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+# Scaling Up and Distilling Down: Language-Guided Robot Skill Acquisition
+
+Huy $\mathrm { H a } ^ { 1 }$ , Pete Florence2, and Shuran Song1
+
+1Columbia University   
+2Google DeepMind
+
+Abstract: We present a framework for robot skill acquisition, which 1) efficiently scale up data generation of language-labelled robot data and 2) effectively distills this data down into a robust multi-task language-conditioned visuo-motor policy. For (1), we use a large language model (LLM) to guide high-level planning, and sampling-based robot planners (e.g. motion or grasp samplers) for generating diverse and rich manipulation trajectories. To robustify this data-collection process, the LLM also infers a code-snippet for the success condition of each task, simultaneously enabling the data-collection process to detect failure and retry as well as the automatic labeling of trajectories with success/failure. For (2), we extend the diffusion policy single-task behavior-cloning approach to multi-task settings with language conditioning. Finally, we propose a new multi-task benchmark with 18 tasks across five domains to test long-horizon behavior, common-sense reasoning, tool-use, and intuitive physics. We find that our distilled policy successfully learned the robust retrying behavior in its data collection procedure, while improving absolute success rates by $3 \mathrm { { \bar { 3 } } . 2 \mathrm { { \bar { \% } } } }$ on average across five domains. All code, data, and qualitative policy results are available at our project website.
+
+![](images/2a01ee3c09f480b88ac8375b319968f15d3d2a0a747b2937e52ed97ed42cfdf5.jpg)  
+Figure 1: Language-guided Skill Acquisition enables scalable robot learning. In the data generation stage, a LLM takes as input task descriptions (a) and uses sampling-based robotic planners and privileged simulation information (b) to perform task-directed exploration. This enables the scaling up of language and task-success labeled dataset generation (c). In the second stage, the dataset is filtered for success and distilled down into a closed-loop language-conditioned visuomotor policy for real world deployment (d).
+
+# 1 Introduction
+
+How can we scalably acquire robust, reusable, real-world manipulation skills? This question has been the driving force behind extensive research in robot learning. Attempts in the field have focused on two primary aspects: First, how to scale up the data collection for a diverse range of manipulation skills, which involves efforts such as improving the hardware [1, 2] and software [3, 4] which support demonstration collection, utilization of non-robotics datasets [5, 6], or trial-and-error explorations [7]. The second aspect of this question concerns effective learning from the collected data, which delves into exploring effective action representations [8–10] and policy formulations [11, 12] that can robustly model the training data and generalize to novel scenarios.
+
+This paper proposes a new framework that provides a comprehensive solution for both aspects by leveraging language guidance, while using no expert demonstrations or reward specification/engineering. We contribute two key components with our framework:
+
+• Scaling Up Language-Guided Data Generation: Our data-collection policy is a large language model (LLM) which has access to a suite of 6DoF exploration primitives (i.e., sampling-based robot planners and utilities). Given an input task description, this policy first simplifies the task by recursively decomposing it into subtasks, resulting in a hierarchical plan (i.e., task tree). Next, this plan is grounded into a sequence of 6DoF exploration primitives, which generates diverse robot trajectories for the task. Finally, the data collection policy verifies the trajectories’ success with an inferred success function and retries the task until it succeeds. This verify & retry step not only improves the data-collection policy’s success, but also adds robot experience on how to recover from failure, an important trait for downstream policy distillation. This data generation approach is scalable, enabling significantly more efficient autonomous task-directed exploration than unguided alternatives (i.e., reinforcement learning) while not being limited by the lack of low-level understanding of the LLM-only solution.
+
+• Distilling Down to Language-Conditioned Visuomotor Policy: We distill these robot experiences into a visuo-linguo-motor policy that infers control sequences from visual observations and a natural language task description. To enable effective learning of high entropy, diverse robot trajectories, we extend the diffusion policy [12] to handle language-based conditioning for multi-task learning. This allows the learned policy to be reused and recomposed through language-based planners. We found that our distilled policy successfully learned the robust retrying behavior from its data collection policy, while improving upon its absolute success rate across five domains by $3 3 . 2 \%$ . Further, we demonstrate that our policy directly transfers to the real-world without fine-tuning using domain randomization.
+
+Our framework combines these two components to get the best of both worlds – leverage LLM’s common-sense reasoning abilities for efficient exploration while learning robust and re-usable 6DoF skills for real-world deployment. In summary, the key contribution of this paper is a new framework for visuo-linguo-motor policy learning that is enabled by three novel components:
+
+• A new language-guided data collection framework that combines language-based task planner with 6DoF robot utilities (e.g. motion planning, grasp sampling).   
+• New formulation of diffusion-based policy that effectively learns multi-task language-conditioned closed-loop control policies.   
+• In addition to our algorithmic contributions, we also contribute a new multi-task benchmark that includes 18 tasks across five domains, requiring long-horizon $\approx 8 0 0$ control cycles), common sense, tool-use, and intuitive physics understanding – capabilities lacking in existing manipulation benchmarks.
+
+# 2 Related Works
+
+Scaling visuo-linguo-motor data. In learning vision-and-language-conditioned motor policies for real-world deployment [9, 10, 13–18], one of the most important questions is how to scale up “robot-complete data” – data that has robot sensory inputs (e.g. vision), action labels (e.g. target end-effector & gripper commands), and task labels (e.g. language description, success). The most prevalent paradigm is to use humans to annotate both actions (e.g. teleoperation) and language [9, 10, 13–18]. When providing action labels, humans can either provide task-specific [9, 10, 15, 18], or task-agnostic (“play”) data [13, 14, 16, 19]. A primary limitation, however, is that data scalability is human-limited.
+
+Other prior works have proposed strategies to enable more-autonomously-scalable data. To scale language annotation, prior works study using visual-language models [20, 21], or procedurally post-hoc provided in simulation [19]. To scale action labels, methods study how to use autonomous sub-optimal policies from random [7] to learned [22] policies. Human egocentric videos [6, 23, 24] has also been shown to be relevant to robot learning [5, 25], but is not robot-complete (lacks action labels), and requires cross-embodiment transfer. Towards unsupervised exploration, prior works have also investigated evolving environments [26, 27] and embodiments [28], automatic task generation [29], leveraging language guidance [30, 31] and world-model error [32], but have not been demonstrated to scale to 6 DoF robotic skill learning. While these approaches reduce human efforts, they are still limited in optimality, generality, and/or completeness of robot data labels.
+
+Another option for the autonomous data collection policy is to use a model-based policy, e.g. task and motion planning (TAMP) [33]. Our approach extends such methods in terms of flexibility and task generality by leveraging LLM’s common-sense knowledge. However, in contrast to recent works which use LLMs as the final policy [34–40], we use the LLM-based planner as a suboptimal data-collection policy. We then distill only successful trajectories into an observable-information [41–43] policy, allowing the distilled policy to improve upon its LLM data collection policy’s performance.
+
+Policy Representations and Multi-task Policy Distillation. One primary question in visuo-motor learning [44] has been how to represent the policy for effective learning, i.e. to enable high precision, multi-modal robot behavior [2, 11, 12, 45, 46]. Another related question has been how to best train multi-task policies [47, 48], including those conditioned on language [9, 10, 13, 15, 16, 18]. Our work presents the novel formulation of bringing diffusion-based [49, 50] policies [12] into the language-conditioned [51, 52] visuomotor domain. Additionally, prior works in multi-task language-conditioning typically focus on cloning policies from experts, meanwhile we study distilling data from a success-filtered suboptimal policy. Success-filtering [11, 53] can be viewed as the simplest form of offline RL [54].
+
+![](images/30b00f83487263cbecc9212528670563a1ad3f986a7739c3026049373ee3b1f0.jpg)  
+Figure 2: Benchmark. We validate our approach on a new multi-task benchmark addressing challenging long-horizon tasks (i.e., 800 control cycles) requiring language understanding (e.g., put [object] to [top] drawer), common sense knowledge (e.g., send a package for return requires raising the mailbox flag), tool-use (e.g., catapult), and intuitive physics (e.g., balance the bus). The tasks are best viewed on our our project website.
+
+# 3 Approach
+
+We propose a new framework for robot learning that performs automatic data collection and policy learning from only a task description. Our design is grounded on four key observations:
+
+• We recognize the importance of random exploration in reinforcement learning, but aim to not be constrained by its inefficiency for long-horizon, sparse reward tasks.   
+• We acknowledge the usefulness of LLM’s common-sense and zero-shot capabilities, but believe language is not by itself the ideal representation for robust, rich, and precise robotic manipulation.   
+• We are inspired by the effectiveness of robotic planning methods, e.g. TAMP, but wish to be flexible to novel tasks and domains and non-reliant on ground truth state during policy inference.   
+• We aim to achieve the simplicity and effectiveness of behavior cloning in distilling collected robot experience into a policy for real-world deployment, while side-stepping the requirement for costly human demonstrations or play data collection.
+
+Using no human demonstration or manually specified reward, our framework combines the strengths of these four areas into a unified framework for both efficient task-directed exploration and multi-task visuo-linguo-motor policy learning.
+
+Method Overview. In the data generation phase, we use an LLM to recursively decompose (§3.1) tasks into a hierachical plan (i.e., task tree) for exploration and ground the plan into sampling-based robot utilities and motion primitives (§3.2). Next, the LLM infers success-detection functions for each task in the plan (§3.3), providing success-labeling. This autonomous data generation process outputs a replay buffer of task-directed exploration experience, labeled with language descriptions and success labels. In the training phase (§3.4), we filter this data for success according to the LLM inferred success condition and distill it into a multi-task vision-and-language-conditioned diffusion policy [12].
+
+# 3.1 Simplify: Task Planning and Decomposition
+
+Given a task description, the first step is to generate a high-level task plan. To improve the flexibility to work with any tasks and 3D assets, we opted for an LLM-based planner to leverage their common-sense and zero-shot reasoning skills. Unlike classical TAMP planners, our framework does not require domain-specific engineering and transition function design to work with new tasks.
+
+Concretely, our recursive LLM planner takes as input the task description, the simulation state, and outputs a plan in the form of a task tree (Fig. 3a). To do so, the LLM first checks whether the task description involves the robot interacting with multiple or only one object. For instance, “move the package into the mailbox” involves opening the mailbox before picking up the package and putting the mailbox in, and should be considered a multi-object task. Meanwhile, “with the mailbox opened, move the package into the mailbox” should be a single-object task. For the base case of single-object tasks, we prompt the LLM to which object part name to to interact. For the case of multi-object tasks, we prompt the LLM to decompose the task into subtasks, and recurse down each subtask.
+
+![](images/8b84c8bcafb3c5523a3a5292076f5a24e44c545a54376ab9b4b74b865087bd27.jpg)  
+Figure 3: Language-Driven Robot Data Generation takes as input the task description and simulation state, and outputs a replay buffer, labelled with language descriptions and success. It starts by using an LLM to simplify tasks recursively (a) until the task involves only one object, resulting in a hierarchical exploration plan. Next, the plan is grounded (b) into a sequence of 6 DOF exploration primitives (e.g. grasp samplers, motion planners, etc.) and rolled out in simulation to give an unlabelled robot trajectory. Finally, an LLM infers a success function code-snippet, and uses it to verify (c) and label it with succeeded or failed. If the trajectory failed, the LLM retries the exploration plan with a different random seed (e.g. a different grasp pose from the grasp sampler). If the robot succeeds or run out of time, the labeled trajectory is returned.
+
+# 3.2 Ground: Compiling a Plan into Robot Utilities
+
+With the generated task tree $\ S 3 . 1$ , the next step is to ground the high-level plan into physical actions. Here, the choice of the low-level robot $A P I$ critically defines the system’s capability and, therefore, becomes a key differentiating factor between different systems. In principle, there are three desired properties we want to see in the action space design:
+
+• Flexibility. Planar actions [10, 37] aren’t flexible enough to manipulate prismatic and revolute joints.   
+• Scalable. Namely, actions should not require human demonstrations to acquire [9, 10, 13–16, 35].   
+• Language-friendly. While joint sequences can encode any action, it is not language-friendly.
+
+We propose to ground the LLM’s plan with API calls into a set of robot utility functions, which include a sampling-based motion planner, a geometry-based grasp and placement sampler, and motion primitives for articulated manipulation. We refer to these utilities as 6 DOF Exploration Primitives (Fig 3b) because, by virtue of being pseudo-random, the sampling-based utilities generate diverse robot trajectories, enabling effective exploration for rich 6 DoF manipulation settings. For instance, our grasp and placement samplers samples uniformly amongst all points in the object part’s point cloud to find good grasps and placements poses, respectively, which are used as input into a rapidly-exploring random trees [55] motion planner that samples uniformly in joint space. This results in diverse grasps, placements, and motion trajectories connecting grasps and placements.
+
+For each leaf node in the inferred task tree (§ 3.1), the grounding process takes as input the node’s task description (e.g. “open the mailbox”), its associated object part name (e.g. “mailbox lid”), and the simulation state, and outputs a sequence of 6 DoF Exploration Primitive API calls. Using the object part name, we can parse the object’s kinematic structure from the simulation state and handle articulated and non-articulated (i.e., rigid, deformable) objects separately. For non-articulated objects, the LLM is prompted to choose the pick & place object names, used to sample grasp and placement pose candidates. For articulated objects (with either revolute or prismatic joints), the leaf node’s associated object part name is used to sample a grasp candidate followed by a rotation or translation primitive conditioned on its joint parameters (i.e., joint type, axis, and origin).
+
+Exploration Plan Rollout. Each node in the exploration plan is grounded only when it is being executed, where the order of execution follows a pre-order tree traversal. By keeping track of the subtask’s state, sub-segments of robot trajectory can be labelled with the subtask’s description, thereby providing dense and automatic text labels for the trajectory. For instance, all actions taken during the inferred subtask “open the mailbox” can be labeled with both the subtask’s description “open the mailbox” and the root task description “move the package into the mailbox”.
+
+Since grounding happens only when a task node is visited, each node’s grounding process is independent of the other leaf nodes, depending only on the simulation state when it is evaluated. While this simplifies planning significantly, it also means that failed execution can occur. For instance, a grasp candidate may render all placement candidates infeasible.
+
+# 3.3 Verify & Retry: Robustifying the Data Collection Policy
+
+Recall, the planning and grounding step can fail, especially when we consider long-horizon tasks. To address this, we propose a verify & retry (Fig. 3c) scheme, which uses environment feedback to detect failed execution.
+
+Verify. For each task, the LLM infers a success function code snippet given the task description, simulation state, and API functions to for query simulation state (e.g., checking contact or joint values, etc). This amounts to prompting the LLM to complete a task success function definition that outputs a boolean value, indicating task success. For instance, given the task “raise the mailbox flag”, the LLM’s inferred code snippet should check whether the mailbox’s flag hinge is raised (Fig. 3c, highlighted green).
+
+Retry. When a trajectory is labeled failed, the robot retries the same sequence of robot utilities with a different random seed (i.e., for the sampling-based robotic utilities) without resetting the simulation state until the task succeeds. For instance, in the bus balance task (Fig. 2, top left), the robot would repeatedly try different grasp and place candidates until the bus is balanced. In the tree traversal process $\ S 3 . 2$ , nodes only yield execution to its parent task when the node’s inferred success condition returns true. This design not only leads to higher success rates in data generation but also provides useful demonstrations on how to recover from failure. In the output replay buffer, the only failed trajectories are ones which timed-out or led to invalid states (e.g. object dropped on the floor).
+
+# 3.4 Language-conditioned Policy Distillation
+
+We extend diffusion policy [12], a state-of-the-art approach for single-task behavior cloning, to the multitask domain by adding language-conditioning. This policy takes as input a task description CLIP [56] feature, proprioception history, and visual observations, and outputs a sequence of end effector control commands. Following Robomimic [4]’s findings, we use a wrist-mounted view in addition to a global (workspace) view to help with tasks requiring precise manipulation. We use their ResNet18-based [57] vision encoders, one for each view. We found that
+
+![](images/f63bc4948ea74ccafeed52fb6956ad8708828d0027d13f9da9fa68a447de7324.jpg)  
+Figure 4: Language-Conditioned Policy Distillation. The policy takes as input a task description, two RGB camera views, and gripper proprioception data, and outputs a sequence of gripper poses and closing command.
+
+using only the latest visual observation along with the full observation horizon of proprioception maintains the policy’s high performance while reducing training time. When used in conjunction with the DDIM [58] noise scheduler, we found that we could use a $1 0 \times$ shorter diffusion process at inference (5 timesteps at inference, 50 timesteps at training) while retaining a comparable performance. Quantitatively, when using a 10 dimensional action space\*, our policy can be run at ${ \approx } 3 5 H z$ on an NVIDIA RTX3080.
+
+# 4 Evaluation
+
+Our experiments try to validate two questions: 1) Can our data generation approach efficiently perform task-directed exploration? 2) Can our policy learning approach effectively distill a multi-modal, multi-task dataset into a generalizable and robust visuo-linguo-motor policy?
+
+Table 1: Benchmark Suite.   
+
+<table><tr><td>Domain</td><td>Complex Artic- Common Tool Multi- Long geometry ulation</td><td></td><td>1sense </td><td></td><td>use task horizon</td><td></td></tr><tr><td>Balance</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Catapult</td><td>xx</td><td>×</td><td>×</td><td>×</td><td>×</td><td>xxx</td></tr><tr><td>Transport</td><td></td><td>×</td><td>×</td><td>×</td><td>X</td><td></td></tr><tr><td>Mailbox</td><td></td><td></td><td></td><td>X</td><td>X</td><td></td></tr><tr><td>Drawer</td><td></td><td></td><td>X</td><td>X</td><td></td><td></td></tr></table>
+
+Our Benchmark contains 18 tasks across 5 domains (Fig. 2 Tab. 1), with the following properties:
+
+• 6DoF & articulated manipulation, for deadling with complex object geometry and articulation.   
+• Geometry Generalization. In our bin transport domain, the robot must generalize its bin transport skill to unseen object instances, with novel shapes, sizes, and colors.   
+• Intuitive physics. Robots should understand the physical properties of the world and use this knowledge to perform tasks. In the bus balance domain, the robot needs to learn the precise grasping and placement to balance a large bus toy on a small block. In the catapult domain, where the block is placed along a catapult arm determines how far the block will be launched, and, thus, which bin (if any) the block will land in.   
+• Common-sense reasoning & Tool-use. Natural language task description is user-friendly but often under-specifies the task. Common-sense can help to fill in the gaps. In the mailbox domain, given the task “send the package for return”, the robot should understand that it not only needs put the package inside, but also raise the mailbox flag to indicate that the package is ready for pickup. In the catapult domain, the robot needs to understand that pressing the catapult’s button will activate the catapult, and that the block needs to be placed on the catapult arm to be launched.
+
+![](images/22c7522a90c85b5b94f32cb97df5872220f10ae2c946c31d19d6e3eb392b4902.jpg)  
+Figure 5: High Entropy yet Precise Language-Guided Action Sequences. Running the pseudorandom languageconditioned diffusion process with different seeds on the same observations yields language-consistent (a-c, different colors for different task descriptions), high entropy actions when possible (a-f, object grasping, transports, & placements) and precise actions when necessary (d, narrow mailbox with large package). Further, domain randomization enables a simulation trained policy (e) to generalize to the real world (f).
+
+• Multi-task conditioning. Given the same visual observations but different task description, the robot should perform different and task-relevant actions. The catapult domain has 3 tasks for three target bins, and the drawer domain has 12 tasks.
+
+• Long horizon behaviour. Our longest horizon domain, mailbox, takes at least 4 subtasks to complete (open the mailbox, put the package in the mailbox while its opened, close the mailbox, then raise the mailbox flag) which can require up to 800 control cycles. In the drawer domain, the robot needs to open the drawer, move the object into the drawer, then close it, which takes about 300 control cycles.
+
+The benchmark is built on top of the MuJoCo [3] simulator, using assets from the Google Scanned dataset [59, 60]. We use a table-top manipulation set-up with a 6DoF robot arm. The task success in evaluation is a manually designed function, instead of LLM generated function used for data collection.
+
+Metrics. We report the success rates $( \% )$ averaged over 200 episodes in Table 2, a task completion efficiency plot in Fig. 6, and qualitative results in Fig. 5. If a domain has multiple tasks then we report the average performance of all tasks. We also compare different LLMs in Table 4 (10 samples per task) and investigate the sources of error in our system for the mailbox domain in Table 3 (200 trials per execution).
+
+Data Generation Baselines. Code-as-Policy [37] is a state-of-the-art approach for using an LLM directly as a robot policy by making state (e.g. query present objects) and action primitive API calls to a robot. Given an LLM-inferred code string, they execute the snippet in an open-loop fashion. Crucially, in their table top manipulation setting, they assume access to planar action primitives. Thus, we introduce the following baselines, which build on top of Code-as-Policy and each other as follows:
+
+• LLM-as-Policy (2D): Similar to code-as-policy using planar pick-and-place, but we use ground truth object segmentation instead of their off-the-shelf object detectors [61, 62].   
+• $( + )$ 6 DOF robot utils: Builds on top of the previous baseline by adding access to 6 DOF robot utilities for grasping, placement, motion planning, and articulated manipulation.   
+• $( + )$ Verify & Retry: Adding to the previous baselines, this baseline uses the LLM’s predicted success condition to label trajectories and retry failed ones. Since the robot utilities involve pseudo-random samplers (e.g. RRT, grasp sampling), retrying the task means running these samplers again using the pseudo-random state and environment state from where failed trajectory left it. Since we use this approach as our data generation policy, it also serves as an ablation of our approach.
+
+Policy Distillation Ablations. We compare against BC-Z [15]’s single-task policies which does not use FiLM conditioning (used in their bin emptying and door opening tasks). To understand the effects of our policy learning design decisions in the single-task regime, we fix training time and dataset size (2 days using at least 500 successful trajectories), and provide the following ablations:
+
+• Action Generation: Instead of using diffusion processes conditioned on the policy input embedding to decode actions, it is typical use multi-layer perceptrons. Following Jang et al. [15], we use one MLP with two hidden layers and ReLU activations for end effector position, one for the orientation, and another for gripper command. This standard policy architecture is deterministic, and is trained with mean-squared error loss for pose and binary cross entropy loss for gripper command.
+
+• Action Space: Besides our absolute end effector pose action space, Delta-Action and velocity control spaces is another popular action space choice [4, 15, 63–65]. We also ablate BC-Z’s execution action horizon (Exec) while keeping their original prediction horizon (Pred).
+
+• Observation Encoder: All approaches encode images using a ResNet18 [57] architecture. Although the original architecture was designed with an average pooling layer, its typical for robotic policies to use a spatial softmax pooling [44] layer instead.
+
+• Data usage: No-Retry trains on successful trajectories generated from the data generation approach without Verify & Retry, so it does not observe any recovery behavior.
+
+# 4.1 Data Collection Policy Evaluation
+
+6DoF exploration is critical. First, we verify different approach’s ability to perform and explore in 6DoF, which is crucial for general manipulation. When 6DoF exploration is introduced, we first observe a drop in the average success rate for simple tasks that could be accomplished with planar actions (Balance, Transport, Tab. 2). However, this ability is critical for exploring complex tasks, providing data to improve upon in the
+
+<table><tr><td rowspan="2">Approach</td><td colspan="3">Planar</td><td colspan="2">6DoF</td><td rowspan="2">Average</td></tr><tr><td>Balance Catapult Transport Mailbox Drawer</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">LLM-as-Policy (2D) (+) 6DoF Robot Utils (+) Verify &amp;Retry</td><td>28.0</td><td>33.3</td><td>21.5</td><td>0.0</td><td>0.0</td><td>27.6</td></tr><tr><td>5.5 45.0</td><td>2.5</td><td>35.0</td><td>0.0</td><td>1.3</td><td>8.8</td></tr><tr><td rowspan="2">Distill No Retry</td><td></td><td>7.3</td><td>82.0</td><td>3.0</td><td>31.8</td><td>33.8</td></tr><tr><td>67.5</td><td>38.5</td><td>32.5</td><td>0.0</td><td>22.7</td><td>32.2</td></tr><tr><td rowspan="2">Distill Ours</td><td>79.0</td><td>58.3</td><td>80.0</td><td>62.0</td><td>55.8</td><td>67.0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 2: Success Rates $( \%$ ) for data generation (top) and distillation approaches (bottom) over 200 trials.
+
+later distilling stage. In particular, we observed that 6DoF actions are important for grasping diverse objects with complex geometry (Transport, Tab. 2), and manipulating articulated objects (Drawer, Mailbox, Tab. 2).
+
+Moreover, 6DoF exploration also helps in diversifying the data collection strategy, which provides the possibility to improve upon in the later distilling stage. For example in the catapult domain, LLM-as-Policy (2D) is only able to solve one of three possible goals (the closest bin) using a deterministic strategy. However, it provides no useful data for learning the other two goals, making it a poor data-collection policy. In contrast, incorporating 6 DOF robot utilities achieves lower but non-zero average success rates in all bins $( 1 6 . 3 \%$ , $3 . 3 \%$ , and $2 . 2 \%$ , full table in appendix), which provide much better exploration data for distillation.
+
+Table 3: Sources & Propagation of Error. Accuracy $( \% )$ of planning, verification, and execution success rate $( \% )$ for each mailbox subtask.   
+
+<table><tr><td>Subtask</td><td>Planning Verify Execution</td></tr><tr><td>Open mailbox</td><td>100</td><td>43.5</td></tr><tr><td>Put package in mailbox</td><td>100</td><td>28.5</td></tr><tr><td>Raise mailbox flag</td><td>100</td><td>62.0</td></tr><tr><td>Close mailbox</td><td>100</td><td>94.2</td></tr></table>
+
+Verify & Retry always helps. In the verify & retry step, the LLM retries all tasks until they are successful. This simple addition improves performance in all domains, with $2 \times , 3 \times$ , $8 \times$ , and $1 3 \times$ in transport, catapult, balance, and drawer domains. Without this crucial step, we observe $0 . 0 \%$ success rate in the mailbox domain, underscoring the difficulty of flawlessly executing long sequences of 6 DOF actions, and the importance of recovery after failure.
+
+Language Model Scaling. In addition to the final task success, we provide more detailed analysis of planning and success condition inference accuracy in Tab. 4. We evaluate on the proprietary GPT3 [66] (175B text-davinci-003) and the open LLAMA2 [67] (7B and 13B). We found that Llama models struggles in complex planning domains because they do not follow instructions provided in the prompts. For instance, in the drawer domain, both models fail to account for drawer opening and closing. However, we observe an upwards trend with respect to Llama model size, with the 13B model outperforming the 7B model by $+ 2 0 . 0 \%$ and $+ 3 8 . 3 \%$ in planning and success verification accuracy respectively.
+
+<table><tr><td>Model</td><td>Size</td><td>Planning</td><td>Success</td></tr><tr><td rowspan="2">LLAMA2</td><td>7B</td><td>42.0</td><td>10.0</td></tr><tr><td>13B</td><td>62.0</td><td>48.3</td></tr><tr><td>GPT3</td><td>175B</td><td>82.0</td><td>91.1</td></tr></table>
+
+Table 4: LLM Evaluation.
+
+# 4.2 Distilled Policy Evaluation
+
+Robustness In, Robustness Out. By filtering trajectories with LLM’s inferred success condition, distilled policies inherit the robustness of their data collection policies while improving upon success rates $( + 2 3 . 4 \%$ and $+ 3 3 . 2 \%$ for no-retry and ours, Tab. 2). Since our distilled policy learned from a robust data collection policy, it also recovers from failures (e.g. failed grasps or placements) and continuously retries a task until it succeeds. Meanwhile, since the no-retry distilled policy learned from a data collection policy which did not retry upon failure, it is sensitive and brittle, leading to $- 3 4 . 8 \%$ lower average success rate across all domains compared to ours (Tab. 2).
+
+High Performance From Diverse Retry Attempts. Plotting how long policies take to solve the balance task (Fig. 6), we observed that our policy and its data collection policy continuously tries a diverse set of grasps and placements after each failed attempt until it succeeds. This results in higher success rates as the policy is given more time, and is reflected in their monotonically increasing success rates.
+
+In contrast, baselines plateau after their first grasp/platement attempts. This highlights the synergy of two design decisions. First, the verify & retry step $( \ S 3 . 3 )$ is crucial for demonstrating retrying behavior, but is by itself insufficient if each retrying action is the identical as the previous one. Instead, opting for a diffusion policy $( \ S \ 3 . 4 )$ for learning from and generating high-entropy, diverse retry attempts (Fig 5) is also essential for high performance.
+
+Policy Learning Baselines. We investigate policy learning design decisions on the single-task balance domain, and remove language conditioning. While BC-Z found spatial softmax hurt their performance and opted for a mean pool, we observed using spatial softmax improved performance by $+ 5 . 0 \%$ . Further, we found that switching from delta to absolute action spaces improved success rates $+ 6 . 5 \%$ and $+ 9 . 5 \%$ when using the MLP action decoder and our diffusion action decoder, respectively, confirming Chi et al. [12]’s findings. Lastly, we find that using our pseudo-random diffusion-based action encoder consistently outperforms a deterministic MLP action mappings, regardless of other design decisions.
+
+![](images/4744b196017d0467fd75e21b1770f0e42c8b6ad74eed9c78004a816824d44cc1.jpg)  
+Figure 6: Distilled Robustness. Our policy inherits robust recovery from failure behavior from its data collection policy, while improving upon success rate.
+
+Sim2Real Transfer. We evaluated a policy trained on domain randomized synthetic data in a real world transport task with five novel objects (Fig. 5e). Averaging across ten episodes per object, our policy achieved $76 \%$ success rate, demonstrating the effectiveness of our approach in Sim2Real transfer.
+
+# 4.3 Limitations
+
+By using priviledged simulation state information, the LLM can infer success conditions which uses ground truth contact, joint information, and object poses. This means our implementation of the data generation phase is limited to simulation environments, and our policy requires sim2real transfer. Further, Our data generation method relies on existing 3D assets and environments, which presents a further opportunity for scaling up with assets from 3D generative models or procedural generation. Finally, while our approach’s dataset contains text labels and success labels for all subtasks, we have only evaluated its effectiveness in learning the root task. Learning from all subtasks and growing a robot’s set of learned, reusable sub-skills over time to enable compositional generalization is left for future work.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Output</td><td colspan="2">Input</td><td rowspan="2">Success (%)</td></tr><tr><td>Generation</td><td>Rep.Exec Pred Pool</td><td></td><td>Proprio</td></tr><tr><td>BC-Z</td><td>FeedForward Delta</td><td>1</td><td>10 Avg</td><td>xxx</td><td>0.0</td></tr><tr><td></td><td>FeedForward Delta</td><td>4</td><td>10 Avg</td><td></td><td>15.0</td></tr><tr><td></td><td>FeedForward Delta</td><td>8</td><td>10 Avg</td><td></td><td>18.5</td></tr><tr><td>Ours</td><td>FeedForward Delta</td><td>8</td><td>16 Spatial</td><td></td><td>29.0</td></tr><tr><td></td><td>FeedForward Abs</td><td>8</td><td>16 Spatial</td><td>ν/√√</td><td>35.5</td></tr><tr><td></td><td>Diffusion</td><td>Delta 8</td><td>16 Spatial</td><td></td><td>69.5</td></tr><tr><td></td><td>Diffusion</td><td>Abs 8</td><td>16Avg</td><td></td><td>76.5</td></tr><tr><td></td><td>Diffusion</td><td>Abs 8</td><td>16 Spatial</td><td></td><td>79.0</td></tr></table>
+
+Table 5: Policy Learning Ablations. Action generation using diffusion models [50] robustly outperforms feed-forward models across other policy design decisions.
+
+# 5 Conclusion
+
+We proposed “Scaling Up and Distilling Down”, a framework that combines the strengths of LLMs, samplingbased planners, and policy learning into a single system that automatically generates, labels, and distills diverse robot-complete exploration experience into a multi-task visuo-linguo-motor policy. The distilled policy inherits long-horizon behaviour, rich low-level manipulation skills, and robustness from its data collection policy while improving upon performance beyond its training distribution. We believe that this integrated approach is a step towards putting robotics on the same scaling trend as that of LLM development while not compromising on the rich low-level control.
+
+# Acknowledgments
+
+We would like to thank Cheng Chi, Zeyi Liu, Samir Yitzhak Gadre, Mengda Xu, Zhenjia Xu, Mandi Zhao and Dominik Bauer for their helpful feedback and fruitful discussions. This work was supported in part by Google Research Award, NSF Award #2143601, and #2132519. We would like to thank Google for the UR5 robot hardware. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the sponsors.
+
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+        "text": "Huy $\\mathrm { H a } ^ { 1 }$ , Pete Florence2, and Shuran Song1 ",
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+        "text": "Abstract: We present a framework for robot skill acquisition, which 1) efficiently scale up data generation of language-labelled robot data and 2) effectively distills this data down into a robust multi-task language-conditioned visuo-motor policy. For (1), we use a large language model (LLM) to guide high-level planning, and sampling-based robot planners (e.g. motion or grasp samplers) for generating diverse and rich manipulation trajectories. To robustify this data-collection process, the LLM also infers a code-snippet for the success condition of each task, simultaneously enabling the data-collection process to detect failure and retry as well as the automatic labeling of trajectories with success/failure. For (2), we extend the diffusion policy single-task behavior-cloning approach to multi-task settings with language conditioning. Finally, we propose a new multi-task benchmark with 18 tasks across five domains to test long-horizon behavior, common-sense reasoning, tool-use, and intuitive physics. We find that our distilled policy successfully learned the robust retrying behavior in its data collection procedure, while improving absolute success rates by $3 \\mathrm { { \\bar { 3 } } . 2 \\mathrm { { \\bar { \\% } } } }$ on average across five domains. All code, data, and qualitative policy results are available at our project website. ",
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+        "text": "1 Introduction ",
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+        "text": "How can we scalably acquire robust, reusable, real-world manipulation skills? This question has been the driving force behind extensive research in robot learning. Attempts in the field have focused on two primary aspects: First, how to scale up the data collection for a diverse range of manipulation skills, which involves efforts such as improving the hardware [1, 2] and software [3, 4] which support demonstration collection, utilization of non-robotics datasets [5, 6], or trial-and-error explorations [7]. The second aspect of this question concerns effective learning from the collected data, which delves into exploring effective action representations [8–10] and policy formulations [11, 12] that can robustly model the training data and generalize to novel scenarios. ",
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+        "text": "• Scaling Up Language-Guided Data Generation: Our data-collection policy is a large language model (LLM) which has access to a suite of 6DoF exploration primitives (i.e., sampling-based robot planners and utilities). Given an input task description, this policy first simplifies the task by recursively decomposing it into subtasks, resulting in a hierarchical plan (i.e., task tree). Next, this plan is grounded into a sequence of 6DoF exploration primitives, which generates diverse robot trajectories for the task. Finally, the data collection policy verifies the trajectories’ success with an inferred success function and retries the task until it succeeds. This verify & retry step not only improves the data-collection policy’s success, but also adds robot experience on how to recover from failure, an important trait for downstream policy distillation. This data generation approach is scalable, enabling significantly more efficient autonomous task-directed exploration than unguided alternatives (i.e., reinforcement learning) while not being limited by the lack of low-level understanding of the LLM-only solution. ",
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+        "text": "• Distilling Down to Language-Conditioned Visuomotor Policy: We distill these robot experiences into a visuo-linguo-motor policy that infers control sequences from visual observations and a natural language task description. To enable effective learning of high entropy, diverse robot trajectories, we extend the diffusion policy [12] to handle language-based conditioning for multi-task learning. This allows the learned policy to be reused and recomposed through language-based planners. We found that our distilled policy successfully learned the robust retrying behavior from its data collection policy, while improving upon its absolute success rate across five domains by $3 3 . 2 \\%$ . Further, we demonstrate that our policy directly transfers to the real-world without fine-tuning using domain randomization. ",
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+        "text": "Our framework combines these two components to get the best of both worlds – leverage LLM’s common-sense reasoning abilities for efficient exploration while learning robust and re-usable 6DoF skills for real-world deployment. In summary, the key contribution of this paper is a new framework for visuo-linguo-motor policy learning that is enabled by three novel components: ",
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+        "text": "• A new language-guided data collection framework that combines language-based task planner with 6DoF robot utilities (e.g. motion planning, grasp sampling).   \n• New formulation of diffusion-based policy that effectively learns multi-task language-conditioned closed-loop control policies.   \n• In addition to our algorithmic contributions, we also contribute a new multi-task benchmark that includes 18 tasks across five domains, requiring long-horizon $\\approx 8 0 0$ control cycles), common sense, tool-use, and intuitive physics understanding – capabilities lacking in existing manipulation benchmarks. ",
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+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+        "text": "Scaling visuo-linguo-motor data. In learning vision-and-language-conditioned motor policies for real-world deployment [9, 10, 13–18], one of the most important questions is how to scale up “robot-complete data” – data that has robot sensory inputs (e.g. vision), action labels (e.g. target end-effector & gripper commands), and task labels (e.g. language description, success). The most prevalent paradigm is to use humans to annotate both actions (e.g. teleoperation) and language [9, 10, 13–18]. When providing action labels, humans can either provide task-specific [9, 10, 15, 18], or task-agnostic (“play”) data [13, 14, 16, 19]. A primary limitation, however, is that data scalability is human-limited. ",
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+        "text": "Other prior works have proposed strategies to enable more-autonomously-scalable data. To scale language annotation, prior works study using visual-language models [20, 21], or procedurally post-hoc provided in simulation [19]. To scale action labels, methods study how to use autonomous sub-optimal policies from random [7] to learned [22] policies. Human egocentric videos [6, 23, 24] has also been shown to be relevant to robot learning [5, 25], but is not robot-complete (lacks action labels), and requires cross-embodiment transfer. Towards unsupervised exploration, prior works have also investigated evolving environments [26, 27] and embodiments [28], automatic task generation [29], leveraging language guidance [30, 31] and world-model error [32], but have not been demonstrated to scale to 6 DoF robotic skill learning. While these approaches reduce human efforts, they are still limited in optimality, generality, and/or completeness of robot data labels. ",
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+        "text": "Another option for the autonomous data collection policy is to use a model-based policy, e.g. task and motion planning (TAMP) [33]. Our approach extends such methods in terms of flexibility and task generality by leveraging LLM’s common-sense knowledge. However, in contrast to recent works which use LLMs as the final policy [34–40], we use the LLM-based planner as a suboptimal data-collection policy. We then distill only successful trajectories into an observable-information [41–43] policy, allowing the distilled policy to improve upon its LLM data collection policy’s performance. ",
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+        "text": "Policy Representations and Multi-task Policy Distillation. One primary question in visuo-motor learning [44] has been how to represent the policy for effective learning, i.e. to enable high precision, multi-modal robot behavior [2, 11, 12, 45, 46]. Another related question has been how to best train multi-task policies [47, 48], including those conditioned on language [9, 10, 13, 15, 16, 18]. Our work presents the novel formulation of bringing diffusion-based [49, 50] policies [12] into the language-conditioned [51, 52] visuomotor domain. Additionally, prior works in multi-task language-conditioning typically focus on cloning policies from experts, meanwhile we study distilling data from a success-filtered suboptimal policy. Success-filtering [11, 53] can be viewed as the simplest form of offline RL [54]. ",
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+        "image_caption": [
+            "Figure 2: Benchmark. We validate our approach on a new multi-task benchmark addressing challenging long-horizon tasks (i.e., 800 control cycles) requiring language understanding (e.g., put [object] to [top] drawer), common sense knowledge (e.g., send a package for return requires raising the mailbox flag), tool-use (e.g., catapult), and intuitive physics (e.g., balance the bus). The tasks are best viewed on our our project website. "
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+        "type": "text",
+        "text": "3 Approach ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose a new framework for robot learning that performs automatic data collection and policy learning from only a task description. Our design is grounded on four key observations: ",
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+        "text": "• We recognize the importance of random exploration in reinforcement learning, but aim to not be constrained by its inefficiency for long-horizon, sparse reward tasks.   \n• We acknowledge the usefulness of LLM’s common-sense and zero-shot capabilities, but believe language is not by itself the ideal representation for robust, rich, and precise robotic manipulation.   \n• We are inspired by the effectiveness of robotic planning methods, e.g. TAMP, but wish to be flexible to novel tasks and domains and non-reliant on ground truth state during policy inference.   \n• We aim to achieve the simplicity and effectiveness of behavior cloning in distilling collected robot experience into a policy for real-world deployment, while side-stepping the requirement for costly human demonstrations or play data collection. ",
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+        "type": "text",
+        "text": "Using no human demonstration or manually specified reward, our framework combines the strengths of these four areas into a unified framework for both efficient task-directed exploration and multi-task visuo-linguo-motor policy learning. ",
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+        "text": "Method Overview. In the data generation phase, we use an LLM to recursively decompose (§3.1) tasks into a hierachical plan (i.e., task tree) for exploration and ground the plan into sampling-based robot utilities and motion primitives (§3.2). Next, the LLM infers success-detection functions for each task in the plan (§3.3), providing success-labeling. This autonomous data generation process outputs a replay buffer of task-directed exploration experience, labeled with language descriptions and success labels. In the training phase (§3.4), we filter this data for success according to the LLM inferred success condition and distill it into a multi-task vision-and-language-conditioned diffusion policy [12]. ",
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+        "type": "text",
+        "text": "3.1 Simplify: Task Planning and Decomposition ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given a task description, the first step is to generate a high-level task plan. To improve the flexibility to work with any tasks and 3D assets, we opted for an LLM-based planner to leverage their common-sense and zero-shot reasoning skills. Unlike classical TAMP planners, our framework does not require domain-specific engineering and transition function design to work with new tasks. ",
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+        "type": "text",
+        "text": "Concretely, our recursive LLM planner takes as input the task description, the simulation state, and outputs a plan in the form of a task tree (Fig. 3a). To do so, the LLM first checks whether the task description involves the robot interacting with multiple or only one object. For instance, “move the package into the mailbox” involves opening the mailbox before picking up the package and putting the mailbox in, and should be considered a multi-object task. Meanwhile, “with the mailbox opened, move the package into the mailbox” should be a single-object task. For the base case of single-object tasks, we prompt the LLM to which object part name to to interact. For the case of multi-object tasks, we prompt the LLM to decompose the task into subtasks, and recurse down each subtask. ",
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+        "image_caption": [
+            "Figure 3: Language-Driven Robot Data Generation takes as input the task description and simulation state, and outputs a replay buffer, labelled with language descriptions and success. It starts by using an LLM to simplify tasks recursively (a) until the task involves only one object, resulting in a hierarchical exploration plan. Next, the plan is grounded (b) into a sequence of 6 DOF exploration primitives (e.g. grasp samplers, motion planners, etc.) and rolled out in simulation to give an unlabelled robot trajectory. Finally, an LLM infers a success function code-snippet, and uses it to verify (c) and label it with succeeded or failed. If the trajectory failed, the LLM retries the exploration plan with a different random seed (e.g. a different grasp pose from the grasp sampler). If the robot succeeds or run out of time, the labeled trajectory is returned. "
+        ],
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+    {
+        "type": "text",
+        "text": "3.2 Ground: Compiling a Plan into Robot Utilities ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With the generated task tree $\\ S 3 . 1$ , the next step is to ground the high-level plan into physical actions. Here, the choice of the low-level robot $A P I$ critically defines the system’s capability and, therefore, becomes a key differentiating factor between different systems. In principle, there are three desired properties we want to see in the action space design: ",
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+        "type": "text",
+        "text": "• Flexibility. Planar actions [10, 37] aren��t flexible enough to manipulate prismatic and revolute joints.   \n• Scalable. Namely, actions should not require human demonstrations to acquire [9, 10, 13–16, 35].   \n• Language-friendly. While joint sequences can encode any action, it is not language-friendly. ",
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+    {
+        "type": "text",
+        "text": "We propose to ground the LLM’s plan with API calls into a set of robot utility functions, which include a sampling-based motion planner, a geometry-based grasp and placement sampler, and motion primitives for articulated manipulation. We refer to these utilities as 6 DOF Exploration Primitives (Fig 3b) because, by virtue of being pseudo-random, the sampling-based utilities generate diverse robot trajectories, enabling effective exploration for rich 6 DoF manipulation settings. For instance, our grasp and placement samplers samples uniformly amongst all points in the object part’s point cloud to find good grasps and placements poses, respectively, which are used as input into a rapidly-exploring random trees [55] motion planner that samples uniformly in joint space. This results in diverse grasps, placements, and motion trajectories connecting grasps and placements. ",
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+        "type": "text",
+        "text": "For each leaf node in the inferred task tree (§ 3.1), the grounding process takes as input the node’s task description (e.g. “open the mailbox”), its associated object part name (e.g. “mailbox lid”), and the simulation state, and outputs a sequence of 6 DoF Exploration Primitive API calls. Using the object part name, we can parse the object’s kinematic structure from the simulation state and handle articulated and non-articulated (i.e., rigid, deformable) objects separately. For non-articulated objects, the LLM is prompted to choose the pick & place object names, used to sample grasp and placement pose candidates. For articulated objects (with either revolute or prismatic joints), the leaf node’s associated object part name is used to sample a grasp candidate followed by a rotation or translation primitive conditioned on its joint parameters (i.e., joint type, axis, and origin). ",
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+        "type": "text",
+        "text": "Exploration Plan Rollout. Each node in the exploration plan is grounded only when it is being executed, where the order of execution follows a pre-order tree traversal. By keeping track of the subtask’s state, sub-segments of robot trajectory can be labelled with the subtask’s description, thereby providing dense and automatic text labels for the trajectory. For instance, all actions taken during the inferred subtask “open the mailbox” can be labeled with both the subtask’s description “open the mailbox” and the root task description “move the package into the mailbox”. ",
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+    {
+        "type": "text",
+        "text": "Since grounding happens only when a task node is visited, each node’s grounding process is independent of the other leaf nodes, depending only on the simulation state when it is evaluated. While this simplifies planning significantly, it also means that failed execution can occur. For instance, a grasp candidate may render all placement candidates infeasible. ",
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+    {
+        "type": "text",
+        "text": "3.3 Verify & Retry: Robustifying the Data Collection Policy ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Recall, the planning and grounding step can fail, especially when we consider long-horizon tasks. To address this, we propose a verify & retry (Fig. 3c) scheme, which uses environment feedback to detect failed execution. ",
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+    {
+        "type": "text",
+        "text": "Verify. For each task, the LLM infers a success function code snippet given the task description, simulation state, and API functions to for query simulation state (e.g., checking contact or joint values, etc). This amounts to prompting the LLM to complete a task success function definition that outputs a boolean value, indicating task success. For instance, given the task “raise the mailbox flag”, the LLM’s inferred code snippet should check whether the mailbox’s flag hinge is raised (Fig. 3c, highlighted green). ",
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+    {
+        "type": "text",
+        "text": "Retry. When a trajectory is labeled failed, the robot retries the same sequence of robot utilities with a different random seed (i.e., for the sampling-based robotic utilities) without resetting the simulation state until the task succeeds. For instance, in the bus balance task (Fig. 2, top left), the robot would repeatedly try different grasp and place candidates until the bus is balanced. In the tree traversal process $\\ S 3 . 2$ , nodes only yield execution to its parent task when the node’s inferred success condition returns true. This design not only leads to higher success rates in data generation but also provides useful demonstrations on how to recover from failure. In the output replay buffer, the only failed trajectories are ones which timed-out or led to invalid states (e.g. object dropped on the floor). ",
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+    {
+        "type": "text",
+        "text": "3.4 Language-conditioned Policy Distillation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We extend diffusion policy [12], a state-of-the-art approach for single-task behavior cloning, to the multitask domain by adding language-conditioning. This policy takes as input a task description CLIP [56] feature, proprioception history, and visual observations, and outputs a sequence of end effector control commands. Following Robomimic [4]’s findings, we use a wrist-mounted view in addition to a global (workspace) view to help with tasks requiring precise manipulation. We use their ResNet18-based [57] vision encoders, one for each view. We found that ",
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+        "img_path": "images/f63bc4948ea74ccafeed52fb6956ad8708828d0027d13f9da9fa68a447de7324.jpg",
+        "image_caption": [
+            "Figure 4: Language-Conditioned Policy Distillation. The policy takes as input a task description, two RGB camera views, and gripper proprioception data, and outputs a sequence of gripper poses and closing command. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "using only the latest visual observation along with the full observation horizon of proprioception maintains the policy’s high performance while reducing training time. When used in conjunction with the DDIM [58] noise scheduler, we found that we could use a $1 0 \\times$ shorter diffusion process at inference (5 timesteps at inference, 50 timesteps at training) while retaining a comparable performance. Quantitatively, when using a 10 dimensional action space\\*, our policy can be run at ${ \\approx } 3 5 H z$ on an NVIDIA RTX3080. ",
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+        "type": "text",
+        "text": "4 Evaluation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our experiments try to validate two questions: 1) Can our data generation approach efficiently perform task-directed exploration? 2) Can our policy learning approach effectively distill a multi-modal, multi-task dataset into a generalizable and robust visuo-linguo-motor policy? ",
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+    {
+        "type": "table",
+        "img_path": "images/aac2af01ab14e71b284d50ba38a37b40f4d7e2ba7bae0cef76d32b765dba60cb.jpg",
+        "table_caption": [
+            "Table 1: Benchmark Suite. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Domain</td><td>Complex Artic- Common Tool Multi- Long geometry ulation</td><td></td><td>1sense </td><td></td><td>use task horizon</td><td></td></tr><tr><td>Balance</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Catapult</td><td>xx</td><td>×</td><td>×</td><td>×</td><td>×</td><td>xxx</td></tr><tr><td>Transport</td><td></td><td>×</td><td>×</td><td>×</td><td>X</td><td></td></tr><tr><td>Mailbox</td><td></td><td></td><td></td><td>X</td><td>X</td><td></td></tr><tr><td>Drawer</td><td></td><td></td><td>X</td><td>X</td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Our Benchmark contains 18 tasks across 5 domains (Fig. 2 Tab. 1), with the following properties: ",
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+        "type": "text",
+        "text": "• 6DoF & articulated manipulation, for deadling with complex object geometry and articulation.   \n• Geometry Generalization. In our bin transport domain, the robot must generalize its bin transport skill to unseen object instances, with novel shapes, sizes, and colors.   \n• Intuitive physics. Robots should understand the physical properties of the world and use this knowledge to perform tasks. In the bus balance domain, the robot needs to learn the precise grasping and placement to balance a large bus toy on a small block. In the catapult domain, where the block is placed along a catapult arm determines how far the block will be launched, and, thus, which bin (if any) the block will land in.   \n• Common-sense reasoning & Tool-use. Natural language task description is user-friendly but often under-specifies the task. Common-sense can help to fill in the gaps. In the mailbox domain, given the task “send the package for return”, the robot should understand that it not only needs put the package inside, but also raise the mailbox flag to indicate that the package is ready for pickup. In the catapult domain, the robot needs to understand that pressing the catapult’s button will activate the catapult, and that the block needs to be placed on the catapult arm to be launched. ",
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+    {
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+        "img_path": "images/22c7522a90c85b5b94f32cb97df5872220f10ae2c946c31d19d6e3eb392b4902.jpg",
+        "image_caption": [
+            "Figure 5: High Entropy yet Precise Language-Guided Action Sequences. Running the pseudorandom languageconditioned diffusion process with different seeds on the same observations yields language-consistent (a-c, different colors for different task descriptions), high entropy actions when possible (a-f, object grasping, transports, & placements) and precise actions when necessary (d, narrow mailbox with large package). Further, domain randomization enables a simulation trained policy (e) to generalize to the real world (f). "
+        ],
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+    {
+        "type": "text",
+        "text": "• Multi-task conditioning. Given the same visual observations but different task description, the robot should perform different and task-relevant actions. The catapult domain has 3 tasks for three target bins, and the drawer domain has 12 tasks. ",
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+    {
+        "type": "text",
+        "text": "• Long horizon behaviour. Our longest horizon domain, mailbox, takes at least 4 subtasks to complete (open the mailbox, put the package in the mailbox while its opened, close the mailbox, then raise the mailbox flag) which can require up to 800 control cycles. In the drawer domain, the robot needs to open the drawer, move the object into the drawer, then close it, which takes about 300 control cycles. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The benchmark is built on top of the MuJoCo [3] simulator, using assets from the Google Scanned dataset [59, 60]. We use a table-top manipulation set-up with a 6DoF robot arm. The task success in evaluation is a manually designed function, instead of LLM generated function used for data collection. ",
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+    {
+        "type": "text",
+        "text": "Metrics. We report the success rates $( \\% )$ averaged over 200 episodes in Table 2, a task completion efficiency plot in Fig. 6, and qualitative results in Fig. 5. If a domain has multiple tasks then we report the average performance of all tasks. We also compare different LLMs in Table 4 (10 samples per task) and investigate the sources of error in our system for the mailbox domain in Table 3 (200 trials per execution). ",
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+    {
+        "type": "text",
+        "text": "Data Generation Baselines. Code-as-Policy [37] is a state-of-the-art approach for using an LLM directly as a robot policy by making state (e.g. query present objects) and action primitive API calls to a robot. Given an LLM-inferred code string, they execute the snippet in an open-loop fashion. Crucially, in their table top manipulation setting, they assume access to planar action primitives. Thus, we introduce the following baselines, which build on top of Code-as-Policy and each other as follows: ",
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+        "type": "text",
+        "text": "• LLM-as-Policy (2D): Similar to code-as-policy using planar pick-and-place, but we use ground truth object segmentation instead of their off-the-shelf object detectors [61, 62].   \n• $( + )$ 6 DOF robot utils: Builds on top of the previous baseline by adding access to 6 DOF robot utilities for grasping, placement, motion planning, and articulated manipulation.   \n• $( + )$ Verify & Retry: Adding to the previous baselines, this baseline uses the LLM’s predicted success condition to label trajectories and retry failed ones. Since the robot utilities involve pseudo-random samplers (e.g. RRT, grasp sampling), retrying the task means running these samplers again using the pseudo-random state and environment state from where failed trajectory left it. Since we use this approach as our data generation policy, it also serves as an ablation of our approach. ",
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+    {
+        "type": "text",
+        "text": "Policy Distillation Ablations. We compare against BC-Z [15]’s single-task policies which does not use FiLM conditioning (used in their bin emptying and door opening tasks). To understand the effects of our policy learning design decisions in the single-task regime, we fix training time and dataset size (2 days using at least 500 successful trajectories), and provide the following ablations: ",
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+    {
+        "type": "text",
+        "text": "• Action Generation: Instead of using diffusion processes conditioned on the policy input embedding to decode actions, it is typical use multi-layer perceptrons. Following Jang et al. [15], we use one MLP with two hidden layers and ReLU activations for end effector position, one for the orientation, and another for gripper command. This standard policy architecture is deterministic, and is trained with mean-squared error loss for pose and binary cross entropy loss for gripper command. ",
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+    {
+        "type": "text",
+        "text": "• Action Space: Besides our absolute end effector pose action space, Delta-Action and velocity control spaces is another popular action space choice [4, 15, 63–65]. We also ablate BC-Z’s execution action horizon (Exec) while keeping their original prediction horizon (Pred). ",
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+    {
+        "type": "text",
+        "text": "• Observation Encoder: All approaches encode images using a ResNet18 [57] architecture. Although the original architecture was designed with an average pooling layer, its typical for robotic policies to use a spatial softmax pooling [44] layer instead. ",
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+        "type": "text",
+        "text": "• Data usage: No-Retry trains on successful trajectories generated from the data generation approach without Verify & Retry, so it does not observe any recovery behavior. ",
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+    {
+        "type": "text",
+        "text": "4.1 Data Collection Policy Evaluation ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "6DoF exploration is critical. First, we verify different approach’s ability to perform and explore in 6DoF, which is crucial for general manipulation. When 6DoF exploration is introduced, we first observe a drop in the average success rate for simple tasks that could be accomplished with planar actions (Balance, Transport, Tab. 2). However, this ability is critical for exploring complex tasks, providing data to improve upon in the ",
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+    {
+        "type": "table",
+        "img_path": "images/0add43e1b5e7daf184ac991319dfa3f26c966b50cb187a76cf9a0f2f3edf5792.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Success Rates $( \\%$ ) for data generation (top) and distillation approaches (bottom) over 200 trials. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Approach</td><td colspan=\"3\">Planar</td><td colspan=\"2\">6DoF</td><td rowspan=\"2\">Average</td></tr><tr><td>Balance Catapult Transport Mailbox Drawer</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"2\">LLM-as-Policy (2D) (+) 6DoF Robot Utils (+) Verify &amp;Retry</td><td>28.0</td><td>33.3</td><td>21.5</td><td>0.0</td><td>0.0</td><td>27.6</td></tr><tr><td>5.5 45.0</td><td>2.5</td><td>35.0</td><td>0.0</td><td>1.3</td><td>8.8</td></tr><tr><td rowspan=\"2\">Distill No Retry</td><td></td><td>7.3</td><td>82.0</td><td>3.0</td><td>31.8</td><td>33.8</td></tr><tr><td>67.5</td><td>38.5</td><td>32.5</td><td>0.0</td><td>22.7</td><td>32.2</td></tr><tr><td rowspan=\"2\">Distill Ours</td><td>79.0</td><td>58.3</td><td>80.0</td><td>62.0</td><td>55.8</td><td>67.0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "later distilling stage. In particular, we observed that 6DoF actions are important for grasping diverse objects with complex geometry (Transport, Tab. 2), and manipulating articulated objects (Drawer, Mailbox, Tab. 2). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Moreover, 6DoF exploration also helps in diversifying the data collection strategy, which provides the possibility to improve upon in the later distilling stage. For example in the catapult domain, LLM-as-Policy (2D) is only able to solve one of three possible goals (the closest bin) using a deterministic strategy. However, it provides no useful data for learning the other two goals, making it a poor data-collection policy. In contrast, incorporating 6 DOF robot utilities achieves lower but non-zero average success rates in all bins $( 1 6 . 3 \\%$ , $3 . 3 \\%$ , and $2 . 2 \\%$ , full table in appendix), which provide much better exploration data for distillation. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/78a03ced9845e5e36be93f05d891ac4d3e7a6d2c26453a50febf301bf9995670.jpg",
+        "table_caption": [
+            "Table 3: Sources & Propagation of Error. Accuracy $( \\% )$ of planning, verification, and execution success rate $( \\% )$ for each mailbox subtask. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Subtask</td><td>Planning Verify Execution</td></tr><tr><td>Open mailbox</td><td>100</td><td>43.5</td></tr><tr><td>Put package in mailbox</td><td>100</td><td>28.5</td></tr><tr><td>Raise mailbox flag</td><td>100</td><td>62.0</td></tr><tr><td>Close mailbox</td><td>100</td><td>94.2</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Verify & Retry always helps. In the verify & retry step, the LLM retries all tasks until they are successful. This simple addition improves performance in all domains, with $2 \\times , 3 \\times$ , $8 \\times$ , and $1 3 \\times$ in transport, catapult, balance, and drawer domains. Without this crucial step, we observe $0 . 0 \\%$ success rate in the mailbox domain, underscoring the difficulty of flawlessly executing long sequences of 6 DOF actions, and the importance of recovery after failure. ",
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+    {
+        "type": "text",
+        "text": "Language Model Scaling. In addition to the final task success, we provide more detailed analysis of planning and success condition inference accuracy in Tab. 4. We evaluate on the proprietary GPT3 [66] (175B text-davinci-003) and the open LLAMA2 [67] (7B and 13B). We found that Llama models struggles in complex planning domains because they do not follow instructions provided in the prompts. For instance, in the drawer domain, both models fail to account for drawer opening and closing. However, we observe an upwards trend with respect to Llama model size, with the 13B model outperforming the 7B model by $+ 2 0 . 0 \\%$ and $+ 3 8 . 3 \\%$ in planning and success verification accuracy respectively. ",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: LLM Evaluation. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Size</td><td>Planning</td><td>Success</td></tr><tr><td rowspan=\"2\">LLAMA2</td><td>7B</td><td>42.0</td><td>10.0</td></tr><tr><td>13B</td><td>62.0</td><td>48.3</td></tr><tr><td>GPT3</td><td>175B</td><td>82.0</td><td>91.1</td></tr></table>",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "4.2 Distilled Policy Evaluation ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Robustness In, Robustness Out. By filtering trajectories with LLM’s inferred success condition, distilled policies inherit the robustness of their data collection policies while improving upon success rates $( + 2 3 . 4 \\%$ and $+ 3 3 . 2 \\%$ for no-retry and ours, Tab. 2). Since our distilled policy learned from a robust data collection policy, it also recovers from failures (e.g. failed grasps or placements) and continuously retries a task until it succeeds. Meanwhile, since the no-retry distilled policy learned from a data collection policy which did not retry upon failure, it is sensitive and brittle, leading to $- 3 4 . 8 \\%$ lower average success rate across all domains compared to ours (Tab. 2). ",
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+    {
+        "type": "text",
+        "text": "High Performance From Diverse Retry Attempts. Plotting how long policies take to solve the balance task (Fig. 6), we observed that our policy and its data collection policy continuously tries a diverse set of grasps and placements after each failed attempt until it succeeds. This results in higher success rates as the policy is given more time, and is reflected in their monotonically increasing success rates. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In contrast, baselines plateau after their first grasp/platement attempts. This highlights the synergy of two design decisions. First, the verify & retry step $( \\ S 3 . 3 )$ is crucial for demonstrating retrying behavior, but is by itself insufficient if each retrying action is the identical as the previous one. Instead, opting for a diffusion policy $( \\ S \\ 3 . 4 )$ for learning from and generating high-entropy, diverse retry attempts (Fig 5) is also essential for high performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Policy Learning Baselines. We investigate policy learning design decisions on the single-task balance domain, and remove language conditioning. While BC-Z found spatial softmax hurt their performance and opted for a mean pool, we observed using spatial softmax improved performance by $+ 5 . 0 \\%$ . Further, we found that switching from delta to absolute action spaces improved success rates $+ 6 . 5 \\%$ and $+ 9 . 5 \\%$ when using the MLP action decoder and our diffusion action decoder, respectively, confirming Chi et al. [12]’s findings. Lastly, we find that using our pseudo-random diffusion-based action encoder consistently outperforms a deterministic MLP action mappings, regardless of other design decisions. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4744b196017d0467fd75e21b1770f0e42c8b6ad74eed9c78004a816824d44cc1.jpg",
+        "image_caption": [
+            "Figure 6: Distilled Robustness. Our policy inherits robust recovery from failure behavior from its data collection policy, while improving upon success rate. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sim2Real Transfer. We evaluated a policy trained on domain randomized synthetic data in a real world transport task with five novel objects (Fig. 5e). Averaging across ten episodes per object, our policy achieved $76 \\%$ success rate, demonstrating the effectiveness of our approach in Sim2Real transfer. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 Limitations ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "By using priviledged simulation state information, the LLM can infer success conditions which uses ground truth contact, joint information, and object poses. This means our implementation of the data generation phase is limited to simulation environments, and our policy requires sim2real transfer. Further, Our data generation method relies on existing 3D assets and environments, which presents a further opportunity for scaling up with assets from 3D generative models or procedural generation. Finally, while our approach’s dataset contains text labels and success labels for all subtasks, we have only evaluated its effectiveness in learning the root task. Learning from all subtasks and growing a robot’s set of learned, reusable sub-skills over time to enable compositional generalization is left for future work. ",
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+    {
+        "type": "table",
+        "img_path": "images/9ae4afdd3193c47d2f2b4a04951d6281d21f9cc3a160399a0b424263980e3765.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 5: Policy Learning Ablations. Action generation using diffusion models [50] robustly outperforms feed-forward models across other policy design decisions. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Output</td><td colspan=\"2\">Input</td><td rowspan=\"2\">Success (%)</td></tr><tr><td>Generation</td><td>Rep.Exec Pred Pool</td><td></td><td>Proprio</td></tr><tr><td>BC-Z</td><td>FeedForward Delta</td><td>1</td><td>10 Avg</td><td>xxx</td><td>0.0</td></tr><tr><td></td><td>FeedForward Delta</td><td>4</td><td>10 Avg</td><td></td><td>15.0</td></tr><tr><td></td><td>FeedForward Delta</td><td>8</td><td>10 Avg</td><td></td><td>18.5</td></tr><tr><td>Ours</td><td>FeedForward Delta</td><td>8</td><td>16 Spatial</td><td></td><td>29.0</td></tr><tr><td></td><td>FeedForward Abs</td><td>8</td><td>16 Spatial</td><td>ν/√√</td><td>35.5</td></tr><tr><td></td><td>Diffusion</td><td>Delta 8</td><td>16 Spatial</td><td></td><td>69.5</td></tr><tr><td></td><td>Diffusion</td><td>Abs 8</td><td>16Avg</td><td></td><td>76.5</td></tr><tr><td></td><td>Diffusion</td><td>Abs 8</td><td>16 Spatial</td><td></td><td>79.0</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We proposed “Scaling Up and Distilling Down”, a framework that combines the strengths of LLMs, samplingbased planners, and policy learning into a single system that automatically generates, labels, and distills diverse robot-complete exploration experience into a multi-task visuo-linguo-motor policy. The distilled policy inherits long-horizon behaviour, rich low-level manipulation skills, and robustness from its data collection policy while improving upon performance beyond its training distribution. We believe that this integrated approach is a step towards putting robotics on the same scaling trend as that of LLM development while not compromising on the rich low-level control. ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Cheng Chi, Zeyi Liu, Samir Yitzhak Gadre, Mengda Xu, Zhenjia Xu, Mandi Zhao and Dominik Bauer for their helpful feedback and fruitful discussions. This work was supported in part by Google Research Award, NSF Award #2143601, and #2132519. We would like to thank Google for the UR5 robot hardware. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the sponsors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References   \n[1] S. Song, A. Zeng, J. Lee, and T. Funkhouser. Grasping in the wild: Learning 6dof closed-loop grasping from low-cost demonstrations. IEEE Robotics and Automation Letters, 5(3):4978–4985, 2020.   \n[2] T. Z. Zhao, V. Kumar, S. Levine, and C. Finn. Learning fine-grained bimanual manipulation with low-cost hardware. arXiv preprint arXiv:2304.13705, 2023.   \n[3] E. Todorov, T. Erez, and Y. Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026–5033. IEEE, 2012. doi:10.1109/IROS.2012.6386109.   \n[4] A. Mandlekar, D. Xu, J. Wong, S. Nasiriany, C. Wang, R. Kulkarni, L. Fei-Fei, S. Savarese, Y. Zhu, and R. Mart´ın-Mart´ın. What matters in learning from offline human demonstrations for robot manipulation. In arXiv preprint arXiv:2108.03298, 2021.   \n[5] S. Nair, A. Rajeswaran, V. Kumar, C. Finn, and A. Gupta. R3m: A universal visual representation for robot manipulation. arXiv preprint arXiv:2203.12601, 2022.   \n[6] K. Grauman, A. Westbury, E. Byrne, Z. Chavis, A. Furnari, R. Girdhar, J. Hamburger, H. Jiang, M. Liu, X. Liu, et al. Ego4d: Around the world in 3,000 hours of egocentric video. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18995–19012, 2022.   \n[7] J. Fu, A. Kumar, O. Nachum, G. Tucker, and S. Levine. D4rl: Datasets for deep data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020.   \n[8] J. Wu, X. Sun, A. Zeng, S. Song, J. Lee, S. Rusinkiewicz, and T. Funkhouser. Spatial action maps for mobile manipulation. arXiv preprint arXiv:2004.09141, 2020.   \n[9] M. Shridhar, L. Manuelli, and D. Fox. Perceiver-actor: A multi-task transformer for robotic manipulation. In Proceedings of the 6th Conference on Robot Learning (CoRL), 2022.   \n[10] M. Shridhar, L. Manuelli, and D. Fox. 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Garg, and D. Fox. Iris: Implicit reinforcement without interaction at scale for learning control from offline robot manipulation data. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 4414–4420. IEEE, 2020.   \n[66] T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020.   \n[67] H. Touvron, L. Martin, K. Stone, P. Albert, A. Almahairi, Y. Babaei, N. Bashlykov, S. Batra, P. Bhargava, S. Bhosale, et al. Llama 2: Open foundation and fine-tuned chat models. arXiv preprint arXiv:2307.09288, 2023. ",
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parse/dev/Nayau9fwXU/Nayau9fwXU.md

@@ -0,0 +1,483 @@
+# DIFFUSION-BASED IMAGE TRANSLATION USING DIS-ENTANGLED STYLE AND CONTENT REPRESENTATION
+
+Gihyun Kwon1, Jong Chul $\mathbf { Y } \mathbf { e } ^ { 2 , 1 }$   
+Department of Bio and Brain Engineering1, Kim Jaechul Graduate School of $\mathsf { A I } ^ { 2 }$ , KAIST   
+cyclomon,jong.ye@kaist.ac.kr
+
+![](images/53ecacf54a3c121a6934bef6ee1398844f4e613366779d29467147a64b79a6a4.jpg)  
+Figure 1: Image translation results by DiffuseIT. Our model can generate high-quality translation outputs using both text and image conditions. More results can be found in the experiment section.
