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+# Inner Monologue: Embodied Reasoning through Planning with Language Models
+
+Wenlong Huang†, Fei $\mathbf { X _ { i } } \mathbf { a } ^ { \dagger }$ , Ted Xiao†, Harris Chan, Jacky Liang, Pete Florence,   
+Andy Zeng, Jonathan Tompson, Igor Mordatch, Yevgen Chebotar, Pierre Sermanet,   
+Noah Brown, Tomas Jackson, Linda Luu, Sergey Levine, Karol Hausman, Brian Ichter Robotics at Google, † equal contribution and alphabetically listed Project website: https://innermonologue.github.io
+
+Abstract: Recent works have shown how the reasoning capabilities of Large Language Models (LLMs) can be applied to domains beyond natural language processing, such as planning and interaction for robots. These embodied problems require an agent to understand many semantic aspects of the world: the repertoire of skills available, how these skills influence the world, and how changes to the world map back to the language. LLMs planning in embodied environments need to consider not just what skills to do, but also how and when to do them - answers that change over time in response to the agent’s own choices. In this work, we investigate to what extent LLMs used in such embodied contexts can reason over sources of feedback provided through natural language, without any additional training. We propose that by leveraging environment feedback, LLMs are able to form an inner monologue that allows them to more richly process and plan in robotic control scenarios. We investigate a variety of sources of feedback, such as success detection, scene description, and human interaction. We find that closed-loop language feedback significantly improves high-level instruction completion on three domains, including simulated and real table top rearrangement tasks and long-horizon mobile manipulation tasks in a kitchen environment in the real world.
+
+# 1 Introduction
+
+Intelligent and flexible embodied interaction requires robots to be able to deploy large repertoires of basic behaviors in appropriate ways, sequence these behaviors as needed for long horizon tasks, and also recognize when to switch to a different approach if a particular behavior or plan is unsuccessful. High-level planning, perceptual feedback, and low-level control are just a few of the sub-tasks that would need to be seamlessly combined together to perform the sort of reasoning required for an embodied agent, such as a robot, to intelligently act in the world. While conventionally these challenges have been approached from the perspective of planning (e.g., TAMP [1]) or hierarchical learning (e.g., HRL [2]), effective high-level reasoning about complex tasks also requires semantic knowledge and understanding of the world.
+
+One of the remarkable observations in recent machine learning research is that large language models (LLMs) can not only generate fluent textual descriptions, but also appear to have rich internalized knowledge about the world [3, 4, 5, 6, 7]. When appropriately conditioned (e.g., prompted), they can even carry out some degree of deduction and respond to questions that appear to require reasoning and inference [8, 9, 10, 11, 12, 13]. This raises an intriguing possibility: beyond their ability to interpret natural language instructions, can language models further serve as reasoning models that combine multiple sources of feedback and become interactive problem solvers for embodied tasks, such as robotic manipulation?
+
+Prior studies show that language helps humans internalize our knowledge and perform complex relational reasoning through thinking in language [14, 15, 16, 17, 18]. Imagine the “inner monologue” that happens when a person tries to solve some task: “I have to unlock the door; let me try to pick up the key and put it in the lock... no, wait, it doesn’t fit, I’ll try another one... that one worked, now I can turn the key.” The thought process in this case involves choices about the best immediate action to solve the high-level task (“pick up the key”), observations about the outcomes of attempted actions (“it doesn’t fit”), and corrective actions that are taken in response to these observations (“I’ll try another one”). Inspired by the human thought process, we propose that such an inner monologue is a natural framework for incorporating feedback for LLMs.
+
+6th Conference on Robot Learning (CoRL 2022), Auckland, New Zealand.
+
+![](images/bbccad41cc5e20aea1abb341ed8f534bf72ee44935a7f38a47d65f5334398156.jpg)  
+Figure 1: Inner Monologue enables grounded closed-loop feedback for robot planning with large language models by leveraging a collection of perception models (e.g., scene descriptors and success detectors) in tandem with pretrained language-conditioned robot skills. Experiments show our system can reason and replan to accomplish complex long-horizon tasks for (a) mobile manipulation and (b,c) tabletop manipulation in both simulated and real settings.
+
+Our work studies these questions by combining LLMs with various sources of textual feedback, only utilizing few-shot prompting without any additional training. We observe that similarly to recent work [19], natural language provides a universal and interpretable interface for such grounding of model communication and allows them to incorporate their conclusions in an overarching inner monologue driven by a language model. While prior work has investigated using language models as planners [20, 21] or incorporating multimodal-informed perception through language [19], to the best of our knowledge no work has studied the critical link of not only planning with language, but also informing embodied feedback with language, which we investigate in this work.
+
+Specifically, we study methods and sources of feedback for closing the agent-environment loop via an inner monologue and their impact on downstream execution success and new capabilities arising from such interaction. In particular, we combine multiple perception models that perform various tasks such as language-conditioned semantic classification or language-based scene description, together with feedback provided by a human user that the robot is cooperating with. To execute the commands given by a user, the actions are chosen from a set of pre-trained robotic manipulation skills together with their textual descriptions that can be invoked by a language model. Our proposed system Inner Monologue chains together these various components (perception models, robotic skills, and human feedback) in a shared language prompt, enabling it to successfully perform user instructions.
+
+Finally, we show that Inner Monologue, without requiring additional training beyond a frozen language model and pre-trained robotic skills, can accomplish complex, long-horizon, and unseen tasks in simulation as well as on two real-world robotic platforms. Notably, we show that it can efficiently retry under observed stochastic failure, replan under systematic infeasibility, or request human feedback for ambiguous queries, resulting in significantly improved performance in dynamical environments. As a demonstration of the versatility of LLMs and grounded closed-loop feedback, we additionally show several surprising capabilities emerging from the inner monologue formulation, including continued adaptation to new instructions, self-proposed goals, interactive scene understanding, multilingual interactions, and more.
+
+# 2 Related Work
+
+Task and Motion Planning. Task and motion planning [22, 23] requires simultaneously solving a high-level, discrete task planning problem [24, 25, 26], and a low-level, continuous motion planning problem [27]. Traditionally, this problem has been solved through optimization [28, 29] or symbolic reasoning [24, 26], but more recently machine learning has been applied to aspects of the problem via learned representations, learned task-primitives, and more [30, 31, 32, 33, 34, 35, 36, 37, 38]. Some works utilize language for planning and grounding [39, 40, 41, 42, 43, 44]. Others have approached the problem through hierarchical learning [45, 46, 34, 47, 48, 49, 50]. In this work, we leverage pre-trained LLMs and their semantic knowledge, along with trained low-level skills, to find feasible plans.
+
+Task Planning with Language Models. Various prior works have explored using language as a space for planning [51, 52, 53, 20, 54, 21]. Some methods use prompt structure, self-talk, or discussion between experts to reason about plans or semantic concepts [55, 19, 10, 11]. Similar to ours are recent task planning approaches that leverage pre-trained autoregressive LLMs to decompose abstract, high-level instructions into a sequence of low-level steps executable by an agent [20, 21] in a zero-shot manner. Specifically, Huang et al. [20] prompt GPT-3 [9] and Codex [56] to generate action plans for embodied agents, where each action step is semantically translated to an admissible action with a Sentence-RoBERTa model [57, 58]. SayCan [21] instead grounds the actions by multiplying each candidate action’s probability under FLAN [59] with the action’s value function, which serves as a proxy for affordance [34]. However, both approaches effectively produce the plan while assuming that each proposed step is executed successfully by the agent. As a result, these approaches may not be robust in handling intermediate failures in dynamic environments or with poor lower level policies. We explore in Inner Monologue ways to incorporate grounded feedback from the environment into the LLM as we produce each step in the plan.
+
+![](images/042f8fad029b3eccb9b2940378efa3272789deb428476874bd69fbe6d8a34faf.jpg)  
+Figure 2: Various types of textual feedback. Success Detection gives task-specific task completion information, Passive Scene Description gives structured semantic scene information at every planning step, and Active Scene Description gives unstructured semantic information only when queried by the LLM planner.
+
+Fusing Vision, Language, and Control in Robotics. Various works have investigated strategies for the challenging problem of fusing vision, language, and control [60, 61, 62, 63, 64, 65, 66]. Some works have been trained directly for language-based interaction in robotic tasks [67, 68, 69, 70]. Recent large visuallanguage models (e.g., CLIP [71]) have been trained on joint image(s) and corresponding text captions via variants of a masked language modeling objective [72, 73, 74, 75], a contrastive loss [76, 77, 71] or other supervised objectives[78, 79]. CLIP has been employed in several robotics and embodied settings in zero-shot manner [80], or combined with Transporter networks [81] as in CLIPort [82]. Finally, Socratic Models [19] proposes the combination of different foundation models (e.g., GPT-3 [9], ViLD [83]) and language-conditioned policies, using language as the common interface. While Socratic Models has been demonstrated on a tabletop object manipulation task, Inner Monologue examines additional challenges for robots operating in dynamic environments, which require closed-loop feedback to the planner.
+
+# 3 Leveraging Embodied Language Feedback with Inner Monologue
+
+We consider the setting where an embodied robotic agent attempts to perform a high-level natural language instruction $i$ . This robotic agent is only capable of executing short-horizon skills from a library of previously trained policies $\pi _ { k } \in \Pi$ with short language descriptions $\ell _ { k }$ , which may be trained with reinforcement learning or behavioral cloning. The “planner,” which is a pretrained LLM [20, 21], attempts to find a sequence of skills to accomplish the instruction. To observe the environment, the planner has access to textual feedback $o$ from the environment that can be appended to the instruction or requested by the planner. Our work studies to what extent the LLM planner is able to reason over and utilize such feedback to “close the loop” with the environment and improve planning.
+
+# 3.1 Inner Monologue
+
+We formulate an “inner monologue” by continually injecting information from the various sources of feedback into the LLM planning language prompts as the robot interacts with the environment. While LLMs have demonstrated exceptional planning capabilities for embodied control tasks [20], prior works have found it crucial to ground LLM predictions with external components such as affordance functions [21] in order to produce useful plans that are executable by robots. However, LLMs used in this context have thus far remained one-directional – providing a list of skills, without making corrections or leveraging opportunities to replan accordingly. In contrast, Inner Monologue studies settings where grounded environment feedback is provided directly to the LLM in a closed-loop fashion. This promotes improved LLM reasoning in complex long-horizon settings, even before any external affordance-based grounding methods are applied.
+
+![](images/b751ebf5a9eeb4eacad63238d86487141852922195f38985cb0dd571d16bf2ea.jpg)  
+Figure 3: Different instantiations of Inner Monologue in three distinct domains – simulated tabletop rearrangement (top), real-world tabletop rearrangement (middle), and real-world kitchen mobile manipulation (bottom). Each domain uses different prompts and different feedback models. Sharing across the domains is the same Inner Monologue formulation that uses a pre-trained langauge model to take in a human instruction and decompose it into a sequence of actionable steps by the agent, while accounting for injected embodied feedback from different models, such as object recognizers and success detectors. In real-world kitchen mobile manipulation domain (bottom), we additionally ground the actions using pre-trained affordance functions built in [21], which do not communicate back to the language model.
+
+Our analysis assumes textual feedback is provided to the planner, but does not assume a single specific method of fusing LLM planning with low-level robotic control or a specific method of extracting environment feedback into language. Rather than focusing on a particular algorithmic implementation, our aim is to provide a case study on the value of incorporating different types of feedback into closed-loop LLM-based planning. Thus, Inner Monologue in Sec 4 utilizes language feedback within separate systems that incorporate different LLMs, different methods of fusing planning with control, different environments and tasks, and different methods of acquiring control policies. We note that in our specific implementations of Inner Monologue, we use pre-trained LLMs for planning that are not finetuned, but rather evaluated solely with few-shot prompting; the full prompts can be found in the Appendix.
+
+# 3.2 Sources of Feedback
+
+In theory any type of environment feedback can inform the LLM planner, as long as it can be expressed through language. We focus on the specific forms of feedback shown in Fig 2: (1) task-specific feedback, such as success detection, and (2) scene-specific feedback (either “passive” or “active”), which describes the scene. Specific instantiations and implementation details of each type of feedback can be found in Sec 4.1, Sec 4.2, and Sec 4.3 respectively for each domain.
+
+Success Detection. The Success feedback gives binary “yes” or “no” response in language form, specifying whether the low-level skill $\pi _ { k }$ has succeeded. Engineered success detectors can operate on ground-truth state in simulation, while learned success detectors can be trained on real examples of successes and failures in the real world [84, 85, 86, 87, 88].
+
+Passive Scene Description. We refer broadly to any sources of scene feedback that are consistently and automatically injected into the LLM prompt as Passive Scene Description, which also typically follow some structure. One common type of such feedback is object recognition [89, 90, 91, 92] that returns a list of present objects, to which we refer as Object feedback. We also demonstrate the use of a task-progress scene description in the simulated tabletop rearrangement environment, to which we refer as Scene feedback.
+
+Active Scene Description. As the proactive counterpart, Active Scene Description encompasses sources of feedback that are provided directly in response to active queries by the LLM planner, which are answered either by a person, or by another pretrained model, such as a Visual Question Answering (VQA) model [93, 94, 95, 96]. Unlike the passive counterpart which are strictly structured and narrow in their scope, this feedback allows the planner to actively gather information relevant to the scene, the task, or even preferences of the user. The combined output we send to the LLM planner includes both the LLM-generated question along with the response. As we aim to investigate whether and how a LLM planner can incorporate such feedback and wish to study both structured VQA-style human feedback as well as unstructured human preferences feedback, we only consider human-provided response in this work, which we refer to as Human feedback.
+
+# 4 Experimental Results
+
+In order to study how different sources of environment feedback can support a rich inner monologue that enables complex robotic control, we study different Inner Monologue implementations in three environments, each with different LLM and different sources of feedback from the environment: 1) simulated tabletop manipulation (Sec 4.1), 2) real-world tabletop manipulation (Sec 4.2), and 3) real-world mobile manipulation in an office kitchen (Sec 4.3). For more details about the experiment setup and results, please refer to the Appendix.
+
+# 4.1 Simulated Tabletop Rearrangement
+
+We experiment with Ravens-based [81] environment, where a robotic arm with a gripper is tasked with rearranging blocks and bowls in some desired configuration, specified by natural language. We evaluate each method on four seen tasks and four unseen tasks, where seen tasks may be used for training (in the case of supervised baseline) or used as few-shot prompting.
+
+This instantiation of Inner Monologue uses (i) InstructGPT [9, 97] for planning [20, 21], (ii) scripted modules to provide language feedback in the form of object recognition (Object), success detection (Success), and task-progress scene description (Scene), and (iii) a pre-trained language-conditioned pick-and-place primitive (similar to CLIPort [82] and Transporter Nets [81]). Object feedback informs the list of present objects and Success feedback informs the success/failure of the most recent action. However, consider the task of stacking multiple blocks, because the unfinished tower of blocks may be knocked over by the robot, it is also critical to reason about overall task progress. Therefore, task-progress scene description (Scene) describes the semantic sub-goals inferred by the LLM towards completing the high-level instruction that are achieved by the agent so far.
+
+We additionally compare to a multi-task CLIPort directly trained on long-horizon task instructions. Because CLIPort is a single-step policy and does not terminate spontaneously during policy rollout, we report CLIPort evaluations with oracle termination (i.e., repeat until oracle indicates task completion) and fixed-step termination (i.e., repeat for 15 steps). To simulate real-world disturbances and evaluate the system’s robustness to disturbances, we add Gaussian noise to multiple levels of the system at test time: $\mathcal { N } ( 0 { , } 3 )$ for pixel observation, $\mathcal { N } ( 0 , 2 . 5 )$ for policy primitive (i.e., pick-place pixel heatmaps), $\mathcal { N } ( 0 , 0 . 0 2 m )$ for place locations.
+
+<table><tr><td rowspan="2" colspan="3"></td><td></td><td>+LLM</td><td colspan="2">+Inner Monologue</td></tr><tr><td>CLIPort</td><td>+oracle Object</td><td>Object + Success</td><td>Object + Scene</td></tr><tr><td rowspan="4">Seen Tasks</td><td>“Pick and place”</td><td>24.0%</td><td>74.0%</td><td>80.0%</td><td>90.0%</td><td>94.0%</td></tr><tr><td>“Stack all the blocks”</td><td>2.0%</td><td>32.0%</td><td>4.0%</td><td>10.0%</td><td>26.0%</td></tr><tr><td>“Put all the blocks on the [x] corner/side&quot;</td><td>2.0%</td><td>32.0%</td><td>30.0%</td><td>28.0%</td><td>30.0%</td></tr><tr><td>“Put all the blocks in the [x] bowl&quot;</td><td>32.0%</td><td>94.0%</td><td>52.0%</td><td>46.0%</td><td>56.0%</td></tr><tr><td rowspan="4">Unseen Tasks</td><td>“Put all the blocks in different corners&quot;</td><td>0.0%</td><td>0.0%</td><td>20.0%</td><td>20.0%</td><td>26.0%</td></tr><tr><td>“Put the blocks in their matching bowls”</td><td>0.0%</td><td>0.0%</td><td>56.0%</td><td>70.0%</td><td>82.0%</td></tr><tr><td>“Put the blocks on mismatched bowls&quot;</td><td>0.0%</td><td>0.0%</td><td>62.0%</td><td>76.0%</td><td>86.0%</td></tr><tr><td>“Stack all the blocks on the [x] corner/side”</td><td>0.0%</td><td>0.0%</td><td>0.0%</td><td>4.0%</td><td>6.0%</td></tr></table>
+
+Table 1: Success rates averaged across 50 episodes in simulated pick-and-place. CLIPort $^ +$ oracle indicates that CLIPort was provided a “termination” oracle. LLM-informed feedback effectively enable retrying/replanning in the presence of test-time disturbances, while enjoying the generalization benefits of LLMs to unseen tasks.
+
+Analysis. As shown in Table 1, Inner Monologue effectively enables retrying and replanning in the face of test-time disturbances, where Object $^ +$ Scene performs the best because of its ability to keep track of sub-goal conditions. Furthermore, this performance directly translates to unseen tasks by leveraging rich semantic knowledge of LLM. Finally, we observe that non-hierarchical and solitary systems such as CLIPort (i) struggle at generalizing to unseen long-horizon tasks under test-time disturbances, and (ii) on training tasks, an oracle is also often required to indicate task completion for good performance.
+
+# 4.2 Real-World Tabletop Rearrangement
+
+We evaluate Inner Monologue on a real-world robot platform designed to resemble the simulation experiments. This instantiation uses (i) InstructGPT [9, 97] for planning, (ii) MDETR [98] for open-vocab object recognition (Object) (iii) heuristics on the object bounding box predictions from MDETR for Success Detection (Success), and (iv) a suction-based pick-and-place motion primitive that uses an LLM to parse target objects from a language command (e.g., given by the planner).
+
+We investigate two tasks: (i) a 3-block stacking task where 2 blocks are already pre-stacked, and (ii) a long-horizon sorting task to place food in one plate and condiments in another (where categorizing food versus condiments is autonomously done by the LLM planner). In additional to additional challenges of real-world perception and clutter, we artificially inject Gaussian noise into the policy actions (i.e., add standard deviation $\sigma { = } 4 \mathrm { m m }$ clipped at $2 \sigma$ ) to stress test recovery from failures via replanning with grounded closed-loop feedback. Results are presented in Table 2.
+
+<table><tr><td></td><td>LLM</td><td colspan="3">+Inner Monologue</td></tr><tr><td>Task Family</td><td>Object</td><td>Object</td><td>Success</td><td>Object + Success</td></tr><tr><td>Finish 3-block stacking</td><td>20%</td><td>40%</td><td>40%</td><td>100%</td></tr><tr><td>Sort fruits from bottles</td><td>20%</td><td>50%</td><td>40%</td><td>80%</td></tr><tr><td>Total</td><td>20%</td><td>45%</td><td>40%</td><td>90%</td></tr></table>
+
+Table 2: Success rates averaged across 10 runs in real-world pick-and-place. We observe significant improvement in Inner Monologue with Object and Success feedback, with the two feedback being complementary to each other.
+
+Analysis. We compare to variants with only Object or Success feedback, as well as an open-loop variant (“LLM Object”) that only runs object recognition once at the beginning of the task (similar to the system demonstrated in [19]). The partial 3-block stacking task highlights an immediate failure mode of the open-loop baseline, where the initial scene description struggles to capture a complete representation of the scene (due to clutter and occlusion) to provide as input to the multi-step planner. As a result, the system only executes one pick-and-place action – and cannot recover from mistakes. To address these shortcomings, Inner Monologue $( O b j e c t + S u c c e s s )$ ) leverages closed-loop scene description and success detection after each step, which allows it to successfully replan and recover from policy mistakes.
+
+# 4.3 Real-World Mobile Manipulator in a Kitchen Setting
+
+We implement Inner Monologue in a robotic system using the kitchen environment and task definitions described in SayCan [21]. The Everyday Robots robot, a mobile manipulator with RGB observations, is placed in an office kitchen to interact with common objects using concurrent [99] continuous closed-loop control.
+
+The baseline, SayCan [21], is a method that plans and acts in diverse real world scenarios by combining an LLM with value functions of control policies. While SayCan creates plans that are grounded by the affordances of value functions, the LLM predictions in isolation are never given any closed-loop feedback.
+
+We use an instantiation of Inner Monologue that uses (i) PALM [8] for planning, (ii) value functions from pre-trained control policies for affordance grounding [21], (iii) a learned visual classification model for Success feedback, (iv) human-provided Object feedback, and (v) pre-trained control policies for relevant skills in the scene. We also perform a case study where we allow the agent to ask questions and source Human feedback directly; results are shown in Fig 5a and the Appendix.
+
+We evaluate on 120 runs over three task families: (1) four manipulation tasks, (2) two dexterous manipulation tasks utilizing drawers, and (3) two long-horizon combined manipulation and navigation tasks. We consider both cases with and without manually-added adversarial disturbances during control policy executions that cause skill policy rollouts to fail. While these failures occur naturally even without perturbances, the adversarial disturbances creates a consistent comparison between methods that requires retrying or replanning to accomplish the original instruction.
+
+Table 3: Averaged success rate across 120 evaluations on several task families in our real-world mobile manipulation environment. We consider a standard setting and adversarial setting with external human disturbances. In all cases, LLM-informed embodied feedback is shown to be effective in improving robustness of the system, especially when low-level policies are prone to failures.   
+
+<table><tr><td rowspan="2">Task Family</td><td rowspan="2">SayCan</td><td colspan="2">+Inner Monologue</td></tr><tr><td>Success</td><td>Object + Success</td></tr><tr><td>No Disturbances</td><td></td><td></td><td></td></tr><tr><td>Manipulation</td><td>50.0%</td><td>62.5%</td><td>75.0%</td></tr><tr><td>Mobile Manipulation</td><td>50.0%</td><td>50.0%</td><td>75.0%</td></tr><tr><td>Drawers</td><td>83.3%</td><td>83.3%</td><td>100.0%</td></tr><tr><td>With Disturbances</td><td></td><td></td><td></td></tr><tr><td>Manipulation</td><td>12.5%</td><td>25.0%</td><td>33.3%</td></tr><tr><td>Mobile Manipulation</td><td>0.0%</td><td>25.0%</td><td>75.0%</td></tr><tr><td>Drawers</td><td>0.0%</td><td>44.4%</td><td>44.4%</td></tr><tr><td>Total</td><td>30.8%</td><td>48.7%</td><td>60.4%</td></tr></table>
+
+![](images/b69f1e6523a1bf91127c8424584ad2b72679a7d90508907785b1eab73cac9df5.jpg)  
+Figure 4: Failure causes on 120 evaluations. When disturbances are added (red), only the Inner Monologue variants consistently complete the instructions.
+
+Analysis. Without adversarial disturbances, the baseline SayCan performs reasonably on all tasks, yet incorporating LLM-informed feedback in Inner Monologue allows further improvement by effectively retrying or replanning under natural failures. The most notable difference is in the cases with adversarial disturbances. Without any LLM-informed feedback SayCan has success rate close to $0 \%$ since LLM always assume successful execution of previous skills. Inner Monologue significantly outperforms SayCan because of its ability to invoke appropriate recovery modes depending on the environment feedback. Analysis on the failure causes indicates that Success and Object feedback can reduce LLM planning failures and thus overall failure rate, albeit at the cost of introducing new failure modes to the system.
+
+# 4.4 Plan Generalization Capabilities
+
+Although LLMs can generate fluent continuation from the prompted examples, we surprisingly find that, Inner Monologue demonstrates many impressive reasoning and replanning behaviors beyond the examples given in the prompt. Using a pre-trained LLM as the backbone, the method also inherits many of the appealing properties from its versatility and general-purpose language understanding. In this section, we demonstrate a few of these capabilities; additional capabilities are shown in Appendix (Fig ?? and Fig ??).
+
+Continued Adaptation to New Instructions. Although not explicitly prompted, the LLM planner can react to human interaction that changes the high-level goal mid-task. Fig 5a demonstrates a challenging case, where Human feedback changes the goal during the plan execution, and then changes the goal yet again by saying “finish the previous task”. In another instance, despite not being explicitly prompted to terminate after a human says “please stop”, the LLM planner generalizes to this scenario and predicts a “done” action.
+
+Self-Proposing Goals under Infeasibility. Instead of mindlessly following human-given instructions, Inner Monologue can also propose alternative goals to achieve when the previous goal becomes infeasible. In Fig 5b, to solve the task “put any two blocks inside the purple bowl”, while the first attempted block is intentionally made too heavy for the robot, Inner Monologue proposes to “find a lighter block” and successfully solves the task.
+
+Multilingual Interaction. Pre-trained LLMs are known to be able to translate from one language to another, without any finetuning. We observe that such multilingual understanding also transfers to the embodied settings. Fig 5c shows a case when an instruction is in Chinese, the LLM planner can still correctly interpret it, re-narrate it as a concrete goal to execute in English, and accordingly replan its future actions. Occasionally, we find that this capability even extends to symbols and emojis.
+
+Retrospective Scene Understanding. We also observe that Inner Monologue demonstrates retrospective scene understanding based on past actions and environment feedback, which requires temporal and embodied reasoning. In Fig 5d, after series of actions, we can turn to ask questions about the resulting scene, again a structure that has not appeared in the prompt.
+
+![](images/6e4d56d21ac1d79c836010878e48898181199d009ab6fd9d5a4387fba16cf9ab.jpg)  
+Figure 5: Informing LLM with embodied feedback enables many generalization capabilities, all of which are achieved without similar prompted examples. For instance, Inner Monologue can continually adapt to new instructions given by humans, propose new goals to achieve when faced with infeasibility for the previous plan, interact with humans in different natural languages, and answer questions about the current scene given past actions and feedback.
+
+Despite the appealing findings about these generalization capabilities, we observe that they are of varying levels of consistency when no similar examples have been provided in the prompt, likely limited by the current capabilities of the language models. However, we believe that further investigations into these behaviors and addressing their limitations would each lead to exciting future directions.
+
+# 5 Conclusions, Limitations & Future Works
+
+In this work, we investigated the role that environment feedback plays for LLMs reasoning in tasks involving embodied robotic planning and interaction. We presented a general formulation Inner Monologue that combines different sources of environment feedback with methods fusing LLM planning with robotic control policies and studied its instantiations in three distinct domains. We found that environment feedback significantly improves high-level instruction completion, especially in challenging scenarios with adversarial disturbances. Finally, we analyze generalization capabilities of Inner Monologue that highlight how closed-loop language feedback enables replanning even in complex unseen settings.
+
+Limitations. In $\mathrm { S e c } ~ 4 . 1$ and Sec 4.3, we assume access to oracle scene descriptors in the form of human observers or scripted systems to provide textual description back to the LLM planner. We study the viability of learned systems scene description and object recognition in Appendix Table ??. As for failure modes, Inner Monologue may fail due to several sources of errors: (1) success detections, (2) LLM planning errors, and (3) control errors. False negative predictions from the success detector lead to additional retry attempts, while false positive predictions add adversarial partial observability to the environment. In some instances, we found that the LLM planners ignored the environment feedback and still proposed policy skills involving objects not present in the scene. Additionally, the performance of low-level control policies limits not only overall high-level instruction completion performance, but also limits the scope of tasks that the LLM is able to plan actions for.
+
+Future Works. Several fronts can be improved by future works. First, with advances in image/video captioning and visual-question answering, a fully automated system of Inner Monologue can be implemented without a human in the loop as an oracle. Second, improvements can be made on how to aggregate potentially inaccurate sources of information, such as using text to describe the uncertainty of the feedback modules, or including additional feedback modules for safety and ethics for the proposed plans. Finally, enabling low-level control policies to take as input the textual feedback by LLM also leads to exciting future directions.
+
+# Acknowledgments
+
+The authors would like to thank Kanishka Rao and Vincent Vanhoucke for valuable feedback and discussions. In addition, the authors would like to acknowledge the large team who built [21], upon which we construct our Kitchen Mobile Manipulation experiments.
+
+References   
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+        "text": "Inner Monologue: Embodied Reasoning through Planning with Language Models ",
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+        "text": "Wenlong Huang†, Fei $\\mathbf { X _ { i } } \\mathbf { a } ^ { \\dagger }$ , Ted Xiao†, Harris Chan, Jacky Liang, Pete Florence,   \nAndy Zeng, Jonathan Tompson, Igor Mordatch, Yevgen Chebotar, Pierre Sermanet,   \nNoah Brown, Tomas Jackson, Linda Luu, Sergey Levine, Karol Hausman, Brian Ichter Robotics at Google, † equal contribution and alphabetically listed Project website: https://innermonologue.github.io ",
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+        "text": "Abstract: Recent works have shown how the reasoning capabilities of Large Language Models (LLMs) can be applied to domains beyond natural language processing, such as planning and interaction for robots. These embodied problems require an agent to understand many semantic aspects of the world: the repertoire of skills available, how these skills influence the world, and how changes to the world map back to the language. LLMs planning in embodied environments need to consider not just what skills to do, but also how and when to do them - answers that change over time in response to the agent’s own choices. In this work, we investigate to what extent LLMs used in such embodied contexts can reason over sources of feedback provided through natural language, without any additional training. We propose that by leveraging environment feedback, LLMs are able to form an inner monologue that allows them to more richly process and plan in robotic control scenarios. We investigate a variety of sources of feedback, such as success detection, scene description, and human interaction. We find that closed-loop language feedback significantly improves high-level instruction completion on three domains, including simulated and real table top rearrangement tasks and long-horizon mobile manipulation tasks in a kitchen environment in the real world. ",
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+        "text": "1 Introduction ",
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+        "text": "Intelligent and flexible embodied interaction requires robots to be able to deploy large repertoires of basic behaviors in appropriate ways, sequence these behaviors as needed for long horizon tasks, and also recognize when to switch to a different approach if a particular behavior or plan is unsuccessful. High-level planning, perceptual feedback, and low-level control are just a few of the sub-tasks that would need to be seamlessly combined together to perform the sort of reasoning required for an embodied agent, such as a robot, to intelligently act in the world. While conventionally these challenges have been approached from the perspective of planning (e.g., TAMP [1]) or hierarchical learning (e.g., HRL [2]), effective high-level reasoning about complex tasks also requires semantic knowledge and understanding of the world. ",
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+        "text": "One of the remarkable observations in recent machine learning research is that large language models (LLMs) can not only generate fluent textual descriptions, but also appear to have rich internalized knowledge about the world [3, 4, 5, 6, 7]. When appropriately conditioned (e.g., prompted), they can even carry out some degree of deduction and respond to questions that appear to require reasoning and inference [8, 9, 10, 11, 12, 13]. This raises an intriguing possibility: beyond their ability to interpret natural language instructions, can language models further serve as reasoning models that combine multiple sources of feedback and become interactive problem solvers for embodied tasks, such as robotic manipulation? ",
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+        "text": "Prior studies show that language helps humans internalize our knowledge and perform complex relational reasoning through thinking in language [14, 15, 16, 17, 18]. Imagine the “inner monologue” that happens when a person tries to solve some task: “I have to unlock the door; let me try to pick up the key and put it in the lock... no, wait, it doesn’t fit, I’ll try another one... that one worked, now I can turn the key.” The thought process in this case involves choices about the best immediate action to solve the high-level task (“pick up the key”), observations about the outcomes of attempted actions (“it doesn’t fit”), and corrective actions that are taken in response to these observations (“I’ll try another one”). Inspired by the human thought process, we propose that such an inner monologue is a natural framework for incorporating feedback for LLMs. ",
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+            "Figure 1: Inner Monologue enables grounded closed-loop feedback for robot planning with large language models by leveraging a collection of perception models (e.g., scene descriptors and success detectors) in tandem with pretrained language-conditioned robot skills. Experiments show our system can reason and replan to accomplish complex long-horizon tasks for (a) mobile manipulation and (b,c) tabletop manipulation in both simulated and real settings. "
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+        "text": "Our work studies these questions by combining LLMs with various sources of textual feedback, only utilizing few-shot prompting without any additional training. We observe that similarly to recent work [19], natural language provides a universal and interpretable interface for such grounding of model communication and allows them to incorporate their conclusions in an overarching inner monologue driven by a language model. While prior work has investigated using language models as planners [20, 21] or incorporating multimodal-informed perception through language [19], to the best of our knowledge no work has studied the critical link of not only planning with language, but also informing embodied feedback with language, which we investigate in this work. ",
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+        "text": "Specifically, we study methods and sources of feedback for closing the agent-environment loop via an inner monologue and their impact on downstream execution success and new capabilities arising from such interaction. In particular, we combine multiple perception models that perform various tasks such as language-conditioned semantic classification or language-based scene description, together with feedback provided by a human user that the robot is cooperating with. To execute the commands given by a user, the actions are chosen from a set of pre-trained robotic manipulation skills together with their textual descriptions that can be invoked by a language model. Our proposed system Inner Monologue chains together these various components (perception models, robotic skills, and human feedback) in a shared language prompt, enabling it to successfully perform user instructions. ",
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+        "text": "Finally, we show that Inner Monologue, without requiring additional training beyond a frozen language model and pre-trained robotic skills, can accomplish complex, long-horizon, and unseen tasks in simulation as well as on two real-world robotic platforms. Notably, we show that it can efficiently retry under observed stochastic failure, replan under systematic infeasibility, or request human feedback for ambiguous queries, resulting in significantly improved performance in dynamical environments. As a demonstration of the versatility of LLMs and grounded closed-loop feedback, we additionally show several surprising capabilities emerging from the inner monologue formulation, including continued adaptation to new instructions, self-proposed goals, interactive scene understanding, multilingual interactions, and more. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "Task and Motion Planning. Task and motion planning [22, 23] requires simultaneously solving a high-level, discrete task planning problem [24, 25, 26], and a low-level, continuous motion planning problem [27]. Traditionally, this problem has been solved through optimization [28, 29] or symbolic reasoning [24, 26], but more recently machine learning has been applied to aspects of the problem via learned representations, learned task-primitives, and more [30, 31, 32, 33, 34, 35, 36, 37, 38]. Some works utilize language for planning and grounding [39, 40, 41, 42, 43, 44]. Others have approached the problem through hierarchical learning [45, 46, 34, 47, 48, 49, 50]. In this work, we leverage pre-trained LLMs and their semantic knowledge, along with trained low-level skills, to find feasible plans. ",
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+        "text": "Task Planning with Language Models. Various prior works have explored using language as a space for planning [51, 52, 53, 20, 54, 21]. Some methods use prompt structure, self-talk, or discussion between experts to reason about plans or semantic concepts [55, 19, 10, 11]. Similar to ours are recent task planning approaches that leverage pre-trained autoregressive LLMs to decompose abstract, high-level instructions into a sequence of low-level steps executable by an agent [20, 21] in a zero-shot manner. Specifically, Huang et al. [20] prompt GPT-3 [9] and Codex [56] to generate action plans for embodied agents, where each action step is semantically translated to an admissible action with a Sentence-RoBERTa model [57, 58]. SayCan [21] instead grounds the actions by multiplying each candidate action’s probability under FLAN [59] with the action’s value function, which serves as a proxy for affordance [34]. However, both approaches effectively produce the plan while assuming that each proposed step is executed successfully by the agent. As a result, these approaches may not be robust in handling intermediate failures in dynamic environments or with poor lower level policies. We explore in Inner Monologue ways to incorporate grounded feedback from the environment into the LLM as we produce each step in the plan. ",
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+        "image_caption": [
+            "Figure 2: Various types of textual feedback. Success Detection gives task-specific task completion information, Passive Scene Description gives structured semantic scene information at every planning step, and Active Scene Description gives unstructured semantic information only when queried by the LLM planner. "
+        ],
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+    {
+        "type": "text",
+        "text": "Fusing Vision, Language, and Control in Robotics. Various works have investigated strategies for the challenging problem of fusing vision, language, and control [60, 61, 62, 63, 64, 65, 66]. Some works have been trained directly for language-based interaction in robotic tasks [67, 68, 69, 70]. Recent large visuallanguage models (e.g., CLIP [71]) have been trained on joint image(s) and corresponding text captions via variants of a masked language modeling objective [72, 73, 74, 75], a contrastive loss [76, 77, 71] or other supervised objectives[78, 79]. CLIP has been employed in several robotics and embodied settings in zero-shot manner [80], or combined with Transporter networks [81] as in CLIPort [82]. Finally, Socratic Models [19] proposes the combination of different foundation models (e.g., GPT-3 [9], ViLD [83]) and language-conditioned policies, using language as the common interface. While Socratic Models has been demonstrated on a tabletop object manipulation task, Inner Monologue examines additional challenges for robots operating in dynamic environments, which require closed-loop feedback to the planner. ",
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+        "type": "text",
+        "text": "3 Leveraging Embodied Language Feedback with Inner Monologue ",
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+        "text": "We consider the setting where an embodied robotic agent attempts to perform a high-level natural language instruction $i$ . This robotic agent is only capable of executing short-horizon skills from a library of previously trained policies $\\pi _ { k } \\in \\Pi$ with short language descriptions $\\ell _ { k }$ , which may be trained with reinforcement learning or behavioral cloning. The “planner,” which is a pretrained LLM [20, 21], attempts to find a sequence of skills to accomplish the instruction. To observe the environment, the planner has access to textual feedback $o$ from the environment that can be appended to the instruction or requested by the planner. Our work studies to what extent the LLM planner is able to reason over and utilize such feedback to “close the loop” with the environment and improve planning. ",
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+        "text": "3.1 Inner Monologue ",
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+        "type": "text",
+        "text": "We formulate an “inner monologue” by continually injecting information from the various sources of feedback into the LLM planning language prompts as the robot interacts with the environment. While LLMs have demonstrated exceptional planning capabilities for embodied control tasks [20], prior works have found it crucial to ground LLM predictions with external components such as affordance functions [21] in order to produce useful plans that are executable by robots. However, LLMs used in this context have thus far remained one-directional – providing a list of skills, without making corrections or leveraging opportunities to replan accordingly. In contrast, Inner Monologue studies settings where grounded environment feedback is provided directly to the LLM in a closed-loop fashion. This promotes improved LLM reasoning in complex long-horizon settings, even before any external affordance-based grounding methods are applied. ",
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+        "image_caption": [
+            "Figure 3: Different instantiations of Inner Monologue in three distinct domains – simulated tabletop rearrangement (top), real-world tabletop rearrangement (middle), and real-world kitchen mobile manipulation (bottom). Each domain uses different prompts and different feedback models. Sharing across the domains is the same Inner Monologue formulation that uses a pre-trained langauge model to take in a human instruction and decompose it into a sequence of actionable steps by the agent, while accounting for injected embodied feedback from different models, such as object recognizers and success detectors. In real-world kitchen mobile manipulation domain (bottom), we additionally ground the actions using pre-trained affordance functions built in [21], which do not communicate back to the language model. "
+        ],
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+        "type": "text",
+        "text": "Our analysis assumes textual feedback is provided to the planner, but does not assume a single specific method of fusing LLM planning with low-level robotic control or a specific method of extracting environment feedback into language. Rather than focusing on a particular algorithmic implementation, our aim is to provide a case study on the value of incorporating different types of feedback into closed-loop LLM-based planning. Thus, Inner Monologue in Sec 4 utilizes language feedback within separate systems that incorporate different LLMs, different methods of fusing planning with control, different environments and tasks, and different methods of acquiring control policies. We note that in our specific implementations of Inner Monologue, we use pre-trained LLMs for planning that are not finetuned, but rather evaluated solely with few-shot prompting; the full prompts can be found in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "3.2 Sources of Feedback ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In theory any type of environment feedback can inform the LLM planner, as long as it can be expressed through language. We focus on the specific forms of feedback shown in Fig 2: (1) task-specific feedback, such as success detection, and (2) scene-specific feedback (either “passive” or “active”), which describes the scene. Specific instantiations and implementation details of each type of feedback can be found in Sec 4.1, Sec 4.2, and Sec 4.3 respectively for each domain. ",
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+        "text": "Success Detection. The Success feedback gives binary “yes” or “no” response in language form, specifying whether the low-level skill $\\pi _ { k }$ has succeeded. Engineered success detectors can operate on ground-truth state in simulation, while learned success detectors can be trained on real examples of successes and failures in the real world [84, 85, 86, 87, 88]. ",
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+        "type": "text",
+        "text": "Passive Scene Description. We refer broadly to any sources of scene feedback that are consistently and automatically injected into the LLM prompt as Passive Scene Description, which also typically follow some structure. One common type of such feedback is object recognition [89, 90, 91, 92] that returns a list of present objects, to which we refer as Object feedback. We also demonstrate the use of a task-progress scene description in the simulated tabletop rearrangement environment, to which we refer as Scene feedback. ",
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+        "text": "Active Scene Description. As the proactive counterpart, Active Scene Description encompasses sources of feedback that are provided directly in response to active queries by the LLM planner, which are answered either by a person, or by another pretrained model, such as a Visual Question Answering (VQA) model [93, 94, 95, 96]. Unlike the passive counterpart which are strictly structured and narrow in their scope, this feedback allows the planner to actively gather information relevant to the scene, the task, or even preferences of the user. The combined output we send to the LLM planner includes both the LLM-generated question along with the response. As we aim to investigate whether and how a LLM planner can incorporate such feedback and wish to study both structured VQA-style human feedback as well as unstructured human preferences feedback, we only consider human-provided response in this work, which we refer to as Human feedback. ",
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+        "type": "text",
+        "text": "4 Experimental Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In order to study how different sources of environment feedback can support a rich inner monologue that enables complex robotic control, we study different Inner Monologue implementations in three environments, each with different LLM and different sources of feedback from the environment: 1) simulated tabletop manipulation (Sec 4.1), 2) real-world tabletop manipulation (Sec 4.2), and 3) real-world mobile manipulation in an office kitchen (Sec 4.3). For more details about the experiment setup and results, please refer to the Appendix. ",
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+    {
+        "type": "text",
+        "text": "4.1 Simulated Tabletop Rearrangement ",
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+        "type": "text",
+        "text": "We experiment with Ravens-based [81] environment, where a robotic arm with a gripper is tasked with rearranging blocks and bowls in some desired configuration, specified by natural language. We evaluate each method on four seen tasks and four unseen tasks, where seen tasks may be used for training (in the case of supervised baseline) or used as few-shot prompting. ",
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+        "type": "text",
+        "text": "This instantiation of Inner Monologue uses (i) InstructGPT [9, 97] for planning [20, 21], (ii) scripted modules to provide language feedback in the form of object recognition (Object), success detection (Success), and task-progress scene description (Scene), and (iii) a pre-trained language-conditioned pick-and-place primitive (similar to CLIPort [82] and Transporter Nets [81]). Object feedback informs the list of present objects and Success feedback informs the success/failure of the most recent action. However, consider the task of stacking multiple blocks, because the unfinished tower of blocks may be knocked over by the robot, it is also critical to reason about overall task progress. Therefore, task-progress scene description (Scene) describes the semantic sub-goals inferred by the LLM towards completing the high-level instruction that are achieved by the agent so far. ",
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+        "text": "We additionally compare to a multi-task CLIPort directly trained on long-horizon task instructions. Because CLIPort is a single-step policy and does not terminate spontaneously during policy rollout, we report CLIPort evaluations with oracle termination (i.e., repeat until oracle indicates task completion) and fixed-step termination (i.e., repeat for 15 steps). To simulate real-world disturbances and evaluate the system’s robustness to disturbances, we add Gaussian noise to multiple levels of the system at test time: $\\mathcal { N } ( 0 { , } 3 )$ for pixel observation, $\\mathcal { N } ( 0 , 2 . 5 )$ for policy primitive (i.e., pick-place pixel heatmaps), $\\mathcal { N } ( 0 , 0 . 0 2 m )$ for place locations. ",
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+    {
+        "type": "table",
+        "img_path": "images/23f63105548f5a25f6be3e43c2c0827df5c8ada1ffc79fc8b275dd8d4dc37c46.jpg",
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+        "table_footnote": [
+            "Table 1: Success rates averaged across 50 episodes in simulated pick-and-place. CLIPort $^ +$ oracle indicates that CLIPort was provided a “termination” oracle. LLM-informed feedback effectively enable retrying/replanning in the presence of test-time disturbances, while enjoying the generalization benefits of LLMs to unseen tasks. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"3\"></td><td></td><td>+LLM</td><td colspan=\"2\">+Inner Monologue</td></tr><tr><td>CLIPort</td><td>+oracle Object</td><td>Object + Success</td><td>Object + Scene</td></tr><tr><td rowspan=\"4\">Seen Tasks</td><td>“Pick and place”</td><td>24.0%</td><td>74.0%</td><td>80.0%</td><td>90.0%</td><td>94.0%</td></tr><tr><td>“Stack all the blocks”</td><td>2.0%</td><td>32.0%</td><td>4.0%</td><td>10.0%</td><td>26.0%</td></tr><tr><td>“Put all the blocks on the [x] corner/side&quot;</td><td>2.0%</td><td>32.0%</td><td>30.0%</td><td>28.0%</td><td>30.0%</td></tr><tr><td>“Put all the blocks in the [x] bowl&quot;</td><td>32.0%</td><td>94.0%</td><td>52.0%</td><td>46.0%</td><td>56.0%</td></tr><tr><td rowspan=\"4\">Unseen Tasks</td><td>“Put all the blocks in different corners&quot;</td><td>0.0%</td><td>0.0%</td><td>20.0%</td><td>20.0%</td><td>26.0%</td></tr><tr><td>“Put the blocks in their matching bowls”</td><td>0.0%</td><td>0.0%</td><td>56.0%</td><td>70.0%</td><td>82.0%</td></tr><tr><td>“Put the blocks on mismatched bowls&quot;</td><td>0.0%</td><td>0.0%</td><td>62.0%</td><td>76.0%</td><td>86.0%</td></tr><tr><td>“Stack all the blocks on the [x] corner/side”</td><td>0.0%</td><td>0.0%</td><td>0.0%</td><td>4.0%</td><td>6.0%</td></tr></table>",
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+        "type": "text",
+        "text": "Analysis. As shown in Table 1, Inner Monologue effectively enables retrying and replanning in the face of test-time disturbances, where Object $^ +$ Scene performs the best because of its ability to keep track of sub-goal conditions. Furthermore, this performance directly translates to unseen tasks by leveraging rich semantic knowledge of LLM. Finally, we observe that non-hierarchical and solitary systems such as CLIPort (i) struggle at generalizing to unseen long-horizon tasks under test-time disturbances, and (ii) on training tasks, an oracle is also often required to indicate task completion for good performance. ",
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+    {
+        "type": "text",
+        "text": "4.2 Real-World Tabletop Rearrangement ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate Inner Monologue on a real-world robot platform designed to resemble the simulation experiments. This instantiation uses (i) InstructGPT [9, 97] for planning, (ii) MDETR [98] for open-vocab object recognition (Object) (iii) heuristics on the object bounding box predictions from MDETR for Success Detection (Success), and (iv) a suction-based pick-and-place motion primitive that uses an LLM to parse target objects from a language command (e.g., given by the planner). ",
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+        "type": "text",
+        "text": "We investigate two tasks: (i) a 3-block stacking task where 2 blocks are already pre-stacked, and (ii) a long-horizon sorting task to place food in one plate and condiments in another (where categorizing food versus condiments is autonomously done by the LLM planner). In additional to additional challenges of real-world perception and clutter, we artificially inject Gaussian noise into the policy actions (i.e., add standard deviation $\\sigma { = } 4 \\mathrm { m m }$ clipped at $2 \\sigma$ ) to stress test recovery from failures via replanning with grounded closed-loop feedback. Results are presented in Table 2. ",
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+        "table_caption": [],
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+        "table_body": "<table><tr><td></td><td>LLM</td><td colspan=\"3\">+Inner Monologue</td></tr><tr><td>Task Family</td><td>Object</td><td>Object</td><td>Success</td><td>Object + Success</td></tr><tr><td>Finish 3-block stacking</td><td>20%</td><td>40%</td><td>40%</td><td>100%</td></tr><tr><td>Sort fruits from bottles</td><td>20%</td><td>50%</td><td>40%</td><td>80%</td></tr><tr><td>Total</td><td>20%</td><td>45%</td><td>40%</td><td>90%</td></tr></table>",
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+        "text": "Table 2: Success rates averaged across 10 runs in real-world pick-and-place. We observe significant improvement in Inner Monologue with Object and Success feedback, with the two feedback being complementary to each other. ",
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+        "text": "Analysis. We compare to variants with only Object or Success feedback, as well as an open-loop variant (“LLM Object”) that only runs object recognition once at the beginning of the task (similar to the system demonstrated in [19]). The partial 3-block stacking task highlights an immediate failure mode of the open-loop baseline, where the initial scene description struggles to capture a complete representation of the scene (due to clutter and occlusion) to provide as input to the multi-step planner. As a result, the system only executes one pick-and-place action – and cannot recover from mistakes. To address these shortcomings, Inner Monologue $( O b j e c t + S u c c e s s )$ ) leverages closed-loop scene description and success detection after each step, which allows it to successfully replan and recover from policy mistakes. ",
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+        "type": "text",
+        "text": "4.3 Real-World Mobile Manipulator in a Kitchen Setting ",
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+        "type": "text",
+        "text": "We implement Inner Monologue in a robotic system using the kitchen environment and task definitions described in SayCan [21]. The Everyday Robots robot, a mobile manipulator with RGB observations, is placed in an office kitchen to interact with common objects using concurrent [99] continuous closed-loop control. ",
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+        "type": "text",
+        "text": "The baseline, SayCan [21], is a method that plans and acts in diverse real world scenarios by combining an LLM with value functions of control policies. While SayCan creates plans that are grounded by the affordances of value functions, the LLM predictions in isolation are never given any closed-loop feedback. ",
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+        "type": "text",
+        "text": "We use an instantiation of Inner Monologue that uses (i) PALM [8] for planning, (ii) value functions from pre-trained control policies for affordance grounding [21], (iii) a learned visual classification model for Success feedback, (iv) human-provided Object feedback, and (v) pre-trained control policies for relevant skills in the scene. We also perform a case study where we allow the agent to ask questions and source Human feedback directly; results are shown in Fig 5a and the Appendix. ",
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+        "text": "We evaluate on 120 runs over three task families: (1) four manipulation tasks, (2) two dexterous manipulation tasks utilizing drawers, and (3) two long-horizon combined manipulation and navigation tasks. We consider both cases with and without manually-added adversarial disturbances during control policy executions that cause skill policy rollouts to fail. While these failures occur naturally even without perturbances, the adversarial disturbances creates a consistent comparison between methods that requires retrying or replanning to accomplish the original instruction. ",
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+        "type": "table",
+        "img_path": "images/7ed3f15bf2a139e412449021d4a4d3a56239547a742e782baae455d0285b4c5e.jpg",
+        "table_caption": [
+            "Table 3: Averaged success rate across 120 evaluations on several task families in our real-world mobile manipulation environment. We consider a standard setting and adversarial setting with external human disturbances. In all cases, LLM-informed embodied feedback is shown to be effective in improving robustness of the system, especially when low-level policies are prone to failures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task Family</td><td rowspan=\"2\">SayCan</td><td colspan=\"2\">+Inner Monologue</td></tr><tr><td>Success</td><td>Object + Success</td></tr><tr><td>No Disturbances</td><td></td><td></td><td></td></tr><tr><td>Manipulation</td><td>50.0%</td><td>62.5%</td><td>75.0%</td></tr><tr><td>Mobile Manipulation</td><td>50.0%</td><td>50.0%</td><td>75.0%</td></tr><tr><td>Drawers</td><td>83.3%</td><td>83.3%</td><td>100.0%</td></tr><tr><td>With Disturbances</td><td></td><td></td><td></td></tr><tr><td>Manipulation</td><td>12.5%</td><td>25.0%</td><td>33.3%</td></tr><tr><td>Mobile Manipulation</td><td>0.0%</td><td>25.0%</td><td>75.0%</td></tr><tr><td>Drawers</td><td>0.0%</td><td>44.4%</td><td>44.4%</td></tr><tr><td>Total</td><td>30.8%</td><td>48.7%</td><td>60.4%</td></tr></table>",
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+        "image_caption": [
+            "Figure 4: Failure causes on 120 evaluations. When disturbances are added (red), only the Inner Monologue variants consistently complete the instructions. "
+        ],
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+        "text": "Analysis. Without adversarial disturbances, the baseline SayCan performs reasonably on all tasks, yet incorporating LLM-informed feedback in Inner Monologue allows further improvement by effectively retrying or replanning under natural failures. The most notable difference is in the cases with adversarial disturbances. Without any LLM-informed feedback SayCan has success rate close to $0 \\%$ since LLM always assume successful execution of previous skills. Inner Monologue significantly outperforms SayCan because of its ability to invoke appropriate recovery modes depending on the environment feedback. Analysis on the failure causes indicates that Success and Object feedback can reduce LLM planning failures and thus overall failure rate, albeit at the cost of introducing new failure modes to the system. ",
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+        "type": "text",
+        "text": "4.4 Plan Generalization Capabilities ",
+        "text_level": 1,
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+        "text": "Although LLMs can generate fluent continuation from the prompted examples, we surprisingly find that, Inner Monologue demonstrates many impressive reasoning and replanning behaviors beyond the examples given in the prompt. Using a pre-trained LLM as the backbone, the method also inherits many of the appealing properties from its versatility and general-purpose language understanding. In this section, we demonstrate a few of these capabilities; additional capabilities are shown in Appendix (Fig ?? and Fig ??). ",
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+        "type": "text",
+        "text": "Continued Adaptation to New Instructions. Although not explicitly prompted, the LLM planner can react to human interaction that changes the high-level goal mid-task. Fig 5a demonstrates a challenging case, where Human feedback changes the goal during the plan execution, and then changes the goal yet again by saying “finish the previous task”. In another instance, despite not being explicitly prompted to terminate after a human says “please stop”, the LLM planner generalizes to this scenario and predicts a “done” action. ",
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+        "type": "text",
+        "text": "Self-Proposing Goals under Infeasibility. Instead of mindlessly following human-given instructions, Inner Monologue can also propose alternative goals to achieve when the previous goal becomes infeasible. In Fig 5b, to solve the task “put any two blocks inside the purple bowl”, while the first attempted block is intentionally made too heavy for the robot, Inner Monologue proposes to “find a lighter block” and successfully solves the task. ",
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+        "type": "text",
+        "text": "Multilingual Interaction. Pre-trained LLMs are known to be able to translate from one language to another, without any finetuning. We observe that such multilingual understanding also transfers to the embodied settings. Fig 5c shows a case when an instruction is in Chinese, the LLM planner can still correctly interpret it, re-narrate it as a concrete goal to execute in English, and accordingly replan its future actions. Occasionally, we find that this capability even extends to symbols and emojis. ",
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+        "type": "text",
+        "text": "Retrospective Scene Understanding. We also observe that Inner Monologue demonstrates retrospective scene understanding based on past actions and environment feedback, which requires temporal and embodied reasoning. In Fig 5d, after series of actions, we can turn to ask questions about the resulting scene, again a structure that has not appeared in the prompt. ",
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+        "img_path": "images/6e4d56d21ac1d79c836010878e48898181199d009ab6fd9d5a4387fba16cf9ab.jpg",
+        "image_caption": [
+            "Figure 5: Informing LLM with embodied feedback enables many generalization capabilities, all of which are achieved without similar prompted examples. For instance, Inner Monologue can continually adapt to new instructions given by humans, propose new goals to achieve when faced with infeasibility for the previous plan, interact with humans in different natural languages, and answer questions about the current scene given past actions and feedback. "
+        ],
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+        "type": "text",
+        "text": "Despite the appealing findings about these generalization capabilities, we observe that they are of varying levels of consistency when no similar examples have been provided in the prompt, likely limited by the current capabilities of the language models. However, we believe that further investigations into these behaviors and addressing their limitations would each lead to exciting future directions. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusions, Limitations & Future Works ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we investigated the role that environment feedback plays for LLMs reasoning in tasks involving embodied robotic planning and interaction. We presented a general formulation Inner Monologue that combines different sources of environment feedback with methods fusing LLM planning with robotic control policies and studied its instantiations in three distinct domains. We found that environment feedback significantly improves high-level instruction completion, especially in challenging scenarios with adversarial disturbances. Finally, we analyze generalization capabilities of Inner Monologue that highlight how closed-loop language feedback enables replanning even in complex unseen settings. ",
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+        "type": "text",
+        "text": "Limitations. In $\\mathrm { S e c } ~ 4 . 1$ and Sec 4.3, we assume access to oracle scene descriptors in the form of human observers or scripted systems to provide textual description back to the LLM planner. We study the viability of learned systems scene description and object recognition in Appendix Table ??. As for failure modes, Inner Monologue may fail due to several sources of errors: (1) success detections, (2) LLM planning errors, and (3) control errors. False negative predictions from the success detector lead to additional retry attempts, while false positive predictions add adversarial partial observability to the environment. In some instances, we found that the LLM planners ignored the environment feedback and still proposed policy skills involving objects not present in the scene. Additionally, the performance of low-level control policies limits not only overall high-level instruction completion performance, but also limits the scope of tasks that the LLM is able to plan actions for. ",
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+        "type": "text",
+        "text": "Future Works. Several fronts can be improved by future works. First, with advances in image/video captioning and visual-question answering, a fully automated system of Inner Monologue can be implemented without a human in the loop as an oracle. Second, improvements can be made on how to aggregate potentially inaccurate sources of information, such as using text to describe the uncertainty of the feedback modules, or including additional feedback modules for safety and ethics for the proposed plans. Finally, enabling low-level control policies to take as input the textual feedback by LLM also leads to exciting future directions. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+    },
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+        "type": "text",
+        "text": "The authors would like to thank Kanishka Rao and Vincent Vanhoucke for valuable feedback and discussions. In addition, the authors would like to acknowledge the large team who built [21], upon which we construct our Kitchen Mobile Manipulation experiments. ",
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+        "type": "text",
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In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 13041–13049, 2020.   \n[95] H. Cai, C. Gan, T. Wang, Z. Zhang, and S. Han. Once for all: Train one network and specialize it for efficient deployment. In International Conference on Learning Representations, 2020. URL https://arxiv.org/pdf/1908.09791.pdf.   \n[96] J.-B. Alayrac, J. Donahue, P. Luc, A. Miech, I. Barr, Y. Hasson, K. Lenc, A. Mensch, K. Millican, M. Reynolds, et al. Flamingo: a visual language model for few-shot learning. arXiv preprint arXiv:2204.14198, 2022.   \n[97] L. Ouyang, J. Wu, X. Jiang, D. Almeida, C. L. Wainwright, P. Mishkin, C. Zhang, S. Agarwal, K. Slama, A. Ray, et al. Training language models to follow instructions with human feedback. arXiv preprint arXiv:2203.02155, 2022.   \n[98] A. Kamath, M. Singh, Y. LeCun, G. Synnaeve, I. Misra, and N. Carion. Mdetr-modulated detection for end-to-end multi-modal understanding. 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parse/dev/6u6N8WWwYSM/6u6N8WWwYSM.md

@@ -0,0 +1,351 @@
+# BOOTSTRAPPING SEMANTIC SEGMENTATION WITH REGIONAL CONTRAST
+
+Shikun Liu1, Shuaifeng $\mathbf { Z } \mathbf { h } \mathbf { i } ^ { 1 }$ , Edward Johns2, and Andrew J. Davison1 1Dyson Robotics Lab, Imperial College London 2Robot Learning Lab, Imperial College London shikun.liu17@imperial.ac.uk
+
+# ABSTRACT
+
+We present ReCo, a contrastive learning framework designed at a regional level to assist learning in semantic segmentation. ReCo performs pixel-level contrastive learning on a sparse set of hard negative pixels, with minimal additional memory footprint. ReCo is easy to implement, being built on top of off-the-shelf segmentation networks, and consistently improves performance, achieving more accurate segmentation boundaries and faster convergence. The strongest effect is in semisupervised learning with very few labels. With ReCo, we achieve high quality semantic segmentation model, requiring only 5 examples of each semantic class.
