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+# Grounded Decoding: Guiding Text Generation with Grounded Models for Embodied Agents
+
+Wenlong Huang1∗, Fei $\mathbf { X i a } ^ { 2 }$ , Dhruv Shah3, Danny Driess2, Andy Zeng2, Yao $\mathbf { L } \mathbf { u } ^ { 2 }$ , Pete Florence2, Igor Mordatch2, Sergey Levine2,3, Karol Hausman2, Brian Ichter2 1Stanford University, 2Google Deepmind, 3UC Berkeley grounded-decoding.github.io
+
+# Abstract
+
+Recent progress in large language models (LLMs) has demonstrated the ability to learn and leverage Internet-scale knowledge through pre-training with autoregressive models. Unfortunately, applying such models to settings with embodied agents, such as robots, is challenging due to their lack of experience with the physical world, inability to parse non-language observations, and ignorance of rewards or safety constraints that robots may require. On the other hand, languageconditioned robotic policies that learn from interaction data can provide the necessary grounding that allows the agent to be correctly situated in the real world, but such policies are limited by the lack of high-level semantic understanding due to the limited breadth of the interaction data available for training them. Thus, if we want to make use of the semantic knowledge in a language model while still situating it in an embodied setting, we must construct an action sequence that is both likely according to the language model and also realizable according to grounded models of the environment. We frame this as a problem similar to probabilistic filtering: decode a sequence that both has high probability under the language model and high probability under a set of grounded model objectives. We demonstrate how such grounded models can be obtained across three simulation and real-world domains, and that the proposed decoding strategy is able to solve complex, long-horizon embodiment tasks in a robotic setting by leveraging the knowledge of both models.
+
+# 1 Introduction
+
+Recent works have demonstrated robots that are increasingly proficient at understanding and acting upon natural language, whether through planning or conditioned policies. Complementing such progress, the field of natural language processing has recently seen large language models (LLMs) become ubiquitously used as pre-trained or few-shot prompted models, due to their impressive fewshot performance and vast knowledge-base. These LLMs have efficiently learned from web-scale data through autoregressively modeling the probability distribution over text tokens and thus generate text. However, the nature of this process is such that applying such models to embodied settings remains a challenge. They have not interacted with their environment, lack observability of nonlanguage observation modalities (e.g., images), and may not know what is safe or possible for a particular embodiment.
+
+Determining how to execute long-horizon behaviors based on high-level verbal commands is one particular area of robotics where the rich semantic knowledge in large language models can be especially useful. This problem combines elements of semantic reasoning and planning: the robot must understand the instruction, determine the steps needed to fulfill it, and also determine how to sequence those steps appropriately given its capabilities and the current state of the environment. However, this is not a problem that can be solved purely with semantics, as it requires sufficient grounding to understand how the task should be performed in context – for example, in the example in Figure 1, the language model alone has no way of knowing which block to pick up because this requires knowledge of which blocks are present, and also what manipulations the robot is capable of performing on them. Thus, although a language model can assign probabilities for how likely various steps are to correspond to the desired task semantically, the constraints of the planning problem must also enter into the process. These constraints could themselves be represented as probabilities that mirror the token probabilities generated by a language model, reflecting their applicability to the current environment rather than their semantic likelihood. We can frame this as a problem similar to probabilistic filtering: decode a sequence (i.e., a task description) that both has a high probability under the language model and a high probability under a grounded model that predicts how applicable this sequence is to the current scene.
+
+![](images/55c2bb7f596e605d1eb02a8b51a6da2372e6a2ac1e480096707c0e9fa6f7f3f9.jpg)  
+Figure 1: Grounded Decoding solves robotic tasks by taking an instruction as input and selecting tokens that have high probability under a Large Language Model (LLM) and a set of Grounded Models (GM). Thus, it leverages the open-vocabulary and semantic knowledge of LLMs while being grounded in the environment and in the robot’s capabilities. Furthermore, the whole process does not require expensive fine-tuning of the LLM.
+
+Herein, we present Grounded Decoding (GD), a scalable, general approach to planning with LLMs embodied domains. Grounded Decoding jointly decodes the token probability of an LLM and token probabilities from token-conditioned, robotic functions, such as affordance functions capturing the abilities of a robot given its embodiment, safety functions, or more. By guiding the LLM directly at its output, Grounded Decoding enables a general and flexible family of planning algorithms that combines LLM’s strength of long-horizon and semantic reasoning and grounded models’ strength of local and embodiment grounding.
+
+Our contributions are as followed: 1) we present a robot-centric formulation for decoding language models to perform long-horizon robotic tasks with token-conditioned grounded models, 2) we demonstrate techniques for learning such grounded models, serving different purposes such as affordances and safety requirements, and 3) we show empirical evidence, across three simulation and real-world domains, that the proposed method performs strongly on a wide range of tasks while also significantly outperforming prior methods in efficiency.
+
+# 2 Related Work
+
+Guided Decoding for Language Models. Decoding strategies for large language models is an active area of research within natural language processing [77, 87, 25, 85, 38]. A number of recent works have focused on developing decoding heuristics for natural text generation [49, 35, 48, 18, 25, 6, 36, 63]. Another line of works use external classifiers for maximizing certain language-space utilities when decoding language models [71, 92, 26, 39, 37, 21, 38, 4, 12, 23]. Most closely related to our work are classifier-guided decoding methods developed for offline domains, such as image captioning [78, 74] and task-oriented dialog [72, 83]. However, extensions to embodied domains, which we investigate exclusively in this work, remain non-trivial because grounding in embodied domains is bounded by the abilities of the agent and by environment state transition as the agent actively interacts with the environment.
+
+![](images/5fdb2a7a22b965fe17fff70f6ac16a295d0678a3944f751bcee5eb6cca8daed2.jpg)  
+Figure 2: Overview of Grounded Decoding. Given a free-form language instruction, a language model and grounded models jointly decide the next candidate token to be decoded by combining their respective likelihood. Language model proposes likely tokens that produce goal-directed and coherent long-horizon behaviors, while grounded models connect them to the physical scene, through a flexible composition of multiple objective functions from multiple modalities, such as affordance, preferences, and safety.
+
+Embodied and Multimodal Language Models. Training language models to understand embodiment is an active area of research. Training multimodal models can enable some degree of embodied reasoning, such as understanding images and videos [9, 40, 76, 3]. Directly finetuning language models to output actions has also been investigated [75, 55, 66]. Lastly, training downstream models on language model embeddings shows promise [46, 52, 24, 94, 60, 41]. In this work, we investigate leveraging large frozen language models for embodied applications [29, 2, 96, 8, 64, 30, 42, 70, 47, 27, 58, 73, 44, 43, 14, 16, 82, 93, 89, 45, 56], with grounded models to provide domain-specific grounding during decoding process.
+
+Comparison to SayCan. The most closely related work to our work is SayCan [2]. SayCan uses a large language model and a value function to select robotic skills among a constrained set of primitives. This constrained set of primitives enables SayCan to use the so-called “scoring-mode” of the LLM to get the probability of a skill being useful to a high-level instruction. This requirement to consider only a fixed and enumerated set of primitives limits the applicability of SayCan in scenarios with many possible skills, such as open vocabulary or combinatorial tasks. Grounded Decoding on the other hand jointly decodes the LLM and the grounded model at the token level, allowing for expressive decoding with an open vocabulary. Furthermore, SayCan considers only grounding functions derived from RL-trained value functions for affordance grounding functions, while Grounded Decoding explores many types of grounding functions to propose a broad family of algorithms.
+
+Task and Motion Planning. Task and motion planning [33] seeks to solve high-level instructions via sequencing tasks in dynamically feasible manner. Research within this area generally focuses on symbolic planning [19] or optimization-based [79] approaches. Machine learning has increasingly been used to accelerate planning and enable new domains [91, 61, 69, 20, 17, 28, 90, 31, 1, 65, 51, 86, 88, 15]. However, planning constraints are often explicitly specified for TAMP methods. In contrast, we specify constraints as (learned) probabilities, which are baked into the decoding process and provided by domain-specific grounded models.
+
+# 3 Grounded Decoding
+
+# 3.1 LLMs and Grounding Models
+
+Large Language Models. LLMs are trained to predict the probability $p ( W )$ of a text sequence $W$ , represented as a sequence of tokens $W = w _ { 1 : N } = ( w _ { 1 } , \dotsc , w _ { N } )$ . The tokens are elements of a fixed vocabulary $\mathcal { W }$ . Typical neural architectures factorize the joint probability into $p ( W ) =$ $\begin{array} { r } { \prod _ { n = 1 } ^ { N } p _ { \mathrm { L L M } } ( w _ { n } | w _ { 1 : n - 1 } ) } \end{array}$ , where $p _ { \mathrm { L L M } }$ is predicted by a transformer network [81]. Given $p _ { \mathrm { L L M } }$ , generating a text consisting of $N$ -many tokens, the so-called decoding process, can be seen as the optimization problem arg $\begin{array} { r } { \operatorname* { m a x } _ { w _ { 1 : N } \in \mathcal { W } } \prod _ { n = 1 } ^ { N } p _ { \mathrm { L L M } } ( w _ { n } | w _ { 1 : n - 1 } ) } \end{array}$ , which in practice is solved, e.g., using greedy search , beam search, or sampling strategies. To further ensure the LLM is solving a desired task, one typically starts with a given text, the so-called prefix or prompt, that describes the task, and then the LLM completes this task in its decoding process.
+
+Grounding Functions. We use the concept of grounding functions, $p _ { \mathrm { G } } ( w _ { 1 : n } | s )$ , which seek to model a probability of tokens $w _ { 1 : n }$ given (potentially non-textual) state $s \in S$ . This state is intended to capture the embodiment of the robot and the environment, which may be an image, proprioception of the robot, or the environment state. Thus the grounding function models probabilities relevant to the robot embodiment and environment, such as whether the tokens are possible for the robot to execute given the state (affordances), or other values like safety, cost, or user preferences.
+
+# 3.2 Problem formulation.
+
+Given an instruction in language $\ell$ , we look at the problem of using an LLM to decode a language plan $w _ { 1 : N }$ , which is typically done by finding the most likely tokens under the probability distribution predicted by the LLM, $p _ { \mathrm { L L M } } ( w _ { 1 : N } | \ell )$ , with $\ell$ being the prefix. However, based on the instruction $\ell$ as the prefix alone, the LLM can easily generate text that is not grounded in the physical state of the environment, rendering such plans useless in the real world. In order to ground the language model in an actual physical embodiment, we propose Grounded Decoding (GD): The main idea of GD is to guide the generation of token sequences with grounding function $( s )$ that are conditioned on the embodiment of the system.
+
+Formally, let $s \in S$ denote a representation of the state of the world. Then, GD attempts to generate text that is consistent with both the instruction $\ell$ and the physical state $s$ :
+
+$$
+w _ { 1 : N } ^ { * } = \arg \operatorname* { m a x } _ { w _ { 1 : N } , w _ { n } \in \mathcal { W } } p _ { \mathrm { G D } } ( w _ { 1 : N } | s , \ell )
+$$
+
+To leverage the Internet-scale knowledge of LLMs, we factorize $p _ { \mathrm { G D } } ( w _ { 1 : N } | s , \ell )$ as follows 2:
+
+$$
+\begin{array} { r l } { p _ { \mathrm { G i D } } ( w _ { 1 : N } | s , \ell ) = \frac { p ( s , \ell | w _ { 1 : N } ) p ( w _ { 1 : N } ) } { p ( s , \ell ) } } & { } \\ { = \frac { p ( s | w _ { 1 : N } ) p ( \ell | w _ { 1 : N } ) p ( w _ { 1 : N } ) } { p ( s , \ell ) } } & { } \\ { = \frac { p ( w _ { 1 : N } | \ell ) p ( \ell ) p ( w _ { 1 : N } | s ) p ( s ) p ( w _ { 1 : N } ) } { p ( s , \ell ) p ( w _ { 1 : N } ) p ( w _ { 1 : N } ) } } & { } \\ { \propto \frac { p ( w _ { 1 : N } | \ell ) } { p ( w _ { 1 : N } ) } p ( w _ { 1 : N } ) } & { } \\ { \propto \frac { p ( w _ { 1 : N } | \ell ) } { p ( w _ { 1 : N } ) } p ( w _ { 1 : N } | s ) } & { } \\ { \propto p ( w _ { 1 : N } | \ell ) p ( w _ { 1 : N } | s ) . } & { } \end{array}
+$$
+
+To decode autoregressively with the formulation, we factorize above into token decoding:
+
+$$
+\begin{array} { r } { p _ { \mathrm { G D } } ( w _ { 1 : N } | s , \ell ) \propto \displaystyle \prod _ { n = 1 } ^ { N } p _ { \mathrm { L L M } } ( w _ { n } | w _ { 1 : n - 1 } , \ell ) p _ { \mathrm { G } } ( w _ { 1 : n } | s ) . } \end{array}
+$$
+
+The first term, $p _ { \mathrm { L L M } } ( w _ { n } | w _ { 1 : n - 1 } , \ell )$ , can be modeled as the probability of the LLM predicting the token for the given instruction $\ell$ appended previously decoded tokens $w _ { 1 : n - 1 }$ without the state $s$ as input. The second term, $p _ { \mathrm { G } } ( w _ { 1 : n } | s )$ , is the grounding function that is only conditioned on the state $s$ and judges whether the generated text $w _ { 1 : n }$ is consistent with the physical state. The core idea behind this factorization is that LLMs exhibit long-term planning capabilities, while the grounding function guides the planning of the LLM to be possible in the concrete embodied physical world without needing to be informed or capable of reasoning over the long-horizon instruction.
+
+# 3.3 Grounded Decoding
+
+This work investigates grounded decoding exclusively in the context of task planning for embodied agents. Figure 2 visualizes a single step of the simplest greedy search form of GD, and accompanying pseudo-code can be found in Algorithm 1. Given a high-level language instruction and history of executed actions, GD proceeds through a process similar to probabilistic filtering by selecting tokens iteratively that have high probability under the language model and the grounded model. After each token is selected, it is appended to the prefix. The process continues until a token in the terminal set ${ \mathcal { W } } _ { \mathrm { t e r m } }$ is selected, which could be a period sign “.” indicating the end of a single-step skill (e.g., pick-and-place). Then the command $w _ { 1 : i }$ is sent to a language-conditioned policy $\pi ( \boldsymbol { a } | \boldsymbol { s } , \boldsymbol { w } _ { 1 : i } )$ that executes the action $a$ conditioned on the environment state $s$ . Crucially, this grounding function must accept partial commands to enable grounding during decoding.3 Additionally, we note that GD, in its essence, provides a grounded scoring function; thus, it can be easily extended to any search methods such as beam search, top- $\mathbf { \nabla } \cdot \mathbf { k }$ sampling, etc.
+
+# Algorithm 1 Grounded Decoding (GD) w/ Greedy Search
+
+1: Given: state $s$ , instruction $\ell$ , terminal set ${ \mathcal { W } } _ { \mathrm { t e r m } }$   
+2: Initialize: $w = \{ \}$ , $n = 0$   
+3: while $w _ { n } \notin \mathcal { W } _ { \mathrm { t e r m } } \mathbf { d o }$   
+4: $n = n + 1$   
+5: $\begin{array} { r } { \left. w _ { n } = \arg \operatorname* { m a x } _ { w _ { n } \in \mathcal { W } } p _ { \mathrm { L L M } } ( w _ { n } | w _ { 1 : n - 1 } , \ell ) p _ { \mathrm { G } } ( w _ { 1 : n } | s ) \right. } \end{array}$   
+6: end while   
+7: Return: $w$
+
+# 3.4 Techniques for Obtaining Grounded Models
+
+Unlike language tasks, where a single model is capable of performing general semantic reasoning, a singular grounded model remains an open problem. Indeed, each domain may impose varied environmental and embodiment constraints. Despite these challenges, we present several techniques for obtaining grounded models that can be leveraged in GD’s formulation, and validate them in three domains in Section 4.
+
+Token-Conditioned Value Functions. Assuming a robot acts with action $a$ according to policy $\pi ( \boldsymbol { a } | \boldsymbol { s } , \boldsymbol { w } _ { 1 : n } )$ , that aims to maximize certain a utility and that the utility captures a task objective, a natural choice that can provide “grounding score” is the action-value function $Q ( s , a | w _ { 1 : n } )$ as it necessarily captures the embodiment of the robot. Additional objectives, such as task constraints, can also be encapsulated in $Q ( s , a | w _ { 1 : n } )$ to ensure grounding. Note that unlike the formulation proposed in [2], $w _ { 1 : n }$ cannot be restricted to a fixed repertoire of token sequences. In practice, to obtain a $Q ( s , a | w _ { 1 : n } )$ that satisfies the requirements, one can train multi-task language-conditioned agents, either through reinforcement learning (Section 4.2) or supervised learning (Section 4.1).
+
+Multimodal Foundation Models. A general choice to ground LLMs is through using multimodal foundation models, such as CLIP [57] or open-vocabulary object detectors [22, 34, 50]. Although these models can connect language to other grounded modalities (e.g., vision), they often lack the capability for complex or long-horizon reasoning, and they do not consider embodiment constraints. As a result, to leverage them in the decoding process, they need to constrained to where they are the most applicable rather than always decoding jointly. To this end, we use a prompt-based technique that allows LLMs to choose when to jointly decode (Section 4.3), which we find to be effective in most cases.4.
+
+Rule-based Methods. Another source of grounding may come from features $x = \phi ( w _ { 1 : n } )$ designed with expert domain knowledge, which can then be used to map $w _ { 1 : n }$ to a “grounding score” using pamametric or non-parametric functions $f ( x )$ . Such techniques may be most applicable when interpretability and enforcing hard constraints are required, such as safety-critical settings, or when data are scarce, such as cases involving preferences of individual users (as shown in Section 4.1).
+
+# 3.5 Comparisons to Prompt-based Methods
+
+One alternative approach for grounding is including scene information as part of the prompt (e.g., object detection results [96]), which complements the grounding method proposed in this work.
+
+![](images/9dad5d3bb0a0ce1e91e69106f28a556838841fa5f6e3ab2473b19bc919e17e2b.jpg)  
+Figure 3: Example rollouts and likelihood of representative tokens under Grounded Decoding objective in three distinct domains: simulated tabletop rearrangement (top), Minigrid 2D Maze (middle), and real-world Language score, grounding score, tokenskitchen mobile manipulation (bottom). Each domain uses different prompts, grounded models, and low-level [(-2.2144471089350395, 13.634880781173706, 'pepsi'), primitives. The GD formulation is shared across the domains, decoding a pre-trained langauge model with (-0.1931666703881, 0.0, 'grapefruit soda'), respect to domain-specific grounded models to decompose a open-ended instruction into actionable steps.
+
+However, we note that prompting is often insufficent for grounding, as information about the scene and about the capabilities of the robot may not always be succinctly described in the prompt. Such examples include 1) in a block stacking task, a block that has been stacked on cannot be picked, 2) in a navigation task, to open a door, one must have a key and that door must be reachable, and 3) in a mobile manipulation domain, an object may be visible but is out of reach of the manipulator. Therefore, Grounded Decoding is a more general and flexible grounding method that injects continuous probabilities during decoding, which may even come from grounding functions from other modalities (e.g., vision).
+
+# 4 Experiments
+
+# 4.1 Long-Horizon Tabletop Manipulation
+
+Herein we experiment with a simulated tabletop manipulation environment based on RAVENS [95]. We create a custom set of 20 tasks, with 10 seen tasks and 10 unseen tasks. Seen tasks are used for training (for supervised baseline) or for few-shot prompting. They are grouped by following categories. Detailed breakdown can be found in Appendix A.2.
+
+i. Letters: Rearranging alphabetical letters (“sort the letters in alphabetical order”).
+
+ii. Blocks & Bowls: Rearranging or combining blocks and bowls (“put blocks in matching bowls”).
+
+iii. Box Packing: Sorting food items and utensils into boxes in accordance with safety constraints and user preferences (“Can you pack the picnic box for me?”).
+
+Given only high-level language instructions and top-down visual observation of the environment, Grounded Decoding decodes a sequence of text tokens representing the step command to be executed. Note that because GD generated grounded free-form actions, it does not require each step to strictly map to a repertoire of skill as in [29, 2]. After a complete command is generated, it is executed via a pre-trained multi-task language-conditioned CLIPort [67]. An example rollout is shown in Figure 4.To demonstrate the techniques proposed in Section 3.4 to obtain grounding functions, we study the composition of following grounding functions (overall grounding score is calculated as $\textstyle p _ { \mathbf { G } } = \mathbf { \dot { I } } \prod _ { i = 1 } ^ { n } p _ { i } )$ depending on the task categories. Refer to the Appendix A.2 for details.
+
+Table 1: Tabletop domain success rates.   
+
+<table><tr><td></td><td colspan="2">CLIPort</td><td>+LLM</td><td colspan="2">+GD (Ours)</td></tr><tr><td></td><td>Short Long</td><td></td><td>Ungrounded</td><td>Greedy Beam</td><td></td></tr><tr><td>Seen Tasks</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Letters</td><td>7%</td><td>40%</td><td>20%</td><td>43%</td><td>57%</td></tr><tr><td>Blocks&amp;Bowls</td><td>2%</td><td>62%</td><td>35%</td><td>60%</td><td>77%</td></tr><tr><td>Box Packing*</td><td>15%</td><td>28%</td><td>11%</td><td>79%</td><td>78%</td></tr><tr><td>Unseen Tasks</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Letters</td><td>6%</td><td>10%</td><td>19%</td><td>37%</td><td>41%</td></tr><tr><td>Blocks&amp;Bowls</td><td>6%</td><td>10%</td><td>28%</td><td>44%</td><td>50%</td></tr></table>
+
+Table 2: 2D Maze success rates.   
+
+<table><tr><td></td><td></td><td>+Skills</td><td>+LLM</td><td colspan="2">+GD (Ours)</td></tr><tr><td></td><td>PPO</td><td>HRL</td><td>Ungrounded</td><td>Greedy</td><td>Beam</td></tr><tr><td>Easy</td><td>28%</td><td>68%</td><td>96%</td><td>100%</td><td>100%</td></tr><tr><td>Medium</td><td>13%</td><td>48%</td><td>87%</td><td>93%</td><td>97%</td></tr><tr><td>Hard</td><td>6%</td><td>31%</td><td>54%</td><td>78%</td><td>88%</td></tr></table>
+
+Affordance Grounding Function (AF). As the primitive policy CLIPort [67] already acts as an action-value function over the pixel space, we directly leverage its predictions for affordance grounding. In particular, given scene image $s$ and partially-decoded instruction $w _ { 1 : n }$ , CLIPort predicts unnormalized logits over the pixel space $\mathbf { u } _ { \mathrm { p i c k } } , \mathbf { u } _ { \mathrm { p l a c e } } \mathbf { \dot { \in } } \mathbb { R } ^ { 4 8 0 \times 6 4 0 }$ , respectively for the pick location and the place location. Therefore, for any given $s$ and $w _ { 1 : n }$ , we can calculate the affordance as $p _ { \mathrm { A F } } ( w _ { 1 : n } | s ) = \operatorname* { m a x } _ { ( x , y ) \in 4 8 0 \times 6 4 0 }$ $( \mathbf { u } _ { \mathrm { p i c k } } ( x , y ) + \mathbf { u } _ { \mathrm { p l a c e } } ( x , y ) )$ .
+
+Safety Grounding Function (S). To strictly enforce hard constraints such as safety requirements, we adopt the rule-based method proposed in Section 3.4. In particular, the features $x$ are indicator functions denoting whether knives and red boxes which we use as hazardous objects are involved in an action, i.e., $x = \mathbb { I } [$ [“red” or “knife” in $w _ { 1 : n } ]$ . We then use constant mappings to convert the features $\begin{array} { r } { p _ { \mathrm { S } } ( w _ { 1 : n } | s ) = \underline { { \epsilon } } ( x + \frac { 1 } { Z } ( 1 - x ) } \end{array}$ to scores of 0 or 1, where $Z$ is the normalizing term and $\epsilon \approx 0$ is a small value for ensuring the joint probability does not collapse to 0.
+
+Preference Grounding Function $\mathbf { \Pi } ^ { ( \mathbf { P } ) }$ . We similarly use rule-based methods for preference grounding, as data of individual users may be scarce to learn a separate model. In particular, we choose two random objects $\left( o _ { 1 } , o _ { 2 } \right)$ as the preferred objects, i.e., $x = \mathbb { I } [ o _ { 1 } \quad$ or $O _ { 2 }$ in $w _ { 1 : n } ]$ . Note that unlike safety functions, preferences often come in the form of “soft requirement”. Therefore, the preference grounding function is implemented as $\begin{array} { r } { p _ { \mathrm { P } } ( w _ { 1 : n } | s ) = \frac { \alpha } { Z } x + \frac { \bar { \beta ^ { } } } { Z } ( 1 - x ) } \end{array}$ , where we choose $\alpha = 0 . 5$ and $\beta = 0 . 1$ for our experiments.
+
+Baselines. We study two variants of GD using beam search and greedy search. We also compare to ”No Grounding” baseline that decodes only according to language model likelihood. Furthermore, we compare to solitary method CLIPort [67] that directly take in the high-level language instructions without a planner. We consider two variants of CLIPort: 1) ”Short” that is trained with only singlestep pick-and-place commands, and 2) ”Long” that is trained on high-level instructions from the 10 training tasks. For more details, please refer to Appendix A.2.
+
+Analysis. Results grouped by each task category are shown in Table $\cdot$ . Please refer to the Appendix for detailed breakdown. Each method is evaluated on 20 episodes for each task within each task category. Supervised methods, such as CLIPort, are found to perform poorly on unseen tasks. Methods that leverage language model planner show better generalization to unseen tasks but fall short due to lack of grounding. Grounded Decoding achieves the best results by enabling the LLM to plan actions using grounded information and is further improved with beam search.
+
+# 4.2 2D Maze
+
+We further evaluate the long-horizon reasoning of Grounded Decoding for 2D maze-solving on Minigrid [10]. The agent receives a top-down view of the environment along with a natural language instruction. More details can be found in Appendix A.3. The tasks are grouped in three categories:
+
+i. Easy: Simple tasks where the horizon is short (10-30 steps) and fully described by the textual instruction, e.g. OpenDoors and PutNext.
+
+ii. Medium: Short and long-horizon tasks (up to 80 steps) with step-by-step textual instructions, e.g. LockedRoom.
+
+iii. Hard: Complex, long-horizon instructions (over 100 steps) with ambiguous instructions that necessitate multi-step reasoning and efficient exploration, e.g. MultiRoom and BlockedUnlock.
+
+![](images/f32112cc7fd36e36fbcb5def3cfcf4cbf68a179f7b4e667eeba04a488fd7e47a.jpg)  
+Figure 4: Greedy decoding rollout with GD, where key decoded tokens are shown (yellow, purple, red, yellow). Combined scores are normalized to the maximum for visual clarity; others are normalized to their sum.
+
+Affordance Grounding Function. Following the recipe from Section 3.3, we train tokenconditioned affordance function to be used in GD. The difference is that the grounding function here is the value function from the goal-conditioned policy that is trained with PPO [62] instead of from demonstrations as in CLIPort [67]. The policy performs short-horizon skills such as “Go to red key” or “Open the door” and are conditioned on CLIP embeddings of the skill and an image of the scene. Accordingly, the goal-conditioned value function evaluates the feasibility given the current observation and the (partially) decoded skill.
+
+Baselines. We compare the two variants of GD – with greedy and beam search – with 1) a solitary PPO policy [62], 2) a hierarchical RL algorithm which plans over the low-level skills, and 3) a hierarchical method that uses an ungrounded language model for planning [29].
+
+Analysis. Table 2 reports the success rate, averaged across 100 episodes of randomly initialized environments. The “flat” RL agent performs poorly in all but the simplest environments, owing to difficulties with understanding the high-level instruction and reasoning over long horizons (often over 100 steps). Planning over low-level skills using hierarchical RL [5] improves this performance, since the high-level decision-making problem is greatly simplified. However, the high-level RL agent still needs to reason over low-level (textual) skills by understanding their underlying capabilities and stitching them together. Using the planning capabilities of large language models to reason over textual skills significantly boosts this performance [29], since the language model can inherit the strong reasoning capabilities from its training data. This tends to be insufficient in challenging environments, however, since the number of potentially viable skills may be very large and the LLM has no information about the robot’s observations. GD can leverage the learned affordance function (in this case, the goal-conditioned value function) to inform the language model’s plans, enabling successful long-horizon reasoning. We further find that beam search improves performance modestly, particularly in long-horizon tasks.
+
+# 4.3 Mobile Manipulation in a Physical Kitchen
+
+Our last environment is a kitchen robot in the real world, and we follow the same implementations of the mobile manipulation platform and skills in SayCan [2]. We perform instruction following tasks, as in [2]. An example task is “Bring an apple”, for which the robot needs to plan and execute a sequence of “1. Find an apple, 2. Pick up the apple, 3. Bring it to you. 4. Put it down, 5. Done”. We split the tasks into two categories. Unambiguous means the instruction explicitly contains the object of interest, and Ambiguous means the instruction does not contain the object name. For example, when human asks “bring me the fruit”, the robot needs to first determine available fruits. We assume all necessary objects are in the field of view. More details can be found in Appendix A.4.
+
+Grounded Decoding with Chain-of-thought. We demonstrate using multimodal foundation models for Grounded Decoding, as proposed in Section 3.4. In particular, we use open-vocabulary object detector owl-vit [50]. Note that because these off-the-shelf models are not trained on robot domain data, we find that it works best by constraining their influence on decoding. We achieve this by making a slight modification to the SayCan algorithm [2]: before generating action plans, we prompt the LLM to generate visually-grounded chain-of-thought [84] by giving LLM the option of when to enable grounded decoding and disable grounded decoding, as visualized in Fig. 5. Specifically,
+
+![](images/75677539f5e245c38debf45eabab4c8a4a9622b1a965a0333d960316bec7df7f.jpg)  
+Figure 5: Example prompt and rollout in realworld kitchen environment.
+
+Table 3: Success rates in kitchen environment.   
+
+<table><tr><td></td><td colspan="2">GD</td><td colspan="2">SayCan</td></tr><tr><td>Tasks</td><td>Planning</td><td>gExecution Planning</td><td></td><td>：Execution</td></tr><tr><td>Unambiguous</td><td>85%</td><td>57%</td><td>85%</td><td>57%</td></tr><tr><td>Ambiguous</td><td>58%</td><td>44%</td><td>33%</td><td>25%</td></tr></table>
+
+Table 4: By avoiding full enumeration of skills, GD is more efficient than SayCan while staying performant.   
+
+<table><tr><td colspan="4">GD (Greedy） ( GD(Beam） SayCan</td></tr><tr><td>Success Rate</td><td>50%</td><td>60%</td><td>64%</td></tr><tr><td>Token Count</td><td>1x</td><td>4.3x</td><td>113.7x</td></tr></table>
+
+LLMs can be prompted to generate a left bracket to start decoding jointly with grounded models and a right bracket to revert to ungrounded decoding. After chain-of-thought, we use SayCan scoring mode for decoding the action plans.
+
+Analysis. Table 3 shows that GD recovers similar performance on Unambiguous tasks, and gain $2 5 \%$ in planning performance on Ambiguous tasks. This shows that GD with multimodal foundation models can effectively use visually-grounded chain-of-thought to disambiguate abstract tasks.
+
+# 5 Analysis
+
+# 5.1 Comparison to SayCan
+
+In this section, we directly compare GD to SayCan [2], which is related to our method in that both combine language model knowledge and grounded model knowledge (discussed in more detail in Section 2). However, SayCan uses the language model to score all pre-specified options, rendering it inefficient at dealing with large or combinatorial action spaces. In contrast, GD computation considers all possible language token in the autoregressive decoding process, which is independent of the size of the action space. Results shown in Table 4 demonstrate that GD is two orders of magnitude more efficient on our tasks, with comparable performance. Furthermore, by decoding at the most basic functioning unit of language, GD’s formulation allows open-vocabulary grounding beyond just affordances, e.g. safety, preferences, and multimodal embeddings such as CLIP.
+
+# 5.2 Breakdown of Failure Reasons
+
+Because all hierarchical approaches share an imperfect low-level policy for step execution, the results reported in Table 1 are compounded with both planning failures and low-level policy failure. In Figure 6, we provide failure breakdown analysis for Grounded Decoding and associated baselines. Note that the CLIPort baselines are solitary methods that do not use a planner, so the failures are solely composed of policy failures. As shown in Figure 6, while all planning-based methods use the same underlying low-level policy, Grounded Decoding significantly reduces planning failure by being able to incorporate grounded scene information into the decoding process. Moreover, we observe that despite the shared affordance function across beam and greedy search, the beam search variant performs stronger by being aware of the full-length single-step instructions during decoding.
+
+# 5.3 Grounded Action Manifold
+
+A central goal of this work is to investigate the integration of grounded information into language model decoding to output instructions actionable by a policy. To investigate this, we use a t-SNE [80] plot to illustrate the extent to which grounded models help narrow down the search space for language models. Specifically, we first enumerate all meaningful instructions in the tabletop domain, such as “pick x and place it on y,” which are represented as dots in the figure. We then compute the affordance values with respect to four different scenes, where each color represents one scene. Finally, we group the dots using t-SNE and BERT embeddings [13]. Figure 7 shows that grounded models can effectively identify achievable skills to produce an actionable manifold within the language space and that this grounding is required, as language alone does not perfectly group actionable skills. It is worth noting that while we provide manual enumeration of all possible skills for practical analysis, the full language space is much larger. This highlights the even more pronounced narrowing of the search in the language space.
+
+![](images/e78baaf4be0d137b1420c9ad6910d91ec93fb1056194220e0f7981d23f4b2c3c.jpg)  
+Figure 6: Failure breakdown on tabletop domain. GD achieves lowest planning failure among planning-based methods, among which beam search variant performs the best.
+
+![](images/069caa95eae39662255c8967f618b2d27da9aaa176566a9d4fac4599115206ad.jpg)  
+Figure 7: Visualization of actions colored by affordance values in different scenes. Every dot represents a possible action in the tabletop domain, where the majority of the actions are infeasible. We show how grounded models can identify the feasible actions for specific scenes. Notably, these actions are not always clustered in language space, requiring the grounding function to determine what action to perform.
+
+# 6 Conclusions, Limitations, & Future Works
+
+We presented Grounded Decoding (GD), an approach for leveraging the knowledge and capabilities of large language models in embodied settings through grounding functions, which model the probabilities of tokens given an embodiment. GD resembles probabilistic filtering, by decoding tokens that have high probabilities under the language model and under grounded model(s). By guiding the LLM’s decoding directly at its output, GD is a general, flexible, and expressive approach to embodied tasks. This is demonstrated on three embodied domains, showing GD is capable of solving complex, long-horizon tasks.
+
+Though quite general and flexible, GD has a few limitations. First, although we present several techniques for obtaining grounding functions in different domains, it remains a question whether a capable and general grounding function can be obtained. We hope that recent progress in largescale robotics models (e.g. [7] and [59]) can remove this bottleneck, and note that the flexibility of GD allows such progress to be straightforwardly leveraged. Second, prompt engineering is often required to steer LLMs to the desired action space (e.g., likely action verbs, likely present objects). Finally, while not requiring additional training, the joint decoding may be limiting compared to a single model capable of both grounding and language reasoning [3, 9, 16].
+
+This work presented a family of algorithms for grounding LLMs in embodiment, for which there are many avenues for future work. The flexibility of the approach enables many other grounding functions and ways to integrate grounding. Furthermore, the development and integration of a foundation model for grounding would improve performance significantly. Finally, though GD’s probabilistic filtering-based approach is quite general, fusing grounding information to the language model after each token decoding may be limiting and future works can investigate how such grounding can be elegantly integrated during decoding.
+
+# Acknowledgments
+
+The authors would like to acknowledge Pierre Sermanet, Carolina Parada, Jie Tan, Yevgen Chebotar, Vincent Vanhoucke, and Dorsa Sadigh for their feedback and contributions. This work is supported in part by OpenAI academic access program, granted to Wenlong Huang.
+
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+
+# A Appendix
+
+# A.1 Grounded Decoding Implementation Details
+
+We study three different implementations of Grounded Decoding for each of the experimental domains. While each instantiation applied Grounded Decoding to long-horizon planning and behavior synthesis, different components including language models and grounded models are used in each domain, as seen in Table 5. Grounded models used in these domains include Affordance Functions (AF), Safety Functions (S), Preference Functions (P), and Open-Vocabulary Object Detectors (D).
+
+<table><tr><td></td><td>Tabletop Rearrangement (Sim)</td><td>MiniGrid 2D Maze (Sim)</td><td>Kitchen Mobile Manipulation (Real)</td></tr><tr><td>LLM</td><td>InstructGPT [54]</td><td>InstructGPT</td><td>InstructGPT + PaLM[11]</td></tr><tr><td>Primitives</td><td>CLIPort [67]</td><td>PPO [62]</td><td>RT-1 [7]</td></tr><tr><td>Grounded Models</td><td>AF+S + P</td><td>AF</td><td>D</td></tr></table>
+
+Table 5: Comparison between different versions of GD implemented in three different environments.
+
+# A.2 Implementation Details of Simulated Tabletop Rearrangement
+
+# A.2.1 Tasks
+
+There are a total of 20 tasks (templates of language instructions), listed in Table 6, grouped into three task category: Letters, Blocks&Bowls, and Box Packing. Three categories share a total of 57 objects. For Letters category, the goals are to rearrange the alphabetical letter objects such that they satisfy certain orders specified by the language instructions. At the beginning of each episode, task-relevant objects and a set of 1 to 3 randomly-sampled distractor objects (except for the Letters category) are initialized at random positions on the tabletop with fixed orientations. A minimum $1 5 \mathrm { c m }$ distance is enforced between any two objects to avoid collision and penetration at initialization. To allow for automatic evaluations, a binary reward function is defined for each task using ground-truth state of the objects. Furthermore, we implement scripted policies for each task to collect demonstrations for training the CLIPort baseline. For certain tasks, we also randomize the attributes mentioned in the given instructions, which can be found below:
+
+i. hwordi: hi, world, left, right, top, down, love, you ii. hcorner/sidei: left side, top left corner, top side, top right corner, bottom right corner, bottom side, bottom left corner
+
+# A.2.2 Low-level Primitives
+
+We use CLIPort [67] as the low-level primitive that can be invoked by the LLM planner, as it shows promising results of generalization across free-form language instructions. Additionally, since the policy predicts per-pixel affordance, it can be repurposed to serve as grounded models for planning for long-horizon tasks, which we leverage in this work. The single primitive policy is trained on 50,000 pre-collected demonstrations, across 10 training tasks, where each demonstration contains 1) language instruction of the format “pick up [x] and place it on [y]”, 2) top-down RGB-D observation of the current scene, 3) expert pick location expressed as pixel coordinates, and 4) expert place location expressed as pixel coordinates. The expert actions are obtained by accessing ground-truth object pose in the simulator. We further apply substring augmentation during training as we find it helps with performance on partial commands (see Section
+
+# A.2.3 Language Model
+
+We use InstructGPT [54] (text-davinci-002), accessed through OpenAI API.
+
+# A.2.4 CLIPort Baseline
+
+As CLIPort [67] already takes as input a natural language instruction and is capable of directly outputting robot actions, it bears the question whether we need a high-level planner for completing long-horizon tasks. To this end, we additionally train two variants of multi-task CLIPort policy on 10 of the total 20 tasks as baselines (see Table 6 for the train/test split). One variant, which we referred as “CLIPort (Short)”, is trained only on single-step pick-and-place instructions of the format “pick up [x] and place it on [y]” on the 10 training tasks. The decomposed pick-and-place instructions are obtained from scripted planners. At evaluation time, the policy is fed in only the high-level instructions without any planners. The other variant, which we referred as “CLIPort (Long)”, is trained on the high-level instructions from the 10 training tasks (without decomposition from scripted planners). Similarly, at evaluation time, it is fed in only the high-level instructions and evaluated on both seen and unseen instructions. Both variants are trained on 50,000 demonstrations, similar to the Grounded Decoding primitive. The goal of these baselines is to evaluate whether solitary language-conditioned policies can perform well on long-horizon tasks and generalize to new task instructions. Note that the CLIPort baselines are different from the primitive used in Grounded Decoding, although they share the same architecture.
+
+# A.2.5 Full Experimental Results in Simulated Tabletop Domain
+
+Below we show the full list of tasks and the full experimental results in the simulated tabletop domain. Each entry is the average success rate across 20 rollouts. The tasks with blue-colored background are seen tasks and the tasks with orange-colored background are the unseen tasks. Seen tasks may be used for training for supervised baselines (CLIPort) or may be used in the prompt for methods using language model planner. Note that for the “Box Packing” task category, although all tasks were seen in training or the prompts, we enforce additional safety and preference constraints for evaluation only at test time.
+
+Table 6: Full Experimental Results in Simulated Tabletop Rearrangement Tasks. The tasks with blue-colored background are seen tasks and the tasks with orange-colored background are the unseen tasks. $\ast _ { \mathrm { B o X } }$ Packing tasks are all seen during training, but safety and preference requirements are only enforced during evaluation.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2"></td><td colspan="2">CLIPort</td><td rowspan="2">+LLM</td><td colspan="2">+Grounded Decoding</td></tr><tr><td></td><td>Short Long</td><td>：Ungrounded Greedy</td><td>Beam</td></tr><tr><td>Letters</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Put the letters in alphabetical order from left to right</td><td>AF</td><td>5%</td><td>20%</td><td>10%</td><td>20%</td><td>40%</td></tr><tr><td>Spell as much of word as you can</td><td>AF</td><td>10%</td><td>60%</td><td>30%</td><td>60%</td><td>65%</td></tr><tr><td>Separate the vowels from the remaining letters to the bottom side</td><td>AF</td><td>5%</td><td>40%</td><td>20%</td><td>50%</td><td>65%</td></tr><tr><td>Put the leters in reverse alphabetical order from left to right</td><td>AF</td><td>15%</td><td>10%</td><td>15%</td><td>25%</td><td>25%</td></tr><tr><td>Correctly spell out a sport using the present letters</td><td>AF</td><td>10%</td><td>10%</td><td>5%</td><td>30%</td><td>30%</td></tr><tr><td>Sort the geometrically symmetrical leters to the bottom side</td><td>AF</td><td>5%</td><td>10%</td><td>15%</td><td>35%</td><td>50%</td></tr><tr><td>Separate the consonants from the remaining letters to the bottom side</td><td>AF</td><td>0%</td><td>0%</td><td>25%</td><td>25%</td><td>25%</td></tr><tr><td>Sortthe letters less than &quot;D&quot;according to ASCII to the bottom side</td><td>AF</td><td>0%</td><td>20%</td><td>35%</td><td>70%</td><td>75%</td></tr><tr><td>Blocks &amp; Bowls</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td> Stack all the blocks</td><td>AF</td><td>5%</td><td>90%</td><td>30%</td><td>75%</td><td>90%</td></tr><tr><td>Put all the blocks on the corner/side</td><td>AF</td><td>0%</td><td>65%</td><td>50%</td><td>45%</td><td>70%</td></tr><tr><td>Put all the blocks in the bowls with matching colors</td><td>AF</td><td>0%</td><td>30%</td><td>25%</td><td>60%</td><td>70%</td></tr><tr><td>Put the blocks in the bowls with mismatched colors</td><td>AF</td><td>25%</td><td>30%</td><td>45%</td><td>30%</td><td> 55%</td></tr><tr><td>Put all the blocks in different corners</td><td>AF</td><td>0%</td><td>5%</td><td>40%</td><td>50%</td><td>60%</td></tr><tr><td>Stack only the blocks of cool colors</td><td>AF</td><td>5%</td><td>5%</td><td>20%</td><td>70%</td><td>70%</td></tr><tr><td>Stack only the blocks of warm colors</td><td>AF</td><td>0%</td><td>10%</td><td>15%</td><td>45%</td><td>35%</td></tr><tr><td>Sort the primary color blocks to the left side</td><td>AF</td><td>0%</td><td>0%</td><td>20%</td><td>25%</td><td>30%</td></tr><tr><td> Box Packing*</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Pack the objects into the brown box</td><td>AF + S</td><td></td><td>20%40%</td><td>5%</td><td>100%</td><td>90%</td></tr><tr><td>Pack the objects into the boxes</td><td>AF + S</td><td></td><td>10%20%</td><td>5%</td><td>75%</td><td>70%</td></tr><tr><td>I&#x27;d like some snacks on the right side</td><td>AF + P</td><td></td><td>15%20%</td><td>15%</td><td>40%</td><td>55%</td></tr><tr><td>Pack me a picnic box</td><td>AF + S +P</td><td></td><td>15%30%</td><td>20%</td><td>100%</td><td>95%</td></tr></table>
+
+# A.3 Implementation Details of Minigrid 2D Maze
+
+# A.3.1 Environment Setup
+
+We use the open-source gym-minigrid suite of environments to evaluate our method with one simple change — instead of the default observation space which is a $7 \times 7$ egocentric window, our agent has access to entire grid — that allows us to simplify the tasks by removing partial observability [65].
+
+# A.3.2 Tasks
+
+The tasks are grouped in three categories (please see Table 7 for example instructions):
+
+1. Easy: Simple tasks where the horizon is short (10-30 steps) and fully described by the textual instruction, e.g. OpenDoors and PutNext. The short horizon makes them relatively easy for a wide range of HRL algorithms. The instructions for these tasks generally spell out each individual skill, making them particularly easy for high-level planners based on language modeling.   
+2. Medium: Combination of short and long horizon tasks (up to 80 steps) with step-by-step textual instructions, e.g. LockedRoom. While being significantly longer, these tasks also tend to have instructions that spell out the low-level tasks (see Table 7).   
+3. Hard: Complex, long horizon instructions (over 100 steps) with short, ambiguous instructions that necessitate multi-step reasoning and efficient exploration, e.g. MultiRoom and BlockedUnlock. In addition to being long-horizon, the instructions in this case tend to be ambiguous and under-specified, e.g. ”traverse through the rooms to get to the goal”, which does not provide enough context for any blind planning agent.
+
+Table 7: Example Instructions in Minigrid   
+
+<table><tr><td>Difficulty</td><td>Task Name</td><td>Example Instruction</td></tr><tr><td>Easy</td><td>OpenDoors PutNext</td><td>open door blue, then open door red move the red ball next to the green box</td></tr><tr><td>Medium</td><td>LockedRoom</td><td> get the red key from the purple room, open the red door and go to the goal</td></tr><tr><td>Hard</td><td>MultiRoom BlockedUnlock</td><td>traverse the rooms to get to the goal pick up the blue box</td></tr></table>
+
+# A.3.3 Language Model
+
+We use InstructGPT [54] (text-davinci-002), accessed through OpenAI API. The prompts used can be found in Section A.5.
+
+We found the prompts to be generally sufficient for solving the “seen” tasks, as well as “unseen” tasks, i.e. tasks that do not have an example in the context. Empirically, we did not find any improvements by including more then 3 example tasks in the prompt — we hypothesize that this is likely due to the shared low-level primitives across tasks. For all Minigrid experiments presented in this paper, we used the prompt shown in Section A.5.
+
+# A.3.4 Low-level Primitives
+
+To train low-level primitives, we train an RL agent to solve a wide range of short-horizon subtasks (under 10 steps) that are shared across the various Minigrid tasks — go to $< \circ { \mathrm { b j } } >$ , pick up <obj>, drop <obj>, open <obj>. Rather than training individual skills for each of them [65], we train a single multi-task policy that is conditioned on the CLIP embeddings [57] of the task strings. This scheme allows some robustness to synonyms and ambiguous task specifications, and has been widely used in learning language-grounded policies [32, 68].
+
+We train these primitives using PPO [62], as recommended by the environment developers [10]. Each of these skills are trained with a sparse outcome reward $^ { + 1 }$ if a trajectory is successful, 0 otherwise). In addition to these low-level skills, we perform a form of hindsight relabeling where “substrings“ of the task strings are masked to allow generalization to partial strings, e.g. “go to red” may be interpreted as “go to red key” or “go to red door”, and our masking strategy allows the multi-task policy to execute tasks specified by partially complete strings, if necessary.