+
+# ABSTRACT
+
+Diffusion-based image translation guided by semantic texts or a single target image has enabled flexible style transfer which is not limited to the specific domains. Unfortunately, due to the stochastic nature of diffusion models, it is often difficult to maintain the original content of the image during the reverse diffusion. To address this, here we present a novel diffusion-based unsupervised image translation method, dubbed as DiffuseIT, using disentangled style and content representation. Specifically, inspired by the slicing Vision Transformer (Tumanyan et al., 2022), we extract intermediate keys of multihead self attention layer from ViT model and used them as the content preservation loss. Then, an image guided style transfer is performed by matching the [CLS] classification token from the denoised samples and target image, whereas additional CLIP loss is used for the text-driven style transfer. To further accelerate the semantic change during the reverse diffusion, we also propose a novel semantic divergence loss and resampling strategy. Our experimental results show that the proposed method outperforms state-of-the-art baseline models in both text-guided and image-guided translation tasks.
+
+# 1 INTRODUCTION
+
+Image translation is a task in which the model receives an input image and converts it into a target domain. Early image translation approaches (Zhu et al., 2017; Park et al., 2020; Isola et al., 2017) were mainly designed for single domain translation, but soon extended to multi-domain translation (Choi et al., 2018; Lee et al., 2019). As these methods demand large training set for each domain, image translation approaches using only a single image pairs have been studied, which include the one-to-one image translation using multiscale training (Lin et al., 2020), or patch matching strategy (Granot et al., 2022; Kolkin et al., 2019). Most recently, Splicing ViT (Tumanyan et al., 2022) exploits a pre-trained DINO ViT (Caron et al., 2021) to convert the semantic appearance of a given image into a target domain while maintaining the structure of input image.
+
+On the other hand, by employing the recent text-to-image embedding model such as CLIP (Radford et al., 2021), several approaches have attempted to generate images conditioned on text prompts (Patashnik et al., 2021; Gal et al., 2021; Crowson et al., 2022; Couairon et al., 2022). As these methods rely on Generative Adversarial Networks (GAN) as a backbone generative model, the semantic changes are not often properly controlled when applied to an out-of-data (OOD) image generation.
+
+Recently, score-based generative models (Ho et al., 2020; Song et al., 2020b; Nichol & Dhariwal, 2021) have demonstrated state-of-the-art performance in text-conditioned image generation (Ramesh et al., 2022; Saharia et al., 2022b; Crowson, 2022; Avrahami et al., 2022). However, when it comes to the image translation scenario in which multiple conditions (e.g. input image, text condition) are given to the score based model, disentangling and separately controlling the components still remains as an open problem.
+
+In fact, one of the most important open questions in image translation by diffusion models is to transform only the semantic information (or style) while maintaining the structure information (or content) of the input image. Although this could not be an issue with the conditional diffusion models trained with matched input and target domain images (Saharia et al., 2022a), such training is impractical in many image translation tasks (e.g. summer-to-winter, horse-to-zebra translation). On the other hand, existing methods using unconditional diffusion models often fail to preserve content information due to the entanglement problems in which semantic and content change at the same time (Avrahami et al., 2022; Crowson, 2022). DiffusionCLIP (Kim et al., 2022) tried to address this problem using denoising diffusion implicit models (DDIM) (Song et al., 2020a) and pixel-wise loss, but the score function needs to be fine-tuned for a novel target domain, which is computationally expensive.
+
+In order to control the diffusion process in such a way that it produces the output that simultaneously retain the content of the input image and follow the semantics of the target text or image, here we introduce a loss function using a pre-trained Vision Transformer (ViT) (Dosovitskiy et al., 2020). Specifically, inspired by the recent idea (Tumanyan et al., 2022), we extract intermediate keys of multihead self attention layer and [CLS] classification tokens of the last layer from the DINO ViT model and used them as our content and style regularization, respectively. More specifically, to preserve the structural information, we use the similarity and contrastive loss between intermediate keys of the input and denoised image during the sampling. Then, an image guided style transfer is performed by matching the [CLS] token between the denoised sample and the target domain, whereas additional CLIP loss is used for the text-driven style transfer. To further improve the sampling speed, we propose a novel semantic divergence loss and resampling strategy.
+
+Extensive experimental results including Fig. 1 confirmed that our method provide state-of-the-art performance in both text- and image- guided style transfer tasks quantitatively and qualitatively. To our best knowledge, this is the first unconditional diffusion model-based image translation method that allows both text- and image- guided style transfer without altering input image content.
+
+# 2 RELATED WORK
+
+Text-guided image synthesis. Thanks to the outstanding performance of text-to-image alignment in the feature space, CLIP has been widely used in various text-related computer vision tasks including object generation (Liu et al., 2021; Wang et al., 2022a), style transfer (Kwon & Ye, 2021; Fu et al., 2021), object segmentation (Luddecke & Ecker, 2022; Wang et al., 2022b), etc. Several recent ¨ approaches also demonstrated state-of-the-art performance in text-guided image manipulation task by combining the CLIP with image generation models. Previous approaches leverage pre-trained StyleGAN (Karras et al., 2020) for image manipulation with a text condition (Patashnik et al., 2021; Gal et al., 2021; Wei et al., 2022). However, StyleGAN-based methods cannot be used in arbitrary natural images since it is restricted to the pre-trained data domain. Pre-trained VQGAN (Esser et al., 2021) was proposed for better generalization capability in the image manipulation, but it often suffers from poor image quality due to limited power of the backbone model.
+
+With the advance of score-based generative models such as Denoising Diffusion Probabilistic Model (DDPM) (Ho et al., 2020), several methods (Ramesh et al., 2022; Saharia et al., 2022b) tried to generate photo-realistic image samples with given text conditions. However, these approaches are not adequate for image translation framework as the text condition and input image are not usually disentangled. Although DiffusionCLIP (Kim et al., 2022) partially solves the problem using DDIM sampling and pixelwise regularization during the reverse diffusion, it has major disadvantage in that it requires fine-tuning process of score models. As a concurrent work, DDIB(Su et al., 2022) proposed diffusion model based image translation using deterministic probability flow ODE formulation.
+
+Single-shot Image Translation. In image translation using single target image, early models mainly focused on image style transfer (Gatys et al., 2016; Huang & Belongie, 2017; Park & Lee, 2019; Yoo et al., 2019). Afterwards, methods using StyleGAN adaptation (Ojha et al., 2021; Zhu et al., 2021; Kwon & Ye, 2022; Chong & Forsyth, 2021) showed great performance, but there are limitations as the models are domain-specific (e.g. human faces). In order to overcome this, methods for converting unseen image into a semantic of target (Lin et al., 2020; Kolkin et al., 2019; Granot et al., 2022) have been proposed, but these methods often suffer from degraded image quality. Recently, Splicing ViT (Tumanyan et al., 2022) successfully exploited pre-trained DINO ViT(Caron et al., 2021) to convert the semantic appearance of given image into target domain while preserving the structure of input.
+
+# 3 PROPOSED METHOD
+
+# 3.1 DDPM SAMPLING WITH MANIFOLD CONSTRAINT
+
+In DDPMs (Ho et al., 2020), starting from a clean image ${ \pmb x } _ { 0 } \sim { \pmb q } ( { \pmb x } _ { 0 } )$ , a forward diffusion process $q ( { \pmb x } _ { t } | { \pmb x } _ { t - 1 } )$ is described as a Markov chain that gradually adds Gaussian noise at every time steps $t$ :
+
+$$
+q ( \pmb { x } _ { T } | \pmb { x } _ { 0 } ) : = \prod _ { t = 1 } ^ { T } q ( \pmb { x } _ { t } | \pmb { x } _ { t - 1 } ) , \quad \mathrm { w h e r e } \quad q ( \pmb { x } _ { t } | \pmb { x } _ { t - 1 } ) : = \mathcal { N } ( \pmb { x } _ { t } ; \sqrt { 1 - \beta _ { t } } \pmb { x } _ { t - 1 } , \beta _ { t } I ) ,
+$$
+
+where $\{ \beta \} _ { t = 0 } ^ { T }$ is a variance schedule. By denoting $\alpha _ { t } : = 1 - \beta _ { t }$ and $\textstyle { \bar { \alpha _ { t } } } : = \prod _ { s = 1 } ^ { t } \alpha _ { s }$ , the forward diffused sample at $t$ , i.e. $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ , can be sampled in one step as:
+
+$$
+\begin{array} { r } { \pmb { x } _ { t } = \sqrt { \bar { \alpha } _ { t } } \pmb { x } _ { 0 } + \sqrt { 1 - \bar { \alpha } _ { t } } \pmb { \epsilon } , \quad \mathrm { w h e r e } \quad \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I ) . } \end{array}
+$$
+
+As the reverse of the forward step $q ( \pmb { x } _ { t - 1 } | \pmb { x } _ { t } )$ is intractable, DDPM learns to maximize the variational lowerbound through a parameterized Gaussian transitions $p _ { \theta } ( \pmb { x } _ { t - 1 } | \pmb { x } _ { t } )$ with the parameter $\theta$ . Accordingly, the reverse process is approximated as Markov chain with learned mean and fixed variance, starting from $p ( { \bf x } _ { T } ) = \mathcal { N } ( { \bf x } _ { T } ; \bar { \bf 0 } , I )$ :
+
+$$
+p _ { \theta } ( \boldsymbol { x } _ { 0 : T } ) : = p _ { \theta } ( \boldsymbol { x } _ { T } ) \prod _ { t = 1 } ^ { T } p _ { \theta } ( \boldsymbol { x } _ { t - 1 } | \boldsymbol { x } _ { t } ) , \quad \mathrm { w h e r e } \quad p _ { \theta } ( \boldsymbol { x } _ { t - 1 } | \boldsymbol { x } _ { t } ) : = \mathcal { N } ( \boldsymbol { x } _ { t - 1 } ; \boldsymbol { \mu } _ { \theta } ( \boldsymbol { x } _ { t } , t ) , \sigma _ { t } ^ { 2 } I ) .
+$$
+
+where
+
+$$
+\pmb { \mu } _ { \theta } ( \pmb { x } _ { t } , t ) : = \frac { 1 } { \sqrt { \alpha _ { t } } } \bigg ( \pmb { x } _ { t } - \frac { 1 - \alpha _ { t } } { \sqrt { 1 - \bar { \alpha } _ { t } } } \pmb { \epsilon } _ { \theta } ( \pmb { x } _ { t } , t ) \bigg ) ,
+$$
+
+Here, $\epsilon _ { \theta } ( x _ { t } , t )$ is the diffusion model trained by optimizing the objective:
+
+$$
+\operatorname* { m i n } _ { \theta } L ( \theta ) , \quad \mathrm { w h e r e } \quad L ( \theta ) : = \mathbb { E } _ { t , x _ { 0 } , \epsilon } \Bigl [ \| \epsilon - \epsilon _ { \theta } \bigl ( \sqrt { \bar { \alpha } _ { t } } x _ { 0 } + \sqrt { 1 - \bar { \alpha } _ { t } } \epsilon , t \bigr ) \| ^ { 2 } \Bigr ] .
+$$
+
+After the optimization, by plugging learned score function into the generative (or reverse) diffusion process, one can simply sample from $p _ { \theta } ( \pmb { x } _ { t - 1 } | \pmb { x } _ { t } )$ by
+
+$$
+{ \pmb x } _ { t - 1 } = { \pmb \mu } _ { \theta } ( { \pmb x } _ { t } , t ) + \sigma _ { t } { \pmb \epsilon } = \frac { 1 } { \sqrt { \alpha _ { t } } } \bigg ( { \pmb x } _ { t } - \frac { 1 - \alpha _ { t } } { \sqrt { 1 - \bar { \alpha } _ { t } } } { \pmb \epsilon } _ { \theta } ( { \pmb x } _ { t } , t ) \bigg ) + \sigma _ { t } { \pmb \epsilon }
+$$
+
+In image translation using conditional diffusion models (Saharia et al., 2022a; Sasaki et al., 2021),√ the diffusion model $\epsilon _ { \theta }$ in (5) and (6) should be replaced with $\epsilon _ { \theta } ( { \pmb y } , \sqrt { \bar { \alpha } _ { t } } { \pmb x } _ { 0 } + \sqrt { 1 - \bar { \alpha } _ { t } } { \epsilon } , t )$ where $\textbf {  { y } }$ denotes the matched target image. Accordingly, the sample generation is tightly controlled by the matched target in a supervised manner, so that the image content change rarely happen. Unfortunately, the requirement of the matched targets for the training makes this approach impractical.
+
+To address this, Dhariwal & Nichol (2021) proposed classifier-guided image translation using the unconditional diffusion model training as in (5) and a pre-trained classifier $p _ { \phi } ( \pmb { y } | \pmb { x } _ { t } )$ . Specifically, $\mu _ { \theta } ( x _ { t } , t )$ in (4) and (6) are supplemented with the gradient of the classifier, i.e. $\hat { \mu } _ { \boldsymbol { \theta } } ( \mathbf { { x } } _ { t } , t ) : =$ $\begin{array} { r } { \pmb { \mu _ { \boldsymbol { \theta } } } ( \pmb { x } _ { t } , t ) + \sigma _ { t } \nabla _ { \pmb { x } _ { t } } \log p _ { \phi } ( \pmb { y } | \pmb { x } _ { t } ) } \end{array}$ . However, most of the classifiers, which should be separately trained, are not usually sufficient to control the content of the samples from the reverse diffusion process.
+
+![](images/0149e98efe21671497c4a5b0926d07a81c5bb04b5eb13420addf7d160d9d803f.jpg)  
+Figure 2: Given the input image $\pmb { x } _ { s r c }$ , we guide the reverse diffusion process $\{ \pmb { x } _ { t } \} _ { t = T } ^ { 0 }$ using various losses. (a) $\ell _ { c o n t }$ : the structural similarity loss between input and outputs in terms of contrastive loss between extracted keys from ViT. (b) $\ell _ { C L I P }$ : relative distance to the target text $\mathbf { \delta } d _ { t r g }$ in CLIP space in terms of $\mathbf { \delta } _ { \mathbf { \mathcal { X } } _ { s r c } }$ and $\pmb { d } _ { s r c }$ . (c) $\ell _ { s t y }$ : the [CLS] token distances between the outputs and target $\mathbf { \Delta } _ { \pmb { x } _ { t r g } }$ . (d) $\ell _ { s e m }$ : dissimilarity between the [CLS] token from the present and past denoised samples.
+
+Inspired by the recent manifold constrained gradient (MCG) for inverse problems (Chung et al., 2022a), here we formulate our content and style guidance problem as an inverse problem, which can be solved by minimizing the following total cost function with respect to the sample $_ { \textbf { \em x } }$ :
+
+$$
+\begin{array} { r } { \ell _ { t o t a l } ( { \pmb x } ; { \pmb x } _ { t r g } , { \pmb x } _ { s r c } ) , \quad \mathrm { o r } \quad \ell _ { t o t a l } ( { \pmb x } ; { \pmb d } _ { t r g } , { \pmb x } _ { s r c } , { \pmb d } _ { s r c } ) } \end{array}
+$$
+
+where $\pmb { x } _ { s r c }$ and $\pmb { x } _ { t r g }$ refer to the source and target images, respectively; and $\pmb { d } _ { s r c }$ and $\mathbf { \delta } d _ { t r g }$ refer to the source and target text, respectively. In our paper, the first form of the total loss in (7) is used for image-guided translation, where the second form is for the text-guided translation. Then, the sampling from the reverse diffusion with MCG is given by
+
+$$
+\begin{array} { l l l } { \displaystyle { \boldsymbol { x } _ { t - 1 } ^ { \prime } = \frac { 1 } { \sqrt { \alpha _ { t } } } \Big ( \boldsymbol { x } _ { t } - \frac { 1 - \alpha _ { t } } { \sqrt { 1 - \bar { \alpha } _ { t } } } \epsilon _ { \theta } \big ( \boldsymbol { x } _ { t } , t \big ) \Big ) + \sigma _ { t } \epsilon } } \\ { \displaystyle { \boldsymbol { x } _ { t - 1 } = \boldsymbol { x } _ { t - 1 } ^ { \prime } - \nabla _ { \boldsymbol { x } _ { t } } \ell _ { t o t a l } \big ( \hat { \boldsymbol { x } } _ { 0 } \big ( \boldsymbol { x } _ { t } \big ) \big ) } } \end{array}
+$$
+
+where $\hat { \pmb { x } } _ { 0 } ( { \pmb x } _ { t } )$ refers to the estimated clean image from the sample $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ using the Tweedie’s formula (Kim & Ye, 2021):
+
+$$
+\hat { \pmb { x } } _ { 0 } ( \pmb { x } _ { t } ) : = \frac { \pmb { x } _ { t } } { \sqrt { \bar { \alpha } _ { t } } } - \frac { \sqrt { 1 - \bar { \alpha } _ { t } } } { \sqrt { \bar { \alpha } _ { t } } } \pmb { \epsilon } _ { \theta } ( \pmb { x } _ { t } , t ) .
+$$
+
+In the following, we describe how the total loss $\ell _ { t o t a l }$ is defined. For brevity, we notate $\hat { \pmb { x } } _ { 0 } ( { \pmb x } _ { t } )$ as $_ { \textbf { \em x } }$ in the following sections.
+
+# 3.2 STRUCTURE LOSS
+
+As previously mentioned, the main objective of image translation is maintaining the content structure between output and the input image, while guiding the output to follow semantic of target condition. Existing methods (Couairon et al., 2022; Kim et al., 2022) use pixel-wise loss or the perceptual loss for the content preservation. However, the pixel space does not explicitly discriminate content and semantic components: too strong pixel loss hinders the semantic change of output, whereas weak pixel loss alters the structural component along with semantic changes. To address the problem, we need to separately process the semantic and structure information of the image.
+
+Recently, (Tumanyan et al., 2022) demonstrated successful disentanglement of both components using a pre-trained DINO ViT (Caron et al., 2021). They showed that in ViT, the keys $k ^ { l }$ of multi-head self attention (MSA) layer contain structure information, and [CLS] token of last layer contains the semantic information. With above features, they proposed a loss for maintaining structure between input and network output with matching the self similarity matrix $S ^ { l }$ of the keys, which can be represented in the following form for our problem:
+
+$$
+\ell _ { s s i m } ( \pmb { x } _ { s r c } , \pmb { x } ) = \| S ^ { l } ( \pmb { x } _ { s r c } ) - S ^ { l } ( \pmb { x } ) \| _ { F } , \quad \mathrm { w h e r e } \quad \big [ S ^ { l } ( \pmb { x } ) \big ] _ { i , j } = \cos ( k _ { i } ^ { l } ( \pmb { x } ) , k _ { j } ^ { l } ( \pmb { x } ) ) ,
+$$
+
+where $k _ { i } ^ { l } ( { \pmb x } )$ and $k _ { j } ^ { l } ( \pmb { x } )$ indicate $i , j$ th key in the $l$ -th MSA layer extracted from ViT with image $_ { \textbf { \em x } }$ . The self-similarity loss can maintain the content information between input and output, but we found that only using this loss results in a weak regularization in our DDPM framework. Since the key $k _ { i }$ contains the spatial information corresponding the $i$ -th patch location, we use additional regularization with contrastive learning as shown in Fig. 2(a), inspired by the idea of using both of relation consistency and contrastive learning(Jung et al., 2022). Specifically, leveraging the idea of patch contrastive loss (Park et al., 2020), we define the infoNCE loss using the DINO ViT keys:
+
+$$
+\ L _ { c o n t } ( x _ { s r c } , x ) = - \sum _ { i } ! \log \left( \frac { \exp ( \sin ( k _ { i } ^ { l } ( x ) , k _ { i } ^ { l } ( x _ { s r c } ) ) / \tau ) } { \exp ( \sin ( k _ { i } ^ { l } ( x ) , k _ { i } ^ { l } ( x _ { s r c } ) ) / \tau + \sum _ { j \ne i } \exp ( \sin ( k _ { i } ^ { l } ( x ) , k _ { j } ^ { l } ( x _ { s r c } ) ) / \tau ) } \right) ,
+$$
+
+where $\tau$ is temperature, and $\mathrm { s i m } ( \cdot , \cdot )$ represents the normalized cosine similarity. With this loss, we regularize the key of same positions to have closer distance, while maximizing the distances between the keys at different positions.
+
+# 3.3 STYLE LOSS
+
+CLIP Loss for Text-guided Image Translation Based on the previous work of (Dhariwal & Nichol, 2021), CLIP-guided diffusion (Crowson, 2022) proposed to guide the reverse diffusion using pre-trained CLIP model using the following loss function:
+
+$$
+\ell _ { C L I P } ( \boldsymbol { d } _ { t r g } , \boldsymbol { x } ) : = - \mathrm { s i m } \left( E _ { T } ( \boldsymbol { d } _ { t r g } ) , E _ { I } ( \boldsymbol { x } ) \right) ,
+$$
+
+where $\mathbf { \delta } d _ { t r g }$ is the target text prompt, and $E _ { I } , E _ { T }$ refer to the image and text encoder of CLIP, respectively. Although this loss can give text-guidance to diffusion model, the results often suffer from poor image quality.
+
+Instead, we propose to use input-aware directional CLIP loss (Gal et al. (2021)) which matches the CLIP embedding of the output image to the target vector in terms of $\mathbf { \delta } d _ { t r g }$ , $d _ { s r c }$ , and $\mathbf { \delta } _ { \mathbf { \mathcal { X } } _ { s r c } }$ . More specifically, our CLIP-based semantic loss is described as (see also Fig. 2(b)):
+
+$$
+\ell _ { C L I P } ( \pmb { x } ; d _ { t r g } , \pmb { x } _ { s r c } , d _ { s r c } ) : = - \mathrm { s i m } ( \pmb { v } _ { t r g } , \pmb { v } _ { s r c } )
+$$
+
+where
+
+$$
+\pmb { v } _ { t r g } : = E _ { T } ( d _ { t r g } ) + \lambda _ { i } E _ { I } ( \pmb { x } _ { s r c } ) - \lambda _ { s } E _ { T } ( d _ { s r c } ) , \quad \pmb { v } _ { s r c } : = E _ { I } ( \mathrm { a u g } ( \pmb { x } ) )
+$$
+
+where $\mathrm { a u g } ( \cdot )$ denotes the augmentation for preventing adversarial artifacts from CLIP. Here, we simultaneously remove the source domain information $- \lambda _ { s } E _ { T } ( { \pmb d } _ { s r c } )$ and reflect the source image information to output $+ \lambda _ { i } E _ { I } ( { \pmb x } _ { s r c } )$ according to the values of $\lambda _ { s }$ and $\lambda _ { i }$ . Therefore it is possible to obtain stable outputs compared to using the conventional loss.
+
+Furthermore, in contrast to the existing methods using only single pre-trained CLIP model (e.g. ViT/B-32), we improve the text-image embedding performance by using the recently proposed CLIP model ensemble method (Couairon et al. (2022)). Specifically, instead of using a single embedding, we concatenate the multiple embedding vectors from multiple pre-trained CLIP models and used the it as our final embedding.
+
+Semantic Style Loss for Image-guided Image Translation In the case of image-guide translation, we propose to use [CLS] token of ViT as our style guidance. As explained in the previous part 3.2, the [CLS] token contains the semantic style information of the image. Therefore, we can guide the diffusion process to match the semantic of the samples to that of target image by minimizing the [CLS] token distances as shown in Fig. 2(c). Also, we found that using only [CLS] tokens often results in misaligned color values. To prevent this, we guide the output to follow the overall color statistic of target image with weak MSE loss between the images. Therefore, our loss function is described as follows:
+
+$$
+\ell _ { s t y } ( { \pmb x } _ { t r g } , { \pmb x } ) = | | e _ { [ C L S ] } ^ { L } ( { \pmb x } _ { t r g } ) - e _ { [ C L S ] } ^ { L } ( { \pmb x } ) | | _ { 2 } + \lambda _ { m s e } | | { \pmb x } _ { t r g } - { \pmb x } | | _ { 2 } .
+$$
+
+where $e _ { [ C L S ] } ^ { L }$ denotes the last layer [CLS] token.
+
+# 3.4 ACCELERATION STRATEGY
+
+Semantic Divergence Loss With the proposed loss functions, we can achieve text- or imageguided image translation outputs. However, we empirically observed that the generation process requires large steps to reach the the desired output. To solve the problem, we propose a simple approach to accelerate the diffusion process. As explained before, the [CLS] token of ViT contains the overall semantic information of the image. Since our purpose is to make the semantic information as different from the original as possible while maintaining the structure, we conjecture that we can achieve our desired purpose by maximizing the distance between the [CLS] tokens of the previous step and the current output during the generation process as described in Fig. 2(d). Therefore, our loss function at time $t$ is given by
+
+$$
+\ell _ { s e m } ( \pmb { x } _ { t } ; \pmb { x } _ { t + 1 } ) = - | | e _ { [ C L S ] } ^ { L } ( \hat { \pmb { x } } _ { 0 } ( \pmb { x } _ { t } ) ) - e _ { [ C L S ] } ^ { L } ( \hat { \pmb { x } } _ { 0 } ( \pmb { x } _ { t + 1 } ) ) | | _ { 2 } ,
+$$
+
+Specifically, we maximize the distance between the denoised output of the present time and the previous time, so that next step sample has different semantic from the previous step. One could think of alternatives to maximize pixel-wise or perceptual distance, but we have experimentally found that in these cases, the content structure is greatly harmed. In contrast, our proposed loss has advantages in terms of image quality because it can control only the semantic appearance.
+
+Resampling Strategy As shown in CCDF acceleration strategy (Chung et al., 2022b), a better initialization leads to an accelerated reverse diffusion for inverse problem. Empirically, in our image translation problem we also find that finding the good starting point at time step $T$ for the reverse diffusion affects the overall image quality. Specifically, in order to guide the initial estimate $\mathbf { \nabla } _ { \mathbf { x } _ { T } }$ to be sufficiently good, we perform $N$ repetition of one reverse sampling ${ \mathbf { } } { \mathbf { } } { \mathbf { } } x _ { T - 1 }$ followed by one forward step ${ \pmb x } _ { T } = \sqrt { 1 - \beta _ { T - 1 } } { \pmb x } _ { T - 1 } + \beta _ { T - 1 } { \pmb \epsilon }$ to find the $\mathbf { \nabla } _ { \mathbf { x } _ { T } }$ whose gradient for the next step is easily affected by the loss. With this initial resampling strategy, we can empirically found the initial $\mathbf { \nabla } _ { \mathbf { \mathcal { X } } \mathcal { T } }$ that can reduce the number of reverse steps. The overall process is in our algorithm in Appendix.
+
+# 3.5 TOTAL LOSS
+
+Putting all together, the final loss in (7) for the text-guided reverse diffusion is given by
+
+$$
+\ell _ { t o t a l } = \lambda _ { 1 } \ell _ { c o n t } + \lambda _ { 2 } \ell _ { s s i m } + \lambda _ { 3 } \ell _ { C L I P } + \lambda _ { 4 } \ell _ { s e m } + \lambda _ { 5 } \ell _ { r n g } ,
+$$
+
+where $\ell _ { r n g }$ is a regularization loss to prevent the irregular step of reverse diffusion process suggested in (Crowson (2022)). If the target style image $\pmb { x } _ { t r g }$ is given instead of text conditions $\pmb { d } _ { s r c }$ and $\mathbf { \delta } d _ { t r g }$ , then $\ell _ { C L I P }$ is simply substituted for $\ell _ { s t y }$ .
+
+# 4 EXPERIMENT
+
+# 4.1 EXPERIMENTAL DETAILS
+
+For implementation, we refer to the official source code of blended diffusion (Avrahami et al. (2022)). All experiments were performed using unconditional score model pre-trained with Imagenet $2 5 6 \times 2 5 6$ resolution datasets (Dhariwal $\&$ Nichol (2021)). In all the experiments, we used diffusion step of $T = 6 0$ and the resampling repetition of $N = 1 0$ ; therefore, the total of 70 diffusion reverse steps are used. The generation process takes 40 seconds per image in single RTX 3090 unit. In $\ell _ { C L I P }$ , we used the ensemble of 5 pre-trained CLIP models (RN50, RN50x4, ViTB/32, RN50x16, ViT-B/16) for the text-guidance, following the setup of Couairon et al. (2022). Our detailed experimental settings are elaborated in Appendix.
+
+# 4.2 TEXT-GUIDED SEMANTIC IMAGE TRANSLATION
+
+To evaluate the performance of our text-guided image translation, we conducted comparisons with state-of-the-art baseline models. For baseline methods, we selected the recently proposed models which use pre-trained CLIP for text-guided image manipulation: VQGAN-CLIP (Crowson et al. (2022)), CLIP-guided diffusion (CGD) (Crowson (2022)), DiffusionCLIP (Kim et al. (2022)), and FlexIT (Couairon et al. (2022)). For all baseline methods, we referenced the official source codes.
+
+![](images/1cf571a652c5e3bcefacb285259f44d4c0dd479de1bceb0320b09d1beb20cdac.jpg)  
+Figure 3: Qualitative comparison of text-guided translation on Animals dataset. Our model generates realistic samples that reflects the text condition, with better perceptual quality than the baselines.
+
+![](images/e3a8a1ed8dec40cb1ec6c123cd202e894167a2f8b252b162ef59f6d8c18f0549.jpg)  
+Figure 4: Qualitative comparison of text-guided image translation on Landscape dataset. Our model generates outputs with better perceptual quality than the baselines.
+
+Since our framework can be applied to arbitrary text semantics, we tried quantitative and qualitative evaluation on various kinds of natural image datasets. We tested our translation performance using two different datasets: animal faces (Si & Zhu (2012)) and landscapes (Chen et al. (2018)). The animal face dataset contains 14 classes of animal face images, and the landscapes dataset consists of 7 classes of various natural landscape images.
+
+Table 1: Quantitative comparison in the text-guided image translation. Our model outperforms baselines in overall scores for both of Animals and Landscapes datasets as well as user study.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Animals</td><td colspan="3">Landscapes</td><td colspan="3">User Study</td></tr><tr><td>SFID↓</td><td>CSFID↓</td><td>LPIPS↓</td><td>SFID↓</td><td>CSFID↓</td><td>LPIPS↓</td><td>Text个</td><td>Realism↑</td><td>Content</td></tr><tr><td>VQGAN-CLIP</td><td>30.01</td><td>65.51</td><td>0.462</td><td>33.31</td><td>82.92</td><td>0.571</td><td>2.78</td><td>2.05</td><td>2.16</td></tr><tr><td>CLIP-GD</td><td>12.50</td><td>53.05</td><td>0.468</td><td>18.13</td><td>62.19</td><td>0.458</td><td>2.61</td><td>2.24</td><td>2.28</td></tr><tr><td>DiffusionCLIP</td><td>25.09</td><td>66.50</td><td>0.379</td><td>29.85</td><td>76.29</td><td>0.568</td><td>2.50</td><td>2.54</td><td>3.06</td></tr><tr><td>FlexIT</td><td>32.71</td><td>57.87</td><td>0.215</td><td>18.04</td><td>60.04</td><td>0.243</td><td>2.22</td><td>3.15</td><td>3.89</td></tr><tr><td>Ours</td><td>9.98</td><td>41.07</td><td>0.372</td><td>16.86</td><td>54.48</td><td>0.417</td><td>3.68</td><td>4.28</td><td>4.11</td></tr></table>
+
+To measure the performance of the generated images, we measured the FID score (Heusel et al. (2017)). However, when using the basic FID score measurement, the output value is not stable because our number of generated images is not large. To compensate for this, we measure the performance using a simplified FID (Kim et al. (2020)) that does not consider the diagonal term of the feature distributions. Also, we additionally showed a class-wise SFID score that measures the SFID for each class of the converted output because it is necessary to measure whether the converted output accurately reflects the semantic information of the target class. Finally, we used the averaged LPIPS score between input and output to verify the content preservation performance of our method. Further experimental settings can be found in our Appendix.
+
+In Table 1, we show the quantitative comparison results. In image quality measurement using SFID and CSFID, our model showed the best performance among all baseline methods. Especially for Animals dataset, our SFID value outperformed others in large gain. In the content preservation by LPIPS score, our method scored the second best. In case of FlexIT, it showed the best score in LPIPS since the model is directly trained with LPIPS loss. However, too low value of LPIPS is undesired as it means that the model failed in proper semantic change. This can be also seen in qualitative result of Figs. 3 and 4, where our results have proper semantic features of target texts with content preservation, whereas the results from FlexIT failed in semantic change as it is too strongly confined to the source images. In other baseline methods, most of the methods failed in proper content preservation. Since our method is based on DDPM, our model can generate diverse images as shown in the additional outputs in our Appendix.
+
+To further evaluate the perceptual quality of generated samples, we conducted user study. In order to measure the detailed opinions, we used custom-made opinion scoring system. We asked the users in three different parts: 1) Are the output have correct semantic of target text? (Text-match), 2) are the generated images realistic? (Realism), 3) do the outputs contains the content information of source images? (Content). Detailed user-study settings are in our Appendix. In Table 1, our model showed the best performance, which further shows the superiority of our method.
+
+# 4.3 IMAGE-GUIDED SEMANTIC IMAGE TRANSLATION
+
+Since our method can be easily adapted to the image translation guided by target images, we evaluate the performance of our model with comparison experiments. We compare our model with appearance transfer models of Splicing ViT (Tumanyan et al. (2022)), STROTSS (Kolkin et al. (2019)), and style transfer methods WCT2 (Yoo et al. (2019)) and SANet (Park & Lee (2019)).
+
+Fig. 5 is a qualitative comparison result of image guided translation task. Our model successfully generated outputs that follow the semantic styles of the target images while maintaining the content of the source images. In
+
+<table><tr><td>Method</td><td>Style↑</td><td>Realism↑</td><td>Content个</td></tr><tr><td>SANet</td><td>2.75</td><td>4.08</td><td>4.37</td></tr><tr><td>WCT2</td><td>2.59</td><td>4.64</td><td>4.90</td></tr><tr><td>STROTSS</td><td>3.92</td><td>2.91</td><td>3.17</td></tr><tr><td>SplicingViT</td><td>3.50</td><td>2.08</td><td>2.15</td></tr><tr><td>Ours</td><td>4.23</td><td>4.25</td><td>4.51</td></tr></table>
+
+Table 2: User study comparison of image-guided translation tasks. Our model outperforms baseline methods in overall perceptual quality.
+
+the case of other models, we can see that the content was severely deformed or the semantic style was not properly reflected. We also measured the overall perceptual quality through a user study. As with text-guided translation, we investigated user opinion through three different questions. In Table 2, our model obtained the best score in style matching score and the second best in realism and content preservation scores. Baseline WCT2 showed the best in realism and content scores, but it shows the worst score in style matching because the outputs are hardly changed from the inputs except for overall colors. The opinions scores confirm that our model outperforms the baselines. More details are in our Appendix.
+
+![](images/91a3d6e135ef1149b0c93af10fb5561e26b3a4b67f08efeca9bbcdec3b16dfdf.jpg)  
+Figure 5: Qualitative comparison of image-guided image translation. Our results have better perceptual quality than the baseline outputs.
+
+![](images/8d61b3fa1f9e7b8174060133dfaf3929cbc61b8e910cd1f6a12a1f76681e70fd.jpg)  
+Figure 6: Qualitative comparison on ablation study. Our full setting shows the best results.
+
+# 4.4 ABLATION STUDY
+
+To verify the proposed components in our framework, we compare the generation performance with different settings. In Fig. 6, we show that (a) the outputs from our best setting have the correct semantic of target text, with preserving the content of the source; (b) by removing $\ell _ { s e m }$ , the results still have the appearance of source images, suggesting that images are not fully converted to the target domain; (c) without $\ell _ { c o n t }$ , the output images totally failed to capture the content of source images; (d) by using LPIPS perceptual loss instead of proposed $\ell _ { c o n t }$ , the results can only capture the approximate content of source images; (e) using pixel-wise $l _ { 2 }$ maximization loss instead of proposed $\ell _ { s e m }$ , the outputs suffer from irregular artifacts; (f) without using our proposed resampling trick, the results cannot fully reflect the semantic information of target texts. (g) With using VGG16 network instead of DINO ViT, the output structure is severely degraded with artifacts. Overall, we can obtain the best generation outputs by using all of our proposed components. For further evaluation, we will show the quantitative results of ablation study in our Appendix.
+
+# 5 CONCLUSION
+
+In conclusion, we proposed a novel loss function which utilizes a pre-trained ViT model to guide the generation process of DDPM models in terms of content preservation and semantic changes. We further propose a novel strategy of resampling technique for better initialization of diffusion process. For evaluation, our extensive experimental results show that our proposed framework has superior performance compared to baselines in both of text- and image-guided semantic image translation tasks. Despite the successful results, our method often fails to translate the image styles when there is large domain gap between source and target. With respect to this, we show the failure cases and discussions on limitations in Appendix.
+
+# Acknowledgement
+
+This research was supported by Field-oriented Technology Development Project for Customs Administration through National Research Foundation of Korea(NRF) funded by the Ministry of Science & ICT and Korea Customs Service(NRF-2021M3I1A1097938), the KAIST Key Research Institute (Interdisciplinary Research Group) Project, and the National Research Foundation of Korea under Grant (NRF-2020R1A2B5B03001980).
+
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+
+# A EXPERIMENTAL DETAILS
+
+# A.1 IMPLEMENTATION DETAILS
+
+For implementation, in case of using text-guided image manipulation, our initial sampling numbers are set as $T = 1 0 0$ , but we skipped the initial 40 steps to maintain the abstract content of input image. Therefore, the total number of sampling steps is $T = 6 0$ . With resampling step of $N = 1 0$ , we use total of 70 iterations for single image output. We found that using more resampling steps does not show meaningful performance improvement. In image-guided manipulation, we set initial sampling number $T = 2 0 0$ , and skipped the initial 80 steps. We used resampling step $N = 1 0$ . Therefore, we use total of 130 iterations. Although we used more iterations than text-guided translation, it takes about 40 seconds.
+
+For hyperparameters, we use $\lambda _ { 1 } = 2 0 0$ , $\lambda _ { 2 } = 1 0 0$ , $\lambda _ { 3 } = 2 0 0 0$ , $\lambda _ { 4 } = 1 0 0 0$ , $\lambda _ { 5 } = 2 0 0$ . For imageguided translation, we set $\lambda _ { m s e } = 1 . 5$ . For our CLIP loss, we set $\lambda _ { s } = 0 . 4$ and $\lambda _ { i } = 0 . 2$ . For our ViT backbone model, we used pre-trained DINO ViT that follows the baseline of Splicing ViT (Tumanyan et al., 2022). For extracting keys of intermediate layer, we use layer of $l = 1 1$ , and for [CLS] token, we used last layer output. Since ViT and CLIP model only take $2 2 4 \times 2 2 4$ resolution images, we resized all images before calculating the losses with ViT and CLIP.
+
+To further improve the sample quality of our qualitative results, we used restarting trick in which we check the $\ell _ { r e g }$ loss calculated at initial time step $T$ , and restart the whole process if the loss value is too high. If the initial loss $\ell _ { r e g } > 0 . 0 1$ , we restarted the process. For quantitative result, we did not use the restart trick for fair comparison.
+
+For augmentation, we use the same geometrical augmentations proposed in FlexIT(Couairon et al., 2022). Also, following the setting from CLIP-guided diffusion(Crowson, 2022), we included noise augmentation in which we mix the noisy image to $\hat { x } _ { 0 } ( x _ { t } )$ as it further removes the artifacts.
+
+In our image-guided image translation on natural landscape images, we matched the color distribution of output image to that of target image with (Hahne & Aggoun, 2021), as it showed better perceptual quality. Our detailed implementation can be found in our official GitHub repository.1
+
+For baseline experiments, we followed the official source codes in all of the models2345. For diffusion-based models (DiffusionCLIP, CLIP-guided diffusion), we used unconditional score model pre-trained on $2 5 6 \times 2 5 6$ resolutions. In DiffusionCLIP, we fine-tuned the score model longer than suggested training iteration, as it showed better quality. In CLIP-guided diffusion, we set the CLIP-guided loss as 2000, and also set initial sampling number as $T = 1 0 0$ with skipping initial 40 steps. For VQGAN-based models (FlexIT, VQGAN-CLIP), we used VQGAN trained on imagenet $2 5 6 \times 2 5 6$ resolutions datasets. In VQGAN-CLIP, as using longer iteration results in extremely degraded images, therefore we optimized only 30 iterations, which is smaller than suggested iterations $( \geq 8 0 )$ . In the experiments of FlexIT, we followed the exactly same settings suggested in the original paper.
+
+For baselines of image-guided style transfer tasks, we also referenced the original source codes6789.   
+In all of the experiments, we followed the suggested settings from the original papers.
+
+# A.2 DATASET DETAILS
+
+For our quantitative results using text-guided image translation, we used two different datasets Animals and Landscapes. In Animals dataset, the original dataset contains 21 different classes, but we filtered out the images from 14 classes (bear, cat, cow, deer, dog, lion, monkey, mouse, panda, pig, rabbit, sheep, tiger, wolf) which can be classified as mammals. Remaining classes (e.g. human, chicken, etc.) are removed since they have far different semantics from the mammal faces.Therefore we reported quantitative scores only with filtered datasets for fair comparison. The dataset contains 100-300 images per each class, and we selected 4 testing images from each class in order to use them as our content source images. With selected samples, we calculated the metrics using the outputs of translating the 4 images from a source class into all the remaining classes. Therefore, in our animal face dataset, total of 676 generated images are used for evaluation.
+
+In Landscapes dataset, we manually classified the images into 7 different classes (beach, desert, forest, grass field, mountain, sea, snow). Each class has 300 different images except for desert class which have 100 different images. Since some classes have not enough number of images, we borrowed images from seasons (Anoosheh et al., 2018) dataset. For metric calculation, we selected 8 testing images from each class, and used them as our content source images. Again, we translated the 8 images from source class into all the remaining classes. Therefore, a total of 336 generated images are used for our quantitative evaluation.
+
+For single image guided translation, we selected random images from AFHQ dataset for animal face translation; and for natural image generation, we selected random images from our Landscapes datasets.
+
+# A.3 USER STUDY DETAILS
+
+For our user study in text-guided image translation task, we generated 130 different images using 13 different text conditions with our proposed and baseline models. Then we randomly selected 65 images and made 6 different questions. More specifically, we asked the participants question about three different parts: 1) Are the outputs have correct semantic of target text? (Text-match), 2) Are the generated images realistic? (Realism), 3) Do the outputs contain the content information of source images (Content). We randomly recruited a total of 30 users, and provided them the questions using Google Form. The 30 different users come from age group 20s and 50s. We set the minimum score as 1, and the maximum score is 5. The users can score among 5 different options : 1-Very Unlikely, 2-Unlikely, 3-Normal, 4-Likely, 5-Very Likely.
+
+For the user study on image-guided translation task, we generated 40 different images using 8 different images conditions. Then we followed the same protocol to user study on text-guided image translation tasks, except for the content of questions. We asked the users in three different parts: 1) Are the outputs have correct semantic of target style image? (Style-match), 2) Are the generated images realistic? (Realism), 3) Do the outputs contain the content information of source images (Content).
+
+# A.4 ALGORITHM
+
+For detailed explanation, we include Algorithm of our proposed image translation mathods in Algorithm 1.
