of-embeddings or bag-of-words models, but does not capture sequence-level effects. Bengio et al. (2015) also make an empirical observation that the method of randomly replacing words with fixed probability with a draw from the uniform distribution improved performance slightly for an image captioning task; however, they do not examine why performance improved. + +# 3 METHOD + +# 3.1 PRELIMINARIES + +We consider language models where given a sequence of indices $X = ( x _ { 1 } , x _ { 2 } , \cdot \cdot \cdot , x _ { T } )$ , over the vocabulary $V$ , we model + +$$ +p ( X ) = \prod _ { t = 1 } ^ { T } p ( x _ { t } | x _ { < t } ) +$$ + +In $n$ -gram models, it is not feasible to model the full context $x _ { < t }$ for large $t$ due to the exponential number of possible histories. Recurrent neural network (RNN) language models can (in theory) model longer dependencies, since they operate over distributed hidden states instead of modeling an exponential number of discrete counts (Bengio et al., 2003; Mikolov, 2012). + +An $L$ -layer recurrent neural network is modeled as $h _ { t } ^ { ( l ) } = f _ { \theta } ( h _ { t - 1 } ^ { ( l ) } , h _ { t } ^ { ( l - 1 ) } )$ , where $l$ denotes the layer index, $h ^ { ( 0 ) }$ contains the one-hot encoding of $X$ , and in its simplest form $f _ { \theta }$ applies an affine transformation followed by a nonlinearity. In this work, we use RNNs with a more complex form of $f _ { \theta }$ , namely long short-term memory (LSTM) units (Hochreiter & Schmidhuber, 1997), which have been shown to ease training and allow RNNs to capture longer dependencies. The output distribution over the vocabulary V at time t is pθ(xt|xNoisedγ(x1:2)q(x）Analoguex120q(“_")=1interpolationx170unigraminterpolationx120N1+(x1,.)/c(x1)unigramabsolute discountingx1,x220N1+(x1,·)/c(x1)q(x) x Ni+(•,x)Kneser-Ney + +Table 1: Noising schemes Example noising schemes and their bigram smoothing analogues. Here we consider the bigram probability $p ( x _ { 1 } , x _ { 2 } ) \ : = \ : p ( x _ { 2 } \vert x _ { 1 } ) p ( x _ { 1 } )$ . Notation: $\gamma ( \boldsymbol { x } _ { 1 : t } )$ denotes the noising probability for a given input sequence $x _ { 1 : t }$ , $q ( x )$ denotes the proposal distribution, and $N _ { 1 + } ( x , \bullet )$ denotes the number of distinct bigrams in the training set where $x$ is the first unigram. In all but the last case we only noise the context $x _ { 1 }$ and not the target prediction $x _ { 2 }$ . + +Let $N _ { 1 + } ( x _ { 1 } , \bullet ) \ { \stackrel { \mathrm { d e f } } { = } } \ | \{ x _ { 2 } : c ( x _ { 1 } , x _ { 2 } ) > 0 \} |$ be the number of distinct continutions following $x _ { 1 }$ , or equivalently the number of bigram types beginning with $x _ { 1 }$ (Chen $\&$ Goodman, 1996). From the above intuitions, we arrive at the absolute discounting noising probability + +$$ +\gamma _ { \mathrm { A D } } ( x _ { 1 } ) = \gamma _ { 0 } \frac { N _ { 1 + } ( x _ { 1 } , \bullet ) } { \sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) } +$$ + +where for $0 \leq \gamma _ { 0 } \leq 1$ we have $0 \leq \gamma _ { \mathrm { A D } } \leq 1$ , though in practice we can also clip larger noising probabilities to 1. Note that this encourages noising of unigrams that precede many possible other tokens while discouraging noising of common unigrams, since if we ignore the final token, $\begin{array} { r } { \sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) = c ( x _ { 1 } ) } \end{array}$ . + +Proposal Distribution While choosing the unigram distribution as the proposal distribution $q ( x )$ preserves unigram frequencies, by borrowing from the smoothing literature we find another distribution performs better. We again begin with two motivating examples: + +“San Francisco” + +“New York” + +Both bigrams appear frequently in text corpora. As a direct consequence, the unigrams “Francisco” and “York” also appear frequently. However, since “Francisco” and “York” typically follow “San” and “New”, respectively, they should not have high probability in the proposal distribution as they might if we use unigram frequencies (Chen $\&$ Goodman, 1996). Instead, it would be better to increase the proposal probability of unigrams with diverse histories, or more precisely unigrams that complete a large number of bigram types. Thus instead of drawing from the unigram distribution, we consider drawing from + +$$ +q ( x ) \propto N _ { 1 + } ( \bullet , x ) +$$ + +Note that we now noise the prediction $x _ { t }$ in addition to the context $x _ { 1 : t - 1 }$ . Combining this new proposal distribution with the discounted $\gamma _ { \mathrm { A D } } ( x _ { 1 } )$ from the previous section, we obtain the noising analogue of Kneser-Ney smoothing. + +Table 1 summarizes the discussed noising schemes. + +# 3.5 TRAINING AND TESTING + +During training, noising is performed per batch and is done online such that each epoch of training sees a different noised version of the training data. At test time, to match the training objective we should sample multiple corrupted versions of the test data, then average the predictions (Srivastava et al., 2014). In practice, however, we find that simply using the maximum likelihood (uncorrupted) input sequence works well; evaluation runtime remains unchanged. + +# 3.6 EXTENSIONS + +The schemes described are for the language model setting. To extend them to the sequence-tosequence or encoder-decoder setting, we noise both $x _ { < t }$ as well as $y _ { < t }$ . While in the decoder we + +Table 2: Single-model perplexity on Penn Treebank with different noising schemes. We also compare to the variational method of Gal (2015), who also train LSTM models with the same hidden dimension. Note that performing Monte Carlo dropout at test time is significantly more expensive than our approach, where test time is unchanged. + +
Noising schemeValidationTest
Medium models (512 hidden size)
none (dropout only) blank84.380.4
82.778.8
unigram83.180.1
bigram Kneser-Ney79.976.9
Large models (1500 hidden size)
none (dropout only) blank unigram bigram Kneser-Ney81.677.5
79.475.5
79.4 76.276.1 73.4
Zaremba et al. (2014)82.2
Gal (2015) variational dropout (tied weights)78.4
Gal (2015) (untied weights,Monte Carlo)77.3 丨75.0 73.4
+ +Table 3: Perplexity on Text8 with different noising schemes. + +
Noising schemeValidationTest
none94.3123.6
blank85.0110.7
unigram85.2111.3
bigram Kneser-Ney84.5110.6
+ +have $y _ { < t }$ and $y _ { t }$ as analogues to language model context and target prediction, it is unclear whether noising $x _ { < t }$ should be beneficial. Empirically, however, we find this to be the case (Table 4). + +# 4 EXPERIMENTS + +# 4.1 LANGUAGE MODELING + +Penn Treebank We train networks for word-level language modeling on the Penn Treebank dataset, using the standard preprocessed splits with a 10K size vocabulary (Mikolov, 2012). The PTB dataset contains $9 2 9 \mathrm { k }$ training tokens, 73k validation tokens, and ${ 8 2 } \mathrm { k }$ test tokens. Following Zaremba et al. (2014), we use minibatches of size 20 and unroll for 35 time steps when performing backpropagation through time. All models have two hidden layers and use LSTM units. Weights are initialized uniformly in the range $[ - 0 . 1 , 0 . 1 ]$ . We consider models with hidden sizes of 512 and 1500. + +We train using stochastic gradient descent with an initial learning rate of 1.0, clipping the gradient if its norm exceeds 5.0. When the validation cross entropy does not decrease after a training epoch, we halve the learning rate. We anneal the learning rate 8 times before stopping training, and pick the model with the lowest perplexity on the validation set. + +For regularization, we apply feed-forward dropout (Pham et al., 2014) in combination with our noising schemes. We report results in Table 2 for the best setting of the dropout rate (which we find to match the settings reported in Zaremba et al. (2014)) as well as the best setting of noising + +![](images/a74791f89bc2ae43fdc9a304f47b23fda338e2c01b613335fe659ced2c4f1495.jpg) +Figure 1: Example training and validation curves for an unnoised model and model regularized using the bigram Kneser-Ney noising scheme. + +
SchemePerplexityBLEU
dropout, no noising8.8424.6
blank noising8.2825.3 (+0.7)
unigram noising8.1525.5 (+0.9)
bigram Kneser-Ney7.9226.0 (+1.4)
source only8.7424.8 (+0.2)
target only8.1425.6 (+1.0)