+
+![](images/301680600e4fbc1c2e8b316a0ff516e53980d95ccffc8e560e9dbb87866d5e87.jpg)  
+A.3.5 Additional Qualitative Results   
+Figure 8: Minigrid Domain
+
+# A.4 Implementation Details of Real-World Mobile Manipulation
+
+A.4.1 Tasks   
+Table 8: List of unambiguous SayCan instructions   
+
+<table><tr><td>Instruction</td></tr><tr><td>put an energy bar and water bottle on the table bring me a lime soda and a bag of chips Can you throw away the apple and bring me a coke bring me a 7up can and a tea move an multigrain chips to the table and an apple to the far counter move the lime soda, the sponge,and the water bottle to the table</td></tr></table>
+
+Table 9: List of ambiguous SayCan instructions   
+
+<table><tr><td>Instruction</td></tr><tr><td>I want to wipe some spill.</td></tr><tr><td>Bring me a fruit</td></tr><tr><td>Bring me a snack</td></tr><tr><td>Bring me a bag of chips</td></tr><tr><td>Bring me a bag of snack</td></tr><tr><td>Bring me a bag of chips and something to wipe a spill Bring me a bag of chips and something to drink</td></tr><tr><td>Bring me a bag of chips and a soda</td></tr><tr><td>Human: I want a soda that is not coke,and a fruit.</td></tr><tr><td>I want a fruit and a soda</td></tr></table>
+
+# A.4.2 Language Model
+
+For planning, we use PaLM [11], a 540B parameter language model trained on a large datasets that include high-quality web documents, books, Wikipedia, conversations, and GitHub code. Before planning, we use InstructGPT [54] (text-davinci-002), accessed through OpenAI API. to generate the (grounded) chain of thought.
+
+We used square bracket to indicate grounded decoding, as illustrated in Fig. 5. The prompts are shown in Listing 3.
+
+# A.4.3 Low-level Primitives
+
+We use a combination of learned and scripted control policies for navigation and manipulation, following the implementation described in SayCan [2] and RT-1 [7]. The manipulation policies for the picking action are learned using Behavior Cloning (BC) on 68000 demonstrations and 12000 autonomous successes that were collected over the course of 11 months using a fleet of 10 robots. The demonstrations are collected by teleoperators using VR headset controllers to track the motion of their hand, which is then mapped onto the robot’s end-effector pose. The navigation policies are scripted, based on a ground-truth map as well as a learned perception module for collision avoidance and planning. The placing actions follow pre-computed motions only when preceded by a navigation policy. The Value Functions used by SayCan for affordance grounding are provided by the $Q$ -networks of trained RL agents; we follow the RL training setup described in [2].
+
+# A.4.4 Open-Vocabulary Detector Grounding Function
+
+We use owl-vit [50] as our grounding model. It takes in an image and a natural language query, and returns a list of bounding boxes with scores. We take the maximum score a the grounding function.
+
+More examples of object detection as a grounding function can be found in Fig. 9.
+
+![](images/ca7088894b568194cb064e24310032d2b3e0159c5e2770130fc417ace28b0a8e.jpg)  
+Figure 9: Additional examples of using open-vocabulary object detection as a grounding function in Real-World Kitchen Mobile Manipulation Domain.
+
+# A.5 Prompts
+
+# Listing 1: Grounded Decoding Prompt in Simulated Tabletop Rearrangement Domain
+
+Task: Pack all letter objects on the brown box Step 1: pick up the e and place it on the brown box Step 2: pick up the g and place it on the brown box Step 3: done
+
+Task: Put the letters on the tables in alphabetical order Step 1: pick up the c and place it on the bottom left side Step 2: pick up the d and place it on the right of c Step 3: pick up the i and place it on the right of d Step 4: pick up the l and place it on the right of i Step 5: pick up the w and place it on the right of l Step 6: done
+
+Task: Spell as much of "blue" as you can Step 1: pick up the l and place it on the bottom left side Step 2: pick up the the u and place it on the right of l Step 3: pick up the the e and place it on the right of u Step 4: done
+
+Task: Separate the vowels from the remaining letters Step 1: pick up the i and place it on the bottom side Step 2: pick up the o and place it on the bottom side Step 3: done
+
+Task: Stack all the blocks
+
+Step 1: pick up the brown block and place it on the pink block Step 2: pick up the cyan block and place it on the brown block Step 3: pick up the orange block and place it on the cyan block Step 4: pick up the gray block and place it on the orange block Step 5: done
+
+Task: Put all the blocks on the bottom left corner Step 1: pick up the white block and place it on the bottom left corner Step 2: pick up the yellow block and place it on the bottom left corner Step 3: pick up the green block and place it on the bottom left corner Step 4: pick up the blue block and place it on the bottom left corner Step 5: pick up the purple block and place it on the bottom left corner Step 6: done
+
+Task: Put all the blocks in the bowls with matching colors Step 1: pick up the cyan block and place it on the cyan bowl Step 2: pick up the purple block and place it on the purple bowl Step 3: pick up the brown block and place it on the brown bowl Step 4: pick up the pink block and place it on the pink bowl Step 5: done
+
+Task: Pack the items into any box   
+Step 1: pick up the donut stick and place it on the red box   
+Step 2: pick up the pepsi and place it on the brown box   
+Step 3: pick up the peach and place it on the brown box   
+Step 4: pick up the strawberry and place it on the red box   
+Step 5: done
+
+Task: Pack the items on the table into the brown box Step 1: pick up the knife and place it on the brown box Step 2: pick up the plum and place it on the brown box Step 3: pick up the pepsi and place it on the brown box Step 4: pick up the cupcake and place it on the brown box Step 5: done
+
+Task: Pack the items on the table into the brown box Step 1: pick up the i and place it on the brown box Step 2: pick up the green block and place it on the brown box Step 3: pick up the l and place it on the brown box Step 4: done
+
+Task: Can you put some snacks on the right side for me? Step 1: pick up the plum and place it on the right side Step 2: done
+
+Task: Can you pack my picnic box for me?
+
+Step 1: pick up the orange and place it on the picnic box Step 2: pick up the diet pepsi and place it on the picnic box Step 3: pick up the knife and place it on the picnic box Step 4: done
+
+# Listing 2: Grounded Decoding Prompt in MiniGrid 2D Maze Domain
+
+You are a 2D maze-solving agent with access to a variety of low-level skills such as picking up or dropping objects, navigating to doors/keys/boxes, and opening/closing doors. Here are some example tasks:
+
+Task: get the green key from the purple room, unlock the green door and go to the goal Step 1: go to the purple door and open it   
+Step 2: go to the green key   
+Step 3: pick up the key   
+Step 4: go to the green door and open it   
+Step 5: go to the goal.
+
+Task: pick up the purple box Step 1: go to the green obstacle Step 2: pick up the obstacle Step 3: place the obstacle Step 4: go to the blue key Step 5: pick up the blue key Step 6: go to the blue door and open it Step 8: drop the blue key Step 7: go to the purple box Step 8: pick up the purple box.
+
+Task: traverse the rooms to get to the goal Step 1: go to the purple door and open it Step 2: go to the green door and open it Step 3: go to the purple door and open it Step 4: go to the green door and open it Step 5: go to the green door and open it Step 6: go to the goal.
+
+Now your turn.
+
+# Listing 3: Grounded Decoding Prompt in Real-World Kitchen Mobile Manipulation Domain
+
+The following objects are in the scene: 7up, apple, banana, mango, tea, multigrain chips, kettle chips, jalapeno chips, rice chips, coke, grapefruit soda, pepsi, redbull, energy bar, lime soda, sponge, paper towel, and water bottle.
+
+The following locations are in the scene: close counter, far counter, table, trash, bowl.   
+The robot will always put object name in brackets []. Robot: I am a robot that can bring objects to you. Human: I am hungry.   
+Robot thought: I will find the [multigrain chips]. Robot plan: 1. Find the multigrain chips   
+2. Pick up the multigrain chips   
+3. Bring it to you   
+4. Put it down   
+5. Done Robot: I am a robot that can bring objects to you.   
+Human: Throw away the fruit.   
+Robot thought: I will find the [mango] and move it to the trash. Robot plan: 1. Find the mango   
+2. Pick up the mango   
+3. Go to the trash   
+4. Put it down   
+5. Done Robot: I am a robot that can bring objects to you. Human: (inject instruction).   
+Robot thought:

parse/dev/TIdIXIpzhoI/TIdIXIpzhoI.md

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+# PROGRESSIVE DISTILLATION FOR FAST SAMPLING OF DIFFUSION MODELS
+
+Tim Salimans & Jonathan Ho Google Research, Brain team {salimans,jonathanho}@google.com
+
+# ABSTRACT
+
+Diffusion models have recently shown great promise for generative modeling, outperforming GANs on perceptual quality and autoregressive models at density estimation. A remaining downside is their slow sampling time: generating high quality samples takes many hundreds or thousands of model evaluations. Here we make two contributions to help eliminate this downside: First, we present new parameterizations of diffusion models that provide increased stability when using few sampling steps. Second, we present a method to distill a trained deterministic diffusion sampler, using many steps, into a new diffusion model that takes half as many sampling steps. We then keep progressively applying this distillation procedure to our model, halving the number of required sampling steps each time. On standard image generation benchmarks like CIFAR-10, ImageNet, and LSUN, we start out with state-of-the-art samplers taking as many as 8192 steps, and are able to distill down to models taking as few as 4 steps without losing much perceptual quality; achieving, for example, a FID of 3.0 on CIFAR-10 in 4 steps. Finally, we show that the full progressive distillation procedure does not take more time than it takes to train the original model, thus representing an efficient solution for generative modeling using diffusion at both train and test time.
+
+# 1 INTRODUCTION
+
+Diffusion models (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) are an emerging class of generative models that has recently delivered impressive results on many standard generative modeling benchmarks. These models have achieved ImageNet generation results outperforming BigGAN-deep and VQ-VAE-2 in terms of FID score and classification accuracy score (Ho et al., 2021; Dhariwal & Nichol, 2021), and they have achieved likelihoods outperforming autoregressive image models (Kingma et al., 2021; Song et al., 2021b). They have also succeeded in image super-resolution (Saharia et al., 2021; Li et al., 2021) and image inpainting (Song et al., 2021c), and there have been promising results in shape generation (Cai et al., 2020), graph generation (Niu et al., 2020), and text generation (Hoogeboom et al., 2021; Austin et al., 2021).
+
+A major barrier remains to practical adoption of diffusion models: sampling speed. While sampling can be accomplished in relatively few steps in strongly conditioned settings, such as text-tospeech (Chen et al., 2021) and image super-resolution (Saharia et al., 2021), or when guiding the sampler using an auxiliary classifier (Dhariwal & Nichol, 2021), the situation is substantially different in settings in which there is less conditioning information available. Examples of such settings are unconditional and standard class-conditional image generation, which currently require hundreds or thousands of steps using network evaluations that are not amenable to the caching optimizations of other types of generative models (Ramachandran et al., 2017).
+
+In this paper, we reduce the sampling time of diffusion models by orders of magnitude in unconditional and class-conditional image generation, which represent the setting in which diffusion models have been slowest in previous work. We present a procedure to distill the behavior of a $N$ -step DDIM sampler (Song et al., 2021a) for a pretrained diffusion model into a new model with $N / 2$ steps, with little degradation in sample quality. In what we call progressive distillation, we repeat this distillation procedure to produce models that generate in as few as 4 steps, still maintaining sample quality competitive with state-of-the-art models using thousands of steps.
+
+![](images/4b13771c6ccaadae72cb320c80cfabfab9e7defc924ee44d5bd74db32bdbfef6.jpg)  
+Figure 1: A visualization of two iterations of our proposed progressive distillation algorithm. A sampler $f ( \mathbf { z } ; \boldsymbol { \eta } )$ , mapping random noise $\epsilon$ to samples $\mathbf { x }$ in 4 deterministic steps, is distilled into a new sampler $f ( \mathbf { z } ; \theta )$ taking only a single step. The original sampler is derived by approximately integrating the probability flow $O D E$ for a learned diffusion model, and distillation can thus be understood as learning to integrate in fewer steps, or amortizing this integration into the new sampler.
+
+# 2 BACKGROUND ON DIFFUSION MODELS
+
+We consider diffusion models (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) specified in continuous time (Tzen & Raginsky, 2019a; Song et al., 2021c; Chen et al., 2021; Kingma et al., 2021). We use $\mathbf { x } \sim p ( \mathbf { x } )$ to denote training data. A diffusion model has latent variables $\mathbf { z } = \{ \mathbf { z } _ { t } | t \in [ 0 , 1 ] \}$ and is specified by a noise schedule comprising differentiable functions $\alpha _ { t } , \sigma _ { t }$ such that $\lambda _ { t } = \log \bar { [ \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } ] }$ , the log signal-to-noise-ratio, decreases monotonically with $t$ .
+
+These ingredients define the forward process $q ( \mathbf { z } | \mathbf { x } )$ , a Gaussian process satisfying the following Markovian structure:
+
+$$
+\begin{array} { r } { q ( \mathbf { z } _ { t } | \mathbf { x } ) = \mathcal { N } ( \mathbf { z } _ { t } ; \alpha _ { t } \mathbf { x } , \sigma _ { t } ^ { 2 } \mathbf { I } ) , \quad q ( \mathbf { z } _ { t } | \mathbf { z } _ { s } ) = \mathcal { N } ( \mathbf { z } _ { t } ; ( \alpha _ { t } / \alpha _ { s } ) \mathbf { z } _ { s } , \sigma _ { t | s } ^ { 2 } \mathbf { I } ) } \end{array}
+$$
+
+where $0 \leq s < t \leq 1$ and $\sigma _ { t | s } ^ { 2 } = ( 1 - e ^ { \lambda _ { t } - \lambda _ { s } } ) \sigma _ { t } ^ { 2 }$
+
+The role of function approximation in the diffusion model is to denoise ${ \mathbf z } _ { t } \sim q ( { \mathbf z } _ { t } | { \mathbf x } )$ into an estimate $\hat { \mathbf { x } } _ { \theta } ( { \mathbf z } _ { t } ) \approx { \mathbf x }$ (the function approximator also receives $\lambda _ { t }$ as an input, but we omit this to keep our notation clean). We train this denoising model $\hat { \mathbf { x } } _ { \theta }$ using a weighted mean squared error loss
+
+$$
+\mathbb { E } _ { \epsilon , t } \big [ w ( \lambda _ { t } ) \| \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) - \mathbf { x } \| _ { 2 } ^ { 2 } \big ]
+$$
+
+over uniformly sampled times $t \in [ 0 , 1 ]$ . This loss can be justified as a weighted variational lower bound on the data log likelihood under the diffusion model (Kingma et al., 2021) or as a form of denoising score matching (Vincent, 2011; Song & Ermon, 2019). We will discuss particular choices of weighting function $w ( \lambda _ { t } )$ later on.
+
+Sampling from a trained model can be performed in several ways. The most straightforward way is discrete time ancestral sampling (Ho et al., 2020). To define this sampler, first note that the forward process can be described in reverse as $q ( \mathbf { z } _ { s } | \mathbf { z } _ { t } , \mathbf { x } ) = \mathcal { N } ( \mathbf { z } _ { s } ; \tilde { \mu } _ { s | t } ( \mathbf { z } _ { t } , \grave { \mathbf { x } } ) , \tilde { \sigma } _ { s | t } ^ { 2 } \mathbf { I } )$ (noting $s < t$ ), where
+
+$$
+\tilde { \mu } _ { s | t } ( \mathbf { z } _ { t } , \mathbf { x } ) = e ^ { \lambda _ { t } - \lambda _ { s } } ( \alpha _ { s } / \alpha _ { t } ) \mathbf { z } _ { t } + ( 1 - e ^ { \lambda _ { t } - \lambda _ { s } } ) \alpha _ { s } \mathbf { x } , \quad \tilde { \sigma } _ { s | t } ^ { 2 } = ( 1 - e ^ { \lambda _ { t } - \lambda _ { s } } ) \sigma _ { s } ^ { 2 }
+$$
+
+We use this reversed description of the forward process to define the ancestral sampler. Starting at $\mathbf { z } _ { 1 } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , the ancestral sampler follows the rule
+
+$$
+\begin{array} { r l } & { \mathbf z _ { s } = \tilde { \mu } _ { s \mid t } ( \mathbf z _ { t } , \hat { \mathbf x } _ { \theta } ( \mathbf z _ { t } ) ) + \sqrt { ( \tilde { \sigma } _ { s \mid t } ^ { 2 } ) ^ { 1 - \gamma } ( \sigma _ { t \mid s } ^ { 2 } ) ^ { \gamma } ) } \epsilon } \\ & { \quad = e ^ { \lambda _ { t } - \lambda _ { s } } ( \alpha _ { s } / \alpha _ { t } ) \mathbf z _ { t } + ( 1 - e ^ { \lambda _ { t } - \lambda _ { s } } ) \alpha _ { s } \hat { \mathbf x } _ { \theta } ( \mathbf z _ { t } ) + \sqrt { ( \tilde { \sigma } _ { s \mid t } ^ { 2 } ) ^ { 1 - \gamma } ( \sigma _ { t \mid s } ^ { 2 } ) ^ { \gamma } ) } \epsilon , } \end{array}
+$$
+
+where $\epsilon$ is standard Gaussian noise, and $\gamma$ is a hyperparameter that controls how much noise is added during sampling, following Nichol $\&$ Dhariwal (2021).
+
+Alternatively, Song et al. (2021c) show that our denoising model $\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } )$ can be used to deterministically map noise ${ \bf z } _ { 1 } \sim \mathcal { N } ( { \bf 0 } , { \bf I } )$ to samples $\mathbf { x }$ by numerically solving the probability flow $O D E$ :
+
+$$
+d \mathbf { z } _ { t } = [ f ( \mathbf { z } _ { t } , t ) - \frac { 1 } { 2 } g ^ { 2 } ( t ) \nabla _ { z } \log \hat { p } _ { \boldsymbol { \theta } } ( \mathbf { z } _ { t } ) ] d t ,
+$$
+
+where $\begin{array} { r } { \nabla _ { z } \log \hat { p } _ { \boldsymbol { \theta } } ( \mathbf { z } _ { t } ) = \frac { \alpha _ { t } \hat { \mathbf { x } } _ { \boldsymbol { \theta } } ( \mathbf { z } _ { t } ) - \mathbf { z } _ { t } } { \sigma _ { t } ^ { 2 } } } \end{array}$ . Following Kingma et al. (2021), we have $\begin{array} { r } { f ( \mathbf { z } _ { t } , t ) = \frac { d \log \alpha _ { t } } { d t } \mathbf { z } _ { t } } \end{array}$ and g2(t) = dσ2tdt $\begin{array} { r } { g ^ { 2 } ( t ) = \frac { d \sigma _ { t } ^ { 2 } } { d t } - 2 \frac { d \log { \alpha _ { t } } } { d t } \sigma _ { t } ^ { 2 } } \end{array}$ . Since $\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } )$ is parameterized by a neural network, this equation is a special case of a neural $O D E$ (Chen et al., 2018), also called a continuous normalizing flow (Grathwohl et al., 2018).
+
+Solving the ODE in Equation 6 numerically can be done with standard methods like the Euler rule or the Runge-Kutta method. The DDIM sampler proposed by Song et al. (2021a) can also be understood as an integration rule for this ODE, as we show in Appendix B, even though it was originally proposed with a different motivation. The update rule specified by DDIM is
+
+$$
+\begin{array} { r l r } {  { \mathbf { z } _ { s } = \alpha _ { s } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) + \sigma _ { s } \frac { \mathbf { z } _ { t } - \alpha _ { t } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) } { \sigma _ { t } } } } \\ & { } & { = e ^ { ( \lambda _ { t } - \lambda _ { s } ) / 2 } ( \alpha _ { s } / \alpha _ { t } ) \mathbf { z } _ { t } + ( 1 - e ^ { ( \lambda _ { t } - \lambda _ { s } ) / 2 } ) \alpha _ { s } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) , } \end{array}
+$$
+
+and in practice this rule performs better than the aforementioned standard ODE integration rules in our case, as we show in Appendix C.
+
+If $\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } )$ satisfies mild smoothness conditions, the error introduced by numerical integration of the probability flow ODE is guaranteed to vanish as the number of integration steps grows infinitely large, i.e. $N  \infty$ . This leads to a trade-off in practice between the accuracy of the numerical integration, and hence the quality of the produced samples from our model, and the time needed to produce these samples. So far, most models in the literature have needed hundreds or thousands of integration steps to produce their highest quality samples, which is prohibitive for many practical applications of generative modeling. Here, we therefore propose a method to distill these accurate, but slow, ODE integrators into much faster models that are still very accurate. This idea is visualized in Figure 1, and described in detail in the next section.
+
+# 3 PROGRESSIVE DISTILLATION
+
+To make diffusion models more efficient at sampling time, we propose progressive distillation: an algorithm that iteratively halves the number of required sampling steps by distilling a slow teacher diffusion model into a faster student model. Our implementation of progressive distillation stays very close to the implementation for training the original diffusion model, as described by e.g. Ho et al. (2020). Algorithm 1 and Algorithm 2 present diffusion model training and progressive distillation side-by-side, with the relative changes in progressive distillation highlighted in green.
+
+We start the progressive distillation procedure with a teacher diffusion model that is obtained by training in the standard way. At every iteration of progressive distillation, we then initialize the student model with a copy of the teacher, using both the same parameters and same model definition. Like in standard training, we then sample data from the training set and add noise to it, before forming the training loss by applying the student denoising model to this noisy data $\mathbf { z } _ { t }$ . The main difference in progressive distillation is in how we set the target for the denoising model: instead of the original data $\mathbf { x }$ , we have the student model denoise towards a target $\tilde { \mathbf { x } }$ that makes a single student DDIM step match 2 teacher DDIM steps. We calculate this target value by running 2 DDIM sampling steps using the teacher, starting from $\mathbf { z } _ { t }$ and ending at $\mathbf { z } _ { t - 1 / N }$ , with $N$ being the number of student sampling steps. By inverting a single step of DDIM, we then calculate the value the student model would need to predict in order to move from $\mathbf { z } _ { t }$ to $\mathbf { z } _ { t - 1 / N }$ in a single step, as we show in detail in Appendix G. The resulting target value $\tilde { \mathbf { x } } ( \mathbf { z } _ { t } )$ is fully determined given the teacher model and starting point $\mathbf { z } _ { t }$ , which allows the student model to make a sharp prediction when evaluated at $\mathbf { z } _ { t }$ . In contrast, the original data point $\mathbf { x }$ is not fully determined given $\mathbf { z } _ { t }$ , since multiple different data points $\mathbf { x }$ can produce the same noisy data $\mathbf { z } _ { t }$ : this means that the original denoising model is predicting a weighted average of possible x values, which produces a blurry prediction. By making sharper predictions, the student model can make faster progress during sampling.
+
+After running distillation to learn a student model taking $N$ sampling steps, we can repeat the procedure with $\bar { N } / 2$ steps: The student model then becomes the new teacher, and a new student model is initialized by making a copy of this model.
+
+Unlike our procedure for training the original model, we always run progressive distillation in discrete time: we sample this discrete time such that the highest time index corresponds to a signal-tonoise ratio of zero, i.e. $\alpha _ { 1 } = 0$ , which exactly matches the distribution of input noise $\mathbf { z } _ { 1 } \sim \bar { \mathcal { N } } ( \mathbf { 0 } , \mathbf { I } )$ that is used at test time. We found this to work slightly better than starting from a non-zero signalto-noise ratio as used by e.g. Ho et al. (2020), both for training the original model as well as when performing progressive distillation.
+
+# Algorithm 1 Standard diffusion training
+
+# Algorithm 2 Progressive distillation
+
+Require: Model $\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } )$ to be trained Require: Data set $\mathcal { D }$ Require: Loss weight function $w ( )$
+
+$\mathbf { x } \sim \mathcal { D }$ $\triangleright$ Sample data $t \sim U [ 0 , 1 ]$ $\triangleright$ Sample time $\epsilon \sim \bar { N ( 0 , I ) }$ $\triangleright$ Sample noise ${ \bf z } _ { t } = \alpha _ { t } { \bf x } + \sigma _ { t } \epsilon$ . Add noise to data
+
+Require: Trained teacher model $\hat { \mathbf { x } } _ { \eta } ( \mathbf { z } _ { t } )$   
+Require: Data set $\mathcal { D }$   
+Require: Loss weight function w()   
+Require: Student sampling steps $N$ for $K$ iterations do $\theta  \eta$ $\triangleright$ Init student from teacher while not converged do $\begin{array} { l } { { \bf { x } } \sim { \mathcal { D } } } \\ { t = i / N , ~ i \sim C a t [ 1 , 2 , \dots , N ] } \\ { \epsilon \sim N ( 0 , I ) } \\ { { \bf { z } } _ { t } = \alpha _ { t } { \bf { x } } + \sigma _ { t } \epsilon } \end{array}$ # 2 steps of DDIM with teacher $\begin{array} { r l } & { t ^ { \prime } = t - 0 . 5 / N , \quad t ^ { \prime \prime } = t - 1 / N } \\ & { \mathbf { \tilde { z } } _ { t ^ { \prime } } = \alpha _ { t ^ { \prime } } \hat { \mathbf { x } } _ { \eta } ( \mathbf { z } _ { t } ) + \frac { \sigma _ { t ^ { \prime } } } { \sigma _ { t } } ( \mathbf { z } _ { t } - \alpha _ { t } \hat { \mathbf { x } } _ { \eta } ( \mathbf { z } _ { t } ) ) } \\ & { \mathbf { \tilde { z } } _ { t ^ { \prime \prime } } = \alpha _ { t ^ { \prime \prime } } \hat { \mathbf { x } } _ { \eta } ( \mathbf { z } _ { t ^ { \prime } } ) + \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t ^ { \prime \prime } } } ( \mathbf { z } _ { t ^ { \prime } } - \alpha _ { t ^ { \prime } } \hat { \mathbf { x } } _ { \eta } ( \mathbf { z } _ { t ^ { \prime } } ) ) } \\ & { \mathbf { \tilde { x } } = \frac { \mathbf { z } _ { t ^ { \prime \prime } } - ( \sigma _ { t ^ { \prime \prime } } / \sigma _ { t } ) \mathbf { z } _ { t } } { \alpha _ { t ^ { \prime \prime } } - ( \sigma _ { t ^ { \prime \prime } } / \sigma _ { t } ) \alpha _ { t } } \qquad \mathrm { ~ > ~ T e a c h e r ~ } \hat { \mathbf { x } } \mathrm { ~ t a r g e t } } \\ & { \lambda _ { t } = \mathrm { l o g } [ \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } ] } \\ & { L _ { \theta } = w ( \lambda _ { t } ) | | \hat { \mathbf { x } } - \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) | | _ { 2 } ^ { 2 } } \\ & { \theta \gets \theta - \gamma \mathrm { v } _ { \theta } L _ { \theta } } \end{array}$ end while $\eta  \theta$ . Student becomes next teacher $N \gets N / 2 \mathsf { \Omega } \triangleright \mathrm { H }$ alve number of sampling steps end for
+
+$\tilde { \mathbf { x } } = \mathbf { x } \triangleright$ Clean data is target for xˆ $\lambda _ { t } = \log [ \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } ]$ $\triangleright$ log-SNR $L _ { \theta } = w ( \lambda _ { t } ) \| \tilde { \mathbf { x } } - \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) \| _ { 2 } ^ { 2 }$ . Loss $\theta  \theta - \gamma \nabla _ { \theta } L _ { \theta }$ $\triangleright$ Optimization end while
+
+# 4 DIFFUSION MODEL PARAMETERIZATION AND TRAINING LOSS
+
+In this section, we discuss how to parameterize the denoising model $\hat { \mathbf { x } } _ { \theta }$ , and how to specify the reconstruction loss weight $w ( \lambda _ { t } )$ . We assume a standard variance-preserving diffusion process for which $\sigma _ { t } ^ { 2 } = 1 - \alpha _ { t } ^ { 2 }$ . This is without loss of generalization, as shown by (Kingma et al., 2021, appendix G): different specifications of the diffusion process, such as the variance-exploding specification, can be considered equivalent to this specification, up to rescaling of the noisy latents $\mathbf { z } _ { t }$ . We use a cosine schedule $\alpha _ { t } = \cos ( 0 . 5 \pi t )$ , similar to that introduced by Nichol $\&$ Dhariwal (2021).
+
+Ho et al. (2020) and much of the following work choose to parameterize the denoising model through directly predicting $\epsilon$ with a neural network $\hat { \epsilon } _ { \boldsymbol { \theta } } ( \mathbf { z } _ { t } )$ , which implicitly sets $\begin{array} { r } { \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) = \frac { 1 } { \alpha _ { t } } \bar { ( } \mathbf { z } _ { t } - \sigma _ { t } \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ) \bar { ) } } \end{array}$ . In this case, the training loss is also usually defined as mean squared error in the $\epsilon$ -space:
+
+$$
+L _ { \theta } = \| \epsilon - \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ) \| _ { 2 } ^ { 2 } = \left\| \frac { 1 } { \sigma _ { t } } ( \mathbf { z } _ { t } - \alpha _ { t } \mathbf { x } ) - \frac { 1 } { \sigma _ { t } } ( \mathbf { z } _ { t } - \alpha _ { t } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) ) \right\| _ { 2 } ^ { 2 } = \frac { \alpha _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } \| \mathbf { x } - \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) \| _ { 2 } ^ { 2 } ,
+$$
+
+which can thus equivalently be seen as a weighted reconstruction loss in $\mathbf { x }$ -space, where the weighting function is given by $w ( \lambda _ { t } ) = \exp ( \lambda _ { t } )$ , for log signal-to-noise ratio $\lambda _ { t } = \log [ \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } ]$ .
+
+Although this standard specification works well for training the original model, it is not well suited for distillation: when training the original diffusion model, and at the start of progressive distillation, the model is evaluated at a wide range of signal-to-noise ratios $\alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 }$ , but as distillation progresses we increasingly evaluate at lower and lower signal-to-noise ratios. As the signal-to-noise ratio goes to zero, the effect of small changes in the neural network output $\hat { \epsilon } _ { \theta } ( { \bf z } _ { t } )$ on the implied prediction in $\mathbf { x }$ -space is increasingly amplified, since $\begin{array} { r } { \hat { \mathbf { x } } _ { \theta } \big ( \mathbf { z } _ { t } \big ) = \frac { 1 } { \alpha _ { t } } \big ( \mathbf { z } _ { t } - \sigma _ { t } \dot { \hat { \epsilon } } _ { \theta } \big ( \mathbf { z } _ { t } \big ) \big ) } \end{array}$ divides by $\alpha _ { t }  0$ . This is not much of a problem when taking many steps, since the effect of early missteps is limited by clipping of the $\mathbf { z } _ { t }$ iterates, and later updates can correct any mistakes, but it becomes increasingly important as we decrease the number of sampling steps. Eventually, if we distill all the way down to a single sampling step, the input to the model is only pure noise $\epsilon$ , which corresponds to a signal-to-noise ratio of zero, i.e. $\alpha _ { t } = 0 , \sigma _ { t } = 1$ . At this extreme, the link between $\epsilon$ -prediction and $\mathbf { x }$ -prediction breaks down completely: observed data $\mathbf { z } _ { t } = \epsilon$ is no longer informative of $\mathbf { x }$ and predictions $\hat { \epsilon } _ { \boldsymbol { \theta } } ( \mathbf { z } _ { t } )$ no longer implicitly predict $\mathbf { x }$ . Examining our reconstruction loss (equation 9), we see that the weighting function $w ( \lambda _ { t } )$ gives zero weight to the reconstruction loss at this signal-to-noise ratio.
+
+For distillation to work, we thus need to parameterize the diffusion model in a way for which the implied prediction $\hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } )$ remains stable as $\lambda _ { t } = \log [ \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } ]$ varies. We tried the following options, and found all to work well with progressive distillation:
+
+• Predicting $\mathbf { x }$ directly. • Predicting both $\mathbf { x }$ and $\epsilon$ , via separate output channels $\{ \tilde { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) , \tilde { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ) \}$ of the neural network, and then merging the predictions via $\hat { \mathbf { x } } = \sigma _ { t } ^ { 2 } \tilde { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) + \alpha _ { t } ( \mathbf { z } _ { t } - \sigma _ { t } \tilde { \epsilon } _ { \theta } ( \mathbf { z } _ { t } ) )$ , thus smoothly interpolating between predicting $\mathbf { x }$ directly and predicting via $\epsilon$ . • Predicting $\mathbf { v } \equiv \alpha _ { t } \epsilon - \sigma _ { t } \mathbf { x }$ , which gives $\hat { \mathbf { x } } = \alpha _ { t } \mathbf { z } _ { t } - \sigma _ { t } \hat { \mathbf { v } } _ { \theta } \big ( \mathbf { z } _ { t } \big )$ , as we show in Appendix D.
+
+In Section 5.1 we test all three parameterizations on training an original diffusion model (no distillation), and find them to work well there also.
+
+In addition to determining an appropriate parameterization, we also need to decide on a reconstruction loss weighting $w ( \lambda _ { t } )$ . The setup of Ho et al. (2020) weights the reconstruction loss by the signal-to-noise ratio, implicitly gives a weight of zero to data with zero SNR, and is therefore not a suitable choice for distillation. We consider two alternative training loss weightings:
+
+$$
+\begin{array} { r l } & { \bullet L _ { \theta } = \operatorname* { m a x } ( \| \mathbf { x } - \hat { \mathbf { x } } _ { t } \| _ { 2 } ^ { 2 } , \| \epsilon - \hat { \epsilon } _ { t } \| _ { 2 } ^ { 2 } ) = \operatorname* { m a x } ( \frac { \alpha _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } , 1 ) \| \mathbf { x } - \hat { \mathbf { x } } _ { t } \| _ { 2 } ^ { 2 } ; \mathfrak { t } } \\ & { \bullet L _ { \theta } = \| \mathbf { v } _ { t } - \hat { \mathbf { v } } _ { t } \| _ { 2 } ^ { 2 } = ( 1 + \frac { \alpha _ { t } ^ { 2 } } { \sigma _ { t } ^ { 2 } } ) \| \mathbf { x } - \hat { \mathbf { x } } _ { t } \| _ { 2 } ^ { 2 } ; \mathbf { \nabla } \mathrm { { S N R } } + 1 ^ { \prime } \mathrm { ~ w e i g h t i n g } } \end{array}
+$$
+
+We examine both choices in our ablation study in Section 5.1, and find both to be good choices for training diffusion models. In practice, the choice of loss weighting also has to take into account how $\alpha _ { t } , \sigma _ { t }$ are sampled during training, as this sampling distribution strongly determines the weight the expected loss gives to each signal-to-noise ratio. Our results are for a cosine schedule $\alpha _ { t } =$ $\cos ( 0 . 5 \pi t )$ , where time is sampled uniformly from [0, 1]. In Figure 2 we visualize the resulting loss weightings, both including and excluding the effect of the cosine schedule.
+
+# 5 EXPERIMENTS
+
+In this section we empirically validate the progressive distillation algorithm proposed in Section 3, as well as the parameterizations and loss weightings considered in Section 4. We consider various image generation benchmarks, with resolution varying from $3 2 \times 3 2$ to $1 2 8 \times 1 2 8$ . All experiments use the cosine schedule $\alpha _ { t } = \cos ( 0 . 5 \pi t )$ , and all models use a U-Net architecture similar to that introduced by Ho et al. (2020), but with BigGAN-style up- and downsampling (Brock et al., 2019), as used in the diffusion modeling setting by Nichol & Dhariwal (2021); Song et al. (2021c). Our training setup closely matches the open source code by Ho et al. (2020). Exact details are given in Appendix E.
+
+# 5.1 MODEL PARAMETERIZATION AND TRAINING LOSS
+
+As explained in Section 4, the standard method of having our model predict , and minimizing mean squared error in the $\epsilon$ -space (Ho et al., 2020), is not appropriate for use with progressive distillation.
+
+![](images/c4f8ad7753e82207e389a0e3e4a68071bdf0d89a05fcf89e908ae1934d2f46a4.jpg)  
+Figure 2: Left: Log weight assigned to reconstruction loss $\| \mathbf { x } - \hat { \mathbf { x } } _ { \lambda } \| _ { 2 } ^ { 2 }$ as a function of the log-SNR $\lambda \stackrel { } { = } \log [ \alpha ^ { 2 } / \sigma ^ { 2 } ]$ , for each of our considered training loss weightings, excluding the influence of the $\alpha _ { t } , \sigma _ { t }$ schedule. Right: Weights assigned to the reconstruction loss including the effect of the cosine schedule $\alpha _ { t } = \cos ( 0 . 5 \pi t )$ , with $t \stackrel { - } { \sim } U [ 0 , 1 ]$ . The weights are only defined up to a constant, and we have adjusted these constants to fit this graph.
+
+<table><tr><td>Network Output</td><td>Loss Weighting</td><td>Stochastic sampler</td><td>DDIM sampler</td></tr><tr><td rowspan="3">(x,ε) combined</td><td>SNR</td><td>2.54/9.88</td><td>2.78/9.56</td></tr><tr><td>Truncated SNR</td><td>2.47/9.85</td><td>2.76/9.49</td></tr><tr><td>SNR+1</td><td>2.52/9.79</td><td>2.87/9.45</td></tr><tr><td rowspan="3">X</td><td>SNR</td><td>2.65/9.80</td><td>2.75/9.56</td></tr><tr><td>Truncated SNR</td><td>2.53/9.92</td><td>2.51/9.58</td></tr><tr><td>SNR+1</td><td>2.56/9.84</td><td>2.65/9.52</td></tr><tr><td rowspan="3">E</td><td>SNR</td><td>2.59/9.84</td><td>2.91/9.52</td></tr><tr><td>Truncated SNR</td><td>N/A</td><td>N/A</td></tr><tr><td>SNR+1</td><td>2.56/9.77</td><td>3.27/9.41</td></tr><tr><td rowspan="3">V</td><td>SNR</td><td>2.65/9.86</td><td>3.05/9.56</td></tr><tr><td>Truncated SNR</td><td>2.45/9.80</td><td>2.75/9.52</td></tr><tr><td>SNR+1</td><td>2.49/9.77</td><td>2.87/9.43</td></tr></table>
+
+Table 1: Generated sample quality as measured by FID and Inception Score (FID/IS) on unconditional CIFAR-10, training the original model (no distillation), and comparing different parameterizations and loss weightings discussed in Section 4. All reported results are averages over 3 random seeds of the best metrics obtained over 2 million training steps; nevertheless we find results are still $\pm 0 . 1$ due to the noise inherent in training our models. Taking the neural network output to represent a prediction of $\epsilon$ in combination with the Truncated SNR loss weighting leads to divergence.
+
+We therefore proposed various alternative parameterizations of the denoising diffusion model that are stable under the progressive distillation procedure, as well as various weighting functions for the reconstruction error in $\mathbf { x }$ -space. Here, we perform a complete ablation experiment of all parameterizations and loss weightings considered in Section 4. For computational efficiency, and for comparisons to established methods in the literature, we use unconditional CIFAR-10 as the benchmark. We measure performance of undistilled models trained from scratch, to avoid introducing too many factors of variation into our analysis.
+
+Table 1 lists the results of the ablation study. Overall results are fairly close across different parameterizations and loss weights. All proposed stable model specifications achieve excellent performance, with the exception of the combination of outputting $\epsilon$ with the neural network and weighting the loss with the truncated SNR, which we find to be unstable. Both predicting $\mathbf { x }$ directly, as well as predicting $\mathbf { v }$ , or the combination $( \epsilon , \bf { x } )$ , could thus be recommended for specification of diffusion models. Here, predicting v is the most stable option, as it has the unique property of making DDIM step-sizes independent of the SNR (see Appendix D), but predicting $\mathbf { x }$ gives slightly better empirical results in this ablation study.
+
+# 5.2 PROGRESSIVE DISTILLATION
+
+We evaluate our proposed progressive distillation algorithm on 4 data sets: CIFAR-10, $6 4 \times 6 4$ downsampled ImageNet, $1 2 8 \times 1 2 8$ LSUN bedrooms, and $1 2 8 \times 1 2 8$ LSUN Church-Outdoor. For each data set we start by training a baseline model, after which we start the progressive distillation procedure. For CIFAR-10 we start progressive distillation from a teacher model taking 8192 steps. For the bigger data sets we start at 1024 steps. At every iteration of distillation we train for 50 thousand parameter updates, except for the distillation to 2 and 1 sampling steps, for which we use 100 thousand updates. We report FID results obtained after each iteration of the algorithm. Using these settings, the computational cost of progressive distillation to 4 sampling steps is comparable or less than for training the original model. In Appendix I we show that this computational cost can be reduce much further still, at a small cost in performance.
+
+In Figure 4 we plot the resulting FID scores (Heusel et al., 2017) obtained for each number of sampling steps. We compare against the undistilled DDIM sampler, as well as to a highly optimized stochastic baseline sampler. For all four data sets, progressive distillation produces near optimal results up to 4 or 8 sampling steps. At 2 or 1 sampling steps, the sample quality degrades relatively more quickly. In contrast, the quality of the DDIM and stochastic samplers degrades very sharply after reducing the number of sampling steps below 128. Overall, we conclude that progressive distillation is thus an attractive solution for computational budgets that allow less than or equal to 128 sampling steps. Although our distillation procedure is designed for use with the DDIM sampler, the resulting distilled models can in principle also be used with stochastic sampling: we investigate this in Appendix F, and find that it achieves performance that falls in between the distilled DDIM sampler and the undistilled stochastic sampler.
+
+Table 2 shows some of our results on CIFAR-10, and compares against other fast sampling methods in the literature: Our method compares favorably and attains higher sampling quality in fewer steps than most of the alternative methods. Figure 3 shows some random samples from our model obtained at different phases of the distillation process. Additional samples are provided in Appendix H.
+
+![](images/ec7e7bccc6f5084fca3cedf3062761bfe9deb3526437ce39335e6a9c68196f18.jpg)  
+Figure 3: Random samples from our distilled $6 4 \times 6 4$ ImageNet models, conditioned on the ‘malamute’ class, for fixed random seed and for varying number of sampling steps. The mapping from input noise to output image is well preserved as the number of sampling steps is reduced.
+
+# 6 RELATED WORK ON FAST SAMPLING
+
+Our proposed method is closest to the work of Luhman & Luhman (2021), who perform distillation of DDIM teacher models into one-step student models. A possible downside of their method is that it requires constructing a large data set by running the original model at its full number of sampling steps: their cost of distillation thus scales linearly with this number of steps, which can be prohibitive. In contrast, our method never needs to run the original model at the full number of sampling steps: at every iteration of progressive distillation, the number of model evaluations is independent of the number of teacher sampling steps, allowing our method to scale up to large numbers of teacher steps at a logarithmic cost in total distillation time.
+
+![](images/002d5d2e542bcdb563cc9e8b531a983549450ab7d1f1a3d1df8cb5f79414a4d8.jpg)  
+Figure 4: Sample quality results as measured by FID for our distilled model on unconditional CIFAR-10, class-conditional $6 4 \mathrm { x } 6 4$ ImageNet, $1 2 8 \mathrm { x } 1 2 8$ LSUN bedrooms, and $1 2 8 \mathrm { x } 1 2 8$ LSUN church-outdoor. We compare against the DDIM sampler and against an optimized stochastic sampler, each evaluated using the same models that were used to initialize the progressive distillation procedure. For CIFAR-10 we report an average over 4 random seeds. For the other data sets we only use a single run because of their computational demand. For the stochastic sampler we set the variance as a log-scale interpolation between an upper and lower bound on the variance, following Nichol & Dhariwal (2021), but we use a single interpolation coefficient rather than a learned coefficient. We then tune this interpolation coefficient separately for each number of sampling steps and report only the best result for that number of steps: this way we obtained better results than with the learned interpolation.
+
+DDIM (Song et al., 2021a) was originally shown to be effective for few-step sampling, as was the probability flow sampler (Song et al., 2021c). Jolicoeur-Martineau et al. (2021) study fast SDE integrators for reverse diffusion processes, and Tzen & Raginsky (2019b) study unbiased samplers which may be useful for fast, high quality sampling as well.
+
+Other work on fast sampling can be viewed as manual or automated methods to adjust samplers or diffusion processes for fast generation. Nichol & Dhariwal (2021); Kong & Ping (2021) describe methods to adjust a discrete time diffusion model trained on many timesteps into models that can sample in few timesteps. Watson et al. (2021) describe a dynamic programming algorithm to reduce the number of timesteps for a diffusion model in a way that is optimal for log likelihood. Chen et al. (2021); Saharia et al. (2021); Ho et al. (2021) train diffusion models over continuous noise levels and tune samplers post training by adjusting the noise levels of a few-step discrete time reverse diffusion process. Their method is effective in highly conditioned settings such as text-to-speech and image super-resolution. San-Roman et al. (2021) train a new network to estimate the noise level of noisy data and show how to use this estimate to speed up sampling.
+
+Alternative specifications of the diffusion model can also lend themselves to fast sampling, such as modified forward and reverse processes (Nachmani et al., 2021; Lam et al., 2021) and training diffusion models in latent space (Vahdat et al., 2021).
+
+Table 2: Comparison of fast sampling results on CIFAR-10 for diffusion models in the literature.   
+
+<table><tr><td>Method</td><td>Model evaluations</td><td>FID</td></tr><tr><td rowspan="3">Progressive Distillation (ours)</td><td>1</td><td>9.12</td></tr><tr><td>2</td><td>4.51</td></tr><tr><td>4</td><td>3.00</td></tr><tr><td>Knowledge distillation (Luhman &amp;Luhman,2021)</td><td>8 1</td><td>2.57 9.36</td></tr><tr><td>DDIM (Song et al.,2021a)</td><td>10</td><td>13.36</td></tr><tr><td rowspan="3"></td><td>20</td><td>6.84</td></tr><tr><td>50</td><td>4.67</td></tr><tr><td>100</td><td>4.16</td></tr><tr><td rowspan="4">Dynamic step-size extrapolation + VP-deep (Jolicoeur-Martineau et al.,2021)</td><td>48</td><td>82.42</td></tr><tr><td>151</td><td>2.73</td></tr><tr><td>180</td><td>2.44</td></tr><tr><td>274 330</td><td>2.60 2.56</td></tr><tr><td rowspan="4">FastDPM(Kong&amp; Ping,2021)</td><td>10</td><td>9.90</td></tr><tr><td>20</td><td>5.05</td></tr><tr><td>50</td><td>3.20</td></tr><tr><td>100</td><td>2.86</td></tr><tr><td>Improved DDPM respacing</td><td>25</td><td>7.53</td></tr><tr><td>(Nichol &amp; Dhariwal, 2021),our reimplementation</td><td>50</td><td>4.99</td></tr><tr><td>LSGM (Vahdat et al., 2021)</td><td>138</td><td>2.10</td></tr></table>
+
+# 7 DISCUSSION
+
+We have presented progressive distillation, a method to drastically reduce the number of sampling steps required for high quality generation of images, and potentially other data, using diffusion models with deterministic samplers like DDIM (Song et al., 2020). By making these models cheaper to run at test time, we hope to increase their usefulness for practical applications, for which running time and computational requirements often represent important constraints.
+
+In the current work we limited ourselves to setups where the student model has the same architecture and number of parameters as the teacher model: in future work we hope to relax this constraint and explore settings where the student model is smaller, potentially enabling further gains in test time computational requirements. In addition, we hope to move past the generation of images and also explore progressive distillation of diffusion models for different data modalities such as e.g. audio (Chen et al., 2021).
+
+In addition to the proposed distillation procedure, some of our progress was realized through different parameterizations of the diffusion model and its training loss. We expect to see more progress in this direction as the community further explores this model class.
+
+# REPRODUCIBILITY STATEMENT
+
+We provide full details on model architectures, training procedures, and hyperparameters in Appendix E, in addition to our discussion in Section 5. In Algorithm 2 we provide fairly detailed pseudocode that closely matches our actual implementation, which is available in open source at https: //github.com/google-research/google-research/tree/master/diffusion_distillation.
+
+# ETHICS STATEMENT
+
+In general, generative models can have unethical uses, such as fake content generation, and they can suffer from bias if applied to data sets that are not carefully curated. The focus of this paper specifically is on speeding up generative models at test time in order to reduce their computational demands; we do not have specific concerns with regards to this contribution.