+
+# B QUANTITATIVE ABLATION STUDY
+
+For more thorough evaluation of our proposed components, we report ablation study on quantitative metrics. In this experiment, we only used Animals dataset due to the time limit. In Table 3, we show the quantitative results on various settings. When we remove one of our acceleration strategies, in setting (b) and (f), we can see that the fid score is degraded as the outputs are not properly changed from the original source images. (e) When we use L2 maximization instead of our proposed $\ell _ { s e m }$ , FID scores are improved from setting (b), but still the performance is not on par with our best settings. (d) When we use weak content regularization using LPIPS, we can see that the overall scores are degraded. When we remove our proposed $\ell _ { c o n t }$ , we can observe that SFID and CSFID scores are lower than other settings. However, we can see that LPIPS score is severely high as the model hardly reflect the content information of original source images. (g) we use pre-trained VGG instead of using ViT for ablation study. Instead of ViT keys for structure loss, we substitute it with features extracted from VGG16 relu3 1 activation layer. Also, we substitute ViT [CLS] token with VGG16 relu5 1 feature as it contains high-level semantic features. We can see that the model
+
+Algorithm 1 Semantic image translation: given a diffusion score model $\epsilon _ { \theta } ( \pmb { x } _ { t } , t )$ , CLIP model, and VIT model
+
+Input: source image $\pmb { x } _ { s r c }$ , diffusion steps $T$ , resampling steps $N$ , target text $\mathbf { \delta } d _ { t r g }$ , source text $\pmb { d } _ { s r c }$ or target image $\mathbf { \boldsymbol { x } } _ { t r g }$
+
+Output: translated image √ $\hat { \pmb x }$ which has semantic of $\mathbf { \delta } d _ { t r g }$ (or $\pmb { x } _ { t r g }$ ) and content of $\mathbf { \delta } _ { \mathbf { \mathcal { X } } _ { s r c } }$
+
+1: for all $t$ from $T$ to 0 do   
+2: $\begin{array} { r l } & { \epsilon  \epsilon _ { \theta } ( x _ { t } , t ) } \\ & { \hat { x } _ { 0 } ( x _ { t } )  \frac { x _ { t } } { \sqrt { \bar { \alpha } _ { t } } } - \frac { \sqrt { 1 - \bar { \alpha } _ { t } } } { \sqrt { \bar { \alpha } _ { t } } } \epsilon } \end{array}$   
+3:   
+4: if text-guided then   
+5: $\nabla _ { t o t a l }  \nabla _ { x _ { t } } \ell _ { t o t a l } ( \hat { { \boldsymbol x } } _ { 0 } ( x _ { t } ) ; d _ { t r g } , { \boldsymbol x } _ { s r c } , d _ { s r c } )$   
+6: else if image-guided then   
+7: $\nabla _ { t o t a l }  \mathsf { \bar { V } } _ { \pmb { x } _ { t } } \ell _ { t o t a l } ( \hat { \pmb { x } } _ { 0 } ( \pmb { x } _ { t } ) ; \pmb { x } _ { t r g } , \pmb { x } _ { s r c } )$   
+8: end if   
+9: $\begin{array} { r l } & { z \sim \mathcal { N } ( 0 , { \bf I } ) } \\ & { { \bf x } _ { t - 1 } ^ { \prime } = \frac { 1 } { \sqrt { \alpha _ { t } } } \Big ( { \bf x } _ { t } - \frac { 1 - \alpha _ { t } } { \sqrt { 1 - \bar { \alpha } _ { t } } } \epsilon \Big ) + \sigma _ { t } z } \\ & { { \bf x } _ { t - 1 } = { \bf x } _ { t - 1 } ^ { \prime } - \nabla _ { t o t a l } } \end{array}$   
+10:   
+11:   
+12: if t = T and $n < N$ then   
+13: $\begin{array} { l } { { \pmb x } _ { t }  { \mathcal { N } } ( \sqrt { 1 - \beta _ { t - 1 } } { \pmb x } _ { t - 1 } , \beta _ { t - 1 } { \bf I } ) } \\ { { \pmb n }  { \ b n } + 1 } \end{array}$   
+14:   
+15: go to 2   
+16: end if   
+17: end for   
+18: return ${ \pmb x } _ { - 1 }$
+
+Table 3: Quantitative comparison of ablation studies.   
+
+<table><tr><td rowspan="2"> Settings</td><td colspan="3">Animals</td></tr><tr><td>SFID↓</td><td>CSFID↓</td><td>LPIPS↓</td></tr><tr><td>VGG instead of ViT (g)</td><td>9.72</td><td>43.08</td><td>0.518</td></tr><tr><td>No resampling (f)</td><td>11.88</td><td>59.09</td><td>0.316</td></tr><tr><td>L2 Max instead of lsem (e)</td><td>13.18</td><td>49.47</td><td>0.324</td></tr><tr><td>LPIPs instead of lcont (d)</td><td>11.15</td><td>58.67</td><td>0.400</td></tr><tr><td>No lcont (c)</td><td>9.90</td><td>33.07</td><td>0.477</td></tr><tr><td>No lsem (b)</td><td>15.00</td><td>53.43</td><td>0.347</td></tr><tr><td>Ours (a)</td><td>9.98</td><td>41.07</td><td>0.372</td></tr></table>
+
+shows decent SFID and CSFID scores, but the LPIPS score is very high. The result show that using VGG does not properly operate as regularization tool, rather it degrades the generation process with damaging the structural consistency. Overall, when using our best setting, we can obtain the best output considering all of the scores.
+
+# C ARTISTIC STYLE TRANSFER
+
+With our framework, we can easily adapt our method to artistic style transfer. With simply changing the text conditions, or using artistic paintings as our image conditions, we can obtain the artistic style transfer results as shown in Fig. 7.
+
+# D FACE IMAGE TRANSLATION
+
+Instead of using score mode pre-trained on Imagenet dataset, we can use pre-trained score model on FFHQ human face dataset. In order to keep the face identity between source and output images, we include $\lambda _ { i d } \ell _ { i d }$ which leverage pre-trained face identification model ArcFace(Deng et al., 2019). We calculate identity loss between $\pmb { x } _ { s r c }$ and denoised image $\hat { \pmb { x } } _ { 0 } ( { \pmb x } _ { t } )$ . We use $\lambda _ { i d } = 1 0 0$ .
+
+![](images/de5f035d2242c8c5c5f1031a7221c9b9597f5c36db3e0a8359f4eb3e689c8ba4.jpg)  
+Figure 7: Various outputs of artistic style transfer. We can translation natural images into artistic style paintings with both of text or image conditions.
+
+![](images/9cd3f4687ea51fdbfb0bf9967d443e408ca5470fd5c3315cbc9bb851370f6ce3.jpg)  
+Figure 8: Outputs from face image translation models. The outputs from our model successfully translated the human face images with proper target domain semantic information.
+
+In Fig. 8, we show that our method also can be used in face image translation tasks. For comparison, we included baseline models of face editing method StyleCLIP (Patashnik et al., 2021), and oneshot face stylization model of JojoGAN (Chong & Forsyth, 2021). The results show that our method can translate the source faces into target domain with proper semantic change. In baseline models, although some images show high quality outputs, in most cases the image failed in translating the images. Also, since the baseline models rely on pre-trained StyleGAN, they require additional GAN inversion process to translate the source image. Therefore, the content information is not perfectly matched to the source image due to the limitation of GAN inversion methods.
+
+# E INFERENCE TIME COMPARISON
+
+To evaluate the time-efficiency of our method, we calculate the inference times of the various imageguided translation models. All experiments are conducted with single RTX3090 GPU, on the same hardware and software environment. We use the images of resolution $2 5 6 \times 2 5 6$ for experiments. In Table 4, we compare the times taken for single image translation. For single-shot semantic transfer models of Splicing ViT, the inference time is relatively long as we need to optimize large U-Net model for each image translation. In STROTSS, it requires texture matching calculation for single image translation, so it takes long time. For arbitrary style transfer models of WCT2 and SANet, the inference is done with only single-step network forward process, as the model is already trained with large dataset. Our model takes about 40 seconds, which is moderate when compared to the one-shot semantic transfer models (SplicingVit,STROTSS). However, the time is still longer than the style transfer models, as our model need multiple reverse DDPM steps for inference. In the future work, we are planning to improve the inference time with leveraging recent approaches.
+
+Table 4: Quantitative comparison on inference times of image-guided translation models.   
+
+<table><tr><td></td><td>Ours</td><td>Splicing Vit</td><td>STROTSS</td><td>WCT2</td><td>SANET</td></tr><tr><td>time</td><td>37s</td><td>25m 30s</td><td>53s</td><td>0.18s</td><td>0.12s</td></tr></table>
+
+![](images/73edf21e67d349939b36ba9aa57d652b0555fbf8949c96b896166007b2423294.jpg)  
+Figure 9: Comparison results on semantic segmentation maps from baseline outputs. When comparing segmentation maps, our model outputs show high structural consistency with the source images.
+
+# F SEMANTIC SEGMENTATION OUTPUTS
+
+To further verify the structural consistency between output and source images, we compared the semantic segmentation maps from outputs and source images. For experiment, we use semantic segmentation model (Zhou et al., 2017) which is pre-trained on ADE20K dataset. We referenced the official source code10 for segmentation model. Figure 9 shows the comparison results. In case of the baseline models VQGAN-CLIP, CLIP-guided diffusion, we can see that the segmentation maps are not properly aligned to the source maps, which means the model cannot keep the structure of source images. In case of FlexIT, the model outputs maps have high similarity to the source maps, but the semantic change is not properly applied. In our model, we can see the output maps have high similarity to the source maps, while semantic information is properly changed.
+
+![](images/c74cbbaa0eb1d0061d3e1b799d8c764a5dfa0c5d6f9cc11372d8717d7266f2b5.jpg)  
+Figure 10: Additional comparison on image-guided translation. For fair experiment conditioning, we trained the baseline SANet with ViT-based losses.
+
+![](images/7d2a8a34bf513770a88969f830f917a3648e033e5295e69d5ea228f810207983.jpg)  
+Figure 11: Ablation study results on pixel-wise l2 loss. Without pixel loss, the output image color is not matched to the color scale of the target images.
+
+# G ADDITIONAL COMPARISON ON IMAGE-GUIDED TRANSLATION
+
+For fair comparison with the baseline models, we trained the baseline SANet with our proposed ViTbased loss functions. When we train SANet with replacing the existing style and content loss with our $\ell _ { c o n t }$ and $\ell _ { s t y }$ , we found that the training is not properly working. Therefore, we simultaneously used existing VGG-based style and content loss with our proposed ViT-based losses. In Fig. 10, we can see that when training SANet with ViT, the results still show incomplete semantic transfer results. Although the output seems to contain more complex textures than basic model, the model performance is still confined to simple color transformation.
+
+![](images/820fbbd5245336d9f05b260580958ed4894d400c522a64fb446d5aea0ad6ef19.jpg)  
+Figure 12: Failure case outputs. If the semantic distance between source and target conditions are extremely far, semantic translation sometimes fails.
+
+To further evaluate the effect of pixel-wixe l2 loss for image-guided translation, we conducted additional experiments in Fig. 11. When we remove the pixel-wise l2 loss in our image-guided translation task, we can see the semantic of output images follow the target images, but the overall color of the output images are slightly unaligned with the target image color. The result show that using weak l2 loss help the model to accurately apply the color of target images to outputs.
+
+# H LIMITATION AND FUTURE WORK
+
+Although our method has shown successful performance in image conversion, it still has limitations to solve. First, if the semantic distance between the source image and the target domain is too far (e.g building Tiger), the output is not translated properly as shown in Fig. 12. We conjecture that this occurs when the text-image embedding space in the CLIP model is not accurately aligned, therefore it can be solved by using the advanced text-to-image embedding model. Second, our method has limitation that the image generation quality heavily relies on the performance of the pre-trained score model. This can also be solved if we use a diffusion model backbone with better performance. In future work, we plan to improve our proposed method in these two directions.
+
+# I ADDITIONAL RESULTS
+
+For additional results , in Fig. 13 we show the image translation outputs using text conditions. In Fig. 14, we additionally show the results from our image-guided image translation. We can successfully change the semantic of various natural images with text and image conditions.
+
+![](images/936b6fba4101d4ca8bbdbe7dde5b23c57161a091bab935864a599aef6983d958.jpg)  
+Figure 13: Qualitative results of text-guided image translation.
+
+![](images/719c2f2e4fc3aa6e8e9475fd282cb232608c9a180ce59af5d89a527cea3c8792.jpg)  
+Figure 14: Qualitative results of image-guided image translation.

parse/dev/nO5caZwFwYu/nO5caZwFwYu.md

@@ -0,0 +1,267 @@
+# EFFICIENT ACTIVE SEARCH FOR COMBINATORIAL OPTIMIZATION PROBLEMS
+
+André Hottung Bielefeld University, Germany andre.hottung@uni-bielefeld.de
+
+Yeong-Dae Kwon Samsung SDS, Korea y.d.kwon@samsung.com
+
+Kevin Tierney Bielefeld University, Germany kevin.tierney@uni-bielefeld.de
+
+# ABSTRACT
+
+Recently, numerous machine learning based methods for combinatorial optimization problems have been proposed that learn to construct solutions in a sequential decision process via reinforcement learning. While these methods can be easily combined with search strategies like sampling and beam search, it is not straightforward to integrate them into a high-level search procedure offering strong search guidance. Bello et al. (2016) propose active search, which adjusts the weights of a (trained) model with respect to a single instance at test time using reinforcement learning. While active search is simple to implement, it is not competitive with state-of-the-art methods because adjusting all model weights for each test instance is very time and memory intensive. Instead of updating all model weights, we propose and evaluate three efficient active search strategies that only update a subset of parameters during the search. The proposed methods offer a simple way to significantly improve the search performance of a given model and outperform state-of-the-art machine learning based methods on combinatorial problems, even surpassing the well-known heuristic solver LKH3 on the capacitated vehicle routing problem. Finally, we show that (efficient) active search enables learned models to effectively solve instances that are much larger than those seen during training.
+
+# 1 INTRODUCTION
+
+In recent years, a wide variety of machine learning (ML) based methods for combinatorial optimization problems have been proposed (e.g., Kool et al. (2019); Hottung et al. (2020)) . While early approaches failed to outperform traditional operations research methods, the gap between handcrafted and learned heuristics has been steadily closing. However, the main potential of ML-based methods lies not only in their ability to outperform existing methods, but in automating the design of customized heuristics in situations where no handcrafted heuristics have yet been developed. We hence focus on developing approaches that require as little additional problem-specific knowledge as possible.
+
+Existing ML based methods for combinatorial optimization problems can be classified into construction methods and improvement methods. Improvement methods search the space of complete solutions by iteratively refining a given start solution. They allow for a guided exploration of the search space and are able to find high-quality solutions. However, they usually rely on problemspecific components. In contrast, construction methods create a solution sequentially starting from an empty solution (i.e., they consider a search space consisting of incomplete solutions). At test time, they can be used to either greedily construct a single solution or to sample multiple solutions from the probability distribution encoded in the trained neural network. Furthermore, the sequential solution generation process can be easily integrated into a beam search without requiring any problem-specific components. However, search methods like sampling and beam search offer no (or very limited) search guidance. Additionally, these methods do not react towards the solutions seen so far, i.e., the underlying distribution from which solutions are sampled is never changed throughout the search.
+
+Bello et al. (2016) propose a generic search strategy called active search that allows an extensive, guided search for construction methods without requiring any problem specific components. Active search is an iterative search method that at each iteration samples solutions for a single test instance using a given model and then adjusts the parameters of that model with the objective to increase the likelihood of generating high-quality solutions in future iterations. They report improved performance over random sampling when starting the search from an already trained model. Despite promising results, active search has not seen adaption in the literature. The reason for this is its resource requirements, as adjusting all model parameters separately for each test instance is very time intensive, especially compared to methods that can sample solutions to multiple different instances in one batch.
+
+We extend the idea of active search as follows. (1) We propose to only adjust a subset of (model) parameters to a single instance during the search, while keeping all other parameters fixed. We show that this efficient active search (EAS) drastically reduces the runtime of active search without impairing the solution quality. (2) We implement and evaluate three different implementations of EAS and show that all offer significantly improved performance over pure sampling approaches.
+
+In our EAS implementations, the majority of (model) parameters are not updated during the search, which drastically reduces the runtime, because gradients only need to be computed for a subset of model weights, and most operations can be applied identically across a batch of different instances. Furthermore, we show that for some problems, EAS finds even better solutions than the original active search. All EAS implementations can be easily applied to existing ML construction methods.
+
+We evaluate the proposed EAS approaches on the traveling salesperson problem (TSP), the capacitated vehicle routing problem (CVRP) and the job shop scheduling problem (JSSP). For all problems, we build upon already existing construction approaches that only offer limited search capabilities. In all experiments, EAS leads to significantly improved performance over sampling approaches. For the CVRP and the JSSP, the EAS approaches outperform all state-of-the-art ML based approaches, and even the well-known heuristic solver LKH3 for the CVRP. Furthermore, EAS approaches assists in model generalization, resulting in drastically improved performance when searching for solutions to instances that are much larger than the instances seen during model training.
+
+# 2 LITERATURE REVIEW
+
+Construction methods Hopfield (1982) first used a neural network (a Hopfield network) to solve small TSP instances with up to 30 cities. The development of recent neural network architectures has paved the way for ML approaches that are able to solve large instances. The pointer network architecture proposed by Vinyals et al. (2015) efficiently learns the conditional probability of a permutation of a given input sequence, e.g., a permutation of cities for a TSP solution. The authors solve TSP instances with up to 50 cities via supervised learning. Bello et al. (2016) report that training a pointer network via actor-critic RL instead results in a better performance on TSP instances with 50 and 100 cities. Furthermore, graph neural networks are used to solve the TSP, e.g., a graph embedding network in Khalil et al. (2017) and a graph attention network in Deudon et al. (2018).
+
+The first applications of neural network based methods to the CVRP are reported by Nazari et al. (2018) and Kool et al. (2019). Nazari et al. (2018) propose a model with an attention mechanism and a recurrent neural network (RNN) decoder that can be trained via actor-critic RL. Kool et al. (2019) propose an attention model that uses an encoder that is similar to the encoder used in the transformer architecture Vaswani et al. (2017). Peng et al. (2019) and Xin et al. (2021) extend the attention model to update the node embeddings throughout the search, resulting in improved performance at the cost of longer runtimes for the CVRP. Falkner & Schmidt-Thieme (2020) propose an attention-based model that constructs tours in parallel for the CVRP with time windows.
+
+While ML-based construction methods have mainly focused on routing problems, there are some notable exceptions. For example, Khalil et al. (2017) use a graph embedding network approach to solve the minimum vertex cover and the maximum cut problems (in addition to the TSP). Zhang et al. (2020) propose a graph neural network based approach for the job shop scheduling problem (JSSP). Li et al. (2018) use a guided tree search enhanced ML approach to solve the maximal independent set, minimum vertex cover, and the maximal clique problems. For a more detailed review of ML methods on different combinatorial optimization problems, we refer to Vesselinova et al. (2020).
+
+While most approaches construct routing problem solutions autoregressively, some approaches predict a heat-map that describes which edges will likely be part of a good solution. The heat-map is then used in a post-hoc search to construct solutions. Joshi et al. (2019) use a graph convolutional network to create a heat-map and a beam search to search for solutions. Similarly, Fu et al. (2020) use a graph convolutional residual network with Monte Carlo tree search to solve large TSP instances. Kool et al. (2021) use the model from Joshi et al. (2019) to generate the heat-map and use it to search for solution to TSP and CVRP instances with a dynamic programming based approach.
+
+Improvement methods Improvement methods integrate ML based methods into high-level search heuristics or try to learn improvement operators directly. In general, they often invest more time into solving an instance than construction based methods (and usually find better solutions). Chen & Tian (2019) propose an approach that iteratively changes a local part of the solution. At each iteration, the trainable region picking policy selects a part of the solution that should be changed and a trainable rule picking policy selects an action from a given set of possible modification operations. Hottung & Tierney (2020) propose a method for the CVRP that iteratively destroys parts of a solution using predefined, handcrafted operators and then reconstructs them with a learned repair operator. Wu et al. (2021) and de O. da Costa et al. (2020) propose to use RL to pick an improving solution from a specified local neighborhood (e.g., the 2-Opt neighborhood) to solve routing problems. Hottung et al. (2021) learn a continuous representation of discrete routing problem solutions using conditional variational autoencoders and search for solutions using a generic, continuous optimizer.
+
+# 3 SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS WITH EAS
+
+We propose three EAS implementations that adjust a small subset of (model) parameters in an iterative search process. Given an already trained model, we investigate adjusting (1) the normally static embeddings of the problem instance that are generated by the encoder model, (2) the weights of additional instance-specific residual layers added to the decoder, and (3) the parameters of a lookup table that directly affect the probability distribution returned by model. In each iteration, multiple solutions are sampled for one instance and the dynamic (model) parameters are adjusted with the goal of increasing the probability of generating high quality solutions (as during model training). This allows the search to sample solutions of higher quality in subsequent iterations, i.e., the search can focus on the more promising areas of the search space. Once a high-quality solution for an instance is found, the adjusted parameters are discarded, so that the search process can be repeated on other instances. All strategies efficiently generate solutions to a batch of instances in parallel, because the network layers not updated during the search are applied identically to all instances of the batch.
+
+Background RL based approaches for combinatorial problems aim to learn a neural network based model $p _ { \theta } ( \pi | l )$ with weights $\theta$ that can be used to generate a solution $\pi$ given an instance l. State-ofthe-art approaches usually use a model that consists of an encoder and a decoder unit. The encoder usually creates static embeddings $\omega$ that describe the instance $l$ using a computationally expensive encoding process (e.g., Kool et al. (2019); Kwon et al. (2020)). The static embeddings are then used to autoregessively construct solutions using the decoder over $T$ time steps. At each step $t$ , the decoder $q _ { \phi } ( a | s _ { t } , \bar { \omega } )$ , with weights $\phi \subset \theta$ , outputs a probability value for each possible action $a$ in the state $s _ { t }$ (e.g., for the TSP, each action corresponds to visiting a different city next). The starting state $s _ { 1 }$ describes the problem instance $l$ (e.g., the positions of the cities for the TSP and the starting city) and the state $s _ { t + 1 }$ is obtained by applying the action $a _ { t }$ selected at time step $t$ to the state $s _ { t }$ . The (partial) solution $\pi _ { t }$ is defined by the sequence of selected actions $a _ { 1 } , a _ { 2 } , \ldots , a _ { t }$ . Once a complete solution, $\pi _ { T }$ , fulfilling all constraints of the problem is constructed, the objective function value $\mathbf { \bar { \boldsymbol { C } } } ( \pi , l )$ of the solution can be computed (e.g., the tour length for the TSP).
+
+Figure 1 shows the solution generation for a TSP instance with a model that uses static embeddings. The static embeddings $\omega$ are used at each decoding step to generate a probability distribution over all possible next actions and the selected action is provided to the decoder in the next decoding step. During testing, solutions can be constructed by either selecting actions greedily or by sampling each action according to $q _ { \phi } ( a | s _ { t } , \omega )$ . Since the static embeddings are not updated during solution generation they only need to be computed once per instance, which allows to quickly sample multiple solutions per instance. We note that not all models use static embeddings. Some approaches update all instance embeddings after each action (e.g., Zhang et al. (2020)), which allows the embeddings to contain information on the current solution state $s _ { t }$ .
+
+![](images/3abde8df1e97587e52664ac4987fda3a29e2025873cd251133e478eda9740eb0.jpg)  
+Figure 1: Sampling a solution for the TSP with a model $p _ { \theta } ( \pi | l )$ that uses static instance embeddings.
+
+# 3.1 EMBEDDING UPDATES
+
+Our first proposed strategy, called EAS-Emb, updates the embeddings $\omega$ generated by an encoder using a loss function consisting of an RL component $\mathcal { L } _ { R L }$ and an imitation learning $\left( \operatorname { I L } \right)$ component $\mathcal { L } _ { I L }$ . The loss $\mathcal { L } _ { R L }$ is based on REINFORCE (Williams, 1992) and is the expected cost of the generated solutions, $\mathbb { E } \left[ C ( \pi ) \right]$ . We aim to adjust the embedding parameters to increase the likelihood of generating solutions with lower costs (e.g., a shorter tour length for the TSP). The loss $\mathcal { L } _ { I L }$ is the negation of the log-probability of (re-)generating the best solution seen so far. We adjust the embedding parameters to increase this probability.
+
+More formally, for an instance $l$ we generate the embeddings $\omega$ using a given encoder. Based on $\omega$ we can (repeatedly) sample a solution $\pi$ whose cost is $C ( \pi )$ . A subset of the embeddings $\hat { \omega } \subseteq \omega$ is adjusted to minimize $\mathcal { L } _ { R L }$ using the gradient
+
+$$
+\nabla _ { \hat { \omega } } \mathcal { L } _ { R L } ( \hat { \omega } ) = \mathbb { E } _ { \boldsymbol \pi } \left[ ( C ( \boldsymbol \pi ) - b _ { \circ } ) \nabla _ { \hat { \omega } } \log q _ { \phi } ( \boldsymbol \pi \mid \hat { \omega } ) \right]
+$$
+
+where $\begin{array} { r } { q _ { \phi } ( \pi \mid \hat { \omega } ) \equiv \prod _ { t = 1 } ^ { T } q _ { \phi } ( a _ { t } \mid s _ { t } , \hat { \omega } ) } \end{array}$ , and $b _ { \circ }$ is a baseline (we use the baseline proposed in Kwon et al. (2020) for our experiments).
+
+For the second loss $\mathcal { L } _ { I L }$ , let $\bar { \pi }$ be the best solution found so far for the instance $l$ , that consists of the actions $\bar { a } _ { 1 } , \dots , \bar { a } _ { T }$ . We use teacher forcing to make the decoder $q _ { \phi } ( \cdot | s _ { t } , \hat { \omega } )$ generate the solution $\bar { \pi }$ , during which we obtain the probability values associated with the actions $\bar { a } _ { 1 } , \dots , \bar { a } _ { T }$ . We increase the log-likelihood of generating $\bar { \pi }$ by adjusting $\hat { \omega }$ using the gradient
+
+$$
+\nabla _ { \boldsymbol { \hat { \omega } } } \mathcal { L } _ { I L } ( \boldsymbol { \hat { \omega } } ) = - \nabla _ { \boldsymbol { \hat { \omega } } } \log q _ { \phi } ( \bar { \pi } \mid \boldsymbol { \hat { \omega } } ) \equiv - \nabla _ { \boldsymbol { \hat { \omega } } } \log \prod _ { t = 1 } ^ { T } q _ { \phi } ( \bar { a } _ { t } | s _ { t } , \boldsymbol { \hat { \omega } } ) .
+$$
+
+The gradient of the overall loss $\mathcal { L } _ { R I L }$ is defined as $\nabla _ { \hat { \omega } } \mathcal { L } _ { R I L } ( \hat { \omega } ) = \nabla _ { \hat { \omega } } \mathcal { L } _ { R L } ( \hat { \omega } ) + \lambda \cdot \nabla _ { \hat { \omega } } \mathcal { L } _ { I L } ( \hat { \omega } )$ , where $\lambda$ is a tunable parameter. If a high value for $\lambda$ is selected, the search focuses on generating solutions that are similar to the incumbent solution. This accelerates the convergence of the search policy, which is useful when the number of search iterations is limited.
+
+We note that both decoding processes required for RL and $\mathrm { I L }$ can be carried out in parallel, using the same forward pass through the network. Furthermore, only the parameters $\hat { \omega }$ are instance specific, while all other model parameters are identical for all instances. This makes parallelization of multiple instances in a batch more efficient both in time and memory.
+
+# 3.2 ADDED-LAYER UPDATES
+
+We next propose EAS-Lay, which adds an instance-specific residual layer to a trained model. During the search, the weights in the added layer are updated, while the weights of all other original layers are held fixed. We use both RL and $\mathrm { I L }$ , similarly to EAS-Emb in Section 3.1.
+
+We formalize EAS-Lay as follows. For each instance $l$ we insert a layer
+
+$$
+\operatorname { L } ^ { \star } ( h ) = h + ( ( \operatorname { R e L u } ( h W ^ { 1 } + b ^ { 1 } ) W ^ { 2 } + b ^ { 2 } )
+$$
+
+into the given decoder $q _ { \phi }$ , resulting in a slightly modified model $\tilde { q } _ { \phi , \psi }$ , where $\psi = \{ W ^ { 1 } , b ^ { 1 } , W ^ { 2 } , b ^ { 2 } \}$ . The layer takes in the input $h$ and applies two linear transformations with a ReLu activation function in between. The weight matrices $W ^ { \bar { 1 } }$ and $W ^ { 2 }$ and the bias vectors $b ^ { 1 }$ and $b ^ { 2 }$ are adjusted throughout the search via gradient descent. The weights in the matrix $W ^ { 2 }$ and the vector $b ^ { 2 }$ are initialized to zero so that the added layer does not affect the output of the model during the first iteration of the search. The gradient for $\mathcal { L } _ { R L }$ is given as
+
+$$
+\nabla _ { \psi } \mathcal { L } _ { R L } ( \psi ) = \mathbb { E } _ { \pi } \big [ ( C ( \pi ) - b _ { \circ } ) \nabla _ { \psi } \log \tilde { q } _ { \phi , \psi } ( \pi ) \big ] ,
+$$
+
+with $\begin{array} { r } { \tilde { q } _ { \phi , \psi } ( \pi ) \equiv \prod _ { t = 1 } ^ { T } \tilde { q } _ { \phi , \psi } ( a _ { t } \mid s _ { t } , \omega ) } \end{array}$ , and $b _ { \circ }$ is a baseline. The gradient for $\mathcal { L } _ { I L }$ is defined similarly.
+
+Note that the majority of the network operations are not instance specific. They can be applied identically to all instances running in parallel as a batch, resulting in significantly lower runtime during search. The position at which the new layer is inserted has an impact on the performance of EAS-Lay, and identifying the best position usually requires testing. In general, the memory requirement of this approach can be reduced by inserting the additional layer closer towards the output layer of the network. This decreases the number of layers to be considered during backpropagation. We noticed for transformer-based architectures that applying the residual layer $\mathrm { L } ^ { \star } ( \cdot )$ to the query vector $q$ before it is passed to the single attention head usually results in a good performance.
+
+# 3.3 TABULAR UPDATES
+
+EAS-Emb and EAS-Lay require significantly less memory per instance than the original active search. However, they still need to store many gradient weights associated with multiple layers for the purpose of backpropagation. This significantly limits the number of solutions one can generate in parallel. We hence propose EAS-Tab, which does not require backpropagation, but instead uses a simple lookup table to modify the policy of the given model. For each action at a given state, the table provides a guide on how to change its probability, so that the sampled solution has a higher chance at being similar to the best solution found in the past.
+
+Formally, at each step $t$ during the sequential generation of a solution, we redefine the probability of selecting action $a _ { t }$ in the state $s _ { t }$ as $q _ { \phi } ( a | \bar { s _ { t } } , \omega ) ^ { \alpha } \cdot Q _ { g ( s _ { t } , a _ { t } ) }$ and renormalize over all possible actions using the softmax function. Here, $\alpha$ is a hyperparameter, and $g$ is a function that maps each possible state and action pair to an entry in the table $Q$ . The network parameters $\theta$ remain unchanged, resulting in fast and memory efficient solution generation. The hyperparameter $\alpha$ is similar to the temperature value proposed in Bello et al. (2016) and modifies the steepness of the probability distribution returned by the model (lower values increase the exploration of the search). During search, the table $Q$ is updated with the objective of increasing the quality of the generated solutions. More precisely, after each iteration, $Q$ is updated based on the best solution $\bar { \pi }$ found so far consisting of the actions $\bar { a } _ { 1 } , \dots , \bar { a } _ { T }$ at states $\bar { s } _ { 1 } , \dots , \bar { s } _ { T }$ , respectively, with
+
+$$
+Q _ { g ( s _ { t } , a _ { t } ) } = \left\{ \begin{array} { l l } { \operatorname* { m a x } ( 1 , \frac { \sigma } { q _ { \phi } ( a | s _ { t } , \omega ) ^ { \alpha } } ) , } & { \mathrm { i f } g ( s _ { t } , a _ { t } ) \in \{ g ( \bar { a } _ { 1 } , \bar { s } _ { 1 } ) , \dots , g ( \bar { a } _ { T } , \bar { s } _ { T } ) \} } \\ { 1 , } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+The hyperparameter $\sigma$ defines the degree of exploitation of the search. If a higher value of $\sigma$ is used, the probabilities for actions that generate the incumbent solution are increased.
+
+In contrast to embedding or added-layer updates, this EAS method requires deeper understanding of the addressed combinatorial optimization problem to design the function $g ( s _ { t } , a _ { t } )$ . For example, for the TSP with $n$ nodes we use a table $Q$ of size $n \times n$ in which each entry $Q _ { i , j }$ corresponds to a directed edge $e _ { i , j }$ of the problem instance. The probability increases for the same directed edge that was used in the incumbent solution. This definition of $g ( s _ { t } , a _ { t } )$ effectively ignores the information on all the previous visits stored in state $s _ { t }$ , focusing instead on the current location (city) in choosing the next move. We note that this EAS approach is similar to the ant colony optimization algorithm (Dorigo et al., 2006), which has been applied to a wide variety of combinatorial optimization problems.
+
+# 4 EXPERIMENTS
+
+We evaluate all EAS strategies using existing, state-of-the-art RL based methods for three different combinatorial optimization problems. For the first two, the TSP and the CVRP, we implement EAS for the POMO approach (Kwon et al., 2020). For the third problem, the JSSP, we use the L2D method from Zhang et al. (2020). We extend the code made available by the authors of POMO (MIT license) and L2D (no license) with our EAS strategies to ensure a fair evaluation. Note that we only make minor modifications to these methods, and we use the models trained by the authors when available.
+
+We run all experiments on a GPU cluster using a single Nvidia Tesla V100 GPU and a single core of an Intel Xeon 4114 CPU at $2 . 2 \ : \mathrm { G H z }$ for each experiment. Our source code is available at https://github.com/ahottung/EAS. We use the Adam optimizer (Kingma & Ba, 2014) for all EAS approaches. The hyperparameters $\lambda , \sigma , \alpha$ , and the learning rate for the optimizer are tuned via Bayesian optimization using scikit-optimize (Head et al., 2020) on separate validation instances, which are sampled from the same distribution as the test instances. The hyperparameters are not adjusted for larger instances used to evaluate the generalization performance.
+
+# 4.1 TSP
+
+The TSP is a well-known routing problem involving finding the shortest tour between a set of $n$ nodes (i.e., cities) that visits each node exactly once and returns to the starting node. We assume that the distance matrix obeys the triangle inequality.
+
+Implementation POMO uses a model that is very similar to the AM model from Kool et al. (2019). The model generates instance embeddings only once per instance and does not update them during construction. The probability distribution over all actions are generated by a decoder, whose last layer is a single-headed attention layer. This last layer calculates the compatibility of a query vector $q$ to the key vector $k _ { i }$ for each node $i$ . In this operation, the key vector $k _ { i }$ is an embedding that has been computed separately, but identically for each input (i.e., node $i$ ) during the instance encoding process. For EAS-Emb, we only update the set of single-head keys $k _ { i }$ $( i = 1 , \ldots , n )$ . For EAS-Lay we apply the residual layer $\mathrm { L } ^ { \star } ( \cdot )$ described in Equation 3 to the query vector $q$ before it is passed to the single attention head. For EAS-Tab, we use a table $Q$ of size $n \times n$ and the mapping function $g ( s _ { t } , a _ { t } )$ such that each entry $Q _ { i , j }$ corespondents to a directed edge $e _ { i , j }$ of the problem instance.
+
+Setup We use the 10,000 TSP instances with $n = 1 0 0$ from Kool et al. (2019) for testing and three additional sets of 1,000 instances to evaluate generalization performance. We evaluate the EAS approaches against just using POMO with greedy action selection, random sampling, and active search as in Bello et al. (2016). In all cases, we use the model trained on instances with $n = 1 0 0$ made available by the POMO authors. For greedy action selection, POMO generates $8 \cdot n$ solutions for an instance of size $n$ (using 8 augmentations and $n$ different starting cities). In all other cases, we generate $2 0 0 \cdot 8 \cdot n$ solutions per instance (over the course of 200 iterations for the (E)AS approaches). The batch size (the number of instances solved in parallel) is selected for each method individually to fully utilize the available GPU memory. We compare to the exact solver Concorde (Applegate et al., 2006), the heuristic solver LKH3 (Helsgaun, 2017), the graph convolutional neural network with beam search (GCN-BS) from Joshi et al. (2019), the 2-Opt based deep learning (2-Opt-DL) approach from de O. da Costa et al. (2020), the learning improvement heuristics (LIH) method from Wu et al. (2021), the conditional variational autoencoder (CVAE-Opt) approach (Hottung et al., 2021), and deep policy dynamic programming (DPDP) (Kool et al., 2021).
+
+Results Table 1 shows the average costs, average gap and the total runtime (wall-clock time) for each instance set. The exact solver Concorde performs best overall, as it is a highly specialized TSP solver. Of the POMO-based approaches, the original active search offers the best gap to optimality, but requires 5 days of runtime. EAS significantly lowers the runtime while the gap is only marginally larger. DPDP performs best among ML-based approaches. However, DPDP relies on a handcrafted and problem-specific beam search, whereas EAS methods are completely problem-independent. On the larger instances, EAS significantly improves generalization performance, reducing the gap over sampling by up to $3 . 6 \mathrm { x }$ . We also evaluate active search using the imitation learning loss, but observe no impact on the search performance (see Appendix B).
+
+Table 1: Results for the TSP   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Testing (10k inst.)</td><td colspan="10">Generalization (1k instances)</td></tr><tr><td colspan="3">n=100</td><td colspan="3">n=125</td><td colspan="3">n=150</td><td colspan="3"></td><td>n = 200</td></tr><tr><td>Method</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td></td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td></tr><tr><td>Concorde</td><td></td><td>[7.765 0.000%</td><td></td><td></td><td>82M|8.583 0.000%</td><td></td><td></td><td>12M|9.346 0.000%</td><td></td><td></td><td>17M|10.687 0.000%</td><td></td><td>31M</td></tr><tr><td>LKH3</td><td>7.765</td><td>0.000%</td><td>8H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.5830.000%73M9.3460.000% 99M10.6870.000%</td><td>3H</td></tr><tr><td>GCN-BS</td><td>7.87</td><td>1.39%</td><td>40M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>-</td></tr><tr><td>2-Opt-DL</td><td>7.83</td><td>0.87%</td><td>41M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>LIH</td><td>7.87</td><td>1.42%</td><td>2H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>1</td><td>0.343%</td><td>6D</td><td></td><td>8.646 0.736%</td><td>21H</td><td></td><td>9.482 1.454%</td><td></td><td>30H</td><td></td><td></td><td></td></tr><tr><td>DPDP</td><td>7.765</td><td>0.004%</td><td>2H</td><td></td><td>8.589 0.070%</td><td>31M</td><td></td><td>9.4340.942%</td><td></td><td>44M</td><td></td><td>11.154 4.370%</td><td>74M</td></tr><tr><td>Greedy</td><td>7.776</td><td>0.146%</td><td>1M|</td><td>8.607</td><td>0.278%</td><td></td><td>&lt;1M|9.397</td><td></td><td>0.542%</td><td>&lt;1M|</td><td>10.843</td><td>1.457%</td><td>1M</td></tr><tr><td>Sampling</td><td>7.770</td><td>0.074%</td><td>4H</td><td>8.595</td><td>0.145%</td><td>45M</td><td></td><td>9.378</td><td>0.334%</td><td>78M</td><td></td><td>10.8381.416%</td><td>3H</td></tr><tr><td>Active S.</td><td></td><td>7.768 0.046%</td><td>5D</td><td>8.591</td><td>0.095%</td><td>15H</td><td>9.364</td><td></td><td>0.192%</td><td>19H</td><td>10.735</td><td>50.447%</td><td>24H</td></tr><tr><td>0 EAS-Emb</td><td></td><td>7.769 0.063%</td><td>5H</td><td>8.591</td><td>0.092%</td><td>57M</td><td></td><td>9.363</td><td>0.174%</td><td>2H</td><td>10.730 0.400%</td><td></td><td>4H</td></tr><tr><td>P EAS-Lay</td><td></td><td>7.769 0.053%</td><td>7H</td><td>8.591</td><td>0.089%</td><td>74M</td><td></td><td>9.363 0.176%</td><td></td><td>2H</td><td>10.737 0.471%</td><td></td><td>4H</td></tr><tr><td>EAS-Tab</td><td></td><td>7.768 0.048%</td><td>5H</td><td>8.591</td><td>0.091%49M</td><td></td><td></td><td>9.3650.196%</td><td></td><td>1H</td><td>10.756 0.650%</td><td></td><td>3H</td></tr></table>
+
+# 4.2 CVRP
+
+The goal of the CVRP is to find the shortest routes for a set of vehicles with limited capacity that must deliver goods to a set of $n$ customers. We again use the POMO approach as a basis for our EAS strategies. As is standard in the ML literature, we evaluate all approaches on instance sets where the locations and demands are sampled uniformly at random. Additionally, we consider the more realistic instance sets proposed in Hottung & Tierney (2020) with up to 297 customers (see Appendix A).
+
+Implementation We use the same EAS implementation for the CVRP as for the TSP.
+
+Setup We use the 10,000 CVRP instances from Kool et al. (2019) for testing and additional sets of 1,000 instances to evaluate the generalization performance. Again, we compare the EAS approaches to POMO using greedy action selection, sampling and active search. We generate the same number of solutions per instance as for the TSP. We compare to LIH, CAVE-Opt, DPDP, NeuRewriter (Chen & Tian, 2019) and neural large neighborhood search (NLNS) from Hottung & Tierney (2020).
+
+Results Table 2 shows the average costs, the average gap to LKH3 and the total wall-clock time for all instance sets. EAS-Lay outperforms all other approaches on the test instances, including approaches that rely on problem-specific knowledge, with a gap that beats LKH3. Both other EAS methods also find solutions of better quality than LKH3, which is quite an accomplishment given the many years of work on the LKH3 approach. We note it is difficult to provide a fair comparison between a single-core, CPU-bound technique like LKH3 and our approaches that use a GPU. Nonetheless, assuming a linear speedup, at least 18 CPU cores would be needed for LKH3 to match the runtime of EAS-Tab. On the generalization instance sets with $n = 1 2 5$ and $n = 1 5 0$ , the EAS approaches also outperform LKH3 and CVAE-Opt while being significantly faster than active search. On the instances with $n = 2 0 0$ , active search finds the best solutions of all POMO based approaches with a gap of $0 . 2 2 \%$ to LKH3, albeit with a long runtime of 36 hours. We hypothesize that significant changes to the learned policy are necessary to generate high-quality solutions for instances that are very different to those seen during training. Active search’s ability to modify all model parameters makes it easier to make those changes. EAS-Tab offers the worst performance on the instances with $n = 2 0 0$ with a gap of $1 1 . 8 \%$ . This is because EAS-Tab is very sensitive to the selection of the hyperparameter $\alpha$ , meaning that EAS-Tab requires hyperparameter tuning on some problems to generalize more effectively. Adjusting $\alpha$ for the $n = 2 0 0$ case improves EAS-Tab’s gap to at least $3 . 5 4 \%$ , making it slightly better than greedy or sampling.
+
+Table 2: Results for the CVRP on instances with uniformly sampled locations and demands   
+
+<table><tr><td rowspan="2"></td><td colspan="3">Testing (10k inst.) n=100</td><td colspan="10">Generalization (1k instances)</td></tr><tr><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>n =125 Gap</td><td>Time</td><td>Obj.</td><td>n=150 Gap</td><td>Time</td><td>Obj.</td><td>n = 200 Gap</td><td></td><td>Time</td></tr><tr><td>Method</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LKH3</td><td>|15.65</td><td>0.00%</td><td></td><td>6D|17.50</td><td>0.00%</td><td></td><td>19H|19.22</td><td></td><td>0.00%</td><td></td><td>20H|22.00</td><td>0.00%</td><td>25H</td></tr><tr><td>NLNS NeuRewriter</td><td>15.99</td><td>2.23%</td><td>62M</td><td>[|18.07</td><td>3.23%</td><td></td><td>9M|19.96</td><td></td><td>3.86%</td><td>12M|23.02</td><td></td><td>4.66%</td><td>24M</td></tr><tr><td>LIH</td><td>16.10</td><td></td><td>66M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>16.03</td><td>2.47% 1.36%</td><td>5H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>=</td></tr><tr><td>DPDP</td><td>15.63</td><td>-0.13%</td><td>11D 23H</td><td>17.87 17.51</td><td>2.08% 0.07%</td><td>36H 3H</td><td>19.84 19.31</td><td></td><td>3.24% 0.48%</td><td>46H 5H</td><td>22.26</td><td>51.20%</td><td>= 9H</td></tr><tr><td>Greedy</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>15.76</td><td>0.76%</td><td>2M</td><td>17.73</td><td>1.29%</td><td>&lt;1M</td><td>19.64</td><td></td><td>2.18%</td><td>1M</td><td>|22.90</td><td>4.12%</td><td>1M</td></tr><tr><td>Sampling 0</td><td>15.67</td><td>0.17%</td><td>7H</td><td>17.60</td><td>0.54%</td><td>73M</td><td>19.48</td><td></td><td>1.35%</td><td>2H</td><td>23.18</td><td>5.35%</td><td>5H</td></tr><tr><td>Active S.</td><td>15.63</td><td>-0.07%</td><td>8D</td><td>17.47</td><td>-0.21%</td><td>25H</td><td>19.21</td><td></td><td>-0.03%</td><td>29H</td><td>22.05</td><td>0.22%</td><td>36H</td></tr><tr><td>EAS-Emb</td><td>15.63 15.61</td><td>-0.08%</td><td>9H 12H</td><td>17.47</td><td>-0.21%</td><td>93M</td><td>19.22</td><td></td><td>0.03%</td><td>3H</td><td>22.19</td><td>0.88%</td><td>6H</td></tr><tr><td>EAS-Lay EAS-Tab</td><td>15.62</td><td>-0.23% -0.14%</td><td>8H</td><td>17.50</td><td>17.46 -0.24% 0.00%</td><td>2H 80M</td><td>19.21 19.36</td><td></td><td>-0.04% 0.72%</td><td>3H 2H</td><td>22.10 24.56</td><td>0.45% 11.8%</td><td>8H 5H</td></tr></table>
+
+# 4.3 JSSP
+
+The JSSP is a scheduling problem involving assigning jobs to a set of heterogeneous machines. Each job consists of multiple operations that are run sequentially on the set of machines. The objective is to minimize the time needed to complete all jobs, called the makespan. We evaluate EAS using the L2D approach, which is a state-of-the-art ML based construction method using a graph neural network.
+
+Implementation L2D represents JSSP instances as disjunctive graphs in which each operation of an instance is represented by a node in the graph. To create a schedule, L2D sequentially selects the operation that should be scheduled next. To this end, an embedding $h _ { v }$ is created for each node $v$ in a step-wise encoding process. In contrast to POMO, the embeddings $h _ { v }$ are recomputed after each decision step $t$ . Since EAS-Emb requires static embeddings, we modify the network to use $\tilde { h } _ { v } ^ { t } = h _ { v } ^ { t } + h _ { v } ^ { S T }$ as an embedding for node t to zero. During the searc $v$ at step  with E $t$ , where S-Emb $h _ { v } ^ { S T }$ is a vector that is initialized winly adjust the static component $h _ { v } ^ { S T }$ of the embedding with gradient descent. For EAS-Lay, we insert the residual layer $\mathrm { L } ^ { \star } ( \cdot )$ described in Equation 3 to each embedding $h _ { v }$ separately and identically. Finally, for EAS-Tab, we use a table $Q$ of size $| O | \times | O |$ , where $| O |$ is the number of operations, and we design the function $g ( s _ { t } , a _ { t } )$ so that the entry $Q _ { i , j }$ corresponds to selecting the operation $o _ { j }$ directly after the operation $o _ { i }$ .
+
+Setup We use three instance sets with 100 instances from Zhang et al. (2020) for testing and to evaluate the generalization performance. We use the exact solver Google OR-Tools (Perron & Furnon) as a baseline, allowing it a maximum runtime of 1 hour per instance. Furthermore, we compare to L2D with greedy action selection. Note that the performance of the L2D implementation is CPU bound and does not allow different instances to be batch processed. We hence solve instances sequentially and generate significantly fewer solutions per instance than for the TSP and the CVRP. For sampling, active search and the EAS approaches we sample 8,000 solutions per problem instance over the course of 200 iterations for the (efficient) active search approaches.
+
+Results Table 3 shows the average gap to the OR-Tools solution and the total wall-clock time per instance set. EAS-Emb offers the best performance for all three instance sets. On the $1 0 \times 1 0$ instances, EAS-Emb reduces the gap by $50 \%$ in comparison to pure sampling. Even on the $2 0 \times 1 5$ instances it reduces the gap to $1 6 . 8 \%$ from $2 0 . 8 \%$ for pure sampling, despite the low number of sampled solutions per instance. EAS-Lay offers performance that is comparable to active search. We note that if L2D were to more heavily use the GPU, instances could be solved in batches, thus drastically reducing the runtime of EAS-Lay and EAS-Tab. While EAS-Tab shows similar performance to active search on the test instance set, it is unable to generalize effectively to the larger instances.
+
+# 4.4 SEARCH TRAJECTORY ANALYSIS
+
+To get a better understanding of how efficient active search improves performance, we monitor the quality of the solutions sampled at each of the 200 iterations of the search. Figure 2 reports the average quality over all test instances for the JSSP and over the first 1,000 test instances for the TSP and CVRP. As expected, the quality of solutions generated via pure sampling does not change over the course of the search for all three problems. For all other methods, the quality of the generated solutions improves throughout the search. Thus, all active search variants successfully modify the (model) parameters in a way that increases the likelihood of generating high-quality solutions.
+
+Table 3: Results for the JSSP   
+
+<table><tr><td colspan="2" rowspan="2"></td><td colspan="3">Testing(100 inst.）</td><td colspan="4">Generalization (1OO instances)</td></tr><tr><td colspan="2">10×10</td><td colspan="3">15×15</td><td colspan="3">20×15</td></tr><tr><td colspan="2" rowspan="2">Method</td><td rowspan="2">Obj. Gap</td><td rowspan="2">Time</td><td rowspan="2">Obj.</td><td>Gap</td><td>Time</td><td>Obj. Gap</td><td>Time</td></tr><tr><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2"></td><td rowspan="2">0.0%</td></tr><tr><td rowspan="2">OR-Tools Greedy</td><td rowspan="2">1807.6 0.0%</td></tr><tr><td>37S|1</td><td rowspan="2">[1188.0</td><td colspan="2" rowspan="2">0.0%</td><td rowspan="2">3H| |1345.5</td><td rowspan="2"></td><td rowspan="2">80H</td></tr><tr><td rowspan="2"></td><td rowspan="2">1988.6</td></tr><tr><td>22.3% 871.7</td><td>20S 1528.3</td><td>28.6%</td><td>44S</td><td>1738.0</td><td>29.2%</td><td>60S</td></tr><tr><td rowspan="4">Sampling L</td><td>854.2</td><td>8.0% 5.8%</td><td>8H 8H</td><td>1378.3 16.0% 1345.2 13.2%</td><td>25H 32H</td><td>1624.6 1576.5</td><td>20.8% 17.2%</td><td>40H</td></tr><tr><td>EAS-Emb</td><td>837.0 3.7%</td><td>7H</td><td>1326.4 11.7%</td><td>22H</td><td>1570.8</td><td>16.8%</td><td>50H 37H</td></tr><tr><td>EAS-Lay</td><td>859.6 6.5%</td><td>7H</td><td>1352.6</td><td>13.8%</td><td>25H 1581.8</td><td>17.6%</td><td>46H</td></tr><tr><td>EAS-Tab</td><td>860.2 6.5%</td><td>8H</td><td>1376.8</td><td>15.9% 29H</td><td></td><td>1623.420.7%</td><td>51H</td></tr></table>
+
+![](images/aeff2be7dc644f08b6349a6f6cabfb88e1ed829fcc964cb99457f18da9f01b79.jpg)  
+Figure 2: Average costs of sampled solutions at each iteration (best viewed in color).
+
+![](images/3814f29ab52dc4112005668d02a29044fe9d34e4442175ae8284d12c9a420a11.jpg)  
+Figure 3: Influence of $\lambda$ on the solution quality for EAS-Emb and EAS-Lay.
+
+For the TSP, EAS-Emb and EAS-Lay offer nearly identical performance, with EAS-Tab outperforming both by a very slight margin. The original active search is significantly more unstable, which is likely the result of the learning rate being too high. Note that the learning rate has been tuned on an independent validation set. These results indicate that selecting a suitable learning rate is significantly more difficult for the original active search than for our efficient active search variants where only a subset of (model) parameters are changed. For the CVRP, all EAS variants find better solutions on average than the original search after only a few iterations. Keeping most parameters fixed seems to simplify the underlying learning problem and allows for faster convergence. For the JSSP, EAS-Emb offers significantly better performance than all other methods. The reason for this is that the L2D approach uses only two node features and has a complex node embedding generation procedure. While the original active search must fine tune the entire embedding generation process to modify the generated solutions, EAS-Emb can just modify the node embedding directly.
+
+# 4.5 ABLATION STUDY: IMITATION LEARNING LOSS
+
+We evaluate the impact of the imitation learning loss $\mathcal { L } _ { I L }$ of EAS-Emb and EAS-Lay with a sensitivity and ablation analysis for the hyperparameter $\lambda$ . We solve the first 500 test instances (to reduce the computational costs) for the TSP and CVRP, and all test instances for the JSSP using EAS-Emb and EAS-Lay with different $\lambda$ values. The learning rate remains fixed to a value determined in independent tuning runs in which $\lambda$ is fixed to zero. Figure 3 shows the results for all three problems. For the TSP and the CVRP, the results show that $\mathcal { L } _ { I L }$ can significantly improve performance. When $\lambda$ is set to 0 or very small values, $\mathcal { L } _ { I L }$ is disabled, thus including $\mathcal { L } _ { I L }$ is clearly beneficial on the TSP and CVRP. For the JSSP, the inclusion of $\mathcal { L } _ { I L }$ does not greatly improve performance, but it does not hurt it, either. Naturally, $\lambda$ should not be selected too low or too high as either too little or too much intensification can hurt search performance.
+
+# 5 CONCLUSION
+
+We presented a simple technique that can be used to extend ML-based construction heuristics by an extensive search. Our proposed modification of active search fine tunes a small subset of (model) parameters to a single instance at test time. We evaluate three example implementations of EAS that all result in significantly improved model performance in both testing and generalization experiments on three different, difficult combinatorial optimization problems. Our approach of course comes with some key limitations. Search requires time, thus for applications needing extremely fast (or practically instant) solutions, greedy construction remains a better option. Furthermore, while the problems we experiment on have the same computational complexity as real-world optimization problems, additional work may be needed to handle complex side constraints as often seen in industrial problems.
+
+# ACKNOWLEDGMENTS
+
+The computational experiments in this work have been performed using the Bielefeld GPU Cluster.
+
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+Liang Xin, Wen Song, Zhiguang Cao, and Jie Zhang. Step-wise deep learning models for solving routing problems. IEEE Transactions on Industrial Informatics, 17(7):4861–4871, 2021.
+
+Cong Zhang, Wen Song, Zhiguang Cao, Jie Zhang, Puay Siew Tan, and Xu Chi. Learning to dispatch for job shop scheduling via deep reinforcement learning. In Advances in Neural Information Processing Systems, volume 33, pp. 1621–1632, 2020.
+
+# A EXPERIMENTS FOR MORE REALISTIC CVRP INSTANCES
+
+We provide results from additional experiments for the CVRP on more realistic instances to show that our approach is effective at solving instances with a wide range of structures. Our EAS methods are implemented in the same way as in Section 4.2
+
+Setup We evaluate EAS on 9 instance sets from Hottung & Tierney (2020) (consisting of 20 instances each) that have been generated based on the instances from Uchoa et al. (2017) performing 3 runs per instance. The characteristics of the instances vary significantly between sets, but all instances in the same set have been sampled from an identical distribution. For each instance set we train a new model for 3 weeks on a separate, corresponding training set. For testing, we run all (efficient) active search approaches for 200 iterations using the newly trained models. Additionally, we test the generalization performance by solving all instance sets with EAS-Lay using the CVRP model of Section 4.2 that has been trained on the uniform instances (with $n = 1 0 0$ ) from Kool et al. (2019) and call this Lay\*. We only evaluate the generalization performance of EAS-Lay (the best performing EAS approach from Section 4.2) to keep the computational costs low. In all experiments, we use hyperparameters tuned for the uniform CVRP instances. We compare to NLNS, LKH3 and the state-of-the-art unified hybrid genetic search (GS) from Vidal et al. (2014). As is standard in the operations research literature, we round the distances between customers to the nearest integer. Furthermore, we solve instances sequentially and not in batches of different instances. However, to make better use of the available GPU memory, we solve up to 10 copies of the same instance in parallel for the EAS approaches and for POMO with sampling. The best solution found so far is shared between all runs, which has an impact on the imitation learning loss $\mathcal { L } _ { I L }$ for EAS-Emb and EAS-Lay. For EAS-Tab we set $\tilde { Q } = ( 1 - \bar { \beta } ) \cdot Q + \beta \cdot Q ^ { g l o b }$ , where $Q ^ { g l o b }$ is the lookup table for the best solution over all runs and $\beta$ is linearly increased from 0 to 1 over the course of the search.
