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+# BENCHMARKING THE SPECTRUM OF AGENT CAPABILITIES
+
+# Danijar Hafner
+
+Google Research, Brain Team University of Toronto mail@danijar.com
+
+# ABSTRACT
+
+Evaluating the general abilities of intelligent agents requires complex simulation environments. Existing benchmarks typically evaluate only one narrow task per environment, requiring researchers to perform expensive training runs on many different environments. We introduce Crafter, an open world survival game with visual inputs that evaluates a wide range of general abilities within a single environment. Agents either learn from the provided reward signal or through intrinsic objectives and are evaluated by semantically meaningful achievements that can be unlocked during each episode, such as discovering resources and crafting tools. Consistently unlocking all achievements requires strong generalization, deep exploration, and long-term reasoning. We experimentally verify that Crafter is of appropriate difficulty to drive future research and provide baselines scores of reward agents and unsupervised agents. Furthermore, we observe sophisticated behaviors emerging from maximizing the reward signal, such as building tunnel systems, bridges, houses, and plantations. We hope that Crafter will accelerate research progress by quickly evaluating a wide spectrum of abilities.
+
+# 1 INTRODUCTION
+
+Crafter is an open world survival game for reinforcement learning research. Shown in Figure 1, Crafter features randomly generated 2D worlds with forests, lakes, mountains, and caves. The player needs to forage for food and water, find shelter to sleep, defend against monsters, collect materials, and build tools. The game mechanics are inspired by the popular game Minecraft and were simplified and optimized for research productivity. Crafter aims to be a fruitful benchmark for reinforcement learning by focusing on the following design goals:
+
+Research challenges Crafter poses substantial challenges to current methods. Procedural generation requires strong generalization, the technology tree evaluates wide and deep exploration, image observations calls for representation learning, repeated subtasks and sparse rewards evaluate long-term reasoning and credit assignment.
+
+Meaningful evaluation Agents are evaluated by a range of achievements that can be unlocked in each episode. The achievements correspond to meaningful milestones in behavior, offering insights into ability spectrum of both reward agents and unsupervised agents.
+
+Iteration speed Crafter evaluates many agent abilities within a single environment, vastly reducing the computational requirements over benchmarks suites that require training on many separate environments from scratch, while making it more likely that the measured performance is representative of new domains.
+
+![](images/927083a984075d7f618fcdcdfd9c0e2627a41223e2bbb0f4e892d8cd75b47a70.jpg)  
+Figure 1: Agent view of a procedurally generated world in Crafter, showing terrain types, resources, and creatures. Agents learn from image inputs and aim to unlock a range of semantically meaningful achievements during each episode. The achievements evaluate strong generalization, wide and deep exploration, and long-term reasoning.
+
+![](images/4ff4aac35ce68fcec48907c7d4ec4e105662b82d357033b534ebdbc6dea74b40.jpg)  
+Figure 2: Play Crafter yourself through the human interface.
+
+# 2 RELATED WORK
+
+Benchmarks have been a driving force behind the progress and successes of reinforcement learning as a field (Bellemare et al., 2013; Brockman et al., 2016; Kempka et al., 2016; Beattie et al., 2016; Tassa et al., 2018; Juliani et al., 2018). Benchmarks often require a large amount of computational resources and yet only test a small fraction of the abilities that a general agent should master (Cobbe et al., 2020). This section directly compares Crafter to four particularly related benchmarks.
+
+Minecraft Crafter is inspired by the successful 3D video game Minecraft, which is available to researchers via Malmo (Johnson et al., 2016) and MineRL (Guss et al., 2019). Minecraft features diverse open worlds with randomly generated and modifiable terrain, as well as many different resources, tools, and monsters. However, Minecraft is too complex to be solved by current methods (Milani et al., 2020), it is unclear by what metric agents should be evaluated by, the environment is slow, and can be difficult to use because it requires Java and a window server. In comparison, Crafter captures many principles of Minecraft in a simple and fast environment, where results can be obtained in a matter of hours, and where a large number of semantically meaningful evaluation metrics are available for reinforcement learning with or without extrinsic reward. The goal of Crafter is not to replace Minecraft but progress faster towards it.
+
+Atari The Atari Learning Environment (Bellemare et al., 2013) has been the gold standard benchmark in reinforcement learning. It comprises around 54 individual games, depending on the evaluation protocol (Mnih et al., 2015; Schulman et al., 2017; Badia et al., 2020; Hafner et al., 2020). While the large number of games tests different abilities of agents, they require a large amount of computation. The recommended protocol of training the agent with 5 random seeds on each game for 200M steps requires over 2000 GPU days (Castro et al., 2018; Hessel et al., 2018). This substantially slows down experimentation and makes the complete benchmark infeasible for most academic labs. Moreover, Atari games are nearly deterministic, so agents can approximately memorize their action sequences and are not required to generalize to new situations (Machado et al., 2018).
+
+ProcGen ProcGen (Cobbe et al., 2020) provides a benchmark that is similar to Atari but explicitly addresses the determinism present in Atari through the use of procedural generation and randomized textures. It consists of 16 games, where each episode features a randomly generated level layout. Similarly, Crafter relies on procedural generation to provide a different world map with different distribution of resources and monsters for every episode. However, ProcGen still requires training methods on 16 individual games for 200M environment steps, which each focus on a narrow aspect of an agent’s general abilities. In comparison, Crafter evaluates many different abilities of an agent by training only on a single environment for 5M steps, substantially accelerating experimentation.
+
+NetHack NetHack (Küttler et al., 2020) is a text-based game, where the player traverses a randomly generated system of dungeons with many different items and creatures. Unlike the other discussed environments, NetHack uses symbolic inputs and thus does not evaluate an agent’s ability to learn representations of high-dimensional inputs. The game is challenging due to the large amount of knowledge required about the many different items and their effects, even for human players. As a result, NetHack requires many environment steps for agents to acquire this domain-specific knowledge; 1B steps were used in the original paper. In contrast, Crafter generates diverse complex worlds from simple underlying rules, focusing more on generalization than memorization of facts.
+
+# 3 CRAFTER BENCHMARK
+
+We introduce Crafter, a benchmark that evaluates a variety of agent abilities in a single environment. This section describes the game mechanics of the environment, the interface of agent inputs and actions, the evaluation protocol that is based on a range of semantically meaningful achievements, and the open challenges that Crafter poses for future research.
+
+![](images/427b06e7de5d61130b9b23d25f5e328a4c7a56adcafda3ff8cce2bce7ef89914.jpg)  
+Figure 3: Crafter procedurally generates a unique world for every episode that features several terrain types: grasslands, forests, lakes, mountains, caves. Memorizing action sequences is thus not a viable strategy and agents are forced to learn behaviors that generalize to new situations.
+
+# 3.1 GAME MECHANICS
+
+This section describes the game mechanics of Crafter, namely its randomly generated world maps, the levels of health and other internal quantities that the player has to maintain, the resources it can collect and objects and tools it can make from them, as well as the creatures and how they are influenced by the time of day. The images of all materials and objects are shown in Figure E.1. All randomness in the environment is uniquely determined by an integer seed that is derived from the initial seed passed to the environment and the episode number.
+
+Terrain generation A unique world is generated for every episode, shown in Figure 3. The world leverages an underlying grid of $6 4 \times 6 4$ cells but the agent only observes the world through pixel images. The terrain features grasslands, lakes, and mountains. Lakes can have shores, grasslands can have forests, and mountains can have caves, ores, and lava. These are determined by OpenSimplex noise (Spencer, 2014), a form of locally smooth noise. Within the areas determined by noise, objects appear with equal probability at any location, such as trees in forests and skeletons in caves.
+
+Health and survival The player has levels of health, food, water, and rest that it must prevent from reaching zero. The levels for food, water, and rest decrease over time and are restored by drinking from a lake, chasing cows or growing fruits to eat, and sleeping in places where monsters cannot attack. Once one of the three levels reaches zero, the player starts losing health points. It can also lose health points when attacked by monsters. When the health points reach zero, the player dies. Health points regenerate over time when the player is not hungry, thirsty, or sleepy.
+
+Resources and crafting There are many resources, such as saplings, wood, stone, coal, iron, and diamonds, the player can collect in its inventory and use to build tools and place objects in the world. Many of the resources require tools that the place must first build from more basic resources, leading to a technology tree with several levels. Standing nearby a table enables the player to craft wood pickaxes and swords, as well as stone pickaxes and stone swords. Crafting a furnace from stone enables crafting iron pickaxes and iron swords from both iron, coal, and wood.
+
+Creatures and night Creatures are initialized in random locations and move randomly. Zombies and cows live in grasslands and are automatically spawned and despawned to ensure a given amount of creatures. At night, the agent’s view is restricted and noisy and a larger number of zombies is spawned. This makes it difficult to survive without securing a shelter, such as a cave. Skeletons live in caves and try to keep the player at a distance to shoot arrows at the player. The player can interact with creatures to decrease their health points. Cows move randomly and offer a food source.
+
+# 3.2 ENVIRONMENT INTERFACE
+
+This section defines the specification of the environment, explains the available actions, agent inputs, episode termination, and additional information provided by the environment. The design goal of these is to make the environment easy to use and inspect. The environment uses the Gym interface (Brockman et al., 2016) with visual agent inputs and flat categorical actions.
+
+![](images/963f582bd37aa5dd9aa4ebdcb613e4e02e884f41702752f577e51296fcb57fba.jpg)  
+Figure 4: The 22 achievements that can be unlocked within each episode. The arrows indicate which achievements will be completed along the way of working toward more challenging achievements. Several of the earlier tasks have to be repeated multiple times, such as collecting resources, to progress further. A reward is only given when an achievement is unlocked for the first time during the episode.
+
+Observations Agent receive color images of size $6 4 \times 6 4 \times 3$ as their only inputs. The image shows a local top-down view of the map, reaching 4 cells west and east and 3 cells north and south of the player position. Below this view of the world, the image shows the current inventory state of the player, including its health points, food, water, and rest levels, collected materials, and crafted tools. The agent needs to learn to read its inventory state out of the image.
+
+Actions The action space is a flat categorical space with 17 actions, represented by integer indices. The actions allow the player to move in all 4 directions along the grid, interact with the object in front of it, go to sleep, place objects, and make tools. Each object and tool has a separate action associated with it. Tools are kept in the inventory whereas objects are automatically placed in front of the player. If the agent does not hold the required materials for making an object or tool, the action has no effect.
+
+Termination Each episode terminates when the player’s health points reach 0. This can happen when the player dies out of hunger, thirst, or tiredness, when attacked by a zombie or skeleton, or when falling into lava. Health points automatically regenerate, as long as the agent is not too hungry, thirsty, or sleepy. There is no negative reward for dying, as the reward signal already includes a penalty for losing health points. Episodes also end when reaching the time limit of 10,000 steps.
+
+Additional information The environment allows access to privileged information about the world state that the agent is forbidden to observe. This includes numeric inventory counts, achievement counts, the current coordinate of the player on the grid, and a semantic grid representation of the map. These can be used for debugging purposes or for other research scenarios, such as predicting the underlying environment state to evaluate representation learning or video prediction models.
+
+# 3.3 EVALUATION PROTOCOL
+
+To evaluate the diverse abilities of artificial agents on Crafter, we define two benchmarks. The first benchmark allows agents to access a provided reward signal, while the second benchmark does not and requires agents to purely learn from intrinsic objectives. Besides access to the provided reward signal, the evaluation protocols are identical. An agent is granted a budget of 1M environment steps to interact with the environment. The agent performance is evaluated through success rates of the individual achievements throughout its training, as well as an aggregated score.
+
+Achievements To evaluate a wide spectrum of agent abilities, Crafter defines 22 achievements. The achievements are shown in Figure 4 and correspond to semantically meaningful behaviors, such as collecting various resources, building objects and tools, finding food and water, defeating monsters, and waking up safely after sleeping. The achievements cover a wide range of difficulties, making them suitable to evaluate both weak and strong players and providing continuous feedback throughout the development process of new methods. Some achievements are independent of each other to test for breadth of exploration, while others depend on each other to test for deep exploration.
+
+Reward Crafter provides a sparse reward signal that is the sum of two components. The main component is a reward of $+ 1$ every time the agent unlocks each achievement for the first time during the current episode. The second component is a reward of $- 0 . 1$ for every health point lost and a reward of $+ 0 . 1$ for every health point that is regenerated. Because the maximum number of health points is 9, the second reward component only affects the first decimal of the episode return, and ceiling the episode return yields the number of achievements unlocked during the episode.
+
+Success rates The success rates offer insights into the breadth of abilities learned by an agent. The success rates are computed separately for each of the achievements, as the fraction of training episodes during which the agent has unlocked the achievement at least once. It is computed across all episodes that lead up to the budget of 1M environment steps, requiring agents to be data-efficient.1 Note that the number of environment steps is fixed but the number of episodes can differ between agents. Unlocking an achievement more than once per episode does not affect the success rate.
+
+Score The score summarizes the agent abilities into a single number. It is computed by aggregating the success rates for the individual achievements. Unlocking difficult achievements, even if it happens rarely, should contribute more than increasing the success rate of achievements that are already unlocked frequently even further. To account for the range of difficulties of the achievements, we average the success rates in log-space, known as the geometric mean.2 Unlike the reward, the score thus takes the achievement’s difficulties into account, without having to know them beforehand.
+
+Discussion Aggregating across tasks via a geometric mean weighs tasks based on their difficulty to the agent, resulting in higher scores for agents that explore more broadly. For example, collecting a diamond $1 \%$ of the time instead of $0 \%$ is a meaningful improvement, whereas collecting wood $9 5 \%$ of the time instead of $90 \%$ is not. This allows distinguishing how broadly agents have explored their environment even if they achieve similar rewards. The geometric mean also establishes a meaningful metric for unsupervised agents, which may get bored of tasks after performing them a few times and then move on to new tasks. A caveat of the geometric mean is that agents with rewards are evaluated by something they only indirectly optimize for, which can change their ranking order. Increasing reward and score is generally correlated, but capacity-limited agents may choose to optimize reward by mastering easy tasks and ignoring hard tasks, which only slowly increases the geometric mean.
+
+# 3.4 RESEARCH CHALLENGES
+
+Crafter aims to evaluate a diverse range of agent abilities within a single environment. Thus, if a method performs well on Crafter there should be a high chance that it also handles the challenges of other environments. The challenges also make Crafter suitable for evaluating progress on open research questions, such as strong generalization, wide and deep exploration, discovering reusable skills, and long-term memory and reasoning. Crafter is designed to pose the following challenges:
+
+Exploration Independent achievements evaluate wide exploration, without offering a linear path for the agent to follow. Dependent achievements evaluate deep exploration of the technology tree. Collecting a diamond requires an iron pickaxe, which in turn requires a furnace, table, coal, iron, and wood. The furnace requires collecting stone, which requires building a wood pickaxe at a table.
+
+Generalization Every episode is situated in a unique world that is procedurally generated. Moreover, many aspects of the game reoccur in different contexts, such as creatures and resources that can be found in different landscapes and times of day. This forces successful agents to recognize similar situations in different circumstances and be robust to changes in irrelevant details.
+
+![](images/07995ac8b8af22f31c0b8ac4cad048aa54297cdbb8fd679443eed524925b0785.jpg)  
+Figure 5: Crafter Benchmark Scores for various agents with and without rewards. Current top methods achieve scores of up to $10 \%$ that are far from the $50 \%$ of human experts, posing a substantial challenge for future research. Crafter scores are computed as the geometric mean across achievements of their success rates within the budget of 1M environment steps. Numbers in Table 1.
+
+Reusable skills Advancing in the game requires the agent to repeat several behaviors over long horizons, such as finding food, defending against monsters, and collecting common materials that are needed many times. The behavior of a successful agent naturally decomposes into sub-tasks, making Crafter suitable for studying hierarchical reinforcement learning.
+
+Credit assignment Only sparse rewards are given for unlocking an achievement for the first time during each episode. Moreover, several achievements require long-term reasoning, such as collecting the necessary resources for crafting a particular tool or planting saplings that can be harvested many hundred time steps later. This makes Crafter a challenge for temporal credit assignment.
+
+Memory The agent inputs only show the player’s immediate surroundings, making Crafter partially observed. To survive for a long time, agents need to remember where to find lakes to drink and open grasslands to hunt. Moreover, to effectively find rare resources, such as iron and diamonds, the agent needs to remember what parts of the map it has already searched.
+
+Representation The agent observes its environment via high-dimensional images, from which it has to extract entities that are meaningful for decision making. Similar to applications in the real world, the reward signal is sparse and the amount of environment interaction limited. As a result, successful agents will likely rely on explicit representation learning techniques.
+
+Survival In previous environments, the player can often survive by doing nothing. This allows for degenerate solutions to intrinsic objectives, unlike the real world where animals are forced to adapt to survive and maintain homeostasis and allostasis. In Crafter, the player struggles to survive through the constant pressure of maintaining enough water, food, rest, and defending against zombies.
+
+<table><tr><td>Method</td><td>Score (%)</td><td>Return</td></tr><tr><td>Human Experts</td><td>50.5±6.8</td><td>14.3±2.3</td></tr><tr><td>DreamerV2</td><td>10.0±1.2</td><td>9.0±1.7</td></tr><tr><td>PPO</td><td>4.6±0.3</td><td>4.2±1.2</td></tr><tr><td>Rainbow</td><td>4.3±0.2</td><td>5.0±1.3</td></tr><tr><td>Plan2Explore (Unsup)</td><td>2.1±0.1</td><td>2.1±1.5</td></tr><tr><td>RND (Unsup)</td><td>2.0±0.1</td><td>0.7±1.3</td></tr><tr><td>Random</td><td>1.6±0.0</td><td>2.1±1.3</td></tr></table>
+
+Table 1: Crafter benchmark scores. The Crafter score is computed as the geometric mean of success rates for all 22 achievements available in the environment. The score prefers general agents that unlock a wide range of achievements over those that unlock a small number of achievements very frequently. For example, an agent that explores many different achievements over the course of training achieves a higher score than one that only performs same simple tasks over an over. The score thus establishes a meaningful metric both for agents with and without reward.
+
+![](images/afbd1f04deffd2ccdfcebecd668b62d441d5ed684c326b9a42c003932a5d2fd6.jpg)  
+Figure 6: Agent ability spectrum showing the success rates of agents with rewards. These are unlocking percentages for all 22 achievements, computed over all training episodes. Rainbow manages to drink water and forage for food. PPO additionally rarely collects coal and builds stone tools. DreamerV2 achieves these more frequently and additionally sometimes grows and eats fruits. Numbers in Appendix A.
+
+# 4 EXPERIMENTS
+
+To established baselines for future work, we train various reinforcement learning methods on Crafter either with and without rewards. The two benchmarks follow the evaluation protocol in Section 3.3, which grants each agent a budget of 1M environment frames and computes the success rates of the individual achievements across all training episodes, as well as an aggregate score for the agent. Furthermore, we analyze the emergent agent behaviors qualitatively and record a dataset of human expert players to estimate the difficulty of the environment. The environment, code for the baseline agents and figures in this paper, and the human dataset are available on the project website. 3
+
+# 4.1 BENCHMARK WITH REWARDS
+
+We provide baselines scores for three reinforcement learning algorithms on Crafter with rewards. DreamerV2 (Hafner et al., 2020) learns a world model and optimizes a policy through planning in latent space. We used its default hyper parameters for Atari and increased the model size. PPO (Schulman et al., 2017) is a popular method that learns to map input images to actions through policy gradients. We use a convolutional neural network policy with hyper parameters that were tuned for Atari (Hill et al., 2018). Rainbow (Hessel et al., 2018) is based on Q-Learning and combines several advances, including for exploration. The defaults for Atari did not work well, so we tuned the hyper parameters for Crafter and found a compromise between Atari defaults and the data-efficient version of the method (van Hasselt et al., 2019) to be ideal. All agents trained for 1M environment steps in under 24 hours on a single GPU and we repeated the training for 10 random seeds per method. The training reward curves are included in Appendix D.
+
+The scores are listed in Table 1 and visualized in Figure 5. DreamerV2 achieves a score of $1 0 . 0 \%$ , followed by PPO with $4 . 6 \%$ and Rainbow of $4 . 3 \%$ . Despite these being top reinforcement learning methods, they lack behind the score of expert human players of $5 0 . 5 \%$ , which we describe in further detail in Section 4.3. We conclude that Crafter is a challenging benchmark, where current methods make learning progress but future research is needed to achieve high performance. For comparison, we report the episode returns in Table 1, computed over the episodes within the last $1 0 ^ { 5 }$ environment steps of training. We find a trend similar to the scores but notice that the methods are harder to tell apart, because differences on hard tasks that are rarely achieved affect the return less. Moreover, the scores are more meaningful for unsupervised agents, which should explore many achievements over time, but not necessarily remain interested in them until the end of training. The success rates for individual achievements are visualized in Figure 7, which offer insights into the breadth and depth of agent abilities. Rainbow displays high success rates on easier achievements. PPO learned to additionally make stone tools and furnaces. DreamerV2 achieved these more frequently and discovered growing and harvesting plants. None of the agents learned to collect and use iron for tools or to collect diamonds, or to achieve high success rates on many of the achievements.
+
+![](images/8317bb62cc63fd883c757ada4f4b08f83669a451ab64b8b482bcd7ae2d0299cf.jpg)  
+Figure 7: Agent ability spectrum showing the success rates for Crafter without rewards. Random actions unlock the 6 easiest achievements sometimes, such as drinking water and collecting wood. Plan2Explore forages for food and defeats monsters more frequently, to ensure longer survival. RND additionally collects stones and rarely even collects coal and builds furnaces. Numbers in Appendix B.
+
+# 4.2 UNSUPERVISED BENCHMARK
+
+We provide baselines scores for two unsupervised reinforcement learning agents on Crafter without rewards. We also include a baseline that simply chooses random actions. RND (Burda et al., 2018b) is a popular exploration method that seeks out novel inputs, estimated as the prediction error of a network that aims to predict fixed random embeddings of the input images. We use its default parameters for Atari. Plan2Explore (Sekar et al., 2020) learns a world model to plan for the expected information gain of imagined trajectories, allowing it to directly seek out imagined states that have not been experienced before. We implement Plan2Explore on top of DreamerV2 and keep the same hyper parameters. We use a non-episodic value function as RND does, which helps exploration in episodic environments (Burda et al., 2018b). All agents trained for 1M environment steps in under 24 hours on a single GPU and we repeated the training for 10 random seeds per method.
+
+The scores are listed in Table 1 and in Figure 5. Plan2Explore achieves a score of $2 . 1 \%$ , followed by RND at $2 . 0 \%$ , both ahead of the random agent at $1 . 6 \%$ . Despite these being top unsupervised reinforcement learning methods, they lack far behind optimal performance or even the performance of agents that learn with rewards, posing a substantial challenge for future research. The results are encouraging, showing that unsupervised objectives by themselves can lead to meaningful behaviors (Burda et al., 2018a) in Crafter. Inspecting the success rates for individual achievements in Figure 6 confirms that Plan2Explore and RND make progress in exploring the different behaviors compared to the random agent, including occasionally collecting coal, placing furnaces, and making stone swords, which are several steps deep into the technology tree.
+
+# 4.3 EMERGENT BEHAVIORS
+
+To better understand the potential of the environment, we train DreamerV2 for 50M steps and investigate the behaviors qualitatively. In this amount of time, the agent learns to build stone tools and even iron tools on individual occurrences. Interestingly, we observe a range of sophisticated emergent behaviors, such as building tunnel systems, building bridges to cross lakes, and outsmarting skeletons by dodging arrows, blocking arrows with stones, and digging through walls to surprise skeletons from the side. Furthermore, DreamerV2 learns to seek shelter to protect itself from the zombies at night by hiding in caves and even digging its own caves and closing the entrances with stones. Finally, we find that the agent sometimes manages to build plantations of many saplings, defends them against monsters, and eats the growing fruits in order to ensure a reliable and steady food supply. A video of the emergent behaviors is available on the project website.
+
+# 4.4 HUMAN EXPERTS DATASET
+
+Crafter includes a graphical user interface that allows humans to play the game via the keyboard and record the trajectories of the game. The human interface can be installed via the command shown in Figure 2. Through the human interface, we recorded the games of 5 human experts for a combined total of 100 episodes. The experts were given the instructions of the game and allowed several hours of practice. Out of the 100 episodes, 5 episodes unlock all 22 achievements. The human experts achieved a score of $5 0 . 5 \%$ , unlocking all achievements as shown in Table C.1. The achievements most difficult to humans were to collect diamonds and grow and harvest plans, with success rates of $12 \%$ and $8 \%$ , respectively. While the human dataset is separate from the Crafter benchmark, it provides an estimate of human performance and can be used for research on learning from demonstrations and imitation learning. The human dataset is available on the project website.
+
+# 5 DISCUSSION
+
+Future work We selected the difficulty of Crafter to be challenging yet not hopeless for current methods. As research progresses towards solving the challenges that are currently present, it may become necessary to extend Crafter by new enemies, resources, items, and achievements. Being written purely in Python, Crafter can easily be extended in this way. Moreover, grouping the 22 achievements into categories, such as memory, generalization, and exploration, would allow us to summarize agent abilities more abstractly (Osband et al., 2019). We did not attempt such a categorization because it is subjective and will become clearer as more researchers use the environment.
+
+Summary We introduced Crafter, a benchmark with visual inputs that evaluates a variety of general agent abilities in a single environment. We described the game mechanics, evaluation protocol, and open challenges posed by the benchmark, and performed experiments with several agents with and without rewards to provide baseline scores. Agents are evaluated based on how frequently they manage to unlock achievements that correspond to semantically meaningful milestones of behavior. We conclude that Crafter is well suited and of appropriate difficulty to guide future research on intelligent agents, both for learning from extrinsic rewards and purely from intrinsic objectives.
+
+Acknowledgements We would like to thank Oleh Rybkin, Ben Eysenbach, Sherjil Ozair, Julius Kunze, Feryal Behbahani, Timothy Lillicrap, Jimmy Ba, Nicolas Heess, Kory Mathewson, Mohammad Norouzi, Hamza Merzic, and Sergey Levine for discussions and feedback.
+
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+
+Heinrich Küttler, Nantas Nardelli, Alexander H Miller, Roberta Raileanu, Marco Selvatici, Edward Grefenstette, and Tim Rocktäschel. The nethack learning environment. arXiv preprint arXiv:2006.13760, 2020.
+
+Marlos C Machado, Marc G Bellemare, Erik Talvitie, Joel Veness, Matthew Hausknecht, and Michael Bowling. Revisiting the arcade learning environment: Evaluation protocols and open problems for general agents. Journal of Artificial Intelligence Research, 61:523–562, 2018.   
+Stephanie Milani, Nicholay Topin, Brandon Houghton, William H Guss, Sharada P Mohanty, Keisuke Nakata, Oriol Vinyals, and Noboru Sean Kuno. Retrospective analysis of the 2019 minerl competition on sample efficient reinforcement learning. In NeurIPS 2019 Competition and Demonstration Track, pp. 203–214. PMLR, 2020.   
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015.   
+Ian Osband, Yotam Doron, Matteo Hessel, John Aslanides, Eren Sezener, Andre Saraiva, Katrina McKinney, Tor Lattimore, Csaba Szepesvari, Satinder Singh, et al. Behaviour suite for reinforcement learning. arXiv preprint arXiv:1908.03568, 2019.   
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.   
+Ramanan Sekar, Oleh Rybkin, Kostas Daniilidis, Pieter Abbeel, Danijar Hafner, and Deepak Pathak. Planning to explore via self-supervised world models. arXiv preprint arXiv:2005.05960, 2020.   
+Kurt Spencer. Noise!, 2014. URL https://uniblock.tumblr.com/post/ 97868843242/noise.   
+Yuval Tassa, Yotam Doron, Alistair Muldal, Tom Erez, Yazhe Li, Diego de Las Casas, David Budden, Abbas Abdolmaleki, Josh Merel, Andrew Lefrancq, et al. Deepmind control suite. arXiv preprint arXiv:1801.00690, 2018.   
+Hado $\mathrm { \bf P }$ van Hasselt, Matteo Hessel, and John Aslanides. When to use parametric models in reinforcement learning? Advances in Neural Information Processing Systems, 32:14322–14333, 2019.
+
+# A SUCCESS RATES WITH REWARDS
+
+Table A.1: Success rates on Crafter with rewards. Success rates are computed as the fraction of episodes during which the achievement has been unlocked at least once. It is computed across all training episodes within the budget of 1M environment steps. The score is the geometric mean of success rates over all achievements, as described in Section 3.3. Note that the score is computed for each seed separately before averaging over seeds and not the other way around. Numbers within $9 5 \%$ of the best number in each row are highlighted in bold.   
+
+<table><tr><td>Achievement</td><td>Rainbow</td><td>PPO</td><td>DreamerV2</td></tr><tr><td>Collect Coal</td><td>0.0%</td><td>0.4%</td><td>14.7%</td></tr><tr><td>Collect Diamond</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Drink</td><td>24.0%</td><td>30.3%</td><td>80.0%</td></tr><tr><td>Collect Iron</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Sapling</td><td>97.4%</td><td>66.7%</td><td>86.6%</td></tr><tr><td>Collect Stone</td><td>0.2%</td><td>3.0%</td><td>42.7%</td></tr><tr><td>Collect Wood</td><td>74.9%</td><td>83.0%</td><td>92.7%</td></tr><tr><td>Defeat Skeleton</td><td>0.7%</td><td>0.2%</td><td>2.6%</td></tr><tr><td>Defeat Zombie</td><td>39.6%</td><td>2.0%</td><td>53.1%</td></tr><tr><td>Eat Cow Eat Plant</td><td>26.1%</td><td>12.0%</td><td>17.1%</td></tr><tr><td>Make Iron Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.1%</td></tr><tr><td>Make Iron Sword</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Stone Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td></td><td>0.0%</td><td>0.0%</td><td>0.2%</td></tr><tr><td>Make Stone Sword</td><td>0.0%</td><td>0.0%</td><td>0.3%</td></tr><tr><td>MakeWood Pickaxe</td><td>4.8%</td><td>21.1%</td><td>59.6%</td></tr><tr><td>MakeWood Sword</td><td>9.8%</td><td>20.1%</td><td>40.2%</td></tr><tr><td>Place Furnace</td><td>0.0%</td><td>0.1%</td><td>1.8%</td></tr><tr><td>Place Plant</td><td>94.2%</td><td>65.0%</td><td>84.4%</td></tr><tr><td>Place Stone</td><td>0.0%</td><td>1.7%</td><td>29.0%</td></tr><tr><td>Place Table Wake Up</td><td>52.3%</td><td>66.1%</td><td>85.7%</td></tr><tr><td></td><td>93.3%</td><td>92.5%</td><td>92.8%</td></tr><tr><td>Score</td><td>4.3%</td><td>4.6%</td><td>10.0%</td></tr></table>
+
+# B SUCCESS RATES WITHOUT REWARDS
+
+Table B.1: Success rates on Crafter without rewards. Success rates are computed as the fraction of episodes during which the achievement has been unlocked at least once. It is computed across all training episodes within the budget of 1M environment steps. The score is the geometric mean of success rates over all achievements, as described in Section 3.3. Note that the score is computed for each seed separately before averaging over seeds and not the other way around. Numbers within $9 5 \%$ of the best number in each row are highlighted in bold.   
+
+<table><tr><td>Achievement</td><td>Random</td><td>RND</td><td>Plan2Explore</td></tr><tr><td>Collect Coal</td><td>0.0%</td><td>0.1%</td><td>0.1%</td></tr><tr><td>Collect Diamond</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Drink</td><td>9.3%</td><td>52.1%</td><td>48.7%</td></tr><tr><td>Collect Iron</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Sapling</td><td>50.2%</td><td>34.1%</td><td>25.5%</td></tr><tr><td>Collect Stone</td><td>0.0%</td><td>0.6%</td><td>0.5%</td></tr><tr><td>Collect Wood</td><td>24.4%</td><td>49.6%</td><td>46.8%</td></tr><tr><td>Defeat Skeleton</td><td>0.0%</td><td>0.3%</td><td>0.2%</td></tr><tr><td>Defeat Zombie</td><td>0.1%</td><td>0.3%</td><td>0.2%</td></tr><tr><td>Eat Cow</td><td>0.4%</td><td>0.9%</td><td>0.7%</td></tr><tr><td>Eat Plant</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Iron Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Iron Sword</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Stone Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Stone Sword</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Wood Pickaxe</td><td>0.3%</td><td>2.5%</td><td>3.3%</td></tr><tr><td>Make Wood Sword</td><td>0.3%</td><td>2.6%</td><td>3.3%</td></tr><tr><td>Place Furnace</td><td>0.0%</td><td>0.1%</td><td>0.0%</td></tr><tr><td>Place Plant</td><td>44.6%</td><td>21.4%</td><td>14.0%</td></tr><tr><td>Place Stone</td><td>0.0%</td><td>0.4%</td><td>0.3%</td></tr><tr><td>Place Table</td><td>4.4%</td><td>16.7%</td><td>16.3%</td></tr><tr><td>Wake Up</td><td>93.6%</td><td>7.8%</td><td>47.8%</td></tr><tr><td>Score</td><td>1.6%</td><td>2.0%</td><td>2.1%</td></tr></table>
+
+C SUCCESS RATES OF HUMAN EXPERTS   
+
+<table><tr><td rowspan=1 colspan=1>Achievement</td><td rowspan=1 colspan=1>Human Experts</td></tr><tr><td rowspan=2 colspan=1>Collect CoalCollect DiamondCollect DrinkCollect IronCollect SaplingCollect StoneCollect WoodDefeat SkeletonDefeat ZombieEat CowEat PlantMake Iron PickaxeMake Iron SwordMake Stone PickaxeMake Stone SwordMake Wood PickaxeMake Wood SwordPlace FurnacePlace PlantPlace StonePlace TableWake Up</td><td rowspan=1 colspan=1>86.0%</td></tr><tr><td rowspan=1 colspan=1>12.0%92.0%53.0%67.0%100.0%100.0%31.0%84.0%89.0%8.0%26.0%22.0%78.0%78.0%100.0%45.0%32.0%24.0%90.0%100.0%73.0%</td></tr><tr><td rowspan=1 colspan=1>Score</td><td rowspan=1 colspan=1>50.5%</td></tr></table>
+
+Table C.1: Success rates of human experts on Crafter. The success rates of human experts are computed as the fraction of all 100 recorded games during which the achievement has been unlocked at least once. To compute the score analogously to the artificial agents, we randomly split the 100 games into 5 groups that are treated as the different seeds. We then follow the same procedure as for the artificial agents.
+
+# D EPISODE REWARD
+
+![](images/338f346d7a084460e94455d5a4881b2faa5b4f45790ff6f10db2e632f8cad139.jpg)  
+Figure D.1: Total episode reward with shaded standard deviation. The optimal achievable episode reward is 22. While visualizing rewards can be informative for debugging, final performance on Crafter should be reported by computing the score instead. The score takes the different difficulties of the achievements into account and is defined as the geometric mean of the success rates for all achievements, as described in Section 3.3.
+
+# E TEXTURES
+
+![](images/0d8c08cdc16878b50eb4f96d1c275fa25daf0c68d2532ac314e0d75397258e5d.jpg)  
+Figure E.1: Crafter features worlds with several materials, resources, objects, and creatures. The player can interact with these to collect resources, maintain its food and water supplies, craft pickaxes and swords, and defend itself. The textures were specifically created for Crafter.
+
+Table F.1: The action space is a flat categorical space, making Crafter easy to use. The 17 actions enable the agent to move, collect materials, place objects, craft objects, and interact with what is in front of the player. Actions whose requirements are not satisfied have no effect.   
+
+<table><tr><td>Integer</td><td>Name</td><td>Requirement</td></tr><tr><td>0</td><td>Noop</td><td>Always applicable.</td></tr><tr><td>1</td><td>Move Left</td><td>Flat ground left to the agent.</td></tr><tr><td>2</td><td>Move Right</td><td>Flat ground right to the agent.</td></tr><tr><td></td><td>Move Up</td><td>Flat ground above the agent.</td></tr><tr><td>34</td><td>Move Down</td><td>Flat ground below the agent.</td></tr><tr><td>5</td><td>Do</td><td>Facing creature or material; have necessary tool.</td></tr><tr><td>6</td><td>Sleep</td><td>Energy level is below maximum.</td></tr><tr><td>7</td><td>Place Stone</td><td>Stone in inventory.</td></tr><tr><td>8</td><td>Place Table</td><td>Wood in inventory.</td></tr><tr><td>9</td><td>Place Furnace</td><td>Stone in inventory.</td></tr><tr><td>10</td><td>Place Plant</td><td>Sapling in inventory.</td></tr><tr><td>11</td><td>MakeWood Pickaxe</td><td>Nearby table;wood in inventory.</td></tr><tr><td>12</td><td>Make Stone Pickaxe</td><td>Nearby table; wood, stone in inventory.</td></tr><tr><td>13</td><td>Make Iron Pickaxe</td><td>Nearby table, furnace; wood, coal, iron an inventory.</td></tr><tr><td>14</td><td>MakeWoodSword</td><td>Nearby table; wood in inventory.</td></tr><tr><td>15</td><td>Make Stone Sword</td><td>Nearby table; wood, stone in inventory.</td></tr><tr><td>16</td><td>Make Iron Sword</td><td>Nearby table,furnace; wood,coal, iron in inventory.</td></tr></table>
+
+# G ACHIEVEMENT CURVES OF RAINBOW
+
+![](images/d8ff547c8eaea779ee2f87193c3e3ceccae323aa5987a92db81fff15bb63877d.jpg)  
+Figure G.1: Achievement counts of Rainbow with shaded min and max.
+
+# H ACHIEVEMENT CURVES OF PPO
+
+![](images/ec05f24e779a8afca9372d913efee15f22166c1b172a81e9e31d74563be9e746.jpg)  
+Figure H.1: Achievement counts of PPO with shaded min and max.
+
+# I ACHIEVEMENT CURVES OF DREAMERV2
+
+![](images/a5b5b00cceec2d12353949146116927a8f8b38180ea23555fd0c1de881e9d931.jpg)  
+Figure I.1: Achievement counts of DreamerV2 with shaded min and max.
+
+![](images/19a5cebc1da9f12fe2227dbd43278ace8b14cb339620d39d4e8648b7b195328a.jpg)  
+Figure J.1: Achievement counts of random actions with shaded min and max.
+
+# K ACHIEVEMENT CURVES OF UNSUPERVISED RND
+
+![](images/ca65c424ae4225c58fc334aaf6505814883ad5294f8898fbc72b6b73a7467191.jpg)  
+Figure K.1: Achievement counts of unsupervised RND with shaded min and max.
+
+![](images/d172937b53f24365d2be9d54009ea862518c6b9a7a2e08359c69416a505600e9.jpg)  
+Figure L.1: Achievement counts of unsupervised Plan2Explore with shaded min and max.