+ +Table 4: Perplexities and BLEU scores for machine translation task. Results for bigram KN noising on only the source sequence and only the target sequence are given as well. + +probability $\gamma _ { 0 }$ on the validation set.2 Figure 1 shows the training and validation perplexity curves for a noised versus an unnoised run. + +Our large models match the state-of-the-art regularization method for single model performance on this task. In particular, we find that picking $\gamma _ { \mathrm { A D } } ( x _ { 1 } )$ and $q ( x )$ corresponding to Kneser-Ney smoothing yields significant gains in validation perplexity, both for the medium and large size models. Recent work (Merity et al., 2016; Zilly et al., 2016) has also achieved impressive results on this task by proposing different architectures which are orthogonal to our data augmentation schemes. + +Text8 In order to determine whether noising remains effective with a larger dataset, we perform experiments on the Text8 corpus3. The first 90M characters are used for training, the next 5M for validation, and the final 5M for testing, resulting in $1 5 . 3 \mathbf { M }$ training tokens, 848K validation tokens, and 855K test tokens. We preprocess the data by mapping all words which appear 10 or fewer times to the unknown token, resulting in a 42K size vocabulary. Other parameter settings are the same as described in the Penn Treebank experiments, besides that only models with hidden size 512 are considered, and noising is not combined with feed-forward dropout. Results are given in Table 3. + +# 4.2 MACHINE TRANSLATION + +For our machine translation experiments we consider the English-German machine translation track of IWSLT $2 0 1 5 ^ { 4 }$ . The IWSLT 2015 corpus consists of sentence-aligned subtitles of TED and TEDx talks. The training set contains roughly 190K sentence pairs with 5.4M tokens. Following Luong & Manning (2015), we use TED tst2012 as a validation set and report BLEU score results (Papineni et al., 2002) on tst2014. We limit the vocabulary to the top 50K most frequent words for each language. + +![](images/c52239f55d9409031ab76a2c2d420a3bbb70c906f6f5db3a48bd4677b262db5b.jpg) +Figure 2: Perplexity with noising on Penn Treebank while varying the value of $\gamma _ { 0 }$ . Using discounting to scale $\gamma _ { 0 }$ (yielding $\gamma _ { \mathrm { A D . } }$ ) maintains gains for a range of values of noising probability, which is not true for the unscaled case. + +![](images/66596b6cf7d5d61afc0773e432d36885d11d8c1b6493b1dddbc53657d86ae861.jpg) +Figure 3: Mean KL-divergence over validation set between softmax distributions of noised and unnoised models and lower order distributions. Noised model distributions are closer to the uniform and unigram frequency distributions. + +We train a two-layer LSTM encoder-decoder network (Sutskever et al., 2014; Cho et al., 2014) with 512 hidden units in each layer. The decoder uses an attention mechanism (Bahdanau et al., 2014) with the dot alignment function (Luong et al., 2015). The initial learning rate is 1.0 and we start halving the learning rate when the relative difference in perplexity on the validation set between two consecutive epochs is less than $1 \%$ . We follow training protocols as described in Sutskever et al. (2014): (a) LSTM parameters and word embeddings are initialized from a uniform distribution between $[ - 0 . 1 , 0 . 1 ]$ , (b) inputs are reversed, (c) batch size is set to 128, (d) gradient clipping is performed when the norm exceeds a threshold of 5. We set hidden unit dropout rate to 0.2 across all settings as suggested in Luong et al. (2015). We compare unigram, blank, and bigram Kneser-Ney noising. Noising rate $\gamma$ is selected on the validation set. + +Results are shown in Table 4. We observe performance gains for both blank noising and unigram noising, giving roughly $+ 0 . 7$ BLEU score on the test set. The proposed bigram Kneser-Ney noising scheme gives an additional performance boost of $+ 0 . 5 – 0 . 7$ on top of the blank noising and unigram noising models, yielding a total gain of $+ 1 . 4$ BLEU. + +# 5 DISCUSSION + +# 5.1 SCALING $\gamma$ VIA DISCOUNTING + +We now examine whether discounting has the desired effect of noising subsequences according to their uncertainty. If we consider the discounting + +$$ +\gamma _ { \mathrm { A D } } ( x _ { 1 } ) = \gamma _ { 0 } \frac { N _ { 1 + } ( x _ { 1 } , \bullet ) } { c ( x _ { 1 } ) } +$$ + +we observe that the denominator $c ( x _ { 1 } )$ can dominate than the numerator $N _ { 1 + } ( x _ { 1 } , \bullet )$ . Common tokens are often noised infrequently when discounting is used to rescale the noising probability, while rare tokens are noised comparatively much more frequently, where in the extreme case when a token appears exactly once, we have $\gamma _ { \mathrm { A D } } ~ = ~ \gamma _ { 0 }$ . Due to word frequencies following a Zipfian power law distribution, however, common tokens constitute the majority of most texts, and thus discounting leads to significantly less noising. + +We compare the performance of models trained with a fixed $\gamma _ { 0 }$ versus a $\gamma _ { 0 }$ rescaled using discounting. As shown in Figure 2, bigram discounting leads to gains in perplexity for a much broader range of $\gamma _ { 0 }$ . Thus the discounting ratio seems to effectively capture the “right” tokens to noise. + +
NoisingBigramsTrigrams
none (dropout only)2881381
blank noising2760372
unigram noising2612365
+ +Table 5: Perplexity of last unigram for unseen bigrams and trigrams in Penn Treebank validation set. We compare noised and unnoised models with noising probabilities chosen such that models have near-identical perplexity on full validation set. + +# 5.2 NOISED VERSUS UNNOISED MODELS + +Smoothed distributions In order to validate that data noising for RNN models has a similar effect to that of smoothing counts in $n$ -gram models, we consider three models trained with unigram noising as described in Section 4.1 on the Penn Treebank corpus with $\gamma = 0$ (no noising), $\gamma = 0 . 1$ , and $\gamma = 0 . 2 5$ . Using the trained models, we measure the Kullback-Leibler divergence $D _ { \mathrm { K L } } ( p \Vert q ) =$ $\textstyle \sum _ { i } p _ { i } \log ( p _ { i } / q _ { i } )$ over the validation set between the predicted softmax distributions, $\hat { p }$ , and the uniform distribution as well as the unigram frequency distribution. We then take the mean KL divergence over all tokens in the validation set. + +Recall that in interpolation smoothing, a weighted combination of higher and lower order $n$ -gram models is used. As seen in Figure 3, the softmax distributions of noised models are significantly closer to the lower order frequency distributions than unnoised models, in particular in the case of the unigram distribution, thus validating our analysis in Section 3.3. + +Unseen $n$ -grams Smoothing is most beneficial for increasing the probability of unobserved sequences. To measure whether noising has a similar effect, we consider bigrams and trigrams in the validation set that do not appear in the training set. For these unseen bigrams (15062 occurrences) and trigrams (43051 occurrences), we measure the perplexity for noised and unnoised models with near-identical perplexity on the full set. As expected, noising yields lower perplexity for these unseen instances. + +# 6 CONCLUSION + +In this work, we show that data noising is effective for regularizing neural network-based sequence models. By deriving a correspondence between noising and smoothing, we are able to adapt advanced smoothing methods for $n$ -gram models to the neural network setting, thereby incorporating well-understood generative assumptions of language. Possible applications include exploring noising for improving performance in low resource settings, or examining how these techniques generalize to sequence modeling in other domains. + +# ACKNOWLEDGMENTS + +We thank Will Monroe for feedback on a draft of this paper, Anand Avati for help running experiments, and Jimmy Wu for computing support. We also thank the developers of Theano (Theano Development Team, 2016) and Tensorflow (Abadi et al., 2016). Some GPUs used in this work were donated by NVIDIA Corporation. ZX, SW, and JL were supported by an NDSEG Fellowship, NSERC PGS-D Fellowship, and Facebook Fellowship, respectively. This project was funded in part by DARPA MUSE award FA8750-15-C-0242 AFRL/RIKF. + +# REFERENCES + +Martın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. + +Martin Arjovsky, Amar Shah, and Yoshua Bengio. Unitary evolution recurrent neural networks. arXiv preprint arXiv:1511.06464, 2015. + +Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. + +Samy Bengio, Oriol Vinyals, Navdeep Jaitly, and Noam Shazeer. Scheduled sampling for sequence prediction with recurrent neural networks. In Neural Information Processing Systems (NIPS), 2015. + +Yoshua Bengio, Rejean Ducharme, Pascal Vincent, and Christian Jauvin. A neural probabilistic ´ language model. In Journal Of Machine Learning Research, 2003. + +Samuel R Bowman, Luke Vilnis, Oriol Vinyals, Andrew M Dai, Rafal Jozefowicz, and Samy Bengio. Generating sentences from a continuous space. arXiv preprint arXiv:1511.06349, 2015. + +Peter F Brown, Peter V Desouza, Robert L Mercer, Vincent J Della Pietra, and Jenifer C Lai. Classbased n-gram models of natural language. Computational linguistics, 1992. + +Stanley F Chen and Joshua Goodman. An empirical study of smoothing techniques for language modeling. In Association for Computational Linguistics (ACL), 1996. + +Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Hol- ¨ ger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078, 2014. + +Andrew M Dai and Quoc V Le. Semi-supervised sequence learning. In Advances in Neural Information Processing Systems, pp. 3061–3069, 2015. + +Li Deng, Alex Acero, Mike Plumpe, and Xuedong Huang. Large-vocabulary speech recognition under adverse acoustic environments. In ICSLP, 2000. + +Yarin Gal. A theoretically grounded application of dropout in recurrent neural networks. arXiv:1512.05287, 2015. + +Awni Hannun, Carl Case, Jared Casper, Bryan Catanzaro, Greg Diamos, et al. Deep speech: Scaling up end-to-end speech recognition. arXiv preprint arXiv:1412.5567, 2014. + +Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 1997. + +Mohit Iyyer, Varun Manjunatha, Jordan Boyd-Graber, and Hal Daume III. Deep unordered compo- ´ sition rivals syntactic methods for text classification. In Association for Computatonal Linguistics (ACL), 2015. + +A. Krizhevsky, I. Sutskever, and G. E. Hinton. ImageNet Classification with Deep Convolutional Neural Networks. In NIPS, 2012. + +David Krueger and Roland Memisevic. Regularizing rnns by stabilizing activations. arXiv preprint arXiv:1511.08400, 2015. + +Ankit Kumar, Ozan Irsoy, Jonathan Su, James Bradbury, Robert English, Brian Pierce, Peter Ondruska, Ishaan Gulrajani, and Richard Socher. Ask me anything: Dynamic memory networks for natural language processing. arXiv preprint arXiv:1506.07285, 2015. + +Quoc V Le, Navdeep Jaitly, and Geoffrey E Hinton. A simple way to initialize recurrent networks of rectified linear units. arXiv preprint arXiv:1504.00941, 2015. + +Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based Learning Applied to Document Recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. + +Minh-Thang Luong and Christopher D Manning. Stanford neural machine translation systems for spoken language domains. In Proceedings of the International Workshop on Spoken Language Translation, 2015. + +Minh-Thang Luong, Hieu Pham, and Christopher D. Manning. Effective approaches to attentionbased neural machine translation. In Empirical Methods in Natural Language Processing (EMNLP), 2015. + +Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. Pointer sentinel mixture models. arXiv preprint arXiv:1609.07843, 2016. + +Toma´s Mikolov. ˇ Statistical language models based on neural networks. PhD thesis, PhD thesis, Brno University of Technology. 2012.[PDF], 2012. + +Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting on association for computational linguistics, pp. 311–318. Association for Computational Linguistics, 2002. + +Vu Pham, Theodore Bluche, Christopher Kermorvant, and J ´ er´ ome Louradour. Dropout improves ˆ recurrent neural networks for handwriting recognition. In Frontiers in Handwriting Recognition (ICFHR), 2014 14th International Conference on, 2014. + +Stanislau Semeniuta, Aliaksei Severyn, and Erhardt Barth. Recurrent dropout without memory loss. arXiv preprint arXiv:1603.05118, 2016. + +Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 2014. + +Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pp. 3104–3112, 2014. + +Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016. URL http://arxiv.org/abs/ 1605.02688. + +S. Wager, W. Fithian, S. I. Wang, and P. Liang. Altitude training: Strong bounds for single-layer dropout. In Advances in Neural Information Processing Systems (NIPS), 2014. + +Stefan Wager, William Fithian, and Percy Liang. Data augmentation via levy processes. arXiv preprint arXiv:1603.06340, 2016. + +Sida I Wang, Mengqiu Wang, Stefan Wager, Percy Liang, and Christopher D Manning. Feature noising for log-linear structured prediction. In Empirical Methods in Natural Language Processing (EMNLP), 2013. + +Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014. + +Julian Georg Zilly, Rupesh Kumar Srivastava, Jan Koutn´ık, and Jurgen Schmidhuber. Recurrent ¨ highway networks. arXiv preprint arXiv:1607.03474, 2016. + +# A SKETCH OF NOISING ALGORITHM + +We provide pseudocode of the noising algorithm corresponding to bigram Kneser-Ney smoothing for $n$ -grams (In the case of sequence-to-sequence tasks, we estimate the count-based parameters separately for source and target). To simplify, we assume a batch size of one. The noising algorithm is applied to each data batch during training. No noising is applied at test time. + +
Algorithm 1 Bigram KN noising (Language modeling setting)
Require counts c(x), number of distinct continuations N1+(x,·), proposal distribution q(x） N1+(.,x) Inputs X,Y batch of unnoised data indices, scaling factor γo
procedure NOISEBGKN(X, Y) >X=(x1,...,xt),Y=(x2,...,xt+1)
X,Y←X,Y
for j = 1,...,t do γ ←γ0N1+(xj,·)/c(xj)
if ~ Bernoulli(~) then
xj~ Categorical(q) Updates X
yj~ Categorical(q)
end if
end for
return X, Y Run training iteration with noised batch
end procedure
\ No newline at end of file diff --git a/parse/train/H1VyHY9gg/H1VyHY9gg_content_list.json b/parse/train/H1VyHY9gg/H1VyHY9gg_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..7abb3e6b7a62b33259b451978db548be8c29e841 --- /dev/null +++ b/parse/train/H1VyHY9gg/H1VyHY9gg_content_list.json @@ -0,0 +1,1659 @@ +[ + { + "type": "text", + "text": "DATA NOISING AS SMOOTHING IN NEURAL NETWORK LANGUAGE MODELS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ziang Xie, Sida I. Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, Andrew Y. Ng ´ ", + "bbox": [ + 184, + 170, + 802, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Computer Science Department, Stanford University zxie,sidaw,danilevy,anie,ang @cs.stanford.edu, jiweil,jurafsky @stanford.edu ", + "bbox": [ + 186, + 185, + 637, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Data noising is an effective technique for regularizing neural network models. While noising is widely adopted in application domains such as vision and speech, commonly used noising primitives have not been developed for discrete sequencelevel settings such as language modeling. In this paper, we derive a connection between input noising in neural network language models and smoothing in $n$ - gram models. Using this connection, we draw upon ideas from smoothing to develop effective noising schemes. We demonstrate performance gains when applying the proposed schemes to language modeling and machine translation. Finally, we provide empirical analysis validating the relationship between noising and smoothing. ", + "bbox": [ + 233, + 301, + 764, + 441 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 488, + 336, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Language models are a crucial component in many domains, such as autocompletion, machine translation, and speech recognition. A key challenge when performing estimation in language modeling is the data sparsity problem: due to large vocabulary sizes and the exponential number of possible contexts, the majority of possible sequences are rarely or never observed, even for very short subsequences. ", + "bbox": [ + 174, + 526, + 823, + 595 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In other application domains, data augmentation has been key to improving the performance of neural network models in the face of insufficient data. In computer vision, for example, there exist well-established primitives for synthesizing additional image data, such as by rescaling or applying affine distortions to images (LeCun et al., 1998; Krizhevsky et al., 2012). Similarly, in speech recognition adding a background audio track or applying small shifts along the time dimension has been shown to yield significant gains, especially in noisy settings (Deng et al., 2000; Hannun et al., 2014). However, widely-adopted noising primitives have not yet been developed for neural network language models. ", + "bbox": [ + 174, + 603, + 825, + 714 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Classic $n$ -gram models of language cope with rare and unseen sequences by using smoothing methods, such as interpolation or absolute discounting (Chen & Goodman, 1996). Neural network models, however, have no notion of discrete counts, and instead use distributed representations to combat the curse of dimensionality (Bengio et al., 2003). Despite the effectiveness of distributed representations, overfitting due to data sparsity remains an issue. Existing regularization methods, however, are typically applied to weights or hidden units within the network (Srivastava et al., 2014; Le et al., 2015) instead of directly considering the input data. ", + "bbox": [ + 174, + 722, + 825, + 819 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we consider noising primitives as a form of data augmentation for recurrent neural network-based language models. By examining the expected pseudocounts from applying the noising schemes, we draw connections between noising and linear interpolation smoothing. Using this connection, we then derive noising schemes that are analogues of more advanced smoothing methods. We demonstrate the effectiveness of these schemes for regularization through experiments on language modeling and machine translation. Finally, we validate our theoretical claims by examining the empirical effects of noising. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 102, + 344, + 118 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work can be viewed as a form of data augmentation, for which to the best of our knowledge there exists no widely adopted schemes in language modeling with neural networks. Classical regularization methods such as $L _ { 2 }$ -regularization are typically applied to the model parameters, while dropout is applied to activations which can be along the forward as well as the recurrent directions (Zaremba et al., 2014; Semeniuta et al., 2016; Gal, 2015). Others have introduced methods for recurrent neural networks encouraging the hidden activations to remain stable in norm, or constraining the recurrent weight matrix to have eigenvalues close to one (Krueger & Memisevic, 2015; Arjovsky et al., 2015; Le et al., 2015). These methods, however, all consider weights and hidden units instead of the input data, and are motivated by the vanishing and exploding gradient problem. ", + "bbox": [ + 174, + 133, + 825, + 261 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Feature noising has been demonstrated to be effective for structured prediction tasks, and has been interpreted as an explicit regularizer (Wang et al., 2013). Additionally, Wager et al. (2014) show that noising can inject appropriate generative assumptions into discriminative models to reduce their generalization error, but do not consider sequence models (Wager et al., 2016). ", + "bbox": [ + 174, + 266, + 825, + 321 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The technique of randomly zero-masking input word embeddings for learning sentence representations has been proposed by Iyyer et al. (2015), Kumar et al. (2015), and Dai & Le (2015), and adopted by others such as Bowman et al. (2015). However, to the best of our knowledge, no analysis has been provided besides reasoning that zeroing embeddings may result in a model ensembling effect similar to that in standard dropout. This analysis is applicable to classification tasks involving sum-of-embeddings or bag-of-words models, but does not capture sequence-level effects. Bengio et al. (2015) also make an empirical observation that the method of randomly replacing words with fixed probability with a draw from the uniform distribution improved performance slightly for an image captioning task; however, they do not examine why performance improved. ", + "bbox": [ + 174, + 329, + 825, + 455 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 METHOD ", + "text_level": 1, + "bbox": [ + 174, + 474, + 282, + 492 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 174, + 507, + 326, + 522 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We consider language models where given a sequence of indices $X = ( x _ { 1 } , x _ { 2 } , \\cdot \\cdot \\cdot , x _ { T } )$ , over the vocabulary $V$ , we model ", + "bbox": [ + 176, + 534, + 823, + 563 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/c0d2f832e5e905d46da1dbe6fa5ecbb8e2bc7af43c18208d5774ce81ef3d5ebd.jpg", + "text": "$$\np ( X ) = \\prod _ { t = 1 } ^ { T } p ( x _ { t } | x _ { < t } )\n$$", + "text_format": "latex", + "bbox": [ + 423, + 561, + 575, + 606 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In $n$ -gram models, it is not feasible to model the full context $x _ { < t }$ for large $t$ due to the exponential number of possible histories. Recurrent neural network (RNN) language models can (in theory) model longer dependencies, since they operate over distributed hidden states instead of modeling an exponential number of discrete counts (Bengio et al., 2003; Mikolov, 2012). ", + "bbox": [ + 174, + 616, + 825, + 672 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "An $L$ -layer recurrent neural network is modeled as $h _ { t } ^ { ( l ) } = f _ { \\theta } ( h _ { t - 1 } ^ { ( l ) } , h _ { t } ^ { ( l - 1 ) } )$ , where $l$ denotes the layer index, $h ^ { ( 0 ) }$ contains the one-hot encoding of $X$ , and in its simplest form $f _ { \\theta }$ applies an affine transformation followed by a nonlinearity. In this work, we use RNNs with a more complex form of $f _ { \\theta }$ , namely long short-term memory (LSTM) units (Hochreiter & Schmidhuber, 1997), which have been shown to ease training and allow RNNs to capture longer dependencies. The output distribution over the vocabulary V at time t is pθ(xt|xNoisedγ(x1:2)q(x）Analoguex120q(“_")=1interpolationx170unigraminterpolationx120N1+(x1,.)/c(x1)unigramabsolute discountingx1,x220N1+(x1,·)/c(x1)q(x) x Ni+(•,x)Kneser-Ney", + "bbox": [ + 240, + 101, + 758, + 186 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1: Noising schemes Example noising schemes and their bigram smoothing analogues. Here we consider the bigram probability $p ( x _ { 1 } , x _ { 2 } ) \\ : = \\ : p ( x _ { 2 } \\vert x _ { 1 } ) p ( x _ { 1 } )$ . Notation: $\\gamma ( \\boldsymbol { x } _ { 1 : t } )$ denotes the noising probability for a given input sequence $x _ { 1 : t }$ , $q ( x )$ denotes the proposal distribution, and $N _ { 1 + } ( x , \\bullet )$ denotes the number of distinct bigrams in the training set where $x$ is the first unigram. In all but the last case we only noise the context $x _ { 1 }$ and not the target prediction $x _ { 2 }$ . ", + "bbox": [ + 173, + 202, + 825, + 273 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Let $N _ { 1 + } ( x _ { 1 } , \\bullet ) \\ { \\stackrel { \\mathrm { d e f } } { = } } \\ | \\{ x _ { 2 } : c ( x _ { 1 } , x _ { 2 } ) > 0 \\} |$ be the number of distinct continutions following $x _ { 1 }$ , or equivalently the number of bigram types beginning with $x _ { 1 }$ (Chen $\\&$ Goodman, 1996). From the above intuitions, we arrive at the absolute discounting noising probability ", + "bbox": [ + 173, + 299, + 825, + 345 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/103e5fafdc04d38a9dbcd28816e24601025f95aa7ddfec353e823ad8955d335b.jpg", + "text": "$$\n\\gamma _ { \\mathrm { A D } } ( x _ { 1 } ) = \\gamma _ { 0 } \\frac { N _ { 1 + } ( x _ { 1 } , \\bullet ) } { \\sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) }\n$$", + "text_format": "latex", + "bbox": [ + 398, + 354, + 598, + 391 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where for $0 \\leq \\gamma _ { 0 } \\leq 1$ we have $0 \\leq \\gamma _ { \\mathrm { A D } } \\leq 1$ , though in practice we can also clip larger noising probabilities to 1. Note that this encourages noising of unigrams that precede many possible other tokens while discouraging noising of common unigrams, since if we ignore the final token, $\\begin{array} { r } { \\sum _ { x _ { 2 } } c ( x _ { 1 } , x _ { 2 } ) = c ( x _ { 1 } ) } \\end{array}$ . ", + "bbox": [ + 174, + 400, + 825, + 458 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposal Distribution While choosing the unigram distribution as the proposal distribution $q ( x )$ preserves unigram frequencies, by borrowing from the smoothing literature we find another distribution performs better. We again begin with two motivating examples: ", + "bbox": [ + 173, + 474, + 825, + 517 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "“San Francisco” ", + "bbox": [ + 334, + 529, + 442, + 544 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "“New York” ", + "bbox": [ + 580, + 529, + 661, + 544 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Both bigrams appear frequently in text corpora. As a direct consequence, the unigrams “Francisco” and “York” also appear frequently. However, since “Francisco” and “York” typically follow “San” and “New”, respectively, they should not have high probability in the proposal distribution