+
+Results Table 4 shows the gap to the unified hybrid genetic search and the average runtime per instance for all methods. For EAS-Lay we report the performance of the instance set specific models and additionally the generalization performance when using the model trained on uniform CVRP instances (with $n = 1 0 0$ ). The later results are marked with a star. EAS-Emb and EAS-Lay both find better solution than NLNS and LKH3 on 8 out of the 9 instance sets. EAS-Tab outperforms NLNS and LKH3 on all but two instance sets. As a side note, we have found that the original active search (AS) performs surprisingly well, outperforming LKH3 on 3 instance sets, even though it still cannot surpass our newly proposed EAS methods. The version of EAS-Lay (marked with a star) that uses the model trained on uniform instances with $n = 1 0 0$ performs surprisingly well with gaps between $0 . 2 6 \%$ to $4 . 0 8 \%$ to the GS.
+
+Table 4: Results for the CVRP on the instance sets from Hottung & Tierney (2020).   
+
+<table><tr><td></td><td></td><td colspan="4">Gap to GS in %</td><td colspan="2">POMO</td><td colspan="5"> Avg. Runtime in minutes</td></tr><tr><td>Inst.</td><td>n</td><td>POMO Sam. AS</td><td>POMO-EAS Emb Lay Lay* Tab</td><td></td><td>[NLNS LKH|</td><td></td><td></td><td></td><td>POMO-EAS Sam. AS Emb Lay Lay* TabNLNS LKH GS</td><td></td><td></td><td></td></tr><tr><td>XE 1</td><td>100</td><td></td><td></td><td></td><td></td><td>2.12</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>XE</td><td>128</td><td>0.86 0.65 0.76 0.80</td><td>10.23 0.26 0.260.25</td><td>0.61 0.31| 0.26 0.29</td><td>0.32 0.44</td><td>0.9 0.54</td><td>1.3 1.6</td><td>1.3 1.8</td><td>1.4 2.0</td><td>1.5 0.9 2.1</td><td>3.2</td><td>6.2 0.6 2.0</td></tr><tr><td>XE</td><td>180</td><td>0.51 0.20</td><td>0.09 0.09</td><td>0.54 0.13</td><td>0.58 0.16</td><td>1.3 2.5</td><td>2.2</td><td>3.3</td><td>3.7</td><td>24 3.8</td><td>3.2 3.2</td><td>1.2 1.1 1.4</td></tr><tr><td>3.57 XE</td><td>199</td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.2</td><td>4.7</td><td>3.5</td><td>3.2</td><td>3.6 2.4</td></tr><tr><td>XE 9</td><td>213</td><td>1.50 0.88 1.96 1.30</td><td>0.37 0.45</td><td>1.29 0.80 4.08 0.83</td><td>2.03 2.26</td><td>0.72 3.3 1.09</td><td>2.4</td><td></td><td>5.2</td><td>4.9 5.4</td><td>10.2</td><td>1.1 2.4</td></tr><tr><td>XE 11</td><td>236</td><td>1.42 1.22</td><td>0.64 0.71 0.82 0.84</td><td>1.76 0.94</td><td>0.65</td><td>3.7 0.78 4.0</td><td>2.5 2.7</td><td>4.6 5.2</td><td>5.1</td><td>3.9 5.6 4.8</td><td>10.2</td><td>1.1 3.2</td></tr><tr><td>3 XE</td><td></td><td>1.40 0.88</td><td>0.38 0.56</td><td>2.83 0.80</td><td>0.82</td><td>1.55 7.1</td><td>3.5</td><td>8.7</td><td>6</td><td></td><td>10.2</td><td>5.7 3.6</td></tr><tr><td>XE 15</td><td>268</td><td>1.81 2.17</td><td>0.850.96</td><td>2.51 1.27</td><td>1.81</td><td>1.32 7.3</td><td>3.3</td><td>8.8</td><td></td><td>76</td><td>10.3</td><td>5.8 5.5</td></tr><tr><td>XE 17</td><td>297</td><td>1.66 0.97</td><td>0.44 0.65</td><td>2.15 0.92</td><td>1.41</td><td>1.23 8.9</td><td>3.8</td><td>8.8</td><td>6.8</td><td>9.4</td><td>10.3</td><td>2.5 4.2</td></tr></table>
+
+# B ABLATION STUDY: ACTIVE SEARCH LOSS
+
+We evaluate if applying the imitation learning loss component used by EAS-Emb and EAS-Lay to the original active search can significantly improve the performance. To this end, we solve all test instances using active search with and without the imitation learning loss component. Note that the hyperparameters for each approach have been tuned independently on separate validation set instances. Table 5 shows the results. We observe no significant impact of the imitation learning loss $\mathcal { L } _ { \pi }$ on the performance of active search. This means that active search with imitation learning loss is not competitive with EAS-Lay and EAS-Emb across all problems, even when sharing the same loss function.
+
+![](images/41ab96920d4ee9221016be5023bf0d3e37f416c536ccee2088f21bf8bca9448d.jpg)  
+Figure 4: Influence of $\sigma$ on the solution quality for EAS-Tab
+
+# C PARAMETER SWEEP: EAS-TAB INTENSIFICATION
+
+We investigate the impact of the hyperparameter $\sigma$ on EAS-Tab, which controls the degree of exploitation of the search. By setting $\sigma$ to zero (or very small values) we essentially disable the lookup table, thus examining its impact on the search. We again solve all three problems with different values of $\sigma$ on a subset of test instances. We fix $\alpha$ independently based on tuning on a separate set of validation instances. Figure 4 provides the results for adjusting $\sigma$ . For all three problems, $\sigma = 1 0$ provides the best trade-off between exploration and exploitation. Note that low $\sigma$ values (which reduce the impact of the lookup table updates) hurt performance, meaning that the table based adjustments are effective in all cases.

parse/dev/nO5caZwFwYu/nO5caZwFwYu_content_list.json

@@ -0,0 +1,1454 @@
+[
+    {
+        "type": "text",
+        "text": "EFFICIENT ACTIVE SEARCH FOR COMBINATORIAL OPTIMIZATION PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            709,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "André Hottung Bielefeld University, Germany andre.hottung@uni-bielefeld.de ",
+        "bbox": [
+            184,
+            170,
+            480,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yeong-Dae Kwon Samsung SDS, Korea y.d.kwon@samsung.com ",
+        "bbox": [
+            614,
+            170,
+            813,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kevin Tierney Bielefeld University, Germany kevin.tierney@uni-bielefeld.de ",
+        "bbox": [
+            183,
+            233,
+            480,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, numerous machine learning based methods for combinatorial optimization problems have been proposed that learn to construct solutions in a sequential decision process via reinforcement learning. While these methods can be easily combined with search strategies like sampling and beam search, it is not straightforward to integrate them into a high-level search procedure offering strong search guidance. Bello et al. (2016) propose active search, which adjusts the weights of a (trained) model with respect to a single instance at test time using reinforcement learning. While active search is simple to implement, it is not competitive with state-of-the-art methods because adjusting all model weights for each test instance is very time and memory intensive. Instead of updating all model weights, we propose and evaluate three efficient active search strategies that only update a subset of parameters during the search. The proposed methods offer a simple way to significantly improve the search performance of a given model and outperform state-of-the-art machine learning based methods on combinatorial problems, even surpassing the well-known heuristic solver LKH3 on the capacitated vehicle routing problem. Finally, we show that (efficient) active search enables learned models to effectively solve instances that are much larger than those seen during training. ",
+        "bbox": [
+            233,
+            343,
+            766,
+            579
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            607,
+            336,
+            622
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In recent years, a wide variety of machine learning (ML) based methods for combinatorial optimization problems have been proposed (e.g., Kool et al. (2019); Hottung et al. (2020)) . While early approaches failed to outperform traditional operations research methods, the gap between handcrafted and learned heuristics has been steadily closing. However, the main potential of ML-based methods lies not only in their ability to outperform existing methods, but in automating the design of customized heuristics in situations where no handcrafted heuristics have yet been developed. We hence focus on developing approaches that require as little additional problem-specific knowledge as possible. ",
+        "bbox": [
+            174,
+            638,
+            825,
+            750
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing ML based methods for combinatorial optimization problems can be classified into construction methods and improvement methods. Improvement methods search the space of complete solutions by iteratively refining a given start solution. They allow for a guided exploration of the search space and are able to find high-quality solutions. However, they usually rely on problemspecific components. In contrast, construction methods create a solution sequentially starting from an empty solution (i.e., they consider a search space consisting of incomplete solutions). At test time, they can be used to either greedily construct a single solution or to sample multiple solutions from the probability distribution encoded in the trained neural network. Furthermore, the sequential solution generation process can be easily integrated into a beam search without requiring any problem-specific components. However, search methods like sampling and beam search offer no (or very limited) search guidance. Additionally, these methods do not react towards the solutions seen so far, i.e., the underlying distribution from which solutions are sampled is never changed throughout the search. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bello et al. (2016) propose a generic search strategy called active search that allows an extensive, guided search for construction methods without requiring any problem specific components. Active search is an iterative search method that at each iteration samples solutions for a single test instance using a given model and then adjusts the parameters of that model with the objective to increase the likelihood of generating high-quality solutions in future iterations. They report improved performance over random sampling when starting the search from an already trained model. Despite promising results, active search has not seen adaption in the literature. The reason for this is its resource requirements, as adjusting all model parameters separately for each test instance is very time intensive, especially compared to methods that can sample solutions to multiple different instances in one batch. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We extend the idea of active search as follows. (1) We propose to only adjust a subset of (model) parameters to a single instance during the search, while keeping all other parameters fixed. We show that this efficient active search (EAS) drastically reduces the runtime of active search without impairing the solution quality. (2) We implement and evaluate three different implementations of EAS and show that all offer significantly improved performance over pure sampling approaches. ",
+        "bbox": [
+            174,
+            236,
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+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In our EAS implementations, the majority of (model) parameters are not updated during the search, which drastically reduces the runtime, because gradients only need to be computed for a subset of model weights, and most operations can be applied identically across a batch of different instances. Furthermore, we show that for some problems, EAS finds even better solutions than the original active search. All EAS implementations can be easily applied to existing ML construction methods. ",
+        "bbox": [
+            174,
+            313,
+            825,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed EAS approaches on the traveling salesperson problem (TSP), the capacitated vehicle routing problem (CVRP) and the job shop scheduling problem (JSSP). For all problems, we build upon already existing construction approaches that only offer limited search capabilities. In all experiments, EAS leads to significantly improved performance over sampling approaches. For the CVRP and the JSSP, the EAS approaches outperform all state-of-the-art ML based approaches, and even the well-known heuristic solver LKH3 for the CVRP. Furthermore, EAS approaches assists in model generalization, resulting in drastically improved performance when searching for solutions to instances that are much larger than the instances seen during model training. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            501
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 LITERATURE REVIEW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            535,
+            383,
+            551
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Construction methods Hopfield (1982) first used a neural network (a Hopfield network) to solve small TSP instances with up to 30 cities. The development of recent neural network architectures has paved the way for ML approaches that are able to solve large instances. The pointer network architecture proposed by Vinyals et al. (2015) efficiently learns the conditional probability of a permutation of a given input sequence, e.g., a permutation of cities for a TSP solution. The authors solve TSP instances with up to 50 cities via supervised learning. Bello et al. (2016) report that training a pointer network via actor-critic RL instead results in a better performance on TSP instances with 50 and 100 cities. Furthermore, graph neural networks are used to solve the TSP, e.g., a graph embedding network in Khalil et al. (2017) and a graph attention network in Deudon et al. (2018). ",
+        "bbox": [
+            174,
+            575,
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+            702
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The first applications of neural network based methods to the CVRP are reported by Nazari et al. (2018) and Kool et al. (2019). Nazari et al. (2018) propose a model with an attention mechanism and a recurrent neural network (RNN) decoder that can be trained via actor-critic RL. Kool et al. (2019) propose an attention model that uses an encoder that is similar to the encoder used in the transformer architecture Vaswani et al. (2017). Peng et al. (2019) and Xin et al. (2021) extend the attention model to update the node embeddings throughout the search, resulting in improved performance at the cost of longer runtimes for the CVRP. Falkner & Schmidt-Thieme (2020) propose an attention-based model that constructs tours in parallel for the CVRP with time windows. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            819
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While ML-based construction methods have mainly focused on routing problems, there are some notable exceptions. For example, Khalil et al. (2017) use a graph embedding network approach to solve the minimum vertex cover and the maximum cut problems (in addition to the TSP). Zhang et al. (2020) propose a graph neural network based approach for the job shop scheduling problem (JSSP). Li et al. (2018) use a guided tree search enhanced ML approach to solve the maximal independent set, minimum vertex cover, and the maximal clique problems. For a more detailed review of ML methods on different combinatorial optimization problems, we refer to Vesselinova et al. (2020). ",
+        "bbox": [
+            174,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While most approaches construct routing problem solutions autoregressively, some approaches predict a heat-map that describes which edges will likely be part of a good solution. The heat-map is then used in a post-hoc search to construct solutions. Joshi et al. (2019) use a graph convolutional network to create a heat-map and a beam search to search for solutions. Similarly, Fu et al. (2020) use a graph convolutional residual network with Monte Carlo tree search to solve large TSP instances. Kool et al. (2021) use the model from Joshi et al. (2019) to generate the heat-map and use it to search for solution to TSP and CVRP instances with a dynamic programming based approach. ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Improvement methods Improvement methods integrate ML based methods into high-level search heuristics or try to learn improvement operators directly. In general, they often invest more time into solving an instance than construction based methods (and usually find better solutions). Chen & Tian (2019) propose an approach that iteratively changes a local part of the solution. At each iteration, the trainable region picking policy selects a part of the solution that should be changed and a trainable rule picking policy selects an action from a given set of possible modification operations. Hottung & Tierney (2020) propose a method for the CVRP that iteratively destroys parts of a solution using predefined, handcrafted operators and then reconstructs them with a learned repair operator. Wu et al. (2021) and de O. da Costa et al. (2020) propose to use RL to pick an improving solution from a specified local neighborhood (e.g., the 2-Opt neighborhood) to solve routing problems. Hottung et al. (2021) learn a continuous representation of discrete routing problem solutions using conditional variational autoencoders and search for solutions using a generic, continuous optimizer. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 SOLVING COMBINATORIAL OPTIMIZATION PROBLEMS WITH EAS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            404,
+            743,
+            417
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose three EAS implementations that adjust a small subset of (model) parameters in an iterative search process. Given an already trained model, we investigate adjusting (1) the normally static embeddings of the problem instance that are generated by the encoder model, (2) the weights of additional instance-specific residual layers added to the decoder, and (3) the parameters of a lookup table that directly affect the probability distribution returned by model. In each iteration, multiple solutions are sampled for one instance and the dynamic (model) parameters are adjusted with the goal of increasing the probability of generating high quality solutions (as during model training). This allows the search to sample solutions of higher quality in subsequent iterations, i.e., the search can focus on the more promising areas of the search space. Once a high-quality solution for an instance is found, the adjusted parameters are discarded, so that the search process can be repeated on other instances. All strategies efficiently generate solutions to a batch of instances in parallel, because the network layers not updated during the search are applied identically to all instances of the batch. ",
+        "bbox": [
+            174,
+            438,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Background RL based approaches for combinatorial problems aim to learn a neural network based model $p _ { \\theta } ( \\pi | l )$ with weights $\\theta$ that can be used to generate a solution $\\pi$ given an instance l. State-ofthe-art approaches usually use a model that consists of an encoder and a decoder unit. The encoder usually creates static embeddings $\\omega$ that describe the instance $l$ using a computationally expensive encoding process (e.g., Kool et al. (2019); Kwon et al. (2020)). The static embeddings are then used to autoregessively construct solutions using the decoder over $T$ time steps. At each step $t$ , the decoder $q _ { \\phi } ( a | s _ { t } , \\bar { \\omega } )$ , with weights $\\phi \\subset \\theta$ , outputs a probability value for each possible action $a$ in the state $s _ { t }$ (e.g., for the TSP, each action corresponds to visiting a different city next). The starting state $s _ { 1 }$ describes the problem instance $l$ (e.g., the positions of the cities for the TSP and the starting city) and the state $s _ { t + 1 }$ is obtained by applying the action $a _ { t }$ selected at time step $t$ to the state $s _ { t }$ . The (partial) solution $\\pi _ { t }$ is defined by the sequence of selected actions $a _ { 1 } , a _ { 2 } , \\ldots , a _ { t }$ . Once a complete solution, $\\pi _ { T }$ , fulfilling all constraints of the problem is constructed, the objective function value $\\mathbf { \\bar { \\boldsymbol { C } } } ( \\pi , l )$ of the solution can be computed (e.g., the tour length for the TSP). ",
+        "bbox": [
+            174,
+            611,
+            825,
+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows the solution generation for a TSP instance with a model that uses static embeddings. The static embeddings $\\omega$ are used at each decoding step to generate a probability distribution over all possible next actions and the selected action is provided to the decoder in the next decoding step. During testing, solutions can be constructed by either selecting actions greedily or by sampling each action according to $q _ { \\phi } ( a | s _ { t } , \\omega )$ . Since the static embeddings are not updated during solution generation they only need to be computed once per instance, which allows to quickly sample multiple solutions per instance. We note that not all models use static embeddings. Some approaches update all instance embeddings after each action (e.g., Zhang et al. (2020)), which allows the embeddings to contain information on the current solution state $s _ { t }$ . ",
+        "bbox": [
+            174,
+            797,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/3abde8df1e97587e52664ac4987fda3a29e2025873cd251133e478eda9740eb0.jpg",
+        "image_caption": [
+            "Figure 1: Sampling a solution for the TSP with a model $p _ { \\theta } ( \\pi | l )$ that uses static instance embeddings. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            303,
+            98,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 EMBEDDING UPDATES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            237,
+            369,
+            251
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our first proposed strategy, called EAS-Emb, updates the embeddings $\\omega$ generated by an encoder using a loss function consisting of an RL component $\\mathcal { L } _ { R L }$ and an imitation learning $\\left( \\operatorname { I L } \\right)$ component $\\mathcal { L } _ { I L }$ . The loss $\\mathcal { L } _ { R L }$ is based on REINFORCE (Williams, 1992) and is the expected cost of the generated solutions, $\\mathbb { E } \\left[ C ( \\pi ) \\right]$ . We aim to adjust the embedding parameters to increase the likelihood of generating solutions with lower costs (e.g., a shorter tour length for the TSP). The loss $\\mathcal { L } _ { I L }$ is the negation of the log-probability of (re-)generating the best solution seen so far. We adjust the embedding parameters to increase this probability. ",
+        "bbox": [
+            173,
+            263,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "More formally, for an instance $l$ we generate the embeddings $\\omega$ using a given encoder. Based on $\\omega$ we can (repeatedly) sample a solution $\\pi$ whose cost is $C ( \\pi )$ . A subset of the embeddings $\\hat { \\omega } \\subseteq \\omega$ is adjusted to minimize $\\mathcal { L } _ { R L }$ using the gradient ",
+        "bbox": [
+            174,
+            368,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/189abc5d6fb79dfe56b65ebb32b7d500676d329066d509c67e35215326a97182.jpg",
+        "text": "$$\n\\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { R L } ( \\hat { \\omega } ) = \\mathbb { E } _ { \\boldsymbol \\pi } \\left[ ( C ( \\boldsymbol \\pi ) - b _ { \\circ } ) \\nabla _ { \\hat { \\omega } } \\log q _ { \\phi } ( \\boldsymbol \\pi \\mid \\hat { \\omega } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { q _ { \\phi } ( \\pi \\mid \\hat { \\omega } ) \\equiv \\prod _ { t = 1 } ^ { T } q _ { \\phi } ( a _ { t } \\mid s _ { t } , \\hat { \\omega } ) } \\end{array}$ , and $b _ { \\circ }$ is a baseline (we use the baseline proposed in Kwon et al. (2020) for our experiments). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the second loss $\\mathcal { L } _ { I L }$ , let $\\bar { \\pi }$ be the best solution found so far for the instance $l$ , that consists of the actions $\\bar { a } _ { 1 } , \\dots , \\bar { a } _ { T }$ . We use teacher forcing to make the decoder $q _ { \\phi } ( \\cdot | s _ { t } , \\hat { \\omega } )$ generate the solution $\\bar { \\pi }$ , during which we obtain the probability values associated with the actions $\\bar { a } _ { 1 } , \\dots , \\bar { a } _ { T }$ . We increase the log-likelihood of generating $\\bar { \\pi }$ by adjusting $\\hat { \\omega }$ using the gradient ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/02db29bd1a00a2d24674766123c6311de2c249ba381d50637f18cb95eaf93a9c.jpg",
+        "text": "$$\n\\nabla _ { \\boldsymbol { \\hat { \\omega } } } \\mathcal { L } _ { I L } ( \\boldsymbol { \\hat { \\omega } } ) = - \\nabla _ { \\boldsymbol { \\hat { \\omega } } } \\log q _ { \\phi } ( \\bar { \\pi } \\mid \\boldsymbol { \\hat { \\omega } } ) \\equiv - \\nabla _ { \\boldsymbol { \\hat { \\omega } } } \\log \\prod _ { t = 1 } ^ { T } q _ { \\phi } ( \\bar { a } _ { t } | s _ { t } , \\boldsymbol { \\hat { \\omega } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The gradient of the overall loss $\\mathcal { L } _ { R I L }$ is defined as $\\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { R I L } ( \\hat { \\omega } ) = \\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { R L } ( \\hat { \\omega } ) + \\lambda \\cdot \\nabla _ { \\hat { \\omega } } \\mathcal { L } _ { I L } ( \\hat { \\omega } )$ , where $\\lambda$ is a tunable parameter. If a high value for $\\lambda$ is selected, the search focuses on generating solutions that are similar to the incumbent solution. This accelerates the convergence of the search policy, which is useful when the number of search iterations is limited. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We note that both decoding processes required for RL and $\\mathrm { I L }$ can be carried out in parallel, using the same forward pass through the network. Furthermore, only the parameters $\\hat { \\omega }$ are instance specific, while all other model parameters are identical for all instances. This makes parallelization of multiple instances in a batch more efficient both in time and memory. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 ADDED-LAYER UPDATES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We next propose EAS-Lay, which adds an instance-specific residual layer to a trained model. During the search, the weights in the added layer are updated, while the weights of all other original layers are held fixed. We use both RL and $\\mathrm { I L }$ , similarly to EAS-Emb in Section 3.1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We formalize EAS-Lay as follows. For each instance $l$ we insert a layer ",
+        "bbox": [
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+    {
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+        "img_path": "images/a4efe175befc0bb45d9375e239084557e813c70bf10e86d2d14bc5459fc391f8.jpg",
+        "text": "$$\n\\operatorname { L } ^ { \\star } ( h ) = h + ( ( \\operatorname { R e L u } ( h W ^ { 1 } + b ^ { 1 } ) W ^ { 2 } + b ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "into the given decoder $q _ { \\phi }$ , resulting in a slightly modified model $\\tilde { q } _ { \\phi , \\psi }$ , where $\\psi = \\{ W ^ { 1 } , b ^ { 1 } , W ^ { 2 } , b ^ { 2 } \\}$ . The layer takes in the input $h$ and applies two linear transformations with a ReLu activation function in between. The weight matrices $W ^ { \\bar { 1 } }$ and $W ^ { 2 }$ and the bias vectors $b ^ { 1 }$ and $b ^ { 2 }$ are adjusted throughout the search via gradient descent. The weights in the matrix $W ^ { 2 }$ and the vector $b ^ { 2 }$ are initialized to zero so that the added layer does not affect the output of the model during the first iteration of the search. The gradient for $\\mathcal { L } _ { R L }$ is given as ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1609797b2d752b07426872754f67e6261c30974ae005d79b301a2100da2fcbd.jpg",
+        "text": "$$\n\\nabla _ { \\psi } \\mathcal { L } _ { R L } ( \\psi ) = \\mathbb { E } _ { \\pi } \\big [ ( C ( \\pi ) - b _ { \\circ } ) \\nabla _ { \\psi } \\log \\tilde { q } _ { \\phi , \\psi } ( \\pi ) \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            137,
+            660,
+            157
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $\\begin{array} { r } { \\tilde { q } _ { \\phi , \\psi } ( \\pi ) \\equiv \\prod _ { t = 1 } ^ { T } \\tilde { q } _ { \\phi , \\psi } ( a _ { t } \\mid s _ { t } , \\omega ) } \\end{array}$ , and $b _ { \\circ }$ is a baseline. The gradient for $\\mathcal { L } _ { I L }$ is defined similarly. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the majority of the network operations are not instance specific. They can be applied identically to all instances running in parallel as a batch, resulting in significantly lower runtime during search. The position at which the new layer is inserted has an impact on the performance of EAS-Lay, and identifying the best position usually requires testing. In general, the memory requirement of this approach can be reduced by inserting the additional layer closer towards the output layer of the network. This decreases the number of layers to be considered during backpropagation. We noticed for transformer-based architectures that applying the residual layer $\\mathrm { L } ^ { \\star } ( \\cdot )$ to the query vector $q$ before it is passed to the single attention head usually results in a good performance. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 TABULAR UPDATES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "EAS-Emb and EAS-Lay require significantly less memory per instance than the original active search. However, they still need to store many gradient weights associated with multiple layers for the purpose of backpropagation. This significantly limits the number of solutions one can generate in parallel. We hence propose EAS-Tab, which does not require backpropagation, but instead uses a simple lookup table to modify the policy of the given model. For each action at a given state, the table provides a guide on how to change its probability, so that the sampled solution has a higher chance at being similar to the best solution found in the past. ",
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+    {
+        "type": "text",
+        "text": "Formally, at each step $t$ during the sequential generation of a solution, we redefine the probability of selecting action $a _ { t }$ in the state $s _ { t }$ as $q _ { \\phi } ( a | \\bar { s _ { t } } , \\omega ) ^ { \\alpha } \\cdot Q _ { g ( s _ { t } , a _ { t } ) }$ and renormalize over all possible actions using the softmax function. Here, $\\alpha$ is a hyperparameter, and $g$ is a function that maps each possible state and action pair to an entry in the table $Q$ . The network parameters $\\theta$ remain unchanged, resulting in fast and memory efficient solution generation. The hyperparameter $\\alpha$ is similar to the temperature value proposed in Bello et al. (2016) and modifies the steepness of the probability distribution returned by the model (lower values increase the exploration of the search). During search, the table $Q$ is updated with the objective of increasing the quality of the generated solutions. More precisely, after each iteration, $Q$ is updated based on the best solution $\\bar { \\pi }$ found so far consisting of the actions $\\bar { a } _ { 1 } , \\dots , \\bar { a } _ { T }$ at states $\\bar { s } _ { 1 } , \\dots , \\bar { s } _ { T }$ , respectively, with ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/dafda474a5834db9ed417690205a5bb2e7098e1162c71613804c95c313b3d37d.jpg",
+        "text": "$$\nQ _ { g ( s _ { t } , a _ { t } ) } = \\left\\{ \\begin{array} { l l } { \\operatorname* { m a x } ( 1 , \\frac { \\sigma } { q _ { \\phi } ( a | s _ { t } , \\omega ) ^ { \\alpha } } ) , } & { \\mathrm { i f } g ( s _ { t } , a _ { t } ) \\in \\{ g ( \\bar { a } _ { 1 } , \\bar { s } _ { 1 } ) , \\dots , g ( \\bar { a } _ { T } , \\bar { s } _ { T } ) \\} } \\\\ { 1 , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The hyperparameter $\\sigma$ defines the degree of exploitation of the search. If a higher value of $\\sigma$ is used, the probabilities for actions that generate the incumbent solution are increased. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast to embedding or added-layer updates, this EAS method requires deeper understanding of the addressed combinatorial optimization problem to design the function $g ( s _ { t } , a _ { t } )$ . For example, for the TSP with $n$ nodes we use a table $Q$ of size $n \\times n$ in which each entry $Q _ { i , j }$ corresponds to a directed edge $e _ { i , j }$ of the problem instance. The probability increases for the same directed edge that was used in the incumbent solution. This definition of $g ( s _ { t } , a _ { t } )$ effectively ignores the information on all the previous visits stored in state $s _ { t }$ , focusing instead on the current location (city) in choosing the next move. We note that this EAS approach is similar to the ant colony optimization algorithm (Dorigo et al., 2006), which has been applied to a wide variety of combinatorial optimization problems. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate all EAS strategies using existing, state-of-the-art RL based methods for three different combinatorial optimization problems. For the first two, the TSP and the CVRP, we implement EAS for the POMO approach (Kwon et al., 2020). For the third problem, the JSSP, we use the L2D method from Zhang et al. (2020). We extend the code made available by the authors of POMO (MIT license) and L2D (no license) with our EAS strategies to ensure a fair evaluation. Note that we only make minor modifications to these methods, and we use the models trained by the authors when available. ",
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+        "type": "text",
+        "text": "We run all experiments on a GPU cluster using a single Nvidia Tesla V100 GPU and a single core of an Intel Xeon 4114 CPU at $2 . 2 \\ : \\mathrm { G H z }$ for each experiment. Our source code is available at https://github.com/ahottung/EAS. We use the Adam optimizer (Kingma & Ba, 2014) for all EAS approaches. The hyperparameters $\\lambda , \\sigma , \\alpha$ , and the learning rate for the optimizer are tuned via Bayesian optimization using scikit-optimize (Head et al., 2020) on separate validation instances, which are sampled from the same distribution as the test instances. The hyperparameters are not adjusted for larger instances used to evaluate the generalization performance. ",
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+    {
+        "type": "text",
+        "text": "4.1 TSP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The TSP is a well-known routing problem involving finding the shortest tour between a set of $n$ nodes (i.e., cities) that visits each node exactly once and returns to the starting node. We assume that the distance matrix obeys the triangle inequality. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation POMO uses a model that is very similar to the AM model from Kool et al. (2019). The model generates instance embeddings only once per instance and does not update them during construction. The probability distribution over all actions are generated by a decoder, whose last layer is a single-headed attention layer. This last layer calculates the compatibility of a query vector $q$ to the key vector $k _ { i }$ for each node $i$ . In this operation, the key vector $k _ { i }$ is an embedding that has been computed separately, but identically for each input (i.e., node $i$ ) during the instance encoding process. For EAS-Emb, we only update the set of single-head keys $k _ { i }$ $( i = 1 , \\ldots , n )$ . For EAS-Lay we apply the residual layer $\\mathrm { L } ^ { \\star } ( \\cdot )$ described in Equation 3 to the query vector $q$ before it is passed to the single attention head. For EAS-Tab, we use a table $Q$ of size $n \\times n$ and the mapping function $g ( s _ { t } , a _ { t } )$ such that each entry $Q _ { i , j }$ corespondents to a directed edge $e _ { i , j }$ of the problem instance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Setup We use the 10,000 TSP instances with $n = 1 0 0$ from Kool et al. (2019) for testing and three additional sets of 1,000 instances to evaluate generalization performance. We evaluate the EAS approaches against just using POMO with greedy action selection, random sampling, and active search as in Bello et al. (2016). In all cases, we use the model trained on instances with $n = 1 0 0$ made available by the POMO authors. For greedy action selection, POMO generates $8 \\cdot n$ solutions for an instance of size $n$ (using 8 augmentations and $n$ different starting cities). In all other cases, we generate $2 0 0 \\cdot 8 \\cdot n$ solutions per instance (over the course of 200 iterations for the (E)AS approaches). The batch size (the number of instances solved in parallel) is selected for each method individually to fully utilize the available GPU memory. We compare to the exact solver Concorde (Applegate et al., 2006), the heuristic solver LKH3 (Helsgaun, 2017), the graph convolutional neural network with beam search (GCN-BS) from Joshi et al. (2019), the 2-Opt based deep learning (2-Opt-DL) approach from de O. da Costa et al. (2020), the learning improvement heuristics (LIH) method from Wu et al. (2021), the conditional variational autoencoder (CVAE-Opt) approach (Hottung et al., 2021), and deep policy dynamic programming (DPDP) (Kool et al., 2021). ",
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+    {
+        "type": "text",
+        "text": "Results Table 1 shows the average costs, average gap and the total runtime (wall-clock time) for each instance set. The exact solver Concorde performs best overall, as it is a highly specialized TSP solver. Of the POMO-based approaches, the original active search offers the best gap to optimality, but requires 5 days of runtime. EAS significantly lowers the runtime while the gap is only marginally larger. DPDP performs best among ML-based approaches. However, DPDP relies on a handcrafted and problem-specific beam search, whereas EAS methods are completely problem-independent. On the larger instances, EAS significantly improves generalization performance, reducing the gap over sampling by up to $3 . 6 \\mathrm { x }$ . We also evaluate active search using the imitation learning loss, but observe no impact on the search performance (see Appendix B). ",
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+    {
+        "type": "table",
+        "img_path": "images/b850778457b82bd499ebcca9bc2866dca78cef43add4b0a344aaaf470b378364.jpg",
+        "table_caption": [
+            "Table 1: Results for the TSP "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Testing (10k inst.)</td><td colspan=\"10\">Generalization (1k instances)</td></tr><tr><td colspan=\"3\">n=100</td><td colspan=\"3\">n=125</td><td colspan=\"3\">n=150</td><td colspan=\"3\"></td><td>n = 200</td></tr><tr><td>Method</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>Gap</td><td></td><td>Time</td><td>Obj.</td><td>Gap</td><td>Time</td></tr><tr><td>Concorde</td><td></td><td>[7.765 0.000%</td><td></td><td></td><td>82M|8.583 0.000%</td><td></td><td></td><td>12M|9.346 0.000%</td><td></td><td></td><td>17M|10.687 0.000%</td><td></td><td>31M</td></tr><tr><td>LKH3</td><td>7.765</td><td>0.000%</td><td>8H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.5830.000%73M9.3460.000% 99M10.6870.000%</td><td>3H</td></tr><tr><td>GCN-BS</td><td>7.87</td><td>1.39%</td><td>40M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>-</td></tr><tr><td>2-Opt-DL</td><td>7.83</td><td>0.87%</td><td>41M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>LIH</td><td>7.87</td><td>1.42%</td><td>2H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>1</td><td>0.343%</td><td>6D</td><td></td><td>8.646 0.736%</td><td>21H</td><td></td><td>9.482 1.454%</td><td></td><td>30H</td><td></td><td></td><td></td></tr><tr><td>DPDP</td><td>7.765</td><td>0.004%</td><td>2H</td><td></td><td>8.589 0.070%</td><td>31M</td><td></td><td>9.4340.942%</td><td></td><td>44M</td><td></td><td>11.154 4.370%</td><td>74M</td></tr><tr><td>Greedy</td><td>7.776</td><td>0.146%</td><td>1M|</td><td>8.607</td><td>0.278%</td><td></td><td>&lt;1M|9.397</td><td></td><td>0.542%</td><td>&lt;1M|</td><td>10.843</td><td>1.457%</td><td>1M</td></tr><tr><td>Sampling</td><td>7.770</td><td>0.074%</td><td>4H</td><td>8.595</td><td>0.145%</td><td>45M</td><td></td><td>9.378</td><td>0.334%</td><td>78M</td><td></td><td>10.8381.416%</td><td>3H</td></tr><tr><td>Active S.</td><td></td><td>7.768 0.046%</td><td>5D</td><td>8.591</td><td>0.095%</td><td>15H</td><td>9.364</td><td></td><td>0.192%</td><td>19H</td><td>10.735</td><td>50.447%</td><td>24H</td></tr><tr><td>0 EAS-Emb</td><td></td><td>7.769 0.063%</td><td>5H</td><td>8.591</td><td>0.092%</td><td>57M</td><td></td><td>9.363</td><td>0.174%</td><td>2H</td><td>10.730 0.400%</td><td></td><td>4H</td></tr><tr><td>P EAS-Lay</td><td></td><td>7.769 0.053%</td><td>7H</td><td>8.591</td><td>0.089%</td><td>74M</td><td></td><td>9.363 0.176%</td><td></td><td>2H</td><td>10.737 0.471%</td><td></td><td>4H</td></tr><tr><td>EAS-Tab</td><td></td><td>7.768 0.048%</td><td>5H</td><td>8.591</td><td>0.091%49M</td><td></td><td></td><td>9.3650.196%</td><td></td><td>1H</td><td>10.756 0.650%</td><td></td><td>3H</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "4.2 CVRP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The goal of the CVRP is to find the shortest routes for a set of vehicles with limited capacity that must deliver goods to a set of $n$ customers. We again use the POMO approach as a basis for our EAS strategies. As is standard in the ML literature, we evaluate all approaches on instance sets where the locations and demands are sampled uniformly at random. Additionally, we consider the more realistic instance sets proposed in Hottung & Tierney (2020) with up to 297 customers (see Appendix A). ",
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+    {
+        "type": "text",
+        "text": "Implementation We use the same EAS implementation for the CVRP as for the TSP. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Setup We use the 10,000 CVRP instances from Kool et al. (2019) for testing and additional sets of 1,000 instances to evaluate the generalization performance. Again, we compare the EAS approaches to POMO using greedy action selection, sampling and active search. We generate the same number of solutions per instance as for the TSP. We compare to LIH, CAVE-Opt, DPDP, NeuRewriter (Chen & Tian, 2019) and neural large neighborhood search (NLNS) from Hottung & Tierney (2020). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results Table 2 shows the average costs, the average gap to LKH3 and the total wall-clock time for all instance sets. EAS-Lay outperforms all other approaches on the test instances, including approaches that rely on problem-specific knowledge, with a gap that beats LKH3. Both other EAS methods also find solutions of better quality than LKH3, which is quite an accomplishment given the many years of work on the LKH3 approach. We note it is difficult to provide a fair comparison between a single-core, CPU-bound technique like LKH3 and our approaches that use a GPU. Nonetheless, assuming a linear speedup, at least 18 CPU cores would be needed for LKH3 to match the runtime of EAS-Tab. On the generalization instance sets with $n = 1 2 5$ and $n = 1 5 0$ , the EAS approaches also outperform LKH3 and CVAE-Opt while being significantly faster than active search. On the instances with $n = 2 0 0$ , active search finds the best solutions of all POMO based approaches with a gap of $0 . 2 2 \\%$ to LKH3, albeit with a long runtime of 36 hours. We hypothesize that significant changes to the learned policy are necessary to generate high-quality solutions for instances that are very different to those seen during training. Active search’s ability to modify all model parameters makes it easier to make those changes. EAS-Tab offers the worst performance on the instances with $n = 2 0 0$ with a gap of $1 1 . 8 \\%$ . This is because EAS-Tab is very sensitive to the selection of the hyperparameter $\\alpha$ , meaning that EAS-Tab requires hyperparameter tuning on some problems to generalize more effectively. Adjusting $\\alpha$ for the $n = 2 0 0$ case improves EAS-Tab’s gap to at least $3 . 5 4 \\%$ , making it slightly better than greedy or sampling. ",
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/acaba43ae1df91044310bfda809b60ffddcc8cd8cff8313d303fd2d71b71174d.jpg",
+        "table_caption": [
+            "Table 2: Results for the CVRP on instances with uniformly sampled locations and demands "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">Testing (10k inst.) n=100</td><td colspan=\"10\">Generalization (1k instances)</td></tr><tr><td>Obj.</td><td>Gap</td><td>Time</td><td>Obj.</td><td>n =125 Gap</td><td>Time</td><td>Obj.</td><td>n=150 Gap</td><td>Time</td><td>Obj.</td><td>n = 200 Gap</td><td></td><td>Time</td></tr><tr><td>Method</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LKH3</td><td>|15.65</td><td>0.00%</td><td></td><td>6D|17.50</td><td>0.00%</td><td></td><td>19H|19.22</td><td></td><td>0.00%</td><td></td><td>20H|22.00</td><td>0.00%</td><td>25H</td></tr><tr><td>NLNS NeuRewriter</td><td>15.99</td><td>2.23%</td><td>62M</td><td>[|18.07</td><td>3.23%</td><td></td><td>9M|19.96</td><td></td><td>3.86%</td><td>12M|23.02</td><td></td><td>4.66%</td><td>24M</td></tr><tr><td>LIH</td><td>16.10</td><td></td><td>66M</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td></tr><tr><td>CVAE-Opt</td><td>16.03</td><td>2.47% 1.36%</td><td>5H</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>=</td><td>=</td></tr><tr><td>DPDP</td><td>15.63</td><td>-0.13%</td><td>11D 23H</td><td>17.87 17.51</td><td>2.08% 0.07%</td><td>36H 3H</td><td>19.84 19.31</td><td></td><td>3.24% 0.48%</td><td>46H 5H</td><td>22.26</td><td>51.20%</td><td>= 9H</td></tr><tr><td>Greedy</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>15.76</td><td>0.76%</td><td>2M</td><td>17.73</td><td>1.29%</td><td>&lt;1M</td><td>19.64</td><td></td><td>2.18%</td><td>1M</td><td>|22.90</td><td>4.12%</td><td>1M</td></tr><tr><td>Sampling 0</td><td>15.67</td><td>0.17%</td><td>7H</td><td>17.60</td><td>0.54%</td><td>73M</td><td>19.48</td><td></td><td>1.35%</td><td>2H</td><td>23.18</td><td>5.35%</td><td>5H</td></tr><tr><td>Active S.</td><td>15.63</td><td>-0.07%</td><td>8D</td><td>17.47</td><td>-0.21%</td><td>25H</td><td>19.21</td><td></td><td>-0.03%</td><td>29H</td><td>22.05</td><td>0.22%</td><td>36H</td></tr><tr><td>EAS-Emb</td><td>15.63 15.61</td><td>-0.08%</td><td>9H 12H</td><td>17.47</td><td>-0.21%</td><td>93M</td><td>19.22</td><td></td><td>0.03%</td><td>3H</td><td>22.19</td><td>0.88%</td><td>6H</td></tr><tr><td>EAS-Lay EAS-Tab</td><td>15.62</td><td>-0.23% -0.14%</td><td>8H</td><td>17.50</td><td>17.46 -0.24% 0.00%</td><td>2H 80M</td><td>19.21 19.36</td><td></td><td>-0.04% 0.72%</td><td>3H 2H</td><td>22.10 24.56</td><td>0.45% 11.8%</td><td>8H 5H</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 JSSP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The JSSP is a scheduling problem involving assigning jobs to a set of heterogeneous machines. Each job consists of multiple operations that are run sequentially on the set of machines. The objective is to minimize the time needed to complete all jobs, called the makespan. We evaluate EAS using the L2D approach, which is a state-of-the-art ML based construction method using a graph neural network. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Implementation L2D represents JSSP instances as disjunctive graphs in which each operation of an instance is represented by a node in the graph. To create a schedule, L2D sequentially selects the operation that should be scheduled next. To this end, an embedding $h _ { v }$ is created for each node $v$ in a step-wise encoding process. In contrast to POMO, the embeddings $h _ { v }$ are recomputed after each decision step $t$ . Since EAS-Emb requires static embeddings, we modify the network to use $\\tilde { h } _ { v } ^ { t } = h _ { v } ^ { t } + h _ { v } ^ { S T }$ as an embedding for node t to zero. During the searc $v$ at step  with E $t$ , where S-Emb $h _ { v } ^ { S T }$ is a vector that is initialized winly adjust the static component $h _ { v } ^ { S T }$ of the embedding with gradient descent. For EAS-Lay, we insert the residual layer $\\mathrm { L } ^ { \\star } ( \\cdot )$ described in Equation 3 to each embedding $h _ { v }$ separately and identically. Finally, for EAS-Tab, we use a table $Q$ of size $| O | \\times | O |$ , where $| O |$ is the number of operations, and we design the function $g ( s _ { t } , a _ { t } )$ so that the entry $Q _ { i , j }$ corresponds to selecting the operation $o _ { j }$ directly after the operation $o _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Setup We use three instance sets with 100 instances from Zhang et al. (2020) for testing and to evaluate the generalization performance. We use the exact solver Google OR-Tools (Perron & Furnon) as a baseline, allowing it a maximum runtime of 1 hour per instance. Furthermore, we compare to L2D with greedy action selection. Note that the performance of the L2D implementation is CPU bound and does not allow different instances to be batch processed. We hence solve instances sequentially and generate significantly fewer solutions per instance than for the TSP and the CVRP. For sampling, active search and the EAS approaches we sample 8,000 solutions per problem instance over the course of 200 iterations for the (efficient) active search approaches. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Results Table 3 shows the average gap to the OR-Tools solution and the total wall-clock time per instance set. EAS-Emb offers the best performance for all three instance sets. On the $1 0 \\times 1 0$ instances, EAS-Emb reduces the gap by $50 \\%$ in comparison to pure sampling. Even on the $2 0 \\times 1 5$ instances it reduces the gap to $1 6 . 8 \\%$ from $2 0 . 8 \\%$ for pure sampling, despite the low number of sampled solutions per instance. EAS-Lay offers performance that is comparable to active search. We note that if L2D were to more heavily use the GPU, instances could be solved in batches, thus drastically reducing the runtime of EAS-Lay and EAS-Tab. While EAS-Tab shows similar performance to active search on the test instance set, it is unable to generalize effectively to the larger instances. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 SEARCH TRAJECTORY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To get a better understanding of how efficient active search improves performance, we monitor the quality of the solutions sampled at each of the 200 iterations of the search. Figure 2 reports the average quality over all test instances for the JSSP and over the first 1,000 test instances for the TSP and CVRP. As expected, the quality of solutions generated via pure sampling does not change over the course of the search for all three problems. For all other methods, the quality of the generated solutions improves throughout the search. Thus, all active search variants successfully modify the (model) parameters in a way that increases the likelihood of generating high-quality solutions. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f2ce96d91caec68f4b05fe8c69adad675170d3045007f55110ea4d8ab189f66e.jpg",
+        "table_caption": [
+            "Table 3: Results for the JSSP "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\" rowspan=\"2\"></td><td colspan=\"3\">Testing(100 inst.）</td><td colspan=\"4\">Generalization (1OO instances)</td></tr><tr><td colspan=\"2\">10×10</td><td colspan=\"3\">15×15</td><td colspan=\"3\">20×15</td></tr><tr><td colspan=\"2\" rowspan=\"2\">Method</td><td rowspan=\"2\">Obj. Gap</td><td rowspan=\"2\">Time</td><td rowspan=\"2\">Obj.</td><td>Gap</td><td>Time</td><td>Obj. Gap</td><td>Time</td></tr><tr><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\">0.0%</td></tr><tr><td rowspan=\"2\">OR-Tools Greedy</td><td rowspan=\"2\">1807.6 0.0%</td></tr><tr><td>37S|1</td><td rowspan=\"2\">[1188.0</td><td colspan=\"2\" rowspan=\"2\">0.0%</td><td rowspan=\"2\">3H| |1345.5</td><td rowspan=\"2\"></td><td rowspan=\"2\">80H</td></tr><tr><td rowspan=\"2\"></td><td rowspan=\"2\">1988.6</td></tr><tr><td>22.3% 871.7</td><td>20S 1528.3</td><td>28.6%</td><td>44S</td><td>1738.0</td><td>29.2%</td><td>60S</td></tr><tr><td rowspan=\"4\">Sampling L</td><td>854.2</td><td>8.0% 5.8%</td><td>8H 8H</td><td>1378.3 16.0% 1345.2 13.2%</td><td>25H 32H</td><td>1624.6 1576.5</td><td>20.8% 17.2%</td><td>40H</td></tr><tr><td>EAS-Emb</td><td>837.0 3.7%</td><td>7H</td><td>1326.4 11.7%</td><td>22H</td><td>1570.8</td><td>16.8%</td><td>50H 37H</td></tr><tr><td>EAS-Lay</td><td>859.6 6.5%</td><td>7H</td><td>1352.6</td><td>13.8%</td><td>25H 1581.8</td><td>17.6%</td><td>46H</td></tr><tr><td>EAS-Tab</td><td>860.2 6.5%</td><td>8H</td><td>1376.8</td><td>15.9% 29H</td><td></td><td>1623.420.7%</td><td>51H</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/aeff2be7dc644f08b6349a6f6cabfb88e1ed829fcc964cb99457f18da9f01b79.jpg",
+        "image_caption": [
+            "Figure 2: Average costs of sampled solutions at each iteration (best viewed in color). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/3814f29ab52dc4112005668d02a29044fe9d34e4442175ae8284d12c9a420a11.jpg",
+        "image_caption": [
+            "Figure 3: Influence of $\\lambda$ on the solution quality for EAS-Emb and EAS-Lay. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the TSP, EAS-Emb and EAS-Lay offer nearly identical performance, with EAS-Tab outperforming both by a very slight margin. The original active search is significantly more unstable, which is likely the result of the learning rate being too high. Note that the learning rate has been tuned on an independent validation set. These results indicate that selecting a suitable learning rate is significantly more difficult for the original active search than for our efficient active search variants where only a subset of (model) parameters are changed. For the CVRP, all EAS variants find better solutions on average than the original search after only a few iterations. Keeping most parameters fixed seems to simplify the underlying learning problem and allows for faster convergence. For the JSSP, EAS-Emb offers significantly better performance than all other methods. The reason for this is that the L2D approach uses only two node features and has a complex node embedding generation procedure. While the original active search must fine tune the entire embedding generation process to modify the generated solutions, EAS-Emb can just modify the node embedding directly. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 ABLATION STUDY: IMITATION LEARNING LOSS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate the impact of the imitation learning loss $\\mathcal { L } _ { I L }$ of EAS-Emb and EAS-Lay with a sensitivity and ablation analysis for the hyperparameter $\\lambda$ . We solve the first 500 test instances (to reduce the computational costs) for the TSP and CVRP, and all test instances for the JSSP using EAS-Emb and EAS-Lay with different $\\lambda$ values. The learning rate remains fixed to a value determined in independent tuning runs in which $\\lambda$ is fixed to zero. Figure 3 shows the results for all three problems. For the TSP and the CVRP, the results show that $\\mathcal { L } _ { I L }$ can significantly improve performance. When $\\lambda$ is set to 0 or very small values, $\\mathcal { L } _ { I L }$ is disabled, thus including $\\mathcal { L } _ { I L }$ is clearly beneficial on the TSP and CVRP. For the JSSP, the inclusion of $\\mathcal { L } _ { I L }$ does not greatly improve performance, but it does not hurt it, either. Naturally, $\\lambda$ should not be selected too low or too high as either too little or too much intensification can hurt search performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We presented a simple technique that can be used to extend ML-based construction heuristics by an extensive search. Our proposed modification of active search fine tunes a small subset of (model) parameters to a single instance at test time. We evaluate three example implementations of EAS that all result in significantly improved model performance in both testing and generalization experiments on three different, difficult combinatorial optimization problems. Our approach of course comes with some key limitations. Search requires time, thus for applications needing extremely fast (or practically instant) solutions, greedy construction remains a better option. Furthermore, while the problems we experiment on have the same computational complexity as real-world optimization problems, additional work may be needed to handle complex side constraints as often seen in industrial problems. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The computational experiments in this work have been performed using the Bielefeld GPU Cluster. ",
+        "bbox": [
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+            127,
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+            143
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A EXPERIMENTS FOR MORE REALISTIC CVRP INSTANCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            673,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We provide results from additional experiments for the CVRP on more realistic instances to show that our approach is effective at solving instances with a wide range of structures. Our EAS methods are implemented in the same way as in Section 4.2 ",
+        "bbox": [
+            174,
+            136,
+            823,
+            178
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Setup We evaluate EAS on 9 instance sets from Hottung & Tierney (2020) (consisting of 20 instances each) that have been generated based on the instances from Uchoa et al. (2017) performing 3 runs per instance. The characteristics of the instances vary significantly between sets, but all instances in the same set have been sampled from an identical distribution. For each instance set we train a new model for 3 weeks on a separate, corresponding training set. For testing, we run all (efficient) active search approaches for 200 iterations using the newly trained models. Additionally, we test the generalization performance by solving all instance sets with EAS-Lay using the CVRP model of Section 4.2 that has been trained on the uniform instances (with $n = 1 0 0$ ) from Kool et al. (2019) and call this Lay\\*. We only evaluate the generalization performance of EAS-Lay (the best performing EAS approach from Section 4.2) to keep the computational costs low. In all experiments, we use hyperparameters tuned for the uniform CVRP instances. We compare to NLNS, LKH3 and the state-of-the-art unified hybrid genetic search (GS) from Vidal et al. (2014). As is standard in the operations research literature, we round the distances between customers to the nearest integer. Furthermore, we solve instances sequentially and not in batches of different instances. However, to make better use of the available GPU memory, we solve up to 10 copies of the same instance in parallel for the EAS approaches and for POMO with sampling. The best solution found so far is shared between all runs, which has an impact on the imitation learning loss $\\mathcal { L } _ { I L }$ for EAS-Emb and EAS-Lay. For EAS-Tab we set $\\tilde { Q } = ( 1 - \\bar { \\beta } ) \\cdot Q + \\beta \\cdot Q ^ { g l o b }$ , where $Q ^ { g l o b }$ is the lookup table for the best solution over all runs and $\\beta$ is linearly increased from 0 to 1 over the course of the search. ",
+        "bbox": [
+            173,
+            184,
+            825,
+            450
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Results Table 4 shows the gap to the unified hybrid genetic search and the average runtime per instance for all methods. For EAS-Lay we report the performance of the instance set specific models and additionally the generalization performance when using the model trained on uniform CVRP instances (with $n = 1 0 0$ ). The later results are marked with a star. EAS-Emb and EAS-Lay both find better solution than NLNS and LKH3 on 8 out of the 9 instance sets. EAS-Tab outperforms NLNS and LKH3 on all but two instance sets. As a side note, we have found that the original active search (AS) performs surprisingly well, outperforming LKH3 on 3 instance sets, even though it still cannot surpass our newly proposed EAS methods. The version of EAS-Lay (marked with a star) that uses the model trained on uniform instances with $n = 1 0 0$ performs surprisingly well with gaps between $0 . 2 6 \\%$ to $4 . 0 8 \\%$ to the GS. ",
+        "bbox": [
+            174,
+            457,
+            825,
+            595
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b70c77680f3caf606ae0f0a6c0624de6f9421152730ee2f175b081fef75c62a.jpg",
+        "table_caption": [
+            "Table 4: Results for the CVRP on the instance sets from Hottung & Tierney (2020). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"4\">Gap to GS in %</td><td colspan=\"2\">POMO</td><td colspan=\"5\"> Avg. Runtime in minutes</td></tr><tr><td>Inst.</td><td>n</td><td>POMO Sam. AS</td><td>POMO-EAS Emb Lay Lay* Tab</td><td></td><td>[NLNS LKH|</td><td></td><td></td><td></td><td>POMO-EAS Sam. AS Emb Lay Lay* TabNLNS LKH GS</td><td></td><td></td><td></td></tr><tr><td>XE 1</td><td>100</td><td></td><td></td><td></td><td></td><td>2.12</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>XE</td><td>128</td><td>0.86 0.65 0.76 0.80</td><td>10.23 0.26 0.260.25</td><td>0.61 0.31| 0.26 0.29</td><td>0.32 0.44</td><td>0.9 0.54</td><td>1.3 1.6</td><td>1.3 1.8</td><td>1.4 2.0</td><td>1.5 0.9 2.1</td><td>3.2</td><td>6.2 0.6 2.0</td></tr><tr><td>XE</td><td>180</td><td>0.51 0.20</td><td>0.09 0.09</td><td>0.54 0.13</td><td>0.58 0.16</td><td>1.3 2.5</td><td>2.2</td><td>3.3</td><td>3.7</td><td>24 3.8</td><td>3.2 3.2</td><td>1.2 1.1 1.4</td></tr><tr><td>3.57 XE</td><td>199</td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.2</td><td>4.7</td><td>3.5</td><td>3.2</td><td>3.6 2.4</td></tr><tr><td>XE 9</td><td>213</td><td>1.50 0.88 1.96 1.30</td><td>0.37 0.45</td><td>1.29 0.80 4.08 0.83</td><td>2.03 2.26</td><td>0.72 3.3 1.09</td><td>2.4</td><td></td><td>5.2</td><td>4.9 5.4</td><td>10.2</td><td>1.1 2.4</td></tr><tr><td>XE 11</td><td>236</td><td>1.42 1.22</td><td>0.64 0.71 0.82 0.84</td><td>1.76 0.94</td><td>0.65</td><td>3.7 0.78 4.0</td><td>2.5 2.7</td><td>4.6 5.2</td><td>5.1</td><td>3.9 5.6 4.8</td><td>10.2</td><td>1.1 3.2</td></tr><tr><td>3 XE</td><td></td><td>1.40 0.88</td><td>0.38 0.56</td><td>2.83 0.80</td><td>0.82</td><td>1.55 7.1</td><td>3.5</td><td>8.7</td><td>6</td><td></td><td>10.2</td><td>5.7 3.6</td></tr><tr><td>XE 15</td><td>268</td><td>1.81 2.17</td><td>0.850.96</td><td>2.51 1.27</td><td>1.81</td><td>1.32 7.3</td><td>3.3</td><td>8.8</td><td></td><td>76</td><td>10.3</td><td>5.8 5.5</td></tr><tr><td>XE 17</td><td>297</td><td>1.66 0.97</td><td>0.44 0.65</td><td>2.15 0.92</td><td>1.41</td><td>1.23 8.9</td><td>3.8</td><td>8.8</td><td>6.8</td><td>9.4</td><td>10.3</td><td>2.5 4.2</td></tr></table>",
+        "bbox": [
+            183,
+            636,
+            815,
+            777
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ABLATION STUDY: ACTIVE SEARCH LOSS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            806,
+            558,
+            821
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We evaluate if applying the imitation learning loss component used by EAS-Emb and EAS-Lay to the original active search can significantly improve the performance. To this end, we solve all test instances using active search with and without the imitation learning loss component. Note that the hyperparameters for each approach have been tuned independently on separate validation set instances. Table 5 shows the results. We observe no significant impact of the imitation learning loss $\\mathcal { L } _ { \\pi }$ on the performance of active search. This means that active search with imitation learning loss is not competitive with EAS-Lay and EAS-Emb across all problems, even when sharing the same loss function. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/41ab96920d4ee9221016be5023bf0d3e37f416c536ccee2088f21bf8bca9448d.jpg",
+        "image_caption": [
+            "Figure 4: Influence of $\\sigma$ on the solution quality for EAS-Tab "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            125,
+            823,
+            295
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            347,
+            825,
+            376
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C PARAMETER SWEEP: EAS-TAB INTENSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            396,
+            622,
+            412
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We investigate the impact of the hyperparameter $\\sigma$ on EAS-Tab, which controls the degree of exploitation of the search. By setting $\\sigma$ to zero (or very small values) we essentially disable the lookup table, thus examining its impact on the search. We again solve all three problems with different values of $\\sigma$ on a subset of test instances. We fix $\\alpha$ independently based on tuning on a separate set of validation instances. Figure 4 provides the results for adjusting $\\sigma$ . For all three problems, $\\sigma = 1 0$ provides the best trade-off between exploration and exploitation. Note that low $\\sigma$ values (which reduce the impact of the lookup table updates) hurt performance, meaning that the table based adjustments are effective in all cases. ",
+        "bbox": [
+            173,
+            428,
+            825,
+            540
+        ],
+        "page_idx": 12
+    }
+]

parse/dev/wlEOsQ917F/wlEOsQ917F.md

@@ -0,0 +1,362 @@
+# A framework for bilevel optimization that enables stochastic and global variance reduction algorithms
+
+# Mathieu Dagréou
+
+Inria, CEA Université Paris-Saclay Palaiseau, France mathieu.dagreou@inria.fr
+
+Pierre Ablin CNRS Université Paris-Dauphine, PSL-University Paris, France pierre.ablin@cnrs.fr
+
+# Samuel Vaiter
+
+# Thomas Moreau
+
+CNRS Université Côte d’Azur, LJAD Nice, France samuel.vaiter@cnrs.fr
+
+Inria, CEA Université Paris-Saclay Palaiseau, France thomas.moreau@inria.fr
+
+# Abstract
+
+Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O \big ( \frac { 1 } { T } \big )$ convergence rate, and that it achieves linear convergence under Polyak-Łojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.