parse/dev/1W0z96MFEoH/1W0z96MFEoH_content_list.json

@@ -0,0 +1,1331 @@
+[
+    {
+        "type": "text",
+        "text": "BENCHMARKING THE SPECTRUM OF AGENT CAPABILITIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            573,
+            136
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Danijar Hafner ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            157,
+            294,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Google Research, Brain Team University of Toronto mail@danijar.com ",
+        "bbox": [
+            184,
+            175,
+            382,
+            222
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            242,
+            544,
+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Evaluating the general abilities of intelligent agents requires complex simulation environments. Existing benchmarks typically evaluate only one narrow task per environment, requiring researchers to perform expensive training runs on many different environments. We introduce Crafter, an open world survival game with visual inputs that evaluates a wide range of general abilities within a single environment. Agents either learn from the provided reward signal or through intrinsic objectives and are evaluated by semantically meaningful achievements that can be unlocked during each episode, such as discovering resources and crafting tools. Consistently unlocking all achievements requires strong generalization, deep exploration, and long-term reasoning. We experimentally verify that Crafter is of appropriate difficulty to drive future research and provide baselines scores of reward agents and unsupervised agents. Furthermore, we observe sophisticated behaviors emerging from maximizing the reward signal, such as building tunnel systems, bridges, houses, and plantations. We hope that Crafter will accelerate research progress by quickly evaluating a wide spectrum of abilities. ",
+        "bbox": [
+            233,
+            271,
+            766,
+            479
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            503,
+            336,
+            518
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Crafter is an open world survival game for reinforcement learning research. Shown in Figure 1, Crafter features randomly generated 2D worlds with forests, lakes, mountains, and caves. The player needs to forage for food and water, find shelter to sleep, defend against monsters, collect materials, and build tools. The game mechanics are inspired by the popular game Minecraft and were simplified and optimized for research productivity. Crafter aims to be a fruitful benchmark for reinforcement learning by focusing on the following design goals: ",
+        "bbox": [
+            174,
+            532,
+            544,
+            671
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Research challenges Crafter poses substantial challenges to current methods. Procedural generation requires strong generalization, the technology tree evaluates wide and deep exploration, image observations calls for representation learning, repeated subtasks and sparse rewards evaluate long-term reasoning and credit assignment. ",
+        "bbox": [
+            174,
+            676,
+            544,
+            761
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Meaningful evaluation Agents are evaluated by a range of achievements that can be unlocked in each episode. The achievements correspond to meaningful milestones in behavior, offering insights into ability spectrum of both reward agents and unsupervised agents. ",
+        "bbox": [
+            174,
+            766,
+            544,
+            834
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Iteration speed Crafter evaluates many agent abilities within a single environment, vastly reducing the computational requirements over benchmarks suites that require training on many separate environments from scratch, while making it more likely that the measured performance is representative of new domains. ",
+        "bbox": [
+            174,
+            840,
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+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/927083a984075d7f618fcdcdfd9c0e2627a41223e2bbb0f4e892d8cd75b47a70.jpg",
+        "image_caption": [
+            "Figure 1: Agent view of a procedurally generated world in Crafter, showing terrain types, resources, and creatures. Agents learn from image inputs and aim to unlock a range of semantically meaningful achievements during each episode. The achievements evaluate strong generalization, wide and deep exploration, and long-term reasoning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            563,
+            536,
+            823,
+            737
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ff4aac35ce68fcec48907c7d4ec4e105662b82d357033b534ebdbc6dea74b40.jpg",
+        "image_caption": [
+            "Figure 2: Play Crafter yourself through the human interface. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            101,
+            823,
+            169
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            213,
+            344,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Benchmarks have been a driving force behind the progress and successes of reinforcement learning as a field (Bellemare et al., 2013; Brockman et al., 2016; Kempka et al., 2016; Beattie et al., 2016; Tassa et al., 2018; Juliani et al., 2018). Benchmarks often require a large amount of computational resources and yet only test a small fraction of the abilities that a general agent should master (Cobbe et al., 2020). This section directly compares Crafter to four particularly related benchmarks. ",
+        "bbox": [
+            174,
+            243,
+            825,
+            313
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Minecraft Crafter is inspired by the successful 3D video game Minecraft, which is available to researchers via Malmo (Johnson et al., 2016) and MineRL (Guss et al., 2019). Minecraft features diverse open worlds with randomly generated and modifiable terrain, as well as many different resources, tools, and monsters. However, Minecraft is too complex to be solved by current methods (Milani et al., 2020), it is unclear by what metric agents should be evaluated by, the environment is slow, and can be difficult to use because it requires Java and a window server. In comparison, Crafter captures many principles of Minecraft in a simple and fast environment, where results can be obtained in a matter of hours, and where a large number of semantically meaningful evaluation metrics are available for reinforcement learning with or without extrinsic reward. The goal of Crafter is not to replace Minecraft but progress faster towards it. ",
+        "bbox": [
+            174,
+            318,
+            825,
+            457
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Atari The Atari Learning Environment (Bellemare et al., 2013) has been the gold standard benchmark in reinforcement learning. It comprises around 54 individual games, depending on the evaluation protocol (Mnih et al., 2015; Schulman et al., 2017; Badia et al., 2020; Hafner et al., 2020). While the large number of games tests different abilities of agents, they require a large amount of computation. The recommended protocol of training the agent with 5 random seeds on each game for 200M steps requires over 2000 GPU days (Castro et al., 2018; Hessel et al., 2018). This substantially slows down experimentation and makes the complete benchmark infeasible for most academic labs. Moreover, Atari games are nearly deterministic, so agents can approximately memorize their action sequences and are not required to generalize to new situations (Machado et al., 2018). ",
+        "bbox": [
+            174,
+            462,
+            825,
+            587
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "ProcGen ProcGen (Cobbe et al., 2020) provides a benchmark that is similar to Atari but explicitly addresses the determinism present in Atari through the use of procedural generation and randomized textures. It consists of 16 games, where each episode features a randomly generated level layout. Similarly, Crafter relies on procedural generation to provide a different world map with different distribution of resources and monsters for every episode. However, ProcGen still requires training methods on 16 individual games for 200M environment steps, which each focus on a narrow aspect of an agent’s general abilities. In comparison, Crafter evaluates many different abilities of an agent by training only on a single environment for 5M steps, substantially accelerating experimentation. ",
+        "bbox": [
+            173,
+            592,
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+            704
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "NetHack NetHack (Küttler et al., 2020) is a text-based game, where the player traverses a randomly generated system of dungeons with many different items and creatures. Unlike the other discussed environments, NetHack uses symbolic inputs and thus does not evaluate an agent’s ability to learn representations of high-dimensional inputs. The game is challenging due to the large amount of knowledge required about the many different items and their effects, even for human players. As a result, NetHack requires many environment steps for agents to acquire this domain-specific knowledge; 1B steps were used in the original paper. In contrast, Crafter generates diverse complex worlds from simple underlying rules, focusing more on generalization than memorization of facts. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 CRAFTER BENCHMARK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            838,
+            401,
+            854
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We introduce Crafter, a benchmark that evaluates a variety of agent abilities in a single environment. This section describes the game mechanics of the environment, the interface of agent inputs and actions, the evaluation protocol that is based on a range of semantically meaningful achievements, and the open challenges that Crafter poses for future research. ",
+        "bbox": [
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+            867,
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/427b06e7de5d61130b9b23d25f5e328a4c7a56adcafda3ff8cce2bce7ef89914.jpg",
+        "image_caption": [
+            "Figure 3: Crafter procedurally generates a unique world for every episode that features several terrain types: grasslands, forests, lakes, mountains, caves. Memorizing action sequences is thus not a viable strategy and agents are forced to learn behaviors that generalize to new situations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 GAME MECHANICS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This section describes the game mechanics of Crafter, namely its randomly generated world maps, the levels of health and other internal quantities that the player has to maintain, the resources it can collect and objects and tools it can make from them, as well as the creatures and how they are influenced by the time of day. The images of all materials and objects are shown in Figure E.1. All randomness in the environment is uniquely determined by an integer seed that is derived from the initial seed passed to the environment and the episode number. ",
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+    {
+        "type": "text",
+        "text": "Terrain generation A unique world is generated for every episode, shown in Figure 3. The world leverages an underlying grid of $6 4 \\times 6 4$ cells but the agent only observes the world through pixel images. The terrain features grasslands, lakes, and mountains. Lakes can have shores, grasslands can have forests, and mountains can have caves, ores, and lava. These are determined by OpenSimplex noise (Spencer, 2014), a form of locally smooth noise. Within the areas determined by noise, objects appear with equal probability at any location, such as trees in forests and skeletons in caves. ",
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+    {
+        "type": "text",
+        "text": "Health and survival The player has levels of health, food, water, and rest that it must prevent from reaching zero. The levels for food, water, and rest decrease over time and are restored by drinking from a lake, chasing cows or growing fruits to eat, and sleeping in places where monsters cannot attack. Once one of the three levels reaches zero, the player starts losing health points. It can also lose health points when attacked by monsters. When the health points reach zero, the player dies. Health points regenerate over time when the player is not hungry, thirsty, or sleepy. ",
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+    {
+        "type": "text",
+        "text": "Resources and crafting There are many resources, such as saplings, wood, stone, coal, iron, and diamonds, the player can collect in its inventory and use to build tools and place objects in the world. Many of the resources require tools that the place must first build from more basic resources, leading to a technology tree with several levels. Standing nearby a table enables the player to craft wood pickaxes and swords, as well as stone pickaxes and stone swords. Crafting a furnace from stone enables crafting iron pickaxes and iron swords from both iron, coal, and wood. ",
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+    {
+        "type": "text",
+        "text": "Creatures and night Creatures are initialized in random locations and move randomly. Zombies and cows live in grasslands and are automatically spawned and despawned to ensure a given amount of creatures. At night, the agent’s view is restricted and noisy and a larger number of zombies is spawned. This makes it difficult to survive without securing a shelter, such as a cave. Skeletons live in caves and try to keep the player at a distance to shoot arrows at the player. The player can interact with creatures to decrease their health points. Cows move randomly and offer a food source. ",
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+    {
+        "type": "text",
+        "text": "3.2 ENVIRONMENT INTERFACE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This section defines the specification of the environment, explains the available actions, agent inputs, episode termination, and additional information provided by the environment. The design goal of these is to make the environment easy to use and inspect. The environment uses the Gym interface (Brockman et al., 2016) with visual agent inputs and flat categorical actions. ",
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+        "type": "image",
+        "img_path": "images/963f582bd37aa5dd9aa4ebdcb613e4e02e884f41702752f577e51296fcb57fba.jpg",
+        "image_caption": [
+            "Figure 4: The 22 achievements that can be unlocked within each episode. The arrows indicate which achievements will be completed along the way of working toward more challenging achievements. Several of the earlier tasks have to be repeated multiple times, such as collecting resources, to progress further. A reward is only given when an achievement is unlocked for the first time during the episode. "
+        ],
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+    {
+        "type": "text",
+        "text": "Observations Agent receive color images of size $6 4 \\times 6 4 \\times 3$ as their only inputs. The image shows a local top-down view of the map, reaching 4 cells west and east and 3 cells north and south of the player position. Below this view of the world, the image shows the current inventory state of the player, including its health points, food, water, and rest levels, collected materials, and crafted tools. The agent needs to learn to read its inventory state out of the image. ",
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+    {
+        "type": "text",
+        "text": "Actions The action space is a flat categorical space with 17 actions, represented by integer indices. The actions allow the player to move in all 4 directions along the grid, interact with the object in front of it, go to sleep, place objects, and make tools. Each object and tool has a separate action associated with it. Tools are kept in the inventory whereas objects are automatically placed in front of the player. If the agent does not hold the required materials for making an object or tool, the action has no effect. ",
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+    {
+        "type": "text",
+        "text": "Termination Each episode terminates when the player’s health points reach 0. This can happen when the player dies out of hunger, thirst, or tiredness, when attacked by a zombie or skeleton, or when falling into lava. Health points automatically regenerate, as long as the agent is not too hungry, thirsty, or sleepy. There is no negative reward for dying, as the reward signal already includes a penalty for losing health points. Episodes also end when reaching the time limit of 10,000 steps. ",
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+    {
+        "type": "text",
+        "text": "Additional information The environment allows access to privileged information about the world state that the agent is forbidden to observe. This includes numeric inventory counts, achievement counts, the current coordinate of the player on the grid, and a semantic grid representation of the map. These can be used for debugging purposes or for other research scenarios, such as predicting the underlying environment state to evaluate representation learning or video prediction models. ",
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+    {
+        "type": "text",
+        "text": "3.3 EVALUATION PROTOCOL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To evaluate the diverse abilities of artificial agents on Crafter, we define two benchmarks. The first benchmark allows agents to access a provided reward signal, while the second benchmark does not and requires agents to purely learn from intrinsic objectives. Besides access to the provided reward signal, the evaluation protocols are identical. An agent is granted a budget of 1M environment steps to interact with the environment. The agent performance is evaluated through success rates of the individual achievements throughout its training, as well as an aggregated score. ",
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+    {
+        "type": "text",
+        "text": "Achievements To evaluate a wide spectrum of agent abilities, Crafter defines 22 achievements. The achievements are shown in Figure 4 and correspond to semantically meaningful behaviors, such as collecting various resources, building objects and tools, finding food and water, defeating monsters, and waking up safely after sleeping. The achievements cover a wide range of difficulties, making them suitable to evaluate both weak and strong players and providing continuous feedback throughout the development process of new methods. Some achievements are independent of each other to test for breadth of exploration, while others depend on each other to test for deep exploration. ",
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+    {
+        "type": "text",
+        "text": "Reward Crafter provides a sparse reward signal that is the sum of two components. The main component is a reward of $+ 1$ every time the agent unlocks each achievement for the first time during the current episode. The second component is a reward of $- 0 . 1$ for every health point lost and a reward of $+ 0 . 1$ for every health point that is regenerated. Because the maximum number of health points is 9, the second reward component only affects the first decimal of the episode return, and ceiling the episode return yields the number of achievements unlocked during the episode. ",
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+    {
+        "type": "text",
+        "text": "Success rates The success rates offer insights into the breadth of abilities learned by an agent. The success rates are computed separately for each of the achievements, as the fraction of training episodes during which the agent has unlocked the achievement at least once. It is computed across all episodes that lead up to the budget of 1M environment steps, requiring agents to be data-efficient.1 Note that the number of environment steps is fixed but the number of episodes can differ between agents. Unlocking an achievement more than once per episode does not affect the success rate. ",
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+    {
+        "type": "text",
+        "text": "Score The score summarizes the agent abilities into a single number. It is computed by aggregating the success rates for the individual achievements. Unlocking difficult achievements, even if it happens rarely, should contribute more than increasing the success rate of achievements that are already unlocked frequently even further. To account for the range of difficulties of the achievements, we average the success rates in log-space, known as the geometric mean.2 Unlike the reward, the score thus takes the achievement’s difficulties into account, without having to know them beforehand. ",
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+    {
+        "type": "text",
+        "text": "Discussion Aggregating across tasks via a geometric mean weighs tasks based on their difficulty to the agent, resulting in higher scores for agents that explore more broadly. For example, collecting a diamond $1 \\%$ of the time instead of $0 \\%$ is a meaningful improvement, whereas collecting wood $9 5 \\%$ of the time instead of $90 \\%$ is not. This allows distinguishing how broadly agents have explored their environment even if they achieve similar rewards. The geometric mean also establishes a meaningful metric for unsupervised agents, which may get bored of tasks after performing them a few times and then move on to new tasks. A caveat of the geometric mean is that agents with rewards are evaluated by something they only indirectly optimize for, which can change their ranking order. Increasing reward and score is generally correlated, but capacity-limited agents may choose to optimize reward by mastering easy tasks and ignoring hard tasks, which only slowly increases the geometric mean. ",
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+    {
+        "type": "text",
+        "text": "3.4 RESEARCH CHALLENGES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Crafter aims to evaluate a diverse range of agent abilities within a single environment. Thus, if a method performs well on Crafter there should be a high chance that it also handles the challenges of other environments. The challenges also make Crafter suitable for evaluating progress on open research questions, such as strong generalization, wide and deep exploration, discovering reusable skills, and long-term memory and reasoning. Crafter is designed to pose the following challenges: ",
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+    {
+        "type": "text",
+        "text": "Exploration Independent achievements evaluate wide exploration, without offering a linear path for the agent to follow. Dependent achievements evaluate deep exploration of the technology tree. Collecting a diamond requires an iron pickaxe, which in turn requires a furnace, table, coal, iron, and wood. The furnace requires collecting stone, which requires building a wood pickaxe at a table. ",
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+    {
+        "type": "text",
+        "text": "Generalization Every episode is situated in a unique world that is procedurally generated. Moreover, many aspects of the game reoccur in different contexts, such as creatures and resources that can be found in different landscapes and times of day. This forces successful agents to recognize similar situations in different circumstances and be robust to changes in irrelevant details. ",
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+    {
+        "type": "image",
+        "img_path": "images/07995ac8b8af22f31c0b8ac4cad048aa54297cdbb8fd679443eed524925b0785.jpg",
+        "image_caption": [
+            "Figure 5: Crafter Benchmark Scores for various agents with and without rewards. Current top methods achieve scores of up to $10 \\%$ that are far from the $50 \\%$ of human experts, posing a substantial challenge for future research. Crafter scores are computed as the geometric mean across achievements of their success rates within the budget of 1M environment steps. Numbers in Table 1. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Reusable skills Advancing in the game requires the agent to repeat several behaviors over long horizons, such as finding food, defending against monsters, and collecting common materials that are needed many times. The behavior of a successful agent naturally decomposes into sub-tasks, making Crafter suitable for studying hierarchical reinforcement learning. ",
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+    {
+        "type": "text",
+        "text": "Credit assignment Only sparse rewards are given for unlocking an achievement for the first time during each episode. Moreover, several achievements require long-term reasoning, such as collecting the necessary resources for crafting a particular tool or planting saplings that can be harvested many hundred time steps later. This makes Crafter a challenge for temporal credit assignment. ",
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+    {
+        "type": "text",
+        "text": "Memory The agent inputs only show the player’s immediate surroundings, making Crafter partially observed. To survive for a long time, agents need to remember where to find lakes to drink and open grasslands to hunt. Moreover, to effectively find rare resources, such as iron and diamonds, the agent needs to remember what parts of the map it has already searched. ",
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+    {
+        "type": "text",
+        "text": "Representation The agent observes its environment via high-dimensional images, from which it has to extract entities that are meaningful for decision making. Similar to applications in the real world, the reward signal is sparse and the amount of environment interaction limited. As a result, successful agents will likely rely on explicit representation learning techniques. ",
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+    {
+        "type": "text",
+        "text": "Survival In previous environments, the player can often survive by doing nothing. This allows for degenerate solutions to intrinsic objectives, unlike the real world where animals are forced to adapt to survive and maintain homeostasis and allostasis. In Crafter, the player struggles to survive through the constant pressure of maintaining enough water, food, rest, and defending against zombies. ",
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+    {
+        "type": "table",
+        "img_path": "images/cac2a38412e947193c1e675af701879367ca5289b6fb8052ef700cb94d215cf3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Score (%)</td><td>Return</td></tr><tr><td>Human Experts</td><td>50.5±6.8</td><td>14.3±2.3</td></tr><tr><td>DreamerV2</td><td>10.0±1.2</td><td>9.0±1.7</td></tr><tr><td>PPO</td><td>4.6±0.3</td><td>4.2±1.2</td></tr><tr><td>Rainbow</td><td>4.3±0.2</td><td>5.0±1.3</td></tr><tr><td>Plan2Explore (Unsup)</td><td>2.1±0.1</td><td>2.1±1.5</td></tr><tr><td>RND (Unsup)</td><td>2.0±0.1</td><td>0.7±1.3</td></tr><tr><td>Random</td><td>1.6±0.0</td><td>2.1±1.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Crafter benchmark scores. The Crafter score is computed as the geometric mean of success rates for all 22 achievements available in the environment. The score prefers general agents that unlock a wide range of achievements over those that unlock a small number of achievements very frequently. For example, an agent that explores many different achievements over the course of training achieves a higher score than one that only performs same simple tasks over an over. The score thus establishes a meaningful metric both for agents with and without reward. ",
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+    {
+        "type": "image",
+        "img_path": "images/afbd1f04deffd2ccdfcebecd668b62d441d5ed684c326b9a42c003932a5d2fd6.jpg",
+        "image_caption": [
+            "Figure 6: Agent ability spectrum showing the success rates of agents with rewards. These are unlocking percentages for all 22 achievements, computed over all training episodes. Rainbow manages to drink water and forage for food. PPO additionally rarely collects coal and builds stone tools. DreamerV2 achieves these more frequently and additionally sometimes grows and eats fruits. Numbers in Appendix A. "
+        ],
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To established baselines for future work, we train various reinforcement learning methods on Crafter either with and without rewards. The two benchmarks follow the evaluation protocol in Section 3.3, which grants each agent a budget of 1M environment frames and computes the success rates of the individual achievements across all training episodes, as well as an aggregate score for the agent. Furthermore, we analyze the emergent agent behaviors qualitatively and record a dataset of human expert players to estimate the difficulty of the environment. The environment, code for the baseline agents and figures in this paper, and the human dataset are available on the project website. 3 ",
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+    {
+        "type": "text",
+        "text": "4.1 BENCHMARK WITH REWARDS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We provide baselines scores for three reinforcement learning algorithms on Crafter with rewards. DreamerV2 (Hafner et al., 2020) learns a world model and optimizes a policy through planning in latent space. We used its default hyper parameters for Atari and increased the model size. PPO (Schulman et al., 2017) is a popular method that learns to map input images to actions through policy gradients. We use a convolutional neural network policy with hyper parameters that were tuned for Atari (Hill et al., 2018). Rainbow (Hessel et al., 2018) is based on Q-Learning and combines several advances, including for exploration. The defaults for Atari did not work well, so we tuned the hyper parameters for Crafter and found a compromise between Atari defaults and the data-efficient version of the method (van Hasselt et al., 2019) to be ideal. All agents trained for 1M environment steps in under 24 hours on a single GPU and we repeated the training for 10 random seeds per method. The training reward curves are included in Appendix D. ",
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+    {
+        "type": "text",
+        "text": "The scores are listed in Table 1 and visualized in Figure 5. DreamerV2 achieves a score of $1 0 . 0 \\%$ , followed by PPO with $4 . 6 \\%$ and Rainbow of $4 . 3 \\%$ . Despite these being top reinforcement learning methods, they lack behind the score of expert human players of $5 0 . 5 \\%$ , which we describe in further detail in Section 4.3. We conclude that Crafter is a challenging benchmark, where current methods make learning progress but future research is needed to achieve high performance. For comparison, we report the episode returns in Table 1, computed over the episodes within the last $1 0 ^ { 5 }$ environment steps of training. We find a trend similar to the scores but notice that the methods are harder to tell apart, because differences on hard tasks that are rarely achieved affect the return less. Moreover, the scores are more meaningful for unsupervised agents, which should explore many achievements over time, but not necessarily remain interested in them until the end of training. The success rates for individual achievements are visualized in Figure 7, which offer insights into the breadth and depth of agent abilities. Rainbow displays high success rates on easier achievements. PPO learned to additionally make stone tools and furnaces. DreamerV2 achieved these more frequently and discovered growing and harvesting plants. None of the agents learned to collect and use iron for tools or to collect diamonds, or to achieve high success rates on many of the achievements. ",
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+    {
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+        "img_path": "images/8317bb62cc63fd883c757ada4f4b08f83669a451ab64b8b482bcd7ae2d0299cf.jpg",
+        "image_caption": [
+            "Figure 7: Agent ability spectrum showing the success rates for Crafter without rewards. Random actions unlock the 6 easiest achievements sometimes, such as drinking water and collecting wood. Plan2Explore forages for food and defeats monsters more frequently, to ensure longer survival. RND additionally collects stones and rarely even collects coal and builds furnaces. Numbers in Appendix B. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 UNSUPERVISED BENCHMARK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We provide baselines scores for two unsupervised reinforcement learning agents on Crafter without rewards. We also include a baseline that simply chooses random actions. RND (Burda et al., 2018b) is a popular exploration method that seeks out novel inputs, estimated as the prediction error of a network that aims to predict fixed random embeddings of the input images. We use its default parameters for Atari. Plan2Explore (Sekar et al., 2020) learns a world model to plan for the expected information gain of imagined trajectories, allowing it to directly seek out imagined states that have not been experienced before. We implement Plan2Explore on top of DreamerV2 and keep the same hyper parameters. We use a non-episodic value function as RND does, which helps exploration in episodic environments (Burda et al., 2018b). All agents trained for 1M environment steps in under 24 hours on a single GPU and we repeated the training for 10 random seeds per method. ",
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+    },
+    {
+        "type": "text",
+        "text": "The scores are listed in Table 1 and in Figure 5. Plan2Explore achieves a score of $2 . 1 \\%$ , followed by RND at $2 . 0 \\%$ , both ahead of the random agent at $1 . 6 \\%$ . Despite these being top unsupervised reinforcement learning methods, they lack far behind optimal performance or even the performance of agents that learn with rewards, posing a substantial challenge for future research. The results are encouraging, showing that unsupervised objectives by themselves can lead to meaningful behaviors (Burda et al., 2018a) in Crafter. Inspecting the success rates for individual achievements in Figure 6 confirms that Plan2Explore and RND make progress in exploring the different behaviors compared to the random agent, including occasionally collecting coal, placing furnaces, and making stone swords, which are several steps deep into the technology tree. ",
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+    {
+        "type": "text",
+        "text": "4.3 EMERGENT BEHAVIORS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To better understand the potential of the environment, we train DreamerV2 for 50M steps and investigate the behaviors qualitatively. In this amount of time, the agent learns to build stone tools and even iron tools on individual occurrences. Interestingly, we observe a range of sophisticated emergent behaviors, such as building tunnel systems, building bridges to cross lakes, and outsmarting skeletons by dodging arrows, blocking arrows with stones, and digging through walls to surprise skeletons from the side. Furthermore, DreamerV2 learns to seek shelter to protect itself from the zombies at night by hiding in caves and even digging its own caves and closing the entrances with stones. Finally, we find that the agent sometimes manages to build plantations of many saplings, defends them against monsters, and eats the growing fruits in order to ensure a reliable and steady food supply. A video of the emergent behaviors is available on the project website. ",
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+    {
+        "type": "text",
+        "text": "4.4 HUMAN EXPERTS DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            406,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Crafter includes a graphical user interface that allows humans to play the game via the keyboard and record the trajectories of the game. The human interface can be installed via the command shown in Figure 2. Through the human interface, we recorded the games of 5 human experts for a combined total of 100 episodes. The experts were given the instructions of the game and allowed several hours of practice. Out of the 100 episodes, 5 episodes unlock all 22 achievements. The human experts achieved a score of $5 0 . 5 \\%$ , unlocking all achievements as shown in Table C.1. The achievements most difficult to humans were to collect diamonds and grow and harvest plans, with success rates of $12 \\%$ and $8 \\%$ , respectively. While the human dataset is separate from the Crafter benchmark, it provides an estimate of human performance and can be used for research on learning from demonstrations and imitation learning. The human dataset is available on the project website. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            285,
+            310,
+            301
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Future work We selected the difficulty of Crafter to be challenging yet not hopeless for current methods. As research progresses towards solving the challenges that are currently present, it may become necessary to extend Crafter by new enemies, resources, items, and achievements. Being written purely in Python, Crafter can easily be extended in this way. Moreover, grouping the 22 achievements into categories, such as memory, generalization, and exploration, would allow us to summarize agent abilities more abstractly (Osband et al., 2019). We did not attempt such a categorization because it is subjective and will become clearer as more researchers use the environment. ",
+        "bbox": [
+            174,
+            315,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Summary We introduced Crafter, a benchmark with visual inputs that evaluates a variety of general agent abilities in a single environment. We described the game mechanics, evaluation protocol, and open challenges posed by the benchmark, and performed experiments with several agents with and without rewards to provide baseline scores. Agents are evaluated based on how frequently they manage to unlock achievements that correspond to semantically meaningful milestones of behavior. We conclude that Crafter is well suited and of appropriate difficulty to guide future research on intelligent agents, both for learning from extrinsic rewards and purely from intrinsic objectives. ",
+        "bbox": [
+            174,
+            419,
+            825,
+            517
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements We would like to thank Oleh Rybkin, Ben Eysenbach, Sherjil Ozair, Julius Kunze, Feryal Behbahani, Timothy Lillicrap, Jimmy Ba, Nicolas Heess, Kory Mathewson, Mohammad Norouzi, Hamza Merzic, and Sergey Levine for discussions and feedback. ",
+        "bbox": [
+            176,
+            521,
+            825,
+            564
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "bbox": [
+            171,
+            94,
+            826,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A SUCCESS RATES WITH REWARDS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            485,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/39adc8f0461f9d8b10d22a0ebc598033f0ed0ee116afd51834a80fdce66e32ae.jpg",
+        "table_caption": [
+            "Table A.1: Success rates on Crafter with rewards. Success rates are computed as the fraction of episodes during which the achievement has been unlocked at least once. It is computed across all training episodes within the budget of 1M environment steps. The score is the geometric mean of success rates over all achievements, as described in Section 3.3. Note that the score is computed for each seed separately before averaging over seeds and not the other way around. Numbers within $9 5 \\%$ of the best number in each row are highlighted in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Achievement</td><td>Rainbow</td><td>PPO</td><td>DreamerV2</td></tr><tr><td>Collect Coal</td><td>0.0%</td><td>0.4%</td><td>14.7%</td></tr><tr><td>Collect Diamond</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Drink</td><td>24.0%</td><td>30.3%</td><td>80.0%</td></tr><tr><td>Collect Iron</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Sapling</td><td>97.4%</td><td>66.7%</td><td>86.6%</td></tr><tr><td>Collect Stone</td><td>0.2%</td><td>3.0%</td><td>42.7%</td></tr><tr><td>Collect Wood</td><td>74.9%</td><td>83.0%</td><td>92.7%</td></tr><tr><td>Defeat Skeleton</td><td>0.7%</td><td>0.2%</td><td>2.6%</td></tr><tr><td>Defeat Zombie</td><td>39.6%</td><td>2.0%</td><td>53.1%</td></tr><tr><td>Eat Cow Eat Plant</td><td>26.1%</td><td>12.0%</td><td>17.1%</td></tr><tr><td>Make Iron Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.1%</td></tr><tr><td>Make Iron Sword</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Stone Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td></td><td>0.0%</td><td>0.0%</td><td>0.2%</td></tr><tr><td>Make Stone Sword</td><td>0.0%</td><td>0.0%</td><td>0.3%</td></tr><tr><td>MakeWood Pickaxe</td><td>4.8%</td><td>21.1%</td><td>59.6%</td></tr><tr><td>MakeWood Sword</td><td>9.8%</td><td>20.1%</td><td>40.2%</td></tr><tr><td>Place Furnace</td><td>0.0%</td><td>0.1%</td><td>1.8%</td></tr><tr><td>Place Plant</td><td>94.2%</td><td>65.0%</td><td>84.4%</td></tr><tr><td>Place Stone</td><td>0.0%</td><td>1.7%</td><td>29.0%</td></tr><tr><td>Place Table Wake Up</td><td>52.3%</td><td>66.1%</td><td>85.7%</td></tr><tr><td></td><td>93.3%</td><td>92.5%</td><td>92.8%</td></tr><tr><td>Score</td><td>4.3%</td><td>4.6%</td><td>10.0%</td></tr></table>",
+        "bbox": [
+            240,
+            171,
+            756,
+            511
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B SUCCESS RATES WITHOUT REWARDS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            519,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/9f4e10bba4c64ff08f0045550b90258a046a54ab7382f77a4aac610d41e2be54.jpg",
+        "table_caption": [
+            "Table B.1: Success rates on Crafter without rewards. Success rates are computed as the fraction of episodes during which the achievement has been unlocked at least once. It is computed across all training episodes within the budget of 1M environment steps. The score is the geometric mean of success rates over all achievements, as described in Section 3.3. Note that the score is computed for each seed separately before averaging over seeds and not the other way around. Numbers within $9 5 \\%$ of the best number in each row are highlighted in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Achievement</td><td>Random</td><td>RND</td><td>Plan2Explore</td></tr><tr><td>Collect Coal</td><td>0.0%</td><td>0.1%</td><td>0.1%</td></tr><tr><td>Collect Diamond</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Drink</td><td>9.3%</td><td>52.1%</td><td>48.7%</td></tr><tr><td>Collect Iron</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Collect Sapling</td><td>50.2%</td><td>34.1%</td><td>25.5%</td></tr><tr><td>Collect Stone</td><td>0.0%</td><td>0.6%</td><td>0.5%</td></tr><tr><td>Collect Wood</td><td>24.4%</td><td>49.6%</td><td>46.8%</td></tr><tr><td>Defeat Skeleton</td><td>0.0%</td><td>0.3%</td><td>0.2%</td></tr><tr><td>Defeat Zombie</td><td>0.1%</td><td>0.3%</td><td>0.2%</td></tr><tr><td>Eat Cow</td><td>0.4%</td><td>0.9%</td><td>0.7%</td></tr><tr><td>Eat Plant</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Iron Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Iron Sword</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Stone Pickaxe</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Stone Sword</td><td>0.0%</td><td>0.0%</td><td>0.0%</td></tr><tr><td>Make Wood Pickaxe</td><td>0.3%</td><td>2.5%</td><td>3.3%</td></tr><tr><td>Make Wood Sword</td><td>0.3%</td><td>2.6%</td><td>3.3%</td></tr><tr><td>Place Furnace</td><td>0.0%</td><td>0.1%</td><td>0.0%</td></tr><tr><td>Place Plant</td><td>44.6%</td><td>21.4%</td><td>14.0%</td></tr><tr><td>Place Stone</td><td>0.0%</td><td>0.4%</td><td>0.3%</td></tr><tr><td>Place Table</td><td>4.4%</td><td>16.7%</td><td>16.3%</td></tr><tr><td>Wake Up</td><td>93.6%</td><td>7.8%</td><td>47.8%</td></tr><tr><td>Score</td><td>1.6%</td><td>2.0%</td><td>2.1%</td></tr></table>",
+        "bbox": [
+            240,
+            171,
+            756,
+            512
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8975cb506e2dc55b8f0830746eae0fd241417e1c2bf24799ba101e9eff3e0f60.jpg",
+        "table_caption": [
+            "C SUCCESS RATES OF HUMAN EXPERTS "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Achievement</td><td rowspan=1 colspan=1>Human Experts</td></tr><tr><td rowspan=2 colspan=1>Collect CoalCollect DiamondCollect DrinkCollect IronCollect SaplingCollect StoneCollect WoodDefeat SkeletonDefeat ZombieEat CowEat PlantMake Iron PickaxeMake Iron SwordMake Stone PickaxeMake Stone SwordMake Wood PickaxeMake Wood SwordPlace FurnacePlace PlantPlace StonePlace TableWake Up</td><td rowspan=1 colspan=1>86.0%</td></tr><tr><td rowspan=1 colspan=1>12.0%92.0%53.0%67.0%100.0%100.0%31.0%84.0%89.0%8.0%26.0%22.0%78.0%78.0%100.0%45.0%32.0%24.0%90.0%100.0%73.0%</td></tr><tr><td rowspan=1 colspan=1>Score</td><td rowspan=1 colspan=1>50.5%</td></tr></table>",
+        "bbox": [
+            352,
+            143,
+            643,
+            484
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table C.1: Success rates of human experts on Crafter. The success rates of human experts are computed as the fraction of all 100 recorded games during which the achievement has been unlocked at least once. To compute the score analogously to the artificial agents, we randomly split the 100 games into 5 groups that are treated as the different seeds. We then follow the same procedure as for the artificial agents. ",
+        "bbox": [
+            173,
+            494,
+            826,
+            565
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D EPISODE REWARD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            592,
+            362,
+            607
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/338f346d7a084460e94455d5a4881b2faa5b4f45790ff6f10db2e632f8cad139.jpg",
+        "image_caption": [
+            "Figure D.1: Total episode reward with shaded standard deviation. The optimal achievable episode reward is 22. While visualizing rewards can be informative for debugging, final performance on Crafter should be reported by computing the score instead. The score takes the different difficulties of the achievements into account and is defined as the geometric mean of the success rates for all achievements, as described in Section 3.3. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            302,
+            630,
+            696,
+            803
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E TEXTURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/0d8c08cdc16878b50eb4f96d1c275fa25daf0c68d2532ac314e0d75397258e5d.jpg",
+        "image_caption": [
+            "Figure E.1: Crafter features worlds with several materials, resources, objects, and creatures. The player can interact with these to collect resources, maintain its food and water supplies, craft pickaxes and swords, and defend itself. The textures were specifically created for Crafter. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            137,
+            825,
+            420
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/3f367472e7a6ffab9760b8d979fd7542b7f21fd25b67ad0dce5f7e108ac04be1.jpg",
+        "table_caption": [
+            "Table F.1: The action space is a flat categorical space, making Crafter easy to use. The 17 actions enable the agent to move, collect materials, place objects, craft objects, and interact with what is in front of the player. Actions whose requirements are not satisfied have no effect. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Integer</td><td>Name</td><td>Requirement</td></tr><tr><td>0</td><td>Noop</td><td>Always applicable.</td></tr><tr><td>1</td><td>Move Left</td><td>Flat ground left to the agent.</td></tr><tr><td>2</td><td>Move Right</td><td>Flat ground right to the agent.</td></tr><tr><td></td><td>Move Up</td><td>Flat ground above the agent.</td></tr><tr><td>34</td><td>Move Down</td><td>Flat ground below the agent.</td></tr><tr><td>5</td><td>Do</td><td>Facing creature or material; have necessary tool.</td></tr><tr><td>6</td><td>Sleep</td><td>Energy level is below maximum.</td></tr><tr><td>7</td><td>Place Stone</td><td>Stone in inventory.</td></tr><tr><td>8</td><td>Place Table</td><td>Wood in inventory.</td></tr><tr><td>9</td><td>Place Furnace</td><td>Stone in inventory.</td></tr><tr><td>10</td><td>Place Plant</td><td>Sapling in inventory.</td></tr><tr><td>11</td><td>MakeWood Pickaxe</td><td>Nearby table;wood in inventory.</td></tr><tr><td>12</td><td>Make Stone Pickaxe</td><td>Nearby table; wood, stone in inventory.</td></tr><tr><td>13</td><td>Make Iron Pickaxe</td><td>Nearby table, furnace; wood, coal, iron an inventory.</td></tr><tr><td>14</td><td>MakeWoodSword</td><td>Nearby table; wood in inventory.</td></tr><tr><td>15</td><td>Make Stone Sword</td><td>Nearby table; wood, stone in inventory.</td></tr><tr><td>16</td><td>Make Iron Sword</td><td>Nearby table,furnace; wood,coal, iron in inventory.</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "G ACHIEVEMENT CURVES OF RAINBOW ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            526,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/d8ff547c8eaea779ee2f87193c3e3ceccae323aa5987a92db81fff15bb63877d.jpg",
+        "image_caption": [
+            "Figure G.1: Achievement counts of Rainbow with shaded min and max. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "H ACHIEVEMENT CURVES OF PPO ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            480,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/ec05f24e779a8afca9372d913efee15f22166c1b172a81e9e31d74563be9e746.jpg",
+        "image_caption": [
+            "Figure H.1: Achievement counts of PPO with shaded min and max. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "I ACHIEVEMENT CURVES OF DREAMERV2 ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/a5b5b00cceec2d12353949146116927a8f8b38180ea23555fd0c1de881e9d931.jpg",
+        "image_caption": [
+            "Figure I.1: Achievement counts of DreamerV2 with shaded min and max. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/19a5cebc1da9f12fe2227dbd43278ace8b14cb339620d39d4e8648b7b195328a.jpg",
+        "image_caption": [
+            "Figure J.1: Achievement counts of random actions with shaded min and max. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "K ACHIEVEMENT CURVES OF UNSUPERVISED RND ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/ca65c424ae4225c58fc334aaf6505814883ad5294f8898fbc72b6b73a7467191.jpg",
+        "image_caption": [
+            "Figure K.1: Achievement counts of unsupervised RND with shaded min and max. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
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+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/d172937b53f24365d2be9d54009ea862518c6b9a7a2e08359c69416a505600e9.jpg",
+        "image_caption": [
+            "Figure L.1: Achievement counts of unsupervised Plan2Explore with shaded min and max. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 20
+    }
+]

parse/dev/BIpTWmO_BY/BIpTWmO_BY_content_list.json

@@ -0,0 +1,1688 @@
+[
+    {
+        "type": "text",
+        "text": "SLEEPER AGENT: SCALABLE HIDDEN TRIGGER BACKDOORS FOR NEURAL NETWORKS TRAINED FROM SCRATCH ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As the curation of data for machine learning becomes increasingly automated, dataset tampering is a mounting threat. Backdoor attackers tamper with training data to embed a vulnerability in models that are trained on that data. This vulnerability is then activated at inference time by placing a “trigger” into the model’s input. Typical backdoor attacks insert the trigger directly into the training data, although the presence of such an attack may be visible upon inspection. In contrast, the Hidden Trigger Backdoor Attack achieves poisoning without placing a trigger into the training data at all. However, this hidden trigger attack is ineffective at poisoning neural networks trained from scratch. We develop a new hidden trigger attack, Sleeper Agent, which employs gradient matching, data selection, and target model re-training during the crafting process. Sleeper Agent is the first hidden trigger backdoor attack to be effective against neural networks trained from scratch. We demonstrate its effectiveness on ImageNet and in black-box settings. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            512,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "High-performance deep learning systems have grown in scale at a rapid pace. As a result, practitioners seek larger and larger datasets with which to train their data-hungry models. Due to the surging demand for training data along with improved accessibility via the web, the data curation process is increasingly automated. Dataset manipulation attacks exploit vulnerabilities in the curation pipeline to manipulate training data so that downstream machine learning models contain exploitable behaviors. Some attacks degrade inference across samples (Biggio et al., 2012; Fowl et al., 2021), while targeted data poisoning attacks induce a malfunction on a specific target sample (Shafahi et al., 2018; Geiping et al., 2020). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Backdoor attacks are a style of dataset manipulation that induces a model to execute the attacker’s desired behavior when its input contains a backdoor trigger (Gu et al., 2017; Bagdasaryan et al., 2020). To this end, typical backdoor attacks inject the trigger directly into training data so that models trained on this data rely on the trigger to perform inference (Gu et al., 2017; Chen et al., 2017). Such threat models for classification problems typically incorporate label flips as well. However, images poisoned under this style of attack are often easily identifiable since they belong to the incorrect class and contain a visible trigger. One line of work uses only small or realistic-looking triggers, but these may still be visible and are often placed in conspicuous image regions (Chen et al., 2017; Gu et al., 2017; Li et al., 2020). Another recent method, Hidden Trigger Backdoor Attack (HTBD), instead crafts correctly labeled poisons which do not contain the trigger at all, but this feature collision method is not effective on models trained from scratch (Saha et al., 2019; Schwarzschild et al., 2020). The task of crafting backdoor poisons which simultaneously hide the trigger and are also effective at compromising deep models remains an open and challenging problem. This is especially the case in the black-box scenario, where the attacker does not know the victim’s architecture and training routine, and in the clean-label scenario where the attacker cannot flip labels. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we develop the first hidden trigger attack that can reliably backdoor deep neural networks trained from scratch. Our threat model is illustrated in Figure 1. Our attack, Sleeper Agent, contains the following essential features: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/70312f209f976a747db4540c9abe2d4c162c1d1c40fdd8a411e48bf0f4b08231.jpg",