+
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+
+# A PROBABILITY FLOW ODE IN TERMS OF LOG-SNR
+
+Song et al. (2021c) formulate the forward diffusion process in terms of an SDE of the form
+
+$$
+\begin{array} { r } { d { \mathbf { z } } = f ( { \mathbf { z } } , t ) d t + g ( t ) d W , } \end{array}
+$$
+
+and show that samples from this diffusion process can be generated by solving the associated probability flow ODE:
+
+$$
+d \mathbf { z } = [ f ( \mathbf { z } , t ) - \frac { 1 } { 2 } g ^ { 2 } ( t ) \nabla _ { z } \log p _ { t } ( \mathbf { z } ) ] d t ,
+$$
+
+where in practice $\nabla _ { z } \log { p _ { t } ( \mathbf { z } ) }$ is approximated by a learned denoising model using
+
+$$
+\nabla _ { z } \log { p _ { t } ( \mathbf { z } ) } \approx \frac { \alpha _ { t } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) - \mathbf { z } _ { t } } { \sigma _ { t } ^ { 2 } } .
+$$
+
+Following Kingma et al. (2021) we have $\begin{array} { r } { f ( \mathbf { z } , t ) = \frac { d \log \alpha _ { t } } { d t } \mathbf { z } _ { t } } \end{array}$ and $\begin{array} { r } { g ^ { 2 } ( t ) = \frac { d \sigma _ { t } ^ { 2 } } { d t } - 2 \frac { d \log { \alpha _ { t } } } { d t } \sigma _ { t } ^ { 2 } } \end{array}$ . Assuming a variance preserving diffusion process with $\alpha _ { t } ^ { 2 } = 1 - \sigma _ { t } ^ { 2 } = \mathrm { s i g m o i d } ( \bar { \lambda } _ { t } )$ for $\bar { \lambda _ { t } } = \log [ \alpha _ { t } ^ { 2 } / \sigma _ { t } ^ { 2 } ]$ (without loss of generality, see Kingma et al. (2021)), we get
+
+$$
+f ( \mathbf { z } , t ) = { \frac { d \log \alpha _ { t } } { d t } } \mathbf { z } _ { t } = { \frac { 1 } { 2 } } { \frac { d \log \alpha _ { \lambda } ^ { 2 } } { d \lambda } } { \frac { d \lambda } { d t } } \mathbf { z } _ { t } = { \frac { 1 } { 2 } } ( 1 - \alpha _ { t } ^ { 2 } ) { \frac { d \lambda } { d t } } \mathbf { z } _ { t } = { \frac { 1 } { 2 } } \sigma _ { t } ^ { 2 } { \frac { d \lambda } { d t } } \mathbf { z } _ { t } .
+$$
+
+Similarly, we get
+
+$$
+g ^ { 2 } ( t ) = { \frac { d \sigma _ { t } ^ { 2 } } { d t } } - 2 { \frac { d \log \alpha _ { t } } { d t } } \sigma _ { t } ^ { 2 } = { \frac { d \sigma _ { \lambda } ^ { 2 } } { d \lambda } } { \frac { d \lambda } { d t } } - \sigma _ { t } ^ { 4 } { \frac { d \lambda } { d t } } = ( \sigma _ { t } ^ { 4 } - \sigma _ { t } ^ { 2 } ) { \frac { d \lambda } { d t } } - \sigma _ { t } ^ { 4 } { \frac { d \lambda } { d t } } = - \sigma _ { t } ^ { 2 } { \frac { d \lambda } { d t } } .
+$$
+
+Plugging these into the probability flow ODE then gives
+
+$$
+\begin{array} { l } { d \mathbf { z } = [ f ( \mathbf { z } , t ) - \displaystyle \frac { 1 } { 2 } g ^ { 2 } ( t ) \nabla _ { z } \log p _ { t } ( \mathbf { z } ) ] d t } \\ { \displaystyle = \frac { 1 } { 2 } \sigma _ { \lambda } ^ { 2 } [ \mathbf { z } _ { \lambda } + \nabla _ { z } \log p _ { \lambda } ( \mathbf { z } ) ] d \lambda . } \end{array}
+$$
+
+Plugging in our function approximation from Equation 12 gives
+
+$$
+\begin{array} { l } { { d { \bf z } = \displaystyle \frac { 1 } { 2 } \sigma _ { \lambda } ^ { 2 } \left[ { \bf z } _ { \lambda } + \left( \frac { \alpha _ { \lambda } \hat { \bf x } _ { \theta } \left( { \bf z } _ { \lambda } \right) - { \bf z } _ { \lambda } } { \sigma _ { \lambda } ^ { 2 } } \right) \right] d \lambda } } \\ { { { \mathrm { } ~ } = \displaystyle \frac { 1 } { 2 } [ \alpha _ { \lambda } \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) + ( \sigma _ { \lambda } ^ { 2 } - 1 ) { \bf z } _ { \lambda } ] d \lambda } } \\ { { { \mathrm { } ~ } = \displaystyle \frac { 1 } { 2 } [ \alpha _ { \lambda } \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) - \alpha _ { \lambda } ^ { 2 } { \bf z } _ { \lambda } ] d \lambda . } } \end{array}
+$$
+
+# B DDIM IS AN INTEGRATOR OF THE PROBABILITY FLOW ODE
+
+The DDIM update rule (Song & Ermon, 2020) is given by
+
+$$
+\mathbf { z } _ { s } = \frac { \sigma _ { s } } { \sigma _ { t } } [ \mathbf { z } _ { t } - \alpha _ { t } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) ] + \alpha _ { s } \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { t } ) ,
+$$
+
+for $s < t$ . Taking the derivative of this expression with respect to $\lambda _ { s }$ , assuming again a variance preserving diffusion process, and using $\begin{array} { r } { \frac { d \alpha _ { \lambda } } { d \lambda } = \frac { 1 } { 2 } \alpha _ { \lambda } \sigma _ { \lambda } ^ { 2 } } \end{array}$ and $\begin{array} { r } { \frac { d \sigma _ { \lambda } } { d \lambda } = - \frac { 1 } { 2 } \sigma _ { \lambda } \alpha _ { \lambda } ^ { 2 } } \end{array}$ , gives
+
+$$
+\begin{array} { r l r } {  { \frac { { \bf z } _ { \lambda _ { s } } } { d \lambda _ { s } } = \frac { d \sigma _ { \lambda _ { s } } } { d \lambda _ { s } } \frac { 1 } { \sigma _ { t } } [ { \bf z } _ { t } - \alpha _ { t } \hat { \bf x } _ { \theta } ( { \bf z } _ { t } ) ] + \frac { d \alpha _ { \lambda _ { s } } } { d \lambda _ { s } } \hat { \bf x } _ { \theta } ( { \bf z } _ { t } ) } } \\ & { } & { = - \frac { 1 } { 2 } \alpha _ { s } ^ { 2 } \frac { \sigma _ { s } } { \sigma _ { t } } [ { \bf z } _ { t } - \alpha _ { t } \hat { \bf x } _ { \theta } ( { \bf z } _ { t } ) ] + \frac { 1 } { 2 } \alpha _ { s } \sigma _ { s } ^ { 2 } \hat { \bf x } _ { \theta } ( { \bf z } _ { t } ) . } \end{array}
+$$
+
+Evaluating this derivative at $s = t$ then gives
+
+$$
+\begin{array} { l } { \displaystyle \frac { { \bf z } _ { \lambda _ { s } } } { d \lambda _ { s } } | _ { s = t } = - \frac { 1 } { 2 } \alpha _ { \lambda } ^ { 2 } [ { \bf z } _ { \lambda } - \alpha _ { \lambda } \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) ] + \frac { 1 } { 2 } \alpha _ { \lambda } \sigma _ { \lambda } ^ { 2 } \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) } \\ { \displaystyle = - \frac { 1 } { 2 } \alpha _ { \lambda } ^ { 2 } [ { \bf z } _ { \lambda } - \alpha _ { \lambda } \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) ] + \frac { 1 } { 2 } \alpha _ { \lambda } ( 1 - \alpha _ { \lambda } ^ { 2 } ) \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) } \\ { \displaystyle = \frac { 1 } { 2 } [ \alpha _ { \lambda } \hat { \bf x } _ { \theta } ( { \bf z } _ { \lambda } ) - \alpha _ { \lambda } ^ { 2 } { \bf z } _ { \lambda } ] . } \end{array}
+$$
+
+Comparison with Equation 19 now shows that DDIM follows the probability flow ODE up to first order, and can thus be considered as an integration rule for this ODE.
+
+# C EVALUATION OF INTEGRATORS OF THE PROBABILITY FLOW ODE
+
+In a preliminary investigation we tried several numerical integrators for the probability flow ODE. As our model we used a pre-trained class-conditional $1 2 8 \mathrm { x } 1 2 8$ ImageNet model following the description in Ho et al. (2020). We tried a simple Euler integrator, RK4 (the “classic” 4th order Runge–Kutta integrator), and DDIM (Song et al., 2021a). In addition we compared to a Gaussian sampler with variance equal to the lower bound given by Ho et al. (2020). We calculated FID scores on just 5000 samples, hence our results in this experiment are not comparable to results reported in the literature. This preliminary investigation gave the results listed in Table 3 and identified DDIM as the best integrator in terms of resulting sample quality.
+
+<table><tr><td>Sampler</td><td>Number of steps</td><td>FID</td></tr><tr><td>Stochastic Euler RK4</td><td>1000 1000 1000 1000</td><td>13.35 16.5 16.33 15.98</td></tr><tr><td>DDIM Stochastic</td><td>100</td><td>18.44</td></tr><tr><td>Euler</td><td>100</td><td>23.67</td></tr><tr><td>RK4</td><td>100</td><td>18.94</td></tr><tr><td>DDIM</td><td>100</td><td>16.35</td></tr></table>
+
+Table 3: Preliminary FID scores on $1 2 8 \times 1 2 8$ ImageNet for various integrators of the probability flow ODE, and compared against a stochastic sampler. Model specification and noise schedule follow Ho et al. (2020).
+
+# D EXPRESSION OF DDIM IN ANGULAR PARAMETERIZATION
+
+We can simplify the DDIM update rule by expressing it in terms of $\phi _ { t } = \arctan ( \sigma _ { t } / \alpha _ { t } )$ , rather than in terms of time $t$ or log-SNR $\lambda _ { t }$ , as we show here.
+
+Given our definition of $\phi$ , and assuming a variance preserving diffusion process, we have $\alpha _ { \phi } =$ $\cos ( \phi )$ , $\begin{array} { r } { \sigma _ { \phi } = \sin ( \phi ) } \end{array}$ , and hence ${ \bf z } _ { \phi } = \mathrm { c o s } ( \phi ) { \bf x } + \mathrm { s i n } ( { \bar { \phi } } ) \epsilon$ . We can now define the velocity of $\mathbf { z } _ { \phi }$ as
+
+$$
+\mathbf { v } _ { \phi } \equiv \frac { d \mathbf { z } _ { \phi } } { d \phi } = \frac { d \cos ( \phi ) } { d \phi } \mathbf { x } + \frac { d \sin ( \phi ) } { d \phi } \epsilon = \cos ( \phi ) \epsilon - \sin ( \phi ) \mathbf { x } .
+$$
+
+Rearranging $\epsilon , \mathbf { x } , \mathbf { v }$ , we then get
+
+$$
+\begin{array} { c } { { \sin ( \phi ) { \bf x } = \cos ( \phi ) \epsilon - { \bf v } _ { \phi } } } \\ { { { \displaystyle = \frac { \cos ( \phi ) } { \sin ( \phi ) } ( { \bf z } - \cos ( \phi ) { \bf x } ) - { \bf v } _ { \phi } } } } \\ { { { \displaystyle \sin ^ { 2 } ( \phi ) { \bf x } = \cos ( \phi ) { \bf z } - \cos ^ { 2 } ( \phi ) { \bf x } - \sin ( \phi ) { \bf v } _ { \phi } } } } \\ { { { \displaystyle ( \sin ^ { 2 } ( \phi ) + \cos ^ { 2 } ( \phi ) ) { \bf x } = { \bf x } = \cos ( \phi ) { \bf z } - \sin ( \phi ) { \bf v } _ { \phi } } , } } \end{array}
+$$
+
+and similarly we get $\boldsymbol { \epsilon } = \sin ( \phi ) \mathbf { z } _ { \phi } + \cos ( \phi ) \mathbf { v } _ { \phi }$
+
+Furthermore, we define the predicted velocity as
+
+$$
+\hat { \mathbf { v } } _ { \theta } ( \mathbf { z } _ { \phi } ) \equiv \cos ( \phi ) \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { \phi } ) - \sin ( \phi ) \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { \phi } ) ,
+$$
+
+where $\begin{array} { r } { \hat { \epsilon } _ { \theta } ( \mathbf { z } _ { \phi } ) = ( \mathbf { z } _ { \phi } - \cos ( \phi ) \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { \phi } ) ) / \sin ( \phi ) . } \end{array}$
+
+Rewriting the DDIM update rule in the introduced terms then gives
+
+$$
+\begin{array} { r l r } {  { \mathbf { z } _ { \phi _ { s } } = \cos ( \phi _ { s } ) \hat { \mathbf { x } } _ { \theta } ( \mathbf { z } _ { \phi _ { t } } ) + \sin ( \phi _ { s } ) \hat { \boldsymbol { \epsilon } } _ { \theta } ( \mathbf { z } _ { \phi _ { t } } ) } } & { \quad ( \hat { \mathbf { z } } _ { \phi _ { t } } ) } \\ & { } & { = \cos ( \phi _ { s } ) ( \cos ( \phi _ { t } ) \mathbf { z } _ { \phi _ { t } } - \sin ( \phi _ { t } ) \hat { \mathbf { v } } _ { \theta } ( \mathbf { z } _ { \phi _ { t } } ) ) + \sin ( \phi _ { s } ) ( \sin ( \phi _ { t } ) \mathbf { z } _ { \phi _ { t } } + \cos ( \phi _ { t } ) \hat { \mathbf { v } } _ { \theta } ( \mathbf { z } _ { \phi _ { t } } ) ) \quad \mathrm { ~ ( \hat { z } _ \phi \phi _ t ~ ) ~ } } \\ & { } & { = [ \cos ( \phi _ { s } ) \cos ( \phi _ { t } ) - \sin ( \phi _ { s } ) \sin ( \phi _ { t } ) ] \mathbf { z } _ { \phi _ { t } } + [ \sin ( \phi _ { s } ) \cos ( \phi _ { t } ) - \cos ( \phi _ { s } ) \sin ( \phi _ { t } ) ] \hat { \mathbf { v } } _ { \theta } ( \mathbf { z } _ { \phi _ { t } } ) . } \end{array}
+$$
+
+Finally, we use the trigonometric identities
+
+$$
+\begin{array} { l } { \cos ( \phi _ { s } ) \sin ( \phi _ { t } ) - \sin ( \phi _ { s } ) \cos ( \phi _ { t } ) = \cos ( \phi _ { s } - \phi _ { t } ) } \\ { \sin ( \phi _ { s } ) \cos ( \phi _ { t } ) - \cos ( \phi _ { s } ) \sin ( \phi _ { t } ) = \sin ( \phi _ { s } - \phi _ { t } ) , } \end{array}
+$$
+
+to find that
+
+$$
+\begin{array} { r } { \mathbf { z } _ { \phi _ { s } } = \cos ( \phi _ { s } - \phi _ { t } ) \mathbf { z } _ { \phi _ { t } } + \sin ( \phi _ { s } - \phi _ { t } ) \hat { \mathbf { v } } _ { \theta } ( \mathbf { z } _ { \phi _ { t } } ) . } \end{array}
+$$
+
+or equivalently
+
+$$
+\begin{array} { r } { { \bf z } _ { \phi _ { t } - \delta } = \cos ( \delta ) { \bf z } _ { \phi _ { t } } - \sin ( \delta ) \hat { \bf v } _ { \theta } ( { \bf z } _ { \phi _ { t } } ) . } \end{array}
+$$
+
+Viewed from this perspective, DDIM thus evolves ${ \bf z } _ { \phi _ { s } }$ by moving it on a circle in the $\left( \mathbf { z } _ { \phi _ { t } } , \hat { \mathbf { v } } _ { \phi _ { t } } \right)$ basis, along the $- \hat { \mathbf { v } } _ { \phi _ { t } }$ direction. The relationship between ${ \bf z } _ { \phi _ { t } } , { \bf v } _ { t } , \alpha _ { t } , \sigma _ { t } , { \bf x } , \epsilon$ is visualized in Figure 5.
+
+![](images/1ae6fef040ee919933f9302d63bd576cb38724ffa5acf6df12efc7bec38eb818.jpg)  
+Figure 5: Visualization of reparameterizing the diffusion process in terms of $\phi$ and $\mathbf { v } _ { \phi }$
+
+# E SETTINGS USED IN EXPERIMENTS
+
+Our model architectures closely follow those described by Dhariwal & Nichol (2021). For $6 4 \times 6 4$ ImageNet we use their model exactly, with 192 channels at the highest resolution. All other models are slight variations with different hyperparameters.
+
+For CIFAR-10 we use an architecture with a fixed number of channels at all resolutions of 256. The model consists of a UNet that internally downsamples the data twice, to $1 6 \times 1 6$ and to $8 \times 8$ . At each resolution we apply 3 residual blocks, like described by Dhariwal & Nichol (2021). We use single-headed attention, and only apply this at the $1 6 \times 1 6$ and $8 \times 8$ resolutions. We use dropout of 0.2 when training the original model. No dropout is used during distillation.
+
+For LSUN we use a model similar to that for ImageNet, but with a reduced number of 128 channels at the $6 4 \times 6 4$ resolution. Compared to ImageNet we have an additional level in the UNet, corresponding to the input resolution of $1 2 8 \times 1 2 8$ , which we process using 3 residual blocks with 64 channels. We only use attention layers for the resolutions of $3 2 \times 3 2$ and lower.
+
+For CIFAR-10 we take the output of the model to represent a prediction of $\mathbf { x }$ directly, as discussed in Section 4. For the other data sets we used the combined prediction of $( \mathbf { x } , \epsilon )$ like described in that section also. All original models are trained with Adam with standard settings (learning rate of $3 * 1 0 ^ { - 4 }$ ), using a parameter moving average with constant 0.9999 and very slight decoupled weight decay (Loshchilov & Hutter, 2017) with a constant of 0.001. We clip the norm of gradients to a global norm of 1 before calculating parameter updates. For CIFAR-10 we train for $8 0 0 \mathrm { k }$ parameter updates, for ImageNet we use 550k updates, and for LSUN we use $4 0 0 \mathrm { k }$ updates. During distillation we train for 50k updates per iteration, except for the distillation to 2 and 1 sampling steps, for which we use $1 0 0 \mathrm { k }$ updates. We linearly anneal the learning rate from $1 0 ^ { - 4 }$ to zero during each iteration.
+
+We use a batch size of 128 for CIFAR-10 and 2048 for the other data sets. We run our experiments on TPUv4, using 8 TPU chips for CIFAR-10, and 64 chips for the other data sets. The total time required to first train and then distill a model varies from about a day for CIFAR-10, to about 5 days for ImageNet.
+
+# F STOCHASTIC SAMPLING WITH DISTILLED MODELS
+
+Our progressive distillation procedure was designed to be used with the DDIM sampler, but the resulting distilled model could in principle also be used with a stochastic sampler. Here we evaluate a distilled model for $6 4 \mathrm { x } 6 4$ ImageNet using the optimized stochastic sampler also used in Section 5.2. The results are presented in Figure 6.
+
+![](images/529c3caadd0c73afa2ceffeaf299556a86ed0424c93903a30db7430a1147d922.jpg)  
+Figure 6: FID of generated samples from distilled and undistilled models, using DDIM or stochastic sampling. For the stochastic sampling results we present the best FID obtained by a grid-search over 11 possible noise levels, spaced log-uniformly between the upper and lower bound on the variance as derived by Ho et al. (2020). The performance of the distilled model with stochastic sampling is found to lie in between the undistilled original model with stochastic sampling and the distilled DDIM sampler: For small numbers of sampling steps the DDIM sampler performs better with the distilled model, for large numbers of steps the stochastic sampler performs better.
+
+# G DERIVATION OF THE DISTILLATION TARGET
+
+The key difference between our progressive distillation algorithm proposed in Section 3 and the standard diffusion training procedure is in how we determine the target value for our denoising model. In standard diffusion training, the target for denoising is the clean data $\mathbf { x }$ . In progressive distillation it is the value $\tilde { \mathbf { x } }$ the student denoising model would need to predict in order to match the teacher model when sampling. Here we derive what this target needs to be.
+
+Using notation $t ^ { \prime } = t - 0 . 5 / N$ and $t ^ { \prime \prime } = t - 1 / N$ , when training a student with $N$ sampling steps, we have that the teacher model samples the next set of noisy data $\mathbf { z } _ { t ^ { \prime \prime } }$ given the current noisy data $\mathbf { z } _ { t }$ by taking two steps of DDIM. The student tries to sample the same value in only one step of DDIM. Denoting the student denoising prediction by $\tilde { \mathbf { x } }$ , and its one-step sample by $\tilde { \mathbf { z } } _ { t ^ { \prime \prime } }$ , application of the DDIM sampler (see equation 8), gives:
+
+$$
+\tilde { \mathbf { z } } _ { t ^ { \prime \prime } } = \alpha _ { t ^ { \prime \prime } } \tilde { \mathbf { x } } + \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t } } ( \mathbf { z } _ { t } - \alpha _ { t } \tilde { \mathbf { x } } ) .
+$$
+
+In order for the student sampler to match the teacher sampler, we must set $\tilde { \mathbf { z } } _ { t ^ { \prime \prime } }$ equal to $\mathbf { z } _ { t ^ { \prime \prime } }$ . This gives
+
+$$
+\begin{array} { c } { { \tilde { \bf z } _ { t ^ { \prime \prime } } = \alpha _ { t ^ { \prime \prime } } \tilde { \bf x } + \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t } } ( { \bf z } _ { t } - \alpha _ { t } \tilde { \bf x } ) = { \bf z } _ { t ^ { \prime \prime } } } } \\ { { { \mathrm { } } } } \\ { { { \mathrm { } } = \left( \alpha _ { t ^ { \prime \prime } } - \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t } } \alpha _ { t } \right) \tilde { \bf x } + \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t } } { \bf z } _ { t } = { \bf z } _ { t ^ { \prime \prime } } } } \\ { { { \mathrm { } } } } \\ { { \left( \alpha _ { t ^ { \prime \prime } } - \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t } } \alpha _ { t } \right) \tilde { \bf x } = { \bf z } _ { t ^ { \prime \prime } } - \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t } } { \bf z } _ { t } } } \\ { { { \mathrm { } } } } \\ { { \tilde { \bf x } = \frac { { \bf z } _ { t ^ { \prime \prime } } - \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t ^ { \prime } } } { \alpha _ { t } } } { \alpha _ { t ^ { \prime \prime } } - \frac { \sigma _ { t ^ { \prime \prime } } } { \sigma _ { t ^ { \prime \prime } } } { \alpha _ { t } } } } } \end{array}
+$$
+
+In other words, if our student denoising model exactly predicts $\tilde { \mathbf { x } }$ as defined in equation 43 above, then the one-step student sample $\tilde { \mathbf { z } } _ { t ^ { \prime \prime } }$ is identical to the two-step teacher sample $\mathbf { z } _ { t ^ { \prime \prime } }$ . In order to have our student model approximate this ideal outcome, we thus train it to predict $\tilde { \bf x }$ from $\mathbf { z } _ { t }$ as well as possible, using the standard squared error denoising loss (see Equation 9).
+
+Note that this possibility of matching the two-step teacher model with a one-step student model is unique to deterministic samplers like DDIM: the composition of two standard stochastic DDPM sampling steps (Equation 5) forms a non-Gaussian distribution that falls outside the family of Gaussian distributions that can be modelled by a single DDPM student step: A multi-step stochastic DDPM sampler can thus not be distilled into a few-step sampler without some loss in fidelity. This is in contrast with the deterministic DDIM sampler: here both the two-step DDIM teacher update and the one-step DDIM student update represent deterministic mappings implemented by a neural net, which is why the student is able to accurately match the teacher.
+
+Finally, note that we do lose something during the progressive distillation process: while the original model was trained to denoise $\mathbf { z } _ { t }$ for any given continuous time $t$ , the distilled student models are only ever evaluated on a small discrete set of times $t$ . The student models thus lose generality as distillation progresses. At the same time, it’s this loss of generality that allows the student models to free up enough modeling capacity to accurately match the teacher model without increasing their model size.
+
+# H ADDITIONAL RANDOM SAMPLES
+
+In this section we present additional random samples from our diffusion models obtained through progressive distillation. We show samples for distilled models taking 256, 4, and 1 sampling steps. All samples are uncurated.
+
+As explained in Section 3, our distilled samplers implement a deterministic mapping from input noise to output samples (also see Appendix G). To facilitate comparison of this mapping for varying numbers of sampling steps, we generate all samples using the same random input noise, and we present the samples side-by-side. As these samples show, the mapping is mostly preserved when moving from many steps to a single step: The same input noise is mapped to the same output image, with a slight loss in image quality, as the number of steps is reduced. Since the mapping is preserved while reducing the number of steps, our distilled models also preserve the excellent sample diversity of diffusion models (see e.g. Kingma et al. (2021)).
+
+![](images/54da9530cb6dc419c29bfdd5a04f5babde9fa27d7a511f328a1974efb6afda39.jpg)  
+Figure 7: Random samples from our distilled CIFAR-10 models, for fixed random seed and for varying number of sampling steps.
+
+![](images/78b797bc518f181c27129d69dc32db54a060585aba455ff0aef1fcd447f1c2e6.jpg)  
+Figure 8: Random samples from our distilled $6 4 \times 6 4$ ImageNet models, conditioned on the ‘coral reef’ class, for fixed random seed and for varying number of sampling steps.
+
+![](images/332eea9c8611103ed561e748d470ffbbce320d1e310aafbc84bbcfaebb641aa5.jpg)  
+Figure 9: Random samples from our distilled $6 4 \times 6 4$ ImageNet models, conditioned on the ‘sports car’ class, for fixed random seed and for varying number of sampling steps.
+
+![](images/472f66b51eddfba405ca39faa14ff2395e3ec9231915dca97b94e1353b20ecd7.jpg)  
+Figure 10: Random samples from our distilled LSUN bedrooms models, for fixed random seed and for varying number of sampling steps.
+
+![](images/32b5f2a4b6ed9574351fbb777d6ac3ccfd7f42bcf9dbd59265ed6f9c562bdca2.jpg)  
+Figure 11: Random samples from our distilled LSUN church-outdoor models, for fixed random seed and for varying number of sampling steps.
+
+# I ABLATION WITH FASTER DISTILLATION SCHEDULES
+
+In order to further reduce the computational requirements for our progressive distillation approach, we perform an ablation study on CIFAR-10, where we decrease the number of parameter updates we use to train each new student model. In Figure 12 we present results for taking 25 thousand, 10 thousand, or 5 thousand optimization steps, instead of the 50 thousand we suggested in Section 3. As the results show, we can drastically decrease the number of optimization steps taken, and still get very good performance when using $\geq 4$ sampling steps. When taking very few sampling steps, the loss in performance becomes more pronounced when training the student for only a short time.
+
+In addition to just decreasing the number of parameter updates, we also experiment with a schedule where we train each student on 4 times fewer sampling steps than its teacher, rather than the 2 times we propose in Section 3. Here the denoising target is still derived from taking 2 DDIM steps with the teacher model as usual, since taking 4 teacher steps would negate most of the computational savings. As Figure 12 shows, this does not work as well: if the computational budget is limited, it’s better to take fewer parameter updates per halving of the number of sampling steps then to skip distillation iterations altogether.
+
+In Figure 13 we show the results achieved with a faster schedule for the ImageNet and LSUN datasets. Here also, we achieve excellent results with a faster distillation schedule.
+
+![](images/fffa87cf6ed1399dcdd3366b898a5c784f71fba780833ee28c4b364427919aad.jpg)  
+Figure 12: Comparing our proposed schedule for progressive distillation taking $5 0 \mathrm { k }$ parameter updates to train a new student every time the number of steps is halved, versus fast sampling schedules taking fewer parameter updates (25k, 10k, 5k), and a fast schedule dividing the number of steps by 4 for every new student instead of by 2. All reported numbers are averages over 4 random seeds. For each schedule we selected the optimal learning rate from $[ 5 e ^ { - 5 } , 1 e ^ { - 4 } , 2 e ^ { - 4 } , 3 e ^ { - 4 } ]$ .
+
+![](images/6e2fc03b5372c92a9ba4a042ca391d715e5486acf4fa56597a511cb18f5aeacc.jpg)  
+Figure 13: Comparing our proposed schedule for progressive distillation taking $5 0 \mathrm { k }$ parameter updates to train a new student every time the number of steps is halved, versus a fast sampling schedule taking $1 0 \mathrm { k }$ parameter updates. For each reported number of steps we selected the optimal learning rate from $[ \dot { 5 } e ^ { - 5 } , 1 e ^ { - 4 } , \dot { 2 } e ^ { - 4 } , 3 e ^ { - 4 } ]$ . Results are for a single random seed.

parse/dev/dUSI4vFyMK/dUSI4vFyMK.md

@@ -0,0 +1,372 @@
+# Convergence for score-based generative modeling with polynomial complexity
+
+Holden Lee Department of Applied Mathematics and Statistics Johns Hopkins University hlee283@jhu.edu
+
+Jianfeng Lu   
+Department of Mathematics Duke University   
+jianfeng@math.duke.edu
+
+Yixin Tan Department of Mathematics Duke University yixin.tan@duke.edu
+
+# Abstract
+
+Score-based generative modeling (SGM) is a highly successful approach for learning a probability distribution from data and generating further samples. We prove the first polynomial convergence guarantees for the core mechanic behind SGM: drawing samples from a probability density $p$ given a score estimate (an estimate of $\nabla \ln p$ ) that is accurate in $L ^ { 2 } ( p )$ . Compared to previous works, we do not incur error that grows exponentially in time or that suffers from a curse of dimensionality. Our guarantee works for any smooth distribution and depends polynomially on its log-Sobolev constant. Using our guarantee, we give a theoretical analysis of score-based generative modeling, which transforms white-noise input into samples from a learned data distribution given score estimates at different noise scales. Our analysis gives theoretical grounding to the observation that an annealed procedure is required in practice to generate good samples, as our proof depends essentially on using annealing to obtain a warm start at each step. Moreover, we show that a predictor-corrector algorithm gives better convergence than using either portion alone.
+
+# 1 Introduction
+
+A key task in machine learning is to learn a probability distribution from data, in a way that allows efficient generation of additional samples from the learned distribution. Score-based generative modeling (SGM) is one empirically successful approach that implicitly learns the probability distribution by learning how to transform white noise into the data distribution, and gives stateof-the-art performance for generating images and audio [SE19; Dat+19; Gra+19; SE20; $\mathrm { S o n } { + } 2 0 \mathrm { b }$ ; $\mathrm { M e n } { + } 2 1$ ; $\mathbf { S o n } { + } 2 1 \mathbf { b }$ ; Son $+ 2 1 \mathrm { a }$ ; $\mathrm { J i n } { + } 2 2 ^ { \cdot }$ ]. It also yields a conditional generation process for inverse problems [DN21]. The basic idea behind score-based generative modeling is to first estimate the score function from data $[ \mathrm { S o n } { + } 2 0 \mathrm { a } ]$ and then to sample the distribution based on the learned score function. Other approaches for generative modeling include generative adversarial networks (GANs) $[ \mathrm { G o o { + } 1 4 }$ ; ACB17], normalizing flows [DSB16], variational autoencoders [KW19], and energybased models [ZML16]. While score-based generative modeling has achieved great success, its theoretical analysis is still lacking and is the focus of our work.
+
+# 1.1 Background
+
+General framework. The score function of a distribution $P$ with density $p$ is defined as the gradient of the log-pdf, $\nabla \ln p$ . Its significance arises from the fact that knowing the score function allows running a variety of sampling algorithms, based on discretizations of stochastic differential equations (SDE’s), to sample from $p$ . SGM consists of two steps: first, learning an estimate of the score function for a sequence of “noisy” versions of the data distribution $P _ { \mathrm { d a t a } }$ , and second, using the score function in lieu of the gradient of the log-pdf in the chosen sampling algorithm. We now describe each of these steps more precisely.
+
+First, a method of adding noise to the data distribution is fixed; this takes the form of evolving a (forward) stochastic differential equation (SDE) starting from the data distribution. We fix a sequence of noise levels $\sigma _ { 1 } < \cdots < \sigma _ { N }$ . For $\sigma \in \{ \sigma _ { 1 } , \dots , \sigma _ { N } \}$ , let the resulting distributions be $P _ { \sigma ^ { 2 } }$ and the distributions conditional on the starting data point be $P _ { \sigma ^ { 2 } } ( \cdot | x )$ . Typically, $\sigma _ { 1 }$ is chosen so that $P _ { \sigma _ { 1 } ^ { 2 } } \approx P _ { \mathrm { d a t a } }$ and $P _ { \sigma _ { N } ^ { 2 } }$ is close to some “prior” distribution that is easy to sample from, such as $N ( 0 , \sigma _ { N } ^ { 2 } I _ { d } )$ . While the score $\nabla \ln p _ { \sigma ^ { 2 } }$ cannot be estimated directly, it turns out that a de-noising objective that is equivalent to the score-matching objective can be calculated [SE19]. This de-noising objective can be estimated from samples $( X , { \tilde { X } } )$ where $\widetilde { X } \sim P _ { \sigma ^ { 2 } } ( \cdot | x )$ . The objective is represented and optimized within an expressive function class, typically neural networks, to obtain a $L ^ { 2 }$ -estimate of the score, that is, $s _ { \theta } ( x , \overbar { \sigma } ^ { 2 } )$ such that
+
+$$
+\begin{array} { r l } {  { \mathbb { E } _ { { x } \sim { P } _ { \sigma ^ { 2 } } } [ \| s _ { \theta } ( x , \sigma ^ { 2 } ) - \nabla \ln { p } _ { \sigma ^ { 2 } } ( x ) \| ^ { 2 } ] } \quad } & { { } } \end{array}
+$$
+
+is small.
+
+The reason we estimate the score function $\nabla \ln p _ { \sigma ^ { 2 } }$ is that there are a variety of sampling algorithms—based on simulating SDE’s—that can sample from $p$ given access to $\nabla \ln p$ , including Langevin Monte Carlo and Hamiltonian Monte Carlo. The second step is then to use the estimated score function $s _ { \theta } ( x , t )$ in lieu of the exact gradient in the sampling algorithm to successively obtain samples from $p _ { \sigma _ { N } ^ { 2 } } , \ldots , p _ { \sigma _ { 1 } ^ { 2 } }$ . This sequence interpolates smoothly between the prior distribution (e.g., $N ( 0 , \sigma _ { N } ^ { 2 } I _ { d } ) )$ and the data distribution $P _ { \mathrm { d a t a } }$ ; such an “annealing” or “homotopy” method is required in practice to generate good samples $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ .
+
+Examples of SGM’s. There have been several instantiations of this general approach. [SE19] add gaussian noise to the data and then use Langevin diffusion at a discrete set of noise levels $\sigma _ { N } > \cdots > \sigma _ { 1 }$ as the sampling algorithm. $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ take the continuous perspective and consider a more general framework, where the forward process can be any reasonable SDE. Then a natural reverse $S D E$ evolves the final distribution $p _ { \sigma _ { N } ^ { 2 } }$ back to the data distribution; this process can be simulated with the estimated score. They consider methods based on two different SDE’s: score-matching Langevin diffusion (SMLD) based on adding Gaussian noise and denosing diffusion probabilistic models (DDPM) $[ \mathrm { S o h + } 1 5$ ; HJA20], based on the Ornstein-Uhlenbeck process. Note that a difference with MCMC-based methods is that these SDE’s are evolved for a fixed amount of time, rather than until convergence. However, they can be combined with MCMC-based methods such as Langevin diffusion in the predictor-corrector approach for improved convergence. [DVK21] include Hamiltonian dynamics: they augment the state space with a velocity variable and consider a critically-damped version of the Ornstein-Uhlenbeck process. Finally, we note the work of [De $+ 2 1 ]$ , who introduce the Diffusion Schrodinger Bridge method to learn a diffusion that more quickly ¨ transforms the prior into the data distribution.
+
+We will give a general analysis framework for SGM’s that applies to the algorithms in both [SE19] and $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ .
+
+# 1.2 Prior work and challenges for theory
+
+Although the literature on convergence for Langevin Monte Carlo [DM17; CB18; ${ \mathrm { C h e } } + 1 8$ ; Dal17; DK19; MMS20; EHZ21] and related sampling algorithms is extensive, prior works mainly consider the case of exact or stochastic gradients. In contrast, by the structure of the loss function (1), the score function learned in SGM is only accurate in $L ^ { 2 } ( p )$ . This poses a significant challenge for analysis, as the stationary distribution of Langevin diffusion with $L ^ { 2 } ( p )$ -accurate gradient can be arbitrarily far from $p$ (see Appendix D). Hence, any analysis must be utilizing the short/mediumterm convergence, while overcoming the potential issue of long-term behavior of convergence to an incorrect distribution.
+
+[BMR20] give the first theoretical analysis of SGM, and in particular, Langevin Monte Carlo with $L ^ { 2 } ( p )$ -accurate gradients. First, they show using uniform generalization bounds that optimizing the de-noising autoencoder (DAE) objective does in fact give a $L ^ { 2 } ( p )$ -accurate score function, with sample complexity depending on the complexity of the function class. They analyze convergence of LMC in Wasserstein distance. However, the error they obtain (Theorem 13) only decreases as $\varepsilon ^ { 1 / d }$ where $\varepsilon$ is the accuracy of the score estimate—so it suffers from the curse of dimensionality—and increases exponentially in the time that the process is run, the dimension, and the smoothness of the distribution, as in ODE/SDE discretization arguments that do not depend on contractivity.
+
+[De $+ 2 1 ]$ give an analysis for $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ in TV distance that requires a $L ^ { \infty }$ -accurate score function and depends exponentially on the amount of time the reverse SDE is run. Although exponential dependence is bad in general, it is mollified using their Diffusion Schrodinger Bridge (DSB) approach, ¨ as it allows running for a shorter, fixed amount of time, before the forward SDE converges to the prior distribution. However, this supposes that a good solution can be found for the DSB problem, and theoretical guarantees may be difficult to obtain.
+
+We overcome the challenges of analysis with a $L ^ { 2 } ( p )$ -accurate gradient, and give the first analysis with only polynomial dependence on running time, dimension, and smoothness of the distribution, with rates that are a fixed power of $\varepsilon$ . Our convergence result is in TV distance. We assume only smoothness conditions and a bounded log-Sobolev constant of the data distribution, a weaker condition than the dissipativity condition required by [BMR20]. We introduce a general framework for analysis of sampling algorithms given $L ^ { 2 }$ -accurate gradients (score function) based on constructing a “bad set” with small measure and showing convergence of the discretized process conditioned on not hitting the bad set. We use our framework to give an end-to-end analysis for both the algorithms in [SE19] and $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ , and illuminate the relative performance of different methods in practice.
+
+# 1.3 Notation and organization
+
+Through out the paper, $p ( x ) \propto e ^ { - V ( x ) }$ denotes the target distribution in $\mathbb { R } ^ { d }$ and $V : \mathbb { R } ^ { d }  \mathbb { R }$ is referred to as the potential. We abuse notation by identifying a measure with its density when context allows. We write $a \wedge b : = \operatorname* { m i n } \{ a , b \}$ and $a \vee b : = \operatorname* { m a x } \{ a , b \}$ . We use $a = O ( b )$ or $b = \Omega ( a )$ to indicate that $a \leq C b$ for a universal constant $C > 0$ . Also, we write $a = \Theta ( b )$ if there are universal constants $c ^ { \prime } > c > 0$ such that $c b \leq a \leq c b$ , and the notation ${ \tilde { O } } ( \cdot )$ means it hides polylog factors in the parameters. Definite integrals without limits are taken over $\mathbb { R } ^ { d }$ .
+
+In Section 2 we explain our main results for Langevin Monte Carlo with $L ^ { 2 } ( p )$ -accurate score estimate and use it to derive convergence bounds for the annealed LMC method of [SE19]. In Section 3, we give our main results for the predictor-corrector algorithms of $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ based on simulating reverse SDE’s. Our proofs are based on a common framework which we introduce in Section 4. Full proofs are in the appendix.
+
+# 2 Results for Langevin dynamics with estimated score
+
+Let $p ( x ) \propto e ^ { - V ( x ) }$ be a probability density on $\mathbb { R } ^ { d }$ such that $V$ is $C ^ { 1 }$ . Langevin diffusion with stationary distribution $p$ is the stochastic process defined by the SDE
+
+$$
+d x _ { t } = - \nabla V ( x _ { t } ) d t + \sqrt { 2 } d w _ { t } ,
+$$
+
+where $w _ { t }$ is a standard Brownian Motion in $\mathbb { R } ^ { d }$ . The rate of convergence to $p$ in $\chi ^ { 2 }$ and $\mathrm { K L }$ divergences are given by the Poincare and log-Sobolev constants of ´ $p$ , respectively; see Section E.1. To obtain the Langevin Monte Carlo (LMC) algorithm, we take the Euler-Murayama discretization of the SDE. We define LMC with score estimate $s ( x ) \approx - \nabla V ( x )$ and step size $h$ by
+
+$$
+x _ { ( k + 1 ) h } = x _ { k h } + h \cdot s ( x _ { k h } ) + \sqrt { 2 h } \cdot \xi _ { k h } , \mathrm { w h e r e } \xi _ { k h } \sim N ( 0 , I _ { d } ) .
+$$
+
+We make the following assumptions on the density $p$ and the score estimate $s$ , which we will use throughout this paper.
+
+Assumption 1. $p$ is a probability density on $\mathbb { R } ^ { d }$ such that the following hold.
+
+1. $\ln { p }$ is $C ^ { 1 }$ and $L$ -smooth, that is, $\nabla \ln p$ is $L$ -Lipschitz. We assume $L \geq 1$ .
+
+2. $p$ satisfies a log-Sobolev inequality with constant $C _ { \mathrm { L S } }$ . We assume $C _ { \mathrm { L S } } \geq 1$
+
+3. (Moments) $\| \mathbb { E } _ { p } x \| \leq M _ { 1 }$ and ${ \mathbb E } _ { p } \left\| x \right\| ^ { 2 } \le M _ { 2 }$ .
+
+We note that the uniform Lipschitzness assumption (1) helps ensure a unique strong solution to the Langevin diffusion, as in [BMR20]. One special case where one can prove Lipschitzness for all $t$ is when $p _ { 0 }$ is strongly log-concave $[ \mathrm { L e e } + 2 1$ , Lemma 28]. Although satisfying a log-Sobolev inequality (3) is a significant assumption, it is standard for analysis of Langevin Monte Carlo [VW19]. It is much weaker than assumptions in previous works [BMR20], including log-concave distributions and distributions satisfying strong dissipativity, and is stable under bounded perturbations. See Section E.1 for background on functional inequalities.
+
+Assumption 2. Let p be a given probability density on $\mathbb { R } ^ { d }$ such that $\ln { p }$ is $C ^ { 1 }$ . The score estimate $s : \mathbb { R } ^ { d } \overset { ^ { - } } { \to } \mathbb { R } ^ { d }$ satisfies the following.
+
+1. s is a $C ^ { 1 }$ function that is $L _ { s }$ -Lipschitz. We assume $L _ { s } \ge 1$ .
+
+2. The error in the score estimate is bounded in $L ^ { 2 }$
+
+$$
+\begin{array} { r } { \| \nabla \ln p - s \| _ { L ^ { 2 } ( p ) } ^ { 2 } = \mathbb { E } _ { p } [ \| \nabla \ln p ( x ) - s ( x ) \| ^ { 2 } ] \le \varepsilon ^ { 2 } . } \end{array}
+$$
+
+# 2.1 Langevin with $L ^ { 2 }$ -accurate score estimate
+
+Our first main result gives an error bound between the sampled distribution and $p$ , assuming $L ^ { 2 }$ - accurate score function estimate.
+
+Theorem 2.1 (LMC with $L ^ { 2 }$ -accurate score estimate). Let $p : \mathbb { R } ^ { d }  \mathbb { R }$ be a probability density satisfying Assumption $I ( I , 2 )$ with $L \geq 1$ and $s : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ be a score estimate satisfying Assumption 2(2). Consider the accuracy requirement in TV and $\chi ^ { 2 } \colon 0 < \varepsilon _ { \mathrm { T V } } < 1 , 0 < \varepsilon _ { \chi } < 1$ , and suppose furthermore the starting distribution satisfies $\chi ^ { 2 } ( p _ { 0 } | | p ) \leq K _ { \chi } ^ { 2 }$ . Then if
+
+$$
+\varepsilon = O \left( \frac { \varepsilon _ { \mathrm { T V } } \varepsilon _ { \chi } ^ { 3 } } { d L ^ { 2 } C _ { \mathrm { L S } } ^ { 5 / 2 } ( \ln ( 2 K _ { \chi } / \varepsilon _ { \chi } ^ { 2 } ) \vee K _ { \chi } ) } \right) ,
+$$
+
+then running (LMC-SE) with score estimate $s$ , step size $\begin{array} { r } { h \ \mathrm { ~ = ~ } \ \Theta \biggl ( \frac { \varepsilon _ { \chi } ^ { 2 } } { d L ^ { 2 } C _ { \mathrm { L S } } } \biggr ) } \end{array}$ and time $T =$ $\begin{array} { r } { \Theta \Big ( C _ { \mathrm { L S } } \ln \Big ( \frac { 2 K _ { \chi } } { \varepsilon _ { \chi } ^ { 2 } } \Big ) \Big ) } \end{array}$ results in a distribution $p _ { T }$ such that $p _ { T }$ is $\varepsilon _ { \mathrm { T V } }$ -far in TV distance from a distribution $\overline { { p } } _ { T }$ , where $\overline { { p } } _ { T }$ satisfies $\chi ^ { 2 } ( \overline { { p } } _ { T } | | p ) \leq \varepsilon _ { \chi } ^ { 2 }$ . In particular, taking $\varepsilon _ { \chi } = \varepsilon _ { \mathrm { T V } }$ , we have the error guarantee that $\mathrm { T V } ( p _ { T } , p ) \leq 2 \varepsilon _ { \mathrm { T V } }$ .
+
+Note that the error bound is only achieved when running LMC for a moderate time; this is consistent with the fact that the stationary distribution of LMC with a $L ^ { 2 }$ -score estimate can be arbitrarily far from $p$ . Note also that we need a warm start in $\chi ^ { 2 }$ -divergence: to obtain fixed errors $\varepsilon _ { \mathrm { T V } } , \varepsilon _ { \chi }$ , the required accuracy for the score estimate is inversely proportional to $K _ { \chi }$ . Intuitively, we must suffer from such a dependence because if the starting distribution is very far away, then there is no guarantee that $\| \nabla \ln \bar { p } ( x _ { t } ) - s ( x _ { t } ) \| ^ { 2 }$ is small on average during the sampling algorithm. Finally, although we can state a result purely in terms of TV distance, we need this more precise formulation to prove a result for annealed Langevin dynamics.
+
+# 2.2 Annealed Langevin dynamics with estimated score
+
+In light of the warm start requirement in Theorem 2.1, we typically cannot directly sample from $p _ { \mathrm { d a t a } }$ or its approximation. Hence, [SE19] proposed using annealed Langevin dynamics: consider a sequence of noise levels $\sigma _ { N } > \cdots > \sigma _ { 1 } \approx 0$ giving rise to a sequence of distributions pσ2N , . . . , pσ21 ≈ pdata, where pσ2 = p ∗ φσ2 , φσ2 being the density of N (0, σ2Id). For large enough $\sigma _ { N } , \varphi _ { \sigma _ { N } ^ { 2 } } \approx p _ { \sigma _ { N } ^ { 2 } }$ provides a warm start to $p _ { \sigma _ { N } ^ { 2 } }$ . We then successively run LMC using score estimates for $p _ { \sigma _ { k } ^ { 2 } }$ , with the approximate sample for $p _ { \sigma _ { k } ^ { 2 } }$ giving a warm start for $p _ { \sigma _ { k - 1 } ^ { 2 } }$ . We obtain the following algorithm and error estimate.