+
+# 1 INTRODUCTION
+
+Semantic segmentation is an essential part of applications such as scene understanding and autonomous driving, whose goal is to assign a semantic label to each pixel in an image. Significant progress has been achieved by use of large datasets with high quality human annotations. However, labelling images with pixel-level accuracy is time consuming and expensive; for example, labelling a single image in CityScapes can take more than 90 minutes (Cordts et al., 2016). When deploying semantic segmentation models in practical applications where only limited labelled data are available, high quality ground-truth annotation is a significant bottleneck.
+
+To reduce the need for labelled data, there is a recent surge of interest in leveraging unlabelled data for semi-supervised learning. Previous methods include improving segmentation models via adversarial learning (Hung et al., 2019; Mittal et al., 2019) and self-training (Zou et al., 2019; 2018; Zhu et al., 2020). Others focus on designing advanced data augmentation strategies to generate pseudo image-annotation pairs from unlabelled images (Olsson et al., 2021; French et al., 2020).
+
+In both semi-supervised and supervised learning, a segmentation model often predicts smooth label maps, because neighbouring pixels are usually of the same class, and rarer high-frequency regions are typically only found in object boundaries. This learning bias produces blurry contours and regularly mis-labels rare objects. After carefully examining the label predictions, we further observe that wrongly labelled pixels are typically confused with very few other classes; e.g. a pixel labelled as rider has a much higher chance of being wrongly classified as person, compared to train or bus. By understanding this class structure, learning can be actively focused on the challenging pixels to improve overall segmentation quality.
+
+![](images/c392360a05db92b63807900471b937e8fe659774ca8eca14c432b20af0fd9b6c.jpg)  
+Figure 1: ReCo pushes representations within a class closer to the class mean representation, whilst simultaneously pushing these representations away from negative representations sampled in different classes. The sampling distribution from negative classes is adaptive to each query class.
+
+Here we propose ReCo, a contrastive learning framework designed at a regional level. Specifically, ReCo is a new loss function which helps semantic segmentation not only to learn from local context (neighbouring pixels), but also from global semantic class relationships across the entire dataset. ReCo performs contrastive learning on a pixel-level dense representation, as visualised in Fig. 1. For each semantic class in a mini-batch, ReCo samples a set of pixel-level representations (queries), and encourages them to be close to the class mean representation (positive keys), and simultaneously pushes them away from representations sampled from other classes (negative keys).
+
+For pixel-level contrastive learning with high-resolution images, it is impractical to sample all pixels. In ReCo, we actively sample a sparse set of queries and keys, consisting of less than $5 \%$ of all available pixels. We sample negative keys from a learned distribution based on the relative distance between the mean representation of each negative key and the query class. This distribution can be interpreted as a pairwise semantic class relationship, dynamically updated during training. We sample queries for those having a low prediction confidence. Active sampling helps ReCo to rapidly focus on the most confusing pixels for each semantic class, and requires minimal additional memory.
+
+ReCo enables a high-accuracy segmentation model to be trained with very few human annotations. We evaluate ReCo in a semi-supervised setting, with two different modes: i) Partial Dataset Full Labels — a sparse subset of training images, where each image has full ground-truth labels, and the remaining images are unlabelled; ii) Partial Labels Full Dataset — all images have some labels, but covering only a sparse subset of pixels within each image. In both settings, we show that ReCo can consistently improve performance across all methods and datasets.
+
+# 2 RELATED WORK
+
+Semantic Segmentation One recent direction is in designing more effective deep convolutional neural networks. Fully convolutional networks (FCNs) (Long et al., 2015) are the foundation of modern segmentation network design. They were later improved with dilated/atrous convolutions with larger receptive fields, capturing more long range information (Chen et al., 2017; 2018). Alternative approaches include encoder-decoder architectures (Ronneberger et al., 2015; Kirillov et al., 2019), sometimes using skip connections (Ronneberger et al., 2015) to refine filtered details.
+
+A parallel direction is to improve optimisation strategies, by designing loss functions that better respect class imbalance (Lin et al., 2017) or using rendering strategy to refine uncertain pixels from high-frequency regions improving the label quality (Kirillov et al., 2020). ReCo is built upon this line of research, to improve segmentation by providing additional supervision on hard pixels.
+
+Semi-supervised Classification and Segmentation The goal of semi-supervised learning is to improve model performance by taking advantage of a large amount of unlabelled data during training. Here consistency regularisation and entropy minimisation are two common strategies. The intuition is that the network’s output should be invariant to data perturbation and geometric transformation. Based on these strategies, many semi-supervised methods have been developed for image classification (Sohn et al., 2020; Tarvainen & Valpola, 2017; Berthelot et al., 2019; Kuo et al., 2020).
+
+However, for segmentation, generating effective pseudo-labels and well-designed data augmentation are non-trivial. Some solutions improved the quality of pseudo-labelling, using adversarial learning (Hung et al., 2019; Mittal et al., 2019) or enforcing consistency from different augmented images (French et al., 2020; Olsson et al., 2021). In this work, we show that we can improve the performance of current semi-supervised segmentation methods by jointly training with a suitable auxiliary task.
+
+Contrastive Learning Contrastive learning learns a similarity function to bring views of the same data closer in representation space, whilst pushing views of different data apart. Most recent contrastive frameworks learn similarity scores based on global representations of the views, parameterising data with a single vector (He et al., 2020; Chen et al., 2020; Khosla et al., 2020). Dense representations, on the other hand, rely on pixel-level representations and naturally provide additional supervision, capturing fine-grained pixel correspondence. Contrastive pre-training based on dense representations has recently been explored, and shows better performance in dense prediction tasks, such as object detection and keypoint detection (Wang et al., 2021b; O. Pinheiro et al., 2020).
+
+Contrastive Learning for Semantic Segmentation Contrastive learning has been recently studied to improve semantic segmentation, with a number of different design strategies. Zhang et al. (2021) and Zhao et al. (2021) both perform contrastive learning via pre-training, based on the generated auxiliary labels and ground-truth labels respectively, but at the cost of huge memory consumption. In contrast, ours performs contrastive learning whilst requiring much less memory, via active sampling. In concurrent work, (Wang et al., 2021a; Alonso et al., 2021) also perform contrastive learning with active sampling. However, whilst both these methods are applied to a stored feature bank, ours focuses on sampling features on-the-fly. Active sampling in Alonso et al. (2021) is further based on learnable, class-specific attention modules, whilst ours only samples features based on relation graphs and prediction confidence, without introducing any additional computation overhead, which results in a simpler and much more memory-efficient implementation.
+
+# 3 RECO – REGIONAL CONTRAST
+
+# 3.1 PIXEL-LEVEL CONTRASTIVE LEARNING
+
+Let $( X , Y )$ be a training dataset with training images $x \in X$ and their corresponding $C$ -class pixellevel segmentation labels $y \in Y$ , where $y$ can be either provided in the original dataset, or generated automatically as pseudo-labels. A segmentation network $f$ is then optimised to learn a mapping $f _ { \boldsymbol { \theta } } :$ $X \mapsto Y$ , parameterised by network parameters $\theta$ . This segmentation network $f$ can be decomposed into two parts: an encoder network: $\phi : X \mapsto Z$ , and a decoder classification head $\psi _ { c } : Z \mapsto Y$ . To perform pixel-level contrastive learning, we additionally attach a decoder representation head $\psi _ { r }$ on top of the encoder network $\phi$ , parallel to the classification head, mapping the encoded feature into a higher $m$ -dimensional dense representation with the same spatial resolution as the input image: $\psi _ { r } : Z \mapsto R , R \in \mathbb { R } ^ { m }$ . This representation head is only applied during training to guide the classifier using the ReCo loss as an auxiliary task, and is removed during inference.
+
+A pixel-level contrastive loss is a function which encourages queries $r _ { q }$ to be similar to the positive key $r _ { k } ^ { + }$ , and dissimilar to the negative keys $r _ { k } ^ { - }$ . All queries and keys are sampled from the decoder representation head: rq, r+,k $r _ { q } , r _ { k } ^ { + , - } \in R$ . In ReCo, we use a pixel-level contrastive loss across all available semantic classes in each mini-batch, with the distance between keys and queries measured by their normalised dot product. The general formation of the ReCo loss $L _ { \tt r e c o }$ is then defined as:
+
+$$
+L _ { \mathbf { r e c o } } = \sum _ { c \in \mathcal { C } } \sum _ { r _ { q } \sim \mathcal { R } _ { q } ^ { c } } - \log \frac { \exp ( r _ { q } \cdot r _ { k } ^ { c , + } / \tau ) } { \exp ( r _ { q } \cdot r _ { k } ^ { c , + } / \tau ) + \sum _ { r _ { k } ^ { - } \sim \mathcal { R } _ { k } ^ { c } } \exp ( r _ { q } \cdot r _ { k } ^ { - } / \tau ) } ,
+$$
+
+for which $\mathcal { C }$ is a set containing all available classes in the current mini-batch, $\tau$ is the temperature control of the softness of the distribution, $\mathcal { R } _ { q } ^ { c }$ represents a query set containing all representations whose labels belong to class $c$ , $\mathcal { R } _ { k } ^ { c }$ represents a negative key set containing all representations whose labels do not belong to class $c$ , and $r _ { k } ^ { c , + }$ represents the positive key which is the mean representation of class $c$ . Suppose is a set containing all pixel coordinates with the same resolution as , these queries and keys are then defined as:
+
+$$
+\mathcal { R } _ { q } ^ { c } = \bigcup _ { [ u , v ] \in \mathcal { P } } \mathbb { 1 } ( y _ { [ u , v ] } = c ) r _ { [ u , v ] } , \ \mathcal { R } _ { k } ^ { c } = \bigcup _ { [ u , v ] \in \mathcal { P } } \mathbb { 1 } ( y _ { [ u , v ] } \neq c ) r _ { [ u , v ] } , \ r _ { k } ^ { c , + } = \frac { 1 } { | \mathcal { R } _ { q } ^ { c } | } \sum _ { r _ { q } \in \mathcal { R } _ { q } ^ { c } } r _ { q } .
+$$
+
+# 3.2 ACTIVE HARD SAMPLING ON QUERIES AND KEYS
+
+Contrastive learning on all pixels in high-resolution images would be computationally expensive.   
+Here, we introduce active hard sampling strategies to optimise only a sparse set of queries and keys.
+
+Active Key Sampling When classifying a pixel, a semantic network might be uncertain only over a very small number of candidates, among all available classes. The uncertainty from these candidates typically comes from a close spatial (e.g. rider and bicycle) or semantic (e.g. horse and cow) relationship. To reduce this uncertainty, we propose to sample negative keys non-uniformly, based on the relative distance between each negative key class and the query class. This involves building a pair-wise class relationship graph $G$ , with $G \in \mathbb { R } ^ { | \mathcal { C } | \times | \mathcal { C } | }$ , computed and dynamically updated for each mini-batch. This pair-wise relationship is measured by the normalised dot product between the mean representation from a pair of two classes and is defined as:
+
+$$
+G [ p , q ] = \left( r _ { k } ^ { p , + } \cdot r _ { k } ^ { q , + } \right) , \quad \forall p , q \in \mathcal { C } , \ \mathrm { a n d } \ p \neq q .
+$$
+
+We further apply SoftMax to normalise these pair-wise relationships among all negative classes $j$ for each query class $c$ , which produces a probabilistic distribution:
+
+$\textstyle \exp ( G [ c , i ] ) / \sum _ { j \in { \mathcal { C } } , j \neq c } \exp ( G [ c , j ] )$ . We sample negative keys for each class $i$ based on this distribution, to learn the corresponding query class $c$ . This procedure allocates more samples to hard, confusing classes chosen specifically for each query class, helping the segmentation network to learn a more accurate decision boundary.
+
+Active Query Sampling Due to the natural class imbalance in semantic segmentation, it is easy to over-fit on common classes, such as the road and building classes in the CityScapes dataset, or the background class in the Pascal VOC dataset. These common classes contribute to the majority of pixel space in training images, and so randomly sampling queries will under-sample rare classes and provide minimal supervision to these classes.
+
+Therefore, we instead sample hard queries — for those whose corresponding pixel prediction confidence is below a defined threshold. Accordingly, ReCo’s loss would then guide the segmentation network by providing appropriate supervision on these less certain pixels. The easy and hard queries are defined as follows, and visualised in Fig. 2,
+
+$$
+\mathcal { R } _ { q } ^ { c , e a s y } = \bigcup _ { r _ { q } \in \mathcal { R } _ { q } ^ { c } } \mathbb { 1 } ( \hat { y } _ { q } > \delta _ { s } ) r _ { q } , \quad \mathcal { R } _ { q } ^ { c , h a r d } = \bigcup _ { r _ { q } \in \mathcal { R } _ { q } ^ { c } } \mathbb { 1 } ( \hat { y } _ { q } \leq \delta _ { s } ) r _ { q } ,
+$$
+
+where $\hat { y } _ { q }$ is the predicted confidence of label $c$ after the SoftMax operation corresponding to the same pixel location as $r _ { q }$ , and $\delta _ { s }$ is the user-defined confidence threshold.
+
+![](images/1175caa05b585655e133c94e3b0f6dc9ae581471e69922fae7fb62f0686910cb.jpg)  
+Figure 2: Easy and hard queries (shown in white) determined from the predicted confidence map in the Cityscapes dataset. Here we set the confidence threshold $\delta _ { s } = 0 . 9 7$ .
+
+# 3.3 SEMI-SUPERVISED SEMANTIC SEGMENTATION WITH RECO
+
+ReCo can easily be added to modern semi-supervised segmentation methods without changing the training pipeline, with no additional cost at inference time. To incorporate ReCo, we simply add an additional representation head $\psi _ { r }$ as described in Section 3.1, and apply the ReCo loss (in Eq. 1) to this representation using the sampling strategy introduced in Section 3.2.
+
+We apply the Mean Teacher framework (Tarvainen & Valpola, 2017) following prior state-of-the-art semi-supervised segmentation methods (Olsson et al., 2021; Mittal et al., 2019). Instead of using the original segmentation network $f _ { \theta }$ (which we call the student model), we instead use $f _ { \theta ^ { \prime } }$ (which we call the teacher model) to generate pseudo-labels from unlabelled images, where $\theta ^ { \prime }$ is a moving average of the previous state of $\theta$ during training optimisation: $\theta _ { t } ^ { \prime } = \lambda \theta _ { t - 1 } ^ { \prime } \bar { + } ( 1 - \lambda ) \theta _ { t }$ , with a decay parameter $\lambda = 0 . 9 9$ . This teacher model can be treated as a temporal ensemble of student models across training time $t$ , resulting in more stable predictions for unlabelled images. The student model $f _ { \theta }$ is then used to train on the augmented unlabelled images, with pseudo-labels as the ground-truths.
+
+For all pixels with defined ground-truth labels, we apply the ReCo loss on dense representations corresponding to all valid pixels. For all pixels without such labels, we only sample pixels whose predicted pseudo-label confidence is greater than a threshold $\delta _ { w }$ . This avoids sampling pixels which are likely to have incorrect pseudo-labels.
+
+We apply the ReCo loss to a combined set of labelled and unlabelled pixels. The overall training loss for semi-supervised segmentation is then the linear combination of supervised cross-entropy loss (on ground-truth labels), unsupervised cross-entropy loss (on pseudo-labels) and ReCo loss:
+
+$$
+L _ { t o t a l } = L _ { s u p e r v i s e d } + \eta \cdot L _ { u n s u p e r v i s e d } + L _ { r e c o }
+$$
+
+where $\eta$ is defined as the percentage of pixels whose predicted confidence are greater than $\delta _ { s }$ , a scalar re-weighting the contribution for unsupervised loss, following prior methods (Olsson et al., 2021;
+
+![](images/809972c8991576f173e37564b9603381a78cd20d01cc70b40de50090bdd9171b.jpg)  
+Figure 3: Visualisation of the ReCo framework applied to semi-supervised segmentation and trained with three losses. A supervised loss is computed based on labelled data with ground-truth annotations. An unsupervised loss is computed for unlabelled data with generated pseudo-labels. And finally a ReCo loss is computed based on pixel-level dense representation predicted from both labelled and unlabelled images.
+
+Mittal et al., 2019). This makes sure the segmentation network would not be dominated by gradients produced by uncertain pseudo-labels, which typically occur during the early stage of training. Fig. 3 shows a visualisation of the ReCo framework for semi-supervised segmentation.
+
+# 4 EXPERIMENTS
+
+Semi-Supervised Segmentation Benchmark Redesign We propose two modes of semisupervised segmentation tasks, aiming at two different applications.
+
+i) Partial Dataset Full Labels: A small subset of the images is trained with complete ground-truth labels for each image, whilst the remaining training images are unlabelled. This is the de-facto standard of evaluating semi-supervised segmentation in prior works.   
+ii) Partial Labels Full Dataset: All images are trained with partial labels, but only a small percentage of labels are provided for each class in each training image. We create the dataset by first randomly sampling a pixel for each class, and then continuously apply a $[ 5 \times 5 ]$ square kernel for dilation until we meet the percentage criteria.
+
+The Partial Dataset Full Label setting evaluates the ability to generalise semantic classes given a few examples with perfect boundary information. The Partial Label Full Dataset evaluates learning semantic class completion given many examples with no or minimal boundary information.
+
+![](images/01730836a383cf595ab1cb93774d8e67234f48693672101e2fdcb2c8cc7761b2.jpg)  
+Figure 4: Example of training labels for Pascal VOC dataset in Partial Labels Full Dataset setting. (1 Pixel is zoomed 5 times for better visualisation.)
+
+Datasets We experiment on segmentation datasets: Cityscapes (Cordts et al., 2016) and Pascal VOC 2012 (Everingham et al., 2015) in both partial and full label setting. We also
+
+evaluate on a more difficult indoor scene dataset SUN RGB-D (Song et al., 2015) in the full label setting only, mainly due to the low quality annotations making it difficult for fair evaluation in the partial label setting. An example of the partially labelled Pascal VOC is shown in Fig. 4.
+
+Strong Baselines Prior semi-supervised segmentation methods are typically designed with different backbone architectures, and trained with different strategies, which makes it difficult to compare them fairly. In this work, we standardise the baselines and implement four strong semi-supervised segmentation methods ourselves: S4GAN (Mittal et al., 2019): an adversarial learning based semisupervised method; CutOut (French et al., 2020), CutMix (French et al., 2020), ClassMix (Olsson et al., 2021): three image augmentation strategies designed specifically for semi-supervised segmentation. Our implementations for all baselines obtain performance on par with, and most of the time surpassing, the performance reported in each original publication, giving us a set of strong baselines. All baselines and our method were implemented on DeepLab ${ \mathrm { V } } 3 +$ (Chen et al., 2018) with ResNet-101 backbone (He et al., 2016), and all with the same optimisation strategies. Detailed hyper-parameters used for each dataset are provided in the Appendix A.
+
+4.1 RESULTS ON PASCAL VOC, CITYSCAPES, SUN RGB-D (FULL LABELS)
+
+First, we compared our results to semi-supervised baselines in a full label setting. We applied ReCo on top of ClassMix, which consistently outperformed other semi-supervised baselines.
+
+Table 1 shows the mean IoU validation performance on three datasets over three individual runs (different labelled and unlabelled data splits). The number of labelled images shown in the three columns for each dataset, are chosen such that the least-appeared classes have appeared in 5, 15 and 50 images respectively. In the fewest-label setting for each dataset, applying ReCo with ClassMix can improve results by a significant margin, with up to $1 . 5 - 4 . 5 \%$ absolute improvement.
+
+<table><tr><td></td><td colspan="3">Pascal VOC</td><td colspan="3">CityScapes</td><td colspan="3">SUN RGB-D</td></tr><tr><td>Method</td><td></td><td>60 labels 200 labels 600 labels</td><td></td><td>20 labels</td><td>50labels</td><td>150 labels</td><td>50labels</td><td>：150 labels 500 labels</td><td></td></tr><tr><td>Supervised</td><td>37.79</td><td>53.87</td><td>64.04</td><td>38.12</td><td>45.42</td><td>54.93</td><td>19.79</td><td>28.78</td><td>37.73</td></tr><tr><td>S4GAN (Mittal et al.,2019)</td><td>47.95</td><td>61.25</td><td>66.21</td><td>37.65</td><td>47.08</td><td>56.46</td><td>20.53</td><td>29.79</td><td>38.08</td></tr><tr><td>CutOut (French et al., 2020)</td><td>52.96</td><td>63.57</td><td>69.85</td><td>42.52</td><td>50.15</td><td>59.42</td><td>25.94</td><td>34.45</td><td>41.25</td></tr><tr><td>CutMix (French et al.,2020)</td><td>53.71</td><td>66.95</td><td>72.42</td><td>44.02</td><td>54.72</td><td>62.24</td><td>27.60</td><td>37.55</td><td>42.69</td></tr><tr><td>ClassMix (Olsson et al.,2021)</td><td>49.06</td><td>67.95</td><td>72.50</td><td>45.61</td><td>55.56</td><td>63.94</td><td>28.42</td><td>37.55</td><td>42.46</td></tr><tr><td>ReCo +ClassMix</td><td>53.31</td><td>69.81</td><td>72.75</td><td>49.86</td><td>57.69</td><td>65.04</td><td>29.65</td><td>39.14</td><td>44.55</td></tr></table>
+
+Table 1: mean IoU validation performance for Pascal VOC, CityScapes and SUN RGB-D datasets.   
+We report the mean over three independent runs for all methods.
+
+<table><tr><td>Pascal VOC</td><td>1/16 [92] 1/8 [183]1/4 [366] 1/2 [732]</td><td></td><td></td><td></td></tr><tr><td>AdvSemSeg (Hung et al.,2019)</td><td>39.69</td><td>47.58</td><td>59.97</td><td>65.27</td></tr><tr><td>Mean Teacher (Tarvainen &amp; Valpola,2017)</td><td>48.70</td><td>55.81</td><td>63.01</td><td>69.16</td></tr><tr><td>CCT(Ouali et al.,2020)</td><td>33.10</td><td>47.60</td><td>58.80</td><td>62.10</td></tr><tr><td>GCT (Ke et al.,2020)</td><td>46.04</td><td>54.98</td><td>64.71</td><td>70.67</td></tr><tr><td>VAT (Miyato et al.,2018)</td><td>36.92</td><td>49.35</td><td>56.88</td><td>63.34</td></tr><tr><td>CutMix (French et al.,2020)</td><td>55.58</td><td>63.20</td><td>68.36</td><td>69.87</td></tr><tr><td>PseudoSeg (Zou et al.,2021)</td><td>57.60</td><td>65.50</td><td>69.14</td><td>72.41</td></tr><tr><td>ReCo + ClassMix</td><td>64.78</td><td>72.02</td><td>73.14</td><td>74.69</td></tr></table>
+
+To further justify the effectiveness of ReCo, we also include results on existing benchmarks, to compare with other semi-supervised methods in Table 2. Here, all baselines were re-implemented and reported in the PseudoSeg setting (Zou et al., 2021), where the labelled images are sampled from the original PASCAL dataset, with a total of 1.4k images. In both benchmarks, ReCo shows state-of-the-art performance, and specifically is able to reach PseudoSeg’s performance, whilst requir
+
+Table 2: mean IoU validation performance for Pascal VOC with data partition and training strategy proposed in PseudoSeg (Zou et al., 2021). The percentage and the number of labelled data used are listed in the first row.
+
+ing only half the labelled data. Additional results are further shown in Appendix C.
+
+In Fig. 5, we present qualitative results from the semi-supervised setup with the fewest labels: 20 labels for CityScapes and 50 labels SUN RGB-D datasets. The 60 labelled Pascal VOC is further shown in Appendix D. In Fig. 5, we can see the advantage of ReCo, where the edges and boundaries of small objects are clearly more pronounced such as in the person and bicycle classes in CityScapes, and the lamp and pillow classes in SUN RGB-D. Interestingly, we found that in SUN RGB-D, though all methods may confuse ambiguous class pairs such as table and desk or window and curtain, ReCo still produces consistently sharp and accurate object boundaries compared to the Supervised and ClassMix baselines where labels are noisy near object boundaries.
+
+# 4.2 RESULTS ON PASCAL VOC AND CITYSCAPES (PARTIAL LABELS)
+
+In the partial label setting, we evaluated on the CityScapes and Pascal VOC datasets. Table 3 compared ReCo to the two best semi-supervised baselines and a supervised baseline. Again, we see ReCo can improve performance in all cases when applied on top of ClassMix, with around $1 - 3 \%$
+
+![](images/b5fe9e69bd192e091d00e294308b8c1874870f5345a5f8c9859125ce40a6ca8c.jpg)
+
+Figure 5: Visualisation of CityScapes and SUN RGB-D validation set trained on 20 and 50 labelled images respectively. Interesting regions are shown in white arrows.   
+
+<table><tr><td></td><td colspan="4">CityScapes (Partial)</td></tr><tr><td>Method</td><td>1 pixel</td><td>1% labels</td><td>5% labels</td><td>25% labels</td></tr><tr><td>Supervised</td><td>44.08</td><td>52.89</td><td>56.65</td><td>63.43</td></tr><tr><td>CutMix</td><td>46.91</td><td>54.90</td><td>59.69</td><td>65.61</td></tr><tr><td>ClassMix</td><td>47.42</td><td>56.68</td><td>60.96</td><td>66.46</td></tr><tr><td>ReCo + ClassMix</td><td>49.66</td><td>58.97</td><td>62.32</td><td>66.92</td></tr></table>
+
+Table 3: mean IoU validation performance for Pascal VOC and Cityscapes datasets trained on $1 , 1 \%$ , $5 \%$ and $2 5 \%$ labelled pixels per class per image. We report the mean over three independent runs for all methods.   
+
+<table><tr><td></td><td colspan="4">Pascal VOC (Partial)</td></tr><tr><td>Method</td><td>1 pixel</td><td>1% labels</td><td>5% labels</td><td>25% labels</td></tr><tr><td>Supervised</td><td>60.33</td><td>66.17</td><td>69.16</td><td>73.75</td></tr><tr><td>CutMix</td><td>63.50</td><td>70.83</td><td>73.04</td><td>75.64</td></tr><tr><td>ClassMix</td><td>63.69</td><td>71.04</td><td>72.90</td><td>75.79</td></tr><tr><td>ReCo + ClassMix</td><td>66.11</td><td>72.67</td><td>74.09</td><td>75.96</td></tr></table>
+
+absolute improvement. We observe less relative performance improvement than in the full label setting; very sparse ground-truth annotations could confuse ReCo, resulting in inaccurate supervision.
+
+We show qualitative results on Pascal VOC dataset trained on 1 labelled pixel per class per image in Fig. 6. As in the full label setting, we see smoother and more accurate boundary predictions from ReCo. More visualisations from CityScapes are shown in Appendix E.
+
+# 4.3 ABLATIVE ANALYSIS
+
+Next we present an ablative analysis on 20 labelled CityScapes images to understand the behaviour of ReCo with respect to hyper-parameters. We use our default experimental setting from Section 4.1, using ReCo with ClassMix. Additional ablations are further shown in Appendix B.
+
+Number of Queries and Keys We first evaluate the performance by varying the number of queries and keys used in ReCo framework, whilst fixing all other hyper-parameters. In Fig. 7a and 7b, we can observe that performance is better when sampling more queries and keys, but after a certain point, the improvements become marginal. Notably, even in our smallest option of having 32 queries per class in a mini-batch — consisting of less than $0 . 5 \%$ among all available pixel space — this can still improve performance by a non-trivial margin. Compared to a concurrent work (Zhang et al., 2021) which requires $1 0 \mathrm { k }$ queries and $4 0 \mathrm { k }$ keys in each training iteration, ReCo can be optimised with $\times 5 0$ more efficiency in terms of memory footprint.
+
+![](images/0739cc72fbaee2cab769c477c1d34bcdd520f1abbf4dca31741349e102bb6193.jpg)  
+Figure 6: Visualisation of Pascal VOC validation set with ClassMix (left) vs. with ReCo (right) trained on 1 labelled pixel per class per image. Interesting regions are shown in white arrows.
+
+Ratio of Unlabelled Data We evaluate how ReCo can generalise across different levels of unlabelled data. In Fig. 7c, we show that by only training on $10 \%$ unlabelled data in the original setting, we can already surpass the ClassMix baseline. This shows that ReCo can achieve strong generalisation not only in label efficiency but also in data efficiency.
+
+Choice of Semi-Supervised Method Finally, we show that ReCo is robust to the choice of different semi-supervised methods. In Fig. 7d, we can see that ReCo obtains a better performance from a variety choice of semi-supervised baselines.
+
+Effect of Active Sampling In Fig. 7e, we see that randomly sampling queries and keys gives much less improvement compared to active sampling in our default setting. Particularly, hard query sampling has a dominant effect on generalisation: if we instead only sample from easy queries, ReCo only marginally improves on the baseline. This further verifies that most queries are redundant.
+
+![](images/d1528c733884e8502f3353f61a0c9b71cc1164ca318641422551372d3bc937f2.jpg)  
+Figure 7: mean IoU validation performance on 20 labelled CityScapes dataset based on different choices of hyper-parameters. Grey: ClassMix (if not labelled otherwise) in our default setting. Light Blue: ReCo $^ +$ ClassMix (if not labelled otherwise) in a different hyper-parameter setting. Dark Blue: ReCo $^ +$ ClassMix in our default setting.
+
+# 5 VISUALISATIONS AND INTERPRETABILITY OF CLASS RELATIONSHIPS
+
+In this section, we visualise the pair-wise semantic class relation graph defined in Eq. 3, additionally supported by a semantic class dendrogram using the off-the-shelf hierarchical clustering algorithm in SciPy (Virtanen et al., 2020) for better visualisation. The features for each semantic class used in both visualisations are averaged across all available pixel embeddings in each class from the validation set. In all visualisations, we compared features learned with ReCo built on top of supervised learning compared to a standard supervised learning method trained on all data, representing the semantic class relationships of the full dataset.
+
+Using the same definitions in Section 3.1, we first choose such pixel embedding to be the embedding $Z$ predicted from the encoder network $\phi$ in both supervised learning and with ReCo. We also show the visualisation for embedding $R$ which is the actual representation we used for ReCo loss and active sampling.
+
+![](images/6a0eedd3a7abbcf3b91e4d92c859b2e3c046c295a8645eb107a586a2f42f5e73.jpg)  
+Figure 8: Visualisation of semantic class relation graph (top) and its corresponding semantic class dendrogram (bottom) on CityScapes dataset. Brighter colour represents closer (more confused) relationship. Best viewed in zoom.
+
+In Fig. 8, we present the semantic class relationship and dendrogram for the CityScapes dataset by embedding $R$ and $Z$ with and without ReCo. We can clearly see that ReCo helps disentangle features compared to supervised learning where many pairs of semantic classes are similar. In addition, we find that the dendrogram generated by ReCo based on embedding $Z$ is more structured, showing a clear and interpretable semantic tree by grouping semantically similar classes together: for example, all large transportation classes car, truck, bus and train are under the same parent branch. In addition, we find that nearly all classes based on embedding $R$ are perfectly disentangled, except for bus and train, suggesting the CityScapes dataset might not have sufficient bus and train examples to learn a distinctive representation for these two classes.
+
+The pair-wise relation graph helps us to understand the distribution of semantic classes in each dataset, and clarifies the pattern of incorrect predictions from the trained semantic network. We additionally provide a dendrogram based on embedding $R$ for the SUN RGBD dataset, clearly showing ambiguous class pairs, such as night stand and dresser; table and desk; floor
+
+![](images/87a4765d9dd11e7bc7baebbec9ccec6dba349253e537720cdd9a860d8bfadc80.jpg)  
+Figure 9: Visualisation of semantic class dendrogram based on embedding $R$ on SUN RGB-D dataset using ${ \mathrm { R e C o } } + { \mathrm { S } }$ upervised method. Best viewed in zoom.
+
+and floormat, consistent with our results shown in Fig. 5. Complete visualisations of these semantic class relationships are shown in Appendix F.
+
+# 6 CONCLUSION
+
+In this work, we have presented ReCo, a new pixel-level contrastive framework with active sampling, designed specifically for semantic segmentation. ReCo can improve performance in semantic segmentation methods with minimal additional memory footprint. In particular, ReCo has shown its strongest effect in semi-supervised learning with very few labels, where we improved on the stateof-the-art by a large margin. In further work, we aim to design effective contrastive frameworks for video representation learning.
+
+# REPRODUCIBILITY
+
+All of the information for reproducibility is shown in Appendix A. Code is available at https: //github.com/lorenmt/reco.
+
+# ACKNOWLEDGEMENT
+
+This work has been supported by Dyson Technology Ltd. We thank Zhe Lin for the initial discussion and Zhengyang Feng for his help on the evaluation metric design.
+
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+
+# APPENDIX
+
+# A IMPLEMENTATION DETAILS
+
+We trained all methods with SGD optimiser with learning rate $2 . 5 \times 1 0 ^ { - 3 }$ , momentum 0.9, and weight decay $5 \cdot 1 0 ^ { - 4 } $ . We adopted the polynomial annealing policy to schedule the learning rate, which is multiplied by $\begin{array} { r } { ( 1 - \frac { i \bar { t } e r } { t o t a l . i t e r } ) ^ { p o \bar { w } e r } } \end{array}$ with $p o w e r = 0 . 9$ , and trained for $4 0 \mathrm { k }$ iterations for all datasets. Code is attached in the supplementary material.
+
+For CityScapes, we first downsampled all images in the dataset to half resolution $[ 5 1 2 \times 1 0 2 4 ]$ prior to use. We extracted $[ 5 1 2 \times 5 1 2 ]$ random crops and used a batch size of 2 during training.
+
+For Pascal VOC, we extracted $[ 3 2 1 \times 3 2 1 ]$ random crops, applied a random scale between [0.5, 1.5], and used a batch size of 10 during training.
+
+For SUN RGB-D, we first rescaled all images to $[ 3 8 4 \times 5 1 2 ]$ resolution, extracted $[ 3 2 1 \times 3 2 1 ]$ random crops, applied a random scale between [0.5, 1.5], and used a batch size of 5. We additionally re-organised the original training and validation split in SUN RGB-D dataset from 5285 and 5050 to 9860 and 475 samples respectively, to increase the amount of training data which we think is more appropriate for semi-supervised task.
+
+All datasets were additionally augmented with Gaussian blur, colour jittering, and random horizontal flip. The pre-processing for CityScapes and Pascal VOC are consistent with the prior work (Olsson et al., 2021). In Table 2, we extracted $[ 5 1 3 \times 5 1 3 ]$ random crops and applied a random scale between [0.5, 2.0], following PseudoSeg’s training setup (Zou et al., 2021).
+
+In our ReCo framework, we sampled 256 query samples and 512 key samples and used temperature $\tau = 0 . 5$ for each mini-batch, which we found to work well in all datasets. The dimensionality for pixel-level representation was set to $m = 2 5 6$ . The confidence thresholds were set to $\delta _ { w } = 0 . 7$ and $\delta _ { s } = 0 . 9 7$ .
+
+# B ADDITIONAL ABLATIVE STUDIES
+
+ReCo Only Results Table 4 shows ReCo designed with and without data augmentation, trained on 20 and 50 labelled CityScapes dataset. We observe that using pure semisupervised learning with additional unlabelled data will lead to worse performance compared to supervised learning without such labelled data. This shows data augmentation strategies designed for semi-supervised segmentation are the key component to make best use of the unlabelled data. Although the vanilla ReCo still performs better compared to standard semi-supervised learning, the active sam
+
+<table><tr><td>CityScapes</td><td>20 Labels 50 Labels</td></tr><tr><td>Supervised</td><td>38.10 47.10</td></tr><tr><td>Semi-Supervised</td><td>28.59 43.74</td></tr><tr><td>Semi-Supervised +ReCo</td><td>29.16 46.96</td></tr><tr><td>ClassMix</td><td>45.61 55.56</td></tr><tr><td>ClassMix+Reco</td><td>49.86 57.69</td></tr></table>
+
+Table 4: mean IoU validation performance for 20 and 50 labelled CityScapes data for supervised method (top) and semi-supervised methods (bottom).
+
+pling of ReCo based on incorrect pseudo-labels leads to marginal improvement compared to a pure data augmentation method like ClassMix. Therefore, ReCo performs better as an auxiliary framework combined with a strong semi-supervised method.
+
+Compared to Feature Bank Methods We have experimented with ReCo with a stored feature bank framework similar to the design in the concurrent works (Alonso et al., 2021; Wang et al., 2021a). We found that just by replacing our batch-wise sampling method with a feature bank sampling method will achieve a similar performance $4 9 . 3 4 ~ \mathrm { m I o U } )$ ) compared to our original design $( 4 9 . 8 6 ~ \mathrm { m I o U } )$ on 20 labelled CityScapes, but with a slower training speed. This verifies our assumption that batch-wise sampling is an accurate approximation of class distribution over the entire dataset.
+
+# C RESULTS ON SEMI-SUPERVISED SEGMENTATION BENCHMARKS
+
+Here, we present quantitative results for other semi-supervised semantic segmentation benchmarks in CityScapes and Pascal VOC datasets. Note that, this benchmark is much less challenging compared to our proposed benchmark in Section 4.1, evaluated with significantly less number of labelled images. Since some methods applied with different backbones and training strategies, we compared each result with respect to its performance gap compared to its corresponding fully supervised result, as shown in brackets, to ensure fairness following Feng et al. (2020).
+
+In Table 5, we show results for ReCo applied on top of ClassMix, and trained with both DeepLabv2 (Chen et al., 2017) and DeepLabv $^ { 3 + }$ (Chen et al., 2018). We can observe that ReCo achieved the best performances in most cases in both datasets, showing its robustness to different backbone architectures and number of labelled training images.
+
+<table><tr><td>Pascal VOC</td><td>Backbone</td><td>1/106 [100]</td><td>1/50 [212]</td><td>1/20 [529]</td><td>1/8 [1323]</td><td>Full [10582]</td></tr><tr><td>AdvSemSeg (Hung et al.,2019)</td><td>DeepLabv2</td><td></td><td>57.20(17.70)</td><td>64.70(10.20)</td><td>69.50(5.40)</td><td>74.90</td></tr><tr><td>S4GAN (Mittal et al.,2019)</td><td>DeepLabv2</td><td></td><td>63.30(12.30)</td><td>67.20(8.40)</td><td>71.40(4.20)</td><td>75.60</td></tr><tr><td>CutMix (French et al.,2020)</td><td>DeepLabv2</td><td>53.79(18.71)</td><td>64.81(7.73)</td><td>66.48(6.06)</td><td>67.60(4.94)</td><td>72.54</td></tr><tr><td>ClassMix (Olsson et al.,2021)</td><td>DeepLabv2</td><td>54.18(19.95)</td><td>66.15(7.98)</td><td>67.77(6.36)</td><td>71.00(3.13)</td><td>74.13</td></tr><tr><td>CCT (Ouali et al.,2020)</td><td>PSPNet</td><td></td><td></td><td>=</td><td>70.45(4.80)</td><td>75.25</td></tr><tr><td>CAC (Lai et al.,2021)</td><td>PSPNet</td><td></td><td></td><td></td><td>72.50(3.90)</td><td>76.40</td></tr><tr><td>GCT (Ke et al.,2020)</td><td>DeepLabv2</td><td></td><td></td><td></td><td>70.57(3.49)</td><td>74.06</td></tr><tr><td>DMT (Feng et al.,2020)</td><td>DeepLabv2</td><td>63.04(11.71)</td><td>67.15(7.60)</td><td>69.92(4.83)</td><td>72.70(2.05)</td><td>74.75</td></tr><tr><td>ReCo + ClassMix</td><td>DeepLabv2</td><td>63.16(11.20)</td><td>66.41(7.95)</td><td>68.85(5.51)</td><td>71.00(3.36)</td><td>74.36</td></tr><tr><td>ReCo+ClassMix</td><td>DeepLabv3+</td><td>63.60(14.15)</td><td>72.14(5.61)</td><td>73.66(4.09)</td><td>74.62( (3.13)</td><td>77.75</td></tr><tr><td>CityScapes</td><td>Backbone</td><td>1/30 [100]</td><td>1/8 [372]</td><td>1/4 [744]</td><td>1/2 [1488]</td><td>Full [2975]</td></tr><tr><td>AdvSemSeg (Hung et al.,2019)</td><td>DeepLabv2</td><td></td><td>58.80(7.60)</td><td>62.30(4.10)</td><td>65.70(0.70)</td><td>66.40</td></tr><tr><td>S4GAN (Mittal et al., 2019)</td><td>DeepLabv2</td><td></td><td>59.30(6.50)</td><td>61.90(3.90)</td><td></td><td>65.80</td></tr><tr><td>CutMix (French et al.,2020)</td><td>DeepLabv2</td><td>51.20(16.33)</td><td>60.34(7.19)</td><td>63.87(3.66)</td><td></td><td>67.53</td></tr><tr><td>ClassMix (Olsson et al.,2021)</td><td>DeepLabv2</td><td>54.07(12.12)</td><td>61.35(4.84)</td><td>63.63(2.56)</td><td>66.29(-0.10)</td><td>66.19</td></tr><tr><td>DMT (Feng et al.,2020)</td><td>DeepLabv2</td><td>54.81(13.36)</td><td>63.03(5.13)</td><td>=</td><td></td><td>68.16</td></tr><tr><td>ECS*(Mendel et al.,2020)</td><td>DeepLabv3+</td><td></td><td>67.38(7.38)</td><td>70.70(4.06)</td><td>72.89(1.87)</td><td>74.76</td></tr><tr><td>ReCo+ClassMix</td><td>DeepLabv2</td><td>56.53(12.07)</td><td>64.94(3.66)</td><td>67.53(1.07)</td><td>68.69(-0.09)</td><td>68.60</td></tr><tr><td>ReCo +ClassMix</td><td>DeepLabv3+</td><td>60.28(10.20)</td><td>66.44(4.04)</td><td>68.50(1.98)</td><td>70.63(-0.15)</td><td>70.48</td></tr></table>
+
+Table 5: mean IoU validation performance in semi-supervsed Pascal VOC and CityScapes datasets. We list the percentage along with the number of labelled images at the top row. The first and second best performances in each data partition setting are coloured in red and orange respectively. $^ *$ trained images in doubled resolution. All results were taken from the corresponding publications.
+
+# D VISUALISATION ON PASCAL VOC (TRAINED WITH 60 LABELLED IMAGES)
+
+In the full label setting, the baselines Supervised and ClassMix are very prone to completely misclassifying rare objects such as boat, bottle and table, while our method can predict these rare classes accurately.
+
+![](images/080f1e165fde866182ca85eb2ad2769d22d31503a8937a7b391377094cf6925e.jpg)
+
+<table><tr><td rowspan=1 colspan=1>Background Aeroplane  Bicycle</td><td rowspan=1 colspan=1>Bird</td><td rowspan=1 colspan=1>Boat</td><td rowspan=1 colspan=1>Bottle</td><td rowspan=1 colspan=1>Bus</td><td rowspan=1 colspan=1>Car</td><td rowspan=1 colspan=1>Cat</td><td rowspan=1 colspan=1>Chair</td><td rowspan=1 colspan=1>Cow</td></tr><tr><td rowspan=1 colspan=2>Table      Dog     Horse   Motorbike</td><td rowspan=1 colspan=1>Motorbike</td><td rowspan=1 colspan=1>Person</td><td rowspan=1 colspan=1>Plant</td><td rowspan=1 colspan=1>Sheep</td><td rowspan=1 colspan=1>Sofa</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Monitor</td></tr></table>
+
+# E VISUALISATION ON CITYSCAPES (TRAINED WITH $1 \%$ LABELLED PIXEL)
+
+In the partial label setting, the performance improvements are less pronounced compared to the full label setting in CityScapes dataset. The improvements typically come from the more accurate predictions in small object boundaries such as in traffic light and traffic sign. Learning semantics with partial labels with minimal boundary information remains an open research question and still has huge scope for improvements.
+
+![](images/b8a41e6882e35596dfde824f3e6ffcf816d7c94c2b55f9e11bfc357a683da7f7.jpg)
+
+<table><tr><td>Road</td><td>Sidewalk</td><td>Building</td><td>Wall</td><td></td><td>Pole</td><td>TrafficLight</td><td>Traffic Sign</td><td>Vegetation</td><td></td></tr><tr><td>Sky</td><td>Person</td><td>Rider</td><td>Car</td><td>ruck</td><td>Bus</td><td>Train</td><td>Motorcycle</td><td>Bicycle</td><td>Invalid</td></tr></table>
+
+# F VISUALISATION ON SEMANTIC CLASS RELATIONSHIP FROM PASCAL VOC (TOP) AND SUN RGB-D (BOTTOM)
+
+We show features learned by ReCo are more disentangled compared to the Supervised baseline in all datasets, which helps the segmentation model to learn a better decision boundary. Brighter colour represents closer (more confused) relationship. Best viewed in zoom.