+        "image_caption": [
+            "Figure 1: High-level schematic of our attack. A small proportion of slightly perturbed data is added to the training set which “backdoors” the model so that it misclassifies patched images at inference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Gradient matching: our attack is based on recent advances which replace direct solvers for bi-level optimization problems with a gradient alignment objective (Geiping et al., 2020). However, we will see that the following technical additions are necessary to successfully backdoor neural networks (see Table 9).   \n• Data selection: we specifically poison images that have a high impact on training in order to maximize the attack’s effect.   \n• Adaptive retraining: while crafting poisons, we periodically retrain the surrogate models to better reflect how models respond to our poisoned data during training.   \n• Ensembles: Sleeper Agent incorporates an ensemble of distinct surrogate architectures in order to achieve transferability across models.   \n• Black-box: our method succeeds in crafting poisons on a surrogate network or ensemble, knowing nothing about the victim’s architecture and training hyperparameters. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate empirically that Sleeper Agent is effective against a variety of architectures and in the black-box scenario where the attacker does not know the victim’s architecture. The latter scenario has proved very difficult for existing methods (Schwarzschild et al., 2020), although it is more realistic. An added benefit of the gradient matching strategy is that it scales to large tasks. We demonstrate this property by backdooring models on ImageNet (Russakovsky et al., 2015). Some random clean and poisoned samples from the ImageNet dataset are shown in Figure 2. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data poisoning attacks come in many shapes and sizes. For a detailed taxonomy of data poisoning attacks, refer to Goldblum et al. (2020). Early data poisoning attacks often focused simply on degrading clean validation performance on simple models like SVMs, logistic regression models, and linear classifiers (Biggio et al., 2012; Munoz-Gonz ˜ alez et al., 2017; Steinhardt et al., 2017). ´ These methods often relied upon the learning problems being convex in order to exactly anticipate the impact of perturbations to training data. Following these early works, attacks quickly became more specialized in their scope and approach. Modern availability attacks on deep networks degrade overall performance via gradient minimization (Shen et al., 2019), easily learnable patterns (Huang et al., 2021), or adversarial noise generated by autoencoders (Feng et al., 2019). However, these works often perturb the entire training set - an unrealistic assumption for many poisoning settings. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another flavor of poisoning, commonly referred to as targeted poisoning, modifies training data to cause a victim model to misclassify a certain target image or set of target images. Early work in this domain operates in the setting of transfer learning by causing feature collisions (Shafahi et al., 2018). Subsequent work improved results by surrounding a target image in feature space with poisoned features (Zhu et al., 2019). Follow up works further improved targeted poisoning by proposing methods that are effective against from-scratch training regimes (Huang et al., 2020; Geiping et al., 2020). These attacks remain limited in scope, however, and often fail to induce misclassification on more than one target image (Geiping et al., 2020). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/75aec45a53f576b60b6142a15657c76f25add464aa1ddb792aa618ad209a651c.jpg",
+        "image_caption": [
+            "Figure 2: Sample clean source (first column), patched source (second column), clean target (third column), and poisoned target (fourth column) from the ImageNet dataset. The last column is slightly perturbed, but the perturbed and corresponding clean images are hardly distinguishable by the human eye. More visualizations can be found in the Appendix B. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Adjacent to targeted data poisoning are backdoor attacks. Generally speaking, backdoor attacks, sometimes called Trojan attacks, modify training data in order to embed a trigger vulnerability that can then be activated at test time. Crucially, this attack requires the attacker to modify data at inference time. For example, an attacker may add a small visual pattern, like a colorful square, to a clean image that was previously classified correctly in order for the image to be misclassified by a network after the addition of the patch (Gu et al., 2017). However, these works can require training labels to be flipped, and/or a conspicuous patch to be added to training data. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Of particular relevance to this work is a subset of backdoor attacks that are clean label, meaning that modifications to training data must not change the semantic label of that data. This is especially important because an attacker may not control the labeling method of the victim and therefore cannot rely upon techniques like label flipping in order to induce poisoning. One previous work enforces this criterion by applying patches to adversarial examples, but the patches are clearly visible, even when they are not fully opaque, and the attack fails when patches are transparent enough to be unnoticeable (Turner et al., 2019; Schwarzschild et al., 2020). Another work, “Hidden Trigger Backdoor Attacks” enforces an $\\ell _ { \\infty }$ constraint on the entire perturbation (as is common in the adversarial attack literature), but this method is only effective on hand selected class pairs and only works in transfer learning scenarios where the pretrained victim model is both fixed and known to the attacker (Saha et al., 2019; Schwarzschild et al., 2020). Another clean label backdoor attack hides the trigger in training data via steganography (Li et al., 2019), however this attack also assumes access to the pretrained model that a victim will use to fine tune on poisoned data. Moreover, the latter attack uses triggers that cover the entire image, and these triggers cannot be chosen by the user. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In contrast to these existing methods, Sleeper Agent does not require knowledge of the victim model, the perturbations are not visible in poisoned training data, and poisons can be adapted to any patch. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 THREAT MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We follow commonly used threat models used in the backdoor literature (Gu et al., 2017; Saha et al., 2019). We define two parties, the attacker and the victim. We assume that the attacker perturbs and disseminates data. As in Saha et al. (2019); Geiping et al. (2020), we assume the training data modifications are bounded in $\\ell _ { \\infty }$ norm. The victim then trains a model on data - a portion of which has been perturbed by the attacker. Once the victim’s model is trained and deployed, we also assume that the attacker can then apply a patch to select images at test time to trigger the backdoor attack. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, we diverge from Gu et al. (2017); Saha et al. (2019) in our assumptions about the knowledge of the victim. We assume a far more strict threat model wherein the attacker does not have access to the parameters, architecture, or learning procedure of the victim. This represents a realistic scenario wherein a victim trains a randomly initialized deep network from scratch on scraped data. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 PROBLEM SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Formally, we aim to craft perturbations $\\delta = \\{ \\delta _ { i } \\} _ { i = 1 } ^ { N }$ to training data $\\mathcal { T } = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ for a loss function, $\\mathcal { L }$ , and a surrogate network, $F$ , with parameters $\\theta$ that solve the following bilevel problem: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d053972b8ca417895aed2f2091b2f9cddcbd68a022b906cd596170cbcdffbd75.jpg",
+        "text": "$$\n\\begin{array} { r } { \\underset { \\delta \\in \\mathcal { C } } { \\operatorname* { m i n } } \\ \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } } \\bigg [ \\mathcal { L } \\left( F ( x + p ; \\theta ( \\delta ) ) , y _ { t } \\right) \\bigg ] } \\\\ { \\mathrm { s . t . } \\ \\theta ( \\delta ) \\in \\underset { \\theta } { \\arg \\operatorname* { m i n } } \\displaystyle \\sum _ { ( x _ { i } , y _ { i } ) \\in \\mathcal { T } } \\mathcal { L } ( F ( x _ { i } + \\delta _ { i } ; \\theta ) , y _ { i } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $p$ denotes the trigger, $y _ { t }$ denotes the intended target label of the attacker, and $\\mathcal { C }$ denotes a set of constraints on the perturbations. Naive backdoor attacks often solve this bilevel problem by inserting $p$ directly into training data (belonging to class $y _ { t }$ ) so that the network learns to associate the trigger pattern with the desired class label. However, our threat model is more strict, which is reflected in our constraints on $\\delta$ . We require that $\\delta$ is bounded in $\\ell _ { \\infty }$ norm and that $\\delta _ { i } = \\mathbf { 0 }$ for all but a small fraction of indices, $i$ . WLOG, assume that the first $M \\leq N$ perturbations are allowed to be nonzero. In the black-box scenario, the surrogate model, $F$ , may not resemble the victim, in terms of either architecture or training hyperparameters, and yet the attack is effective nonetheless. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We stress that unlike Saha et al. (2019), our primary area of interest is not transfer learning, but rather from-scratch training. This threat model results in a more complex optimization procedure - one where simpler objectives, like feature collision, have failed (Schwarzschild et al., 2020). Due to the inner optimization problem posed in Equation 2, directly computing optimal perturbations is intractable for deep networks as it would require differentiating through the training procedure of $F$ . Thus, heuristics must be used to optimize the poisons. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 OUR APPROACH",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recently, several works have proposed solving bilevel problems for deep networks by utilizing gradient alignment. Gradient alignment modifies training data to align the training gradient with the gradient of some desired objective. It has proven useful for dataset condensation (Zhao et al., 2020), as well as integrity and availability poisoning attacks (Geiping et al., 2020; Fowl et al., 2021). Unlike other heuristics like partial unrolling of the computation graph or feature collision, gradient alignment has proven to be a stable way to solve a bilevel problem that involves training a deep network in the inner objective. However, poisoning approaches utilizing gradient alignment have often come with limitations, such as poor performance on multiple target images (Geiping et al., 2020), or strict requirements about poisoning an entire dataset (Fowl et al., 2021). ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In contrast, we study the behaviour of a class of attacks capable of causing misclassification of a large proportion of unseen patched images of a selected class, all while modifying only a small fraction of training data. We first define the adversarial objective: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6077e6c5d0398d92411af95da71dee8ba10348c7df87eee8e1179ea6b998eed.jpg",
+        "text": "$$\n\\mathcal { L } _ { a d v } = \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } _ { s } } \\bigg [ \\mathcal { L } \\big ( F ( x + p ; \\theta ) , y _ { t } \\big ) \\bigg ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            188
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { s }$ denotes the source class distribution, $p$ is a patch that the attacker uses to trigger misclassification at test-time, and $y _ { t }$ is the intended target label. This objective is minimized when an image becomes misclassified into a desired class after the attacker’s patch is added to it. For example, an attacker may aim for a network to classify images of dogs correctly but to misclassify the same dog images as cats when a patch is added to the dog images. ",
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+    {
+        "type": "text",
+        "text": "To achieve this behavior, we perturb training data by optimizing the following alignment objective: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67e86fa4c63c66958686178e5672da5a94f59a9a12e3c4664f0077e747c3cbcb.jpg",
+        "text": "$$\n\\mathcal { A } = 1 - \\frac { \\nabla _ { \\theta } \\mathcal { L } _ { t r a i n } \\cdot \\nabla _ { \\theta } \\mathcal { L } _ { a d v } } { \\left| \\left| \\nabla _ { \\theta } \\mathcal { L } _ { t r a i n } \\right| \\right| \\cdot \\left| \\left| \\nabla _ { \\theta } \\mathcal { L } _ { a d v } \\right| \\right| } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/54f16d2d4d6a2d5606837650e2f0e9cf71b10b2ac174eafb482d5aca55ce2976.jpg",
+        "text": "$$\n\\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } _ { t r a i n } = \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } \\nabla _ { \\boldsymbol { \\theta } } \\mathcal { L } \\big ( F ( x _ { i } + \\delta _ { i } ; \\boldsymbol { \\theta } ) , y _ { i } \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "is the training gradient involving the nonzero perturbations. We then estimate the expectation in Equation 3 by calculating the average adversarial loss over $K$ training points from the source class: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4384aa14b9980f43388b03d921e3553294b730c3273314c5103c7b8295a62a35.jpg",
+        "text": "$$\n\\nabla _ { \\theta } \\mathcal { L } _ { a d v } = \\frac { 1 } { K } \\sum _ { ( x , y _ { s } ) \\in \\mathcal { T } } \\nabla _ { \\theta } \\bigg ( \\mathcal { L } \\big ( F ( x + p ; \\theta ) , y _ { t } \\big ) \\bigg )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In our most basic attack, we begin optimizing the objective in Equation 4 by fixing a parameter vector $\\theta ^ { * }$ in order to calculate $\\mathcal { A }$ . This parameter vector is trained on clean data and is used to calculate the training and adversarial gradients. We then optimize using 250 steps of signed Adam. Note that while this is not a general constraint for our method, we follow the setup in Saha et al. (2019) where all poisons are drawn from a single target class. That is to say, the $M$ poisons the attacker is allowed to perturb have the form $\\{ ( x _ { i } , y _ { t } ) \\} _ { i = 1 } ^ { \\overline { { M } } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also employ differentiable data augmentation which has shown to improve stability of poisons in Geiping et al. (2020). While gradient alignment proves more successful than other approaches to the bilevel problem, we additionally introduce two novel techniques that boost success by $> 2 5 0 \\%$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Poison Selection: Our threat model assumes the attacker disseminates perturbed images online through avenues such as social media. With this in mind, the attacker can choose which images to perturb. For example, the attacker could choose images of dogs in which to “hide” the trigger. While random selection with our objective does successfully poison victims trained from scratch, we experiment with selection by gradient norm. Because we aim to align the training gradient with our adversarial objective, source images which have larger gradients could prove to be more potent poisons. We find that choosing source poison images by taking images with the maximum training gradient norm at the parameter vector $\\theta ^ { * }$ noticeably improves poison performance (see Tables 3, 9). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Model Retraining: In the most straightforward version of our attack, the attacker optimizes the perturbations using fixed model parameters for a number of steps (usually 250). However, this may lead to perturbations overfitting to a clean-trained model; during a real attack a model is trained on poisoned data, but we optimize the poisons on a model that trained only with clean data. To close the gap, we introduce model retraining during the poison crafting procedure. After retraining our model on the perturbed data, we again take optimization steps on the perturbations, but this time evaluating the training and adversarial losses at the new parameter vector. We repeat this process of retraining/optimizing several times and find that this noticeably improves the success of the poisons - often boosting success by more than $2 0 \\%$ (see Tables 3, 9). ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we empirically test the proposed Sleeper Agent backdoor attack on multiple datasets, against black-box settings, using a benchmark, and against popular defenses. ",
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+    {
+        "type": "table",
+        "img_path": "images/5455e4382fa10d63e00bda5b2abf80eec6d444e16bb2fb128e99e6fdd869263f.jpg",
+        "table_caption": [
+            "Table 1: Baseline evaluations on CIFAR-10. Perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and poison budget is $1 \\%$ of training images. Each number denotes an average (and std. error) over 24 crafting and training runs along with randomly sampled source/target class pairs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>ResNet-18</td><td>MobileNetV2</td><td>VGG11</td></tr><tr><td>Clean validation accuracy(%)</td><td>92.31 (±0.08)</td><td>88.19 (±0.05)</td><td>89.00 (±0.03)</td></tr><tr><td>Poison validation accuracy(%)</td><td>92.16 (±0.05)</td><td>88.03 (±0.05)</td><td>88.70 (±0.04)</td></tr><tr><td>Clean source accuracy(%)</td><td>92.36 (±0.93)</td><td>88.55 (±1.64)</td><td>90.62 (±1.23)</td></tr><tr><td>Poison source accuracy(%)</td><td>91.50 (±0.88)</td><td>87.79 (±1.60)</td><td>89.45 (±1.19)</td></tr><tr><td>Triggered source accuracy(%)</td><td>12.96 (±5.40)</td><td>21.09 (±5.41)</td><td>17.97 (±4.00)</td></tr><tr><td>Attack Success Rate(%)</td><td>85.27 (±5.90)</td><td>72.92 (±6.09)</td><td>75.15 (±5.40)</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 2: The effect of poison budget. Experiments on CIFAR-10 with ResNet-18 models (He et al., 2016). Perturbations have $\\ell _ { \\infty } \\mathrm { - n o r m } \\le 1 6 / 2 5 5$ . Each number denotes an average (and std. error) over 32 crafting and training runs along with randomly sampled source/target class pairs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Poison Budget</td><td>50 (0.1%)</td><td>100 (0.2%)</td><td>250 (0.5%)</td><td>400 (0.6%)</td><td>500 (1%)</td></tr><tr><td>Clean validation accuracy(%)</td><td>92.34 (±0.05)</td><td>92.36 (±0.04)</td><td>92.31 (±0.04)</td><td>92.15 (±0.08)</td><td>92.26 (±0.06)</td></tr><tr><td>Poison validation accuracy(%)</td><td>92.33 (±0.04)</td><td>92.34 (±0.05)</td><td>92.25 (±0.04)</td><td>92.12 (±0.06)</td><td>92.17 (±0.04)</td></tr><tr><td>Clean source accuracy(%)</td><td>93.01 (±0.69)</td><td>91.08 (±0.85)</td><td>92.43 (±0.74)</td><td>92.42 (±0.80)</td><td>92.14 (±0.78)</td></tr><tr><td>Poison source accuracy(%)</td><td>93.03 (±0.67)</td><td>90.61 (±0.86)</td><td>91.83 （±0.75)</td><td>91.88 (±0.79)</td><td>91.56 (±0.77)</td></tr><tr><td>Triggered source accuracy(%)</td><td>61.04 (±4.27)</td><td>40.07 (±5.72)</td><td>22.77 （±4.77)</td><td>15.88 (±4.91)</td><td>13.07 (±4.57)</td></tr><tr><td>Attack Success Rate(%)</td><td>24.71 (±4.10)</td><td>49.76 (±6.21)</td><td>72.48 (±5.24)</td><td>81.44 (±5.25)</td><td>85.11 (±5.04)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 BASELINE EVALUATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Typically, backdoor attacks are considered successful if poisoned models do not suffer from a significant drop in validation accuracy on images without triggers, but they reliably misclassify images from the source class into the target class when a trigger is applied. We begin by testing our method in the gray-box setting. In the gray-box setting, we use the same architecture but different random initialization for crafting poisons and testing. Table 1 depicts the performance of Sleeper Agent on CIFAR-10 when perturbing $1 \\%$ of images in the training set with each perturbation constrained in an $\\ell _ { \\infty }$ -norm ball of radius 16/255. During poison crafting, the surrogate model undergoes four evenly spaced retraining periods $T = 4$ ), and we test the effectiveness of each surrogate model architecture at generating poisons for victim models of the same architecture. In subsequent sections, we will extend these experiments to the black-box setting and to an ensemblized attacker. We observe in these experiments that the poisoned models indeed achieve very similar validation accuracy to their clean counterparts, yet the application of triggers to source class images causes them to be misclassified into the target class as desired. In Table 2, we observe that Sleeper Agent can even be effective when the attacker is only able to poison a very small percentage of the training set. Note that the success of backdoor attacks depends greatly on the choice of source and target classes, especially since some classes contain very large objects which may dominate the image, even when a trigger is inserted. As a result, the variance of attack performance is high since we sample class pairs randomly. The poisoning and victim hyperparameters we use for our experiments can be found in Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The benefits of ensembling: One simple way we can improve the transferability of our backdoor attack across initializations of the same architecture is to craft our poisons on an ensemble of multiple copies of the same architecture but trained using different initializations and different batch sampling during their training procedures. In Table 3, we observe that this ensembling strategy indeed can offer major performance boosts, both with and without retraining. ",
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+    {
+        "type": "text",
+        "text": "The black-box setting: Now that we have established the transferability of Sleeper Agent across models of the same architecture, we test on the hard black-box scenario where the victim’s architecture is completely unknown to the attacker. This setting has proven extremely challenging for existing methods (Schwarzschild et al., 2020). Table 4 contains four settings. In the first row, we simply craft the poisons on a single ResNet-18 and transfer these to other models. Second, we craft poisons on an ensemble consisting of two MobileNet-V2 and two ResNet-34 architectures and transfer to the remaining models. Third, for each architecture, we craft poisons with an ensemble consisting of the other two architectures and test on the remaining one. The second and third scenarios are ensemblized black-box attacks, and we see that Sleeper Agent is effective. In the last row, we perform the same experiment but with the testing model included in the ensemble, and we observe that a single ensemble can craft poisons that are extremely effective on a range of architectures. We choose ResNet-18, MobileNet-V2, and VGG11 as these are common and contain a wide array of structural diversity (He et al., 2016; Sandler et al., 2018; Simonyan & Zisserman, 2014). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/32cae721a4baef29787c923b91b8b83c44364eb813d0093387924577fe8d75e0.jpg",
+        "table_caption": [
+            "Table 3: Ensembles consisting of copies of the same architecture (ResNet-18). $S$ denotes the size of the ensemble, and $T$ denotes the retraining factor. Experiments are conducted on CIFAR-10, perturbations have $\\ell _ { \\infty }$ -norm bounded by 16/255, and the attacker can poison $1 \\%$ of training images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Clean validation (%)</td><td>Poison validation (%)</td><td>Attack Success Rate (%)</td></tr><tr><td>Sleeper Agent (S = 1,T = 0)</td><td>92.36 (±0.05)</td><td>92.08 （±0.08)</td><td>63.49 (±6.13)</td></tr><tr><td>Sleeper Agent (S= 2,T=0)</td><td>92.10 (±0.04)</td><td>92.12 (±0.06)</td><td>64.70 (±5.65)</td></tr><tr><td>Sleeper Agent (S = 4,T= 0)</td><td>92.14 (±0.03)</td><td>91.98(±0.05)</td><td>74.81 (±4.10)</td></tr><tr><td>Sleeper Agent (S=2,T= 4)</td><td>92.11 (±0.07)</td><td>92.08 (±0.13)</td><td>87.40(±6.23)</td></tr><tr><td>Sleeper Agent (S=4,T=4)</td><td>92.17 (±0.03)</td><td>91.81 (±0.06)</td><td>88.45 (±6.00)</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3dfc72c979d5a17517f6978c22c0ea8381c9867670aa1bbc39c6e0edceab6024.jpg",
+        "table_caption": [
+            "Table 4: Black-box attacks: First row: Attacks crafted on a single ResNet-18 and transferred. Second row: attacks crafted on MobileNet-V2 and ResNet-34 and transfered. Third row: attacks crafted on the remaining architectures excluding the victim. The ensemble used in the last row includes the victim architecture. Experiments are conducted on CIFAR-10 and perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and the attacker can poison $1 \\%$ of training images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>ResNet-18</td><td>MobileNet-V2</td><td>VGG11</td><td>Average</td></tr><tr><td>Sleeper Agent (S=1,T=4,ResNet-18)</td><td></td><td>29.10%</td><td>31.96%</td><td>29.86%</td></tr><tr><td>Sleeper Agent (S = 4,T= 0,MobileNet-V2,ResNet-34)</td><td>70.30%</td><td>一</td><td>46.48%</td><td>58.44%</td></tr><tr><td>Sleeper Agent (S= 4,T= 0, victim excluded)</td><td>63.11%</td><td>42.40%</td><td>55.28%</td><td>53.60%</td></tr><tr><td>Sleeper Agent (S= 6,T= O,victim included)</td><td>68.46%</td><td>67.28%</td><td>85.37%</td><td>73.30%</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ImageNet evaluations: In addition to CIFAR-10, we perform experiments on ImageNet. Table 5 contains the performance of Sleeper Agent on ImageNet where attacks are crafted and tested on randomly initialized ResNet-18 models. Perturbations are constrained in an $\\ell _ { \\infty }$ -norm ball of radius 16/255 - a bound seen in prior poisoning works on ImageNet (Fowl et al., 2021; Geiping et al., 2020; Saha et al., 2019). We first study the effect of re-training during poison crafting. Even performing only two equally spaced re-training periods improves the success rate significantly. Additionally, we observe that our data selection technique allows Sleeper Agent to maintain a high success rate even with a lower poison budget. Figure 2 contains visualizations of the patched sources and the crafted targets. The poisoning and victim hyperparameters from experiments can be found in Appendix A. Further visualizations and additional experiments are presented in Appendices B and C. ",
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+    {
+        "type": "text",
+        "text": "4.2 COMPARISON TO OTHER METHODS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There are several existing clean-label hidden-trigger backdoor attacks that claim success in settings different than ours. In order to further demonstrate the success of our method, we compare our poisons to ones generated from these methods in our more strict threat model of from-scratch training. In these experiments, poisons are generated from our attack, clean label backdoor, and hidden trigger backdoor. All poison trials have the same randomly selected source-target class pairs, the same budget, and the same $\\varepsilon$ -bound (Note: clean-label backdoor originally did not use $\\ell _ { \\infty }$ bounds, so we adjust the opacity of their perturbations to ensure the constraint is satisfied). We then train a randomly initialized network from scratch on these poisons and evaluate success over 1000 patched target images. We test three popular network architectures and find that our attack significantly outperforms both methods and is the only backdoor method to exceed single digit success rates, confirming the findings of Schwarzschild et al. (2020) on the fragility of these existing methods. See Table 6 for full results. Note that the difference in results between Table 1 and these results may arise from saving the poisoned images and loading them into this benchmark setup. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f4530d37742e0778c4bd6cc5176582caa160cdc504eb3102c5b986666f734127.jpg",
+        "table_caption": [
+            "Table 5: ImageNet evaluations. Attacks are conducted on ResNet-18 models and perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255. The high standard errors are due to the high variance of the sampling of source/target pairs, and limited number of runs to maintain computational feasibility. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Poison budget</td><td>Clean validation (%)</td><td>Poison validation (%)</td><td>Attack Success Rate (%)</td></tr><tr><td>Sleeper Agent (S=1,T=0)</td><td>0.05%</td><td>69.27 (±0.03)</td><td>67.87 (±0.03)</td><td>22.00 (±5.65)</td></tr><tr><td>Sleeper Agent (S=1,T=0)</td><td>0.10%</td><td>69.23 (±0.03)</td><td>67.80 (±0.04)</td><td>23.25 （±5.50)</td></tr><tr><td>Sleeper Agent (S=1,T=2)</td><td>0.05%</td><td>69.21 (±0.04)</td><td>67.84 (±0.10)</td><td>44.00(±6.73)</td></tr><tr><td>Sleeper Agent (S=1,T=2)</td><td>0.10%</td><td>69.14 (±0.03)</td><td>67.75 (±0.08)</td><td>41.00 (±14.45)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/9dd673a9b00ffc103fdffaff054c37c2e6b177e655a674bae75dba72cb762faf.jpg",
+        "table_caption": [
+            "Table 6: Benchmark results on CIFAR-10. Comparison of our method to popular “clean-label” attacks. Results averaged over the same source/target pairs with $\\epsilon = 1 6 / 2 5 5$ and poison budget $1 \\%$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>ResNet-18</td><td>MobileNetV2</td><td>VGG11</td><td>Average</td></tr><tr><td>Hidden-Trigger Backdoor (Saha et al.,2019)</td><td>3.50%</td><td>3.76%</td><td>5.02%</td><td>4.09%</td></tr><tr><td>Clean-Label Backdoor (Turner et al., 2019)</td><td>2.78%</td><td>3.50%</td><td>4.70%</td><td>3.66%</td></tr><tr><td>Sleeper Agent (Ours)</td><td>50.72%</td><td>58.21%</td><td>57.86%</td><td>55.59%</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 DEFENSES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A selling point for hidden trigger backdoor attacks is that the trigger that is used to induce misclassification at test-time is not present in any training data, thus making inspection based defenses, or automated pattern matching more difficult. However, there exist numerous defenses, aside from visual inspection, that have been proposed to mitigate the effects of poisoning - both backdoor and other attacks. We test our method against a number of popular defenses. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Spectral Signatures: This defense, proposed in Tran et al. (2018), aims to filter a pre-selected amount of training data based upon correlations with singular vectors of the feature covariance matrix. This defense was originally intended to detect triggers used in backdoor attacks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Activation Clustering: Chen et al. (2018) cluster activation patterns to detect anomalous inputs.   \nUnlike the spectral signatures defense, this defense does not filter a pre-selected volume of data. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "DPSGD: Poison defenses based on differentially private SGD (Abadi et al., 2016) have also been proposed (Hong et al., 2020). Differentially private learning inures models to small changes in training data, which provably imbues robustness to poisoned data. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Data Augmentations: Recent work has suggested that strong data augmentations, such as mixup, break data poisoning (Borgnia et al., 2021). This has been confirmed in recent benchmark tests which demonstrate many poisoning techniques are brittle to slight changes in victim training routine (Schwarzschild et al., 2020). We test against mixup augmentation (Zhang et al., 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "STRIP: Gao et al. (2019) propose to add strong perturbations by superimposing input images at test time to detect the backdoored inputs based on the entropy of the predicted class distribution. If the entropy is lower than a predefined threshold, the input is considered backdoored and is rejected. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "NeuralCleanse: Wang et al. (2019) propose a defense designed for traditional backdoor attacks by reconstructing the maximally adversarial trigger used to backdoor a model. While this defense was not designed for hidden trigger backdoor attacks, we experiment with this as a detection defense wherein we test whether NeuralCleanse can detect the backdoored class. This modification is denoted by NeuralCleanse\\*. In our trials, NeuralCleanse\\* does not detect any of the backdoored classes - as determined by taking the maximum mask MAD (see Wang et al. (2019)). ",
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+    {
+        "type": "text",
+        "text": "We find that across the board, all of these defenses exhibit a robustness-accuracy trade-off. Many of these defenses do not reliably nullify the attack, and defenses that do degrade attack success also induce such a large drop in validation accuracy that they are unattractive options for practitioners. For example, to lower the attack success to an average of $1 3 . 1 4 \\%$ , training with DPSGD degrades natural accuracy on CIFAR-10 to $7 0 \\%$ . See Table 7 for the complete results of these experiments. ",
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+    {
+        "type": "text",
+        "text": "4.4 SLEEPER AGENT CAN POISON IMAGES IN ANY CLASS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Typical backdoor attacks which rely on label flips or feature collisions can only function when poisons come from the source and/or target classes (Saha et al., 2019; Turner et al., 2019). This restriction may be a serious limitation in practice. In contrast, we show that Sleeper Agent can be ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b04eb8eaf8a1671075ab744e37a4e7e85c5aceb14af5eb57dd5346a9216339d7.jpg",
+        "table_caption": [
+            "Table 7: Defenses. Experiments are conducted on CIFAR-10 with ResNet-18 models, perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and poison budget is $1 \\%$ of training images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Defense</td><td>Attack Success Rate (%)</td><td>Clean Validation Accuracy (%)</td></tr><tr><td>Spectral Signatures</td><td>37.17 (±10.10)</td><td>89.94 (±0.19)</td></tr><tr><td>Activation Clustering</td><td>15.17 (±5.38)</td><td>72.38 (±0.48)</td></tr><tr><td>DPSGD</td><td>13.14 (±4.49)</td><td>70.00 (±0.17)</td></tr><tr><td>Data Augmentation</td><td>69.75 (±10.77)</td><td>91.32 (±0.12)</td></tr><tr><td>STRIP</td><td>62.68 (±4.90)</td><td>92.23 (±0.05)</td></tr><tr><td>NeuralCleanse*</td><td>53.20 (±10.49)</td><td>91.92 (±0.12)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "effective even when we poison images drawn from all classes. To take advantage of our data selection strategy, we select poisons with maximum gradient norm across all classes. Table 8 contains the performance of Sleeper Agent in the aforementioned setting. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 8: Random poisons. Experiments are conducted on CIFAR-10 with ResNet-18 models. Perturbations have $\\ell _ { \\infty }$ -norm bounded above by $1 6 / 2 5 5$ and poisons are drawn from all classes. Each number denotes an average (and standard error) over 16 independent crafting and training runs along with randomly sampled source/target class pairs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Poison budget</td><td>Attack Success Rate (%)</td></tr><tr><td>Sleeper Agent (S = 1,T= 4)</td><td>1%</td><td>41.90 (±7.16)</td></tr><tr><td>Sleeper Agent (S=1,T= 4)</td><td>3%</td><td>66.51 (±6.90)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.5 ABLATION STUDIES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Here we analyze the importance of each technique in our algorithm via ablation studies. We focus on three aspects of our method: 1) patch location, 2) retraining during poison crafting, 3) poison selection. Table 9 details several combinations and their effects on poison success. We find that randomizing patch location improves poisoning success, and both retraining and data selection based on maximum gradient significantly improve poison performance. Combining all three boosts poison success more than four-fold. See Section 3.3 for a description of these techniques. Additional experiments and more ablation studies can be found in the Appendix C. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0627ef1cfea2ffc92ba698190f901d395c9988f3911c76caebe044d99f9e2441.jpg",
+        "table_caption": [
+            "Table 9: Ablation studies. Investigation the effects of random patch-location, retraining, and data selection. Experiments are conducted on CIFAR-10 with ResNet-18 models, perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and poison budget is $1 \\%$ of training images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack Setup</td><td>Attack Success Rate (%)</td></tr><tr><td>Fix patch-location (bottom-right corner)</td><td>19.25 (±3.01)</td></tr><tr><td>Random patch-location</td><td>33.95 (±4.57)</td></tr><tr><td>Randompatch-location+retraining</td><td>59.42 (±5.78)</td></tr><tr><td>Randompatch-location+ data selection</td><td>63.49 (±6.13)</td></tr><tr><td>Random patch-location + retraining + data selection</td><td>85.27 (±5.90)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we develop the first hidden-trigger backdoor attack that is effective against deep networks trained from scratch. This is a challenging setting for backdoor attacks, and existing attacks typically operate in less strict settings. Nonetheless, we choose the strict setting because practitioners often train networks from scratch in real-world applications, and patched poisons may be easily visible upon human inspection. In order to accomplish the above goal, we use a gradient matching objective as a surrogate for the bilevel optimization problem, and we add features such as re-training and data selection in order to significantly enhance the performance of our method, Sleeper Agent. ",
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+    },
+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our full implementation and instructions needed to reproduce the experimental results are included in the supplementary materials, and we explain the training details, models, hyperparameters, and computational resources in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "ETHICS STATEMENT ",
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+        "page_idx": 9
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+        "type": "text",
+        "text": "In this work, we illuminate a new scalable backdoor attack that could be used to stealthily compromise security-critical systems. We hope that by highlighting the potential danger of this nefarious threat model, our work will give rise to stronger defenses and will encourage caution on the part of practitioners. ",
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+        "text": "APPENDIX ",
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+        "text": "A IMPLEMENTATION DETAILS ",
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+    {
+        "type": "text",
+        "text": "The most challenging setting for evaluating a backdoor attack involves training a model from scratch. It is crucial to compute the average attack success rate on all patched source images in the validation set to evaluate effectiveness reliably. Following the discussion above, for all experiments, we select random source-target pairs. During training, we add our patch to all images from the source class in the training set. To compute the attack success rate, followed by Geiping et al. (2020), we measure the average rate at which patched source images are successfully classified as the target class. To be consistent and to provide a fair comparison to Saha et al. (2019), we use a random patch selected from Saha et al. (2019). Our choice of patch size in the baseline experiments is the same as Saha et al. (2019), which is, $8 \\times 8$ for CIFAR-10 $6 . 2 5 \\%$ of the pixels) and $3 0 \\times 3 0$ for the ImageNet $( 1 . 7 9 \\%$ of the pixels). Figure 3 (right) shows the patch we utilize in all of our experiments. Note that the choice of the patch in our implementation is not essential. To show this, we conduct the same baseline evaluation discussed in 4.1 using a random patch generated using a Bernoulli distribution. From table 10, we observe that the choice of the patch does not affect Sleeper Agent’s success rate. ",
+        "bbox": [
+            173,
+            170,
+            825,
+            349
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b538020ce551a981a529acbe4c29ae93e54dcf906579ac2962fb82b513cd6513.jpg",
+        "image_caption": [
+            "Figure 3: Sample random patch (left) and HTBD patch (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            406,
+            364,
+            584,
+            417
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 10: Baseline evaluations using random patches on CIFAR-10. Perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and poison budget is $1 \\%$ of training images. Each number denotes an average (and standard error) over 24 independent crafting and training runs along with randomly sampled source/target class pairs. Each run has a unique patch generated randomly. Figure 3 (left) shows a sample random patch we use for the experiments presented in this table. ",
+        "bbox": [
+            176,
+            484,
+            823,
+            554
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/6e1beadf8a294fb2612cff44e69fef782695c8aa9bd7eccd0d621999d72076d3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td>ResNet-18</td></tr><tr><td>Clean validation accuracy(%)</td><td>92.16 (±0.08)</td></tr><tr><td>Poison validation accuracy(%)</td><td>92.00 (±0.07)</td></tr><tr><td>Clean source accuracy(%)</td><td>92.55 (±0.98)</td></tr><tr><td>Poison source accuracy(%)</td><td>91.77 (±1.09)</td></tr><tr><td>Triggered source accuracy(%)</td><td>14.86 (±5.06)</td></tr><tr><td>Attack Success Rate(%)</td><td>82.05 (±5.80)</td></tr></table>",
+        "bbox": [
+            346,
+            564,
+            653,
+            661
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 MODELS AND HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            689,
+            460,
+            703
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For our evaluations we use ResNet-18, ResNet-34, MobileNet-v2, and VGG11 (He et al., 2016; Sandler et al., 2018; Simonyan & Zisserman, 2014). For training ResNet-18 and ResNet-34, we use initial learning rate 0.1, and for MobileNet-v2 and VGG11, we use initial learning rate 0.01. We schedule learning rate drops at epochs 14, 24, and 35 by a factor of 0.1. For all models, we employ SGD with Nesterov momentum, and we set the momentum coefficient to 0.9. We use batches of 128 images and weight decay with a coefficient of $4 \\times 1 0 ^ { - 4 }$ . For all CIFAR-10 experiments, we train and retrain for 40 epochs, and for validation, we train the re-initialized model for 80 epochs. For the ImageNet experiments, we employ pre-trained models from torchvision to start crafting, and for retraining and validation, we apply a similar procedure explained: training for 80 epochs for both retraining and validation while we schedule learning rate drops at epochs 30, 50, and 70 by a factor of 0.1. To incorporate data augmentation, for CIFAR-10, we apply horizontal flips with probability 0.5 and random crops of size $3 2 \\times 3 2$ with zero-padding of 4. And for the ImageNet, we use the following data augmentations: 1) resize to $2 5 6 \\times 2 5 6 , 2$ ) central crop of size $2 2 4 \\times 2 2 4$ , 3) horizontal flip with probability 0.5, 4) random crops of size $2 2 4 \\times 2 2 4$ with zero-padding of 28. Our complete implementation code is attached. ",
+        "bbox": [
+            173,
+            715,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf4abcc4d5fc3d85123c52ff5a9ee9334b047b126a9cddd9a10bf21ae27f6497.jpg",
+        "image_caption": [
+            "Figure 4: Average poisoning time for various Sleeper Agent setups. All experiments are conducted on CIFAR-10 with ResNet-18 models. Perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and the poison budget is $1 \\%$ of training images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            131,
+            758,
+            357
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 RUNTIME COST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            448,
+            328,
+            463
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We use two NVIDIA GEFORCE RTX 2080 Ti GPUs for baseline evaluations on CIFAR-10 and four of the aforementioned GPUs for ImageNet baseline evaluations. Figure 4 shows the time cost of the Sleeper Agent with different settings. ",
+        "bbox": [
+            174,
+            474,
+            825,
+            517
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B VISUALIZATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            541,
+            352,
+            558
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present more triggered source and poisoned targets drawn from the ImageNet dataset. Figures 5 and 6 show patched sources and poisoned targets generated by Sleeper Agent. We observe that the generated perturbed images and their corresponding clean images are hardly distinguishable by the human eye. ",
+        "bbox": [
+            174,
+            575,
+            825,
+            631
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            655,
+            442,
+            671
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present additional experiments. ",
+        "bbox": [
+            173,
+            688,
+            501,
+            703
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 PATCH SIZE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
+            300,
+            737
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To further investigate the effect of patch size on the attack success rate, we perform the baseline evaluation discussed in 4.1 using different patch sizes. From Table 11, we observe that by poisoning only $0 . 0 5 \\%$ of the training set and using a larger patch, we can effectively poison ImageNet. Furthermore, by using a proper amount of perturbation, Sleeper Agent works well with the smaller patches. Visualizations of patched sources using patch size of $4 5 \\times 4 5$ are shown in Figure 6. ",
+        "bbox": [
+            174,
+            751,
+            825,
+            820
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            840,
+            330,
+            854
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our experiments show that Sleeper Agent works well on other architectures. To explore this, we conduct our ImageNet baseline experiments on MobileNet-v2. Table 12 depicts the performance of Sleeper Agent on MobileNet-v2 when perturbing $0 . 0 5 \\%$ of images in the ImageNet training set with each perturbation constrained in an $\\ell _ { \\infty }$ -norm ball of radius 16/255. ",
+        "bbox": [
+            176,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/4607f687feb0e305f7242623c1ee6c998dc3e91cb8939487711c6781585e3464.jpg",
+        "image_caption": [
+            "Figure 5: Sample clean source (first column), patched source (second column), clean target (third column), and poisoned target (fourth column) from the ImageNet dataset. Perturbations have $\\ell _ { \\infty }$ - norm bounded above by 16/255, and the patch size is $3 0 \\times 3 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            104,
+            803,
+            338
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/9ec18b1b2f7ec99bb99cc6c35bccc044ee6422f9528580a4846a808823edf98e.jpg",
+        "image_caption": [
+            "Figure 6: Sample clean source (first column), patched source (second column), clean target (third column), and poisoned target (fourth column) from the ImageNet dataset. Perturbations have $\\ell _ { \\infty }$ - norm bounded above by 16/255, and the patch size is $4 5 \\times 4 5$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            426,
+            803,
+            661
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/667edf2dc6165b5575283e8887bab05c21f4cbbbb63f84e41edbfbd74efe03a5.jpg",
+        "table_caption": [
+            "Table 11: The effect of patch size. Experiments are conducted on CIFAR-10 and ImageNet datasets with ResNet-18 models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Dataset</td><td>Poison budget</td><td>Patch size</td><td>loo-norm</td><td>Attack Success Rate (%)</td></tr><tr><td>Sleeper Agent (S = 1,T = 4)</td><td>CIFAR-10</td><td>1%</td><td>6×6</td><td>20/255</td><td>64.78</td></tr><tr><td>Sleeper Agent (S = 1, T= 4)</td><td>CIFAR-10</td><td>1%</td><td>8×8</td><td>16/255</td><td>85.27</td></tr><tr><td>Sleeper Agent (S=1,T=2)</td><td>ImageNet</td><td>0.05%</td><td>25×25</td><td>16/255</td><td>38.00</td></tr><tr><td>Sleeper Agent (S=1,T=2)</td><td>ImageNet</td><td>0.05%</td><td>25×25</td><td>24/255</td><td>52.00</td></tr><tr><td>Sleeper Agent (S=1,T=2)</td><td>ImageNet</td><td>0.05%</td><td>30×30</td><td>16/255</td><td>44.00</td></tr><tr><td>Sleeper Agent (S =1,T= 2)</td><td>ImageNet</td><td>0.05%</td><td>45×45</td><td>16/255</td><td>50.50</td></tr></table>",
+        "bbox": [
+            173,
+            795,
+            823,
+            886
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/5610dfb8b40561bd3e97067753ed02e49a70230f7ff1126a5292f4ae7629ce11.jpg",
+        "table_caption": [
+            "Table 12: ImageNet evaluations on MobileNet-v2. Perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and the patch size is $3 0 \\times 3 0$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td>Poison budget</td><td></td><td>Patch sizeAttack Success Rate (%)</td></tr><tr><td>Sleeper Agent (S=1, T=2)</td><td>0.05%</td><td>30</td><td>41.00</td></tr></table>",
+        "bbox": [
+            204,
+            150,
+            792,
+            188
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C.3 RETRAINING FACTOR ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            212,
+            369,
+            228
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 13 shows the effect of the retraining factor on the attack success rate on the CIFAR-10 dataset. As can be observed from the table, for $T$ larger than 4, we do not see a considerable improvement in the attack success rate. Since increasing $T$ is costly, we choose $T = 4$ as it simultaneously gives us a high success rate and is also significantly faster than $T = 6$ and $T = 8$ . We observe that even with $T = 4$ , the attack success rate is above $9 5 \\%$ in most trials. ",
+        "bbox": [
+            173,
+            238,
+            825,
+            309
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/196d61636da79e61787dff0bbd8b0e3839ea08ddc0a315e20ebd1e064c66ccea.jpg",
+        "table_caption": [
+            "Table 13: Ablation studies. Investigation of the effects of retraining factor $T$ . Experiments are conducted on CIFAR-10 with ResNet-18 models, perturbations have $\\ell _ { \\infty }$ -norm bounded above by 16/255, and the poison budget is $1 \\%$ of the training images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Retraining factor</td><td>T= 2</td><td>T=4</td><td>T=6</td><td>T=8</td></tr><tr><td>Attack Success Rate (%)</td><td>70.66 (±6.66)</td><td>84.64 (±6.64)</td><td>84.95 (±6.42)</td><td>86.48 (±6.26)</td></tr></table>",
+        "bbox": [
+            174,
+            382,
+            823,
+            422
+        ],
+        "page_idx": 15
+    }
+]

parse/dev/oDRQGo8I7P/oDRQGo8I7P.md

@@ -0,0 +1,414 @@
+# Riemannian Score-Based Generative Modelling
+
+Valentin De Bortoli∗†, Émile Mathieu∗‡, Michael Hutchinson∗‡,
+
+James Thornton‡, Yee Whye Teh‡, Arnaud Doucet‡
+
+# Abstract
+
+Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a “noising” stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a “denoising” process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.