as they might if we use unigram frequencies (Chen $\\&$ Goodman, 1996). Instead, it would be better to increase the proposal probability of unigrams with diverse histories, or more precisely unigrams that complete a large number of bigram types. Thus instead of drawing from the unigram distribution, we consider drawing from ", + "bbox": [ + 173, + 556, + 825, + 654 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/980c4872295f29ca37308b17006c5554cd89b053a47ad326e301ec6e7b60fe18.jpg", + "text": "$$\nq ( x ) \\propto N _ { 1 + } ( \\bullet , x )\n$$", + "text_format": "latex", + "bbox": [ + 437, + 652, + 560, + 670 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that we now noise the prediction $x _ { t }$ in addition to the context $x _ { 1 : t - 1 }$ . Combining this new proposal distribution with the discounted $\\gamma _ { \\mathrm { A D } } ( x _ { 1 } )$ from the previous section, we obtain the noising analogue of Kneser-Ney smoothing. ", + "bbox": [ + 174, + 674, + 826, + 717 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 1 summarizes the discussed noising schemes. ", + "bbox": [ + 174, + 723, + 511, + 738 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.5 TRAINING AND TESTING ", + "text_level": 1, + "bbox": [ + 176, + 755, + 387, + 770 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "During training, noising is performed per batch and is done online such that each epoch of training sees a different noised version of the training data. At test time, to match the training objective we should sample multiple corrupted versions of the test data, then average the predictions (Srivastava et al., 2014). In practice, however, we find that simply using the maximum likelihood (uncorrupted) input sequence works well; evaluation runtime remains unchanged. ", + "bbox": [ + 174, + 781, + 825, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.6 EXTENSIONS ", + "text_level": 1, + "bbox": [ + 174, + 868, + 305, + 883 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The schemes described are for the language model setting. To extend them to the sequence-tosequence or encoder-decoder setting, we noise both $x _ { < t }$ as well as $y _ { < t }$ . While in the decoder we ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/433cc1e793e26806f3efe06406a2893eefd2a4b30dcac6547f09e55dd8656961.jpg", + "table_caption": [ + "Table 2: Single-model perplexity on Penn Treebank with different noising schemes. We also compare to the variational method of Gal (2015), who also train LSTM models with the same hidden dimension. Note that performing Monte Carlo dropout at test time is significantly more expensive than our approach, where test time is unchanged. " + ], + "table_footnote": [], + "table_body": "
Noising schemeValidationTest
Medium models (512 hidden size)
none (dropout only) blank84.380.4
82.778.8
unigram83.180.1
bigram Kneser-Ney79.976.9
Large models (1500 hidden size)
none (dropout only) blank unigram bigram Kneser-Ney81.677.5
79.475.5
79.4 76.276.1 73.4
Zaremba et al. (2014)82.2
Gal (2015) variational dropout (tied weights)78.4
Gal (2015) (untied weights,Monte Carlo)77.3 丨75.0 73.4
", + "bbox": [ + 272, + 101, + 725, + 335 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/8bf8a47b4b2c61b0c81790044b83acf9981a8eb13ac5bc06ab7e338f3f44d3b6.jpg", + "table_caption": [ + "Table 3: Perplexity on Text8 with different noising schemes. " + ], + "table_footnote": [], + "table_body": "
Noising schemeValidationTest
none94.3123.6
blank85.0110.7
unigram85.2111.3
bigram Kneser-Ney84.5110.6
", + "bbox": [ + 351, + 433, + 647, + 518 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "have $y _ { < t }$ and $y _ { t }$ as analogues to language model context and target prediction, it is unclear whether noising $x _ { < t }$ should be beneficial. Empirically, however, we find this to be the case (Table 4). ", + "bbox": [ + 173, + 587, + 825, + 614 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 646, + 326, + 661 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 LANGUAGE MODELING ", + "text_level": 1, + "bbox": [ + 176, + 684, + 377, + 698 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Penn Treebank We train networks for word-level language modeling on the Penn Treebank dataset, using the standard preprocessed splits with a 10K size vocabulary (Mikolov, 2012). The PTB dataset contains $9 2 9 \\mathrm { k }$ training tokens, 73k validation tokens, and ${ 8 2 } \\mathrm { k }$ test tokens. Following Zaremba et al. (2014), we use minibatches of size 20 and unroll for 35 time steps when performing backpropagation through time. All models have two hidden layers and use LSTM units. Weights are initialized uniformly in the range $[ - 0 . 1 , 0 . 1 ]$ . We consider models with hidden sizes of 512 and 1500. ", + "bbox": [ + 174, + 713, + 825, + 811 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We train using stochastic gradient descent with an initial learning rate of 1.0, clipping the gradient if its norm exceeds 5.0. When the validation cross entropy does not decrease after a training epoch, we halve the learning rate. We anneal the learning rate 8 times before stopping training, and pick the model with the lowest perplexity on the validation set. ", + "bbox": [ + 174, + 819, + 825, + 875 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For regularization, we apply feed-forward dropout (Pham et al., 2014) in combination with our noising schemes. We report results in Table 2 for the best setting of the dropout rate (which we find to match the settings reported in Zaremba et al. (2014)) as well as the best setting of noising ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/a74791f89bc2ae43fdc9a304f47b23fda338e2c01b613335fe659ced2c4f1495.jpg", + "image_caption": [ + "Figure 1: Example training and validation curves for an unnoised model and model regularized using the bigram Kneser-Ney noising scheme. " + ], + "image_footnote": [], + "bbox": [ + 228, + 102, + 769, + 270 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/85a9023a7951293ef108ad65feeb0560267842bb762db52ddd9b9e963097c913.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
SchemePerplexityBLEU
dropout, no noising8.8424.6
blank noising8.2825.3 (+0.7)
unigram noising8.1525.5 (+0.9)
bigram Kneser-Ney7.9226.0 (+1.4)
source only8.7424.8 (+0.2)
target only8.1425.6 (+1.0)
", + "bbox": [ + 331, + 324, + 666, + 445 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 4: Perplexities and BLEU scores for machine translation task. Results for bigram KN noising on only the source sequence and only the target sequence are given as well. ", + "bbox": [ + 173, + 462, + 825, + 489 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "probability $\\gamma _ { 0 }$ on the validation set.2 Figure 1 shows the training and validation perplexity curves for a noised versus an unnoised run. ", + "bbox": [ + 173, + 518, + 823, + 546 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our large models match the state-of-the-art regularization method for single model performance on this task. In particular, we find that picking $\\gamma _ { \\mathrm { A D } } ( x _ { 1 } )$ and $q ( x )$ corresponding to Kneser-Ney smoothing yields significant gains in validation perplexity, both for the medium and large size models. Recent work (Merity et al., 2016; Zilly et al., 2016) has also achieved impressive results on this task by proposing different architectures which are orthogonal to our data augmentation schemes. ", + "bbox": [ + 174, + 554, + 825, + 625 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Text8 In order to determine whether noising remains effective with a larger dataset, we perform experiments on the Text8 corpus3. The first 90M characters are used for training, the next 5M for validation, and the final 5M for testing, resulting in $1 5 . 