+
+![](images/d7a0d4015766f4796dde34cff6b6e3cc473bbe0377ad9e6741668a650b58f0e5.jpg)  
+(a) Embed. Z (Supervised) (b) Embed. Z (ReCo $^ +$ Supervised) (c) Embed. R (ReCo $^ +$ Supervised)

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+    {
+        "type": "text",
+        "text": "BOOTSTRAPPING SEMANTIC SEGMENTATION WITH REGIONAL CONTRAST ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Shikun Liu1, Shuaifeng $\\mathbf { Z } \\mathbf { h } \\mathbf { i } ^ { 1 }$ , Edward Johns2, and Andrew J. Davison1 1Dyson Robotics Lab, Imperial College London 2Robot Learning Lab, Imperial College London shikun.liu17@imperial.ac.uk ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            279
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present ReCo, a contrastive learning framework designed at a regional level to assist learning in semantic segmentation. ReCo performs pixel-level contrastive learning on a sparse set of hard negative pixels, with minimal additional memory footprint. ReCo is easy to implement, being built on top of off-the-shelf segmentation networks, and consistently improves performance, achieving more accurate segmentation boundaries and faster convergence. The strongest effect is in semisupervised learning with very few labels. With ReCo, we achieve high quality semantic segmentation model, requiring only 5 examples of each semantic class. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            426,
+            336,
+            443
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Semantic segmentation is an essential part of applications such as scene understanding and autonomous driving, whose goal is to assign a semantic label to each pixel in an image. Significant progress has been achieved by use of large datasets with high quality human annotations. However, labelling images with pixel-level accuracy is time consuming and expensive; for example, labelling a single image in CityScapes can take more than 90 minutes (Cordts et al., 2016). When deploying semantic segmentation models in practical applications where only limited labelled data are available, high quality ground-truth annotation is a significant bottleneck. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To reduce the need for labelled data, there is a recent surge of interest in leveraging unlabelled data for semi-supervised learning. Previous methods include improving segmentation models via adversarial learning (Hung et al., 2019; Mittal et al., 2019) and self-training (Zou et al., 2019; 2018; Zhu et al., 2020). Others focus on designing advanced data augmentation strategies to generate pseudo image-annotation pairs from unlabelled images (Olsson et al., 2021; French et al., 2020). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In both semi-supervised and supervised learning, a segmentation model often predicts smooth label maps, because neighbouring pixels are usually of the same class, and rarer high-frequency regions are typically only found in object boundaries. This learning bias produces blurry contours and regularly mis-labels rare objects. After carefully examining the label predictions, we further observe that wrongly labelled pixels are typically confused with very few other classes; e.g. a pixel labelled as rider has a much higher chance of being wrongly classified as person, compared to train or bus. By understanding this class structure, learning can be actively focused on the challenging pixels to improve overall segmentation quality. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/c392360a05db92b63807900471b937e8fe659774ca8eca14c432b20af0fd9b6c.jpg",
+        "image_caption": [
+            "Figure 1: ReCo pushes representations within a class closer to the class mean representation, whilst simultaneously pushing these representations away from negative representations sampled in different classes. The sampling distribution from negative classes is adaptive to each query class. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we propose ReCo, a contrastive learning framework designed at a regional level. Specifically, ReCo is a new loss function which helps semantic segmentation not only to learn from local context (neighbouring pixels), but also from global semantic class relationships across the entire dataset. ReCo performs contrastive learning on a pixel-level dense representation, as visualised in Fig. 1. For each semantic class in a mini-batch, ReCo samples a set of pixel-level representations (queries), and encourages them to be close to the class mean representation (positive keys), and simultaneously pushes them away from representations sampled from other classes (negative keys). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For pixel-level contrastive learning with high-resolution images, it is impractical to sample all pixels. In ReCo, we actively sample a sparse set of queries and keys, consisting of less than $5 \\%$ of all available pixels. We sample negative keys from a learned distribution based on the relative distance between the mean representation of each negative key and the query class. This distribution can be interpreted as a pairwise semantic class relationship, dynamically updated during training. We sample queries for those having a low prediction confidence. Active sampling helps ReCo to rapidly focus on the most confusing pixels for each semantic class, and requires minimal additional memory. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "ReCo enables a high-accuracy segmentation model to be trained with very few human annotations. We evaluate ReCo in a semi-supervised setting, with two different modes: i) Partial Dataset Full Labels — a sparse subset of training images, where each image has full ground-truth labels, and the remaining images are unlabelled; ii) Partial Labels Full Dataset — all images have some labels, but covering only a sparse subset of pixels within each image. In both settings, we show that ReCo can consistently improve performance across all methods and datasets. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            390,
+            344,
+            406
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Semantic Segmentation One recent direction is in designing more effective deep convolutional neural networks. Fully convolutional networks (FCNs) (Long et al., 2015) are the foundation of modern segmentation network design. They were later improved with dilated/atrous convolutions with larger receptive fields, capturing more long range information (Chen et al., 2017; 2018). Alternative approaches include encoder-decoder architectures (Ronneberger et al., 2015; Kirillov et al., 2019), sometimes using skip connections (Ronneberger et al., 2015) to refine filtered details. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A parallel direction is to improve optimisation strategies, by designing loss functions that better respect class imbalance (Lin et al., 2017) or using rendering strategy to refine uncertain pixels from high-frequency regions improving the label quality (Kirillov et al., 2020). ReCo is built upon this line of research, to improve segmentation by providing additional supervision on hard pixels. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Semi-supervised Classification and Segmentation The goal of semi-supervised learning is to improve model performance by taking advantage of a large amount of unlabelled data during training. Here consistency regularisation and entropy minimisation are two common strategies. The intuition is that the network’s output should be invariant to data perturbation and geometric transformation. Based on these strategies, many semi-supervised methods have been developed for image classification (Sohn et al., 2020; Tarvainen & Valpola, 2017; Berthelot et al., 2019; Kuo et al., 2020). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "However, for segmentation, generating effective pseudo-labels and well-designed data augmentation are non-trivial. Some solutions improved the quality of pseudo-labelling, using adversarial learning (Hung et al., 2019; Mittal et al., 2019) or enforcing consistency from different augmented images (French et al., 2020; Olsson et al., 2021). In this work, we show that we can improve the performance of current semi-supervised segmentation methods by jointly training with a suitable auxiliary task. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contrastive Learning Contrastive learning learns a similarity function to bring views of the same data closer in representation space, whilst pushing views of different data apart. Most recent contrastive frameworks learn similarity scores based on global representations of the views, parameterising data with a single vector (He et al., 2020; Chen et al., 2020; Khosla et al., 2020). Dense representations, on the other hand, rely on pixel-level representations and naturally provide additional supervision, capturing fine-grained pixel correspondence. Contrastive pre-training based on dense representations has recently been explored, and shows better performance in dense prediction tasks, such as object detection and keypoint detection (Wang et al., 2021b; O. Pinheiro et al., 2020). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contrastive Learning for Semantic Segmentation Contrastive learning has been recently studied to improve semantic segmentation, with a number of different design strategies. Zhang et al. (2021) and Zhao et al. (2021) both perform contrastive learning via pre-training, based on the generated auxiliary labels and ground-truth labels respectively, but at the cost of huge memory consumption. In contrast, ours performs contrastive learning whilst requiring much less memory, via active sampling. In concurrent work, (Wang et al., 2021a; Alonso et al., 2021) also perform contrastive learning with active sampling. However, whilst both these methods are applied to a stored feature bank, ours focuses on sampling features on-the-fly. Active sampling in Alonso et al. (2021) is further based on learnable, class-specific attention modules, whilst ours only samples features based on relation graphs and prediction confidence, without introducing any additional computation overhead, which results in a simpler and much more memory-efficient implementation. ",
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+    {
+        "type": "text",
+        "text": "3 RECO – REGIONAL CONTRAST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3.1 PIXEL-LEVEL CONTRASTIVE LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $( X , Y )$ be a training dataset with training images $x \\in X$ and their corresponding $C$ -class pixellevel segmentation labels $y \\in Y$ , where $y$ can be either provided in the original dataset, or generated automatically as pseudo-labels. A segmentation network $f$ is then optimised to learn a mapping $f _ { \\boldsymbol { \\theta } } :$ $X \\mapsto Y$ , parameterised by network parameters $\\theta$ . This segmentation network $f$ can be decomposed into two parts: an encoder network: $\\phi : X \\mapsto Z$ , and a decoder classification head $\\psi _ { c } : Z \\mapsto Y$ . To perform pixel-level contrastive learning, we additionally attach a decoder representation head $\\psi _ { r }$ on top of the encoder network $\\phi$ , parallel to the classification head, mapping the encoded feature into a higher $m$ -dimensional dense representation with the same spatial resolution as the input image: $\\psi _ { r } : Z \\mapsto R , R \\in \\mathbb { R } ^ { m }$ . This representation head is only applied during training to guide the classifier using the ReCo loss as an auxiliary task, and is removed during inference. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A pixel-level contrastive loss is a function which encourages queries $r _ { q }$ to be similar to the positive key $r _ { k } ^ { + }$ , and dissimilar to the negative keys $r _ { k } ^ { - }$ . All queries and keys are sampled from the decoder representation head: rq, r+,k $r _ { q } , r _ { k } ^ { + , - } \\in R$ . In ReCo, we use a pixel-level contrastive loss across all available semantic classes in each mini-batch, with the distance between keys and queries measured by their normalised dot product. The general formation of the ReCo loss $L _ { \\tt r e c o }$ is then defined as: ",
+        "bbox": [
+            173,
+            422,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/68b0547fe0941acb03fda9253e486c4915d291ecb3bfee98130687549e23e0e2.jpg",
+        "text": "$$\nL _ { \\mathbf { r e c o } } = \\sum _ { c \\in \\mathcal { C } } \\sum _ { r _ { q } \\sim \\mathcal { R } _ { q } ^ { c } } - \\log \\frac { \\exp ( r _ { q } \\cdot r _ { k } ^ { c , + } / \\tau ) } { \\exp ( r _ { q } \\cdot r _ { k } ^ { c , + } / \\tau ) + \\sum _ { r _ { k } ^ { - } \\sim \\mathcal { R } _ { k } ^ { c } } \\exp ( r _ { q } \\cdot r _ { k } ^ { - } / \\tau ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            501,
+            735,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for which $\\mathcal { C }$ is a set containing all available classes in the current mini-batch, $\\tau$ is the temperature control of the softness of the distribution, $\\mathcal { R } _ { q } ^ { c }$ represents a query set containing all representations whose labels belong to class $c$ , $\\mathcal { R } _ { k } ^ { c }$ represents a negative key set containing all representations whose labels do not belong to class $c$ , and $r _ { k } ^ { c , + }$ represents the positive key which is the mean representation of class $c$ . Suppose is a set containing all pixel coordinates with the same resolution as , these queries and keys are then defined as: ",
+        "bbox": [
+            173,
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+            632
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af38e8ba44a2c1b5d9a0e115e404f68144368852fbabe582024d6a7fc0f8fde5.jpg",
+        "text": "$$\n\\mathcal { R } _ { q } ^ { c } = \\bigcup _ { [ u , v ] \\in \\mathcal { P } } \\mathbb { 1 } ( y _ { [ u , v ] } = c ) r _ { [ u , v ] } , \\ \\mathcal { R } _ { k } ^ { c } = \\bigcup _ { [ u , v ] \\in \\mathcal { P } } \\mathbb { 1 } ( y _ { [ u , v ] } \\neq c ) r _ { [ u , v ] } , \\ r _ { k } ^ { c , + } = \\frac { 1 } { | \\mathcal { R } _ { q } ^ { c } | } \\sum _ { r _ { q } \\in \\mathcal { R } _ { q } ^ { c } } r _ { q } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 ACTIVE HARD SAMPLING ON QUERIES AND KEYS ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Contrastive learning on all pixels in high-resolution images would be computationally expensive.   \nHere, we introduce active hard sampling strategies to optimise only a sparse set of queries and keys. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Active Key Sampling When classifying a pixel, a semantic network might be uncertain only over a very small number of candidates, among all available classes. The uncertainty from these candidates typically comes from a close spatial (e.g. rider and bicycle) or semantic (e.g. horse and cow) relationship. To reduce this uncertainty, we propose to sample negative keys non-uniformly, based on the relative distance between each negative key class and the query class. This involves building a pair-wise class relationship graph $G$ , with $G \\in \\mathbb { R } ^ { | \\mathcal { C } | \\times | \\mathcal { C } | }$ , computed and dynamically updated for each mini-batch. This pair-wise relationship is measured by the normalised dot product between the mean representation from a pair of two classes and is defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2de69cd8fe5affc54c5eea18f83dffe6076343a4a726d39127761a9440d728d2.jpg",
+        "text": "$$\nG [ p , q ] = \\left( r _ { k } ^ { p , + } \\cdot r _ { k } ^ { q , + } \\right) , \\quad \\forall p , q \\in \\mathcal { C } , \\ \\mathrm { a n d } \\ p \\neq q .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We further apply SoftMax to normalise these pair-wise relationships among all negative classes $j$ for each query class $c$ , which produces a probabilistic distribution: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\textstyle \\exp ( G [ c , i ] ) / \\sum _ { j \\in { \\mathcal { C } } , j \\neq c } \\exp ( G [ c , j ] )$ . We sample negative keys for each class $i$ based on this distribution, to learn the corresponding query class $c$ . This procedure allocates more samples to hard, confusing classes chosen specifically for each query class, helping the segmentation network to learn a more accurate decision boundary. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Active Query Sampling Due to the natural class imbalance in semantic segmentation, it is easy to over-fit on common classes, such as the road and building classes in the CityScapes dataset, or the background class in the Pascal VOC dataset. These common classes contribute to the majority of pixel space in training images, and so randomly sampling queries will under-sample rare classes and provide minimal supervision to these classes. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we instead sample hard queries — for those whose corresponding pixel prediction confidence is below a defined threshold. Accordingly, ReCo’s loss would then guide the segmentation network by providing appropriate supervision on these less certain pixels. The easy and hard queries are defined as follows, and visualised in Fig. 2, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6a481df57ef8250e889be80e66df9374a91bf090642c75370927ce0b0489792.jpg",
+        "text": "$$\n\\mathcal { R } _ { q } ^ { c , e a s y } = \\bigcup _ { r _ { q } \\in \\mathcal { R } _ { q } ^ { c } } \\mathbb { 1 } ( \\hat { y } _ { q } > \\delta _ { s } ) r _ { q } , \\quad \\mathcal { R } _ { q } ^ { c , h a r d } = \\bigcup _ { r _ { q } \\in \\mathcal { R } _ { q } ^ { c } } \\mathbb { 1 } ( \\hat { y } _ { q } \\leq \\delta _ { s } ) r _ { q } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { y } _ { q }$ is the predicted confidence of label $c$ after the SoftMax operation corresponding to the same pixel location as $r _ { q }$ , and $\\delta _ { s }$ is the user-defined confidence threshold. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/1175caa05b585655e133c94e3b0f6dc9ae581471e69922fae7fb62f0686910cb.jpg",
+        "image_caption": [
+            "Figure 2: Easy and hard queries (shown in white) determined from the predicted confidence map in the Cityscapes dataset. Here we set the confidence threshold $\\delta _ { s } = 0 . 9 7$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 SEMI-SUPERVISED SEMANTIC SEGMENTATION WITH RECO ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ReCo can easily be added to modern semi-supervised segmentation methods without changing the training pipeline, with no additional cost at inference time. To incorporate ReCo, we simply add an additional representation head $\\psi _ { r }$ as described in Section 3.1, and apply the ReCo loss (in Eq. 1) to this representation using the sampling strategy introduced in Section 3.2. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We apply the Mean Teacher framework (Tarvainen & Valpola, 2017) following prior state-of-the-art semi-supervised segmentation methods (Olsson et al., 2021; Mittal et al., 2019). Instead of using the original segmentation network $f _ { \\theta }$ (which we call the student model), we instead use $f _ { \\theta ^ { \\prime } }$ (which we call the teacher model) to generate pseudo-labels from unlabelled images, where $\\theta ^ { \\prime }$ is a moving average of the previous state of $\\theta$ during training optimisation: $\\theta _ { t } ^ { \\prime } = \\lambda \\theta _ { t - 1 } ^ { \\prime } \\bar { + } ( 1 - \\lambda ) \\theta _ { t }$ , with a decay parameter $\\lambda = 0 . 9 9$ . This teacher model can be treated as a temporal ensemble of student models across training time $t$ , resulting in more stable predictions for unlabelled images. The student model $f _ { \\theta }$ is then used to train on the augmented unlabelled images, with pseudo-labels as the ground-truths. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For all pixels with defined ground-truth labels, we apply the ReCo loss on dense representations corresponding to all valid pixels. For all pixels without such labels, we only sample pixels whose predicted pseudo-label confidence is greater than a threshold $\\delta _ { w }$ . This avoids sampling pixels which are likely to have incorrect pseudo-labels. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We apply the ReCo loss to a combined set of labelled and unlabelled pixels. The overall training loss for semi-supervised segmentation is then the linear combination of supervised cross-entropy loss (on ground-truth labels), unsupervised cross-entropy loss (on pseudo-labels) and ReCo loss: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccedc51e47d33c2e43fe410c17e67c255fd12c5f5f87bb340a2f90fbc047f49c.jpg",
+        "text": "$$\nL _ { t o t a l } = L _ { s u p e r v i s e d } + \\eta \\cdot L _ { u n s u p e r v i s e d } + L _ { r e c o }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is defined as the percentage of pixels whose predicted confidence are greater than $\\delta _ { s }$ , a scalar re-weighting the contribution for unsupervised loss, following prior methods (Olsson et al., 2021; ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/809972c8991576f173e37564b9603381a78cd20d01cc70b40de50090bdd9171b.jpg",
+        "image_caption": [
+            "Figure 3: Visualisation of the ReCo framework applied to semi-supervised segmentation and trained with three losses. A supervised loss is computed based on labelled data with ground-truth annotations. An unsupervised loss is computed for unlabelled data with generated pseudo-labels. And finally a ReCo loss is computed based on pixel-level dense representation predicted from both labelled and unlabelled images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Mittal et al., 2019). This makes sure the segmentation network would not be dominated by gradients produced by uncertain pseudo-labels, which typically occur during the early stage of training. Fig. 3 shows a visualisation of the ReCo framework for semi-supervised segmentation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Semi-Supervised Segmentation Benchmark Redesign We propose two modes of semisupervised segmentation tasks, aiming at two different applications. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "i) Partial Dataset Full Labels: A small subset of the images is trained with complete ground-truth labels for each image, whilst the remaining training images are unlabelled. This is the de-facto standard of evaluating semi-supervised segmentation in prior works.   \nii) Partial Labels Full Dataset: All images are trained with partial labels, but only a small percentage of labels are provided for each class in each training image. We create the dataset by first randomly sampling a pixel for each class, and then continuously apply a $[ 5 \\times 5 ]$ square kernel for dilation until we meet the percentage criteria. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The Partial Dataset Full Label setting evaluates the ability to generalise semantic classes given a few examples with perfect boundary information. The Partial Label Full Dataset evaluates learning semantic class completion given many examples with no or minimal boundary information. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/01730836a383cf595ab1cb93774d8e67234f48693672101e2fdcb2c8cc7761b2.jpg",
+        "image_caption": [
+            "Figure 4: Example of training labels for Pascal VOC dataset in Partial Labels Full Dataset setting. (1 Pixel is zoomed 5 times for better visualisation.) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets We experiment on segmentation datasets: Cityscapes (Cordts et al., 2016) and Pascal VOC 2012 (Everingham et al., 2015) in both partial and full label setting. We also ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "evaluate on a more difficult indoor scene dataset SUN RGB-D (Song et al., 2015) in the full label setting only, mainly due to the low quality annotations making it difficult for fair evaluation in the partial label setting. An example of the partially labelled Pascal VOC is shown in Fig. 4. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Strong Baselines Prior semi-supervised segmentation methods are typically designed with different backbone architectures, and trained with different strategies, which makes it difficult to compare them fairly. In this work, we standardise the baselines and implement four strong semi-supervised segmentation methods ourselves: S4GAN (Mittal et al., 2019): an adversarial learning based semisupervised method; CutOut (French et al., 2020), CutMix (French et al., 2020), ClassMix (Olsson et al., 2021): three image augmentation strategies designed specifically for semi-supervised segmentation. Our implementations for all baselines obtain performance on par with, and most of the time surpassing, the performance reported in each original publication, giving us a set of strong baselines. All baselines and our method were implemented on DeepLab ${ \\mathrm { V } } 3 +$ (Chen et al., 2018) with ResNet-101 backbone (He et al., 2016), and all with the same optimisation strategies. Detailed hyper-parameters used for each dataset are provided in the Appendix A. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 RESULTS ON PASCAL VOC, CITYSCAPES, SUN RGB-D (FULL LABELS) ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "First, we compared our results to semi-supervised baselines in a full label setting. We applied ReCo on top of ClassMix, which consistently outperformed other semi-supervised baselines. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows the mean IoU validation performance on three datasets over three individual runs (different labelled and unlabelled data splits). The number of labelled images shown in the three columns for each dataset, are chosen such that the least-appeared classes have appeared in 5, 15 and 50 images respectively. In the fewest-label setting for each dataset, applying ReCo with ClassMix can improve results by a significant margin, with up to $1 . 5 - 4 . 5 \\%$ absolute improvement. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/603366c355d5ea2d62cecf1e19c5c243f2f550ba816503caab91d72d647512c7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Pascal VOC</td><td colspan=\"3\">CityScapes</td><td colspan=\"3\">SUN RGB-D</td></tr><tr><td>Method</td><td></td><td>60 labels 200 labels 600 labels</td><td></td><td>20 labels</td><td>50labels</td><td>150 labels</td><td>50labels</td><td>：150 labels 500 labels</td><td></td></tr><tr><td>Supervised</td><td>37.79</td><td>53.87</td><td>64.04</td><td>38.12</td><td>45.42</td><td>54.93</td><td>19.79</td><td>28.78</td><td>37.73</td></tr><tr><td>S4GAN (Mittal et al.,2019)</td><td>47.95</td><td>61.25</td><td>66.21</td><td>37.65</td><td>47.08</td><td>56.46</td><td>20.53</td><td>29.79</td><td>38.08</td></tr><tr><td>CutOut (French et al., 2020)</td><td>52.96</td><td>63.57</td><td>69.85</td><td>42.52</td><td>50.15</td><td>59.42</td><td>25.94</td><td>34.45</td><td>41.25</td></tr><tr><td>CutMix (French et al.,2020)</td><td>53.71</td><td>66.95</td><td>72.42</td><td>44.02</td><td>54.72</td><td>62.24</td><td>27.60</td><td>37.55</td><td>42.69</td></tr><tr><td>ClassMix (Olsson et al.,2021)</td><td>49.06</td><td>67.95</td><td>72.50</td><td>45.61</td><td>55.56</td><td>63.94</td><td>28.42</td><td>37.55</td><td>42.46</td></tr><tr><td>ReCo +ClassMix</td><td>53.31</td><td>69.81</td><td>72.75</td><td>49.86</td><td>57.69</td><td>65.04</td><td>29.65</td><td>39.14</td><td>44.55</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: mean IoU validation performance for Pascal VOC, CityScapes and SUN RGB-D datasets.   \nWe report the mean over three independent runs for all methods. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/e86a8ea56e0e97c7c3c88b45f168d27a4fb399e303820baa1abb18992e4612a5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Pascal VOC</td><td>1/16 [92] 1/8 [183]1/4 [366] 1/2 [732]</td><td></td><td></td><td></td></tr><tr><td>AdvSemSeg (Hung et al.,2019)</td><td>39.69</td><td>47.58</td><td>59.97</td><td>65.27</td></tr><tr><td>Mean Teacher (Tarvainen &amp; Valpola,2017)</td><td>48.70</td><td>55.81</td><td>63.01</td><td>69.16</td></tr><tr><td>CCT(Ouali et al.,2020)</td><td>33.10</td><td>47.60</td><td>58.80</td><td>62.10</td></tr><tr><td>GCT (Ke et al.,2020)</td><td>46.04</td><td>54.98</td><td>64.71</td><td>70.67</td></tr><tr><td>VAT (Miyato et al.,2018)</td><td>36.92</td><td>49.35</td><td>56.88</td><td>63.34</td></tr><tr><td>CutMix (French et al.,2020)</td><td>55.58</td><td>63.20</td><td>68.36</td><td>69.87</td></tr><tr><td>PseudoSeg (Zou et al.,2021)</td><td>57.60</td><td>65.50</td><td>69.14</td><td>72.41</td></tr><tr><td>ReCo + ClassMix</td><td>64.78</td><td>72.02</td><td>73.14</td><td>74.69</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To further justify the effectiveness of ReCo, we also include results on existing benchmarks, to compare with other semi-supervised methods in Table 2. Here, all baselines were re-implemented and reported in the PseudoSeg setting (Zou et al., 2021), where the labelled images are sampled from the original PASCAL dataset, with a total of 1.4k images. In both benchmarks, ReCo shows state-of-the-art performance, and specifically is able to reach PseudoSeg’s performance, whilst requir",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: mean IoU validation performance for Pascal VOC with data partition and training strategy proposed in PseudoSeg (Zou et al., 2021). The percentage and the number of labelled data used are listed in the first row. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "ing only half the labelled data. Additional results are further shown in Appendix C. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 5, we present qualitative results from the semi-supervised setup with the fewest labels: 20 labels for CityScapes and 50 labels SUN RGB-D datasets. The 60 labelled Pascal VOC is further shown in Appendix D. In Fig. 5, we can see the advantage of ReCo, where the edges and boundaries of small objects are clearly more pronounced such as in the person and bicycle classes in CityScapes, and the lamp and pillow classes in SUN RGB-D. Interestingly, we found that in SUN RGB-D, though all methods may confuse ambiguous class pairs such as table and desk or window and curtain, ReCo still produces consistently sharp and accurate object boundaries compared to the Supervised and ClassMix baselines where labels are noisy near object boundaries. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 RESULTS ON PASCAL VOC AND CITYSCAPES (PARTIAL LABELS) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            853,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the partial label setting, we evaluated on the CityScapes and Pascal VOC datasets. Table 3 compared ReCo to the two best semi-supervised baselines and a supervised baseline. Again, we see ReCo can improve performance in all cases when applied on top of ClassMix, with around $1 - 3 \\%$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/b5fe9e69bd192e091d00e294308b8c1874870f5345a5f8c9859125ce40a6ca8c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/7aa3fd00b4020c4d45e31341769e04cda9730f7f908bd1bec7abfe00ae47168a.jpg",
+        "table_caption": [
+            "Figure 5: Visualisation of CityScapes and SUN RGB-D validation set trained on 20 and 50 labelled images respectively. Interesting regions are shown in white arrows. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">CityScapes (Partial)</td></tr><tr><td>Method</td><td>1 pixel</td><td>1% labels</td><td>5% labels</td><td>25% labels</td></tr><tr><td>Supervised</td><td>44.08</td><td>52.89</td><td>56.65</td><td>63.43</td></tr><tr><td>CutMix</td><td>46.91</td><td>54.90</td><td>59.69</td><td>65.61</td></tr><tr><td>ClassMix</td><td>47.42</td><td>56.68</td><td>60.96</td><td>66.46</td></tr><tr><td>ReCo + ClassMix</td><td>49.66</td><td>58.97</td><td>62.32</td><td>66.92</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/771442127b8a75cea7cd11e5ae86b9f0e36cdfc473466a89f17776aa3768d645.jpg",
+        "table_caption": [
+            "Table 3: mean IoU validation performance for Pascal VOC and Cityscapes datasets trained on $1 , 1 \\%$ , $5 \\%$ and $2 5 \\%$ labelled pixels per class per image. We report the mean over three independent runs for all methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Pascal VOC (Partial)</td></tr><tr><td>Method</td><td>1 pixel</td><td>1% labels</td><td>5% labels</td><td>25% labels</td></tr><tr><td>Supervised</td><td>60.33</td><td>66.17</td><td>69.16</td><td>73.75</td></tr><tr><td>CutMix</td><td>63.50</td><td>70.83</td><td>73.04</td><td>75.64</td></tr><tr><td>ClassMix</td><td>63.69</td><td>71.04</td><td>72.90</td><td>75.79</td></tr><tr><td>ReCo + ClassMix</td><td>66.11</td><td>72.67</td><td>74.09</td><td>75.96</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "absolute improvement. We observe less relative performance improvement than in the full label setting; very sparse ground-truth annotations could confuse ReCo, resulting in inaccurate supervision. ",
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+    {
+        "type": "text",
+        "text": "We show qualitative results on Pascal VOC dataset trained on 1 labelled pixel per class per image in Fig. 6. As in the full label setting, we see smoother and more accurate boundary predictions from ReCo. More visualisations from CityScapes are shown in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "4.3 ABLATIVE ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Next we present an ablative analysis on 20 labelled CityScapes images to understand the behaviour of ReCo with respect to hyper-parameters. We use our default experimental setting from Section 4.1, using ReCo with ClassMix. Additional ablations are further shown in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "Number of Queries and Keys We first evaluate the performance by varying the number of queries and keys used in ReCo framework, whilst fixing all other hyper-parameters. In Fig. 7a and 7b, we can observe that performance is better when sampling more queries and keys, but after a certain point, the improvements become marginal. Notably, even in our smallest option of having 32 queries per class in a mini-batch — consisting of less than $0 . 5 \\%$ among all available pixel space — this can still improve performance by a non-trivial margin. Compared to a concurrent work (Zhang et al., 2021) which requires $1 0 \\mathrm { k }$ queries and $4 0 \\mathrm { k }$ keys in each training iteration, ReCo can be optimised with $\\times 5 0$ more efficiency in terms of memory footprint. ",
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+    {
+        "type": "image",
+        "img_path": "images/0739cc72fbaee2cab769c477c1d34bcdd520f1abbf4dca31741349e102bb6193.jpg",
+        "image_caption": [
+            "Figure 6: Visualisation of Pascal VOC validation set with ClassMix (left) vs. with ReCo (right) trained on 1 labelled pixel per class per image. Interesting regions are shown in white arrows. "
+        ],
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+    {
+        "type": "text",
+        "text": "Ratio of Unlabelled Data We evaluate how ReCo can generalise across different levels of unlabelled data. In Fig. 7c, we show that by only training on $10 \\%$ unlabelled data in the original setting, we can already surpass the ClassMix baseline. This shows that ReCo can achieve strong generalisation not only in label efficiency but also in data efficiency. ",
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+    {
+        "type": "text",
+        "text": "Choice of Semi-Supervised Method Finally, we show that ReCo is robust to the choice of different semi-supervised methods. In Fig. 7d, we can see that ReCo obtains a better performance from a variety choice of semi-supervised baselines. ",
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+    {
+        "type": "text",
+        "text": "Effect of Active Sampling In Fig. 7e, we see that randomly sampling queries and keys gives much less improvement compared to active sampling in our default setting. Particularly, hard query sampling has a dominant effect on generalisation: if we instead only sample from easy queries, ReCo only marginally improves on the baseline. This further verifies that most queries are redundant. ",
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+    {
+        "type": "image",
+        "img_path": "images/d1528c733884e8502f3353f61a0c9b71cc1164ca318641422551372d3bc937f2.jpg",
+        "image_caption": [
+            "Figure 7: mean IoU validation performance on 20 labelled CityScapes dataset based on different choices of hyper-parameters. Grey: ClassMix (if not labelled otherwise) in our default setting. Light Blue: ReCo $^ +$ ClassMix (if not labelled otherwise) in a different hyper-parameter setting. Dark Blue: ReCo $^ +$ ClassMix in our default setting. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 VISUALISATIONS AND INTERPRETABILITY OF CLASS RELATIONSHIPS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we visualise the pair-wise semantic class relation graph defined in Eq. 3, additionally supported by a semantic class dendrogram using the off-the-shelf hierarchical clustering algorithm in SciPy (Virtanen et al., 2020) for better visualisation. The features for each semantic class used in both visualisations are averaged across all available pixel embeddings in each class from the validation set. In all visualisations, we compared features learned with ReCo built on top of supervised learning compared to a standard supervised learning method trained on all data, representing the semantic class relationships of the full dataset. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Using the same definitions in Section 3.1, we first choose such pixel embedding to be the embedding $Z$ predicted from the encoder network $\\phi$ in both supervised learning and with ReCo. We also show the visualisation for embedding $R$ which is the actual representation we used for ReCo loss and active sampling. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/6a0eedd3a7abbcf3b91e4d92c859b2e3c046c295a8645eb107a586a2f42f5e73.jpg",
+        "image_caption": [
+            "Figure 8: Visualisation of semantic class relation graph (top) and its corresponding semantic class dendrogram (bottom) on CityScapes dataset. Brighter colour represents closer (more confused) relationship. Best viewed in zoom. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "In Fig. 8, we present the semantic class relationship and dendrogram for the CityScapes dataset by embedding $R$ and $Z$ with and without ReCo. We can clearly see that ReCo helps disentangle features compared to supervised learning where many pairs of semantic classes are similar. In addition, we find that the dendrogram generated by ReCo based on embedding $Z$ is more structured, showing a clear and interpretable semantic tree by grouping semantically similar classes together: for example, all large transportation classes car, truck, bus and train are under the same parent branch. In addition, we find that nearly all classes based on embedding $R$ are perfectly disentangled, except for bus and train, suggesting the CityScapes dataset might not have sufficient bus and train examples to learn a distinctive representation for these two classes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The pair-wise relation graph helps us to understand the distribution of semantic classes in each dataset, and clarifies the pattern of incorrect predictions from the trained semantic network. We additionally provide a dendrogram based on embedding $R$ for the SUN RGBD dataset, clearly showing ambiguous class pairs, such as night stand and dresser; table and desk; floor ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/87a4765d9dd11e7bc7baebbec9ccec6dba349253e537720cdd9a860d8bfadc80.jpg",
+        "image_caption": [
+            "Figure 9: Visualisation of semantic class dendrogram based on embedding $R$ on SUN RGB-D dataset using ${ \\mathrm { R e C o } } + { \\mathrm { S } }$ upervised method. Best viewed in zoom. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "and floormat, consistent with our results shown in Fig. 5. Complete visualisations of these semantic class relationships are shown in Appendix F. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we have presented ReCo, a new pixel-level contrastive framework with active sampling, designed specifically for semantic segmentation. ReCo can improve performance in semantic segmentation methods with minimal additional memory footprint. In particular, ReCo has shown its strongest effect in semi-supervised learning with very few labels, where we improved on the stateof-the-art by a large margin. In further work, we aim to design effective contrastive frameworks for video representation learning. ",
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+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "All of the information for reproducibility is shown in Appendix A. Code is available at https: //github.com/lorenmt/reco. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
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+    {
+        "type": "text",
+        "text": "This work has been supported by Dyson Technology Ltd. We thank Zhe Lin for the initial discussion and Zhengyang Feng for his help on the evaluation metric design. ",
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+        "text": "Pauli Virtanen, Ralf Gommers, Travis E Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, et al. Scipy 1.0: fundamental algorithms for scientific computing in python. Nature methods, 2020. ",
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+        "text": "Feihu Zhang, Philip Torr, Rene Ranftl, and Stephan R Richter. Looking beyond single images for contrastive semantic segmentation learning. In Advances in Neural Information Processing Systems (NeurIPS), 2021. ",
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+        "text": "Xiangyun Zhao, Raviteja Vemulapalli, Philip Andrew Mansfield, Boqing Gong, Bradley Green, Lior Shapira, and Ying Wu. Contrastive learning for label efficient semantic segmentation. In Proceedings of the International Conference on Computer Vision (ICCV), 2021. ",
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+        "text": "Yi Zhu, Zhongyue Zhang, Chongruo Wu, Zhi Zhang, Tong He, Hang Zhang, R Manmatha, Mu Li, and Alexander Smola. Improving semantic segmentation via self-training. arXiv preprint arXiv:2004.14960, 2020. ",
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+    {
+        "type": "text",
+        "text": "Yang Zou, Zhiding Yu, BVK Kumar, and Jinsong Wang. Domain adaptation for semantic segmentation via class-balanced self-training. arXiv preprint arXiv:1810.07911, 2018. ",
+        "bbox": [
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+    {
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+        "text": "Yang Zou, Zhiding Yu, Xiaofeng Liu, BVK Kumar, and Jinsong Wang. Confidence regularized self-training. In Proceedings of the International Conference on Computer Vision (ICCV), 2019. ",
+        "bbox": [
+            169,
+            553,
+            823,
+            583
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yuliang Zou, Zizhao Zhang, Han Zhang, Chun-Liang Li, Xiao Bian, Jia-Bin Huang, and Tomas Pfister. Pseudoseg: Designing pseudo labels for semantic segmentation. In Proceedings of the International Conference on Learning Representations (ICLR), 2021. ",
+        "bbox": [
+            176,
+            590,
+            823,
+            635
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            263,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            136,
+            439,
+            151
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We trained all methods with SGD optimiser with learning rate $2 . 5 \\times 1 0 ^ { - 3 }$ , momentum 0.9, and weight decay $5 \\cdot 1 0 ^ { - 4 } $ . We adopted the polynomial annealing policy to schedule the learning rate, which is multiplied by $\\begin{array} { r } { ( 1 - \\frac { i \\bar { t } e r } { t o t a l . i t e r } ) ^ { p o \\bar { w } e r } } \\end{array}$ with $p o w e r = 0 . 9$ , and trained for $4 0 \\mathrm { k }$ iterations for all datasets. Code is attached in the supplementary material. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            223
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For CityScapes, we first downsampled all images in the dataset to half resolution $[ 5 1 2 \\times 1 0 2 4 ]$ prior to use. We extracted $[ 5 1 2 \\times 5 1 2 ]$ random crops and used a batch size of 2 during training. ",
+        "bbox": [
+            171,
+            229,
+            823,
+            258
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For Pascal VOC, we extracted $[ 3 2 1 \\times 3 2 1 ]$ random crops, applied a random scale between [0.5, 1.5], and used a batch size of 10 during training. ",
+        "bbox": [
+            174,
+            265,
+            821,
+            294
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For SUN RGB-D, we first rescaled all images to $[ 3 8 4 \\times 5 1 2 ]$ resolution, extracted $[ 3 2 1 \\times 3 2 1 ]$ random crops, applied a random scale between [0.5, 1.5], and used a batch size of 5. We additionally re-organised the original training and validation split in SUN RGB-D dataset from 5285 and 5050 to 9860 and 475 samples respectively, to increase the amount of training data which we think is more appropriate for semi-supervised task. ",
+        "bbox": [
+            173,
+            300,
+            825,
+            369
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "All datasets were additionally augmented with Gaussian blur, colour jittering, and random horizontal flip. The pre-processing for CityScapes and Pascal VOC are consistent with the prior work (Olsson et al., 2021). In Table 2, we extracted $[ 5 1 3 \\times 5 1 3 ]$ random crops and applied a random scale between [0.5, 2.0], following PseudoSeg’s training setup (Zou et al., 2021). ",
+        "bbox": [
+            174,
+            377,
+            825,
+            434
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In our ReCo framework, we sampled 256 query samples and 512 key samples and used temperature $\\tau = 0 . 5$ for each mini-batch, which we found to work well in all datasets. The dimensionality for pixel-level representation was set to $m = 2 5 6$ . The confidence thresholds were set to $\\delta _ { w } = 0 . 7$ and $\\delta _ { s } = 0 . 9 7$ . ",
+        "bbox": [
+            173,
+            439,
+            825,
+            496
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL ABLATIVE STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            516,
+            486,
+            532
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "ReCo Only Results Table 4 shows ReCo designed with and without data augmentation, trained on 20 and 50 labelled CityScapes dataset. We observe that using pure semisupervised learning with additional unlabelled data will lead to worse performance compared to supervised learning without such labelled data. This shows data augmentation strategies designed for semi-supervised segmentation are the key component to make best use of the unlabelled data. Although the vanilla ReCo still performs better compared to standard semi-supervised learning, the active sam",
+        "bbox": [
+            174,
+            547,
+            483,
+            727
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/b85969c2bd36442199a9700f045f97743d090da7c225d4735d013e2dd9a957c8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CityScapes</td><td>20 Labels 50 Labels</td></tr><tr><td>Supervised</td><td>38.10 47.10</td></tr><tr><td>Semi-Supervised</td><td>28.59 43.74</td></tr><tr><td>Semi-Supervised +ReCo</td><td>29.16 46.96</td></tr><tr><td>ClassMix</td><td>45.61 55.56</td></tr><tr><td>ClassMix+Reco</td><td>49.86 57.69</td></tr></table>",
+        "bbox": [
+            508,
+            550,
+            813,
+            648
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 4: mean IoU validation performance for 20 and 50 labelled CityScapes data for supervised method (top) and semi-supervised methods (bottom). ",
+        "bbox": [
+            496,
+            659,
+            823,
+            714
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "pling of ReCo based on incorrect pseudo-labels leads to marginal improvement compared to a pure data augmentation method like ClassMix. Therefore, ReCo performs better as an auxiliary framework combined with a strong semi-supervised method. ",
+        "bbox": [
+            176,
+            728,
+            820,
+            770
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Compared to Feature Bank Methods We have experimented with ReCo with a stored feature bank framework similar to the design in the concurrent works (Alonso et al., 2021; Wang et al., 2021a). We found that just by replacing our batch-wise sampling method with a feature bank sampling method will achieve a similar performance $4 9 . 3 4 ~ \\mathrm { m I o U } )$ ) compared to our original design $( 4 9 . 8 6 ~ \\mathrm { m I o U } )$ on 20 labelled CityScapes, but with a slower training speed. This verifies our assumption that batch-wise sampling is an accurate approximation of class distribution over the entire dataset. ",
+        "bbox": [
+            173,
+            777,
+            825,
+            875
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C RESULTS ON SEMI-SUPERVISED SEGMENTATION BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            736,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Here, we present quantitative results for other semi-supervised semantic segmentation benchmarks in CityScapes and Pascal VOC datasets. Note that, this benchmark is much less challenging compared to our proposed benchmark in Section 4.1, evaluated with significantly less number of labelled images. Since some methods applied with different backbones and training strategies, we compared each result with respect to its performance gap compared to its corresponding fully supervised result, as shown in brackets, to ensure fairness following Feng et al. (2020). ",
+        "bbox": [
+            174,
+            133,
+            825,
+            217
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Table 5, we show results for ReCo applied on top of ClassMix, and trained with both DeepLabv2 (Chen et al., 2017) and DeepLabv $^ { 3 + }$ (Chen et al., 2018). We can observe that ReCo achieved the best performances in most cases in both datasets, showing its robustness to different backbone architectures and number of labelled training images. ",
+        "bbox": [
+            173,
+            223,
+            825,
+            280
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b9475f7cb7f0241ddf150f447da1afbbb875d847ca5e16caf9d24fe874152f74.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Pascal VOC</td><td>Backbone</td><td>1/106 [100]</td><td>1/50 [212]</td><td>1/20 [529]</td><td>1/8 [1323]</td><td>Full [10582]</td></tr><tr><td>AdvSemSeg (Hung et al.,2019)</td><td>DeepLabv2</td><td></td><td>57.20(17.70)</td><td>64.70(10.20)</td><td>69.50(5.40)</td><td>74.90</td></tr><tr><td>S4GAN (Mittal et al.,2019)</td><td>DeepLabv2</td><td></td><td>63.30(12.30)</td><td>67.20(8.40)</td><td>71.40(4.20)</td><td>75.60</td></tr><tr><td>CutMix (French et al.,2020)</td><td>DeepLabv2</td><td>53.79(18.71)</td><td>64.81(7.73)</td><td>66.48(6.06)</td><td>67.60(4.94)</td><td>72.54</td></tr><tr><td>ClassMix (Olsson et al.,2021)</td><td>DeepLabv2</td><td>54.18(19.95)</td><td>66.15(7.98)</td><td>67.77(6.36)</td><td>71.00(3.13)</td><td>74.13</td></tr><tr><td>CCT (Ouali et al.,2020)</td><td>PSPNet</td><td></td><td></td><td>=</td><td>70.45(4.80)</td><td>75.25</td></tr><tr><td>CAC (Lai et al.,2021)</td><td>PSPNet</td><td></td><td></td><td></td><td>72.50(3.90)</td><td>76.40</td></tr><tr><td>GCT (Ke et al.,2020)</td><td>DeepLabv2</td><td></td><td></td><td></td><td>70.57(3.49)</td><td>74.06</td></tr><tr><td>DMT (Feng et al.,2020)</td><td>DeepLabv2</td><td>63.04(11.71)</td><td>67.15(7.60)</td><td>69.92(4.83)</td><td>72.70(2.05)</td><td>74.75</td></tr><tr><td>ReCo + ClassMix</td><td>DeepLabv2</td><td>63.16(11.20)</td><td>66.41(7.95)</td><td>68.85(5.51)</td><td>71.00(3.36)</td><td>74.36</td></tr><tr><td>ReCo+ClassMix</td><td>DeepLabv3+</td><td>63.60(14.15)</td><td>72.14(5.61)</td><td>73.66(4.09)</td><td>74.62( (3.13)</td><td>77.75</td></tr><tr><td>CityScapes</td><td>Backbone</td><td>1/30 [100]</td><td>1/8 [372]</td><td>1/4 [744]</td><td>1/2 [1488]</td><td>Full [2975]</td></tr><tr><td>AdvSemSeg (Hung et al.,2019)</td><td>DeepLabv2</td><td></td><td>58.80(7.60)</td><td>62.30(4.10)</td><td>65.70(0.70)</td><td>66.40</td></tr><tr><td>S4GAN (Mittal et al., 2019)</td><td>DeepLabv2</td><td></td><td>59.30(6.50)</td><td>61.90(3.90)</td><td></td><td>65.80</td></tr><tr><td>CutMix (French et al.,2020)</td><td>DeepLabv2</td><td>51.20(16.33)</td><td>60.34(7.19)</td><td>63.87(3.66)</td><td></td><td>67.53</td></tr><tr><td>ClassMix (Olsson et al.,2021)</td><td>DeepLabv2</td><td>54.07(12.12)</td><td>61.35(4.84)</td><td>63.63(2.56)</td><td>66.29(-0.10)</td><td>66.19</td></tr><tr><td>DMT (Feng et al.,2020)</td><td>DeepLabv2</td><td>54.81(13.36)</td><td>63.03(5.13)</td><td>=</td><td></td><td>68.16</td></tr><tr><td>ECS*(Mendel et al.,2020)</td><td>DeepLabv3+</td><td></td><td>67.38(7.38)</td><td>70.70(4.06)</td><td>72.89(1.87)</td><td>74.76</td></tr><tr><td>ReCo+ClassMix</td><td>DeepLabv2</td><td>56.53(12.07)</td><td>64.94(3.66)</td><td>67.53(1.07)</td><td>68.69(-0.09)</td><td>68.60</td></tr><tr><td>ReCo +ClassMix</td><td>DeepLabv3+</td><td>60.28(10.20)</td><td>66.44(4.04)</td><td>68.50(1.98)</td><td>70.63(-0.15)</td><td>70.48</td></tr></table>",
+        "bbox": [
+            171,
+            292,
+            802,
+            544
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 5: mean IoU validation performance in semi-supervsed Pascal VOC and CityScapes datasets. We list the percentage along with the number of labelled images at the top row. The first and second best performances in each data partition setting are coloured in red and orange respectively. $^ *$ trained images in doubled resolution. All results were taken from the corresponding publications. ",
+        "bbox": [
+            173,
+            554,
+            825,
+            611
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D VISUALISATION ON PASCAL VOC (TRAINED WITH 60 LABELLED IMAGES) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            754,
+            136
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the full label setting, the baselines Supervised and ClassMix are very prone to completely misclassifying rare objects such as boat, bottle and table, while our method can predict these rare classes accurately. ",
+        "bbox": [
+            173,
+            151,
+            820,
+            193
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/080f1e165fde866182ca85eb2ad2769d22d31503a8937a7b391377094cf6925e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            202,
+            826,
+            765
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/4baebf3db414acb944b4740be2ecea8e45ce61ea41a16fc75f5505a8fb4982f1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Background Aeroplane  Bicycle</td><td rowspan=1 colspan=1>Bird</td><td rowspan=1 colspan=1>Boat</td><td rowspan=1 colspan=1>Bottle</td><td rowspan=1 colspan=1>Bus</td><td rowspan=1 colspan=1>Car</td><td rowspan=1 colspan=1>Cat</td><td rowspan=1 colspan=1>Chair</td><td rowspan=1 colspan=1>Cow</td></tr><tr><td rowspan=1 colspan=2>Table      Dog     Horse   Motorbike</td><td rowspan=1 colspan=1>Motorbike</td><td rowspan=1 colspan=1>Person</td><td rowspan=1 colspan=1>Plant</td><td rowspan=1 colspan=1>Sheep</td><td rowspan=1 colspan=1>Sofa</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Monitor</td></tr></table>",
+        "bbox": [
+            179,
+            756,
+            820,
+            779
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E VISUALISATION ON CITYSCAPES (TRAINED WITH $1 \\%$ LABELLED PIXEL) ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            102,
+            816,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In the partial label setting, the performance improvements are less pronounced compared to the full label setting in CityScapes dataset. The improvements typically come from the more accurate predictions in small object boundaries such as in traffic light and traffic sign. Learning semantics with partial labels with minimal boundary information remains an open research question and still has huge scope for improvements. ",
+        "bbox": [
+            173,
+            133,
+            825,
+            204
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8a41e6882e35596dfde824f3e6ffcf816d7c94c2b55f9e11bfc357a683da7f7.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            215,
+            826,
+            550
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/1d1faee2aebe0d662445de16e655943bbf0508b697b27d5ffb3f0dd6bbe939cf.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Road</td><td>Sidewalk</td><td>Building</td><td>Wall</td><td></td><td>Pole</td><td>TrafficLight</td><td>Traffic Sign</td><td>Vegetation</td><td></td></tr><tr><td>Sky</td><td>Person</td><td>Rider</td><td>Car</td><td>ruck</td><td>Bus</td><td>Train</td><td>Motorcycle</td><td>Bicycle</td><td>Invalid</td></tr></table>",
+        "bbox": [
+            178,
+            549,
+            821,
+            570
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F VISUALISATION ON SEMANTIC CLASS RELATIONSHIP FROM PASCAL VOC (TOP) AND SUN RGB-D (BOTTOM) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            776,
+            136
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We show features learned by ReCo are more disentangled compared to the Supervised baseline in all datasets, which helps the segmentation model to learn a better decision boundary. Brighter colour represents closer (more confused) relationship. Best viewed in zoom. ",
+        "bbox": [
+            173,
+            151,
+            823,
+            193
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/d7a0d4015766f4796dde34cff6b6e3cc473bbe0377ad9e6741668a650b58f0e5.jpg",
+        "image_caption": [
+            "(a) Embed. Z (Supervised) (b) Embed. Z (ReCo $^ +$ Supervised) (c) Embed. R (ReCo $^ +$ Supervised) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            208,
+            825,
+            661
+        ],
+        "page_idx": 16
+    }
+]

parse/dev/9h3KsOVXhLZ/9h3KsOVXhLZ.md

@@ -0,0 +1,243 @@
+# SwinTrack: A Simple and Strong Baseline for Transformer Tracking
+
+Liting Lin1,2∗ Heng $\mathbf { F a n ^ { 3 * } }$ Zhipeng Zhang4 Yong $\mathbf { X } \mathbf { u } ^ { 1 , 2 }$ Haibin Ling5
+
+1School of Computer Science & Engineering, South China Univ. of Tech., Guangzhou, China 2Peng Cheng Laboratory, Shenzhen, China   
+3Department of Computer Science and Engineering, University of North Texas, Denton, USA 4DiDi Chuxing, Beijing, China 5Department of Computer Science, Stony Brook University, Stony Brook, USA l.lt@mail.scut.edu.cn, heng.fan $@$ unt.edu, zhipeng.zhang.cv $@$ outlook.com yxu@scut.edu.cn, hling $@$ cs.stonybrook.edu
+
+# Abstract
+
+Recently Transformer has been largely explored in tracking and shown state-of-theart (SOTA) performance. However, existing efforts mainly focus on fusing and enhancing features generated by convolutional neural networks (CNNs). The potential of Transformer in representation learning remains under-explored. In this paper, we aim to further unleash the power of Transformer by proposing a simple yet efficient fully-attentional tracker, dubbed SwinTrack, within classic Siamese framework. In particular, both representation learning and feature fusion in SwinTrack leverage the Transformer architecture, enabling better feature interactions for tracking than pure CNN or hybrid CNN-Transformer frameworks. Besides, to further enhance robustness, we present a novel motion token that embeds historical target trajectory to improve tracking by providing temporal context. Our motion token is lightweight with negligible computation but brings clear gains. In our thorough experiments, SwinTrack exceeds existing approaches on multiple benchmarks. Particularly, on the challenging LaSOT, SwinTrack sets a new record with 0.713 SUC score. It also achieves SOTA results on other benchmarks. We expect SwinTrack to serve as a solid baseline for Transformer tracking and facilitate future research. Our codes and results are released at https://github.com/LitingLin/SwinTrack.
+
+# 1 Introduction
+
+Visual tracking has seen considerable progress since deep learning. In particular, the recent Transformer [30] has significantly pushed the state-of-the-art in tracking owing to its ability in modeling long-range dependencies. However, existing methods usually leverage Transformer for fusing and enhancing features generated from convolutional neural networks (CNNs), e.g., ResNet [14]. The potential of exploiting Transformer for feature representation learning is largely under-explored.
+
+Recently, Vision Transformer (ViT) [7] has exhibited great potential in robust feature representation learning. Particularly, its extension Swin Transformer [23] has achieved state-of-the-art (SOTA) results on multiple tasks. Taking inspiration from this, we argue, besides the feature fusion, the representation learning in tracking can also benefit from Transformer via attention. Thus motivated, we propose to develop a fully attentional tracking framework based on Siamese architecture. Specifically, both the feature representation learning and the feature fusion of template and search region are realized by Transformer. More concretely, we borrow the architecture of the powerful
+
+![](images/bff5b4163414025c27ee09c6e9bfb93d55e373aa201c681052eec140b863d86c.jpg)  
+Figure 1: Comparison on LaSOT [9]. Our tracker (SwinTrack-B-384) sets a new record with 0.713 SUC score and still runs efficiently at around $4 5 f p s$ . A lighter version (SwinTrack-T-224) achieves 0.672 SUC score and runs at around $9 6 f p s$ , which is on par with existing SOTAs in accuracy but much faster.
+
+Swin Transformer [23] and adapt it to Siamese tracking. Note that, other Transformer architectures can be used. For feature fusion, we introduce a simple homogeneous concatenation-based fusion architecture, without a query-based decoder.
+
+Moreover, taking into consideration that tracking is a temporal task, we propose a novel motion token to improve robustness. Inspired by that the target usually moves smoothly in a short period, motion token is represented by the historical target trajectory within a local temporal window. We incorporate the (single) motion token in the decoder of feature fusion to leverage motion information during tracking. Despite being conceptually simple, our motion token can effectively boost tracking performance, with negligible computation.
+
+We name our framework SwinTrack. As a pure Transformer framework, SwinTrack enables better interactions inside the feature learning of template and search region and their fusion compared to pure CNN-based [1, 20] and hybrid CNN-Transformer [5, 32, 36] frameworks, leading to more robust performance (see Fig. 1). Fig. 2 demonstrates the architecture of SwinTrack. We conduct extensive experiments on five large-scale benchmarks to verify the effectiveness of SwinTrack, including LaSOT [9], $\mathrm { L a S O T _ { \mathrm { e x t } } }$ [8], TrackingNet [26], GOT-10k [15] and TNL2k [34]. On all benchmarks, SwinTrack achieves promising results and meanwhile runs fast at $4 5 f p s$ . In particular, on the challenging LaSOT, SwinTrack sets a new record of 71.3 SUC score, surpassing the strongest prior tracker [36] (to date) by 3.1 absolute percentage points and crossing the 0.7 SUC threshold for the first time (see Fig. 1 again). It also achieves 49.1 SUC, 84.0 SUC, 72.4 AO and 55.9 SUC scores on $\mathrm { L a S O T _ { e x t } }$ , TrackingNet, GOT-10k and TNL2k respectively, which are better than or on par with state-of-the-arts (SoTAs). In addition, we provide a lighter version of SwinTrack that obtains comparable results to SoTAs but runs much faster at around 98 fps.
+
+In summary, our contributions are as follows: (i) We propose SwinTrack, a simple and strong baseline for fully attentional tracking; (ii) We present a simple yet effective motion token, enabling the integration of rich motion context during tracking, further boosting the robustness of SwinTrack, with negligible computation; (iii) Our proposed SwinTrack achieves state-of-the-art performance on multiple benchmarks. We believe SwinTrack further shows the potential of Transformer and expect it to serve as a baseline for future research.
+
+# 2 Related Work
+
+Siamese Tracking. The Siamese tracking methods formulate tracking as a matching problem and aim to offline learn a generic matching function for this task. The seminal method of [1] introduces a fully convolutional Siamese network for tracking and shows a good balance between accuracy and speed. To improve Siamese tracking in handling scale variation, the work of [20] incorporates the region proposal network (RPN) [27] into the Siamese network and proposes the anchor-based tracker, showing higher accuracy with faster speed. Later, numerous extensions have been presented to improve Siamese tracking, including deeper backbone [19], multi-stage architecture [10, 11], anchor-free Siamese trackers [41], deformable attention [37], to name a few.
+
+![](images/ceb1ea2031ef2c2e53af6c2d1dc3d1053361e969b449e58a2d842f8df5d040e9.jpg)  
+Figure 2: Architecture of SwinTrack, which contains three parts including Transformer-based feature representation extraction, Transformer-based feature fusion and prediction head. Our SwinTrack is a simple and neat tracking framework without complex designs such as multi-scale features or temporal template updating, yet demonstrating state-of-the-art performance. Best viewed in color.
+
+Transformer in Vision. Transformer [30] originates from natural language processing (NLP) for machine translation and has been introduced to vision recently and shows great potential. The work of [3] first uses Transformer for object detection and achieved promising results. To explore the capability of Transformer in representation learning, the work of [7] applies Transformer to construct backbone network, and the resulting Vision Transformer (ViT) attains excellent performance compared to convolutional networks while requiring fewer training resources, which encourages many extensions upon ViT[29, 4, 38, 33, 23]. Among them, the Swin Transformer [23] has received extensive attention. It proposes a simple shifted window strategy to replace the fixed-patch method in ViT, which significantly improves efficiency and meanwhile demonstrates state-of-the-art results on multiple image tasks. Our work is inspired by Swin Transformer, but differently, we focus on the video task of visual tracking.
+
+Transformer in Tracking. Inspired by the success in other fields, researchers have leveraged Transformer for tracking. The method of [5] applies Transformer to enhance and fuse features in the Siamese tracking for improvement. The approach of [32] uses Transformer to exploit temporal features to improve tracking robustness. The work of [36] introduces a new transformer architecture dedicated to visual tracking, explores the Spatio-temporal Transformer by integrating the model updating operations into a Transformer module.
+
+Our SwinTrack is related to but significantly different from the above Transformer-based trackers. Specifically, the aforementioned methods mainly apply Transformer to fuse convolutional features and belong to the hybrid CNN-Transformer architecture. Unlike them, SwinTrack is a pure Transformerbased tracking architecture where both representation learning and feature fusion are realized with Transformer, enabling the exploration of better features for robust tracking.
+
+# 3 Tracking via Vision-Motion Transformer
+
+We present SwinTrack, a vision-motion integrated Transformer for object tracking, in Fig. 2. The proposed framework contains three main components, i.e., the Swin-Transformer backbone for feature extraction, the encoder-decoder network for mixing vision-motion cues, and the head network for localizing targets. In the following sections, we first shortly describe the Swin-Transformer backbone network, then elaborate on the proposed vision-motion encoder-decoder. Afterward, we give a discussion about our method and shortly describe the network head and training loss.
+
+# 3.1 Swin-Transformer for Feature Extraction
+
+The deep convolutional neural network has significantly improved the performance of trackers. Along with the advancement of trackers, the backbone network has evolved twice: AlexNet [17] and ResNet [14]. Swin-Transformer [23], in comparison to ResNet, can give a more compact feature representation and richer semantic information to assist succeeding networks in better localizing the target objects (demonstrate in the ablation study demonstrated in the ablation study), which is thus chosen for basic feature extraction in our model.
+
+Our tracker, following Siamese tracking framework [1], requires a pair of image patches as inputs, i.e., template image $\mathbf { z } \in \overline { { \mathbb { R } } } ^ { H _ { z } \times W _ { z } \times 3 }$ and search region image $\mathbf { x } \in \mathbb { R } ^ { \mathbf { \hat { H } } _ { x } \times W _ { x } \times 3 }$ . As in the typical SwinTransformer procedure, template and search region images are divided to non-overlapped patches and sent to the network, which generates template tokens (dubbed T-tokens) φ(z) ∈ R Hzs Wzs ×C a nd search region tokens (dubbed S-tokens) $\varphi ( \mathbf { x } ) \in \mathbb { R } ^ { \frac { H _ { x } } { s } \frac { W _ { x } } { s } \times C }$ . $s$ is the stride of the backbone network. Since there is no dimension projection in our model, $C$ is the hidden dimension of the whole model.
+
+# 3.2 Vision-Motion Representation Learning
+
+The essential step for matching-based visual tracking is injecting the template information into the search region. In our framework, we adopt an encoder to fuse the features from the template and the search region, meanwhile, a decoder is arranged to achieve vision-motion representation learning, as illustrated in Fig. 2.
+
+Encoder for fusing template and search tokens. The encoder contains a sequence of Transformer blocks where each consists of a multi-head self-attention (MSA) module and a feed-forward network (FFN). FFN contains a two-layers multi-layer perceptron (MLP), GELU activation layer is inserted after the first linear layer. Layer normalization (LN) is always conducted before every module (MSA and FFN). Residual connection is applied to MSA and FFN modules.
+
+Before feeding the features into the encoder, the template and search region tokens are concatenated along spatial dimensions to generate a mixing representation $\mathbf { f } _ { m }$ . For each block, the MSA module computes self-attention over mixing union representation, which equals to separately conducting self-attention on T-tokens/S-tokens and meanwhile performing cross-attention between T-tokens and S-tokens, but more efficient. FFN refines the features generated by MSA. When the tokens get out of the encoder, a de-concatenation operation is arranged to decouple the template and search region tokens. The process of encoder can be expressed as:
+
+$$
+\begin{array} { r l } & { \mathbf { f } _ { m } ^ { 1 } = \mathrm { C o n c a t } ( \boldsymbol { \varphi } ( \mathbf { z } ) , \boldsymbol { \varphi } ( \mathbf { x } ) ) } \\ & { \qquad \cdots \cdot } \\ & { \mathbf { f } _ { m } ^ { l ^ { \prime } } = \mathbf { f } _ { m } ^ { l } + \mathrm { M S A } ( \mathrm { L N } ( \mathbf { f } _ { m } ^ { l } ) ) } \\ & { \mathbf { f } _ { m } ^ { l + 1 } = \mathbf { f } _ { m } ^ { l ^ { \prime } } + \mathrm { F F N } ( \mathrm { L N } ( \mathbf { f } _ { m } ^ { l ^ { \prime } } ) ) } \\ & { \qquad \cdots } \\ & { \mathbf { f } _ { z } ^ { L } , \mathbf { f } _ { x } ^ { L } = \mathrm { D e C o n c a t } ( \mathbf { f } ^ { L } ) , } \end{array}
+$$
+
+where $l$ denotes the $l$ -th layer and $L$ denotes the number of blocks.
+
+Decoder for fusing vision and motion information. Before describing the architecture of decoder, we first detail how to generate a motion token (dubbed M-token). Motion token is the embedding of the historical trajectory of the target object. The past object trajectory is represented as a set of target object box coordinates, $T = \left\{ \Phi _ { 1 } , \Phi _ { 2 } , . . . , \Phi _ { t } \right\}$ , where $t$ represents the frame index, o is the bounding box of target object. o is defined by the top-left and bottom-right corners of the target object, denotes as $\Phi _ { t } ^ { - } = \bigl ( o _ { t } ^ { - x _ { 1 } } , o _ { t } ^ { y _ { 1 } } , o _ { t } ^ { x _ { 2 } } , o _ { t } ^ { y _ { 2 } } \bigr )$ . For flexible modeling, a sampling process is required to ensure the following properties: 1) fixed length, 2) focusing on the latest trajectories and 3) reducing redundancy. In our method, we sample object trajectory as:
+
+$$
+\begin{array} { r } { \mathcal { T } = \big \{ \Phi _ { s ( 1 ) } , \Phi _ { s ( 2 ) } , . . . , \Phi _ { s ( n ) } \big \} , \mathrm { w h e r e } s ( i ) = m a x ( t - i \times \Delta , 1 ) , } \end{array}
+$$
+
+$n$ is the number of sampled object trajectories, $\Delta$ is the fixed sampling interval. For Siamese tracker, the search region is cropped from the input image. In detail, a cropping with resizing operation can be used to describe the process. Giving the point in the input image as $\left( \mathbf { x } _ { i } , \mathbf { y } _ { i } \right)$ , the corresponding point in the search region as $\left( \mathbf { x } _ { o } , \mathbf { y } _ { o } \right)$ , we can formulate the cropping process employed in pre-processing of the Siamese Tracker as $\mathbf { x } _ { o } = ( \mathbf { x } _ { i } - i _ { x } ) s _ { x } + o _ { x }$ and ${ \bf y } _ { o } = ( { \bf y } _ { i } - i _ { y } ) s _ { y } + o _ { y }$ , where $( i _ { x } , i _ { y } )$ is the center of the cropping window in the input image, $( s _ { x } , s _ { y } )$ is the scaling factor, $( o _ { x } , o _ { y } )$ is the center of cropped and scaled window in the search region. We apply the same transformation on the sampled object trajectory to make the object trajectory invariant to the cropping, denoting $\bar { \mathcal { T } } = \{ \bar { \mathbb { D } } _ { s ( 1 ) } , \bar { \mathbb { D } } _ { s ( 2 ) } , . . . , \bar { \mathbb { D } } _ { s ( n ) } \}$ as the result.