+
+# 1 Introduction
+
+Bilevel optimization is attracting more and more attention in the machine learning community thanks to its wide range of applications. Typical examples are hyperparameters selection [5, 38, 17, 6], data augmentation [11, 42], implicit deep learning [3] or neural architecture search [33]. Bilevel optimization aims at minimizing a function whose value depends on the result of another optimization problem:
+
+$$
+\operatorname* { m i n } _ { x \in \mathbb { R } ^ { d } } h ( x ) = F ( z ^ { * } ( x ) , x ) , \quad \mathrm { s u c h t h a t } z ^ { * } ( x ) \in \arg \operatorname* { m i n } _ { z \in \mathbb { R } ^ { p } } G ( z , x ) ,
+$$
+
+where $F$ and $G$ are two real valued functions defined on $\mathbb { R } ^ { p } \times \mathbb { R } ^ { d }$ . $G$ is called the inner function, $F$ is the outer function and $h$ is the value function. Similarly, $z$ is the inner variable and $x$ is the outer variable. In most cases, the function $z ^ { * }$ can only be approximated by an optimization algorithm, which makes bilevel optimization problems challenging. Under appropriate hypotheses, the function $h$ is differentiable, and the chain rule and implicit function theorem give for any $\boldsymbol { x } \in \mathbb { R } ^ { d }$
+
+$$
+\nabla h ( x ) = \nabla _ { 2 } F ( z ^ { * } ( x ) , x ) + \nabla _ { 2 1 } ^ { 2 } G ( z ^ { * } ( x ) , x ) v ^ { * } ( x ) \ ,
+$$
+
+36th Conference on Neural Information Processing Systems (NeurIPS 2022).
+
+where $v ^ { \ast } ( x ) \in \mathbb { R } ^ { p }$ is the solution of a linear system
+
+$$
+\boldsymbol { v } ^ { * } ( \boldsymbol { x } ) = - \left[ \nabla _ { 1 1 } ^ { 2 } G ( \boldsymbol { z } ^ { * } ( \boldsymbol { x } ) , \boldsymbol { x } ) \right] ^ { - 1 } \nabla _ { 1 } F ( \boldsymbol { z } ^ { * } ( \boldsymbol { x } ) , \boldsymbol { x } ) \ .
+$$
+
+In the light of (2) and (3), it turns out that the derivation of the gradient of $h$ at each iteration is cumbersome because it involves two subproblems: the resolution of the inner problem to find an approximation of $z ^ { * } ( x )$ and the resolution of a linear system to find an approximation of $v ^ { * } ( x )$ . It makes the practical implementation of first order methods like gradient descent for (1) challenging.
+
+As is the case in many machine learning problems, we suppose in this paper that $F$ and $G$ are empirical means:
+
+$$
+F ( z , x ) = \frac { 1 } { m } \sum _ { j = 1 } ^ { m } F _ { j } ( z , x ) , \quad G ( z , x ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } G _ { i } ( z , x )
+$$
+
+This structure suggests the use of stochastic methods to solve (1). For single-level problems (that is, classical optimization problems where one function should be minimized), using Stochastic Gradient Descent (SGD; [41, 7]) and variants is natural because individual gradients are straightforward unbiased estimators of the gradient. In the bilevel framework, we want to develop algorithms that make progress on problem (1) by using only a few functions $F _ { j }$ and $G _ { i }$ at a time. However, since $\nabla h$ involves the inverse of the Hessian of $G$ , building such stochastic algorithms is quite challenging, one of the difficulties being that there is no straightforward unbiased estimator of $\nabla h$ . Still, in settings where $m$ or $n$ are large, where computing even a single evaluation of $F$ or $G$ is extremely expensive, stochastic methods are the only scalable algorithms.
+
+![](images/5b8583b8f4e3127a6f57e87a8eec2bdb05bebdf958e042d951a51cd676a39614.jpg)  
+Figure 1: Convergence curves of the two proposed methods on a toy problem. SABA is a stochastic method that achieves fast convergence on the value function.
+
+Variance reduction [27, 13, 43, 15, 12] is a popular technique to obtain fast stochastic algorithms. In a single-level setting, these methods build an approximation of the gradient of the objective function using only stochastic gradients. Contrary to SGD, the variance of the approximation goes to 0 as the algorithm progresses, allowing for faster convergence. For instance, the SAGA method [13] achieves linear convergence if the objective function satisfies a Polyak-Łojasciewicz inequality, and $O ( { \textstyle { \frac { 1 } { T } } } )$ convergence rate on smooth non-convex functions [40]. The extension of these methods to bilevel optimization is a natural idea to develop faster algorithms. However, this idea is hard to implement because it is hard to derive unbiased estimators of $\nabla h$ , let alone variance reduction ones.
+
+Contributions. We introduce a novel framework for bilevel optimization in Section 2, where the inner variable, the solution of the linear system (3) and the outer variable evolve jointly. The evolution directions are written as sums of derivatives of $F _ { j }$ and $G _ { i }$ , which allows us to derive simple unbiased stochastic estimators. In this framework, we propose SOBA, an extension of SGD (Section 2.1), and SABA (Section 2.2), an extension of the variance reduction algorithm SAGA [13]. In Section 3 we analyse the convergence of our methods. SOBA is shown to achieve $\begin{array} { r } { \operatorname* { i n f } _ { t \leq T } \mathbb { E } [ \| \nabla h ( x ^ { t } ) \| ^ { 2 } ] = O ( \log ( T ) T ^ { - \frac { 1 } { 2 } } ) } \end{array}$ with decreasing step sizes. We prove that SABA with fixed step sizes achieves $\begin{array} { r } { \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \| \nabla h ( x ^ { t } ) \| ^ { 2 } ] = O ( \frac { 1 } { T } ) } \end{array}$ . SABA is therefore, to the best of our knowledge, the first stochastic bilevel algorithm that matches the convergence rate of gradient descent on $h$ . We also prove that SABA achieves linear convergence under the assumption that $h$ satisfies a Polyak-Łojasciewicz inequality. To the best of our knowledge, SABA is also the first stochastic bilevel algorithm to feature such a property. Importantly, these rates match the rates of the single level counterparts of each algorithm in non-convex setting (SGD for SOBA and SAGA for SABA). Finally, in Section 4, we provide an extensive benchmark of many stochastic bilevel methods on hyperparameters selection and data hyper-cleaning, and illustrate the usefulness of our approach.
+
+Related work. The bilevel optimization problem has a strong history in the optimization community, taking root in game theory [45]. Gradient-based algorithms to solve (1) can be mainly classified in two different categories depending on how $\nabla h$ is computed, by automatic or implicit differentiation.
+
+Since the solution of the inner problem $z ^ { * } ( x )$ is approximated by the output of an iterative algorithm, it is possible to use automatic differentiation [46, 31] to approximate $\nabla h ( x )$ . It consists in differentiating the different steps of the inner optimization algorithm – see [4] for a review – and has been applied successfully to several bilevel problems arising in machine learning [14, 16]. One of the main drawbacks of this approach is that it requires to store in memory each iterate of the inner optimization algorithm, although this problem can sometimes be overcome using invertible optimization algorithms [34] or truncated backpropagation [44].
+
+The use of the implicit function theorem to obtain (2) and (3) is known as implicit differentiation [5]. While the cost of computing exactly (2) can be prohibitive for large scale problems, Pedregosa [38] showed that we can still converge to a stationary point of the problem by using approximate solutions of the inner problem and linear system (3), if the approximation error goes to 0 sufficiently quickly. The complexity of approximate implicit differentiation has been studied in [20]. Ramzi et al. [39] propose to reuse the computations done in the forward pass to approximate the solution of the linear system (3) when the inner problem is solved thanks to a quasi-Newton method.
+
+In the last few years, several works have proposed different strategies to solve (1) in a stochastic fashion. A first set of methods relies on two nested loops: one inner loop to solve the inner problem with a stochastic method, and one outer loop to update the outer variable with an approximate gradient direction. In [19, 26, 9] the authors use several SGD iterations for the inner problem and then use stochastic Neumann approximations to get an estimate solution of the linear system, which provides them with an approximation of $\nabla h$ used to update $x$ . The analysis of this kind of method was refined by Chen et al. [9], allowing to achieve the same convergence rates as those of SGD. The convergence of the hypergradient when using stochastic solvers for the inner problem and the linear system has been studied in [21]. Arbel and Mairal [2] replace the Neumann approximation by SGD steps to estimate (3). Other authors have proposed single loop algorithms, alternating steps in the inner and the outer problem. Hong et al. [24] propose to perform Neumann approximations of the inverse Hessian and use a single SGD step for the inner problem. It was refined in [23] and [47] where the optimization procedure uses a momentum acceleration. Other variations around this idea include [25, 28, 10, 22, 30]. We refer to Table 1 in appendix for a detailed comparison of these methods.
+
+Notation. The set of integers between 1 and $n$ (included) is denoted $[ n ]$ . For $f : \mathbb { R } ^ { p } \times \mathbb { R } ^ { d } \to \mathbb { R }$ we denote $\nabla _ { i } f ( z , x )$ its gradient w.r.t. the $i ^ { \mathrm { { t h } } }$ variable. The Hessian of $f$ with respect to the first variable is denoted $\nabla _ { 1 1 } ^ { 2 } f ( z , x ) \in \mathbb { R } ^ { p \times p }$ , and the cross-derivatives matrix is $\nabla _ { 2 1 } ^ { 2 } f ( z , x ) \in \mathbb { R } ^ { d \times p }$ . If $v$ is a vector, $\lVert v \rVert$ is its Euclidean norm. If $M$ is a matrix, $\lVert M \rVert$ is its spectral norm. A function is said to be $L$ -smooth, for $L > 0$ , if it is differentiable, and its gradient is $L$ -Lipschitz.
+
+# 2 Proposed framework
+
+In this section, we introduce our framework in which the solution of the inner problem, the solution of the linear system (3) and the outer variable all evolve at the same time, following directions that are written as a sum of derivatives of $F _ { j }$ and $G _ { i }$ . We define
+
+# Algorithm 1 General framework
+
+Input: initializations $z _ { 0 } \in \mathbb { R } ^ { p }$ , $\overline { { x _ { 0 } \in \mathbb { R } ^ { d } } }$ $v _ { 0 } \in \mathbb { R } ^ { p }$ , number of iterations $T$ , step size sequences $( \rho ^ { t } ) _ { t < T }$ and $( \gamma ^ { t } ) _ { t < T }$ .
+
+$$
+D _ { z } ( z , v , x ) = \nabla _ { 1 } G ( z , x ) ,
+$$
+
+$$
+D _ { v } ( z , v , x ) = \nabla _ { 1 1 } ^ { 2 } G ( z , x ) v + \nabla _ { 1 } F ( z , x ) ,
+$$
+
+$$
+D _ { x } ( z , v , x ) = \nabla _ { 2 1 } ^ { 2 } G ( z , x ) v + \nabla _ { 2 } F ( z , x ) .
+$$
+
+These directions are motivated by the fact that we have $\nabla h ( x ) ~ = ~ D _ { x } ( z ^ { * } ( x ) , v ^ { * } ( \bar { x } ) , x )$ , with $z ^ { * } ( x )$ the minimizer of $G ( \cdot , x )$ and $v ^ { * } ( x )$ the solution of $\nabla _ { 1 1 } ^ { 2 } G ( z ^ { * } ( x ) , x ) v \ : = \ : - \nabla _ { 1 } F ( z ^ { * } ( x ) , x )$ . When $x$ is
+
+# end for
+
+fixed, we approximate $z ^ { * }$ by doing a gradient descent on $G$ , following the direction $- D _ { z } ( z , v , x )$ . Finally, when $z$ and $x$ are fixed, we find $v ^ { * }$ by following the direction $- D _ { v } ( z , v , x )$ , which corresponds to a gradient descent on $\begin{array} { r } { v \mapsto \frac { 1 } { 2 } \langle \nabla _ { 1 1 } ^ { 2 } G ( z , x ) v , v \rangle + \langle \nabla _ { 1 } F ( z , x ) , v \rangle } \end{array}$ . The rest of the paper is devoted to the study of the global dynamics where the three variables $z , v$ and $x$ evolve at the same time, following stochastic approximations of $D _ { z } , D _ { v }$ and $D _ { x }$ . The next proposition motivates the choice of these directions.
+
+Proposition 2.1. Assume that for all $x \in \mathbb { R } ^ { d } , G ( \cdot , x )$ is strongly convex. If $( z , v , x )$ is a zero of $( D _ { z } , D _ { v } , D _ { x } )$ , then $z = z ^ { * } ( x )$ , $v = v ^ { * } ( x )$ and $\nabla h ( x ) = ~ 0$ .
+
+We also note that the computation of these directions does not require to compute the matrices $\nabla _ { 1 1 } ^ { 2 } G ( z , x )$ and $\nabla _ { 2 1 } ^ { 2 } G ( z , \bar { x } )$ : we only need to compute their product with a vector, which can be computed at a cost similar to that of computing a gradient.
+
+The framework we propose is summarized in Algorithm 1. It consists in following a joint update rule in $( z , v , x )$ that follows directions $D _ { z } ^ { t } , D _ { v } ^ { t }$ and $D _ { x } ^ { t }$ that are unbiased estimators of $D _ { z } , D _ { v } , D _ { x }$ The first and most important remark is that whereas $\nabla h$ cannot be written as a sum over samples, the directions $D _ { z } , D _ { v }$ and $D _ { x }$ involve only simple sums, since their expressions are “linear” in $F$ and $G$ :
+
+$$
+\begin{array} { r l } & { D _ { z } ( z , v , x ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla _ { 1 } G _ { i } ( z , x ) ~ , } \\ & { D _ { v } ( z , v , x ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla _ { 1 1 } ^ { 2 } G _ { i } ( z , x ) v + \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \nabla _ { 1 } F _ { j } ( z , x ) ~ , } \\ & { D _ { x } ( z , v , x ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \nabla _ { 2 1 } ^ { 2 } G _ { i } ( z , x ) v + \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \nabla _ { 2 } F _ { j } ( z , x ) ~ . } \end{array}
+$$
+
+It is therefore straightforward to derive unbiased estimators of these directions. In [30], the authors considered one particular case of our framework, where each direction is estimated by using the STORM variance reduction technique (see [12]). Taking a step back by proposing the framework summarized in Algorithm 1 opens the way to potential new algorithms that implement other techniques that exist in stochastic single level optimization. In what follows, we study two of them.
+
+# 2.1 First example: the SOBA algorithm
+
+The simplest unbiased estimator is obtained by replacing each mean by one of its terms chosen uniformly at random, akin to what is done in classical single-level SGD. We call the resulting algorithm SOBA (StOchastic Bilevel Algorithm). To do so, we choose two independent random indices $i \in [ n ]$ and $j \in [ m ]$ uniformly and estimate each term coming from $G$ using $G _ { i }$ and each term coming from $F$ using $F _ { j }$ . This gives the unbiased SOBA directions
+
+$$
+\boxed { \begin{array} { r l } & { D _ { z } ^ { t } = \nabla _ { 1 } G _ { i } ( z ^ { t } , x ^ { t } ) ~ , } \\ & { D _ { v } ^ { t } = \nabla _ { 1 1 } ^ { 2 } G _ { i } ( z ^ { t } , x ^ { t } ) v ^ { t } + \nabla _ { 1 } F _ { j } ( z ^ { t } , x ^ { t } ) ~ , } \\ & { D _ { x } ^ { t } = \nabla _ { 2 1 } ^ { 2 } G _ { i } ( z ^ { t } , x ^ { t } ) v ^ { t } + \nabla _ { 2 } F _ { j } ( z ^ { t } , x ^ { t } ) ~ . } \end{array} }
+$$
+
+This provides us with a first algorithm, SOBA, where we plug Equations (10a) to (10c) in Algorithm 1. We defer its analysis to the next section. Importantly, we use different step sizes for the update in $( z , v )$ and for the update in $x$ . We use the same step size in $z$ and in $v$ since the inner problem and the linear system have similar conditioning, which is that of $\nabla _ { 1 1 } ^ { 2 } G ( z ^ { t } , x ^ { t } )$ . The need for a different step size for the outer and inner problem is clear: both problems can have a different conditioning.
+
+An important remark for SOBA is that all the stochastic directions used are computed at the same point $\bar { z } ^ { t } , v ^ { t }$ and $x ^ { t }$ with the same indices $( i , j )$ . The update of $z$ , $v$ and $x$ can thus be performed in parallel instead of sequentially, benefiting from hardware parallelism. Moreover, this enables to share the computations between the different directions. This is the case in hyperparameters selection where $\begin{array} { r } { \bar { G } _ { i } ( z , x ) = \ell _ { i } ( \langle z , d _ { i } \rangle ) + \frac { x } { 2 } \| z \| ^ { 2 } } \end{array}$ , with $d _ { i }$ a training sample, and $\ell _ { i }$ that measures how good is the prediction $\langle z , d _ { i } \rangle$ . In this setting, we have $\nabla _ { 1 } G _ { i } ( z , x ) = \ell _ { i } ^ { \prime } ( \langle z , d _ { i } \rangle ) d _ { i } + x z$ and $\nabla _ { 1 1 } ^ { 2 } \bar { G } _ { i } ( z , x ) v \stackrel { - } { = } \ell _ { i } ^ { \prime \prime } ( \langle z , d _ { i } \rangle ) \langle v , d _ { i } \rangle d _ { i }$ . The prediction $\langle z , d _ { i } \rangle$ can thus be computed only once to obtain both quantities. For more complicated models, where automatic differentiation is used to compute the different derivatives and Jacobian-vector products, we can store the computational graph only once to compute at the same time $\nabla _ { 1 } G _ { i } ( z , x ) , \dot { \nabla _ { 1 1 } ^ { 2 } } G _ { i } ( z , x ) v$ and $\nabla _ { 2 1 } ^ { 2 } G _ { i } ( z , x ) \dot { v }$ , requiring only one backward pass, thanks to the $\mathcal { R }$ technique [37].
+
+Finally, like all single loop bilevel algorithms, our method updates at the same time the inner and outer variable, avoiding unnecessary optimization of the inner problem when $x$ is far from the optimum.
+
+# 2.2 Global variance reduction with the SABA algorithm
+
+In classical optimization, SGD fails to reach optimal rates because of the variance of the gradient estimator. Variance reduction algorithms aim at reducing this variance, in order to follow directions that are closer to the true gradient, and to achieve superior practical and theoretical convergence.
+
+In our framework, since the directions $D _ { z } , D _ { v }$ and $D _ { x }$ are all written as sums of derivatives of $F _ { j }$ and $G _ { i }$ , it is easy to adapt most classical variance reduction algorithms. We focus on the celebrated SAGA algorithm [13]. The extension we propose is called SABA (Stochastic Average Bilevel Algorithm). The general idea is to replace each sum in the directions $D$ by a sum over a memory, updating only one term at each iteration. To help the exposition, we denote $y = ( z , x , v )$ the vector of joint variables. Since we have sums over $i$ and over $j$ , we have two memories for each variable: $\boldsymbol { w } _ { i } ^ { t }$ for $i \in [ n ]$ and $\tilde { w } _ { j } ^ { t }$ for $j \in [ m ]$ , which keep track of the previous values of the variable $y$ .
+
+At each iteration $t$ , we draw two random independent indices $i \in [ n ]$ and $j \in [ m ]$ uniformly and update the memories. To do so, we put $w _ { i } ^ { t + 1 } = y ^ { t }$ and $w _ { i ^ { \prime } } ^ { t + 1 } = w _ { i ^ { \prime } } ^ { t }$ for $i ^ { \prime } \neq i$ , and $\tilde { w } _ { j } ^ { t + 1 } = y ^ { t }$ and $\tilde { w } _ { j ^ { \prime } } ^ { t + 1 } = \tilde { w } _ { j ^ { \prime } } ^ { t }$ for $\boldsymbol { j ^ { \prime } } \neq \boldsymbol { j }$ . Each sum in the directions $D$ is then approximated using SAGA-like rules: given $n$ functions $\phi _ { i ^ { \prime } }$ for $i ^ { \prime } \in [ n ]$ , we define $\begin{array} { r } { S [ \phi , w ] _ { i } ^ { t } = \phi _ { i } ( w _ { i } ^ { t + 1 } ) - \phi _ { i } ( w _ { i } ^ { t } ) + \frac { 1 } { n } \sum _ { i ^ { \prime } = 1 } ^ { n } \phi _ { i ^ { \prime } } ( w _ { i ^ { \prime } } ^ { t } ) . } \end{array}$ This is an unbiased estimators of the average of the $\phi$ ’s since $\begin{array} { r } { \mathbb E _ { i } \Big [ S [ \phi , w ] _ { i } ^ { t } \Big ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \phi _ { i } ( y ^ { t } ) } \end{array}$ .
+
+With a slight abuse of notation, we call $\nabla _ { 1 1 } ^ { 2 } G v$ the sequence of functions $( y \mapsto \nabla _ { 1 1 } ^ { 2 } G _ { i } ( z , x ) v ) _ { i \in [ n ] }$ and $\nabla _ { 2 1 } ^ { 2 } G v$ the sequence of functions $( y \mapsto \nabla _ { 2 1 } ^ { 2 } G _ { i } ( z , x ) v ) _ { i \in [ n ] }$ . We define the SABA directions as
+
+$$
+\begin{array} { r c l } { { } } & { { } } & { { D _ { z } ^ { t } = S [ \nabla _ { 1 } G , w ] _ { i } ^ { t } ~ , } } \\ { { } } & { { } } & { { D _ { v } ^ { t } = S [ \nabla _ { 1 1 } ^ { 2 } G v , w ] _ { i } ^ { t } + S [ \nabla _ { 1 } F , \tilde { w } ] _ { j } ^ { t } ~ , } } \\ { { } } & { { } } & { { D _ { x } ^ { t } = S [ \nabla _ { 2 1 } ^ { 2 } G v , w ] _ { i } ^ { t } + S [ \nabla _ { 2 } F , \tilde { w } ] _ { j } ^ { t } ~ . } } \end{array}
+$$
+
+These estimators are unbiased estimators of the directions $D _ { z } , D _ { v }$ and $D _ { x }$ . The SABA algorithm corresponds to Algorithm 1 where we use Equations (11a) to (11c) as update directions. When taking a step size $\gamma ^ { t } = \bar { 0 }$ in the outer problem, hereby stopping progress in $x$ , we recover the iterations of the SAGA algorithm on the inner problem. In practice, the sum in $S$ is computed by doing a rolling average (see Appendix B for precision), and the quantities $\phi _ { i } ( w _ { i } ^ { t } )$ are stored rather than recomputed: the cost of computing the SABA directions is the same as that of SGD. It requires an additional memory for the five quantities, of total size $n \times p + ( n + m ) \times ( p + d )$ floats that can be reduced by using larger batch sizes. Indeed, if $b _ { \mathrm { i n } }$ and $b _ { \mathrm { o u t } }$ are respectively the inner and the outer batch sizes, the memory load is reduced to nb × p + (nb + mb) × (p × d) with nb = ⌈ nbinn ⌉ and $\begin{array} { r } { m _ { b } = \left\lceil \frac { m } { b _ { \mathrm { o u t } } } \right\rceil } \end{array}$ which are smaller than the number of samples. This memory load can also be reduced in specific cases, for instance when $G$ and $F$ correspond to linear models, where the individual gradients and Hessian-vector products are proportional to the samples. In this case, we only store the proportionality ratio, reducing the memory load to $3 n + 2 m$ floats. Like for SOBA, the computations of the new quantities $\phi _ { i } ( w _ { i } ^ { t + 1 } )$ are done in parallel, thus benefiting from hardware acceleration and shared computations. Despite this memory load, using SAGA-like variance reduction instead of STORM as done in [30, 47, 28] has the advantage to bring the variance of the estimate directions to zero, enabling faster $O ( { \textstyle { \frac { 1 } { T } } } )$ convergence.
+
+In the next section, we show that SABA is fast. It essentially has the same properties as SAGA: despite being stochastic, it converges with fixed step sizes, and reaches the same rate of convergence as gradient descent on $h$ .
+
+# 3 Theoretical analysis
+
+In this section, we provide convergence rates of SOBA and SABA under some classical assumptions. Note that, unlike most of the stochastic bilevel optimization papers, we work in finite sample setting rather than the more general expectation setting. Actually, SABA does not make any sense for functions that don’t have a finite sum structure. However, we stress that SOBA could be studied in a more general setting to obtain the same bounds as here. Also, the finite sum setting is still interesting since doing empirical risk minimization is very common in practice in machine learning. The proofs and the constants in big- $O$ are deferred in Appendix C.
+
+# 3.1 Background and assumptions
+
+We start by stating some regularity assumptions on the functions $F$ and $G$ .
+
+Assumption 3.1. The function $F$ is twice differentiable. The derivatives $\nabla F$ and $\nabla ^ { 2 } F$ are Lipschitz continuous in $( z , x )$ with respective Lipschitz constants $L _ { 1 } ^ { F }$ and $L _ { 2 } ^ { F }$ .
+
+Note that the above assumption is typically verified in the machine learning context, e.g., when $F$ is the ordinary least squares (OLS) loss or the logistic loss.
+
+Assumption 3.2. The function $G$ is three times continuously differentiable on $\mathbb { R } ^ { p } \times \mathbb { R } ^ { d }$ . For any $x \in \mathbb { R } ^ { d } , G ( \cdot , x )$ is $\mu _ { G }$ -strongly convex. The derivatives $\overrightarrow { \nabla G }$ , $\nabla ^ { 2 } G$ and $\nabla ^ { 3 } G$ are Lipschitz continuous in $( z , x )$ with respective Lipschitz constants $L _ { 1 } ^ { G }$ , $L _ { 2 } ^ { G }$ and $L _ { 3 } ^ { G }$ .
+
+Strong convexity and smoothness with respect to $z$ of $G$ are verified when $G$ is a regularized leastsquares/logistic regression with a full rank design matrix, when the data is not separable for the logistic regression. Moreover, the strong convexity ensures the existence and uniqueness of the inner optimization problem for any $x \in \mathbb { R } ^ { d }$ .
+
+Assumption 3.3. There exists $C _ { F } ~ > ~ 0$ such that for any $x$ we have $\| \nabla _ { 1 } F ( z ^ { * } ( x ) , x ) \| \leq C _ { F } .$
+
+This assumption, combined with the strong convexity of $G ( \cdot , x )$ , shows boundedness of $v ^ { * }$ . This assumption holds, for instance, in the case of hyperparameters selection for a Ridge regression problem. Note that in Assumptions 3.1 and 3.2, we assume more regularity of $F$ and $G$ than in stochastic bilevel optimization literature (see for instance [19, 24, 26, 2]). It is necessary to get the smoothness of $v ^ { * }$ which will allow to adapt the proof of Chen et al. [9] and get tight convergence rates. The following lemma gives us some smoothness properties of the considered directions that will be useful to derive convergence rates of our methods.
+
+Lemma 3.4. Under the Assumptions 3.1 to 3.3, there exist constants $L _ { z }$ , $L _ { v }$ and $L _ { x }$ such that $\begin{array} { r } { \| D _ { z } ( z , v , x ) \| ^ { 2 } \leq L _ { z } ^ { 2 } \| z - z ^ { * } ( \hat { x } ) \| ^ { 2 } , \ \| D _ { v } ( z , v , x ) \| ^ { 2 } \leq L _ { v } ^ { 2 } ( \| z - z ^ { * } ( x ) \| ^ { 2 } + \| v - v ^ { * } ( x ) \| ^ { 2 } ) } \end{array}$ and $\| D _ { x } ( z , v , x ) - \nabla h ( x ) \| ^ { 2 } \leq L _ { x } ^ { 2 } ( \| z - z ^ { * } ( x ) \| ^ { 2 } + \| v - v ^ { * } ( x ) \| ^ { 2 } )$ .
+
+In first order optimization, a fundamental assumption on the objective function is the smoothness assumption. In the case of vanilla gradient descent applied to a function $f$ , it allows to get a convergence rate of $\| \nabla f ( x ^ { t } ) \| ^ { 2 }$ in $O ( \bar { 1 } / T )$ , i.e. convergence to a stationary point [36]. The following lemma proved by Ghadimi and Wang [19, Lemma 2.2] ensures the smoothness of $h$ .
+
+Lemma 3.5. Under the Assumptions 3.1 to 3.3, the function $h$ is $L ^ { h }$ -smooth for some $L ^ { h } > 0$ .
+
+The constant $L ^ { h }$ is specified in Appendix C.3. As usual with the analysis of stochastic methods, we define the expected norms of the directions $V _ { z } ^ { t } = \mathbb { E } [ \| D _ { z } ^ { t } \| ^ { 2 } ]$ , $V _ { v } ^ { t } = \dot { \mathbb { E } } [ \| D _ { v } ^ { t } \| ^ { 2 } ]$ and $V _ { x } ^ { t } = \mathbb { E } [ \| D _ { x } ^ { t } \| ^ { 2 } ]$ , where the expectation is taken over the past. Thanks to variance-bias decomposition, they are the sum of the variance of the stochastic direction and the squared-norm of the unbiased direction. For SOBA, we use classical bounds on variances like those found for instance in [24]:
+
+Assumption 3.6. There exists $B _ { z }$ , $B _ { v }$ and $B _ { x }$ such that for all $t$ , $\begin{array} { r l r } { \mathbb { E } _ { t } [ \| D _ { z } ^ { t } \| ^ { 2 } ] } & { { } \le } & { B _ { z } ^ { 2 } ( 1 + \| D _ { z } ( z ^ { t } , v ^ { t } , x ^ { t } ) \| ^ { 2 } ) } \end{array}$ and $\begin{array} { r } { \mathbb { E } _ { t } [ \| D _ { v } ^ { t } \| ^ { 2 } ] \ \leq \ B _ { v } ^ { 2 } ( 1 + \| D _ { v } ( z ^ { t } , v ^ { t } , x ^ { t } ) \| ^ { 2 } ) } \end{array}$ where $\mathbb { E } _ { t }$ denotes the expectation conditionally to $( z ^ { t } , v ^ { t } , \ddot { x } ^ { t } )$ .
+
+For SOBA and SABA, we need to bound the expected norm of $D _ { x } ^ { t }$ . For SABA, this assumption allows to get a the same sample complexity as SAGA for single level problems.
+
+Assumption 3.7. There exists $B _ { x }$ such that for all $t$ , $\mathbb { E } _ { t } [ \| D _ { x } ^ { t } \| ^ { 2 } ] \leq B _ { x } ^ { 2 }$ .
+
+Assumptions 3.6 and 3.7 are verified for instance, if all the $G _ { i }$ and $\nabla _ { 1 } G _ { i }$ have at most quadratic growth, and if $F$ has bounded gradients. They are also verified if the iterates remain in a compact set. Note that we do not assume that $G$ has bounded gradients, as this would contradict its strong-convexity. Finally, for the analysis of SABA, we need regularity on each $G _ { i }$ and $F _ { j }$ :
+
+Assumption 3.8. For all $i \in [ n ]$ and $j \in [ m ]$ , the functions $\nabla G _ { i }$ , $\nabla F _ { j }$ , $\nabla _ { 1 1 } ^ { 2 } G _ { i }$ and $\nabla _ { 2 1 } ^ { 2 } G _ { i }$ are Lipschitz continuous in $( z , x )$ .
+
+# 3.2 Fundamental descent lemmas
+
+Our analysis for SOBA and SABA is based on the control of both $\delta _ { z } ^ { t } = \mathbb { E } [ \| z ^ { t } - z ^ { * } ( x ^ { t } ) \| ^ { 2 } ]$ and $\delta _ { v } ^ { t } = \mathbb E [ | | \bar { v } ^ { t } - v ^ { * } ( x ^ { t } ) | | ^ { 2 } ]$ , Strong convexity of $G$ and smoothness of $z ^ { * } ( x )$ and $v ^ { * } ( x )$ allow to obtain the following lemma by adapting the proof of Chen et al. [9]. In what follows, we drop the dependency of the step sizes $\rho$ and $\gamma$ in $t$ for clarity.
+
+Lemma 3.9. Assume that $\begin{array} { r } { \gamma ^ { 2 } \leq \operatorname* { m i n } \left( \frac { \mu _ { G } L _ { * } ^ { 2 } } { 4 B _ { x } ^ { 2 } L _ { z x } ^ { 2 } } , \frac { \mu _ { G } L _ { * } ^ { 2 } } { 8 B _ { x } ^ { 2 } L _ { v x } ^ { 2 } } \right) \rho } \end{array}$ . We have:
+
+$$
+\begin{array} { r l } & { \delta _ { z } ^ { t + 1 } \leq \left( 1 - \frac { \rho \mu _ { G } } { 4 } \right) \delta _ { z } ^ { t } + 2 \rho ^ { 2 } V _ { z } ^ { t } + \beta _ { z x } \gamma ^ { 2 } V _ { x } ^ { t } + \overline { { \beta } } _ { z x } \frac { \gamma ^ { 2 } } { \rho } \mathbb { E } [ \| D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) \| ^ { 2 } ] } \\ & { \delta _ { v } ^ { t + 1 } \leq \left( 1 - \frac { \rho \mu _ { G } } { 8 } \right) \delta _ { v } ^ { t } + \beta _ { v z } \rho \delta _ { z } ^ { t } + 2 \rho ^ { 2 } V _ { v } ^ { t } + \beta _ { v x } \gamma ^ { 2 } V _ { x } ^ { t } + \overline { { \beta } } _ { z x } \frac { \gamma ^ { 2 } } { \rho } \mathbb { E } [ \| D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) \| ^ { 2 } ] } \end{array}
+$$
+
+where $\beta _ { z x } = \beta _ { v x } = 3 L _ { * } ^ { 2 }$ , $\begin{array} { r } { \overline { { \beta } } _ { z x } \ : = \ : \frac { 8 L _ { * } ^ { 2 } } { \mu _ { G } } } \end{array}$ , $\begin{array} { r } { \overline { { \beta } } _ { v x } = { \frac { 1 6 L _ { * } ^ { 2 } } { \mu _ { G } } } } \end{array}$ , $L _ { * }$ is the maximum between the Lipschitz constants of and ∗ zx (see Lemma ), ， , and are respectively $z ^ { * }$ $v ^ { * }$ $C . I$ $\begin{array} { r } { \beta _ { v z } = \frac { \mathrm { ~ i ~ } } { \mu _ { G } ^ { 3 } } ( L _ { 1 } ^ { F } \mu _ { G } + L _ { 2 } ^ { G } ) ^ { 2 } } \end{array}$ $L _ { z x }$ $L _ { v x }$ the smoothness constants of $z ^ { * }$ and $v ^ { * }$ .
+
+We insist that this result is obtained in general for Algorithm 1 with arbitrary unbiased directions. We can therefore invoke this lemma for the analysis of both SOBA and SABA. We use the smoothness of $h$ to get the following lemma, which is similar to [9, Lemma 1].
+
+Lemma 3.10. Let $h ^ { t } = \mathbb { E } [ h ( x ^ { t } ) ]$ and $g ^ { t } = \mathbb { E } [ \| \nabla h ( x ^ { t } ) \| ^ { 2 } ]$ . We have
+
+$$
+h ^ { t + 1 } \leq h ^ { t } - \frac { \gamma } { 2 } g ^ { t } - \frac { \gamma } { 2 } \mathbb { E } [ \| D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) \| ^ { 2 } ] + \frac { \gamma } { 2 } L _ { x } ^ { 2 } ( \delta _ { z } ^ { t } + \delta _ { v } ^ { t } ) + \frac { L ^ { h } } { 2 } \gamma ^ { 2 } V _ { x } ^ { t } ~ .
+$$
+
+If $z ^ { t } = z ^ { * } ( x ^ { t } )$ , $v ^ { t } = v ^ { * } ( x ^ { t } )$ , that is $\delta _ { z } , \delta _ { v }$ both cancel and $D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) = \nabla h ( x ^ { t } )$ , we get an inequality reminiscent of the smoothness inequality for SGD on $h$ .
+
+# 3.3 Analysis of SOBA
+
+The analysis of SOBA is based on Lemmas 3.5 and 3.9. We have the following theorem, with fixed step sizes depending on the number of iterations:
+
+Theorem 1 (Convergence of SOBA, fixed step size). Fix an iteration $T > 1$ and assume that Assumptions 3.1 to 3.7 hold. We consider fixed steps $\begin{array} { r } { \rho ^ { t } = \frac { \overline { { \rho } } } { \sqrt { T } } } \end{array}$ and $\gamma ^ { t } = \xi \rho ^ { t }$ with $\overline { \rho }$ and $\xi$ precised in the appendix. Let $( x ^ { t } ) _ { t \geq 1 }$ the sequence of outer iterates for SOBA. Then,
+
+$$
+\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \| \nabla h ( x ^ { t } ) \| ^ { 2 } ] = O ( T ^ { - \frac { 1 } { 2 } } ) \mathrm { ~ . ~ }
+$$
+
+As opposed to [24], we do not need that the ratio $\frac { \gamma } { \rho }$ goes to 0, which allows to get a complexity (that is, the number of call to oracles to have an $\epsilon$ -stationary solution) in $O ( \epsilon ^ { - 2 } )$ better than the $\tilde { O } ( \epsilon ^ { - \frac { 5 } { 2 } } )$ they have. Also, note that this rate is the same as the one of SGD for non-convex and smooth objective [18, 8]. We obtain a similar rate using decreasing step sizes:
+
+Theorem 2 (Convergence of SOBA, decreasing step size). Assume that Assumptions 3.1 to 3.7 hold. We consider steps $\rho ^ { t } = \overline { { \rho } } t ^ { - \frac { 1 } { 2 } }$ and $\gamma ^ { t } = \xi \rho$ . Let $x ^ { t }$ the sequence of outer iterates for SOBA. Then,
+
+$$
+\operatorname* { i n f } _ { t \leq T } \mathbb { E } [ \| \nabla h ( x ^ { t } ) \| ^ { 2 } ] = O ( \log ( T ) T ^ { - { \frac { 1 } { 2 } } } ) \enspace .
+$$
+
+As for SGD, SOBA suffers from the need of decreasing step sizes to get actual convergence because of the variance of the estimation on each directions. On the other hand, the analysis of SABA leverages the dynamic of all three variables, resulting in fast convergence with fixed step sizes.
+
+# 3.4 SABA: a stochastic method with optimal rates
+
+In what follows, we denote $N = n + m$ the total number of samples. The following theorem shows $O ( N ^ { \frac { 2 } { 3 } } T ^ { - 1 } )$ convergence for the SABA algorithm in the general case where we only assume smoothness of $h$ . Our analysis of SABA is inspired by the analysis of single-level SAGA by Reddi et al. [40].
+
+Theorem 3 (Convergence of SABA, smooth case). Assume that Assumptions 3.1 to 3.3 and 3.7 to 3.8 hold. We suppose $\rho = \rho ^ { \prime } N ^ { - \frac { 2 } { 3 } }$ and $\gamma = \xi \rho$ , where $\rho ^ { \prime }$ and $\xi$ depend only on $F$ and $G$ and are specified in appendix. Let $x ^ { t }$ the iterates of SABA. Then,
+
+$$
+\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \| \nabla h ( x ^ { t } ) \| ^ { 2 } ] = O \left( N ^ { \frac { 2 } { 3 } } T ^ { - 1 } \right) \ .
+$$
+
+To prodefine $\begin{array} { r } { { S } ^ { t } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \| y ^ { t } - w _ { i } ^ { t } \| ^ { 2 } + \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \| y ^ { t } - \tilde { w } _ { j } ^ { t } \| ^ { 2 } } \end{array}$ m the memory to the current variables. We. In appendix, we show that we can find $\phi _ { s } , \phi _ { z } , \phi _ { v } > 0$ such that the quantity $\mathcal { L } ^ { t } = h ^ { t } + \phi _ { s } S ^ { t } + \phi _ { z } \delta _ { z } ^ { t } + \phi _ { v } \delta _ { v } ^ { t }$   
+Summing these inequalities for $t = 1 \dots T$ and using the fact that $\textstyle { \mathcal { L } } ^ { t }$ is lower bounded demonstrates   
+the theorem.
+
+Note that the step sizes are constant with respect to the time, but they scale with $N ^ { - \frac { 2 } { 3 } }$ . As a consequence, the sample complexity is $O ( N ^ { \frac { 2 } { 3 } } \epsilon ^ { - 1 } )$ which is analogous of the one of SAGA for non-convex single level problems [40]. This is better than the sample complexity of Algorithm 1 with full batch directions, which is $O ( N \epsilon ^ { - 1 } )$ . Hence, with SABA, we get the best of both worlds: the stochasticity makes the scaling in $N$ of the sample complexity goes from $N$ in full batch mode to $N ^ { \frac { 2 } { 3 } }$ for SABA, and the variance reduction makes the scaling in $\epsilon$ goes from $\epsilon ^ { - 2 }$ for SOBA to $\epsilon ^ { - 1 }$ for SABA. Our experiments in Section 4 confirm this gain.
+
+Furthermore, if we assume that $h$ satisfies a Polyak-Łojasiewicz (PL) inequality, we recover linear convergence. Recall that $h$ has the PL property if there exists $\mu _ { h } > 0$ such that for all $x \in \mathbb { R } ^ { d }$ $\begin{array} { r } { \frac { 1 } { 2 } \| \nabla h ( \bar { \boldsymbol { x } } ) \| ^ { 2 } \geq \mu _ { h } ( h ( \boldsymbol { x } ) - h ^ { * } ) } \end{array}$ with $h ^ { * }$ the minimum of $h$ .
+
+Theorem 4 (Convergence of SABA, PL case). Assume that $h$ satisfies the PL inequality and that Assumptions 3.1 to 3.3 and 3.7 to 3.8 hold. We suppose $\rho = \rho ^ { \prime } N ^ { - \frac { 2 } { 3 } }$ and $\gamma = \xi \rho ^ { \prime } N ^ { - 1 }$ , where $\rho ^ { \prime }$ and $\xi$ depend only on $F$ and $G$ and are specified in appendix. Let $x ^ { t }$ the iterates of SABA and $\begin{array} { r } { c ^ { \prime } \triangleq \operatorname* { m i n } \left( \mu _ { h } , \frac { 1 } { 1 6 P ^ { \prime } } \right) } \end{array}$ with $P ^ { \prime }$ specified in the appendix. Then,
+
+$$
+\mathbb { E } [ h ^ { T } ] - h ^ { * } = ( 1 - c ^ { \prime } \gamma ) ^ { T } ( h ^ { 0 } - h ^ { * } + C ^ { 0 } )
+$$
+
+where $C ^ { 0 }$ is a constant specified in appendix that depends on the initialization of $z , v , x$ and memory.
+
+The proof is similar to that of the previous theorem: we find coefficients $\phi _ { s } , \phi _ { z } , \phi _ { v }$ such that $\mathcal { L } ^ { t } \stackrel {  } { = } h ^ { t } + \phi _ { s } S ^ { t } + \phi _ { z } \delta _ { z } ^ { t } + \phi _ { v } \delta _ { v } ^ { t }$ satisfies the inequality $\mathcal { L } ^ { t + 1 } \leq ( 1 - c ^ { \prime } \gamma ) \mathcal { L } ^ { t }$ , which is then unrolled. Note that in the case where we initialize $z$ and $v$ with $z ^ { 0 } = z ^ { * } ( x ^ { 0 } )$ , $v ^ { 0 } = v ^ { * } ( x ^ { 0 } )$ , and the memories $w _ { i } ^ { 0 } = w ^ { 0 }$ , $\tilde { w } _ { j } ^ { 0 } = w ^ { 0 }$ for all $i , j$ , the constant $C ^ { 0 }$ cancels and the bound simplifies to $\mathbb { E } [ h ( x ^ { T } ) ] - h ^ { * } \leq ( 1 - c ^ { \prime } \gamma ) ^ { T } ( h ( x ^ { 0 } ) - h ^ { * } )$ .
+
+Just like classical variance reduction methods in single-level optimization, this theorem shows that our method achieves linear convergence under PL assumption on the value function. To the best of our knowledge, our method is the first stochastic bilevel optimization method that enjoys such property. We note that the PL hypothesis is more general than $\mu _ { h }$ -strong convexity of $h - \mathrm { i t }$ is a necessary condition for strong convexity.
+
+We see here the importance of global variance reduction. Indeed, using variance reduction only on $z$ and SGD on $x$ would lead to sub-linear convergence in $x$ . This would be the case even with a perfect estimation of $z ^ { * } ( x )$ . Similarly, using variance reduction only on $x$ and SGD on $z$ would lead to sub-linear convergence in $z$ , and hence in $x$ . Using global variance reduction with respect to each variable as we propose here is the only way to achieve linear convergence. We now turn to experiments, where we find that our method is also promising from a practical point of view.
+
+# 4 Experiments
+
+Here we compare the performances of SOBA and SABA with competitor methods on different tasks. The different methods being compared are stocBiO [26], AmiGO [2], FSLA [30], MRBO [47], TTSA [24], BSA [19] and SUSTAIN [28]. A detailed account of the experiments is provided in Appendix B. 1
+
+# 4.1 Hyperparameters selection
+
+The first task we perform is hyperparameters selection to choose regularization parameters on $\ell ^ { 2 }$ logistic regression. Let us denote $( ( d _ { i } ^ { \mathrm { t r a i n } } , y _ { i } ^ { \mathrm { t r a i n } } ) ) _ { 1 \leq i \leq n }$ and $( ( d _ { i } ^ { \mathrm { v a l } } , y _ { i } ^ { \mathrm { v a l } } ) ) _ { 1 \leq i \leq m }$ the training and the validation sets. In this case, the inner variable $\bar { \theta }$ corresponds to the parameters of the model, and the outer variable $\lambda$ to the regularization. The functions $F$ and $G$ of the problem (1) are the logistic loss, with $\ell ^ { 2 }$ penalty for $G$ , that is to say $\begin{array} { r } { F ( \theta , \lambda ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \varphi ( y _ { i } ^ { \mathrm { v a l } } \langle \bar { d } _ { i } ^ { \mathrm { v a l } } , \theta \rangle ) } \end{array}$ and $\begin{array} { r } { G ( \theta , \lambda ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \varphi ( y _ { i } ^ { \mathrm { t r a i n } } \langle d _ { i } ^ { \mathrm { t r a i n } } , \theta \rangle ) + \frac { 1 } { 2 } \sum _ { k = 1 } ^ { p } e ^ { \lambda _ { k } } \theta _ { k } ^ { 2 } } \end{array}$ where $\varphi ( u ) = \log ( 1 + e ^ { - u } )$ . We fit a binary classification model on the IJCNN $1 ^ { 2 }$ dataset. Here, $n = 4 9 9 9 0$ , $m = 9 1 7 0 1$ and $p = 2 2$ .
+
+The suboptimality gap is plotted in Figure $2 \mathrm { a }$ for each method. The lowest values are reached by SABA. Moreover, SABA is the only single-loop method that reaches a suboptimality below $1 0 ^ { - 5 }$ SOBA reaches a quite high final value but slightly better than TTSA and FSLA. The gap between
+
+![](images/2688ecd25ca7cb86a395fda5ce9b688673c5e37d3422795d1fc2b1b4c85e6aab.jpg)  
+Figure 2: Comparison of SOBA and SABA with other stochastic bilevel optimization methods. For each algorithm, we plot the median performance over 10 runs. In both experiments, SABA achieves the best performance. The dashed lines are for one loop competitor methods, the dotted lines are for two loops methods and the solid lines are the proposed methods. Left: hyperparameter selection for $\ell ^ { 2 }$ penalized logistic regression on IJCNN1 dataset , Right: data hyper-cleaning on MNIST with $p = 0 . 5$ corruption rate.
+
+SOBA and SABA highlights the benefits of variance reduction: it gives us a lower plateau and the fixed step sizes enable faster convergence.
+
+# 4.2 Data hyper-cleaning
+
+The second task we perform is data hyper-cleaning introduced in [16] on the $\mathsf { M N I S T } ^ { 3 }$ dataset. The data is patitioned into a training set $( d _ { i } ^ { \mathrm { t r a i \bar { n } } } , y _ { i } ^ { \mathrm { t r a i n } } )$ , a validation set $( d _ { i } ^ { \mathrm { v a l } } , y _ { i } ^ { \mathrm { v a l } } )$ , and a test set. The training set contains 20000 samples, the validation set 5000 samples and the test set 10000 samples. The targets $y$ take values in $\{ 0 , \ldots , 9 \}$ and the samples $x$ are in dimension 784. Each sample in the training set is corrupted with probability $p$ : a sample is corrupted when we replace its label $y _ { i }$ by a random label in $\{ 0 , \ldots , 9 \}$ . Samples in the validation and test sets are not corrupted. The goal of datacleaning is to train a multinomial logistic regression on the train set and learn a weight per training sample, that should go to 0 for corrupted samples. This is formalized by the bilevel optimization problem (1) with $\begin{array} { r } { F ( \mathbf { \Sigma } \breve { \theta , \lambda } ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \dot { \ell } ( \theta d _ { i } ^ { \mathrm { v a l } } , \dot { y _ { i } ^ { \mathrm { v a l } } } ) } \end{array}$ and $\begin{array} { r } { G ( \theta , \lambda ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n ^ { - } } \sigma ( \lambda _ { i } ) \ell ( \theta d _ { i } ^ { \mathrm { t r a i n } } , y _ { i } ^ { \mathrm { t r a i n } } ) \dot { + } C _ { r } \| \theta \| ^ { 2 } } \end{array}$ where $\ell$ is the cross entropy loss and $\sigma$ is the sigmoid function. The inner variable $\theta$ is a matrix of size $1 0 \times 7 8 4$ , and the outer variable $\lambda$ is a vector in dimension $n _ { \mathrm { t r a i n } } = 2 0 0 0 0$ .
+
+For the estimated parameters $\theta$ during optimization, we report in Figure 2b the test error, i.e., the percent of wrong predictions on the testing data. We use for this experiment a corruption probability $p = 0 . 5$ . In general, the error decreases quickly until it reaches a final value. We observe that our method SABA outperforms all the other methods by reaching faster its smallest error, which is smaller than the ones of the other methods. For SOBA, it reaches a lower final error than stocBiO and BSA. In appendix, we provide other convergence curves, and find that for higher values of $p$ , SABA is still the fastest algorithm to reach its final accuracy. Overall, we find that among all methods, even those that implement variance reduction (that is FSLA, MRBO, SUSTAIN, SABA), SABA is the one that demonstrates the best empirical performance.
+
+# 5 Conclusion
+
+In this paper, we have presented a framework for bilevel optimization that enables the straightforward development of stochastic algorithms. The gist of our framework is that the directions in Equations (4) to (6) are all written as simple sums of samples derivatives. We leveraged this fact to propose SOBA, an extension of SGD to our framework, and SABA, an extension of SAGA to our framework, which both achieve similar convergence rates as their single level counterparts. Finally, we think that our framework opens a large panel of potential methods for stochastic bilevel optimization involving techniques of extrapolation, variance reduction, momentum and so on.
+
+# Acknowledgments and Disclosure of Funding
+
+We thank Othmane Sebbouh, Zaccharie Ramzi and Benoît Malézieux for their precious comments. The authors acknowledge the support of the ANER RAGA BFC. SV acknowledges the support of the ANR GraVa ANR-18-CE40-0005. This work is supported by a public grant overseen by the French National Research Agency (ANR) through the program UDOPIA, project funded by the ANR-20-THIA-0013-01 and DATAIA convergence institute (ANR-17-CONV-0003).
+
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+[46] R. Wengert. A simple automatic derivative evaluation program. Communications of the ACM, 7 (8):463–464, 1964.   
+[47] Junjie Yang, Kaiyi Ji, and Yingbin Liang. Provably Faster Algorithms for Bilevel Optimization. In Advances in Neural Information Processing Systems (NeurIPS). Curran Associates, Inc., 2021.