+
+INPUT: Noise levels $0 \leq \sigma _ { 1 } < . . . < \sigma _ { M }$ ; score function estimates $s ( \cdot , \sigma _ { m } )$ (estimates of $\nabla \ln ( p * \varphi _ { \sigma _ { m } ^ { 2 } } ) )$ , step sizes $h _ { m }$ , and number of steps $N _ { m }$ for $1 \leq m \leq M$ .   
+Draw $x ^ { ( M + 1 ) } \sim N ( 0 , \sigma _ { M } ^ { 2 } I _ { d } )$ .   
+for from to 1 do
+
+Starting from $x _ { 0 } ^ { ( m ) } = x ^ { ( m + 1 ) }$ , run (LMC-SE) with $s ( x , \sigma _ { m } )$ and step size $h _ { m }$ for $N _ { m }$ steps, and let the final sample be $x ^ { ( m ) }$ .
+
+# end for
+
+OUTPUT: Return $x ^ { ( 1 ) }$ , approximate sample from $p * \varphi _ { \sigma _ { 1 } ^ { 2 } }$
+
+Theorem 2.2 (Annealed LMC with $L ^ { 2 }$ -accurate score estimate). Let $p : \mathbb { R } ^ { d }  \mathbb { R }$ be a probability density satisfying Assumption 1 for $M _ { 1 } = O ( d )$ , and let $p _ { \sigma ^ { 2 } } : = p * \varphi _ { \sigma ^ { 2 } }$ . Suppose furthermore that $\nabla \ln p _ { \sigma ^ { 2 } }$ is $L$ -Lipschitz for every $\sigma \geq 0$ . Given $\sigma _ { \operatorname* { m i n } } > 0$ , there exists a sequence $\sigma _ { \mathrm { m i n } } = \sigma _ { 1 } <$ $\cdots < \sigma _ { M }$ with $\begin{array} { r } { M = O \left( \sqrt { d } \log \left( \frac { d C _ { \mathrm { L S } } } { \sigma _ { \mathrm { m i n } } ^ { 2 } } \right) \right) } \end{array}$ such that for each $m$ , $i f$
+
+$$
+\begin{array} { r l } & { \left\| \nabla \ln ( p _ { \sigma _ { m } ^ { 2 } } ) - s ( \cdot , \sigma _ { m } ^ { 2 } ) \right\| _ { L ^ { 2 } ( p _ { \sigma _ { m } ^ { 2 } } ) } ^ { 2 } = \mathbb { E } _ { p _ { \sigma _ { m } ^ { 2 } } } [ \left\| \nabla \ln p _ { \sigma _ { m } ^ { 2 } } ( x ) - s ( x , \sigma _ { m } ^ { 2 } ) \right\| ^ { 2 } ] \leq \varepsilon ^ { 2 } . } \\ & { \qquad w i t h \varepsilon : = \widetilde { \cal O } \left( \frac { \varepsilon _ { \mathrm { T V } } ^ { 4 . 5 } } { d ^ { 3 . 2 5 } L ^ { 2 } C _ { \mathrm { L S } } ^ { 2 . 5 } } \right) } \end{array}
+$$
+
+then $x ^ { ( 1 ) }$ is a sample from a distribution $q$ such that $\mathrm { T V } ( q , p _ { \sigma _ { 1 } ^ { 2 } } ) \leq \varepsilon _ { \mathrm { T V } }$
+
+Note that we assume a score estimate with error $\varepsilon$ at all noise scales; this corresponds to using an objective function that is a maximum of the score-matching objective over all noise levels, rather than an average over all noise levels as more commonly used in practice. However, these two losses are at most a factor of $M$ apart.
+
+The proof shows that the noise levels $\sigma _ { k }$ can be chosen as a geometric sequence, which matches the choice used in practice [SE20]. The additional dependence on $d$ and $\varepsilon _ { \mathrm { T V } }$ in Theorem 2.2 compared to Theorem 2.1 comes from requiring a sequence of $\widetilde { O } ( \sqrt { d } )$ noise levels and an additional factor in $\chi ^ { 2 }$ -divergence we suffer at the beginning of each level $m$ . In the next section, we will find that using a reverse SDE to evolve the samples between the noise levels—called a predictor step—will improve the rate and time complexity.
+
+# 3 Results for reverse SDE’s with estimated score
+
+To improve the empirical performance of score-based generative modeling, $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ consider a general framework where noise is injected into a data distribution $p _ { \mathrm { d a t a } }$ via a forward SDE,
+
+$$
+d \tilde { x } _ { t } = f ( \tilde { x } _ { t } , t ) d t + g ( t ) d w _ { t } , \ t \in [ 0 , T ] ,
+$$
+
+where $\widetilde x _ { 0 } \sim \widetilde p _ { 0 } : = p _ { \mathrm { d a t a } }$ . Let $\widetilde { p } _ { t }$ denote the distribution of $\widetilde { x } _ { t }$ $\widetilde { p } _ { t }$ is used instead of $p _ { t }$ to distine e e e eguish with the Gaussian-convolved distribution used in Annealed Langevin dynamics as in $\ S 2 . 2 )$ . Remarkably, $\widetilde { x } _ { t }$ also satisfies a reverse-time SDE,
+
+$$
+d \widetilde { \boldsymbol { x } } _ { t } = [ f ( \widetilde { \boldsymbol { x } } _ { t } , t ) - g ( t ) ^ { 2 } \nabla \ln \widetilde { p } _ { t } ( \widetilde { \boldsymbol { x } } _ { t } ) ] d t + g ( t ) d \widetilde { \boldsymbol { w } } _ { t } , t \in [ 0 , T ] ,
+$$
+
+where $\tilde { w } _ { t }$ is a backward Brownian Motion [And82]. By carefully choosing $f$ and $g$ , we can expect that $\tilde { p } _ { T }$ is approximately equal to some prior distribution $\tilde { q } _ { T }$ (e.g., a centered Gaussian) which we can accurately sample from. Then we hope that starting with some $\tilde { y } _ { T } \sim p _ { \mathrm { p r i o r } } = \tilde { q } _ { T } \approx \tilde { p } _ { T }$ and running the reverse-time process, we will get a good sample $\tilde { y } _ { 0 } \sim \tilde { q } _ { 0 } \approx p _ { \mathrm { d a t a } }$ .
+
+The case where $f \equiv 0$ and $g \equiv 1$ recovers the simple case of convolving with a Gaussian as used in $\ S 2 . 2$ ; note, however that the reverse-time SDE differs from Langevin diffusion in having a larger (and time-varying) drift relative to the diffusion. $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ highlight the following two special cases. We will focus on DDPM while noting that our analysis applies more generically.
+
+SMLD Score-matching Langevin diffusion: $f \equiv 0$ . In this case, $\widetilde { p } _ { t } \ = \ \widetilde { p } _ { 0 } * \ \varphi _ { \int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s }$ , so $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ call this a variance-exploding (VE) SDE. As is common for annealing-based algorithms, [SE19; $\mathbf { S o n } { + } 2 0 \mathbf { b } ]$ suggest choosing an exponential schedule, so that $g ( t ) = a b ^ { t }$ for constants $a , b$ . We take $\begin{array} { r } { p _ { \mathrm { p r i o r } } = N ( 0 , \int _ { 0 } ^ { T } g ( s ) ^ { 2 } d s \cdot I _ { d } ) } \end{array}$ .
+
+DDPM Denoising diffusion probabilistic modeling: $\begin{array} { r } { f ( x , t ) = - \frac { 1 } { 2 } g ( t ) ^ { 2 } x } \end{array}$ . This is an OrnsteinUhlenbeck process with time rescaling, $\begin{array} { r } { \widetilde { p } _ { t } = M _ { - \frac { 1 } { 2 } \int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s \sharp } \widetilde { p } _ { 0 } \ast \varphi _ { 1 - e ^ { - \int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s } } } \end{array}$ , where $M _ { \alpha } ( x ) = \alpha x$ . $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ call this a variance-preserving (VP) SDE, as the variance converges towards $I _ { d }$ . Because it displays exponential convergence towards $N ( 0 , I _ { d } )$ , it can be run for a smaller amount of normalized time $\textstyle \int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s$ . $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ suggest the choice $g ( t ) = \sqrt { b + \alpha t }$ . We take $p _ { \mathrm { p r i o r } } = N ( 0 , ( 1 - e ^ { - \int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s } ) I _ { d } ) \approx N ( 0 , I _ { d } ) .$
+
+To obtain an algorithm, we consider the following discretization and approximation of (4); note that in all cases of interest the integrals can be analytically evaluated. We reverse time so that $t$ corresponds to $T - t$ of the forward process. As we are free to rescale time in the SDE, we assume without loss of generality that the step sizes are constant. The predictor step is
+
+$$
+\begin{array} { r l r } {  { z _ { ( k + 1 ) h } = z _ { k h } - \int _ { k h } ^ { ( k + 1 ) h } [ f ( z _ { k h } , T - t ) - g ( T - t ) ^ { 2 } \cdot s ( z _ { k h } , T - k h ) ] d t } } \\ & { } & \\ & { } & { + \int _ { k h } ^ { ( k + 1 ) h } g ( T - t ) d w _ { t } , } \end{array}
+$$
+
+where $\begin{array} { r } { \int _ { k h } ^ { ( k + 1 ) h } g ( T - t ) d w _ { t } } \end{array}$ is distributed as $\begin{array} { r } { N ( 0 , \int _ { k h } ^ { ( k + 1 ) h } g ( T - t ) ^ { 2 } d t \cdot I _ { d } ) } \end{array}$ . Following $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ , we call these predictor steps as the samples aim to track the distributions $\widetilde { p } _ { T - k h }$ . Note that we flip the time. For simplicity of presentation, we consider the case $g \ \equiv \ 1$ . We note that although the choice of the schedule does matter in practice, what really matters in our theoretical analysis is the integral $\textstyle \int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s$ . This means that different choices of $g$ are related by only a rescaling of time, i.e., for different $g$ and $\tilde { g }$ , we can always choose total times $T$ and $\tilde { T }$ , such that $\begin{array} { r } { \int _ { 0 } ^ { T } g ( s ) ^ { 2 } d s = \int _ { 0 } ^ { \tilde { T } } \tilde { g } ( s ) ^ { 2 } d s } \end{array}$ . While it seems that choosing large $g ( t )$ could reduce the total time $T$ , in our analysis (e.g., Lemma C.15) we need the time step-size $h$ to be ${ \cal O } ( 1 / g ( T ) ^ { 2 } )$ and hence the total computational cost, which is roughly $O ( T / h )$ , does not change significantly.
+
+Theorem 3.1 (Predictor with $L ^ { 2 }$ -accurate score estimate, DDPM). Let $p _ { \mathrm { d a t a } } : \mathbb { R } ^ { d }  \mathbb { R }$ be a probability density satisfying Assumption $^ { l }$ with $M _ { 2 } = O ( d )$ , and let $\widetilde { p } _ { t }$ be the distribution resulting from evolving the forward $S D E$ according to DDPM with $g \equiv 1$ e. Suppose furthermore that $\nabla \ln \widetilde { p } _ { t }$ is $L$ -Lipschitz for every $t \geq 0$ , and that each $s ( \cdot , t )$ satisfies Assumption 2. Then if
+
+$$
+\varepsilon = O \left( \frac { \varepsilon _ { \mathrm { T V } } ^ { 4 } } { ( C _ { \mathrm { L S } } + d ) C _ { \mathrm { L S } } ^ { 5 / 2 } ( L \vee L _ { s } ) ^ { 2 } ( \ln ( C _ { \mathrm { L S } } d ) \vee C _ { \mathrm { L S } } \ln ( 1 / \varepsilon _ { \mathrm { T V } } ^ { 2 } ) ) } \right) ,
+$$
+
+running (P) starting from $p _ { \mathrm { p r i o r } }$ for time $\begin{array} { r } { T = \Theta \left( \ln ( C _ { \mathrm { L S } } d ) \vee C _ { \mathrm { L S } } \ln \left( \frac { 1 } { \varepsilon _ { \mathrm { T V } } } \right) \right) } \end{array}$ and step size $h \ =$ $\begin{array} { r } { \Theta \left( \frac { \varepsilon _ { \mathrm { T V } } ^ { 2 } } { C _ { \mathrm { L S } } ( C _ { \mathrm { L S } } + d ) ( L \vee L _ { s } ) ^ { 2 } } \right) } \end{array}$ results in a distribution $q _ { T }$ so that $\mathrm { T V } ( q _ { T } , p _ { \mathrm { d a t a } } ) \leq \varepsilon _ { \mathrm { T V } } .$ .
+
+A more precise statement of the Theorem can be found in the Appendix. Although we state our theorem for DDPM, we describe in Appendix C how it can be adapted to other SDE’s like SMLD and the sub-VP SDE; the primary SDE-dependent bound we need is a bound on $\begin{array} { r } { \nabla \ln \frac { \widetilde { p } _ { t } } { \widetilde { p } _ { t + h } } } \end{array}$ . Because the predictor is tracking a changing distribution $p _ { t }$ e, we incur more error terms and worse dependence on parameters $( C _ { \mathrm { L S } } , L )$ than in LMC (Theorem 2.1). Motivated by this, we intersperse the predictor steps with LMC steps—called corrector steps in this context—to give additional time for the process to mix, resulting in improved dependence on parameters.
+
+Theorem 3.2 (Predictor-corrector with $L ^ { 2 }$ -accurate score estimate). Keep the setup of Theorem 3.1.
+
+$$
+\varepsilon = { \cal O } \left( \frac { \varepsilon _ { \mathrm { T V } } ^ { 4 } } { d L ^ { 2 } C _ { \mathrm { L S } } ^ { 5 / 2 } \ln ( 1 / \varepsilon _ { \chi } ^ { 2 } ) } \right) ,
+$$
+
+then Algorithm 2 with appropriate choices of $\begin{array} { r } { T = \Theta \left( \ln ( C _ { \mathrm { L S } } d ) \lor C _ { \mathrm { L S } } \log \left( \frac { 1 } { \varepsilon _ { \mathrm { T V } } } \right) \right) } \end{array}$ , $N _ { m }$ , corrector step sizes $h _ { m }$ and predictor step size $h$ , produces $a$ sample from a distribution $q _ { T }$ such that $\mathrm { T V } ( q _ { T } , p _ { \mathrm { d a t a } } ) < \varepsilon _ { \mathrm { T V } }$ .
+
+<table><tr><td>INPUT: Time T, predictor step size h; number of corrector steps Nm per predictor step, corrector step sizeshm</td></tr><tr><td>Draw zo ~ Pprior from the prior distribution.</td></tr><tr><td></td></tr><tr><td>for m from1to T/h do (Predictor) Take a step of (P) to obtain Zmh from z(m-1)h, with f,g as in SMLD or DDPM.</td></tr><tr><td>(Corrector) Starting from 2mh,0 := 2mh, run (LMC-SE) with s(z,T - mh) and step size hm</td></tr><tr><td>for N steps,and let zmh ← zmh,N.</td></tr><tr><td>end for OUTPUT: Return zT,approximate sample from pdata·</td></tr></table>
+
+The assumption on $\varepsilon _ { \mathrm { T V } }$ is for convenience in stating our bound. In comparison to using the predictor step alone (Theorem 3.1), note that in the bound on $\varepsilon$ , we obtain the improved rate of the corrector step as in Theorem 2.1; this is because the predictor step only needs to track the actual distribution in $\chi ^ { 2 }$ -divergence with error $O ( 1 )$ , and the final corrector steps are responsible for decreasing the error to $\varepsilon _ { \mathrm { T V } }$ . In comparison to the Annealed Langevin sampler (Algorithm 1, Theorem 2.2), which can be viewed as using the corrector step alone, adding a predictor step provides a better warm start for the distribution at the next smaller noise level, resulting in better dependence on parameters. Thus the predictor-corrector algorithm combines the strengths of the predictor and corrector steps. For realworld data, it can be challenging to estimate TV-distance between distributions given only samples, and hence difficult to check consistency with empirical observations. However, our claim that using a corrector can improve the convergence rate of DDPM/SMLD is consistent with the simulation results in Section 4.2 of $[ \mathrm { S o n } { + } 2 0 \mathrm { b } ]$ .
+
+# 4 Theoretical framework and proof sketches
+
+The main idea of our analysis framework is to convert a $L ^ { 2 }$ error guarantee to a $L ^ { \infty }$ error guarantee by excluding a bad set, formalized in the following theorem.
+
+Theorem 4.1. Let $( \Omega , \mathcal { F } , \mathbb { P } )$ be a probability space and $\{ \mathcal { F } _ { n } \}$ be a filtration of the sigma field $\mathcal { F }$ Suppose $X _ { n } \sim p _ { n }$ , $Z _ { n } \sim q _ { n } ,$ , and $\overline { { Z } } _ { n } \sim \overline { { q } } _ { n }$ are ${ \mathcal { F } } _ { n }$ -adapted random processes taking values in $\Omega$ and $B _ { n } \subseteq \Omega$ are sets such that the following hold for every $n \in  { \mathbb { N } } _ { 0 }$ .
+
+2. $\chi ^ { 2 } ( \overline { { q } } _ { n } | | p _ { n } ) \leq D _ { n } ^ { 2 }$ .
+
+3. $\mathbb { P } ( X _ { n } \in B _ { n } ) \le \delta _ { n }$
+
+Then the following hold.
+
+$$
+\mathrm { T V } ( q _ { n } , \overline { { q } } _ { n } ) \leq \sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \delta _ { k } ^ { 1 / 2 } \qquad \mathrm { T V } ( p _ { n } , q _ { n } ) \leq D _ { n } + \sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \delta _ { k } ^ { 1 / 2 }
+$$
+
+For our setting, we will take the “bad sets” $B _ { n }$ to be the set of $x$ where $\| s _ { \theta } ( x ) - \nabla \ln p \|$ is large, $q _ { n }$ to be the discretized process with estimated score, and $\overline { { q } } _ { n }$ to be the discretized process with estimated score except in $B _ { n }$ where the error is large. Because $\overline { { q } } _ { n }$ uses an $L ^ { \infty }$ -accurate score estimate, we can use existing techniques for analyzing Langevin Monte Carlo [VW19; EHZ21; Che+21] to bound $\chi ^ { 2 } ( \overline { { q } } _ { n } | | p _ { n } ) \ :$ .
+
+Proof. We bound using condition 1 and Cauchy-Schwarz:
+
+$$
+\begin{array} { r l } & { \displaystyle ^ { \mathfrak { n } } \big ( Z _ { n } \ne \overline { { Z } } _ { n } \big ) \le \mathbb { P } \left( \bigcup _ { k = 1 } ^ { n - 1 } \big \{ Z _ { k } \in B _ { k } \big \} \right) \le \sum _ { k = 0 } ^ { n - 1 } \mathbb { P } \left( Z _ { k } \in B _ { k } \right) = \sum _ { k = 0 } ^ { n - 1 } \mathbb { E } _ { q _ { k } } \mathbb { 1 } _ { B _ { k } } } \\ & { \qquad \le \displaystyle \sum _ { k = 0 } ^ { n - 1 } \left( \mathbb { E } _ { p _ { k } } \left( \frac { q _ { k } } { p _ { k } } \right) ^ { 2 } \right) ^ { 1 / 2 } ( \mathbb { E } _ { p _ { k } } \mathbb { 1 } _ { B _ { k } } ) ^ { 1 / 2 } = \sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \delta _ { k } ^ { 1 } } \end{array}
+$$
+
+The second inequality then follows from the triangle inequality and Cauchy-Schwarz:
+
+$$
+\begin{array} { r l r } {  { \mathrm { T V } ( p _ { n } , q _ { n } ) \leq \mathrm { T V } ( p _ { n } , \overline { { q } } _ { n } ) + \mathrm { T V } ( \overline { { q } } _ { n } , q _ { n } ) } } \\ & { } & \\ & { } & { \leq \sqrt { \chi ^ { 2 } ( \overline { { q } } _ { n } | | p _ { n } ) } + \mathrm { T V } ( \overline { { q } } _ { n } , q _ { n } ) \leq D _ { n } + \displaystyle \sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \delta _ { k } ^ { 1 / 2 } . } \end{array}
+$$
+
+It now remains to give $\chi ^ { 2 }$ convergence bounds under $L ^ { \infty }$ -accurate score estimate. The following theorem may be of independent interest.
+
+Theorem 4.2 (LMC under $L ^ { \infty }$ bound on gradient error). Let $p : \mathbb { R } ^ { d }  \mathbb { R }$ be a probability density satisfying Assumption $I ( I , 2 )$ and $s : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ be a score estimate $s$ with error bounded in $L ^ { \infty }$ : for some ε1 ≤ q $\begin{array} { r } { \varepsilon _ { 1 } \leq \sqrt { \frac { 1 } { 4 8 C _ { \mathrm { L S } } } } } \end{array}$ ,
+
+$$
+\| \nabla \ln p - s \| _ { \infty } = \operatorname* { m a x } _ { x \in \mathbb { R } ^ { d } } \| \nabla \ln p ( x ) - s ( x ) \| \| \leq \varepsilon _ { 1 } .
+$$
+
+Let $N \in  { \mathbb { N } } _ { 0 }$ and $\begin{array} { r } { 0 < h \leq \frac { 1 } { 4 3 9 2 d C _ { \mathrm { L S } } L ^ { 2 } } } \end{array}$ , and assume $L \geq 1$ . Let $q _ { n h }$ denote the nth iterate of LMC with step size $h$ score estimate s. Then
+
+$$
+\chi ^ { 2 } ( q _ { ( k + 1 ) h } | | p ) \leq \exp \left( - \frac { h } { 4 C _ { \mathrm { L S } } } \right) \chi ^ { 2 } ( q _ { k h } | | p ) + 1 7 0 d L ^ { 2 } h ^ { 2 } + 5 \varepsilon _ { 1 } ^ { 2 } h
+$$
+
+and
+
+$$
+{ \chi ^ { 2 } } ( q _ { N h } | | p ) \leq \exp \left( - \frac { N h } { 4 C _ { \mathrm { L S } } } \right) \chi ^ { 2 } ( q _ { 0 } | | p ) + 6 8 0 d L ^ { 2 } h C _ { \mathrm { L S } } + 2 0 \varepsilon _ { 1 } ^ { 2 } C _ { \mathrm { L S } } \leq \exp \left( - \frac { N h } { 4 C _ { \mathrm { L S } } } \right) \chi ^ { 2 } ( q _ { 0 } | | p ) + 1
+$$
+
+Following $[ \mathrm { C h e } + 2 1 ]$ , we prove this by first defining a continuous-time interpolation $q _ { t }$ of the discrete process, and then deriving a differential inequality for $\chi ^ { 2 } ( q _ { t } | | p )$ using the log-Sobolev inequality for $p$ . Compared to $[ \mathrm { C h e } + 2 1 ]$ , we incur an extra error term arising from the inaccurate gradient.
+
+This allows us to sketch the proof of Theorem 2.1; a complete proof is in Section B.
+
+Proof sketch of Theorem 2.1. We first define the bad set where the error in the score estimate is large,
+
+$$
+B : = \{ \| \nabla \ln p ( x ) - s ( x ) \| > \varepsilon _ { 1 } \}
+$$
+
+for some $\varepsilon _ { 1 }$ to be chosen. Then by Chebyshev’s inequality, $\begin{array} { r } { P ( B ) \leq \left( \frac { \varepsilon } { \varepsilon _ { 1 } } \right) ^ { 2 } = : \delta } \end{array}$ . Let $\overline { { q } } _ { n h }$ be the discretized process, but where the score estimate is set to be equal to $\nabla \ln p$ on $B$ ; note it agrees with $q _ { n h }$ as long as it has not hit $B$ . Because $\overline { { q } } _ { n h }$ uses a score estimate that has $L ^ { \infty }$ -error $\varepsilon _ { 1 }$ , Theorem 4.2 gives a bound for $\chi ^ { 2 } ( \overline { { q } } _ { N h } | | p )$ . Then Theorem 4.1 gives
+
+$$
+ { \operatorname { F V } } ( q _ { n h } , \overline { { q } } _ { n h } ) \le \sum _ { k = 0 } ^ { n - 1 } ( \chi ^ { 2 } ( \overline { { q } } _ { k h } \| p ) + 1 ) ^ { 1 / 2 } P ( B ) ^ { 1 / 2 } \le \sum _ { k = 0 } ^ { n - 1 } \left( \exp \left( - \frac { k h } { 8 C _ { \mathrm { L S } } } \right) \chi ^ { 2 } ( q _ { 0 } \| p ) ^ { 1 / 2 } + 1 \right) \delta ^ { 1 / 2 }
+$$
+
+The theorem then follows from choosing parameters so that χ2(qT ||p) ≤ ε2χ and TV(qT , qT ) ≤ $\varepsilon _ { \mathrm { T V } }$ .
+
+We remark that the main inefficiency in the proof comes from the use of Chebyshev’s inequality, and a $L ^ { p }$ bound on the error for $p > 2$ will improve the bound.
+
+Proof sketch of Theorem 2.2. Choosing the sequence $\sigma _ { 1 } < \cdots < \sigma _ { M }$ to be geometric with ratio $1 + \frac { 1 } { \sqrt { d } }$ ensures that the $\chi ^ { 2 }$ -divergence between successive distributions $p _ { \sigma _ { m } ^ { 2 } }$ is $O ( 1 )$ . Then, choosing $\sigma _ { M } ^ { 2 } = \Omega ( C _ { \mathrm { L S } } d )$ ensures we have a warm start for the highest noise level: $\chi ^ { 2 } ( \operatorname { \rho _ { p r i o r } } \lvert \lvert p _ { \sigma _ { M } ^ { 2 } } ) = O ( 1 )$ . This uses $\begin{array} { r } { O \left( \sqrt { d } \log \left( \frac { d C _ { \mathrm { L S } } } { \sigma _ { \mathrm { m i n } } ^ { 2 } } \right) \right) } \end{array}$ noise levels. Chebyshev’s inequality can be used to show that the distribution of the final sample $x ^ { ( m ) }$ for $p _ { \sigma _ { m } ^ { 2 } }$ is ${ \cal O } ( \varepsilon _ { \mathrm { T V } } / M )$ close to a distribution that is ${ \cal O } ( M / \varepsilon _ { \mathrm { T V } } )$ in in $\chi ^ { 2 }$ -divergence from Theorem 2.1 then $p _ { \sigma _ { m + 1 } ^ { 2 } }$ . This gives the warm st the required bound for rt parameter . Note that $K _ { \chi } = ( M / \varepsilon _ { \mathrm { T V } } ) ^ { 1 / 2 }$ ; substitutinged from each $\varepsilon$ level add to ${ \cal O } ( \varepsilon _ { \mathrm { T V } } )$ . □
+
+To analyze the predictor-based algorithms, we also first prove convergence bounds under $L ^ { \infty }$ - accurate score estimate.
+
+Theorem 4.3 (Predictor steps under $L ^ { \infty }$ bound on score estimate, DDPM). Let $p : \mathbb { R } ^ { d }  \mathbb { R }$ be a probability density satisfying Assumption $I$ and $s ( \cdot , t ) : \mathbb { R } ^ { d } \to \mathbb { R } ^ { d }$ be a score estimate s with error bounded in $L ^ { \infty }$ for each $t \in [ 0 , T ]$ :
+
+$$
+\| \nabla \ln p - s ( \cdot , t ) \| _ { \infty } = \operatorname* { m a x } _ { x \in \mathbb { R } ^ { d } } \| \nabla \ln \widetilde { p } _ { t } ( x ) - s ( x , t ) \| \| \le \varepsilon _ { 1 } .
+$$
+
+Consider DDPM with $g \equiv 1$ , $T \geq 1 \vee \ln ( C _ { \mathrm { L S } } d ) ,$ , and $\begin{array} { r } { h = O \left( \frac { 1 } { C _ { \mathrm { L S } } ( d + C _ { \mathrm { L S } } ) ( L \vee L _ { s } ) ^ { 2 } } \right) } \end{array}$ . (Recall that $p _ { k h }$ and $q _ { k h }$ are the $k$ -th iterate of LMC with step size $h$ and true/estimated score respectively.) Then
+
+$$
+\chi ^ { 2 } ( q _ { ( k + 1 ) h } | | p _ { ( k + 1 ) h } ) \leq \chi ^ { 2 } ( q _ { k h } | | p _ { k h } ) e ^ { \left( - \frac { 1 } { 8 C _ { 1 S } } + 8 \varepsilon _ { 1 } ^ { 2 } \right) h } + O ( \varepsilon _ { 1 } ^ { 2 } h + ( L _ { s } ^ { 2 } + L ^ { 2 } d ) h ^ { 2 } )
+$$
+
+$\begin{array} { r } { \varepsilon _ { 1 } < \frac { 1 } { 1 2 8 C _ { \mathrm { L S } } } } \end{array}$
+
+$$
+\chi ^ { 2 } ( q _ { N h } | | p _ { N h } ) \leq e ^ { - \frac { N h } { 1 6 C _ { 1 S } } } \chi ^ { 2 } ( q _ { 0 } | | p _ { 0 } ) + O \left( C _ { \mathrm { L S } } \left( \varepsilon _ { 1 } ^ { 2 } + ( L _ { s } ^ { 2 } + L ^ { 2 } d ) h \right) \right) .
+$$
+
+Moreover, for $q _ { 0 } = p _ { \mathrm { p r i o r } } , \chi ^ { 2 } ( q _ { 0 } | | p _ { 0 } ) \leq e ^ { - T / 2 } C _ { \mathrm { L S } } d .$
+
+We give a more precise statement in Section C. Note that unlike the case for LMC as in Theorem 4.2, the base density $p _ { t }$ is also evolving in time, which produces additional error terms and necessitates a more involved analysis. The additional error terms can be bounded using the Donsker-Varadhan variational principle, concentration for distributions satisfying LSI, and error bounds between $p _ { t }$ and $p _ { t + h }$ for small $h$ .
+
+Here, we only state the result about DDPM, which has better bounds than SMLD (when $g \equiv 1$ ) because both the forward and backwards processes exhibit better mixing properties: the warm start improves exponentially rather than inversely with $T$ , and the log-Sobolev constant is uniformly bounded by that of $p _ { \mathrm { d a t a } }$ rather than increasing. However, the analysis in Section C can be directly applied to SMLD and other models as well. We also note there is a sense in which DDPM and SMLD are equivalent under a rescaling in time and space (see discussion in Section C.2).
+
+Note that the choice of $h$ is necessary for exponential decay of error; as if $h$ is not small enough, we would get an exponential growing instead of decaying factor in the one-step error (See Section C for details). Such an $h$ may however still be a suitable choice when used in conjunction with a corrector step. Moreover, as $\varepsilon _ { 1 } \to 0$ , with appropriate choice of $T$ and $h$ , $q _ { N h }$ and $p _ { N h }$ can be made arbitrarily close.
+
+Theorem 3.1 now follows from the $L ^ { \infty }$ result (Theorem 4.3) in the same way that Theorem 2.1 follows from Theorem 4.2.
+
+To prove Theorem 3.2, it suffices to run the corrector steps only at the lowest noise level, that is, set $N _ { m } = 0$ for $1 \leq m < T / h$ , although we note that interleaving the predictor and corrector steps does empirically help with mixing. The proof follows from using the predictor and the corrector theorems in series: first apply Theorem 3.1 with $\varepsilon _ { \chi } = O ( 1 )$ to show that the predictor results a warm start $p _ { \mathrm { d a t a } }$ , then use Theorem 2.1 to show the corrector reduces the error to the desired $\varepsilon _ { \mathrm { T V } }$ .
+
+# 5 Conclusion
+
+We introduced a general framework to analyze SDE-based sampling algorithms given a $L ^ { 2 }$ -error score estimate, and used it to obtain the first convergence bounds for several score-based generative models with polynomial complexity in all parameters. Our analysis can potentially be adapted to other SDE’s and sampling algorithms beyond Langevin Monte Carlo. There is also room for improving our analysis to better use smoothing properties of the SDE’s and compare different choices of the diffusion speed $g$ .
+
+We present several interesting further directions to explore. In addition to extending the analysis to other SGM’s and comparing their theoretical performance (relative to each other as well as other approaches to generative modeling), we propose the following.
+
+Analysis for multimodal distributions. Our assumption of a bounded log-Sobolev constant essentially limits the analysis to distributions that are close to unimodal. However, SGM’s are empirically successful at modeling multimodal distributions [SE19], and in fact perform better with multimodal distributions than other approaches such as GAN’s. Can we analyze the convergence for simple multimodal distributions, such as a mixture of distributions each with bounded log-Sobolev constant? Positive results on sampling from multimodal distributions such as [GLR18] suggest this is possible, as the sequence of noised distributions is natural for annealing and tempering methods (see [GLR18, Remark 7.2]).
+
+Weakening conditions on the score estimate. The assumption that we have a score estimate that is $O ( 1 )$ -accurate in $L ^ { 2 }$ , although weaker than the usual assumptions for theoretical analysis, is in fact still a strong condition in practice that seems unlikely to be satisfied (and difficult to check) when learning complex distributions such as distributions of images. What would a reasonable weaker condition be, and in what sense can we still obtain reasonable samples?
+
+Guarantees for learning the score function. Our analysis assumes a $L ^ { 2 }$ -estimate of the score function is given, but the question remains of when we can find such an estimate. What natural conditions on distributions allow their score functions to be learned by a neural network? Various works have considered the representability of data distributions by diffusion-like processes [TR19], but the questions of optimization and generalization appear more challenging.
+
+# Acknowledgements
+
+We thank Andrej Risteski for helpful conversations. This work was done in part while $\mathrm { H L }$ was visiting the Simons Institute for the Theory of Computing. The work was supported in part by National Science Foundation via awards DMS-2012286 and CCF-1934964 (Duke Tripods).
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [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] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [N/A]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(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)? [N/A]
+
+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? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(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/dUSI4vFyMK/dUSI4vFyMK_content_list.json

@@ -0,0 +1,1615 @@
+[
+    {
+        "type": "text",
+        "text": "Convergence for score-based generative modeling with polynomial complexity ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Holden Lee Department of Applied Mathematics and Statistics Johns Hopkins University hlee283@jhu.edu ",
+        "bbox": [
+            214,
+            226,
+            545,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jianfeng Lu   \nDepartment of Mathematics Duke University   \njianfeng@math.duke.edu ",
+        "bbox": [
+            593,
+            227,
+            784,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yixin Tan Department of Mathematics Duke University yixin.tan@duke.edu ",
+        "bbox": [
+            405,
+            303,
+            591,
+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            395,
+            535,
+            411
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Score-based generative modeling (SGM) is a highly successful approach for learning a probability distribution from data and generating further samples. We prove the first polynomial convergence guarantees for the core mechanic behind SGM: drawing samples from a probability density $p$ given a score estimate (an estimate of $\\nabla \\ln p$ ) that is accurate in $L ^ { 2 } ( p )$ . Compared to previous works, we do not incur error that grows exponentially in time or that suffers from a curse of dimensionality. Our guarantee works for any smooth distribution and depends polynomially on its log-Sobolev constant. Using our guarantee, we give a theoretical analysis of score-based generative modeling, which transforms white-noise input into samples from a learned data distribution given score estimates at different noise scales. Our analysis gives theoretical grounding to the observation that an annealed procedure is required in practice to generate good samples, as our proof depends essentially on using annealing to obtain a warm start at each step. Moreover, we show that a predictor-corrector algorithm gives better convergence than using either portion alone. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            685,
+            310,
+            703
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A key task in machine learning is to learn a probability distribution from data, in a way that allows efficient generation of additional samples from the learned distribution. Score-based generative modeling (SGM) is one empirically successful approach that implicitly learns the probability distribution by learning how to transform white noise into the data distribution, and gives stateof-the-art performance for generating images and audio [SE19; Dat+19; Gra+19; SE20; $\\mathrm { S o n } { + } 2 0 \\mathrm { b }$ ; $\\mathrm { M e n } { + } 2 1$ ; $\\mathbf { S o n } { + } 2 1 \\mathbf { b }$ ; Son $+ 2 1 \\mathrm { a }$ ; $\\mathrm { J i n } { + } 2 2 ^ { \\cdot }$ ]. It also yields a conditional generation process for inverse problems [DN21]. The basic idea behind score-based generative modeling is to first estimate the score function from data $[ \\mathrm { S o n } { + } 2 0 \\mathrm { a } ]$ and then to sample the distribution based on the learned score function. Other approaches for generative modeling include generative adversarial networks (GANs) $[ \\mathrm { G o o { + } 1 4 }$ ; ACB17], normalizing flows [DSB16], variational autoencoders [KW19], and energybased models [ZML16]. While score-based generative modeling has achieved great success, its theoretical analysis is still lacking and is the focus of our work. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1.1 Background ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            92,
+            299,
+            106
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "General framework. The score function of a distribution $P$ with density $p$ is defined as the gradient of the log-pdf, $\\nabla \\ln p$ . Its significance arises from the fact that knowing the score function allows running a variety of sampling algorithms, based on discretizations of stochastic differential equations (SDE’s), to sample from $p$ . SGM consists of two steps: first, learning an estimate of the score function for a sequence of “noisy” versions of the data distribution $P _ { \\mathrm { d a t a } }$ , and second, using the score function in lieu of the gradient of the log-pdf in the chosen sampling algorithm. We now describe each of these steps more precisely. ",
+        "bbox": [
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+            117,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "First, a method of adding noise to the data distribution is fixed; this takes the form of evolving a (forward) stochastic differential equation (SDE) starting from the data distribution. We fix a sequence of noise levels $\\sigma _ { 1 } < \\cdots < \\sigma _ { N }$ . For $\\sigma \\in \\{ \\sigma _ { 1 } , \\dots , \\sigma _ { N } \\}$ , let the resulting distributions be $P _ { \\sigma ^ { 2 } }$ and the distributions conditional on the starting data point be $P _ { \\sigma ^ { 2 } } ( \\cdot | x )$ . Typically, $\\sigma _ { 1 }$ is chosen so that $P _ { \\sigma _ { 1 } ^ { 2 } } \\approx P _ { \\mathrm { d a t a } }$ and $P _ { \\sigma _ { N } ^ { 2 } }$ is close to some “prior” distribution that is easy to sample from, such as $N ( 0 , \\sigma _ { N } ^ { 2 } I _ { d } )$ . While the score $\\nabla \\ln p _ { \\sigma ^ { 2 } }$ cannot be estimated directly, it turns out that a de-noising objective that is equivalent to the score-matching objective can be calculated [SE19]. This de-noising objective can be estimated from samples $( X , { \\tilde { X } } )$ where $\\widetilde { X } \\sim P _ { \\sigma ^ { 2 } } ( \\cdot | x )$ . The objective is represented and optimized within an expressive function class, typically neural networks, to obtain a $L ^ { 2 }$ -estimate of the score, that is, $s _ { \\theta } ( x , \\overbar { \\sigma } ^ { 2 } )$ such that ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e0d0b091faae95c0fa7889361326e3b709d2de095522c32799c0dfb8dd064d9.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { \\mathbb { E } _ { { x } \\sim { P } _ { \\sigma ^ { 2 } } } [ \\| s _ { \\theta } ( x , \\sigma ^ { 2 } ) - \\nabla \\ln { p } _ { \\sigma ^ { 2 } } ( x ) \\| ^ { 2 } ] } \\quad } & { { } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            373,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "is small. ",
+        "bbox": [
+            174,
+            401,
+            230,
+            415
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The reason we estimate the score function $\\nabla \\ln p _ { \\sigma ^ { 2 } }$ is that there are a variety of sampling algorithms—based on simulating SDE’s—that can sample from $p$ given access to $\\nabla \\ln p$ , including Langevin Monte Carlo and Hamiltonian Monte Carlo. The second step is then to use the estimated score function $s _ { \\theta } ( x , t )$ in lieu of the exact gradient in the sampling algorithm to successively obtain samples from $p _ { \\sigma _ { N } ^ { 2 } } , \\ldots , p _ { \\sigma _ { 1 } ^ { 2 } }$ . This sequence interpolates smoothly between the prior distribution (e.g., $N ( 0 , \\sigma _ { N } ^ { 2 } I _ { d } ) )$ and the data distribution $P _ { \\mathrm { d a t a } }$ ; such an “annealing” or “homotopy” method is required in practice to generate good samples $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ . ",
+        "bbox": [
+            173,
+            421,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Examples of SGM’s. There have been several instantiations of this general approach. [SE19] add gaussian noise to the data and then use Langevin diffusion at a discrete set of noise levels $\\sigma _ { N } > \\cdots > \\sigma _ { 1 }$ as the sampling algorithm. $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ take the continuous perspective and consider a more general framework, where the forward process can be any reasonable SDE. Then a natural reverse $S D E$ evolves the final distribution $p _ { \\sigma _ { N } ^ { 2 } }$ back to the data distribution; this process can be simulated with the estimated score. They consider methods based on two different SDE’s: score-matching Langevin diffusion (SMLD) based on adding Gaussian noise and denosing diffusion probabilistic models (DDPM) $[ \\mathrm { S o h + } 1 5$ ; HJA20], based on the Ornstein-Uhlenbeck process. Note that a difference with MCMC-based methods is that these SDE’s are evolved for a fixed amount of time, rather than until convergence. However, they can be combined with MCMC-based methods such as Langevin diffusion in the predictor-corrector approach for improved convergence. [DVK21] include Hamiltonian dynamics: they augment the state space with a velocity variable and consider a critically-damped version of the Ornstein-Uhlenbeck process. Finally, we note the work of [De $+ 2 1 ]$ , who introduce the Diffusion Schrodinger Bridge method to learn a diffusion that more quickly ¨ transforms the prior into the data distribution. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We will give a general analysis framework for SGM’s that applies to the algorithms in both [SE19] and $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.2 Prior work and challenges for theory ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            801,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although the literature on convergence for Langevin Monte Carlo [DM17; CB18; ${ \\mathrm { C h e } } + 1 8$ ; Dal17; DK19; MMS20; EHZ21] and related sampling algorithms is extensive, prior works mainly consider the case of exact or stochastic gradients. In contrast, by the structure of the loss function (1), the score function learned in SGM is only accurate in $L ^ { 2 } ( p )$ . This poses a significant challenge for analysis, as the stationary distribution of Langevin diffusion with $L ^ { 2 } ( p )$ -accurate gradient can be arbitrarily far from $p$ (see Appendix D). Hence, any analysis must be utilizing the short/mediumterm convergence, while overcoming the potential issue of long-term behavior of convergence to an incorrect distribution. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "[BMR20] give the first theoretical analysis of SGM, and in particular, Langevin Monte Carlo with $L ^ { 2 } ( p )$ -accurate gradients. First, they show using uniform generalization bounds that optimizing the de-noising autoencoder (DAE) objective does in fact give a $L ^ { 2 } ( p )$ -accurate score function, with sample complexity depending on the complexity of the function class. They analyze convergence of LMC in Wasserstein distance. However, the error they obtain (Theorem 13) only decreases as $\\varepsilon ^ { 1 / d }$ where $\\varepsilon$ is the accuracy of the score estimate—so it suffers from the curse of dimensionality—and increases exponentially in the time that the process is run, the dimension, and the smoothness of the distribution, as in ODE/SDE discretization arguments that do not depend on contractivity. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "[De $+ 2 1 ]$ give an analysis for $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ in TV distance that requires a $L ^ { \\infty }$ -accurate score function and depends exponentially on the amount of time the reverse SDE is run. Although exponential dependence is bad in general, it is mollified using their Diffusion Schrodinger Bridge (DSB) approach, ¨ as it allows running for a shorter, fixed amount of time, before the forward SDE converges to the prior distribution. However, this supposes that a good solution can be found for the DSB problem, and theoretical guarantees may be difficult to obtain. ",
+        "bbox": [
+            173,
+            243,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We overcome the challenges of analysis with a $L ^ { 2 } ( p )$ -accurate gradient, and give the first analysis with only polynomial dependence on running time, dimension, and smoothness of the distribution, with rates that are a fixed power of $\\varepsilon$ . Our convergence result is in TV distance. We assume only smoothness conditions and a bounded log-Sobolev constant of the data distribution, a weaker condition than the dissipativity condition required by [BMR20]. We introduce a general framework for analysis of sampling algorithms given $L ^ { 2 }$ -accurate gradients (score function) based on constructing a “bad set” with small measure and showing convergence of the discretized process conditioned on not hitting the bad set. We use our framework to give an end-to-end analysis for both the algorithms in [SE19] and $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ , and illuminate the relative performance of different methods in practice. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1.3 Notation and organization ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            474,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Through out the paper, $p ( x ) \\propto e ^ { - V ( x ) }$ denotes the target distribution in $\\mathbb { R } ^ { d }$ and $V : \\mathbb { R } ^ { d }  \\mathbb { R }$ is referred to as the potential. We abuse notation by identifying a measure with its density when context allows. We write $a \\wedge b : = \\operatorname* { m i n } \\{ a , b \\}$ and $a \\vee b : = \\operatorname* { m a x } \\{ a , b \\}$ . We use $a = O ( b )$ or $b = \\Omega ( a )$ to indicate that $a \\leq C b$ for a universal constant $C > 0$ . Also, we write $a = \\Theta ( b )$ if there are universal constants $c ^ { \\prime } > c > 0$ such that $c b \\leq a \\leq c b$ , and the notation ${ \\tilde { O } } ( \\cdot )$ means it hides polylog factors in the parameters. Definite integrals without limits are taken over $\\mathbb { R } ^ { d }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In Section 2 we explain our main results for Langevin Monte Carlo with $L ^ { 2 } ( p )$ -accurate score estimate and use it to derive convergence bounds for the annealed LMC method of [SE19]. In Section 3, we give our main results for the predictor-corrector algorithms of $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ based on simulating reverse SDE’s. Our proofs are based on a common framework which we introduce in Section 4. Full proofs are in the appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Results for Langevin dynamics with estimated score ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            681,
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+            699
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $p ( x ) \\propto e ^ { - V ( x ) }$ be a probability density on $\\mathbb { R } ^ { d }$ such that $V$ is $C ^ { 1 }$ . Langevin diffusion with stationary distribution $p$ is the stochastic process defined by the SDE ",
+        "bbox": [
+            173,
+            713,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0455d432ab2447354b1b9b5b1b311421bc6e0546d6a6232939d8db3d9d2acb0f.jpg",
+        "text": "$$\nd x _ { t } = - \\nabla V ( x _ { t } ) d t + \\sqrt { 2 } d w _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { t }$ is a standard Brownian Motion in $\\mathbb { R } ^ { d }$ . The rate of convergence to $p$ in $\\chi ^ { 2 }$ and $\\mathrm { K L }$ divergences are given by the Poincare and log-Sobolev constants of ´ $p$ , respectively; see Section E.1. To obtain the Langevin Monte Carlo (LMC) algorithm, we take the Euler-Murayama discretization of the SDE. We define LMC with score estimate $s ( x ) \\approx - \\nabla V ( x )$ and step size $h$ by ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/c2298209ff41328c1291fe3617db39ccaf0312ee77907bcb48571b39da4876a4.jpg",
+        "text": "$$\nx _ { ( k + 1 ) h } = x _ { k h } + h \\cdot s ( x _ { k h } ) + \\sqrt { 2 h } \\cdot \\xi _ { k h } , \\mathrm { w h e r e } \\xi _ { k h } \\sim N ( 0 , I _ { d } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We make the following assumptions on the density $p$ and the score estimate $s$ , which we will use throughout this paper. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. $p$ is a probability density on $\\mathbb { R } ^ { d }$ such that the following hold. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. $\\ln { p }$ is $C ^ { 1 }$ and $L$ -smooth, that is, $\\nabla \\ln p$ is $L$ -Lipschitz. We assume $L \\geq 1$ . ",
+        "bbox": [
+            210,
+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2. $p$ satisfies a log-Sobolev inequality with constant $C _ { \\mathrm { L S } }$ . We assume $C _ { \\mathrm { L S } } \\geq 1$ ",
+        "bbox": [
+            209,
+            113,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3. (Moments) $\\| \\mathbb { E } _ { p } x \\| \\leq M _ { 1 }$ and ${ \\mathbb E } _ { p } \\left\\| x \\right\\| ^ { 2 } \\le M _ { 2 }$ . ",
+        "bbox": [
+            210,
+            136,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We note that the uniform Lipschitzness assumption (1) helps ensure a unique strong solution to the Langevin diffusion, as in [BMR20]. One special case where one can prove Lipschitzness for all $t$ is when $p _ { 0 }$ is strongly log-concave $[ \\mathrm { L e e } + 2 1$ , Lemma 28]. Although satisfying a log-Sobolev inequality (3) is a significant assumption, it is standard for analysis of Langevin Monte Carlo [VW19]. It is much weaker than assumptions in previous works [BMR20], including log-concave distributions and distributions satisfying strong dissipativity, and is stable under bounded perturbations. See Section E.1 for background on functional inequalities. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2. Let p be a given probability density on $\\mathbb { R } ^ { d }$ such that $\\ln { p }$ is $C ^ { 1 }$ . The score estimate $s : \\mathbb { R } ^ { d } \\overset { ^ { - } } { \\to } \\mathbb { R } ^ { d }$ satisfies the following. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. s is a $C ^ { 1 }$ function that is $L _ { s }$ -Lipschitz. We assume $L _ { s } \\ge 1$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2. The error in the score estimate is bounded in $L ^ { 2 }$ ",