+
+Then, to embed the transformed object trajectory into the network, we adopt four embedding matrices to embed the elements in box coordinates separately. We denotes the embedding matrix as W ∈ R(g+1)×d, $\^ \mathrm { g }$ controls the embedding granularity of the object trajectory, $d$ is the size of each embedding vector. The last entry of the embedding matrix is used as the padding vector, indicating an invalid state, like object absence or out of the search region. Thus, we normalize the sampled target object box coordinates in the range $[ 1 , \mathbb { g } ]$ , and quantize to integers to get the index of embedding vector:
+
+$$
+\begin{array} { r l } & { \hat { T } = \{ \hat { \mathbb { O } } _ { s ( 1 ) } , \hat { \mathbb { O } } _ { s ( 2 ) } , . . . , \hat { \mathbb { O } } _ { s ( n ) } \} , } \\ & { \mathrm { \it ~ \gamma ~ \mathrm { h e r e } ~ } \hat { \mathbb { O } } _ { s ( i ) } = [ \mathrm { n } ( \bar { \mathbb { O } } _ { s ( i ) } ^ { x _ { 1 } } , w ) , \mathrm { n } ( \bar { \mathbb { O } } _ { s ( i ) } ^ { y _ { 1 } } , h ) , \mathrm { n } ( \bar { \mathbb { O } } _ { s ( i ) } ^ { x _ { 2 } } , w ) , \mathrm { n } ( \bar { \mathbb { O } } _ { s ( i ) } ^ { y _ { 2 } } , h ) ] , } \\ & { \mathrm { \it ~ \mathrm { \it ~ \hat { \Omega } ~ } } \mathrm { \ n } ( o , l ) = \left\{ \begin{array} { l l } { \mathrm { \Delta } [ \frac { o } { l } \times \mathbb { g } ] } & { \mathrm { i f ~ } \mathrm { v a l i d } , } \\ { \mathbb { g } + 1 } & { \mathrm { e l s e } , } \end{array} \right. } \end{array}
+$$
+
+$( w , h )$ is the size of search region feature map.
+
+Finally, the motion token $\mathbf { E } _ { m o t i o n } ~ \in ~ \mathbb { R } ^ { 1 \times d }$ is given by a concatenation of all box coordinate embedding of the sampled object trajectory. FLOPs is negligible because the construction of motion token is just a composition of embedding lookups and token concatenation.
+
+The decoder consists of a multi-head cross-attention(MCA) module and a feed-forward network(FFN). The decoder takes the outputsvision-motion representation $\mathbf { f } _ { v m } \in \mathbb { R } ^ { \frac { H _ { x } } { s } \times \frac { W _ { x } } { s } \times C }$ d the motion token as input, generating tof by computing cross-attention over $\mathbf { f } _ { x } ^ { L }$ finaland $\mathrm { C o n c a t } ( { \bf E } _ { m o t i o n } , { \bf f } _ { z } ^ { L } , { \bf f } _ { x } ^ { L } )$ . The decoder is akin to a layer in the encoder, except that the correlation between the template tokens and the search tokens is dropped since we do not need to update the features from the template image in the last layer. The process of the decoder is formulated as:
+
+$$
+\begin{array} { r l } & { \mathbf { f } _ { m } ^ { D } = \mathrm { C o n c a t } ( \mathbf { E } _ { m o t i o n } , \mathbf { f } _ { z } ^ { L } , \mathbf { f } _ { x } ^ { L } ) } \\ & { \mathbf { f } _ { v m } ^ { \prime } = \mathbf { f } _ { x } ^ { L } + \mathrm { M C A } ( \mathrm { L N } ( \mathbf { f } _ { x } ^ { L } ) , \mathrm { L N } ( \mathbf { f } _ { m } ^ { D } ) ) } \\ & { \mathbf { f } _ { v m } = \mathbf { f } _ { v m } ^ { \prime } + \mathrm { F F N } ( \mathrm { L N } ( \mathbf { f } _ { v m } ^ { \prime } ) ) . } \end{array}
+$$
+
+$\mathbf { f } _ { v m }$ will feed to the head network to generate a classification response map and a bounding box regression map.
+
+Positional encoding. Transformer requires a positional encoding to identify the position of the current processing token[30] because the self-attention module is permutation-invariance. We adopt the untied positional encoding [16] as our positional encoding method. The untied positional encoding enhances the expressiveness of the model through untie the positional embeddings from token embeddings with an isolated positional embedding matrix. It also considers the case of special tokens, like the motion token in this paper. We generalize the untied positional encoding to multi-dimensions multi-sources data to comply with concatenated-based fusion in our tracker. See the appendix for the details.
+
+# 3.3 Discussion
+
+Why concatenated attention? To simplify the description, we call the method described above concatenation-based fusion. To fuse and process features from multiple sources, it is intuitive to perform self-attention on the feature from each source separately and then compute cross-attention across features from different sources. We call this method cross-attention-based fusion. Transformer makes fewer assumptions about the spatial structure of data, which provides great modeling flexibility.
+
+In comparison to cross-attention-based fusion, concatenation-based fusion can save computation cost through operation sharing and reduce model parameters through weight sharing. From the perspective of metric learning, weight sharing is an essential design to ensure the metric between two branches of data is symmetric. Through concatenation-based fusion, we implement this property not only in the feature extraction stage but also in the feature fusion stage. In general, concatenation-based fusion improves both efficiency and performance.
+
+Why not window-based self/cross-attention? Since we select stage 3 of the Swin-Transformer as the output, the number of tokens involved is significantly reduced, window-based attention cannot save too many FLOPs. Furthermore, considering the extra latency introduced by the window partition and window reverse operations, window-based attention may even be the slower one.
+
+Why not a query-based decoder? Derivated from vanilla Transformer decoder, many transformerbased models in vision tasks leverage a learnable query to extract the desired objective features from the encoder, like object queries in [3], target query in [36]. However, in our experiment, a query-based decoder suffers from slow convergence and inferior performance. Most Siamese trackers [20, 35, 13] formulate tracking as a foreground-background classification problem, which can better exploit the background information. The vanilla Transformer decoder is a generative model, the generative approaches are considered not suitable for the classification tasks. In another aspect, learning a general target query for any kind of object might cause a bottleneck. In terms of vanilla Transformer encoder-decoder architecture, SwinTrack is an "encoder" only model. Furthermore, quite a little domain knowledge can be easily applied on a classic Siamese tracker to improve the performance, like introducing the smooth movement assumption by using Hanning penalty window on the response map.
+
+Are other forms of motion token feasible? Other forms to construct motion token are possible, such as constructing motion token by summing up the past box coordinate embeddings or representing past object trajectories by one token per box. In our early experiments, we find that the proposed motion token is more effective with the best performance. Summing up the past box coordinate embeddings may result in over-parameterization on the coordinate embeddings. While adding temporal motion representation along with visual features to the single-layer decoder in a multi-token form is ineffective, precise temporal modeling may be required in this form.
+
+# 3.4 Head and Loss
+
+Head. The head network is split into two branches: classification and bounding box regression. Each of them is a three-layer perceptron. And both of them receives the feature map from the decoder as input to predict the classification response map $r _ { c l s } \in \mathbb { R } ^ { ( H _ { x } \times W _ { x } ) \times 1 }$ and bounding box regression map $\bar { r _ { r e g } } \in \mathbb { R } ^ { ( H _ { x } \times W _ { x } ) \times 4 }$ , respectively.
+
+Classification loss. In classification branch, we employ the $I o U$ -aware classification score as the training target and the varifocal loss [39] as the training loss function. IoU-aware design has been very popular recently, but most works consider IoU prediction as an auxiliary branch to assist classification or bounding box regression [41, 2, 35]. To remove the gap between different prediction branches, [39] and [21] replace the hard classification target from the ground-truth value, (i.e., 1 for positive samples, 0 for negative samples), to the IoU between the predicted bounding box and the ground-truth one, which is named the $I o U .$ -aware classification score (IACS). IACS can help the model select a more accurate bounding box prediction candidate from the pool by trying to predict the quality of the bounding box prediction in another branch at the same position. Along with the IACS, the varifocal loss was proposed in [39] to help the IACS approach outperform other IoU-aware designs.
+
+The classification loss can be formulated as:
+
+$$
+\mathbb { L } _ { c l s } = \mathbb { L } _ { \mathrm { V F L } } ( p , \mathrm { I o U } ( b , \hat { b } ) ) ,
+$$
+
+where $p$ denotes the predicted IACS, $b$ denotes the predicted bounding box, and $\hat { b }$ denotes the ground-truth bounding box.
+
+Regression loss. For bounding box regression, we employ the generalized IoU loss[28]. The regression loss function can be formulated as:
+
+$$
+\mathbb { L } _ { r e g } = \sum _ { j } \mathbb { 1 } _ { \left\{ \mathrm { I o U } ( b _ { j } , \hat { b } ) > 0 \right\} } [ p \mathbb { L } _ { \mathrm { G I o U } } ( b _ { j } , \hat { b } ) ] .
+$$
+
+The GIoU loss is weighted by $p$ to emphasize the high classification score samples. The training signals from the negative samples are ignored.
+
+# 4 Experiments
+
+# 4.1 Implementation
+
+Model. We design two variants of SwinTrack with different configurations as follows:
+
+• SwinTrack-T-224. Backbone: Swin Transformer-Tiny [23], pretrained with ImageNet-1k; Template size: $[ 1 1 2 \times 1 1 2 ]$ ; Search region size: $[ 2 2 4 \times 2 2 4 ]$ ; $C = 3 8 4$ ; $N = 4$ ;   
+• SwinTrack-B-384. Backbone: Swin Transformer-Base [23], pretrained with ImageNet-22k; Template size: $[ 1 9 2 \times 1 9 2 ]$ ; Search region size: $[ 3 8 4 \times 3 8 4 ]$ ; $C = 5 1 2$ ; $N = 8$ ;
+
+where $C$ and $N$ are the channel number of the hidden layers in the first stage of Swin Transformer and the number of encoder blocks in feature fusion, respectively. In all variants, we use the output after the third stage of Swin Transformer for feature extraction. Thus, the backbone stride $s$ is 16.
+
+For motion token, the number of sampled object trajectory $n$ is set to 16, the fixed sampling interval $\Delta$ is set to 15. If the frame rate of the video sequence is available, the sampling interval is adjusted according to the frame rate. Suppose the frame rate is $\mathbb { f }$ , the new sampling interval is getting by $\mathrm { { \frac { \Delta } { 3 0 } f } }$ , 30 fps is the standard frame rate we assumed. g, which controls the embedding granularity, is set to the same size as the search region feature map, like 14 for SwinTrack-T-224, and 24 for SwinTrack-B-384. For the model for GOT-10k sequences, $n$ is set to 8, $\Delta$ is set to 8, and no frame rate adjustment is applied.
+
+Training. We train SwinTrack using the training splits of LaSOT [9], TrackingNet [26], GOT-10k [15] (1,000 videos are removed following [36] for fair comparison) and COCO 2017 [22]. In addition, we report the results of SwinTrack-T-224 and SwinTrack-B-384 with GOT-10k training split only to follow the protocol described in [15].
+
+The model is optimized with AdamW [24], with a learning rate of 5e-4, and a weight decay of 1e-4. The learning rate of the backbone is set to 5e-5. We train the network on 8 NVIDIA V100 GPUs for 300 epochs with 131,072 samples per epoch. The learning rate is dropped by a factor of 10 after 210 epochs. A 3-epoch linear warmup is applied to stabilize the training process. DropPath [18] is applied on the backbone and the encoder with a rate of 0.1. For the models trained for the GOT-10k evaluation protocol, to prevent over-fitting, the training epoch is set to 150, and the learning rate is dropped after 120 epochs.
+
+For the motion token, the object trajectory for the Siamese training pair is generated with the method described above. The frames that object annotated as absent or out of the video sequence are marked as invalid, the corresponding box coordinates set to $- \infty$ . Since the coarse granularity of the coordinate embedding in our setting is already can be seen as an augmentation of historical object trajectory, no additional data augmentation is applied.
+
+Inference. We follow the common procedures for Siamese network-based tracking [1]. The template image is cropped from the first frame of the video sequence. The target object is in the center of the image with a background area factor of 2. The search region is cropped from the current tracking frame, and the image center is the target center position predicted in the previous frame. The background area factor for the search region is 4.
+
+Our SwinTrack takes the template image and search region as inputs and output classification map $r _ { c l s }$ and regression map $r _ { r e g }$ . To utilize positional prior in tracking, we apply hanning window penalty on $r _ { c l s }$ , and the final classification map $r _ { c l s } ^ { \prime }$ is obtained via $\boldsymbol { r } _ { c l s } ^ { \prime } = ( 1 - \gamma ) \times \boldsymbol { r } _ { c l s } + \gamma \times \boldsymbol { h }$ , where $\gamma$ is the weight parameter and $h$ is the Hanning window with the same size as $r _ { c l s }$ . The target position is determined by the largest value in $r _ { c l s } ^ { \prime }$ and the scale is estimated based on the corresponding regression results in $r _ { r e g }$ .
+
+For the motion token, the predicted confidence score and bounding box are collected on the fly. A confidence threshold $\theta _ { c o n f }$ is applied, if the confidence score given by the classification branch of the head is lower than the threshold, the target object in the current frame is marked as lost by setting the collected bounding box to $- \infty$ . $\theta _ { c o n f }$ is set to 0.4 for LaSOT, the rests are set to 0.3.
+
+Table 1: Experiments and comparisons on five benchmarks: LaSOT, $\mathrm { L a S O T _ { e x t } }$ , TrackingNet, GOT10k and TNL2k.   
+
+<table><tr><td rowspan="2">Tracker</td><td colspan="2">LaSOT[9]</td><td colspan="2">LaSOText [8]</td><td colspan="2">TrackingNet [26]</td><td colspan="3">GOT-10k [15]</td><td colspan="2">TNL2k [34]</td></tr><tr><td>SUC</td><td>P</td><td>SUC</td><td>P</td><td>SUC</td><td>P</td><td>AO</td><td>SR0.5</td><td>SR0.75</td><td>SUC</td><td>P</td></tr><tr><td>C-RPN [10]</td><td>45.5</td><td>44.3</td><td>27.5</td><td>32.0</td><td>66.9</td><td>61.9</td><td>1</td><td>1</td><td>1</td><td>-</td><td>-</td></tr><tr><td>SiamPRN++ [19]</td><td>49.6</td><td>49.1</td><td>34.0</td><td>39.6</td><td>73.3</td><td>69.4</td><td>51.7</td><td>61.6</td><td>32.5</td><td>41.3</td><td>41.2</td></tr><tr><td>Ocean [41]</td><td>56.0</td><td>56.6</td><td>-</td><td>-</td><td>1</td><td>-</td><td>61.1</td><td>72.1</td><td>47.3</td><td>38.4</td><td>37.7</td></tr><tr><td>DiMP [2]</td><td>56.9</td><td>56.7</td><td>39.2</td><td>45.1</td><td>74.0</td><td>68.7</td><td>61.1</td><td>71.7</td><td>49.2</td><td>44.7</td><td>43.4</td></tr><tr><td>LTMU [6]</td><td>57.2</td><td>57.2</td><td>41.4</td><td>47.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>1</td><td>48.5</td><td>47.3</td></tr><tr><td>SiamR-CNN[31]</td><td>64.8</td><td>1</td><td>1</td><td>1</td><td>81.2</td><td>80.0</td><td>64.9</td><td>72.8</td><td>59.7</td><td>52.3</td><td>52.8</td></tr><tr><td>STMTrack [12]</td><td>60.6</td><td>63.3</td><td>-</td><td>-</td><td>80.3</td><td>76.7</td><td>64.2</td><td>73.7</td><td>57.5</td><td>-</td><td>-</td></tr><tr><td>AutoMatch [40]</td><td>58.3</td><td>59.9</td><td>37.6</td><td>43.0</td><td>76.0</td><td>72.6</td><td>65.2</td><td>76.6</td><td>54.3</td><td>-</td><td>-</td></tr><tr><td>TrDiMP [32]</td><td>63.9</td><td>61.4</td><td>1</td><td>1</td><td>78.4</td><td>73.1</td><td>67.1</td><td>77.7</td><td>58.3</td><td>-</td><td>-</td></tr><tr><td>TransT[5]</td><td>64.9</td><td>69.0</td><td>-</td><td>1</td><td>81.4</td><td>80.3</td><td>67.1</td><td>76.8</td><td>60.9</td><td>51.0</td><td>-</td></tr><tr><td>STARK [36]</td><td>67.1</td><td></td><td></td><td>=</td><td>82.0</td><td>-</td><td>68.8</td><td>78.1</td><td>64.1</td><td>-</td><td>-</td></tr><tr><td>KeepTrack [25]</td><td>67.1</td><td>70.2</td><td>48.2</td><td>-</td><td>-</td><td>-</td><td>1</td><td>-</td><td>1</td><td>-</td><td>-</td></tr><tr><td>SwinTrack-T-224</td><td>67.2</td><td>70.8</td><td>47.6</td><td>53.9</td><td>81.1</td><td>78.4</td><td>71.3</td><td>81.9</td><td>64.5</td><td>53.0</td><td>53.2</td></tr><tr><td>SwinTrack-B-384</td><td>71.3</td><td>76.5</td><td>49.1</td><td>55.6</td><td>84.0</td><td>82.8</td><td>72.4</td><td>80.5</td><td>67.8</td><td>55.9</td><td>57.1</td></tr></table>
+
+# 4.2 Comparisons to State-of-the-arts
+
+We conduct experiments and compare SwinTrack with SoTA trackers on five benchmarks.
+
+LaSOT. LaSOT [9] consists of 280 videos for test. Tab. 1 shows the results and comparisons with SoTAs. From Tab. 1, we can observe that SwinTrack-T-224 with light architecture reaches SoTA performance with 0.672 SUC and 0.708 PRE scores, which is competitive compared with other Transformer-based trackers, including STARK-ST101 (0.671 SUC score) and TransT (0.649 SUC), and other trackers using complicated designs such as KeepTrack (0.671 SUC) and SiamR-CNN (0.648 SUC score). With a larger backbone and input size, our strongest variant SwinTrack-B-384 sets a new record with 0.713 SUC score, surpassing START-ST101 and KeepTrack by 4.2 absolute percentage points.
+
+$\mathbf { L a S O T } _ { \mathrm { e x t } }$ . The recent $\mathrm { L a S O T _ { \mathrm { e x t } } }$ [8] is an extension of LaSOT by adding 150 extra videos. These new sequences are challenging as many similar distractors cause difficulties for tracking. The results of our tracker related to this dataset are an average of three times. KeepTrack uses a complex association technique to handle distractors and achieves a promising 0.482 SUC score as in Tab. 1. Compared with complicated KeepTrack, SwinTrack-T-224 is simple and neat, yet shows comparable performance with 0.476 SUC score. In addition, due to complicated design, KeepTrack runs at less than $2 0 f p s$ , while SwinTrack-T-224 runs in 98 fps, $5 \times$ faster than KeepTrack. When using a larger model, SwinTrack-B-384 shows the best performance with 0.491 SUC score.
+
+TrackingNet. We evaluate different trackers on the test set of TrackingNet [26]. From Tab. 1, we observe that our SwinTrack-T-224 achieves a comparable result with 0.811 SUC score. Using a larger model and input size, SwinTrack-B-384 obtains the best performance with 0.840 SUC score, better than STARK-ST101 with 0.820 SUC score and TransT with 0.814 SUC score.
+
+GOT-10k. GOT-10k [15] offers 180 videos for test and it requires trackers to be trained using GOT-10k train split only. From Tab. 1, we see that SwinTrack-B-384 achieves the best mAO of 0.724, and SwinTrack-T-224 obtains a mAO of 0.713. Both models outperform other Transformer-based counterparts significantly, including START-ST101 (0.688 mAO), TransT (0.671 mAO) and TrDiMP (0.671 mAO).
+
+TNL2k. TNL2k [34] is a newly released tracking dataset with 700 videos for test. As reported in Tab. 1, both models surpass the others. SwinTrack-B-384 set a new state-of-the-art with 0.559 SUC score.
+
+Table 2: Comparison on running speed and # parameters with other Transformer-based trackers.   
+
+<table><tr><td>Tracker</td><td>Speed (fps)</td><td>MACs² (G)</td><td>Params (M)</td></tr><tr><td>TrDiMP [32]</td><td>26</td><td>=</td><td>1</td></tr><tr><td>TransT[5]</td><td>50</td><td>=</td><td>23</td></tr><tr><td>STARK-ST50 [36]</td><td>42</td><td>10.9</td><td>24</td></tr><tr><td>STARK-ST101 [36]</td><td>32</td><td>18.5</td><td>42</td></tr><tr><td>SwinTrack-T-224</td><td>98</td><td>6.4</td><td>23</td></tr><tr><td>SwinTrack-B-384</td><td>45</td><td>69.7</td><td>91</td></tr></table>
+
+Table 3: Ablation experiments of SwinTrack on four benchmarks. The experiments are conducted on SwinTrack-T-224 without the motion token. ❶: baseline method, i.e., SwinTrack-T-224 without motion token; $\otimes$ : replacing Transformer backbone in the baseline method with ResNet-50; ❸: replacing our feature fusion with cross attention-based fusion in the baseline method; $\bullet$ : replacing the decoder in baseline with a target query-based; $\pmb { \ 6 }$ : replacing united positional encoding with absolute sine position encoding in the baseline method; $\pmb { \circledcirc }$ : replacing the IoU-aware classification loss with the plain binary cross entropy loss; $\pmb { \ 6 }$ : removing the Hanning penalty window in the baseline method inference.   
+
+<table><tr><td></td><td>LaSOT SUC (%)</td><td>LaSOText SUC (%)</td><td>TrackingNet SUC (%)</td><td>GOT-10k3 mA0 (%)</td><td>Speed fps</td><td>Params M</td></tr><tr><td>1</td><td>66.7</td><td>46.9</td><td>80.8</td><td>70.9</td><td>98</td><td>22.7</td></tr><tr><td>②</td><td>64.2</td><td>41.8</td><td>79.5</td><td>68.2</td><td>121</td><td>20.0</td></tr><tr><td>③</td><td>66.6</td><td>45.4</td><td>80.2</td><td>69.3</td><td>72</td><td>34.6</td></tr><tr><td>4</td><td>66.6</td><td>43.2</td><td>79.6</td><td>69.0</td><td>91</td><td>25.3</td></tr><tr><td>5</td><td>65.7</td><td>45.0</td><td>80.0</td><td>70.0</td><td>103</td><td>21.6</td></tr><tr><td>6</td><td>66.2</td><td>46.7</td><td>79.4</td><td>68.2</td><td>98</td><td>22.7</td></tr><tr><td>0</td><td>65.7</td><td>46.0</td><td>80.0</td><td>69.6</td><td>98</td><td>22.7</td></tr></table>
+
+Efficiency comparison. We report the comparisons of SwinTrack with other Transformer-based trackers in terms of efficiency and complexity. As displayed in Tab. 2, SwinTrack-T-224 with a small model runs the fastest with a speed of 98 fps. Especially, compared with STARK-ST101 and STARK-ST50 with $3 2 f p s$ and $4 2 f p s$ , SwinTrack-T-224 is $3 \times$ and $2 \times$ faster. Despite using a larger model, our SwinTrack-B-384 is still faster than STARK-ST101 and STARK-ST50.
+
+# 4.3 Ablation Experiment
+
+Comparison with ResNet backbone. We compare the Swin-Transformer backbone with popular ResNet-50 [14]. As shown in Tab. 3 ( $\pmb { \theta }$ vs. $\pmb { \varrho }$ ). The Swin Transformer backbone significantly boosts the performance by $2 . 5 \%$ SUC score in LaSOT, $5 . 1 \%$ SUC score in $\mathrm { L a S O T _ { e x t } }$ . The result shows that the strong appearance modeling capability provided by the Swin Transformer plays a crucial role.
+
+Feature fusion. As displayed in Tab. 3 $\pmb { \theta }$ vs. $\pmb { \Theta }$ ), compared with the concatenation-based fusion, the cross attention-based fusion runs at a slower speed, occupies much more memory, and also has an inferior performance on all datasets. Slower speed can be due to the latency brought by the extra operations. The parameter-sharing strategy not only just reduces the number of parameters but also benefits metric learning.
+
+Comparison with the query-based decoder. Queries is commonly adopted in the decoder of Transformer network in vision tasks, e.g. object query [3] and target query [36]. Nevertheless, our empirical results in Tab. 3 $\pmb { \ 0 }$ vs. $\pmb { \varrho }$ ) show that a target query-based decoder degrades the tracking performance on all benchmarks, even with $2 \times$ training pairs. As discussed, one possible reason is the generative model is not suitable for classification. Besides, learning a general target query for any kind of object may also be difficult.
+
+Position encoding. We compare the united positional encoding used in SwinTrack and the original absolute position encoding in Transformer [30]. Notice, We make a little modification to the original absolute position encoding: Except for the 2D embedding, the index of token source (e.g. 1 for the tokens from the template patch, 2 for the tokens from the search region patch) is also embedded. As shown in Tab. 3 (❶ vs. $\bullet$ ), our method with united positional encoding obtains improvements with 0.8-1.9 absolute percentage points on the benchmarks with negligible loss in speed (98 vs. 103).
+
+Table 4: Ablation experiments on our proposed motion token on the tracking performance on four benchmarks. The experiments are conducted on SwinTrack-T-224. ❶: SwinTrack-T-224; ❷: SwinTrack-B-384; $\otimes$ : SwinTrack-T-224 without motion token; ❹: SwinTrack-B-384 without motion token; $\pmb { \ 6 }$ : replacing the motion token in SwinTrack-T-224 with a learnable embedding token.   
+
+<table><tr><td>SUC (%)</td><td>LaSOT</td><td>LaSOText SUC (%)</td><td>TrackingNet SUC (%)</td><td>GOT-10k mA0 (%)</td><td>Speed fps</td></tr><tr><td>1</td><td>67.2</td><td>47.6</td><td>81.1</td><td>71.3</td><td>96</td></tr><tr><td>②</td><td>71.3</td><td>49.1</td><td>84.0</td><td>72.4</td><td>45</td></tr><tr><td>③</td><td>66.7</td><td>47.0</td><td>80.8</td><td>70.0</td><td>98</td></tr><tr><td>4</td><td>70.2</td><td>48.5</td><td>84.0</td><td>70.7</td><td>45</td></tr><tr><td>6</td><td>66.3</td><td>45.2</td><td>81.2</td><td>70.0</td><td>96</td></tr></table>
+
+Loss function. From Tab. 3 $( \bullet \nu . \bullet )$ , we observe that the model trained with varifocal loss significantly outperforms the one with binary cross entropy (BCE) loss without loss of efficiency. This result indicates that the varifocal loss can assist the classification branch of the head to generate an IoU-aware response map, and thus help the tracker to improve the tracking performance.
+
+Post processing. One may wonder with highly discriminative Transformer architecture and IoUaware classification loss does the hanning penalty window is still functional, which introduces a strong smooth movement assumption. In the experiments, we remove the hanning penalty window in post-processing, as shown in Tab. 3 $( \pmb { \mathbb { 0 } } \nu s . \pmb { \mathbb { \otimes } } )$ , the performance is dropped by 1.0 SUC for LaSOT, $1 . 3 \mathrm { \ A O }$ for GOT-10k in absolute percentage, and less than $1 \%$ in the SUC metric of other datasets. This suggests that the strong smooth movement assumption is still applicable for our tracker. But compared with the former Transformer-based tracker [5], the performance gap between with and without penalty window post-processing is narrowing.
+
+Effectiveness of motion token. We study the effectiveness of the motion token by conducting comparison experiments. As shown in Tab. 4 $\pmb { \mathbb { \otimes } }$ vs. $\pmb { \otimes }$ and $\pmb { \varrho }$ vs. $\pmb { \varrho }$ ), the models with motion token outperforms the models without motion token on all datasets, especially on $\mathrm { L a S O T _ { e x t } }$ and GOT-10k. The results indicate that the motion token can assist the tracker to handle hard similar distractors in $\mathrm { L a S O T _ { e x t } }$ and stabilize the short-term tracking like the sequences in GOT-10k test set. We also study whether the effectiveness of the motion token is simply from the extra embedding vector. We set up an experiment as in Tab. 4 $( \pmb { \Theta } )$ , which replaces the motion token with a learnable embedding token. The result shows that the extra embedding vector has negative impacts indicating the effectiveness of the embedding of object trajectory.
+
+# 5 Conclusion
+
+In this work, we present SwinTrack, a simple and strong baseline for Transformer tracking. In SwinTrack, both representation learning and feature fusion are implemented with the attention mechanism. Extensive experiments demonstrate the effectiveness of such architecture. Besides, we propose the motion token to enhance the robustness of the tracker by providing the historical object trajectory, showing the flexibility of the Transformer model in architectural design. With the power of sequence-to-sequence model architecture, a context-rich tracker is possible, and more contextual cues can be incorporated. Finally, We hope this work can inspire and facilitate future research.
+
+# Acknowledgments and Disclosure of Funding
+
+This work is supported by Peng Cheng Laboratory Research Project No. PCL2021A07. Heng Fan and his employer receive no financial support for the research, authorship, and/or publication of this article.
+
+# References
+
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parse/dev/Ih2bG6h1r4S/Ih2bG6h1r4S.md

@@ -0,0 +1,485 @@
+# ATLAS: Universal Function Approximator for Memory Retention
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Artificial neural networks (ANNs), despite their universal function approximation   
+2 capability and practical success, are subject to catastrophic forgetting. Catastrophic   
+3 forgetting refers to the abrupt unlearning of a previous task when a new task is   
+4 learned. It is an emergent phenomenon that plagues ANNs and hinders continual   
+5 learning. Existing universal function approximation theorems for ANNs guarantee   
+6 function approximation ability, but seldom touch on the model details and do not   
+7 predict catastrophic forgetting. This paper presents a novel universal approximation   
+8 theorem for multi-variable functions using only single-variable functions and   
+9 exponential functions. Furthermore, we present ATLAS—a novel ANN architecture   
+10 based on the exponential approximation theorem and B-splines. It is shown that   
+11 ATLAS is a universal function approximator capable of memory retention and,   
+12 therefore, continual learning. The memory retention of ATLAS is imperfect,   
+13 with some off-target effects during continual learning, but it is well-behaved and   
+14 predictable. An efficient implementation of ATLAS is provided. Experiments   
+15 are conducted to evaluate both the function approximation and memory retention   
+16 capabilities of ATLAS.
+
+# 17 1 Introduction
+
+18 Catastrophic forgetting [7, 13, 23] is an emergent phenomenon where a machine learning model   
+19 such as an artificial neural network (ANN) learns a new task, and the subsequent parameter updates   
+20 interfere with the model’s performance on previously learned tasks. Catastrophic forgetting is also   
+21 called catastrophic interference [19]. If an ANN cannot effectively learn many tasks, it has limited   
+22 utility in the context of continual learning [9, 12]. Catastrophic forgetting is like learning to pick   
+23 up a cup, but simultaneously forgetting how to breathe. Even linear functions are susceptible to   
+24 catastrophic forgetting, as illustrated in Figure 1   
+25 The simple example of a linear regression model being susceptible to catastrophic forgetting might be   
+26 due to the non-linearity of the target function, noise, or parameter sharing across the input. Parameter   
+27 sharing is avoidable with piece-wise defined functions such as splines [27]. ANNs can be explained   
+28 in many ways; a useful analogy is to compare ANNs to very large lookup tables that store information.   
+29 Removing and updating values has off-target effects throughout the table or ANN.   
+30 Universal function approximation theorems are a cornerstone of machine learning, and prove that   
+31 ANNs can approximate any given continuous target function [10, 11, 15] under certain assumptions.   
+32 The theorems do not specify how to find an ANN with sufficient performance for problems in   
+33 practice. Gradient descent optimisation is the convention for finding/training neural networks, but   
+34 other optimisation and learning procedures exist [22]. ATLAS models trained with gradient descent   
+35 methods exhibit desirable properties. However, other optimisation techniques like evolutionary   
+36 algorithms may not elicit the same properties.   
+37 This paper introduces ATLAS—a novel universal function approximator based on B-splines that has   
+38 some intrinsic memory retention, even in the absence of other training and regularisation techniques.   
+39 ATLAS has well-behaved parameter gradients that are sparse, bounded and orthogonal between   
+40 input points that are far enough from each other. The accompanying representation and universal   
+41 approximation theorems are also provided.
+
+![](images/48a779adaf44eba98e2fe1a3bd4690138d9bcace728802639941cafdcd7a32b1.jpg)  
+Figure 1: A linear function is susceptible to catastrophic forgetting.
+
+# 42 2 Relevant Studies
+
+43 It is conjectured that overlapping representations in ANNs lead to catastrophic forgetting [12].   
+44 Catastrophic forgetting occurs when parameters necessary for one task change while training to meet   
+45 the objectives of another task [14, 20]. The least desirable strategy to mitigate catastrophic forgetting   
+46 is retraining a model over all tasks. Regularisation techniques like elastic weight consolidation (EWC)   
+47 have also been employed [14]. Data augmentation approaches such as rehearsal and pseudo-rehearsal   
+48 have also been employed [23]. Other ideas from optimal control theory in combination with dynamic   
+49 programming have also been applied to counteract catastrophic forgetting, with a cost functional   
+50 similar in form to the action integral from physics and Lagrangian mechanics [16].   
+51 Orthogonal Gradient Descent (OGD) is a training augmentation or optimisation technique that   
+52 modifies the gradient updates of subsequent tasks to be orthogonal to previous tasks [6, 1]. One   
+53 can describe data in terms of a distribution defined over the input space, target values, and time (the   
+54 order of data or tasks that are presented during training). OGD attempts to make gradient updates   
+55 orthogonal to each other over time. ATLAS, in contrast, possesses distal orthogonality, meaning that   
+56 if two inputs are far enough from each other in the input space, then corresponding gradient updates   
+57 will be orthogonal. A corollary of this is that if the data distribution between tasks shifts in the input   
+58 space, then the subsequent gradient updates will tend to be orthogonal. ATLAS does not use external   
+59 memory like OGD. Extensions of OGD include PCA-OGD, which compresses gradient updates into   
+60 principal components to reduce memory requirements [4]. The Neural Tangent Kernel (NTK) overlap   
+61 matrices, as discussed by Doan et al. [4], could be a useful tool for analysing ATLAS models.   
+62 The survey by Delange et al. [3] gives an extensive overview of continual learning to address   
+63 catastrophic forgetting. ATLAS is a model that implements parameter isolation, because of its use of   
+64 piece-wise defined splines. Particularly relevant to ATLAS is the work on scale of initialisation and   
+65 extreme memorisation [21]. Increasing the density of basis functions in ATLAS can lead to better   
+66 memorisation, and increases the scale of some parameters in ATLAS which may affect generalisation.   
+67 Pi-sigma neural networks use nodes that compute products instead of sums [26]. Pi-sigma neural   
+68 networks have some similarities with the global structure of ATLAS. B-splines, which form the basis   
+69 of ATLAS, have been applied for machine learning [5]. Scardapane et al. [25] investigated trainable   
+70 activation functions parameterised by splines. Uniform cubic B-splines have basis functions that are   
+71 translates of one another [2]. Uniform cubic B-splines have been tested for memory retention, and   
+72 ATLAS is an improvement on existing spline models [27].   
+73 B-splines, and by extension ATLAS, can be trained to fit lower frequency components, expanded and   
+74 trained again until a network is found with sufficient accuracy and generalisation, similar to other   
+75 techniques [17, 18]. It is not necessary to expand the capacity of an ATLAS model to learn new   
+76 tasks, as with some other approaches [24]. ATLAS does in practice demonstrate something akin to   
+77 "graceful forgetting" as discussed in Golkar et al. [8].   
+79 Vector quantities like $\vec { \bf x }$ are clearly indicated with a bar or arrow for legibility. Parameters, inputs,   
+80 functions etc. without a bar or arrow are scalar quantities like $S ( x )$ . Some scalar quantities with   
+81 indices are the scalar components of a vector like $x _ { j }$ or scalar parameters in the model like $\theta _ { i }$ . The   
+82 gradient operator that acts on a scalar function like $\vec { \nabla } _ { \vec { \theta } } A ( \vec { \bf x } )$ yields a vector-valued function $\vec { \nabla } _ { \vec { \theta } } A ( \vec { \bf x } )$   
+83 as is typical of multi-variable calculus.
+
+# 84 4 Exponential Representation Theorem
+
+85 Any continuous multi-variable function on a compact space can be uniformly approximated with   
+86 multi-variable polynomials by the Stone-Weierstrass Theorem. Let $\mathcal { T }$ denote an index set of tuples of   
+87 natural numbers including zero such that $i _ { j } \in \mathbb { N } ^ { 0 }$ for all $j \in \mathbb N$ with $i = ( i _ { 1 } , . . , i _ { n } ) \in \mathcal { T }$ and $a _ { i } \in \mathbb { R }$ .   
+88 Multi-variable polynomials can be represented as:
+
+$$
+y ( \vec { \bf x } ) = y ( x _ { 1 } , . . , x _ { n } ) = \sum _ { i \in \mathcal { I } } a _ { i } x _ { 1 } ^ { i _ { 1 } } x _ { 2 } ^ { i _ { 2 } } . . . x _ { n } ^ { i _ { n } } = \sum _ { i \in \mathcal { I } } a _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }
+$$
+
+89 Each monomial term $a _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }$ is a product of single-variable functions in each variable. It is   
+90 desirable to rewrite products as sums using exponentials and logarithms.
+
+91 Lemma 1. For any $a _ { i } \in \mathbb { R }$ , there exists $\gamma _ { i } > 0$ and $\beta _ { i } > 0$ , such that: $a _ { i } = \gamma _ { i } - \beta _ { i }$
+
+92 Theorem 1 (Exponential representation theorem). Any multi-variable polynomial function $y ( \vec { \bf x } )$   
+93 of $n$ variables over the positive orthant, can be exactly represented by continuous single-variable   
+94 functions $g _ { i , j } ( x _ { j } )$ and $h _ { i , j } ( x _ { j } )$ in the form:
+
+$$
+y ( \vec { \bf x } ) = \sum _ { i \in \mathcal { I } } \exp \bigl ( \Sigma _ { j = 1 } ^ { n } g _ { i , j } ( x _ { j } ) \bigr ) - \exp \bigl ( \Sigma _ { j = 1 } ^ { n } h _ { i , j } ( x _ { j } ) \bigr )
+$$
+
+Proof. Consider any monomial term 95 $a _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }$ with $a _ { i } \in \mathbb { R }$ , then by Lemma 1 there exist strictly 96 positive numbers $\gamma _ { i } > 0$ and $\beta _ { i } > 0$ , such that:
+
+$$
+\begin{array} { r l } & { a _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } = \gamma _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } - \beta _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } } \\ & { \phantom { a a a } = \exp \Bigl ( \log \Bigl ( \gamma _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } \Bigr ) \Bigr ) - \exp \Bigl ( \log \Bigl ( \beta _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } \Bigr ) \Bigr ) } \\ & { \phantom { a a a a } = \exp \Bigl ( \log ( \gamma _ { i } ) + \Sigma _ { j = 1 } ^ { n } \log \Bigl ( x _ { j } ^ { i _ { j } } \Bigr ) \Bigr ) - \exp \Bigl ( \log ( \beta _ { i } ) + \Sigma _ { j = 1 } ^ { n } \log \Bigl ( x _ { j } ^ { i _ { j } } \Bigr ) \Bigr ) } \end{array}
+$$
+
+97 The argument of each exponential function is a sum of single-variable functions and constants.   
+98 Without loss of generality, a set of single-variable functions can be defined such that:
+
+$$
+a _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } = \exp \bigl ( \Sigma _ { j = 1 } ^ { n } g _ { i , j } ( x _ { j } ) \bigr ) - \exp \bigl ( \Sigma _ { j = 1 } ^ { n } h _ { i , j } ( x _ { j } ) \bigr )
+$$
+
+Since this holds for any 99 $a _ { i } \Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }$ and all $i \in \mathcal { T }$ , it follows that:
+
+$$
+y ( \vec { \bf x } ) = \sum _ { i \in \mathcal { I } } \exp \bigl ( \Sigma _ { j = 1 } ^ { n } g _ { i , j } ( x _ { j } ) \bigr ) - \exp \bigl ( \Sigma _ { j = 1 } ^ { n } h _ { i , j } ( x _ { j } ) \bigr )
+$$
+
+101 This result is fundamental to the paper. Since every continuous function can be approximated with   
+102 multi-variable polynomials, it follows that every continuous function can be approximated with   
+103 positive and negative exponential functions. Single-variable function approximators are pivotal and   
+104 must be reconsidered. Universal function approximation can also be proven with the sub-algebra   
+105 formulation of the Stone-Weierstrass theorem, but it’s not as delightful and simple as the first   
+106 constructive proof given above.   
+108 Splines are piece-wise defined single-variable functions over some interval. Each sub-interval of a   
+109 spline is most often locally given by a low degree polynomial, even though the global structure is not   
+110 a low degree polynomial. B-splines are polynomial splines that are defined in a way that resembles   
+111 other basis function formulations [2]. Each single-variable function in ATLAS is approximated   
+112 with uniform cubic B-spline basis functions, shown in Figure 2. B-splines can approximate any   
+113 single-variable function, similar to using the Fourier basis. With uniform B-splines, each basis   
+114 function is scaled so that the unit interval is uniformly partitioned, as in Figure 2.
+
+![](images/028d13c2b6b7713311944022490eca8111753bed108284b61d20c89e04ca0f30.jpg)  
+Figure 2: If uniformly spaced B-splines are used, then each basis function has the same shape. This makes it possible to use the same activation function by scaling and translating the inputs. This is also true for different densities of uniform cubic B-splines.
+
+115 The activation function to implement B-splines is given by:
+
+$$
+S ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { 6 } x ^ { 3 } } & { 0 \leq x < 1 } \\ { \frac { 1 } { 6 } \left[ - 3 ( x - 1 ) ^ { 3 } + 3 ( x - 1 ) ^ { 2 } + 3 ( x - 1 ) + 1 \right] } & { 1 \leq x < 2 } \\ { \frac { 1 } { 6 } \left[ 3 ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 4 \right] } & { 2 \leq x < 3 } \\ { \frac { 1 } { 6 } ( 4 - x ) ^ { 3 } } & { 3 \leq x < 4 } \\ { 0 } & { o t h e r w i s e } \end{array} \right.
+$$
+
+116 The choice was made to use uniform cubic B-splines due to their excellent performance and robustness   
+117 to catastrophic forgetting, illustrated in Figure 3. Using uniform B-splines instead of arbitrary sub  
+118 interval partitions (also called knots in literature) makes optimisation easier. Optimising partitions is   
+119 non-linear, but optimising only coefficient (also called control points) is linear and thus convex.   
+120 Each basis function is multiplied by a parameter and summed together. The total number of basis   
+121 functions is typically fixed. Cubic B-splines are $3 ^ { \mathrm { r d } }$ order polynomials, and thus require a minimum   
+122 of $3 + 1 = 4$ control points or basis functions.   
+123 Instead of considering arbitrary densities of uniform cubic B-splines, we look at powers of two times   
+124 the minimum number of basis functions, called $\rho$ -density B-spline functions.
+
+![](images/f5b41c215b545eed9c1d0d7f767355817a439db8fe08f105e81662c748816dd4.jpg)  
+Figure 3: Single-variable function
+
+125 Definition 1 ( $\rho$ -density B-spline function). A $\rho$ -density B-spline function is a uniform cubic B-spline function with 126 $2 ^ { \rho + 2 }$ basis functions:
+
+$$
+f ( x ) = \sum _ { i = 1 } ^ { 2 ^ { { \rho } + 2 } } { { \theta } _ { i } S _ { i } ( x ) } = \sum _ { i = 1 } ^ { 2 ^ { { \rho } + 2 } } { { \theta } _ { i } S ( w _ { i } x + b _ { i } ) } = \sum _ { i = 1 } ^ { 2 ^ { { \rho } + 2 } } { { \theta } _ { i } S ( ( 2 ^ { { \rho } + 2 } - 3 ) x + 4 - i ) }
+$$
+
+127 Consider the problem of expanding a single-variable function approximator with more basis functions   
+128 to increase its expressive power. Using the Fourier basis makes it trivially easy by adding higher   
+129 frequency sines and cosines with coefficients initialised to zero. It is trickier to achieve something   
+130 similar with uniform cubic B-splines. There are algorithms for creating new splines from existing   
+131 splines with knot insertion, but the intermediate steps result in non-uniform knots and splines. A   
+132 simple and practical compromise that we propose is to use mixtures of different $\rho$ -density B-spline   
+133 functions, as illustrated in Figure 2.   
+134 Definition 2 (mixed-density B-spline function). A mixed-density B-spline function is a single  
+135 variable function approximator that is obtained by summing together different $\rho$ -density B-spline   
+136 functions. Only the maximum $\rho$ -density $\mathbf { B }$ -spline function has trainable parameters, the others are   
+137 constant. Mixed-density B-spline functions are of the form:
+
+$$
+f ( x ) = \sum _ { \rho = 0 } ^ { r } \sum _ { i = 1 } ^ { 2 ^ { { \rho } + 2 } } \theta _ { \rho , i } S _ { \rho , i } ( x )
+$$
+
+138 Only the maximum $r = \rho$ -density B-spline has trainable coefficients. All lower density $r > \rho$ -   
+139 density B-spline have frozen and constant coefficients. The maximum $r = \rho$ -density B-spline has   
+140 trainable coefficients with gradient updates that are orthogonal if the distance between two inputs is   
+141 large enough.   
+142 Similar to increasing the expressiveness of a Fourier basis function approximator by adding higher   
+143 frequency terms, one can add larger density cubic B-spline functions. Analytically, we can initialise   
+144 all the new scalar parameters $\theta _ { r + 1 , i } = 0 , \forall i \in \mathbf { N }$ such that:
+
+$$
+f ( x ) = \sum _ { \rho = 0 } ^ { r } \sum _ { i = 1 } ^ { 2 ^ { { \rho } + 2 } } { \theta _ { \rho , i } } { S _ { \rho , i } } ( x ) = \sum _ { \rho = 0 } ^ { r + 1 } \sum _ { i = 1 } ^ { 2 ^ { { \rho } + 2 } } { \theta _ { \rho , i } } { S _ { \rho , i } } ( x )
+$$
+
+145 It is therefore possible to create a minimal model with $r = 0$ initialised at zero, and train the model   
+146 until convergence. Then one can create a new model with $r = 1$ , by subsuming the previous model’s   
+147 parameters, and train this more expressive model until convergence. This process of training and   
+148 expansion can be continued indefinitely, and is shown in Figure 8.   
+150 ATLAS is named for carrying the burden of all it must remember, after the Titan god Atlas in Greek   
+151 mythology who was tasked with holding the weight of the world. ATLAS is also an acronym for   
+152 AddiTive exponentiaL Additive Splines.   
+153 Definition 3 (ATLAS). ATLAS is a function approximator of $n$ variables, with mixed-density   
+154 B-spline functions $f _ { j } ( x _ { j } ) , g _ { i , j } ( x _ { j } )$ , and $h _ { i , j } ( x _ { j } )$ in the form:
+
+![](images/37ecf4172c618d313c9c69a080f62cc0f347d6201f000ae3a9355d77e4d4c362.jpg)  
+Figure 4: Doubling densities of basis functions before and after training.
+
+$$
+A ( { \vec { \mathbf { x } } } ) : = \sum _ { j = 1 } ^ { n } f _ { j } ( x _ { j } ) + \sum _ { k = 1 } ^ { M } { \frac { 1 } { k ^ { 2 } } } \exp { \bigl ( } \Sigma _ { j = 1 } ^ { n } g _ { k , j } ( x _ { j } ) { \bigr ) } - { \frac { 1 } { k ^ { 2 } } } \exp { \bigl ( } \Sigma _ { j = 1 } ^ { n } h _ { k , j } ( x _ { j } ) { \bigr ) }
+$$
+
+155 ATLAS is equivalently given by the compact notation:
+
+$$
+A ( \vec { \bf x } ) : = F ( \vec { \bf x } ) + \sum _ { k = 1 } ^ { M } \frac { 1 } { k ^ { 2 } } \exp ( G _ { k } ( \vec { \bf x } ) ) - \frac { 1 } { k ^ { 2 } } \exp ( H _ { k } ( \vec { \bf x } ) )
+$$
+
+156 The absolutely convergent series of scale factors $k ^ { - 2 }$ was chosen for numerical stability and to ensure   
+157 the model is absolutely convergent. Another feature is that the series of scale factors also breaks the   
+158 symmetry that would otherwise exist if all mixed-density B-spline functions were initialised to zero.   
+159 Initialising all the parameters to be zero is a departure from the conventional approach of random   
+160 initialisation. The number of exponential terms can be increased without changing the output of the   
+161 model. We can choose to initialise $G _ { M + 1 } ( \vec { \bf x } ) = 0$ and ${ \cal H } _ { M + 1 } ( \vec { \bf x } ) = 0$ , such that the model capacity   
+162 can be increased at will.   
+163 ATLAS is a universal function approximator with some inherent memory retention. It possesses three   
+164 properties atypical of most universal function approximators:
+
+1. The activity within ATLAS is sparse – most neural units are zero and inactive. 2. The gradient vector with respect to trainable parameters is bounded regardless of the size and capacity of the model, so training is numerically stable for many possible training hyper-parameters. 3. Inputs that are sufficiently far from each other have orthogonal representations.
+
+The proofs of the three properties follows from the single-variable case, the assumption of bounded single-variable functions and parameters, and the absolutely convergent $k ^ { - 2 }$ scale factors.
+
+Property 1 (Sparsity). For any 72 ${ \vec { \bf x } } \in D ( A ) \subset R ^ { n }$ and bounded trainable parameters $\theta _ { i }$ with index 173 set $\Theta$ , the gradient vector of trainable parameters (for ATLAS) is sparse:
+
+$$
+\left\| \vec { \nabla } _ { \vec { \theta } } A ( \vec { \bf x } ) \right\| _ { 0 } = \sum _ { i \in \Theta } d _ { H a m m i n g } \left( \frac { \partial A } { \partial \theta _ { i } } ( \vec { \bf x } ) , 0 \right) \leq 4 n ( 2 M + 1 )
+$$
+
+174 Remark. For a fixed number of variables $n$ , the model has a total of $n 2 ^ { r + 2 } ( 2 M + 1 )$ trainable   
+175 parameters. The gradient vector has a maximum of $4 n ( 2 M { + } 1 )$ non-zero entries, which is independent   
+176 of $r$ . Recall that only the maximum density $( \rho = r )$ ) cubic $\mathbf { B }$ -spline function has trainable parameters.   
+177 The fraction of trainable basis functions that are active is at most $2 ^ { - r }$ . Sparsity entails efficient   
+178 implementation, and suggests possible memory retention and robustness to catastrophic forgetting.   
+179 Property 2 (Gradient flow attenuation). For any ${ \vec { \bf x } } \in D ( A ) \subset R ^ { n }$ and bounded trainable parameters   
+180 $\theta _ { i }$ with index set $\Theta$ : if all the mixed-density $B$ -spline functions are bounded, then the gradient vector   
+181 of trainable parameters for ATLAS is bounded:
+
+$$
+\left\| { \vec { \nabla } } _ { { \vec { \theta } } } A ( { \vec { \mathbf { x } } } ) \right\| _ { 1 } = \sum _ { i \in \Theta } \left| { \frac { \partial A } { \partial \theta _ { i } } } ( { \vec { \mathbf { x } } } ) \right| < U
+$$
+
+182 Remark. For a fixed number of variables $n$ , the model has a total of $n 2 ^ { r + 2 } ( 2 M + 1 )$ trainable   
+183 parameters. The factor of $k ^ { - 2 }$ inside the expression for ATLAS is necessary to ensure the sum is   
+184 convergent in the limit of infinitely many exponential terms $M \to \infty$ . Only the maximum density   
+185 $( \rho = r )$ cubic B-spline function has trainable parameters, so that the gradient vector is bounded in   
+186 the limit of arbitrarily large densities $r  \infty$ . Smaller densities cannot be trainable, otherwise this   
+187 property does not hold. The bounded gradient vector implies that ATLAS is numerically stable during   
+188 training, regardless of its size or parameter count.   
+189 Property 3 (Distal orthogonality). For any ${ \vec { \mathbf { x } } } , { \vec { \mathbf { y } } } \in D ( A ) \subset R ^ { n }$ and bounded trainable parameters   
+190 $\theta _ { i }$ for an ATLAS model $A ( { \vec { \bf x } } )$ :
+
+$$
+\operatorname* { m i n } _ { j = 1 , \dots , n } \{ | x _ { j } - y _ { j } | \} > 2 ^ { - r } \implies \langle \vec { \nabla } _ { \vec { \theta } } A ( \vec { \bf x } ) , \vec { \nabla } _ { \vec { \theta } } A ( \vec { \bf y } ) \rangle = 0
+$$
+
+191 Remark. Two points that sufficiently differ in each input variable have orthogonal parameter gradients.   
+192 Distal orthogonality means ATLAS is reasonably robust to catastrophic forgetting, without other   
+193 regularisation and training techniques. However, memory retention can still potentially be improved   
+194 when used in conjunction with other techniques.   
+195 ATLAS can be implemented with 1D convolution, reshaping, embedding, multiplication and dense   
+196 layers. The same basis functions have to be computed for each input variable, hence 1D convolutions.   
+197 By correctly scaling, shifting, and rounding inputs one can compute only the non-zero basis functions   
+198 with embedding layers. The number of basis functions are chosen from powers of two for convenience,   
+199 with the maximum density B-spline function having exactly $\lambda = 4 \times 2 ^ { r }$ basis functions. Summing   
+200 over all densities the total number of all basis functions in each input variable is at most $2 \lambda$ , because   
+201 a geometric series was used. For every output dimension $p$ , there are $2 M$ exponentials. Each   
+202 exponential has $n$ single variable functions, with at most $2 \lambda$ cubic B-spline basis functions each.   
+203 ATLAS models have time complexity $\mathcal { O } ( p M n \log \lambda )$ , and $\mathcal { O } ( p M n \lambda )$ space complexity.
+
+# 7 Methodology
+
+205 The 1-,2- and 8-dimensional models were considered for evaluation, in combination with a chosen   
+206 width for the update region in Task 2 from 0.1 to 0.9 in 0.1 increments. 30 trials were performed for   
+207 each combination of model dimension and update region width. Mean Absolute Error (MAE) loss   
+208 function, the Adam optimiser, and mini batch sizes of 100 are used throughout all experiments.   
+209 At the beginning of each trial (for a given dimension and update region width) a random learning rate   
+210 was sampled uniformly between $1 0 ^ { = 6 }$ and $0 . 0 1 + 1 0 ^ { - 6 }$ . A random noise level was sampled from an   
+211 exponential distribution with scale parameter equal to one. The Task 1 target function is constructed   
+212 from 1000 Euclidean radial basis functions (RBFs) with locations chosen uniformly over the entire   
+213 input domain, with RBF scale parameters sampled independently from an exponential distribution   
+214 (scale parameter equal to 10). The weights of each radial basis function are sampled from a normal   
+215 distribution with mean zero and standard deviation equal to one. The Task 2 target function is exactly   
+216 the same as the Task 1 target function – except for a square-like region with width equal to update   
+217 region width. The location of the update region is chosen uniformly at random, and such that it is   
+18 completely inside the domain of the model. The updated region masks the Task 1 target function and   
+219 instead replaces the values inside it with another function that is sampled from the same distribution   
+220 as the Task 1 target function, but independently from the Task 1 target function.   
+221 After the generation of the target functions 10000 data points are sampled for training, validation, and   
+222 test sets for Task 1 and Task 2. To simulate the effect of learning unrelated tasks, the training data for   
+223 Task 2 is only sampled from update region - with no training data outside of it being presented again,   
+224 by contrast the validation and test sets for Task 2 were sampled over the entire input domain. Gaussian   
+225 noise with standard deviation equal to the randomly chosen noise level is added to all training data.   
+226 An ATLAS model ( $M = 1 0$ positive and $M = 1 0$ negative exponential functions, maximum basis   
+227 function density $r = 4$ ) with guaranteed distal orthogonality is trained and evaluated on Task 1   
+228 and Task 2. Then a modified ATLAS model ( $M = 1 0$ positive and $M = 1 0$ negative exponential   
+229 functions, maximum basis function density $r = 4$ , trainable lower density basis functions) without   
+230 guaranteed distal orthogonality is trained and evaluated on Task 1 and Task 2 using the same data sets   
+231 as previously mentioned model. The final test errors for Task 2 are presented. A randomly selected   
+232 trial of the 2-dimensional case is shown for visual inspection. The experiments presented in the main   
+233 body of the paper were performed on Google Colab and the relevant code is provided.