+
+# 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] See Section 3 and Section 4   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(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 Section 3.1 (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix C
+
+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] See Appendix B   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix 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]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(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/wlEOsQ917F/wlEOsQ917F_content_list.json

@@ -0,0 +1,1615 @@
+[
+    {
+        "type": "text",
+        "text": "A framework for bilevel optimization that enables stochastic and global variance reduction algorithms ",
+        "text_level": 1,
+        "bbox": [
+            186,
+            122,
+            812,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mathieu Dagréou ",
+        "text_level": 1,
+        "bbox": [
+            266,
+            227,
+            390,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Inria, CEA Université Paris-Saclay Palaiseau, France mathieu.dagreou@inria.fr ",
+        "bbox": [
+            223,
+            241,
+            431,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pierre Ablin CNRS Université Paris-Dauphine, PSL-University Paris, France pierre.ablin@cnrs.fr ",
+        "bbox": [
+            488,
+            226,
+            774,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Samuel Vaiter ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Thomas Moreau ",
+        "text_level": 1,
+        "bbox": [
+            586,
+            318,
+            702,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "CNRS Université Côte d’Azur, LJAD Nice, France samuel.vaiter@cnrs.fr ",
+        "bbox": [
+            258,
+            332,
+            459,
+            386
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Inria, CEA Université Paris-Saclay Palaiseau, France thomas.moreau@inria.fr ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O \\big ( \\frac { 1 } { T } \\big )$ convergence rate, and that it achieves linear convergence under Polyak-Łojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bilevel optimization is attracting more and more attention in the machine learning community thanks to its wide range of applications. Typical examples are hyperparameters selection [5, 38, 17, 6], data augmentation [11, 42], implicit deep learning [3] or neural architecture search [33]. Bilevel optimization aims at minimizing a function whose value depends on the result of another optimization problem: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/673a643345889d4e3248fc0df9e2e6740aec277c118909e511938888551c02a8.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathbb { R } ^ { d } } h ( x ) = F ( z ^ { * } ( x ) , x ) , \\quad \\mathrm { s u c h t h a t } z ^ { * } ( x ) \\in \\arg \\operatorname* { m i n } _ { z \\in \\mathbb { R } ^ { p } } G ( z , x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $F$ and $G$ are two real valued functions defined on $\\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { d }$ . $G$ is called the inner function, $F$ is the outer function and $h$ is the value function. Similarly, $z$ is the inner variable and $x$ is the outer variable. In most cases, the function $z ^ { * }$ can only be approximated by an optimization algorithm, which makes bilevel optimization problems challenging. Under appropriate hypotheses, the function $h$ is differentiable, and the chain rule and implicit function theorem give for any $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90ad97bdf05e07bbf69e0840bb834a6e93d616567a411c7bf450c6ae5a96dfac.jpg",
+        "text": "$$\n\\nabla h ( x ) = \\nabla _ { 2 } F ( z ^ { * } ( x ) , x ) + \\nabla _ { 2 1 } ^ { 2 } G ( z ^ { * } ( x ) , x ) v ^ { * } ( x ) \\ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "36th Conference on Neural Information Processing Systems (NeurIPS 2022). ",
+        "bbox": [
+            173,
+            922,
+            629,
+            936
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $v ^ { \\ast } ( x ) \\in \\mathbb { R } ^ { p }$ is the solution of a linear system ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/753ead0f5c30f5763cf9f2e08983156c9b1f3ab03e9fcb8c1002682f604aed00.jpg",
+        "text": "$$\n\\boldsymbol { v } ^ { * } ( \\boldsymbol { x } ) = - \\left[ \\nabla _ { 1 1 } ^ { 2 } G ( \\boldsymbol { z } ^ { * } ( \\boldsymbol { x } ) , \\boldsymbol { x } ) \\right] ^ { - 1 } \\nabla _ { 1 } F ( \\boldsymbol { z } ^ { * } ( \\boldsymbol { x } ) , \\boldsymbol { x } ) \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the light of (2) and (3), it turns out that the derivation of the gradient of $h$ at each iteration is cumbersome because it involves two subproblems: the resolution of the inner problem to find an approximation of $z ^ { * } ( x )$ and the resolution of a linear system to find an approximation of $v ^ { * } ( x )$ . It makes the practical implementation of first order methods like gradient descent for (1) challenging. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As is the case in many machine learning problems, we suppose in this paper that $F$ and $G$ are empirical means: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48952885d7753c9b07c3aebab5746a0ccea5fa554b21835026246a566fa9b494.jpg",
+        "text": "$$\nF ( z , x ) = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } F _ { j } ( z , x ) , \\quad G ( z , x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } G _ { i } ( z , x )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            173,
+            234,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This structure suggests the use of stochastic methods to solve (1). For single-level problems (that is, classical optimization problems where one function should be minimized), using Stochastic Gradient Descent (SGD; [41, 7]) and variants is natural because individual gradients are straightforward unbiased estimators of the gradient. In the bilevel framework, we want to develop algorithms that make progress on problem (1) by using only a few functions $F _ { j }$ and $G _ { i }$ at a time. However, since $\\nabla h$ involves the inverse of the Hessian of $G$ , building such stochastic algorithms is quite challenging, one of the difficulties being that there is no straightforward unbiased estimator of $\\nabla h$ . Still, in settings where $m$ or $n$ are large, where computing even a single evaluation of $F$ or $G$ is extremely expensive, stochastic methods are the only scalable algorithms. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/5b8583b8f4e3127a6f57e87a8eec2bdb05bebdf958e042d951a51cd676a39614.jpg",
+        "image_caption": [
+            "Figure 1: Convergence curves of the two proposed methods on a toy problem. SABA is a stochastic method that achieves fast convergence on the value function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            560,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Variance reduction [27, 13, 43, 15, 12] is a popular technique to obtain fast stochastic algorithms. In a single-level setting, these methods build an approximation of the gradient of the objective function using only stochastic gradients. Contrary to SGD, the variance of the approximation goes to 0 as the algorithm progresses, allowing for faster convergence. For instance, the SAGA method [13] achieves linear convergence if the objective function satisfies a Polyak-Łojasciewicz inequality, and $O ( { \\textstyle { \\frac { 1 } { T } } } )$ convergence rate on smooth non-convex functions [40]. The extension of these methods to bilevel optimization is a natural idea to develop faster algorithms. However, this idea is hard to implement because it is hard to derive unbiased estimators of $\\nabla h$ , let alone variance reduction ones. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions. We introduce a novel framework for bilevel optimization in Section 2, where the inner variable, the solution of the linear system (3) and the outer variable evolve jointly. The evolution directions are written as sums of derivatives of $F _ { j }$ and $G _ { i }$ , which allows us to derive simple unbiased stochastic estimators. In this framework, we propose SOBA, an extension of SGD (Section 2.1), and SABA (Section 2.2), an extension of the variance reduction algorithm SAGA [13]. In Section 3 we analyse the convergence of our methods. SOBA is shown to achieve $\\begin{array} { r } { \\operatorname* { i n f } _ { t \\leq T } \\mathbb { E } [ \\| \\nabla h ( x ^ { t } ) \\| ^ { 2 } ] = O ( \\log ( T ) T ^ { - \\frac { 1 } { 2 } } ) } \\end{array}$ with decreasing step sizes. We prove that SABA with fixed step sizes achieves $\\begin{array} { r } { \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ \\| \\nabla h ( x ^ { t } ) \\| ^ { 2 } ] = O ( \\frac { 1 } { T } ) } \\end{array}$ . SABA is therefore, to the best of our knowledge, the first stochastic bilevel algorithm that matches the convergence rate of gradient descent on $h$ . We also prove that SABA achieves linear convergence under the assumption that $h$ satisfies a Polyak-Łojasciewicz inequality. To the best of our knowledge, SABA is also the first stochastic bilevel algorithm to feature such a property. Importantly, these rates match the rates of the single level counterparts of each algorithm in non-convex setting (SGD for SOBA and SAGA for SABA). Finally, in Section 4, we provide an extensive benchmark of many stochastic bilevel methods on hyperparameters selection and data hyper-cleaning, and illustrate the usefulness of our approach. ",
+        "bbox": [
+            173,
+            580,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Related work. The bilevel optimization problem has a strong history in the optimization community, taking root in game theory [45]. Gradient-based algorithms to solve (1) can be mainly classified in two different categories depending on how $\\nabla h$ is computed, by automatic or implicit differentiation. ",
+        "bbox": [
+            174,
+            810,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Since the solution of the inner problem $z ^ { * } ( x )$ is approximated by the output of an iterative algorithm, it is possible to use automatic differentiation [46, 31] to approximate $\\nabla h ( x )$ . It consists in differentiating the different steps of the inner optimization algorithm – see [4] for a review – and has been applied successfully to several bilevel problems arising in machine learning [14, 16]. One of the main drawbacks of this approach is that it requires to store in memory each iterate of the inner optimization algorithm, although this problem can sometimes be overcome using invertible optimization algorithms [34] or truncated backpropagation [44]. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The use of the implicit function theorem to obtain (2) and (3) is known as implicit differentiation [5]. While the cost of computing exactly (2) can be prohibitive for large scale problems, Pedregosa [38] showed that we can still converge to a stationary point of the problem by using approximate solutions of the inner problem and linear system (3), if the approximation error goes to 0 sufficiently quickly. The complexity of approximate implicit differentiation has been studied in [20]. Ramzi et al. [39] propose to reuse the computations done in the forward pass to approximate the solution of the linear system (3) when the inner problem is solved thanks to a quasi-Newton method. ",
+        "bbox": [
+            173,
+            136,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the last few years, several works have proposed different strategies to solve (1) in a stochastic fashion. A first set of methods relies on two nested loops: one inner loop to solve the inner problem with a stochastic method, and one outer loop to update the outer variable with an approximate gradient direction. In [19, 26, 9] the authors use several SGD iterations for the inner problem and then use stochastic Neumann approximations to get an estimate solution of the linear system, which provides them with an approximation of $\\nabla h$ used to update $x$ . The analysis of this kind of method was refined by Chen et al. [9], allowing to achieve the same convergence rates as those of SGD. The convergence of the hypergradient when using stochastic solvers for the inner problem and the linear system has been studied in [21]. Arbel and Mairal [2] replace the Neumann approximation by SGD steps to estimate (3). Other authors have proposed single loop algorithms, alternating steps in the inner and the outer problem. Hong et al. [24] propose to perform Neumann approximations of the inverse Hessian and use a single SGD step for the inner problem. It was refined in [23] and [47] where the optimization procedure uses a momentum acceleration. Other variations around this idea include [25, 28, 10, 22, 30]. We refer to Table 1 in appendix for a detailed comparison of these methods. ",
+        "bbox": [
+            173,
+            237,
+            825,
+            444
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notation. The set of integers between 1 and $n$ (included) is denoted $[ n ]$ . For $f : \\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { d } \\to \\mathbb { R }$ we denote $\\nabla _ { i } f ( z , x )$ its gradient w.r.t. the $i ^ { \\mathrm { { t h } } }$ variable. The Hessian of $f$ with respect to the first variable is denoted $\\nabla _ { 1 1 } ^ { 2 } f ( z , x ) \\in \\mathbb { R } ^ { p \\times p }$ , and the cross-derivatives matrix is $\\nabla _ { 2 1 } ^ { 2 } f ( z , x ) \\in \\mathbb { R } ^ { d \\times p }$ . If $v$ is a vector, $\\lVert v \\rVert$ is its Euclidean norm. If $M$ is a matrix, $\\lVert M \\rVert$ is its spectral norm. A function is said to be $L$ -smooth, for $L > 0$ , if it is differentiable, and its gradient is $L$ -Lipschitz. ",
+        "bbox": [
+            173,
+            446,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Proposed framework ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            534,
+            382,
+            551
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce our framework in which the solution of the inner problem, the solution of the linear system (3) and the outer variable all evolve at the same time, following directions that are written as a sum of derivatives of $F _ { j }$ and $G _ { i }$ . We define ",
+        "bbox": [
+            173,
+            563,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 General framework ",
+        "text_level": 1,
+        "bbox": [
+            531,
+            569,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Input: initializations $z _ { 0 } \\in \\mathbb { R } ^ { p }$ , $\\overline { { x _ { 0 } \\in \\mathbb { R } ^ { d } } }$ $v _ { 0 } \\in \\mathbb { R } ^ { p }$ , number of iterations $T$ , step size sequences $( \\rho ^ { t } ) _ { t < T }$ and $( \\gamma ^ { t } ) _ { t < T }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c8ea2e88e87a5719ad7c39dc1f9c5aa334772c691070e1fa10a524f99dac7eab.jpg",
+        "text": "$$\nD _ { z } ( z , v , x ) = \\nabla _ { 1 } G ( z , x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            633,
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/996cecb29016fbe6174834ee4b2af1d24afbda38b250f2689bcf387c5fafa554.jpg",
+        "text": "$$\nD _ { v } ( z , v , x ) = \\nabla _ { 1 1 } ^ { 2 } G ( z , x ) v + \\nabla _ { 1 } F ( z , x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/c4a75ecef04aabb2255a9fa5dcc236b7c35d1ef10951a5f31261e1ea5640cad6.jpg",
+        "text": "$$\nD _ { x } ( z , v , x ) = \\nabla _ { 2 1 } ^ { 2 } G ( z , x ) v + \\nabla _ { 2 } F ( z , x ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "These directions are motivated by the fact that we have $\\nabla h ( x ) ~ = ~ D _ { x } ( z ^ { * } ( x ) , v ^ { * } ( \\bar { x } ) , x )$ , with $z ^ { * } ( x )$ the minimizer of $G ( \\cdot , x )$ and $v ^ { * } ( x )$ the solution of $\\nabla _ { 1 1 } ^ { 2 } G ( z ^ { * } ( x ) , x ) v \\ : = \\ : - \\nabla _ { 1 } F ( z ^ { * } ( x ) , x )$ . When $x$ is ",
+        "bbox": [
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+            694,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "end for ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "fixed, we approximate $z ^ { * }$ by doing a gradient descent on $G$ , following the direction $- D _ { z } ( z , v , x )$ . Finally, when $z$ and $x$ are fixed, we find $v ^ { * }$ by following the direction $- D _ { v } ( z , v , x )$ , which corresponds to a gradient descent on $\\begin{array} { r } { v \\mapsto \\frac { 1 } { 2 } \\langle \\nabla _ { 1 1 } ^ { 2 } G ( z , x ) v , v \\rangle + \\langle \\nabla _ { 1 } F ( z , x ) , v \\rangle } \\end{array}$ . The rest of the paper is devoted to the study of the global dynamics where the three variables $z , v$ and $x$ evolve at the same time, following stochastic approximations of $D _ { z } , D _ { v }$ and $D _ { x }$ . The next proposition motivates the choice of these directions. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Proposition 2.1. Assume that for all $x \\in \\mathbb { R } ^ { d } , G ( \\cdot , x )$ is strongly convex. If $( z , v , x )$ is a zero of $( D _ { z } , D _ { v } , D _ { x } )$ , then $z = z ^ { * } ( x )$ , $v = v ^ { * } ( x )$ and $\\nabla h ( x ) = ~ 0$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We also note that the computation of these directions does not require to compute the matrices $\\nabla _ { 1 1 } ^ { 2 } G ( z , x )$ and $\\nabla _ { 2 1 } ^ { 2 } G ( z , \\bar { x } )$ : we only need to compute their product with a vector, which can be computed at a cost similar to that of computing a gradient. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The framework we propose is summarized in Algorithm 1. It consists in following a joint update rule in $( z , v , x )$ that follows directions $D _ { z } ^ { t } , D _ { v } ^ { t }$ and $D _ { x } ^ { t }$ that are unbiased estimators of $D _ { z } , D _ { v } , D _ { x }$ The first and most important remark is that whereas $\\nabla h$ cannot be written as a sum over samples, the directions $D _ { z } , D _ { v }$ and $D _ { x }$ involve only simple sums, since their expressions are “linear” in $F$ and $G$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2cad3bc99892a0519dcf64b4d51cc53bbeebfb43daa1744f6954f349acaee224.jpg",
+        "text": "$$\n\\begin{array} { r l } & { D _ { z } ( z , v , x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla _ { 1 } G _ { i } ( z , x ) ~ , } \\\\ & { D _ { v } ( z , v , x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla _ { 1 1 } ^ { 2 } G _ { i } ( z , x ) v + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\nabla _ { 1 } F _ { j } ( z , x ) ~ , } \\\\ & { D _ { x } ( z , v , x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla _ { 2 1 } ^ { 2 } G _ { i } ( z , x ) v + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\nabla _ { 2 } F _ { j } ( z , x ) ~ . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is therefore straightforward to derive unbiased estimators of these directions. In [30], the authors considered one particular case of our framework, where each direction is estimated by using the STORM variance reduction technique (see [12]). Taking a step back by proposing the framework summarized in Algorithm 1 opens the way to potential new algorithms that implement other techniques that exist in stochastic single level optimization. In what follows, we study two of them. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.1 First example: the SOBA algorithm ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The simplest unbiased estimator is obtained by replacing each mean by one of its terms chosen uniformly at random, akin to what is done in classical single-level SGD. We call the resulting algorithm SOBA (StOchastic Bilevel Algorithm). To do so, we choose two independent random indices $i \\in [ n ]$ and $j \\in [ m ]$ uniformly and estimate each term coming from $G$ using $G _ { i }$ and each term coming from $F$ using $F _ { j }$ . This gives the unbiased SOBA directions ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a75d9ca6f4e1eb9d69ea5b9354eec61b127209a03944608b0531a9e1cefad59.jpg",
+        "text": "$$\n\\boxed { \\begin{array} { r l } & { D _ { z } ^ { t } = \\nabla _ { 1 } G _ { i } ( z ^ { t } , x ^ { t } ) ~ , } \\\\ & { D _ { v } ^ { t } = \\nabla _ { 1 1 } ^ { 2 } G _ { i } ( z ^ { t } , x ^ { t } ) v ^ { t } + \\nabla _ { 1 } F _ { j } ( z ^ { t } , x ^ { t } ) ~ , } \\\\ & { D _ { x } ^ { t } = \\nabla _ { 2 1 } ^ { 2 } G _ { i } ( z ^ { t } , x ^ { t } ) v ^ { t } + \\nabla _ { 2 } F _ { j } ( z ^ { t } , x ^ { t } ) ~ . } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This provides us with a first algorithm, SOBA, where we plug Equations (10a) to (10c) in Algorithm 1. We defer its analysis to the next section. Importantly, we use different step sizes for the update in $( z , v )$ and for the update in $x$ . We use the same step size in $z$ and in $v$ since the inner problem and the linear system have similar conditioning, which is that of $\\nabla _ { 1 1 } ^ { 2 } G ( z ^ { t } , x ^ { t } )$ . The need for a different step size for the outer and inner problem is clear: both problems can have a different conditioning. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "An important remark for SOBA is that all the stochastic directions used are computed at the same point $\\bar { z } ^ { t } , v ^ { t }$ and $x ^ { t }$ with the same indices $( i , j )$ . The update of $z$ , $v$ and $x$ can thus be performed in parallel instead of sequentially, benefiting from hardware parallelism. Moreover, this enables to share the computations between the different directions. This is the case in hyperparameters selection where $\\begin{array} { r } { \\bar { G } _ { i } ( z , x ) = \\ell _ { i } ( \\langle z , d _ { i } \\rangle ) + \\frac { x } { 2 } \\| z \\| ^ { 2 } } \\end{array}$ , with $d _ { i }$ a training sample, and $\\ell _ { i }$ that measures how good is the prediction $\\langle z , d _ { i } \\rangle$ . In this setting, we have $\\nabla _ { 1 } G _ { i } ( z , x ) = \\ell _ { i } ^ { \\prime } ( \\langle z , d _ { i } \\rangle ) d _ { i } + x z$ and $\\nabla _ { 1 1 } ^ { 2 } \\bar { G } _ { i } ( z , x ) v \\stackrel { - } { = } \\ell _ { i } ^ { \\prime \\prime } ( \\langle z , d _ { i } \\rangle ) \\langle v , d _ { i } \\rangle d _ { i }$ . The prediction $\\langle z , d _ { i } \\rangle$ can thus be computed only once to obtain both quantities. For more complicated models, where automatic differentiation is used to compute the different derivatives and Jacobian-vector products, we can store the computational graph only once to compute at the same time $\\nabla _ { 1 } G _ { i } ( z , x ) , \\dot { \\nabla _ { 1 1 } ^ { 2 } } G _ { i } ( z , x ) v$ and $\\nabla _ { 2 1 } ^ { 2 } G _ { i } ( z , x ) \\dot { v }$ , requiring only one backward pass, thanks to the $\\mathcal { R }$ technique [37]. ",
+        "bbox": [
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+            547,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, like all single loop bilevel algorithms, our method updates at the same time the inner and outer variable, avoiding unnecessary optimization of the inner problem when $x$ is far from the optimum. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 Global variance reduction with the SABA algorithm ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In classical optimization, SGD fails to reach optimal rates because of the variance of the gradient estimator. Variance reduction algorithms aim at reducing this variance, in order to follow directions that are closer to the true gradient, and to achieve superior practical and theoretical convergence. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our framework, since the directions $D _ { z } , D _ { v }$ and $D _ { x }$ are all written as sums of derivatives of $F _ { j }$ and $G _ { i }$ , it is easy to adapt most classical variance reduction algorithms. We focus on the celebrated SAGA algorithm [13]. The extension we propose is called SABA (Stochastic Average Bilevel Algorithm). The general idea is to replace each sum in the directions $D$ by a sum over a memory, updating only one term at each iteration. To help the exposition, we denote $y = ( z , x , v )$ the vector of joint variables. Since we have sums over $i$ and over $j$ , we have two memories for each variable: $\\boldsymbol { w } _ { i } ^ { t }$ for $i \\in [ n ]$ and $\\tilde { w } _ { j } ^ { t }$ for $j \\in [ m ]$ , which keep track of the previous values of the variable $y$ . ",
+        "bbox": [
+            174,
+            814,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "At each iteration $t$ , we draw two random independent indices $i \\in [ n ]$ and $j \\in [ m ]$ uniformly and update the memories. To do so, we put $w _ { i } ^ { t + 1 } = y ^ { t }$ and $w _ { i ^ { \\prime } } ^ { t + 1 } = w _ { i ^ { \\prime } } ^ { t }$ for $i ^ { \\prime } \\neq i$ , and $\\tilde { w } _ { j } ^ { t + 1 } = y ^ { t }$ and $\\tilde { w } _ { j ^ { \\prime } } ^ { t + 1 } = \\tilde { w } _ { j ^ { \\prime } } ^ { t }$ for $\\boldsymbol { j ^ { \\prime } } \\neq \\boldsymbol { j }$ . Each sum in the directions $D$ is then approximated using SAGA-like rules: given $n$ functions $\\phi _ { i ^ { \\prime } }$ for $i ^ { \\prime } \\in [ n ]$ , we define $\\begin{array} { r } { S [ \\phi , w ] _ { i } ^ { t } = \\phi _ { i } ( w _ { i } ^ { t + 1 } ) - \\phi _ { i } ( w _ { i } ^ { t } ) + \\frac { 1 } { n } \\sum _ { i ^ { \\prime } = 1 } ^ { n } \\phi _ { i ^ { \\prime } } ( w _ { i ^ { \\prime } } ^ { t } ) . } \\end{array}$ This is an unbiased estimators of the average of the $\\phi$ ’s since $\\begin{array} { r } { \\mathbb E _ { i } \\Big [ S [ \\phi , w ] _ { i } ^ { t } \\Big ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\phi _ { i } ( y ^ { t } ) } \\end{array}$ . ",
+        "bbox": [
+            173,
+            90,
+            826,
+            181
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With a slight abuse of notation, we call $\\nabla _ { 1 1 } ^ { 2 } G v$ the sequence of functions $( y \\mapsto \\nabla _ { 1 1 } ^ { 2 } G _ { i } ( z , x ) v ) _ { i \\in [ n ] }$ and $\\nabla _ { 2 1 } ^ { 2 } G v$ the sequence of functions $( y \\mapsto \\nabla _ { 2 1 } ^ { 2 } G _ { i } ( z , x ) v ) _ { i \\in [ n ] }$ . We define the SABA directions as ",
+        "bbox": [
+            176,
+            185,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/851627f4797c8bee48086eb1f575679b6b9eb4c71c728ebde4ec66df75a4e654.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { } } & { { } } & { { D _ { z } ^ { t } = S [ \\nabla _ { 1 } G , w ] _ { i } ^ { t } ~ , } } \\\\ { { } } & { { } } & { { D _ { v } ^ { t } = S [ \\nabla _ { 1 1 } ^ { 2 } G v , w ] _ { i } ^ { t } + S [ \\nabla _ { 1 } F , \\tilde { w } ] _ { j } ^ { t } ~ , } } \\\\ { { } } & { { } } & { { D _ { x } ^ { t } = S [ \\nabla _ { 2 1 } ^ { 2 } G v , w ] _ { i } ^ { t } + S [ \\nabla _ { 2 } F , \\tilde { w } ] _ { j } ^ { t } ~ . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "These estimators are unbiased estimators of the directions $D _ { z } , D _ { v }$ and $D _ { x }$ . The SABA algorithm corresponds to Algorithm 1 where we use Equations (11a) to (11c) as update directions. When taking a step size $\\gamma ^ { t } = \\bar { 0 }$ in the outer problem, hereby stopping progress in $x$ , we recover the iterations of the SAGA algorithm on the inner problem. In practice, the sum in $S$ is computed by doing a rolling average (see Appendix B for precision), and the quantities $\\phi _ { i } ( w _ { i } ^ { t } )$ are stored rather than recomputed: the cost of computing the SABA directions is the same as that of SGD. It requires an additional memory for the five quantities, of total size $n \\times p + ( n + m ) \\times ( p + d )$ floats that can be reduced by using larger batch sizes. Indeed, if $b _ { \\mathrm { i n } }$ and $b _ { \\mathrm { o u t } }$ are respectively the inner and the outer batch sizes, the memory load is reduced to nb × p + (nb + mb) × (p × d) with nb = ⌈ nbinn ⌉ and $\\begin{array} { r } { m _ { b } = \\left\\lceil \\frac { m } { b _ { \\mathrm { o u t } } } \\right\\rceil } \\end{array}$ which are smaller than the number of samples. This memory load can also be reduced in specific cases, for instance when $G$ and $F$ correspond to linear models, where the individual gradients and Hessian-vector products are proportional to the samples. In this case, we only store the proportionality ratio, reducing the memory load to $3 n + 2 m$ floats. Like for SOBA, the computations of the new quantities $\\phi _ { i } ( w _ { i } ^ { t + 1 } )$ are done in parallel, thus benefiting from hardware acceleration and shared computations. Despite this memory load, using SAGA-like variance reduction instead of STORM as done in [30, 47, 28] has the advantage to bring the variance of the estimate directions to zero, enabling faster $O ( { \\textstyle { \\frac { 1 } { T } } } )$ convergence. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the next section, we show that SABA is fast. It essentially has the same properties as SAGA: despite being stochastic, it converges with fixed step sizes, and reaches the same rate of convergence as gradient descent on $h$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Theoretical analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            592,
+            372,
+            608
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide convergence rates of SOBA and SABA under some classical assumptions. Note that, unlike most of the stochastic bilevel optimization papers, we work in finite sample setting rather than the more general expectation setting. Actually, SABA does not make any sense for functions that don’t have a finite sum structure. However, we stress that SOBA could be studied in a more general setting to obtain the same bounds as here. Also, the finite sum setting is still interesting since doing empirical risk minimization is very common in practice in machine learning. The proofs and the constants in big- $O$ are deferred in Appendix C. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Background and assumptions ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We start by stating some regularity assumptions on the functions $F$ and $G$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.1. The function $F$ is twice differentiable. The derivatives $\\nabla F$ and $\\nabla ^ { 2 } F$ are Lipschitz continuous in $( z , x )$ with respective Lipschitz constants $L _ { 1 } ^ { F }$ and $L _ { 2 } ^ { F }$ . ",
+        "bbox": [
+            173,
+            768,
+            820,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the above assumption is typically verified in the machine learning context, e.g., when $F$ is the ordinary least squares (OLS) loss or the logistic loss. ",
+        "bbox": [
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+            803,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.2. The function $G$ is three times continuously differentiable on $\\mathbb { R } ^ { p } \\times \\mathbb { R } ^ { d }$ . For any $x \\in \\mathbb { R } ^ { d } , G ( \\cdot , x )$ is $\\mu _ { G }$ -strongly convex. The derivatives $\\overrightarrow { \\nabla G }$ , $\\nabla ^ { 2 } G$ and $\\nabla ^ { 3 } G$ are Lipschitz continuous in $( z , x )$ with respective Lipschitz constants $L _ { 1 } ^ { G }$ , $L _ { 2 } ^ { G }$ and $L _ { 3 } ^ { G }$ . ",
+        "bbox": [
+            174,
+            834,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Strong convexity and smoothness with respect to $z$ of $G$ are verified when $G$ is a regularized leastsquares/logistic regression with a full rank design matrix, when the data is not separable for the logistic regression. Moreover, the strong convexity ensures the existence and uniqueness of the inner optimization problem for any $x \\in \\mathbb { R } ^ { d }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.3. There exists $C _ { F } ~ > ~ 0$ such that for any $x$ we have $\\| \\nabla _ { 1 } F ( z ^ { * } ( x ) , x ) \\| \\leq C _ { F } .$ ",
+        "bbox": [
+            171,
+            123,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This assumption, combined with the strong convexity of $G ( \\cdot , x )$ , shows boundedness of $v ^ { * }$ . This assumption holds, for instance, in the case of hyperparameters selection for a Ridge regression problem. Note that in Assumptions 3.1 and 3.2, we assume more regularity of $F$ and $G$ than in stochastic bilevel optimization literature (see for instance [19, 24, 26, 2]). It is necessary to get the smoothness of $v ^ { * }$ which will allow to adapt the proof of Chen et al. [9] and get tight convergence rates. The following lemma gives us some smoothness properties of the considered directions that will be useful to derive convergence rates of our methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.4. Under the Assumptions 3.1 to 3.3, there exist constants $L _ { z }$ , $L _ { v }$ and $L _ { x }$ such that $\\begin{array} { r } { \\| D _ { z } ( z , v , x ) \\| ^ { 2 } \\leq L _ { z } ^ { 2 } \\| z - z ^ { * } ( \\hat { x } ) \\| ^ { 2 } , \\ \\| D _ { v } ( z , v , x ) \\| ^ { 2 } \\leq L _ { v } ^ { 2 } ( \\| z - z ^ { * } ( x ) \\| ^ { 2 } + \\| v - v ^ { * } ( x ) \\| ^ { 2 } ) } \\end{array}$ and $\\| D _ { x } ( z , v , x ) - \\nabla h ( x ) \\| ^ { 2 } \\leq L _ { x } ^ { 2 } ( \\| z - z ^ { * } ( x ) \\| ^ { 2 } + \\| v - v ^ { * } ( x ) \\| ^ { 2 } )$ . ",
+        "bbox": [
+            173,
+            248,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In first order optimization, a fundamental assumption on the objective function is the smoothness assumption. In the case of vanilla gradient descent applied to a function $f$ , it allows to get a convergence rate of $\\| \\nabla f ( x ^ { t } ) \\| ^ { 2 }$ in $O ( \\bar { 1 } / T )$ , i.e. convergence to a stationary point [36]. The following lemma proved by Ghadimi and Wang [19, Lemma 2.2] ensures the smoothness of $h$ . ",
+        "bbox": [
+            173,
+            299,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.5. Under the Assumptions 3.1 to 3.3, the function $h$ is $L ^ { h }$ -smooth for some $L ^ { h } > 0$ . ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The constant $L ^ { h }$ is specified in Appendix C.3. As usual with the analysis of stochastic methods, we define the expected norms of the directions $V _ { z } ^ { t } = \\mathbb { E } [ \\| D _ { z } ^ { t } \\| ^ { 2 } ]$ , $V _ { v } ^ { t } = \\dot { \\mathbb { E } } [ \\| D _ { v } ^ { t } \\| ^ { 2 } ]$ and $V _ { x } ^ { t } = \\mathbb { E } [ \\| D _ { x } ^ { t } \\| ^ { 2 } ]$ , where the expectation is taken over the past. Thanks to variance-bias decomposition, they are the sum of the variance of the stochastic direction and the squared-norm of the unbiased direction. For SOBA, we use classical bounds on variances like those found for instance in [24]: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.6. There exists $B _ { z }$ , $B _ { v }$ and $B _ { x }$ such that for all $t$ , $\\begin{array} { r l r } { \\mathbb { E } _ { t } [ \\| D _ { z } ^ { t } \\| ^ { 2 } ] } & { { } \\le } & { B _ { z } ^ { 2 } ( 1 + \\| D _ { z } ( z ^ { t } , v ^ { t } , x ^ { t } ) \\| ^ { 2 } ) } \\end{array}$ and $\\begin{array} { r } { \\mathbb { E } _ { t } [ \\| D _ { v } ^ { t } \\| ^ { 2 } ] \\ \\leq \\ B _ { v } ^ { 2 } ( 1 + \\| D _ { v } ( z ^ { t } , v ^ { t } , x ^ { t } ) \\| ^ { 2 } ) } \\end{array}$ where $\\mathbb { E } _ { t }$ denotes the expectation conditionally to $( z ^ { t } , v ^ { t } , \\ddot { x } ^ { t } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For SOBA and SABA, we need to bound the expected norm of $D _ { x } ^ { t }$ . For SABA, this assumption allows to get a the same sample complexity as SAGA for single level problems. ",
+        "bbox": [
+            173,
+            506,
+            823,
+            535
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.7. There exists $B _ { x }$ such that for all $t$ , $\\mathbb { E } _ { t } [ \\| D _ { x } ^ { t } \\| ^ { 2 } ] \\leq B _ { x } ^ { 2 }$ . ",
+        "bbox": [
+            174,
+            537,
+            642,
+            554
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumptions 3.6 and 3.7 are verified for instance, if all the $G _ { i }$ and $\\nabla _ { 1 } G _ { i }$ have at most quadratic growth, and if $F$ has bounded gradients. They are also verified if the iterates remain in a compact set. Note that we do not assume that $G$ has bounded gradients, as this would contradict its strong-convexity. Finally, for the analysis of SABA, we need regularity on each $G _ { i }$ and $F _ { j }$ : ",
+        "bbox": [
+            174,
+            560,
+            826,
+            617
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.8. For all $i \\in [ n ]$ and $j \\in [ m ]$ , the functions $\\nabla G _ { i }$ , $\\nabla F _ { j }$ , $\\nabla _ { 1 1 } ^ { 2 } G _ { i }$ and $\\nabla _ { 2 1 } ^ { 2 } G _ { i }$ are Lipschitz continuous in $( z , x )$ . ",
+        "bbox": [
+            174,
+            622,
+            823,
+            652
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Fundamental descent lemmas ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            664,
+            419,
+            679
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our analysis for SOBA and SABA is based on the control of both $\\delta _ { z } ^ { t } = \\mathbb { E } [ \\| z ^ { t } - z ^ { * } ( x ^ { t } ) \\| ^ { 2 } ]$ and $\\delta _ { v } ^ { t } = \\mathbb E [ | | \\bar { v } ^ { t } - v ^ { * } ( x ^ { t } ) | | ^ { 2 } ]$ , Strong convexity of $G$ and smoothness of $z ^ { * } ( x )$ and $v ^ { * } ( x )$ allow to obtain the following lemma by adapting the proof of Chen et al. [9]. In what follows, we drop the dependency of the step sizes $\\rho$ and $\\gamma$ in $t$ for clarity. ",
+        "bbox": [
+            173,
+            688,
+            825,
+            746
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.9. Assume that $\\begin{array} { r } { \\gamma ^ { 2 } \\leq \\operatorname* { m i n } \\left( \\frac { \\mu _ { G } L _ { * } ^ { 2 } } { 4 B _ { x } ^ { 2 } L _ { z x } ^ { 2 } } , \\frac { \\mu _ { G } L _ { * } ^ { 2 } } { 8 B _ { x } ^ { 2 } L _ { v x } ^ { 2 } } \\right) \\rho } \\end{array}$ . We have: ",
+        "bbox": [
+            173,
+            748,
+            624,
+            775
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f65f526ac4a0f42f4512be04e32029c311e51a7726b022d8c22f66a6cb7df06b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\delta _ { z } ^ { t + 1 } \\leq \\left( 1 - \\frac { \\rho \\mu _ { G } } { 4 } \\right) \\delta _ { z } ^ { t } + 2 \\rho ^ { 2 } V _ { z } ^ { t } + \\beta _ { z x } \\gamma ^ { 2 } V _ { x } ^ { t } + \\overline { { \\beta } } _ { z x } \\frac { \\gamma ^ { 2 } } { \\rho } \\mathbb { E } [ \\| D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) \\| ^ { 2 } ] } \\\\ & { \\delta _ { v } ^ { t + 1 } \\leq \\left( 1 - \\frac { \\rho \\mu _ { G } } { 8 } \\right) \\delta _ { v } ^ { t } + \\beta _ { v z } \\rho \\delta _ { z } ^ { t } + 2 \\rho ^ { 2 } V _ { v } ^ { t } + \\beta _ { v x } \\gamma ^ { 2 } V _ { x } ^ { t } + \\overline { { \\beta } } _ { z x } \\frac { \\gamma ^ { 2 } } { \\rho } \\mathbb { E } [ \\| D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) \\| ^ { 2 } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            215,
+            780,
+            782,
+            851
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta _ { z x } = \\beta _ { v x } = 3 L _ { * } ^ { 2 }$ , $\\begin{array} { r } { \\overline { { \\beta } } _ { z x } \\ : = \\ : \\frac { 8 L _ { * } ^ { 2 } } { \\mu _ { G } } } \\end{array}$ , $\\begin{array} { r } { \\overline { { \\beta } } _ { v x } = { \\frac { 1 6 L _ { * } ^ { 2 } } { \\mu _ { G } } } } \\end{array}$ , $L _ { * }$ is the maximum between the Lipschitz constants of and ∗ zx (see Lemma ), ， , and are respectively $z ^ { * }$ $v ^ { * }$ $C . I$ $\\begin{array} { r } { \\beta _ { v z } = \\frac { \\mathrm { ~ i ~ } } { \\mu _ { G } ^ { 3 } } ( L _ { 1 } ^ { F } \\mu _ { G } + L _ { 2 } ^ { G } ) ^ { 2 } } \\end{array}$ $L _ { z x }$ $L _ { v x }$ the smoothness constants of $z ^ { * }$ and $v ^ { * }$ . ",
+        "bbox": [
+            174,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We insist that this result is obtained in general for Algorithm 1 with arbitrary unbiased directions. We can therefore invoke this lemma for the analysis of both SOBA and SABA. We use the smoothness of $h$ to get the following lemma, which is similar to [9, Lemma 1]. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.10. Let $h ^ { t } = \\mathbb { E } [ h ( x ^ { t } ) ]$ and $g ^ { t } = \\mathbb { E } [ \\| \\nabla h ( x ^ { t } ) \\| ^ { 2 } ]$ . We have ",
+        "bbox": [
+            173,
+            136,
+            620,
+            152
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60a8b5e303f000db113358f398206514b13d198f9e3b9f31ffa2140a730db22f.jpg",
+        "text": "$$\nh ^ { t + 1 } \\leq h ^ { t } - \\frac { \\gamma } { 2 } g ^ { t } - \\frac { \\gamma } { 2 } \\mathbb { E } [ \\| D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) \\| ^ { 2 } ] + \\frac { \\gamma } { 2 } L _ { x } ^ { 2 } ( \\delta _ { z } ^ { t } + \\delta _ { v } ^ { t } ) + \\frac { L ^ { h } } { 2 } \\gamma ^ { 2 } V _ { x } ^ { t } ~ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            157,
+            745,
+            189
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "If $z ^ { t } = z ^ { * } ( x ^ { t } )$ , $v ^ { t } = v ^ { * } ( x ^ { t } )$ , that is $\\delta _ { z } , \\delta _ { v }$ both cancel and $D _ { x } ( z ^ { t } , v ^ { t } , x ^ { t } ) = \\nabla h ( x ^ { t } )$ , we get an inequality reminiscent of the smoothness inequality for SGD on $h$ . ",
+        "bbox": [
+            173,
+            196,
+            825,
+            227
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 Analysis of SOBA ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            238,
+            338,
+            255
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The analysis of SOBA is based on Lemmas 3.5 and 3.9. We have the following theorem, with fixed step sizes depending on the number of iterations: ",
+        "bbox": [
+            173,
+            261,
+            823,
+            290
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 (Convergence of SOBA, fixed step size). Fix an iteration $T > 1$ and assume that Assumptions 3.1 to 3.7 hold. We consider fixed steps $\\begin{array} { r } { \\rho ^ { t } = \\frac { \\overline { { \\rho } } } { \\sqrt { T } } } \\end{array}$ and $\\gamma ^ { t } = \\xi \\rho ^ { t }$ with $\\overline { \\rho }$ and $\\xi$ precised in the appendix. Let $( x ^ { t } ) _ { t \\geq 1 }$ the sequence of outer iterates for SOBA. Then, ",
+        "bbox": [
+            173,
+            299,
+            826,
+            349
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9da4ca1859e2f4486e6a59330ee7e0b44553df77ed6dbacadf22987a95fb4189.jpg",
+        "text": "$$\n\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ \\| \\nabla h ( x ^ { t } ) \\| ^ { 2 } ] = O ( T ^ { - \\frac { 1 } { 2 } } ) \\mathrm { ~ . ~ }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            356,
+            617,
+            398
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As opposed to [24], we do not need that the ratio $\\frac { \\gamma } { \\rho }$ goes to 0, which allows to get a complexity (that is, the number of call to oracles to have an $\\epsilon$ -stationary solution) in $O ( \\epsilon ^ { - 2 } )$ better than the $\\tilde { O } ( \\epsilon ^ { - \\frac { 5 } { 2 } } )$ they have. Also, note that this rate is the same as the one of SGD for non-convex and smooth objective [18, 8]. We obtain a similar rate using decreasing step sizes: ",
+        "bbox": [
+            173,
+            407,
+            826,
+            472
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 (Convergence of SOBA, decreasing step size). Assume that Assumptions 3.1 to 3.7 hold. We consider steps $\\rho ^ { t } = \\overline { { \\rho } } t ^ { - \\frac { 1 } { 2 } }$ and $\\gamma ^ { t } = \\xi \\rho$ . Let $x ^ { t }$ the sequence of outer iterates for SOBA. Then, ",
+        "bbox": [
+            173,
+            481,
+            823,
+            512
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/76826469efea4ca34fad1e18a86a761b926bb4edffb1f2b761a265b1f6c9ee23.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { t \\leq T } \\mathbb { E } [ \\| \\nabla h ( x ^ { t } ) \\| ^ { 2 } ] = O ( \\log ( T ) T ^ { - { \\frac { 1 } { 2 } } } ) \\enspace .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            518,
+            629,
+            545
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As for SGD, SOBA suffers from the need of decreasing step sizes to get actual convergence because of the variance of the estimation on each directions. On the other hand, the analysis of SABA leverages the dynamic of all three variables, resulting in fast convergence with fixed step sizes. ",
+        "bbox": [
+            173,
+            554,
+            826,
+            598
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.4 SABA: a stochastic method with optimal rates ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            609,
+            534,
+            626
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In what follows, we denote $N = n + m$ the total number of samples. The following theorem shows $O ( N ^ { \\frac { 2 } { 3 } } T ^ { - 1 } )$ convergence for the SABA algorithm in the general case where we only assume smoothness of $h$ . Our analysis of SABA is inspired by the analysis of single-level SAGA by Reddi et al. [40]. ",
+        "bbox": [
+            173,
+            632,
+            825,
+            690
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 (Convergence of SABA, smooth case). Assume that Assumptions 3.1 to 3.3 and 3.7 to 3.8 hold. We suppose $\\rho = \\rho ^ { \\prime } N ^ { - \\frac { 2 } { 3 } }$ and $\\gamma = \\xi \\rho$ , where $\\rho ^ { \\prime }$ and $\\xi$ depend only on $F$ and $G$ and are specified in appendix. Let $x ^ { t }$ the iterates of SABA. Then, ",
+        "bbox": [
+            171,
+            699,
+            826,
+            744
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa9814b49bb946cfb9c109ef1f9ae2cd5d2de464d1eeead1b48bf982b7720ed3.jpg",
+        "text": "$$\n\\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ \\| \\nabla h ( x ^ { t } ) \\| ^ { 2 } ] = O \\left( N ^ { \\frac { 2 } { 3 } } T ^ { - 1 } \\right) \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            751,
+            633,
+            794
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To prodefine $\\begin{array} { r } { { S } ^ { t } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\| y ^ { t } - w _ { i } ^ { t } \\| ^ { 2 } + \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\| y ^ { t } - \\tilde { w } _ { j } ^ { t } \\| ^ { 2 } } \\end{array}$ m the memory to the current variables. We. In appendix, we show that we can find $\\phi _ { s } , \\phi _ { z } , \\phi _ { v } > 0$ such that the quantity $\\mathcal { L } ^ { t } = h ^ { t } + \\phi _ { s } S ^ { t } + \\phi _ { z } \\delta _ { z } ^ { t } + \\phi _ { v } \\delta _ { v } ^ { t }$   \nSumming these inequalities for $t = 1 \\dots T$ and using the fact that $\\textstyle { \\mathcal { L } } ^ { t }$ is lower bounded demonstrates   \nthe theorem. ",
+        "bbox": [
+            173,
+            803,
+            826,
+            878
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that the step sizes are constant with respect to the time, but they scale with $N ^ { - \\frac { 2 } { 3 } }$ . As a consequence, the sample complexity is $O ( N ^ { \\frac { 2 } { 3 } } \\epsilon ^ { - 1 } )$ which is analogous of the one of SAGA for non-convex single level problems [40]. This is better than the sample complexity of Algorithm 1 with full batch directions, which is $O ( N \\epsilon ^ { - 1 } )$ . Hence, with SABA, we get the best of both worlds: the stochasticity makes the scaling in $N$ of the sample complexity goes from $N$ in full batch mode to $N ^ { \\frac { 2 } { 3 } }$ for SABA, and the variance reduction makes the scaling in $\\epsilon$ goes from $\\epsilon ^ { - 2 }$ for SOBA to $\\epsilon ^ { - 1 }$ for SABA. Our experiments in Section 4 confirm this gain. ",
+        "bbox": [
+            174,
+            881,
+            823,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            164
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, if we assume that $h$ satisfies a Polyak-Łojasiewicz (PL) inequality, we recover linear convergence. Recall that $h$ has the PL property if there exists $\\mu _ { h } > 0$ such that for all $x \\in \\mathbb { R } ^ { d }$ $\\begin{array} { r } { \\frac { 1 } { 2 } \\| \\nabla h ( \\bar { \\boldsymbol { x } } ) \\| ^ { 2 } \\geq \\mu _ { h } ( h ( \\boldsymbol { x } ) - h ^ { * } ) } \\end{array}$ with $h ^ { * }$ the minimum of $h$ . ",
+        "bbox": [
+            176,
+            166,
+            823,
+            209
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4 (Convergence of SABA, PL case). Assume that $h$ satisfies the PL inequality and that Assumptions 3.1 to 3.3 and 3.7 to 3.8 hold. We suppose $\\rho = \\rho ^ { \\prime } N ^ { - \\frac { 2 } { 3 } }$ and $\\gamma = \\xi \\rho ^ { \\prime } N ^ { - 1 }$ , where $\\rho ^ { \\prime }$ and $\\xi$ depend only on $F$ and $G$ and are specified in appendix. Let $x ^ { t }$ the iterates of SABA and $\\begin{array} { r } { c ^ { \\prime } \\triangleq \\operatorname* { m i n } \\left( \\mu _ { h } , \\frac { 1 } { 1 6 P ^ { \\prime } } \\right) } \\end{array}$ with $P ^ { \\prime }$ specified in the appendix. Then, ",
+        "bbox": [
+            173,
+            219,
+            826,
+            281
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d171f703a0c21f685a2182fd2abaa97c57ea243855b8b097d1c3c03eb9ed610a.jpg",
+        "text": "$$\n\\mathbb { E } [ h ^ { T } ] - h ^ { * } = ( 1 - c ^ { \\prime } \\gamma ) ^ { T } ( h ^ { 0 } - h ^ { * } + C ^ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            287,
+            638,
+            306
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $C ^ { 0 }$ is a constant specified in appendix that depends on the initialization of $z , v , x$ and memory. ",
+        "bbox": [
+            173,
+            313,
+            823,
+            328
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The proof is similar to that of the previous theorem: we find coefficients $\\phi _ { s } , \\phi _ { z } , \\phi _ { v }$ such that $\\mathcal { L } ^ { t } \\stackrel {  } { = } h ^ { t } + \\phi _ { s } S ^ { t } + \\phi _ { z } \\delta _ { z } ^ { t } + \\phi _ { v } \\delta _ { v } ^ { t }$ satisfies the inequality $\\mathcal { L } ^ { t + 1 } \\leq ( 1 - c ^ { \\prime } \\gamma ) \\mathcal { L } ^ { t }$ , which is then unrolled. Note that in the case where we initialize $z$ and $v$ with $z ^ { 0 } = z ^ { * } ( x ^ { 0 } )$ , $v ^ { 0 } = v ^ { * } ( x ^ { 0 } )$ , and the memories $w _ { i } ^ { 0 } = w ^ { 0 }$ , $\\tilde { w } _ { j } ^ { 0 } = w ^ { 0 }$ for all $i , j$ , the constant $C ^ { 0 }$ cancels and the bound simplifies to $\\mathbb { E } [ h ( x ^ { T } ) ] - h ^ { * } \\leq ( 1 - c ^ { \\prime } \\gamma ) ^ { T } ( h ( x ^ { 0 } ) - h ^ { * } )$ . ",
+        "bbox": [
+            173,
+            337,
+            825,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Just like classical variance reduction methods in single-level optimization, this theorem shows that our method achieves linear convergence under PL assumption on the value function. To the best of our knowledge, our method is the first stochastic bilevel optimization method that enjoys such property. We note that the PL hypothesis is more general than $\\mu _ { h }$ -strong convexity of $h - \\mathrm { i t }$ is a necessary condition for strong convexity. ",
+        "bbox": [
+            174,
+            414,
+            825,
+            483
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We see here the importance of global variance reduction. Indeed, using variance reduction only on $z$ and SGD on $x$ would lead to sub-linear convergence in $x$ . This would be the case even with a perfect estimation of $z ^ { * } ( x )$ . Similarly, using variance reduction only on $x$ and SGD on $z$ would lead to sub-linear convergence in $z$ , and hence in $x$ . Using global variance reduction with respect to each variable as we propose here is the only way to achieve linear convergence. We now turn to experiments, where we find that our method is also promising from a practical point of view. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            587,
+            312,
+            603
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here we compare the performances of SOBA and SABA with competitor methods on different tasks. The different methods being compared are stocBiO [26], AmiGO [2], FSLA [30], MRBO [47], TTSA [24], BSA [19] and SUSTAIN [28]. A detailed account of the experiments is provided in Appendix B. 1 ",
+        "bbox": [
+            174,
+            616,
+            826,
+            676
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 Hyperparameters selection ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            690,
+            400,
+            705
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The first task we perform is hyperparameters selection to choose regularization parameters on $\\ell ^ { 2 }$ logistic regression. Let us denote $( ( d _ { i } ^ { \\mathrm { t r a i n } } , y _ { i } ^ { \\mathrm { t r a i n } } ) ) _ { 1 \\leq i \\leq n }$ and $( ( d _ { i } ^ { \\mathrm { v a l } } , y _ { i } ^ { \\mathrm { v a l } } ) ) _ { 1 \\leq i \\leq m }$ the training and the validation sets. In this case, the inner variable $\\bar { \\theta }$ corresponds to the parameters of the model, and the outer variable $\\lambda$ to the regularization. The functions $F$ and $G$ of the problem (1) are the logistic loss, with $\\ell ^ { 2 }$ penalty for $G$ , that is to say $\\begin{array} { r } { F ( \\theta , \\lambda ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\varphi ( y _ { i } ^ { \\mathrm { v a l } } \\langle \\bar { d } _ { i } ^ { \\mathrm { v a l } } , \\theta \\rangle ) } \\end{array}$ and $\\begin{array} { r } { G ( \\theta , \\lambda ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\varphi ( y _ { i } ^ { \\mathrm { t r a i n } } \\langle d _ { i } ^ { \\mathrm { t r a i n } } , \\theta \\rangle ) + \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { p } e ^ { \\lambda _ { k } } \\theta _ { k } ^ { 2 } } \\end{array}$ where $\\varphi ( u ) = \\log ( 1 + e ^ { - u } )$ . We fit a binary classification model on the IJCNN $1 ^ { 2 }$ dataset. Here, $n = 4 9 9 9 0$ , $m = 9 1 7 0 1$ and $p = 2 2$ . ",
+        "bbox": [
+            173,
+            713,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The suboptimality gap is plotted in Figure $2 \\mathrm { a }$ for each method. The lowest values are reached by SABA. Moreover, SABA is the only single-loop method that reaches a suboptimality below $1 0 ^ { - 5 }$ SOBA reaches a quite high final value but slightly better than TTSA and FSLA. The gap between ",
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+        "image_caption": [
+            "Figure 2: Comparison of SOBA and SABA with other stochastic bilevel optimization methods. For each algorithm, we plot the median performance over 10 runs. In both experiments, SABA achieves the best performance. The dashed lines are for one loop competitor methods, the dotted lines are for two loops methods and the solid lines are the proposed methods. Left: hyperparameter selection for $\\ell ^ { 2 }$ penalized logistic regression on IJCNN1 dataset , Right: data hyper-cleaning on MNIST with $p = 0 . 5$ corruption rate. "
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+        "text": "SOBA and SABA highlights the benefits of variance reduction: it gives us a lower plateau and the fixed step sizes enable faster convergence. ",
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+        "text": "4.2 Data hyper-cleaning ",
+        "text_level": 1,
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+        "text": "The second task we perform is data hyper-cleaning introduced in [16] on the $\\mathsf { M N I S T } ^ { 3 }$ dataset. The data is patitioned into a training set $( d _ { i } ^ { \\mathrm { t r a i \\bar { n } } } , y _ { i } ^ { \\mathrm { t r a i n } } )$ , a validation set $( d _ { i } ^ { \\mathrm { v a l } } , y _ { i } ^ { \\mathrm { v a l } } )$ , and a test set. The training set contains 20000 samples, the validation set 5000 samples and the test set 10000 samples. The targets $y$ take values in $\\{ 0 , \\ldots , 9 \\}$ and the samples $x$ are in dimension 784. Each sample in the training set is corrupted with probability $p$ : a sample is corrupted when we replace its label $y _ { i }$ by a random label in $\\{ 0 , \\ldots , 9 \\}$ . Samples in the validation and test sets are not corrupted. The goal of datacleaning is to train a multinomial logistic regression on the train set and learn a weight per training sample, that should go to 0 for corrupted samples. This is formalized by the bilevel optimization problem (1) with $\\begin{array} { r } { F ( \\mathbf { \\Sigma } \\breve { \\theta , \\lambda } ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\dot { \\ell } ( \\theta d _ { i } ^ { \\mathrm { v a l } } , \\dot { y _ { i } ^ { \\mathrm { v a l } } } ) } \\end{array}$ and $\\begin{array} { r } { G ( \\theta , \\lambda ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n ^ { - } } \\sigma ( \\lambda _ { i } ) \\ell ( \\theta d _ { i } ^ { \\mathrm { t r a i n } } , y _ { i } ^ { \\mathrm { t r a i n } } ) \\dot { + } C _ { r } \\| \\theta \\| ^ { 2 } } \\end{array}$ where $\\ell$ is the cross entropy loss and $\\sigma$ is the sigmoid function. The inner variable $\\theta$ is a matrix of size $1 0 \\times 7 8 4$ , and the outer variable $\\lambda$ is a vector in dimension $n _ { \\mathrm { t r a i n } } = 2 0 0 0 0$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "For the estimated parameters $\\theta$ during optimization, we report in Figure 2b the test error, i.e., the percent of wrong predictions on the testing data. We use for this experiment a corruption probability $p = 0 . 5$ . In general, the error decreases quickly until it reaches a final value. We observe that our method SABA outperforms all the other methods by reaching faster its smallest error, which is smaller than the ones of the other methods. For SOBA, it reaches a lower final error than stocBiO and BSA. In appendix, we provide other convergence curves, and find that for higher values of $p$ , SABA is still the fastest algorithm to reach its final accuracy. Overall, we find that among all methods, even those that implement variance reduction (that is FSLA, MRBO, SUSTAIN, SABA), SABA is the one that demonstrates the best empirical performance. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have presented a framework for bilevel optimization that enables the straightforward development of stochastic algorithms. The gist of our framework is that the directions in Equations (4) to (6) are all written as simple sums of samples derivatives. We leveraged this fact to propose SOBA, an extension of SGD to our framework, and SABA, an extension of SAGA to our framework, which both achieve similar convergence rates as their single level counterparts. Finally, we think that our framework opens a large panel of potential methods for stochastic bilevel optimization involving techniques of extrapolation, variance reduction, momentum and so on. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We thank Othmane Sebbouh, Zaccharie Ramzi and Benoît Malézieux for their precious comments. The authors acknowledge the support of the ANER RAGA BFC. SV acknowledges the support of the ANR GraVa ANR-18-CE40-0005. This work is supported by a public grant overseen by the French National Research Agency (ANR) through the program UDOPIA, project funded by the ANR-20-THIA-0013-01 and DATAIA convergence institute (ANR-17-CONV-0003). ",
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+        "text": "References   \n[1] Zeeshan Akhtar, Amrit Singh Bedi, Srujan Teja Thomdapu, and Ketan Rajawat. Projection-Free Algorithm for Stochastic Bi-level Optimization. preprint ArXiv 2110.11721, 2021.   \n[2] Michael Arbel and Julien Mairal. Amortized Implicit Differentiation for Stochastic Bilevel Optimization. In International Conference on Learning Representations (ICLR), 2022.   \n[3] Shaojie Bai, J. Zico Kolter, and Vladlen Koltun. Deep Equilibrium Models. In Advances in Neural Information Processing Systems (NeurIPS). Curran Associates, Inc., 2019.   \n[4] Atilim Gunes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differentiation in Machine Learning: A survey. Journal of Machine Learning Research, 18(153):1–43, 2018.   \n[5] Yoshua Bengio. Gradient-Based Optimization of Hyperparameters. 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Fast exact multiplication by the hessian. Neural computation, 6(1): 147–160, 1994.   \n[38] Fabian Pedregosa. Hyperparameter optimization with approximate gradient. In International Conference on Machine Learning (ICML), pages 737–746. PMLR, 2016.   \n[39] Zaccharie Ramzi, Florian Mannel, Shaojie Bai, Jean-Luc Starck, Philippe Ciuciu, and Thomas Moreau. SHINE: SHaring the INverse Estimate from the forward pass for bi-level optimization and implicit models. In International Conference on Learning Representations (ICLR), 2022.   \n[40] Sashank J. Reddi, Suvrit Sra, Barnabas Poczos, and Alex Smola. Fast Incremental Method for Nonconvex Optimization. In 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, pages 1971–1977, 2016.   \n[41] Herbert Robbins and Sutton Monro. A stochastic approximation method. The Annals of Mathematical Statistics, 22(3):400–407, 1951.   \n[42] Cédric Rommel, Thomas Moreau, Joseph Paillard, and Alexandre Gramfort. CADDA: Classwise Automatic Differentiable Data Augmentation for EEG Signals. In International Conference on Learning Representations (ICLR), 2022.   \n[43] Mark Schmidt, Nicolas Le Roux, and Francis Bach. Minimizing finite sums with the stochastic average gradient. Mathematical Programming, 162(1-2):83–112, 2017.   \n[44] Amirreza Shaban, Ching-An Cheng, Nathan Hatch, and Byron Boots. Truncated Backpropagation for Bilevel Optimization. In Artificial Intelligence and Statistics (AISTAT), pages 1723–1732, Okinawa, Japan, 2019.   \n[45] Heinrich von Stackelberg. Theory of the market economy. Oxford University Press, 1952.   \n[46] R. Wengert. A simple automatic derivative evaluation program. Communications of the ACM, 7 (8):463–464, 1964.   \n[47] Junjie Yang, Kaiyi Ji, and Yingbin Liang. Provably Faster Algorithms for Bilevel Optimization. In Advances in Neural Information Processing Systems (NeurIPS). Curran Associates, Inc., 2021. ",
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parse/dev/wv79UiY5U7/wv79UiY5U7.md

@@ -0,0 +1,437 @@
+# Data Curation for Image Captioning with Text-to-Image Generative Models
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Recent advances in image captioning are driven by increasingly larger-scale vision–   
+2 language pretraining, relying on massive computational resources and increasingly   
+3 large datasets. Instead of solely focusing on scaling pretraining, we ask whether   
+4 it is possible to improve performance by improving the quality of the samples in   
+5 existing datasets. We pursue this question through two approaches to data curation:   
+6 one that assumes that some examples should be avoided due to mismatches between   
+7 the image and caption, and one that assumes that the mismatch can be addressed by   
+8 replacing the image, for which we use the state-of-the-art Stable Diffusion model.   