+
+# 1 Introduction
+
+Score-based Generative Models (SGMs) also called diffusion models (Song and Ermon, 2019; Song et al., 2021; Ho et al., 2020; Dhariwal and Nichol, 2021) formulate generative modelling as a denoising process. Noise is incrementally added to data using a diffusion process until it becomes approximately Gaussian. The generative model is then obtained by simulating an approximation of the corresponding time-reversal process, which progressively denoises a Gaussian sample to obtain a data sample. This process is also a diffusion whose drift depends on the logarithmic gradients of the noised data densities, i.e. the Stein scores, estimated using a neural network via score matching (Hyvärinen, 2005; Vincent, 2011).
+
+SGMs have been primarily applied to data living on Euclidean spaces, i.e. manifolds with flat geometry. However, in a large number of scientific domains the distributions of interest are supported on Riemannian manifolds. These include, to name a few, protein modelling (Shapovalov and Dunbrack Jr, 2011), cell development (Klimovskaia et al., 2020), image recognition (Lui, 2012), geological sciences (Karpatne et al., 2018; Peel et al., 2001), graph-structured and hierarchical data (Roy et al., 2007; Steyvers and Tenenbaum, 2005), robotics (Feiten et al., 2013; Senanayake and Ramos, 2018) and high-energy physics (Brehmer and Cranmer, 2020).
+
+We introduce in this work Riemannian Score-based Generative Models (RSGMs), an extension of SGMs to Riemannian manifolds which incorporate the geometry of the data by defining the forward diffusion process directly on the Riemannian manifold, inducing a manifold-valued reverse process. This requires constructing a noising process on the manifold that converges to an easy-to-sample reference distribution. We establish that, as in the Euclidean case, the corresponding time-reversal process is also a diffusion whose drift includes the Stein score which is intractable but can similarly be estimated via score matching. Methodological extensions are required as in most cases the transition kernel of the noising process cannot be sampled exactly. For example on compact manifolds it is typically only available as an infinite sum through the Sturm–Liouville decomposition (Chavel, 1984). To this end, we develop non-standard techniques for score estimation and rely on the use of Geodesic Random Walks for sampling (Jørgensen, 1975). We provide theoretical convergence bounds for RSGMs on compact manifolds and demonstrate our approach on a range of manifolds and tasks, including modelling a number of natural disaster occurrence datasets collected by Mathieu and Nickel (2020). We show that RGSMs achieve better performance than recent baselines (Mathieu and Nickel, 2020; Rozen et al., 2021) and scale better to high-dimensional manifolds.
+
+# 2 Euclidean Score-based Generative Modelling
+
+We recall here briefly the key concepts behind SGMs on the Euclidean space $\mathbb { R } ^ { d }$ and refer the readers to Song et al. (2021) for a more detailed introduction. We consider a forward noising process $( \mathbf { X } _ { t } ) _ { t \geq 0 }$ defined by the following Stochastic Differential Equation (SDE)
+
+$$
+\mathrm { d } \mathbf { X } _ { t } = - \mathbf { X } _ { t } \mathrm { d } t + \sqrt { 2 } \mathrm { d } \mathbf { B } _ { t } , \quad \mathbf { X } _ { 0 } \sim p _ { 0 } ,
+$$
+
+where $( \mathbf { B } _ { t } ) _ { t \geq 0 }$ is a $d$ -dimensional Brownian motion and $p _ { 0 }$ is the data distribution. The available data gives us an empirical approximation of $p _ { 0 }$ . The process $( \mathbf { X } _ { t } ) _ { t \geq 0 }$ is simply an Ornstein–Ulhenbeck (OU) process which converges with geometric rate to $\mathrm { { N } } ( 0 , \mathrm { { I d } ) }$ . Under mild conditions on $p _ { 0 }$ , the timereversed process $( \mathbf { Y } _ { t } ) _ { t \geq 0 } \stackrel { - } { = } ( \mathbf { X } _ { T - t } ) _ { t \in [ 0 , T ] }$ also satisfies an SDE (Cattiaux et al., 2021; Haussmann and Pardoux, 1986) given by
+
+$$
+\mathrm { d } \mathbf { Y } _ { t } = \{ \mathbf { Y } _ { t } + 2 \nabla \log p _ { T - t } ( \mathbf { Y } _ { t } ) \} \mathrm { d } t + \sqrt { 2 } \mathrm { d } \mathbf { B } _ { t } , \quad \mathbf { Y } _ { 0 } \sim p _ { T } ,
+$$
+
+where $p _ { t }$ denotes the density of $\mathbf { X } _ { t }$ . By construction, the law of $\mathbf { Y } _ { T - t }$ is equal to the law of $\mathbf { X } _ { t }$ for $t \in [ 0 , T ]$ and in particular $\mathbf { Y } _ { T } \sim p _ { 0 }$ . Hence, if one could sample from $( \mathbf { Y } _ { t } ) _ { t \in [ 0 , T ] }$ then its final distribution would be the data distribution $p _ { 0 }$ ∈. Unfortunately we cannot sample exactly from (2) as $p _ { T }$ and the scores $( \nabla \log p _ { t } ( x ) ) _ { t \in [ 0 , T ] }$ are intractable. Hence SGMs rely on a few approximations. First, $p _ { T }$ is replaced by the reference distribution $\mathrm { { N } } ( 0 , \mathrm { { I d } ) }$ as we know that $p _ { T }$ converges geometrically towards it. Second, the following denoising score matching identity is exploited to estimate the scores
+
+$$
+\begin{array} { r } { \nabla _ { x _ { t } } \log p _ { t } ( x _ { t } ) = \int _ { \mathbb { R } ^ { d } } \nabla _ { x _ { t } } \log p _ { t | 0 } ( x _ { t } | x _ { 0 } ) p _ { 0 | t } ( x _ { 0 } | x _ { t } ) \mathrm { d } x _ { 0 } , } \end{array}
+$$
+
+where $p _ { t \vert 0 } ( x _ { t } \vert x _ { 0 } )$ is the transition density of the OU process (1) which is available in closed-form. It follows directly that $\nabla \log p _ { t }$ is the minimizer of $\ell _ { t } ( \mathbf { s } ) = \mathbb { E } [ \| \mathbf { s } ( \mathbf { X } _ { t } ) - \nabla _ { x _ { t } } \log p _ { t | 0 } ( \mathbf { X } _ { t } | \mathbf { X } _ { 0 } ) \| ^ { 2 } ]$ over functions s where the expectation is over the joint distribution of $\mathbf { X } _ { 0 } , \mathbf { X } _ { t }$ . This result can be leveraged by considering a neural network $\mathbf { s } _ { \theta } : [ 0 , T ] \times \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ trained by minimizing the loss function $\begin{array} { r } { \ell ( \theta ) = \int _ { 0 } ^ { T } \lambda _ { t } \ell _ { t } ( \mathbf { s } _ { \theta } ( t , \cdot ) ) \mathrm { d } t } \end{array}$ for some weighting function $\lambda _ { t } ~ > ~ 0$ . Finally, an Euler–Maruyama discretization of (2) is performed using a discretization step $\gamma$ such that $T = \gamma N$ for $N \in \mathbb N$
+
+$$
+Y _ { n + 1 } = Y _ { n } + \gamma \{ Y _ { n } + 2 \mathbf { s } _ { \theta } ( T - n \gamma , Y _ { n } ) \} + { \sqrt { 2 \gamma } } Z _ { n + 1 } , \quad Y _ { 0 } \sim \mathrm { N ( 0 , I d ) } , \quad Z _ { n } \sim \mathrm { N ( 0 , I d ) } .
+$$
+
+The above showcases the basics of SGMs but we highlight that many improvements have been proposed; see (e.g. Song and Ermon, 2020; Jolicoeur-Martineau et al., 2021; Dhariwal and Nichol, 2021). In particular, selecting an adaptive stepsize $( \gamma _ { n } ) _ { n \in \mathbb { N } }$ (Bao et al., 2022; Watson et al., 2021) and using a predictor-corrector scheme (Song et al., 2021) instead of a simple Euler–Maruyama discretization drastically improves performance.
+
+# 3 Riemannian Score-based Generative Modelling
+
+We now move to the Riemannian manifold setting, and more specifically assume that $\mathcal { M }$ is a complete, orientable connected and boundaryless Riemannian manifold, endowed with a Riemannian metric $g ^ { \ 4 }$ . Four components are required to extend SGMs to this setting: i) a forward noising process on $\mathcal { M }$ which converges to an easy-to-sample reference distribution, ii) a time-reversal formula on $\mathcal { M }$ which defines a backward generative process, iii) a method for approximating samples of SDEs on manifolds, iv) a method to efficiently approximate the drift of the time-reversal process. Notation are gathered in App. B.
+
+# 3.1 Noising processes on manifolds
+
+The first necessary component is a suitable generic noising process on manifolds that will converge to a convenient stationary distribution. A simple choice is to use Langevin dynamics described by
+
+$$
+\begin{array} { r } { \mathrm { d } \mathbf { X } _ { t } = - \frac { 1 } { 2 } \nabla _ { \mathbf { X } _ { t } } U ( \mathbf { X } _ { t } ) \mathrm { d } t + \mathrm { d } \mathbf { B } _ { t } ^ { \mathcal { M } } , } \end{array}
+$$
+
+which admits the invariant density (w.r.t. the volume form) given by $\mathrm { d } p _ { \mathrm { r e f } } / \mathrm { d } { \mathsf { V o l } } _ { \mathcal { M } } ( x ) \propto \mathrm { e } ^ { - U ( x ) }$ (Durmus, 2016, Section 2.4), where $\nabla$ is the Riemannian gradient5.
+
+Two simple choices for $U ( x )$ present themselves. Firstly, setting $U ( x ) = d _ { \mathcal { M } } ( x , \mu ) ^ { 2 } / ( 2 \gamma ^ { 2 } )$ , where $d _ { \mathcal { M } }$ is the geodesic distance and $\mu \in \mathcal { M }$ is an arbitrary mean location, induces the drift $\nabla _ { \mathbf { X } _ { t } } { \tilde { U } } ( \mathbf { X } _ { t } ) =$ $- \exp _ { \mathbf { X } _ { t } } ^ { - 1 } ( \mu ) / \gamma ^ { 2 }$ 6. This is the potential of the ‘Riemannian normal’ (Pennec, 2006) distribution, from which it is in general neither trivial to sample nor to compute the normalisation constant (Hauberg, 2018; Mathieu et al., 2019). An alternative is to target the ’exponential wrapped’ Gaussian. This is the pushforward of a Gaussian distribution in the tangent space at the mean location along the exponential map. The potential is given by $U ( x ) = d _ { \mathcal { M } } ( \bar { x _ { , } } \mu ) ^ { 2 } \bar { / ( 2 \gamma ^ { 2 } ) } + \log | \partial \exp _ { \mu } ^ { - 1 } ( x ) | ^ { 7 }$ . In contrast to the Riemannian normal, sampling and evaluating the density of this distribution is easy.
+
+One recovers the standard Ornstein–Uhlenbeck noising process (Song et al., 2021) for both of these target distributions when $\mathcal { M } = \mathbb { R } ^ { d }$ and $\mu = 0$ since then the drift $\begin{array} { r } { b ( t , { \bf \bar { X } } _ { t } ) = \frac { 1 } { 2 } \exp _ { { \bf X } _ { t } } ^ { - 1 } ( 0 ) = - \frac { 1 } { 2 } { \bf X } _ { t } } \end{array}$ On compact manifolds, the invariant measure $\mathrm { V o l } _ { \mathcal { M } }$ has finite volume, thus a natural choice is to target the uniform distribution which is given by $\mathrm { V o l } _ { \mathcal { M } } / | \mathcal { M } |$ . In this case, $\nabla _ { \mathbf { X } _ { t } } U ( \mathbf { X } _ { t } ) = 0$ and the noising process is simply a Brownian motion on $\mathcal { M }$ .
+
+# 3.2 Time-reversal on Riemannian manifolds
+
+In order to use these noising processes we prove the time-reversal formula for manifolds, a generalisation of the results in the Euclidean case, e.g. see Cattiaux et al. (2021, Theorem 4.9). Consider an SDE of the form $\mathrm { d } \mathbf { X } _ { t } = b ( \mathbf { X } _ { t } ) \mathrm { d } t + \mathrm { d } \mathbf { B } _ { t } ^ { \mathcal { M } }$ where $\mathbf { B } _ { t } ^ { \mathcal { M } }$ is a Brownian motion on $\mathcal { M }$ . We refer to App. C.3 for an introduction to Brownian motions on manifolds. This result shows that if $( \mathbf { X } _ { t } ) _ { t \in [ 0 , T ] }$ is a diffusion process then $( \mathbf { X } _ { T - t } ) _ { t \in [ 0 , T ] }$ is also a diffusion process w.r.t. the backward filtration whose coefficients can be computed, and are shown in Eq. (4). The proof relies on an extension of Cattiaux et al. (2021, Theorem 4.9) to the Riemannian manifold case and is postponed to App. H.
+
+Theorem 1 (Time-reversed diffusion). Let $T \geq 0$ and $( \mathbf { B } _ { t } ^ { \mathcal { M } } ) _ { t \geq 0 }$ be a Brownian motion on $\mathcal { M }$ such that $\mathbf { B } _ { 0 } ^ { \mathcal { M } }$ has distribution the volume form ${ p _ { \mathrm { r e f } } } ^ { 8 }$ . Let $( \mathbf { X } _ { t } ) _ { t \in [ 0 , T ] }$ be associated with the $S D E$ $\mathrm { d } \mathbf { X } _ { t } = b ( \mathbf { X } _ { t } ) \mathrm { d } t + \mathrm { d } \mathbf { B } _ { t } ^ { \mathcal { M } }$ . Let $( \mathbf { Y } _ { t } ) _ { t \in [ 0 , T ] } = ( \mathbf { X } _ { T - t } ) _ { t \in [ 0 , T ] }$ and assume that $\mathrm { K L } ( \mathbb { P } | \mathbb { Q } ) < + \infty ,$ where $\mathbb { Q }$ is the distribution of $( \mathbf { B } _ { t } ^ { \mathcal { M } } ) _ { t \in [ 0 , T ] }$ and $\mathbb { P }$ the distribution of $( \mathbf { X } _ { t } ) _ { t \in [ 0 , T ] }$ . In addition, assume that $\mathbb { P } _ { t } = \mathcal { L } ( \mathbf { X } _ { t } )$ , the distribution of $\mathbf { X } _ { t }$ , admits a smooth positive density $p _ { t }$ w.r.t. $p _ { \mathrm { r e f } }$ for any $t \in [ 0 , T ]$ . Then, $( \mathbf { Y } _ { t } ) _ { t \in [ 0 , T ] }$ is associated with the SDE
+
+$$
+\mathrm { d } \mathbf { Y } _ { t } = \{ - b ( \mathbf { Y } _ { t } ) + \nabla \log p _ { T - t } ( \mathbf { Y } _ { t } ) \} \mathrm { d } t + \mathrm { d } \mathbf { B } _ { t } ^ { \mathcal { M } } .
+$$
+
+# 3.3 Approximate sampling of diffusions
+
+Obtaining samples from SDEs on a manifold is non-trivial in general. If $\mathcal { M }$ is isometrically embedded into $\mathbb { R } ^ { p }$ (with $p \geq d$ ) one can define $( \mathbf { B } _ { t } ^ { \mathcal { M } } ) _ { t \geq 0 }$ as a $\mathbb { R } ^ { p }$ -valued process, see App. C.3. However, this approach is extrinsic, as it requires the knowledge of the projection operator to place points back on the manifold at each step which can accumulate errors. Here we consider an intrisic approach based on Geodesic Random Walks (GRWs), see Jørgensen (1975) for a review of their properties. GRWs can approximate any well-behaved diffusion on $\mathcal { M }$ . Hence, we introduce GRWs in a general framework and consider a discrete-time process $( X _ { n } ^ { \gamma } ) _ { n \in \mathbb { N } }$ which approximates the diffusion $( \mathbf { X } _ { t } ) _ { t \geq 0 }$ defined by
+
+$$
+\mathrm { d } \mathbf { X } _ { t } = b ( t , \mathbf { X } _ { t } ) \mathrm { d } t + \sigma ( t , \mathbf { X } _ { t } ) \mathrm { d } \mathbf { B } _ { t } ^ { \mathcal { M } } .
+$$
+
+This generalisation is key to sampling the backward diffusion process defined in Theorem 1.
+
+![](images/c4d058f414b61edf69b154a4a699fdd1b1d0ea4c8a429ab4995efd005cef6f77.jpg)  
+single step of a (b) Many steps yield an ap- (c) The density of a single step of Gaussian Random proximate Brownian mo- Walk [Left] and the Brownian motion density [Right] tion trajectory. agree well for small time steps.
+
+Geodesic Random Walk.
+
+Figure 1: Geodesic Random Walks can be used to approximate Brownian motion and more generally SDEs on manifolds. (a) At each step, tangential noise is sampled (red), which is added the drift term (not pictured). This tangent vector is then pushed through the exponential map to produce a geodesics step on the manifold (blue). (b) Iterating this procedure yield approximate sample paths from the process.
+
+# Algorithm 1 GRW (Geodesic Random Walk)
+
+Require: $T , N , X _ { 0 } ^ { \gamma } , b , \sigma , \mathrm { P }$   
+1: $\gamma = T / N$ . Step-size   
+2: for $k \in \{ 0 , \ldots , N - 1 \}$ do   
+3: $Z _ { k + 1 } \sim \mathrm { N } ( 0 , \mathrm { I d } )$ . Sample a Gaussian in the tangent space of $X _ { k } ^ { \gamma }$   
+4: $W _ { k + 1 } = \gamma \dot { b } ( k \gamma , X _ { k } ^ { \gamma } ) + \sqrt { \gamma } \sigma ( k \gamma , X _ { k } ^ { \gamma } ) Z _ { k + 1 } \mathrm { ~ \tiny ~ > ~ } \mathrm { C o r }$ mpute the Euler–Maruyama step on tangent space   
+5: $X _ { k + 1 } ^ { \gamma } = \exp _ { X _ { k } ^ { \gamma } } [ W _ { k + 1 } ]$ . Move along the geodesic defined by $\bar { W _ { k + 1 } }$ and $\bar { X } _ { k } ^ { \gamma }$ on $\mathcal { M }$   
+6: end fo7: return $\{ X _ { k } ^ { \gamma } \} _ { k = 0 } ^ { N }$
+
+Definition 2 (Geodesic Random Walk). Let $X _ { 0 } ^ { \gamma }$ be a $\mathcal { M }$ -valued random variable. For any $\gamma > 0$ , we define $( X _ { n } ^ { \gamma } ) _ { n \in \mathbb { N } }$ such that for any $n \in \mathbb { N }$ , $X _ { n + 1 } ^ { \gamma ^ { \vee } } = \exp _ { X _ { n } ^ { \gamma } } [ \gamma \{ b ( X _ { n } ^ { \gamma } ) + \sqrt { \gamma } V _ { n + 1 } \} ]$ , where $( V _ { n } ) _ { n \in \mathbb { N } }$ is a sequence of $\mathrm { T } \mathcal { M }$ -valued random variables such that for any $n \in \mathbb { N }$ , $\mathbb { E } [ V _ { n + 1 } | \mathcal { F } _ { n } ] = 0$ and $\mathbb { E } [ V _ { n + 1 } V _ { n + 1 } ^ { \top } | \mathcal { F } _ { n } ] = \sigma \sigma ^ { \top } ( X _ { n } ^ { \gamma } ) ,$ , where ${ \mathcal { F } } _ { n }$ is the filtration generated by $\{ X _ { k } ^ { \gamma } \} _ { k = 0 } ^ { n }$ . We say that the $\mathcal { M }$ -valued process $( X _ { n } ^ { \gamma } ) _ { n \in \mathbb { N } }$ is a Geodesic Random Walk.
+
+Algorithm 1 approximately simulates the diffusion $( \mathbf { X } _ { t } ) _ { t \in [ 0 , T ] }$ defined in Eq. (5) using GRWs; see Kuwada (2012); Cheng et al. (2022) for quantitative error bounds in the time-homogeneous case and App. I.2 for a novel extentsion for the time-inhomogeneous case. Fig. 1 provides a graphical illustration of this procedure.
+
+# 3.4 Score approximation on Riemannian manifolds
+
+Score matching and loss functions. The reverse process from Eq. (4) involves the Stein score $\nabla \log p _ { t }$ which is unfortunately intractable. To derive an approximation, we first remark that for any $s , t \in ( 0 , T ]$ with $t > s$ and $\begin{array} { r } { \in \mathcal { M } , p _ { t } ( x _ { t } ) = \int _ { \mathcal { M } } p _ { t | s } \bar { ( x _ { t } | x _ { s } ) } \mathrm { d } \mathbb { P } _ { s } ( x _ { s } ) } \end{array}$ , where $\mathbb { P } _ { s } = \mathcal { L } ( \mathbf { X } _ { s } )$ , the distribution of $\mathbf { X } _ { s }$ . Thus, we have that for any $s , t \in [ 0 , T ]$ with $t > s$ and $x _ { t } \in \mathcal { M }$
+
+$$
+\begin{array} { r } { \nabla _ { x _ { t } } \log p _ { t } ( x _ { t } ) = \int _ { \mathcal { M } } \nabla _ { x _ { t } } \log p _ { t | s } ( x _ { t } | x _ { s } ) \mathbb { P } _ { s | t } ( x _ { t } , \mathrm { d } x _ { s } ) . } \end{array}
+$$
+
+Hence, for any $s , t \in [ 0 , T ]$ with $t > s$ we have that $\begin{array} { r } { \nabla \log p _ { t } = \arg \operatorname* { m i n } \{ \ell _ { t | s } ( \mathbf { s } _ { t } ) : \mathbf { s } _ { t } \in \mathrm { L } ^ { 2 } ( \mathbb { P } _ { t } ) \} , } \end{array}$ where $\begin{array} { r } { \ell _ { t | s } ( \mathbf { s } _ { t } ) = \int _ { \mathcal { M } ^ { 2 } } \| \nabla _ { x } \log p _ { t | s } ( x _ { t } | x _ { s } ) - \mathbf { s } _ { t } ( x _ { t } ) \| ^ { 2 } \mathrm { d } \mathbb { P } _ { s , t } ( x _ { s } , x _ { t } ) } \end{array}$ , which is referred as the Denois Ming Score Matching (DSM) loss. It can also be written in an implicit fashion.
+
+Proposition 3. Let $t , s \in ( 0 , T ]$ with $t > s$ . Then, for any $\mathbf { s } _ { t } \in \mathrm { C } ^ { \infty } ( \mathcal { M } )$ , $\ell _ { t | s } ( \mathbf { s } _ { t } ) = 2 \ell _ { t } ^ { \mathrm { i m } } ( \mathbf { s } _ { t } ) +$ $\begin{array} { r } { \int _ { \mathcal { M } ^ { 2 } } \| \nabla _ { \boldsymbol { x } _ { t } } \log p _ { t | s } ( \boldsymbol { x } _ { t } | \boldsymbol { x } _ { s } ) \| ^ { 2 } \mathrm { d } \mathbb { P } _ { s , t } ( \boldsymbol { x } _ { s } , \boldsymbol { x } _ { t } ) , } \end{array}$ , where $\begin{array} { r } { \ell _ { t } ^ { \mathrm { i m } } ( \mathbf { s } _ { t } ) = \int _ { \mathcal { M } } \{ \frac { 1 } { 2 } \| \mathbf { s } _ { t } ( x _ { t } ) \| ^ { 2 } + \mathrm { d i v } ( \mathbf { s } _ { t } ) ( x _ { t } ) \} \mathrm { d } \mathbb { P } _ { t } ( x _ { t } ) , } \end{array}$
+
+The proof is postponed to App. J. For any $t \in ( 0 , T ]$ the minimizers of the loss $\ell _ { t } ^ { \mathrm { i m } }$ on $\mathcal { X } ( \mathcal { M } )$ (where $\mathcal { X } ( \mathcal { M } )$ is the set of vector fields on $\mathcal { M }$ ) are the same as the ones for $\ell _ { t \mid s }$ . The loss $\ell _ { t } ^ { \mathrm { i m } }$ is referred to as the implicit score matching (ISM) loss (Hyvärinen, 2005). These losses are direct analogous to the versions typically used in Euclidean space.
+
+In the case where we have access to $\{ \nabla \log p _ { t | s } : T \leq t > s \geq 0 \}$ , the forward noising process transition kernels, or an approximation of this family, then we can use the DSM loss to learn $\{ \mathbf { s } _ { t } \in \mathcal { X } ( \mathcal { M } ) : \ t \in [ 0 , t ] \}$ . If this is not the case then we turn to $\ell _ { t } ^ { \mathrm { i m } }$ . Note that $\ell _ { t } ^ { \mathrm { i m } }$ requires the computation of a divergence term which requires $d$ Jacobian-vector calls. In high dimension, a stochastic estimator is necessary (Hutchinson, 1989). Following Song and Ermon (2020); Nichol and Dhariwal (2021) the loss can be weighted with a term $\lambda _ { t } > 0$ .
+
+Table 1: Differences between SGM on Euclidean spaces and RSGM on Riemannian manifolds.   
+
+<table><tr><td colspan="4">Require: e,T,N,{Xm} n}m=1,loss,s,0o,Niter,Pref,P</td></tr><tr><td></td><td>1:/// TRAINING //I</td><td></td><td></td></tr><tr><td>2:</td><td>for n ∈ {0,...,Niter-1} do</td><td></td><td></td></tr><tr><td>3:</td><td>Xo~(1/M）∑m=1δxm</td><td></td><td>Random mini-batch from dataset</td></tr><tr><td>4:</td><td>t ~ U([e,T])</td><td></td><td>Uniform sampling between ε and T</td></tr><tr><td>5:</td><td>Xt =GRW(t,N,Xo,b,Id,P)</td><td></td><td>Approximate forward diffusion with Algorithm 1</td></tr><tr><td>6:</td><td>l(0n)=lt(T,N,Xo,Xt,loss,s0n)</td><td></td><td> Compute score matching loss from Table 2</td></tr><tr><td>7:</td><td>0n+1 =optimizer_update(0n,l(0n))</td><td></td><td>ADAM optimizer step</td></tr><tr><td>8:</td><td>end for</td><td></td><td></td></tr><tr><td>9:</td><td>0*=0Nepoch</td><td></td><td></td></tr><tr><td>10:</td><td>I/ SAMPLING /II</td><td></td><td></td></tr><tr><td>11:</td><td>Yo~pref</td><td></td><td>&gt; Sample from uniform distribution</td></tr><tr><td>12:</td><td>b(t,x)= so*(T-t,x) for any t ∈ [0,T],x ∈M</td><td></td><td>Reverse process drift</td></tr><tr><td>13: 14:</td><td>{Y}=0=GRW(T,N,Yo,bo,d,P)</td><td>&gt;Approximate reverse diffusion with Algorithm 1</td><td></td></tr></table>
+
+<table><tr><td>Ingredient \Space</td><td>Euclidean</td><td>‘Generic&#x27;Manifold</td><td>Compact Manifold</td></tr><tr><td>Forward process dXt =</td><td>-Xtdt+dBM</td><td>VxU(Xt)dt+dBM</td><td>dBM</td></tr><tr><td>Easy-to-sample distribution</td><td>Gaussian</td><td>Wrapped Gaussian</td><td>Uniform</td></tr><tr><td>Time reversal</td><td>Cattiaux et al. (2021)</td><td colspan="2">Theorem 1</td></tr><tr><td>Sampling forward process</td><td>Direct</td><td colspan="2">Geodesic Random Walk (Algorithm 1)</td></tr><tr><td>Sampling backward process</td><td>Euler-Maruyama</td><td colspan="2">Geodesic Random Walk (Algorithm 1)</td></tr></table>
+
+Parametric family of vector fields. We approximate $( \nabla \log p _ { t } ) _ { t \in [ 0 , T ] }$ by a family of functions $\{ \mathbf { s } _ { \theta } \} _ { \theta \in \Theta }$ where $\Theta$ is a set of parameters and $\mathbf { s } _ { \theta } : \ [ 0 , T ] \to \mathcal { X } ( \mathcal { M } )$ . In a Euclidean space, vector fields are simply functions $\mathbf { s } _ { \theta } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ . In manifolds, although for any $x \in \mathcal { M }$ , $\mathrm { T } _ { x } \mathcal { M } \cong \mathbb { R } ^ { d }$ , there does not necessarily exist a set of $d$ smooth vector fields $\{ E _ { i } \} _ { i = 1 } ^ { d }$ such that span $\begin{array} { r l } {  { \bigl ( \{ E _ { i } ( x ) \} _ { i = 1 } ^ { d } \bigr ) = } } \end{array}$ $\mathrm { T } _ { x } \mathcal { M }$ (Chapter 8, page 179, Lee, 2006) 9. Fortunately, one can rely on a larger set of smooth vector fields netwo $\{ E _ { i } ( x ) \} _ { i = 1 } ^ { n }$ $n > d$ at does span the tangent bundle. Theto parametrize the score network as $\mathbf { s } _ { \theta } : [ 0 , T ] \times { \mathcal { M } }  \mathbb { R } ^ { n }$ $\begin{array} { r } { \mathbf { s } _ { \theta } ( t , x ) = \sum _ { i = 1 } ^ { n } \mathbf { s } _ { \theta } ^ { i } ( t , x ) E _ { i } ( x ) } \end{array}$ See App. E for a discussion on the different choices of generating sets $\{ E _ { i } ( x ) \} _ { i = 1 } ^ { n }$ .