+
+# 8 Results
+
+235 As shown in Figure 5 the effect of distal orthogonality is clear and crisp boundaries that limit the   
+236 effect of Task 2 on the memory of Task 1. Without distal orthogonality there are more off-target   
+237 effects that can be visualised.   
+238 The effect of distal orthogonality on the averaged MAE for various trials for 1-,2- and 8-dimensional   
+239 problems are presented as scatter plots of the averaged MAE over 30 trials for different update region   
+240 widths as shown in Figure 9. The expected off-target error depends on the dimension of the problem   
+241 and the width of the updated regions.   
+242 Analytical results to the expected off-target error require simplification, but a reasonable assumption   
+243 in the absence of other evidence is that each input dimension has equal contribution on the unit   
+244 hyper-cube. Assume for a fixed input dimension $n$ and some region of width $0 < \delta < 1$ where the   
+245 target function $Y$ is changed such that $| \Delta Y | = 1$ is one larger than it was originally. The expected   
+246 off-target error depends on $k$ the number of input variables inside the updated region: $\begin{array} { r } { \varepsilon _ { k } \approx \frac { n - k } { n } } \end{array}$ . To   
+247 correctly account for all permutations with the same magnitude of change:
+
+![](images/abc047c8daa2d7d4508d0e7b12e7c5544158182f326a1f2c57252235ea6718be.jpg)  
+(b) No guaranteed distal orthogonality, Off-target effects deviate from Task 2 target.   
+Figure 5: A randomly chosen trial is presented for visual inspection.
+
+![](images/c5f2017dcb9a32255e16051d74b2a19c166c0b5d374771d11f8d2cb9bd66bf5b.jpg)  
+Figure 6: Distal orthogonality guaranteed: All validation MAE curves for Task 1.
+
+![](images/a69d9601804d8eee790f1274360694637d2a944d8cca3e75d34005b28d78ef9d.jpg)  
+Figure 7: No distal orthogonality: Task 2 validation MAE with update region width $\delta = 0 . 1$ .
+
+![](images/da9313910da0a408c010fc9a69dbb2ed85a4c7adf2661ead6e9efba059b37682.jpg)  
+Figure 8: Distal orthogonality guaranteed: Task 2 validation MAE with update region width $\delta = 0 . 1$
+
+![](images/d78c6adad52c47bd1c64ab462593e759f6f7af846c619afef9505aca94d5be52.jpg)  
+Figure 9: The effect of distal orthogonality on the final test error on task 2 for the 1-,2- and 8- dimensional input.
+
+$$
+p ( \varepsilon _ { k } ) = { \binom { n } { k } } \delta ^ { n - k } \left( 1 - \delta \right) ^ { k }
+$$
+
+248 One can calculate expected change values:
+
+$$
+\mathbb { E } [ \varepsilon ] = \sum _ { k = 0 } ^ { n } \varepsilon _ { k } p ( \varepsilon _ { k } ) \approx \sum _ { k = 0 } ^ { n } { \binom { n - k } { n } } \binom { n } { k } \delta ^ { n - k } \left( 1 - \delta \right) ^ { k } = \delta
+$$
+
+249 However if one assumes that the target function inside the updated region of width $\delta$ is correct, with probability 250 $\delta ^ { n }$ of sampling from the entire input-domain, then the expected off-target error should be:
+
+$$
+{ \mathrm { E x p e c t e d ~ o f f - t a r g e t ~ e r r o r } } \approx \delta - \delta ^ { n }
+$$
+
+251 This seems consistent with some of the experimental results, but further investigation is needed.
+
+# 9 Conclusion
+
+The main contribution of the paper is theoretical and technical. A representation theorem is presented that outlines how to approximate multi-variable functions with single-variable functions (splines and exponential functions). ATLAS approximates all arbitrary single-variable functions with mixtures of B-spline functions. ATLAS is constructed in such a way that the gradient vector with respect to trainable parameters is bounded, regardless of how large an ATLAS model is. The activation of units in ATLAS is sparse, and allowed for an efficient implementation that only computes non-zero activation values with the aid of embedding layers. The gradient update vector with respect to trainable parameters is orthogonal for different inputs as long as the inputs are sufficiently different from each other.
+
+262 For every output dimension $p$ in an ATLAS model, there are $2 M$ exponentials. Each exponential has   
+263 $n$ single variable functions, with at most $2 \lambda$ cubic B-spline basis functions each. ATLAS models   
+264 have time complexity $\mathcal { O } ( p M n \log \lambda )$ , and $\mathcal { O } ( p M n \lambda )$ space complexity.   
+265 ATLAS was shown to exhibit some memory retention, without the assistance of other techniques.   
+266 This is a good indication of the potential for combining it with other techniques and models for   
+267 continual learning. The chosen experiments demonstrated the theoretically derived predictions and   
+268 contrasted two models, incuding a variant of ATLAS without distal orthogonality guarantees.   
+269 As far as societal impacts are concerned: It is possible that ATLAS could allow for the creation of   
+270 more powerful machine learning algorithms, that require less resources to train and deploy. Further   
+271 testing is needed to make any concrete claim.   
+272 References   
+273 [1] M. A. Bennani, T. Doan, and M. Sugiyama. Generalisation guarantees for continual learn  
+274 ing with orthogonal gradient descent, 2021. URL https://openreview.net/forum?id=   
+275 hecuSLbL_vC.   
+276 [2] K. Branson. A practical review of uniform b-splines, 2004.   
+277 [3] M. Delange, R. Aljundi, M. Masana, S. Parisot, X. Jia, A. Leonardis, G. Slabaugh, and   
+278 T. Tuytelaars. A continual learning survey: Defying forgetting in classification tasks. IEEE   
+279 Transactions on Pattern Analysis and Machine Intelligence, pages 1–1, 2021. doi: 10.1109/   
+280 tpami.2021.3057446. URL https://doi.org/10.1109%2Ftpami.2021.3057446.   
+281 [4] T. Doan, M. Abbana Bennani, B. Mazoure, G. Rabusseau, and P. Alquier. A theoretical analysis   
+282 of catastrophic forgetting through the ntk overlap matrix. In A. Banerjee and K. Fukumizu, edi  
+283 tors, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics,   
+284 volume 130 of Proceedings of Machine Learning Research, pages 1072–1080. PMLR, 13–15   
+285 Apr 2021. URL https://proceedings.mlr.press/v130/doan21a.html.   
+286 [5] A. S. Douzette. B-splines in machine learning. Master’s thesis, Department of Mathematics,   
+287 University of Oslo, 2017.   
+288 [6] M. Farajtabar, N. Azizan, A. Mott, and A. Li. Orthogonal gradient descent for continual   
+289 learning. In S. Chiappa and R. Calandra, editors, Proceedings of the Twenty Third International   
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+291 Learning Research, pages 3762–3773. PMLR, 26–28 Aug 2020. URL https://proceedings.   
+292 mlr.press/v108/farajtabar20a.html.   
+293 [7] R. M. French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences,   
+294 3(4):128–135, 1999.   
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+296 //arxiv.org/abs/1903.04476.   
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+298 neural networks. Trends in cognitive sciences, 24(12):1028–1040, 2020.   
+299 [10] B. Hanin. Universal function approximation by deep neural nets with bounded width and relu   
+300 activations. Mathematics, 7(10):992, 2019.   
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+302 approximators. Neural networks, 2(5):359–366, 1989.   
+303 [12] P. Kaushik, A. Gain, A. Kortylewski, and A. Yuille. Understanding catastrophic forgetting and   
+304 remembering in continual learning with optimal relevance mapping. CoRR, abs/2102.11343,   
+305 2021. URL https://arxiv.org/abs/2102.11343.   
+306 [13] R. Kemker, M. McClure, A. Abitino, T. Hayes, and C. Kanan. Measuring catastrophic forgetting   
+307 in neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32,   
+308 2018.   
+309 [14] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan,   
+310 J. Quan, T. Ramalho, A. Grabska-Barwinska, D. Hassabis, C. Clopath, D. Kumaran, and   
+311 R. Hadsell. Overcoming catastrophic forgetting in neural networks. Proceedings of the   
+312 National Academy of Sciences, 114(13):3521–3526, 2017. ISSN 0027-8424. doi: 10.1073/pnas.   
+313 1611835114. URL https://www.pnas.org/content/114/13/3521.   
+314 [15] A. Kratsios. The universal approximation property: Characterization, construction, representa  
+315 tion, and existence. Annals of Mathematics and Artificial Intelligence, 89(5):435–469, 2021.   
+316 doi: 10.1007/s10472-020-09723-1.   
+317 [16] R. Krishnan and P. Balaprakash. Meta continual learning via dynamic programming, 2020.   
+318 URL https://arxiv.org/abs/2008.02219.
+
+[17] S. H. Lane, M. Flax, D. Handelman, and J. Gelfand. Multi-layer perceptrons with b-spline receptive field functions. In Advances in Neural Information Processing Systems, pages 684–692, 1991.   
+[18] A. C. Li and D. Pathak. Functional regularization for reinforcement learning via learned fourier features, 2021. URL https://arxiv.org/abs/2112.03257.   
+[19] M. McCloskey and N. J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pages 109–165. Elsevier, 1989.   
+[20] K. McRae and P. A. Hetherington. Catastrophic interference is eliminated in pretrained networks. In Proceedings of the 15h Annual Conference of the Cognitive Science Society, pages 723–728, 1993.   
+[21] H. Mehta, A. Cutkosky, and B. Neyshabur. Extreme memorization via scale of initialization, 2020. URL https://arxiv.org/abs/2008.13363.   
+[22] A. Meulemans, F. Carzaniga, J. Suykens, J. a. Sacramento, and B. F. Grewe. A theoretical framework for target propagation. In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 20024– 20036. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/ 2020/file/e7a425c6ece20cbc9056f98699b53c6f-Paper.pdf.   
+[23] A. Robins. Catastrophic forgetting, rehearsal and pseudorehearsal. Connection Science, 7(2): 123–146, 1995.   
+[24] A. A. Rusu, N. C. Rabinowitz, G. Desjardins, H. Soyer, J. Kirkpatrick, K. Kavukcuoglu, R. Pascanu, and R. Hadsell. Progressive neural networks, 2016. URL https://arxiv.org/ abs/1606.04671.   
+[25] S. Scardapane, M. Scarpiniti, D. Comminiello, and A. Uncini. Learning activation functions from data using cubic spline interpolation. In Italian Workshop on Neural Nets, pages 73–83. Springer, 2017.   
+[26] Y. Shin and J. Ghosh. The pi-sigma network: an efficient higher-order neural network for pattern classification and function approximation. In IJCNN-91-Seattle International Joint Conference on Neural Networks, volume i, pages 13–18 vol.1, 1991. doi: 10.1109/IJCNN.1991.155142.   
+[27] H. van Deventer, P. J. van Rensburg, and A. Bosman. Kasam: Spline additive models for function approximation, 2022. URL https://arxiv.org/abs/2205.06376.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+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]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [TODO]
+
+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? [TODO]   
+(b) Did you mention the license of the assets? [TODO]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [TODO]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/Ih2bG6h1r4S/Ih2bG6h1r4S_content_list.json

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+    {
+        "type": "text",
+        "text": "ATLAS: Universal Function Approximator for Memory Retention ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Artificial neural networks (ANNs), despite their universal function approximation   \n2 capability and practical success, are subject to catastrophic forgetting. Catastrophic   \n3 forgetting refers to the abrupt unlearning of a previous task when a new task is   \n4 learned. It is an emergent phenomenon that plagues ANNs and hinders continual   \n5 learning. Existing universal function approximation theorems for ANNs guarantee   \n6 function approximation ability, but seldom touch on the model details and do not   \n7 predict catastrophic forgetting. This paper presents a novel universal approximation   \n8 theorem for multi-variable functions using only single-variable functions and   \n9 exponential functions. Furthermore, we present ATLAS—a novel ANN architecture   \n10 based on the exponential approximation theorem and B-splines. It is shown that   \n11 ATLAS is a universal function approximator capable of memory retention and,   \n12 therefore, continual learning. The memory retention of ATLAS is imperfect,   \n13 with some off-target effects during continual learning, but it is well-behaved and   \n14 predictable. An efficient implementation of ATLAS is provided. Experiments   \n15 are conducted to evaluate both the function approximation and memory retention   \n16 capabilities of ATLAS. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "17 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            607,
+            312,
+            625
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "18 Catastrophic forgetting [7, 13, 23] is an emergent phenomenon where a machine learning model   \n19 such as an artificial neural network (ANN) learns a new task, and the subsequent parameter updates   \n20 interfere with the model’s performance on previously learned tasks. Catastrophic forgetting is also   \n21 called catastrophic interference [19]. If an ANN cannot effectively learn many tasks, it has limited   \n22 utility in the context of continual learning [9, 12]. Catastrophic forgetting is like learning to pick   \n23 up a cup, but simultaneously forgetting how to breathe. Even linear functions are susceptible to   \n24 catastrophic forgetting, as illustrated in Figure 1   \n25 The simple example of a linear regression model being susceptible to catastrophic forgetting might be   \n26 due to the non-linearity of the target function, noise, or parameter sharing across the input. Parameter   \n27 sharing is avoidable with piece-wise defined functions such as splines [27]. ANNs can be explained   \n28 in many ways; a useful analogy is to compare ANNs to very large lookup tables that store information.   \n29 Removing and updating values has off-target effects throughout the table or ANN.   \n30 Universal function approximation theorems are a cornerstone of machine learning, and prove that   \n31 ANNs can approximate any given continuous target function [10, 11, 15] under certain assumptions.   \n32 The theorems do not specify how to find an ANN with sufficient performance for problems in   \n33 practice. Gradient descent optimisation is the convention for finding/training neural networks, but   \n34 other optimisation and learning procedures exist [22]. ATLAS models trained with gradient descent   \n35 methods exhibit desirable properties. However, other optimisation techniques like evolutionary   \n36 algorithms may not elicit the same properties.   \n37 This paper introduces ATLAS—a novel universal function approximator based on B-splines that has   \n38 some intrinsic memory retention, even in the absence of other training and regularisation techniques.   \n39 ATLAS has well-behaved parameter gradients that are sparse, bounded and orthogonal between   \n40 input points that are far enough from each other. The accompanying representation and universal   \n41 approximation theorems are also provided. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/48a779adaf44eba98e2fe1a3bd4690138d9bcace728802639941cafdcd7a32b1.jpg",
+        "image_caption": [
+            "Figure 1: A linear function is susceptible to catastrophic forgetting. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "42 2 Relevant Studies ",
+        "text_level": 1,
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+        "text": "43 It is conjectured that overlapping representations in ANNs lead to catastrophic forgetting [12].   \n44 Catastrophic forgetting occurs when parameters necessary for one task change while training to meet   \n45 the objectives of another task [14, 20]. The least desirable strategy to mitigate catastrophic forgetting   \n46 is retraining a model over all tasks. Regularisation techniques like elastic weight consolidation (EWC)   \n47 have also been employed [14]. Data augmentation approaches such as rehearsal and pseudo-rehearsal   \n48 have also been employed [23]. Other ideas from optimal control theory in combination with dynamic   \n49 programming have also been applied to counteract catastrophic forgetting, with a cost functional   \n50 similar in form to the action integral from physics and Lagrangian mechanics [16].   \n51 Orthogonal Gradient Descent (OGD) is a training augmentation or optimisation technique that   \n52 modifies the gradient updates of subsequent tasks to be orthogonal to previous tasks [6, 1]. One   \n53 can describe data in terms of a distribution defined over the input space, target values, and time (the   \n54 order of data or tasks that are presented during training). OGD attempts to make gradient updates   \n55 orthogonal to each other over time. ATLAS, in contrast, possesses distal orthogonality, meaning that   \n56 if two inputs are far enough from each other in the input space, then corresponding gradient updates   \n57 will be orthogonal. A corollary of this is that if the data distribution between tasks shifts in the input   \n58 space, then the subsequent gradient updates will tend to be orthogonal. ATLAS does not use external   \n59 memory like OGD. Extensions of OGD include PCA-OGD, which compresses gradient updates into   \n60 principal components to reduce memory requirements [4]. The Neural Tangent Kernel (NTK) overlap   \n61 matrices, as discussed by Doan et al. [4], could be a useful tool for analysing ATLAS models.   \n62 The survey by Delange et al. [3] gives an extensive overview of continual learning to address   \n63 catastrophic forgetting. ATLAS is a model that implements parameter isolation, because of its use of   \n64 piece-wise defined splines. Particularly relevant to ATLAS is the work on scale of initialisation and   \n65 extreme memorisation [21]. Increasing the density of basis functions in ATLAS can lead to better   \n66 memorisation, and increases the scale of some parameters in ATLAS which may affect generalisation.   \n67 Pi-sigma neural networks use nodes that compute products instead of sums [26]. Pi-sigma neural   \n68 networks have some similarities with the global structure of ATLAS. B-splines, which form the basis   \n69 of ATLAS, have been applied for machine learning [5]. Scardapane et al. [25] investigated trainable   \n70 activation functions parameterised by splines. Uniform cubic B-splines have basis functions that are   \n71 translates of one another [2]. Uniform cubic B-splines have been tested for memory retention, and   \n72 ATLAS is an improvement on existing spline models [27].   \n73 B-splines, and by extension ATLAS, can be trained to fit lower frequency components, expanded and   \n74 trained again until a network is found with sufficient accuracy and generalisation, similar to other   \n75 techniques [17, 18]. It is not necessary to expand the capacity of an ATLAS model to learn new   \n76 tasks, as with some other approaches [24]. ATLAS does in practice demonstrate something akin to   \n77 \"graceful forgetting\" as discussed in Golkar et al. [8].   \n79 Vector quantities like $\\vec { \\bf x }$ are clearly indicated with a bar or arrow for legibility. Parameters, inputs,   \n80 functions etc. without a bar or arrow are scalar quantities like $S ( x )$ . Some scalar quantities with   \n81 indices are the scalar components of a vector like $x _ { j }$ or scalar parameters in the model like $\\theta _ { i }$ . The   \n82 gradient operator that acts on a scalar function like $\\vec { \\nabla } _ { \\vec { \\theta } } A ( \\vec { \\bf x } )$ yields a vector-valued function $\\vec { \\nabla } _ { \\vec { \\theta } } A ( \\vec { \\bf x } )$   \n83 as is typical of multi-variable calculus. ",
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+    {
+        "type": "text",
+        "text": "84 4 Exponential Representation Theorem ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "85 Any continuous multi-variable function on a compact space can be uniformly approximated with   \n86 multi-variable polynomials by the Stone-Weierstrass Theorem. Let $\\mathcal { T }$ denote an index set of tuples of   \n87 natural numbers including zero such that $i _ { j } \\in \\mathbb { N } ^ { 0 }$ for all $j \\in \\mathbb N$ with $i = ( i _ { 1 } , . . , i _ { n } ) \\in \\mathcal { T }$ and $a _ { i } \\in \\mathbb { R }$ .   \n88 Multi-variable polynomials can be represented as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/172d3f5b955b90be7aafbce26e43f6e6536ef2fde2ba78db92373afd0148de66.jpg",
+        "text": "$$\ny ( \\vec { \\bf x } ) = y ( x _ { 1 } , . . , x _ { n } ) = \\sum _ { i \\in \\mathcal { I } } a _ { i } x _ { 1 } ^ { i _ { 1 } } x _ { 2 } ^ { i _ { 2 } } . . . x _ { n } ^ { i _ { n } } = \\sum _ { i \\in \\mathcal { I } } a _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }\n$$",
+        "text_format": "latex",
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+    {
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+        "text": "89 Each monomial term $a _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }$ is a product of single-variable functions in each variable. It is   \n90 desirable to rewrite products as sums using exponentials and logarithms. ",
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+    {
+        "type": "text",
+        "text": "91 Lemma 1. For any $a _ { i } \\in \\mathbb { R }$ , there exists $\\gamma _ { i } > 0$ and $\\beta _ { i } > 0$ , such that: $a _ { i } = \\gamma _ { i } - \\beta _ { i }$ ",
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+    {
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+        "text": "92 Theorem 1 (Exponential representation theorem). Any multi-variable polynomial function $y ( \\vec { \\bf x } )$   \n93 of $n$ variables over the positive orthant, can be exactly represented by continuous single-variable   \n94 functions $g _ { i , j } ( x _ { j } )$ and $h _ { i , j } ( x _ { j } )$ in the form: ",
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+        "type": "equation",
+        "img_path": "images/caa86e7cdb535c32b2b4442e1e8d257886bd9cea5bc4b88e38f938a6dc1900cc.jpg",
+        "text": "$$\ny ( \\vec { \\bf x } ) = \\sum _ { i \\in \\mathcal { I } } \\exp \\bigl ( \\Sigma _ { j = 1 } ^ { n } g _ { i , j } ( x _ { j } ) \\bigr ) - \\exp \\bigl ( \\Sigma _ { j = 1 } ^ { n } h _ { i , j } ( x _ { j } ) \\bigr )\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "Proof. Consider any monomial term 95 $a _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }$ with $a _ { i } \\in \\mathbb { R }$ , then by Lemma 1 there exist strictly 96 positive numbers $\\gamma _ { i } > 0$ and $\\beta _ { i } > 0$ , such that: ",
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+    },
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+        "text": "$$\n\\begin{array} { r l } & { a _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } = \\gamma _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } - \\beta _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } } \\\\ & { \\phantom { a a a } = \\exp \\Bigl ( \\log \\Bigl ( \\gamma _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } \\Bigr ) \\Bigr ) - \\exp \\Bigl ( \\log \\Bigl ( \\beta _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } \\Bigr ) \\Bigr ) } \\\\ & { \\phantom { a a a a } = \\exp \\Bigl ( \\log ( \\gamma _ { i } ) + \\Sigma _ { j = 1 } ^ { n } \\log \\Bigl ( x _ { j } ^ { i _ { j } } \\Bigr ) \\Bigr ) - \\exp \\Bigl ( \\log ( \\beta _ { i } ) + \\Sigma _ { j = 1 } ^ { n } \\log \\Bigl ( x _ { j } ^ { i _ { j } } \\Bigr ) \\Bigr ) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "97 The argument of each exponential function is a sum of single-variable functions and constants.   \n98 Without loss of generality, a set of single-variable functions can be defined such that: ",
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+        "img_path": "images/d5c683e4c0b8820ab8e7c3767f90ccdcf1f8f27347d12e201184689b8a39a761.jpg",
+        "text": "$$\na _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } } = \\exp \\bigl ( \\Sigma _ { j = 1 } ^ { n } g _ { i , j } ( x _ { j } ) \\bigr ) - \\exp \\bigl ( \\Sigma _ { j = 1 } ^ { n } h _ { i , j } ( x _ { j } ) \\bigr )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Since this holds for any 99 $a _ { i } \\Pi _ { j = 1 } ^ { n } x _ { j } ^ { i _ { j } }$ and all $i \\in \\mathcal { T }$ , it follows that: ",
+        "bbox": [
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+    {
+        "type": "equation",
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+        "text": "$$\ny ( \\vec { \\bf x } ) = \\sum _ { i \\in \\mathcal { I } } \\exp \\bigl ( \\Sigma _ { j = 1 } ^ { n } g _ { i , j } ( x _ { j } ) \\bigr ) - \\exp \\bigl ( \\Sigma _ { j = 1 } ^ { n } h _ { i , j } ( x _ { j } ) \\bigr )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "101 This result is fundamental to the paper. Since every continuous function can be approximated with   \n102 multi-variable polynomials, it follows that every continuous function can be approximated with   \n103 positive and negative exponential functions. Single-variable function approximators are pivotal and   \n104 must be reconsidered. Universal function approximation can also be proven with the sub-algebra   \n105 formulation of the Stone-Weierstrass theorem, but it’s not as delightful and simple as the first   \n106 constructive proof given above.   \n108 Splines are piece-wise defined single-variable functions over some interval. Each sub-interval of a   \n109 spline is most often locally given by a low degree polynomial, even though the global structure is not   \n110 a low degree polynomial. B-splines are polynomial splines that are defined in a way that resembles   \n111 other basis function formulations [2]. Each single-variable function in ATLAS is approximated   \n112 with uniform cubic B-spline basis functions, shown in Figure 2. B-splines can approximate any   \n113 single-variable function, similar to using the Fourier basis. With uniform B-splines, each basis   \n114 function is scaled so that the unit interval is uniformly partitioned, as in Figure 2. ",
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+        "img_path": "images/028d13c2b6b7713311944022490eca8111753bed108284b61d20c89e04ca0f30.jpg",
+        "image_caption": [
+            "Figure 2: If uniformly spaced B-splines are used, then each basis function has the same shape. This makes it possible to use the same activation function by scaling and translating the inputs. This is also true for different densities of uniform cubic B-splines. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "115 The activation function to implement B-splines is given by: ",
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+        "img_path": "images/0f60cba105e6fa7a084f3ea7ede061ce07b11c9f6731613603081524fe88a5d7.jpg",
+        "text": "$$\nS ( x ) = \\left\\{ \\begin{array} { l l } { \\frac { 1 } { 6 } x ^ { 3 } } & { 0 \\leq x < 1 } \\\\ { \\frac { 1 } { 6 } \\left[ - 3 ( x - 1 ) ^ { 3 } + 3 ( x - 1 ) ^ { 2 } + 3 ( x - 1 ) + 1 \\right] } & { 1 \\leq x < 2 } \\\\ { \\frac { 1 } { 6 } \\left[ 3 ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 4 \\right] } & { 2 \\leq x < 3 } \\\\ { \\frac { 1 } { 6 } ( 4 - x ) ^ { 3 } } & { 3 \\leq x < 4 } \\\\ { 0 } & { o t h e r w i s e } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "116 The choice was made to use uniform cubic B-splines due to their excellent performance and robustness   \n117 to catastrophic forgetting, illustrated in Figure 3. Using uniform B-splines instead of arbitrary sub  \n118 interval partitions (also called knots in literature) makes optimisation easier. Optimising partitions is   \n119 non-linear, but optimising only coefficient (also called control points) is linear and thus convex.   \n120 Each basis function is multiplied by a parameter and summed together. The total number of basis   \n121 functions is typically fixed. Cubic B-splines are $3 ^ { \\mathrm { r d } }$ order polynomials, and thus require a minimum   \n122 of $3 + 1 = 4$ control points or basis functions.   \n123 Instead of considering arbitrary densities of uniform cubic B-splines, we look at powers of two times   \n124 the minimum number of basis functions, called $\\rho$ -density B-spline functions. ",
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+        "image_caption": [
+            "Figure 3: Single-variable function "
+        ],
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+        "type": "text",
+        "text": "125 Definition 1 ( $\\rho$ -density B-spline function). A $\\rho$ -density B-spline function is a uniform cubic B-spline function with 126 $2 ^ { \\rho + 2 }$ basis functions: ",
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+        "img_path": "images/c962bae81aa58e95b5d3e1c0a7d5ccbeaa795b9215016102e5552b539f5e1b50.jpg",
+        "text": "$$\nf ( x ) = \\sum _ { i = 1 } ^ { 2 ^ { { \\rho } + 2 } } { { \\theta } _ { i } S _ { i } ( x ) } = \\sum _ { i = 1 } ^ { 2 ^ { { \\rho } + 2 } } { { \\theta } _ { i } S ( w _ { i } x + b _ { i } ) } = \\sum _ { i = 1 } ^ { 2 ^ { { \\rho } + 2 } } { { \\theta } _ { i } S ( ( 2 ^ { { \\rho } + 2 } - 3 ) x + 4 - i ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "127 Consider the problem of expanding a single-variable function approximator with more basis functions   \n128 to increase its expressive power. Using the Fourier basis makes it trivially easy by adding higher   \n129 frequency sines and cosines with coefficients initialised to zero. It is trickier to achieve something   \n130 similar with uniform cubic B-splines. There are algorithms for creating new splines from existing   \n131 splines with knot insertion, but the intermediate steps result in non-uniform knots and splines. A   \n132 simple and practical compromise that we propose is to use mixtures of different $\\rho$ -density B-spline   \n133 functions, as illustrated in Figure 2.   \n134 Definition 2 (mixed-density B-spline function). A mixed-density B-spline function is a single  \n135 variable function approximator that is obtained by summing together different $\\rho$ -density B-spline   \n136 functions. Only the maximum $\\rho$ -density $\\mathbf { B }$ -spline function has trainable parameters, the others are   \n137 constant. Mixed-density B-spline functions are of the form: ",
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+        "img_path": "images/c77a7e07edeec0d0abb8e373e1dd0afc01f91476965d858ad5c7886ad07d469c.jpg",
+        "text": "$$\nf ( x ) = \\sum _ { \\rho = 0 } ^ { r } \\sum _ { i = 1 } ^ { 2 ^ { { \\rho } + 2 } } \\theta _ { \\rho , i } S _ { \\rho , i } ( x )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "138 Only the maximum $r = \\rho$ -density B-spline has trainable coefficients. All lower density $r > \\rho$ -   \n139 density B-spline have frozen and constant coefficients. The maximum $r = \\rho$ -density B-spline has   \n140 trainable coefficients with gradient updates that are orthogonal if the distance between two inputs is   \n141 large enough.   \n142 Similar to increasing the expressiveness of a Fourier basis function approximator by adding higher   \n143 frequency terms, one can add larger density cubic B-spline functions. Analytically, we can initialise   \n144 all the new scalar parameters $\\theta _ { r + 1 , i } = 0 , \\forall i \\in \\mathbf { N }$ such that: ",
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+        "type": "text",
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+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/1dc0147e883359e959f6be2022d913184433c48f1218f6f1d9bf3d66d296daf5.jpg",
+        "text": "$$\nf ( x ) = \\sum _ { \\rho = 0 } ^ { r } \\sum _ { i = 1 } ^ { 2 ^ { { \\rho } + 2 } } { \\theta _ { \\rho , i } } { S _ { \\rho , i } } ( x ) = \\sum _ { \\rho = 0 } ^ { r + 1 } \\sum _ { i = 1 } ^ { 2 ^ { { \\rho } + 2 } } { \\theta _ { \\rho , i } } { S _ { \\rho , i } } ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "145 It is therefore possible to create a minimal model with $r = 0$ initialised at zero, and train the model   \n146 until convergence. Then one can create a new model with $r = 1$ , by subsuming the previous model’s   \n147 parameters, and train this more expressive model until convergence. This process of training and   \n148 expansion can be continued indefinitely, and is shown in Figure 8.   \n150 ATLAS is named for carrying the burden of all it must remember, after the Titan god Atlas in Greek   \n151 mythology who was tasked with holding the weight of the world. ATLAS is also an acronym for   \n152 AddiTive exponentiaL Additive Splines.   \n153 Definition 3 (ATLAS). ATLAS is a function approximator of $n$ variables, with mixed-density   \n154 B-spline functions $f _ { j } ( x _ { j } ) , g _ { i , j } ( x _ { j } )$ , and $h _ { i , j } ( x _ { j } )$ in the form: ",
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+    {
+        "type": "image",
+        "img_path": "images/37ecf4172c618d313c9c69a080f62cc0f347d6201f000ae3a9355d77e4d4c362.jpg",
+        "image_caption": [
+            "Figure 4: Doubling densities of basis functions before and after training. "
+        ],
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+        "img_path": "images/69fd83e684314d7b31ae10ecb0bc79671538bdac7605d3ff1ef68c887e93f4ad.jpg",
+        "text": "$$\nA ( { \\vec { \\mathbf { x } } } ) : = \\sum _ { j = 1 } ^ { n } f _ { j } ( x _ { j } ) + \\sum _ { k = 1 } ^ { M } { \\frac { 1 } { k ^ { 2 } } } \\exp { \\bigl ( } \\Sigma _ { j = 1 } ^ { n } g _ { k , j } ( x _ { j } ) { \\bigr ) } - { \\frac { 1 } { k ^ { 2 } } } \\exp { \\bigl ( } \\Sigma _ { j = 1 } ^ { n } h _ { k , j } ( x _ { j } ) { \\bigr ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "155 ATLAS is equivalently given by the compact notation: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/21400ea6e71ae3699c131c816277d737f1eaa31e57fdaa0b327c33a825e1ab57.jpg",
+        "text": "$$\nA ( \\vec { \\bf x } ) : = F ( \\vec { \\bf x } ) + \\sum _ { k = 1 } ^ { M } \\frac { 1 } { k ^ { 2 } } \\exp ( G _ { k } ( \\vec { \\bf x } ) ) - \\frac { 1 } { k ^ { 2 } } \\exp ( H _ { k } ( \\vec { \\bf x } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "156 The absolutely convergent series of scale factors $k ^ { - 2 }$ was chosen for numerical stability and to ensure   \n157 the model is absolutely convergent. Another feature is that the series of scale factors also breaks the   \n158 symmetry that would otherwise exist if all mixed-density B-spline functions were initialised to zero.   \n159 Initialising all the parameters to be zero is a departure from the conventional approach of random   \n160 initialisation. The number of exponential terms can be increased without changing the output of the   \n161 model. We can choose to initialise $G _ { M + 1 } ( \\vec { \\bf x } ) = 0$ and ${ \\cal H } _ { M + 1 } ( \\vec { \\bf x } ) = 0$ , such that the model capacity   \n162 can be increased at will.   \n163 ATLAS is a universal function approximator with some inherent memory retention. It possesses three   \n164 properties atypical of most universal function approximators: ",
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+    {
+        "type": "text",
+        "text": "1. The activity within ATLAS is sparse – most neural units are zero and inactive. 2. The gradient vector with respect to trainable parameters is bounded regardless of the size and capacity of the model, so training is numerically stable for many possible training hyper-parameters. 3. Inputs that are sufficiently far from each other have orthogonal representations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The proofs of the three properties follows from the single-variable case, the assumption of bounded single-variable functions and parameters, and the absolutely convergent $k ^ { - 2 }$ scale factors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Property 1 (Sparsity). For any 72 ${ \\vec { \\bf x } } \\in D ( A ) \\subset R ^ { n }$ and bounded trainable parameters $\\theta _ { i }$ with index 173 set $\\Theta$ , the gradient vector of trainable parameters (for ATLAS) is sparse: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/cf84bbc72c04c479ea06bbf48f63219072ab102b5c79bd6c876f7ed2bd38ae92.jpg",
+        "text": "$$\n\\left\\| \\vec { \\nabla } _ { \\vec { \\theta } } A ( \\vec { \\bf x } ) \\right\\| _ { 0 } = \\sum _ { i \\in \\Theta } d _ { H a m m i n g } \\left( \\frac { \\partial A } { \\partial \\theta _ { i } } ( \\vec { \\bf x } ) , 0 \\right) \\leq 4 n ( 2 M + 1 )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "174 Remark. For a fixed number of variables $n$ , the model has a total of $n 2 ^ { r + 2 } ( 2 M + 1 )$ trainable   \n175 parameters. The gradient vector has a maximum of $4 n ( 2 M { + } 1 )$ non-zero entries, which is independent   \n176 of $r$ . Recall that only the maximum density $( \\rho = r )$ ) cubic $\\mathbf { B }$ -spline function has trainable parameters.   \n177 The fraction of trainable basis functions that are active is at most $2 ^ { - r }$ . Sparsity entails efficient   \n178 implementation, and suggests possible memory retention and robustness to catastrophic forgetting.   \n179 Property 2 (Gradient flow attenuation). For any ${ \\vec { \\bf x } } \\in D ( A ) \\subset R ^ { n }$ and bounded trainable parameters   \n180 $\\theta _ { i }$ with index set $\\Theta$ : if all the mixed-density $B$ -spline functions are bounded, then the gradient vector   \n181 of trainable parameters for ATLAS is bounded: ",
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+        "img_path": "images/f5fd907a7fc8180b698937683a16d888a3d1dfdea2c78db1bd7b746b2e82ec33.jpg",
+        "text": "$$\n\\left\\| { \\vec { \\nabla } } _ { { \\vec { \\theta } } } A ( { \\vec { \\mathbf { x } } } ) \\right\\| _ { 1 } = \\sum _ { i \\in \\Theta } \\left| { \\frac { \\partial A } { \\partial \\theta _ { i } } } ( { \\vec { \\mathbf { x } } } ) \\right| < U\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "182 Remark. For a fixed number of variables $n$ , the model has a total of $n 2 ^ { r + 2 } ( 2 M + 1 )$ trainable   \n183 parameters. The factor of $k ^ { - 2 }$ inside the expression for ATLAS is necessary to ensure the sum is   \n184 convergent in the limit of infinitely many exponential terms $M \\to \\infty$ . Only the maximum density   \n185 $( \\rho = r )$ cubic B-spline function has trainable parameters, so that the gradient vector is bounded in   \n186 the limit of arbitrarily large densities $r  \\infty$ . Smaller densities cannot be trainable, otherwise this   \n187 property does not hold. The bounded gradient vector implies that ATLAS is numerically stable during   \n188 training, regardless of its size or parameter count.   \n189 Property 3 (Distal orthogonality). For any ${ \\vec { \\mathbf { x } } } , { \\vec { \\mathbf { y } } } \\in D ( A ) \\subset R ^ { n }$ and bounded trainable parameters   \n190 $\\theta _ { i }$ for an ATLAS model $A ( { \\vec { \\bf x } } )$ : ",
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+        "img_path": "images/17e5c01c81a8228b695cb473c4143b12814436d94c854560a38cb0d03395832e.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { j = 1 , \\dots , n } \\{ | x _ { j } - y _ { j } | \\} > 2 ^ { - r } \\implies \\langle \\vec { \\nabla } _ { \\vec { \\theta } } A ( \\vec { \\bf x } ) , \\vec { \\nabla } _ { \\vec { \\theta } } A ( \\vec { \\bf y } ) \\rangle = 0\n$$",
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+        "text": "191 Remark. Two points that sufficiently differ in each input variable have orthogonal parameter gradients.   \n192 Distal orthogonality means ATLAS is reasonably robust to catastrophic forgetting, without other   \n193 regularisation and training techniques. However, memory retention can still potentially be improved   \n194 when used in conjunction with other techniques.   \n195 ATLAS can be implemented with 1D convolution, reshaping, embedding, multiplication and dense   \n196 layers. The same basis functions have to be computed for each input variable, hence 1D convolutions.   \n197 By correctly scaling, shifting, and rounding inputs one can compute only the non-zero basis functions   \n198 with embedding layers. The number of basis functions are chosen from powers of two for convenience,   \n199 with the maximum density B-spline function having exactly $\\lambda = 4 \\times 2 ^ { r }$ basis functions. Summing   \n200 over all densities the total number of all basis functions in each input variable is at most $2 \\lambda$ , because   \n201 a geometric series was used. For every output dimension $p$ , there are $2 M$ exponentials. Each   \n202 exponential has $n$ single variable functions, with at most $2 \\lambda$ cubic B-spline basis functions each.   \n203 ATLAS models have time complexity $\\mathcal { O } ( p M n \\log \\lambda )$ , and $\\mathcal { O } ( p M n \\lambda )$ space complexity. ",
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+        "type": "text",
+        "text": "7 Methodology ",
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+        "text": "205 The 1-,2- and 8-dimensional models were considered for evaluation, in combination with a chosen   \n206 width for the update region in Task 2 from 0.1 to 0.9 in 0.1 increments. 30 trials were performed for   \n207 each combination of model dimension and update region width. Mean Absolute Error (MAE) loss   \n208 function, the Adam optimiser, and mini batch sizes of 100 are used throughout all experiments.   \n209 At the beginning of each trial (for a given dimension and update region width) a random learning rate   \n210 was sampled uniformly between $1 0 ^ { = 6 }$ and $0 . 0 1 + 1 0 ^ { - 6 }$ . A random noise level was sampled from an   \n211 exponential distribution with scale parameter equal to one. The Task 1 target function is constructed   \n212 from 1000 Euclidean radial basis functions (RBFs) with locations chosen uniformly over the entire   \n213 input domain, with RBF scale parameters sampled independently from an exponential distribution   \n214 (scale parameter equal to 10). The weights of each radial basis function are sampled from a normal   \n215 distribution with mean zero and standard deviation equal to one. The Task 2 target function is exactly   \n216 the same as the Task 1 target function – except for a square-like region with width equal to update   \n217 region width. The location of the update region is chosen uniformly at random, and such that it is   \n18 completely inside the domain of the model. The updated region masks the Task 1 target function and   \n219 instead replaces the values inside it with another function that is sampled from the same distribution   \n220 as the Task 1 target function, but independently from the Task 1 target function.   \n221 After the generation of the target functions 10000 data points are sampled for training, validation, and   \n222 test sets for Task 1 and Task 2. To simulate the effect of learning unrelated tasks, the training data for   \n223 Task 2 is only sampled from update region - with no training data outside of it being presented again,   \n224 by contrast the validation and test sets for Task 2 were sampled over the entire input domain. Gaussian   \n225 noise with standard deviation equal to the randomly chosen noise level is added to all training data.   \n226 An ATLAS model ( $M = 1 0$ positive and $M = 1 0$ negative exponential functions, maximum basis   \n227 function density $r = 4$ ) with guaranteed distal orthogonality is trained and evaluated on Task 1   \n228 and Task 2. Then a modified ATLAS model ( $M = 1 0$ positive and $M = 1 0$ negative exponential   \n229 functions, maximum basis function density $r = 4$ , trainable lower density basis functions) without   \n230 guaranteed distal orthogonality is trained and evaluated on Task 1 and Task 2 using the same data sets   \n231 as previously mentioned model. The final test errors for Task 2 are presented. A randomly selected   \n232 trial of the 2-dimensional case is shown for visual inspection. The experiments presented in the main   \n233 body of the paper were performed on Google Colab and the relevant code is provided. ",
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+        "text": "8 Results ",
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+        "text": "235 As shown in Figure 5 the effect of distal orthogonality is clear and crisp boundaries that limit the   \n236 effect of Task 2 on the memory of Task 1. Without distal orthogonality there are more off-target   \n237 effects that can be visualised.   \n238 The effect of distal orthogonality on the averaged MAE for various trials for 1-,2- and 8-dimensional   \n239 problems are presented as scatter plots of the averaged MAE over 30 trials for different update region   \n240 widths as shown in Figure 9. The expected off-target error depends on the dimension of the problem   \n241 and the width of the updated regions.   \n242 Analytical results to the expected off-target error require simplification, but a reasonable assumption   \n243 in the absence of other evidence is that each input dimension has equal contribution on the unit   \n244 hyper-cube. Assume for a fixed input dimension $n$ and some region of width $0 < \\delta < 1$ where the   \n245 target function $Y$ is changed such that $| \\Delta Y | = 1$ is one larger than it was originally. The expected   \n246 off-target error depends on $k$ the number of input variables inside the updated region: $\\begin{array} { r } { \\varepsilon _ { k } \\approx \\frac { n - k } { n } } \\end{array}$ . To   \n247 correctly account for all permutations with the same magnitude of change: ",
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+            "(b) No guaranteed distal orthogonality, Off-target effects deviate from Task 2 target. ",
+            "Figure 5: A randomly chosen trial is presented for visual inspection. "
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+        "image_caption": [
+            "Figure 6: Distal orthogonality guaranteed: All validation MAE curves for Task 1. "
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+            "Figure 7: No distal orthogonality: Task 2 validation MAE with update region width $\\delta = 0 . 1$ . "
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+            "Figure 8: Distal orthogonality guaranteed: Task 2 validation MAE with update region width $\\delta = 0 . 1$ "
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+        "image_caption": [
+            "Figure 9: The effect of distal orthogonality on the final test error on task 2 for the 1-,2- and 8- dimensional input. "
+        ],
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+    {
+        "type": "equation",
+        "img_path": "images/62dab8029220c676c3074b985a2ab8483df5ee915ca30fac5e960ad5618cf15b.jpg",
+        "text": "$$\np ( \\varepsilon _ { k } ) = { \\binom { n } { k } } \\delta ^ { n - k } \\left( 1 - \\delta \\right) ^ { k }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "248 One can calculate expected change values: ",
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+        "type": "equation",
+        "img_path": "images/af43d47f42b3aa8d8e5003e0391e47b8901ff329dc0d6a4498d6ca58c85159d1.jpg",
+        "text": "$$\n\\mathbb { E } [ \\varepsilon ] = \\sum _ { k = 0 } ^ { n } \\varepsilon _ { k } p ( \\varepsilon _ { k } ) \\approx \\sum _ { k = 0 } ^ { n } { \\binom { n - k } { n } } \\binom { n } { k } \\delta ^ { n - k } \\left( 1 - \\delta \\right) ^ { k } = \\delta\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "249 However if one assumes that the target function inside the updated region of width $\\delta$ is correct, with probability 250 $\\delta ^ { n }$ of sampling from the entire input-domain, then the expected off-target error should be: ",
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+        "type": "equation",
+        "img_path": "images/f6440d186ac7e09e72087844941b503a7a7dd8e137813e52eb5f03041577d570.jpg",
+        "text": "$$\n{ \\mathrm { E x p e c t e d ~ o f f - t a r g e t ~ e r r o r } } \\approx \\delta - \\delta ^ { n }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "251 This seems consistent with some of the experimental results, but further investigation is needed. ",
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+    {
+        "type": "text",
+        "text": "9 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The main contribution of the paper is theoretical and technical. A representation theorem is presented that outlines how to approximate multi-variable functions with single-variable functions (splines and exponential functions). ATLAS approximates all arbitrary single-variable functions with mixtures of B-spline functions. ATLAS is constructed in such a way that the gradient vector with respect to trainable parameters is bounded, regardless of how large an ATLAS model is. The activation of units in ATLAS is sparse, and allowed for an efficient implementation that only computes non-zero activation values with the aid of embedding layers. The gradient update vector with respect to trainable parameters is orthogonal for different inputs as long as the inputs are sufficiently different from each other. ",
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+    {
+        "type": "text",
+        "text": "262 For every output dimension $p$ in an ATLAS model, there are $2 M$ exponentials. Each exponential has   \n263 $n$ single variable functions, with at most $2 \\lambda$ cubic B-spline basis functions each. ATLAS models   \n264 have time complexity $\\mathcal { O } ( p M n \\log \\lambda )$ , and $\\mathcal { O } ( p M n \\lambda )$ space complexity.   \n265 ATLAS was shown to exhibit some memory retention, without the assistance of other techniques.   \n266 This is a good indication of the potential for combining it with other techniques and models for   \n267 continual learning. The chosen experiments demonstrated the theoretically derived predictions and   \n268 contrasted two models, incuding a variant of ATLAS without distal orthogonality guarantees.   \n269 As far as societal impacts are concerned: It is possible that ATLAS could allow for the creation of   \n270 more powerful machine learning algorithms, that require less resources to train and deploy. Further   \n271 testing is needed to make any concrete claim.   \n272 References   \n273 [1] M. A. Bennani, T. Doan, and M. Sugiyama. Generalisation guarantees for continual learn  \n274 ing with orthogonal gradient descent, 2021. URL https://openreview.net/forum?id=   \n275 hecuSLbL_vC.   \n276 [2] K. Branson. A practical review of uniform b-splines, 2004.   \n277 [3] M. Delange, R. Aljundi, M. Masana, S. Parisot, X. Jia, A. Leonardis, G. Slabaugh, and   \n278 T. Tuytelaars. A continual learning survey: Defying forgetting in classification tasks. IEEE   \n279 Transactions on Pattern Analysis and Machine Intelligence, pages 1–1, 2021. doi: 10.1109/   \n280 tpami.2021.3057446. URL https://doi.org/10.1109%2Ftpami.2021.3057446.   \n281 [4] T. Doan, M. Abbana Bennani, B. Mazoure, G. Rabusseau, and P. Alquier. A theoretical analysis   \n282 of catastrophic forgetting through the ntk overlap matrix. In A. Banerjee and K. Fukumizu, edi  \n283 tors, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics,   \n284 volume 130 of Proceedings of Machine Learning Research, pages 1072–1080. PMLR, 13–15   \n285 Apr 2021. URL https://proceedings.mlr.press/v130/doan21a.html.   \n286 [5] A. S. Douzette. B-splines in machine learning. Master’s thesis, Department of Mathematics,   \n287 University of Oslo, 2017.   \n288 [6] M. Farajtabar, N. Azizan, A. Mott, and A. Li. Orthogonal gradient descent for continual   \n289 learning. In S. Chiappa and R. Calandra, editors, Proceedings of the Twenty Third International   \n290 Conference on Artificial Intelligence and Statistics, volume 108 of Proceedings of Machine   \n291 Learning Research, pages 3762–3773. PMLR, 26–28 Aug 2020. URL https://proceedings.   \n292 mlr.press/v108/farajtabar20a.html.   \n293 [7] R. M. French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences,   \n294 3(4):128–135, 1999.   \n295 [8] S. Golkar, M. Kagan, and K. Cho. Continual learning via neural pruning, 2019. URL https:   \n296 //arxiv.org/abs/1903.04476.   \n297 [9] R. Hadsell, D. Rao, A. A. Rusu, and R. Pascanu. Embracing change: Continual learning in deep   \n298 neural networks. Trends in cognitive sciences, 24(12):1028–1040, 2020.   \n299 [10] B. Hanin. Universal function approximation by deep neural nets with bounded width and relu   \n300 activations. Mathematics, 7(10):992, 2019.   \n301 [11] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal   \n302 approximators. Neural networks, 2(5):359–366, 1989.   \n303 [12] P. Kaushik, A. Gain, A. Kortylewski, and A. Yuille. Understanding catastrophic forgetting and   \n304 remembering in continual learning with optimal relevance mapping. CoRR, abs/2102.11343,   \n305 2021. URL https://arxiv.org/abs/2102.11343.   \n306 [13] R. Kemker, M. McClure, A. Abitino, T. Hayes, and C. Kanan. Measuring catastrophic forgetting   \n307 in neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32,   \n308 2018.   \n309 [14] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan,   \n310 J. Quan, T. Ramalho, A. Grabska-Barwinska, D. Hassabis, C. Clopath, D. Kumaran, and   \n311 R. Hadsell. Overcoming catastrophic forgetting in neural networks. Proceedings of the   \n312 National Academy of Sciences, 114(13):3521–3526, 2017. ISSN 0027-8424. doi: 10.1073/pnas.   \n313 1611835114. URL https://www.pnas.org/content/114/13/3521.   \n314 [15] A. Kratsios. The universal approximation property: Characterization, construction, representa  \n315 tion, and existence. Annals of Mathematics and Artificial Intelligence, 89(5):435–469, 2021.   \n316 doi: 10.1007/s10472-020-09723-1.   \n317 [16] R. Krishnan and P. Balaprakash. Meta continual learning via dynamic programming, 2020.   \n318 URL https://arxiv.org/abs/2008.02219. ",
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+    {
+        "type": "text",
+        "text": "[17] S. H. Lane, M. Flax, D. Handelman, and J. Gelfand. Multi-layer perceptrons with b-spline receptive field functions. In Advances in Neural Information Processing Systems, pages 684–692, 1991.   \n[18] A. C. Li and D. Pathak. Functional regularization for reinforcement learning via learned fourier features, 2021. URL https://arxiv.org/abs/2112.03257.   \n[19] M. McCloskey and N. J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pages 109–165. Elsevier, 1989.   \n[20] K. McRae and P. A. Hetherington. Catastrophic interference is eliminated in pretrained networks. In Proceedings of the 15h Annual Conference of the Cognitive Science Society, pages 723–728, 1993.   \n[21] H. Mehta, A. Cutkosky, and B. Neyshabur. Extreme memorization via scale of initialization, 2020. URL https://arxiv.org/abs/2008.13363.   \n[22] A. Meulemans, F. Carzaniga, J. Suykens, J. a. Sacramento, and B. F. Grewe. A theoretical framework for target propagation. In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 20024– 20036. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/ 2020/file/e7a425c6ece20cbc9056f98699b53c6f-Paper.pdf.   \n[23] A. Robins. Catastrophic forgetting, rehearsal and pseudorehearsal. Connection Science, 7(2): 123–146, 1995.   \n[24] A. A. Rusu, N. C. Rabinowitz, G. Desjardins, H. Soyer, J. Kirkpatrick, K. Kavukcuoglu, R. Pascanu, and R. Hadsell. Progressive neural networks, 2016. URL https://arxiv.org/ abs/1606.04671.   \n[25] S. Scardapane, M. Scarpiniti, D. Comminiello, and A. Uncini. Learning activation functions from data using cubic spline interpolation. In Italian Workshop on Neural Nets, pages 73–83. Springer, 2017.   \n[26] Y. Shin and J. Ghosh. The pi-sigma network: an efficient higher-order neural network for pattern classification and function approximation. In IJCNN-91-Seattle International Joint Conference on Neural Networks, volume i, pages 13–18 vol.1, 1991. doi: 10.1109/IJCNN.1991.155142.   \n[27] H. van Deventer, P. J. van Rensburg, and A. Bosman. Kasam: Spline additive models for function approximation, 2022. URL https://arxiv.org/abs/2205.06376. ",
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+        "text": "Checklist ",
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+        "type": "text",
+        "text": "1. For all authors... ",
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+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [TODO] ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [TODO]   \n(b) Did you mention the license of the assets? [TODO]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [TODO]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
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parse/dev/TVHS5Y4dNvM/TVHS5Y4dNvM.md

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+# CONVOLUTIONS ATTENTION MLPs Patches Are All You Need?
+
+Anonymous authors Paper under double-blind review
+
+# Abstract
+
+Although convolutional networks have been the dominant architecture for vision tasks for many years, recent experiments have shown that Transformer-based models, most notably the Vision Transformer (ViT), may exceed their performance in some settings. However, due to the quadratic runtime of the self-attention layers in Transformers, ViTs require the use of patch embeddings, which group together small regions of the image into single input features, in order to be applied to larger image sizes. This raises a question: Is the performance of ViTs due to the inherently-more-powerful Transformer architecture, or is it at least partly due to using patches as the input representation? In this paper, we present some evidence for the latter: specifically, we propose the ConvMixer, an extremely simple model that is similar in spirit to the ViT and the even-more-basic MLP-Mixer in that it operates directly on patches as input, separates the mixing of spatial and channel dimensions, and maintains equal size and resolution throughout the network. In contrast, however, the ConvMixer uses only standard convolutions to achieve the mixing steps. Despite its simplicity, we show that the ConvMixer outperforms the ViT, MLP-Mixer, and some of their variants for similar parameter counts and data set sizes, in addition to outperforming classical vision models such as the ResNet. Our code is available at https://github.com/tmp-iclr/convmixer.
+
+# 1 Introduction
+
+For many years, convolutional neural networks have been the dominant architecture for deep learning systems applied to computer vision tasks. But recently, architectures based upon Transformer models, e.g., the so-called Vision Transformer architecture (Dosovitskiy et al., 2020), have demonstrated compelling performance in many of these tasks, often outperforming classical convolutional architectures, especially for large data sets. An understandable assumption, then, is that it is only a matter of time before Transformers become the dominant architecture for vision domains, just as they have for language processing. In order to apply Transformers to images, however, the representation had to be changed: because the computational cost of the self-attention layers used in Transformers would scale quadratically with the number of pixels per image if applied naively at the per-pixel level, the compromise was to first split the image into multiple “patches”, linearly embed them, and then apply the transformer directly to this collection of patches.
+
+![](images/1173f087f8ac39da4f86d64104c20dd719f3a66b7c4817498a0ed0bbddaddd8e.jpg)  
+Figure 1: Accuracy vs. parameters, trained and evaluated on ImageNet-1k.
+
+In this work, we explore the question of whether, fundamentally, the strong performance of vision transformers may result more from this patch-based representation than from the Transformer architecture itself. We develop a very simple convolutional architecture which we dub the “ConvMixer” due to its similarity to the recently-proposed MLP-Mixer (Tolstikhin et al., 2021). This architecture is similar to the Vision Transformer (and MLP-Mixer) in many respects: it directly operates on patches, it maintains an equal-resolution-and-size representation throughout all layers, it does no downsampling of the representation at successive layers, and it separates “channel-wise mixing”
+
+![](images/229107206062b568c8ab98c917943152041566e7e38345a854151e830ba98de1.jpg)  
+Figure 2: ConvMixer uses “tensor layout” patch embeddings to preserve locality, and then applies $d$ copies of a simple fully-convolutional block consisting of large-kernel depthwise convolution followed by pointwise convolution, before finishing with global pooling and a simple linear classifier.   
+Figure 3: Implementation of ConvMixer in PyTorch; see Appendix D for more implementations.
+
+def ConvMixer(h, depth, kernel_size $^ { ! = 9 }$ , patch_size $^ { - 7 }$ , n_classes $\scriptstyle \left. = 1 0 0 0 \right|$ ):   
+2 Seq, ActBn $= \mathsf { n n }$ .Sequential, lambda x: Seq(x, nn.GELU(), nn.BatchNorm2d(h))   
+3 Residual $=$ type('Residual', (Seq,), {'forward': lambda self, x: self[0](x) + x})   
+4 return Seq(ActBn(nn.Conv2d(3, h, patch_size, stride=patch_size)),   
+5 \*[Seq(Residual(ActBn(nn.Conv2d(h, h, kernel_size, groups=h, padding $=$ "same"))),   
+6 ActBn(nn.Conv2d(h, h, 1))) for i in range(depth)],   
+7 nn.AdaptiveAvgPool2d((1,1)), nn.Flatten(), nn.Linear(h, n_classes))
+
+from the “spatial mixing” of information. But unlike the Vision Transformer and MLP-Mixer, our architecture does all these operations via only standard convolutions.