+9 These approaches are evaluated using the BLIP model on the COCO and Flickr30K   
+10 datasets. Models trained with our data curation approaches consistently outperform   
+11 their baselines, indicating that better image captioning models can be trained by   
+12 curating existing resources. Finally, we conduct a human study to understand the   
+13 errors made by the Stable Diffusion model and highlight directions for future work   
+14 in text-to-image generation.
+
+# 15 1 Introduction
+
+16 Large-scale vision–language pretraining has been the driving force behind recent advances in image   
+17 captioning [14]. The amount of image–text data needed to pretrain recent generative language   
+18 models [28, 23, 53] has made it necessary to train on “noisy” samples harvested from the web   
+19 [46, 45], as opposed to crowdsourced captions [32]. This emerging reliance on harvested data has   
+20 made it important to perform additional filtering steps to remove low-quality data [28], in addition to   
+21 more resource-intensive pretraining. Given that computing resources are not equally distributed [21],   
+22 there is a need to also pursue less resource-intensive research directions.   
+23 We show how to improve image captioning by improving the quality of the downstream task data   
+24 through data curation: the process of dynamically updating the samples during training. We devise   
+25 three techniques for data curation that are designed to prevent the total size of the dataset from   
+26 increasing: the complete removal of an image–caption sample from a dataset; replacing a caption   
+27 with another caption; and replacing images using a text-to-image generation model [41]. These   
+28 curation techniques are used to update image–caption samples that have outlier losses, with respect   
+29 to the rest of a training dataset, under the current model parameters. In other words, the samples that   
+30 are proving difficult to model. Also, the synthesis of completely new images is radically different   
+31 from standard data augmentation techniques, such as random cropping or color manipulation [47], or   
+32 swapping and mask words in text [12].   
+33 We conduct experiments using BLIP [28], a strong image captioning model, on the Flickr30K [56]   
+34 and MS COCO datasets [32]. The results show that the sample removal and image replacement   
+35 techniques lead to consistent improvements of 1–3 CIDEr points compared to not curating the   
+36 dataset. Our analyses show that Flickr30K benefits from more curation than COCO due to differences   
+37 in the distribution of long captions in each dataset. Finally, we find that it is better to curate the   
+38 data dynamically while training instead of replacing images before starting to train the model.   
+39 Taken together, these findings show the promise of model-in-the-loop text-to-image generation for   
+40 multimodal learning, while highlighting that improvements in text-to-image generation are likely to   
+41 further enhance the effectiveness of data curation.
+
+![](images/ec55786e780ac343012b7e186f7424d0f65b7f4749737d805de26a12cfaadb2f.jpg)  
+Figure 1: Overview of our data curation approaches. For dynamic removal or replacement of captions, high loss image-text pairs are either removed or the image is paired with an alternative caption in the following training epoch. For image replacement, captions of original images are used as prompts for text-to-image generation to synthesize new image–text pairs. We experiment with both options of replacing the image only, or pair another relevant caption to the synthesized image.
+
+# 42 2 Related work
+
+43 Image Captioning Image Captioning is the task of describing images with syntactically and   
+44 semantically sentences. Current deep learning-based image captioning models have evolved as   
+45 the encode-decoder frameworks with multi-modal connection [8, 9], attentive [24, 16] and fusion   
+46 strategies [58]. Standard captioning datasets contain Flickr30K [56] and the commonly used MS   
+47 COCO [32], which consisting of images with events, objects and scenes. Each image is paired with   
+48 five captions. Some works have demonstrated the benefits of training on synthetic captions [29, 3] or   
+49 datasets collected from other vision-and-language learning tasks [38, 7].   
+50 Data Augmentation Data augmentation [13] has achieved increasing attention in both natural   
+51 language processing [33] and vision-and-language learning [27]. Early methods generate augmented   
+52 examples in the model’s feature space [54] or interpolate the inputs and labels of few examples [57].   
+53 For downstream tasks in the text domain, Yang et al. [55] and Anaby-Tavor et al. [1] generate   
+54 synthetic text examples through state-of-the-art pretrained language models and show improved   
+55 performance on common-sense reasoning and text-classification. For image captioning, BERT [11]   
+56 has been used to generate additional captions to improve the diversity of the captioning datasets [3].   
+57 Hossain et al. [22] used GAN-synthesized images as additional augmentation training set to improve   
+58 image captioning models.   
+59 Diffusion Models and Application Diffusion models [49, 35] have grown rapidly and become   
+60 the powerful deep generative models. They have shown potential in a variety of applications,   
+61 including text-to-image generation [36, 15], image-to-image translation [42], as well as semantic   
+62 segmentation [26, 5] and video generation [20, 48, 52]. While recent large scale latent diffusion   
+63 models have shown strong capability in generating both artistic and photo-realistic high-resolution   
+64 images [41, 34, 39, 43], applying large-scale stable diffusion models in vision-language downstream   
+65 tasks remains under-explored. Concurrently, Azizi et al. [4] and Jain et al. [25] show that image   
+66 classifiers can be improved by learning from augmentation images generated by finetuned stable  
+67 diffusion models. To the best of our knowledge, we are the first to explore how image captioning   
+68 models can benefit from simple data curation without scaling up existing datasets, and how stable  
+69 diffusion text-to-image models can be applied and contribute in the process.
+
+# 70 3 Data Curation for Captioning
+
+71 Our goal is to improve image captioning models by preventing the model from training on difficult   
+72 samples. There are many reasons for the possible existence of these difficult samples, including   
+73 mismatches or inconsistencies between the image and caption [3]. More formally, given an image   
+74 captioning training dataset $\mathcal { D }$ with $K$ images, let $\mathrm { I } _ { k }$ be the $k$ -th image. Each image is paired with   
+75 $J$ captions; let $\mathrm { C } _ { k } ^ { j }$ be $j$ th caption of image $k$ , and thus, let $( \boldsymbol { \mathrm { I } } _ { k } , \boldsymbol { \mathrm { C } } _ { k } ^ { j } )$ be an image–caption sample in   
+76 the dataset. Assume the existence of model $\mathcal { M }$ , which is being trained on dataset $\mathcal { D }$ , from which we   
+77 can calculate the loss of each sample at each epoch $t$ : $\mathcal { L } _ { \mathcal { M } } ^ { t } ( \mathrm { I } _ { k } , \mathrm { C } _ { k } ^ { j } )$ , which can be used to track the   
+78 difficult samples. At the end of each epoch, the difficult samples are candidates for our data curation   
+79 techniques, resulting in dynamic updates to the training dataset $\mathcal { D } \to \mathcal { D } _ { 1 } \to \cdots \to \mathcal { D } _ { T }$ .
+
+# 3.1 Identifying the difficult samples
+
+81 Difficult training samples may contain mismatches or inconsis  
+82 tencies between the image and the caption [3]. We propose to   
+83 use the captioning model that is being trained to automatically   
+84 identify such samples. After each epoch, we compute the loss   
+85 of each sample in the current training dataset, given the current   
+86 model parameters. The highest loss samples are targets for our   
+87 data curation methods; more specifically, we focus on samples   
+88 with losses that are either two standard deviations from the mean,   
+89 or a fixed $X \%$ away e.g. $10 \%$ , $20 \%$ , etc. In this way, the training   
+90 dataset is dynamically updated at the end of each epoch according   
+91 to the model’s captioning capability. The adjacent figure shows   
+92 the empirical distribution of losses in the training samples of   
+93 the Flickr30K dataset. It is clear that, without data curation, the   
+94 high-loss samples remain high-loss during five epochs of training.
+
+![](images/f93276185cbc74760b08a71a948ec69ff823864460eba48e32ddb37a732b76e3.jpg)  
+Figure 2: Distribution of persample losses in Flickr30K.
+
+# 95 3.2 Sample Removal / Caption Replacement
+
+The simplest approach to data curation is to remove or replace the high-loss samples. In REMOVE, the high-loss samples are completely removed from the remainder of the training process, reducing the total number of image–caption training samples. In REPLACECAP, we simply replace the caption in the image–caption sample with a different caption taken from the other captions that describe the image, effectively creating a duplicate. With the caption replacement method, the total number of samples used to train the model remains the same, as well as the total number of the unique images. This creates a clean control condition for the subsequent experiments.
+
+# 103 3.3 Image Generation-based Replacement
+
+An alternative to removing difficult samples or replacing captions is to pair an existing caption with a new image. This has the benefit of training the model on the same total number of samples while exposing it to more unique images. The new image could be found by humans, in a long-running human-in-the-loop cycle. Instead, we use a text-to-image generation model, in a rapid model-inthe-loop step, to synthesize images based on the other sentences that describe the image. Some representative examples of images generated using this technique can be seen in Figure 10.
+
+Our methodology is based on the open source Stable Diffusion model [41], which can generate images given a textual prompt. 1 We integrate this into training as follows: Given an image $I _ { k }$ in the training data and its captions $\{ ( I _ { k } , C _ { k } ^ { 1 } ) , \ldots , ( I _ { k } , C _ { k } ^ { J } ) \}$ , we synthesize a new image $\hat { I } _ { k }$ without increasing the total number of samples in the original dataset. Instead, we replace the original image in the sample with the generated image. Specifically, for image $I _ { k }$ , we replace a high-loss sample $( I _ { k } , C _ { k } ^ { j } )$ with the synthesized image-text pair $( \hat { I } _ { k } , C _ { k } ^ { j } )$ .
+
+# 116 Round-trip captioning evaluation
+
+117 In order to effectively use a text-to-image generation model for data curation, we need an objec  
+118 tive measure that can estimate the expected quality of a generated image. Most previous work   
+119 uses image-oriented measures like FID [19] or CLIPScore [17] but these measures are claimed   
+120 to lack alignment with perceptual quality [44]. We also found they were not suitable for our   
+121 purpose, and that CLIPScore cannot distinguish between low- and high-loss samples in the cap  
+122 tioning model (Figure 9). Here, we propose an alternative that is directly related to our task: given   
+123 the generated image, measure the quality of the caption that can be generated by a fixed model.   
+124 Our assumption is that if the generated images   
+125 are of a similar quality to the original images,   
+126 the resulting captions should be similar to each   
+127 other. We call this a round-trip captioning evalu  
+128 ation, which comprises three steps illustrated in   
+129 Figure 3. In Step (1), we use the captions in the   
+130 validation set to generate images using a text-to  
+131 image generation model. In Step (2), we use an   
+132 existing image-captioning model to predict cap  
+133 tions for the generated images. Specifically, we   
+134 use BLIP fine-tuned on the COCO dataset but   
+135 any other strong captioning model could be used   
+136 instead. Finally, in Step (3), we compare the pre  
+137 dicted captions against the original captions. We   
+138 now discuss the the factors that we found make   
+139 a difference when generating images.
+
+![](images/dd71c6cf0bdb2bfce4295a9727ef57802e3b4792986d2731663ccef0b54643af.jpg)  
+Figure 3: Round-trip captioning evaluation.
+
+# 140 Prompt engineering matters
+
+Recall that text-to-image generation models produce images based on a textual prompts. Given a set of five captions that describe an image, there are several options for how to prompt the image generation model. We experiment with three options:
+
+• Single caption: Each caption is used in isolation to generate a new image. • Sentence-BERT selection: There is a lot of variety in how different captions describe the same image. Instead of using all captions, we can use a representative caption from the set. This is achieved using the Sentence-BERT [40] model to find the caption that is closest to the average embedding of all captions. • Concatenation: All five captions are concatenated as the text prompt for generation.
+
+For all three approaches mentioned above, we can append an additional string to the prompt as a styler to force a specific style in the generated image $+ \cal S$ tyler). The styler used here is: "national geographic, high quality photography, Canon EOS R3, Flickr".2
+
+# Finetuning improves image relevance
+
+Table 1 shows the results of the round-trip captioning evaluation on the Flickr30K dataset using different textual prompts and whether or not to fine-tune the diffusion model. When we fine-tune StableDiffusion, we use the MS COCO [32] dataset with a prompt consisting of a concatenation of all 5 captions, for 15,000 steps with a constant learning rate of $1 e { - } 5$ and a batch size of 32. The best performance is clearly found by fine-tuning Stable Diffusion 1.5 and using a prompt with a concatenation of the captions and the styler. We use this configuration in the remainder of the paper.
+
+Table 1: Round-trip captioning evaluation on Flickr30K with different Stable Diffusion models, prompts, and fine-tuning. BLEU, CIDEr, Meteor.   
+
+<table><tr><td>Model</td><td>FT</td><td>Prompt</td><td>B</td><td>C</td><td>M</td></tr><tr><td>Upper-bound</td><td></td><td></td><td>37.6</td><td>27.2</td><td>57.1</td></tr><tr><td>SD 1.5</td><td>=</td><td>concat</td><td>31.0</td><td>24.7</td><td>52.5</td></tr><tr><td>SD 1.5</td><td>-</td><td>+ styler</td><td>30.8</td><td>24.2</td><td>52.5</td></tr><tr><td>SD 1.5</td><td>F</td><td>+ styler</td><td>33.5</td><td>25.0</td><td>53.5</td></tr><tr><td>SD 1.5</td><td>F</td><td>SBERT + styler</td><td>30.6</td><td>24.1</td><td>52.0</td></tr><tr><td>SD 2.0</td><td>-</td><td>concat + styler</td><td>31.2</td><td>24.8</td><td>52.0</td></tr></table>
+
+![](images/5880a1e3bd6cd3c1d7c81a867381c35d10f2f48c5a760f279138df1cf0bdf3a4.jpg)  
+Figure 4: Qualitative examples from the COCO dataset of captions generated by the BLIP model (top), and the same model trained using our REPLACEIMG data curation (bottom). The errors made by the BLIP model (shown in red) are avoided by REPLACEIMG curation (shown in blue).
+
+# 167 4 Experiments
+
+168 We evaluate our data curation methods on the MS COCO and Flickr30K datasets when finetuning the   
+169 pretrained BLIP [28] model. We evaluate the captions using BLEU [37], METEOR [10], ROUGE   
+170 [31], CIDEr [51], SPICE [2], CLIPScore, and RefCLIPScore [18].   
+171 We use the ViT-based BLIP model [28] as our captioning model. We note that BLIP has a captioning   
+172 and filtering (CapFilt) data augmentation process during its pretraining, where both components were   
+173 finetuned on the COCO dataset. Therefore we use pretrained checkpoint $\mathrm { B L I P } _ { C a p F i l t }$ for Flick $3 0 \mathrm { k }$   
+174 and $\mathrm { B L I P } _ { b a s e }$ for COCO in our experiment, removing the effects from the CapFilt process. We   
+175 finetune BLIP using a batch size of 128 for 5 epochs on $4 \times$ A100 GPUs.
+
+# 4.1 Results
+
+Removal/Caption Replacement As shown in Table 2, dynamically removing mismatched imagetext pairs or replacing captions can effectively improve performance on both datasets over baselines on all metrics. For Flickr30K, the dynamic updates work best when apply to the top $1 \%$ of high-loss samples for REPLACECAP, and to samples whose loss are two standard deviations higher than the mean for REMOVE. For COCO, both REPLACECAP and REMOVE works best when curating the top $1 \%$ of high-loss samples. We repeat that during the curation process, no additional data samples or computation cost is introduced. We further study the effect of the amount of curation in Section 5.
+
+184 Image Generation-based Replacement We evaluate Image Generation-based Replacement on   
+185 both the Flickr30K and COCO dataset. During finetuning, we replace images in the original text  
+186 image pairs with Stable Diffusion-synthesized images (ReplaceImg in Table 2). The results show   
+187 improvements compared to the baseline in every evaluation measure with best performance obtained   
+188 at replacement ratio of $40 \%$ for Flickr30K and at $10 \%$ for COCO. We show qualitative examples in   
+189 Figure 4, where models finetuned with our proposed curation method can generate better captions for   
+190 some scenes that may confuse the standard finetuned model. In Section 5.1, we analyze the effects of   
+191 varying the amount of synthetic images replaced, and in Section 5.2, we conduct a human study of   
+192 the types of errors found in the generated images.
+
+![](images/bd0d9934e6787c5f94594cb8179e809d60cfc4a1ff3bb7ada286ba3543b5e6bd.jpg)  
+Figure 5: Effects of the amount of data curated when finetuning the captioning model. We can observe that Flickr30K needs more curation $40 \%$ REPLACEIMG or 2 std REMOVE) than COCO ( $10 \%$ REPLACEIMG or $1 \%$ REPLACECAP). Flickr30K benefits more from removing high-loss training samples, indicating the original dataset may be noisier than MS COCO. For the 2 std approach, the number of samples curated is not fixed after each epoch and varies between $5 \%$ to $10 \%$ .
+
+# 193 5 Analysis and Discussion
+
+# 5.1 Data Curation: how much and when?
+
+We analyze how the amount of curation affects image captioning performance. We examine different ratios of training samples that are removed, replaced with an alternative caption, or replaced with a synthesized image. For REMOVE and REPLACECAP, we consider curation ratio of $1 \%$ , $5 \%$ and $10 \%$ of high-loss samples. For REPLACEIMG, we consider $10 \% { - } 8 0 \%$ curation ratio. In addition to fixed $X \%$ ratios, we also intereven on samples that have losses two standard deviations worse than the mean.
+
+201 Flickr30K needs more curation than COCO. The results of this analysis are shown in Figure 5.   
+202 The best improvement in performance for Flickr30K is achieved either through removing high loss   
+203 samples that are two standard deviations away, or replacing images for $40 \%$ of the high loss samples.   
+204 In the COCO dataset, replacing images for $10 \%$ of the   
+205 high loss samples gives the best improvement compared   
+206 to no data curation. The second best performing method   
+207 for COCO is removing or replacing captions of only $1 \%$   
+208 of the high loss samples. This indicates that Flickr30K   
+209 may contain more noisy samples than the MS COCO   
+210 dataset. Compared to MS COCO, Flickr30K contains   
+211 more samples with long captions (Figure 6), which may   
+212 include overly-specific details that are inconsistent with   
+213 other captions and are hard for the model to learn. See   
+214 more examples in our supplemental materials. Through   
+215 our curation-based finetuning, these samples can be effec  
+216 tively identified, removed or replaced, which indicates that   
+217 our method is efficient when training with noisy datasets. We note that curating more than $50 \%$ of   
+218 the data does not benefit training and actually harms performance.   
+219 Static image replacement versus dynamic replacement In REPLACEIMG (Section 3.3), we   
+220 dynamically replace images for the difficult training samples. Another static approach is to replace   
+221 the identical images, i.e. $I _ { k }$ in $\{ ( I _ { k } , C _ { k } ^ { 1 } ) , \ldots , ( I _ { k } , \check { C } _ { k } ^ { J } ) \}$ , with unique SD-synthesized images before   
+222 training, instead of updating the training samples while training. With static image replacement, for   
+223 each of the reference captions, we replace their original image with a SD-synthesized image. Static   
+224 replacement with $20 \% { - } 8 0 \%$ curation ratio corresponds to replacing images for one–four captions of
+
+![](images/a6f75623bf6c27b9b39c97e30c9dccdd5ceeb53188a88c396f0b67988df825f7.jpg)  
+Figure 6: Distribution of caption lengths.
+
+![](images/b724073f5d31a2a624b4d55823d3db3c56d82fc197f5992637a482159ecf6e7b.jpg)  
+Figure 7: Dynamic image replacement against static replacement, as a function of the number of samples replaced.
+
+![](images/9109d63e5c77bff9c3902a83a05ce5457cdefa8e3688c476533290a693e205f7.jpg)  
+Figure 8: Loss distribution of training samples across epochs with different curation methods.
+
+![](images/b7f9ce0a993f619a4c9e733eba30581534866055483bee3d24f984c1a430c635.jpg)  
+(a) Distribution of text-to-image generation errors.
+
+![](images/b19f84fd0a3b52e2e475622b0bb0a3e1d862d65ede861c5eb52a84372a704122.jpg)  
+(b) Human evaluation versus CLIPScore.
+
+Figure 9: Results of the human study of the errors made by the Stable Diffusion model in 100 images. The images used in the study were chosen to represent either low or high model loss. (a) Histogram of the number of errors annotated in each category. The most frequently occurring annotations concern weird deformations in the expected objects or humans. (b) Relationship between average number of identified errors by human annotations for each synthesized image and its captioning loss with regard to original captions. More errors are identified in images of higher loss. However, CLIPScore appears to fail in validating qualities of the synthesized images, as the score ranges are almost identical for samples that contain more errors.
+
+225 the original five. The $50 \%$ replacement ratio mimics a fair coin-flip, where for each of the text-image   
+226 samples, there is $50 \%$ probability for the image to be replaced by a synthesized image.   
+227 We compare the efficacy of these two approaches in Figure 7. When evaluating on the original   
+228 1k validation set, we see that for both approaches, incorporating synthesized images of $20 \%$ or   
+229 $40 \%$ can assist finetuning and achieves higher BLEU4 and CIDEr scores. Nevertheless, dynamic   
+230 image replacement consistently performs better than the static method, showing focusing on the hard   
+231 samples is effective. For both replacement methods, performance starts to decrease when the curation   
+232 ratio is too high. This may indicate that when incorporating too many images from the synthetic   
+233 distribution, the gap increases between the training and evaluation sets.   
+234 Figure 8 shows the effect of the curation techniques in the training loss distributions across epochs.   
+235 For the REMOVE approach, training samples with loss that are two standard deviations worse than the   
+236 mean are dynamically removed during training, leading to the shrinking tail of the loss distribution.   
+237 SD-based image replacement gradually reduces losses through learning from a mixture of Gaussian   
+238 distribution from original image-text pairs and the ones contain synthesized images.
+
+<table><tr><td>Image</td><td>Caption</td><td>CLIPScore Loss</td><td></td><td>Categorized Errors</td></tr><tr><td></td><td>A picture of two women with one in lacy white dress with handbag and leggings and the other with a tall red hat, black mid-dress,and frame like plastic dress on top.</td><td>84.1</td><td>181.0</td><td>type/color of clothing, color-clothing, weird-face</td></tr><tr><td></td><td>A pedicab driver waiting on his bike.</td><td>89.3</td><td>169.2</td><td>weird-main-object, weird-other-object, weird-body-parts, stance</td></tr><tr><td></td><td>A man in a black suit with tie and corsage smiles77.6 at a girl who smiles back,both are sitting at a tableat a semi formal event such as a wedding or reunion.</td><td></td><td>163.5</td><td>color-clothing, weird-body-parts, wrong-main-object, scene/event/location</td></tr><tr><td></td><td>Two men are playing guitars and one man is singing into a microphone on a stage with the spotlight on them.</td><td>74.7</td><td>26.0</td><td>weird-face, weird-body-parts, weird-main-object, weird-other-object</td></tr><tr><td></td><td>There a several people in a dark bar-type room，84.9 including one girl on a stool.</td><td></td><td>26.5</td><td>number, weird-face, weird-main-object, weird-body-parts</td></tr><tr><td></td><td>Many children are playing and swimming in the water.</td><td>78.2</td><td>26.9</td><td>weird-face, weird-body-parts</td></tr></table>
+
+Figure 10: Examples of synthesized images that are of high losses (top) and examples of synthesized images that are of low losses (bottom). Human annotations show that consistent error types have been recognized for the high loss samples while CLIPScore fails to align with human judgement. The low loss synthesized images are visually less complicated than the higher loss ones, but can still often look weird and contain errors in color or objects.
+
+# 239 5.2 Human Study: Errors made by SD models
+
+Finally, we conduct a human study of the errors present in the SD-synthesized images. This will serve to better understand any shortcomings with this approach that is not captured by automatic evaluation measures.
+
+243 We first ranked SD-synthesized images by model loss from the 1K images in the validation set. This   
+244 validation set of synthesized images was generated using the best performing configuration of the   
+245 Stable Diffusion model (see Section 3.3). We then sampled a subset for human annotation using the   
+246 top and bottom 50 images based on their loss using our fine-tuned captioning model. These images   
+247 are uniformly divided into 5 sets, each containing 20 images with equal number of the high loss   
+248 ones and the low loss ones. The data was annotated by 12 people, members of a university research   
+249 lab with a basic understanding of Stable Diffusion but no knowledge of the bi-modal distribution   
+250 of images. The annotators were asked to categorize the errors they observed in the synthesized   
+251 images, given both the image and the reference sentences that were used to generate the images. Each   
+252 participant annotated one set of 20 images.   
+253 Starting from the categories defined by van Miltenburg and Elliott [50], we predefined 25 categories   
+254 including general errors such as color, or number mismatches, and errors related to people and   
+255 objects in the images. Please see the user interface in supplemental materials. We analyze the human   
+256 judgements for the images that have at least three annotations, yielding 74 unique images.   
+257 As shown in Figure 9a, the most common problem of SD-synthesized images are that they often   
+258 generate weird face or body parts, which makes the images less natural or pleasant. The Stable   
+259 Diffusion model is also weak at generating the correct number of people or objects. From Figure 9b   
+260 we confirm the quality of our collected annotations that high loss figures often contain more errors   
+261 on average. Furthermore, we note that CLIPScore does not appear to align with human judgements,   
+262 indicating its weak capability of evaluating quality of generated images. Please see more concrete   
+263 examples in Figure 10.
+
+# 264 6 Conclusion
+
+65 In this paper, we have shown a simple, yet effective, data curation framework that can improve the   
+66 performance of image captioning models. We investigated three approaches to data curation that   
+67 dynamically update the training dataset based on high-loss image-caption samples. The methods   
+68 involved either removing a sample, replacing the caption in a sample, or generating a new image   
+69 from existing captions. Experimental results on the Flickr30K and MS COCO datasets show the   
+70 effectiveness of these approaches to data curation without increasing the total size of the training   
+71 dataset. A deeper analysis of the images synthesized by Stable Diffusion shows frequent errors on   
+72 generating objects of a certain amount or color, and struggles with human body features. A human   
+73 evaluation of the errors in those images shows a clear difference in images with high or low losses.   
+274 In the future, we expect that better text-to-image generation models will lead to further improvements   
+275 from using synthesized images for difficult captions in existing training datasets. We plan on   
+276 verifying whether these findings extend to other image captioning models, which was not possible   
+277 here due to computational issues. Finally, we are interested in applying the same framework to other   
+278 multimodal tasks, especially those with undercomplete datasets that cannot comprehensively cover   
+279 the distributional space due to the cost of crowdsourcing enough data, e.g. visual question answering,   
+280 or visually-grounded dialog.
+
+# 281 Limitations
+
+While our curation methods being effective on image-captioning in the finetuning and fewshotlearning settings, it is not clear if the same strategy can be scaled and adapted also to vision-language pretraining. Currently our data curation methods also rely on state-of-the art pretrained models for both image understanding and text-to-image generation. In pretraining, models will often be trained from scratch and pretraining data are often collected from multiple datasets and resources.
+
+Moreover, while we take an online approach to data curation, our current approach is upper bounded in speed and performance of the text-to-image generation model. This might be a large bottle neck for adapting the strategy for more complicated vision-and-language tasks.
+
+# 290 Ethics Statement
+
+Text-to-image generation with Stable Diffusion is controversial in the broader AI and ethics community[6]. For example, it can generate images according to gender or racial stereotypes, which may prove harmful to members of those communities [30]. In this paper, we use Stable Diffusion to improve the quality of an image captioning model, given a specific set of crowdsourced captions. Those captions may themselves contain harmful stereotypes that would become more prevalent in our dynamically updated training datasets. As we dynamically update the model with new images based on loss values, we remove the water-marker in our generated images to prevent information leak to the model. Use of the synthesized images will strictly follow community guidelines.
+
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+        "text": "Data Curation for Image Captioning with Text-to-Image Generative Models ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 Recent advances in image captioning are driven by increasingly larger-scale vision–   \n2 language pretraining, relying on massive computational resources and increasingly   \n3 large datasets. Instead of solely focusing on scaling pretraining, we ask whether   \n4 it is possible to improve performance by improving the quality of the samples in   \n5 existing datasets. We pursue this question through two approaches to data curation:   \n6 one that assumes that some examples should be avoided due to mismatches between   \n7 the image and caption, and one that assumes that the mismatch can be addressed by   \n8 replacing the image, for which we use the state-of-the-art Stable Diffusion model.   \n9 These approaches are evaluated using the BLIP model on the COCO and Flickr30K   \n10 datasets. Models trained with our data curation approaches consistently outperform   \n11 their baselines, indicating that better image captioning models can be trained by   \n12 curating existing resources. Finally, we conduct a human study to understand the   \n13 errors made by the Stable Diffusion model and highlight directions for future work   \n14 in text-to-image generation. ",
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+        "text": "15 1 Introduction ",
+        "text_level": 1,
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+        "text": "16 Large-scale vision–language pretraining has been the driving force behind recent advances in image   \n17 captioning [14]. The amount of image–text data needed to pretrain recent generative language   \n18 models [28, 23, 53] has made it necessary to train on “noisy” samples harvested from the web   \n19 [46, 45], as opposed to crowdsourced captions [32]. This emerging reliance on harvested data has   \n20 made it important to perform additional filtering steps to remove low-quality data [28], in addition to   \n21 more resource-intensive pretraining. Given that computing resources are not equally distributed [21],   \n22 there is a need to also pursue less resource-intensive research directions.   \n23 We show how to improve image captioning by improving the quality of the downstream task data   \n24 through data curation: the process of dynamically updating the samples during training. We devise   \n25 three techniques for data curation that are designed to prevent the total size of the dataset from   \n26 increasing: the complete removal of an image–caption sample from a dataset; replacing a caption   \n27 with another caption; and replacing images using a text-to-image generation model [41]. These   \n28 curation techniques are used to update image–caption samples that have outlier losses, with respect   \n29 to the rest of a training dataset, under the current model parameters. In other words, the samples that   \n30 are proving difficult to model. Also, the synthesis of completely new images is radically different   \n31 from standard data augmentation techniques, such as random cropping or color manipulation [47], or   \n32 swapping and mask words in text [12].   \n33 We conduct experiments using BLIP [28], a strong image captioning model, on the Flickr30K [56]   \n34 and MS COCO datasets [32]. The results show that the sample removal and image replacement   \n35 techniques lead to consistent improvements of 1–3 CIDEr points compared to not curating the   \n36 dataset. Our analyses show that Flickr30K benefits from more curation than COCO due to differences   \n37 in the distribution of long captions in each dataset. Finally, we find that it is better to curate the   \n38 data dynamically while training instead of replacing images before starting to train the model.   \n39 Taken together, these findings show the promise of model-in-the-loop text-to-image generation for   \n40 multimodal learning, while highlighting that improvements in text-to-image generation are likely to   \n41 further enhance the effectiveness of data curation. ",
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+        "image_caption": [
+            "Figure 1: Overview of our data curation approaches. For dynamic removal or replacement of captions, high loss image-text pairs are either removed or the image is paired with an alternative caption in the following training epoch. For image replacement, captions of original images are used as prompts for text-to-image generation to synthesize new image–text pairs. We experiment with both options of replacing the image only, or pair another relevant caption to the synthesized image. "
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+        "text": "42 2 Related work ",
+        "text_level": 1,
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+        "text": "43 Image Captioning Image Captioning is the task of describing images with syntactically and   \n44 semantically sentences. Current deep learning-based image captioning models have evolved as   \n45 the encode-decoder frameworks with multi-modal connection [8, 9], attentive [24, 16] and fusion   \n46 strategies [58]. Standard captioning datasets contain Flickr30K [56] and the commonly used MS   \n47 COCO [32], which consisting of images with events, objects and scenes. Each image is paired with   \n48 five captions. Some works have demonstrated the benefits of training on synthetic captions [29, 3] or   \n49 datasets collected from other vision-and-language learning tasks [38, 7].   \n50 Data Augmentation Data augmentation [13] has achieved increasing attention in both natural   \n51 language processing [33] and vision-and-language learning [27]. Early methods generate augmented   \n52 examples in the model’s feature space [54] or interpolate the inputs and labels of few examples [57].   \n53 For downstream tasks in the text domain, Yang et al. [55] and Anaby-Tavor et al. [1] generate   \n54 synthetic text examples through state-of-the-art pretrained language models and show improved   \n55 performance on common-sense reasoning and text-classification. For image captioning, BERT [11]   \n56 has been used to generate additional captions to improve the diversity of the captioning datasets [3].   \n57 Hossain et al. [22] used GAN-synthesized images as additional augmentation training set to improve   \n58 image captioning models.   \n59 Diffusion Models and Application Diffusion models [49, 35] have grown rapidly and become   \n60 the powerful deep generative models. They have shown potential in a variety of applications,   \n61 including text-to-image generation [36, 15], image-to-image translation [42], as well as semantic   \n62 segmentation [26, 5] and video generation [20, 48, 52]. While recent large scale latent diffusion   \n63 models have shown strong capability in generating both artistic and photo-realistic high-resolution   \n64 images [41, 34, 39, 43], applying large-scale stable diffusion models in vision-language downstream   \n65 tasks remains under-explored. Concurrently, Azizi et al. [4] and Jain et al. [25] show that image   \n66 classifiers can be improved by learning from augmentation images generated by finetuned stable  \n67 diffusion models. To the best of our knowledge, we are the first to explore how image captioning   \n68 models can benefit from simple data curation without scaling up existing datasets, and how stable  \n69 diffusion text-to-image models can be applied and contribute in the process. ",
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+    {
+        "type": "text",
+        "text": "70 3 Data Curation for Captioning ",
+        "text_level": 1,
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+        "text": "71 Our goal is to improve image captioning models by preventing the model from training on difficult   \n72 samples. There are many reasons for the possible existence of these difficult samples, including   \n73 mismatches or inconsistencies between the image and caption [3]. More formally, given an image   \n74 captioning training dataset $\\mathcal { D }$ with $K$ images, let $\\mathrm { I } _ { k }$ be the $k$ -th image. Each image is paired with   \n75 $J$ captions; let $\\mathrm { C } _ { k } ^ { j }$ be $j$ th caption of image $k$ , and thus, let $( \\boldsymbol { \\mathrm { I } } _ { k } , \\boldsymbol { \\mathrm { C } } _ { k } ^ { j } )$ be an image–caption sample in   \n76 the dataset. Assume the existence of model $\\mathcal { M }$ , which is being trained on dataset $\\mathcal { D }$ , from which we   \n77 can calculate the loss of each sample at each epoch $t$ : $\\mathcal { L } _ { \\mathcal { M } } ^ { t } ( \\mathrm { I } _ { k } , \\mathrm { C } _ { k } ^ { j } )$ , which can be used to track the   \n78 difficult samples. At the end of each epoch, the difficult samples are candidates for our data curation   \n79 techniques, resulting in dynamic updates to the training dataset $\\mathcal { D } \\to \\mathcal { D } _ { 1 } \\to \\cdots \\to \\mathcal { D } _ { T }$ . ",
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+    {
+        "type": "text",
+        "text": "3.1 Identifying the difficult samples ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "81 Difficult training samples may contain mismatches or inconsis  \n82 tencies between the image and the caption [3]. We propose to   \n83 use the captioning model that is being trained to automatically   \n84 identify such samples. After each epoch, we compute the loss   \n85 of each sample in the current training dataset, given the current   \n86 model parameters. The highest loss samples are targets for our   \n87 data curation methods; more specifically, we focus on samples   \n88 with losses that are either two standard deviations from the mean,   \n89 or a fixed $X \\%$ away e.g. $10 \\%$ , $20 \\%$ , etc. In this way, the training   \n90 dataset is dynamically updated at the end of each epoch according   \n91 to the model’s captioning capability. The adjacent figure shows   \n92 the empirical distribution of losses in the training samples of   \n93 the Flickr30K dataset. It is clear that, without data curation, the   \n94 high-loss samples remain high-loss during five epochs of training. ",
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+    {
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+        "img_path": "images/f93276185cbc74760b08a71a948ec69ff823864460eba48e32ddb37a732b76e3.jpg",
+        "image_caption": [
+            "Figure 2: Distribution of persample losses in Flickr30K. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "95 3.2 Sample Removal / Caption Replacement ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The simplest approach to data curation is to remove or replace the high-loss samples. In REMOVE, the high-loss samples are completely removed from the remainder of the training process, reducing the total number of image–caption training samples. In REPLACECAP, we simply replace the caption in the image–caption sample with a different caption taken from the other captions that describe the image, effectively creating a duplicate. With the caption replacement method, the total number of samples used to train the model remains the same, as well as the total number of the unique images. This creates a clean control condition for the subsequent experiments. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "103 3.3 Image Generation-based Replacement ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "An alternative to removing difficult samples or replacing captions is to pair an existing caption with a new image. This has the benefit of training the model on the same total number of samples while exposing it to more unique images. The new image could be found by humans, in a long-running human-in-the-loop cycle. Instead, we use a text-to-image generation model, in a rapid model-inthe-loop step, to synthesize images based on the other sentences that describe the image. Some representative examples of images generated using this technique can be seen in Figure 10. ",
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+    {
+        "type": "text",
+        "text": "Our methodology is based on the open source Stable Diffusion model [41], which can generate images given a textual prompt. 1 We integrate this into training as follows: Given an image $I _ { k }$ in the training data and its captions $\\{ ( I _ { k } , C _ { k } ^ { 1 } ) , \\ldots , ( I _ { k } , C _ { k } ^ { J } ) \\}$ , we synthesize a new image $\\hat { I } _ { k }$ without increasing the total number of samples in the original dataset. Instead, we replace the original image in the sample with the generated image. Specifically, for image $I _ { k }$ , we replace a high-loss sample $( I _ { k } , C _ { k } ^ { j } )$ with the synthesized image-text pair $( \\hat { I } _ { k } , C _ { k } ^ { j } )$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "116 Round-trip captioning evaluation ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "117 In order to effectively use a text-to-image generation model for data curation, we need an objec  \n118 tive measure that can estimate the expected quality of a generated image. Most previous work   \n119 uses image-oriented measures like FID [19] or CLIPScore [17] but these measures are claimed   \n120 to lack alignment with perceptual quality [44]. We also found they were not suitable for our   \n121 purpose, and that CLIPScore cannot distinguish between low- and high-loss samples in the cap  \n122 tioning model (Figure 9). Here, we propose an alternative that is directly related to our task: given   \n123 the generated image, measure the quality of the caption that can be generated by a fixed model.   \n124 Our assumption is that if the generated images   \n125 are of a similar quality to the original images,   \n126 the resulting captions should be similar to each   \n127 other. We call this a round-trip captioning evalu  \n128 ation, which comprises three steps illustrated in   \n129 Figure 3. In Step (1), we use the captions in the   \n130 validation set to generate images using a text-to  \n131 image generation model. In Step (2), we use an   \n132 existing image-captioning model to predict cap  \n133 tions for the generated images. Specifically, we   \n134 use BLIP fine-tuned on the COCO dataset but   \n135 any other strong captioning model could be used   \n136 instead. Finally, in Step (3), we compare the pre  \n137 dicted captions against the original captions. We   \n138 now discuss the the factors that we found make   \n139 a difference when generating images. ",
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+        "img_path": "images/dd71c6cf0bdb2bfce4295a9727ef57802e3b4792986d2731663ccef0b54643af.jpg",
+        "image_caption": [
+            "Figure 3: Round-trip captioning evaluation. "
+        ],
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+    {
+        "type": "text",
+        "text": "140 Prompt engineering matters ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Recall that text-to-image generation models produce images based on a textual prompts. Given a set of five captions that describe an image, there are several options for how to prompt the image generation model. We experiment with three options: ",
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+    {
+        "type": "text",
+        "text": "• Single caption: Each caption is used in isolation to generate a new image. • Sentence-BERT selection: There is a lot of variety in how different captions describe the same image. Instead of using all captions, we can use a representative caption from the set. This is achieved using the Sentence-BERT [40] model to find the caption that is closest to the average embedding of all captions. • Concatenation: All five captions are concatenated as the text prompt for generation. ",
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+    {
+        "type": "text",
+        "text": "For all three approaches mentioned above, we can append an additional string to the prompt as a styler to force a specific style in the generated image $+ \\cal S$ tyler). The styler used here is: \"national geographic, high quality photography, Canon EOS R3, Flickr\".2 ",
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+    },
+    {
+        "type": "text",
+        "text": "Finetuning improves image relevance ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1 shows the results of the round-trip captioning evaluation on the Flickr30K dataset using different textual prompts and whether or not to fine-tune the diffusion model. When we fine-tune StableDiffusion, we use the MS COCO [32] dataset with a prompt consisting of a concatenation of all 5 captions, for 15,000 steps with a constant learning rate of $1 e { - } 5$ and a batch size of 32. The best performance is clearly found by fine-tuning Stable Diffusion 1.5 and using a prompt with a concatenation of the captions and the styler. We use this configuration in the remainder of the paper. ",
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+        "type": "table",
+        "img_path": "images/2679cf42c5ca7afde4a76a523590adcad544a86c1101f2b35addc5e22e0d29de.jpg",
+        "table_caption": [
+            "Table 1: Round-trip captioning evaluation on Flickr30K with different Stable Diffusion models, prompts, and fine-tuning. BLEU, CIDEr, Meteor. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>FT</td><td>Prompt</td><td>B</td><td>C</td><td>M</td></tr><tr><td>Upper-bound</td><td></td><td></td><td>37.6</td><td>27.2</td><td>57.1</td></tr><tr><td>SD 1.5</td><td>=</td><td>concat</td><td>31.0</td><td>24.7</td><td>52.5</td></tr><tr><td>SD 1.5</td><td>-</td><td>+ styler</td><td>30.8</td><td>24.2</td><td>52.5</td></tr><tr><td>SD 1.5</td><td>F</td><td>+ styler</td><td>33.5</td><td>25.0</td><td>53.5</td></tr><tr><td>SD 1.5</td><td>F</td><td>SBERT + styler</td><td>30.6</td><td>24.1</td><td>52.0</td></tr><tr><td>SD 2.0</td><td>-</td><td>concat + styler</td><td>31.2</td><td>24.8</td><td>52.0</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/5880a1e3bd6cd3c1d7c81a867381c35d10f2f48c5a760f279138df1cf0bdf3a4.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative examples from the COCO dataset of captions generated by the BLIP model (top), and the same model trained using our REPLACEIMG data curation (bottom). The errors made by the BLIP model (shown in red) are avoided by REPLACEIMG curation (shown in blue). "
+        ],
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+    {
+        "type": "text",
+        "text": "167 4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "168 We evaluate our data curation methods on the MS COCO and Flickr30K datasets when finetuning the   \n169 pretrained BLIP [28] model. We evaluate the captions using BLEU [37], METEOR [10], ROUGE   \n170 [31], CIDEr [51], SPICE [2], CLIPScore, and RefCLIPScore [18].   \n171 We use the ViT-based BLIP model [28] as our captioning model. We note that BLIP has a captioning   \n172 and filtering (CapFilt) data augmentation process during its pretraining, where both components were   \n173 finetuned on the COCO dataset. Therefore we use pretrained checkpoint $\\mathrm { B L I P } _ { C a p F i l t }$ for Flick $3 0 \\mathrm { k }$   \n174 and $\\mathrm { B L I P } _ { b a s e }$ for COCO in our experiment, removing the effects from the CapFilt process. We   \n175 finetune BLIP using a batch size of 128 for 5 epochs on $4 \\times$ A100 GPUs. ",
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+    {
+        "type": "text",
+        "text": "4.1 Results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Removal/Caption Replacement As shown in Table 2, dynamically removing mismatched imagetext pairs or replacing captions can effectively improve performance on both datasets over baselines on all metrics. For Flickr30K, the dynamic updates work best when apply to the top $1 \\%$ of high-loss samples for REPLACECAP, and to samples whose loss are two standard deviations higher than the mean for REMOVE. For COCO, both REPLACECAP and REMOVE works best when curating the top $1 \\%$ of high-loss samples. We repeat that during the curation process, no additional data samples or computation cost is introduced. We further study the effect of the amount of curation in Section 5. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "184 Image Generation-based Replacement We evaluate Image Generation-based Replacement on   \n185 both the Flickr30K and COCO dataset. During finetuning, we replace images in the original text  \n186 image pairs with Stable Diffusion-synthesized images (ReplaceImg in Table 2). The results show   \n187 improvements compared to the baseline in every evaluation measure with best performance obtained   \n188 at replacement ratio of $40 \\%$ for Flickr30K and at $10 \\%$ for COCO. We show qualitative examples in   \n189 Figure 4, where models finetuned with our proposed curation method can generate better captions for   \n190 some scenes that may confuse the standard finetuned model. In Section 5.1, we analyze the effects of   \n191 varying the amount of synthetic images replaced, and in Section 5.2, we conduct a human study of   \n192 the types of errors found in the generated images. ",
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+    {
+        "type": "image",
+        "img_path": "images/bd0d9934e6787c5f94594cb8179e809d60cfc4a1ff3bb7ada286ba3543b5e6bd.jpg",
+        "image_caption": [
+            "Figure 5: Effects of the amount of data curated when finetuning the captioning model. We can observe that Flickr30K needs more curation $40 \\%$ REPLACEIMG or 2 std REMOVE) than COCO ( $10 \\%$ REPLACEIMG or $1 \\%$ REPLACECAP). Flickr30K benefits more from removing high-loss training samples, indicating the original dataset may be noisier than MS COCO. For the 2 std approach, the number of samples curated is not fixed after each epoch and varies between $5 \\%$ to $10 \\%$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "193 5 Analysis and Discussion ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "5.1 Data Curation: how much and when? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We analyze how the amount of curation affects image captioning performance. We examine different ratios of training samples that are removed, replaced with an alternative caption, or replaced with a synthesized image. For REMOVE and REPLACECAP, we consider curation ratio of $1 \\%$ , $5 \\%$ and $10 \\%$ of high-loss samples. For REPLACEIMG, we consider $10 \\% { - } 8 0 \\%$ curation ratio. In addition to fixed $X \\%$ ratios, we also intereven on samples that have losses two standard deviations worse than the mean. ",
+        "bbox": [
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+        "type": "text",
+        "text": "201 Flickr30K needs more curation than COCO. The results of this analysis are shown in Figure 5.   \n202 The best improvement in performance for Flickr30K is achieved either through removing high loss   \n203 samples that are two standard deviations away, or replacing images for $40 \\%$ of the high loss samples.   \n204 In the COCO dataset, replacing images for $10 \\%$ of the   \n205 high loss samples gives the best improvement compared   \n206 to no data curation. The second best performing method   \n207 for COCO is removing or replacing captions of only $1 \\%$   \n208 of the high loss samples. This indicates that Flickr30K   \n209 may contain more noisy samples than the MS COCO   \n210 dataset. Compared to MS COCO, Flickr30K contains   \n211 more samples with long captions (Figure 6), which may   \n212 include overly-specific details that are inconsistent with   \n213 other captions and are hard for the model to learn. See   \n214 more examples in our supplemental materials. Through   \n215 our curation-based finetuning, these samples can be effec  \n216 tively identified, removed or replaced, which indicates that   \n217 our method is efficient when training with noisy datasets. We note that curating more than $50 \\%$ of   \n218 the data does not benefit training and actually harms performance.   \n219 Static image replacement versus dynamic replacement In REPLACEIMG (Section 3.3), we   \n220 dynamically replace images for the difficult training samples. Another static approach is to replace   \n221 the identical images, i.e. $I _ { k }$ in $\\{ ( I _ { k } , C _ { k } ^ { 1 } ) , \\ldots , ( I _ { k } , \\check { C } _ { k } ^ { J } ) \\}$ , with unique SD-synthesized images before   \n222 training, instead of updating the training samples while training. With static image replacement, for   \n223 each of the reference captions, we replace their original image with a SD-synthesized image. Static   \n224 replacement with $20 \\% { - } 8 0 \\%$ curation ratio corresponds to replacing images for one–four captions of ",
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+        "image_caption": [
+            "Figure 6: Distribution of caption lengths. "
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+        "image_caption": [
+            "Figure 7: Dynamic image replacement against static replacement, as a function of the number of samples replaced. "
+        ],
+        "image_footnote": [],
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+    {
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+        "image_caption": [
+            "Figure 8: Loss distribution of training samples across epochs with different curation methods. "
+        ],
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+    {
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+        "image_caption": [
+            "(a) Distribution of text-to-image generation errors. "
+        ],
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+    {
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+        "image_caption": [
+            "(b) Human evaluation versus CLIPScore. "
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+    {
+        "type": "text",
+        "text": "Figure 9: Results of the human study of the errors made by the Stable Diffusion model in 100 images. The images used in the study were chosen to represent either low or high model loss. (a) Histogram of the number of errors annotated in each category. The most frequently occurring annotations concern weird deformations in the expected objects or humans. (b) Relationship between average number of identified errors by human annotations for each synthesized image and its captioning loss with regard to original captions. More errors are identified in images of higher loss. However, CLIPScore appears to fail in validating qualities of the synthesized images, as the score ranges are almost identical for samples that contain more errors. ",
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+        "type": "text",
+        "text": "225 the original five. The $50 \\%$ replacement ratio mimics a fair coin-flip, where for each of the text-image   \n226 samples, there is $50 \\%$ probability for the image to be replaced by a synthesized image.   \n227 We compare the efficacy of these two approaches in Figure 7. When evaluating on the original   \n228 1k validation set, we see that for both approaches, incorporating synthesized images of $20 \\%$ or   \n229 $40 \\%$ can assist finetuning and achieves higher BLEU4 and CIDEr scores. Nevertheless, dynamic   \n230 image replacement consistently performs better than the static method, showing focusing on the hard   \n231 samples is effective. For both replacement methods, performance starts to decrease when the curation   \n232 ratio is too high. This may indicate that when incorporating too many images from the synthetic   \n233 distribution, the gap increases between the training and evaluation sets.   \n234 Figure 8 shows the effect of the curation techniques in the training loss distributions across epochs.   \n235 For the REMOVE approach, training samples with loss that are two standard deviations worse than the   \n236 mean are dynamically removed during training, leading to the shrinking tail of the loss distribution.   \n237 SD-based image replacement gradually reduces losses through learning from a mixture of Gaussian   \n238 distribution from original image-text pairs and the ones contain synthesized images. ",
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+    {
+        "type": "table",
+        "img_path": "images/59cb387eaa11efcf1fe9990004718392ee957360c7b38bc20eb8747030037cea.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Image</td><td>Caption</td><td>CLIPScore Loss</td><td></td><td>Categorized Errors</td></tr><tr><td></td><td>A picture of two women with one in lacy white dress with handbag and leggings and the other with a tall red hat, black mid-dress,and frame like plastic dress on top.</td><td>84.1</td><td>181.0</td><td>type/color of clothing, color-clothing, weird-face</td></tr><tr><td></td><td>A pedicab driver waiting on his bike.</td><td>89.3</td><td>169.2</td><td>weird-main-object, weird-other-object, weird-body-parts, stance</td></tr><tr><td></td><td>A man in a black suit with tie and corsage smiles77.6 at a girl who smiles back,both are sitting at a tableat a semi formal event such as a wedding or reunion.</td><td></td><td>163.5</td><td>color-clothing, weird-body-parts, wrong-main-object, scene/event/location</td></tr><tr><td></td><td>Two men are playing guitars and one man is singing into a microphone on a stage with the spotlight on them.</td><td>74.7</td><td>26.0</td><td>weird-face, weird-body-parts, weird-main-object, weird-other-object</td></tr><tr><td></td><td>There a several people in a dark bar-type room，84.9 including one girl on a stool.</td><td></td><td>26.5</td><td>number, weird-face, weird-main-object, weird-body-parts</td></tr><tr><td></td><td>Many children are playing and swimming in the water.</td><td>78.2</td><td>26.9</td><td>weird-face, weird-body-parts</td></tr></table>",
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+        "image_caption": [
+            "Figure 10: Examples of synthesized images that are of high losses (top) and examples of synthesized images that are of low losses (bottom). Human annotations show that consistent error types have been recognized for the high loss samples while CLIPScore fails to align with human judgement. The low loss synthesized images are visually less complicated than the higher loss ones, but can still often look weird and contain errors in color or objects. "
+        ],
+        "image_footnote": [],
+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "239 5.2 Human Study: Errors made by SD models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Finally, we conduct a human study of the errors present in the SD-synthesized images. This will serve to better understand any shortcomings with this approach that is not captured by automatic evaluation measures. ",