+
+Combining this parameterization with the score matching losses, the time-reversal formula of Theorem 1 and the sampling of forward and backward processes described in Sec. 3.3, we define our RGSM algorithm in Algorithm 2. This algorithm can also benefit from a predictor-corrector scheme as in (Song et al., 2021), see App. G.
+
+# 4 RSGMs on compact manifolds
+
+Assuming compactness of the manifold $\mathcal { M }$ , we can leverage a number of special properties to implement a specific case of our algorithm. In particular we benefit from the fact that on compact manifolds we have a proper uniform distribution over the manifold, and have access to a variety of approximations of the heat kernel. As highlighted in Sec. 3.1, in the compact setting we use Brownian motion as the noising SDE, which targets the uniform distribution as the stationary distribution. Table 1 highlights the main differences between RSGMs on compact manifolds, generic manifolds and Euclidean score-based models.
+
+Heat kernel on compact Riemannian manifolds. For any $x _ { 0 } \in \mathcal { M }$ and $t \geq s \geq 0$ , the heat kernel $p _ { t | s } ( \cdot | x _ { s } )$ is defined as the density of $\mathbf { B } _ { t } ^ { \mathcal { M } }$ w.r.t. the uniform measure on the manifold.
+
+Contrary to the Gaussian transition density of the OU process (or the Brownian motion) in the Euclidean setting, it is typically only available as an infinite series. In order to circumvent this issue we consider two techniques: i) a truncation approach, ii) a Taylor expansion around $t = 0$ called a Varadhan asymptotics. First, we recall that in the case of compact manifolds the heat kernel is given by the Sturm–Liouville decomposition (Chavel, 1984) given for any $t > 0$ and $x _ { 0 } , x _ { t } \in \mathcal { M }$ by
+
+$$
+\begin{array} { r } { p _ { t | 0 } ( x _ { t } | \boldsymbol { x } _ { 0 } ) = \sum _ { j \in \mathbb { N } } \mathrm { e } ^ { - \lambda _ { j } t } \phi _ { j } ( x _ { 0 } ) \phi _ { j } ( x _ { t } ) , } \end{array}
+$$
+
+where the convergence occurs in $\mathrm { L } ^ { 2 } ( p _ { \mathrm { r e f } } \otimes p _ { \mathrm { r e f } } )$ , $( \lambda _ { j } ) _ { j \in \mathbb { N } }$ and $( \phi _ { j } ) _ { j \in \mathbb { N } }$ are the eigenvalues, respectively the eigenvectors, of $- \Delta _ { \mathcal { M } }$ , the Laplace-Beltrami operator in the manifold, in $\mathrm { L } ^ { 2 } ( p _ { \mathrm { r e f } } )$ (see Saloff-Coste, 1994, Section 2). When the eigenvalues and eigenvectors are known, we rely on an approximation of the logarithmic gradient of $p _ { t | 0 }$ by truncating the sum in Eq. (S8) with $J \in \mathbb N$ terms to obtain for any $t > 0$ and $x _ { 0 } , x _ { t } \in \mathcal { M }$
+
+$$
+\begin{array} { r } { \nabla _ { x _ { t } } \log p _ { t | 0 } ( x _ { t } | x _ { 0 } ) \approx S _ { J , t } ( x _ { 0 } , x _ { t } ) \triangleq \nabla _ { x _ { t } } \log \sum _ { j = 0 } ^ { J } \mathrm { e } ^ { - \lambda _ { j } t } \phi _ { j } ( x _ { 0 } ) \phi _ { j } ( x _ { t } ) . } \end{array}
+$$
+
+Under regularity conditions on $\mathcal { M }$ it can be shown that for any $x , y \in \mathcal { M }$ and $t ~ \geq ~ 0$ , $\begin{array} { r } { \operatorname* { l i m } _ { J \to + \infty } \bar { S } _ { J , t } ( \bar { x } _ { 0 } , x _ { t } ) = \nabla _ { x _ { t } } \log p _ { t | 0 } ( x _ { t } | x _ { 0 } ) } \end{array}$ (see Jones et al., 2008, Lemma 1). In the case of the $d$ -dimensional torus or sphere the eigenvalues and eigenvectors are computable (see Saloff-Coste, 1994, Section 2) and we can apply this method to approximate $p _ { t | 0 }$ for any $t > 0$ , see App. F
+
+When the eigenvalues and eigenvectors are unknown or not tractable, we can still derive an approximation of the heat kernel for small times $t$ . Using Varadhan’s asymptotics—see Bismut (1984, Theorem 3.8) or Chen et al. (2021, Theorem 2.1)—for any $x , y \in { \mathcal { M } }$ with $y \not \in \operatorname { C u t } ( x )$ (where $\operatorname { C u t } ( x )$ is the cut-locus of $x$ in $\mathcal { M }$ (see Lee, 2018, Chapter 10)) we have that
+
+$$
+\begin{array} { r } { \operatorname* { l i m } _ { t  0 } t \nabla _ { x _ { t } } \log p _ { t | 0 } ( x _ { t } | x _ { 0 } ) = \exp _ { x _ { t } } ^ { - 1 } ( x _ { 0 } ) . } \end{array}
+$$
+
+Using the previously defined score-matching losses and the approximations to the heat kernel above, we highlight three methods to compute $\nabla \log p _ { t }$ in Table 2.
+
+Table 2: Computational complexity of score matching losses w.r.t. score network forward and backward passes.   
+$\varepsilon$ is a random variable on $\mathrm { T } _ { \mathbf { X } _ { t } } \mathcal { M }$ such that $\mathbb { E } [ \varepsilon ] = 0$ and $\mathbb { E } [ \varepsilon \varepsilon ^ { \top } ] = \operatorname { I d }$ .
+
+<table><tr><td>Loss</td><td>Approximation</td><td>Loss function</td><td>Requirements Pt10</td><td>expx</td><td>Complexity</td></tr><tr><td rowspan="3">lt1o (DSM)</td><td>None</td><td>1E[|ls(Xt)-Vlog pt|o(Xt|Xo)l2]</td><td>√</td><td>×</td><td>0(1)</td></tr><tr><td>Truncation (7)</td><td>1E[|ls(Xt)- SJ,t(Xo,Xt)|l2]</td><td>asymptotic expansion</td><td>×</td><td>0(1)</td></tr><tr><td>Varhadan (8)</td><td>1E[|ls(Xt）-expx¹(Xo)/tl|²2]</td><td>X</td><td>√</td><td>0(1)</td></tr><tr><td>lt|s (DSM)</td><td>Varhadan (8)</td><td>E[|s(Xt)-expx1(Xs)/(t-s)l²]</td><td>×</td><td>√</td><td>0(1)</td></tr><tr><td rowspan="2">ei (ISM)</td><td>Deterministic</td><td>E[¹|ls(Xt)||² + div(s)(Xt)]</td><td>×</td><td>×</td><td>O(d)</td></tr><tr><td>Stochastic</td><td>E[|s(Xt)|² +æT∂s(Xt)e]</td><td>X</td><td>X</td><td>0(1)</td></tr></table>
+
+Convergence results in the compact setting We now provide a theoretical analysis of RSGM under the assumption that $\mathcal { M }$ is compact. The following result ensures that RSGM generates samples whose distribution is close to the data distribution $p _ { 0 }$ . Let us denote $\{ Y _ { k } \} _ { n \in \{ 0 , \dots , N \} }$ the sequence generated by Algorithm 2. This result relies on the following assumption, which is satisfied for a large class of manifolds $\mathcal { M }$ such as the $d$ -dimensional sphere and torus, compact matrix groups and products of these manifolds.
+
+A1. There exist $C , \alpha > 0$ such that for any $t \in ( 0 , 1 ]$ and $x \in \mathcal { M }$ , $p _ { t | 0 } ( x | x ) \leq C t ^ { - \alpha / 2 }$ , where $p _ { t | 0 } ( \cdot | x _ { 0 } )$ is the density of the heat kernel, i.e. the density of $\mathbf { B } _ { t } ^ { \mathcal { M } }$ with initial condition $x _ { 0 }$ 10 .
+
+Theorem 4. Assume A1, that $p _ { 0 }$ is smooth and positive and that there exists $\mathsf { M } \geq 0$ such that for any $t \in [ 0 , T ]$ and $x \in \mathcal { M }$ , $\| \mathbf { s } _ { \theta ^ { \star } } ( t , x ) - \nabla \log p _ { t } ( x ) \| \leq \mathtt { M } ,$ , with $\mathbf { s } _ { \theta ^ { \star } } \in \mathrm { C } ( [ 0 , T ] , \mathcal { X } ( \mathcal { M } ) )$ . Then $i f$ $T > 1 / 2$ , there exists $C \geq 0$ independent on $T$ such that
+
+$$
+\mathbf { W } _ { 1 } ( \mathcal { L } ( Y _ { N } ) , p _ { 0 } ) = C ( \mathrm { e } ^ { - \lambda _ { 1 } T } + \sqrt { T / 2 } \mathsf { M } + \mathrm { e } ^ { T } \gamma ^ { 1 / 2 } ) ,
+$$
+
+where $\mathbf { W } _ { 1 }$ is the Wasserstein distance of order one on the probability measures on $\mathcal { M }$
+
+The proof is postponed to App. I. In particular, for any $\varepsilon > 0$ , choosing $T > 0$ large enough, M small enough (which can be achieved using the universal property of neural networks) and $\gamma$ small enough, we get that $\mathbf { W } _ { 1 } ( { \mathcal { L } } ( Y _ { N } ) , p _ { 0 } ) \leq \varepsilon$ . This result might seem weaker than the result obtained for Moser flows in (Rozen et al., 2021, Theorem 3), but we emphasize that our bound takes into account the time-discretization contrary to Rozen et al. (2021) which considers the continuous-time flow. If we consider the time-reversed continuous-time SDE then we recover a bound in total variation distance, see App. I. Note that the upper bound $\mathbb { M }$ encompasses both the bias introduced by the use of a neural network and the bias introduced by the use of an approximation of the score.
+
+# 5 Related work
+
+In this section we discuss previous work on parametrizing family of distributions for manifold-valued data. Here, the manifold structure is considered to be prescribed, in contrast with methods that jointly learn the manifold structure and density (e.g. Brehmer and Cranmer, 2020; Caterini et al., 2021).
+
+Parametric family of distributions. The various parametric families of manifold-valued distributions that have been proposed can be categorized into three main approaches (Navarro et al., 2017): wrapping, projecting and conditioning. Wrapped distributions consider a parametric distribution on $\mathbb { R } ^ { n }$ that is pushed-forward along a surjective map $\psi : \mathbb { R } ^ { n } \to { \mathcal { M } }$ . Projected distributions are defined by marginalizing out some distribution along the normal bundle of $\mathcal { M }$ . Conditioning distributions encompass the von Mises-Fisher and Kent distributions (Fisher, 1953; Kent, 1982). Considering mixtures of these distributions is key to increase flexibility (Peel et al., 2001; Mardia et al., 2008).
+
+Push-forward of Euclidean normalizing flows. More recently, approaches leveraging the flexibility of normalizing flows (Papamakarios et al., 2019) have been proposed. Following the wrapping method described above, these methods parametrize a normalizing flow in $\mathbb { R } ^ { n }$ before being pushed along an invertible map $\psi : \mathbb { R } ^ { n } \to { \mathcal { M } }$ . However, to globally represent the manifold, the map $\psi$ needs to be a homeomorphism, which can only happen if $\mathcal { M }$ is topologically equivalent to $\mathbb { R } ^ { n }$ , hence limiting the scope of that approach. One natural choice for this map is the exponential map $\exp _ { x } : { \mathrm { T } } _ { x } { \mathcal { M } } \tilde { \cong } \mathbb { R } ^ { d }$ This approach has been taken, for instance, by Falorsi et al. (2019) and Bose et al. (2020), respectively parametrizing distributions on Lie groups and hyperbolic space.
+
+Neural ODE on manifolds. To avoid artifacts or numerical instabilities due to the manifold embedding, another line of work uses tools from Riemannian geometry to define flows directly on the manifold of interest (Falorsi and Forré, 2020; Mathieu and Nickel, 2020; Falorsi, 2021). Since these methods do not require a specific embedding mapping, they are referred as Riemannian. They extend continuous normalizing flows (CNFs) (Grathwohl et al., 2019) to the manifold setting, by implicity parametrizing flows as solutions of Ordinary Differential Equations (ODEs). As such, the parametric flow is a continuous function of time. This approach has recently been extended by Rozen et al. (2021) introducing Moser flows, whose main appeal being that it circumvents the need to solve an ODE in the training process. We refer to App. K for an in-depth discussion on the links between our work and Moser flows.
+
+Optimal transport on manifolds. Another line of work has developed flows on manifolds using tools from optimal transport. Sei (2013) introduced a flow that is given by $f _ { \theta } : x \mapsto \exp _ { x } ( \nabla \psi _ { \theta } ^ { c } )$ with $\psi _ { \theta } ^ { c }$ a $c$ -convex function and $c = d _ { \mathcal { M } } ^ { 2 }$ the squared geodesic distance. This approach is motivated by the fact that the optimal transport map takes such an expression (Ambrosio, 2003). These methods operate directly on the manifold, similarly to CNFs, yet in contrast they are discrete in time. The benefits of this approach depend on the specific choice of parametric family of $c$ -convex functions (Rezende and Racanière, 2021; Cohen et al., 2021), trading-off expressivity with scalability.
+
+Table 3: Summary of computational complexity (w.r.t. neural network forward and backward passes) for different methods. $d$ is the manifold dimension, $k$ the number of Monte Carlo batches in Moser flow’s regularizer, $N$ is the number of steps in the (adaptive) ODE solver, whereas $N ^ { * }$ is the number of steps in the SDE Euler-Maruyama solver–which can usually be lower than $N$ . Moser flow and RSGM training complexity varies if the Hutchinson stochastic estimator is used. See Table 2 for score matching losses complexity.   
+
+<table><tr><td>Method</td><td>Training</td><td>Likelihood evaluation</td><td>Sampling</td></tr><tr><td>RCNF</td><td>Solving ODE O(dN)</td><td>Solving augmented ODE O(dN)</td><td>Solving ODE O(N)</td></tr><tr><td>Moser flow</td><td>Computing div O(dk) or O(k)</td><td>Solving augmented ODE O(dN)</td><td> Solving ODE O(N)</td></tr><tr><td>RSGM</td><td>Score matching O(d) or O(1)</td><td>Solving augmented ODE O(dN)</td><td>Solving SDE O(N*)</td></tr></table>
+
+# 6 Experiments
+
+In this section we benchmark the empirical performance of RSGMs along with other manifold-valued methods introduced in Sec. 5. We also compare to a ‘Stereographic‘ score-based model, introduced in App. N. First, we assess their modelling capacity on earth and climate science spherical data. Then, we test the methods scalability with respect to manifold dimensions with a synthetic experiment on the torus $\mathbb { T } ^ { d }$ . Eventually, we evaluate the models’ regularity and time complexity with a synthetic $\mathrm { S O _ { 3 } ( \mathbb { R } ) }$ target. Experimental details are provided in App. O.
+
+# 6.1 Earth and climate science datasets on the sphere
+
+We start by evaluating RSGMs on a collection of simple datasets, each containing an empirical distribution of occurrences of earth and climate science events on the surface of the earth. These events are: volcanic eruptions (NGDC/WDS), earthquakes (NGDC/WDS), floods (Brakenridge, 2017) and wild fires (EOSDIS, 2020). We compare to previous baseline methods: Riemannian Continuous Normalizing Flows (Mathieu and Nickel, 2020), Moser Flows (Rozen et al., 2021) and a mixture of Kent distributions (Peel et al., 2001). Additionally, we consider a standard SGM on the 2D plane followed by the inverse stereographic projection which induces a density on the sphere (Gemici et al., 2016). We evaluate the log-likelihood of each model, extending to the manifold setting the likelihood computation techniques of SGMs, see App. D. We observe from Table 4, that all benchmarked methods have comparable performance when evaluated on these simple tasks with RSGM performing marginally better on most datasets. However, we empirically notice that Moser flows are slow to train and additionally that both Moser flows and stereographic SGMs are computationally expensive to evaluate.
+
+# 6.2 Synthetic data on tori
+
+We now move to another manifold, that is the torus $ { \mathbb { T } } ^ { d } = \mathbb { S } ^ { 1 } \times \cdots \times \mathbb { S } ^ { 1 }$ , so as to assess the scalability of the different methods with respect to the dimension $d$ . We consider a wrapped Gaussian target distribution on $\mathbb { T } ^ { d }$ with a random mean and unit variance. Moser flows’ (Rozen et al., 2021) loss involves a regularization term which involves an integral over the manifold, approximated by a Monte Carlo (MC) estimator with uniform proposal. This term regularizes Moser flows towards probability measures, i.e. with unit volume. We thus expect Moser flows to fail in high-dimension as the number of samples $K$ required for the MC estimator to be accurate will grows as $\mathcal { O } ( \mathrm { e } ^ { d } )$ , and the memory required to compute this estimator grows either in $\mathcal { O } ( K d )$ for exact divergences or $\mathcal O ( K )$ for approximated divergences (see Table 3).
+
+Table 4: Negative log-likelihood scores for each method on the earth and climate science datasets. Bold indicates best results (up to statistical significance). Means and confidence intervals are computed over 5 different runs. Novel methods are shown with blue shading.   
+
+<table><tr><td>Method</td><td>Volcano</td><td>Earthquake</td><td>Flood</td><td>Fire</td></tr><tr><td>Mixture of Kent</td><td>-0.80±0.47</td><td>0.33±0.05</td><td>0.73±0.07</td><td>-1.18±0.06</td></tr><tr><td>Riemannian CNF</td><td>-6.05±0.61</td><td>0.14±0.23</td><td>1.11±0.19</td><td>-0.80±0.54</td></tr><tr><td>Moser Flow</td><td>-4.21±0.17</td><td>-0.16±0.06</td><td>0.57±0.10</td><td>-1.28±0.05</td></tr><tr><td>Stereographic Score-Based</td><td>-3.80±0.27</td><td>-0.19±0.05</td><td>0.59±0.07</td><td>-1.28±0.12</td></tr><tr><td>Riemannian Score-Based</td><td>-4.92±0.25</td><td>-0.19±0.07</td><td>0.45±0.17</td><td>−1.33±0.06</td></tr><tr><td>Dataset size</td><td>827</td><td>6120</td><td>4875</td><td>12809</td></tr></table>
+
+![](images/7e25baa8fb0a4cf65890feb7c2d1dfd331025836c5af84626fe9776d60ff651e.jpg)  
+Figure 2: Trained score-based generative models on earth sciences data. The learned density is colored green-blue. Blue and red dots represent training and testing datapoints, respectively.
+
+In Fig. 3, we observe that RSGMs are able to fit well the target distribution even in high dimension, with a linear or constant computational cost—depending on the divergence estimator. In contrast, Moser flows scale poorly with the dimension, to the extent that we are unable to train them for $d \geq 1 0$ . This is due to the combination of the complexity which grows linearly with both the dimension $d$ and the number of MC samples $K$ , which itself ought to grow exponentially with $d$ —as discussed in the previous paragraph. This is illustrated by the gap between the ‘Moser’ and ‘ODE’ likelihoods which increases with the manifold dimension (see left Fig. 3).
+
+# 6.3 Synthetic data on the Special Orthogonal group
+
+In order to demonstrate the broad range of applicability of our model we now turn to the task of density estimation on the special orthogonal group $\operatorname { S O } _ { d } ( \mathbb { R } ) = \{ \mathrm { Q } \in \mathrm { M } _ { d } ( \mathbb { R } ) : \mathrm { Q Q } ^ { \top } = \mathrm { I d }$ , $\operatorname* { d e t } ( \mathrm { Q } ) = 1 \dot { } \dot { } \mathrm { J }$ . We consider the synthetic dataset consisting of samples in $\mathrm { S O _ { 3 } ( \mathbb { R } ) }$ from a mixture of wrapped normal distributions with $M$ components.
+
+We compare RSGMs against Moser flows and a wrapped-exponential baseline inspired by Falorsi et al. (2019)—where we parametrize a standard Euclidean SGM on $\mathfrak { s o } ( 3 )$ that is then pushed-forward on $\mathrm { S O _ { 3 } ( \mathbb { R } ) }$ . RSGMs are trained using the $\ell _ { t \mid 0 }$ (DSM) loss with the Varadhan approximation (see Table 2). From Table 5 we observe that, RSGMs perform consistently, whether the target distribution has few or many mixture components $M$ , as opposed to Exp-wrapped SGMs and Moser flows which only perform well in some range of $M$ . Similarly to Sec. 6.2, we find Moser flows to be much slower to train due to the large number of Monte Carlo samples needed in the reguralizer $\mathcal { K } = 1 0 ^ { 4 }$ ). We also note from Table 5 that the number of score network evaluations (NFE) is significantly lower for RSGMs, and is particularly detrimental for Moser flows $\mathrm { ( \gg 1 0 ^ { 3 } }$ ).
+
+![](images/0e3b1a9c46720a2a9d57c333a86b38ea08aec67b7eea4b99853c2569dd68cd60.jpg)  
+Figure 3: Comparison of Moser flows and RSGMs training speed and performance on the synthetic highdimension torus task. Moser flows trained with $\lambda _ { \operatorname* { m i n } } = 1$ . We report two likelihoods, the ‘Moser’ closed form density—not guaranteed to be normalized—and the ‘ODE’ likelihood given by solving an augmented ODE (as in CNFs) with the vector field induced by the Moser flow density—which is guaranteed to have unit volume.
+
+![](images/954eedf17879fa1b17338f807d99ef5cc0a27bb8a284de2d0d00724a1feff6ce.jpg)  
+(a) Histograms of $\mathrm { S O } _ { 3 } ( \mathbb { R } )$ samples from a target mix- (b) RSGMs are much more robust to hyperparameters ture distribution with $M = 4$ components, represented than Exp-wrapped SGMs. The diffusion coefficient is via their Euler angles. given by $\sigma ( t , { \bf X } _ { t } ) = \sqrt { \beta ( t ) } .$ , $\beta ( t ) = \beta _ { 0 } + ( \beta _ { f } - \beta _ { 0 } ) t$ .   
+Figure 4: Trained score-based generative models on synthetic $\mathrm { S O } _ { 3 } ( \mathbb { R } )$ data.
+
+Table 5: Test log-likelihood and associated number of function evaluations (NFE) in $1 0 ^ { 3 }$ on the synthetic mixture distribution with $M$ components on $\mathrm { S O } _ { 3 } ( \mathbb { R } )$ . Bold indicates best results (up to statistical significance). Means and standard deviations are computed over 5 different runs. Novel methods are shown with blue shading.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">M=16</td><td colspan="2">M=32</td><td colspan="2">M=64</td></tr><tr><td>log-likelihood</td><td>NFE</td><td>log-likelihood</td><td>NFE</td><td>log-likelihood</td><td>NFE</td></tr><tr><td>Moser Flow</td><td>0.85±0.03</td><td>2.3±0.5</td><td>0.17±0.03</td><td>2.3±0.9</td><td>-0.49±0.02</td><td>7.3±1.4</td></tr><tr><td>Exp-wrapped SGM</td><td>0.87±0.04</td><td>0.5±0.1</td><td>0.16±0.03</td><td>0.5±0.0</td><td>-0.58±0.04</td><td>0.5±0.0</td></tr><tr><td>RSGM</td><td>0.89±0.03</td><td>0.1±0.0</td><td>0.20±0.03</td><td>0.1±0.0</td><td>-0.49±0.02</td><td>0.1±0.0</td></tr></table>
+
+# 6.4 Synthetic data on hyperbolic space
+
+Finally we demonstrate RSGM on a non-compact manifold: the two dimensional hyperbolic space $\mathbb { H } ^ { 2 }$ , which is defined as the simply connected space of constant negative curvature. We use Langevin dynamics as the noising process (Eq. (3)) and target a wrapped Gaussian as the invariant distribution. We again consider a synthetic dataset of samples from a mixture of exp-wrapped normal distribution. From Fig. 5, we can qualitatively see that both score-based models are able to fit the target distribution.
+
+![](images/97731b70c25c65c7a2bd88c1b8878395496cc03283e3c664d8bf663fb68d54a2.jpg)  
+Figure 5: Samples from different probability distributions on $\mathbb { H } ^ { 2 }$ coloured w.r.t their density.
+
+# 7 Discussion and limitations
+
+In this paper we introduced Riemannian Score-Based Generative Models (RSGMs), a class of deep generative models that represent target densities supported on manifolds, as the time-reversal of Langevin dynamics. The main benefits of our method stems from its scalability to high dimensions, its applicability to a broad class of manifolds due to the diversity of available loss functions, its robustness and crucially its capacity to model complex datasets. We also provided theoretical guarantees on the convergence of RSGMs. In future work, we would like explore more generic classes of manifolds, such a ones with a boundary, along with alternative noising processes. Another promising extension concerns stochastic control on manifolds and more precisely, deriving efficient algorithms to solve Schrödinger bridges in the same spirit as De Bortoli et al. (2021) on Euclidean state spaces.
+
+# Acknowledgments
+
+We are grateful to the anonymous reviewers for their insightful comments and the for fruitful discussion more generally. We thank the geomstat team Miolane et al. (2020a). and Engineering and Physical Research Council (EPSRC) under grant EP/R013616/1. This is part of the collaboration between US DOD, UK MOD and UK EPSRC under the Multidisciplinary University Research Initiative. AD is also partially supported by the EPSRC grant EP/R034710/1 CoSines.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] Our main contribution is the extension of diffusion models on Riemannian manifolds.   
+(b) Did you describe the limitations of your work? [Yes] See Sec. 7.   
+(c) Did you discuss any potential negative societal impacts of your work? [No] The work presented in this paper focuses on the learning of score-based models on manifold. We do not foresee any immediate societal impact of such a study.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have read the ethics review guidelines and our paper conforms to them.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Yes, see A1. (b) Did you include complete proofs of all theoretical results? [Yes] Yes, proofs are postponed to the supplementary material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Experimental details are given in App. O.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Experimental details are given in App. O.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Error bars are reported for each experiment.
+
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Experimental details are given in App. O.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Sec. 6.1.   
+(b) Did you mention the license of the assets? [Yes] See App. O.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] Not applicable.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] Not applicable.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] Not applicable.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] Not applicable.   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No] Not applicable.   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No] Not applicable.

parse/dev/vqGi8Kp0wM/vqGi8Kp0wM.md

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+# MIND THE GAP: DOMAIN GAP CONTROL FOR SINGLESHOT DOMAIN ADAPTATION FOR GENERATIVE AD-VERSARIAL NETWORKS
+
+Peihao Zhu KAUST
+
+Rameen Abdal KAUST
+
+John Femiani Miami University
+
+Peter Wonka KAUST
+
+![](images/c7dacbbd371536f541fe5cd3c6da8ffd96becf37e8d964556034271441fd9bcb.jpg)  
+Figure 1: One-shot domain adaptation: (left) a single reference image from domain $B$ is used to refine a GAN $G _ { A }$ to learn $G _ { B }$ ; (center) every image in domain $A$ has an analog in domain $B$ that shares a latent code and many salient attributes; (right) because salient attributes are preserved in the new domain, many latent-edits are meaningful in the new domain.
+
+# ABSTRACT
+
+We present a new method for one shot domain adaptation. The input to our method is a trained GAN that can produce images in domain $A$ and a single reference image $I _ { B }$ from domain $B$ . The proposed algorithm can translate any output of the trained GAN from domain $A$ to domain $B$ . There are two main advantages of our method compared to the current state of the art: First, our solution achieves higher visual quality, e.g. by noticeably reducing overfitting. Second, our solution allows for more degrees of freedom to control the domain gap, i.e. what aspects of the image $I _ { B }$ are used to define the domain $B$ . Technically, we realize the new method by building on a pre-trained StyleGAN generator as GAN and a pre-trained CLIP model for representing the domain gap. We propose several new regularizers for controlling the domain gap to optimize the weights of the pre-trained StyleGAN generator so that it will output images in domain $B$ instead of domain $A$ . The regularizers prevent the optimization from taking on too many attributes of the single reference image. Our results show significant visual improvements over the state of the art as well as multiple applications that highlight improved control1.
+
+# 1 INTRODUCTION
+
+We propose a new method for domain adaptation based on a single target image. As shown in Fig. 1, given a trained GAN for domain $A$ , and a single image $I _ { B }$ from domain $B$ , our approach learns to find a corresponding image in domain $B$ for any image in domain $A$ . We can achieve this by finetuning the GAN for domain $A$ to obtain a second GAN that generates images in domain $B$ . The two GANs share a latent space so that a single latent code will generate two corresponding images, one in domain $A$ and one in domain $B$ . The main selling point of our method is that it achieves superior quality than the state of the art in single shot domain adaption. Our method is computationally lightweight and only takes a few minutes on a single GPU, so that it can be widely applied.
+
+In order to do this, we leverage multiple existing components, including two excellent pre-trained networks: First, we use StyleGAN2 (Karras et al., 2020b) as a pre-trained GAN. A follow-up version has been published on arXiv (Karras et al., 2021), but the code only became available after we finished all experiments. Second, we use a pre-trained network for image embedding, CLIP (Radford et al., 2021), to encode images as vectors. Third, we use the pioneering idea of StyleGANNADA (Gal et al., 2021), which builds upon StyleCLIP (Patashnik et al., 2021), to encode a domain gap (or domain shift) as vector in CLIP embedding space. Fourth, we leverage II2S (Zhu et al., 2020b) as GAN embedding method to transfer image $I _ { B }$ into domain $A$ to obtain a better estimation of the domain gap.
+
+Even though the visual quality of StyleGAN-NADA is already impressive when used as a single image domain adaption method, we identified multiple technical issues that can be improved to achieve another large jump in visual quality. First, and most importantly, StyleGAN-NADA was designed for zero-shot domain adaptation, and does not have a good solution to model the domain gap based on a single example image. Their reference implementation models the domain gap as a vector from the average image in domain $A$ to the given image $I _ { B }$ in CLIP embedding space. However, this leads to overfitting in practice and the transfer results lose attributes of the input images, so that input images from domain $A$ get mapped to images that are all too similar to $I _ { B }$ in domain $B$ . We identify a better solution to this problem. In fact, the domain gap should be modeled as a vector from the image $I _ { B }$ to its analog in domain $A$ , so that the image in domain $A$ shares salient within-domain attributes with the reference image. We therefore need to solve an inverse $B$ -to- $A$ domain-transfer problem, which we propose to tackle using the state-of-the-art GAN embedding method II2S (Zhu et al., 2020b). A key insight is that we can use a heavily regularized version of the II2S GAN inversion method to do the reverse problem of transferring any related image (from a domain $B$ ) into the domain $A$ , helping to characterize the semantic domain gap better than previous work. Further extensions enable us to fine tune the modeling of the domain gap to explicitly model which attributes of the input image should be kept. Second, we propose multiple new regularizers to improve the quality. Third, we propose a technical improvement to the heuristic layer selection proposed in StyleGAN-NADA that is more straightforward and robust.
+
+In summary, we make the following contributions:
+
+1. We reduce the mode collapse/overfitting problem which often occurs in one-shot and fewshot domain adaptation. Our results look similar to the target domain images with fewer artifacts. These results are also faithful to the identities of the source domain images and able to capture fine details.   
+2. Our domain adaptation provides more freedom to control the “similarity” between images across domains that share a common latent-code, which makes a large number of downstream applications possible, e.g., pose adaptation, lighting adaptation, expression adaptation, texture adaptation, interpolation, and layer mixing, using state-of-the-art image editing frameworks.
+
+# 2 RELATED WORK
+
+Domain adaptation. Domain adaptation is the task of adapting a model to different domains. Different works in this area (Bousmalis et al., 2016; 2017; Na et al., 2020; Wang & Breckon, 2020; Kang et al., 2019) try to learn diverse domain independent representations using the source domain to make predictions, such as image classification, in the target domains. More importantly, generating diverse representations of images by combining natural language supervision has been of interest to the computer vision and NLP research communities (Frome et al., 2013). Recently, OpenAI’s Contrastive Language-Image Pretraining (CLIP) (Radford et al., 2021) work established that transformer, and large datasets, could generate transferable visual models. In CLIP, both images and text are represented by high dimensional semantic-embedding vectors, which can then be used for zero-shot learning.
+
+GAN-based domain adaptation. In the GAN domain, various models and training strategies have been proposed for few-shot domain adaptation tasks (Bousmalis et al., 2017; ZHANG et al., 2018; Li et al., 2020; Liu et al., 2019). Most relevant to our work, the domain adaptation methods (Patashnik et al., 2021; Gal et al., 2021; Jang et al., 2021; Song et al., 2021) that build upon StyleGAN (Karras et al., 2019; 2020b;a) demonstrate impressive visual quality and semantic interpretability in the target domain. These methods can be broadly classified into few-shot and single-shot domain adaptation methods.
+
+A notable few-shot method, StyleGAN-ADA (Karras et al., 2020a) proposes an adaptive discriminator augmentation method to train StyleGAN on limited data. Another work, DiffAug (Zhao et al., 2020), applies differentiable transformations to the real and generated images for robust training. A discriminator related approach, FreezeD (Mo et al., 2020), freezes lower layers of the discriminator to achieve domain adaptation. Toonify (justinpinkney/toonify) interpolates between the modelweights of different generators to generate samples from a novel domain. A more recent work (Ojha et al., 2021), reduces overfitting on limited data by preserving the relative similarities and differences in the instances of samples in the source domain using cross domain correspondence.
+
+Latent space interpretation and semantic editing. GAN interpretation and understanding of the latent space has been a topic of interest since the advent of GANs. Some notable works in this domain (Bau et al., 2018; 2019; Hark ¨ onen et al., 2020; Shen et al., 2020; Tewari et al., 2020a) ¨ have led to many GAN-based image editing applications. More recent studies into the activation space of StyleGAN have demonstrated that the GAN can be exploited for downstream tasks like unsupervised and few-shot part segmentation (Zhang et al., 2021; Tritrong et al., 2021; Abdal et al., 2021a; Collins et al., 2020; Bielski & Favaro, 2019), extracting 3D models of the objects (Pan et al., 2021; Chan et al., 2020) and other semantic image editing applications (Zhu et al., 2021; Tan et al., 2020; Wu et al., 2020; Patashnik et al., 2021).
+
+Image embedding is one of the approaches used to study the interpretability of the GANs. To enable the semantic editing of a given image using GANs, one needs to embed/project the image into its latent space. Image2StyleGAN (Abdal et al., 2019) embeds images into the extended StyleGAN space called $W +$ space. Some followup works (Zhu et al., 2020a; Richardson et al., 2020; Tewari et al., 2020b) introduce regularizers and encoders to keep the latent code faithful to the original space of the StyleGAN. Improved-Image2StyleGAN (II2S) (Zhu et al., 2020b) uses $P _ { N }$ space to regularize the embeddings for high-quality image reconstruction and image editing. We use this method to embed real images into the StyleGAN and show that our domain adaptation preserves the properties of the original StyleGAN in Sec 4.
+
+Image editing is another tool to identify the concepts learned by a GAN. In the StyleGAN domain, recent works (Hark ¨ onen et al., 2020; Shen et al., 2020; Tewari et al., 2020a; Abdal et al., ¨ 2021b) extract meaningful linear and non-linear paths in the latent space. InterfaceGAN (Shen et al., 2020) finds linear directions to edit latent-codes in a supervised manner. On the other hand, GANSpace (Hark ¨ onen et al., 2020) extracts unsupervised linear directions for editing using PCA ¨ in the $W$ space. Another framework, StyleRig (Tewari et al., 2020a), maps the latent space of the GAN to a 3D model. StyleFlow (Abdal et al., 2021b) extracts non-linear paths in the latent space to enable sequential image editing. In this work, we will use StyleFlow to test the semantic editing of our domain adapted images.
+
+In the area of text-based image editing, StyleCLIP (Patashnik et al., 2021) extends CLIP to perform GAN-based image editing. StyleCLIP uses the CLIP embedding vector to traverse the StyleGAN manifold, by adjusting the latent-codes of a GAN, in order to make a generated image’s CLIP embedding similar to the target vector, while remaining close to the input in latent space. A downside to this approach is that these edits are unable to shift the domain of a GAN outside its original manifold. However, their use of CLIP embeddings inspired StyleGAN-NADA (Gal et al., 2021), which creates a new GAN using refinement learning to do zero-shot domain adaptation. Although unpublished, they also demonstrate one-shot domain adaptation in their accompanying code. The original and target domain are represented by CLIP text embeddings. The difference of the embeddings represents a direction used to shift the domains. Although in the accompanying source-code (rinongal/StyleGAN NADA), they use a bootstrap-estimate of the mean CLIP image embedding of the original domain, and use a reference image or its CLIP image embedding to represent the new domain.
+
+# 3 METHOD
+
+Our approach involves fine-tuning a GAN trained for some original domain $A$ , e.g. FFHQ faces, to adapt it to a new related domain $B$ . In our approach, the images in $A$ and the images in $B$ are related to each-other by a common latent code. Any image which can be generated or embedded in domain $A$ can be transferred to a corresponding and similar image in $B$ . We use the CLIP embeddings as a semantic-space in order to model the difference between domains $A$ and $B$ , and we use StyleGAN (Karras et al., 2018; 2020b) as the image generator. A key to our approach is to preserve directions within and across domains as illustrated in Fig. 3. Before fine-tuning the GAN for domain $A$ (to obtain the GAN for domain $B$ ), we determine a domain-gap direction. This direction, called $v ^ { \mathrm { r e f } }$ , is a vector in CLIP embedding space which points towards a reference image $I _ { B }$ which is in domain $B$ from its corresponding image $I _ { A }$ in which is in domain $A$ . We use the CLIP image-embedding model $E _ { I }$ to find
+
+$$
+v ^ { \mathrm { r e f } } = E _ { I } ( I _ { B } ) - E _ { I } ( I _ { A } ) .
+$$
+
+Finding $I _ { A }$ in domain A for a given image in domain B is a significant limitation in the current state of the art, StyleGAN-NADA (Gal et al., 2021), as they use the mean of domain $A$ . The mean of domain A is a very crude approximation for $I _ { A }$ . Instead, we propose an inverse domain adaption step, by projecting the image $I _ { B }$ into the domain $A$ to find a sample that is more similar and specific to the reference image than the mean of domain $A$ . In principle, this problem is also a domain adaption problem similar to the problem we are trying to solve, just in the inverse direction. The major difference is that we have a pre-trained GAN available in domain A.
+
+We use the II2S GAN-inversion method (Zhu et al., 2020b) in order to find a latent code for an image similar to $I _ { B }$ that is plausibly within domain $A$ . The I2S and II2S methods use an extended version of $W$ space from StyleGAN2. The $W$ code is used 18 times, once for each style block in StyleGAN2. When allowing each element to vary independently, the resulting latent space is called $W +$ space Abdal et al. (2019; 2020); Zhu et al. (2020b). I2S showed that the additional degrees of freedom allow GAN inversion for a wider set of images with very detailed reconstruction capabilities, and II2S showed that an additional regularization term to keep the latent codes close to their original distribution made latent-code manipulation more robust. II2S uses a hyperparameter, $\lambda$ , which can be increased in order to generate latent codes using more regularization, and therefore in higher density regions of the $W +$ latent space. The effect of this parameter is shown in Fig. 2. The value suggested in the original work was $\lambda = 0 . 0 0 1$ , however, low values of lambda allow II2S to find latent codes that are too far away from the latent-codes produced by the mapping network of the original GAN and thus produce images that are less plausible to have come from domain $A$ , underestimating the gap between domains. In the context of domain shift we find it is useful to use $\lambda = 0 . 0 1$ as illustrated in Fig. 2. The result is a latent code $w ^ { \mathrm { r e f } }$ in $W +$ space which is shifted towards a high-density portion of the domain $A$ . Then the image generated from that code, $I _ { A }$ , is an image in domain $A$ that corresponds to $I _ { B }$ .