+
+The chief result we show in this paper is that this ConvMixer architecture, despite its extreme simplicity (it can be implemented in $\approx 6$ lines of dense PyTorch code), outperforms both “standard” computer vision models such as ResNets of similar parameter counts and some corresponding Vision Transformer and MLP-Mixer variants, even with a slate of additions intended to make those architectures more performant on smaller data sets. Importantly, this is despite the fact that we did not design our experiments to maximize accuracy nor speed, in contrast to the models we compared against. Our results suggest that, at least to some extent, the patch representation itself may be a critical component to the “superior” performance of newer architectures like Vision Transformers. While these results are naturally just a snapshot, and more experiments are required to exactly disentangle the effect of patch embeddings from other factors, we believe that this provides a strong “convolutionalbut-patch-based” baseline to compare against for more advanced architectures in the future.
+
+# 2 A Simple Model: ConvMixer
+
+Our model, dubbed ConvMixer, consists of a patch embedding layer followed by repeated applications of a simple fully-convolutional block. We maintain the spatial structure of the patch embeddings, as illustrated in Fig. 2. Patch embeddings with patch size $p$ and embedding dimension $h$ can be implemented as convolution with $c _ { \mathsf { i n } }$ input channels, $h$ output channels, kernel size $p$ , and stride $p$ :
+
+$$
+z _ { 0 } = \mathsf { B N } ( \sigma \{ \mathsf { C o n v } _ { c _ { \mathrm { i n } }  h } ( X , \mathsf { s t r i d e } { = } p , \mathsf { k e r n e l } _ { - } \mathsf { s i z e } { = } p ) \} )
+$$
+
+The ConvMixer block itself consists of depthwise convolution (i.e., grouped convolution with groups equal to the number of channels, $h$ ) followed by pointwise (i.e., kernel size $1 \times 1$ ) convolution. As we will explain in Sec. 3, ConvMixers work best with unusually large kernel sizes for the depthwise convolution. Each of the convolutions is followed by an activation and post-activation BatchNorm:
+
+$$
+\begin{array} { r } { z _ { l } ^ { \prime } = \mathsf { B N } \left( \sigma \{ \mathsf { C o n v D e p t h w i s e } ( z _ { l - 1 } ) \} \right) + z _ { l - 1 } } \\ { z _ { l + 1 } = \mathsf { B N } \left( \sigma \{ \mathsf { C o n v P o i n t w i s e } ( z _ { l } ^ { \prime } ) \} \right) } \end{array}
+$$
+
+After many applications of this block, we perform global pooling to get a feature vector of size $h$ , which we pass to a softmax classifier. See Fig. 3 for an implementation of ConvMixer in PyTorch.
+
+Design parameters. An instantiation of ConvMixer depends on four parameters: (1) the “width” or hidden dimension $h$ (i.e., the dimension of the patch embeddings), (2) the depth $d$ , or the number of repetitions of the ConvMixer layer, (3) the patch size $p$ which controls the internal resolution of the model, (4) the kernel size $k$ of the depthwise convolutional layer. We name ConvMixers after their hidden dimension and depth, like ConvMixer- $h / d$ . We refer to the original input size $n$ divided by the patch size $p$ as the internal resolution; note, however, that ConvMixers support variable-sized inputs.
+
+<table><tr><td colspan="8">Current “Most Interesting” ConvMixer Configurations vs. Other Simple Models</td></tr><tr><td>Network</td><td>Patch Size</td><td>Kernel Size</td><td># Params (×106）</td><td>Throughput (img/sec)</td><td>Act. Fn.</td><td>#Epochs</td><td>ImNet top-1 (%)</td></tr><tr><td>ConvMixer-1536/20 ConvMixer-768/32</td><td>7 7</td><td>9 7</td><td>51.6 21.1</td><td>89 203</td><td>G R</td><td>150 300</td><td>81.37 80.16</td></tr><tr><td>ResNet-152</td><td></td><td>3</td><td>60.2</td><td>872</td><td>R</td><td>150</td><td>79.64</td></tr><tr><td>DeiT-B</td><td>1 16</td><td></td><td>86</td><td>703</td><td>G</td><td>300</td><td>81.8</td></tr><tr><td></td><td>8</td><td></td><td>129</td><td>140</td><td>G</td><td>400</td><td>81.0</td></tr><tr><td>ResMLP-B24/8</td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 1: Models trained and evaluated on $2 2 4 \times 2 2 4$ ImageNet-1k only. See more in Appendix A.
+
+Motivation. Our architecture is based on the idea of mixing, as in Tolstikhin et al. (2021). In particular, we chose depthwise convolution to mix spatial locations and pointwise convolution to mix channel locations. A key idea from previous work is that MLPs and self-attention can mix distant spatial locations, i.e., they can have an arbitrarily large receptive field. Consequently, we used convolutions with an unusually large kernel size to mix distant spatial locations.
+
+While self-attention and MLPs are theoretically more flexible, allowing for large receptive fields and content-aware behavior, the inductive bias of convolution is well-suited to vision tasks and leads to high data efficiency. By using such a standard operation, we also get a glimpse into the effect of the patch representation itself in contrast to the conventional pyramid-shaped, progressivelydownsampling design of convolutional networks.
+
+# 3 Experiments
+
+Training setup. We primarily evaluate ConvMixers on ImageNet-1k classification without any pretraining or additional data. We added ConvMixer to the timm framework (Wightman, 2019) and trained it with nearly-standard settings: we used RandAugment (Cubuk et al., 2020), mixup (Zhang et al., 2017), CutMix (Yun et al., 2019), random erasing (Zhong et al., 2020), and gradient norm clipping in addition to default timm augmentation. We used the AdamW (Loshchilov & Hutter, 2018) optimizer and a simple triangular learning rate schedule. Due to limited compute, we did absolutely no hyperparameter tuning on ImageNet and trained for fewer epochs than competitors. Consequently, our models could be over- or under-regularized, and the accuracies we report likely underestimate the capabilities of our model.
+
+Results. A ConvMixer-1536/20 with 52M parameters can achieve $8 1 . 4 \%$ top-1 accuracy on ImageNet, and a ConvMixer-768/32 with 21M parameters $8 0 . 2 \%$ (see Table 1). Wider ConvMixers seem to converge in fewer epochs, but are memory- and compute-hungry. They also work best with large kernel sizes: ConvMixer- $1 5 3 6 / 2 0 \log \approx 1 \%$ accuracy when reducing the kernel size from $k = 9$ to $k = 3$ (we discuss kernel sizes more in Appendix A & B). ConvMixers with smaller patches are substantially better in our experiments, similarly to Sandler et al. (2019); we believe larger patches require deeper ConvMixers. With everything held equal except increasing the patch size from 7 to 14, ConvMixer-1536/20 achieves $7 8 . 9 \%$ top-1 accuracy but is around $4 \times$ faster. We trained one model with ReLU to demonstrate that GELU (Hendrycks & Gimpel, 2016), which is popular in recent isotropic models, isn’t necessary.
+
+Comparisons. Our model and ImageNet1k-only training setup closely resemble that of recent patchbased models like DeiT (Touvron et al., 2020). Due to ConvMixer’s simplicity, we focus on comparing to only the most basic isotropic patch-based architectures adapted to the ImageNet-1k setting, namely DeiT and ResMLP. Attempting a fair comparison with a standard baseline, we trained ResNets using exactly the same parameters as ConvMixers; while this choice of parameters is suboptimal (Wightman et al., 2021), it is likely also suboptimal for ConvMixers, since we did no hyperparameter tuning.
+
+Looking at Table 1 and Fig. 1, ConvMixers achieve competitive accuracies for a given parameter budget: ConvMixer-1536/20 outperforms both ResNet-152 and ResMLP-B24 despite having substantially fewer parameters and is competitive with DeiT-B. ConvMixer-768/32 uses just a third of the parameters of ResNet-152, but is similarly accurate. Note that unlike ConvMixer, the DeiT and ResMLP results involved hyperparameter tuning, and when substantial resources are dedicated to tuning ResNets, including training for twice as many epochs, they only outperform an equivalently-sized ConvMixer by $\approx 0 . 2 \%$ (Wightman et al., 2021). However, ConvMixers are substantially slower at inference than the competitors, likely due to their smaller patch size; hyperparameter tuning and optimizations could narrow this gap. For more discussion and comparisons, see Table 2 and Appendix A.
+
+CIFAR-10 Experiments. We also performed smaller-scale experiments on CIFAR-10, where ConvMixers achieve over $9 6 \%$ accuracy with as few as $0 . 7 { \bf M }$ parameters, demonstrating the data efficiency of the convolutional inductive bias. Details of these experiments are presented in Appendix B.
+
+# 4 Related Work
+
+Isotropic architectures. Vision transformers have inspired a new paradigm of “isotropic” architectures, i.e., those with equal size and shape throughout the network, which use patch embeddings for the first layer. These models look similar to repeated transformer-encoder blocks (Vaswani et al., 2017) with different operations replacing the self-attention and MLP operations. For example, MLP-Mixer (Tolstikhin et al., 2021) replaces them both with MLPs applied across different dimensions (i.e., spatial and channel location mixing); ResMLP (Touvron et al., 2021a) is a data-efficient variation on this theme. CycleMLP (Chen et al., 2021), gMLP (Liu et al., 2021a), and vision permutator (Hou et al., 2021), replace one or both blocks with various novel operations. These are all quite performant, which is typically attributed to the novel choice of operations. In contrast, Melas-Kyriazi (2021) proposed an MLP-based isotropic vision model, and also hypothesized patch embeddings could be behind its performance. ResMLP tried replacing its linear interaction layer with (small-kernel) convolution and achieved good performance, but kept its MLP-based cross-channel layer and did not explore convolutions further. As our investigation of ConvMixers suggests, these works may conflate the effect of the new operations (like self-attention and MLPs) with the effect of the use of patch embeddings and the resulting isotropic architecture.
+
+A study predating vision transformers investigates isotropic (or “isometric”) MobileNets (Sandler et al., 2019), and even implements patch embeddings under another name. Their architecture simply repeats an isotropic MobileNetv3 block. They identify a tradeoff between patch size and accuracy that matches our experience, and train similarly performant models (see Appendix A, Table 2). However, their block is substantially more complex than ours; simplicity and motivation sets our work apart.
+
+Patches aren’t all you need. Several papers have increased vision transformer performance by replacing standard patch embeddings with a different stem: Xiao et al. (2021) and Yuan et al. (2021a) use a standard convolutional stem, while Yuan et al. (2021b) repeatedly combines nearby patch embeddings. However, this conflates the effect of using patch embeddings with the effect of adding convolution or similar inductive biases e.g., locality. We attempt to focus on the use of patches.
+
+CNNs meet ViTs. Many efforts have been made to incorporate features of convolutional networks into vision transformers and vice versa. Self-attention can emulate convolution (Cordonnier et al., 2019) and can be initialized or regularized to be like it (d’Ascoli et al., 2021); other works simply add convolution operations to transformers (Dai et al., 2021; Guo et al., 2021), or include downsampling to be more like traditional pyramid-shaped convolutional networks (Wang et al., 2021). Conversely, self-attention or attention-like operations can supplement or replace convolution in ResNet-style models (Bello et al., 2019; Ramachandran et al., 2019; Bello, 2021). While all of these attempts have been successful in one way or another, they are orthogonal to this work, which aims to emphasize the effect of the architecture common to most ViTs by showcasing it with a less-expressive operation.
+
+# 5 Conclusion
+
+We presented ConvMixers, an extremely simple class of models that independently mixes the spatial and channel locations of patch embeddings using only standard convolutions. We also highlighted that using large kernel sizes, inspired by the large receptive fields of ViTs and MLP-Mixers, provides a substantial performance boost. While neither our model nor our experiments were designed to maximize accuracy or speed, i.e., we did not search for good hyperparameters, ConvMixers outperform the Vision Transformer and MLP-Mixer, and are competitive with ResNets, DeiTs, and ResMLPs.
+
+We provided evidence that the increasingly common “isotropic” architecture with a simple patch embedding stem is itself a powerful template for deep learning. Patch embeddings allow all the downsampling to happen at once, immediately decreasing the internal resolution and thus increasing the effective receptive field size, making it easier to mix distant spatial information. Our title, while an exaggeration, points out that attention isn’t the only export from language processing into computer vision: tokenizing inputs, i.e., using patch embeddings, is also a powerful and important takeaway.
+
+While our model is not state-of-the-art, we find its simple patch-mixing design to be compelling. We hope that ConvMixers can serve as a baseline for future patch-based architectures with novel operations, or that they can provide a basic template for new conceptually simple and performant models.
+
+Future work. We are optimistic that a deeper ConvMixer with larger patches could reach a desirable tradeoff between accuracy, parameters, and throughput after longer training and more regularization and hyperparameter tuning, similarly to how Wightman et al. (2021) enhanced ResNet performance through carefully-designed training regimens. Low-level optimization of large-kernel depthwise convolution could substantially increase throughput, and small enhancements to our architecture like the addition of bottlenecks or a more expressive classifier could trade simplicity for performance.
+
+Due to its large internal resolution and isotropic design, ConvMixer may be especially well-suited for semantic segmentation, and it would be useful to run experiments on this task with a ConvMixer-like model and on other tasks such as object detection. More experiments could be designed to more clearly extricate the effect of patch embeddings from other architectural choices. In particular, for a more in-depth comparison to ViTs and MLP-Mixers, which excel when trained on very large data sets, it is important to investigate the performance of ConvMixers in the regime of large-scale pre-training.
+
+A note on paper length. Expecting more text in this paper? Wondering if it’s a workshop paper we hastily submitted to ICLR? No. This paper presents a simple idea, one where we genuinely believe that a short paper presentation is more effective. Do we really need exactly 8 (now 9? 10?) pages to describe every machine learning architecture and algorithm in existence? We proposed an incredibly simple architecture and made a very simple point that we think is worth more discussion: patches work well in convolutional architectures. We think that four and a half pages is more than enough space for this. The details of the experiments and architectures are in the appendix for those who want to read through it all.
+
+# References
+
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+
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+
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+
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+
+Prajit Ramachandran, Niki Parmar, Ashish Vaswani, Irwan Bello, Anselm Levskaya, and Jonathon Shlens. Stand-alone self-attention in vision models. arXiv preprint arXiv:1906.05909, 2019.
+
+Mark Sandler, Jonathan Baccash, Andrey Zhmoginov, and Andrew Howard. Non-discriminative data or weak model? on the relative importance of data and model resolution. In Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops, pp. 0–0, 2019.
+
+Ilya Tolstikhin, Neil Houlsby, Alexander Kolesnikov, Lucas Beyer, Xiaohua Zhai, Thomas Unterthiner, Jessica Yung, Daniel Keysers, Jakob Uszkoreit, Mario Lucic, et al. Mlp-mixer: An all-mlp architecture for vision. arXiv preprint arXiv:2105.01601, 2021.
+
+Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. Training data-efficient image transformers & distillation through attention. arXiv preprint arXiv:2012.12877, 2020.
+
+Hugo Touvron, Piotr Bojanowski, Mathilde Caron, Matthieu Cord, Alaaeldin El-Nouby, Edouard Grave, Armand Joulin, Gabriel Synnaeve, Jakob Verbeek, and Hervé Jégou. Resmlp: Feedforward networks for image classification with data-efficient training. arXiv preprint arXiv:2105.03404, 2021a.
+
+Hugo Touvron, Matthieu Cord, Alexandre Sablayrolles, Gabriel Synnaeve, and Hervé Jégou. Going deeper with image transformers. arXiv preprint arXiv:2103.17239, 2021b.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017.
+
+Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao. Pyramid vision transformer: A versatile backbone for dense prediction without convolutions. arXiv preprint arXiv:2102.12122, 2021.
+
+Ross Wightman. Pytorch image models. https://github.com/rwightman/ pytorch-image-models, 2019.   
+Ross Wightman, Hugo Touvron, and Hervé Jégou. Resnet strikes back: An improved training procedure in timm, 2021.   
+Tete Xiao, Mannat Singh, Eric Mintun, Trevor Darrell, Piotr Dollár, and Ross Girshick. Early convolutions help transformers see better. arXiv preprint arXiv:2106.14881, 2021.   
+Kun Yuan, Shaopeng Guo, Ziwei Liu, Aojun Zhou, Fengwei Yu, and Wei Wu. Incorporating convolution designs into visual transformers. arXiv preprint arXiv:2103.11816, 2021a.   
+Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zihang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan. Tokens-to-token vit: Training vision transformers from scratch on imagenet. arXiv preprint arXiv:2101.11986, 2021b.   
+Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 6023–6032, 2019.   
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+
+# A Comparison to other models
+
+Table 2: Throughputs measured on an RTX8000 GPU using batch size 64 and fp16. ConvMixers and ResNets trained ourselves. Other statistics: DeiT (Touvron et al., 2020), ResMLP (Touvron et al., 2021a), Swin (Liu et al., 2021b), ViT (Dosovitskiy et al., 2020), MLP-Mixer (Tolstikhin et al., 2021), Isotropic MobileNets (Sandler et al., 2019). We think models with matching colored dots $( \bullet )$ are informative to compare with each other. †Throughput tested, but not trained. Activations: ReLU, GELU.   
+
+<table><tr><td colspan="8">Comparison with other simple models trained on ImageNet-1k only with input size 224.</td></tr><tr><td>Network</td><td>Patch Size</td><td>Kernel Size</td><td># Params (×106）</td><td>Throughput (img/sec)</td><td>Act. Fn.</td><td>#Epochs</td><td>ImNet top-1 (%)</td></tr><tr><td>ConvMixer-1536/20 ：</td><td>7</td><td>9</td><td>51.6</td><td>89</td><td>G G</td><td>150 150</td><td>81.37</td></tr><tr><td>ConvMixer-1536/20</td><td>7</td><td>3</td><td>49.4</td><td>136</td><td>G</td><td></td><td>80.43</td></tr><tr><td>ConvMixer-1536/20</td><td>14</td><td>9</td><td>52.3</td><td>334</td><td></td><td>150 300</td><td>78.92</td></tr><tr><td>ConvMixer-768/32·</td><td>7</td><td>7</td><td>21.1</td><td>203</td><td>R</td><td></td><td>80.16</td></tr><tr><td>ConvMixer-1024/16</td><td>7</td><td>9</td><td>19.4</td><td>173</td><td>G</td><td>100</td><td>79.45</td></tr><tr><td>ConvMixer-1024/12</td><td>7</td><td>8</td><td>14.6</td><td>248</td><td>G</td><td>90</td><td>77.75</td></tr><tr><td>ConvMixer-512/16</td><td>7</td><td>8</td><td>5.4</td><td>403</td><td>G</td><td>90</td><td>73.76</td></tr><tr><td>ConvMixer-512/12</td><td>7</td><td>8</td><td>4.2</td><td>532</td><td>G</td><td>90</td><td>72.59</td></tr><tr><td>ConvMixer-768/32</td><td>14</td><td>3</td><td>20.2</td><td>1258</td><td>R</td><td>300</td><td>74.93</td></tr><tr><td>ConvMixer-1024/20 ·</td><td>14</td><td>9</td><td>24.4</td><td>520</td><td>G</td><td>150</td><td>76.94</td></tr><tr><td>ResNet-152 .</td><td>1</td><td>3</td><td>60.2</td><td>872</td><td>R</td><td>150</td><td>79.64</td></tr><tr><td>ResNet-101</td><td></td><td>3</td><td>44.6</td><td>1040</td><td>R</td><td>150</td><td>78.33</td></tr><tr><td>ResNet-50</td><td>1</td><td>3</td><td>25.6</td><td>1942</td><td>R</td><td>150</td><td>76.32</td></tr><tr><td>DeiT-Bt</td><td>7</td><td>1</td><td>86.7</td><td>77</td><td>G</td><td>1</td><td></td></tr><tr><td>DeiT-St</td><td>7</td><td>1</td><td>22.1</td><td>164</td><td>G</td><td></td><td>1</td></tr><tr><td>DeiT-Tit</td><td>7</td><td>1</td><td>5.7</td><td>327</td><td>G</td><td>1</td><td>一</td></tr><tr><td>DeiT-B·</td><td>16</td><td>1</td><td>86</td><td>703</td><td>G</td><td>300</td><td>1 81.8</td></tr><tr><td>DeiT-S·</td><td>16</td><td></td><td>22</td><td>1491</td><td>G</td><td>300</td><td>79.8</td></tr><tr><td>DeiT-Ti·</td><td>16</td><td>一</td><td>5.7</td><td>2727</td><td>G</td><td>300</td><td>72.2</td></tr><tr><td>ResMLP-S12/8·</td><td>8</td><td></td><td>22.1</td><td>638</td><td>G</td><td>400</td><td></td></tr><tr><td>ResMLP-B24/8·</td><td>8</td><td>一</td><td>129</td><td>140</td><td>G</td><td>400</td><td>79.1 81.0</td></tr><tr><td>ResMLP-B24</td><td>16</td><td>一</td><td>116</td><td>1191</td><td>G</td><td>400</td><td>81.0</td></tr><tr><td>Swin-S ·</td><td>4</td><td></td><td>50</td><td>566</td><td>G</td><td>300</td><td>83.0</td></tr><tr><td>Swin-T ·</td><td>4</td><td></td><td>29</td><td>884</td><td>G</td><td>300</td><td>81.3</td></tr><tr><td>ViT-B/16·</td><td>16</td><td></td><td>86</td><td>704</td><td>G</td><td>300</td><td>77.9</td></tr><tr><td></td><td></td><td>一</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixer-B/16·</td><td>16</td><td>1</td><td>59</td><td>816</td><td>G</td><td>300</td><td>76.44</td></tr><tr><td>Isotropic MobileNetv3 ·</td><td>8</td><td>3</td><td>20</td><td></td><td>R</td><td>1</td><td>80.6</td></tr><tr><td>Isotropic MobileNetv3 ·</td><td>16</td><td>3</td><td>20</td><td>一</td><td>R</td><td>1</td><td>77.6</td></tr></table>
+
+Experiment overview. We did not design our experiments to maximize accuracy: We chose “common sense” parameters for timm and its augmentation settings, found that it worked well for a ConvMixer-1024/12, and stuck with them for the proceeding experiments. We admit this is not an optimal strategy, however, we were aware from our early experiments on CIFAR-10 that results seemed robust to various small changes. We did not have access to sufficient compute to attempt to tune hyperparameters for each model: e.g., larger ConvMixers could probably benefit from more regularization than we chose, and smaller ones from less regularization. Keeping the parameters the same across ConvMixer instances seemed more reasonable than guessing for each.
+
+However, to some extent, we changed the number of epochs per model: for earlier experiments, we merely wanted a “proof of concept”, and used only 90–100 epochs. Once we saw potential, we increased this to 150 epochs and trained some larger models, namely ConvMixer-1024/20 with $p = 1 4$ patches and ConvMixer-1536/20 with $p = 7$ patches. Then, believing that we should explore deeper-but-less-wide ConvMixers, and knowing from CIFAR-10 that the deeper models converged more slowly, we trained ConvMixer-768/32s with $p = 1 4$ and $p = 7$ for 300 epochs. Of course, training time was a consideration: ConvMixer-1536/20 took about 9 days to train (on $1 0 \times \mathrm { R T X 8 0 0 0 s } )$ 150 epochs, and ConvMixer-768/32 is over twice as fast, making 300 epochs more feasible.
+
+If anything, we believe that in the worst case, the lack of parameter tuning in our experiments resulted in underestimating the accuracies of ConvMixers. Further, due to our limited compute and the fact that large models (particularly ConvMixers) are expensive to train on large data sets, we generally trained our models for fewer epochs than competition like DeiT and ResMLP (see Table 2).
+
+A note on throughput. We measured throughput using batches of 64 images in half precision on a single RTX8000 GPU, averaged over 20 such batches. In particular, we measured CUDA execution time rather than “wall-clock” time. We noticed discrepancies in the relative throughputs of models, e.g., Touvron et al. (2020) reports that ResNet-152 is $2 \times$ faster than DeiT-B, but our measurements show that it is only $1 . 2 5 \times$ faster. We therefore speculate that our throughputs may underestimate the performance of ResNets and ConvMixers relative to the transformers. The difference may be due to using RTX8000 rather than V100 GPUs, or other low-level differences. Our throughputs were similar for batch sizes 32 and 128.
+
+ResNets. As a simple baseline to which to compare ConvMixers, we trained three standard ResNets using exactly the same training setup and parameters as ConvMixer-1536/20. Despite having fewer parameters and being architecturally much simpler, ConvMixers substantially outperform these ResNets in terms of accuracy. A possible confounding factor is that ConvMixers use GELU, which may boost performance, while ResNets use ReLU. In an attempt to rule out this confound, we used ReLU in a later ConvMixer-768/32 experiment and found that it still achieved competitive accuracy. We also note that the choice of ReLU vs. GELU was not important on CIFAR-10 experiments (see Table 3). However, ConvMixers do have substantially less throughput.
+
+DeiTs. We believe that DeiT is the most reasonable comparison in terms of vision transformers: It only adds additional regularization, as opposed to architectural additions in the case of CaiT (Touvron et al., 2021b), and is then essentially a “vanilla” ViT modulo the distillation token (we don’t consider distilled architectures). In terms of a fixed parameter budget, ConvMixers generally outperform DeiTs. For example, ConvMixer-1536/20 is only $0 . 4 3 \%$ less accurate than DeiT-B despite having over 30M fewer parameters; ConvMixer-768/32 is $0 . 3 6 \%$ more accurate than DeiT-S despite having 0.9M fewer parameters; and ConvMixer-512/16 is $0 . 3 9 \%$ more accurate than DeiT-Ti for nearly the same number of parameters. Admittedly, none of the ConvMixers are very competitive in terms of throughput, with the closest being the ConvMixer-512/16 which is $5 \times$ slower than DeiT-Ti.
+
+A confounding factor is the difference in patch size between DeiT and ConvMixer; DeiT uses $p = 1 6$ while ConvMixer uses $p = 7$ . This means DeiT is substantially faster. However, ConvMixers using larger patches are not as competitive. While we were not able to train DeiTs with larger patch sizes, it is possible that they would outperform ConvMixers on the parameter count vs. accuracy curve; however, we tested their throughput for $p = 7$ , and they are even slower than ConvMixers. Given the difference between convolution and self-attention, we are not sure it is salient to control for patch size differences.
+
+DeiTs were subject to more hyperparameter tuning than ConvMixers, as well as longer training times. They also used stochastic depth while we did not, which can in some cases contribute percent differences in model accuracy (Touvron et al., 2021a). It is therefore possible that further hyperparameter tuning and more epochs for ConvMixers could close the gap between the two architectures for large patches, e.g., $p = 1 6$ .
+
+ResMLPs. Similarly to DeiT for ViT, we believe that ResMLP is the most relevant MLP-Mixer variant to compare against. Unlike DeiT, we can compare against instances of ResMLP with similar patch size: ResMLP-B24/8 has $p = 8$ patches, and underperforms ConvMixer-1536/20 by $0 . 3 7 \%$ , despite having over twice the number of parameters; it also has similarly low throughput. ConvMixer-768/32 also outperforms ResMLP-S12/8 for millions fewer parameters, but $3 \times$ less throughput.
+
+ResMLP did not significantly improve in terms of accuracy for halving the patch size from 16 to 8, which shows that smaller patches do not always lead to better accuracy for a fixed architecture and regularization strategy (e.g., training a $p = 8$ DeiT may be challenging).
+
+Swin Transformers. While we intend to focus on the most basic isotropic, patch-based architectures for fair comparisons with ConvMixer, it is also interesting to compare to a more complicated model that is closer to state-of-the-art. For a similar parameter budget, ConvMixer is around $1 . 2 \mathrm { - } 1 . 6 \%$ less accurate than the Swin Transformer, while also being $4 { - } 6 \times$ slower. However, considering we did not attempt to tune or optimize our model in any way, we find it surprising that an exceedingly simple patch-based model that uses only plain convolution does not lag too far behind Swin Transformer.
+
+Isotropic MobileNets. These models are closest in design to ours, despite using a repeating block that is substantially more complex than the ConvMixer one. Despite this, for a similar number of parameters, we can get similar performance. Notably, isotropic MobileNets seem to suffer less from larger patch sizes than ConvMixers, which makes us optimistic that sufficient parameter tuning could lead to more performant large-patch ConvMixers.
+
+Other models. We included ViT and MLP-Mixer instances in our table, though they are not competitive with ConvMixer, DeiT, or ResMLP, even though MLP-Mixer has comparable regularization to ConvMixer. That is, ConvMixer seems to outperform MLP-Mixer and ViT, while being closer to complexity to them in terms of design and training regime than the other competitors, DeiT and ResMLP.
+
+Kernel size. While we found some evidence that larger kernels are better on CIFAR-10, we wanted to see if this finding transferred to ImageNet. Consequently, we trained our best-performing model, ConvMixer-1536/20, with kernel size $k = 3$ rather than $k = 9$ . This resulted in a decrease of $0 . 9 4 \%$ top-1 accuracy, which we believe is quite significant relative to the mere 2.2M additional parameters. However, $k = 3$ is substantially faster than $k = 9$ for spatial-domain convolution; we speculate that low-level optimizations could close the performance gap to some extent, e.g., by using implicit instead of explicit padding. Since large-kernel convolutions throughout a model are unconventional, there has likely been low demand for such optimizations.
+
+# B Experiments on CIFAR-10
+
+Residual connections. We experimented with leaving out one, the other, or both residual connections before settling on the current configuration, and consequently chose to leave out the second residual connection. Our baseline model without the connection achieves $9 5 . 8 8 \%$ accuracy, while including the connection reduces it to $9 4 . 7 8 \%$ . Surprisingly, we see only a $0 . 3 1 \%$ decrease in accuracy for removing all residual connections. We acknowledge that these findings for residual connections may not generalize to deeper ConvMixers trained on larger data sets.
+
+Table 3: Small ablation study of training a ConvMixer-256/8 on CIFAR-10.   
+
+<table><tr><td colspan="2">Ablation of ConvMixer-256/8 on CIFAR-10</td></tr><tr><td>Ablation</td><td>CIFAR-10 Acc. (%)</td></tr><tr><td>Baseline</td><td>95.88</td></tr><tr><td>- Residual in Eq. 2</td><td>95.57</td></tr><tr><td>+ Residual in Eq. 3</td><td>94.78</td></tr><tr><td>BatchNorm→LayerNorm GELU→ReLU</td><td>94.44</td></tr><tr><td></td><td>95.51</td></tr><tr><td>- Mixup and CutMix</td><td>95.92</td></tr><tr><td>-Random Erasing</td><td>95.24</td></tr><tr><td>- RandAug</td><td>92.86</td></tr><tr><td>-Random Scaling</td><td>86.24</td></tr><tr><td>- Gradient Norm Clipping</td><td>86.33</td></tr></table>
+
+Normalization. Our model is conceptually similar to the vision transformer and MLP-Mixer, both of which use LayerNorm instead of BatchNorm. We attempted to use LayerNorm instead, and saw a decrease in performance of around $1 \%$ as well as slower convergence (see Table 3). However, this was for a relatively shallow model, and we cannot guarantee that LayerNorm would not hinder ImageNet-scale models to an even larger degree. We note that the authors of ResMLP also saw a relatively small increase in accuracy for replacing LayerNorm with BatchNorm, but for a largerscale experiment (Touvron et al., 2021a). We conclude that BatchNorm is no more crucial to our architecture than other regularizations or parameter settings (e.g., kernel size).
+
+Having settled on an architecture, we proceeded to adjust its parameters $h , d , p , k$ as well as weight decay on CIFAR-10 experiments. (Initially, we took the unconventional approach of excluding weight decay since we were already using strong regularization in the form of RandAug and mixup.) We acknowledge that tuning our architecture on CIFAR-10 does not necessarily generalize to performance on larger data sets, and that this is a limitation of our study.
+
+# B.1 Results
+
+ConvMixers are quite performant on CIFAR-10, easily achieving $> 9 1 \%$ accuracy for as little as 100, 000 parameters, or $> 9 6 \%$ accuracy for only 887, 000 parameters (see Table 4). With additional refinements e.g., a more expressive classifier or bottlenecks, we think that ConvMixer could be even more competitive. For all experiments, we trained for 200 epochs on CIFAR-10 with RandAug, mixup, cutmix, random erasing, gradient norm clipping, and the standard augmentations in timm. We remove some of these augmentations in Table 3, finding that RandAug and random scaling (“default” in timm) are very important, each accounting for over $3 \%$ of the accuracy.
+
+Scaling ConvMixer. We adjusted the hidden dimension $h$ and the depth $d$ , finding that deeper networks take longer to converge while wider networks converge faster. That said, increasing the width or the depth is an effective way to increase accuracy; a doubling of depth incurs less compute than a doubling of width. The number of parameters in a ConvMixer is given exactly by:
+
+$$
+\# \mathsf { p a r a m s } = h [ d ( k ^ { 2 } + h + 6 ) + c _ { \mathsf { i n } } p ^ { 2 } + n _ { \mathsf { c l a s s e s } } + 3 ] + n _ { \mathsf { c l a s s e s } } ,
+$$
+
+including affine scaling parameters in BatchNorm layers, convolutional kernels, and the classifier.
+
+Kernel size. We initially hypothesized that large kernels would be important for ConvMixers, as they would allow the mixing of distant spatial information similarly to unconstrained MLPs or selfattention layers. We tried to investigate the effect of kernel size on CIFAR-10: we fixed the model to be a ConvMixer-256/8, and increased the kernel size by 2s from 3 to 15.
+
+Using a kernel size of 3, the ConvMixer only achieves $9 3 . 6 1 \%$ accuracy. Simply increasing it to 5 gives an additional $1 . 5 0 \%$ accuracy, and further to 7 an additional $0 . 6 1 \%$ . The gains afterwards are relatively marginal, with kernel size 15 giving an additional $0 . 2 8 \%$ accuracy. It could be that with more training iterations or more regularization, the effect of larger kernels would be more pronounced. Nonetheless, we concluded that ConvMixers benefit from larger-than-usual kernels, and thus used kernel sizes 7 or 9 in most of our later experiments.
+
+It is conventional wisdom that large-kernel convolutions can be “decomposed” into stacked smallkernel convolutions with activations between them, and it is therefore standard practice to use $k = 3$ convolutions, stacking more of them to increase the receptive field size with additional benefits from nonlinearities. This raises a question: is the benefit of larger kernels in ConvMixer actually better than simply increasing the depth with small kernels? First, we note that deeper networks are generally harder to train, so by increasing the kernel size independently of the depth, we may recover some of the benefits of depth without making it harder for signals to “propagate back” through the network. To test this, we trained a ConvMixer-256/10 with $k = 3$ (698K parameters) in the same setting as a ConvMixer-256/8 with $k = 9$ (707K parameters), i.e., we increased depth in a smallkernel model to roughly match the parameters of a large-kernel model. The ConvMixer-256/10 achieved $9 4 . 2 9 \%$ accuracy ( $1 . 5 \%$ less), which provides more evidence for the importance of larger kernels in ConvMixers. Next, instead of fixing the parameter budget, we tripled the depth (using the intuition that 3 stacked $k = 3$ convolutions have the receptive field of a $k = 9$ convolution), giving a ConvMixer-256/24 with 1670K parameters, and got $9 5 . 1 6 \%$ accuracy, i.e., still less.
+
+Patch size. CIFAR-10 inputs are so small that we initially only used $p = 1$ , i.e., the patch embedding layer does little more than compute $h$ linear combinations of the input image. Using $p = 2$ , we see a reduction in accuracy of about $0 . 8 0 \%$ ; this is a worthy tradeoff in terms of training and inference time. Further increasing the patch size leads to rapid decreases in accuracy, with only $9 2 . 6 1 \%$ for $p = 4$ .
+
+Since the “internal resolution” is decreased by a factor of $p$ when increasing the patch size, we assumed that larger kernels would be less important for larger $p$ . We investigated this by again increasing the kernel size from 3 to 11 for ConvMixer-256/8 with $p = 2$ : however, this time, the improvement going from 3 to 5 is only $1 . 1 3 \%$ , and larger kernels than 5 provide only marginal benefit.
+
+Weight decay. We did many of our initial experiments with minimal weight decay. However, this was not optimal: by tuning weight decay, we can get an additional $0 . 1 5 \%$ of accuracy for no cost. Consequently, we used weight decay (without tuning) for our larger-scale experiments on ImageNet.
+
+Table 4: An investigation of ConvMixer design parameters $h , d , p , k$ and weight decay on CIFAR-10   
+
+<table><tr><td colspan="7">Tiny ConvMixers trained on CIFAR-10.</td></tr><tr><td>Width h</td><td>Depth d</td><td>Patch Size p</td><td>Kernel Size k</td><td># Params （×103）</td><td>Weight Decay</td><td>CIFAR-10 Acc. (%)</td></tr><tr><td>128</td><td>4</td><td>1</td><td>8</td><td>103</td><td>0</td><td>91.26</td></tr><tr><td>128</td><td>8</td><td>1</td><td>8</td><td>205</td><td>0</td><td>93.83</td></tr><tr><td>128</td><td>12</td><td>1</td><td>8</td><td>306</td><td>0</td><td>94.83</td></tr><tr><td>256</td><td>4</td><td>1</td><td>8</td><td>338</td><td>0</td><td>93.37</td></tr><tr><td>256</td><td>8</td><td>1</td><td>8</td><td>672</td><td>0</td><td>95.60</td></tr><tr><td>256</td><td>12</td><td>1</td><td>8</td><td>1006</td><td>0</td><td>96.39</td></tr><tr><td>256</td><td>16</td><td>1</td><td>8</td><td>1339</td><td>0</td><td>96.74</td></tr><tr><td>256</td><td>20</td><td>1</td><td>8</td><td>1673</td><td>0</td><td>96.67</td></tr><tr><td>↓Kernel adjustments</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="7">256</td></tr><tr><td>256 256</td><td>8 8</td><td>1 1</td><td>3 5</td><td>559 592</td><td>0 0</td><td>93.61 95.19</td></tr><tr><td>256</td><td>8</td><td>1 1</td><td>7</td><td>641</td><td>0</td><td>95.80</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>0</td><td>95.88</td></tr><tr><td>256</td><td>8</td><td>1</td><td>11 13</td><td>788</td><td>0</td><td>95.70</td></tr><tr><td>256</td><td>8</td><td></td><td></td><td>887</td><td>0</td><td>96.04</td></tr><tr><td></td><td>8</td><td>1</td><td>15</td><td>1001</td><td>0</td><td>96.08</td></tr><tr><td colspan="7">↓Patch adjustments</td></tr><tr><td>256</td><td>8</td><td>2</td><td>9</td><td>709</td><td>0</td><td>95.00</td></tr><tr><td>256</td><td>8</td><td>4</td><td>9</td><td>718</td><td>0</td><td>92.61</td></tr><tr><td>256</td><td>8</td><td>8</td><td>9</td><td>755</td><td>0</td><td>85.57</td></tr><tr><td colspan="7">↓Weight decay adjustments</td></tr><tr><td>256 256</td><td>8 8</td><td>1 1</td><td>9 9</td><td>707 707</td><td>1×10-1 1×10-2</td><td>95.88 96.03</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>1×10-3</td><td>95.76</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>1×10-4</td><td>95.63</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>1×10-5</td><td>95.88</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="7">↓Kernel size adjustments when p = 2</td></tr><tr><td>256 256</td><td>8</td><td>2</td><td>3</td><td>561</td><td>0</td><td>94.08 95.21</td></tr><tr><td>256</td><td>8 8</td><td>2</td><td>5</td><td>594</td><td>0</td><td>95.35</td></tr><tr><td>256</td><td>8</td><td>2</td><td>7</td><td>643</td><td>0</td><td></td></tr><tr><td>256</td><td></td><td>2</td><td>9</td><td>709</td><td>0</td><td>95.00</td></tr><tr><td></td><td>8</td><td>2</td><td>11</td><td>791</td><td>0</td><td>95.14</td></tr><tr><td colspan="7">↓ Adding weight decay to the above</td></tr><tr><td>256 256</td><td>8</td><td>2</td><td>3</td><td>561</td><td>1×10-2</td><td>94.69</td></tr><tr><td>256</td><td>8</td><td>2</td><td>5</td><td>594</td><td>1×10-2</td><td>95.26</td></tr><tr><td></td><td>8</td><td>2</td><td>7</td><td>643</td><td>1×10-2</td><td>95.25</td></tr><tr><td>256</td><td>8</td><td>2</td><td>9</td><td>709</td><td>1×10-2</td><td>95.06</td></tr><tr><td>256</td><td>8</td><td>2</td><td>11</td><td>791</td><td>1×10-2</td><td>95.17</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+# C Weight Visualizations
+
+![](images/1c4da216843f97335f558406d3efba1df3af76b19a16585394c9d4b95d29b4e1.jpg)  
+Figure 4: Patch embedding weights for a ConvMixer-1024/20 with patch size 14 (see Table 2).
+
+![](images/230831c56cc1f8ec2f0d13a282afad95e436e131d9529eff9246e5e65ae7c491.jpg)  
+Figure 5: Patch embedding weights for a ConvMixer-768/32 with patch size 7 (see Table 2).
+
+![](images/779437de5095255212075157ce801d7f55b20fbf8578f9e001a1c2cd67368c3f.jpg)  
+Figure 6: Random subsets of 64 depthwise convolutional kernels from progressively deeper layers of ConvMixer-1536/20 (see Table 1).
+
+In Figure 4 and 5, we visualize the (complete) weights of the patch embedding layers of a ConvMixer1536/20 with $p = 1 4$ and a ConvMixer-768/32 with $p = 7$ , respectively. Much like Sandler et al. (2019), the layer consists of Gabor-like filters as well as “colorful globs” or rough edge detectors. The filters seem to be more structured than those learned by MLP-Mixer (Tolstikhin et al., 2021); also unlike MLP-Mixer, the weights look much the same going from $p = 1 4$ to $p = 7$ : the latter simply looks like a downsampled version of the former. It is unclear, then, why we see such a drop in accuracy for larger patches. However, some of the filters essentially look like noise, maybe suggesting a need for more regularization or longer training, or even more data. Ultimately, we cannot read too much into the learned representations here.
+
+In Figure 6, we plot the hidden convolutional kernels for successive layers of a ConvMixer. Initially, the kernels seem to be relatively small, but make use of their allowed full size in later layers; there is a clear hierarchy of features as one would expect from a standard convolutional architecture. Interestingly, Touvron et al. (2021a) saw a similar effect for ResMLP, where earlier layers look like small-kernel convolution, while later layers were more diffuse, despite these layers being representated by an unconstrained matrix multiplication rather than convolution.
+
+# D Implementation
+
+import torch.nn as nn   
+2   
+3 class Residual(nn.Module):   
+4 def __init__(self, fn):   
+5 super().__init__()   
+6 self.fn $=$ fn   
+7   
+8 def forward(self, x):   
+9 return self.fn $( { \bf x } ) ~ + ~ { \bf x }$   
+10   
+11 def ConvMixer(dim, depth, kernel_size $^ { = 9 }$ , patch_size $^ { - 7 }$ , n_classes=1000):   
+12 return nn.Sequential(   
+13 nn.Conv2d(3, dim, kernel_size=patch_size, stride=patch_size),   
+14 nn.GELU(),   
+15 nn.BatchNorm2d(dim),   
+16 \*[nn.Sequential(   
+17 Residual(nn.Sequential(   
+18 nn.Conv2d(dim, dim, kernel_size, groups $=$ dim, padding $=$ "same"),   
+19 nn.GELU(),   
+20 nn.BatchNorm2d(dim)   
+21 )),   
+22 nn.Conv2d(dim, dim, kernel_size $^ { = 1 }$ ),   
+23 nn.GELU(),   
+24 nn.BatchNorm2d(dim)   
+25 ) for i in range(depth)],   
+26 nn.AdaptiveAvgPool2d((1,1)),   
+27 nn.Flatten(),   
+28 nn.Linear(dim, n_classes)   
+29 )
+
+def ConvMixr(h,d,k,p,n):   
+2 S,C, $\mathsf { A } \mathop { = }$ Sequential,Conv2d,lambda x:S(x,GELU(),BatchNorm2d(h))   
+3 R=type('',(S,),{'forward':lambda s $, \mathbf { x } : \mathbf { s } \left[ \left. \Theta \right] \left( \mathbf { x } \right) + \mathbf { x } \right\} \mathrm { . }$ )   
+4 return S(A(C(3,h,p,p)),\*[S(R(A(C(h,h,k,groups $\mathtt { \Gamma } = \mathtt { h }$ ,padding $\mathbf { \tau } _ { | = \mathbf { k } / \mathbf { \Omega } / 2 }$ ))),A(C(h,h,1))) for i $\hookrightarrow$ in range(d)],AdaptiveAvgPool2d((1,1)),Flatten(),Linear $( \mathtt { h } , \mathtt { n } )$ )
+
+Figure 8: An implementation of our model in exactly 280 characters, in case you happen to know of any means of disseminating information that could benefit from such a length. All you need to do to run this is from torch.nn import \*.
+
+This section presents an expanded (but still quite compact) version of the terse ConvMixer implementation that we presented in the paper. The code is given in Figure 7. We also present an even more terse implementation in Figure 8, which to the best of our knowledge is the first model that achieves the elusive dual goals of $8 0 \% +$ ImageNet top-1 accuracy while also fitting into a tweet.