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+        "text": "243 We first ranked SD-synthesized images by model loss from the 1K images in the validation set. This   \n244 validation set of synthesized images was generated using the best performing configuration of the   \n245 Stable Diffusion model (see Section 3.3). We then sampled a subset for human annotation using the   \n246 top and bottom 50 images based on their loss using our fine-tuned captioning model. These images   \n247 are uniformly divided into 5 sets, each containing 20 images with equal number of the high loss   \n248 ones and the low loss ones. The data was annotated by 12 people, members of a university research   \n249 lab with a basic understanding of Stable Diffusion but no knowledge of the bi-modal distribution   \n250 of images. The annotators were asked to categorize the errors they observed in the synthesized   \n251 images, given both the image and the reference sentences that were used to generate the images. Each   \n252 participant annotated one set of 20 images.   \n253 Starting from the categories defined by van Miltenburg and Elliott [50], we predefined 25 categories   \n254 including general errors such as color, or number mismatches, and errors related to people and   \n255 objects in the images. Please see the user interface in supplemental materials. We analyze the human   \n256 judgements for the images that have at least three annotations, yielding 74 unique images.   \n257 As shown in Figure 9a, the most common problem of SD-synthesized images are that they often   \n258 generate weird face or body parts, which makes the images less natural or pleasant. The Stable   \n259 Diffusion model is also weak at generating the correct number of people or objects. From Figure 9b   \n260 we confirm the quality of our collected annotations that high loss figures often contain more errors   \n261 on average. Furthermore, we note that CLIPScore does not appear to align with human judgements,   \n262 indicating its weak capability of evaluating quality of generated images. Please see more concrete   \n263 examples in Figure 10. ",
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+    {
+        "type": "text",
+        "text": "264 6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "65 In this paper, we have shown a simple, yet effective, data curation framework that can improve the   \n66 performance of image captioning models. We investigated three approaches to data curation that   \n67 dynamically update the training dataset based on high-loss image-caption samples. The methods   \n68 involved either removing a sample, replacing the caption in a sample, or generating a new image   \n69 from existing captions. Experimental results on the Flickr30K and MS COCO datasets show the   \n70 effectiveness of these approaches to data curation without increasing the total size of the training   \n71 dataset. A deeper analysis of the images synthesized by Stable Diffusion shows frequent errors on   \n72 generating objects of a certain amount or color, and struggles with human body features. A human   \n73 evaluation of the errors in those images shows a clear difference in images with high or low losses.   \n274 In the future, we expect that better text-to-image generation models will lead to further improvements   \n275 from using synthesized images for difficult captions in existing training datasets. We plan on   \n276 verifying whether these findings extend to other image captioning models, which was not possible   \n277 here due to computational issues. Finally, we are interested in applying the same framework to other   \n278 multimodal tasks, especially those with undercomplete datasets that cannot comprehensively cover   \n279 the distributional space due to the cost of crowdsourcing enough data, e.g. visual question answering,   \n280 or visually-grounded dialog. ",
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+    {
+        "type": "text",
+        "text": "281 Limitations ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "While our curation methods being effective on image-captioning in the finetuning and fewshotlearning settings, it is not clear if the same strategy can be scaled and adapted also to vision-language pretraining. Currently our data curation methods also rely on state-of-the art pretrained models for both image understanding and text-to-image generation. In pretraining, models will often be trained from scratch and pretraining data are often collected from multiple datasets and resources. ",
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+        "type": "text",
+        "text": "Moreover, while we take an online approach to data curation, our current approach is upper bounded in speed and performance of the text-to-image generation model. This might be a large bottle neck for adapting the strategy for more complicated vision-and-language tasks. ",
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+        "type": "text",
+        "text": "290 Ethics Statement ",
+        "text_level": 1,
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+        "text": "Text-to-image generation with Stable Diffusion is controversial in the broader AI and ethics community[6]. For example, it can generate images according to gender or racial stereotypes, which may prove harmful to members of those communities [30]. In this paper, we use Stable Diffusion to improve the quality of an image captioning model, given a specific set of crowdsourced captions. Those captions may themselves contain harmful stereotypes that would become more prevalent in our dynamically updated training datasets. As we dynamically update the model with new images based on loss values, we remove the water-marker in our generated images to prevent information leak to the model. Use of the synthesized images will strictly follow community guidelines. ",
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+        "type": "text",
+        "text": "References ",
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+        "text": "1SenseTime Research   \n2The University of Texas at Austin   \n3University of Sydney   \n{liyangguang,zhaolichen,cuiyufeng}@sensetime.com   \njeffliang@utexas.edu ",
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+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Recently, large-scale Contrastive Language-Image Pre-training (CLIP) (Radford et al., 2021) has attracted unprecedented attention for its impressive zero-shot recognition ability and excellent transferability to downstream tasks. However, CLIP is quite data-hungry and requires 400M image-text pairs for pre-training, thereby restricting its adoption. This work proposes a novel training paradigm, Data efficient CLIP (DeCLIP), to alleviate this limitation. We demonstrate that by carefully utilizing the widespread supervision among the image-text pairs, our DeCLIP can learn generic visual features more efficiently. Instead of using the single image-text contrastive supervision, we fully exploit data potential through the use of (1) self-supervision within each modality; (2) multi-view supervision across modalities; (3) nearest-neighbor supervision from other similar pairs. Benefiting from these intrinsic supervision, our DeCLIP-ResNet50 can achieve $6 0 . 4 \\%$ zeroshot top1 accuracy on ImageNet, which is $0 . 8 \\%$ above the CLIP-ResNet50 while using $7 . 1 \\times$ fewer data. Our DeCLIP-ResNet50 outperforms its counterpart in 8 out of 11 visual datasets when transferred to downstream tasks. Moreover, Scaling up the model and computing also works well in our framework. Our code, dataset and models are released at: https://github.com/Sense-GVT/DeCLIP ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Over the last few years, pre-trained models have greatly revolutionized computer vision (CV) and natural language processing (NLP). The first wave of exploring pre-trained models took place in the field of CV. Deep convolutional neural nets (Krizhevsky et al., 2012; Simonyan & Zisserman, 2014; He et al., 2016) are pre-trained on well-labeled ImageNet (Deng et al., 2009) and then transferred to downstream CV tasks (Girshick et al., 2014; Long et al., 2015; Vinyals et al., 2015). Standardly, CV models are pre-trained to predict a fixed set of pre-defined object categories, e.g., 1000 classes in ImageNet. However, this supervised pre-training is hard to scale since we need arduous human labeling to specify new visual concepts. ",
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+        "text": "When pre-training meets NLP, the intrinsic supervision within the natural language makes the pretraining more scalable (Devlin et al., 2018; Radford et al., 2019; Brown et al., 2020). Witnessing the progress in NLP, researchers use natural language supervision to learn visual features. The language-image pre-training can scale up to a very large size, benefiting from abundant image-text pairs on the Internet. For instance, CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021) adopt the contrastive loss to push the embedding of matched image-text pairs together while pushing those of non-matched pairs apart. They achieve prestigious performance by learning from an enormous dataset that contains 400M/1B image-text pairs. However, these methods also require huge storage and computing resources, which is not affordable for most laboratories and companies. We argue that these prior arts only use the single image-text contrastive supervision while overlooking the widespread supervision within the pairs, thus is inefficient. ",
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+        "img_path": "images/0ce1d9446a4c27c1beac26f91cc23c8f585923fe59a3017bc43c6e9fe2aab4fe.jpg",
+        "image_caption": [
+            "Figure 1: Zero-shot performance of CLIP- Figure 2: Transfer the DeCLIP-ResNet50 (abbr. ResNet50 and our DeCLIP-ResNet50 when us- as DeC) and CLIP-ResNet50 (abbr. as C) to 11 ing different amounts of data. (88M, $6 2 . 5 \\% )$ ) de- downstream visual datasets using linear probe notes the use of 88M data with top-1 accuracy verification. Our DeCLIP achieves better results $6 2 . 5 \\%$ on the ImageNet-1K validation dataset. in 8 out of 11 datasets. Our model has much better data efficiency. "
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+        "text": "",
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+        "type": "text",
+        "text": "Firstly, there underlies rich structural information within each modality itself (LeCun & Misra, 2021). We can tweak some words/pixels in a sentence/image while retaining a similar semantic meaning. This sort of self-supervision can be exploited to learn a more common-sense representation for each modality (Devlin et al., 2018; He et al., 2020; Chen et al., 2020a). Moreover, inspired by contrasting multi-crops in an image (Caron et al., 2020), we further extend the multi-view 1 supervision into our multi-modality setting. Specifically, each image is paired with multiple textual descriptions obtained via stochastic augmentations, vice versa. The benefit is intuitive: this auxiliary multi-view supervision brings more invariant and robust information. ",
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+        "text": "Besides these overlooked supervision, we propose a novel nearest-neighbor (NN) supervision from other similar pairs. This NN supervision is mainly based on the intuition that one image is likely to have other similar text descriptions among the dataset. As shown in right figure, the image with the text ’going to see a lot of vintage tractors this week’ can also be described by ’vintage at tractors a gathering’. For this reason, we sample the NN in the embedding space and utilize them as additional supervisory signals. Aggregating these supervision leads to our novel training paradigm DeCLIP, which stands for Data efficient Contrastive Language-Image Pretraining. ",
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+        "text": "Extensive experiments show the effectiveness and efficiency of our DeCLIP. As shown in Fig. 1, with a ResNet50 image encoder and a Transformer text encoder, our model can achieve $6 0 . 4 \\%$ zero-shot top1 accuracy on ImageNet, which is $0 . 8 \\%$ above the CLIPResNet50 while using $7 . 1 \\times$ fewer data. Using only 88M image-text pairs, our best ResNet50/ViTB32 models boost the zero-shot performance to $6 2 . 5 \\%$ and $6 6 . 2 \\%$ , nearly $3 . 0 \\%$ higher than the best number reported for these two architectures. We further verify the transferability of our models on downstream tasks. As indicated in Fig. 2, our DeCLIP-ResNet50 outperforms its counterpart in 8 out of 11 visual datasets. Moreover, Scaling up the model and computing also works well in our framework. Using $4 . 5 \\times$ fewer data, our DeCLIP-RegNetY-64GF achieves $7 3 . 7 \\%$ zero-shot ImageNet top1 accuracy, which is on-pair with CLIP- ${ \\mathrm { R 5 0 } } \\times 6 4$ . Pre-trained models, code, and datasets shall be released to the community. The contributions are summarized as follows: ",
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+        "image_caption": [
+            "Figure 3: Examples of Nearest Neighbor from Conceptual Captions dataset. "
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+        "text": "• To the best of our knowledge, this is the first work to study self-supervision and cross-modal multi-view supervision in the million-scale image-text pre-training task. Our work opens a new direction to fully exploit the intrinsic supervision within the multi-modal data instead of scaling up data naively. ",
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+        "text": "• We propose novel cross-modal Nearest-Neighbor Supervision (NNS) to harness information from other similar pairs. The NNS can also be regarded as a semantic-level augmentation. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 PRE-TRAINED MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The critical idea of pre-training is to first extract general knowledge implicitly from the massive amount of data and then transfer the knowledge to versatile downstream tasks (Han et al., 2021). Big NLP models (Devlin et al., 2018; Brown et al., 2020) yield unprecedented performance via learning from tremendous language data over the Internet and labor-free supervision within the language itself. In the field of CV, supervised pre-training on ImageNet is still the standard practice. While achieving great success on downstream CV tasks (Girshick et al., 2014; Long et al., 2015; Vinyals et al., 2015) , this supervised manner is hard to scale. To address this challenge, our DeCLIP learns directly from image-text pairs that are abundant across the Internet. More importantly, by exploiting the widespread supervision within the pairs, our DeCLIP is more data-efficient than the prior art. ",
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+    {
+        "type": "text",
+        "text": "2.2 SUPERVISION WITHIN DATA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Language supervision Joulin et al. (2016); Gomez et al. (2017); Zhang et al. (2020); Sariyildiz et al. (2020); Desai & Johnson (2021) demonstrate the effectiveness of learning transferable visual features from language supervision. Pioneering work CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021) achieve prestigious performance via learning from 400M/1B image-text pairs. We are following these two works to improve their data efficiency. ",
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+    {
+        "type": "text",
+        "text": "Relevant concurrent work including: SLIP Mu et al. (2021) introduces self-supervision to CLIP. FILIP Yao et al. (2021) leverages the finer-grained alignment between image patches and textual words. OTTER Wu et al. (2021); Cheng et al. (2021) uses online entropic optimal transport to find a soft image-text match as labels to mitigate the noise within the dataset. ",
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+        "type": "text",
+        "text": "Visual self-supervision Our work is also highly related to self-supervised learning (SSL) (LeCun & Misra, 2021). Contrastive learning, as a pretext task of SSL, has achieved remarkable success in visual representation learning (He et al., 2020; Chen et al., 2020a; Caron et al., 2020; Grill et al., 2020; Chen et al., 2020a). Researchers also extend contrastive learning into multi-modal settings (Yuan et al., 2021). However, it is only limited to a small COCO dataset (Yuan et al., 2021). ",
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+    {
+        "type": "text",
+        "text": "Nearest-neighbor supervision Recently, researchers have exploited nearest-neighbor supervision to learn visual features (Dwibedi et al., 2021; Van Gansbeke et al., 2021). They find that using nearest-neighbor as positive samples in the contrastive loss improves the performances on multiple downstream tasks. However, they mainly focus on the single visual modality pretraining on relatively small datasets, such as ImageNet. We propose novel nearest-neighbor supervision for multi-modal learning to harness information from other similar pairs. ",
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+        "type": "text",
+        "text": "2.3 MULTI-MODAL LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Most vision-language models Chen et al. (2020b); Lu et al. (2019); Li et al. (2020) use a bunch of cross-modal transformers to fuse and align the information between text and image. These methods either need an off-the-shelf object detector to extract region features or dedicated cross-modal transformer layers, significantly hindering their scalability. Our DeCLIP, by contrast, uses a simple yet effective two-tower framework with multi-modal interaction only at the top. Moreover, this series of models (Radford et al., 2021; Jia et al., 2021; Huo et al., 2021) can perform zero-shot recognition, adapting to new categories with no seen labeled data. Shen et al. (2021) also shows that the pretrained CLIP model can significantly benefit the downstream VQA and image caption tasks. Our DeCLIP is supposed to be compatible with more modalities, e.g., acoustic signals (Akbari et al., 2021). More modalities included, more correlated supervision are expected to be exploited. ",
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+    {
+        "type": "image",
+        "img_path": "images/5122ea60d58db20432ff26bdbb4bbe9a554035589c526eeff41d31ea91e0641f.jpg",
+        "image_caption": [
+            "Figure 4: (a) CLIP and ALIGN jointly train an image encoder and a text encoder to predict the correct pairings of a batch of (image, text) training examples. (b) Our DeCLIP overview. $\\textcircled{1}$ means Self-Supervision(SS). For image SS, we maximize the similarity between two augmented views of the same instance. For text SS, we leverage Masked Language Modeling(MLM) within a text sentence. $\\textcircled{2}$ represents cross-modal Multi-View Supervision(MVS). We first have two augmented views of both image and text, then contrast the $2 \\times 2$ image-text pairs. $\\textcircled{3}$ indicates NearestNeighbor Supervision(NNS). We sample text NN in the embedding space to serve as additional supervision. The combination of the three supervision leads to efficient multi-modal learning. "
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+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we first revisit CLIP and denote some basic concepts, such as image-text contrastive supervision (i.e., the InfoNCE loss). Next, we present the overview of our DeCLIP framework. Then we introduce every auxiliary supervision: Self-Supervision(SS), Multi-View Supervision(MVS), and Nearest-Neighbor Supervision(NNS). ",
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+    {
+        "type": "text",
+        "text": "3.1 REVISITING CLIP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Contrastive Language-Image Pre-training (CLIP) (Radford et al., 2021) aims to learn directly from the raw text about images. They use a dual-encoder architecture as in Fig. 4(a). The model consists of an image encoder (e.g., CNN (He et al., 2016) or ViT (Dosovitskiy et al., 2020)) and a text encoder(e.g., Transformer (Vaswani et al., 2017) or its variants (Radford et al., 2019)), with a multimodal interaction at the top. The image and text features are projected to the same dimension and followed by L2 normalization before interaction. At the training phase, a contrastive objective pushes the embeddings of matched image-text pairs together while pushing those of non-matched pairs apart. In a batch of $N$ image-text pairs $\\{ ( \\pmb { x } _ { i } ^ { I } , \\pmb { x } _ { i } ^ { T } ) \\}$ , we denote $\\bar { \\mathbf { \\boldsymbol { x } } } _ { i } ^ { I }$ and $\\mathbf { \\bar { \\mathbf { x } } } _ { i } ^ { T }$ as image and text of the $i _ { t h }$ pair. Let $z _ { i } ^ { I }$ and $z _ { j } ^ { T }$ be the normalized embedding of the $i _ { t h }$ image and $j _ { t h }$ text, respectively. CLIP uses InfoNCE loss (Oord et al., 2018). The loss for the image encoder can be denoted as Eq. 1. ",
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+    {
+        "type": "equation",
+        "img_path": "images/24124e253e38c6704603134c4ef33a12601e11c766185c26503b711bcee1bced.jpg",
+        "text": "$$\nL _ { I } = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\frac { \\exp ( \\sin ( z _ { i } ^ { I } , z _ { i } ^ { T } ) / \\tau ) } { \\sum _ { j = 1 } ^ { N } \\exp ( \\sin ( z _ { i } ^ { I } , z _ { j } ^ { T } ) / \\tau ) }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here, the similarity function $\\mathrm { s i m } ( , )$ is measured by dot product, and $\\tau$ is a learnable temperature variable to scale the logits. We have a symmetrical loss for image and text encoder, thus the overall loss function $L _ { C L I P }$ is the average of $L _ { I }$ and $L _ { T }$ . ",
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+        "type": "text",
+        "text": "At the test phase, the learned text encoder synthesizes a zero-shot linear classifier by embedding the arbitrary categories of the test dataset. Because it is rare in the dataset that image caption is just ",
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+        "type": "text",
+        "text": "a single word, CLIP uses prompts to make up the context of the category $\\{ 1 \\mathsf { a b e 1 } \\}$ , such as $\" a$ photo of $\\mathrm { ~ a ~ } \\left\\{ 1 \\mathrm { a b } { \\in } 1 \\right\\} \"$ . Unless otherwise specified, we use the same prompt engineering and ensembling techniques as CLIP. Details can be found in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "3.2 OVERVIEW OF DECLIP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As shown in Fig. 4(b), our DeCLIP has three additional supervisory signals. ",
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+        "text": "$\\textcircled{1}$ We first use existing methods to exploit image and text Self-Supervision (SS) within its modality. For image SS, we adopt the simple yet effective SimSiam (Chen & He, 2021). The objective is to maximize the similarity between two augmented image features. For text SS, we adopt the most widely used Masked Language Modeling (MLM) (Devlin et al., 2018) as the pre-text task. We believe other kinds of self-supervised learning algorithms (e.g. MoCo (He et al., 2020), SimCSE (Gao et al., 2021) are orthogonal with our framework. ",
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+        "text": "$\\textcircled{2}$ While SS only focuses on a single modality, we further propose cross-modal Multi-View Supervision (MVS). We apply stochastic data augmentations for both images and texts, resulting in two correlated views1 of each example. Then, the image-text contrastive loss is calculated for all the $2 \\times 2$ pairs. Worth mentioning, the original CLIP does not use text augmentations and only uses random square crop image augmentations, thereby is data-hungry. The extension is instinctive and straightforward. Specifically, we contrast the $2 \\times 2$ pairs and resulting in $3 \\times$ more additional supervision. ",
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+        "type": "text",
+        "text": "$\\textcircled{3}$ We also propose novel Nearest-Neighbor Supervision (NNS) mined in embedding space to make better use of similar text descriptions among the dataset. In detail, we maintain a first-in-first-out feature queue that is representative of the whole data distribution. We use the nearest-neighbor search in embedding space to get the semantically similar text descriptions. Then we use the imagetext contrastive loss to get additional supervision. ",
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+    {
+        "type": "text",
+        "text": "3.3 SUPERVISION EXISTS EVERYWHERE",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Self-Supervision within each modality Following SimSiam (Chen & He, 2021) (depicted in Fig 5(a)), we first have two augmented views $( x ^ { I } , \\bar { \\tilde { x } } ^ { I } )$ for each image. These two views are sent to the image encoder (weights are shared between views). We also use the popularized nonlinear predictor module, which is typically a 2-layer MLP, to improve the representation quality in the encoder (Chen et al., 2020a). The objective is to maximize the similarity between $\\tilde { z } ^ { I }$ and $p ^ { I }$ , which is a negative cosine similarity in this paper. To avoid the trivial “collapsing” solution, we follow (Chen & He, 2021) to adopt a stop-grad technique. ",
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+        "type": "text",
+        "text": "As shown in Fig. 5(b), we follow the method in BERT (Devlin et al., 2018) for our text self-supervision. In detail, we first randomly choose $1 5 \\%$ of all tokens in each sequence. Then the token is replaced with (1) the [mask] token $8 0 \\%$ of the time (2) a random token $1 0 \\%$ of the time (3) the unchanged token $1 0 \\%$ of the time. Then, the output of the language module for the corresponding token is used to predict the original token with cross-entropy loss. ",
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+        "type": "image",
+        "img_path": "images/efca79d801146a93e7ca9561d754ff7a39aecdc8ab47135af616f0ab59f5eb72.jpg",
+        "image_caption": [
+            "Figure 5: Self-Supervision with each modality. We adopt SimSiam and MLM for image and text SS. "
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+        "text": "Multi-View Supervision The authors only contrast the original text (w/o augmentation) with a single ‘global view’ of the image in the original CLIP. However, the text annotation of the image might not describe the whole picture, instead depicts a small local view of this image. For instance, as shown in the image with the text \"a cute white cat\" in Fig. 4, the central concept (cat) only occupies a small part of the picture. To mitigate this discrepancy, we get a closer look at the local region and utilize it as our auxiliary supervision, as shown in the augmented view in Fig. 4(b). This intuitive idea is akin to the successful Multi-crop transformation (Caron et al., 2020; Van Gansbeke et al., 2021) in image SSL. We further extend it into the multi-modal setting. More specifically, we reuse the two image views introduced in SS, which contains the RandomResizedCrop policy to obtain a small local view. For text, as our goal is to understand the overall semantic meaning of a sentence, we adopt text classification augmentation EDA (Wei & Zou, 2019) to generate two text views. Besides the original contrastive loss between $( z ^ { I } , \\dot { z } ^ { T } )$ , we can contrast $( \\breve { z } ^ { I } , \\tilde { z } ^ { T } )$ , $( \\tilde { z } ^ { I } , z ^ { T } )$ and $( \\tilde { z } ^ { I } , \\tilde { z } ^ { T } )$ , leading to $3 \\times$ diverse and high-quality additional supervision. More conveniently, they are naturally compatible with the image-text contrastive loss as denoted in Eq. 1. ",
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+        "page_idx": 4
+    },
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Nearest-Neighbor Supervision As shown in Fig 3, one image is likely to have other similar text descriptions among the dataset. To harness the information from other pairs and go beyond single pairs, we propose using nearest-neighbor (NN) to obtain more diverse supervision. More formally, we aim to find the NN feature $z ^ { T ^ { \\prime } }$ of text feature $z ^ { T }$ in the embedding space. The distance between two features could be measured by a simple cosine similarity. It is infeasible to search NN in the whole million-scale dataset. Thus, we maintain a FIFO queue $Q$ to simulate the whole data distribution. The size of $Q$ is 64K in our implementation. As shown in Fig. 6, we further get the contrastive loss between $( z ^ { I } , z ^ { T ^ { \\prime } } )$ . Since there are two augmented image features, we also calculate the contrastive loss between $( \\tilde { z } ^ { I } , z ^ { T ^ { \\prime } } )$ . Fortunately, NNS is also compatible with Eq. 1. ",
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+    },
+    {
+        "type": "text",
+        "text": "In summary, we denote $L _ { I S S }$ and $L _ { T S S }$ as the loss function of image SS and text SS, respectively. ${ \\cal L } _ { M V S }$ is multi-view loss, and $L _ { N N S }$ is nearest-neighbor loss. We have the overall loss function of our DeCLIP as in Eq. 2. ",
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+        "img_path": "images/05899bf9ce8b7ecbf860f3ef3c370d4df60211f7f85b8b68a64cbf2208b142fb.jpg",
+        "image_caption": [
+            "Figure 6: Nearest-Neighbor Supervision. $z ^ { T ^ { \\prime } }$ is the NN of feature $z ^ { T }$ in the embedding space. $z ^ { T ^ { \\prime } }$ will serve as an additional objective for $z ^ { I }$ . We use the feature-level nearest neighbor for the text descriptions as the supervision. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "equation",
+        "img_path": "images/099e973feb436fd6ca674d188bd97dc50d7da4c983895863c106332cf6dab9c0.jpg",
+        "text": "$$\nL _ { D e C L I P } = ( 1 - \\alpha - \\beta - \\gamma ) L _ { C L I P } + \\alpha ( L _ { I S S } + L _ { T S S } ) + \\beta L _ { M V S } + \\gamma L _ { N N S }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Pre-training datasets We summarize our pre-training dataset as in Tab. 1. Our DeCLIP full data consists of two parts: open-source data and web-crawled data. The open-source data comes from three different datasets: Conceptual Captions (CC3M) (Sharma et al., 2018), Conceptual 12M (CC12M) (Changpinyo et al., 2021), and YFCC (Thomee et al., 2016). Worth mentioning, due to the download failure or nonEnglish caption, we do not obtain the complete data for these datasets. We further use the YFCC15M query to crawl about 59M filtered ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0a212f39308583b1cf69a8c17dd30e076d9672d7286f2297521508281ad81028.jpg",
+        "table_caption": [
+            "Table 1: Details of DeCLIP pre-training datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td>TRAINING SIZE</td></tr><tr><td>CLIP (RADFORD ET AL.,2021)</td><td>400M</td></tr><tr><td>ALIGN (JIA ET AL., 2021)</td><td>1.8B</td></tr><tr><td>CC (SHARMA ET AL., 2018)</td><td>3M</td></tr><tr><td>CC-12M(CHANGPINYO ET AL.,2021)</td><td>11M</td></tr><tr><td>YFCC (THOMEE ET AL.,2016)</td><td>15M</td></tr><tr><td>DECLIP OPEN-SOURCE DATA</td><td>29M</td></tr><tr><td>DECLIP WEB-CRAWLED DATA</td><td>59M</td></tr><tr><td>DECLIP FULL DATA</td><td>88M</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "web data on the Internet, together with open-source data to form the 88M DeCLIP full pre-training dataset. More details can be seen in Appendix C. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Downstream datasets We assess our model performances in a wider variety of distributions and tasks. Following the CLIP (Radford et al., 2021), we evaluate the image encoder transferability on 11 widely used downstream datasets, such as Food-101, CIFAR-10, etc. To conduct a fair comparison, the metric and division for each dataset that can be collected are consistent with those of CLIP. More details of downstream datasets can be found in Tabel 6 of Appendix D. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9cf73d38023a7786008f4c6ac9a8a422d164c277356e8beea2be9652a80ddb72.jpg",
+        "table_caption": [
+            "Table 2: Zero-shot top1 accuracy on ImageNet. Our DeCLIP shows great data-efficency. "
+        ],
+        "table_footnote": [
+            "† OUR REIMPLEMENTATION. "
+        ],
+        "table_body": "<table><tr><td>METHOD</td><td>IMAGE ENCODER</td><td>#PARAMS</td><td>TRAINING SIZE</td><td>ZERO-SHOT TOP1 ACC.</td></tr><tr><td>CLIP+</td><td>REsNET50</td><td>24M</td><td>88M</td><td>56.9</td></tr><tr><td>CLIP</td><td>REsNET50</td><td>24M</td><td>400M</td><td>59.6</td></tr><tr><td>DECLIP</td><td>REsNET50</td><td>24M</td><td>88M(↓4.5×)</td><td>62.5(个 +2.9)</td></tr><tr><td>CLIP</td><td>REsNET101</td><td>42M</td><td>400M</td><td>62.2</td></tr><tr><td>CLIP†</td><td>VIT-B/32</td><td>88M</td><td>88M</td><td>57.4</td></tr><tr><td>CLIP</td><td>VIT-B/32</td><td>88M</td><td>400M</td><td>63.2</td></tr><tr><td>DECLIP</td><td>VIT-B/32</td><td>88M</td><td>88M(↓4.5×)</td><td>66.2(↑ +3.0)</td></tr><tr><td>CLIP</td><td>ResNET50×64</td><td>291M</td><td>400M</td><td>73.6</td></tr><tr><td>DECLIP</td><td>REGNETY-64GF</td><td>276M</td><td>88M(↓4.5×)</td><td>73.7(个 +0.1)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 EXPERIMENTS SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Network architectures Following CLIP, we first consider two different architectures for the image encoder: a modified version of ResNet50 (Radford et al., 2021) and ViT-B/32 (Dosovitskiy et al., 2020). The text encoder is a Transformer (Vaswani et al., 2017) with the architecture modifications described in Radford et al. (2019). The image and text features are projected to the same 1024 dimension, followed by L2 normalization before interaction. Benefiting from the rapid progress of large CV models, we further scale up our model. Our largest model is a RegNetY-64GF (Radosavovic et al., 2020; Goyal et al., 2021) image encoder with a BERT (Devlin et al., 2018) text encoder, which is on-pair with the largest CLIP- $. { \\mathrm { R 5 0 } } { \\times } 6 4$ model. ",
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+    },
+    {
+        "type": "text",
+        "text": "Pre-training setup For a fair comparison with CLIP, we train our DeCLIP-ResNet50 and DeCLIP-ViT-B/32 from scratch for 32 epochs. Unless otherwise specified, we use full data, i.e., 88M image-text pairs, to obtain the best performance. The input resolution of the image encoder is $2 2 4 \\times 2 2 4$ , and the maximum context length of the text encoder is 76. The learnable temperature parameter $\\tau$ is initialized to 0.07. The loss weights of additional supervision $\\alpha$ , $\\beta$ and $\\gamma$ are all set to 0.2. More details can be found in Appendix C. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Downstream evaluation setup We evaluate our model transferability by performing linear classification on frozen features, i.e., the pre-trained image encoder is fixed and serves as a feature extractor. After feature extraction, we train the linear classifier with the L-BFGS optimizer as the same in Radford et al. (2021). ",
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+    {
+        "type": "text",
+        "text": "4.3 MAIN RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Zero-shot recognition on ImageNet After pre-training, we use natural language to refer to visual concepts enabling the zero-shot ability of our model. As shown in Fig. 1, our DeCLIP-ResNet50 consistently outperforms the CLIP-ResNet50 across all dataset sizes. When the data amount reaches 56M (29M open-source $\\mathbf { + \\ 2 7 M }$ web-crawled), our model can achieve $6 0 . 4 \\%$ accuracy, $0 . 8 \\%$ above the CLIP-ResNet50, while using $7 . 1 \\times$ fewer data. As described in Tab. 2, with our full data, our best ResNet50/ViT-B32 models boost the zero-shot performance to $6 2 . 5 \\%$ and $6 6 . 2 \\%$ , nearly $3 . 0 \\%$ higher than the best number reported for these two architectures. Moreover, our DeCLIP-ResNet50 is even $0 . 3 \\%$ better than CLIP-ResNet101, revealing the effectiveness and efficiency of our framework. Scaling up model capacity works in our framework as well. Our biggest DeCLIP-RegNetY64GF achieves $7 3 . 7 \\%$ accuracy, which is $0 . 1 \\%$ above CLIP ResNet $5 0 \\times 6 4$ with fewer parameters. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Downstream evaluation results We report our linear probe performance on 11 downstream datasets in Tab. 3. Our DeCLIP-ResNet50 outperforms its CLIP counterpart in 8 out of 11 datasets, with a $0 . 8 \\%$ average improvement. There are some datasets that our models perform worse than CLIP models, such as SUN and Food101. We conjecture that this is caused by the different distribution of the pre-trained dataset. Interestingly, our ResNet50 and ViT-B/32 models might have distinct performance on several datasets, such as Pets and Aircraft. We infer that these two types of neural networks might have different data preferences, i.e., different feature extraction capacities when they meet the same data. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/3d2b4004d15eb6f48d3f5e80d9d1488c241c13b497c0d9fef0536ead1748173d.jpg",
+        "table_caption": [
+            "Table 3: Linear probe performance on 11 downstream datasets. There are some abbreviations. C10/100 is CIFAR10/100. F101 is Food101. Flow is Flowers. Cal is Caltech. Air is Aircraft. IN is ImageNet. Our DeCLIP models achieve higher average accuracy over 11 datasets. "
+        ],
+        "table_footnote": [
+            "† OUR REIMPLEMENTATED CLIP MODEL TRAINED ON 88M DATA. "
+        ],
+        "table_body": "<table><tr><td>MODEL</td><td></td><td>50</td><td></td><td></td><td>10</td><td>MOTI</td><td></td><td></td><td>美</td><td></td><td>N</td><td>U</td></tr><tr><td>CLIP-RESNET50+ CLIP-RESNET50</td><td>85.1 88.2</td><td>87.3</td><td>65.0 70.3</td><td>71.3 73.3</td><td>80.8 86.4</td><td>98.2 96.1</td><td>77.2 78.3</td><td>89.6 89.6</td><td>44.8 49.1</td><td>71.0 76.4</td><td>70.8 73.3</td><td>76.5 79.1</td></tr><tr><td>DECLIP-RESNET50</td><td>88.7</td><td>88.7 89.8</td><td>71.2</td><td>72.8</td><td>82.7</td><td>99.2</td><td>81.7</td><td>93.9</td><td>48.4</td><td>76.8</td><td>74.0</td><td>79.9(个 +0.8)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CLIP-VIT-B/32+</td><td>85.3 90.0</td><td>91.6</td><td>74.2 80.5</td><td>72.3 76.6</td><td>80.8 88.8</td><td>97.9 96.9</td><td>77.6 81.8</td><td>93.2 93.0</td><td>47.0 52.0</td><td>72.5 76.5</td><td>70.3 76.1</td><td>78.4 82.5</td></tr><tr><td>CLIP-VIT-B/32</td><td></td><td>95.1</td><td></td><td></td><td></td><td>99.1</td><td></td><td>94.8</td><td></td><td></td><td></td><td></td></tr><tr><td>DECLIP-VIT-B/32</td><td>89.2</td><td>96.5</td><td>84.7</td><td>75.0</td><td>85.0</td><td></td><td>81.6</td><td></td><td>53.5</td><td>78.5</td><td>75.3</td><td>83.0 (个 +0.5)</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/796081aa69bcfc094c57ebf682e4ad8fe85bc76f651c50febc1462a8b3dfd2f4.jpg",
+        "table_caption": [
+            "Table 4: Ablation on additional supervision. SS/MVS/NNS denotes Self-Supervision, Multi-View Supervision and NearestNeighbor Supervision, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CLIP</td><td>MVS</td><td>Ss</td><td>NNS</td><td>ZERO-SHOT</td></tr><tr><td>广</td><td>×</td><td>×</td><td>×</td><td>20.6</td></tr><tr><td></td><td>√</td><td>×</td><td>×</td><td>24.8(个 +4.2)</td></tr><tr><td>√</td><td>广</td><td></td><td>×</td><td>25.4(个 +4.8)</td></tr><tr><td>√</td><td></td><td>&lt;</td><td>√</td><td>27.2(个 +6.6)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/bc5e47ac7e18c08e37f2312818f8ebdaedd2ea57aaef0cfcf10e0f1d8daa11ee.jpg",
+        "image_caption": [
+            "Figure 7: Ablation on pre-training cost on CC3M dataset. The proposed method performs better with less training time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ablation on additional supervision In order to understand the effectiveness of each additional supervision, we conduct an ablation study as indicated in Tab. 4. We follow the DeCLIP-ResNet50 protocol in the pre-training setup except for a smaller 1024 batch size on a smaller CC3M dataset. The single image-text contrastive supervision (CLIP) results in $2 0 . 6 \\%$ zero-shot top1 accuracy on ImageNet. We can observe that MVS boost amazing $4 . 2 \\%$ improvement over the original CLIP. As discussed in Sec. 3.3, the benefits might come from two sides: 1). the MVS can look at a small local view of an image which might be a better fit with the text description. 2). $2 \\times 2$ augmented views can provide $3 \\times$ diverse and high-quality additional supervision, thus leading to more robust representations. SS can further contribute an additional $0 . 6 \\%$ improvement on the basis of MVS. We believe SS could bring more improvements with proper dedicated SSL methods. NNS further brings $1 . 8 \\%$ improvement on the high basis of SS. We will discuss more about NNS at Sec. 4.5. On the YFCC15M dataset, our DeCLIP using full additional supervision gets significant effect improvement, Appendix F shows the details. ",
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+    },
+    {
+        "type": "text",
+        "text": "Ablation on training cost Since we need to encode twice for each image-text pair, we admit our DeCLIP needs a higher training cost than CLIP. Regarding the training time, one DeCLIP iteration equals $1 . 5 \\times$ CLIP iteration. In this ablation, we train the original CLIP-ResNet50 longer than our DeCLIP-ResNet50. As shown in Fig. 7, longer 64 epochs training can bring about $1 . 1 \\%$ improvement. However, our model has $2 7 . 2 \\%$ top1 accuracy, which is still $5 . 3 \\%$ higher than the time-equivalent CLIP. It reveals that our framework can extract richer and more representative features through our proposed supervision. We also include memory usage ablation in Appendix F. ",
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+    {
+        "type": "text",
+        "text": "4.5 ANALYSIS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Class activation maps We try to understand what renders our DeCLIP effective. As shown in Fig. 8, we visualize the class activation maps (CAM) (Zhou et al., 2016) of different models trained on the YFCC dataset. The results validate that the proposed method learns more representative features with the aid of multiple supervision. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2419d1a278807a209c104291cc45f9107ffa7faa3350cc9fbaf8bbe17baf2a26.jpg",
+        "image_caption": [
+            "Figure 8: Class activation maps (CAM) for the CLIP vs. our DeCLIP model trained on the YFCC dataset. The CAMs of our model segment the complete object, while the CLIP model only looks at a few components. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/f61344fb96af452036cbd5e7de88d22ad6f3e802c2d5da0e26f0d73a004c59c8.jpg",
+        "image_caption": [
+            "Figure 9: Nearest neighbor samples from different datasets. As we can see, the NN pair is very similar to the original pair, thus it can provide high-quality supervision. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Nearest neighbor samples In Fig 9, we show some nearest neighbor (NN) samples from different datasets. The first row is the original image-text pair, while the second row is the NN pair. In general, we can see that the texts have similar intellectual meanings. Therefore, the NN pair can provide high-quality supervision. When taking datasets into consideration, the matching performs very well in well-filtered datasets, such as CC3M and CC12M. This matching might be a little worse in more noisy datasets, such as YFCC and web-crawled, but it can still provide some beneficial guidance. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This paper introduces DeCLIP, a Data efficient Contrastive Language-Image Pre-training paradigm. Our goal is to learn visual representations through the use of broader and scalable supervision. Specifically, instead of using the single image-text contrastive supervision, we fully exploit data potential through the use of (1) self-supervision within each modality; (2) multi-view supervision across modalities; (3) nearest-neighbor supervision from other similar pairs. Experimentally, DeCLIP shows superior effectiveness and efficiency with different types of neural nets(CNN and ViT) and different amounts of data. We hope our work could bring insights about exploiting the multimodal data. ",
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+    {
+        "type": "text",
+        "text": "A PSEUDO CODE OF DECLIP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The training pseudo code of DeCLIP is as follows: ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 DeCLIP ",
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+    {
+        "type": "text",
+        "text": "Input: $I , \\tilde { I } , T , \\tilde { T }$ , image encoder, text encoder, F eature Queue   \n1: function BATCH-UPDATING(I, ˜I, T , T˜)   \n2: # Get The Features of The Current Batch.   \n3: $I _ { f } , \\tilde { I } _ { f } \\gets i m a g e \\_ e n c o d e r ( I )$ , image encoder( ˜I)   \n4: $T _ { f } , \\tilde { T } _ { f } \\gets t e x t . e n c o d e r ( T )$ , text encoder $( \\tilde { T } )$   \n5: TNN ← NEAREST-NEIGHBOR(F eature Queue, Tf )   \n6:   \n7: # Calculate The losses.   \n8: $L _ { C L I P } \\gets \\mathrm { I N F O N C E - L O S S } ( I _ { f } , T _ { f } )$   \n9: $L _ { S S } \\gets \\mathrm { I M A G E - S S - L O s s } ( I _ { f } , I _ { f } ) + \\mathrm { T E X T - S S - L O s s } ( T _ { f } )$   \n10: ${ \\cal L } _ { M V S } \\gets \\mathrm { I N F O N C E - L O s s } ( I _ { f } , \\bar { T } _ { f } ) + \\mathrm { I N F O N C E - L O s s } ( \\bar { I } _ { f } , T _ { f } ) + \\mathrm { I N F O N C E - L O s s } ( \\bar { I } _ { f } , \\bar { T } _ { f } )$   \n11: $L _ { N N S } \\gets \\mathrm { I N F O N C E - L O s s } ( \\bar { I } _ { f } , \\bar { T } _ { N N } )$   \n12: $L _ { D e C L I P } \\gets ( 1 - \\alpha - \\beta - \\gamma ) L _ { C L I P } + \\alpha L _ { S S } + \\beta L _ { M V S } + \\gamma L _ { N N S }$   \n13:   \n14: # Update The Network.   \n15: image encoder $\\gets$ BACKWARD-UPDATE(image encoder, LDeCLIP )   \n16: text encoder $\\gets$ BACKWARD-UPDATE(text encoder, LDeCLIP )   \n17: F eature Queue FIFO-UPDATE(F eature Queue, $T _ { f }$ )   \n18: end function   \n19:   \n20: function IMAGE-SS-LOSS(If , ˜If )   \n21: $z , \\tilde { z } \\gets$ image encoder.proj $\\left( I _ { f } \\right)$ , image encoder.proj( ˜If )   \n22: p, p˜ ← image encoder.pred(z), image encoder.pred(˜z)   \n23: $z , \\tilde { z } \\gets z . d e t a c h ( ) , \\tilde { z } . d e t a c h ( )$   \n24: # Calculate the Negative Cosine Similarity loss.   \n25: $L _ { I m a g e - S S }  N C S ( p , \\tilde { z } ) / 2 + N C S ( \\tilde { p } , z ) / 2$   \n26: return LImage−SS   \n27: end function   \n28:   \n29: function TEXT-SS-LOSS $( T _ { f } )$ )   \n30: # Get The Masking GT when performing the text-encoder.   \n31: Wf $, W _ { g t } \\gets T _ { f }$ .word feat, $T _ { f }$ .mask id   \n32: $W _ { p r e d }  t e x t . e n c o d e r . p r e d ( W _ { f } )$   \n33: # Calculate the Cross-Entropy loss.   \n34: $L _ { T e x t - S S } \\gets C E ( W _ { g t } , W _ { p r e d } )$   \n35: return LT ext−SS   \n36: end function ",
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+    },
+    {
+        "type": "text",
+        "text": "B DATA AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Details of SS The image SS uses SimSiam (Chen & He, 2021) method as the image selfsupervision. The prediction module is a 2-layer MLP, in which the hidden dimensions are 512 and output dimensions are 1024, the projection module is a 3-layer MLP, in which the hidden and output dimensions are both 1024. The text SS uses the Masked Language Model (Devlin et al., 2018) as the pretext task. In detail, we first randomly choose $1 5 \\%$ of all tokens in each sequence. Then the token is replaced with (1) the [mask] token $8 0 \\%$ of the time (2) a random token $\\bar { 1 } 0 \\%$ of the time (3) the unchanged token $1 0 \\%$ of the time. ",
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+    },
+    {
+        "type": "text",
+        "text": "Image augmentations The augmentation policy includes: RandomResizedCrop with scale in [0.2,1.0] (Wu et al., 2018), ColorJitter containing $\\{$ {brightness, contrast, saturation, hue} strength of $\\{ 0 . 4 , 0 . 4 , 0 . 4 , 0 . 1 \\}$ with an applying probability of 0.8, RandomGrayscale with an applying probability of 0.2. Blurring augmentation (Chen et al., 2020a) has a Gaussian kernel with std in [0.1, 2.0], and RandomHorizontalFlip. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Text augmentations We use the EDA (Wei & Zou, 2019) method as our text augmentation strategy, which contains three types of text augmentation strategies: synonym replacement, random swap, and random deletion. Each text will randomly select one of these three types for text augmentation. ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "C PRE-TRAINING DATASETS & IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+            256,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Open-source data. Conceptual Captions (Sharma et al., 2018) is a $3 . 3 \\ \\mathrm { M }$ image caption opensource data. Due to the failure of the download link, we only download about 3M data(CC3M). Conceptual 12M (Sharma et al., 2018) contains approximately 12M of image-text pairs(CC12M), which is larger than the CC3M and covers a more diverse set of visual concepts. Also, due to the failure of the download link, we only download about 11M data. YFCC (Thomee et al., 2016), the Yahoo Flickr Creative Commons 100M data, is a dataset for vision language tasks. We download about $8 6 . 5 \\mathrm { M }$ data from the YFCC website and use four filtering rules to filter the DeCLIP YFCC15M dataset to benchmark against CLIP YFCC15M. The four rules are: filtering data with damaged images, filtering data without the caption, filtering data with a caption English word ratio less than 0.8, filtering data with a caption only including one part of speech. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Web-crawled data. We use the user tags and machine tags of YFCC15M to form a tag list, and use WordNet (Miller, 1995) to find synonyms for each tag in the tag list to form a synonym tag list. The tag list and synonym tag list form a query list. Then we use the query list to crawl images from the Internet, after filtering data with smaller images, filtering data with damaged images, filtering data without the caption, and filtering data with Chinese in the caption, we collect 59M web crawled data. Tabel 5 shows the source link of pre-training datasets, and Figure 10 shows some cases random sampled from each dataset. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ff256bd190f5fa5cb1563d8163a4724b11aa7664d82de5778c462e23266912f7.jpg",
+        "table_caption": [
+            "Table 5: The source link of DeCLIP pre-training datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td>DATASET API</td></tr><tr><td>CONCEPTUALCAPTIONS</td><td>HTTPS://AI.GOOGLE.COM/RESEARCH/CONCEPTUALCAPTIONS</td></tr><tr><td>CONCEPTUAL12M</td><td>HTTPS://GITHUB.COM/GOOGLE-RESEARCH-DATASETS/CONCEPTUAL-12M</td></tr><tr><td>YFCC</td><td>HTTP://PROJECTS.DFKI.UNI-KL.DE/YFCC100M</td></tr><tr><td>GOOGLE</td><td>HTTPS://WWW.GOOGLE.COM.HK</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Implementation details We train DeCLIP-ResNet50 (abbr. as R50) and DeCLIP-ViT-B/32 (abbr. as V-B32) from scratch for 32 epochs. For R50, we use the FP16-SGD optimizer with the batch size of 10,240 $( 1 2 8 \\times 8 0 )$ ). Starting with an 0.01 learning rate (lr), we first linearly increasing the lr to 0.2 (a.k.a warm-up) in one epoch. Then we use cosine anneal lr decay to decrease the lr. The weight decay is set to 0.0001. For V-B32, we use a hybrid FP16-AdamW-SGD optimizer with bacth size $1 0 , 2 4 0 ( 1 2 8 \\times 8 0 )$ ). For the ViT image encoder, we use AdamW optimizer with the lr warming up from 1e-4 to 1e-3 in one epoch. The weight decay is set to 0.05. For the text encoder, we use the SGD optimizer with the lr warming up from 1e-3 to 0.02 in one epoch. The weight decay is set to 1e-4. ",
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+    },
+    {
+        "type": "text",
+        "text": "Pre-training cost Our R50 and V-B32 took 8/10 days to train on 80 V100 GPUs, respectively. Our largest DeCLIP-RegNetY-64GF took 21 days on 160 V100 GPUs, while the largest CLIP- ${ \\mathrm { R 5 0 } } \\times 6 4$ from (Radford et al., 2021) spent 18 days on 592 V100 GPUs. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/7597d0083fb5b3c88dc12cc3288664828aa98bb8384c0b068b46b4ce50440dfa.jpg",
+        "image_caption": [
+            "Figure 10: Example image-text pairs randomly sampled from the training dataset. (a) Conceptual Captions, (b) YFCC, (c) Conceptual 12M, (d) Web-crawled. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D DOWNSTREAM DATASETS & IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Downstream data. We begin with the 12 datasets from the well-studied evaluation suite introduced by Kornblith et al. (2019). Within these 12 datasets, Birdsnap can not be downloaded, and PASCAL VOC 2007 classification is replaced by more challenging ImageNet-1K, resulting in our 11 datasets. They are: Food-101, CIFAR-10, CIFAR-100, SUN397, Stanford Cars, FGVC Aircraft, Describable Textures, Oxford-IIIT Pets, Caltech-101, Oxford Flowers 102. Tab. 6 is the detailed information of these datasets. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/8662742de9a8a9ec0d5242984fd33aa7e005d4a664aa011675880d70caa2e1cd.jpg",
+        "table_caption": [
+            "Table 6: Details of DeCLIP downstream datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td>CLASSES</td><td>TRAIN SIZE</td><td>TEST SIZE</td><td>EVALUATIONMETRIC</td></tr><tr><td>CIFAR10</td><td>10</td><td>50,000</td><td>10,000</td><td>ACCURACY</td></tr><tr><td>CIFAR100</td><td>100</td><td>50.000</td><td>10,000</td><td>ACCURACY</td></tr><tr><td>F00D-101</td><td>101</td><td>75,750</td><td>25,250</td><td>ACCURACY</td></tr><tr><td>OXFORDIIIT-PETS</td><td>37</td><td>3,680</td><td>3,669</td><td>MEAN PER CLASS</td></tr><tr><td>OXFORD 102 FLOWERS</td><td>102</td><td>2.040</td><td>6,149</td><td>MEAN PER CLASS</td></tr><tr><td>SUN</td><td>397</td><td>19,850</td><td>19,850</td><td>ACCURACY</td></tr><tr><td>STANFORD CARS</td><td>196</td><td>8,144</td><td>8,041</td><td>ACCURACY</td></tr><tr><td>DTD</td><td>47</td><td>3,760</td><td>1,880</td><td>ACCURACY</td></tr><tr><td>CALTECH-101</td><td>102</td><td>3,060</td><td>6,085</td><td>MEAN PER</td></tr><tr><td>FGVC AIRCRAFT</td><td>100</td><td>6,667</td><td>3,333</td><td>MEAN PER CLASS</td></tr><tr><td>IMAGENET1K</td><td>1000</td><td>1,281,167</td><td>50,000</td><td>ACCURACY</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Implementation details We follow CLIP (Radford et al., 2021) to train a logistic regression classifier using L-BFGS, with maximum 1,000 iterations, and report the corresponding metric for each dataset. We determine the L2 regularization strength $\\lambda$ using a hyperparameter sweep on the validation sets over the range between $1 0 ^ { - 6 }$ and $1 0 ^ { 6 }$ , with 96 logarithmically spaced steps. To save compute required for the sweeps, we perform a parametric binary search that starts with $\\lambda = [ 1 0 ^ { - 6 }$ , $1 0 ^ { - \\dot { 4 } }$ , $1 0 ^ { - 2 }$ , 1, $1 0 ^ { 2 }$ , $1 0 ^ { 4 }$ , $1 0 ^ { 6 } ]$ and iteratively halves the interval around the peak until it reaches a resolution of 8 steps per decade. The hyperparameter sweeps are performed on a validation split of each dataset. For the datasets that contains a validation split in addition to from the test split, we use the provided validation set to perform the hyperparameter search, and for the datasets that do not provide a validation split or have not published labels for the test data, we split the training dataset to perform the hyperparameter search and report the performance on the validation data. ",
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E PROMPT ENGINEERING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            277,
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+            294
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Due to the reason that it’s relatively rare in the dataset for the text to be a single word, we use prompts such as $\" a$ photo of a $\\{ \\mathrm { 1 a b e 1 } \\} \"$ for zero-shot classification. For a fair comparison, we use the same prompts as proposed in Radford et al. (2021) for the ImageNet dataset. As shown in Fig 11, the prompts reduce the domain gap between the training dataset and testset, and fully consider the different situations for the picture. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "a bad photo of a {label}.   \na photo of many {label}.   \na sculpture of a {label}.   \na photo of the hard to see {label}.   \na low resolution photo of the {label}.   \na rendering of a {label}.   \ngraffiti of a {label}.   \na bad photo of the {label}.   \na cropped photo of the {label}.   \na tattoo of a {label}.   \nthe embroidered {label}.   \na photo of a hard to see {label}.   \na bright photo of a {label}.   \na photo of a clean {label}.   \na photo of a dirty {label}.   \na dark photo of the {label}.   \na drawing of a {label}.   \na photo of my {label}.   \nthe plastic {label}.   \na photo of the cool {label}. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/e685a6a711b7ad54e01e952638e18b72f3ce6a258ddcd267d907ad4e151b4ac2.jpg",
+        "image_caption": [
+            "Figure 11: The prompts for zero-shot testing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F ADDITIONAL STUDY ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/a1c88e67ffd3076e07cf6bd1348a5c72a05f0560bd752c84a44e89c351ccc327.jpg",
+        "table_caption": [
+            "Table 7: DeCLIP zero-shot performance of ImageNet top1 on different training datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>BACKBONE</td><td>DATASET</td><td>DATA SIZE</td><td>BATCH SIZE</td><td>ZERO-SHOT</td></tr><tr><td>REsNET50</td><td>CONCEPTUALCAPTIONS</td><td>3M</td><td>2,048</td><td>27.8</td></tr><tr><td>REsNET50</td><td>CONCEPTUAL12M</td><td>11M</td><td>4,096</td><td>41.0</td></tr><tr><td>RESNET50</td><td>YFCC</td><td>15M</td><td>4,096</td><td>41.9</td></tr><tr><td>REsNET50</td><td>DECLIP OPEN-SOURCE DATA</td><td>29M</td><td>6,144</td><td>49.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Different Pre-training Datasets Data is critical for language-image pre-training task. As shown in Tab. 7, we evaluate our DeCLIP on different sources of datasets. Combining Tab.7 and Fig.1, we can see that when the amount of training data continues to scale up, the zero-shot recognition ability continues to improve as well. In addition, we can see that the open source data has high quality. The 29M open source data can achieve $4 9 . 3 \\%$ zero-shot top1 accuracy on ImageNet through the DeCLIP training paradigm. Our open-source data is an affordable benchmark, which would be beneficial for explorations. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Our YFCC re-implementation Although we use the same number of image-text pairs as the CLIP YFCC-15M, our YFCC data is different from CLIP. We also reproduce the naive CLIP on our YFCC-15M data, which results in $3 5 . 9 \\%$ zero-shot top1 accuracy on ImageNet-1K (see Fig. 8). It is relatively higher than the number in CLIP paper $( 3 1 . 1 \\% )$ . We conjecture the improvements might be caused by the different data cleaning strategies. However, our DeCLIP can achieve $4 1 . 9 \\%$ zero-shot accuracy which is also $6 . 0 \\%$ higher than our CLIP re-implementation. ",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/53ff2d3a432b12e62a157fe31dc337181959190e86b844012553188c822353c6.jpg",
+        "table_caption": [
+            "Table 8: DeCLIP zero-shot performance of ImageNet top1 on YFCC datasets. Although we use the same amount of data, our YFCC is different with CLIP YFCC due to the different data cleaning strategies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>BACKBONE</td><td>DATASET</td><td>DATA SIZE</td><td>BATCH SIZE</td><td>ZERO-SHOT</td></tr><tr><td>CLIP</td><td>REsNET50</td><td>YFCC</td><td>15M</td><td></td><td>31.3</td></tr><tr><td>CLIP (OUR REIMP.)</td><td>REsNET50</td><td>YFCC*</td><td>15M</td><td>4,096</td><td>35.9</td></tr><tr><td>DECLIP</td><td>REsNET50</td><td>YFCC*</td><td>15M</td><td>4,096</td><td>41.9(↑+6.0)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Memory usage Because of the additional views, our DeCLIP is more memory-consuming. Thanks to the ICLR anonymous review comments: a fairer comparison might be doubling the batch size of CLIP. Flowing the ablation study in Fig. 7, we double the batch size and train CLIP-ResNet50 for 64 epochs. The final result is $2 2 . 3 \\%$ which is still $4 . 9 \\%$ lower than our DeCLIP model. We summarize the memory usage, training cost, and the final accuracy as below. All experiments are conducted on CC-3M with 16 V100 GPUs. ",
+        "bbox": [
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+            356,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/a199ca3e8ed6e778fb5e867454a918ea7578fff4a0d309b8192582da8d2d66b6.jpg",
+        "table_caption": [
+            "Table 9: Ablation on Memory usage. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>BATCH SIZE PER GPU</td><td>MEMORY (GB)</td><td>EPOCHS</td><td>COST (GPU HOURS)</td><td>ZERO-SHOT</td></tr><tr><td>CLIP-RESNET50</td><td>128</td><td>15.8</td><td>64</td><td>416</td><td>21.7</td></tr><tr><td>CLIP-RESNET50</td><td>256</td><td>24.0</td><td>64</td><td>399</td><td>22.3</td></tr><tr><td>DECLIP-RESNET50</td><td>128</td><td>22.7</td><td>32</td><td>304</td><td>27.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 16
+    }
+]