+
+![](images/33664606250168b7c256a1b040c0d5a66237efd23e7565538a93668f87fff82d.jpg)  
+Figure 2: An illustration showing how II2S embeds $I _ { B }$ in the original StyleGAN domain $A$ , shown for two different values of $\lambda$ . Reference images from other domains are shown in the top row. The value recommended by Zhu et al. (2020b) is shown in the second row, and the value used in this work is shown in the third row. Although there is some subjectivity involved, we believe that the large value $\lambda = 1 \mathrm { e } { - 2 }$ is needed for II2S to find images that plausibly could belong to the domain $A$ , which in this case is FFHQ faces.
+
+Training As illustrated in Fig. 2, we use II2S to find an image $I _ { A }$ which we consider to be similar to $I _ { B }$ but still plausibly within a domain $A$ . In principle, it is possible that II2S finds $I _ { A }$ so that $I _ { B }$ is similar enough to be considered the same, in which case the two domains overlap. However, we are concerned with the cases where the domains are different, and the vector $v ^ { \mathrm { r e f } }$ indicates the direction of a gap, or shift, between domain $A$ and domain $B$ . We use refinement learning to train a new generator, $G _ { B }$ , so that images generated from $G _ { B }$ are shifted parallel to $v ^ { \mathrm { r e f } }$ in CLIP space, relative to images from $G _ { A }$ . The desired shift is indicated by the red arrows in Fig. 3. During training, latent codes $w$ are generated using the mapping network of StyleGAN2. Both $G _ { A }$ and $G _ { B }$ are used to generate images from the same latent code, but the weights of $G _ { A }$ are frozen and only $G _ { B }$ is updated during training. The goal of refinement learning is that $G _ { B }$ will preserve semantic information that is within domain $A$ but also that it will generate image shifted across a gap between domains. When refining the generator for domain $B$ , we freeze the weights of the StyleGAN2 ‘ToRGB’ layers, and the mapping network is also frozen. The overall process of training is illustrated in Fig. 4.
+
+![](images/ee65bd1d60143097769d365a0c96e0e16bd71be15187ce0e62eb7aa5cf10da93.jpg)  
+Figure 3: The vectors in the CLIP image embedding space, $E _ { I }$ , which control domain adaptation. Each domain is depicted here as a dashed outline; the vectors $v ^ { \mathrm { r e f } }$ and $v ^ { \mathrm { s a m p } }$ cross between the two domains and are used to refine a generator for domain $B$ . Corresponding images should be shifted in the same direction. The vectors $v _ { A }$ and $v _ { B }$ model important semantic differences within each domain that should also be preserved by domain transfer. $G _ { A } ( w )$ and $G _ { B } ( w )$ are corresponding images for an arbitrary latent-code $w$ encountered during training. Style mixing (shown on the right) shifts a part of the latent code towards the reference image effecting the result in both domains.
+
+The goal of training is to shift CLIP embeddings from domain $A$ in a direction parallel to $v ^ { \mathrm { r e f } }$ . We use the vector $v ^ { \mathrm { s a m p } }$ to represent the current domain shift of the network $G _ { B }$ during training, on a single sample. We have
+
+$$
+v ^ { \mathrm { s a m p } } = E _ { I } ( G _ { B } ( w ) ) - E _ { I } ( G _ { A } ( w ) )
+$$
+
+as a cross-domain vector for corresponding images generated from the same $w$ latent code using the two generators. We use the loss
+
+$$
+L _ { \mathrm { c l i p . a c r o s s } } = 1 - \mathrm { s i m } ( v ^ { \mathrm { r e f } } , v ^ { \mathrm { s a m p } } ) ,
+$$
+
+where $\begin{array} { r } { \mathrm { s i m } ( \mathbf { a } , \mathbf { b } ) = \frac { \mathbf { a } ^ { T } \mathbf { b } } { \| \mathbf { a } \| \| \mathbf { b } \| } } \end{array}$ is the cosine similarity score. This loss term is minimized when the domain shift vectors are parallel.
+
+It is important that the reference image $I _ { B }$ matches the generated image, $G _ { B } ( \boldsymbol { w } ^ { \mathrm { r e f } } )$ , both in a semantic sense, as measured by the similarity of the CLIP embeddings, and also in a visual sense. We accomplish this using two losses: $L _ { \mathrm { r e f \_ c l i p } }$ and $L _ { \mathrm { r e f . r e c } }$ . The first loss measures the change in the CLIP-embeddings of the original and reconstructed reference image,
+
+$$
+{ \cal L } _ { \mathrm { r e f . c l i p } } = 1 - \sin \left( E _ { I } \left( I _ { B } \right) , E _ { I } \left( G _ { B } ( w ^ { \mathrm { r e f } } ) \right) \right) ,
+$$
+
+ensuring that the $G _ { B }$ can reconstruct the embedding. Unlike $L _ { \mathrm { c l i p . a c c r o s s } }$ , this loss term is not based on a change in embeddings between the two domains, instead it guides $G _ { B }$ by aligning it to a global embedding in CLIP space, ensuring that $I _ { B }$ remains fixed in the domain of $G _ { B }$ .
+
+The second loss term is a reconstruction loss based on perceptual and pixel-level accuracy,
+
+$$
+{ \cal L } _ { \mathrm { r e f . r e c } } = { \cal L } _ { \mathrm { P I P S } } \left( I _ { B } , G _ { B } ( w ^ { \mathrm { r e f } } ) \right) + { \cal L } _ { 2 } \left( I _ { B } , G _ { B } ( w ^ { \mathrm { r e f } } ) \right)
+$$
+
+![](images/02e37c75b921ad9c4ef97d7d6614bf3b7fdc869f20ca1d013edd59a4407f6e80.jpg)  
+Figure 4: A process diagram for domain transfer. White rectangles indicate calculations, computed values are shown on the connecting lines. The four loss-calculations are indicated by blue rectangles, and the learnable weights of StyleGAN2 (all weights except the mapping network and the ToRGB layers) are indicated in green.
+
+where $L _ { \mathrm { P I P S } }$ is the perceptual loss from Zhang et al. (2018), and $L _ { 2 }$ is the squared euclidean difference between pixels. The purpose of this loss is to ensure that the visual, and not just the semantic, qualities of the image are preserved. This is necessary in addition to $L _ { \mathrm { r e f \_ c l i p } }$ because, while the CLIP embeddings do capture many semantic and visual qualities of the image, there are still many perceptually distinct images that could produce the same CLIP embedding. This is visible in Fig. 6, without the reconstruction loss $G _ { B }$ fails to preserve some important visual qualities (such as symmetry) of the input.
+
+There is a tendency for GANs to reduce the variation during training, especially in few-shot finetuning. We combat this by preserving the semantic information that is not related to the domain gap. A semantic change that is not related to the change in domains should not be affected by $G _ { B }$ . Therefore, the vector connecting the reference and sample images within the domain $A$ should be parallel to the corresponding vector in domain $B$ . Let $v _ { A } = { \dot { E } } _ { I } ( G _ { A } ( w ) ) - E _ { I } ( I _ { A } )$ be a vector connecting a sample image with latent-code $w$ to the reference image in the CLIP space. This vector represents semantic changes that are within domain $A$ , and we want the matching semantic changes to occur within the domain $B$ . Let $v _ { B } = E _ { I } ( G _ { B } ( w ) ) - E _ { I } ( I _ { B } )$ denote the corresponding vector in domain $B$ . We introduce the loss
+
+$$
+{ \cal L } _ { \mathrm { c l i p . w i t h i n } } = 1 - \mathrm { s i m } ( v _ { A } , v _ { B } ) ,
+$$
+
+which is minimized when the two within-domain changes are parallel.
+
+The final loss is then a weighted sum of losses
+
+$$
+L = L _ { \mathrm { c l i p . a c r o s s } } + \lambda _ { \mathrm { c l i p . w i t h i n } } L _ { \mathrm { c l i p . w i t h i n } } + \lambda _ { \mathrm { r e f . c l i p } } L _ { \mathrm { r e f . c l i p } } + \lambda _ { \mathrm { r e f . r e c } } L _ { \mathrm { r e f . r e c } } ,
+$$
+
+with empirically determined weights of $\lambda _ { \mathrm { c l i p . w i t h i n } } = 0 . 5$ , $\lambda _ { \mathrm { { r e f . c l i p } } } = 3 0$ , and $\lambda _ { \mathrm { { r e f . r e c } } } = 1 0$ . Together, these four loss terms guide the refinement process for $G _ { B }$ . Among these losses, $L _ { \mathrm { c l i p . a c r o s s } }$ was proposed by StyleGAN-NADA (Gal et al., 2021). The other losses are novel contributions of this work.
+
+Style Mixing After the training step, the generator $G _ { B }$ generates images that are semantically similar to the reference image $I _ { B }$ . However, we have observed that the visual style may not be sufficiently similar. We attribute this to the idea that the target domain may be a subset of the images produced by the new generator $G _ { B }$ . This issue was addressed in StyleGAN-NADA (Gal et al., 2021) using a second latent-mining network in order to identify a distribution of latent codes within the domain of $G _ { B }$ that better match the reference image. Our approach exploits the structure of latent codes in $W +$ space. Latent vectors in $W +$ space can be divided into 18 blocks of 512 elements, each impacting a different layer of StyleGAN2. Empirically, the latter blocks of the $W +$ code have been shown to have more effect on the style (e.g. texture and color) of the image whereas the earlier layers impact the coarse-structure or content (Zhu et al., 2021) of the image. We partition the latent code in the image into $w = ( w _ { C } , w _ { S } )$ where $w _ { C }$ consists of the first $m$ blocks of the $W +$ latent code that capture the content of the image, and $w _ { S }$ consists of the remaining blocks and captures the style. In this work, we will use $m = 7$ unless otherwise specified. Then we transfer the style from a reference image using linear interpolation, to form $\hat { w } = ( w _ { C } , \hat { w } _ { S } )$ where
+
+$$
+\hat { w } _ { S } = ( 1 - \alpha ) w _ { S } + \alpha ( w _ { S } ^ { \mathrm { r e f } } ) ,
+$$
+
+d d $w _ { S } ^ { \mathrm { r e f } }$ is last  accor $( 1 8 - m )$ blocks of e distribut $w ^ { \mathrm { r e f } }$ . Consider the distribution of images generated from randomof latent codes from the mapping network of StyleGAN2. If $w$   
+$\alpha = 0$ , then the distribution of images $G _ { B } ( \hat { w } )$ includes the reference image, but encompasses a wide variety of other fine visual styles. If $\alpha = 1$ , then the images $G _ { B } ( \hat { w } )$ will still have a diverse content, but they will all very closely follow the visual style of $I _ { B }$ . An important application of this method is in conditional editing of real photographs. To achieve that, first we take a real input image $I _ { \mathrm { r e a l } }$ and invert it in domain $A$ using II2S on the generator $G _ { A }$ in order to find a $W +$ latent code $w _ { \mathrm { r e a l } }$ . Then $G _ { B } ( w _ { \mathrm { r e a l } } )$ generates a corresponding image in domain $B$ . We can then compute $\hat { w } _ { \mathrm { r e a l } }$ by interpolating the style codes (8) so that the final image $G _ { B } ( \hat { w } _ { \mathrm { r e a l } } )$ is similar to $I _ { \mathrm { r e a l } }$ but has both content and the visual style shifted towards domain $B$ .
+
+# 4 RESULTS
+
+In this section, we will show qualitative and quantitative results of our work. The only other published method that accomplishes similar one-shot GAN domain adaptation which we are aware of is Ojha et al. (2021). They focus on few-shot domain adaptation, but they also demonstrate a capability to solve the one-shot problem. The most closely related work to our approach is StyleGANNADA (Gal et al., 2021), which is unpublished at the time of submission, however we compare to it as the main competitor. The paper mainly discusses zero-shot domain adaptation, but the approach can also accomplish one-shot domain adaptation, as demonstrated in their accompanying sourcecode. Moreover, it demonstrates impressive improvements over the state of the art and even beats many SOTA few-shot methods considering the visual quality. As our method can still significantly improve upon the results shown in StyleGAN-NADA, this underlines the importance of our idea in reducing overfitting. We compare against additional approaches in the appendix.
+
+Training and Inference Time. Given a reference image, the training time for our method is about 15 minutes for 600 iterations on a single Titan XP GPU using ADAM as an optimizer with the same settings as Gal et al. (2021). We use a batch size of 4. At inference time, there are different applications. In a basic operation, GAN generated images can be transferred with a single forward pass through a GAN generator network, which works in 0.34 seconds. Considering a more advanced operation, where existing photographs are embedded into a GAN latent space, the additional embedding time has to be considered. This embedding time is only 0.22 seconds using e4e (Tov et al., 2021) and about two minutes using II2S (Zhu et al., 2020b).
+
+Visual Evaluation. In Fig. 5, we show a comparison of our results on faces against the two most relevant competing methods – StyleGAN-NADA (Gal et al., 2021) and few-shot-domainadaptation (Ojha et al., 2021). The results show that our method remains faithful to the original identity of the embedded images in domain $A$ , while the other two methods suffer from overfitting, i.e., collapsing to narrow distributions which do not preserve salient features (for example the identity of a person). We show additional visual results in the supplemental materials, including results on cars and dogs and results for fine-tuning the domain adaptation.
+
+User Study. We also perform a user study by collecting 187 responses from Amazon Mechanical Turk in order to compare the visual quality and the domain transfer capabilities of our framework compared to the competing methods. When asked which method generates higher quality images from domain $B$ , $73 \%$ of users preferred our approach to StyleGAN-NADA, and $7 7 \%$ selected ours over Few-shot (Ojha et al., 2021). When asked which method is better at maintaining the similarity to a corresponding source image in domain $A$ , we found that $80 \%$ of the responses chose our approach over StyleGAN-NADA, and $91 \%$ preferred our approach to Few-shot. Our method outperforms the competing works in terms of the quality of the generated image, and the similarity of the generated image to the source image from domain $A$ . According to the user study, the other methods produced images that are more similar to $I _ { B }$ , but that is also an indication of overfitting and mode collapse.
+
+![](images/c03080d563567e8683fe2cda77f8c5f47d53fc6cbefffa61b3a4686c8893d9c7.jpg)  
+Figure 5: Comparison of our framework with state-of-the-art frameworks for StyleGAN domain adaptation. We compare with StyleGAN-NADA (Gal et al., 2021) and the few-shot method of Ojha et al. (2021). Each row corresponds to a different reference image $I _ { B }$ , and each column is a different real image $I _ { \mathrm { r e a l } }$ from domain $A$ . Notice that our method is able to match the styles of the reference images, while StyleGAN-NADA fails to maintain the content of the images in domain $A$ (for example the identity of a person is lost). On the other hand, the few-shot method suffers from severe mode collapse.
+
+Ablation study. We perform an ablation study to evaluate each component of our framework. In Fig. 6, we show the effect of II2S embedding, different losses and style mixing/interpolation on the output.
+
+![](images/0af535aec3c15978491267893aee93be4c4980f2b0f7a6b7c2f91697f062e707.jpg)  
+Figure 6: Ablation study of the losses and style mixing used in our framework. From left to right: the reference image $I _ { A }$ and several images from domain $A$ , the baseline approach (StyleGAN-NADA), adding II2S instead of using the mean of domain $A$ , adding $L _ { \mathrm { r e f \_ c l i p } }$ , $L _ { \mathrm { c l i p . w i t h i n } }$ , and then using style mixing. The top row shows reconstructions of the image $I _ { A }$ using $\mathbf { \dot { \boldsymbol { G } } } _ { B }$ .
+
+Image editing capabilities. Another important aspect of our method is that we are able to preserve the semantic properties of the original StyleGAN (domain $A$ ) in domain $B$ . We can make edits to the images in domain $B$ via the learned generator $G _ { B }$ without retraining the image editing frameworks on the new domain. Fig. 7 shows image editing capabilities in the new domain $B$ . We use StyleFlow edits such as lighting, pose, gender etc. to show the fine-grained edits possible in the new domain.
+
+![](images/dd78bb9f0093e609d00b183ab7fa4bde091065a1d5e750b4eca5a1bcd03b6afc.jpg)  
+Figure 7: Image editing capabilities of the new domain $B$ using StyleFlow (Abdal et al., 2021b). This figure shows the editing results of the embedded real image $I _ { r e a l }$ transferred to domain $B$ . Notice that our method preserves the semantic properties of the original StyleGAN.
+
+Limitations Our method has several limitations (See Fig. 8). Some of these limitations are inherent due to the challenging nature of the problem of single-shot domain adaptation. Other limitations can be addressed in future work. First, when we find the initial image in domain $A$ that corresponds to the input in domain $B$ , we do not attempt to control for the semantic similarity. Future work should encourage the images to have similar semantics. Second, we can only transfer between related domains. For example transferring FFHQ faces into the domain of cars is not explored in this paper. Third, also relevant to the original distribution of the StyleGAN, embeddings into the StyleGAN work best when the objects are transformed to the canonical positions (for example face poses that are the same as FFHQ). Extreme poses of the objects in the reference images sometimes fail.
+
+![](images/6cd2a83bc60d6c09e50cb7ff667b7eb517740ab10ebd9e7b29a345392f1891e0.jpg)
+
+Figure 8: Some failure cases of our method. In these examples, we observe that the identity of the face is compromised a bit more than in typical examples of our method.
+
+# 5 CONCLUSIONS
+
+We propose a novel method for single shot domain adaption. The main achievement of this work is to obtain results of unprecedented quality while reducing overfitting observed in previous work. The technical key components of our work are a method to model the domain gap as vector in CLIP embedding space, a way to preserve within-domain variation, and several extensions for fine-grained attribute-based control. We also introduce several new regularizers and a style mixing approach.
+
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+# A APPENDIX: ADDITIONAL RESULTS
+
+A.1 VISUAL EVALUATION OF STYLE TRANSFER.
+
+We provide additional visual evaluation of the results. In Fig. 9 and 10 we show results of domain adaptation applied to faces. The input photographs are in the top row and the reference images are in the first column. We can see that the results take on the style of the reference image, even though the reference image is far outside the original GAN’s latent space. Also, we notice that overfitting is successfully limited, as each result maintains several important aspects of the input image. In Fig. 13 and 14 we show results for cars, cats, and dogs on the same task. This shows that our method is consistent across different StyleGAN objects/datasets.
+
+# A.2 QUANTITATIVE COMPARISON OF SKETCH IMAGES.
+
+We calculate the FID (Heusel et al., 2017) between 1,000 generated images and the entire sketch dataset. Additionally, we report the precision and recall metric (Kynka��anniemi et al., 2019) to ¨ measure the quality and diversity respectively. As shown in Tab. 1, our method outperforms the contemporary methods Few-Shot (Ojha et al., 2021), and StyleGAN-NADA (Gal et al., 2021) on both metrics.
+
+Another contemporary method, TargetCLIP (Chefer et al., 2021), is capable of one-shot ‘essence transfer’ using a latent-edit, however as the weights of the generator are not modified their approach is restricted to the manifold of the original generator. Because it cannot shift to a completely new domain, TargetCLIP failed to produce any sketch images and has a precision ${ } = 0$ . Because the images it did generate are in the original space of StyleGAN it has high recall (0.29), but this number is not meaningful.
+
+Unsurprisingly, all one-shot domain transfer methods have low recall (low diversity) but it is significant that ours is the only approach with positive recall to within 2 significant digits.
+
+Table 1: Quantitative comparison on one-shot adaptation between few-shot-domain-adaptation, StyleGAN-NADA, and our method. Evaluation metrics include FID, precision, and recall (higher means higher diversity).   
+
+<table><tr><td>One Shot Method</td><td>FID↓</td><td>precision↑</td><td>recall个</td></tr><tr><td>Few-shot (Ojha et al., 2021)</td><td>158.86</td><td>0.00</td><td>0.00</td></tr><tr><td>SG-NADA (Gal et al., 2021)</td><td>124.55</td><td>0.12</td><td>0.00</td></tr><tr><td>Ours</td><td>78.35</td><td>0.33</td><td>0.02</td></tr></table>
+
+# A.3 MULTI-SHOT DOMAIN ADAPTATION.
+
+Although it was designed for one-shot domain adaptation, our method can be extended to few-shot domain adaptation by using multiple input/reference image pairs $( I _ { A } , I _ { B } )$ . In Fig. 11, We show the visual improvement obtained using 3-shot reference images.
+
+# A.4 CONTROLLING THE STYLE GAP
+
+Our method provides a way to control the domain gap between the domain $A$ and domain $B$ by explicitly controlling the style of the images sampled from or embedded in domain $A$ . Fig. 12 shows that we can control the degree to which style from the reference image is preserved by increasing the style-mixing parameter $\alpha$ , which is not possible with any of the competing methods. This gives users more control over content generation and editing.
+
+# A.5 ADDITIONAL COMPARISON
+
+In addition to our comparison with StyleGAN-NADA (Gal et al., 2021) and few-shot domain adaptation (Ojha et al., 2021), we compare against three additional methods in Fig. 15. These include one concurrently developed method called TargetCLIP (Chefer et al., 2021) as well as two other methods that work on lower resolution images for one-shot domain transfer. These are the method of Gatys et. al Gatys et al. (2016) and the the AdaIN approach (Huang & Belongie, 2017). Our visual results compare favorably against the new methods in Fig. 15 with respect to preserving the identity of the original image while also generating images that belong to the new domain.
+
+![](images/f6d521cee1abe9205b74a3b4eceff9db85bcbf0114bc6baeff2253793778b714.jpg)  
+Figure 9: Style transfer results obtained by our method after style interpolation in domain $B$ . The top row represents the real images embedded in the latent space of $G _ { A }$ (domain $A$ ) whose latent codes are then used by $G _ { B }$ (domain $B$ ). The first column represents the reference images $I _ { B }$ which are input to our domain adaptation framework.
+
+# A.6 INFERENCE AND EDITING TIME
+
+Our proposed approach uses II2S for training and inference and StyleFlow (Abdal et al., 2021b) for editing in the new domain. GAN inversion using II2S on HD $( 1 0 2 4 \times 1 0 2 4 )$ images takes 150 seconds on average, and each latent-code edit operation takes 0.47 seconds. Generating the images afterwards takes an addition 0.34 seconds. Note that the run-time is dominated by GAN -inversion using II2S, however as we show in Fig. 16 once training is completed, we can use other GAN inversion methods to accomplish the edits. With e4e (Tov et al., 2021) inversion is only 0.22 seconds and the entire process of inversion, editing, and generating the edited image can be accomplished in approximately one second.
+
+![](images/1ab5697c70e36b56b78ff4b0138887938cf9b80475354bb0e466eba264831096.jpg)  
+Figure 10: The structure of rows and columns is the same as in Fig. 9. Note: our method also works well when the reference images are real face images.
+
+single-shot
+
+![](images/48976bfef222ed63404411d895a71d93328ba4685f52f09ad62dfed72cf1e606.jpg)  
+Figure 11: Our method extends to deal with multiple reference images. The figure compares the results using 3 reference images and using single reference image. It can be observed that our method can better catch the general style and achieve more stable results when given multiple reference images.
+
+![](images/a146406cb2501b677b29f849488c371b3a8cb1e40b215db528a71df4f3f4fb9d.jpg)  
+Figure 12: Style interpolation results achieved by our framework. Unlike the competing methods, our method has an explicit control over the styles in the domain $B$ . Each sub figure shows a reference image and images embedded in domain $A$ . Notice that we can control the amount of variation in style depending on a parameter $\alpha$ that can be specified by a user.
+
+![](images/76778aedb90dd70d7229bfa173dcf3e22f7c1f08db614d8e8f8bfb295cbc7815.jpg)  
+Figure 13: Our domain transfer results on cars. The structure of rows and columns is the same as in Fig. 9.
+
+![](images/ffbd88f8fe672433ddd978ad422bd54460c462b1a1f03362df0c0cc6c360067a.jpg)  
+Figure 14: Our domain transfer results on cats and dogs. The structure of rows and columns is the same as in Fig. 9.
+
+![](images/f45952289b221e6c67d15499ae7160ee3802f5b158271bc0daf501f7a26dc4df.jpg)  
+Figure 15: Additional comparisons with other baseline methods including the concurrent method TargetCLIP (Chefer et al., 2021) as well as two lower-resolution methods from Gatys et al. (2016) and AdaIN (Huang & Belongie, 2017). One-shot reference images from domain $B$ are shown in the left column. Each image is the result of transferring the image in the top row into the new domain. Compare these images to our method in Fig. 7, our proposed approach has fewer artifacts while preserving the identity of the image in domain $A$ .
+
+![](images/96cbed344ca85425e5d1f46802b3109da67e173fb3ba972b37a77f807b8aa484.jpg)  
+Figure 16: Comparison domain-transfer and editing using II2S vs e4e. The new GAN is always trained using II2S, but once training is complete, e4e can be used to transfer images into the new domain. II2S takes 2.5 minutes to embed the image, while e4e needs about 0.22 seconds. StyleFlow editing takes 0.47 seconds, and StyleGAN image generation takes about 0.34 seconds.

parse/dev/vqGi8Kp0wM/vqGi8Kp0wM_content_list.json

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+    {
+        "type": "text",
+        "text": "MIND THE GAP: DOMAIN GAP CONTROL FOR SINGLESHOT DOMAIN ADAPTATION FOR GENERATIVE AD-VERSARIAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Peihao Zhu KAUST ",
+        "bbox": [
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+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Rameen Abdal KAUST ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "John Femiani Miami University ",
+        "bbox": [
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+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Peter Wonka KAUST ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7dacbbd371536f541fe5cd3c6da8ffd96becf37e8d964556034271441fd9bcb.jpg",
+        "image_caption": [
+            "Figure 1: One-shot domain adaptation: (left) a single reference image from domain $B$ is used to refine a GAN $G _ { A }$ to learn $G _ { B }$ ; (center) every image in domain $A$ has an analog in domain $B$ that shares a latent code and many salient attributes; (right) because salient attributes are preserved in the new domain, many latent-edits are meaningful in the new domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We present a new method for one shot domain adaptation. The input to our method is a trained GAN that can produce images in domain $A$ and a single reference image $I _ { B }$ from domain $B$ . The proposed algorithm can translate any output of the trained GAN from domain $A$ to domain $B$ . There are two main advantages of our method compared to the current state of the art: First, our solution achieves higher visual quality, e.g. by noticeably reducing overfitting. Second, our solution allows for more degrees of freedom to control the domain gap, i.e. what aspects of the image $I _ { B }$ are used to define the domain $B$ . Technically, we realize the new method by building on a pre-trained StyleGAN generator as GAN and a pre-trained CLIP model for representing the domain gap. We propose several new regularizers for controlling the domain gap to optimize the weights of the pre-trained StyleGAN generator so that it will output images in domain $B$ instead of domain $A$ . The regularizers prevent the optimization from taking on too many attributes of the single reference image. Our results show significant visual improvements over the state of the art as well as multiple applications that highlight improved control1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a new method for domain adaptation based on a single target image. As shown in Fig. 1, given a trained GAN for domain $A$ , and a single image $I _ { B }$ from domain $B$ , our approach learns to find a corresponding image in domain $B$ for any image in domain $A$ . We can achieve this by finetuning the GAN for domain $A$ to obtain a second GAN that generates images in domain $B$ . The two GANs share a latent space so that a single latent code will generate two corresponding images, one in domain $A$ and one in domain $B$ . The main selling point of our method is that it achieves superior quality than the state of the art in single shot domain adaption. Our method is computationally lightweight and only takes a few minutes on a single GPU, so that it can be widely applied. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In order to do this, we leverage multiple existing components, including two excellent pre-trained networks: First, we use StyleGAN2 (Karras et al., 2020b) as a pre-trained GAN. A follow-up version has been published on arXiv (Karras et al., 2021), but the code only became available after we finished all experiments. Second, we use a pre-trained network for image embedding, CLIP (Radford et al., 2021), to encode images as vectors. Third, we use the pioneering idea of StyleGANNADA (Gal et al., 2021), which builds upon StyleCLIP (Patashnik et al., 2021), to encode a domain gap (or domain shift) as vector in CLIP embedding space. Fourth, we leverage II2S (Zhu et al., 2020b) as GAN embedding method to transfer image $I _ { B }$ into domain $A$ to obtain a better estimation of the domain gap. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Even though the visual quality of StyleGAN-NADA is already impressive when used as a single image domain adaption method, we identified multiple technical issues that can be improved to achieve another large jump in visual quality. First, and most importantly, StyleGAN-NADA was designed for zero-shot domain adaptation, and does not have a good solution to model the domain gap based on a single example image. Their reference implementation models the domain gap as a vector from the average image in domain $A$ to the given image $I _ { B }$ in CLIP embedding space. However, this leads to overfitting in practice and the transfer results lose attributes of the input images, so that input images from domain $A$ get mapped to images that are all too similar to $I _ { B }$ in domain $B$ . We identify a better solution to this problem. In fact, the domain gap should be modeled as a vector from the image $I _ { B }$ to its analog in domain $A$ , so that the image in domain $A$ shares salient within-domain attributes with the reference image. We therefore need to solve an inverse $B$ -to- $A$ domain-transfer problem, which we propose to tackle using the state-of-the-art GAN embedding method II2S (Zhu et al., 2020b). A key insight is that we can use a heavily regularized version of the II2S GAN inversion method to do the reverse problem of transferring any related image (from a domain $B$ ) into the domain $A$ , helping to characterize the semantic domain gap better than previous work. Further extensions enable us to fine tune the modeling of the domain gap to explicitly model which attributes of the input image should be kept. Second, we propose multiple new regularizers to improve the quality. Third, we propose a technical improvement to the heuristic layer selection proposed in StyleGAN-NADA that is more straightforward and robust. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, we make the following contributions: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1. We reduce the mode collapse/overfitting problem which often occurs in one-shot and fewshot domain adaptation. Our results look similar to the target domain images with fewer artifacts. These results are also faithful to the identities of the source domain images and able to capture fine details.   \n2. Our domain adaptation provides more freedom to control the “similarity” between images across domains that share a common latent-code, which makes a large number of downstream applications possible, e.g., pose adaptation, lighting adaptation, expression adaptation, texture adaptation, interpolation, and layer mixing, using state-of-the-art image editing frameworks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Domain adaptation. Domain adaptation is the task of adapting a model to different domains. Different works in this area (Bousmalis et al., 2016; 2017; Na et al., 2020; Wang & Breckon, 2020; Kang et al., 2019) try to learn diverse domain independent representations using the source domain to make predictions, such as image classification, in the target domains. More importantly, generating diverse representations of images by combining natural language supervision has been of interest to the computer vision and NLP research communities (Frome et al., 2013). Recently, OpenAI’s Contrastive Language-Image Pretraining (CLIP) (Radford et al., 2021) work established that transformer, and large datasets, could generate transferable visual models. In CLIP, both images and text are represented by high dimensional semantic-embedding vectors, which can then be used for zero-shot learning. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "GAN-based domain adaptation. In the GAN domain, various models and training strategies have been proposed for few-shot domain adaptation tasks (Bousmalis et al., 2017; ZHANG et al., 2018; Li et al., 2020; Liu et al., 2019). Most relevant to our work, the domain adaptation methods (Patashnik et al., 2021; Gal et al., 2021; Jang et al., 2021; Song et al., 2021) that build upon StyleGAN (Karras et al., 2019; 2020b;a) demonstrate impressive visual quality and semantic interpretability in the target domain. These methods can be broadly classified into few-shot and single-shot domain adaptation methods. ",
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+    {
+        "type": "text",
+        "text": "A notable few-shot method, StyleGAN-ADA (Karras et al., 2020a) proposes an adaptive discriminator augmentation method to train StyleGAN on limited data. Another work, DiffAug (Zhao et al., 2020), applies differentiable transformations to the real and generated images for robust training. A discriminator related approach, FreezeD (Mo et al., 2020), freezes lower layers of the discriminator to achieve domain adaptation. Toonify (justinpinkney/toonify) interpolates between the modelweights of different generators to generate samples from a novel domain. A more recent work (Ojha et al., 2021), reduces overfitting on limited data by preserving the relative similarities and differences in the instances of samples in the source domain using cross domain correspondence. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Latent space interpretation and semantic editing. GAN interpretation and understanding of the latent space has been a topic of interest since the advent of GANs. Some notable works in this domain (Bau et al., 2018; 2019; Hark ¨ onen et al., 2020; Shen et al., 2020; Tewari et al., 2020a) ¨ have led to many GAN-based image editing applications. More recent studies into the activation space of StyleGAN have demonstrated that the GAN can be exploited for downstream tasks like unsupervised and few-shot part segmentation (Zhang et al., 2021; Tritrong et al., 2021; Abdal et al., 2021a; Collins et al., 2020; Bielski & Favaro, 2019), extracting 3D models of the objects (Pan et al., 2021; Chan et al., 2020) and other semantic image editing applications (Zhu et al., 2021; Tan et al., 2020; Wu et al., 2020; Patashnik et al., 2021). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Image embedding is one of the approaches used to study the interpretability of the GANs. To enable the semantic editing of a given image using GANs, one needs to embed/project the image into its latent space. Image2StyleGAN (Abdal et al., 2019) embeds images into the extended StyleGAN space called $W +$ space. Some followup works (Zhu et al., 2020a; Richardson et al., 2020; Tewari et al., 2020b) introduce regularizers and encoders to keep the latent code faithful to the original space of the StyleGAN. Improved-Image2StyleGAN (II2S) (Zhu et al., 2020b) uses $P _ { N }$ space to regularize the embeddings for high-quality image reconstruction and image editing. We use this method to embed real images into the StyleGAN and show that our domain adaptation preserves the properties of the original StyleGAN in Sec 4. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Image editing is another tool to identify the concepts learned by a GAN. In the StyleGAN domain, recent works (Hark ¨ onen et al., 2020; Shen et al., 2020; Tewari et al., 2020a; Abdal et al., ¨ 2021b) extract meaningful linear and non-linear paths in the latent space. InterfaceGAN (Shen et al., 2020) finds linear directions to edit latent-codes in a supervised manner. On the other hand, GANSpace (Hark ¨ onen et al., 2020) extracts unsupervised linear directions for editing using PCA ¨ in the $W$ space. Another framework, StyleRig (Tewari et al., 2020a), maps the latent space of the GAN to a 3D model. StyleFlow (Abdal et al., 2021b) extracts non-linear paths in the latent space to enable sequential image editing. In this work, we will use StyleFlow to test the semantic editing of our domain adapted images. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the area of text-based image editing, StyleCLIP (Patashnik et al., 2021) extends CLIP to perform GAN-based image editing. StyleCLIP uses the CLIP embedding vector to traverse the StyleGAN manifold, by adjusting the latent-codes of a GAN, in order to make a generated image’s CLIP embedding similar to the target vector, while remaining close to the input in latent space. A downside to this approach is that these edits are unable to shift the domain of a GAN outside its original manifold. However, their use of CLIP embeddings inspired StyleGAN-NADA (Gal et al., 2021), which creates a new GAN using refinement learning to do zero-shot domain adaptation. Although unpublished, they also demonstrate one-shot domain adaptation in their accompanying code. The original and target domain are represented by CLIP text embeddings. The difference of the embeddings represents a direction used to shift the domains. Although in the accompanying source-code (rinongal/StyleGAN NADA), they use a bootstrap-estimate of the mean CLIP image embedding of the original domain, and use a reference image or its CLIP image embedding to represent the new domain. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our approach involves fine-tuning a GAN trained for some original domain $A$ , e.g. FFHQ faces, to adapt it to a new related domain $B$ . In our approach, the images in $A$ and the images in $B$ are related to each-other by a common latent code. Any image which can be generated or embedded in domain $A$ can be transferred to a corresponding and similar image in $B$ . We use the CLIP embeddings as a semantic-space in order to model the difference between domains $A$ and $B$ , and we use StyleGAN (Karras et al., 2018; 2020b) as the image generator. A key to our approach is to preserve directions within and across domains as illustrated in Fig. 3. Before fine-tuning the GAN for domain $A$ (to obtain the GAN for domain $B$ ), we determine a domain-gap direction. This direction, called $v ^ { \\mathrm { r e f } }$ , is a vector in CLIP embedding space which points towards a reference image $I _ { B }$ which is in domain $B$ from its corresponding image $I _ { A }$ in which is in domain $A$ . We use the CLIP image-embedding model $E _ { I }$ to find ",
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+    {
+        "type": "equation",
+        "img_path": "images/9006d7ded0e0851a5dec81dce6cc91f763d0c2862ec4ed76b78d7fc9f3c47ee4.jpg",
+        "text": "$$\nv ^ { \\mathrm { r e f } } = E _ { I } ( I _ { B } ) - E _ { I } ( I _ { A } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finding $I _ { A }$ in domain A for a given image in domain B is a significant limitation in the current state of the art, StyleGAN-NADA (Gal et al., 2021), as they use the mean of domain $A$ . The mean of domain A is a very crude approximation for $I _ { A }$ . Instead, we propose an inverse domain adaption step, by projecting the image $I _ { B }$ into the domain $A$ to find a sample that is more similar and specific to the reference image than the mean of domain $A$ . In principle, this problem is also a domain adaption problem similar to the problem we are trying to solve, just in the inverse direction. The major difference is that we have a pre-trained GAN available in domain A. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use the II2S GAN-inversion method (Zhu et al., 2020b) in order to find a latent code for an image similar to $I _ { B }$ that is plausibly within domain $A$ . The I2S and II2S methods use an extended version of $W$ space from StyleGAN2. The $W$ code is used 18 times, once for each style block in StyleGAN2. When allowing each element to vary independently, the resulting latent space is called $W +$ space Abdal et al. (2019; 2020); Zhu et al. (2020b). I2S showed that the additional degrees of freedom allow GAN inversion for a wider set of images with very detailed reconstruction capabilities, and II2S showed that an additional regularization term to keep the latent codes close to their original distribution made latent-code manipulation more robust. II2S uses a hyperparameter, $\\lambda$ , which can be increased in order to generate latent codes using more regularization, and therefore in higher density regions of the $W +$ latent space. The effect of this parameter is shown in Fig. 2. The value suggested in the original work was $\\lambda = 0 . 0 0 1$ , however, low values of lambda allow II2S to find latent codes that are too far away from the latent-codes produced by the mapping network of the original GAN and thus produce images that are less plausible to have come from domain $A$ , underestimating the gap between domains. In the context of domain shift we find it is useful to use $\\lambda = 0 . 0 1$ as illustrated in Fig. 2. The result is a latent code $w ^ { \\mathrm { r e f } }$ in $W +$ space which is shifted towards a high-density portion of the domain $A$ . Then the image generated from that code, $I _ { A }$ , is an image in domain $A$ that corresponds to $I _ { B }$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/33664606250168b7c256a1b040c0d5a66237efd23e7565538a93668f87fff82d.jpg",
+        "image_caption": [
+            "Figure 2: An illustration showing how II2S embeds $I _ { B }$ in the original StyleGAN domain $A$ , shown for two different values of $\\lambda$ . Reference images from other domains are shown in the top row. The value recommended by Zhu et al. (2020b) is shown in the second row, and the value used in this work is shown in the third row. Although there is some subjectivity involved, we believe that the large value $\\lambda = 1 \\mathrm { e } { - 2 }$ is needed for II2S to find images that plausibly could belong to the domain $A$ , which in this case is FFHQ faces. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Training As illustrated in Fig. 2, we use II2S to find an image $I _ { A }$ which we consider to be similar to $I _ { B }$ but still plausibly within a domain $A$ . In principle, it is possible that II2S finds $I _ { A }$ so that $I _ { B }$ is similar enough to be considered the same, in which case the two domains overlap. However, we are concerned with the cases where the domains are different, and the vector $v ^ { \\mathrm { r e f } }$ indicates the direction of a gap, or shift, between domain $A$ and domain $B$ . We use refinement learning to train a new generator, $G _ { B }$ , so that images generated from $G _ { B }$ are shifted parallel to $v ^ { \\mathrm { r e f } }$ in CLIP space, relative to images from $G _ { A }$ . The desired shift is indicated by the red arrows in Fig. 3. During training, latent codes $w$ are generated using the mapping network of StyleGAN2. Both $G _ { A }$ and $G _ { B }$ are used to generate images from the same latent code, but the weights of $G _ { A }$ are frozen and only $G _ { B }$ is updated during training. The goal of refinement learning is that $G _ { B }$ will preserve semantic information that is within domain $A$ but also that it will generate image shifted across a gap between domains. When refining the generator for domain $B$ , we freeze the weights of the StyleGAN2 ‘ToRGB’ layers, and the mapping network is also frozen. The overall process of training is illustrated in Fig. 4. ",