parse/dev/TVHS5Y4dNvM/TVHS5Y4dNvM_content_list.json

@@ -0,0 +1,1283 @@
+[
+    {
+        "type": "text",
+        "text": "CONVOLUTIONS ATTENTION MLPs Patches Are All You Need? ",
+        "text_level": 1,
+        "bbox": [
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+            99,
+            558,
+            150
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            174,
+            398,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            459,
+            238,
+            539,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although convolutional networks have been the dominant architecture for vision tasks for many years, recent experiments have shown that Transformer-based models, most notably the Vision Transformer (ViT), may exceed their performance in some settings. However, due to the quadratic runtime of the self-attention layers in Transformers, ViTs require the use of patch embeddings, which group together small regions of the image into single input features, in order to be applied to larger image sizes. This raises a question: Is the performance of ViTs due to the inherently-more-powerful Transformer architecture, or is it at least partly due to using patches as the input representation? In this paper, we present some evidence for the latter: specifically, we propose the ConvMixer, an extremely simple model that is similar in spirit to the ViT and the even-more-basic MLP-Mixer in that it operates directly on patches as input, separates the mixing of spatial and channel dimensions, and maintains equal size and resolution throughout the network. In contrast, however, the ConvMixer uses only standard convolutions to achieve the mixing steps. Despite its simplicity, we show that the ConvMixer outperforms the ViT, MLP-Mixer, and some of their variants for similar parameter counts and data set sizes, in addition to outperforming classical vision models such as the ResNet. Our code is available at https://github.com/tmp-iclr/convmixer. ",
+        "bbox": [
+            233,
+            270,
+            766,
+            521
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            550,
+            318,
+            566
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For many years, convolutional neural networks have been the dominant architecture for deep learning systems applied to computer vision tasks. But recently, architectures based upon Transformer models, e.g., the so-called Vision Transformer architecture (Dosovitskiy et al., 2020), have demonstrated compelling performance in many of these tasks, often outperforming classical convolutional architectures, especially for large data sets. An understandable assumption, then, is that it is only a matter of time before Transformers become the dominant architecture for vision domains, just as they have for language processing. In order to apply Transformers to images, however, the representation had to be changed: because the computational cost of the self-attention layers used in Transformers would scale quadratically with the number of pixels per image if applied naively at the per-pixel level, the compromise was to first split the image into multiple “patches”, linearly embed them, and then apply the transformer directly to this collection of patches. ",
+        "bbox": [
+            176,
+            584,
+            549,
+            791
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/1173f087f8ac39da4f86d64104c20dd719f3a66b7c4817498a0ed0bbddaddd8e.jpg",
+        "image_caption": [
+            "Figure 1: Accuracy vs. parameters, trained and evaluated on ImageNet-1k. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            560,
+            590,
+            821,
+            757
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            791,
+            823,
+            820
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we explore the question of whether, fundamentally, the strong performance of vision transformers may result more from this patch-based representation than from the Transformer architecture itself. We develop a very simple convolutional architecture which we dub the “ConvMixer” due to its similarity to the recently-proposed MLP-Mixer (Tolstikhin et al., 2021). This architecture is similar to the Vision Transformer (and MLP-Mixer) in many respects: it directly operates on patches, it maintains an equal-resolution-and-size representation throughout all layers, it does no downsampling of the representation at successive layers, and it separates “channel-wise mixing” ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/229107206062b568c8ab98c917943152041566e7e38345a854151e830ba98de1.jpg",
+        "image_caption": [
+            "Figure 2: ConvMixer uses “tensor layout” patch embeddings to preserve locality, and then applies $d$ copies of a simple fully-convolutional block consisting of large-kernel depthwise convolution followed by pointwise convolution, before finishing with global pooling and a simple linear classifier. ",
+            "Figure 3: Implementation of ConvMixer in PyTorch; see Appendix D for more implementations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            99,
+            808,
+            239
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "def ConvMixer(h, depth, kernel_size $^ { ! = 9 }$ , patch_size $^ { - 7 }$ , n_classes $\\scriptstyle \\left. = 1 0 0 0 \\right|$ ):   \n2 Seq, ActBn $= \\mathsf { n n }$ .Sequential, lambda x: Seq(x, nn.GELU(), nn.BatchNorm2d(h))   \n3 Residual $=$ type('Residual', (Seq,), {'forward': lambda self, x: self[0](x) + x})   \n4 return Seq(ActBn(nn.Conv2d(3, h, patch_size, stride=patch_size)),   \n5 \\*[Seq(Residual(ActBn(nn.Conv2d(h, h, kernel_size, groups=h, padding $=$ \"same\"))),   \n6 ActBn(nn.Conv2d(h, h, 1))) for i in range(depth)],   \n7 nn.AdaptiveAvgPool2d((1,1)), nn.Flatten(), nn.Linear(h, n_classes)) ",
+        "bbox": [
+            148,
+            297,
+            834,
+            388
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "from the “spatial mixing” of information. But unlike the Vision Transformer and MLP-Mixer, our architecture does all these operations via only standard convolutions. ",
+        "bbox": [
+            176,
+            420,
+            820,
+            448
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The chief result we show in this paper is that this ConvMixer architecture, despite its extreme simplicity (it can be implemented in $\\approx 6$ lines of dense PyTorch code), outperforms both “standard” computer vision models such as ResNets of similar parameter counts and some corresponding Vision Transformer and MLP-Mixer variants, even with a slate of additions intended to make those architectures more performant on smaller data sets. Importantly, this is despite the fact that we did not design our experiments to maximize accuracy nor speed, in contrast to the models we compared against. Our results suggest that, at least to some extent, the patch representation itself may be a critical component to the “superior” performance of newer architectures like Vision Transformers. While these results are naturally just a snapshot, and more experiments are required to exactly disentangle the effect of patch embeddings from other factors, we believe that this provides a strong “convolutionalbut-patch-based” baseline to compare against for more advanced architectures in the future. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 A Simple Model: ConvMixer ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            627,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our model, dubbed ConvMixer, consists of a patch embedding layer followed by repeated applications of a simple fully-convolutional block. We maintain the spatial structure of the patch embeddings, as illustrated in Fig. 2. Patch embeddings with patch size $p$ and embedding dimension $h$ can be implemented as convolution with $c _ { \\mathsf { i n } }$ input channels, $h$ output channels, kernel size $p$ , and stride $p$ : ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b53aef0e001df37c43f066cfd5e52a2cea6115a7c4565ff7a27624946265b5bd.jpg",
+        "text": "$$\nz _ { 0 } = \\mathsf { B N } ( \\sigma \\{ \\mathsf { C o n v } _ { c _ { \\mathrm { i n } }  h } ( X , \\mathsf { s t r i d e } { = } p , \\mathsf { k e r n e l } _ { - } \\mathsf { s i z e } { = } p ) \\} )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The ConvMixer block itself consists of depthwise convolution (i.e., grouped convolution with groups equal to the number of channels, $h$ ) followed by pointwise (i.e., kernel size $1 \\times 1$ ) convolution. As we will explain in Sec. 3, ConvMixers work best with unusually large kernel sizes for the depthwise convolution. Each of the convolutions is followed by an activation and post-activation BatchNorm: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/2d61b33dc4bb29ff022bf93ccc4363c5e27d59286434e2cd3b6018c657544200.jpg",
+        "text": "$$\n\\begin{array} { r } { z _ { l } ^ { \\prime } = \\mathsf { B N } \\left( \\sigma \\{ \\mathsf { C o n v D e p t h w i s e } ( z _ { l - 1 } ) \\} \\right) + z _ { l - 1 } } \\\\ { z _ { l + 1 } = \\mathsf { B N } \\left( \\sigma \\{ \\mathsf { C o n v P o i n t w i s e } ( z _ { l } ^ { \\prime } ) \\} \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "After many applications of this block, we perform global pooling to get a feature vector of size $h$ , which we pass to a softmax classifier. See Fig. 3 for an implementation of ConvMixer in PyTorch. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Design parameters. An instantiation of ConvMixer depends on four parameters: (1) the “width” or hidden dimension $h$ (i.e., the dimension of the patch embeddings), (2) the depth $d$ , or the number of repetitions of the ConvMixer layer, (3) the patch size $p$ which controls the internal resolution of the model, (4) the kernel size $k$ of the depthwise convolutional layer. We name ConvMixers after their hidden dimension and depth, like ConvMixer- $h / d$ . We refer to the original input size $n$ divided by the patch size $p$ as the internal resolution; note, however, that ConvMixers support variable-sized inputs. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/797fbf518f473a175687db54082c266c03238b89096fc21db54819251795847d.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Models trained and evaluated on $2 2 4 \\times 2 2 4$ ImageNet-1k only. See more in Appendix A. "
+        ],
+        "table_body": "<table><tr><td colspan=\"8\">Current “Most Interesting” ConvMixer Configurations vs. Other Simple Models</td></tr><tr><td>Network</td><td>Patch Size</td><td>Kernel Size</td><td># Params (×106）</td><td>Throughput (img/sec)</td><td>Act. Fn.</td><td>#Epochs</td><td>ImNet top-1 (%)</td></tr><tr><td>ConvMixer-1536/20 ConvMixer-768/32</td><td>7 7</td><td>9 7</td><td>51.6 21.1</td><td>89 203</td><td>G R</td><td>150 300</td><td>81.37 80.16</td></tr><tr><td>ResNet-152</td><td></td><td>3</td><td>60.2</td><td>872</td><td>R</td><td>150</td><td>79.64</td></tr><tr><td>DeiT-B</td><td>1 16</td><td></td><td>86</td><td>703</td><td>G</td><td>300</td><td>81.8</td></tr><tr><td></td><td>8</td><td></td><td>129</td><td>140</td><td>G</td><td>400</td><td>81.0</td></tr><tr><td>ResMLP-B24/8</td><td></td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Motivation. Our architecture is based on the idea of mixing, as in Tolstikhin et al. (2021). In particular, we chose depthwise convolution to mix spatial locations and pointwise convolution to mix channel locations. A key idea from previous work is that MLPs and self-attention can mix distant spatial locations, i.e., they can have an arbitrarily large receptive field. Consequently, we used convolutions with an unusually large kernel size to mix distant spatial locations. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While self-attention and MLPs are theoretically more flexible, allowing for large receptive fields and content-aware behavior, the inductive bias of convolution is well-suited to vision tasks and leads to high data efficiency. By using such a standard operation, we also get a glimpse into the effect of the patch representation itself in contrast to the conventional pyramid-shaped, progressivelydownsampling design of convolutional networks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            523,
+            312,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Training setup. We primarily evaluate ConvMixers on ImageNet-1k classification without any pretraining or additional data. We added ConvMixer to the timm framework (Wightman, 2019) and trained it with nearly-standard settings: we used RandAugment (Cubuk et al., 2020), mixup (Zhang et al., 2017), CutMix (Yun et al., 2019), random erasing (Zhong et al., 2020), and gradient norm clipping in addition to default timm augmentation. We used the AdamW (Loshchilov & Hutter, 2018) optimizer and a simple triangular learning rate schedule. Due to limited compute, we did absolutely no hyperparameter tuning on ImageNet and trained for fewer epochs than competitors. Consequently, our models could be over- or under-regularized, and the accuracies we report likely underestimate the capabilities of our model. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Results. A ConvMixer-1536/20 with 52M parameters can achieve $8 1 . 4 \\%$ top-1 accuracy on ImageNet, and a ConvMixer-768/32 with 21M parameters $8 0 . 2 \\%$ (see Table 1). Wider ConvMixers seem to converge in fewer epochs, but are memory- and compute-hungry. They also work best with large kernel sizes: ConvMixer- $1 5 3 6 / 2 0 \\log \\approx 1 \\%$ accuracy when reducing the kernel size from $k = 9$ to $k = 3$ (we discuss kernel sizes more in Appendix A & B). ConvMixers with smaller patches are substantially better in our experiments, similarly to Sandler et al. (2019); we believe larger patches require deeper ConvMixers. With everything held equal except increasing the patch size from 7 to 14, ConvMixer-1536/20 achieves $7 8 . 9 \\%$ top-1 accuracy but is around $4 \\times$ faster. We trained one model with ReLU to demonstrate that GELU (Hendrycks & Gimpel, 2016), which is popular in recent isotropic models, isn’t necessary. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Comparisons. Our model and ImageNet1k-only training setup closely resemble that of recent patchbased models like DeiT (Touvron et al., 2020). Due to ConvMixer’s simplicity, we focus on comparing to only the most basic isotropic patch-based architectures adapted to the ImageNet-1k setting, namely DeiT and ResMLP. Attempting a fair comparison with a standard baseline, we trained ResNets using exactly the same parameters as ConvMixers; while this choice of parameters is suboptimal (Wightman et al., 2021), it is likely also suboptimal for ConvMixers, since we did no hyperparameter tuning. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Looking at Table 1 and Fig. 1, ConvMixers achieve competitive accuracies for a given parameter budget: ConvMixer-1536/20 outperforms both ResNet-152 and ResMLP-B24 despite having substantially fewer parameters and is competitive with DeiT-B. ConvMixer-768/32 uses just a third of the parameters of ResNet-152, but is similarly accurate. Note that unlike ConvMixer, the DeiT and ResMLP results involved hyperparameter tuning, and when substantial resources are dedicated to tuning ResNets, including training for twice as many epochs, they only outperform an equivalently-sized ConvMixer by $\\approx 0 . 2 \\%$ (Wightman et al., 2021). However, ConvMixers are substantially slower at inference than the competitors, likely due to their smaller patch size; hyperparameter tuning and optimizations could narrow this gap. For more discussion and comparisons, see Table 2 and Appendix A. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10 Experiments. We also performed smaller-scale experiments on CIFAR-10, where ConvMixers achieve over $9 6 \\%$ accuracy with as few as $0 . 7 { \\bf M }$ parameters, demonstrating the data efficiency of the convolutional inductive bias. Details of these experiments are presented in Appendix B. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            300,
+            330,
+            315
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Isotropic architectures. Vision transformers have inspired a new paradigm of “isotropic” architectures, i.e., those with equal size and shape throughout the network, which use patch embeddings for the first layer. These models look similar to repeated transformer-encoder blocks (Vaswani et al., 2017) with different operations replacing the self-attention and MLP operations. For example, MLP-Mixer (Tolstikhin et al., 2021) replaces them both with MLPs applied across different dimensions (i.e., spatial and channel location mixing); ResMLP (Touvron et al., 2021a) is a data-efficient variation on this theme. CycleMLP (Chen et al., 2021), gMLP (Liu et al., 2021a), and vision permutator (Hou et al., 2021), replace one or both blocks with various novel operations. These are all quite performant, which is typically attributed to the novel choice of operations. In contrast, Melas-Kyriazi (2021) proposed an MLP-based isotropic vision model, and also hypothesized patch embeddings could be behind its performance. ResMLP tried replacing its linear interaction layer with (small-kernel) convolution and achieved good performance, but kept its MLP-based cross-channel layer and did not explore convolutions further. As our investigation of ConvMixers suggests, these works may conflate the effect of the new operations (like self-attention and MLPs) with the effect of the use of patch embeddings and the resulting isotropic architecture. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A study predating vision transformers investigates isotropic (or “isometric”) MobileNets (Sandler et al., 2019), and even implements patch embeddings under another name. Their architecture simply repeats an isotropic MobileNetv3 block. They identify a tradeoff between patch size and accuracy that matches our experience, and train similarly performant models (see Appendix A, Table 2). However, their block is substantially more complex than ours; simplicity and motivation sets our work apart. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Patches aren’t all you need. Several papers have increased vision transformer performance by replacing standard patch embeddings with a different stem: Xiao et al. (2021) and Yuan et al. (2021a) use a standard convolutional stem, while Yuan et al. (2021b) repeatedly combines nearby patch embeddings. However, this conflates the effect of using patch embeddings with the effect of adding convolution or similar inductive biases e.g., locality. We attempt to focus on the use of patches. ",
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+    },
+    {
+        "type": "text",
+        "text": "CNNs meet ViTs. Many efforts have been made to incorporate features of convolutional networks into vision transformers and vice versa. Self-attention can emulate convolution (Cordonnier et al., 2019) and can be initialized or regularized to be like it (d’Ascoli et al., 2021); other works simply add convolution operations to transformers (Dai et al., 2021; Guo et al., 2021), or include downsampling to be more like traditional pyramid-shaped convolutional networks (Wang et al., 2021). Conversely, self-attention or attention-like operations can supplement or replace convolution in ResNet-style models (Bello et al., 2019; Ramachandran et al., 2019; Bello, 2021). While all of these attempts have been successful in one way or another, they are orthogonal to this work, which aims to emphasize the effect of the architecture common to most ViTs by showcasing it with a less-expressive operation. ",
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+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "text": "We presented ConvMixers, an extremely simple class of models that independently mixes the spatial and channel locations of patch embeddings using only standard convolutions. We also highlighted that using large kernel sizes, inspired by the large receptive fields of ViTs and MLP-Mixers, provides a substantial performance boost. While neither our model nor our experiments were designed to maximize accuracy or speed, i.e., we did not search for good hyperparameters, ConvMixers outperform the Vision Transformer and MLP-Mixer, and are competitive with ResNets, DeiTs, and ResMLPs. ",
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+        "text": "",
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+        "type": "text",
+        "text": "We provided evidence that the increasingly common “isotropic” architecture with a simple patch embedding stem is itself a powerful template for deep learning. Patch embeddings allow all the downsampling to happen at once, immediately decreasing the internal resolution and thus increasing the effective receptive field size, making it easier to mix distant spatial information. Our title, while an exaggeration, points out that attention isn’t the only export from language processing into computer vision: tokenizing inputs, i.e., using patch embeddings, is also a powerful and important takeaway. ",
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+        "type": "text",
+        "text": "While our model is not state-of-the-art, we find its simple patch-mixing design to be compelling. We hope that ConvMixers can serve as a baseline for future patch-based architectures with novel operations, or that they can provide a basic template for new conceptually simple and performant models. ",
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+        "type": "text",
+        "text": "Future work. We are optimistic that a deeper ConvMixer with larger patches could reach a desirable tradeoff between accuracy, parameters, and throughput after longer training and more regularization and hyperparameter tuning, similarly to how Wightman et al. (2021) enhanced ResNet performance through carefully-designed training regimens. Low-level optimization of large-kernel depthwise convolution could substantially increase throughput, and small enhancements to our architecture like the addition of bottlenecks or a more expressive classifier could trade simplicity for performance. ",
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+        "type": "text",
+        "text": "Due to its large internal resolution and isotropic design, ConvMixer may be especially well-suited for semantic segmentation, and it would be useful to run experiments on this task with a ConvMixer-like model and on other tasks such as object detection. More experiments could be designed to more clearly extricate the effect of patch embeddings from other architectural choices. In particular, for a more in-depth comparison to ViTs and MLP-Mixers, which excel when trained on very large data sets, it is important to investigate the performance of ConvMixers in the regime of large-scale pre-training. ",
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+    {
+        "type": "text",
+        "text": "A note on paper length. Expecting more text in this paper? Wondering if it’s a workshop paper we hastily submitted to ICLR? No. This paper presents a simple idea, one where we genuinely believe that a short paper presentation is more effective. Do we really need exactly 8 (now 9? 10?) pages to describe every machine learning architecture and algorithm in existence? We proposed an incredibly simple architecture and made a very simple point that we think is worth more discussion: patches work well in convolutional architectures. We think that four and a half pages is more than enough space for this. The details of the experiments and architectures are in the appendix for those who want to read through it all. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+    {
+        "type": "text",
+        "text": "A Comparison to other models ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/fcb37e0ed9b504e72eb64c6a8a1c3e33b83ed1243bd24c381623827ba8ec5578.jpg",
+        "table_caption": [
+            "Table 2: Throughputs measured on an RTX8000 GPU using batch size 64 and fp16. ConvMixers and ResNets trained ourselves. Other statistics: DeiT (Touvron et al., 2020), ResMLP (Touvron et al., 2021a), Swin (Liu et al., 2021b), ViT (Dosovitskiy et al., 2020), MLP-Mixer (Tolstikhin et al., 2021), Isotropic MobileNets (Sandler et al., 2019). We think models with matching colored dots $( \\bullet )$ are informative to compare with each other. †Throughput tested, but not trained. Activations: ReLU, GELU. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">Comparison with other simple models trained on ImageNet-1k only with input size 224.</td></tr><tr><td>Network</td><td>Patch Size</td><td>Kernel Size</td><td># Params (×106）</td><td>Throughput (img/sec)</td><td>Act. Fn.</td><td>#Epochs</td><td>ImNet top-1 (%)</td></tr><tr><td>ConvMixer-1536/20 ：</td><td>7</td><td>9</td><td>51.6</td><td>89</td><td>G G</td><td>150 150</td><td>81.37</td></tr><tr><td>ConvMixer-1536/20</td><td>7</td><td>3</td><td>49.4</td><td>136</td><td>G</td><td></td><td>80.43</td></tr><tr><td>ConvMixer-1536/20</td><td>14</td><td>9</td><td>52.3</td><td>334</td><td></td><td>150 300</td><td>78.92</td></tr><tr><td>ConvMixer-768/32·</td><td>7</td><td>7</td><td>21.1</td><td>203</td><td>R</td><td></td><td>80.16</td></tr><tr><td>ConvMixer-1024/16</td><td>7</td><td>9</td><td>19.4</td><td>173</td><td>G</td><td>100</td><td>79.45</td></tr><tr><td>ConvMixer-1024/12</td><td>7</td><td>8</td><td>14.6</td><td>248</td><td>G</td><td>90</td><td>77.75</td></tr><tr><td>ConvMixer-512/16</td><td>7</td><td>8</td><td>5.4</td><td>403</td><td>G</td><td>90</td><td>73.76</td></tr><tr><td>ConvMixer-512/12</td><td>7</td><td>8</td><td>4.2</td><td>532</td><td>G</td><td>90</td><td>72.59</td></tr><tr><td>ConvMixer-768/32</td><td>14</td><td>3</td><td>20.2</td><td>1258</td><td>R</td><td>300</td><td>74.93</td></tr><tr><td>ConvMixer-1024/20 ·</td><td>14</td><td>9</td><td>24.4</td><td>520</td><td>G</td><td>150</td><td>76.94</td></tr><tr><td>ResNet-152 .</td><td>1</td><td>3</td><td>60.2</td><td>872</td><td>R</td><td>150</td><td>79.64</td></tr><tr><td>ResNet-101</td><td></td><td>3</td><td>44.6</td><td>1040</td><td>R</td><td>150</td><td>78.33</td></tr><tr><td>ResNet-50</td><td>1</td><td>3</td><td>25.6</td><td>1942</td><td>R</td><td>150</td><td>76.32</td></tr><tr><td>DeiT-Bt</td><td>7</td><td>1</td><td>86.7</td><td>77</td><td>G</td><td>1</td><td></td></tr><tr><td>DeiT-St</td><td>7</td><td>1</td><td>22.1</td><td>164</td><td>G</td><td></td><td>1</td></tr><tr><td>DeiT-Tit</td><td>7</td><td>1</td><td>5.7</td><td>327</td><td>G</td><td>1</td><td>一</td></tr><tr><td>DeiT-B·</td><td>16</td><td>1</td><td>86</td><td>703</td><td>G</td><td>300</td><td>1 81.8</td></tr><tr><td>DeiT-S·</td><td>16</td><td></td><td>22</td><td>1491</td><td>G</td><td>300</td><td>79.8</td></tr><tr><td>DeiT-Ti·</td><td>16</td><td>一</td><td>5.7</td><td>2727</td><td>G</td><td>300</td><td>72.2</td></tr><tr><td>ResMLP-S12/8·</td><td>8</td><td></td><td>22.1</td><td>638</td><td>G</td><td>400</td><td></td></tr><tr><td>ResMLP-B24/8·</td><td>8</td><td>一</td><td>129</td><td>140</td><td>G</td><td>400</td><td>79.1 81.0</td></tr><tr><td>ResMLP-B24</td><td>16</td><td>一</td><td>116</td><td>1191</td><td>G</td><td>400</td><td>81.0</td></tr><tr><td>Swin-S ·</td><td>4</td><td></td><td>50</td><td>566</td><td>G</td><td>300</td><td>83.0</td></tr><tr><td>Swin-T ·</td><td>4</td><td></td><td>29</td><td>884</td><td>G</td><td>300</td><td>81.3</td></tr><tr><td>ViT-B/16·</td><td>16</td><td></td><td>86</td><td>704</td><td>G</td><td>300</td><td>77.9</td></tr><tr><td></td><td></td><td>一</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Mixer-B/16·</td><td>16</td><td>1</td><td>59</td><td>816</td><td>G</td><td>300</td><td>76.44</td></tr><tr><td>Isotropic MobileNetv3 ·</td><td>8</td><td>3</td><td>20</td><td></td><td>R</td><td>1</td><td>80.6</td></tr><tr><td>Isotropic MobileNetv3 ·</td><td>16</td><td>3</td><td>20</td><td>一</td><td>R</td><td>1</td><td>77.6</td></tr></table>",
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+        "type": "text",
+        "text": "Experiment overview. We did not design our experiments to maximize accuracy: We chose “common sense” parameters for timm and its augmentation settings, found that it worked well for a ConvMixer-1024/12, and stuck with them for the proceeding experiments. We admit this is not an optimal strategy, however, we were aware from our early experiments on CIFAR-10 that results seemed robust to various small changes. We did not have access to sufficient compute to attempt to tune hyperparameters for each model: e.g., larger ConvMixers could probably benefit from more regularization than we chose, and smaller ones from less regularization. Keeping the parameters the same across ConvMixer instances seemed more reasonable than guessing for each. ",
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+        "type": "text",
+        "text": "However, to some extent, we changed the number of epochs per model: for earlier experiments, we merely wanted a “proof of concept”, and used only 90–100 epochs. Once we saw potential, we increased this to 150 epochs and trained some larger models, namely ConvMixer-1024/20 with $p = 1 4$ patches and ConvMixer-1536/20 with $p = 7$ patches. Then, believing that we should explore deeper-but-less-wide ConvMixers, and knowing from CIFAR-10 that the deeper models converged more slowly, we trained ConvMixer-768/32s with $p = 1 4$ and $p = 7$ for 300 epochs. Of course, training time was a consideration: ConvMixer-1536/20 took about 9 days to train (on $1 0 \\times \\mathrm { R T X 8 0 0 0 s } )$ 150 epochs, and ConvMixer-768/32 is over twice as fast, making 300 epochs more feasible. ",
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+    {
+        "type": "text",
+        "text": "If anything, we believe that in the worst case, the lack of parameter tuning in our experiments resulted in underestimating the accuracies of ConvMixers. Further, due to our limited compute and the fact that large models (particularly ConvMixers) are expensive to train on large data sets, we generally trained our models for fewer epochs than competition like DeiT and ResMLP (see Table 2). ",
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+    {
+        "type": "text",
+        "text": "A note on throughput. We measured throughput using batches of 64 images in half precision on a single RTX8000 GPU, averaged over 20 such batches. In particular, we measured CUDA execution time rather than “wall-clock” time. We noticed discrepancies in the relative throughputs of models, e.g., Touvron et al. (2020) reports that ResNet-152 is $2 \\times$ faster than DeiT-B, but our measurements show that it is only $1 . 2 5 \\times$ faster. We therefore speculate that our throughputs may underestimate the performance of ResNets and ConvMixers relative to the transformers. The difference may be due to using RTX8000 rather than V100 GPUs, or other low-level differences. Our throughputs were similar for batch sizes 32 and 128. ",
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+    },
+    {
+        "type": "text",
+        "text": "ResNets. As a simple baseline to which to compare ConvMixers, we trained three standard ResNets using exactly the same training setup and parameters as ConvMixer-1536/20. Despite having fewer parameters and being architecturally much simpler, ConvMixers substantially outperform these ResNets in terms of accuracy. A possible confounding factor is that ConvMixers use GELU, which may boost performance, while ResNets use ReLU. In an attempt to rule out this confound, we used ReLU in a later ConvMixer-768/32 experiment and found that it still achieved competitive accuracy. We also note that the choice of ReLU vs. GELU was not important on CIFAR-10 experiments (see Table 3). However, ConvMixers do have substantially less throughput. ",
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+    {
+        "type": "text",
+        "text": "DeiTs. We believe that DeiT is the most reasonable comparison in terms of vision transformers: It only adds additional regularization, as opposed to architectural additions in the case of CaiT (Touvron et al., 2021b), and is then essentially a “vanilla” ViT modulo the distillation token (we don’t consider distilled architectures). In terms of a fixed parameter budget, ConvMixers generally outperform DeiTs. For example, ConvMixer-1536/20 is only $0 . 4 3 \\%$ less accurate than DeiT-B despite having over 30M fewer parameters; ConvMixer-768/32 is $0 . 3 6 \\%$ more accurate than DeiT-S despite having 0.9M fewer parameters; and ConvMixer-512/16 is $0 . 3 9 \\%$ more accurate than DeiT-Ti for nearly the same number of parameters. Admittedly, none of the ConvMixers are very competitive in terms of throughput, with the closest being the ConvMixer-512/16 which is $5 \\times$ slower than DeiT-Ti. ",
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+    {
+        "type": "text",
+        "text": "A confounding factor is the difference in patch size between DeiT and ConvMixer; DeiT uses $p = 1 6$ while ConvMixer uses $p = 7$ . This means DeiT is substantially faster. However, ConvMixers using larger patches are not as competitive. While we were not able to train DeiTs with larger patch sizes, it is possible that they would outperform ConvMixers on the parameter count vs. accuracy curve; however, we tested their throughput for $p = 7$ , and they are even slower than ConvMixers. Given the difference between convolution and self-attention, we are not sure it is salient to control for patch size differences. ",
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+        "type": "text",
+        "text": "DeiTs were subject to more hyperparameter tuning than ConvMixers, as well as longer training times. They also used stochastic depth while we did not, which can in some cases contribute percent differences in model accuracy (Touvron et al., 2021a). It is therefore possible that further hyperparameter tuning and more epochs for ConvMixers could close the gap between the two architectures for large patches, e.g., $p = 1 6$ . ",
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+    {
+        "type": "text",
+        "text": "ResMLPs. Similarly to DeiT for ViT, we believe that ResMLP is the most relevant MLP-Mixer variant to compare against. Unlike DeiT, we can compare against instances of ResMLP with similar patch size: ResMLP-B24/8 has $p = 8$ patches, and underperforms ConvMixer-1536/20 by $0 . 3 7 \\%$ , despite having over twice the number of parameters; it also has similarly low throughput. ConvMixer-768/32 also outperforms ResMLP-S12/8 for millions fewer parameters, but $3 \\times$ less throughput. ",
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+    {
+        "type": "text",
+        "text": "ResMLP did not significantly improve in terms of accuracy for halving the patch size from 16 to 8, which shows that smaller patches do not always lead to better accuracy for a fixed architecture and regularization strategy (e.g., training a $p = 8$ DeiT may be challenging). ",
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+    {
+        "type": "text",
+        "text": "Swin Transformers. While we intend to focus on the most basic isotropic, patch-based architectures for fair comparisons with ConvMixer, it is also interesting to compare to a more complicated model that is closer to state-of-the-art. For a similar parameter budget, ConvMixer is around $1 . 2 \\mathrm { - } 1 . 6 \\%$ less accurate than the Swin Transformer, while also being $4 { - } 6 \\times$ slower. However, considering we did not attempt to tune or optimize our model in any way, we find it surprising that an exceedingly simple patch-based model that uses only plain convolution does not lag too far behind Swin Transformer. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Isotropic MobileNets. These models are closest in design to ours, despite using a repeating block that is substantially more complex than the ConvMixer one. Despite this, for a similar number of parameters, we can get similar performance. Notably, isotropic MobileNets seem to suffer less from larger patch sizes than ConvMixers, which makes us optimistic that sufficient parameter tuning could lead to more performant large-patch ConvMixers. ",
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+    {
+        "type": "text",
+        "text": "Other models. We included ViT and MLP-Mixer instances in our table, though they are not competitive with ConvMixer, DeiT, or ResMLP, even though MLP-Mixer has comparable regularization to ConvMixer. That is, ConvMixer seems to outperform MLP-Mixer and ViT, while being closer to complexity to them in terms of design and training regime than the other competitors, DeiT and ResMLP. ",
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+    {
+        "type": "text",
+        "text": "Kernel size. While we found some evidence that larger kernels are better on CIFAR-10, we wanted to see if this finding transferred to ImageNet. Consequently, we trained our best-performing model, ConvMixer-1536/20, with kernel size $k = 3$ rather than $k = 9$ . This resulted in a decrease of $0 . 9 4 \\%$ top-1 accuracy, which we believe is quite significant relative to the mere 2.2M additional parameters. However, $k = 3$ is substantially faster than $k = 9$ for spatial-domain convolution; we speculate that low-level optimizations could close the performance gap to some extent, e.g., by using implicit instead of explicit padding. Since large-kernel convolutions throughout a model are unconventional, there has likely been low demand for such optimizations. ",
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+    {
+        "type": "text",
+        "text": "B Experiments on CIFAR-10 ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Residual connections. We experimented with leaving out one, the other, or both residual connections before settling on the current configuration, and consequently chose to leave out the second residual connection. Our baseline model without the connection achieves $9 5 . 8 8 \\%$ accuracy, while including the connection reduces it to $9 4 . 7 8 \\%$ . Surprisingly, we see only a $0 . 3 1 \\%$ decrease in accuracy for removing all residual connections. We acknowledge that these findings for residual connections may not generalize to deeper ConvMixers trained on larger data sets. ",
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+    {
+        "type": "table",
+        "img_path": "images/982752939924044d5a67fee33296ece8c4b2d38c4a09fd36a00027068d65ab9b.jpg",
+        "table_caption": [
+            "Table 3: Small ablation study of training a ConvMixer-256/8 on CIFAR-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Ablation of ConvMixer-256/8 on CIFAR-10</td></tr><tr><td>Ablation</td><td>CIFAR-10 Acc. (%)</td></tr><tr><td>Baseline</td><td>95.88</td></tr><tr><td>- Residual in Eq. 2</td><td>95.57</td></tr><tr><td>+ Residual in Eq. 3</td><td>94.78</td></tr><tr><td>BatchNorm→LayerNorm GELU→ReLU</td><td>94.44</td></tr><tr><td></td><td>95.51</td></tr><tr><td>- Mixup and CutMix</td><td>95.92</td></tr><tr><td>-Random Erasing</td><td>95.24</td></tr><tr><td>- RandAug</td><td>92.86</td></tr><tr><td>-Random Scaling</td><td>86.24</td></tr><tr><td>- Gradient Norm Clipping</td><td>86.33</td></tr></table>",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "Normalization. Our model is conceptually similar to the vision transformer and MLP-Mixer, both of which use LayerNorm instead of BatchNorm. We attempted to use LayerNorm instead, and saw a decrease in performance of around $1 \\%$ as well as slower convergence (see Table 3). However, this was for a relatively shallow model, and we cannot guarantee that LayerNorm would not hinder ImageNet-scale models to an even larger degree. We note that the authors of ResMLP also saw a relatively small increase in accuracy for replacing LayerNorm with BatchNorm, but for a largerscale experiment (Touvron et al., 2021a). We conclude that BatchNorm is no more crucial to our architecture than other regularizations or parameter settings (e.g., kernel size). ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Having settled on an architecture, we proceeded to adjust its parameters $h , d , p , k$ as well as weight decay on CIFAR-10 experiments. (Initially, we took the unconventional approach of excluding weight decay since we were already using strong regularization in the form of RandAug and mixup.) We acknowledge that tuning our architecture on CIFAR-10 does not necessarily generalize to performance on larger data sets, and that this is a limitation of our study. ",
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+    {
+        "type": "text",
+        "text": "B.1 Results ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ConvMixers are quite performant on CIFAR-10, easily achieving $> 9 1 \\%$ accuracy for as little as 100, 000 parameters, or $> 9 6 \\%$ accuracy for only 887, 000 parameters (see Table 4). With additional refinements e.g., a more expressive classifier or bottlenecks, we think that ConvMixer could be even more competitive. For all experiments, we trained for 200 epochs on CIFAR-10 with RandAug, mixup, cutmix, random erasing, gradient norm clipping, and the standard augmentations in timm. We remove some of these augmentations in Table 3, finding that RandAug and random scaling (“default” in timm) are very important, each accounting for over $3 \\%$ of the accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Scaling ConvMixer. We adjusted the hidden dimension $h$ and the depth $d$ , finding that deeper networks take longer to converge while wider networks converge faster. That said, increasing the width or the depth is an effective way to increase accuracy; a doubling of depth incurs less compute than a doubling of width. The number of parameters in a ConvMixer is given exactly by: ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d4a99dd02992889056285fc90d3a6cee9d88cf2a3adfc1ec113a76bab422e89.jpg",
+        "text": "$$\n\\# \\mathsf { p a r a m s } = h [ d ( k ^ { 2 } + h + 6 ) + c _ { \\mathsf { i n } } p ^ { 2 } + n _ { \\mathsf { c l a s s e s } } + 3 ] + n _ { \\mathsf { c l a s s e s } } ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "including affine scaling parameters in BatchNorm layers, convolutional kernels, and the classifier. ",
+        "bbox": [
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "Kernel size. We initially hypothesized that large kernels would be important for ConvMixers, as they would allow the mixing of distant spatial information similarly to unconstrained MLPs or selfattention layers. We tried to investigate the effect of kernel size on CIFAR-10: we fixed the model to be a ConvMixer-256/8, and increased the kernel size by 2s from 3 to 15. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using a kernel size of 3, the ConvMixer only achieves $9 3 . 6 1 \\%$ accuracy. Simply increasing it to 5 gives an additional $1 . 5 0 \\%$ accuracy, and further to 7 an additional $0 . 6 1 \\%$ . The gains afterwards are relatively marginal, with kernel size 15 giving an additional $0 . 2 8 \\%$ accuracy. It could be that with more training iterations or more regularization, the effect of larger kernels would be more pronounced. Nonetheless, we concluded that ConvMixers benefit from larger-than-usual kernels, and thus used kernel sizes 7 or 9 in most of our later experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It is conventional wisdom that large-kernel convolutions can be “decomposed” into stacked smallkernel convolutions with activations between them, and it is therefore standard practice to use $k = 3$ convolutions, stacking more of them to increase the receptive field size with additional benefits from nonlinearities. This raises a question: is the benefit of larger kernels in ConvMixer actually better than simply increasing the depth with small kernels? First, we note that deeper networks are generally harder to train, so by increasing the kernel size independently of the depth, we may recover some of the benefits of depth without making it harder for signals to “propagate back” through the network. To test this, we trained a ConvMixer-256/10 with $k = 3$ (698K parameters) in the same setting as a ConvMixer-256/8 with $k = 9$ (707K parameters), i.e., we increased depth in a smallkernel model to roughly match the parameters of a large-kernel model. The ConvMixer-256/10 achieved $9 4 . 2 9 \\%$ accuracy ( $1 . 5 \\%$ less), which provides more evidence for the importance of larger kernels in ConvMixers. Next, instead of fixing the parameter budget, we tripled the depth (using the intuition that 3 stacked $k = 3$ convolutions have the receptive field of a $k = 9$ convolution), giving a ConvMixer-256/24 with 1670K parameters, and got $9 5 . 1 6 \\%$ accuracy, i.e., still less. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Patch size. CIFAR-10 inputs are so small that we initially only used $p = 1$ , i.e., the patch embedding layer does little more than compute $h$ linear combinations of the input image. Using $p = 2$ , we see a reduction in accuracy of about $0 . 8 0 \\%$ ; this is a worthy tradeoff in terms of training and inference time. Further increasing the patch size leads to rapid decreases in accuracy, with only $9 2 . 6 1 \\%$ for $p = 4$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since the “internal resolution” is decreased by a factor of $p$ when increasing the patch size, we assumed that larger kernels would be less important for larger $p$ . We investigated this by again increasing the kernel size from 3 to 11 for ConvMixer-256/8 with $p = 2$ : however, this time, the improvement going from 3 to 5 is only $1 . 1 3 \\%$ , and larger kernels than 5 provide only marginal benefit. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Weight decay. We did many of our initial experiments with minimal weight decay. However, this was not optimal: by tuning weight decay, we can get an additional $0 . 1 5 \\%$ of accuracy for no cost. Consequently, we used weight decay (without tuning) for our larger-scale experiments on ImageNet. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5c08c23f8b04f26c1fa8e99a411f1d052266832a058b073aac20279a527c5a50.jpg",
+        "table_caption": [
+            "Table 4: An investigation of ConvMixer design parameters $h , d , p , k$ and weight decay on CIFAR-10 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"7\">Tiny ConvMixers trained on CIFAR-10.</td></tr><tr><td>Width h</td><td>Depth d</td><td>Patch Size p</td><td>Kernel Size k</td><td># Params （×103）</td><td>Weight Decay</td><td>CIFAR-10 Acc. (%)</td></tr><tr><td>128</td><td>4</td><td>1</td><td>8</td><td>103</td><td>0</td><td>91.26</td></tr><tr><td>128</td><td>8</td><td>1</td><td>8</td><td>205</td><td>0</td><td>93.83</td></tr><tr><td>128</td><td>12</td><td>1</td><td>8</td><td>306</td><td>0</td><td>94.83</td></tr><tr><td>256</td><td>4</td><td>1</td><td>8</td><td>338</td><td>0</td><td>93.37</td></tr><tr><td>256</td><td>8</td><td>1</td><td>8</td><td>672</td><td>0</td><td>95.60</td></tr><tr><td>256</td><td>12</td><td>1</td><td>8</td><td>1006</td><td>0</td><td>96.39</td></tr><tr><td>256</td><td>16</td><td>1</td><td>8</td><td>1339</td><td>0</td><td>96.74</td></tr><tr><td>256</td><td>20</td><td>1</td><td>8</td><td>1673</td><td>0</td><td>96.67</td></tr><tr><td>↓Kernel adjustments</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"7\">256</td></tr><tr><td>256 256</td><td>8 8</td><td>1 1</td><td>3 5</td><td>559 592</td><td>0 0</td><td>93.61 95.19</td></tr><tr><td>256</td><td>8</td><td>1 1</td><td>7</td><td>641</td><td>0</td><td>95.80</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>0</td><td>95.88</td></tr><tr><td>256</td><td>8</td><td>1</td><td>11 13</td><td>788</td><td>0</td><td>95.70</td></tr><tr><td>256</td><td>8</td><td></td><td></td><td>887</td><td>0</td><td>96.04</td></tr><tr><td></td><td>8</td><td>1</td><td>15</td><td>1001</td><td>0</td><td>96.08</td></tr><tr><td colspan=\"7\">↓Patch adjustments</td></tr><tr><td>256</td><td>8</td><td>2</td><td>9</td><td>709</td><td>0</td><td>95.00</td></tr><tr><td>256</td><td>8</td><td>4</td><td>9</td><td>718</td><td>0</td><td>92.61</td></tr><tr><td>256</td><td>8</td><td>8</td><td>9</td><td>755</td><td>0</td><td>85.57</td></tr><tr><td colspan=\"7\">↓Weight decay adjustments</td></tr><tr><td>256 256</td><td>8 8</td><td>1 1</td><td>9 9</td><td>707 707</td><td>1×10-1 1×10-2</td><td>95.88 96.03</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>1×10-3</td><td>95.76</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>1×10-4</td><td>95.63</td></tr><tr><td>256</td><td>8</td><td>1</td><td>9</td><td>707</td><td>1×10-5</td><td>95.88</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"7\">↓Kernel size adjustments when p = 2</td></tr><tr><td>256 256</td><td>8</td><td>2</td><td>3</td><td>561</td><td>0</td><td>94.08 95.21</td></tr><tr><td>256</td><td>8 8</td><td>2</td><td>5</td><td>594</td><td>0</td><td>95.35</td></tr><tr><td>256</td><td>8</td><td>2</td><td>7</td><td>643</td><td>0</td><td></td></tr><tr><td>256</td><td></td><td>2</td><td>9</td><td>709</td><td>0</td><td>95.00</td></tr><tr><td></td><td>8</td><td>2</td><td>11</td><td>791</td><td>0</td><td>95.14</td></tr><tr><td colspan=\"7\">↓ Adding weight decay to the above</td></tr><tr><td>256 256</td><td>8</td><td>2</td><td>3</td><td>561</td><td>1×10-2</td><td>94.69</td></tr><tr><td>256</td><td>8</td><td>2</td><td>5</td><td>594</td><td>1×10-2</td><td>95.26</td></tr><tr><td></td><td>8</td><td>2</td><td>7</td><td>643</td><td>1×10-2</td><td>95.25</td></tr><tr><td>256</td><td>8</td><td>2</td><td>9</td><td>709</td><td>1×10-2</td><td>95.06</td></tr><tr><td>256</td><td>8</td><td>2</td><td>11</td><td>791</td><td>1×10-2</td><td>95.17</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "C Weight Visualizations ",
+        "text_level": 1,
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+    },
+    {
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+        "img_path": "images/1c4da216843f97335f558406d3efba1df3af76b19a16585394c9d4b95d29b4e1.jpg",
+        "image_caption": [
+            "Figure 4: Patch embedding weights for a ConvMixer-1024/20 with patch size 14 (see Table 2). "
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/230831c56cc1f8ec2f0d13a282afad95e436e131d9529eff9246e5e65ae7c491.jpg",
+        "image_caption": [
+            "Figure 5: Patch embedding weights for a ConvMixer-768/32 with patch size 7 (see Table 2). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
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+        "img_path": "images/779437de5095255212075157ce801d7f55b20fbf8578f9e001a1c2cd67368c3f.jpg",
+        "image_caption": [
+            "Figure 6: Random subsets of 64 depthwise convolutional kernels from progressively deeper layers of ConvMixer-1536/20 (see Table 1). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            103,
+            792,
+            640
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figure 4 and 5, we visualize the (complete) weights of the patch embedding layers of a ConvMixer1536/20 with $p = 1 4$ and a ConvMixer-768/32 with $p = 7$ , respectively. Much like Sandler et al. (2019), the layer consists of Gabor-like filters as well as “colorful globs” or rough edge detectors. The filters seem to be more structured than those learned by MLP-Mixer (Tolstikhin et al., 2021); also unlike MLP-Mixer, the weights look much the same going from $p = 1 4$ to $p = 7$ : the latter simply looks like a downsampled version of the former. It is unclear, then, why we see such a drop in accuracy for larger patches. However, some of the filters essentially look like noise, maybe suggesting a need for more regularization or longer training, or even more data. Ultimately, we cannot read too much into the learned representations here. ",
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figure 6, we plot the hidden convolutional kernels for successive layers of a ConvMixer. Initially, the kernels seem to be relatively small, but make use of their allowed full size in later layers; there is a clear hierarchy of features as one would expect from a standard convolutional architecture. Interestingly, Touvron et al. (2021a) saw a similar effect for ResMLP, where earlier layers look like small-kernel convolution, while later layers were more diffuse, despite these layers being representated by an unconstrained matrix multiplication rather than convolution. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D Implementation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            344,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "import torch.nn as nn   \n2   \n3 class Residual(nn.Module):   \n4 def __init__(self, fn):   \n5 super().__init__()   \n6 self.fn $=$ fn   \n7   \n8 def forward(self, x):   \n9 return self.fn $( { \\bf x } ) ~ + ~ { \\bf x }$   \n10   \n11 def ConvMixer(dim, depth, kernel_size $^ { = 9 }$ , patch_size $^ { - 7 }$ , n_classes=1000):   \n12 return nn.Sequential(   \n13 nn.Conv2d(3, dim, kernel_size=patch_size, stride=patch_size),   \n14 nn.GELU(),   \n15 nn.BatchNorm2d(dim),   \n16 \\*[nn.Sequential(   \n17 Residual(nn.Sequential(   \n18 nn.Conv2d(dim, dim, kernel_size, groups $=$ dim, padding $=$ \"same\"),   \n19 nn.GELU(),   \n20 nn.BatchNorm2d(dim)   \n21 )),   \n22 nn.Conv2d(dim, dim, kernel_size $^ { = 1 }$ ),   \n23 nn.GELU(),   \n24 nn.BatchNorm2d(dim)   \n25 ) for i in range(depth)],   \n26 nn.AdaptiveAvgPool2d((1,1)),   \n27 nn.Flatten(),   \n28 nn.Linear(dim, n_classes)   \n29 ) ",
+        "bbox": [
+            143,
+            137,
+            802,
+            502
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "def ConvMixr(h,d,k,p,n):   \n2 S,C, $\\mathsf { A } \\mathop { = }$ Sequential,Conv2d,lambda x:S(x,GELU(),BatchNorm2d(h))   \n3 R=type('',(S,),{'forward':lambda s $, \\mathbf { x } : \\mathbf { s } \\left[ \\left. \\Theta \\right] \\left( \\mathbf { x } \\right) + \\mathbf { x } \\right\\} \\mathrm { . }$ )   \n4 return S(A(C(3,h,p,p)),\\*[S(R(A(C(h,h,k,groups $\\mathtt { \\Gamma } = \\mathtt { h }$ ,padding $\\mathbf { \\tau } _ { | = \\mathbf { k } / \\mathbf { \\Omega } / 2 }$ ))),A(C(h,h,1))) for i $\\hookrightarrow$ in range(d)],AdaptiveAvgPool2d((1,1)),Flatten(),Linear $( \\mathtt { h } , \\mathtt { n } )$ ) ",
+        "bbox": [
+            148,
+            556,
+            820,
+            621
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 8: An implementation of our model in exactly 280 characters, in case you happen to know of any means of disseminating information that could benefit from such a length. All you need to do to run this is from torch.nn import \\*. ",
+        "bbox": [
+            171,
+            626,
+            823,
+            667
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "This section presents an expanded (but still quite compact) version of the terse ConvMixer implementation that we presented in the paper. The code is given in Figure 7. We also present an even more terse implementation in Figure 8, which to the best of our knowledge is the first model that achieves the elusive dual goals of $8 0 \\% +$ ImageNet top-1 accuracy while also fitting into a tweet. ",
+        "bbox": [
+            174,
+            696,
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+            753
+        ],
+        "page_idx": 14
+    }
+]

parse/dev/UmvSlP-PyV/UmvSlP-PyV.md

@@ -0,0 +1,224 @@
+# Beyond neural scaling laws: beating power law scaling via data pruning
+
+# Ben Sorscher∗1
+
+Robert Geirhos∗2
+
+Shashank Shekhar3
+
+# Surya Ganguli1,3§
+
+Ari S. Morcos3§
+
+∗equal contribution 1Department of Applied Physics, Stanford University 2University of Tübingen 3Meta AI (FAIR) §Joint senior authors
+
+# Abstract
+
+Widely observed neural scaling laws, in which error falls off as a power of the training set size, model size, or both, have driven substantial performance improvements in deep learning. However, these improvements through scaling alone require considerable costs in compute and energy. Here we focus on the scaling of error with dataset size and show how in theory we can break beyond power law scaling and potentially even reduce it to exponential scaling instead if we have access to a high-quality data pruning metric that ranks the order in which training examples should be discarded to achieve any pruned dataset size. We then test this improved scaling prediction with pruned dataset size empirically, and indeed observe better than power law scaling in practice on ResNets trained on CIFAR-10, SVHN, and ImageNet. Next, given the importance of finding high-quality pruning metrics, we perform the first large-scale benchmarking study of ten different data pruning metrics on ImageNet. We find most existing high performing metrics scale poorly to ImageNet, while the best are computationally intensive and require labels for every image. We therefore developed a new simple, cheap and scalable self-supervised pruning metric that demonstrates comparable performance to the best supervised metrics. Overall, our work suggests that the discovery of good data-pruning metrics may provide a viable path forward to substantially improved neural scaling laws, thereby reducing the resource costs of modern deep learning.
+
+# 1 Introduction
+
+Empirically observed neural scaling laws [1, 2, 3, 4, 5, 6, 7, 8] in many domains of machine learning, including vision, language, and speech, demonstrate that test error often falls off as a power law with either the amount of training data, model size, or compute. Such power law scaling has motivated significant societal investments in data collection, compute, and associated energy consumption. However, power law scaling is extremely weak and unsustainable. For example, a drop in error from $3 \%$ to $2 \%$ might require an order of magnitude more data, compute, or energy. In language modeling with large transformers, a drop in cross entropy loss from about 3.4 to $2 . 8 \mathrm { n a t s } ^ { 2 }$ requires $I O$ times more training data (Fig. 1 in [2]). Also, for large vision transformers, an additional 2 billion pre-training data points (starting from 1 billion) leads to an accuracy gain on ImageNet of a few percentage points (Fig. 1 in [7]). Here we ask whether we might be able to do better. For example, can we achieve exponential scaling instead, with a good strategy for selecting training examples? Such vastly superior scaling would mean that we could go from $3 \%$ to $2 \%$ error by only adding a few carefully chosen training examples, rather than collecting $1 0 \times$ more random ones.
+
+![](images/f4951b0b0544dc88c37572ef6cd5e8307cf709eff161b62fe14d4d6cb443efed.jpg)  
+Figure 1: Our analytic theory of data pruning predicts that power law scaling of test error with respect to dataset size can be beaten. A: Test error as a function of $\alpha _ { \mathrm { p r u n e } } = f \alpha _ { \mathrm { t o t } }$ with $\theta = 0$ . We observe an excellent match between our analytic theory (solid curves) and numerical simulations (dots) of perceptron learning at parameters ${ \bf N } = 2 0 0$ (here: ${ \bf N } = 2 0 0$ constant throughout figure). The red curve indicates the Pareto optimal test error $\varepsilon$ achievable from a tradeoff between $\alpha _ { \mathrm { t o t } }$ and $f$ at fixed $\alpha _ { \mathrm { p r u n e } }$ B: We find that when data is abundant (scarce) corresponding to large (small) $\alpha _ { \mathrm { t o t } }$ , the better pruning strategy is to keep the hard (easy) examples. C: Color indicates difference in test error in keeping hard versus easy examples, revealing the change in strategy in (B). D: We tested this prediction on a ResNet18 trained on CIFAR-10, finding remarkably the same shift in optimal pruning strategy under the EL2N metric. E: Test accuracy as a function of $f$ and $\alpha _ { \mathrm { p r u n e } }$ . For every fixed $\alpha _ { \mathrm { p r u n e } }$ , there is an optimal $f _ { \mathrm { o p t } }$ (purple curve). F: $I ( \dot { \alpha } _ { \mathrm { p r u n e } } )$ for different $f$ .
+
+Focusing on scaling of performance with training dataset size, we demonstrate that exponential scaling is possible, both in theory and practice. The key idea is that power law scaling of error with respect to data suggests that many training examples are highly redundant. Thus one should in principle be able to prune training datasets to much smaller sizes and train on the smaller pruned datasets without sacrificing performance. Indeed some recent works [9, 10, 11] have demonstrated this possibility by suggesting various metrics to sort training examples in order of their difficulty or importance, ranging from easy or redundant examples to hard or important ones, and pruning datasets by retaining some fraction of the hardest examples. However, these works leave open fundamental theoretical and empirical questions: When and why is successful data pruning possible? What are good metrics and strategies for data pruning? Can such strategies beat power law scaling? Can they scale to ImageNet? Can we leverage large unlabeled datasets to successfully prune labeled datasets? We address these questions through both theory and experiment. Our main contributions are:
+
+1. Employing statistical mechanics, we develop a new analytic theory of data pruning in the student-teacher setting for perceptron learning, where examples are pruned based on their teacher margin, with large (small) margins corresponding to easy (hard) examples. Our theory quantitatively matches numerical experiments and reveals two striking predictions:
+
+(a) The optimal pruning strategy changes depending on the amount of initial data; with abundant (scarce) initial data, one should retain only hard (easy) examples.   
+(b) Exponential scaling is possible with respect to pruned dataset size provided one chooses an increasing Pareto optimal pruning fraction as a function of initial dataset size.
+
+2. We show that the two striking predictions derived from theory hold also in practice in much more general settings. Indeed we empirically demonstrate signatures of exponential scaling of error with respect to pruned dataset size for ResNets trained from scratch on SVHN, CIFAR-10 and ImageNet, and Vision Transformers fine-tuned on CIFAR-10.
+
+3. Motivated by the importance of finding good quality metrics for data pruning, we perform a large scale benchmarking study of 10 different data pruning metrics at scale on ImageNet, finding that most perform poorly, with the exception of the most compute intensive metrics.
+
+4. We leveraged self-supervised learning (SSL) to developed a new, cheap unsupervised data pruning metric that does not require labels, unlike prior metrics. We show this unsupervised metric performs comparably to the best supervised pruning metrics that require labels and much more compute. This result opens the door to the exciting possibility of leveraging pre-trained foundation models to prune new datasets even before they are labeled.
+
+Overall these results shed theoretical and empirical insights into the nature of data in deep learning and our ability to prune it, and suggest our current practice of collecting extremely large datasets may be highly inefficient. Our initial results in beating power law scaling motivate further studies and investments in not just inefficently collecting large amounts of random data, but rather, intelligently collecting much smaller amounts of carefully selected data, potentially leading to the creation and dissemination of foundation datasets, in addition to foundation models [12].
+
+# 2 Background and related work
+
+Our work brings together 3 largely disparate strands of intellectual inquiry in machine learning: (1) explorations of different metrics for quantifying differences between individual training examples; (2) the empirical observation of neural scaling laws; and (3) the statistical mechanics of learning.
+
+# 2.1 Pruning metrics: not all training examples are created equal
+
+Several recent works have explored various metrics for quantifying individual differences between data points. To describe these metrics in a uniform manner, we will think of all of them as ordering data points by their difficulty, ranging from “easiest” to “hardest.” When these metrics have been used for data pruning, the hardest examples are retained, while the easiest ones are pruned away.
+
+EL2N scores. For example [10] trained small ensembles (of about 10) networks for a very short time (about 10 epochs) and computed for every training example the average $L _ { 2 }$ norm of the error vector (EL2N score). Data pruning by retaining only the hardest examples with largest error enabled training from scratch on only $5 0 \%$ and $7 5 \%$ of CIFAR-10 and CIFAR-100 respectively without any loss in final test accuracy. However the performance of EL2N on ImageNet has not yet been explored.
+
+Forgetting scores and classification margins. [9] noticed that over the entire course of training, some examples are learned early and never forgotten, while others can be learned and unlearned (i.e. forgotten) repeatedly. They developed a forgetting score which measures the degree of forgetting of each example. Intuitively examples with low (high) forgetting scores can be thought of as easy (hard) examples. [9] explored data pruning using these metrics, but not at ImageNet scale.
+
+Memorization and influence. [13] defined a memorization score for each example, corresponding to how much the probability of predicting the correct label for the example increases when it is present in the training set relative to when it is absent; a large increase means the example must be memorized (i.e. the remaining training data do not suffice to correctly learn this example). Additionally [13] also considered an influence score that quantifies how much adding a particular example to the training set increases the probability of the correct class label of a test example. Intuitively, low memorization and influence scores correspond to easy examples that are redundant with the rest of the data, while high scores correspond to hard examples that must be individually learned. [13] did not use these scores for data pruning as their computation is expensive. We note since memorization explicitly approximates the increase in test loss due to removing each individual example, it is likely to be a good pruning metric (though it does not consider interactions).
+
+Ensemble active learning. Active learning iterates between training a model and selecting new inputs to be labeled [14, 15, 16, 17, 18]. In contrast, we focus on data pruning: one-shot selection of a data subset sufficient to train to high accuracy from scratch. A variety of coreset algorithms (e.g. [19]) have been proposed for this, but their computation is expensive, and so data-pruning has been less explored at scale on ImageNet. An early clustering approach [20] allowed training on $9 0 \%$ of ImageNet without sacrificing accuracy. Notably [11] reduced this to $8 0 \%$ by training a large ensemble of networks on ImageNet and using ensemble uncertainty to define the difficulty of each example, with low (high) uncertainty corresponding to easy (hard) examples. We will show how to achieve similar pruning performance without labels or the need to train a large ensemble.
+
+Diverse ensembles (DDD). [21] assigned a score to every ImageNet image, given by the number of models in a diverse ensemble (10 models) that misclassified the image. Intuitively, low (high) scores correspond to easy (hard) examples. The pruning performance of this metric remains unexplored.
+
+Summary. We note: (1) only one of these metrics has tested well for its efficacy in data pruning at scale on ImageNet; (2) all of these metrics require label information; (3) there is no theory of when and why data pruning is possible for any of these metrics; and (4) none of these works suggest the possibility of exponential scaling. We thus go beyond this prior work by benchmarking the data pruning efficacy of not only these metrics but also a new unsupervised metric we introduce that does not require label information, all at scale on ImageNet. We also develop an analytic theory for data-pruning for the margin metric that predicts not only the possibility of exponential scaling but also the novel finding that retaining easy instead of hard examples is better when data is scarce.
+
+# 2.2 Neural scaling laws and their potential inefficiency
+
+Recent work [1, 2, 3, 4, 5, 6, 7, 8] has demonstrated that test loss $\mathcal { L }$ often falls off as a power law with different resources like model parameters $( N )$ , number of training examples $( P )$ , and amount of compute $( C )$ . However, the exponents $\nu$ of these power laws are often close to 0, suggesting potentially inefficient use of resources. For example, for large models with lots of compute, so that the amount of training data constitutes a performance bottleneck, the loss scales as $ { \mathcal { L } } \approx P ^ { - \nu }$ Specifically for a large transformer based language model, $\nu = 0 . 0 9 5$ , which implies an order of magnitude increase in training data drops cross-entropy loss by only about 0.6 nats (Fig. 1 in [2]). In neural machine translation experiments $\nu$ varies across language pairs from 0.35 to 0.48 (Table 1 in [5]). Interestingly, [8] explored a fixed computation budget $C$ and optimized jointly over model size $N$ and training set size $P$ , revealing that scaling both $N$ and $P$ commensurately as $C$ increases is compute optimal, and can yield smaller high performing models (trained on more data) than previous work. Nevertheless, for a transformer based language model, a $1 0 0 \times$ increase in compute, corresponding to $1 0 \times$ increases in both model size and training set size, leads to a drop in cross-entropy loss of only about 0.5 nats (Fig. 2 in [8]). Similar slow scaling holds for large vision transformers where adding 2 billion pre-training images reduces ImageNet performance by a few percentage points (Fig. 1 in [7]). While all of these results constitute significant improvements in performance, they do come at a substantial resource cost whose fundamental origin arises from power law scaling with small exponents. Recent theoretical works [22, 23, 24] have argued that the power law exponent is governed by the dimension of a data manifold from which training examples are uniformly drawn. Here we explore whether we can beat power law scaling through careful data selection.
+
+# 2.3 Statistical mechanics of perceptron learning
+
+Statistical mechanics has long played a role in analyzing machine learning problems (see e.g. [25, 26, 27, 28] for reviews). One of the most fundamental applications is perceptron learning in the student-teacher setting [29, 30], in which random i.i.d. Gaussian inputs are labeled by a teacher perceptron to construct a training set. The test error for another student perceptron learning from this training set then scales as a power law with exponent $- 1$ for such data. Such perceptrons have also been analyzed in an active learning setting where the learner is free to design any new input to be labeled [31, 32], rather than choose from a fixed set of inputs, as in data-pruning. Recent work [33] has analyzed this scenario but focused on message passing algorithms that are tailored to the case of Gaussian inputs and perceptrons, and are hard to generalize to real world settings. In contrast we analyze margin based pruning algorithms that are used in practice in diverse settings, as in [9, 10].
+
+# 3 An analytic theory of data pruning
+
+To better understand data pruning, we employed the replica method from statistical mechanics [34] to develop an analytic theory of pruning for the perceptron in the student-teacher setting [25] (see App. A for detailed derivations of all results). Consider a training dataset of $P$ examples $\{ { \bf x } ^ { \mu } , y ^ { \mu } \} _ { \mu = 1 , \dots , P }$ where $\mathbf { x } ^ { \mu } \in \mathbb { R } ^ { N }$ are i.i.d. zero mean unit variance random Gaussian inputs and $y ^ { \mu } = \operatorname { s i g n } ( \mathbf { T } \cdot \mathbf { x } ^ { \mu } )$ are labels generated by a teacher perceptron with weight vector $\mathbf { T } \in \mathbf { \mathbb { R } } ^ { N }$ . We work in the high dimensional statistics limit where $N , P \to \infty$ but the ratio $\begin{array} { r } { \alpha _ { \mathrm { t o t } } = \frac { P } { N } } \end{array}$ of the number of total training examples to parameters remains $O ( 1 )$ . We then consider a pruning algorithm used in [9, 10], namely: (1) train a probe student perceptron for very few epochs on the training data, obtaining weights $\mathbf { J _ { \mathrm { p r o b e } } }$ ; (2) compute the margin $m ^ { \mu } = { \bf J } _ { \mathrm { p r o b e } } \cdot \left( y ^ { \mu } { \bf x } ^ { \mu } \right)$ of each training example, where large (small) margins correspond to easy (hard) examples; (3) construct a pruned dataset of size $P _ { \mathrm { p r u n e } } = f P$ , where $f$ is the fraction of examples kept, by retaining the $P _ { \mathrm { p r u n e } }$ hardest examples, (4) train a new perceptron to completion on the smaller dataset with a smaller ratio $\begin{array} { r } { \alpha _ { \mathrm { p r u n e } } = \frac { P _ { \mathrm { p r u n e } } } { N } } \end{array}$ of examples to parameters.
+
+We are interested in the test error $\varepsilon$ of this final perceptron as a function of $\alpha _ { \mathrm { t o t } } , f$ , and the angle $\theta$ between the probe student $\mathbf { J _ { \mathrm { { p r o b e } } } }$ and the teacher $\mathbf { T }$ . Our theory approximates $\mathbf { J _ { \mathrm { { p r o b e } } } }$ as simply a random Gaussian vector conditioned to have angle $\theta$ with the teacher $\mathbf { T }$ . Under this approximation we obtain an analytic theory for $\varepsilon ( \alpha _ { \mathrm { t o t } } , f , \theta )$ that is asymptotically exact in the high dimensional limit (App. A). We first examine results when $\theta = 0$ , so we are pruning training examples according to their veridical margins with respect to the teacher (Fig. 1A). We find two striking phenomena, each of which constitute predictions in real-world settings that we will successfully confirm empirically.
+
+The best pruning strategy depends on the amount of initial data. First, we note the test error curve for $f = 1$ in Fig. 1A corresponding to no pruning, or equivalently to randomly pruning a larger dataset of size $\alpha _ { \mathrm { t o t } }$ down to a size $\alpha _ { \mathrm { p r u n e } }$ , exhibits the well known classical perceptron learning power law scaling $\varepsilon \propto \alpha _ { \mathrm { p r u n e } } ^ { - 1 }$ . Interestingly though, for small $\alpha _ { \mathrm { t o t } }$ , keeping the hardest examples performs worse than random pruning (lighter curves above darkest curve for small $\alpha _ { \mathrm { p r u n e } }$ in Fig. 1A). However, for large $\alpha _ { \mathrm { t o t } }$ , keeping the hardest examples performs substantially better than random pruning (lighter curves below darkest curve for large $\alpha _ { \mathrm { p r u n e } }$ in Fig. 1A). It turns out keeping the easiest rather than hardest examples is a better pruning strategy when $\alpha _ { \mathrm { t o t } }$ is small (Fig. 1C). If one does not have much data to start with, it is better to keep the easiest examples with largest margins (i.e. the blue regions of Fig. 1B) to avoid overfitting. The easiest examples provide coarse-grained information about the target function, while the hard examples provide fine-grained information about the target function which can prevent the model from learning if one starts with lots of data. In cases where overfitting is less of an issue, it is best to keep the hardest examples with smallest margin that provide more information about the teacher’s decision boundary (i.e. the green region of Fig. 1B). Intuitively, in the limited data regime, it is challenging to model outliers since the basics are not adequately captured; hence, it is more important to keep easy examples so that the model can get to moderate error. However, with a larger dataset, the easy examples can be learned without difficulty, making modeling outliers the fundamental challenge.
+
+Fig. 1C reveals which pruning strategy is best as a joint function of $\alpha _ { \mathrm { t o t } }$ and $f$ . Note the transition between optimal strategies becomes sharper at small fractions $f$ of data kept. This transition between optimal pruning strategies can be viewed as a prediction in more general settings. To test this prediction we trained a ResNet18 on pruned subsets of the CIFAR-10 dataset (Fig. 1D), and observed strikingly similar behavior, indicating the prediction can hold far more generally, beyond perceptron learning. Interestingly, [9, 10] missed this transition, likely because they started pruning from large datasets.