3 \\mathbf { M }$ training tokens, 848K validation tokens, and 855K test tokens. We preprocess the data by mapping all words which appear 10 or fewer times to the unknown token, resulting in a 42K size vocabulary. Other parameter settings are the same as described in the Penn Treebank experiments, besides that only models with hidden size 512 are considered, and noising is not combined with feed-forward dropout. Results are given in Table 3. ", + "bbox": [ + 174, + 641, + 825, + 739 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 MACHINE TRANSLATION ", + "text_level": 1, + "bbox": [ + 174, + 758, + 388, + 772 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For our machine translation experiments we consider the English-German machine translation track of IWSLT $2 0 1 5 ^ { 4 }$ . The IWSLT 2015 corpus consists of sentence-aligned subtitles of TED and TEDx talks. The training set contains roughly 190K sentence pairs with 5.4M tokens. Following Luong & Manning (2015), we use TED tst2012 as a validation set and report BLEU score results (Papineni et al., 2002) on tst2014. We limit the vocabulary to the top 50K most frequent words for each language. ", + "bbox": [ + 174, + 785, + 825, + 869 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c52239f55d9409031ab76a2c2d420a3bbb70c906f6f5db3a48bd4677b262db5b.jpg", + "image_caption": [ + "Figure 2: Perplexity with noising on Penn Treebank while varying the value of $\\gamma _ { 0 }$ . Using discounting to scale $\\gamma _ { 0 }$ (yielding $\\gamma _ { \\mathrm { A D . } }$ ) maintains gains for a range of values of noising probability, which is not true for the unscaled case. " + ], + "image_footnote": [], + "bbox": [ + 176, + 103, + 454, + 255 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/66596b6cf7d5d61afc0773e432d36885d11d8c1b6493b1dddbc53657d86ae861.jpg", + "image_caption": [ + "Figure 3: Mean KL-divergence over validation set between softmax distributions of noised and unnoised models and lower order distributions. Noised model distributions are closer to the uniform and unigram frequency distributions. " + ], + "image_footnote": [], + "bbox": [ + 524, + 104, + 803, + 251 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We train a two-layer LSTM encoder-decoder network (Sutskever et al., 2014; Cho et al., 2014) with 512 hidden units in each layer. The decoder uses an attention mechanism (Bahdanau et al., 2014) with the dot alignment function (Luong et al., 2015). The initial learning rate is 1.0 and we start halving the learning rate when the relative difference in perplexity on the validation set between two consecutive epochs is less than $1 \\%$ . We follow training protocols as described in Sutskever et al. (2014): (a) LSTM parameters and word embeddings are initialized from a uniform distribution between $[ - 0 . 1 , 0 . 1 ]$ , (b) inputs are reversed, (c) batch size is set to 128, (d) gradient clipping is performed when the norm exceeds a threshold of 5. We set hidden unit dropout rate to 0.2 across all settings as suggested in Luong et al. (2015). We compare unigram, blank, and bigram Kneser-Ney noising. Noising rate $\\gamma$ is selected on the validation set. ", + "bbox": [ + 173, + 383, + 825, + 523 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results are shown in Table 4. We observe performance gains for both blank noising and unigram noising, giving roughly $+ 0 . 7$ BLEU score on the test set. The proposed bigram Kneser-Ney noising scheme gives an additional performance boost of $+ 0 . 5 – 0 . 7$ on top of the blank noising and unigram noising models, yielding a total gain of $+ 1 . 4$ BLEU. ", + "bbox": [ + 174, + 530, + 825, + 587 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 DISCUSSION ", + "text_level": 1, + "bbox": [ + 174, + 627, + 310, + 645 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.1 SCALING $\\gamma$ VIA DISCOUNTING ", + "text_level": 1, + "bbox": [ + 176, + 672, + 426, + 688 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now examine whether discounting has the desired effect of noising subsequences according to their uncertainty. If we consider the discounting ", + "bbox": [ + 174, + 707, + 823, + 736 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/47c0323ecd79554a5f1c5395c87014c0e560225c8e275b0bb72df7cb9486da72.jpg", + "text": "$$\n\\gamma _ { \\mathrm { A D } } ( x _ { 1 } ) = \\gamma _ { 0 } \\frac { N _ { 1 + } ( x _ { 1 } , \\bullet ) } { c ( x _ { 1 } ) }\n$$", + "text_format": "latex", + "bbox": [ + 408, + 746, + 588, + 781 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "we observe that the denominator $c ( x _ { 1 } )$ can dominate than the numerator $N _ { 1 + } ( x _ { 1 } , \\bullet )$ . Common tokens are often noised infrequently when discounting is used to rescale the noising probability, while rare tokens are noised comparatively much more frequently, where in the extreme case when a token appears exactly once, we have $\\gamma _ { \\mathrm { A D } } ~ = ~ \\gamma _ { 0 }$ . Due to word frequencies following a Zipfian power law distribution, however, common tokens constitute the majority of most texts, and thus discounting leads to significantly less noising. ", + "bbox": [ + 173, + 790, + 825, + 875 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare the performance of models trained with a fixed $\\gamma _ { 0 }$ versus a $\\gamma _ { 0 }$ rescaled using discounting. As shown in Figure 2, bigram discounting leads to gains in perplexity for a much broader range of $\\gamma _ { 0 }$ . Thus the discounting ratio seems to effectively capture the “right” tokens to noise. ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/3aaade1cbb46aa669668f26488ab20ff05510266f5c3d019e55c0935efcf7dbc.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
NoisingBigramsTrigrams
none (dropout only)2881381
blank noising2760372
unigram noising2612365
", + "bbox": [ + 346, + 101, + 651, + 172 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 5: Perplexity of last unigram for unseen bigrams and trigrams in Penn Treebank validation set. We compare noised and unnoised models with noising probabilities chosen such that models have near-identical perplexity on full validation set. ", + "bbox": [ + 174, + 189, + 825, + 231 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.2 NOISED VERSUS UNNOISED MODELS ", + "text_level": 1, + "bbox": [ + 176, + 261, + 473, + 275 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Smoothed distributions In order to validate that data noising for RNN models has a similar effect to that of smoothing counts in $n$ -gram models, we consider three models trained with unigram noising as described in Section 4.1 on the Penn Treebank corpus with $\\gamma = 0$ (no noising), $\\gamma = 0 . 1$ , and $\\gamma = 0 . 2 5$ . Using the trained models, we measure the Kullback-Leibler divergence $D _ { \\mathrm { K L } } ( p \\Vert q ) =$ $\\textstyle \\sum _ { i } p _ { i } \\log ( p _ { i } / q _ { i } )$ over the validation set between the predicted softmax distributions, $\\hat { p }$ , and the uniform distribution as well as the unigram frequency distribution. We then take the mean KL divergence over all tokens in the validation set. ", + "bbox": [ + 174, + 287, + 825, + 386 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Recall that in interpolation smoothing, a weighted combination of higher and lower order $n$ -gram models is used. As seen in Figure 3, the softmax distributions of noised models are significantly closer to the lower order frequency distributions than unnoised models, in particular in the case of the unigram distribution, thus validating our analysis in Section 3.3. ", + "bbox": [ + 174, + 392, + 825, + 449 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Unseen $n$ -grams Smoothing is most beneficial for increasing the probability of unobserved sequences. To measure whether noising has a similar effect, we consider bigrams and trigrams in the validation set that do not appear in the training set. For these unseen bigrams (15062 occurrences) and trigrams (43051 occurrences), we measure the perplexity for noised and unnoised models with near-identical perplexity on the full set. As expected, noising yields lower perplexity for these unseen instances. ", + "bbox": [ + 174, + 467, + 825, + 551 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 574, + 318, + 590 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we show that data noising is effective for regularizing neural network-based sequence models. By deriving a correspondence between noising and smoothing, we are able to adapt advanced smoothing methods for $n$ -gram models to the neural network setting, thereby incorporating well-understood generative assumptions of language. Possible applications include exploring noising for improving performance in low resource settings, or examining how these techniques generalize to sequence modeling in other domains. ", + "bbox": [ + 174, + 608, + 825, + 691 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 717, + 356, + 731 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We thank Will Monroe for feedback on a draft of this paper, Anand Avati for help running experiments, and Jimmy Wu for computing support. We also thank the developers of Theano (Theano Development Team, 2016) and Tensorflow (Abadi et al., 2016). Some GPUs used in this work were donated by NVIDIA Corporation. 