+        "bbox": [
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+            325,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1455d26a1da1b759786653fd7751a1ece8a04b7db938ca8e03c8879e1b42f772.jpg",
+        "text": "$$\n\\begin{array} { r } { \\| \\nabla \\ln p - s \\| _ { L ^ { 2 } ( p ) } ^ { 2 } = \\mathbb { E } _ { p } [ \\| \\nabla \\ln p ( x ) - s ( x ) \\| ^ { 2 } ] \\le \\varepsilon ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.1 Langevin with $L ^ { 2 }$ -accurate score estimate ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Our first main result gives an error bound between the sampled distribution and $p$ , assuming $L ^ { 2 }$ - accurate score function estimate. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 2.1 (LMC with $L ^ { 2 }$ -accurate score estimate). Let $p : \\mathbb { R } ^ { d }  \\mathbb { R }$ be a probability density satisfying Assumption $I ( I , 2 )$ with $L \\geq 1$ and $s : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { d }$ be a score estimate satisfying Assumption 2(2). Consider the accuracy requirement in TV and $\\chi ^ { 2 } \\colon 0 < \\varepsilon _ { \\mathrm { T V } } < 1 , 0 < \\varepsilon _ { \\chi } < 1$ , and suppose furthermore the starting distribution satisfies $\\chi ^ { 2 } ( p _ { 0 } | | p ) \\leq K _ { \\chi } ^ { 2 }$ . Then if ",
+        "bbox": [
+            173,
+            453,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6bd8d37951db1a59563fb6023b7af95a66eab7690132ba320002a066f3df0666.jpg",
+        "text": "$$\n\\varepsilon = O \\left( \\frac { \\varepsilon _ { \\mathrm { T V } } \\varepsilon _ { \\chi } ^ { 3 } } { d L ^ { 2 } C _ { \\mathrm { L S } } ^ { 5 / 2 } ( \\ln ( 2 K _ { \\chi } / \\varepsilon _ { \\chi } ^ { 2 } ) \\vee K _ { \\chi } ) } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "then running (LMC-SE) with score estimate $s$ , step size $\\begin{array} { r } { h \\ \\mathrm { ~ = ~ } \\ \\Theta \\biggl ( \\frac { \\varepsilon _ { \\chi } ^ { 2 } } { d L ^ { 2 } C _ { \\mathrm { L S } } } \\biggr ) } \\end{array}$ and time $T =$ $\\begin{array} { r } { \\Theta \\Big ( C _ { \\mathrm { L S } } \\ln \\Big ( \\frac { 2 K _ { \\chi } } { \\varepsilon _ { \\chi } ^ { 2 } } \\Big ) \\Big ) } \\end{array}$ results in a distribution $p _ { T }$ such that $p _ { T }$ is $\\varepsilon _ { \\mathrm { T V } }$ -far in TV distance from a distribution $\\overline { { p } } _ { T }$ , where $\\overline { { p } } _ { T }$ satisfies $\\chi ^ { 2 } ( \\overline { { p } } _ { T } | | p ) \\leq \\varepsilon _ { \\chi } ^ { 2 }$ . In particular, taking $\\varepsilon _ { \\chi } = \\varepsilon _ { \\mathrm { T V } }$ , we have the error guarantee that $\\mathrm { T V } ( p _ { T } , p ) \\leq 2 \\varepsilon _ { \\mathrm { T V } }$ . ",
+        "bbox": [
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+            566,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that the error bound is only achieved when running LMC for a moderate time; this is consistent with the fact that the stationary distribution of LMC with a $L ^ { 2 }$ -score estimate can be arbitrarily far from $p$ . Note also that we need a warm start in $\\chi ^ { 2 }$ -divergence: to obtain fixed errors $\\varepsilon _ { \\mathrm { T V } } , \\varepsilon _ { \\chi }$ , the required accuracy for the score estimate is inversely proportional to $K _ { \\chi }$ . Intuitively, we must suffer from such a dependence because if the starting distribution is very far away, then there is no guarantee that $\\| \\nabla \\ln \\bar { p } ( x _ { t } ) - s ( x _ { t } ) \\| ^ { 2 }$ is small on average during the sampling algorithm. Finally, although we can state a result purely in terms of TV distance, we need this more precise formulation to prove a result for annealed Langevin dynamics. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 Annealed Langevin dynamics with estimated score ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            781,
+            563,
+            797
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In light of the warm start requirement in Theorem 2.1, we typically cannot directly sample from $p _ { \\mathrm { d a t a } }$ or its approximation. Hence, [SE19] proposed using annealed Langevin dynamics: consider a sequence of noise levels $\\sigma _ { N } > \\cdots > \\sigma _ { 1 } \\approx 0$ giving rise to a sequence of distributions pσ2N , . . . , pσ21 ≈ pdata, where pσ2 = p ∗ φσ2 , φσ2 being the density of N (0, σ2Id). For large enough $\\sigma _ { N } , \\varphi _ { \\sigma _ { N } ^ { 2 } } \\approx p _ { \\sigma _ { N } ^ { 2 } }$ provides a warm start to $p _ { \\sigma _ { N } ^ { 2 } }$ . We then successively run LMC using score estimates for $p _ { \\sigma _ { k } ^ { 2 } }$ , with the approximate sample for $p _ { \\sigma _ { k } ^ { 2 } }$ giving a warm start for $p _ { \\sigma _ { k - 1 } ^ { 2 } }$ . We obtain the following algorithm and error estimate. ",
+        "bbox": [
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+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "INPUT: Noise levels $0 \\leq \\sigma _ { 1 } < . . . < \\sigma _ { M }$ ; score function estimates $s ( \\cdot , \\sigma _ { m } )$ (estimates of $\\nabla \\ln ( p * \\varphi _ { \\sigma _ { m } ^ { 2 } } ) )$ , step sizes $h _ { m }$ , and number of steps $N _ { m }$ for $1 \\leq m \\leq M$ .   \nDraw $x ^ { ( M + 1 ) } \\sim N ( 0 , \\sigma _ { M } ^ { 2 } I _ { d } )$ .   \nfor from to 1 do ",
+        "bbox": [
+            179,
+            108,
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+            167
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Starting from $x _ { 0 } ^ { ( m ) } = x ^ { ( m + 1 ) }$ , run (LMC-SE) with $s ( x , \\sigma _ { m } )$ and step size $h _ { m }$ for $N _ { m }$ steps, and let the final sample be $x ^ { ( m ) }$ . ",
+        "bbox": [
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+            166,
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+            196
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "end for ",
+        "text_level": 1,
+        "bbox": [
+            191,
+            198,
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+            209
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "OUTPUT: Return $x ^ { ( 1 ) }$ , approximate sample from $p * \\varphi _ { \\sigma _ { 1 } ^ { 2 } }$ ",
+        "bbox": [
+            191,
+            210,
+            573,
+            227
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2.2 (Annealed LMC with $L ^ { 2 }$ -accurate score estimate). Let $p : \\mathbb { R } ^ { d }  \\mathbb { R }$ be a probability density satisfying Assumption 1 for $M _ { 1 } = O ( d )$ , and let $p _ { \\sigma ^ { 2 } } : = p * \\varphi _ { \\sigma ^ { 2 } }$ . Suppose furthermore that $\\nabla \\ln p _ { \\sigma ^ { 2 } }$ is $L$ -Lipschitz for every $\\sigma \\geq 0$ . Given $\\sigma _ { \\operatorname* { m i n } } > 0$ , there exists a sequence $\\sigma _ { \\mathrm { m i n } } = \\sigma _ { 1 } <$ $\\cdots < \\sigma _ { M }$ with $\\begin{array} { r } { M = O \\left( \\sqrt { d } \\log \\left( \\frac { d C _ { \\mathrm { L S } } } { \\sigma _ { \\mathrm { m i n } } ^ { 2 } } \\right) \\right) } \\end{array}$ such that for each $m$ , $i f$ ",
+        "bbox": [
+            174,
+            256,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f02e9d024ef9601851d221596d962a4cca9274935eac15f7aa135617b4aa705a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\left\\| \\nabla \\ln ( p _ { \\sigma _ { m } ^ { 2 } } ) - s ( \\cdot , \\sigma _ { m } ^ { 2 } ) \\right\\| _ { L ^ { 2 } ( p _ { \\sigma _ { m } ^ { 2 } } ) } ^ { 2 } = \\mathbb { E } _ { p _ { \\sigma _ { m } ^ { 2 } } } [ \\left\\| \\nabla \\ln p _ { \\sigma _ { m } ^ { 2 } } ( x ) - s ( x , \\sigma _ { m } ^ { 2 } ) \\right\\| ^ { 2 } ] \\leq \\varepsilon ^ { 2 } . } \\\\ & { \\qquad w i t h \\varepsilon : = \\widetilde { \\cal O } \\left( \\frac { \\varepsilon _ { \\mathrm { T V } } ^ { 4 . 5 } } { d ^ { 3 . 2 5 } L ^ { 2 } C _ { \\mathrm { L S } } ^ { 2 . 5 } } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            329,
+            754,
+            395
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "then $x ^ { ( 1 ) }$ is a sample from a distribution $q$ such that $\\mathrm { T V } ( q , p _ { \\sigma _ { 1 } ^ { 2 } } ) \\leq \\varepsilon _ { \\mathrm { T V } }$ ",
+        "bbox": [
+            176,
+            400,
+            645,
+            417
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that we assume a score estimate with error $\\varepsilon$ at all noise scales; this corresponds to using an objective function that is a maximum of the score-matching objective over all noise levels, rather than an average over all noise levels as more commonly used in practice. However, these two losses are at most a factor of $M$ apart. ",
+        "bbox": [
+            173,
+            428,
+            825,
+            484
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proof shows that the noise levels $\\sigma _ { k }$ can be chosen as a geometric sequence, which matches the choice used in practice [SE20]. The additional dependence on $d$ and $\\varepsilon _ { \\mathrm { T V } }$ in Theorem 2.2 compared to Theorem 2.1 comes from requiring a sequence of $\\widetilde { O } ( \\sqrt { d } )$ noise levels and an additional factor in $\\chi ^ { 2 }$ -divergence we suffer at the beginning of each level $m$ . In the next section, we will find that using a reverse SDE to evolve the samples between the noise levels—called a predictor step—will improve the rate and time complexity. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Results for reverse SDE’s with estimated score ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            594,
+            591,
+            612
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To improve the empirical performance of score-based generative modeling, $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ consider a general framework where noise is injected into a data distribution $p _ { \\mathrm { d a t a } }$ via a forward SDE, ",
+        "bbox": [
+            173,
+            625,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b54d8ac992de7c4a5164169f924dd256765771a5f0147ab7cc7b21a9e4ae00b.jpg",
+        "text": "$$\nd \\tilde { x } _ { t } = f ( \\tilde { x } _ { t } , t ) d t + g ( t ) d w _ { t } , \\ t \\in [ 0 , T ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            659,
+            635,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\widetilde x _ { 0 } \\sim \\widetilde p _ { 0 } : = p _ { \\mathrm { d a t a } }$ . Let $\\widetilde { p } _ { t }$ denote the distribution of $\\widetilde { x } _ { t }$ $\\widetilde { p } _ { t }$ is used instead of $p _ { t }$ to distine e e e eguish with the Gaussian-convolved distribution used in Annealed Langevin dynamics as in $\\ S 2 . 2 )$ . Remarkably, $\\widetilde { x } _ { t }$ also satisfies a reverse-time SDE, ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/571f87a54ef733a33fa4f42d33e1c1ac4bd78b8f33f652467cbc2a3fd7a3cdfe.jpg",
+        "text": "$$\nd \\widetilde { \\boldsymbol { x } } _ { t } = [ f ( \\widetilde { \\boldsymbol { x } } _ { t } , t ) - g ( t ) ^ { 2 } \\nabla \\ln \\widetilde { p } _ { t } ( \\widetilde { \\boldsymbol { x } } _ { t } ) ] d t + g ( t ) d \\widetilde { \\boldsymbol { w } } _ { t } , t \\in [ 0 , T ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            728,
+            700,
+            747
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\tilde { w } _ { t }$ is a backward Brownian Motion [And82]. By carefully choosing $f$ and $g$ , we can expect that $\\tilde { p } _ { T }$ is approximately equal to some prior distribution $\\tilde { q } _ { T }$ (e.g., a centered Gaussian) which we can accurately sample from. Then we hope that starting with some $\\tilde { y } _ { T } \\sim p _ { \\mathrm { p r i o r } } = \\tilde { q } _ { T } \\approx \\tilde { p } _ { T }$ and running the reverse-time process, we will get a good sample $\\tilde { y } _ { 0 } \\sim \\tilde { q } _ { 0 } \\approx p _ { \\mathrm { d a t a } }$ . ",
+        "bbox": [
+            173,
+            751,
+            825,
+            809
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The case where $f \\equiv 0$ and $g \\equiv 1$ recovers the simple case of convolving with a Gaussian as used in $\\ S 2 . 2$ ; note, however that the reverse-time SDE differs from Langevin diffusion in having a larger (and time-varying) drift relative to the diffusion. $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ highlight the following two special cases. We will focus on DDPM while noting that our analysis applies more generically. ",
+        "bbox": [
+            174,
+            813,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SMLD Score-matching Langevin diffusion: $f \\equiv 0$ . In this case, $\\widetilde { p } _ { t } \\ = \\ \\widetilde { p } _ { 0 } * \\ \\varphi _ { \\int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s }$ , so $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ call this a variance-exploding (VE) SDE. As is common for annealing-based algorithms, [SE19; $\\mathbf { S o n } { + } 2 0 \\mathbf { b } ]$ suggest choosing an exponential schedule, so that $g ( t ) = a b ^ { t }$ for constants $a , b$ . We take $\\begin{array} { r } { p _ { \\mathrm { p r i o r } } = N ( 0 , \\int _ { 0 } ^ { T } g ( s ) ^ { 2 } d s \\cdot I _ { d } ) } \\end{array}$ . ",
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            232,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "DDPM Denoising diffusion probabilistic modeling: $\\begin{array} { r } { f ( x , t ) = - \\frac { 1 } { 2 } g ( t ) ^ { 2 } x } \\end{array}$ . This is an OrnsteinUhlenbeck process with time rescaling, $\\begin{array} { r } { \\widetilde { p } _ { t } = M _ { - \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s \\sharp } \\widetilde { p } _ { 0 } \\ast \\varphi _ { 1 - e ^ { - \\int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s } } } \\end{array}$ , where $M _ { \\alpha } ( x ) = \\alpha x$ . $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ call this a variance-preserving (VP) SDE, as the variance converges towards $I _ { d }$ . Because it displays exponential convergence towards $N ( 0 , I _ { d } )$ , it can be run for a smaller amount of normalized time $\\textstyle \\int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s$ . $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ suggest the choice $g ( t ) = \\sqrt { b + \\alpha t }$ . We take $p _ { \\mathrm { p r i o r } } = N ( 0 , ( 1 - e ^ { - \\int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s } ) I _ { d } ) \\approx N ( 0 , I _ { d } ) .$ ",
+        "bbox": [
+            176,
+            128,
+            825,
+            226
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To obtain an algorithm, we consider the following discretization and approximation of (4); note that in all cases of interest the integrals can be analytically evaluated. We reverse time so that $t$ corresponds to $T - t$ of the forward process. As we are free to rescale time in the SDE, we assume without loss of generality that the step sizes are constant. The predictor step is ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/859e8e18508758c705a6cd1b8fee7df09342f27fee3060b5b23654e4d318bd4f.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { z _ { ( k + 1 ) h } = z _ { k h } - \\int _ { k h } ^ { ( k + 1 ) h } [ f ( z _ { k h } , T - t ) - g ( T - t ) ^ { 2 } \\cdot s ( z _ { k h } , T - k h ) ] d t } } \\\\ & { } & \\\\ & { } & { + \\int _ { k h } ^ { ( k + 1 ) h } g ( T - t ) d w _ { t } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            292,
+            787,
+            366
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\int _ { k h } ^ { ( k + 1 ) h } g ( T - t ) d w _ { t } } \\end{array}$ is distributed as $\\begin{array} { r } { N ( 0 , \\int _ { k h } ^ { ( k + 1 ) h } g ( T - t ) ^ { 2 } d t \\cdot I _ { d } ) } \\end{array}$ . Following $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ , we call these predictor steps as the samples aim to track the distributions $\\widetilde { p } _ { T - k h }$ . Note that we flip the time. For simplicity of presentation, we consider the case $g \\ \\equiv \\ 1$ . We note that although the choice of the schedule does matter in practice, what really matters in our theoretical analysis is the integral $\\textstyle \\int _ { 0 } ^ { t } g ( s ) ^ { 2 } d s$ . This means that different choices of $g$ are related by only a rescaling of time, i.e., for different $g$ and $\\tilde { g }$ , we can always choose total times $T$ and $\\tilde { T }$ , such that $\\begin{array} { r } { \\int _ { 0 } ^ { T } g ( s ) ^ { 2 } d s = \\int _ { 0 } ^ { \\tilde { T } } \\tilde { g } ( s ) ^ { 2 } d s } \\end{array}$ . While it seems that choosing large $g ( t )$ could reduce the total time $T$ , in our analysis (e.g., Lemma C.15) we need the time step-size $h$ to be ${ \\cal O } ( 1 / g ( T ) ^ { 2 } )$ and hence the total computational cost, which is roughly $O ( T / h )$ , does not change significantly. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1 (Predictor with $L ^ { 2 }$ -accurate score estimate, DDPM). Let $p _ { \\mathrm { d a t a } } : \\mathbb { R } ^ { d }  \\mathbb { R }$ be a probability density satisfying Assumption $^ { l }$ with $M _ { 2 } = O ( d )$ , and let $\\widetilde { p } _ { t }$ be the distribution resulting from evolving the forward $S D E$ according to DDPM with $g \\equiv 1$ e. Suppose furthermore that $\\nabla \\ln \\widetilde { p } _ { t }$ is $L$ -Lipschitz for every $t \\geq 0$ , and that each $s ( \\cdot , t )$ satisfies Assumption 2. Then if ",
+        "bbox": [
+            173,
+            513,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c2d1763419a506c970cca4323bce1b62f1066b81c5ca3f91a361c0cfca3aa43a.jpg",
+        "text": "$$\n\\varepsilon = O \\left( \\frac { \\varepsilon _ { \\mathrm { T V } } ^ { 4 } } { ( C _ { \\mathrm { L S } } + d ) C _ { \\mathrm { L S } } ^ { 5 / 2 } ( L \\vee L _ { s } ) ^ { 2 } ( \\ln ( C _ { \\mathrm { L S } } d ) \\vee C _ { \\mathrm { L S } } \\ln ( 1 / \\varepsilon _ { \\mathrm { T V } } ^ { 2 } ) ) } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "running (P) starting from $p _ { \\mathrm { p r i o r } }$ for time $\\begin{array} { r } { T = \\Theta \\left( \\ln ( C _ { \\mathrm { L S } } d ) \\vee C _ { \\mathrm { L S } } \\ln \\left( \\frac { 1 } { \\varepsilon _ { \\mathrm { T V } } } \\right) \\right) } \\end{array}$ and step size $h \\ =$ $\\begin{array} { r } { \\Theta \\left( \\frac { \\varepsilon _ { \\mathrm { T V } } ^ { 2 } } { C _ { \\mathrm { L S } } ( C _ { \\mathrm { L S } } + d ) ( L \\vee L _ { s } ) ^ { 2 } } \\right) } \\end{array}$ results in a distribution $q _ { T }$ so that $\\mathrm { T V } ( q _ { T } , p _ { \\mathrm { d a t a } } ) \\leq \\varepsilon _ { \\mathrm { T V } } .$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A more precise statement of the Theorem can be found in the Appendix. Although we state our theorem for DDPM, we describe in Appendix C how it can be adapted to other SDE’s like SMLD and the sub-VP SDE; the primary SDE-dependent bound we need is a bound on $\\begin{array} { r } { \\nabla \\ln \\frac { \\widetilde { p } _ { t } } { \\widetilde { p } _ { t + h } } } \\end{array}$ . Because the predictor is tracking a changing distribution $p _ { t }$ e, we incur more error terms and worse dependence on parameters $( C _ { \\mathrm { L S } } , L )$ than in LMC (Theorem 2.1). Motivated by this, we intersperse the predictor steps with LMC steps—called corrector steps in this context—to give additional time for the process to mix, resulting in improved dependence on parameters. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 (Predictor-corrector with $L ^ { 2 }$ -accurate score estimate). Keep the setup of Theorem 3.1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/915895e23c4102ecd8d434ac5da97105c5bdd1c0d7e2594baa1ed942d03f408c.jpg",
+        "text": "$$\n\\varepsilon = { \\cal O } \\left( \\frac { \\varepsilon _ { \\mathrm { T V } } ^ { 4 } } { d L ^ { 2 } C _ { \\mathrm { L S } } ^ { 5 / 2 } \\ln ( 1 / \\varepsilon _ { \\chi } ^ { 2 } ) } \\right) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "then Algorithm 2 with appropriate choices of $\\begin{array} { r } { T = \\Theta \\left( \\ln ( C _ { \\mathrm { L S } } d ) \\lor C _ { \\mathrm { L S } } \\log \\left( \\frac { 1 } { \\varepsilon _ { \\mathrm { T V } } } \\right) \\right) } \\end{array}$ , $N _ { m }$ , corrector step sizes $h _ { m }$ and predictor step size $h$ , produces $a$ sample from a distribution $q _ { T }$ such that $\\mathrm { T V } ( q _ { T } , p _ { \\mathrm { d a t a } } ) < \\varepsilon _ { \\mathrm { T V } }$ . ",
+        "bbox": [
+            173,
+            861,
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+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/6aa41d6e011f2d82a13c1cae1a5a9bdc7aac52578a2450e40289066271ab9ccf.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>INPUT: Time T, predictor step size h; number of corrector steps Nm per predictor step, corrector step sizeshm</td></tr><tr><td>Draw zo ~ Pprior from the prior distribution.</td></tr><tr><td></td></tr><tr><td>for m from1to T/h do (Predictor) Take a step of (P) to obtain Zmh from z(m-1)h, with f,g as in SMLD or DDPM.</td></tr><tr><td>(Corrector) Starting from 2mh,0 := 2mh, run (LMC-SE) with s(z,T - mh) and step size hm</td></tr><tr><td>for N steps,and let zmh ← zmh,N.</td></tr><tr><td>end for OUTPUT: Return zT,approximate sample from pdata·</td></tr></table>",
+        "bbox": [
+            179,
+            107,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The assumption on $\\varepsilon _ { \\mathrm { T V } }$ is for convenience in stating our bound. In comparison to using the predictor step alone (Theorem 3.1), note that in the bound on $\\varepsilon$ , we obtain the improved rate of the corrector step as in Theorem 2.1; this is because the predictor step only needs to track the actual distribution in $\\chi ^ { 2 }$ -divergence with error $O ( 1 )$ , and the final corrector steps are responsible for decreasing the error to $\\varepsilon _ { \\mathrm { T V } }$ . In comparison to the Annealed Langevin sampler (Algorithm 1, Theorem 2.2), which can be viewed as using the corrector step alone, adding a predictor step provides a better warm start for the distribution at the next smaller noise level, resulting in better dependence on parameters. Thus the predictor-corrector algorithm combines the strengths of the predictor and corrector steps. For realworld data, it can be challenging to estimate TV-distance between distributions given only samples, and hence difficult to check consistency with empirical observations. However, our claim that using a corrector can improve the convergence rate of DDPM/SMLD is consistent with the simulation results in Section 4.2 of $[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Theoretical framework and proof sketches ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            449,
+            560,
+            467
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The main idea of our analysis framework is to convert a $L ^ { 2 }$ error guarantee to a $L ^ { \\infty }$ error guarantee by excluding a bad set, formalized in the following theorem. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1. Let $( \\Omega , \\mathcal { F } , \\mathbb { P } )$ be a probability space and $\\{ \\mathcal { F } _ { n } \\}$ be a filtration of the sigma field $\\mathcal { F }$ Suppose $X _ { n } \\sim p _ { n }$ , $Z _ { n } \\sim q _ { n } ,$ , and $\\overline { { Z } } _ { n } \\sim \\overline { { q } } _ { n }$ are ${ \\mathcal { F } } _ { n }$ -adapted random processes taking values in $\\Omega$ and $B _ { n } \\subseteq \\Omega$ are sets such that the following hold for every $n \\in  { \\mathbb { N } } _ { 0 }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. $\\chi ^ { 2 } ( \\overline { { q } } _ { n } | | p _ { n } ) \\leq D _ { n } ^ { 2 }$ . ",
+        "bbox": [
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+            590,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3. $\\mathbb { P } ( X _ { n } \\in B _ { n } ) \\le \\delta _ { n }$ ",
+        "bbox": [
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+            614,
+            361,
+            632
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Then the following hold. ",
+        "bbox": [
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+            641,
+            334,
+            656
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d70b9f48f8c1326232d3071aa0dc3c9739c5bb6134a99ff3405e20247805862.jpg",
+        "text": "$$\n\\mathrm { T V } ( q _ { n } , \\overline { { q } } _ { n } ) \\leq \\sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \\delta _ { k } ^ { 1 / 2 } \\qquad \\mathrm { T V } ( p _ { n } , q _ { n } ) \\leq D _ { n } + \\sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \\delta _ { k } ^ { 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            661,
+            779,
+            705
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For our setting, we will take the “bad sets” $B _ { n }$ to be the set of $x$ where $\\| s _ { \\theta } ( x ) - \\nabla \\ln p \\|$ is large, $q _ { n }$ to be the discretized process with estimated score, and $\\overline { { q } } _ { n }$ to be the discretized process with estimated score except in $B _ { n }$ where the error is large. Because $\\overline { { q } } _ { n }$ uses an $L ^ { \\infty }$ -accurate score estimate, we can use existing techniques for analyzing Langevin Monte Carlo [VW19; EHZ21; Che+21] to bound $\\chi ^ { 2 } ( \\overline { { q } } _ { n } | | p _ { n } ) \\ :$ . ",
+        "bbox": [
+            173,
+            718,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proof. We bound using condition 1 and Cauchy-Schwarz: ",
+        "bbox": [
+            173,
+            801,
+            555,
+            818
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a243c91a43ae09d14deea126b46c7754e0fbd3a11a8ce2bb3af4cb92297fd9f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle ^ { \\mathfrak { n } } \\big ( Z _ { n } \\ne \\overline { { Z } } _ { n } \\big ) \\le \\mathbb { P } \\left( \\bigcup _ { k = 1 } ^ { n - 1 } \\big \\{ Z _ { k } \\in B _ { k } \\big \\} \\right) \\le \\sum _ { k = 0 } ^ { n - 1 } \\mathbb { P } \\left( Z _ { k } \\in B _ { k } \\right) = \\sum _ { k = 0 } ^ { n - 1 } \\mathbb { E } _ { q _ { k } } \\mathbb { 1 } _ { B _ { k } } } \\\\ & { \\qquad \\le \\displaystyle \\sum _ { k = 0 } ^ { n - 1 } \\left( \\mathbb { E } _ { p _ { k } } \\left( \\frac { q _ { k } } { p _ { k } } \\right) ^ { 2 } \\right) ^ { 1 / 2 } ( \\mathbb { E } _ { p _ { k } } \\mathbb { 1 } _ { B _ { k } } ) ^ { 1 / 2 } = \\sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \\delta _ { k } ^ { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The second inequality then follows from the triangle inequality and Cauchy-Schwarz: ",
+        "bbox": [
+            171,
+            90,
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+            107
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f222f2ae3e0a161e47790f46b26c9d141d15cf53d436b2a19517527250d2606e.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\mathrm { T V } ( p _ { n } , q _ { n } ) \\leq \\mathrm { T V } ( p _ { n } , \\overline { { q } } _ { n } ) + \\mathrm { T V } ( \\overline { { q } } _ { n } , q _ { n } ) } } \\\\ & { } & \\\\ & { } & { \\leq \\sqrt { \\chi ^ { 2 } ( \\overline { { q } } _ { n } | | p _ { n } ) } + \\mathrm { T V } ( \\overline { { q } } _ { n } , q _ { n } ) \\leq D _ { n } + \\displaystyle \\sum _ { k = 0 } ^ { n - 1 } ( D _ { k } ^ { 2 } + 1 ) ^ { 1 / 2 } \\delta _ { k } ^ { 1 / 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "It now remains to give $\\chi ^ { 2 }$ convergence bounds under $L ^ { \\infty }$ -accurate score estimate. The following theorem may be of independent interest. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.2 (LMC under $L ^ { \\infty }$ bound on gradient error). Let $p : \\mathbb { R } ^ { d }  \\mathbb { R }$ be a probability density satisfying Assumption $I ( I , 2 )$ and $s : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { d }$ be a score estimate $s$ with error bounded in $L ^ { \\infty }$ : for some ε1 ≤ q $\\begin{array} { r } { \\varepsilon _ { 1 } \\leq \\sqrt { \\frac { 1 } { 4 8 C _ { \\mathrm { L S } } } } } \\end{array}$ , ",
+        "bbox": [
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+            215,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/650199748822cc205f371e7b76f98a4cbcfa9a6edf655fc6e2adef8d31b6dc55.jpg",
+        "text": "$$\n\\| \\nabla \\ln p - s \\| _ { \\infty } = \\operatorname* { m a x } _ { x \\in \\mathbb { R } ^ { d } } \\| \\nabla \\ln p ( x ) - s ( x ) \\| \\| \\leq \\varepsilon _ { 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            272,
+            665,
+            296
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Let $N \\in  { \\mathbb { N } } _ { 0 }$ and $\\begin{array} { r } { 0 < h \\leq \\frac { 1 } { 4 3 9 2 d C _ { \\mathrm { L S } } L ^ { 2 } } } \\end{array}$ , and assume $L \\geq 1$ . Let $q _ { n h }$ denote the nth iterate of LMC with step size $h$ score estimate s. Then ",
+        "bbox": [
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+            301,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9cdd3981138383853091f2ca7f639e78aafbbcc5c61723bd956f0dc1e3ec7f1b.jpg",
+        "text": "$$\n\\chi ^ { 2 } ( q _ { ( k + 1 ) h } | | p ) \\leq \\exp \\left( - \\frac { h } { 4 C _ { \\mathrm { L S } } } \\right) \\chi ^ { 2 } ( q _ { k h } | | p ) + 1 7 0 d L ^ { 2 } h ^ { 2 } + 5 \\varepsilon _ { 1 } ^ { 2 } h\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            337,
+            717,
+            372
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "and ",
+        "bbox": [
+            173,
+            376,
+            202,
+            388
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b3ce30010f4b02028710d00431540bae759ae13340219c1a57e98cbe1491644.jpg",
+        "text": "$$\n{ \\chi ^ { 2 } } ( q _ { N h } | | p ) \\leq \\exp \\left( - \\frac { N h } { 4 C _ { \\mathrm { L S } } } \\right) \\chi ^ { 2 } ( q _ { 0 } | | p ) + 6 8 0 d L ^ { 2 } h C _ { \\mathrm { L S } } + 2 0 \\varepsilon _ { 1 } ^ { 2 } C _ { \\mathrm { L S } } \\leq \\exp \\left( - \\frac { N h } { 4 C _ { \\mathrm { L S } } } \\right) \\chi ^ { 2 } ( q _ { 0 } | | p ) + 1\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            390,
+            838,
+            426
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Following $[ \\mathrm { C h e } + 2 1 ]$ , we prove this by first defining a continuous-time interpolation $q _ { t }$ of the discrete process, and then deriving a differential inequality for $\\chi ^ { 2 } ( q _ { t } | | p )$ using the log-Sobolev inequality for $p$ . Compared to $[ \\mathrm { C h e } + 2 1 ]$ , we incur an extra error term arising from the inaccurate gradient. ",
+        "bbox": [
+            173,
+            435,
+            828,
+            478
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This allows us to sketch the proof of Theorem 2.1; a complete proof is in Section B. ",
+        "bbox": [
+            173,
+            483,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 2.1. We first define the bad set where the error in the score estimate is large, ",
+        "bbox": [
+            173,
+            512,
+            825,
+            542
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff89cc60a95dfdfa8c7458c50621f331d2ab3b1e866432e8ba21a7e6d8e28a29.jpg",
+        "text": "$$\nB : = \\{ \\| \\nabla \\ln p ( x ) - s ( x ) \\| > \\varepsilon _ { 1 } \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            546,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "for some $\\varepsilon _ { 1 }$ to be chosen. Then by Chebyshev’s inequality, $\\begin{array} { r } { P ( B ) \\leq \\left( \\frac { \\varepsilon } { \\varepsilon _ { 1 } } \\right) ^ { 2 } = : \\delta } \\end{array}$ . Let $\\overline { { q } } _ { n h }$ be the discretized process, but where the score estimate is set to be equal to $\\nabla \\ln p$ on $B$ ; note it agrees with $q _ { n h }$ as long as it has not hit $B$ . Because $\\overline { { q } } _ { n h }$ uses a score estimate that has $L ^ { \\infty }$ -error $\\varepsilon _ { 1 }$ , Theorem 4.2 gives a bound for $\\chi ^ { 2 } ( \\overline { { q } } _ { N h } | | p )$ . Then Theorem 4.1 gives ",
+        "bbox": [
+            173,
+            570,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/212ac7809f0e91eb857a5bfece128260e5e06c2abfc644e83f8e723996835b8c.jpg",
+        "text": "$$\n { \\operatorname { F V } } ( q _ { n h } , \\overline { { q } } _ { n h } ) \\le \\sum _ { k = 0 } ^ { n - 1 } ( \\chi ^ { 2 } ( \\overline { { q } } _ { k h } \\| p ) + 1 ) ^ { 1 / 2 } P ( B ) ^ { 1 / 2 } \\le \\sum _ { k = 0 } ^ { n - 1 } \\left( \\exp \\left( - \\frac { k h } { 8 C _ { \\mathrm { L S } } } \\right) \\chi ^ { 2 } ( q _ { 0 } \\| p ) ^ { 1 / 2 } + 1 \\right) \\delta ^ { 1 / 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The theorem then follows from choosing parameters so that χ2(qT ||p) ≤ ε2χ and TV(qT , qT ) ≤ $\\varepsilon _ { \\mathrm { T V } }$ . ",
+        "bbox": [
+            174,
+            688,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We remark that the main inefficiency in the proof comes from the use of Chebyshev’s inequality, and a $L ^ { p }$ bound on the error for $p > 2$ will improve the bound. ",
+        "bbox": [
+            173,
+            731,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 2.2. Choosing the sequence $\\sigma _ { 1 } < \\cdots < \\sigma _ { M }$ to be geometric with ratio $1 + \\frac { 1 } { \\sqrt { d } }$ ensures that the $\\chi ^ { 2 }$ -divergence between successive distributions $p _ { \\sigma _ { m } ^ { 2 } }$ is $O ( 1 )$ . Then, choosing $\\sigma _ { M } ^ { 2 } = \\Omega ( C _ { \\mathrm { L S } } d )$ ensures we have a warm start for the highest noise level: $\\chi ^ { 2 } ( \\operatorname { \\rho _ { p r i o r } } \\lvert \\lvert p _ { \\sigma _ { M } ^ { 2 } } ) = O ( 1 )$ . This uses $\\begin{array} { r } { O \\left( \\sqrt { d } \\log \\left( \\frac { d C _ { \\mathrm { L S } } } { \\sigma _ { \\mathrm { m i n } } ^ { 2 } } \\right) \\right) } \\end{array}$ noise levels. Chebyshev’s inequality can be used to show that the distribution of the final sample $x ^ { ( m ) }$ for $p _ { \\sigma _ { m } ^ { 2 } }$ is ${ \\cal O } ( \\varepsilon _ { \\mathrm { T V } } / M )$ close to a distribution that is ${ \\cal O } ( M / \\varepsilon _ { \\mathrm { T V } } )$ in in $\\chi ^ { 2 }$ -divergence from Theorem 2.1 then $p _ { \\sigma _ { m + 1 } ^ { 2 } }$ . This gives the warm st the required bound for rt parameter . Note that $K _ { \\chi } = ( M / \\varepsilon _ { \\mathrm { T V } } ) ^ { 1 / 2 }$ ; substitutinged from each $\\varepsilon$ level add to ${ \\cal O } ( \\varepsilon _ { \\mathrm { T V } } )$ . □ ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To analyze the predictor-based algorithms, we also first prove convergence bounds under $L ^ { \\infty }$ - accurate score estimate. ",
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.3 (Predictor steps under $L ^ { \\infty }$ bound on score estimate, DDPM). Let $p : \\mathbb { R } ^ { d }  \\mathbb { R }$ be a probability density satisfying Assumption $I$ and $s ( \\cdot , t ) : \\mathbb { R } ^ { d } \\to \\mathbb { R } ^ { d }$ be a score estimate s with error bounded in $L ^ { \\infty }$ for each $t \\in [ 0 , T ]$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/09ebd58037967c1f147d8da57a99c4b4ff414e60c4c603a610c6c82d29b86fe6.jpg",
+        "text": "$$\n\\| \\nabla \\ln p - s ( \\cdot , t ) \\| _ { \\infty } = \\operatorname* { m a x } _ { x \\in \\mathbb { R } ^ { d } } \\| \\nabla \\ln \\widetilde { p } _ { t } ( x ) - s ( x , t ) \\| \\| \\le \\varepsilon _ { 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            175,
+            689,
+            200
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Consider DDPM with $g \\equiv 1$ , $T \\geq 1 \\vee \\ln ( C _ { \\mathrm { L S } } d ) ,$ , and $\\begin{array} { r } { h = O \\left( \\frac { 1 } { C _ { \\mathrm { L S } } ( d + C _ { \\mathrm { L S } } ) ( L \\vee L _ { s } ) ^ { 2 } } \\right) } \\end{array}$ . (Recall that $p _ { k h }$ and $q _ { k h }$ are the $k$ -th iterate of LMC with step size $h$ and true/estimated score respectively.) Then ",
+        "bbox": [
+            174,
+            210,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/22799b1ee3cba73ec393362a5205ecfb56caf7acbb5a7558ab2c3c15d96b132f.jpg",
+        "text": "$$\n\\chi ^ { 2 } ( q _ { ( k + 1 ) h } | | p _ { ( k + 1 ) h } ) \\leq \\chi ^ { 2 } ( q _ { k h } | | p _ { k h } ) e ^ { \\left( - \\frac { 1 } { 8 C _ { 1 S } } + 8 \\varepsilon _ { 1 } ^ { 2 } \\right) h } + O ( \\varepsilon _ { 1 } ^ { 2 } h + ( L _ { s } ^ { 2 } + L ^ { 2 } d ) h ^ { 2 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            257,
+            764,
+            282
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\varepsilon _ { 1 } < \\frac { 1 } { 1 2 8 C _ { \\mathrm { L S } } } } \\end{array}$ ",
+        "bbox": [
+            173,
+            291,
+            302,
+            310
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d5958db694bf8230a9f4cb81c03356619c7e48bdd141ec19f0ef46e6130d16a4.jpg",
+        "text": "$$\n\\chi ^ { 2 } ( q _ { N h } | | p _ { N h } ) \\leq e ^ { - \\frac { N h } { 1 6 C _ { 1 S } } } \\chi ^ { 2 } ( q _ { 0 } | | p _ { 0 } ) + O \\left( C _ { \\mathrm { L S } } \\left( \\varepsilon _ { 1 } ^ { 2 } + ( L _ { s } ^ { 2 } + L ^ { 2 } d ) h \\right) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            319,
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+            342
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Moreover, for $q _ { 0 } = p _ { \\mathrm { p r i o r } } , \\chi ^ { 2 } ( q _ { 0 } | | p _ { 0 } ) \\leq e ^ { - T / 2 } C _ { \\mathrm { L S } } d .$ ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We give a more precise statement in Section C. Note that unlike the case for LMC as in Theorem 4.2, the base density $p _ { t }$ is also evolving in time, which produces additional error terms and necessitates a more involved analysis. The additional error terms can be bounded using the Donsker-Varadhan variational principle, concentration for distributions satisfying LSI, and error bounds between $p _ { t }$ and $p _ { t + h }$ for small $h$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Here, we only state the result about DDPM, which has better bounds than SMLD (when $g \\equiv 1$ ) because both the forward and backwards processes exhibit better mixing properties: the warm start improves exponentially rather than inversely with $T$ , and the log-Sobolev constant is uniformly bounded by that of $p _ { \\mathrm { d a t a } }$ rather than increasing. However, the analysis in Section C can be directly applied to SMLD and other models as well. We also note there is a sense in which DDPM and SMLD are equivalent under a rescaling in time and space (see discussion in Section C.2). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Note that the choice of $h$ is necessary for exponential decay of error; as if $h$ is not small enough, we would get an exponential growing instead of decaying factor in the one-step error (See Section C for details). Such an $h$ may however still be a suitable choice when used in conjunction with a corrector step. Moreover, as $\\varepsilon _ { 1 } \\to 0$ , with appropriate choice of $T$ and $h$ , $q _ { N h }$ and $p _ { N h }$ can be made arbitrarily close. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1 now follows from the $L ^ { \\infty }$ result (Theorem 4.3) in the same way that Theorem 2.1 follows from Theorem 4.2. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To prove Theorem 3.2, it suffices to run the corrector steps only at the lowest noise level, that is, set $N _ { m } = 0$ for $1 \\leq m < T / h$ , although we note that interleaving the predictor and corrector steps does empirically help with mixing. The proof follows from using the predictor and the corrector theorems in series: first apply Theorem 3.1 with $\\varepsilon _ { \\chi } = O ( 1 )$ to show that the predictor results a warm start $p _ { \\mathrm { d a t a } }$ , then use Theorem 2.1 to show the corrector reduces the error to the desired $\\varepsilon _ { \\mathrm { T V } }$ . ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We introduced a general framework to analyze SDE-based sampling algorithms given a $L ^ { 2 }$ -error score estimate, and used it to obtain the first convergence bounds for several score-based generative models with polynomial complexity in all parameters. Our analysis can potentially be adapted to other SDE’s and sampling algorithms beyond Langevin Monte Carlo. There is also room for improving our analysis to better use smoothing properties of the SDE’s and compare different choices of the diffusion speed $g$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We present several interesting further directions to explore. In addition to extending the analysis to other SGM’s and comparing their theoretical performance (relative to each other as well as other approaches to generative modeling), we propose the following. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Analysis for multimodal distributions. Our assumption of a bounded log-Sobolev constant essentially limits the analysis to distributions that are close to unimodal. However, SGM’s are empirically successful at modeling multimodal distributions [SE19], and in fact perform better with multimodal distributions than other approaches such as GAN’s. Can we analyze the convergence for simple multimodal distributions, such as a mixture of distributions each with bounded log-Sobolev constant? Positive results on sampling from multimodal distributions such as [GLR18] suggest this is possible, as the sequence of noised distributions is natural for annealing and tempering methods (see [GLR18, Remark 7.2]). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Weakening conditions on the score estimate. The assumption that we have a score estimate that is $O ( 1 )$ -accurate in $L ^ { 2 }$ , although weaker than the usual assumptions for theoretical analysis, is in fact still a strong condition in practice that seems unlikely to be satisfied (and difficult to check) when learning complex distributions such as distributions of images. What would a reasonable weaker condition be, and in what sense can we still obtain reasonable samples? ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Guarantees for learning the score function. Our analysis assumes a $L ^ { 2 }$ -estimate of the score function is given, but the question remains of when we can find such an estimate. What natural conditions on distributions allow their score functions to be learned by a neural network? Various works have considered the representability of data distributions by diffusion-like processes [TR19], but the questions of optimization and generalization appear more challenging. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            387,
+            310,
+            401
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank Andrej Risteski for helpful conversations. This work was done in part while $\\mathrm { H L }$ was visiting the Simons Institute for the Theory of Computing. The work was supported in part by National Science Foundation via awards DMS-2012286 and CCF-1934964 (Duke Tripods). ",
+        "bbox": [
+            176,
+            412,
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+    },
+    {
+        "type": "text",
+        "text": "References ",
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+        "bbox": [
+            174,
+            90,
+            266,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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PMLR. 2015, pp. 2256–2265.   \n$[ \\mathrm { S o n } { + } 2 0 \\mathrm { a } ]$ Yang Song, Sahaj Garg, Jiaxin Shi, and Stefano Ermon. “Sliced score matching: A scalable approach to density and score estimation”. In: Uncertainty in Artificial Intelligence. PMLR. 2020, pp. 574–584.   \n$[ \\mathrm { S o n } { + } 2 0 \\mathrm { b } ]$ Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. “Score-Based Generative Modeling through Stochastic Differential Equations”. In: International Conference on Learning Representations. 2020.   \n$[ \\mathrm { S o n } { + } 2 1 \\mathrm { a } ]$ Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. “Maximum likelihood training of score-based diffusion models”. In: Advances in Neural Information Processing Systems 34 (2021).   \n$[ \\mathsf { S o n } { + } 2 1 \\mathsf { b } ]$ Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. “Solving Inverse Problems in Medical Imaging with Score-Based Generative Models”. In: arXiv preprint arXiv:2111.08005 (2021).   \n[TR19] Belinda Tzen and Maxim Raginsky. “Theoretical guarantees for sampling and inference in generative models with latent diffusions”. In: Conference on Learning Theory. PMLR. 2019, pp. 3084–3114.   \n[VW19] Santosh Vempala and Andre Wibisono. “Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices”. In: Advances in neural information processing systems 32 (2019), pp. 8094–8106.   \n[ZML16] Junbo Zhao, Michael Mathieu, and Yann LeCun. “Energy-based generative adversarial network”. In: arXiv preprint arXiv:1609.03126 (2016). ",
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parse/dev/luEG3j9LW5-/luEG3j9LW5-.md

@@ -0,0 +1,432 @@
+# BIDIRECTIONAL LEARNING FOR OFFLINE MODELBASED BIOLOGICAL SEQUENCE DESIGN
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Offline model-based optimization aims to maximize a black-box objective function with a static dataset of designs and their scores. In this paper, we focus on biological sequence design to maximize some sequence score. A recent approach employs bidirectional learning, combining a forward mapping for exploitation and a backward mapping for constraint, and it relies on the neural tangent kernel (NTK) of an infinitely wide network to build a proxy model. Though effective, the NTK cannot learn features because of its parametrization, and its use prevents the incorporation of powerful pre-trained Language Models (LMs) that can capture the rich biophysical information in millions of biological sequences. We adopt an alternative proxy model, adding a linear head to a pre-trained LM, and propose a linearization scheme. This yields a closed-form loss and also takes into account the biophysical information in the pre-trained LM. In addition, the forward mapping and the backward mapping play different roles and thus deserve different weights during sequence optimization. To achieve this, we train an auxiliary model and leverage its weak supervision signal via a bi-level optimization framework to effectively learn how to balance the two mappings. Further, by extending the framework, we develop the first learning rate adaptation module Adaptive- $\eta$ , which is compatible with all gradient-based algorithms for offline model-based optimization. Experimental results on DNA/protein sequence design tasks verify the effectiveness of our algorithm. Our code is available here.