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+        "image_caption": [
+            "Figure 3: The vectors in the CLIP image embedding space, $E _ { I }$ , which control domain adaptation. Each domain is depicted here as a dashed outline; the vectors $v ^ { \\mathrm { r e f } }$ and $v ^ { \\mathrm { s a m p } }$ cross between the two domains and are used to refine a generator for domain $B$ . Corresponding images should be shifted in the same direction. The vectors $v _ { A }$ and $v _ { B }$ model important semantic differences within each domain that should also be preserved by domain transfer. $G _ { A } ( w )$ and $G _ { B } ( w )$ are corresponding images for an arbitrary latent-code $w$ encountered during training. Style mixing (shown on the right) shifts a part of the latent code towards the reference image effecting the result in both domains. "
+        ],
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+        "type": "text",
+        "text": "The goal of training is to shift CLIP embeddings from domain $A$ in a direction parallel to $v ^ { \\mathrm { r e f } }$ . We use the vector $v ^ { \\mathrm { s a m p } }$ to represent the current domain shift of the network $G _ { B }$ during training, on a single sample. We have ",
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+        "type": "equation",
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+        "text": "$$\nv ^ { \\mathrm { s a m p } } = E _ { I } ( G _ { B } ( w ) ) - E _ { I } ( G _ { A } ( w ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "as a cross-domain vector for corresponding images generated from the same $w$ latent code using the two generators. We use the loss ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4fcb8701d2b063f107e5e0da83bea68a1db3fc46f59991761b2d3c2df1b4aec6.jpg",
+        "text": "$$\nL _ { \\mathrm { c l i p . a c r o s s } } = 1 - \\mathrm { s i m } ( v ^ { \\mathrm { r e f } } , v ^ { \\mathrm { s a m p } } ) ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\mathrm { s i m } ( \\mathbf { a } , \\mathbf { b } ) = \\frac { \\mathbf { a } ^ { T } \\mathbf { b } } { \\| \\mathbf { a } \\| \\| \\mathbf { b } \\| } } \\end{array}$ is the cosine similarity score. This loss term is minimized when the domain shift vectors are parallel. ",
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+    {
+        "type": "text",
+        "text": "It is important that the reference image $I _ { B }$ matches the generated image, $G _ { B } ( \\boldsymbol { w } ^ { \\mathrm { r e f } } )$ , both in a semantic sense, as measured by the similarity of the CLIP embeddings, and also in a visual sense. We accomplish this using two losses: $L _ { \\mathrm { r e f \\_ c l i p } }$ and $L _ { \\mathrm { r e f . r e c } }$ . The first loss measures the change in the CLIP-embeddings of the original and reconstructed reference image, ",
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+    {
+        "type": "equation",
+        "img_path": "images/1b4d27ee324f6b0c2d824871d0920d5c4801fe1bfef0c8adc4eb03e80b7f4222.jpg",
+        "text": "$$\n{ \\cal L } _ { \\mathrm { r e f . c l i p } } = 1 - \\sin \\left( E _ { I } \\left( I _ { B } \\right) , E _ { I } \\left( G _ { B } ( w ^ { \\mathrm { r e f } } ) \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "ensuring that the $G _ { B }$ can reconstruct the embedding. Unlike $L _ { \\mathrm { c l i p . a c c r o s s } }$ , this loss term is not based on a change in embeddings between the two domains, instead it guides $G _ { B }$ by aligning it to a global embedding in CLIP space, ensuring that $I _ { B }$ remains fixed in the domain of $G _ { B }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The second loss term is a reconstruction loss based on perceptual and pixel-level accuracy, ",
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+    {
+        "type": "equation",
+        "img_path": "images/336a8cd0e0e585b59eab46c5c0d795abed80d7d2153d87348aee91cb1aa4e99c.jpg",
+        "text": "$$\n{ \\cal L } _ { \\mathrm { r e f . r e c } } = { \\cal L } _ { \\mathrm { P I P S } } \\left( I _ { B } , G _ { B } ( w ^ { \\mathrm { r e f } } ) \\right) + { \\cal L } _ { 2 } \\left( I _ { B } , G _ { B } ( w ^ { \\mathrm { r e f } } ) \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "image",
+        "img_path": "images/02e37c75b921ad9c4ef97d7d6614bf3b7fdc869f20ca1d013edd59a4407f6e80.jpg",
+        "image_caption": [
+            "Figure 4: A process diagram for domain transfer. White rectangles indicate calculations, computed values are shown on the connecting lines. The four loss-calculations are indicated by blue rectangles, and the learnable weights of StyleGAN2 (all weights except the mapping network and the ToRGB layers) are indicated in green. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
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+    {
+        "type": "text",
+        "text": "where $L _ { \\mathrm { P I P S } }$ is the perceptual loss from Zhang et al. (2018), and $L _ { 2 }$ is the squared euclidean difference between pixels. The purpose of this loss is to ensure that the visual, and not just the semantic, qualities of the image are preserved. This is necessary in addition to $L _ { \\mathrm { r e f \\_ c l i p } }$ because, while the CLIP embeddings do capture many semantic and visual qualities of the image, there are still many perceptually distinct images that could produce the same CLIP embedding. This is visible in Fig. 6, without the reconstruction loss $G _ { B }$ fails to preserve some important visual qualities (such as symmetry) of the input. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There is a tendency for GANs to reduce the variation during training, especially in few-shot finetuning. We combat this by preserving the semantic information that is not related to the domain gap. A semantic change that is not related to the change in domains should not be affected by $G _ { B }$ . Therefore, the vector connecting the reference and sample images within the domain $A$ should be parallel to the corresponding vector in domain $B$ . Let $v _ { A } = { \\dot { E } } _ { I } ( G _ { A } ( w ) ) - E _ { I } ( I _ { A } )$ be a vector connecting a sample image with latent-code $w$ to the reference image in the CLIP space. This vector represents semantic changes that are within domain $A$ , and we want the matching semantic changes to occur within the domain $B$ . Let $v _ { B } = E _ { I } ( G _ { B } ( w ) ) - E _ { I } ( I _ { B } )$ denote the corresponding vector in domain $B$ . We introduce the loss ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8309f0cd4059fd68d60a4de0c6af860ddf914cb6c37367006ecd1a79d543a66a.jpg",
+        "text": "$$\n{ \\cal L } _ { \\mathrm { c l i p . w i t h i n } } = 1 - \\mathrm { s i m } ( v _ { A } , v _ { B } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which is minimized when the two within-domain changes are parallel. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The final loss is then a weighted sum of losses ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ae037d41098f581b58dc9dcbd0f2f7f195a68b252322cab3441e489b15081da7.jpg",
+        "text": "$$\nL = L _ { \\mathrm { c l i p . a c r o s s } } + \\lambda _ { \\mathrm { c l i p . w i t h i n } } L _ { \\mathrm { c l i p . w i t h i n } } + \\lambda _ { \\mathrm { r e f . c l i p } } L _ { \\mathrm { r e f . c l i p } } + \\lambda _ { \\mathrm { r e f . r e c } } L _ { \\mathrm { r e f . r e c } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "with empirically determined weights of $\\lambda _ { \\mathrm { c l i p . w i t h i n } } = 0 . 5$ , $\\lambda _ { \\mathrm { { r e f . c l i p } } } = 3 0$ , and $\\lambda _ { \\mathrm { { r e f . r e c } } } = 1 0$ . Together, these four loss terms guide the refinement process for $G _ { B }$ . Among these losses, $L _ { \\mathrm { c l i p . a c r o s s } }$ was proposed by StyleGAN-NADA (Gal et al., 2021). The other losses are novel contributions of this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Style Mixing After the training step, the generator $G _ { B }$ generates images that are semantically similar to the reference image $I _ { B }$ . However, we have observed that the visual style may not be sufficiently similar. We attribute this to the idea that the target domain may be a subset of the images produced by the new generator $G _ { B }$ . This issue was addressed in StyleGAN-NADA (Gal et al., 2021) using a second latent-mining network in order to identify a distribution of latent codes within the domain of $G _ { B }$ that better match the reference image. Our approach exploits the structure of latent codes in $W +$ space. Latent vectors in $W +$ space can be divided into 18 blocks of 512 elements, each impacting a different layer of StyleGAN2. Empirically, the latter blocks of the $W +$ code have been shown to have more effect on the style (e.g. texture and color) of the image whereas the earlier layers impact the coarse-structure or content (Zhu et al., 2021) of the image. We partition the latent code in the image into $w = ( w _ { C } , w _ { S } )$ where $w _ { C }$ consists of the first $m$ blocks of the $W +$ latent code that capture the content of the image, and $w _ { S }$ consists of the remaining blocks and captures the style. In this work, we will use $m = 7$ unless otherwise specified. Then we transfer the style from a reference image using linear interpolation, to form $\\hat { w } = ( w _ { C } , \\hat { w } _ { S } )$ where ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0abfbda9cb2d4d1df1a403a925b46f5f67f8d563b0749573c460a0a1655f220d.jpg",
+        "text": "$$\n\\hat { w } _ { S } = ( 1 - \\alpha ) w _ { S } + \\alpha ( w _ { S } ^ { \\mathrm { r e f } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "d d $w _ { S } ^ { \\mathrm { r e f } }$ is last  accor $( 1 8 - m )$ blocks of e distribut $w ^ { \\mathrm { r e f } }$ . Consider the distribution of images generated from randomof latent codes from the mapping network of StyleGAN2. If $w$   \n$\\alpha = 0$ , then the distribution of images $G _ { B } ( \\hat { w } )$ includes the reference image, but encompasses a wide variety of other fine visual styles. If $\\alpha = 1$ , then the images $G _ { B } ( \\hat { w } )$ will still have a diverse content, but they will all very closely follow the visual style of $I _ { B }$ . An important application of this method is in conditional editing of real photographs. To achieve that, first we take a real input image $I _ { \\mathrm { r e a l } }$ and invert it in domain $A$ using II2S on the generator $G _ { A }$ in order to find a $W +$ latent code $w _ { \\mathrm { r e a l } }$ . Then $G _ { B } ( w _ { \\mathrm { r e a l } } )$ generates a corresponding image in domain $B$ . We can then compute $\\hat { w } _ { \\mathrm { r e a l } }$ by interpolating the style codes (8) so that the final image $G _ { B } ( \\hat { w } _ { \\mathrm { r e a l } } )$ is similar to $I _ { \\mathrm { r e a l } }$ but has both content and the visual style shifted towards domain $B$ . ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we will show qualitative and quantitative results of our work. The only other published method that accomplishes similar one-shot GAN domain adaptation which we are aware of is Ojha et al. (2021). They focus on few-shot domain adaptation, but they also demonstrate a capability to solve the one-shot problem. The most closely related work to our approach is StyleGANNADA (Gal et al., 2021), which is unpublished at the time of submission, however we compare to it as the main competitor. The paper mainly discusses zero-shot domain adaptation, but the approach can also accomplish one-shot domain adaptation, as demonstrated in their accompanying sourcecode. Moreover, it demonstrates impressive improvements over the state of the art and even beats many SOTA few-shot methods considering the visual quality. As our method can still significantly improve upon the results shown in StyleGAN-NADA, this underlines the importance of our idea in reducing overfitting. We compare against additional approaches in the appendix. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training and Inference Time. Given a reference image, the training time for our method is about 15 minutes for 600 iterations on a single Titan XP GPU using ADAM as an optimizer with the same settings as Gal et al. (2021). We use a batch size of 4. At inference time, there are different applications. In a basic operation, GAN generated images can be transferred with a single forward pass through a GAN generator network, which works in 0.34 seconds. Considering a more advanced operation, where existing photographs are embedded into a GAN latent space, the additional embedding time has to be considered. This embedding time is only 0.22 seconds using e4e (Tov et al., 2021) and about two minutes using II2S (Zhu et al., 2020b). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Visual Evaluation. In Fig. 5, we show a comparison of our results on faces against the two most relevant competing methods – StyleGAN-NADA (Gal et al., 2021) and few-shot-domainadaptation (Ojha et al., 2021). The results show that our method remains faithful to the original identity of the embedded images in domain $A$ , while the other two methods suffer from overfitting, i.e., collapsing to narrow distributions which do not preserve salient features (for example the identity of a person). We show additional visual results in the supplemental materials, including results on cars and dogs and results for fine-tuning the domain adaptation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "User Study. We also perform a user study by collecting 187 responses from Amazon Mechanical Turk in order to compare the visual quality and the domain transfer capabilities of our framework compared to the competing methods. When asked which method generates higher quality images from domain $B$ , $73 \\%$ of users preferred our approach to StyleGAN-NADA, and $7 7 \\%$ selected ours over Few-shot (Ojha et al., 2021). When asked which method is better at maintaining the similarity to a corresponding source image in domain $A$ , we found that $80 \\%$ of the responses chose our approach over StyleGAN-NADA, and $91 \\%$ preferred our approach to Few-shot. Our method outperforms the competing works in terms of the quality of the generated image, and the similarity of the generated image to the source image from domain $A$ . According to the user study, the other methods produced images that are more similar to $I _ { B }$ , but that is also an indication of overfitting and mode collapse. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/c03080d563567e8683fe2cda77f8c5f47d53fc6cbefffa61b3a4686c8893d9c7.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of our framework with state-of-the-art frameworks for StyleGAN domain adaptation. We compare with StyleGAN-NADA (Gal et al., 2021) and the few-shot method of Ojha et al. (2021). Each row corresponds to a different reference image $I _ { B }$ , and each column is a different real image $I _ { \\mathrm { r e a l } }$ from domain $A$ . Notice that our method is able to match the styles of the reference images, while StyleGAN-NADA fails to maintain the content of the images in domain $A$ (for example the identity of a person is lost). On the other hand, the few-shot method suffers from severe mode collapse. "
+        ],
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+    {
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+    {
+        "type": "text",
+        "text": "Ablation study. We perform an ablation study to evaluate each component of our framework. In Fig. 6, we show the effect of II2S embedding, different losses and style mixing/interpolation on the output. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/0af535aec3c15978491267893aee93be4c4980f2b0f7a6b7c2f91697f062e707.jpg",
+        "image_caption": [
+            "Figure 6: Ablation study of the losses and style mixing used in our framework. From left to right: the reference image $I _ { A }$ and several images from domain $A$ , the baseline approach (StyleGAN-NADA), adding II2S instead of using the mean of domain $A$ , adding $L _ { \\mathrm { r e f \\_ c l i p } }$ , $L _ { \\mathrm { c l i p . w i t h i n } }$ , and then using style mixing. The top row shows reconstructions of the image $I _ { A }$ using $\\mathbf { \\dot { \\boldsymbol { G } } } _ { B }$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Image editing capabilities. Another important aspect of our method is that we are able to preserve the semantic properties of the original StyleGAN (domain $A$ ) in domain $B$ . We can make edits to the images in domain $B$ via the learned generator $G _ { B }$ without retraining the image editing frameworks on the new domain. Fig. 7 shows image editing capabilities in the new domain $B$ . We use StyleFlow edits such as lighting, pose, gender etc. to show the fine-grained edits possible in the new domain. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/dd78bb9f0093e609d00b183ab7fa4bde091065a1d5e750b4eca5a1bcd03b6afc.jpg",
+        "image_caption": [
+            "Figure 7: Image editing capabilities of the new domain $B$ using StyleFlow (Abdal et al., 2021b). This figure shows the editing results of the embedded real image $I _ { r e a l }$ transferred to domain $B$ . Notice that our method preserves the semantic properties of the original StyleGAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Limitations Our method has several limitations (See Fig. 8). Some of these limitations are inherent due to the challenging nature of the problem of single-shot domain adaptation. Other limitations can be addressed in future work. First, when we find the initial image in domain $A$ that corresponds to the input in domain $B$ , we do not attempt to control for the semantic similarity. Future work should encourage the images to have similar semantics. Second, we can only transfer between related domains. For example transferring FFHQ faces into the domain of cars is not explored in this paper. Third, also relevant to the original distribution of the StyleGAN, embeddings into the StyleGAN work best when the objects are transformed to the canonical positions (for example face poses that are the same as FFHQ). Extreme poses of the objects in the reference images sometimes fail. ",
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+    {
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+        "img_path": "images/6cd2a83bc60d6c09e50cb7ff667b7eb517740ab10ebd9e7b29a345392f1891e0.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "Figure 8: Some failure cases of our method. In these examples, we observe that the identity of the face is compromised a bit more than in typical examples of our method. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel method for single shot domain adaption. The main achievement of this work is to obtain results of unprecedented quality while reducing overfitting observed in previous work. The technical key components of our work are a method to model the domain gap as vector in CLIP embedding space, a way to preserve within-domain variation, and several extensions for fine-grained attribute-based control. We also introduce several new regularizers and a style mixing approach. ",
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+    },
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+        "text": "A APPENDIX: ADDITIONAL RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A.1 VISUAL EVALUATION OF STYLE TRANSFER. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We provide additional visual evaluation of the results. In Fig. 9 and 10 we show results of domain adaptation applied to faces. The input photographs are in the top row and the reference images are in the first column. We can see that the results take on the style of the reference image, even though the reference image is far outside the original GAN’s latent space. Also, we notice that overfitting is successfully limited, as each result maintains several important aspects of the input image. In Fig. 13 and 14 we show results for cars, cats, and dogs on the same task. This shows that our method is consistent across different StyleGAN objects/datasets. ",
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+        "type": "text",
+        "text": "A.2 QUANTITATIVE COMPARISON OF SKETCH IMAGES. ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We calculate the FID (Heusel et al., 2017) between 1,000 generated images and the entire sketch dataset. Additionally, we report the precision and recall metric (Kynka¨anniemi et al., 2019) to ¨ measure the quality and diversity respectively. As shown in Tab. 1, our method outperforms the contemporary methods Few-Shot (Ojha et al., 2021), and StyleGAN-NADA (Gal et al., 2021) on both metrics. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Another contemporary method, TargetCLIP (Chefer et al., 2021), is capable of one-shot ‘essence transfer’ using a latent-edit, however as the weights of the generator are not modified their approach is restricted to the manifold of the original generator. Because it cannot shift to a completely new domain, TargetCLIP failed to produce any sketch images and has a precision ${ } = 0$ . Because the images it did generate are in the original space of StyleGAN it has high recall (0.29), but this number is not meaningful. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Unsurprisingly, all one-shot domain transfer methods have low recall (low diversity) but it is significant that ours is the only approach with positive recall to within 2 significant digits. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be0afe5d287fc39652ae4e267629b1176d001525096cc07e30a10b79873f67c7.jpg",
+        "table_caption": [
+            "Table 1: Quantitative comparison on one-shot adaptation between few-shot-domain-adaptation, StyleGAN-NADA, and our method. Evaluation metrics include FID, precision, and recall (higher means higher diversity). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>One Shot Method</td><td>FID↓</td><td>precision↑</td><td>recall个</td></tr><tr><td>Few-shot (Ojha et al., 2021)</td><td>158.86</td><td>0.00</td><td>0.00</td></tr><tr><td>SG-NADA (Gal et al., 2021)</td><td>124.55</td><td>0.12</td><td>0.00</td></tr><tr><td>Ours</td><td>78.35</td><td>0.33</td><td>0.02</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "A.3 MULTI-SHOT DOMAIN ADAPTATION. ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Although it was designed for one-shot domain adaptation, our method can be extended to few-shot domain adaptation by using multiple input/reference image pairs $( I _ { A } , I _ { B } )$ . In Fig. 11, We show the visual improvement obtained using 3-shot reference images. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 CONTROLLING THE STYLE GAP ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our method provides a way to control the domain gap between the domain $A$ and domain $B$ by explicitly controlling the style of the images sampled from or embedded in domain $A$ . Fig. 12 shows that we can control the degree to which style from the reference image is preserved by increasing the style-mixing parameter $\\alpha$ , which is not possible with any of the competing methods. This gives users more control over content generation and editing. ",
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+    },
+    {
+        "type": "text",
+        "text": "A.5 ADDITIONAL COMPARISON ",
+        "text_level": 1,
+        "bbox": [
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+            842,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In addition to our comparison with StyleGAN-NADA (Gal et al., 2021) and few-shot domain adaptation (Ojha et al., 2021), we compare against three additional methods in Fig. 15. These include one concurrently developed method called TargetCLIP (Chefer et al., 2021) as well as two other methods that work on lower resolution images for one-shot domain transfer. These are the method of Gatys et. al Gatys et al. (2016) and the the AdaIN approach (Huang & Belongie, 2017). Our visual results compare favorably against the new methods in Fig. 15 with respect to preserving the identity of the original image while also generating images that belong to the new domain. ",
+        "bbox": [
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+        "type": "image",
+        "img_path": "images/f6d521cee1abe9205b74a3b4eceff9db85bcbf0114bc6baeff2253793778b714.jpg",
+        "image_caption": [
+            "Figure 9: Style transfer results obtained by our method after style interpolation in domain $B$ . The top row represents the real images embedded in the latent space of $G _ { A }$ (domain $A$ ) whose latent codes are then used by $G _ { B }$ (domain $B$ ). The first column represents the reference images $I _ { B }$ which are input to our domain adaptation framework. "
+        ],
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+    {
+        "type": "text",
+        "text": "A.6 INFERENCE AND EDITING TIME ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Our proposed approach uses II2S for training and inference and StyleFlow (Abdal et al., 2021b) for editing in the new domain. GAN inversion using II2S on HD $( 1 0 2 4 \\times 1 0 2 4 )$ images takes 150 seconds on average, and each latent-code edit operation takes 0.47 seconds. Generating the images afterwards takes an addition 0.34 seconds. Note that the run-time is dominated by GAN -inversion using II2S, however as we show in Fig. 16 once training is completed, we can use other GAN inversion methods to accomplish the edits. With e4e (Tov et al., 2021) inversion is only 0.22 seconds and the entire process of inversion, editing, and generating the edited image can be accomplished in approximately one second. ",
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+        "img_path": "images/1ab5697c70e36b56b78ff4b0138887938cf9b80475354bb0e466eba264831096.jpg",
+        "image_caption": [
+            "Figure 10: The structure of rows and columns is the same as in Fig. 9. Note: our method also works well when the reference images are real face images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "single-shot ",
+        "bbox": [
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+        "page_idx": 16
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+    {
+        "type": "image",
+        "img_path": "images/48976bfef222ed63404411d895a71d93328ba4685f52f09ad62dfed72cf1e606.jpg",
+        "image_caption": [
+            "Figure 11: Our method extends to deal with multiple reference images. The figure compares the results using 3 reference images and using single reference image. It can be observed that our method can better catch the general style and achieve more stable results when given multiple reference images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/a146406cb2501b677b29f849488c371b3a8cb1e40b215db528a71df4f3f4fb9d.jpg",
+        "image_caption": [
+            "Figure 12: Style interpolation results achieved by our framework. Unlike the competing methods, our method has an explicit control over the styles in the domain $B$ . Each sub figure shows a reference image and images embedded in domain $A$ . Notice that we can control the amount of variation in style depending on a parameter $\\alpha$ that can be specified by a user. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 17
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+    {
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+        "img_path": "images/76778aedb90dd70d7229bfa173dcf3e22f7c1f08db614d8e8f8bfb295cbc7815.jpg",
+        "image_caption": [
+            "Figure 13: Our domain transfer results on cars. The structure of rows and columns is the same as in Fig. 9. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 17
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+    {
+        "type": "image",
+        "img_path": "images/ffbd88f8fe672433ddd978ad422bd54460c462b1a1f03362df0c0cc6c360067a.jpg",
+        "image_caption": [
+            "Figure 14: Our domain transfer results on cats and dogs. The structure of rows and columns is the same as in Fig. 9. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 18
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+    {
+        "type": "image",
+        "img_path": "images/f45952289b221e6c67d15499ae7160ee3802f5b158271bc0daf501f7a26dc4df.jpg",
+        "image_caption": [
+            "Figure 15: Additional comparisons with other baseline methods including the concurrent method TargetCLIP (Chefer et al., 2021) as well as two lower-resolution methods from Gatys et al. (2016) and AdaIN (Huang & Belongie, 2017). One-shot reference images from domain $B$ are shown in the left column. Each image is the result of transferring the image in the top row into the new domain. Compare these images to our method in Fig. 7, our proposed approach has fewer artifacts while preserving the identity of the image in domain $A$ . "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 16: Comparison domain-transfer and editing using II2S vs e4e. The new GAN is always trained using II2S, but once training is complete, e4e can be used to transfer images into the new domain. II2S takes 2.5 minutes to embed the image, while e4e needs about 0.22 seconds. StyleFlow editing takes 0.47 seconds, and StyleGAN image generation takes about 0.34 seconds. "
+        ],
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parse/dev/zDiHoIWa0q1/zDiHoIWa0q1_content_list.json

@@ -0,0 +1,1653 @@
+[
+    {
+        "type": "text",
+        "text": "OMNIGROK: GROKKING BEYOND ALGORITHMIC DATA ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ziming Liu, Eric J. Michaud & Max Tegmark Department of Physics, Institute for AI and Fundamental Interactions, MIT {zmliu,ericjm,tegmark}@mit.edu ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Grokking, the unusual phenomenon for algorithmic datasets where generalization happens long after overfitting the training data, has remained elusive. We aim to understand grokking by analyzing the loss landscapes of neural networks, identifying the dependence of the generalization gap on model weight norm as a cause of grokking. We refer to this as the \"LU mechanism\" because training and test losses (against model weight norm) typically resemble \"L\" and \"U\", respectively. This mechanism can explain many aspects of grokking: data size dependence, weight decay dependence, the emergence of representations, etc. Guided by the intuitive picture, we are able to induce grokking on tasks involving images, language and molecules, although the grokking signals are sometimes less dramatic. We attribute the dramatic nature of grokking for algorithmic datasets to representation learning. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            445
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generalization lies at the heart of machine learning. A good machine learning model should arguably be able to generalize fast, and behave in a smooth/predictable way under changes of (hyper)parameters. Grokking, the phenomenon where the model generalizes long after overfitting the training set, has raised interesting questions after it was observed on algorithmic datasets by Power et al. (2022): ",
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+    },
+    {
+        "type": "text",
+        "text": "Q1 The origin of grokking: Why is generalization much delayed after overfitting? Q2 The prevalence of grokking: Can grokking occur on datasets other than algorithmic datasets? ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper aims to answer these questions by analyzing neural loss landscapes: ",
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+    },
+    {
+        "type": "text",
+        "text": "A1 Grokking can result from a mismatch between training and test loss against model weight norm. Specifically, (reduced) training and test losses plotted against model weight norm resemble \"L\" and \"U\", respectively, as shown in Figure 1b. We refer to this phenomenon as the \"LU mechanism\", which we elaborate on in Section 2 and 3.   \nA2 Yes. Indeed, we demonstrate grokking for a wide range of machine learning tasks in Section 4, including image classification, sentiment analysis and molecule property prediction. Grokking signals observed for these tasks are usually less dramatic than for algorithmic datasets, which we attribute to representation learning in Section 5. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Partial answers to Q1 are provided in recent studies: Liu et al. (2022) attribute grokking to the slow formation of good representations, Thilak et al. (2022) attempts to link grokking to the slingshot mechanism of adaptive optimizers, and Barak et al. (2022) uses Fourier gap to describe hidden progress. This paper aims to understand grokking through the lens of neural loss landscapes. Our landscape analysis is able to explain many aspects of grokking: data size dependence, weight decay dependence, emergence of representations, etc. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The paper is organized as follows: In Section 2, we review background on generalization, and introduce the $L U$ mechanism. In Section 3, we show how the LU mechanism leads to grokking for a toy teacher-student setup. In Section 4, we show that the intuition gained from the toy problem can transfer to realistic datasets (MNIST, IMDb reviews and QM9), for which we also observe grokking, although in a slightly non-standard setup where it is relatively weak. In Section 5, we discuss why grokking is more dramatic for algorithmic datasets than on others (e.g., MNIST), by comparing their loss landscapes. We review related work in Section 6 and summarize our conclusions in Section 7. Code is available at https://github.com/KindXiaoming/Omnigrok. ",
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+    {
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+        "img_path": "images/2ee983ed3c17dd831a42da66ffb0e1c74379c0386d8253a6e31d78a1d9886819.jpg",
+        "image_caption": [
+            "Figure 1: (a) $w \\colon L _ { 2 }$ norm of model weights. Generalizing solutions (green stars) are concentrated around a sphere in the weight space where $w \\approx w _ { c }$ (green). Overfitting solutions (orange) populate the $w \\gtrsim w _ { c }$ region. (b) The training loss (orange) and test loss (gray) have the shape of $\\mathrm { L }$ and $\\mathrm { U }$ , respectively. Their mismatch in the $w > w _ { c }$ region leads to fast-slow dynamics, resulting in grokking. "
+        ],
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+    {
+        "type": "text",
+        "text": "2 THE LU MECHANISM FOR GROKKING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Weight norm and reduced loss Letting w denote the weights of a model, any function $f ( \\mathbf { w } )$ (e.g, train/test loss/accuracy) depends on both the weight norm $w \\equiv | | \\mathbf { w } | | _ { 2 }$ and the angular direction $\\hat { \\mathbf { w } } \\equiv \\mathbf { w } / w$ . Similar to Fort and Scherlis (2019), we define a reduced function $\\tilde { f } ( w )$ by minimizing training loss $l _ { \\mathrm { t r a i n } } ( \\mathbf { w } )$ over angular directions, i.e., ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c0ed8f8dcf24b8373a829e211c38cbbe873dd2a79565b0c9e553b324d42ae0e.jpg",
+        "text": "$$\n\\tilde { f } ( w ) \\equiv f ( \\mathbf { w } ^ { * } ( w ) ) , \\quad \\mathrm { w h e r e } \\ \\mathbf { w } ^ { * } ( w ) \\equiv \\underset { | | \\mathbf { w } | | _ { 2 } = w } { \\mathrm { a r g m i n } } \\ l _ { \\mathrm { t r a i n } } ( \\mathbf { w } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we set $f$ as train/test loss/error, but it also applies to other metrics of interest. In practice, we perform the constrained minimization by rescaling the model weights back to their original norm after each unconstrained optimization step. We will see that this reduced 1D loss landscape, which is easy to visualize, captures important features related to grokking. Throughout the paper, our model is initialized by multiplying a factor $\\alpha \\equiv w / w _ { 0 }$ to the standard initialization 1, where $w _ { 0 }$ and $w$ are the weight norm of the network before and after multiplying by $\\alpha$ , respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "LU mechanism Although the loss landscapes of neural networks are nonlinear, Fort and Scherlis (2019) reveal a simple landscape picture: There is a spherical shell in the weight space (the \"Goldilocks\" zone), where generalization is better than outside this zone. We illustrate the Goldilocks zone as the green area with average radius $w _ { c }$ in Figure 1a; the green stars are the generalizing solutions. The test loss is thus higher either both when $w > w _ { c }$ and $w < w _ { c }$ , forming a U-shape against $w$ in Figure 1b (gray curve). By contrast, the training loss has an L-shape against weight norm . There are many solutions which overfit training data for $w > w _ { c }$ , but high training losses are incurred for $w < w _ { c }$ . This corresponds to the L-shaped curve seen in Figure 1b (orange curve, no regularization). In summary, the (reduced) training loss and test loss are L-shaped and U-shaped against weight norm, respectively, which we will refer to as the LU mechanism throughout the paper. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "It is well known in statistics that generalization error has a \"U\" shape against model capacity, which is usually attributed to the bias-variance trade-off. Although this common wisdom was challenged by the observation of double descent (Nakkiran et al., 2021), the \"U\" curve can be recovered from a double descent simply by changing the $\\mathbf { X }$ -axis from the number of model parameters $N$ to the 2-norm of model parameters $w \\equiv | | \\mathbf { w } | | _ { 2 }$ , at least for linear regression $\\mathrm { N g }$ and Ma, 2022). Although the LU mechanism may remind readers of related phenomena (Schoenholz et al., 2016; Yang and Schoenholz, 2017; Nakkiran et al., 2021), their setups are not exactly the same as ours. More importantly, our focus and contribution is to understand grokking, a brand new generalization puzzle. ",
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+    },
+    {
+        "type": "text",
+        "text": "Grokking dynamics We identify the \"LU mechanism\" as the cause of grokking. If the weight norm is initialized to be large (e.g., the black square in the $w > w _ { c }$ region), the model first quickly moves to a nearby overfitting solution by minimizing the training loss. Without any regularization, the model will stay where it is, because the gradient of the training loss is almost zero along the valley of overfitting solutions, so generalization does not happen. Fortunately, there are usually explicit and/or implicit regularizations that can drive the weight vector towards the Goldilocks zone $w \\approx w _ { c }$ . When the regularization magnitude is non-zero but small, the radial motion can be (arbitrarily) slow. If weight decay is the only source of regularization, and training loss is negligible after overfitting, then weight decay $\\gamma$ causes $w ( t ) \\approx \\exp ( - \\gamma t ) w _ { 0 }$ , when $w _ { 0 } > w _ { c }$ , so it takes time $t \\approx \\ln ( w _ { 0 } / w _ { c } ) / \\gamma \\propto \\gamma ^ { - 1 }$ to generalize. A small $\\gamma$ results in a huge generalization delay (i.e., grokking). The dependence on regularization magnitudes is illustrated in Figure 1b: no generalization at all happens for $\\gamma = 0$ small $\\gamma$ leads to slow generalization (grokking), and large $\\gamma$ leads to faster generalization 2. The above analysis only applies to large initializations $w > w _ { c }$ . Small initializations $w < w _ { c }$ can always generalize fast 3, regardless of regularization. ",
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+    {
+        "type": "text",
+        "text": "Why isn’t grokking commonly observed? The standard initialization schemes typically initialize $w$ no larger than $w _ { c }$ . However, if we increase initialization scales (explicitly or implicitly), grokking can appear. In Section 3 and 4, we find that explicitly increasing initialization weight norm can induce grokking. In Section 5, we argue for algorithmic datasets because (shown in Figure 6d) ",
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+        "img_path": "images/617048da6aa257709f5bfa0d635d933aef763743772286e3c17841c9e763e4a3.jpg",
+        "text": "$$\nw _ { c } ( \\mathrm { b a d ~ r e p r e s e n t a t i o n } ) > w _ { c } ( \\mathrm { g o o d ~ r e p r e s e n t a t i o n } ) ,\n$$",
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+        "type": "text",
+        "text": "i.e., a proper initialization for a bad representation is effectively too large for a good representation, leading to grokking. Take the addition (base $p$ ) for example: with the good (linear) representation or a bad (random) representation, the decoder needs to learn to classify $O ( p )$ or ${ \\dot { O } } ( p ^ { 2 } )$ examples, respectively. ",
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+        "type": "text",
+        "text": "3 GROKKING FOR A TEACHER-STUDENT SETUP ",
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+        "image_caption": [
+            "Figure 2: Teacher-student setup. $\\alpha$ : student initialization scale, $\\gamma$ : weight decay. (a) The reduced training loss and test loss have the shape of “L\" and “U\", respectively. (b) Top row: large initialization ${ \\ ' } \\alpha = 2 . 0$ ) can demonstrate no generalization (no reg), grokking (small reg) and fast generalization (large reg). Bottom: small initialization $( \\alpha = 0 . 5$ ) always generalizes fast, regardless of weight deacy. (c) $\\alpha = 2$ . The steps to overfitting is independent of weight decay, while the steps to generalization scale inversely with the weight decay. "
+        ],
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+    {
+        "type": "text",
+        "text": "To illustrate how the LU mechanism results in grokking, we employ a toy teacher-student setup. The teacher and the student share the same architecture (a 5-100-100-5 MLP with tanh activation), but are initialized with different seeds. The student network is initialized with the standard initialization (the default one in PyTorch) but each weight is rescaled by the same factor $\\alpha \\equiv w / w _ { 0 }$ , where $w _ { 0 }$ and $w$ are the weight norm of the student network before and after rescaling. The teacher network is initialized standardly, i.e., $\\alpha _ { \\mathrm { t e a c h e r } } = 1$ . Inputs and outputs have dimensions $d _ { \\mathrm { i n } } = 5$ and $d _ { \\mathrm { o u t } } = 5$ respectively. We generate $N _ { \\mathrm { t r a i n } } = 1 0 0$ training and $N _ { \\mathrm { t e s t } } = 1 0 0$ test samples by first drawing inputs from the standard Gaussian distribution ${ \\cal N } ( 0 , { \\bf I } _ { d _ { \\mathrm { i n } } \\times d _ { \\mathrm { i n } } } )$ , and then feed the input data to the teacher to generate output labels. The student network is trained with the Adam optimizer (learning rate $3 \\check { \\times } 1 0 ^ { - 4 } )$ for $1 { \\dot { 0 } } ^ { 5 }$ steps. ",