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Data augmentation via levy processes. arXiv preprint arXiv:1603.06340, 2016. ", + "bbox": [ + 174, + 549, + 821, + 579 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Sida I Wang, Mengqiu Wang, Stefan Wager, Percy Liang, and Christopher D Manning. Feature noising for log-linear structured prediction. In Empirical Methods in Natural Language Processing (EMNLP), 2013. ", + "bbox": [ + 173, + 587, + 823, + 630 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014. ", + "bbox": [ + 171, + 638, + 823, + 667 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Julian Georg Zilly, Rupesh Kumar Srivastava, Jan Koutn´ık, and Jurgen Schmidhuber. Recurrent ¨ highway networks. arXiv preprint arXiv:1607.03474, 2016. ", + "bbox": [ + 168, + 676, + 823, + 705 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A SKETCH OF NOISING ALGORITHM ", + "text_level": 1, + "bbox": [ + 176, + 102, + 496, + 118 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We provide pseudocode of the noising algorithm corresponding to bigram Kneser-Ney smoothing for $n$ -grams (In the case of sequence-to-sequence tasks, we estimate the count-based parameters separately for source and target). To simplify, we assume a batch size of one. The noising algorithm is applied to each data batch during training. No noising is applied at test time. ", + "bbox": [ + 174, + 133, + 825, + 190 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/ba25aa9492fc996e662fbbadff1ac3507e55932b0ceb6cf05e1b211c9da77e81.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Algorithm 1 Bigram KN noising (Language modeling setting)
Require counts c(x), number of distinct continuations N1+(x,·), proposal distribution q(x） N1+(.,x) Inputs X,Y batch of unnoised data indices, scaling factor γo
procedure NOISEBGKN(X, Y) >X=(x1,...,xt),Y=(x2,...,xt+1)
X,Y←X,Y
for j = 1,...,t do γ ←γ0N1+(xj,·)/c(xj)
if ~ Bernoulli(~) then
xj~ Categorical(q) Updates X
yj~ Categorical(q)
end if
end for
return X, Y Run training iteration with noised batch
end procedure
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Fi-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 140, + 326, + 470, + 342 + ], + "spans": [ + { + "bbox": [ + 140, + 326, + 470, + 342 + ], + "score": 1.0, + "content": "nally, we provide empirical analysis validating the relationship between noising", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 338, + 206, + 352 + ], + "spans": [ + { + "bbox": [ + 141, + 338, + 206, + 352 + ], + "score": 1.0, + "content": "and smoothing.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 387, + 206, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 208, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 208, + 402 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 504, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "Language models are a crucial component in many domains, such as autocompletion, machine trans-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "lation, and speech recognition. A key challenge when performing estimation in language modeling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "is the data sparsity problem: due to large vocabulary sizes and the exponential number of possi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "ble contexts, the majority of possible sequences are rarely or never observed, even for very short", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 165, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 165, + 474 + ], + "score": 1.0, + "content": "subsequences.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "In other application domains, data augmentation has been key to improving the performance of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "neural network models in the face of insufficient data. In computer vision, for example, there exist", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "well-established primitives for synthesizing additional image data, such as by rescaling or applying", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "affine distortions to images (LeCun et al., 1998; Krizhevsky et al., 2012). 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Neural network mod-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "els, however, have no notion of discrete counts, and instead use distributed representations to combat", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "the curse of dimensionality (Bengio et al., 2003). Despite the effectiveness of distributed represen-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "tations, overfitting due to data sparsity remains an issue. Existing regularization methods, however,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "are typically applied to weights or hidden units within the network (Srivastava et al., 2014; Le et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 638, + 315, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 315, + 650 + ], + "score": 1.0, + "content": "2015) instead of directly considering the input data.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this work, we consider noising primitives as a form of data augmentation for recurrent neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "score": 1.0, + "content": "network-based language models. 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Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, Andrew Y. Ng ´", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 112, + 134, + 493, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 114, + 147, + 390, + 180 + ], + "lines": [ + { + "bbox": [ + 112, + 145, + 321, + 158 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 321, + 158 + ], + "score": 1.0, + "content": "Computer Science Department, Stanford University", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 113, + 157, + 391, + 169 + ], + "spans": [ + { + "bbox": [ + 113, + 157, + 391, + 169 + ], + "score": 1.0, + "content": "zxie,sidaw,danilevy,anie,ang @cs.stanford.edu,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 112, + 168, + 293, + 180 + ], + "spans": [ + { + "bbox": [ + 112, + 168, + 293, + 180 + ], + "score": 1.0, + "content": "jiweil,jurafsky @stanford.edu", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 112, + 145, + 391, + 180 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 208, + 333, + 220 + ], + "lines": [ + { + "bbox": [ + 276, + 208, + 335, + 221 + ], + "spans": [ + { + "bbox": [ + 276, + 208, + 335, + 221 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 143, + 239, + 468, + 350 + ], + "lines": [ + { + "bbox": [ + 142, + 240, + 469, + 252 + ], + "spans": [ + { + "bbox": [ + 142, + 240, + 469, + 252 + ], + "score": 1.0, + "content": "Data noising is an effective technique for regularizing neural network models.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 250, + 469, + 264 + ], + "spans": [ + { + "bbox": [ + 141, + 250, + 469, + 264 + ], + "score": 1.0, + "content": "While noising is widely adopted in application domains such as vision and speech,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 262, + 469, + 274 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 469, + 274 + ], + "score": 1.0, + "content": "commonly used noising primitives have not been developed for discrete sequence-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 470, + 286 + ], + "score": 1.0, + "content": "level settings such as language modeling. In this paper, we derive a connection", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 284, + 469, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 457, + 296 + ], + "score": 1.0, + "content": "between input noising in neural network language models and smoothing in", + "type": "text" + }, + { + "bbox": [ + 457, + 286, + 465, + 294 + ], + "score": 0.51, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 284, + 469, + 296 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 295, + 470, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 295, + 470, + 308 + ], + "score": 1.0, + "content": "gram models. Using this connection, we draw upon ideas from smoothing to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 306, + 469, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 469, + 318 + ], + "score": 1.0, + "content": "develop effective noising schemes. We demonstrate performance gains when ap-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 317, + 469, + 328 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 469, + 328 + ], + "score": 1.0, + "content": "plying the proposed schemes to language modeling and machine translation. Fi-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 140, + 326, + 470, + 342 + ], + "spans": [ + { + "bbox": [ + 140, + 326, + 470, + 342 + ], + "score": 1.0, + "content": "nally, we provide empirical analysis validating the relationship between noising", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 338, + 206, + 352 + ], + "spans": [ + { + "bbox": [ + 141, + 338, + 206, + 352 + ], + "score": 1.0, + "content": "and smoothing.