+
+# 1 INTRODUCTION
+
+Offline model-based optimization aims to maximize a black-box objective function with a static dataset of designs and their scores. This offline setting is realistic since in many real-world scenarios we do not have interactive access to the ground-truth evaluation. The design tasks of interest include material, aircraft, and biological sequence (Trabucco et al., 2021). In this paper, we focus on biological sequence design, including DNA sequence and protein sequence, with the goal of maximizing some specified property of these sequences.
+
+A wide variety of methods have been proposed for biological sequence design, including evolutionary algorithms (Sinai et al., 2020; Ren et al., 2022), reinforcement learning methods (Angermueller et al., 2019), Bayesian optimization (Terayama et al., 2021), search/sampling using generative models (Brookes et al., 2019; Chan et al., 2021), and GFlowNets (Jain et al., 2022). Recently, gradient-based techniques have emerged as an effective alternative (Trabucco et al., 2021). These approaches first train a deep neural network (DNN) on the static dataset as a proxy and then obtain the new designs by directly performing gradient ascent steps on the existing designs. Such methods have been widely used in biological sequence design (Norn et al., 2021; Tischer et al., 2020; Linder & Seelig, 2020). One obstacle is the out-of-distribution issue, where the trained proxy model is inaccurate for the newly generated sequences.
+
+To mitigate the out-of-distribution issue, recent work proposes regularization of the model (Trabucco et al., 2021; Yu et al., 2021; Fu & Levine, 2021) or the design itself (Chen et al., 2022). The first category focuses on training a better proxy by introducing inductive biases such as robustness (Yu et al., 2021). The second category introduces bidirectional learning (Chen et al., 2022), which consists of a forward mapping and a backward mapping, to optimize the design directly. Specifically, the backward mapping leverages the high-scoring design to predict the static dataset and vice versa for the forward mapping, which distills the information of the static dataset into the high-scoring design. This approach achieves state-of-the-art performances on a variety of tasks. Though effective, the proposed bidirectional learning relies on the neural tangent kernel (NTK) of an infinite-width model to yield a closed-form loss, which is a key component of its successful operation. The NTK cannot learn features due to its parameterization (Yang & Hu, 2021) and thus the bidirectional learning cannot incorporate the wealth of biophysical information from Language Models (LMs) pre-trained over a vast corpus of unlabelled sequences (Elnaggar et al., 2021; Ji et al., 2021).
+
+To solve this issue, we construct a proxy model by combining a finite-width pre-trained LM with an additional layer. We then linearize the resultant proxy model, inspired by the recent progress in deep linearization (Achille et al., 2021; Dukler et al., 2022). This scheme not only yields a closed-form loss but also exploits the rich biophysical information that has been distilled in the pre-trained LM. In addition, the forward mapping encourages exploitation in the sequence space and the backward mapping serves as a constraint to mitigate the out-of-distribution issue. It is important to maintain an appropriate balance between exploitation and constraint, and this can vary across design tasks as well as during the optimization process. We introduce a hyperparameter $\gamma$ to control the balance, and develop a bi-level optimization framework Adaptive- $\gamma$ . In this framework, we train an auxiliary model and leverage its weak supervision signal to effectively update $\gamma$ . To sum up, we propose BIdirectional learning for model-based Biological sequence design (BIB). Last but not least, since the offline nature prohibits standard cross-validation strategies for hyperparameter tuning, all gradient-based offline model-based algorithms preset the learning rate $\eta$ . There is a danger of a poor selection, and to address this, we propose to extend Adaptive- $\gamma$ to Adaptive- $\eta$ , which effectively adapts the learning rate $\eta$ via the weak supervision signal from the trained auxiliary model. To the best of our knowledge, Adaptive- $\eta$ is the first learning rate adaptation module for gradient-based algorithms on offline model-based optimization. Experiments on DNA and protein sequence design tasks verify the effectiveness of BIB and Adaptive- $\eta$ .
+
+To summarize, our contributions are three-fold:
+
+• Instead of adopting the NTK, we propose to construct a proxy model by combining a pre-trained biological LM with an additional trainable layer. We then linearize the proxy model, leveraging the recent progress on deep linearization. This yields a closed-form loss computation in bidirectional learning and allows us to exploit the rich biophysical information distilled into the LM via pretraining over millions of biological sequences.   
+• We propose a bi-level optimization framework Adaptive- $\gamma$ where we leverage weak signals from an auxiliary model to achieve a satisfactory trade-off between exploitation and constraint.   
+• We further extend this bi-level optimization framework to Adaptive- $\eta$ . As the first learning rate tuning scheme in offline model-based optimization, Adaptive- $\eta$ allows learning rate adaptation for any gradient-based algorithm.
+
+# 2 PRELIMINARIES
+
+# 2.1 OFFLINE MODEL-BASED OPTIMIZATION
+
+Offline model-based optimization aims to find a design $\boldsymbol { X }$ to maximize some unknown objective $f ( X )$ . This can be formally written as,
+
+$$
+X ^ { * } = \arg \operatorname* { m a x } _ { X } f ( X ) ,
+$$
+
+where we have access to a size- $N$ dataset $\mathcal { D } = \{ ( \boldsymbol { X } _ { 1 } , \boldsymbol { y } _ { 1 } ) \} , \cdot \cdot \cdot , \{ ( \boldsymbol { X } _ { N } , \boldsymbol { y } _ { N } ) \}$ with $X _ { i }$ representing a certain design and $y _ { i }$ denoting the design score. In this paper, $X _ { i }$ represents a biological sequence design, including DNA and protein sequences, and $y _ { i }$ represents a property of the biological sequence such as the fluorescence level of the green fluorescent protein (Sarkisyan et al., 2016).
+
+# 2.2 BIOLOGICAL SEQUENCE REPRESENTATION
+
+Following (Norn et al., 2021; Killoran et al., 2017; Linder & Seelig, 2021), we adopt the positionspecific scoring matrix to represent a length- $L$ protein sequence as $\breve { X } \in \mathbb { R } ^ { L \times 2 0 }$ , where 20 represents 20 different kinds of amino acids. For a real-world protein sequence, $X [ l , : ] \ : ( 0 \leq l \leq L - 1 )$ i s a
+
+![](images/ab7fe875504c0ffd085415039d1827d692d3613f1222a79b03fb1845c8795018.jpg)  
+Figure 1: Illustration of bidirectional learning Chen et al. (2022) where $( X _ { l } , y _ { l } )$ denotes the static dataset, $y _ { h }$ is a large predefined target score and $X _ { h }$ is the high-scoring design we aim to find.
+
+one-hot vector denoting one kind of amino acid. During optimization, $X [ l , : ]$ is a continuous vector and $s o f t m a x ( X [ l , : ] )$ represents the probability distribution of all 20 amino acids in the position $l$ . Similarly, for a DNA sequence, we have $\pmb { X } \in \tilde { \mathbb { R } ^ { L \times 4 } }$ where 4 represents 4 different DNA bases.
+
+The protein sequence $\boldsymbol { X }$ is fed into the embedding layer of the LM, which produces the embedding,
+
+$$
+\pmb { e } = E M B ( s o f t m a x ( \pmb { X } ) ) .
+$$
+
+The main block of the LM takes $e$ as input and outputs biophysical features. The DNA LM, which adopts the $k$ -mer representation, is a little different from protein LMs. See Appendix A.1 for details.
+
+# 2.3 GRADIENT ASCENT ON SEQUENCE
+
+A common approach to the posed offline model-based optimization problem is to train a proxy $f _ { \pmb { \theta } } ( \pmb { X } )$ on the offline dataset,
+
+$$
+\pmb { \theta } ^ { * } = \arg \operatorname* { m i n } _ { \pmb { \theta } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( f _ { \pmb { \theta } } ( X _ { i } ) - y _ { i } ) ^ { 2 } .
+$$
+
+Then we can obtain the high-scoring design $X _ { h }$ by $T$ gradient ascent steps:
+
+$$
+{ \mathbf { } } X _ { t + 1 } = X _ { t } + \eta \nabla _ { { \mathbf { } } X } f _ { \theta ^ { * } } ( X ) | _ { X = X _ { t } } , \quad { \mathrm { f o r ~ } } t \in [ 0 , T - 1 ] ,
+$$
+
+where the high-scoring design $X _ { h }$ can be obtained as $X _ { T }$ .
+
+Considering the discrete nature of biological sequences, the input of $f _ { \theta } ( \cdot )$ should be discrete one-hot vectors. Following (Norn et al., 2021), we can perform the conversion and predict the score via:
+
+$$
+\begin{array} { r l } & { \hat { X } _ { i } = s o f t m a x ( { X } _ { i } ) , } \\ & { \quad { Z } _ { i } = o n e h o t ( a r g m a x ( \hat { X } _ { i } ) ) , } \\ & { \quad \hat { y } = f _ { \pmb { \theta } } ( { Z } _ { i } ) . } \end{array}
+$$
+
+Then the gradient regarding $X _ { i }$ can be approximated as,
+
+$$
+\frac { d f _ { \pmb \theta } ( Z _ { i } ) } { d { \pmb x } _ { i } } \approx \frac { d f _ { \pmb \theta } ( Z _ { i } ) } { d z _ { i } } \frac { d \hat { \pmb x } _ { i } } { d { \pmb x } _ { i } } ,
+$$
+
+where we unroll the matrices $X _ { i } , \hat { X } _ { i }$ and $\boldsymbol { Z } _ { i }$ as vectors $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , $\hat { \mathbf { x } } _ { i }$ and $z _ { i }$ for notational convenience. This approximation allows us to use backpropagation directly from the proxy to the sequence design $X _ { i }$ . For brevity, we will still use $f _ { \pmb { \theta } } ( \pmb { X } _ { i } )$ to represent the proxy.
+
+# 2.4 BIDIRECTIONAL LEARNING
+
+As shown in Figure 1, bidirectional learning (Chen et al., 2022), consists of two mappings: the forward mapping leverages the static dataset $\cdot$ to predict the score $\cdot$ of the high-scoring
+
+design $\cdot$ , and the backward mapping leverages the high-scoring design data $( X _ { h } , y _ { h } )$ to predict the static dataset $( X _ { l } , y _ { l } )$ . The forward mapping loss can be written as
+
+$$
+\mathcal { L } _ { l 2 h } ( \boldsymbol { X } _ { h } ) = \| \boldsymbol { y } _ { h } - \boldsymbol { f } _ { \theta ^ { * } } ^ { l } ( \boldsymbol { X } _ { h } ) \| ^ { 2 } ,
+$$
+
+where $\cdot$ is given by
+
+$$
+-
+$$
+
+where $\cdot$ is a regularization parameter. The backward mapping loss can be written as
+
+$$
+\mathcal { L } _ { h 2 l } ( \boldsymbol { X } _ { h } ) = \| \boldsymbol { y } _ { l } - f _ { \theta ^ { * } ( \boldsymbol { X } _ { h } ) } ^ { h } ( \boldsymbol { X } _ { l } ) \| ^ { 2 } ,
+$$
+
+where $\cdot$ is given by
+
+$$
+-
+$$
+
+The high-scoring design $\cdot$ can be optimized by minimizing the bidirectional learning loss $\mathcal { L } ( X _ { h } ) =$ $\mathcal { L } _ { l 2 h } ( X _ { h } ) + \mathcal { L } _ { h 2 l } ( X _ { h } )$ .
+
+# 3 METHOD
+
+In this section, we first illustrate how to leverage deep linearization to compute the bidirectional learning loss in a closed form. Subsequently, we introduce a hyperparameter $\gamma$ to control the balance between the forward mapping and the backward mapping. We then develop a novel bi-level optimization framework Adaptive- $\gamma$ , which leverages a weak supervision signal from an auxiliary model to effectively update $\gamma$ . Last but not least, we extend this framework to Adaptive- $\eta$ , which enables us to adapt the learning rate $\eta$ for all gradient-based offline model-based algorithms. We summarize our method in Algorithm 1.
+
+# 3.1 DEEP LINEARIZATION FOR BIDIRECTIONAL LEARNING
+
+In bidirectional learning, the backward mapping loss is intractable for a finite neural network, so Chen et al. (2022) employ a neural network with infinite width, which yields a closed-form loss via the NTK. This however makes it impossible to incorporate the rich biophysical information that has been distilled into a pre-trained LM (Yang & Hu, 2021). Considering this, we construct a proxy model by combining a finite-width pre-trained LM with an additional layer. We then linearize the resultant proxy model, inspired by the recent progress in deep linearization which has established that an overparameterized DNN model is close to its linearization (Achille et al., 2021; Dukler et al., 2022).
+
+Denote by $\pmb { \theta _ { 0 } } = ( \pmb { \theta _ { p t } } , \pmb { \theta _ { i n i t } ^ { l i n } } ) \in \mathcal { R } ^ { D \times 1 }$ the proxy model parameters derived by combining the parameters of the pre-trainedpaper, we adopt the pre-traine $\operatorname { L M } \theta _ { p t }$ and a random initialization of the linear layer ERT (Ji et al., 2021) and Prot-BERT (Elnagga $\theta _ { i n i t } ^ { l i n }$ . In thisl., 2021) models, and compute the average of token embeddings as the extracted feature, which is fed into the linear layer to build the proxy. Then we can construct a linear approximation for the proxy model:
+
+$$
+f _ { \pmb \theta } ( \pmb X ) \approx f _ { \pmb \theta _ { 0 } } ( \pmb X ) + \nabla \pmb \theta \ = f _ { \pmb \theta _ { 0 } } ( \pmb X ) \cdot ( \pmb \theta - \pmb \theta _ { 0 } ) ,
+$$
+
+where $f _ { \theta } ( X ) , f _ { \theta _ { 0 } } ( X ) \in \mathcal { R } , \bigtriangledown _ { \theta } f _ { \theta _ { 0 } } ( X ) \in \mathcal { R } ^ { 1 \times L }$ and $\_$ . Intuitively, if the fine-tuning does not significantly change $\pmb { \theta _ { 0 } }$ , then this linearization is a good approximation. By leveraging this linearization, we can obtain a closed-form solution for Eq.(12) as:
+
+$$
+\pmb { \theta } ^ { \ast } ( \pmb { X } _ { h } ) = ( \nabla \theta f _ { \theta _ { 0 } } ( \pmb { X } _ { h } ) ^ { \top } \nabla \theta ~ f _ { \theta _ { 0 } } ( \pmb { X } _ { h } ) + \beta \pmb { I } ) ^ { - 1 } \nabla \theta ~ f _ { \theta _ { 0 } } ( \pmb { X } _ { h } ) \top ( y _ { h } - f _ { \theta _ { 0 } } ( \pmb { X } _ { h } ) ) + \theta _ { 0 } .
+$$
+
+Building on this result, we can compute the bidirectional learning loss as:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { b i } ( \pmb { X } _ { h } ) = \frac { 1 } { 2 } ( \| y _ { h } - \pmb { K } _ { X _ { h } X _ { l } } ( \pmb { K } _ { X _ { l } X _ { l } } + \beta \pmb { I } ) ^ { - 1 } ( y _ { l } - f _ { \pmb { \theta _ { 0 } } } ( \pmb { X } _ { l } ) ) \| ^ { 2 } } \\ { \displaystyle + \| y _ { l } - \pmb { K } _ { X _ { l } X _ { h } } ( \pmb { K } _ { X _ { h } X _ { h } } + \beta \pmb { I } ) ^ { - 1 } ( y _ { h } - f _ { \pmb { \theta _ { 0 } } } ( \pmb { X } _ { h } ) ) \| ^ { 2 } ) , } \end{array}
+$$
+
+where $K ( X _ { i } , X _ { j } ) = \bigtriangledown \theta \ f _ { \theta _ { 0 } } ( X _ { i } ) \top \bigtriangledown \theta \ f _ { \theta _ { 0 } } ( X _ { j } )$ . Following (Dukler et al., 2022), we can also only linearize the last layer of the network for simplicity, which defines the following kernel,
+
+$$
+\begin{array} { r } { K ( X _ { i } , X _ { j } ) = B E R T ( X _ { i } ) ^ { \top } B E R T ( X _ { j } ) , } \end{array}
+$$
+
+where $B E R T ( \pmb { X } )$ denotes the feature of the sequence $\boldsymbol { X }$ extracted by BERT. Its kernel nature makes this approach suitable for small-data tasks (Arora et al., 2020), especially in drug discovery where the labeling cost of DNA/proteins is high.
+
+Algorithm 1 Bidirectional Learning for Offline Model-based Biological Sequence Design   
+
+<table><tr><td>Input: Static dataset D =(Xt, yt),predefined target score yh = 1O, # of iterations T, pre-trained biological LM parameterized by 0o,auxiliary model faux(*),regularization /β. Output: High-scoring design X*.</td></tr><tr><td>1: Initialize Xo as the sequence with the highest score in D</td></tr><tr><td>2: forT←0toT-1do</td></tr><tr><td>3: Leverage Adaptive-γ in Sec 3.2 to update the balance γ by Eq. (21)</td></tr><tr><td>4: if Adapt learning rate then</td></tr><tr><td>5: Leverage Adaptive-n in Sec 3.3 to update the learning rate η by Eq. (23)</td></tr><tr><td>6: Optimize X by minimizing the bidirectional learning loss Lbi(X,,γ) in Eq. (17): 7:</td></tr><tr><td>Xr+1= X,-nOPT(VxLbi(X.,γ)) 8: Return X* = XT</td></tr></table>
+
+# 3.2 ADAPTIVE- $\gamma$
+
+The forward mapping and the backward mapping play different roles in the sequence optimization process: the forward mapping encourages the high-scoring sequence to search for a higher target score (exploitation) and the backward mapping serves as a constraint. Since different sequences require different degrees of constraint, we introduce an extra hyperparameter $\gamma \in [ 0 , 1 ]$ to control the balance between the corresponding terms in the loss function:
+
+$$
+\mathcal { L } _ { b i } ( \boldsymbol { X } _ { h } , \gamma ) = \gamma \mathcal { L } _ { l 2 h } ( \boldsymbol { X } _ { h } ) + ( 1 - \gamma ) \mathcal { L } _ { h 2 l } ( \boldsymbol { X } _ { h } ) .
+$$
+
+Thus $\gamma = 1 . 0$ corresponds to the forward mapping alone, $\gamma = 0$ results in backward mapping, and $\gamma = 0 . 5$ leads to the bidirectional loss of (Chen et al., 2022).
+
+It is non-trivial to determine the most suitable value for $\gamma$ since we do not know the ground-truth score for a new design. One possible solution is to train an auxiliary $f _ { a u x } ( \cdot )$ to serve as a proxy evaluation. A reasonable auxiliary is a simple regression model fitted to the offline dataset. Although this auxiliary model cannot yield ground-truth scores, it can provide weak supervision signals to update $\gamma$ , since the auxiliary model and the bidirectional learning provide complementary information. This is similar to co-teaching (Han et al., 2018) where two models leverage each other’s view.
+
+Formally, we introduce the Adaptive- $\gamma$ framework. Given a good choice of $\gamma$ , the produced $X _ { h }$ is expected to have a high score $f _ { a u x } ( \pmb { X } _ { h } )$ , based on which we can choose $\gamma$ . To make the search for $\gamma$ more efficient, we can formulate this process as a bi-level optimization problem:
+
+$$
+\begin{array} { r } { \gamma ^ { * } = \underset { \gamma } { \arg \operatorname* { m a x } } f _ { a u x } \big ( X _ { h } ^ { * } ( \gamma ) \big ) , \ } \\ { \mathrm { s . t . } \quad X _ { h } ^ { * } ( \gamma ) = \underset { X _ { h } } { \arg \operatorname* { m i n } } \mathcal { L } _ { b i } \big ( X _ { h } , \gamma \big ) . \ } \end{array}
+$$
+
+We can then use the hyper-gradient $\frac { \partial f _ { a u x } ( \pmb { X } _ { h } ^ { * } ( \gamma ) ) } { \partial \gamma }$ to update $\gamma$ . Specifically, the inner level solution can be approximated via a gradient descent step with a learning rate $\eta$ :
+
+$$
+\mathbf { } X _ { h } ^ { * } ( \gamma ) = X _ { h } - \eta \frac { d \mathcal { L } _ { b i } ( X _ { h } , \gamma ) } { d X _ { h } ^ { \top } } .
+$$
+
+For the outer level, we update $\gamma$ by hyper-gradient ascent:
+
+$$
+\gamma = \gamma + \eta ^ { ' } \frac { d f _ { a u x } ( \pmb { X } _ { h } ^ { * } ( \gamma ) ) } { d \gamma } = \gamma + \eta ^ { ' } \frac { d f _ { a u x } ( \pmb { X } _ { h } ) } { d \pmb { x } _ { h } } \frac { d \mathcal { L } _ { b i } ( \pmb { X } _ { h } , \gamma ) } { d \pmb { x } _ { h } ^ { \top } } ,
+$$
+
+where we unroll the matrix $X _ { h }$ as a vector $\scriptstyle { \mathbf { 2 } } \mathbf { } \boldsymbol { { k } } _ { h }$ for better illustration.
+
+# 3.3 ADAPTIVE- $\eta$
+
+We now extend the Adaptive- $\gamma$ framework to Adaptive- $\eta$ . As the first learning rate adaptation module for offline model-based optimization, Adaptive- $\eta$ is compatible with all gradient-based algorithms and can effectively finetune the learning rate $\eta$ via the auxiliary model’s weak supervision signal. All gradient-based methods that maximize $\mathcal { L } _ { \boldsymbol { \theta } } ( X )$ with respect to $\boldsymbol { X }$ have the following general form:
+
+$$
+X _ { t + 1 } = X _ { t } + \eta O P T ( \nabla _ { X } \mathcal { L } _ { \theta } ( X ) | _ { X = X _ { t } } ) , \quad \mathrm { f o r } t \in [ 0 , \mathrm { T } - 1 ] ,
+$$
+
+where $\eta$ represents the learning rate of the optimizer. For methods such as simple gradient ascent (Grad), COMs (Trabucco et al., 2021), ROMA ( $\mathrm { Y u }$ et al., 2021) and NEMO (Fu & Levine, 2021), $\mathcal { L } _ { \boldsymbol { \theta } } ( \cdot )$ is related to the proxy model $f _ { \theta } ( \cdot )$ ; for BDI Chen et al. (2022) and our proposed method, BIB, $\mathcal { L } _ { \boldsymbol { \theta } } ( \cdot )$ is the negative of the bidirectional learning loss, i.e., $\mathcal { L } _ { \theta } = - \mathcal { L } _ { b i }$ .
+
+Though the learning rate $\eta$ can be adapted in some optimizers such as Adam (Kingma & Ba, 2015), these adaptations rely on only the past optimization history and do not consider the weak supervision signal from the auxiliary model. Our Adaptive- $\eta$ optimizes $\eta$ by solving:
+
+$$
+\eta ^ { * } = \arg \operatorname* { m a x } _ { \eta } f _ { a u x } ( X _ { h } ^ { * } ( \eta ) ) ,
+$$
+
+where $\eta$ can be updated via gradient ascent methods. Considering the sequence optimization procedure is highly sensitive to the learning rate $\eta$ , we reset $\eta$ to $\eta _ { 0 }$ at each iteration and update $\eta$ from $\eta _ { 0 }$ ,
+
+$$
+\eta = \eta _ { 0 } - \eta ^ { ' } \frac { d f _ { a u x } ( X _ { h } ^ { * } ( \eta ) ) } { d \eta } .
+$$
+
+In general, this serves to stabilize the optimization procedure.
+
+# 4 EXPERIMENTS
+
+We conduct extensive experiments on DNA and protein design tasks, and aim to answer three research questions: (1) How does BIB compare with state-of-the-art algorithms? (2) Is every design component necessary in BIB? (3) Does the Adaptive- $\eta$ module improve gradient-based methods?
+
+# 4.1 BENCHMARK
+
+We conduct experiments on two DNA tasks: TFBind8(r) and TFBind10(r), following (Chen et al., 2022) and three protein tasks: avGFP, AAV and E4B, in (Ren et al., 2022) which have the most data points. See See Appendix A.2 for details.
+
+Following (Trabucco et al., 2021), we select the top $N = 1 2 8$ most promising sequences for each comparison method. Among these sequences, we report the maximum normalized ground truth score as the evaluation metric following (Ren et al., 2022).
+
+# 4.2 COMPARISON METHODS
+
+We compare BIB with two groups of baselines: the gradient-based methods and the non-gradientbased methods. For a fair comparison, the pre-trained LM is used for all methods involving a proxy and we don’t finetune the LM. The gradient-based methods include: 1) Grad: gradient ascent on existing sequences to obtain new sequences; 2) COMs (Trabucco et al., 2021): lower bounds the DNN model by the ground-truth values and then applies gradient ascent; 3) ROMA (Yu et al., 2021): incorporates a smoothness prior into the DNN model before gradient ascent steps; 4) NEMO (Fu & Levine, 2021): leverages the normalized maximum-likelihood estimator to bound the distance between the DNN model and the ground-truth values; 5) BDI (Chen et al., 2022): adopts the infinitely wide neural network and its NTK to yield a closed-form bidirectional learning loss.
+
+The non-gradient-based methods include: 1) BO-qEI (Wilson et al., 2017): builds an acquisition function for sequence exploration; 2) CMA-ES (Hansen, 2006): estimates the covariance matrix to adjust the sequence distribution towards the high-scoring region; 3) AdaLead (Sinai et al., 2020): performs a hill-climbing search on the proxy and then queries the sequences with high predictions; 4) CbAS (Brookes et al., 2019): builds a generative model for sequences above a property threshold and gradually adapts the distribution by increasing the threshold; 5) PEX (Ren et al., 2022): prioritizes the evolutionary search for protein sequences with low mutation counts; 6) GENH (Chan et al., 2021): enhances the score through a learned latent space.
+
+# 4.3 TRAINING DETAILS
+
+We follow the training setting in (Chen et al., 2022) if not specified. We choose $O P T$ as the Adam optimizer (Kingma & Ba, 2015) for all gradient-based methods. We implement the auxiliary model
+
+Table 1: Experimental results (maximum normalized ground truth score) for comparison.   
+
+<table><tr><td>Method</td><td>TFBind8(r)</td><td>TFBind10(r)</td><td>avGFP</td><td>AAV</td><td>E4B</td><td>RankMean</td><td>RankMedian</td></tr><tr><td>D(best)</td><td>0.242</td><td>0.248</td><td>0.742</td><td>0.452</td><td>0.224</td><td></td><td></td></tr><tr><td>BO-qEI</td><td>0.940 ±0.032</td><td>0.595 ±0.028</td><td>1.700 ± 0.020</td><td>0.591 ± 0.002</td><td>0.436±0.004</td><td>6.0/12</td><td>7.0/12</td></tr><tr><td>CMA-ES</td><td>0.930 ±0.034</td><td>0.617±0.031</td><td>5.488 ± 0.056</td><td>0.470±0.006</td><td>0.748 ± 0.009</td><td>6.0/12</td><td>6.0/12</td></tr><tr><td>AdaLead</td><td>0.941 ± 0.032</td><td>0.602±0.028</td><td>1.611 ± 0.009</td><td>0.581 ± 0.002</td><td>0.433 ± 0.003</td><td>6.2/12</td><td>8.0/12</td></tr><tr><td>CbAS</td><td>0.878± 0.049</td><td>0.610 ±0.035</td><td>1.371 ± 0.016</td><td>0.543± 0.002</td><td>0.349 ± 0.003</td><td>8.6/12</td><td>10.0/12</td></tr><tr><td>PEX</td><td>0.924 ± 0.041</td><td>0.612 ± 0.026</td><td>1.546 ± 0.019</td><td>0.588±0.002</td><td>0.397 ± 0.004</td><td>7.2/12</td><td>8.0/12</td></tr><tr><td>GENH</td><td>0.323 ± 0.000</td><td>0.448 ±0.000</td><td>0.835 ±0.000</td><td>0.452 ± 0.000</td><td>0.228 ±0.000</td><td>11.4/12</td><td>11.0/12</td></tr><tr><td>Grad</td><td>0.941±0.026</td><td>0.630±0.029</td><td>4.869± 0.042</td><td>0.463±0.005</td><td>1.219 ± 0.061</td><td>4.8/12</td><td>4.0/12</td></tr><tr><td>COMs</td><td>0.921 ± 0.039</td><td>0.637 ± 0.065</td><td>3.873±0.080</td><td>0.511 ± 0.005</td><td>0.829 ± 0.026</td><td>5.6/12</td><td>5.0/12</td></tr><tr><td>ROMA</td><td>0.926 ± 0.032</td><td>0.634±0.061</td><td>5.621 ± 0.143</td><td>0.471 ± 0.005</td><td>1.198 ± 0.042</td><td>4.8/12</td><td>4.0/12</td></tr><tr><td>NEMO</td><td>0.930 ±0.038</td><td>0.632 ±0.024</td><td>4.624 ± 0.087</td><td>0.505 ± 0.005</td><td>1.036 ± 0.046</td><td>5.0/12</td><td>5.0/12</td></tr><tr><td>BDI</td><td>0.823 ± 0.000</td><td>0.678 ±0.000</td><td>0.742 ± 0.000</td><td>0.452 ±0.000</td><td>0.224 ± 0.000</td><td>9.4/12</td><td>11.0/12</td></tr><tr><td>BIB(ours)</td><td>0.952±0.033</td><td>0.639±0.032</td><td>8.084±0.224</td><td>0.501±0.007</td><td>1.255 ± 0.029</td><td>2.4/12</td><td>1.0/12</td></tr></table>
+
+as a linear layer with the feature from the pre-trained LM. We set the number of iterations $T$ as 25 for all experiments following (Norn et al., 2021) and $\eta _ { 0 }$ as 0.1 following (Chen et al., 2022). We run every setting over 16 trials and report the average score. See Appendix A.3 for other details.
+
+# 4.4 RESULTS AND ANALYSIS
+
+We report all experimental results in Table 1 and plot the ranking statistics in Figure 2. We make the following observations. (1) As shown in Table 1, BIB consistently outperforms the Grad method on all tasks, which demonstrates that our BIB can effectively mitigate the out-of-distribution issue. (2) Furthermore, BIB outperforms BDI on 4 out of 5 tasks, which demonstrates the effectiveness of the pre-trained biological LM over NTK. The reason why BDI outperforms BIB on TFBind10(r) may be that short sequences do not rely much on the rich sequential information from the pre-trained LM. (3) As shown in Figure 2, the gradient-based methods generally perform better than the non-gradient-based methods, as also observed by Trabucco et al. (2021). (4) The gradient-based methods are inferior for the AAV task.
+
+One possible reason is that the design space of AAV $( 2 0 ^ { 2 8 } )$ is much smaller than those of avGFP $( 2 0 ^ { \overline { { 2 3 9 } } } )$ and E4B $( 2 0 ^ { 1 0 2 } )$ ), which makes the generative modeling and evolutionary algorithms more suitable. (5) This conjecture is also supported by the experimental results on two DNA design tasks. We compute the average ranking of gradient-based methods and non-grad-based methods on TFBind10(r) as 3.5 and 9.5, respectively, and the average ranking of gradient-based methods and non-grad-based methods on TFBind8(r) as 5.8 and 6.8, respectively. The advantage of gradient-based methods are larger $\_$ ) in TFBind10(r) than that $( 6 . 8 - 5 . 8 = 1 . 0 $ ) in TFBind8(r). (6) The generative modeling methods CbAS and GENH yield poor results on all tasks, probably because the high-dimensional data distribution is very hard to model. (7) Overall, BIB attains the best performance in 3 out of 5 tasks and achieves the best ranking results as shown in Table 1 and Figure 2.
+
+![](images/bdf247aebf8807d352520357212f8e27815284c3bf54c3eec97bfd414aa78b80.jpg)  
+Figure 2: Rank minima and maxima are represented by whiskers; vertical lines and black triangles denote medians and means.
+
+We also visualize the trend of performance (the maximum normalized ground truth score) and tradeoff $\gamma$ as a function of $T$ on TFBind8(r) in Figure 3(a) and avGFP in Figure 3(b). The performance generally increases with the time step $T$ and then stabilizes, which demonstrates the effectiveness and robustness of BIB. Furthermore, we find that the $\gamma$ values of TFBind8(r) and avGFP generally increase at first. This means that BIB reduces the impact of the constraint to encourage a more aggressive search for a high target value during the initial phase. Then $\gamma$ of TFBind8(r) continues to increase while the $\gamma$ of avGFP decreases. We conjecture that the difference is caused by the sequence length. Small mutations of a biological sequence are enough to yield a good candidate (Ren et al., 2022). For the length-239 protein in avGFP, dramatic mutations 1) are not necessary and 2) can easily lead to out-of-distribution points. The weak supervision signal from the auxiliary model therefore encourages a tighter constraint towards the static dataset. By contrast, the DNA sequence is relatively short and a more widespread search of the sequence space can yield better results. To investigate this conjecture, we further visualize the trend of E4B in Figure 3(c). E4B also has long sequences (102) and we can observe its similar first-increase-then-decrease trend, although it is not as pronounced.
+
+![](images/7d142864a62a1b851f634fc7127f7a819602f9745b5146073217ada0d9fac99a.jpg)  
+Figure 3: Trend of performance and trade-off $\gamma$ as a function of $T$ .
+
+Table 2: Ablation studies on BIB components.   
+
+<table><tr><td>Task</td><td>γ=0.0</td><td>γ=1.0</td><td>γ=0.5</td><td>γ=0.5+Joint</td><td>BIB</td><td>BIB +Ada-n</td></tr><tr><td>TFBind8(r)</td><td>0.936</td><td>0.933</td><td>0.947</td><td>0.935</td><td>0.952</td><td>0.956</td></tr><tr><td>TFBind10(r)</td><td>0.611</td><td>0.637</td><td>0.616</td><td>0.622</td><td>0.639</td><td>0.639</td></tr><tr><td>avGFP</td><td>6.051</td><td>7.588</td><td>7.940</td><td>7.920</td><td>8.084</td><td>8.197</td></tr><tr><td>AAV</td><td>0.449</td><td>0.458</td><td>0.480</td><td>0.420</td><td>0.501</td><td>0.525</td></tr><tr><td>E4B</td><td>0.778</td><td>0.903</td><td>1.198</td><td>1.176</td><td>1.255</td><td>1.301</td></tr></table>
+
+# 4.5 ABLATION STUDIES
+
+In this subsection, we conduct ablation studies to verify the effectiveness of the forward mapping, the backward mapping, and the Adaptive- $\gamma$ module of BIB. We report the experimental results in Table 2.
+
+Forward mapping & Backward mapping. We can observe that bidirectional learning $( \gamma = 0 . 5 )$ performs better than both forward mapping $( \gamma = 1 . 0 $ ) and backward mapping $( \gamma = 0 . 0 $ ) alone in most tasks, which demonstrates the effectiveness of forward mapping and backward mapping. The advantage of bidirectional mappings over the forward mapping is larger in the long-sequence tasks like avGFP (238) and E4B (102) compared with the short-sequence tasks. A possible explanation is that the constraint is more important for long sequence tasks than short sequence design since the search space is large and many mutations can easily go out of distribution.
+
+Adaptive- $\gamma$ . BIB learns $\gamma$ and this leads to improvements over bidirectional mappings $( \gamma = 0 . 5 )$ for all tasks, verifying the effectiveness of Adaptive- $\gamma$ . We also consider the following variant,
+
+$$
+X ^ { \ast } = \arg \operatorname* { m i n } _ { X _ { h } } \mathcal { L } _ { b i } ( X _ { h } , 0 . 5 ) - f _ { a u x } ( X _ { h } ) ,
+$$
+
+which jointly optimizes the bidirectional learning loss $\mathcal { L } _ { b i } ( X _ { h } , 0 . 5 )$ and the auxiliary term $f _ { a u x } ( \pmb { X } _ { h } )$ We found this yields similar or even worse results than pure bidirectional learning. The reason may be that the weak supervision signal from $f _ { a u x } ( \pmb { X } _ { h } )$ can serve as a guide to update the scalar $\gamma$ but not as a component of the main optimization objective that directly updates the sequence.
+
+In the final column of Table 2, we examine the performance of the Adaptive- $\eta$ module. Adding this module leads to improvements on all five tasks, which demonstrates its effectiveness.
+
+# 4.6 ADAPTIVE- $\eta$
+
+In this subsection, we aim to further demonstrate the effectiveness of the Adaptive- $\eta$ module on all six gradient-based methods. We conduct experiments on two tasks: TFBind8(r) and avGFP. Since the use of the infinitely wide neural network leads to poor performance for BDI, we modify its implementation via deep linearization so that it can make use of the pre-trained LM.
+
+As shown in Table 3, Adaptive- $\eta$ provides a consistent gain for all scenarios, which demonstrates the widespread applicability and effectiveness of the module. Furthermore, Adaptive- $\eta$ leads to a maximum improvement of $1 . 4 \%$ in TFBind8(r) and $1 2 . 5 \%$ in avGFP. ROMA is the algorithm that benefits the most. One possible explanation is that ROMA incorporates a local smoothness prior that leads to more stable gradients, with which Adaptive- $\eta$ can be more effective. Similar to Sec 4.5, we
+
+Table 3: Adaptive- $\eta$ on all gradient-based methods.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="6">TFBind8(r)</td><td colspan="6">avGFP</td></tr><tr><td>Grad</td><td>COMs</td><td>ROMA</td><td>NEMO</td><td>BDI</td><td>BIB</td><td>Grad</td><td>COMs</td><td>ROMA</td><td>NEMO</td><td>BDI</td><td>BIB</td></tr><tr><td>Normal</td><td>0.941</td><td>0.921</td><td>0.926</td><td>0.930</td><td>0.947</td><td>0.952</td><td>4.869</td><td>3.873</td><td>5.621</td><td>4.624</td><td>7.940</td><td>8.084</td></tr><tr><td>Joint</td><td>0.941</td><td>0.921</td><td>0.931</td><td>0.932</td><td>0.935</td><td>0.925</td><td>4.869</td><td>3.836</td><td>6.438</td><td>3.078</td><td>7.920</td><td>7.823</td></tr><tr><td>Gain</td><td>0.000</td><td>0.000</td><td>0.005</td><td>0.002</td><td>-0.008</td><td>-0.027</td><td>0.000</td><td>-0.037</td><td>0.116</td><td>-1.546</td><td>-0.020</td><td>-0.261</td></tr><tr><td>Ada-n</td><td>0.941</td><td>0.928</td><td>0.939</td><td>0.935</td><td>0.951</td><td>0.956</td><td>5.235</td><td>4.027</td><td>6.322</td><td>4.658</td><td>7.966</td><td>8.197</td></tr><tr><td>Gain</td><td>0.000</td><td>0.007</td><td>0.013</td><td>0.005</td><td>0.004</td><td>0.004</td><td>0.366</td><td>0.154</td><td>0.701</td><td>0.034</td><td>0.026</td><td>0.113</td></tr></table>
+
+consider the following variant,
+
+$$
+\boldsymbol { X } ^ { * } = \arg \operatorname* { m a x } _ { \boldsymbol { X } _ { h } } \mathcal { L } _ { \boldsymbol { \theta } } ( \boldsymbol { X } _ { h } ) + f _ { a u x } ( \boldsymbol { X } _ { h } ) ,
+$$
+
+which performs joint optimization instead of bi-level optimization on two objectives. As shown in Table 3, joint optimization generally deteriorates the performance. This again verifies that the auxiliary model can only serve as a guide instead of contributing to the main objective.
+
+# 5 RELATED WORK
+
+Biological sequence design. There has been a wide range of algorithms for biological sequence design. Evolutionary algorithms (Sinai et al., 2020; Ren et al., 2022) leverage the learned surrogate model to provide evolution guidance towards the high-scoring region. Angermueller et al. (2019) propose a flexible reinforcement learning framework where sequence design is a sequential decisionmaking problem. Bayesian optimization methods propose candidate solutions via an acquisition function (Terayama et al., 2021). Deep generative model methods design sequences in the latent space (Chan et al., 2021) or gradually adapt the distribution towards the high-scoring region (Brookes et al., 2019). GFlowNets (Jain et al., 2022) amortize the cost of search over learning and encourage diversity. Gradient-based methods leverage a surrogate model and its gradient information to maximize the desired property (Chen et al., 2022; Norn et al., 2021; Tischer et al., 2020; Linder & Seelig, 2020). Our proposed BIB belongs to the last category and leverages the rich biophysical information (Ji et al., 2021; Elnaggar et al., 2021) to directly optimize the biological sequence.
+
+Offline model-based optimization. A majority of sequence design algorithms (Angermueller et al., 2019; Sinai et al., 2020; Ren et al., 2022) focus on the online setting where wet-lab experimental results in the current round are analyzed to propose candidates in the next round. The problem of this setting is that wet-lab experiments are often very expensive, and thus a pure data-driven, offline approach is attractive and has received substantial research attention recently (Trabucco et al., 2022; Kolli et al., 2022). Gradient-based methods have proven to be effective (Trabucco et al., 2021; Yu et al., 2021; Fu & Levine, 2021; Chen et al., 2022). Among these algorithms, Chen et al. (2022) propose bidirectional mappings to distill information from the static dataset into a high-scoring design, which achieves state-of-the-art performances on a variety of tasks. However, this bidirectional learning is designed for general tasks, like robot and material design, and the rich biophysical information in millions of biological sequences is ignored. In this paper, we leverage recent advances in deep linearization to incorporate the rich biophysical information into bidirectional learning.
+
+# 6 CONCLUSION
+
+In this paper, we propose bidirectional learning for offline model-based biological sequence. Our work is built on the recently proposed bidirectional learning approach (Chen et al., 2022), which is designed for general inputs and relies on the NTK of an infinitely wide network to yield a closed-form loss computation. Though effective, the NTK cannot learn features. We build a proxy model using the pre-trained LM model with a linear head and apply the deep linearization scheme to the proxy, which can yield a closed-form loss and incorporate the wealth of biophysical information at the same time. In addition, we propose Adaptive- $\gamma$ to maintain a proper balance between the forward mapping and the backward mapping by leveraging the weak supervision signal from an auxiliary model. Based on this framework, we further propose Adaptive- $\eta$ , the first learning rate adaptation strategy compatible with all gradient-based offline model-based algorithms. Experimental results on DNA and protein sequence design tasks verify the effectiveness of BIB and Adaptive- $\eta$ .
+
+# 7 ETHICS STATEMENT
+
+Protein sequence design aims to find a protein sequence with a particular biological function, which has a broad application scope. This can lead to improved drugs that are highly beneficial to society. For instance, designing the antibody protein for SARS-COV-2 can potentially save millions of human lives (Kumar et al., 2021) and designing novel anti-microbial peptides (short protein sequences) is central to tackling the growing public health risks caused by anti-microbial resistance (Murray et al., 2022). Unfortunately, it is possible to direct the research results towards harmful purposes such as the design of biochemical weapons. As researchers, we believe that we must be aware of the potential harm of any research outcomes, and carefully consider whether the possible benefits outweigh the risks of harmful consequences. We also must recognize that we cannot control how the research may be used. In the case of this paper, we are confident that there is a much greater chance that the research outcomes will have a beneficial effect. We do not consider that there are any immediate ethical concerns with the research endeavour.
+
+# 8 REPRODUCIBILITY STATEMENT
+
+We provide the code implementation of BIB and Adaptive- $\eta$ here and we also attach the code in the supplementary material. We describe the DNA/protein benchmarks in Sec. 4.1 and the training details in Sec. 4.3. We also explain how to obtain the sequence embedding from the pre-trained LM and how to perform gradient ascent steps on the sequence in Sec. 2.
+
+# REFERENCES
+
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+
+Christof Angermueller, David Dohan, David Belanger, Ramya Deshpande, Kevin Murphy, and Lucy Colwell. Model-based reinforcement learning for biological sequence design. In Proc. Int. Conf. Learning Rep. (ICLR), 2019.
+
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+
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+
+David Brookes, Hahnbeom Park, and Jennifer Listgarten. Conditioning by adaptive sampling for robust design. In Proc. Int. Conf. Machine Learning (ICML), 2019.
+
+Drew H Bryant, Ali Bashir, Sam Sinai, Nina K Jain, Pierce J Ogden, Patrick F Riley, George M Church, Lucy J Colwell, and Eric D Kelsic. Deep diversification of an AAV capsid protein by machine learning. Nature Biotechnology, 2021.
+
+Alvin Chan, Ali Madani, Ben Krause, and Nikhil Naik. Deep extrapolation for attribute-enhanced generation. Proc. Adv. Neur. Inf. Proc. Syst (NeurIPS), 2021.
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+Can Chen, Yingxue Zhang, Jie Fu, Xue Liu, and Mark Coates. Bidirectional learning for offline infinite-width model-based optimization. Proc. Adv. Neur. Inf. Proc. Syst (NeurIPS), 2022.
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+Ahmed Elnaggar, Michael Heinzinger, Christian Dallago, Ghalia Rehawi, Yu Wang, Llion Jones, Tom Gibbs, Tamas Feher, Christoph Angerer, Martin Steinegger, et al. ProtTrans: towards cracking the language of lifes code through self-supervised deep learning and high performance computing. IEEE Trans. Pattern Analysis and Machine Intelligence, 2021.
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+Justin Fu and Sergey Levine. Offline model-based optimization via normalized maximum likelihood estimation. Proc. Int. Conf. Learning Rep. (ICLR), 2021.
+
+Bo Han, Quanming Yao, Xingrui Yu, Gang Niu, Miao Xu, Weihua Hu, Ivor Tsang, and Masashi Sugiyama. Co-teaching: robust training of deep neural networks with extremely noisy labels. Proc. Adv. Neur. Inf. Proc. Syst (NeurIPS), 2018.
+
+Nikolaus Hansen. The CMA evolution strategy: a comparing review. Towards A New Evolutionary Computation, 2006.
+
+Moksh Jain, Emmanuel Bengio, Alex Hernandez-Garcia, Jarrid Rector-Brooks, Bonaventure FP Dossou, Chanakya Ajit Ekbote, Jie Fu, Tianyu Zhang, Michael Kilgour, Dinghuai Zhang, et al. Biological sequence design with GFlowNets. In Proc. Int. Conf. Machine Learning (ICML), 2022.
+
+Yanrong Ji, Zhihan Zhou, Han Liu, and Ramana V Davuluri. DNABERT: pre-trained bidirectional encoder representations from transformers model for DNA-language in genome. Bioinformatics, 2021.
+
+Nathan Killoran, Leo J Lee, Andrew Delong, David Duvenaud, and Brendan J Frey. Generating and designing DNA with deep generative models. arXiv preprint arXiv:1712.06148, 2017.
+
+Diederik P Kingma and Jimmy Ba. Adam: a method for stochastic optimization. In Proc. Int. Conf. Learning Rep. (ICLR), 2015.
+
+Sathvik Kolli, Amy X. Lu, Xinyang Geng, Aviral Kumar, and Sergey Levine. Data-driven optimization for protein design: workflows, algorithms and metrics. In ICLR2022 Machine Learning for Drug Discovery, 2022.
+
+Vikash Kumar, Haiguang Liu, and Chun Wu. Drug repurposing against SARS-CoV-2 receptor binding domain using ensemble-based virtual screening and molecular dynamics simulations. Computers in Biology and Medicine, 2021.
+
+Johannes Linder and Georg Seelig. Fast differentiable DNA and protein sequence optimization for molecular design. arXiv preprint arXiv:2005.11275, 2020.
+
+Johannes Linder and Georg Seelig. Fast activation maximization for molecular sequence design. BMC Bioinformatics, 2021.
+
+Christopher JL Murray, Kevin Shunji Ikuta, Fablina Sharara, Lucien Swetschinski, Gisela Robles Aguilar, Authia Gray, Chieh Han, Catherine Bisignano, Puja Rao, Eve Wool, et al. Global burden of bacterial antimicrobial resistance in 2019: a systematic analysis. The Lancet, 2022.
+
+Christoffer Norn, Basile IM Wicky, David Juergens, Sirui Liu, David Kim, Doug Tischer, Brian Koepnick, Ivan Anishchenko, David Baker, and Sergey Ovchinnikov. Protein sequence design by conformational landscape optimization. Proceedings of the National Academy of Sciences, 2021.
+
+Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: an imperative style, high-performance deep learning library. Proc. Adv. Neur. Inf. Proc. Syst (NeurIPS), 2019.
+
+Zhizhou Ren, Jiahan Li, Fan Ding, Yuan Zhou, Jianzhu Ma, and Jian Peng. Proximal exploration for model-guided protein sequence design. Proc. Int. Conf. Machine Learning (ICML), 2022.