+
+Pareto optimal data pruning can beat power law scaling. A second prediction of our theory is that when keeping a fixed fraction $f$ of the hardest examples as $\alpha _ { \mathrm { t o t } }$ increases (i.e. constant color curves in Fig. 1A), the error initially drops exponentially in $\alpha _ { \mathrm { p r u n e } } = f \alpha _ { \mathrm { t o t } }$ , but then settles into the universal power law $\varepsilon \propto \alpha _ { \mathrm { p r u n e } } ^ { - 1 }$ for all fixed $f$ . Thus there is no asymptotic advantage to data pruning at a fixed $f$ . However, by pruning more aggressively (smaller $f$ ) when given more initial data (larger $\alpha _ { \mathrm { t o t . } }$ ), one can achieve a Pareto optimal test error as a function of pruned dataset size $\alpha _ { \mathrm { p r u n e } }$ that remarkably traces out at least an exponential scaling law (Fig. 1A, purple curve). Indeed our theory predicts for each $\alpha _ { \mathrm { p r u n e } }$ a Pareto optimal point in $\alpha _ { \mathrm { t o t } }$ and $f$ (subject to $\alpha _ { \mathrm { p r u n e } } = f \alpha _ { \mathrm { t o t } } )$ , yielding for every fixed $\alpha _ { \mathrm { p r u n e } }$ an optimal $f _ { \mathrm { o p t } }$ , plotted in Fig. 1E. Note $f _ { \mathrm { o p t } }$ decreases with $\alpha _ { \mathrm { p r u n e } }$ indicating more aggressive pruning (smaller $f _ { \mathrm { o p t } } )$ ) of original datasets of larger size $\alpha _ { \mathrm { t o t } }$ is required to obtain larger Pareto optimal pruned datasets of size $\alpha _ { \mathrm { p r u n e } }$ . We will test this striking scaling prediction in Fig. 3.
+
+![](images/57a2549d134949b92c470a9c19a0c10bb40524dc376bec35d42755f606757537.jpg)  
+Figure 2: Data pruning with an imperfect metric. A: Weight vectors and decision boundaries for a teacher (black) and probe student (red) separated by angle $\theta$ . The black point has margin $0 \left( \kappa \right)$ w.r.t. the probe (teacher). B–D: Test error as a function of $\alpha _ { \mathrm { p r u n e } }$ for different $f$ and different $\theta$ .
+
+Beating power law scaling: an information-theoretic perspective. Classical randomly selected data generates slow power law error scaling because each extra training example provides less new information about the correct decision boundary than the previous example. More formally, let $S { \left( \alpha _ { \mathrm { t o t } } \right) }$ denote the typical entropy of the posterior distribution over student perceptron weights consistent with a training set of size $\alpha _ { \mathrm { t o t } }$ . The information gain $I ( \alpha _ { \mathrm { t o t } } )$ due to additional examples beyond $\alpha _ { \mathrm { t o t } }$ can be defined as the rate at which the posterior entropy is reduced: $\begin{array} { r } { I ( \alpha _ { \mathrm { t o t } } ) = - \frac { d } { d \alpha _ { \mathrm { t o t } } } S ( \alpha _ { \mathrm { t o t } } ) } \end{array}$ . In classical perceptron learning $I ( \alpha _ { \mathrm { t o t } } )$ decays to zero as a power law in $\alpha _ { \mathrm { t o t } }$ , reflecting a vanishing amount of information per each new example, leading to the slow power law decay of test error $\varepsilon \propto \alpha _ { \mathrm { t o t } } ^ { - 1 }$ . However, data pruning can increase the information gained per example by pruning away the uninformative examples. To show this, we generalized the replica calculation of the posterior entropy $S$ and information gain $I$ from random datasets of size $\alpha _ { \mathrm { t o t } }$ to pruned datasets of size $\alpha _ { \mathrm { p r u n e } }$ (App. A). We plot the resulting information gain $I ( \alpha _ { \mathrm { p r u n e } } )$ for different $f$ in Fig. 1F. For any fixed $f$ $\dot { \mathbf { \rho } } , I ( \alpha _ { \mathrm { p r u n e } } )$ will eventually decay as a power law as $\alpha _ { \mathrm { p r u n e } } ^ { - 1 }$ . However, by more aggressively pruning (smaller $f$ ) datasets of larger size $\alpha _ { \mathrm { t o t } }$ , $I ( \alpha _ { \mathrm { p r u n e } } )$ can converge to a finite value $I ( \infty ) = 1$ nat/example, resulting in larger pruned datasets only adding useful non-redundant information. Since each new example under Pareto optimal data pruning conveys finite information about the target decision boundary, as seen in Fig. 1F, the test error can decay at least exponentially in pruned dataset size as in Fig. 1A. Classical results [30] have shown that training examples chosen by maximizing the disagreement of a committee of student perceptrons can provide an asymptotically finite information rate, leading to exponential decay in test error. Intriguingly, the Pareto-optimal data pruning strategy we study in this work leads to faster than exponential decay, because it includes (partial) information about the target function provided by the probe student (Fig. 11).
+
+An imperfect pruning metric yields a cross over from exponential to power law scaling. We next examine the case of nonzero angle $\theta$ between the probe student $\mathbf { J _ { \mathrm { p r o b e } } }$ and the teacher $\mathbf { T }$ , such that the ranking of training examples by margin is no longer completely accurate (Fig. 2A). Retaining the hard examples with smallest margin with respect to the probe student will always result in pruned datasets lying near the probe’s decision boundary. But if $\theta$ is large, such examples might be far from the teacher’s decision boundary, and therefore could be less informative about the teacher (Fig. 2A). As a result our theory, confirmed by simulations, predicts that under nonzero angles $\theta$ , the Pareto optimal lower envelope of test error over both $\alpha _ { \mathrm { t o t } }$ and $f$ initially scales exponentially as a function of $\alpha _ { \mathrm { p r u n e } } = f \alpha _ { \mathrm { t o t } }$ but then crosses over to a power law (Fig. 2BCD). Indeed, at any given nonzero $\theta$ , our theory reveals that as $\alpha _ { \mathrm { t o t } }$ (and therefore $\alpha _ { \mathrm { p r u n e . } }$ ) becomes large, one cannot decrease test error any further by retaining less than a minimum fraction $f _ { \mathrm { m i n } } ( \theta )$ of all available data. For example when $\theta = 1 0 ^ { \circ } \overset { \cdot } { ( } \theta = 2 0 ^ { \circ } )$ one can do no better asymptotically than pruning down to $24 \%$ $( 4 6 \% )$ of the total data (Fig. 2CD). As $\theta$ approaches 0, $f _ { \mathrm { m i n } } ( \theta )$ approaches 0, indicating that one can prune extremely aggressively to arbitrarily small $f$ while still improving performance, leading to at least exponential scaling for arbitrarily large $\alpha _ { \mathrm { p r u n e } }$ in Fig. 2B. However, for nonzero $\theta$ , the lack of improvement for $f < f _ { \mathrm { m i n } } ( \theta )$ at large $\alpha _ { \mathrm { p r u n e } }$ renders aggressive pruning ineffective. This result highlights the importance of finding high quality pruning metrics with $\theta \approx 0$ . Such metrics can delay the cross over from exponential to power law scaling as pruned dataset size $\alpha _ { \mathrm { p r u n e } }$ increases, by making aggressive pruning with very small $f$ highly effective. Strikingly, in App. Fig. 10 we demonstrate this cross-over in a real-world setting by showing that the test error on SVHN is bounded below by a power law when the dataset is pruned by a probe ResNet18 under the EL2N metric, trained for 4 epochs (weak pruning metric) but not a probe ResNet18 trained for 40 epochs (strong pruning metric).
+
+![](images/cacca5571c5984f8cd8e6159d713e3d9b0c958faf7b5c2d74cdd483350fe861e.jpg)  
+Figure 3: Beating power law scaling in practice. A–D: Curves of test error against pruned dataset size in 4 settings. Pruning scores were EL2N [10] for CIFAR-10 and SVHN and memorization [13] for ImageNet. See App. B for all pruning/training details and App. D for similar ImageNet plots with EL2N. Note solid curves reflect performance with a fixed total dataset size; if we prune more aggressively with even larger datasets, scaling could improve further (e.g., dashed lines in A). Error curves with no data pruning $( f = 1 )$ are labeled with their best-fit power law scaling $\sim \alpha ^ { - \nu }$ . (Note that for SVHN in B an asymptotic constant error $E ( P  \infty ) = 1 . \bar { 1 } \%$ is subtracted from each of the curves to visualize the power law scaling more clearly.)
+
+![](images/750369d82ba6a10dff6b9975fe4186c91eb279da3bf5bc0fcb89abff4cdcfb6c.jpg)  
+Figure 4: Data pruning improves transfer learning. A: CIFAR-10 performance of a ViT pre-trained on all of ImageNet21K and fine-tuned on different pruned subsets of CIFAR-10 under the EL2N metric. B: CIFAR-10 performance of ResNet50s pretrained on different pruned subsets of ImageNet1K and fine-tuned on all of CIFAR10.
+
+# 4 Data pruning can beat power law scaling in practice
+
+Our theory of data pruning for the perceptron makes three striking predictions which can be tested in more general settings, such as deep neural networks trained on benchmark datasets: (1) relative to random data pruning, keeping only the hardest examples should help when the initial dataset size is large, but hurt when it is small; (2) data pruning by retaining a fixed fraction $f$ of the hardest examples should yield power law scaling, with exponent equal to that of random pruning, as the initial dataset size increases; (3) the test error optimized over both initial data set size and fraction of data kept can trace out a Pareto optimal lower envelope that beats power law scaling of test error as a function of pruned dataset size, through more aggressive pruning at larger initial dataset size. We verified all three of these predictions on ResNets trained on SVHN, CIFAR-10, and ImageNet using varying amounts of initial dataset size and fractions of data kept under data pruning (compare theory in Fig. 3A with deep learning experiments in Fig. 3BCD). In each experimental setting we see better than power law scaling at larger initial data set sizes and more aggressive pruning. Moreover we would likely see even better scaling with even larger initial datasets (as in Fig.3A dashed lines).
+
+Data pruning improves transfer learning. Modern foundation models are pre-trained on a large initial dataset, and then transferred to other downstream tasks by fine-tuning on them. We therefore examined whether data-pruning can be effective for both reducing the amount of fine-tuning data and the amount of pre-training data. To this end, we first analyzed a vision transformer (ViT) pre-trained on ImageNet21K and then fine-tuned on different pruned subsets of CIFAR-10. Interestingly, pre-trained models allow for far more aggressive data pruning; fine-tuning on only $10 \%$ of CIFAR-10 can match or exceed performance obtained by fine tuning on all of CIFAR-10 (Fig. 4A). Furthermore Fig. 4A provides a new example of beating power law scaling in the setting of fine-tuning. Additionally, we examined the efficacy of pruning pre-training data by pre-training ResNet50s on different pruned subsets of ImageNet1K (exactly as in Fig. 3D) and then fine-tuning them on all of CIFAR-10. Fig. 4B demonstrates pre-training on as little as $5 0 \%$ of ImageNet can match or exceed CIFAR-10 performance obtained by pre-training on all of ImageNet. Thus intriguingly pruning pre-training data on an upstream task can still maintain high performance on a different downstream task. Overall these results demonstrate the promise of data pruning in transfer learning for both the pre-training and fine-tuning phases.
+
+![](images/a18df9bfde9ab270a6c571efccafab9598f464bed0af5ea554b36d734f2c5f70.jpg)  
+Figure 5: Dataset pruning at ImageNet scale. A: Spearman’s rank correlation between all pairs of ImageNet metric scores, along with hierarchical clustering (as provided by seaborn.clustermap). B: Benchmarking existing supervised metrics on ImageNet (top-5 validation accuracy). C: Comparing top-5 performance on ImageNet when pruning according to the best existing supervised metric (memorization) and our supervised and self-supervised prototype metrics. In all 3 cases, training on $80 \%$ of ImageNet approximates training on $100 \%$ . See App. B for pruning and training details.
+
+# 5 Benchmarking supervised pruning metrics on ImageNet
+
+We note that the majority of data pruning experiments have been performed on small-scale datasets (i.e. variants of MNIST and CIFAR), while the few pruning metrics proposed for ImageNet have rarely been compared against baselines designed on smaller datasets. Therefore, it is currently unclear how most pruning methods scale to ImageNet and which method is best. Motivated by how strongly the quality of a pruning metric can impact performance in theory (Fig. 2), we decided to fill this knowledge gap by performing a systematic evaluation of 8 different supervised pruning metrics on ImageNet: two variants of influence scores [13], two variants of EL2N [10], DDD [21], memorization [13], ensemble active learning [11], and forgetting [9]. See Section 2 for a review of these metrics. Additionally, we include two new prototypicality metrics that we introduce in the next section.
+
+We first asked how consistent the rankings induced by different metrics are by computing the Spearman rank correlation between each pair of metrics (Fig. 5A). Interestingly, we found substantial diversity across metrics, though some (EL2N, DDD, and memorization) were fairly similar with rank correlations above 0.7. However, we observed marked performance differences between metrics: Fig 5BC shows test performance when a fraction $f$ of the hardest examples under each metric are kept in the training set. Despite the success of many of these metrics on smaller datasets, only a few still match performance obtained by training on the full dataset, when selecting a significantly smaller training subset (i.e. about $8 0 \%$ of ImageNet). Nonetheless, most metrics continue to beat random pruning, with memorization in particular demonstrating strong performance (Fig. 5C). We note that data pruning on ImageNet may be more difficult than data pruning on other datasets, because ImageNet is already carefully curated to filter out uninformative examples.
+
+We found that all pruning metrics amplify class imbalance, which results in degraded performance. To solve this we used a simple $5 0 \%$ class balancing ratio for all ImageNet experiments. Further details and baselines without class balancing are shown in App. H.
+
+# 6 Self-supervised data pruning through a prototypicality metric
+
+Fig. 5 shows many data pruning metrics do not scale well to ImageNet, while the few that do require substantial amounts of compute. Furthermore, all these metrics require labels, thereby limiting their ability to prune data for large-scale foundation models trained on massive unlabeled datasets [12]. Thus there is a clear need for simple, scalable, self-supervised pruning metrics.
+
+To compute a self-supervised pruning metric for ImageNet, we perform $k$ -means clustering in the embedding space of an ImageNet pre-trained self-supervised model (here: SWaV [35]), and define the difficulty of each data point by the Euclidean distance to its nearest cluster centroid, or prototype. Thus easy (hard) examples are the most (least) prototypical. Encouragingly, in Fig. 5C, we find our self-supervised prototype metric matches or exceeds the performance of the best supervised metric, memorization, until only $70 \mathrm { - } 8 0 \%$ of the data is kept, despite the fact that our metric does not use labels and is much simpler and cheaper to compute than many previously proposed supervised metrics. See App. Fig. 9 for further scaling experiments using the self-supervised metric.
+
+To assess whether the clusters found by our metric align with ImageNet classes, we compared their overlaps in Fig. 6A. Interestingly, we found alignment for some but not all classes. For example, class categories such as snakes were largely aligned to a small number of unsupervised clusters, while other classes were dispersed across many such clusters. If class information is available, we can enforce alignment between clusters and classes by simply computing a single prototype for each class (by averaging the embeddings of all examples of this class). While originally intended to be an additional baseline metric (called supervised prototypes, light blue in Fig 5C), this metric remarkably outperforms other supervised metrics and largely matches the performance of memorization, which is prohibitively expensive to compute. Moreover, the performance of the best self-supervised and supervised metrics are similar, demonstrating the promise of self-supervised pruning.
+
+One important choice for the self-supervised prototype metric is the number of clusters $k$ . We found, reassuringly, our results were robust to this choice: $k$ can deviate one order of magnitude more or less than the true number of classes (i.e. 1000 for ImageNet) without affecting performance (App. F).
+
+To better understand example difficulty under various metrics, we visualize extremal images for our self-supervised prototype metric and the memorization metric for one class (Fig 6B,C). Qualitatively, easy examples correspond to highly similar, redundant images, while hard examples look like idiosyncratic outliers. See App. E, Figs. 12,13,14,15,16,17,18,19 for more classes and metrics.
+
+![](images/c6388f541d225b33d830efef18469e869ad3adaa127067232e64f07f1eb633af.jpg)  
+Figure 6: A: Heat map where each row denotes the probability that images in a given cluster come from each ImageNet class. B: The four easiest and hardest images under our self-supervised pruning metric and the best previously published supervised metric (memorization, shown in C) for ImageNet class 100 (black swan).
+
+# 7 Discussion
+
+Summary. We have shown, both in theory and practice, how to break beyond slow power law scaling of error versus dataset size to faster exponential scaling, through data pruning. Additionally we have developed a simple self-supervised pruning metric that enables us to discard $20 \%$ of ImageNet without sacrificing performance, on par with the best and most compute intensive supervised metric.
+
+Limitations. The most notable limitation is that achieving exponential scaling requires a high quality data pruning metric. Since most metrics developed for smaller datasets scale poorly to ImageNet, our results emphasize the importance of future work in identifying high quality, scalable metrics. Our self-supervised metric provides a strong initial baseline. Moreover, a key advantage of data pruning is reduced computational cost due to training on a smaller dataset for the same number of epochs as the full dataset (see App. C). However, we found that performance often increased when training on the pruned dataset for the same number of iterations as on the full dataset, resulting in the same training time, but additional training epochs. However, this performance gain saturated before training time on the pruned dataset approached that on the whole dataset (App. J) thereby still yielding a computational efficiency gain. Overall this tradeoff between accuracy and training time on pruned data is important to consider in evaluating potential gains due to data pruning. Finally, we found that class-balancing was essential to maintain performance on data subsets (App. H). Future work will be required to identify ways to effectively select the appropriate amount of class-balancing.
+
+Ethical considerations. A potential negative societal impact could be that data-pruning leads to unfair outcomes for certain groups. We have done a preliminary analysis of how data-pruning affects performance on individual ImageNet classes (App. I), finding no substantial differential effects across classes. However proper fairness tests specific to deployment settings should always be conducted on every model, whether trained on pruned data or not. Additionally, we analyzed the impact of pruning on OOD performance (App. K).
+
+Outlook: Towards foundation datasets. We believe the most promising future direction is the further development of scalable, unsupervised data pruning metrics. Indeed our theory predicts that the application of pruning metrics on larger scale datasets should yield larger gains by allowing more aggressive pruning. This makes data pruning especially exciting for use on the massive unlabeled datasets used to train large foundation models (e.g. 400M image-text pairs for CLIP [36], 3.5B Instagram images [37], 650M images for the DALLE-2 encoder [38], 780B tokens for PALM [39]). If highly pruned versions of these datasets can be used to train a large number of different models, one can conceive of such carefully chosen data subsets as foundation datasets in which the initial computational cost of data pruning can be amortized across efficiency gains in training many downstream models, just at the initial computational cost of training foundation models is amortized across the efficiency gains of fine-tuning across many downstream tasks. Together, our results demonstrate the promise and potential of data pruning for large-scale training and pretraining.
+
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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]   
+(b) Did you describe the limitations of your work? [Yes] See discussion section 7.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See discussion section 7 and App. I.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See App. A (b) Did you include complete proofs of all theoretical results? [Yes] See App. A
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] All code for theory plots and numerical perceptron simulations is packaged in a reproducible colab notebook.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See App. B.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Since we train a very large (order 100) number of models on computationally intensive tasks (e.g. ImageNet), many of our plots contain only results from a single random seed. However, variability due to random seeds can be inferred by the smooth progression of the relevant quantity on each plot.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See App. B
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes] See App. B   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See App. B   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] See App. B
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/UmvSlP-PyV/UmvSlP-PyV_content_list.json

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+    {
+        "type": "text",
+        "text": "Beyond neural scaling laws: beating power law scaling via data pruning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ben Sorscher∗1 ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Robert Geirhos∗2 ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Shashank Shekhar3 ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Surya Ganguli1,3§ ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ari S. Morcos3§ ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "∗equal contribution 1Department of Applied Physics, Stanford University 2University of Tübingen 3Meta AI (FAIR) §Joint senior authors ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Widely observed neural scaling laws, in which error falls off as a power of the training set size, model size, or both, have driven substantial performance improvements in deep learning. However, these improvements through scaling alone require considerable costs in compute and energy. Here we focus on the scaling of error with dataset size and show how in theory we can break beyond power law scaling and potentially even reduce it to exponential scaling instead if we have access to a high-quality data pruning metric that ranks the order in which training examples should be discarded to achieve any pruned dataset size. We then test this improved scaling prediction with pruned dataset size empirically, and indeed observe better than power law scaling in practice on ResNets trained on CIFAR-10, SVHN, and ImageNet. Next, given the importance of finding high-quality pruning metrics, we perform the first large-scale benchmarking study of ten different data pruning metrics on ImageNet. We find most existing high performing metrics scale poorly to ImageNet, while the best are computationally intensive and require labels for every image. We therefore developed a new simple, cheap and scalable self-supervised pruning metric that demonstrates comparable performance to the best supervised metrics. Overall, our work suggests that the discovery of good data-pruning metrics may provide a viable path forward to substantially improved neural scaling laws, thereby reducing the resource costs of modern deep learning. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Empirically observed neural scaling laws [1, 2, 3, 4, 5, 6, 7, 8] in many domains of machine learning, including vision, language, and speech, demonstrate that test error often falls off as a power law with either the amount of training data, model size, or compute. Such power law scaling has motivated significant societal investments in data collection, compute, and associated energy consumption. However, power law scaling is extremely weak and unsustainable. For example, a drop in error from $3 \\%$ to $2 \\%$ might require an order of magnitude more data, compute, or energy. In language modeling with large transformers, a drop in cross entropy loss from about 3.4 to $2 . 8 \\mathrm { n a t s } ^ { 2 }$ requires $I O$ times more training data (Fig. 1 in [2]). Also, for large vision transformers, an additional 2 billion pre-training data points (starting from 1 billion) leads to an accuracy gain on ImageNet of a few percentage points (Fig. 1 in [7]). Here we ask whether we might be able to do better. For example, can we achieve exponential scaling instead, with a good strategy for selecting training examples? Such vastly superior scaling would mean that we could go from $3 \\%$ to $2 \\%$ error by only adding a few carefully chosen training examples, rather than collecting $1 0 \\times$ more random ones. ",
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+    {
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+        "img_path": "images/f4951b0b0544dc88c37572ef6cd5e8307cf709eff161b62fe14d4d6cb443efed.jpg",
+        "image_caption": [
+            "Figure 1: Our analytic theory of data pruning predicts that power law scaling of test error with respect to dataset size can be beaten. A: Test error as a function of $\\alpha _ { \\mathrm { p r u n e } } = f \\alpha _ { \\mathrm { t o t } }$ with $\\theta = 0$ . We observe an excellent match between our analytic theory (solid curves) and numerical simulations (dots) of perceptron learning at parameters ${ \\bf N } = 2 0 0$ (here: ${ \\bf N } = 2 0 0$ constant throughout figure). The red curve indicates the Pareto optimal test error $\\varepsilon$ achievable from a tradeoff between $\\alpha _ { \\mathrm { t o t } }$ and $f$ at fixed $\\alpha _ { \\mathrm { p r u n e } }$ B: We find that when data is abundant (scarce) corresponding to large (small) $\\alpha _ { \\mathrm { t o t } }$ , the better pruning strategy is to keep the hard (easy) examples. C: Color indicates difference in test error in keeping hard versus easy examples, revealing the change in strategy in (B). D: We tested this prediction on a ResNet18 trained on CIFAR-10, finding remarkably the same shift in optimal pruning strategy under the EL2N metric. E: Test accuracy as a function of $f$ and $\\alpha _ { \\mathrm { p r u n e } }$ . For every fixed $\\alpha _ { \\mathrm { p r u n e } }$ , there is an optimal $f _ { \\mathrm { o p t } }$ (purple curve). F: $I ( \\dot { \\alpha } _ { \\mathrm { p r u n e } } )$ for different $f$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "Focusing on scaling of performance with training dataset size, we demonstrate that exponential scaling is possible, both in theory and practice. The key idea is that power law scaling of error with respect to data suggests that many training examples are highly redundant. Thus one should in principle be able to prune training datasets to much smaller sizes and train on the smaller pruned datasets without sacrificing performance. Indeed some recent works [9, 10, 11] have demonstrated this possibility by suggesting various metrics to sort training examples in order of their difficulty or importance, ranging from easy or redundant examples to hard or important ones, and pruning datasets by retaining some fraction of the hardest examples. However, these works leave open fundamental theoretical and empirical questions: When and why is successful data pruning possible? What are good metrics and strategies for data pruning? Can such strategies beat power law scaling? Can they scale to ImageNet? Can we leverage large unlabeled datasets to successfully prune labeled datasets? We address these questions through both theory and experiment. Our main contributions are: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. Employing statistical mechanics, we develop a new analytic theory of data pruning in the student-teacher setting for perceptron learning, where examples are pruned based on their teacher margin, with large (small) margins corresponding to easy (hard) examples. Our theory quantitatively matches numerical experiments and reveals two striking predictions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(a) The optimal pruning strategy changes depending on the amount of initial data; with abundant (scarce) initial data, one should retain only hard (easy) examples.   \n(b) Exponential scaling is possible with respect to pruned dataset size provided one chooses an increasing Pareto optimal pruning fraction as a function of initial dataset size. ",
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+    {
+        "type": "text",
+        "text": "2. We show that the two striking predictions derived from theory hold also in practice in much more general settings. Indeed we empirically demonstrate signatures of exponential scaling of error with respect to pruned dataset size for ResNets trained from scratch on SVHN, CIFAR-10 and ImageNet, and Vision Transformers fine-tuned on CIFAR-10. ",
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+    {
+        "type": "text",
+        "text": "3. Motivated by the importance of finding good quality metrics for data pruning, we perform a large scale benchmarking study of 10 different data pruning metrics at scale on ImageNet, finding that most perform poorly, with the exception of the most compute intensive metrics. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4. We leveraged self-supervised learning (SSL) to developed a new, cheap unsupervised data pruning metric that does not require labels, unlike prior metrics. We show this unsupervised metric performs comparably to the best supervised pruning metrics that require labels and much more compute. This result opens the door to the exciting possibility of leveraging pre-trained foundation models to prune new datasets even before they are labeled. ",
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+    {
+        "type": "text",
+        "text": "Overall these results shed theoretical and empirical insights into the nature of data in deep learning and our ability to prune it, and suggest our current practice of collecting extremely large datasets may be highly inefficient. Our initial results in beating power law scaling motivate further studies and investments in not just inefficently collecting large amounts of random data, but rather, intelligently collecting much smaller amounts of carefully selected data, potentially leading to the creation and dissemination of foundation datasets, in addition to foundation models [12]. ",
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+    {
+        "type": "text",
+        "text": "2 Background and related work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our work brings together 3 largely disparate strands of intellectual inquiry in machine learning: (1) explorations of different metrics for quantifying differences between individual training examples; (2) the empirical observation of neural scaling laws; and (3) the statistical mechanics of learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 Pruning metrics: not all training examples are created equal ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Several recent works have explored various metrics for quantifying individual differences between data points. To describe these metrics in a uniform manner, we will think of all of them as ordering data points by their difficulty, ranging from “easiest” to “hardest.” When these metrics have been used for data pruning, the hardest examples are retained, while the easiest ones are pruned away. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "EL2N scores. For example [10] trained small ensembles (of about 10) networks for a very short time (about 10 epochs) and computed for every training example the average $L _ { 2 }$ norm of the error vector (EL2N score). Data pruning by retaining only the hardest examples with largest error enabled training from scratch on only $5 0 \\%$ and $7 5 \\%$ of CIFAR-10 and CIFAR-100 respectively without any loss in final test accuracy. However the performance of EL2N on ImageNet has not yet been explored. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Forgetting scores and classification margins. [9] noticed that over the entire course of training, some examples are learned early and never forgotten, while others can be learned and unlearned (i.e. forgotten) repeatedly. They developed a forgetting score which measures the degree of forgetting of each example. Intuitively examples with low (high) forgetting scores can be thought of as easy (hard) examples. [9] explored data pruning using these metrics, but not at ImageNet scale. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Memorization and influence. [13] defined a memorization score for each example, corresponding to how much the probability of predicting the correct label for the example increases when it is present in the training set relative to when it is absent; a large increase means the example must be memorized (i.e. the remaining training data do not suffice to correctly learn this example). Additionally [13] also considered an influence score that quantifies how much adding a particular example to the training set increases the probability of the correct class label of a test example. Intuitively, low memorization and influence scores correspond to easy examples that are redundant with the rest of the data, while high scores correspond to hard examples that must be individually learned. [13] did not use these scores for data pruning as their computation is expensive. We note since memorization explicitly approximates the increase in test loss due to removing each individual example, it is likely to be a good pruning metric (though it does not consider interactions). ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Ensemble active learning. Active learning iterates between training a model and selecting new inputs to be labeled [14, 15, 16, 17, 18]. In contrast, we focus on data pruning: one-shot selection of a data subset sufficient to train to high accuracy from scratch. A variety of coreset algorithms (e.g. [19]) have been proposed for this, but their computation is expensive, and so data-pruning has been less explored at scale on ImageNet. An early clustering approach [20] allowed training on $9 0 \\%$ of ImageNet without sacrificing accuracy. Notably [11] reduced this to $8 0 \\%$ by training a large ensemble of networks on ImageNet and using ensemble uncertainty to define the difficulty of each example, with low (high) uncertainty corresponding to easy (hard) examples. We will show how to achieve similar pruning performance without labels or the need to train a large ensemble. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Diverse ensembles (DDD). [21] assigned a score to every ImageNet image, given by the number of models in a diverse ensemble (10 models) that misclassified the image. Intuitively, low (high) scores correspond to easy (hard) examples. The pruning performance of this metric remains unexplored. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Summary. We note: (1) only one of these metrics has tested well for its efficacy in data pruning at scale on ImageNet; (2) all of these metrics require label information; (3) there is no theory of when and why data pruning is possible for any of these metrics; and (4) none of these works suggest the possibility of exponential scaling. We thus go beyond this prior work by benchmarking the data pruning efficacy of not only these metrics but also a new unsupervised metric we introduce that does not require label information, all at scale on ImageNet. We also develop an analytic theory for data-pruning for the margin metric that predicts not only the possibility of exponential scaling but also the novel finding that retaining easy instead of hard examples is better when data is scarce. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 Neural scaling laws and their potential inefficiency ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Recent work [1, 2, 3, 4, 5, 6, 7, 8] has demonstrated that test loss $\\mathcal { L }$ often falls off as a power law with different resources like model parameters $( N )$ , number of training examples $( P )$ , and amount of compute $( C )$ . However, the exponents $\\nu$ of these power laws are often close to 0, suggesting potentially inefficient use of resources. For example, for large models with lots of compute, so that the amount of training data constitutes a performance bottleneck, the loss scales as $ { \\mathcal { L } } \\approx P ^ { - \\nu }$ Specifically for a large transformer based language model, $\\nu = 0 . 0 9 5$ , which implies an order of magnitude increase in training data drops cross-entropy loss by only about 0.6 nats (Fig. 1 in [2]). In neural machine translation experiments $\\nu$ varies across language pairs from 0.35 to 0.48 (Table 1 in [5]). Interestingly, [8] explored a fixed computation budget $C$ and optimized jointly over model size $N$ and training set size $P$ , revealing that scaling both $N$ and $P$ commensurately as $C$ increases is compute optimal, and can yield smaller high performing models (trained on more data) than previous work. Nevertheless, for a transformer based language model, a $1 0 0 \\times$ increase in compute, corresponding to $1 0 \\times$ increases in both model size and training set size, leads to a drop in cross-entropy loss of only about 0.5 nats (Fig. 2 in [8]). Similar slow scaling holds for large vision transformers where adding 2 billion pre-training images reduces ImageNet performance by a few percentage points (Fig. 1 in [7]). While all of these results constitute significant improvements in performance, they do come at a substantial resource cost whose fundamental origin arises from power law scaling with small exponents. Recent theoretical works [22, 23, 24] have argued that the power law exponent is governed by the dimension of a data manifold from which training examples are uniformly drawn. Here we explore whether we can beat power law scaling through careful data selection. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 Statistical mechanics of perceptron learning ",
+        "text_level": 1,
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+    {
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+        "text": "Statistical mechanics has long played a role in analyzing machine learning problems (see e.g. [25, 26, 27, 28] for reviews). One of the most fundamental applications is perceptron learning in the student-teacher setting [29, 30], in which random i.i.d. Gaussian inputs are labeled by a teacher perceptron to construct a training set. The test error for another student perceptron learning from this training set then scales as a power law with exponent $- 1$ for such data. Such perceptrons have also been analyzed in an active learning setting where the learner is free to design any new input to be labeled [31, 32], rather than choose from a fixed set of inputs, as in data-pruning. Recent work [33] has analyzed this scenario but focused on message passing algorithms that are tailored to the case of Gaussian inputs and perceptrons, and are hard to generalize to real world settings. In contrast we analyze margin based pruning algorithms that are used in practice in diverse settings, as in [9, 10]. ",
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+    {
+        "type": "text",
+        "text": "3 An analytic theory of data pruning ",
+        "text_level": 1,
+        "bbox": [
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+    {
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+        "text": "To better understand data pruning, we employed the replica method from statistical mechanics [34] to develop an analytic theory of pruning for the perceptron in the student-teacher setting [25] (see App. A for detailed derivations of all results). Consider a training dataset of $P$ examples $\\{ { \\bf x } ^ { \\mu } , y ^ { \\mu } \\} _ { \\mu = 1 , \\dots , P }$ where $\\mathbf { x } ^ { \\mu } \\in \\mathbb { R } ^ { N }$ are i.i.d. zero mean unit variance random Gaussian inputs and $y ^ { \\mu } = \\operatorname { s i g n } ( \\mathbf { T } \\cdot \\mathbf { x } ^ { \\mu } )$ are labels generated by a teacher perceptron with weight vector $\\mathbf { T } \\in \\mathbf { \\mathbb { R } } ^ { N }$ . We work in the high dimensional statistics limit where $N , P \\to \\infty$ but the ratio $\\begin{array} { r } { \\alpha _ { \\mathrm { t o t } } = \\frac { P } { N } } \\end{array}$ of the number of total training examples to parameters remains $O ( 1 )$ . We then consider a pruning algorithm used in [9, 10], namely: (1) train a probe student perceptron for very few epochs on the training data, obtaining weights $\\mathbf { J _ { \\mathrm { p r o b e } } }$ ; (2) compute the margin $m ^ { \\mu } = { \\bf J } _ { \\mathrm { p r o b e } } \\cdot \\left( y ^ { \\mu } { \\bf x } ^ { \\mu } \\right)$ of each training example, where large (small) margins correspond to easy (hard) examples; (3) construct a pruned dataset of size $P _ { \\mathrm { p r u n e } } = f P$ , where $f$ is the fraction of examples kept, by retaining the $P _ { \\mathrm { p r u n e } }$ hardest examples, (4) train a new perceptron to completion on the smaller dataset with a smaller ratio $\\begin{array} { r } { \\alpha _ { \\mathrm { p r u n e } } = \\frac { P _ { \\mathrm { p r u n e } } } { N } } \\end{array}$ of examples to parameters. ",
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+    {
+        "type": "text",
+        "text": "We are interested in the test error $\\varepsilon$ of this final perceptron as a function of $\\alpha _ { \\mathrm { t o t } } , f$ , and the angle $\\theta$ between the probe student $\\mathbf { J _ { \\mathrm { { p r o b e } } } }$ and the teacher $\\mathbf { T }$ . Our theory approximates $\\mathbf { J _ { \\mathrm { { p r o b e } } } }$ as simply a random Gaussian vector conditioned to have angle $\\theta$ with the teacher $\\mathbf { T }$ . Under this approximation we obtain an analytic theory for $\\varepsilon ( \\alpha _ { \\mathrm { t o t } } , f , \\theta )$ that is asymptotically exact in the high dimensional limit (App. A). We first examine results when $\\theta = 0$ , so we are pruning training examples according to their veridical margins with respect to the teacher (Fig. 1A). We find two striking phenomena, each of which constitute predictions in real-world settings that we will successfully confirm empirically. ",
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+        "text": "The best pruning strategy depends on the amount of initial data. First, we note the test error curve for $f = 1$ in Fig. 1A corresponding to no pruning, or equivalently to randomly pruning a larger dataset of size $\\alpha _ { \\mathrm { t o t } }$ down to a size $\\alpha _ { \\mathrm { p r u n e } }$ , exhibits the well known classical perceptron learning power law scaling $\\varepsilon \\propto \\alpha _ { \\mathrm { p r u n e } } ^ { - 1 }$ . Interestingly though, for small $\\alpha _ { \\mathrm { t o t } }$ , keeping the hardest examples performs worse than random pruning (lighter curves above darkest curve for small $\\alpha _ { \\mathrm { p r u n e } }$ in Fig. 1A). However, for large $\\alpha _ { \\mathrm { t o t } }$ , keeping the hardest examples performs substantially better than random pruning (lighter curves below darkest curve for large $\\alpha _ { \\mathrm { p r u n e } }$ in Fig. 1A). It turns out keeping the easiest rather than hardest examples is a better pruning strategy when $\\alpha _ { \\mathrm { t o t } }$ is small (Fig. 1C). If one does not have much data to start with, it is better to keep the easiest examples with largest margins (i.e. the blue regions of Fig. 1B) to avoid overfitting. The easiest examples provide coarse-grained information about the target function, while the hard examples provide fine-grained information about the target function which can prevent the model from learning if one starts with lots of data. In cases where overfitting is less of an issue, it is best to keep the hardest examples with smallest margin that provide more information about the teacher’s decision boundary (i.e. the green region of Fig. 1B). Intuitively, in the limited data regime, it is challenging to model outliers since the basics are not adequately captured; hence, it is more important to keep easy examples so that the model can get to moderate error. However, with a larger dataset, the easy examples can be learned without difficulty, making modeling outliers the fundamental challenge. ",
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+        "text": "Fig. 1C reveals which pruning strategy is best as a joint function of $\\alpha _ { \\mathrm { t o t } }$ and $f$ . Note the transition between optimal strategies becomes sharper at small fractions $f$ of data kept. This transition between optimal pruning strategies can be viewed as a prediction in more general settings. To test this prediction we trained a ResNet18 on pruned subsets of the CIFAR-10 dataset (Fig. 1D), and observed strikingly similar behavior, indicating the prediction can hold far more generally, beyond perceptron learning. Interestingly, [9, 10] missed this transition, likely because they started pruning from large datasets. ",
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+        "text": "Pareto optimal data pruning can beat power law scaling. A second prediction of our theory is that when keeping a fixed fraction $f$ of the hardest examples as $\\alpha _ { \\mathrm { t o t } }$ increases (i.e. constant color curves in Fig. 1A), the error initially drops exponentially in $\\alpha _ { \\mathrm { p r u n e } } = f \\alpha _ { \\mathrm { t o t } }$ , but then settles into the universal power law $\\varepsilon \\propto \\alpha _ { \\mathrm { p r u n e } } ^ { - 1 }$ for all fixed $f$ . Thus there is no asymptotic advantage to data pruning at a fixed $f$ . However, by pruning more aggressively (smaller $f$ ) when given more initial data (larger $\\alpha _ { \\mathrm { t o t . } }$ ), one can achieve a Pareto optimal test error as a function of pruned dataset size $\\alpha _ { \\mathrm { p r u n e } }$ that remarkably traces out at least an exponential scaling law (Fig. 1A, purple curve). Indeed our theory predicts for each $\\alpha _ { \\mathrm { p r u n e } }$ a Pareto optimal point in $\\alpha _ { \\mathrm { t o t } }$ and $f$ (subject to $\\alpha _ { \\mathrm { p r u n e } } = f \\alpha _ { \\mathrm { t o t } } )$ , yielding for every fixed $\\alpha _ { \\mathrm { p r u n e } }$ an optimal $f _ { \\mathrm { o p t } }$ , plotted in Fig. 1E. Note $f _ { \\mathrm { o p t } }$ decreases with $\\alpha _ { \\mathrm { p r u n e } }$ indicating more aggressive pruning (smaller $f _ { \\mathrm { o p t } } )$ ) of original datasets of larger size $\\alpha _ { \\mathrm { t o t } }$ is required to obtain larger Pareto optimal pruned datasets of size $\\alpha _ { \\mathrm { p r u n e } }$ . We will test this striking scaling prediction in Fig. 3. ",
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+            "Figure 2: Data pruning with an imperfect metric. A: Weight vectors and decision boundaries for a teacher (black) and probe student (red) separated by angle $\\theta$ . The black point has margin $0 \\left( \\kappa \\right)$ w.r.t. the probe (teacher). B–D: Test error as a function of $\\alpha _ { \\mathrm { p r u n e } }$ for different $f$ and different $\\theta$ . "
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+        "text": "Beating power law scaling: an information-theoretic perspective. Classical randomly selected data generates slow power law error scaling because each extra training example provides less new information about the correct decision boundary than the previous example. More formally, let $S { \\left( \\alpha _ { \\mathrm { t o t } } \\right) }$ denote the typical entropy of the posterior distribution over student perceptron weights consistent with a training set of size $\\alpha _ { \\mathrm { t o t } }$ . The information gain $I ( \\alpha _ { \\mathrm { t o t } } )$ due to additional examples beyond $\\alpha _ { \\mathrm { t o t } }$ can be defined as the rate at which the posterior entropy is reduced: $\\begin{array} { r } { I ( \\alpha _ { \\mathrm { t o t } } ) = - \\frac { d } { d \\alpha _ { \\mathrm { t o t } } } S ( \\alpha _ { \\mathrm { t o t } } ) } \\end{array}$ . In classical perceptron learning $I ( \\alpha _ { \\mathrm { t o t } } )$ decays to zero as a power law in $\\alpha _ { \\mathrm { t o t } }$ , reflecting a vanishing amount of information per each new example, leading to the slow power law decay of test error $\\varepsilon \\propto \\alpha _ { \\mathrm { t o t } } ^ { - 1 }$ . However, data pruning can increase the information gained per example by pruning away the uninformative examples. To show this, we generalized the replica calculation of the posterior entropy $S$ and information gain $I$ from random datasets of size $\\alpha _ { \\mathrm { t o t } }$ to pruned datasets of size $\\alpha _ { \\mathrm { p r u n e } }$ (App. A). We plot the resulting information gain $I ( \\alpha _ { \\mathrm { p r u n e } } )$ for different $f$ in Fig. 1F. For any fixed $f$ $\\dot { \\mathbf { \\rho } } , I ( \\alpha _ { \\mathrm { p r u n e } } )$ will eventually decay as a power law as $\\alpha _ { \\mathrm { p r u n e } } ^ { - 1 }$ . However, by more aggressively pruning (smaller $f$ ) datasets of larger size $\\alpha _ { \\mathrm { t o t } }$ , $I ( \\alpha _ { \\mathrm { p r u n e } } )$ can converge to a finite value $I ( \\infty ) = 1$ nat/example, resulting in larger pruned datasets only adding useful non-redundant information. Since each new example under Pareto optimal data pruning conveys finite information about the target decision boundary, as seen in Fig. 1F, the test error can decay at least exponentially in pruned dataset size as in Fig. 1A. Classical results [30] have shown that training examples chosen by maximizing the disagreement of a committee of student perceptrons can provide an asymptotically finite information rate, leading to exponential decay in test error. Intriguingly, the Pareto-optimal data pruning strategy we study in this work leads to faster than exponential decay, because it includes (partial) information about the target function provided by the probe student (Fig. 11). ",
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+        "text": "An imperfect pruning metric yields a cross over from exponential to power law scaling. We next examine the case of nonzero angle $\\theta$ between the probe student $\\mathbf { J _ { \\mathrm { p r o b e } } }$ and the teacher $\\mathbf { T }$ , such that the ranking of training examples by margin is no longer completely accurate (Fig. 2A). Retaining the hard examples with smallest margin with respect to the probe student will always result in pruned datasets lying near the probe’s decision boundary. But if $\\theta$ is large, such examples might be far from the teacher’s decision boundary, and therefore could be less informative about the teacher (Fig. 2A). As a result our theory, confirmed by simulations, predicts that under nonzero angles $\\theta$ , the Pareto optimal lower envelope of test error over both $\\alpha _ { \\mathrm { t o t } }$ and $f$ initially scales exponentially as a function of $\\alpha _ { \\mathrm { p r u n e } } = f \\alpha _ { \\mathrm { t o t } }$ but then crosses over to a power law (Fig. 2BCD). Indeed, at any given nonzero $\\theta$ , our theory reveals that as $\\alpha _ { \\mathrm { t o t } }$ (and therefore $\\alpha _ { \\mathrm { p r u n e . } }$ ) becomes large, one cannot decrease test error any further by retaining less than a minimum fraction $f _ { \\mathrm { m i n } } ( \\theta )$ of all available data. For example when $\\theta = 1 0 ^ { \\circ } \\overset { \\cdot } { ( } \\theta = 2 0 ^ { \\circ } )$ one can do no better asymptotically than pruning down to $24 \\%$ $( 4 6 \\% )$ of the total data (Fig. 2CD). As $\\theta$ approaches 0, $f _ { \\mathrm { m i n } } ( \\theta )$ approaches 0, indicating that one can prune extremely aggressively to arbitrarily small $f$ while still improving performance, leading to at least exponential scaling for arbitrarily large $\\alpha _ { \\mathrm { p r u n e } }$ in Fig. 2B. However, for nonzero $\\theta$ , the lack of improvement for $f < f _ { \\mathrm { m i n } } ( \\theta )$ at large $\\alpha _ { \\mathrm { p r u n e } }$ renders aggressive pruning ineffective. This result highlights the importance of finding high quality pruning metrics with $\\theta \\approx 0$ . Such metrics can delay the cross over from exponential to power law scaling as pruned dataset size $\\alpha _ { \\mathrm { p r u n e } }$ increases, by making aggressive pruning with very small $f$ highly effective. Strikingly, in App. Fig. 10 we demonstrate this cross-over in a real-world setting by showing that the test error on SVHN is bounded below by a power law when the dataset is pruned by a probe ResNet18 under the EL2N metric, trained for 4 epochs (weak pruning metric) but not a probe ResNet18 trained for 40 epochs (strong pruning metric). ",
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+        "image_caption": [
+            "Figure 3: Beating power law scaling in practice. A–D: Curves of test error against pruned dataset size in 4 settings. Pruning scores were EL2N [10] for CIFAR-10 and SVHN and memorization [13] for ImageNet. See App. B for all pruning/training details and App. D for similar ImageNet plots with EL2N. Note solid curves reflect performance with a fixed total dataset size; if we prune more aggressively with even larger datasets, scaling could improve further (e.g., dashed lines in A). Error curves with no data pruning $( f = 1 )$ are labeled with their best-fit power law scaling $\\sim \\alpha ^ { - \\nu }$ . (Note that for SVHN in B an asymptotic constant error $E ( P  \\infty ) = 1 . \\bar { 1 } \\%$ is subtracted from each of the curves to visualize the power law scaling more clearly.) "
+        ],
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+        "image_caption": [
+            "Figure 4: Data pruning improves transfer learning. A: CIFAR-10 performance of a ViT pre-trained on all of ImageNet21K and fine-tuned on different pruned subsets of CIFAR-10 under the EL2N metric. B: CIFAR-10 performance of ResNet50s pretrained on different pruned subsets of ImageNet1K and fine-tuned on all of CIFAR10. "
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+        "text": "4 Data pruning can beat power law scaling in practice ",
+        "text_level": 1,
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+        "text": "Our theory of data pruning for the perceptron makes three striking predictions which can be tested in more general settings, such as deep neural networks trained on benchmark datasets: (1) relative to random data pruning, keeping only the hardest examples should help when the initial dataset size is large, but hurt when it is small; (2) data pruning by retaining a fixed fraction $f$ of the hardest examples should yield power law scaling, with exponent equal to that of random pruning, as the initial dataset size increases; (3) the test error optimized over both initial data set size and fraction of data kept can trace out a Pareto optimal lower envelope that beats power law scaling of test error as a function of pruned dataset size, through more aggressive pruning at larger initial dataset size. We verified all three of these predictions on ResNets trained on SVHN, CIFAR-10, and ImageNet using varying amounts of initial dataset size and fractions of data kept under data pruning (compare theory in Fig. 3A with deep learning experiments in Fig. 3BCD). In each experimental setting we see better than power law scaling at larger initial data set sizes and more aggressive pruning. Moreover we would likely see even better scaling with even larger initial datasets (as in Fig.3A dashed lines). ",
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+        "text": "Data pruning improves transfer learning. Modern foundation models are pre-trained on a large initial dataset, and then transferred to other downstream tasks by fine-tuning on them. We therefore examined whether data-pruning can be effective for both reducing the amount of fine-tuning data and the amount of pre-training data. To this end, we first analyzed a vision transformer (ViT) pre-trained on ImageNet21K and then fine-tuned on different pruned subsets of CIFAR-10. Interestingly, pre-trained models allow for far more aggressive data pruning; fine-tuning on only $10 \\%$ of CIFAR-10 can match or exceed performance obtained by fine tuning on all of CIFAR-10 (Fig. 4A). Furthermore Fig. 4A provides a new example of beating power law scaling in the setting of fine-tuning. Additionally, we examined the efficacy of pruning pre-training data by pre-training ResNet50s on different pruned subsets of ImageNet1K (exactly as in Fig. 3D) and then fine-tuning them on all of CIFAR-10. Fig. 4B demonstrates pre-training on as little as $5 0 \\%$ of ImageNet can match or exceed CIFAR-10 performance obtained by pre-training on all of ImageNet. Thus intriguingly pruning pre-training data on an upstream task can still maintain high performance on a different downstream task. Overall these results demonstrate the promise of data pruning in transfer learning for both the pre-training and fine-tuning phases. ",
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+        "image_caption": [
+            "Figure 5: Dataset pruning at ImageNet scale. A: Spearman’s rank correlation between all pairs of ImageNet metric scores, along with hierarchical clustering (as provided by seaborn.clustermap). B: Benchmarking existing supervised metrics on ImageNet (top-5 validation accuracy). C: Comparing top-5 performance on ImageNet when pruning according to the best existing supervised metric (memorization) and our supervised and self-supervised prototype metrics. In all 3 cases, training on $80 \\%$ of ImageNet approximates training on $100 \\%$ . See App. B for pruning and training details. "
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+        "type": "text",
+        "text": "5 Benchmarking supervised pruning metrics on ImageNet ",
+        "text_level": 1,
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+        "text": "We note that the majority of data pruning experiments have been performed on small-scale datasets (i.e. variants of MNIST and CIFAR), while the few pruning metrics proposed for ImageNet have rarely been compared against baselines designed on smaller datasets. Therefore, it is currently unclear how most pruning methods scale to ImageNet and which method is best. Motivated by how strongly the quality of a pruning metric can impact performance in theory (Fig. 2), we decided to fill this knowledge gap by performing a systematic evaluation of 8 different supervised pruning metrics on ImageNet: two variants of influence scores [13], two variants of EL2N [10], DDD [21], memorization [13], ensemble active learning [11], and forgetting [9]. See Section 2 for a review of these metrics. Additionally, we include two new prototypicality metrics that we introduce in the next section. ",
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+        "text": "We first asked how consistent the rankings induced by different metrics are by computing the Spearman rank correlation between each pair of metrics (Fig. 5A). Interestingly, we found substantial diversity across metrics, though some (EL2N, DDD, and memorization) were fairly similar with rank correlations above 0.7. However, we observed marked performance differences between metrics: Fig 5BC shows test performance when a fraction $f$ of the hardest examples under each metric are kept in the training set. Despite the success of many of these metrics on smaller datasets, only a few still match performance obtained by training on the full dataset, when selecting a significantly smaller training subset (i.e. about $8 0 \\%$ of ImageNet). Nonetheless, most metrics continue to beat random pruning, with memorization in particular demonstrating strong performance (Fig. 5C). We note that data pruning on ImageNet may be more difficult than data pruning on other datasets, because ImageNet is already carefully curated to filter out uninformative examples. ",
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+        "text": "We found that all pruning metrics amplify class imbalance, which results in degraded performance. To solve this we used a simple $5 0 \\%$ class balancing ratio for all ImageNet experiments. Further details and baselines without class balancing are shown in App. H. ",
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+        "text": "6 Self-supervised data pruning through a prototypicality metric ",
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+        "type": "text",
+        "text": "Fig. 5 shows many data pruning metrics do not scale well to ImageNet, while the few that do require substantial amounts of compute. Furthermore, all these metrics require labels, thereby limiting their ability to prune data for large-scale foundation models trained on massive unlabeled datasets [12]. Thus there is a clear need for simple, scalable, self-supervised pruning metrics. ",
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+        "type": "text",
+        "text": "To compute a self-supervised pruning metric for ImageNet, we perform $k$ -means clustering in the embedding space of an ImageNet pre-trained self-supervised model (here: SWaV [35]), and define the difficulty of each data point by the Euclidean distance to its nearest cluster centroid, or prototype. Thus easy (hard) examples are the most (least) prototypical. Encouragingly, in Fig. 5C, we find our self-supervised prototype metric matches or exceeds the performance of the best supervised metric, memorization, until only $70 \\mathrm { - } 8 0 \\%$ of the data is kept, despite the fact that our metric does not use labels and is much simpler and cheaper to compute than many previously proposed supervised metrics. See App. Fig. 9 for further scaling experiments using the self-supervised metric. ",
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+        "type": "text",
+        "text": "To assess whether the clusters found by our metric align with ImageNet classes, we compared their overlaps in Fig. 6A. Interestingly, we found alignment for some but not all classes. For example, class categories such as snakes were largely aligned to a small number of unsupervised clusters, while other classes were dispersed across many such clusters. If class information is available, we can enforce alignment between clusters and classes by simply computing a single prototype for each class (by averaging the embeddings of all examples of this class). While originally intended to be an additional baseline metric (called supervised prototypes, light blue in Fig 5C), this metric remarkably outperforms other supervised metrics and largely matches the performance of memorization, which is prohibitively expensive to compute. Moreover, the performance of the best self-supervised and supervised metrics are similar, demonstrating the promise of self-supervised pruning. ",
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+        "text": "One important choice for the self-supervised prototype metric is the number of clusters $k$ . We found, reassuringly, our results were robust to this choice: $k$ can deviate one order of magnitude more or less than the true number of classes (i.e. 1000 for ImageNet) without affecting performance (App. F). ",
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+        "type": "text",
+        "text": "To better understand example difficulty under various metrics, we visualize extremal images for our self-supervised prototype metric and the memorization metric for one class (Fig 6B,C). Qualitatively, easy examples correspond to highly similar, redundant images, while hard examples look like idiosyncratic outliers. See App. E, Figs. 12,13,14,15,16,17,18,19 for more classes and metrics. ",
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+        "img_path": "images/c6388f541d225b33d830efef18469e869ad3adaa127067232e64f07f1eb633af.jpg",
+        "image_caption": [
+            "Figure 6: A: Heat map where each row denotes the probability that images in a given cluster come from each ImageNet class. B: The four easiest and hardest images under our self-supervised pruning metric and the best previously published supervised metric (memorization, shown in C) for ImageNet class 100 (black swan). "
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+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
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+        "text": "Summary. We have shown, both in theory and practice, how to break beyond slow power law scaling of error versus dataset size to faster exponential scaling, through data pruning. Additionally we have developed a simple self-supervised pruning metric that enables us to discard $20 \\%$ of ImageNet without sacrificing performance, on par with the best and most compute intensive supervised metric. ",
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+        "text": "Limitations. The most notable limitation is that achieving exponential scaling requires a high quality data pruning metric. Since most metrics developed for smaller datasets scale poorly to ImageNet, our results emphasize the importance of future work in identifying high quality, scalable metrics. Our self-supervised metric provides a strong initial baseline. Moreover, a key advantage of data pruning is reduced computational cost due to training on a smaller dataset for the same number of epochs as the full dataset (see App. C). However, we found that performance often increased when training on the pruned dataset for the same number of iterations as on the full dataset, resulting in the same training time, but additional training epochs. However, this performance gain saturated before training time on the pruned dataset approached that on the whole dataset (App. J) thereby still yielding a computational efficiency gain. Overall this tradeoff between accuracy and training time on pruned data is important to consider in evaluating potential gains due to data pruning. Finally, we found that class-balancing was essential to maintain performance on data subsets (App. H). Future work will be required to identify ways to effectively select the appropriate amount of class-balancing. ",
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+        "text": "Ethical considerations. A potential negative societal impact could be that data-pruning leads to unfair outcomes for certain groups. We have done a preliminary analysis of how data-pruning affects performance on individual ImageNet classes (App. I), finding no substantial differential effects across classes. However proper fairness tests specific to deployment settings should always be conducted on every model, whether trained on pruned data or not. Additionally, we analyzed the impact of pruning on OOD performance (App. K). ",
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+        "text": "Outlook: Towards foundation datasets. We believe the most promising future direction is the further development of scalable, unsupervised data pruning metrics. Indeed our theory predicts that the application of pruning metrics on larger scale datasets should yield larger gains by allowing more aggressive pruning. This makes data pruning especially exciting for use on the massive unlabeled datasets used to train large foundation models (e.g. 400M image-text pairs for CLIP [36], 3.5B Instagram images [37], 650M images for the DALLE-2 encoder [38], 780B tokens for PALM [39]). If highly pruned versions of these datasets can be used to train a large number of different models, one can conceive of such carefully chosen data subsets as foundation datasets in which the initial computational cost of data pruning can be amortized across efficiency gains in training many downstream models, just at the initial computational cost of training foundation models is amortized across the efficiency gains of fine-tuning across many downstream tasks. Together, our results demonstrate the promise and potential of data pruning for large-scale training and pretraining. ",
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