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+        "text": "LU landscapes Firstly, we compute the reduced losses by minimizing the training loss (excluding weight decay) while constraining the weight norm of the student network to be constant. We assume the converging point after training as the global minimum on the spherical surface 4, which explicitly defines the reduced losses $\\tilde { l } _ { \\mathrm { t r a i n } } ( \\bar { \\alpha } )$ and $\\bar { l } _ { \\mathrm { t e s t } } ( \\alpha )$ . As shown in Figure 2a, $\\tilde { l } _ { \\mathrm { t e s t } } ( \\alpha )$ first decreases and then increases as $\\alpha$ increases, displaying a U-shape with a minimum at $\\alpha \\approx 1$ . By contrast, $\\tilde { l } _ { \\mathrm { t r a i n } } ( \\alpha )$ decreases when $\\alpha < 1$ and remains flat near zero when $\\alpha \\geq 1$ , forming an L-shape. When weight decay $\\gamma$ is present, the training landscape becomes $\\tilde { l } _ { \\mathrm { t r a i n } } ( \\alpha , \\gamma ) = \\tilde { l } _ { \\mathrm { t r a i n } } ( \\alpha ) + \\gamma \\alpha ^ { 2 } C ^ { 2 }$ where $C$ is the average parameter magnitude determined by the standard initialization. ",
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+    {
+        "type": "text",
+        "text": "Training dynamics Our problem is a regression task, but we can imitate the behavior of a classification task by manually setting a threshold $\\beta = 0 . 0 1$ and defining a sample to be correctly “classified\" if the prediction error is less than $\\beta$ . We study the dynamics of training and test accuracy. Note that this is the normal training setup where the weight norm is not constrained, although with two non-standard initializations $\\alpha = 0 . 5$ (small) and $\\alpha = 2 . 0$ (large), and three weight decays $\\gamma = 0$ (no reg), $\\gamma = 0 . 0 3$ (small reg) and $\\gamma = 1$ (large reg). As shown in Figure 2b (bottom), small initialization runs always generalize fast regardless of regularization. Large initialzation runs (top) dependend on weight decay: no regularization fails to generalize, small regularization generalizes slowly (grokking), while large regularization generalizes faster. ",
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+        "type": "text",
+        "text": "For the large initialization $\\alpha = 2 . 0$ , we do a finer sweep of $\\gamma$ in [0.03, 1]. We compute the number of steps and weight norm $w$ when training or test accuracy reaches $9 5 \\%$ . As shown in Figure 2c, the time (number of steps) to reach $9 5 \\%$ training accuracy is independent of weight decay $\\gamma$ , while the time to reach $9 5 \\%$ test accuracy is inversely proportional to the weight decay, as we derived above for the LU mechanism. ",
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+    {
+        "type": "text",
+        "text": "4 OMNIGROK: GROKKING FOR MORE INTERESTING TASKS ",
+        "text_level": 1,
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+        "text": "We now analyze loss landscapes and search for grokking for several more interesting datasets, and see that the insights obtained from our toy model can transfer to these datasets. We report the main results here, with experiment details included in Appendix A. ",
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+        "text": "Image classification We visualize loss landscapes of MNIST (Deng, 2012) to verify the LU mechanism, and study the dependence on training data size. Similar to the teacher-student case, we reduce losses and errors (one minus accuracy) to two variables (weight norm $w$ and data size $N$ ) by minimizing over angular directions of weights, i.e., ",
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+    {
+        "type": "equation",
+        "img_path": "images/3403495cddce67ee1478d08cf5e6d9a8745ee8cc95cef62c71ed174227fc4a93.jpg",
+        "text": "$$\n\\tilde { l } _ { \\mathrm { t r a i n } } ( w , N ) \\equiv l _ { \\mathrm { t r a i n } } ( \\mathbf { w } ^ { * } , N ) , \\quad \\tilde { l } _ { \\mathrm { t e s t } } ( w , N ) \\equiv l _ { \\mathrm { t e s t } } ( \\mathbf { w } ^ { * } , N ) , \\quad \\mathbf { w } ^ { * } ( w , N ) \\equiv \\operatorname * { a r g m i n } l _ { \\mathrm { t r a i n } } ( \\mathbf { w } , N ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "shown in Figure 3 (a)(b). The reduced loss landscape reveals three things: (1) Larger initializations lead to grokking. Point $\\mathbf { A }$ in Figure 3 corresponds to the standard initialization $( \\alpha = 1 )$ ), which has low training and test errors, hence no grokking. When increasing the weight norm from $\\mathbf { A }$ to $\\mathbf { B }$ , training error is seen to remain low while test error rises. To generalize, weight decay must be in place to bring the weight norm down, leading to grokking if weight decay is small. (2) Larger datasets lead to de-grokking. Comparing $\\mathbf { B }$ and C in Figure 3, $\\mathbf { C }$ is seen to have larger training size than $\\mathbf { B }$ and lower test error. Larger data size $N$ makes the Goldilocks zone broader, reducing or eliminating grokking even for large weight initializations. (3) Critical data size can be defined. As reported in Power et al. (2022); Liu et al. (2022), we see that there exists a critical training set size below which generalization is impossible. The effective theory analysis in Liu et al. (2022) only applies to algorithmic datasets, but not to other datasets with unknown optimal representations. The loss landscape analysis presented is this work can apply to all supervised-learning tasks. As shown in Figure 3 (b), the contours of constant test error are thumb-like, and the tip of the thumb determines the minimum amount of data required for generalization. ",
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+        "type": "text",
+        "text": "Guided by the landscape analysis, we make two nonstandard decisions to induce grokking on MNIST: (1) we reduce the size of the training set from $6 0 \\mathrm { k }$ to 1k samples (by taking a random subset) and (2) we increase the scale of the weight initialization distribution (by multiplying the initial weights, sampled with Kaiming uniform initialization, by a constant $\\alpha > 1$ ). With these modifications to the training set size and initialization scale, we train a depth-3 width-200 MLP with ReLU activations with the AdamW optimizer using MSE loss with one-hot targets. We find that the network quickly fits the training set, and test accuracy improves much later, as shown in Figure 3d, just as in the stereotypical grokking learning first observed in algorithmic datasets. Figure 3e shows the effect of training set size on time to generalization for MNIST. We find a result similar to what Power et al. (2022) observed, namely that generalization time increases rapidly once one approaches a certain critical data set size. The conclusions still hold for the cross entropy loss (see Appendix F), although with quantitatively milder effects. ",
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+        "img_path": "images/00ced667532473eb9f803757731dd6872a617ba561fe9bb35de5e7b76e35dd6b.jpg",
+        "image_caption": [
+            "Figure 3: MNIST. (a) reduced training error, (b) reduced test error. Comparing A and B: larger weight norm makes learning grok (delay generalization). Comparing B and C: a larger training data size makes learning de-grok (speed up generalization). (c) \"LU\" holds truer for smaller data. (d) Accuracy curves for MNIST in the setting where we observe grokking. (e) Time to generalize as a function of training set size $N$ , replicating Liu et al. (2022). "
+        ],
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+        "img_path": "images/5ff73724264ff45c416ebdd43376d3a15faa43a50a9fd84cb26508f832ccf454.jpg",
+        "image_caption": [
+            "Figure 4: We use an LSTM to predict IMDb reviews. (a) training error; (b) test error; (c) reduced losses for data size 1k (top) and $5 0 \\mathrm { k }$ (bottom); (d) With 1k data, a (weak) grokking signal is observed for large initializations $\\langle \\alpha = 6 \\rangle$ ), while no grokking is observed for standard initializations $( \\alpha = 1 )$ ). "
+        ],
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+        "image_caption": [
+            "Figure 5: We use a GCNN to predict isotropic polarizability of molecules in the QM9 dataset. (a) training loss; (b) test loss; (c) reduced losses for data size 100 (top) and 3000 (bottom); (d) with 200 training samples, grokking is observed for large initialization $( \\alpha = 3$ ). "
+        ],
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+        "type": "text",
+        "text": "Sentiment analysis of text We look for grokking using LSTMs (Hochreiter and Schmidhuber, 1997) for IMDb dataset (Maas et al., 2011). Similar to Eq. (3), we reduce training and test losses to depend on only the weight norm $w$ and data size $N$ . We show the reduced training and test error in Figure 4 (a)(b). For large data size, e.g., the full dataset, training and test errors have similar \"U\" shapes 5, so one cannot create grokking via the \"LU\" mechanism. For small data size, say 1k, however, the mismatch between training and test errors makes it possible to create grokking via large initializations. In Figure 4 (c), we initialize weights larger $( \\alpha = 6$ ) with weight decay 1, overfitting is complete within $\\mathrm { i 0 ^ { 2 } }$ steps, but generalization does not start until around $\\mathrm { 1 0 ^ { 3 } }$ steps. Note that the generalization \"jump\" is not as sharp as on algorithmic datasets (Power et al., 2022) or MNIST, but at least generalization is delayed here. By contrast, if we use the standard initialization $( \\alpha = 1$ ) with no weight decay, generalization happens early on during training, and does not improve much after overfitting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Molecules We search for grokking using the graph convolutional neural network (GCNN) for QM9 dataset (Ramakrishnan et al., 2014). Similar to Eq. (3), we define the reduced training/test losses, which are only dependent on weight norm $w$ and data size $N$ . As shown in Figure 5(a)(b), when data size is large, training and test losses have similar \"U\" shapes, hence grokking is impossible via the \"LU mechanism\". When data size is small, training and test losses mismatch somewhere in the region $\\alpha = w / w _ { 0 } > 1$ , making grokking possible. Indeed, shown in Figure 5(d), there is a sharp drop in test loss around $1 0 ^ { 4 }$ steps if initialization is 3 times larger than standard, while standard initialization does not lead to grokking. Note that zero weight decay is applied in both cases, implying the existence of implicit regularizations. ",
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+    {
+        "type": "text",
+        "text": "5 REPRESENTATION IS KEY TO GROKKING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Section 4, we showed that increasing initialization scales can make grokking happen for standard ML tasks. However, this seems a bit artificial and does not explain why standard initialization leads to grokking on algorithmic datasets, but not on standard ML datasets, say MNIST. The key difference is how much the task relies on representation learning. For the MNIST dataset, the quality of representation determines whether the test accuracy is $9 5 \\%$ or $100 \\%$ ; by contrast in algorithmic datasets, the quality of representation determines whether test accuracy is random guess (bad representation) or $100 \\%$ (good representation). So overfitting (under a bad representation) has a more dramatic effect on algorithmic datasets, i.e., the model weights increase quickly during overfitting but test accuracy remains low. During overfitting, model weight norm is much larger than at initialization, but then drops below the initialization norm when the model generalizes, shown in Figure 9 (see Appendix C), and also observed by Nanda et al. (2023). ",
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+    {
+        "type": "text",
+        "text": "In the following, we will compare algorithmic datasets (Section 5.1) to MNIST (Section 5.2). We show how their loss landscapes depend on representations differently, and how the difference leads to different outcomes (grokking or not). ",
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+    {
+        "type": "text",
+        "text": "5.1 ALGORITHMIC DATASETS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Setup Algorithmic datasets are the task of learning a binary operation $a \\circ b = c ( a , b , c$ are symbols) with neural networks, which aim to predict $c$ from input $( a , b )$ . We take the toy addition setup in (Liu et al., 2022), where each input digit $0 \\leq i \\leq p - 1$ (output label $0 \\leq k \\leq 2 ( q - 1 ) )$ is embedded as a vector $\\mathbf { E } _ { i }$ $( \\mathbf { Y } _ { k } )$ . A decoder MLP is employed to predict $\\mathbf { Y } _ { k } = \\operatorname { D e c } ( \\mathbf { E } _ { i } + \\mathbf { E } _ { j } ) \\left( k = i + j \\right)$ . In the setup of grokking, both the decoder and the input representations ${ \\bf R } \\equiv \\{ { \\bf E } _ { i } \\}$ are trainable, with learning rates $\\eta _ { D }$ and $\\eta _ { R }$ , respectively; in the setup of landscape analysis, only decoder is trainable, as we explain below. Training and test losses depend on three factors: (i) representation $\\mathbf { R }$ , (ii) weight norm $w$ and (iii) weight direction wˆ . As in previous sections, we can optimize wˆ by minimizing the training loss on constant weight norm spheres. We further reduce the high-dimensional representations to 1D by interpolating in a particular direction: ",
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+    {
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+        "img_path": "images/5044fa7a4e05047b235862edf382ebff5d496fc9150df57809bb8963ff8aa1f0.jpg",
+        "image_caption": [
+            "Figure 6: Loss landscapes on the 2D $( w , m )$ plane. (a) Training loss splits the plane into two regions: large loss small $w$ (fast dynamics) and small loss large $w$ (slow dynamics). (b) Test loss; the green star is the generalizing solution. (c) Losses along an illustrative path $\\mathrm { A }  \\mathrm { E }$ , demonstrating multiple descent; (d) zoom-in of the training loss highlighting the gradients on the boundary. (e) the boundary depends on training data size; (f) a simple illustration of grokking dynamics. "
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+    {
+        "type": "equation",
+        "img_path": "images/97d339a168bd77b461159633a3a8650c8ee2c9b8eeeb9ddaf78fb33ad3b5793b.jpg",
+        "text": "$$\n{ \\bf R } = m { \\bf R } _ { \\mathrm { r a n d o m } } + ( 1 - m ) { \\bf R } _ { \\mathrm { l i n e a r } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { R } _ { \\mathrm { l i n e a r } }$ refers to the linear representation in which number $k$ is embedded to $\\mathbf { E } _ { k } = [ k , 0 , \\cdots , 0 ]$ , $\\mathbf { R } _ { \\mathrm { r a n d o m } }$ is the initialized representation drawn from Gaussian distributions, i.e, $\\mathbf { E } _ { k } \\sim { \\cal N } ( \\mathbf { 0 } , \\mathbf { I } )$ , and $m \\in [ 0 , 1 ]$ is a scalar interpolating between $\\mathbf { R } _ { \\mathrm { l i n e a r } }$ and $\\mathbf { R } _ { \\mathrm { r a n d o m } }$ , that we term representation messiness because $\\mathbf { R } = \\mathbf { R } _ { \\mathrm { l i n e a r } }$ when $m = 0$ , and $\\mathbf { R } = \\mathbf { R } _ { \\mathrm { r a n d o m } }$ when $m = 1$ . After these reductions, both training and test losses become functions of two variables, representation messiness $m$ and weight norm $w$ : ",
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+        "type": "equation",
+        "img_path": "images/d1aef9c3883a6cdf64fb0fe003382417f00c56dd06aba4dc7046c00d2e1bbec6.jpg",
+        "text": "$$\n\\mathbf { w } ^ { * } ( w , m ) \\equiv \\operatorname * { a r g m i n } _ { | | \\mathbf { w } | | _ { 2 } = w } l _ { \\mathrm { t r a i n } } ( \\mathbf { w } , m ) , \\quad \\tilde { l } _ { \\mathrm { t r a i n } } ( w , m ) \\equiv l _ { \\mathrm { t r a i n } } ( \\mathbf { w } ^ { * } , m ) , \\quad \\tilde { l } _ { \\mathrm { t e s t } } ( w , m ) \\equiv l _ { \\mathrm { t e s t } } ( \\mathbf { w } ^ { * } , m )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Note that our definition of $\\tilde { l } _ { \\mathrm { t r a i n } } ( w , m )$ excludes the weight decay term $\\begin{array} { r } { \\ell _ { \\mathrm { r e g } } = \\frac { 1 } { 2 } \\gamma w ^ { 2 } } \\end{array}$ , but we should be aware of its presence when we analyze the dynamics of $( w , m )$ , which is governed by the gradient flow on $\\tilde { l } _ { \\mathrm { t r a i n } } ( w , m )$ plus weight decay $( \\eta _ { R } / \\eta _ { D }$ are learning rates of representation/decoder): ",
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+    {
+        "type": "equation",
+        "img_path": "images/f4d3dc738cea25abbfa378e7e927ba364d218974e305f50b4dd47b7713431e42.jpg",
+        "text": "$$\n\\frac { d w } { d t } = - \\eta _ { D } \\left( { \\frac { \\partial \\tilde { l } _ { \\mathrm { t r a i n } } } { \\partial w } } + \\gamma w \\right) , \\quad \\frac { d m } { d t } = - \\eta _ { R } \\frac { \\partial \\tilde { l } _ { \\mathrm { t r a i n } } } { \\partial m } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "More experimental details are included in Appendix E. ",
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+        "type": "text",
+        "text": "Landscape We show $\\tilde { l } _ { \\mathrm { t r a i n } } ( w , m )$ and $\\tilde { l } _ { \\mathrm { t e s t } } ( w , m )$ in Figures 6a and $^ \\mathrm { 6 b }$ , indicating the generalizing solution with a green star. Based on the reduced training loss (Figure 6a), we can divide the 2D plane into two regions I and $\\mathbf { I I }$ , separated by a dashed yellow line (the contour of training $\\mathrm { l o s s } = 0 . 0 5 )$ ): (I): The darker region, with high training losses/gradients and small weight norm. $( \\mathbf { I I } )$ : The lighter region, with low training losses/gradients and large weight norm. Comparing Figures 6a and 6b reveals that training and test loss landscapes differ, especially in region II. Moreover, while the training loss depends weakly on $m$ , the test loss depends strongly on $m$ . As we will see, the (weak) dependence of training loss on representation drives the model to the generalizing solution. However, the driving force is small because the dependence is weak, leading to grokking. We elaborate below how these particular loss landscapes lead to grokking. ",
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+        "text": "Grokking dynamics In region $\\mathbf { I I }$ , the dynamics is slow (for small $\\gamma$ ) due to nearly vanishing gradients. By contrast, the dynamics in region I is relatively fast. As we will explain, dynamics is also slow on the boundary of I and $\\mathbf { I I }$ , and grokking is the consequence of traversing region $\\mathbf { I I }$ and/or the boundary. ",
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+    {
+        "type": "text",
+        "text": "Let us analyze a typical path $\\mathbf { A }$ to $\\mathbf { E }$ shown in Figure 6(a)(b). A rolls \"downhill\" to $\\mathbf { B }$ following training gradients, possibly continuing to $\\mathbf { C }$ due to momentum. $\\mathbf { C }$ is located in $\\mathbf { I I }$ where $\\tilde { l } _ { \\mathrm { t r a i n } } \\approx \\bar { 0 }$ , so according to Eq. (6), $d m / d t \\approx 0$ and $d w / d t \\approx - \\eta _ { D } \\gamma w$ or, equivalently, $d ( \\log w ) / d t \\approx - \\eta _ { D } \\gamma$ . So $( \\log w , m )$ moves with a constant speed $v = \\eta _ { D } \\gamma$ in the $- w$ direction from C to $\\mathbf { D }$ , a point near the boundary. Negative gradients around the boundary point towards larger $w$ and smaller $m$ , shown in Figure 6d (a zoom-in of Figure 6a). The gradients become increasingly large as the model goes deeper inside region I, and at some point, the gradient totally cancels out $v$ in the gradient direction, making the model start to drift along the boundary, as illustrated in Figure 6f. Then the model moves along the boundary with a new velocity $v ^ { \\prime } = v \\mathrm { { c o s } } \\theta ^ { 6 }$ , until it reaches the generalizing solution $\\mathbf { E }$ . The above picture is supported by empirical experiments in Appendix C and also Nanda et al. (2023). Based on the picture, we also show the ability to eliminate grokking in Appendix C. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The slow dynamics from $\\mathbf { C }$ to $\\mathbf { E }$ is the origin of grokking. During this period, the model first moves in the $- w$ direction with a velocity $v$ over the distance $L _ { 1 } = L - h \\mathrm { c o t } \\theta$ , and then moves along the boundary with a velocity $v ^ { \\prime }$ over the distance $L _ { 2 } = h / \\mathrm { s i n } \\theta$ . So the total time is $t = L _ { 1 } / v + L _ { 2 } / v ^ { \\prime } =$ $( L + h \\mathrm { t a n } \\theta ) / ( \\eta _ { D } \\gamma )$ . This formula agrees with the observation that large weight decays $\\gamma$ and/or larger decoder learning rates $\\eta _ { D }$ can make generalization happen faster (Power et al., 2022; Liu et al., 2022). Besides, the path manifests intriguing multiple descent of test loss, shown in Figure 6c. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Dependence of grokking on training data size Another important observation in Power et al. (2022) is that grokking happens faster for larger training size. Our landscape analysis can also explain the data size dependence. In Figure 6e, we show the contours (training $\\mathrm { l o s s } = 0 . 0 2 $ ) for different training sizes (25, 35, 45, 55). The contours of training size 45 and 55 both connect to the green star, meaning that generalization will eventually happen. However, the slopes of the contours are different, i.e., $\\theta _ { 5 5 } < \\theta _ { 4 5 }$ . Since $t = ( L + h \\mathrm { t a n } \\bar { \\theta } ) / ( \\bar { \\eta _ { D } } \\gamma )$ increases as $\\theta$ increases, we have $t _ { 5 5 } < t _ { 4 5 }$ , i.e, more training data leads to faster grokking. For training size 35 and 25, the contours do not connect to the green star, so generalization will not happen, no matter how long the training will be run. ",
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+    {
+        "type": "text",
+        "text": "5.2 MNIST ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We now study how training and test losses depend on representation messiness in the MNIST dataset. We denote the $2 8 \\times 2 8$ images as the raw representation $\\mathbf { R } _ { \\mathrm { r a w } }$ . We construct a linearly separable representation $\\mathbf { R } _ { \\mathrm { l i n e a r } }$ by assigning input representations proportional to their label $y _ { i }$ , for example, an image of a 2 is represented by a matrix with all elements being 2. Similar to Eq. (4), we use $m \\in [ 0 , 1 ]$ to interpolated between $\\mathbf { R } _ { \\mathrm { r a w } }$ and $\\mathbf { R } _ { \\mathrm { l i n e a r } }$ : ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b410570c4650e6fe162ba80728790c650177b4b72c42c37954b1e80ae72ba1b6.jpg",
+        "text": "$$\n\\mathbf { R } = m \\mathbf { R } _ { \\mathrm { r a w } } + ( 1 - m ) \\mathbf { R } _ { \\mathrm { l i n e a r } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            755,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Similarly to Eq. (5), we define and plot $\\tilde { l } _ { \\mathrm { t r a i n } } ( w , m )$ and $\\tilde { l } _ { \\mathrm { t e s t } } ( w , m )$ in Figure 7, using the full dataset $N = 6 0 0 0 0$ . Comparing Figures $\\cdot$ and 7b reveals two things: (1) The training and test losses behave similarly; (2) Both training and test losses depend very weakly on $m$ . This implies that the raw image representation is already quite close to being optimal, so decent test accuracy can be obtained even without learning optimal representations. As a result, grokking does not occur (Figure 7c). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparing Figure 6 and 7, we see that the (strong) dependence of test performance on the representation is the key to grokking: the dependence on representation is strong for algorithmic datasets, so grokking happens. By contrast, the dependence is weak for MNIST, so grokking does not happen. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2ee0b07ac45e7dd2ec29cba69192f54a34e70fb89fc3314f3012c6d161cc627b.jpg",
+        "image_caption": [
+            "Figure 7: MNIST landscapes as functions of representation messiness $m$ and weight norm $w$ : (a) training loss, and (b) test loss. Training and test losses do not have significant mismatch, and neither of them depend on representation strongly, which is in stark contrast to algorithmic datasets (Figure 6). (c) an illustrative path $\\mathrm { A \\to B \\to C }$ does not manifest grokking. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATION TO RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Grokking was first observed for algorithmic datasets by Power et al. (2022). Several attempts have been made to understand grokking: (a) Liu et al. (2022) attributes grokking to the slow formation of good representations. (b) Shah (2021) suggests that generalizable solutions achieve lower loss than overfitting solutions, providing a training signal encouraging generalization. (c) Nanda et al. (2023) suggests grokking is a phase change due to limited data and regularization. (d) Barak et al. (2022) suggests that generalization is due not to random search, but to hidden progress of SGD to gradually amplify a Fourier gap. (e) Thilak et al. (2022) links grokking to the \"Slingshot mechanism\" specific to adaptive optimizers. (f) Millidge (2022) describes training as a random walk over parameters. Our conclusion supports (a)(b)(c)(d), but does not necessarily negate (e)(f). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Double descent is the phenomenon that performance first gets worse and then gets better as we increase the model size, data size, training epochs or regularization (Nakkiran et al., 2021; Yilmaz and Heckel, 2022; Nakkiran, 2019). The typical \"U\" shape of test loss in this paper does not conflict with double descent, because we are plotting the weight norm instead of the number of model parameters $\\mathrm { N g }$ and Ma, 2022). However, the \"U\"-shape should better be considered as empirically common rather than provably universal. In fact, the interaction between properties of data and inductive biases of learning algorithms can be more complicated than double descent (Chen et al., 2021; d’Ascoli et al., 2020). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Initialization From the optimization perspective, initializations are usually based on the \"edge of chaos\" idea such that variance of features and gradients should be preserved in the forward and backward pass (Glorot and Bengio, 2010; He et al., 2015; Bahri et al., 2020; Yang and Schoenholz, 2017; Jing et al., 2017), or based on analyzing Jacobians and/or Hessians (Skorski et al., 2020). From the generalization perspective, it was shown that large initializations overfit data easily but result in poor generalization (Xu et al., 2019; Zhang et al., 2020), which agrees with our LU mechanism. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Weight decay regularization is a standard trick in machine learning and has various effects on optimization and generalization (Zhang et al., 2018; Van Laarhoven, 2017). In particular, Lewkowycz and Gur-Ari (2020) observes that it takes $t \\propto 1 / \\lambda$ training steps to reach maximum test performance. This is strikingly similar to the grokking time $t \\propto 1 / \\lambda$ we derived from the LU mechanism. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This study elucidates the grokking phenomenon from the perspective of loss landscapes. Our conclusions are: (i) grokking originates from the mismatch between training and test losses at high model weight norm (\"LU\" mechanism). (ii) grokking can happen in various models for a wide range of datasets, although the grokking signature is usually most dramatic for algorithmic datasets. (iii) The severity of grokking depends on how much the task relies on learning representations. This work not only reveals the mechanism of grokking, but also shows that reduced landscape analysis is a useful tool for characterizing data-model interaction and representation learning. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "Appendix ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "A EXPERIMENT DETAILS ",
+        "bbox": [
+            176,
+            143,
+            397,
+            160
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sentiment analysis of text IMDb (Maas et al., 2011) includes $5 0 \\mathrm { k }$ movie reviews to be classified as being positive or negative. To pre-process the data, we extract the 1000 most frequent words and tokenize each review into an array of token indices. Less frequent words are ignored, and each review array is padded to length 500. We adopt the LSTM model (Hochreiter and Schmidhuber, 1997) to perform the classification, with two layers, embedding dimension 64, and hidden dimension 128. We use the Adam optimizer (Kingma and Ba, 2014) with learning rate 0.001 to minimize the binary cross entropy loss. We hold back $2 5 \\%$ of the dataset for testing. ",
+        "bbox": [
+            173,
+            175,
+            825,
+            273
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Molecules QM9 is a database for small molecules and their properties. We use a graph convolutional neural network (GCNN) to predict the isotropic polarizability. The GCNN contains 2 convolutional layers with ReLU activation, followed by a linear layer. We use the Adam optimizer with learning rate 0.001 to minimize the MSE loss. We split the dataset into 50/50 train/test. ",
+        "bbox": [
+            174,
+            280,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "MNIST We train width-200 depth-3 ReLU MLPs on the MNIST dataset with MSE loss. We use the AdamW optimizer with a learning rate of 0.001 and a batch size of 200. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B REDUCED LOSS FOR MODULAR ADDITION WITH TRANSFORMERS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8 we show reduced loss landscape plots for transformers trained on modular addition. We use the setup of Nanda et al. (2023) and train a 1-layer transformer on modular addition $( p = 1 1 3 )$ with $d _ { \\mathrm { m o d e l } } = 1 2 8$ , 4 attention heads, and $d _ { \\mathrm { m l p } } = 5 1 2$ with ReLU activations. We train with a learning rate of 0.001 while constraining model weight norm, for a variety of $\\alpha$ and a variety of train set fractions. The LU shape holds for $\\alpha \\in [ 0 . 1 , 4 ]$ (some optimization issue may be responsible for the rise in train loss for $\\alpha > 4$ ). We see the critical train set size is approximately 0.25, in line with earlier studies on grokking. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2714e43f1c8deee347e84c566adac8a2b66defac65bcb419eff15152224526df.jpg",
+        "image_caption": [
+            "Figure 8: Reduced loss landscapes for transformers trained on modular addition, the original setting where grokking was observed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "C WEIGHT NORM EVOLUTION OVER TIME ON ALGORITHMIC TASKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Evolution of weight norm As mentioned in Section 5, the dynamics of model weight norm over the course of training, on algorithmic tasks, support the LU mechanism picture of grokking. Figure 9a, shows how model norm changes over time and we see that there is an initial increase in weight norm, which peaks during overfitting, but then drops during the period of generalization to be lower than the initialization norm. For this experiment, we again used the setup of (Nanda et al., 2023). We train with AdamW with a learning rate of 0.001 and weight decay $\\gamma = 1$ . ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Constraining a small weight norm eliminates grokking As shown in Figure $^ \\mathrm { 9 b }$ , reducing the initialization scale $\\alpha = 0 . 8$ ) and constraining optimization to hold model weight norm constant over training brings train accuracy and test accuracy learning curves together, almost eliminating grokking. We would like to investigate in future works whether this training trick can be helpful for more standard machine learning tasks. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/07dd3f5ae03b6fe9348084b6bdeab48f74d1959c83536156144a01a9bd753640.jpg",
+        "image_caption": [
+            "Figure 9: Training 1L transformer on modular addition $( p = 1 1 3 )$ . (a) Weight norm, train accuracy, and test accuracy over time, initialized and trained normally. Weight norm first increases, and is highest during the period of overfitting, but then drops to become lower than initial weight norm when the model generalizes. (b) Constrained optimization at constant weight norm ( $\\alpha = 0 . 8$ ) largely eliminates grokking, with test and train accuracy improving almost concurrently. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D TIME TO GENERALIZE VERSUS WEIGHT DECAY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In our discussion of the “LU mechanism” as an explanation for grokking in Section 2, we predicted that the training time required for a model to generalize should be $t \\propto \\gamma ^ { - 1 }$ where $\\gamma$ is the weight decay. To test this, we perform a grid search over weight decays $\\gamma$ and plot the number of training steps required for models to reach a specified level of test accuracy in Figure 10a-10b. We also show full training curves for these runs in Figure 10c-10d. We perform experiments in two setups: ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Transformer on modular addition: We use the replication of grokking from Nanda et al. (2023) and train a 1-layer transformer on modular addition $p = 1 1 3$ and a train set fraction of 0.3) where $d _ { \\mathrm { m o d e l } } = 1 2 8$ , with 4 attention heads, $d _ { \\mathrm { m l p } } = 5 1 2$ , ReLU activations, and an AdamW learning rate of 0.001. From Figure 10a, we find that $t \\propto \\gamma ^ { - 1 }$ holds across roughly two orders of magnitude of $t$ and $\\gamma$ . There is some seed dependence on the generalization time (some seeds consistently require longer to generalize), but for each seed (corresponding to a particular model initialization) the relation $\\bar { t } \\propto \\gamma ^ { - 1 }$ appears to fit the data well. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(b) ReLU MLP on MNIST: We train ReLU MLPs on MNIST as described in Appendix A. We use an $\\alpha = 9 . 0$ and train on a reduced training set of 1000 samples to delay generalization / induce grokking. From Figure 10b, we find that for $\\gamma$ roughly between 0.1 and 1.0 the relation $t \\propto \\gamma ^ { - \\bar { 1 } }$ holds. Very high values of weight decay seem to mess with optimization. On the other hand, with very low weight decay the model generalizes faster than naively expected, perhaps due to implicit regularization. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E SECTION 5.1 SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Architecture Similar to Liu et al. (2022), the decoder architecture is an MLP with hard coded addition. Each input symbol $i$ is encoded to a scalar $E _ { i }$ . Each output symbol $k$ is represented by a 30D random vector $\\hat { \\mathbf Y } _ { k }$ . We consider addition with base $p$ , so input $0 \\leq i , j \\leq p - 1$ and output $0 \\leq k = i + j \\leq 2 ( p - 1 )$ . We denote representation as $\\mathbf { R } = \\{ E _ { 0 } , E _ { 1 } \\cdots , E _ { p - 1 } \\}$ . The MLP has two hidden layers, with neurons 1-200-200-30 in each layer and ReLU activations. Given a training sample $( E _ { i } , E _ { j } )  \\mathbf { Y } _ { k }$ where $i + j = k$ , the prediction of the MLP decoder is ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f189d3c9d09121260977a6b566182c40ac2c9cea376778c3b7dc83c66e1babc2.jpg",
+        "text": "$$\n\\mathbf { Y } _ { k } = \\operatorname { D e c } _ { \\mathbf { w } } ( E _ { i } + E _ { j } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            873,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "and the loss function being the mean squared error (MSE) between ${ \\bf Y } _ { k }$ and $\\hat { \\mathbf Y } _ { k }$ , and w being the decoder weight. Although the common setup of grokking is to make both the representation $\\mathbf { R }$ and ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/b67a56b90cc36d049e1a73248d25ef11a4226b4c04955c99390ba84c6d26b36a.jpg",
+        "image_caption": [
+            "Figure 10: Time to generalize as a function of weight decay: we investigate to what extent the relation $t \\stackrel { \\bf { \\breve { \\mathbf { \\alpha } } } } { \\propto } \\gamma ^ { - 1 }$ holds, where $t$ is number of training steps needed for the model to generalize and $\\gamma$ is the AdamW weight decay. When a lower weight decay is used, models spend longer in the period of overfitting before eventually generalizing. We show the generalization time $t$ as a function of $\\gamma$ in (a)-(b) and full training curves for these runs in (c)-(d). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/475dca052fa71d19e05de3c7234dcee63c9797e4f2f4ec8ae2747aad33c68493.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Trainability</td><td rowspan=1 colspan=2>decoderweightw</td><td rowspan=1 colspan=2>Representation R</td></tr><tr><td rowspan=1 colspan=1>norm w=w2</td><td rowspan=1 colspan=1>direction w=w/w</td><td rowspan=1 colspan=1>messiness m</td><td rowspan=1 colspan=1>Other</td></tr><tr><td rowspan=1 colspan=1>Landscape analysis</td><td rowspan=1 colspan=1>No,w</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>No, m</td><td rowspan=1 colspan=1>No,0</td></tr><tr><td rowspan=1 colspan=1>Reduced trajectory</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>No,w*(w,m)</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>No,0</td></tr><tr><td rowspan=1 colspan=1>Full trajectory</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Yes</td></tr></table>",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Threes setups used in this paper, with different set of parameters trainable. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "the decoder w trainable, we will freeze part of them for easier analysis. This is where it could be a bit confusing, so we explicitly distinguish three setups: landscape analysis, reduced trajectory analysis and full trajectory analysis. Each setup have different subset of trainable parameters, as shown in Table 1. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Landscape analysis Both the representation $\\mathbf { R }$ and weight norm $w$ are fixed. Only the weight direction wˆ is trainable. The representation $\\mathbf { R }$ is fixed according to Eq. (4), which is dependent on $m$ the representation messiness. The decoder has fixed weight norm $w$ , but the weight direction wˆ is trainable. For each fixed $( w , m )$ , we minimize training loss over $\\hat { \\mathbf { w } }$ to get ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5706904877ef42197007496e83d06e6ab520cc8ec913284f55ddd542f01828bb.jpg",
+        "text": "$$\n\\hat { \\mathbf { w } } ^ { * } ( w , m ) = \\mathop { \\mathrm { a r g m i n } } _ { \\hat { \\mathbf { w } } } \\ell _ { \\mathrm { t r a i n } } ( w , m , \\hat { \\mathbf { w } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            839,
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+            864
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "and define reduced training and test loss, as in Eq. (5). The minimization is implemented by the Adam optimizer with learning rate $1 0 ^ { - 3 }$ for $1 0 ^ { 4 }$ steps. Although $( w , m )$ are not trainable, we repeat the above minimization independently for different $( w , m )$ . In Figure 6 (a)(b)(d), the background heatmaps belong to landscape analysis. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Reduced trajectory analysis is a “thought experiment\" based on landscape analysis. Since full trajectory analysis can be intractable due to too high dimensions, we try to reduce the trajectory anaysis to 2D, by making two assumptions about the real dynamics: (1) Scale separation: the dynamics of $\\hat { \\mathbf { w } }$ is much faster than the dynamics along $w$ and along $m$ , such that $\\hat { \\mathbf { w } } ( t ) = \\hat { \\mathbf { w } } ^ { * } ( w ( t ) , \\dot { m } ( t ) )$ is valid at every moment during training. (2) Representation evolution is linear, i.e., interpolating between initial random Gaussian and final linear representation. With these two assumptions, the training dynamics is effectively reduced to 2D, depending only on $( w , m )$ , obeying Eq. (6). In Figure 6 (a)(b)(c), the path $\\mathrm { A }  \\mathrm { E }$ belongs to reduced trajectory analysis. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Admittedly the reduced trajectory may deviate from the full trajectory since the assumptions may not be met, but it can shed light on the full trajectory: the weight norm first increases and then increases, and the decrease of weight norm is highly correlated with generalization (please see Appendix C and Figure 9. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F MNIST EXPERIMENTS WITH CROSS ENTROPY LOSS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 14
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+    {
+        "type": "text",
+        "text": "To respond to a reviewer’s concern that our use of the MSE loss is the “secret\" to get grokking on MNIST (Figure 3), we reran our experiments with the cross entropy (CE) loss. The results are qualitatively similar, with some quantitative differences. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/db9721eb609abca8eb117b1014efd2dff107deb304924587fdfe1755304595c4.jpg",
+        "image_caption": [
+            "Figure 11: MNIST with the cross entropy loss (as opposed to the MSE loss used in Figure 3). (a) reduced training error, (b) reduced test error. (c) \"LU\" still holds for the cross entropy loss, but the effect is milder than the MSE loss. In particular, the “Goldilocks zone\" (the weight range where generalization happens) is broader. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Landscape analysis ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Comparing Figure 3 (MSE) and Figure 11 (CE), we notice the they are qualitatively similar: (1) for small datasets, the reduced training error and test error resemble an “L\" and “U\" against the weight norm, respectively; (2) for large datasets, the “U\" becomes more like “L\", i.e., the mismatch between the reduced training and test error is small. However, a quantitative difference exist: CE produces a broader “Goldilocks zone\" (the weight range where generalization happens) than MSE. This implies that to induce grokking with CE, we need to increase the weight norm to a larger value (say $\\alpha = 1 0 0 $ ). ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Training dynamics ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We are able to observe delayed generalization during trianing on MNIST with cross entropy loss, but doing so requires a higher $\\alpha$ than was necessary when using MSE loss, as predicted by the reduced loss landscapes in Figure 11. Figure 12 shows training trajectories from a 3-layer ReLU MLP on MNIST trained with cross entropy loss with $\\alpha = 1 0 0$ and $D = 2 0 0$ . We see that test accuracy rises to $3 0 { - } 4 0 \\%$ early in training, then plateaus for an extended period, before increasing to ${ \\approx } 7 5 \\%$ while train accuracy remains at $100 \\%$ . While the dynamics are not as clean as with MSE loss, since test accuracy first plateaus at better-than-random accuracy, we think it is still fair to classify these dynamics as “grokking” due to the improvement in generalization late in training after a plateau. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/143d781ccd66b69fbf10338bde642e2a12651fb009344773ae800af906cbe994.jpg",
+        "image_caption": [
+            "Figure 12: Training curves using cross entropy loss on MNIST. We are still able to observe delayed generalization on MNIST using cross entropy loss, though test accuracy first plateaus at higher than random-guess accuracy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 15
+    }
+]