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5, + "bbox_fs": [ + 140, + 240, + 470, + 352 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 387, + 206, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 208, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 208, + 402 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 417, + 504, + 472 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "Language models are a crucial component in many domains, such as autocompletion, machine trans-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "lation, and speech recognition. A key challenge when performing estimation in language modeling", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "is the data sparsity problem: due to large vocabulary sizes and the exponential number of possi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "ble contexts, the majority of possible sequences are rarely or never observed, even for very short", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 461, + 165, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 165, + 474 + ], + "score": 1.0, + "content": "subsequences.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 418, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "In other application domains, data augmentation has been key to improving the performance of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "neural network models in the face of insufficient data. In computer vision, for example, there exist", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "well-established primitives for synthesizing additional image data, such as by rescaling or applying", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "affine distortions to images (LeCun et al., 1998; Krizhevsky et al., 2012). Similarly, in speech", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "recognition adding a background audio track or applying small shifts along the time dimension has", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "been shown to yield significant gains, especially in noisy settings (Deng et al., 2000; Hannun et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "2014). However, widely-adopted noising primitives have not yet been developed for neural network", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 555, + 179, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 179, + 568 + ], + "score": 1.0, + "content": "language models.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 478, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 572, + 504, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 137, + 585 + ], + "score": 1.0, + "content": "Classic", + "type": "text" + }, + { + "bbox": [ + 137, + 574, + 145, + 582 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 572, + 504, + 585 + ], + "score": 1.0, + "content": "-gram models of language cope with rare and unseen sequences by using smoothing meth-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "ods, such as interpolation or absolute discounting (Chen & Goodman, 1996). Neural network mod-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 606 + ], + "score": 1.0, + "content": "els, however, have no notion of discrete counts, and instead use distributed representations to combat", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "the curse of dimensionality (Bengio et al., 2003). Despite the effectiveness of distributed represen-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "tations, overfitting due to data sparsity remains an issue. Existing regularization methods, however,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "are typically applied to weights or hidden units within the network (Srivastava et al., 2014; Le et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 638, + 315, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 315, + 650 + ], + "score": 1.0, + "content": "2015) instead of directly considering the input data.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 572, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "In this work, we consider noising primitives as a form of data augmentation for recurrent neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 504, + 678 + ], + "score": 1.0, + "content": "network-based language models. By examining the expected pseudocounts from applying the nois-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "ing schemes, we draw connections between noising and linear interpolation smoothing. Using this", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "connection, we then derive noising schemes that are analogues of more advanced smoothing meth-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "ods. We demonstrate the effectiveness of these schemes for regularization through experiments on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 721 + ], + "score": 1.0, + "content": "language modeling and machine translation. Finally, we validate our theoretical claims by examin-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 720, + 250, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 250, + 734 + ], + "score": 1.0, + "content": "ing the empirical effects of noising.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 655, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 211, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 213, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 213, + 96 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 207 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 505, + 119 + ], + "score": 1.0, + "content": "Our work can be viewed as a form of data augmentation, for which to the best of our knowledge there", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "score": 1.0, + "content": "exists no widely adopted schemes in language modeling with neural networks. Classical regulariza-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 191, + 141 + ], + "score": 1.0, + "content": "tion methods such as", + "type": "text" + }, + { + "bbox": [ + 191, + 129, + 204, + 140 + ], + "score": 0.89, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "-regularization are typically applied to the model parameters, while dropout", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 506, + 152 + ], + "score": 1.0, + "content": "is applied to activations which can be along the forward as well as the recurrent directions (Zaremba", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 505, + 163 + ], + "score": 1.0, + "content": "et al., 2014; Semeniuta et al., 2016; Gal, 2015). Others have introduced methods for recurrent neural", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "networks encouraging the hidden activations to remain stable in norm, or constraining the recurrent", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "weight matrix to have eigenvalues close to one (Krueger & Memisevic, 2015; Arjovsky et al., 2015;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "Le et al., 2015). These methods, however, all consider weights and hidden units instead of the input", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 401, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 401, + 208 + ], + "score": 1.0, + "content": "data, and are motivated by the vanishing and exploding gradient problem.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 211, + 505, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 505, + 223 + ], + "score": 1.0, + "content": "Feature noising has been demonstrated to be effective for structured prediction tasks, and has been", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "interpreted as an explicit regularizer (Wang et al., 2013). Additionally, Wager et al. (2014) show", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 247 + ], + "score": 1.0, + "content": "that noising can inject appropriate generative assumptions into discriminative models to reduce their", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 423, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 423, + 257 + ], + "score": 1.0, + "content": "generalization error, but do not consider sequence models (Wager et al., 2016).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 261, + 505, + 361 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 274 + ], + "score": 1.0, + "content": "The technique of randomly zero-masking input word embeddings for learning sentence represen-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "tations has been proposed by Iyyer et al. 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