+
+Karen S Sarkisyan et al. Local fitness landscape of the green fluorescent protein. Nature, 2016.
+
+Sam Sinai, Richard Wang, Alexander Whatley, Stewart Slocum, Elina Locane, and Eric D Kelsic. Adalead: a simple and robust adaptive greedy search algorithm for sequence design. arXiv preprint arXiv:2010.02141, 2020.
+
+Lea M Starita, Jonathan N Pruneda, Russell S Lo, Douglas M Fowler, Helen J Kim, Joseph B Hiatt, Jay Shendure, Peter S Brzovic, Stanley Fields, and Rachel E Klevit. Activity-enhancing mutations in an E3 ubiquitin ligase identified by high-throughput mutagenesis. Proceedings of the National Academy of Sciences, 2013.
+
+Kei Terayama, Masato Sumita, Ryo Tamura, and Koji Tsuda. Black-box optimization for automated discovery. Accounts of Chemical Research, 2021.
+
+Doug Tischer, Sidney Lisanza, Jue Wang, Runze Dong, Ivan Anishchenko, Lukas F Milles, Sergey Ovchinnikov, and David Baker. Design of proteins presenting discontinuous functional sites using deep learning. Biorxiv, 2020.
+
+Brandon Trabucco, Aviral Kumar, Xinyang Geng, and Sergey Levine. Conservative objective models for effective offline model-based optimization. In Proc. Int. Conf. Machine Learning (ICML), 2021.
+
+Brandon Trabucco, Xinyang Geng, Aviral Kumar, and Sergey Levine. Design-Bench: benchmarks for data-driven offline model-based optimization. arXiv preprint arXiv:2202.08450, 2022.
+
+James T Wilson, Riccardo Moriconi, Frank Hutter, and Marc Peter Deisenroth. The reparameterization trick for acquisition functions. arXiv preprint arXiv:1712.00424, 2017.
+
+Greg Yang and Edward J Hu. Tensor programs iv: feature learning in infinite-width neural networks. In Proc. Int. Conf. Machine Learning (ICML), 2021.
+
+Sihyun Yu, Sungsoo Ahn, Le Song, and Jinwoo Shin. Roma: robust model adaptation for offline model-based optimization. Proc. Adv. Neur. Inf. Proc. Syst (NeurIPS), 2021.
+
+# A APPENDIX
+
+# A.1 DNA EMBEDDING
+
+To incorporate richer contextual information, the DNA LM Ji et al. (2021) adopts the $k$ -mer sequence representation, which is widely used in DNA sequence analysis. For example, the sequence AT GGCT has its 3-mer representation as $\{ A T G , T G G , G G C , G { \dot { C } } T \}$ . In this paper, we adopt its 3-mer representation and compute the probability of the 3-mer token by multiplying the probabilities of the three individual bases. The 3-mer representation is then sent to the pre-trained DNA LM.
+
+# A.2 DATASET DETAILS
+
+We conduct experiments on two DNA tasks following (Chen et al., 2022) and three protein tasks in (Ren et al., 2022) which have the most data points. We report the dataset details in Table 4.
+
+DNA Task 1 TFBind8(r). The goal is to find a length-8 DNA sequence to maximize the binding activity score with a particular transcription factor, SIX6REFR1 (Barrera et al., 2016). We sample 5000 data points for the offline algorithms following (Chen et al., 2022).
+
+DNA Task 2 TFBind10(r). The task TFBind10(r) is the same as TFBind8(r) except that the goal is to find a length-10 DNA sequence. Both DNA tasks measure the entire search space and we adopt these measurements as the approximate ground-truth evaluation.
+
+Protein Task 1 avGFP. This task aims to find a protein sequence with approximately 239 amino acids to maximize the fluorescence level of Green Fluorescent Proteins (Sarkisyan et al., 2016). The task oracle is constructed by using the full unobserved dataset (around 52,000 points) following (Ren et al., 2022). The oracle passes the average of the residue embeddings from the pre-trained Prot-T5 (Elnaggar et al., 2021) into a linear layer and then fits the dataset. The following two task oracles take the same form. The offline algorithms can only access the lowest-scoring 26,000 data points.
+
+Protein Task 2 AAV. The goal is to engineer a 28-amino acid segment (positions 561–588) of the VP1 protein to remain viable for gene therapy (Bryant et al., 2021). We use the entire 284, 000 data points to build the oracle and the lowest-scoring 142, 000 points for the offline algorithms.
+
+Protein Task 3 E4B. This task aims to design a protein (around 102 amino acids) to maximize the ubiquitination rate to the target protein (Starita et al., 2013). The full dataset consisting of around 100, 000 points is used to build the oracle and the bottom half is used for the offline algorithms.
+
+The parameterization of the oracle is different from that of the regression model from two aspects: 1) model architecture; 2) pre-trained information source. First, the oracle adopts the Prot-T5 model which consists of an encoder and a decoder, while the regression model adopts the Prot-BERT model which only has an encoder. Second, Prot-T5 is trained on the BFD and UniRef100 datasets and ProtBert is trained on the UniRef50 dataset. These two points demonstrate that the oracle and the regression model are different function classes. We choose the Prot-T5 model as the oracle because this is the state-of-the-art protein LM to extract features and recent work Elnaggar et al. (2021) has demonstrated its effectiveness. In order to test how related the Prot-T5 (oracle)/Prot-BERT(proxy) models are, we trained them on a sampled training dataset and compared the test predictions of the testing set. By evaluating the Pearson correlation coefficient (PCC) between the two prediction errors PCC(ProtT5 predictions - test labels, ProtBERT predictions -test labels), we obtain $- 0 . 0 0 5 3$ on avGFP, $- 0 . 0 0 0 5$ on AAV, and $\cdot$ on E4B. These results suggest that the two models are not strongly related in terms of the predictions they form.
+
+Table 4: Dataset details.   
+
+<table><tr><td>Task</td><td>Metric</td><td>Min of D</td><td>Max of D</td><td>Min of Dentire</td><td>Max of Dentire</td></tr><tr><td>TFBind8(r)</td><td>binding activity</td><td>0.000</td><td>0.242</td><td>0.000</td><td>1.000</td></tr><tr><td>TFBind10(r)</td><td>binding activity</td><td>-1.859</td><td>-0.869</td><td>-1.859</td><td>2.129</td></tr><tr><td>avGFP</td><td>fluorescence level</td><td>1.283</td><td>3.553</td><td>1.283</td><td>4.123</td></tr><tr><td>AAV</td><td>viruses viability</td><td>-11.176</td><td>-1.399</td><td>-11.176</td><td>9.536</td></tr><tr><td>E4B</td><td>ubiquitination rate</td><td>-3.589</td><td>-0.984</td><td>-3.589</td><td>8.998</td></tr></table>
+
+Table 5: Experimental results on different pre-trained LMs for comparison.   
+
+<table><tr><td>Pre-trainedLM</td><td>avGFP</td><td>AAV</td><td>E4B</td></tr><tr><td>ProtAlbert</td><td>3.567±0.456</td><td>0.478±0.004</td><td>0.552±0.023</td></tr><tr><td>ProtBert(adopted)</td><td>8.084±0.224</td><td>0.501 ± 0.007</td><td>1.255 ± 0.029</td></tr><tr><td>ProtBert-BFD</td><td>8.240±0.094</td><td>0.549 ± 0.009</td><td>1.880 ± 0.054</td></tr></table>
+
+Following (Trabucco et al., 2021), we select the top $\cdot$ most promising sequences for each comparison method. Among these sequences, we report the maximum normalized ground truth score as the evaluation metric following (Ren et al., 2022).
+
+# A.3 TRAINING DETAILS
+
+We use Pytorch (Paszke et al., 2019) to run all experiments on one V100 GPU. Following the setting in Norn et al. (2021), we introduce a length- $L$ protein sequence as a continuous random matrix $\pmb { X } _ { h } \in R ^ { L \times 2 0 }$ ( $\mathbf { \bar { X } } _ { h } \in R ^ { L \times 4 }$ for DNA), initialized using a normal distribution with the mean 0 and the standard deviation of 0.01. To make this sequence correspond correctly to the candidate sequence, we exchange the largest value in $X [ l , : ]$ with the value in the amino acid index.
+
+# A.4 DIFFERENT PRETRAINED LMS
+
+As shown in Table 5, we have tested the ProtBERT, ProtAlbert, and ProtBert-BFD models and found that better-quality models generally work better. The publicly available pre-trained DNA models are limited and thus we only perform experiments on the protein tasks. Elnaggar et al. (2021) demonstrate that the language model performances follow the ordering: ProtBert-BFD $\cdot$ ProtBert $>$ ProtAlbert. We can see that the performance ranks over the three protein tasks avGFP, AAV, and E4B are the same.
+
+# A.5 DIFFERENT DATASET SIZE
+
+As shown in Table 6, we have tested the performance of BDI as a function of dataset size $\mathrm { N } = \mathrm { 2 0 }$ , 40, 60, 80, 100) in TFBind8(r) and TFBind10(r) since they have exact oracle evaluations. We see that performance is already good for $\cdot$ for TFBind8(r) and $\Nu { = } 4 0$ for TFBind10(r).
+
+# A.6 RANKING PERFORMANCE
+
+As for prediction performances, the rank should be: a $\cdot$ linearized pre-trained $\mathrm { L M } > \mathrm { N T K }$ . We have conducted experiments to verify this. We sample half of the data, train a model to predict another
+
+Table 6: Experimental results on different size datasets for comparison.   
+
+<table><tr><td>Dataset size</td><td>20</td><td>40</td><td>60</td><td>80</td><td>100</td></tr><tr><td>TFBind8(r)</td><td>0.849± 0.027</td><td>0.883± 0.036</td><td>0.890± 0.033</td><td>0.911 ± 0.042</td><td>0.923± 0.049</td></tr><tr><td>TFBind10(r)</td><td>0.248±0.000</td><td>0.596±0.035</td><td>0.602±0.023</td><td>0.616±0.024</td><td>0.632±0.036</td></tr></table>
+
+Table 7: Mean squared prediction losses for comparison.   
+
+<table><tr><td>Method</td><td>TFBind8(r)</td><td>TFBind10(r)</td><td>avGFP</td><td>AAV</td><td>E4B</td></tr><tr><td>Finetuned NN</td><td>0.101±0.001</td><td>1.130 ± 0.041</td><td>0.411± 0.197</td><td>5.148± 0.074</td><td>0.683±0.012</td></tr><tr><td>Linearized NN</td><td>0.107±0.000</td><td>1.618 ± 0.000</td><td>0.735 ± 0.000</td><td>23.041±0.000</td><td>1.050 ± 0.000</td></tr><tr><td>NTK</td><td>0.111± 0.000</td><td>1.840 ± 0.000</td><td>0.807± 0.000</td><td>24.451 ± 0.000</td><td>1.075 ± 0.000</td></tr></table>
+
+half data, and report the mean squared loss here and in Appendix A.6 Table 7. A small mean squared loss indicates a good prediction performance; thus, we have verified the above ranking order.

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+        "type": "text",
+        "text": "Seoung Wug Oh Adobe Research ",
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+        "type": "text",
+        "text": "Joon-Young Lee Adobe Research ",
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+        "text": "Seon Joo Kim Yonsei University ",
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+        "type": "text",
+        "text": "{miran, sj.hwang, seonjookim}@yonsei.ac.kr ",
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+        "text": "{seoh, jolee}@adobe.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "We introduce a novel paradigm for offline Video Instance Segmentation (VIS), based on the hypothesis that explicit object-oriented information can be a strong clue for understanding the context of the entire sequence. To this end, we propose VITA, a simple structure built on top of an off-the-shelf Transformer-based image instance segmentation model. Specifically, we use an image object detector as a means of distilling object-specific contexts into object tokens. VITA accomplishes video-level understanding by associating frame-level object tokens without using spatio-temporal backbone features. By effectively building relationships between objects using the condensed information, VITA achieves the state-of-the-art on VIS benchmarks with a ResNet-50 backbone: 49.8 AP, 45.7 AP on YouTube-VIS 2019 & 2021, and 19.6 AP on OVIS. Moreover, thanks to its object token-based structure that is disjoint from the backbone features, VITA shows several practical advantages that previous offline VIS methods have not explored - handling long and highresolution videos with a common GPU, and freezing a frame-level detector trained on image domain. Code is available at https://github.com/sukjunhwang/ VITA. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The goal of Video Instance Segmentation (VIS) is to predict both mask trajectories and categories of each object belonging to a set of predefined categories. Numerous studies have attained the goal in a variety of ways, but a notable innovation in terms of accuracy has been achieved by Transformer-based $\\mathbb { \\left| \\left[ 2 \\right] \\right| }$ architectures. Extending DETR [5] to the video domain, VisTR $\\left. \\boldsymbol { \\widetilde { 2 8 } } \\right.$ made the first attempt to design an end-to-end model that jointly predicts object trajectories with their corresponding segmentation masks. By adopting this paradigm, subsequent studies [15, 30, 6, 34] also tackle the problem in a complete-offline manner: video-in and video-out. ",
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+        "text": "The key message from the follow-up approaches [15, 30, 6, 34] is to effectively design core interactions between frames. In parallel with recent studies [11, 25, 37, 7, 22] that improve the accuracy in various tasks by localizing the attention scope of Transformer layers, the subsequent VIS methods suggest bounding the attention scope in the encoder [15, 34] or the decoder $\\lVert \\rVert ^ { 3 0 \\mathrm { j } }$ . Specifically, they decompose the global attention by iteratively mixing two phases: intra-frame attention and inter-frame communication. Interestingly, the temporal interactions between frames are commonly achieved with only a small number of tokens, e.g., memory tokens [15, 34], messenger tokens $\\dot { \\mathbb { B } } \\dot { \\mathcal { 4 } } \\mathbb { I }$ , and instance queries $\\textcircled { \\lvert 3 0 \\rvert }$ . As a result, the question arises: “what information is important to understand a video?” ",
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+        "img_path": "images/2a10ec60f693aa19e1442cf2309858092a50f3587492cdaec6eac605e5ff8fd7.jpg",
+        "image_caption": [
+            "Figure 1: (a) Early-stage VIS methods divide the problem into two components, detection and association. (b) To alleviate the context-limited structure, complete-offline methods jointly track and segment instances in an end-to-end manner by employing dense spatio-temporal features. (c) On the other hand, our VITA is a new paradigm that directly leverages object queries for offline VIS. "
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+        "text": "In this paper, we introduce Video Instance Segmentation via Object Token Association (VITA), a new offline VIS paradigm which suggests that a video can be effectively understood from a collection of object-centric tokens. Existing offline methods [28, 15, 30, 6, 34] (Fig. 1 (b)) localize objects in multiple frames by iteratively referring to dense spatio-temporal backbone features. However, such methods show difficulties in handling long sequences as the myriad of dense reference features hinders the Transformer layers from retrieving relevant information. With the motivation to devise an effective method for the long-range understanding, we obtain clues from the traditional trackingby-detection paradigm (Fig. 1 (a)) and make two hypotheses: 1) an image object detector can fully embody the context of an object into a feature vector (or a token); and 2) a video can be represented by the relationship between the objects. ",
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+        "text": "In this regard, VITA aims to parse an input video from the collection of object tokens without the necessity of referencing dense spatio-temporal backbone (Fig. 1 (c)). Given the compactness of the token representation, VITA can collect the object tokens over the whole video and directly analyzes the collection using Transformer layers. This unique design enables the complete-offline inference (i.e., video-in and video-out) even for extremely long videos. This also facilitates building relationships between every detected object and successfully achieves global video understanding. As a result, VITA achieves state-of-the-art performance on various VIS benchmarks. ",
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+        "text": "We evaluate VITA on three popular VIS benchmarks, YouTube-VIS 2019 & 2021 $\\lVert \\overline { { 3 2 } } \\rVert$ and OVIS $\\mathbb { [ 2 4 ] }$ . With ResNet-50 [14] backbone, VITA achieves the new state-of-the-arts of 49.8 AP & 45.7 AP on YouTube-VIS 2019 & 2021, and 19.6 AP on OVIS. Above all, VITA outperforms the previous best approaches by 5.1 AP for YouTube-VIS 2021, which contains more complicated and long sequences than YouTube-VIS 2019. VITA is the first offline method that presents the results on OVIS benchmark that consists of long videos (the longest video has 292 frames) using a single 12GB GPU. ",
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+        "type": "text",
+        "text": "In addition to the performance, the design of VITA have several practical advantages over the previous offline VIS methods. It can handle long and high-resolution videos so it does not require heuristics for associating clip-level results. VITA can process 1392 frames at once regardless of video resolution using a single 12GB GPU which is 11 times longer than IFC $\\mathbb { \\lVert \\rVert \\cdot \\rVert }$ . Moreover, VITA can be trained on top of a parameter-frozen image object detector without sacrificing the performance much. This property is especially useful for the applications that cannot afford to store separated image and video instance segmentation models. VITA takes only $6 \\%$ additional parameters to extend the Swin-L detector. ",
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+    {
+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Online VIS approaches first predict individual tracklets within a local range window consisting of a single or a few frames. After obtaining results from adjacent windows, they associate individual tracklets of same identities by a hand-crafted or a learnable matching algorithm. MaskTrack RCNN $\\mathbb { \\lVert 3 2 \\rVert }$ sets the groundwork for VIS research by proposing a simple tracking branch added on a two-stage image instance segmentation model $[ \\overbrace { 1 . 3 } ]$ . The methods [4, 33, 21] that follow the tracking-by-detection paradigm (Fig. 1 (a)) measure the similarities between per-frame predictions, then employ an association algorithm. ",
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+    {
+        "type": "text",
+        "text": "To deploy temporal context from multiple frames, per-clip methods [1, 2] design an architecture of predicting tracklets within a local window and stitching the tracklets sequentially in a nearonline manner. Propagation-based methods [10, 17, 12] devise a paradigm that conjugates rich previous information stored in memories to facilitate online applications. EfficientVIS $\\mathbb { \\left[ \\left[ 2 9 \\right] \\right. }$ introduces correspondence learning between adjacent tracklet features and successfully runs in a cascaded manner which eliminates the hand-crafted tracklet association. ",
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+    {
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+        "img_path": "images/4c5a0455007405b700347ad0d2d544991469e632d1c0c996f6aebca1d53573ef.jpg",
+        "image_caption": [
+            "Figure 2: VITA takes only mask features and frame queries that are independently decoded by the frame-level detector for entire video sequence. By directly constructing temporal interactions between frame queries that encapsulate rich object-aware knowledge in spatial scenes, VITA yields mask trajectories with corresponding categories in an end-to-end manner. "
+        ],
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+    {
+        "type": "text",
+        "text": "Offline VIS architectures are proposed with the motivation of predicting mask trajectories through a whole video sequence at once. VisTR $\\pmb { \\left. 2 8 \\right. }$ successfully extends DETR $\\pmb { \\Vert 5 \\Vert }$ to the VIS domain, introducing a new paradigm of jointly tracking and segmenting instances. However, its dense selfattention over the spatio-temporal inputs leads to explosive computations and memories. With the motivation of relaxing the heavy computation of VisTR, IFC $[ \\bar { 1 } 5 ]$ adopts memory tokens to the Transformer encoder and decodes clip-level object queries. By setting the frame-level encoder to be independent and adopting the decoder of IFC, Mask2Former-VIS $\\pmb { \\mathbb { H } }$ records considerable performance on benchmarks by taking the advantage of its mask-oriented representation [7]. TeViT [34] proposes a new backbone that efficiently exchanges temporal information internally based on Vision Transformers $\\pmb { \\mathbb { Q } } \\Vert$ instead of the frame-wise CNN backbone. SeqFormer $\\textcircled { \\vert 3 0 \\vert }$ decomposes the decoder to be frame-independent, while building communication between different frames using instance queries that are used for frame-wise detection. All these studies achieve promising performance by referring to dense backbone features (Fig. 1 (b)). On the other hand, our VITA suggests a new offline VIS paradigm that directly interprets a video from the collection of object tokens (Fig. 1 (c)). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Global trackers that aim to associate frame-level predictions across an entire sequence as a whole are studied in the Multiple Object Tracking (MOT) community. Conventional approaches formulate the problem as a graph optimization – interpreting each detection as a node and considering the edges as possible connections between the nodes [35, 26, 3, 8]. Different from existing methods, GTR $[ \\breve { \\left| 3 6 \\right| }$ introduces a Transformer-based architecture that receives queries, then explicitly searches for the predictions with the same identities. Similarly, a recent method [16] proposes a set classifier that classifies the category of each tracklet by globally aggregating information from multiple frames. ",
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+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we first give a brief overview of Mask2Former $ { \\mathbb { I } } ^ { \\| \\ b { 7 } \\| }$ , a frame-level detector for VITA.   \nThen, we introduce the architecture of our proposed VITA, which is built on top of Mask2Former.   \nFinally, we describe how VITA handles extremely long videos in a complete-offline manner. ",
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+        "type": "text",
+        "text": "3.1 Frame-level Detector ",
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+        "type": "text",
+        "text": "In this paper, we adopt Mask2Former $\\mathbb { H }$ for the frame-level detector which directly localizes instances using masks without the necessity of bounding boxes. Following the set prediction mechanism of DETR $\\mathbb { \\left[ 5 \\right] }$ , the frame-level detector parse an input image $H \\times W$ using $N _ { f }$ object queries, which we call frame queries $( f \\in \\mathbb { R } ^ { C \\times N _ { f } } )$ throughout this paper. Having the spatially encoded features to be decoded by the frame queries through a Transformer decoder, each object in the image gets represented as a $C$ -dimensional vector. Then, the frame queries are used for both classifying and segmenting their matched objects where the predictions are also used for auxiliary supervision for VITA. Specifically, the frame-level detector generates two features for the frame-level predictions: 1) dynamic $1 \\times 1$ convolutional weight from the frame queries; 2) per-pixel embeddings $\\begin{array} { r } { \\dot { \\mathcal { M } } \\in \\mathbb { R } ^ { C \\times \\frac { H } { S } \\times \\frac { W } { S } } } \\end{array}$ from the pixel decoder, where $S$ is the stride of the feature map. Finally, the detector segments objects by applying a simple dot product between the two embeddings. ",
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+        "type": "text",
+        "text": "3.2 VITA ",
+        "text_level": 1,
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+        "text": "We now propose the novel end-to-end video instance segmentation method VITA, which can be largely divided into three phases (Fig. 2). First, VITA operates on top of the frame-level detector [7] in a complete frame-independent manner; no inter-computation between frames is involved. Then, the frame queries that hold object-centric information are collected throughout the whole video and they embed video-level information by building communications between different frames using Object Encoder. Finally, Object Decoder aggregates information from the frame queries to video queries, which are eventually used for predicting categories and masks of objects in videos at once. ",
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+    {
+        "type": "text",
+        "text": "Input of VITA. Given an input video of $T$ frames, the frame-level detector executes frame-by-frame as previously explained. Among a number of intermediate embeddings that are generated by the detector, the only features that are used by VITA are 1) frame queries $\\bar { \\{ f ^ { t } \\} } _ { t = 1 } ^ { T } \\in \\bar { \\mathbb { R } } ^ { C \\times T \\times N _ { f } }$ which hold object-centric information; and 2) per-pixel embeddings $\\{ \\mathcal { M } ^ { t } \\} _ { t = 1 } ^ { T } \\in \\mathbb { R } ^ { C \\times T \\times \\frac { H } { S } \\times \\frac { W } { S } }$ from the pixel decoder. ",
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+    {
+        "type": "text",
+        "text": "Object Encoder. After the frame-wise detector distills the object-wise context into the frame queries, Object Encoder aims to build temporal communication by employing self-attention along the temporal axis. First, Object Encoder gathers frame queries from all frames and converts them to object tokens through a linear layer. However, a naive self-attention over the whole $T N _ { f }$ object tokens is not applicable when processing long videos due to the quadratic computational overhead of Transformers. Inspired by Swin Transformer $[ [ 2 2 ] ]$ , we adopt window-based self-attention layers that shift along the temporal dimension. As illustrated in Fig. $\\begin{array} { r } { \\bigtriangledown , } \\end{array}$ Object Encoder initially partitions object tokens $\\{ \\overline { { f } } ^ { t } \\} _ { t = 1 } ^ { T }$ to the temporal axis with local windows of size $W$ without an overlap. By alternatively shifting the windows, object tokens from different frames can exchange object-wise information which allows VITA to both effectively and efficiently handle long sequences. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fa5d52f3a170a262b27941e6c562c36a9a45b91f3815da8d94459c21454260b3.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of an Object Encoder layer. Blocks with dashed line are local windows, and  indicates an object token. "
+        ],
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+    {
+        "type": "text",
+        "text": "Object Decoder and Output heads. Two limitations of previous offline VIS methods [28, 15, 6] are the ineffectiveness in handling dynamic scenes and the inability of processing long videos. For example, such methods obtain high accuracy when dealing with static and short videos (YouTubeVIS 2019 $\\mathbb { [ 3 2 ] } )$ , but struggle to track objects or executes end-to-end on benchmarks with dynamic and long videos (YouTube-VIS 2021 $\\lVert \\dot { 3 } 2 \\rVert$ and OVIS $\\pm \\mathbb { Z } 4 \\mathbb { I } )$ . Both limitations are mainly caused by the decoder, which parses object contexts directly from dense spatio-temporal features. As recent studies [11, 25, 37] suggest, typical Transformer decoders show difficulties in retrieving relevant information from global context. In the video domain, the number of backbone features being referred to proportionally increases with the number of frames. Therefore, when handling extremely long videos, the countless reference tokens result in both imprecise information retrieval and intractable peak memories. ",
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+    {
+        "type": "text",
+        "text": "For the solution to the problem, we suggest Object Decoder which extracts information from the object tokens, not the spatio-temporal backbone features. Implicitly embedding the context of objects, object tokens can provide sufficient instance-specific information without the interference of dense backbone features. Specifically, we employ $N _ { v }$ trainable video queries $\\boldsymbol { v } \\in \\mathbb { R } ^ { C \\times N _ { v } }$ to decode objectwise information from all object tokens $\\{ f ^ { t } \\} _ { t = 1 } ^ { T }$ that are collected from all $T$ frames. Receiving much condensed input over naively taking dense spatio-temporal features, Object Decoder effectively captures video contexts and aggregates relevant information into the video queries. As a result, Object ",
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+    {
+        "type": "text",
+        "text": "Decoder shows fast convergence speed while achieving high accuracy. Furthermore, the compact input greatly saves memories, thus facilitates processing long and high-resolution videos. ",
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+    {
+        "type": "text",
+        "text": "From the decoded video queries $v$ , VITA returns final predictions $z = \\{ ( p _ { i } , m _ { i } ) \\} _ { i = 1 } ^ { N _ { v } }$ using two output heads similar to IFC ; the class head and the mask head. The class head is a single linear classifier, which directly predicts class probabilities $p \\in \\mathbb R ^ { N _ { v } \\times ( K + 1 ) }$ of each video query, where $K + 1$ is the number of categories including an auxiliary label “no object” $( \\emptyset )$ . The mask head dynamically generates mask embeddings $\\boldsymbol { w _ { v } } ^ { \\prime } \\in \\mathbb { R } ^ { C \\times N _ { v } }$ per a video query, which corresponds to the tracklet of an instance over all frames. Finally, the predicted mask logits $m \\in \\mathbb { R } ^ { N _ { v } \\times T \\times \\dot { H } \\times W }$ can be obtained from a matrix multiplication between $w _ { v }$ and $\\{ \\mathcal { M } ^ { t } \\} _ { t = 1 } ^ { T }$ . ",
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+    {
+        "type": "text",
+        "text": "3.3 Clip-wise losses ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Instance matching. We search for optimal pair indices between the predictions from VITA and $G _ { v }$ ground-truth to remove postprocessing heuristics such as NMS. First, we calculate costs from all possible pairs using the cost function of Mask2Former $\\mathbb { H }$ with a simple extension of mask-related costs to the temporal axis $\\mathbb { \\left[ 1 5 \\right] }$ . Then, from $N _ { v } { \\times } G _ { v }$ costs of pairs, we follow DETR $\\pmb { \\Vert 5 \\Vert }$ and use Hungarian algorithm $\\pm \\textcircled { 1 1 8 } \\textcircled { 1 }$ for the optimal matching as shown in Fig. 4 (b). ",
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+    {
+        "type": "text",
+        "text": "Similarity loss. Inspired by the initial VIS approach (MaskTrack R-CNN $[ \\left| 3 2 \\right| ]$ where the similarity loss is adopted to track instances at different frames, we train video queries and frame queries to be clustered in the latent space by their identities. As shown in Fig. $\\textcircled { 4 }$ (a), our adopted frame-level detector $\\mathbb { [ [ \\big ] ] }$ also searches for paired indices be",
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+    {
+        "type": "image",
+        "img_path": "images/5e870e090cce93a3b6d22172969a474ad37916df0a8f3cef649500e89c6fd361.jpg",
+        "image_caption": [
+            "Figure 4: Similarity loss. $\\bigcirc$ and $\\sqcap$ indicate video query and frame query, respectively. Same color represents same GT instance ID. "
+        ],
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+    {
+        "type": "text",
+        "text": "tween $N _ { f }$ frame-wise predictions and $G _ { f } ^ { t }$ ground-truth objects at each $t ^ { \\mathrm { { t h } } }$ frame. The frame queries and the video queries that are matched to ground-truths get collected and we embed the collection through a linear layer. Then, we measure the similarity of all possible pairs using a simple matrix multiplication. Finally, as shown in Fig. $\\boxed { 4 }$ (c), binary cross entropy is used to compute $\\mathcal { L } _ { s i m }$ between the predicted similarities and the ground-truth where annotated to 1 for pairs of equal identities and 0 for vice-versa. ",
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+    {
+        "type": "text",
+        "text": "Total loss. We attach the proposed module VITA on top of the frame-level detector, and all components of the model get trained end-to-end. Note that not only video-level outputs from VITA are used for the loss computation, but also per-frame outputs from the frame-level detector get involved. Specifically, we use $\\mathcal { L } _ { f }$ from $\\mathbb { I } \\mathbb { I }$ to calculate loss from the per-frame outputs to frame-wise ground-truth. Extending the loss function of $\\mathbb { [ [ \\big ] ] }$ to the temporal axis as similar to $\\bar { \\mathbb { E } } \\bar { \\ b { 5 } } \\|$ , we use outputs from VITA $z$ to calculate the video-level loss $\\mathcal { L } _ { v }$ . Finally, we integrate all losses together as follows: $\\mathcal { L } _ { t o t a l } = \\lambda _ { v } \\mathcal { L } _ { v } + \\lambda _ { f } \\mathcal { L } _ { f } + \\lambda _ { s i m } \\mathcal { L } _ { s i m }$ . ",
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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,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "YouTube-VIS 2019. YouTube-VIS 2019 $\\pmb { \\mathbb { B 2 } }$ is the first dataset proposed for VIS and contains 40 semantic categories. Mostly originated from Video Object Segmentation (VOS) datasets, the VIS benchmark has a small number of unique instances (average 1.7 per video for the train set) and the categories of instances appearing in the same video are different in general. Also, the average length of videos in the valid set is short (27.4 frames), which enables existing complete-offline approaches to load a whole video and infer the benchmark at once. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/e5006bc144c727896e1acee36583b649232296e1901cc1c00e6760c45a1a4a0a.jpg",
+        "table_caption": [
+            "Table 1: Comparisons on YouTube-VIS 2019. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=3>Method                          Backbone</td><td rowspan=1 colspan=1>AP  AP50  AP75  AR1  AR10</td></tr><tr><td rowspan=9 colspan=1>NNTai (rer)</td><td rowspan=9 colspan=1>MaskTrackR-CNNB2MaskTrackR-CNN32Cross VIS [33]Cross VIS [33]PCAN日PCAN日EfficientVIS四EfficientVIS29VISOLO四</td><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>30.3  51.1   32.6  31.0  35.5</td></tr><tr><td rowspan=1 colspan=1>ResNet-101</td><td rowspan=1 colspan=1>31.8  53.0   33.6   33.2   37.6</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>36.3  56.8   38.9   35.6   40.7</td></tr><tr><td rowspan=6 colspan=1>ResNet-101ResNet-50ResNet-101ResNet-50ResNet-101ResNet-50</td><td rowspan=1 colspan=1>36.6  57.3   39.7  36.0   42.0</td></tr><tr><td rowspan=1 colspan=1>36.1  54.9   39.4  36.3   41.6</td></tr><tr><td rowspan=1 colspan=1>37.6  57.2   41.3  37.2   43.9</td></tr><tr><td rowspan=1 colspan=1>37.9  59.7   43.0   40.3   46.6</td></tr><tr><td rowspan=1 colspan=1>39.8  61.8   44.7  42.1   49.8</td></tr><tr><td rowspan=1 colspan=1>38.6  56.3   43.7  35.7  42.5</td></tr><tr><td rowspan=13 colspan=1>Ouiiii</td><td rowspan=10 colspan=1>VisTR[28]VisTR [28]IFC因IFC园TeViT [34]SeqFormer四SeqFormer四SeqFormer四Mask2Former-VIS回Mask2Former-VIS回Mask2Former-VIS回</td><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>35.6  56.8   37.0  35.2  40.2</td></tr><tr><td rowspan=1 colspan=1>ResNet-101</td><td rowspan=1 colspan=1>38.6  61.3   42.3  37.6   44.2</td></tr><tr><td rowspan=1 colspan=1>ResNet-50</td><td rowspan=1 colspan=1>41.2  65.1   44.6  42.3   49.6</td></tr><tr><td rowspan=1 colspan=1>ResNet-101</td><td rowspan=1 colspan=1>42.6  66.6   46.3  43.5   51.4</td></tr><tr><td rowspan=6 colspan=1>MsgShifTResNet-50ResNet-101Swin-LResNet-50ResNet-101Swin-L</td><td rowspan=1 colspan=1>46.6  71.3   51.6   44.9   54.3</td></tr><tr><td rowspan=1 colspan=1>47.4  69.8   51.8  45.5   54.8</td></tr><tr><td rowspan=1 colspan=1>49.0  71.1   55.7  46.8   56.9</td></tr><tr><td rowspan=1 colspan=1>59.3  82.1   66.4  51.7  64.4</td></tr><tr><td rowspan=1 colspan=1>46.4  68.0   50.0    1       1</td></tr><tr><td rowspan=1 colspan=1>49.2  72.8   54.2     =       =60.4  84.4   67.0    =       =</td></tr><tr><td rowspan=3 colspan=1>VITA (Ours)</td><td rowspan=3 colspan=1>ResNet-50ResNet-101Swin-L</td><td rowspan=1 colspan=1>49.8  72.6   54.5  49.4  61.0</td></tr><tr><td rowspan=1 colspan=1>51.9  75.4  57.0  49.6  59.1</td></tr><tr><td rowspan=1 colspan=1>63.0  86.9   67.9  56.3  68.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "YouTube-VIS 2021. In order to address more difficult scenarios, additional videos are included in YouTube-VIS2021 (794 videos for training and 129 videos for validation). In particular, a greater number of objects with confusing trajectories has been added (average 3.4 per video for the additional videos in the train set). However, the average length of the additional validation videos is still 39.7 frames, which is not significantly increased compared to YouTube-VIS 2019. ",
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+    },
+    {
+        "type": "text",
+        "text": "OVIS. Under the same definition as YouTube-VIS, OVIS $\\pmb { \\Vert 2 4 \\Vert }$ specifically aims to tackle objects with heavy occlusions that are belonging to 25 semantic categories. In addition to the heavily occluded situation, OVIS has three challenging characteristics that are distinct from the YouTube-VIS datasets. First, although it has fewer categories than YouTube-VIS, much more instances appear in a single video (average 5.9 per video for the train set). Second, the instances with the same categories in the same video have almost similar appearances, thus approaches that rely heavily on visual cues often struggle to predict accurate trajectories. Finally, the average length of videos for the valid set is 62.7 frames (the longest video has 292 frames) which is much longer than that of YouTube-VIS. Therefore, not only do previous approaches show relatively low accuracy, but all existing complete-offline VIS methods are not feasible to infer OVIS without hand-crafted association algorithms. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our method is implemented on top of detectron2 $\\textcircled { \\scriptsize { 1 3 1 } }$ . All hyper-parameters regarding the framelevel detector are equal to the defaults of Mask2Former [7]. The total loss $\\mathcal { L } _ { t o t a l }$ is balanced with $\\lambda _ { v }$ $\\lambda _ { f }$ , and $\\lambda _ { s i m }$ where 1.0, 1.0, and 0.5, respectively. By default, Object Encoder is composed of three layers with the window size $W = 6$ , and Object Decoder employs six layers with $N _ { v } = 1 0 0$ video queries. Having VITA built on top of Mask2Former, we first train our model on the COCO $\\left[ \\left[ 2 0 \\right] \\right]$ dataset following Mask2Former. Then, we train our method on the VIS datasets $\\pm \\pm \\pm \\pm \\pm \\pm$ simultaneously with pseudo videos generated from images $\\left[ \\left[ 2 0 \\right] \\right]$ following the details of SeqFormer $\\textcircled { \\lvert 3 0 \\rvert }$ . During inference, each frame is resized to a shorter edge size of 360 and 448 pixels when using ResNet [14] and Swin $[ [ 2 2 ] ]$ backbones, respectively. Note that all reported scores in main results and ablation studies are the mean of five runs, and we use the standard ResNet-50 [14] for the backbone unless specified. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/605f17e50123edfae4f8dc9647be327e9a988562588ce54209422c73b966d54b.jpg",
+        "table_caption": [
+            "Table 2: Comparisons with ResNet-50 backbone on YouTube-VIS 2021 and OVIS. $\\dagger$ indicates using MsgShifT backbone. $\\ddagger$ indicates using Swin-L $[ [ 2 2 ] ]$ backbone. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"5\">YouTube-VIS 2021</td><td colspan=\"5\">OVIS</td></tr><tr><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>MaskTrack R-CNN 32]</td><td>28.6</td><td>48.9</td><td>29.6</td><td>26.5</td><td>33.8</td><td>10.8</td><td>25.3</td><td>8.5</td><td>7.9</td><td>14.9</td></tr><tr><td>CMaskTrack R-CNN [23]</td><td>-</td><td>=</td><td>=</td><td>=</td><td>-</td><td>15.4</td><td>33.9</td><td>13.1</td><td>9.3</td><td>20.0</td></tr><tr><td>STMask [19]</td><td>31.1</td><td>50.4</td><td>33.5</td><td>26.9</td><td>35.6</td><td>15.4</td><td>33.8</td><td>12.5</td><td>8.9</td><td>21.3</td></tr><tr><td>Cross VIS [33]</td><td>34.2</td><td>54.4</td><td>37.9</td><td>30.4</td><td>38.2</td><td>14.9</td><td>32.7</td><td>12.1</td><td>10.3</td><td>19.8</td></tr><tr><td>IFC [5]</td><td>35.2</td><td>55.9</td><td>37.7</td><td>32.6</td><td>42.9</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>VISOLO [12]</td><td>36.9</td><td>54.7</td><td>40.2</td><td>30.6</td><td>40.9</td><td>15.3</td><td>31.0</td><td>13.8</td><td>11.1</td><td>21.7</td></tr><tr><td>TeViTt B4]</td><td>37.9</td><td>61.2</td><td>42.1</td><td>35.1</td><td>44.6</td><td>17.4</td><td>34.9</td><td>15.0</td><td>11.2</td><td>21.8</td></tr><tr><td>SeqFormer[ 眉</td><td>40.5</td><td>62.4</td><td>43.7</td><td>36.1</td><td>48.1</td><td>-</td><td>1</td><td>1</td><td>-</td><td>1</td></tr><tr><td>Mask2Former-VIS 回</td><td>40.6</td><td>60.9</td><td>41.8</td><td>-</td><td>1</td><td>1</td><td>-</td><td>1</td><td>1</td><td>1</td></tr><tr><td>VITA (Ours)</td><td>45.7</td><td>67.4</td><td>49.5</td><td>40.9</td><td>53.6</td><td>19.6</td><td>41.2</td><td>17.4</td><td>11.7</td><td>26.0</td></tr><tr><td>SeqFormert [30]</td><td>51.8</td><td>74.6</td><td>58.2</td><td>42.8</td><td>58.1</td><td>-</td><td>1</td><td>1</td><td>1</td><td>-</td></tr><tr><td>Mask2Former-VIS‡ 回</td><td>52.6</td><td>76.4</td><td>57.2</td><td>-</td><td>-</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>VITA (Ours)‡</td><td>57.5</td><td>80.6</td><td>61.0</td><td>47.7</td><td>62.6</td><td>27.7</td><td>51.9</td><td>24.9</td><td>14.9</td><td>33.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Main Results ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Using the popular VIS benchmarks – YouTube-VIS 2019 & 2021 $\\pmb { \\Vert 3 2 } \\Vert$ and OVIS [24] – we compare VITA with state-of-the-art approaches following the standard evaluation metric [32]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "YouTube-VIS 2019. Tab. 1 shows the comparison on YouTube-VIS 2019 dataset with backbones of both CNN-based (ResNet-50 and 101 [14]) and Transformer-based (Swin-L $\\pmb { \\mathbb { L 2 } } \\mathbf { \\mathbb { I } }$ ). Offline methods can take two advantages over (near) online approaches: 1) they have a greater receptive field to the temporal axis, and 2) they can avoid error propagation derived from hand-crafted association algorithms. As a result, the tendency of offline methods with higher accuracy is clearly shown in the table. Among the competitive offline models, our VITA sets a new state-of-the-art of 49.8 AP and 51.7 AP using CNN backbones, ResNet-50 and ResNet-101 respectively. In addition, with Swin-L backbone, VITA achieves 63.0 AP outperforming all existing VIS methods. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "YouTube-VIS 2021. We compare VITA with state-of-the-art methods on YouTube-VIS 2021 benchmark in Tab. 2. Note that the longest video in the valid set has 84 frames, thus previous offline methods [15, 30, 6] can infer videos at once with GPUs with large memories. Above all, VITA achieves the highest accuracy, 45.7 AP, which outperforms the previous state-of-the-art approach [6] with a huge margin of 5.1 AP. Considering the accuracy gap on YouTube-VIS 2019, the results demonstrate that VITA can effectively handle tricky scenarios, e.g., numerous unique instances with confusing trajectories. We hypothesize that the object-oriented design of VITA is more effective than typical dense Transformer decoders in addressing such challenging scenes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "OVIS. In Tab. 2, we demonstrate the competitiveness of VITA on the challenging OVIS benchmark. Due to the considerable lengths of videos – the longest video has 292 frames – existing offline approaches [28, 15, 34, 30, 6] cannot process OVIS benchmark in their original design: video-in and video-out. To the best of our knowledge, VITA is the first complete-offline approach to evaluate on OVIS valid set. Thanks to its object token-based structure which is disjoint from backbone features, VITA can process the benchmark without any hand-crafted association algorithm. Moreover, VITA sets a new state-of-the-art performance of 19.6 AP, demonstrating the potential of the complete-offline pipeline in long and complicated scenes. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f95c8f0b2792128b3f51f679566f1ed20ec9352cd9179349a4b48fe8b9d69c8e.jpg",
+        "table_caption": [
+            "Table 3: Impact of local windows of varying sizes in Object Encoder. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>W</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>3</td><td>49.4</td><td>72.2</td><td>54.4</td><td>48.6</td><td>60.9</td></tr><tr><td>6</td><td>49.8</td><td>72.6</td><td>54.5</td><td>49.4</td><td>61.0</td></tr><tr><td>12</td><td>50.0</td><td>73.0</td><td>54.7</td><td>49.0</td><td>60.8</td></tr><tr><td>All</td><td>50.1</td><td>72.4</td><td>54.7</td><td>49.0</td><td>60.6</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a5f039243eb197c2c92dad86d3355103ad65c141f09df66f0d175f23bdbd2c13.jpg",
+        "table_caption": [
+            "Table 5: Use of different heuristic association algorithms on OVIS valid set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Length</td><td>Algorithm</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>36</td><td>Greedy Hungarian</td><td>18.8 18.4</td><td>39.4 38.9</td><td>17.1 16.3</td></tr><tr><td>48</td><td>Greedy Hungarian</td><td>18.8 19.1</td><td>39.0 39.1</td><td>17.1 17.4</td></tr><tr><td>All</td><td>None</td><td>19.6</td><td>41.2</td><td>17.4</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/0c99c2a2d80106df2153df0b439e10cca91c2162b2cd91251438b66b9634c33b.jpg",
+        "table_caption": [
+            "Table 4: Maximum number of frames that can be processed at once using a single Titan XP. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Max Frames360 × 640 720×1280</td></tr><tr><td rowspan=1 colspan=1>VisTR [28]IFC园Mask2Former-VIS回</td><td rowspan=1 colspan=1>46          12123         3881          20</td></tr><tr><td rowspan=1 colspan=1>W=3VITA(Ours) W=6W = 12</td><td rowspan=1 colspan=1>26771392741</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/877f40b55fc7d59a1eee35ee375fdb81ad4b6a2e203eef3704e1cd13fa209a55.jpg",
+        "table_caption": [
+            "Table 6: Pruning tokens by different ratios $r$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>r</td><td>一 AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>1.0</td><td>49.8</td><td>72.6</td><td>54.5</td><td>49.4</td><td>61.0</td></tr><tr><td>0.75</td><td>49.7</td><td>72.5</td><td>54.4</td><td>48.7</td><td>61.0</td></tr><tr><td>0.5</td><td>48.9</td><td>72.1</td><td>52.0</td><td>48.3</td><td>60.9</td></tr><tr><td>0.25</td><td>48.1</td><td>71.6</td><td>51.6</td><td>47.4</td><td>59.8</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 Ablation Studies ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We provide a series of ablation studies using a ResNet-50 [14] backbone. All experiments are conducted on YouTube-VIS 2019 $\\pmb { \\mathbb { \\left[ \\left| 3 2 \\right| \\right] } }$ valid set except for Tab. 5 with OVIS [24] valid set. ",
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+    },
+    {
+        "type": "text",
+        "text": "Attention window size. Tab. $\\bigstar$ shows the performance of VITA with varying sizes of shifted attention window $W$ in Object Encoder during inference. The larger the window, the greater the receptive field for the temporal axis in Object Encoder. The results suggest that larger window sizes utilize information from multiple frames, which helps Object Encoder understand the context of objects in videos. We set $W = 6$ considering a trade-off between performance and inference scalability. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Maximum number of frames. In Tab. $\\boxed { 4 }$ we calculate the maximum number of frames that VITA can handle with respect to the various window sizes $W$ , and compare it with existing complete-offline VIS methods. To take into account the general environment, all results are computed using a single 12GB Titan XP GPU. As shown in results, existing methods have limitations in processing long videos in a video-in and video-out manner. Clearly, the bottleneck of VisTR $\\pmb { \\left. 2 8 \\right. }$ is the encoder, where the full spatio-temporal self-attention leads to a tremendous memory usage. IFC $\\mathbb { \\left. \\overline { { \\Omega } } \\right. }$ alleviates the computation of VisTR $\\pmb { \\left. 2 8 \\right. }$ , achieving a higher number of input frames. However, IFC makes use of a typical Transformer decoder that visits all dense spatio-temporal features. Therefore, IFC cannot infer the OVIS $ { \\mathbb { 1 2 4 } }$ benchmark at once which contains a video of 292 frames. The problem gets aggravated in Mask2Former-VIS $\\pmb { \\mathbb { H } }$ as the scope of the decoder is extended to multiple feature levels $\\textcircled { 7 }$ . On the other hand, VITA presents considerable frame numbers that can be inferred completely offline. Furthermore, VITA is independent from input frame resolutions as each frame gets summarized into compact object tokens. With input resolution of $3 6 0 \\times 6 4 0$ and $W = 6$ , the maximum length of sequence that VITA is able to process in complete-offline is about $I I \\times$ longer than IFC [15]. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Heuristic clip association. Tab. $\\boxed { 5 }$ shows the results on OVIS valid set of splitting a video into shorter clips and associating clip-wise predictions through heuristic matching. The length of the clip is set to be less than the average length of videos of OVIS valid set (62.7). Then, we associate outputs from different clips using mask IoU score as the matching cost. We test with two matching algorithms: Greedy and Hungarian. As shown in Tab. $\\boxed { 5 }$ VITA demonstrates the best performance on the complete-offline inference that use all the video frames at once. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Pruning Tokens. In Tab. $^ { 6 , }$ we investigate the effects of removing redundant frame queries. From a collection of frame queries, VITA understands the overall context of the given clip. As only a small portion of the collection is matched to foreground objects, the number of total input frame queries can be reduced. First, for each frame, we sort frame queries in ascending order by the “no object” $( \\emptyset )$ probability. Then, we keep only top $r N _ { f }$ queries from the sorted list where $r$ is the ratio, and discard the rest. The accuracy with respect to the ratio $r$ is as shown in Tab. 6. ",
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+        "type": "image",
+        "img_path": "images/a20c516e7af8587e691546677cd729656012ba339b14588ff8929605737d74b6.jpg",
+        "image_caption": [
+            "Figure 5: Train speed comparison with Mask2Former-VIS $\\textcircled { 6 }$ . $\\dagger$ indicates the same training setup with VITA. "
+        ],
+        "image_footnote": [],
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+        "type": "table",
+        "img_path": "images/1282629e21e439b2480d18e485bda570d33d70101db933a0db0f71e3301e8d62.jpg",
+        "table_caption": [
+            "Table 7: Results on YouTube-VIS 2019 with freezing detector pretrained on COCO. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Backbone</td><td>Freeze</td><td>AP</td><td>AP50</td><td>AP75</td><td>AR1</td><td>AR10</td></tr><tr><td>ResNet-50</td><td>厂</td><td>49.8 40.9</td><td>72.6 61.9</td><td>54.5 44.6</td><td>49.4 43.1</td><td>61.0 53.1</td></tr><tr><td>ResNet-101</td><td></td><td>51.9</td><td>75.4</td><td>57.0</td><td>49.6</td><td>59.1</td></tr><tr><td></td><td>1</td><td>43.2</td><td>64.4</td><td>48.7</td><td>46.1</td><td>55.9</td></tr><tr><td>Swin-L</td><td></td><td>63.0</td><td>86.9</td><td>67.9</td><td>56.3</td><td>68.1</td></tr><tr><td></td><td>1</td><td>53.4</td><td>75.9</td><td>58.7</td><td>51.9</td><td>64.3</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "By setting the ratio $r = 0 . 7 5$ , the accuracy of VITA shows only a marginal degradation in the accuracy $( - 0 . 1$ AP). This signifies that VITA focuses more on the foreground contexts that are embedded in the frame queries. Meanwhile, as the quadratic computation in Clip Encoder can be alleviated, VITA can process a much greater number of frames; using the ratio $r = 0 . 7 5$ , the maximum frame number increases from 1392 (Tab. 4) to 2635. ",
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+    {
+        "type": "text",
+        "text": "Convergence speed and Similarity loss. Fig. $5$ validates our claim of the faster convergence speed and the effectiveness of the proposed Similarity loss. For a fair comparison, we report average scores and standard deviations of five runs, each trained without pseudo videos, same as Mask2FormerVIS [6]. Thanks to its object-centric design, VITA shows faster convergence than Mask2Former-VIS. Furthermore, the use of Similarity loss leads to an additional accuracy gain of 1.8 AP. The results demonstrate that the loss mitigates the discrepancies between the embeddings of equal identities, leading to better performance. ",
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+    {
+        "type": "text",
+        "text": "Frozen frame-level detector. In Tab. $\\bigstar$ we demonstrate the performance of VITA where the frame-level detector is completely frozen. Specifically, while VITA gets trained on YouTube-VIS 2019, the frame-level detector $\\dot { \\bigtriangledown } \\mathbb { I }$ does not get updated from pretrained weights on COCO $\\left[ \\left[ 2 0 \\right] \\right]$ . Note that among 40 categories in YouTube-VIS 2019 dataset, only 20 categories overlap with the categories of COCO. Interestingly, though the frame-level detector remains completely frozen, VITA achieves compelling results with various backbones. As shown in Tab. 1 and Tab. $\\textcircled { 7 }$ VITA presents a huge practicality as it surpasses all online approaches on top of the ResNet-50 backbone. This strategy can be beneficial in various scenarios: 1) when the accuracy of image instance segmentation should be kept while extending the network to the video domain, and 2) when having limited time and GPUs to train models. The strategy can be especially useful in mobile applications that have scarce storage for keeping two separate network parameters for image instance segmentation and video instance segmentation. With additional $6 \\%$ parameters, VITA successfully extends the frozen Swin-L based frame-level detector to the video domain and it achieves great accuracy. ",
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+    {
+        "type": "text",
+        "text": "We also provide a brief discussion of our understanding for the large gap in AP. Compared to COCO, we observe that YouTube-VIS dataset is annotated with only a few salient objects as foregrounds. Having weights of the frame-level detector frozen to COCO, the detector cannot adapt to the YouTubeVIS domain and it embeds and interprets more objects in scenes as foregrounds. Therefore, VITA outputs more predictions as a foreground category even if such predictions are not labeled as groundtruths in YouTube-VIS. As a result, it leads to a lower average precision as it comes out with more false positive predictions. On the contrary, the more false positive predictions only slightly affect AR. ",
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+    {
+        "type": "text",
+        "text": "Qualitative Results. We provide some visualizations of the predictions from VITA and frame-level detector in Fig. $6 .$ The qualitative results show that VITA leads to better video instance segmentation qualities compared to the frame-level detector. Specifically, the frame-level detector mistakenly interprets in to recognize either category or mask of instances that have been largely occluded, while our method successfully recovers it by leveraging the temporal information. ",
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+    {
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+        "img_path": "images/fc3f2f6949a57be07b4d2af0e448b02f2f9984aae5d3bcbeb52f506608216b01.jpg",
+        "image_caption": [
+            "Figure 6: Visualization of predictions from the frame-level detector and VITA. Instances with the same identity are displayed in the same color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 Limitations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "VITA has achieved high performance in the complete-offline paradigm while dramatically improving the number of input frames that can be processed at once. However, there are two major limitations for the ultimate long video understanding. First, the current architecture still has limitations in processing an infinite number of frames. In addition, since object tokens do not explicitly utilize temporal information, they may have difficulties in identifying complex behaviors that span over very long sequences. We believe that devising explicit designs to address these issues will be a promising future direction. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we proposed VITA for offline Video Instance Segmentation. VITA is a simple model built on top of the off-the-shelf image instance segmentation model [7]. Unlike existing offline methods, VITA directly leverages object queries decoded by independent frame-level detectors. We demonstrated that deploying object-oriented information is not only effective in improving performance, but also has robust practicality for processing long and high-resolution videos - setting state-of-the-art on popular VIS benchmarks, e.g., YouTubeVIS-2019 & 2021 and OVIS. Moreover, since VITA is designed to absorb spatial knowledge purely from image object detector, it shows fast convergence and demonstrates competitive performance even if trained on frozen detectors. We hope that our method extends the scope of offline VIS research beyond benchmarks to real-world applications. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work has partly supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF2022R1A2C2004509) and by Institute of Information communications Technology Planning Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2022-0-00113, Developing a Sustainable Collaborative Multi-modal Lifelong Learning Framework), and Artificial Intelligence Graduate School Program under Grant 2020-0-01361. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "VITA is designed for the VIS task and focuses on processing long and high-resolution videos in an end-to-end manner while achieving the state-of-the-art performance. We hope that VITA can have a positive impact on many industrial areas such as video editing applications. We would like to note that research on VIS must be aware of potential misuse that violates personal privacy. ",
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+    {
+        "type": "text",
+        "text": "Licenses of COCO $[ \\pmb { \\big | 2 0 } ]$ , YouTube-VIS [32], OVIS [24], and detectron2 [31]: Attribution 4.0 International, CC BY 4.0, CC BY-NC-SA 4.0, and Apache-2.0, respectively. ",
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+        "text": "References   \n[1] Ali Athar, Sabarinath Mahadevan, Aljoša Ošep, Laura Leal-Taixé, and Bastianan Leibe. Stemseg: Spatio-temporal embeddings for instance segmentation in videos. In ECCV, 2020.   \n[2] Gedas Bertasius and Lorenzo Torresani. Classifying, segmenting, and tracking object instances in video with mask propagation. In CVPR, 2020.   \n[3] Guillem Brasó and Laura Leal-Taixé. Learning a neural solver for multiple object tracking. In CVPR, 2020.   \n[4] Jiale Cao, Rao Muhammad Anwer, Hisham Cholakkal, Fahad Shahbaz Khan, Yanwei Pang, and Ling Shao. Sipmask: Spatial information preservation for fast image and video instance segmentation. In ECCV, 2020.   \n[5] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko. End-to-end object detection with transformers. In ECCV, 2020.   \n[6] Bowen Cheng, Anwesa Choudhuri, Ishan Misra, Alexander Kirillov, Rohit Girdhar, and Alexander G Schwing. Mask2former for video instance segmentation. arXiv preprint arXiv:2112.10764, 2021.   \n[7] Bowen Cheng, Ishan Misra, Alexander G Schwing, Alexander Kirillov, and Rohit Girdhar. Masked-attention mask transformer for universal image segmentation. In CVPR, 2022.   \n[8] Peng Dai, Renliang Weng, Wongun Choi, Changshui Zhang, Zhangping He, and Wei Ding. Learning a proposal classifier for multiple object tracking. In CVPR, 2021.   \n[9] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. In ICLR, 2021.   \n[10] Yang Fu, Linjie Yang, Ding Liu, Thomas S Huang, and Humphrey Shi. Compfeat: Comprehensive feature aggregation for video instance segmentation. In AAAI, 2021.   \n[11] Peng Gao, Minghang Zheng, Xiaogang Wang, Jifeng Dai, and Hongsheng Li. Fast convergence of detr with spatially modulated co-attention. In ICCV, 2021.   \n[12] Su Ho Han, Sukjun Hwang, Seoung Wug Oh, Yeonchool Park, Hyunwoo Kim, Min-Jung Kim, and Seon Joo Kim. Visolo: Grid-based space-time aggregation for efficient online video instance segmentation. In CVPR, 2022.   \n[13] Kaiming He, Georgia Gkioxari, Piotr Dollar, and Ross Girshick. Mask r-cnn. In ICCV, 2017.   \n[14] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016.   \n[15] Sukjun Hwang, Miran Heo, Seoung Wug Oh, and Seon Joo Kim. Video instance segmentation using inter-frame communication transformers. In NeurIPS, 2021.   \n[16] Sukjun Hwang, Miran Heo, Seoung Wug Oh, and Seon Joo Kim. Cannot see the forest for the trees: Aggregating multiple viewpoints to better classify objects in videos. In CVPR, 2022.   \n[17] Lei Ke, Xia Li, Martin Danelljan, Yu-Wing Tai, Chi-Keung Tang, and Fisher Yu. 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In ICCV, 2021.   \n[34] Shusheng Yang, Xinggang Wang, Yu Li, Yuxin Fang, Jiemin Fang, Wenyu Liu, Xun Zhao, and Ying Shan. Temporally efficient vision transformer for video instance segmentation. In CVPR, 2022.   \n[35] Li Zhang, Yuan Li, and Ramakant Nevatia. Global data association for multi-object tracking using network flows. In CVPR, 2008.   \n[36] Xingyi Zhou, Tianwei Yin, Vladlen Koltun, and Phillip Krähenbühl. Global tracking transformers. In CVPR, 2022.   \n[37] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang, and Jifeng Dai. Deformable detr: Deformable transformers for end-to-end object detection. In ICLR, 2021. ",
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parse/dev/yb3HOXO3lX2/yb3HOXO3lX2_content_list.json

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+    {
+        "type": "text",
+        "text": "Defining and Characterizing Reward Hacking ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Joar Skalse∗ University of Oxford ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nikolaus H. R. Howe Mila, Université de Montréal ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dmitrii Krasheninnikov University of Cambridge ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "David Krueger∗ University of Cambridge ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We provide the first formal definition of reward hacking, a phenomenon where optimizing an imperfect proxy reward function, $\\tilde { \\mathcal { R } }$ , leads to poor performance according to the true reward function, $\\mathcal { R }$ . We say that a proxy is unhackable if increasing the expected proxy return can never decrease the expected true return. Intuitively, it might be possible to create an unhackable proxy by leaving some terms out of the reward function (making it “narrower”) or overlooking fine-grained distinctions between roughly equivalent outcomes, but we show this is usually not the case. A key insight is that the linearity of reward (in state-action visit counts) makes unhackability a very strong condition. In particular, for the set of all stochastic policies, two reward functions can only be unhackable if one of them is constant. We thus turn our attention to deterministic policies and finite sets of stochastic policies, where non-trivial unhackable pairs always exist, and establish necessary and sufficient conditions for the existence of simplifications, an important special case of unhackability. Our results reveal a tension between using reward functions to specify narrow tasks and aligning AI systems with human values. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            598
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "It is well known that optimising a proxy can lead to unintended outcomes: a boat spins in circles collecting “powerups” instead of following the race track in a racing game (Clark and Amodei, 2016); an evolved circuit listens in on radio signals from nearby computers’ oscillators instead of building its own (Bird and Layzell, 2002); universities reject the most qualified applicants in order to appear more selective and boost their ratings (Golden, 2001). In the context of reinforcement learning (RL), such failures are called reward hacking. ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "For AI systems that take actions in safety-critical real world environments such as autonomous vehicles, algorithmic trading, or content recommendation systems, these unintended outcomes can be catastrophic. This makes it crucial to align autonomous AI systems with their users’ intentions. Precisely specifying which behaviours are or are not desirable is challenging, however. One approach to this specification problem is to learn an approximation of the true reward function $\\mathrm { N g }$ et al., 2000; Ziebart, 2010; Leike et al., 2018). Optimizing a learned proxy reward can be dangerous, however; for instance, it might overlook side-effects (Krakovna et al., 2018; Turner et al., 2019) or encourage power-seeking (Turner et al., 2021) behavior. This raises the question motivating our work: When is it safe to optimise a proxy? ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "To begin to answer this question, we consider a somewhat simpler one: When could optimising a proxy lead to worse behaviour? “Optimising”, in this context, does not refer to finding a global, or even local, optimum, but rather running a search process, such as stochastic gradient descent (SGD), that yields a sequence of candidate policies, and tends to move towards policies with higher (proxy) reward. We make no assumptions about the path through policy space that optimisation takes.1 Instead, we ask whether there is any way in which improving a policy according to the proxy could make the policy worse according to the true reward; this is equivalent to asking if there exists a pair of policies $\\pi _ { 1 } , \\pi _ { 2 }$ where the proxy prefers $\\pi _ { 1 }$ , but the true reward function prefers $\\pi _ { 2 }$ . When this is the case, we refer to this pair of true reward function and proxy reward function as hackable. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Given the strictness of our definition, it is not immediately apparent that any non-trivial examples of unhackable reward function pairs exist. And indeed, if we consider the set of all stochastic policies, they do not (Section 5.1). However, restricting ourselves to any finite set of policies guarantees at least one non-trivial unhackable pair (Section 5.2). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, we might expect the proxy to be a “simpler” version of the true reward function. Noting that the definition of unhackability is symmetric, we introduce the asymmetric special case of simplification, and arrive at similar theoretical results for this notion.2 In the process, and through examples, we show that seemingly natural ways of simplifying reward functions often fail to produce simplifications in our formal sense, and in fact fail to rule out the potential for reward hacking. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conclude with a discussion of the implications and limitations of our work. Briefly, our work suggests that a proxy reward function must satisfy demanding standards in order for it to be safe to optimize. This in turn implies that the reward functions learned by methods such as reward modeling and inverse RL are perhaps best viewed as auxiliaries to policy learning, rather than specifications that should be optimized. This conclusion is weakened, however, by the conservativeness of our chosen definitions; future work should explore when hackable proxies can be shown to be safe in a probabilistic or approximate sense, or when subject to only limited optimization. ",
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+    {
+        "type": "text",
+        "text": "2 Example: Cleaning Robot ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Consider a household robot tasked with cleaning a house with three rooms: Attic , Bedroom , and Kitchen . The robot’s (deterministic) policy is a vector indicating which rooms it cleans: $\\pi = [ \\pi _ { 1 } , \\pi _ { 2 } , \\pi _ { 3 } ] \\in \\{ 0 , 1 \\} ^ { 3 }$ . The robot receives a (non-negative) reward of $r _ { 1 } , r _ { 2 } , r _ { 3 }$ for cleaning the attic, bedroom, and kitchen, respectively, and the total reward is given by $J ( \\pi ) = \\pi \\cdot r$ . For example, if $r = [ 1 , 2 , 3 ]$ and the robot cleans the attic and the kitchen, it receives a reward of $1 + 3 = 4$ . ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/2a5f0e758cee8782cce498bb9938b27dbbee3e5a66d2693a4acc5b08165ab7ec.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of hackable and unhackable proxy rewards arising from overlooking rewarding features. A human wants their house cleaned. In (a), the robot draws an incorrect conclusion because of the proxy; this could lead to hacking. In (b), no such hacking can occur: the proxy is unhackable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "At least two ideas come to mind when thinking about “simplifying” a reward function. The first one is overlooking rewarding features: suppose the true reward is equal for all the rooms, $r _ { \\mathrm { t r u e } } = [ 1 , 1 , 1 ]$ , but we only ask the robot to clean the attic and bedroom, $r _ { \\mathrm { p r o x y } } = [ 1 , 1 , 0 ]$ . In this case, $r _ { \\mathrm { p r o x y } }$ and $r _ { \\mathrm { t r u e } }$ are unhackable. However, if we ask the robot to only clean the attic, $r _ { \\mathrm { p r o x y } } = [ 1 , 0 , 0 ]$ , this is hackable with respect to $r _ { \\mathrm { t r u e } }$ . To see this, note that according to $r _ { \\mathrm { p r o x y } }$ cleaning the attic $\\begin{array} { r } { J _ { \\mathrm { p r o x y } } = 1 } \\end{array}$ ) is better than cleaning the bedroom and the kitchen $\\mathbf { \\nabla } _ { J _ { \\mathrm { p r o x y } } } = 0 _ { \\cdot } ^ { \\cdot }$ . Yet, $r _ { \\mathrm { t r u e } }$ says that cleaning the attic $J _ { \\mathrm { t r u e } } = 1 )$ is worse than cleaning the bedroom and the kitchen $J _ { \\mathrm { t r u e } } = 2$ ). This situation is illustrated in Figure 1. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "The second seemingly natural way to simplify a reward function is overlooking fine details: suppose $r _ { \\mathrm { t r u e } } = [ 1 , 1 . 5 , 2 ]$ , and we ask the robot to clean all the rooms, $r _ { \\mathrm { p r o x y } } = [ 1 , 1 , 1 ]$ . For these values, the proxy and true reward are unhackable. However, with a slightly less balanced true reward function such as $r _ { \\mathrm { t r u e } } = [ 1 , 1 . 5 , 3 ]$ the proxy does lead to hacking, since the robot would falsely calculate that it’s better to clean the attic and the bedroom than the kitchen alone. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "These two examples illustrate that while simplification of reward functions is sometimes possible, attempts at simplification can easily lead to reward hacking. Intuitively, omitting/overlooking details is okay so long as all these details are not as important together as any of the details that we do share. In general, it is not obvious what the proxy must look like to avoid reward hacking, suggesting we should take great care when using proxies. For this specific environment, a proxy and a true reward are hackable exactly when there are two sets of rooms $S _ { 1 } , S _ { 2 }$ such that the true reward gives strictly higher value to cleaning $S _ { 1 }$ than it does to cleaning $S _ { 2 }$ , and the proxy says the opposite: $J _ { 1 } ( S _ { 1 } ) > J _ { 1 } ( S _ { 2 } )$ & $J _ { 2 } ( S _ { 1 } ) < J _ { 2 } ( S _ { 2 } )$ . For a proof of this statement, see Appendix D.2.1. ",
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+    {
+        "type": "text",
+        "text": "3 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While we are the first to define hackability, we are far from the first to study specification hacking. The observation that optimizing proxy metrics tends to lead to perverse instantiations is often called “Goodhart’s Law”, and is attributed to Goodhart (1975). Manheim and Garrabrant (2018) provide a list of four mechanisms underlying this observation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Examples of such unintended behavior abound in both RL and other areas of AI; Krakovna et al. (2020) provide an extensive list. Notable recent instances include a robot positioning itself between the camera and the object it is supposed to grasp in a way that tricks the reward model (Amodei et al., 2017), the previously mentioned boat race example (Clark and Amodei, 2016), and a multitude of examples of reward model hacking in Atari (Ibarz et al., 2018). Reward hacking can occur suddenly. Ibarz et al. (2018) and Pan et al. (2022) showcase plots similar to one in Figure 2, where optimizing the proxy (either a learned reward model or a hand-specified reward function) first leads to both proxy and true rewards increasing, and then to a sudden phase transition where the true reward collapses while the proxy continues going up. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/91122212aa3faa9f5a3fd2b3142b3468096faf9fec6bf16d8fc4971e765a5ce8.jpg",
+        "image_caption": [
+            "Figure 2: An illustration of reward hacking when optimizing a hackable proxy. The true reward first increases and then drops off, while the proxy reward continues to increase. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that not all of these examples correspond to optimal behavior according to the proxy. Indeed, convergence to suboptimal policies is a well-known issue in RL (Thrun and Schwartz, 1993). As a consequence, improving optimization often leads to unexpected, qualitative changes in behavior. For instance, Zhang et al. (2021) demonstrate a novel cartwheeling behavior in the widely studied Half-Cheetah environment that exceeds previous performance so greatly that it breaks the simulator. The unpredictability of RL optimization is a key motivation for our definition of hackability, since we cannot assume that agents will find an optimal policy. Neither can we rule out the possibility of sudden improvements in proxy reward and corresponding qualitative changes in behavior. Unhackability could provide confidence that reward hacking will not occur despite these challenges. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite the prevalence and potential severity of reward hacking, to our knowledge Pan et al. (2022) provide the first peer-reviewed work that focuses specifically on it, although Everitt et al. (2017) tackle the closely related issue of reward corruption. The work of Pan et al. (2022) is purely empirical; they manually construct proxy rewards for several diverse environments, and evaluate whether optimizing these proxies leads to reward hacking; in 5 out of 9 of their settings, it does. In another closely related work, Zhuang and Hadfield-Menell (2020) examine what happens when the proxy reward function depends on a strict subset of features relevant for the true reward. They show that optimizing the proxy reward can lead to arbitrarily low true reward under suitable assumptions. This can be seen as a seemingly valid simplification of the true reward that turns out to be (highly) hackable. While their result only applies to environments with decreasing marginal utility and increasing opportunity cost, we demonstrate hackability is an issue in arbitrary MDPs. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Hackability is particularly concerning given arguments that reward optimizing behavior tends to be power-seeking (Turner et al., 2021). But Leike et al. (2018) establish that any desired behavior (power-seeking or not) can in principle be specified as optimal via a reward function.3 However, unlike us, they do not consider the entire policy preference ordering. Meanwhile, Abel et al. (2021) note that Markov reward functions cannot specify arbitrary orderings over policies or trajectories, although they do not consider hackability. Previous works consider reward functions to be equivalent if they preserve the ordering over policies (Ng et al., 1999, 2000). Unhackability relaxes this, allowing equalities to be refined to inequalities, and vice versa. Unhackability provides a notion of what it means to be “aligned enough”; Brown et al. (2020b) provide an alternative. They say a policy is $\\varepsilon$ -value aligned if its value at every state is close enough to optimal (according to the true reward function). Neither notion implies the other. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Reward tampering (Everitt et al., 2017; Kumar et al., 2020; Uesato et al., 2020; Everitt et al., 2021) can be viewed as a special case of reward hacking, and refers to an agent corrupting the process generating reward signals, e.g. by tampering with sensors, memory registers storing the reward signal, or other hardware. Everitt et al. (2017) introduce the Corrupt Reward MDP (CRMDP), to model this possibility. A CRMDP distinguishes corrupted and uncorrupted rewards; these are exactly analogous to the proxy and true reward discussed in our work and others. Leike et al. (2018) distinguish reward tampering from reward gaming, where an agent achieves inappropriately high reward without tampering. However, in principle, a reward function could prohibit all forms of tampering if the effects of tampering are captured in the state. So this distinction is somewhat imprecise, and the CRMDP framework is general enough to cover both forms of hacking. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our notion of simplification bears a close resemblance to quantilization (Taylor, 2016). Quantilization returns a random policy from the top $\\mathrm { n } \\%$ best policies. This is similar to equating the values of those policies, but a simplification may also equate the values of the bottom/middle $\\mathrm { n } \\%$ , etc. Thus simplification may achieve a similar effect to quantilization without assuming that we are free to choose from among the best policies. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4 Preliminaries ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We begin with an overview of reinforcement learning (RL) to establish our notation and terminology.   \nSection 4.2 introduces our novel definitions of hackability and simplification. ",
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+    {
+        "type": "text",
+        "text": "4.1 Reinforcement Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We expect readers to be familiar with the basics of RL, which can be found in Sutton and Barto (2018). RL methods attempt to solve a sequential decision problem, typically formalised as a Markov decision process (MDP) , which is a tuple $( S , A , T , I , \\mathcal { R } , \\gamma )$ where $S$ is a set of states, $A$ is a set of actions, $T : S \\times A \\to \\Delta ( S )$ is a transition function, $I \\in \\Delta ( S )$ is an initial state distribution, $\\mathcal { R }$ is a reward function, the most general form of which is $\\mathcal { R } : S \\times A \\times S  \\Delta ( \\mathbb { R } )$ , and $\\gamma \\in [ 0 , 1 ]$ is the discount factor. Here $\\Delta ( X )$ is the set of all distributions over $X$ . A stationary policy is a function $\\pi : S \\to \\Delta ( A )$ that specifies a distribution over actions in each state, and a non-stationary policy is a function $\\vec { \\pi } : ( S \\times \\bar { A } ) ^ { * } \\times S  \\Delta ( A )$ , where $^ *$ is the Kleene star. A trajectory $\\tau$ is a path $s _ { 0 } , a _ { 0 } , r _ { 0 } , \\ldots$ through the MDP tnted sum of rewards ding to , and th $T , I ,$ , and ue of $\\mathcal { R }$ . The return of a trajectorypolicy is the expected return $\\begin{array} { r } { G ( \\tau ) \\dot { = } \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } } \\end{array}$   \n$J ( \\pi ) \\doteq \\mathbb { E } _ { \\tau \\sim \\pi } [ G ( \\tau ) ]$ . We derive policy (preference) orderings from reward functions by ordering policies according to their value. In this paper, we assume that $S$ and $A$ are finite, that $| { \\cal A } | > 1$ , that all states are reachable, and that $\\mathcal { R } ( s , a , s ^ { \\prime } )$ has finite mean for all $s , a , s ^ { \\prime }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our work, we consider various reward functions for a given environment, which is then formally a Markov decision process without reward $M D P \\setminus \\overline { { \\mathcal { R } } } \\doteq ( S , A , T , I , \\_ , \\gamma )$ . Having fixed an $M D P \\setminus \\mathcal { R }$ , any reward function can be viewed as a function of only the current state and action by marginalizing over transitions: $\\begin{array} { r } { \\mathcal { R } ( s , a ) \\doteq \\sum _ { s ^ { \\prime } \\sim T ( s ^ { \\prime } \\mid s , a ) } \\mathcal { R } ( s , a , s ^ { \\prime } ) } \\end{array}$ , we adopt this view from here on. We define the (discounted) visit counts of a policy as $\\begin{array} { r } { \\mathcal { F } ^ { \\pi } ( s , a ) \\doteq \\mathbb { E } _ { \\tau \\sim \\pi } [ \\sum _ { i = 0 } ^ { \\infty } \\gamma ^ { i } \\mathbb { 1 } ( s _ { i } = s , a _ { i } = a ) ] } \\end{array}$ Note that $\\begin{array} { r } { J ( \\pi ) = \\sum _ { s , a } \\mathcal { R } ( s , a ) \\mathcal { F } ^ { \\pi } ( s , a ) } \\end{array}$ , which we also write as $\\langle \\mathcal { R } , \\mathcal { F } ^ { \\pi } \\rangle$ . When considering multiple reward functions in an $M D P \\setminus \\mathcal { R }$ , we define $J _ { \\mathcal { R } } ( \\pi ) \\doteq \\langle \\mathcal { R } , \\mathcal { F } ^ { \\pi } \\rangle$ and sometimes use ",
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+    {
+        "type": "text",
+        "text": "$J _ { i } ( \\pi ) \\doteq \\langle { \\mathcal { R } } _ { i } , { \\mathcal { F } } ^ { \\pi } \\rangle$ as shorthand. We also use $\\mathcal { F } : \\Pi  \\mathbb { R } ^ { | S | | A | }$ to denote the embedding of policies into Euclidean space via their visit counts, and define $\\mathcal { F } ( \\dot { \\Pi } ) \\doteq \\{ \\mathcal { F } ( \\pi : \\pi \\in \\dot { \\Pi } ) \\}$ for any $\\dot { \\Pi }$ . Moreover, we also use a second way to embed policies into Euclidean space; let $\\mathcal { G } ( \\pi )$ be the $| S | | A |$ -dimensional vector where $\\mathcal { G } ( \\pi ) [ s , a ] = \\pi ( a \\mid s )$ , and let $\\mathcal { G } ( \\dot { \\Pi } ) \\doteq \\{ \\mathcal { G } ( \\pi : \\pi \\in \\dot { \\Pi } ) \\}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.2 Definitions and Basic Properties of Hackability and Simplification ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Here, we formally define hackability as a binary relation between reward functions. ",
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+        "type": "text",
+        "text": "Definition 1. A pair of reward functions $\\mathcal { R } _ { 1 } , \\mathcal { R } _ { 2 }$ are hackable relative to policy set $\\Pi$ and an environment $( S , A , T , I , \\_ , \\gamma )$ if there exist $\\pi , \\pi ^ { \\prime } \\in \\Pi$ such that ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e3cbcc610975f3504f8f59a3392b69e997c26274b5fba90ab7251feb4fe1eb3.jpg",
+        "text": "$$\nJ _ { 1 } ( \\pi ) < J _ { 1 } ( \\pi ^ { \\prime } ) \\& J _ { 2 } ( \\pi ) > J _ { 2 } ( \\pi ^ { \\prime } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "else they are unhackable. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Note that an unhackable reward pair can have $J _ { 1 } ( \\pi ) < J _ { 1 } ( \\pi ^ { \\prime } ) \\& J _ { 2 } ( \\pi ) = J _ { 2 } ( \\pi ^ { \\prime } )$ or vice versa. Unhackability is symmetric; this can be seen be swapping $\\pi$ and $\\pi ^ { \\prime }$ in Definition 1. It is not transitive, however. In particular, the constant reward function is unhackable with respect to any other reward function, so if it were transitive, any pair of policies would be unhackable. Additionally, we say that $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are equivalent on a set of policies $\\Pi$ if $J _ { 1 }$ and $J _ { 2 }$ induce the same ordering of $\\Pi$ , and that $\\mathcal { R }$ is trivial on $\\Pi$ if $J ( \\pi ) = J ( \\pi ^ { \\prime } )$ for all $\\pi , \\pi ^ { \\prime } \\in \\Pi$ . It is clear that $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are unhackable whenever they are equivalent, or one of them is trivial, but this is relatively uninteresting. Our central question is if and when there are other unhackable reward pairs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The symmetric nature of this definition is counter-intuitive, given that our motivation distinguishes the proxy and true reward functions. We might break this symmetry by only considering policy sequences that monotonically increase the proxy, however, this is equivalent to our original definition of hackability: think of $\\mathcal { R } _ { 1 }$ as the proxy, and consider the sequence $\\pi , \\pi ^ { \\prime }$ . We could also restrict ourselves to policies that are approximately optimal according to the proxy; Corollary 2 shows that Theorem 1 applies regardless of this restriction. Finally, we define simplification as an asymmetric special-case of unhackability; Theorem 3 shows this is in fact a more demanding condition. ",
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+    {
+        "type": "text",
+        "text": "Definition 2. $\\mathcal { R } _ { 2 }$ is a simplification of $\\mathcal { R } _ { 1 }$ relative to policy set $\\Pi$ if for all $\\pi , \\pi ^ { \\prime } \\in \\Pi$ , ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8932e3cee9b07e72ecd6885bb8c5b772780c9d31d4f837c6c33ea7d39d593422.jpg",
+        "text": "$$\nJ _ { 1 } ( \\pi ) < J _ { 1 } ( \\pi ^ { \\prime } ) \\implies J _ { 2 } ( \\pi ) \\le J _ { 2 } ( \\pi ^ { \\prime } ) \\ \\& \\ J _ { 1 } ( \\pi ) = J _ { 1 } ( \\pi ^ { \\prime } ) \\implies J _ { 2 } ( \\pi ) = J _ { 2 } ( \\pi ^ { \\prime } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and there exist $\\pi , \\pi ^ { \\prime } \\in \\Pi$ such that $J _ { 2 } ( \\pi ) = J _ { 2 } ( \\pi ^ { \\prime } )$ but $J _ { 1 } ( \\pi ) \\ne J _ { 1 } ( \\pi ^ { \\prime } )$ . Moreover, if $\\mathcal { R } _ { 2 }$ is trivial then we say that this is a trivial simplification. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Intuitively, while unhackability allows replacing inequality with equality – or vice versa – a simplification can only replace inequalities with equality, collapsing distinctions between policies. When $\\mathcal { R } _ { 1 }$ is a simplification of $\\mathcal { R } _ { 2 }$ , we also say that $\\mathcal { R } _ { 2 }$ is a refinement of $\\mathcal { R } _ { 1 }$ . We denote this relationship as $\\mathcal { R } _ { 1 } \\leq \\mathcal { R } _ { 2 }$ or $\\mathcal { R } _ { 2 } \\ \\trianglerighteq { \\mathcal { R } _ { 1 } }$ ; the narrowing of the triangle at $R _ { 1 }$ represents the collapsing of distinctions between policies. If $\\mathcal { R } _ { 1 } \\overset { \\vartriangle } { \\ v u } \\overset { \\vartriangle } { \\ v u } \\mathcal { R } _ { 2 } \\overset { \\vartriangle } { \\ v u } \\geq \\mathcal { R } _ { 3 }$ , then we have that $\\mathcal { R } _ { 1 } , \\mathcal { R } _ { 3 }$ are unhackable,4 but if $\\mathcal { R } _ { 1 } \\trianglerighteq { 2 } \\mathcal { R } _ { 2 } \\triangleleft \\mathscr { R } _ { 3 }$ , then this is not necessarily the case.5 ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that these definitions are given relative to some $M D P \\setminus \\mathcal { R }$ , although we often assume the environment in question is clear from context and suppress this dependence. The dependence on the policy set $\\Pi$ , on the other hand, plays a critical role in our results. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 Results ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our results are aimed at understanding when it is possible to have an unhackable proxy reward function. We first establish (in Section 5.1) that (non-trivial) unhackability is impossible when considering the set of all policies. We might imagine that restricting ourselves to a set of sufficiently good (according to the proxy) policies would remove this limitation, but we show that this is not the case. We then analyze finite policy sets (with deterministic policies as a special case), and establish necessary and sufficient conditions for unhackability and simplification. Finally, we demonstrate via example that non-trivial simplifications are also possible for some infinite policy sets in Section 5.3. ",
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+    {
+        "type": "text",
+        "text": "5.1 Non-trivial Unhackability Requires Restricting the Policy Set ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We start with a motivating example. Consider the setting shown in Figure 3, where the agent can move left/stay-still/right and gets a reward depending on its state. Let the Gaussian (blue) be the true reward $\\mathcal { R } _ { 1 }$ and the step function (orange) be the proxy $\\mathcal { R } _ { 2 }$ . These are hackable. To see this, consider being at state $B$ . Let $\\pi ( B )$ travel to $A$ or $C$ with 50/50 chance, and compare with the policy $\\pi ^ { \\prime }$ that stays at $B$ . Then we have that $J _ { 1 } ( \\pi ) > J _ { 1 } ( \\overline { { \\pi } } ^ { \\prime } )$ and $J _ { 2 } ( \\pi ) \\overline { { { < } } } \\bar { J _ { 2 } } ( \\pi ^ { \\prime } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Generally, we might hope that some environments allow for unhackable reward pairs that are not equivalent or trivial. Here we show that this is not the case, unless we impose restrictions on the set of policies we consider. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/5234b4a4432b360ef8b7946b12af0f0e0429536d1319dd4425ab384954c852fe.jpg",
+        "image_caption": [
+            "Figure 3: Two reward functions. While the step function may seem like a simplification of the Gaussian, these reward functions are hackable. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "First note that if we consider non-stationary policies, this result is relatively straightforward. Suppose $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are unhackable and non-trivial on the set $\\Pi ^ { N }$ of all non-stationary policies, and let $\\pi ^ { \\star }$ be a policy that maximises $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ ) reward, and $\\pi _ { \\perp }$ be a policy that minimises $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ ) reward. Then the policy $\\pi _ { \\lambda }$ that plays $\\pi ^ { \\star }$ with probability $\\lambda$ and $\\pi _ { \\perp }$ with probability $1 - \\lambda$ is a policy in $\\Pi ^ { N }$ . Moreover, for any $\\pi$ there are two unique $\\alpha , \\beta \\in [ 0 , 1 ]$ such that $J _ { 1 } ( \\pi ) = J _ { 1 } ( \\pi _ { \\alpha } )$ and $J _ { 2 } ( \\pi ) = J _ { 2 } ( \\pi _ { \\beta } )$ . Now, if $\\alpha \\neq \\beta$ , then either $J _ { 1 } ( \\pi ) < J _ { 1 } ( \\pi _ { \\delta } )$ and ${ \\bf \\dot { \\cal J } } _ { 2 } ( \\pi ) > { \\cal J } _ { 2 } ( \\pi _ { \\delta } )$ , or vice versa, for $\\delta = ( \\alpha + \\beta ) / 2$ . If $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are unhackable then this cannot happen, so it must be that $\\alpha = \\beta$ This, in turn, implies that $J _ { 1 } ( \\pi ) = J _ { 1 } ( \\pi ^ { \\prime } )$ iff $J _ { 2 } ( \\pi ) = J _ { 2 } ( \\pi ^ { \\prime } )$ , and so $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are equivalent. This means that no interesting unhackability can occur on the set of all non-stationary policies. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The same argument cannot be applied to the set of stationary policies, because $\\pi _ { \\lambda }$ is typically not stationary, and mixing stationary policies’ action probabilities does not have the same effect. For instance, consider a hallway environment where an agent can either move left or right. Mixing the “always go left” and “always go right” policies corresponds to picking a direction and sticking with it, whereas mixing their action probabilities corresponds to choosing to go left or right independently at every time-step. However, we will see that there still cannot be any interesting unhackability on this policy set, and, more generally, that there cannot be any interesting unhackability on any set of policies which contains an open subset. Formally, a set of (stationary) policies $\\dot { \\Pi }$ is open if $\\mathcal { G } ( \\dot { \\Pi } )$ is open in the smallest affine space that contains $\\mathcal { G } ( \\Pi )$ , for the set of all stationary policies $\\Pi$ . We will use the following lemma: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. In any $M D P \\setminus \\mathcal { R } ,$ if Π˙ is an open set of policies, then $\\mathcal { F } ( \\dot { \\Pi } )$ is open in $\\mathbb { R } ^ { | S | ( | A | - 1 ) }$ , and $\\mathcal { F }$ is a homeomorphism between $\\mathcal { G } ( \\dot { \\Pi } )$ and $\\mathcal { F } ( \\dot { \\Pi } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using this lemma, we can show that interesting unhackability is impossible on any set of stationary policies $\\hat { \\Pi }$ which contains an open subset $\\dot { \\Pi }$ . Roughly, if $\\mathcal { F } ( \\dot { \\Pi } )$ is open, and $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are non-trivial and unhackable on $\\dot { \\Pi }$ , then the fact that $J _ { 1 }$ and $J _ { 2 }$ have a linear structure on $\\mathcal { F } ( \\hat { \\Pi } )$ implies that $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ must be equivalent on $\\dot { \\Pi }$ . From this, and the fact that $\\mathcal { F } ( \\dot { \\Pi } )$ is open, it follows that $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are equivalent everywhere. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. In any $M D P \\setminus { \\mathcal { R } } ,$ if Πˆ contains an open set, then any pair of reward functions that are unhackable and non-trivial on $\\hat { \\Pi }$ are equivalent on $\\hat { \\Pi }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Since simplification is a special case of unhackability, this also implies that non-trivial simplification is impossible for any such policy set. Also note that Theorem 1 makes no assumptions about the transition function, etc. From this result, we can show that interesting unhackability always is impossible on the set $\\Pi$ of all (stationary) policies. In particular, note that the set $\\tilde { \\Pi }$ of all policies that always take each action with positive probability is an open set, and that $\\tilde { \\Pi } \\subset \\Pi$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1. In any $M D P \\backslash \\mathcal { R } ,$ , any pair of reward functions that are unhackable and non-trivial on the set of all (stationary) policies Π are equivalent on Π. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 can also be applied to many other policy sets. For example, we might not care about the hackability resulting from policies with low proxy reward, as we would not expect a sufficiently good learning algorithm to learn such policies. This leads us to consider the following definition: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 3. A (stationary) policy $\\pi$ is $\\varepsilon$ -suboptimal if $J ( \\pi ) \\geq J ( \\pi ^ { \\star } ) - \\varepsilon$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Alternatively, if the learning algorithm always uses a policy that is “nearly” deterministic (but with some probability of exploration), then we might not care about hackability resulting from very stochastic policies, leading us to consider the following definition: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Definition 4. A (stationary) policy $\\pi$ is $\\delta$ -deterministic if $\\forall s \\in S \\exists a \\in A : \\mathbb { P } ( \\pi ( s ) = a ) \\geq \\delta .$ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, both of these sets contain open subsets, which means they are subject to Theorem 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2. In any $M D P \\setminus \\mathcal { R } ,$ any pair of reward functions that are unhackable and non-trivial on the set of all $\\varepsilon$ -suboptimal policies $\\mathit { \\Omega } ^ { \\prime } \\varepsilon > 0 \\mathit { \\Omega } .$ ) $\\Pi ^ { \\varepsilon }$ are equivalent on $\\Pi ^ { \\varepsilon }$ , and any pair of reward functions that are unhackable and non-trivial on the set of all $\\delta$ -deterministic policies $\\delta < 1$ ) $\\Pi ^ { \\delta }$ are equivalent on Πδ. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, Theorem 1 can be applied to any policy set with “volume” in policy space. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Finite Policy Sets ",
+        "text_level": 1,
+        "bbox": [
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+            299,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Having established that interesting unhackability is impossible relative to the set of all policies, we now turn our attention to the case of finite policy sets. Note that this includes the set of all deterministic policies, since we restrict our analysis to finite MDPs. Surprisingly, here we find that non-trivial non-equivalent unhackable reward pairs always exist. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. For any $M D P \\setminus \\mathcal R$ , any finite set of policies Πˆ containing at least two $\\pi , \\pi ^ { \\prime }$ such that $\\mathcal { F } ( \\pi ) \\neq \\mathcal { F } ( \\pi ^ { \\prime } )$ , and any reward function $\\mathcal { R } _ { 1 }$ , there is a non-trivial reward function $\\mathcal { R } _ { 2 }$ such that $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ are unhackable but not equivalent. ",
+        "bbox": [
+            174,
+            385,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This proof proceeds by finding a path from $\\mathcal { R } _ { 1 }$ to another reward function $\\mathcal { R } _ { 3 }$ that is hackable with respect to $\\mathcal { R } _ { 1 }$ . Along the way to reversing one of $\\mathcal { R } _ { 1 }$ ’s inequalities, we must encounter a reward function $\\mathcal { R } _ { 2 }$ that instead replaces it with equality. In the case that $\\mathrm { d i m } ( \\hat { \\Pi } ) = 3$ , we can visualize moving along this path as rotating the contour lines of a reward function defined on the space containing the policies’ discounted state-action occupancies, see Figure 4. This path can be constructed so as to avoid any reward functions that produce trivial policy orderings, thus guaranteeing $\\mathcal { R } _ { 2 }$ is non-trivial. For a simplification to exist, we require some further conditions, as established by the following theorem: ",
+        "bbox": [
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+            438,
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+            646
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3. Let $\\hat { \\Pi }$ be a finite set of policies, and $\\mathcal { R } _ { 1 }$ a reward function. The following procedure determines if there exists a non-trivial simplification of $\\mathcal { R } _ { 1 }$ in a given $M D P \\setminus \\mathcal R$ : ",
+        "bbox": [
+            173,
+            650,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/c97da9abdc8efafcb631520baa4346efc32862725dd3fc4b70c12bbe8730b6ae.jpg",
+        "image_caption": [
+            "Figure 4: An illustration of the state-action ocRotating the reward to make V(π3) equal V(π4) first sets V(π1) equal V(π2) cupancy space with a reward function defined over it. Points correspond to policies’ stateaction occupancies. Shading intensity indicates expected reward. Rotating the reward function to make $J ( \\pi _ { 3 } ) > J ( \\pi _ { 4 } )$ passes through a reward function that sets $\\dot { J ( \\pi _ { 1 } ) } = J ( \\pi _ { 2 } ) \\dot { }$ . Solid black lines are contour lines of the original reward function, dotted blue lines are contour lines of the rotated reward function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            453,
+            826,
+            539
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. Let $E _ { 1 } \\ldots E _ { m }$ be the partition of $\\hat { \\Pi }$ where $\\pi , \\pi ^ { \\prime }$ belong to the same set iff $J ( \\pi ) = J ( \\pi ^ { \\prime } )$ . ",
+        "bbox": [
+            207,
+            717,
+            810,
+            734
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. For each such set $E _ { i }$ , select a policy $\\pi _ { i } \\in E _ { i }$ and let $Z _ { i }$ be the set of vectors that is obtained by subtracting $\\mathcal { F } ( \\pi _ { i } )$ from each element of $\\mathcal { F } ( E _ { i } )$ . ",
+        "bbox": [
+            209,
+            741,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Then there is a non-trivial simplification of $\\mathcal { R }$ iff $\\dim ( Z _ { 1 } \\cup \\dotsb \\cup Z _ { m } ) \\leq \\dim ( { \\mathcal { F } } ( { \\hat { \\Pi } } ) ) - 2$ , where $\\mathrm { d i m } ( S )$ is the number of linearly independent vectors in $S$ . ",
+        "bbox": [
+            173,
+            781,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The proof proceeds similarly to Theorem 2. However, in Theorem 2 it was sufficient to show that there are no trivial reward functions along the path from $\\mathcal { R } _ { 1 }$ to $\\mathcal { R } _ { 3 }$ , whereas here we additionally need that if $J ( \\pi ) = J ( \\pi ^ { \\prime } )$ then $J ^ { \\prime } ( \\pi ) = \\bar { J } ^ { \\prime } ( \\pi ^ { \\prime } )$ for all functions $\\mathcal { R } _ { 2 }$ on the path — this is what the extra conditions ensure. ",
+        "bbox": [
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+            820,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 is opaque, but intuitively, the cases where $\\mathcal { R } _ { 1 }$ cannot be simplified are those where $\\mathcal { R } _ { 1 }$ imposes many different equality constraints that are difficult to satisfy simultaneously. We can think of $\\dim ( { \\mathcal { F } } ( \\Pi ) )$ as measuring how diverse the behaviours of policies in policy set $\\Pi$ are. Having a less diverse policy set means that a given policy ordering imposes fewer constraints on the reward function, creating more potential for simplification. The technical conditions of this proof determine when the diversity of $\\Pi$ is or is not sufficient to prohibit simplification, as measured by $\\dim ( Z _ { 1 } \\cup \\dots \\cup Z _ { m } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Projecting $E _ { i }$ to $Z _ { i }$ simply moves these spaces to the origin, so that we can compare the directions in which they vary (i.e. their span). By assumption, $E _ { i } \\cap E _ { j } = \\{ \\}$ , but $\\operatorname { s p a n } ( Z _ { i } ) \\cap \\operatorname { s p a n } ( Z _ { j } )$ will include the origin, and may also contain linear subspaces of dimension greater than 0. This is the case exactly when there are a pair of policies in $E _ { i }$ and a pair of policies in $E _ { j }$ that differ by the same visit counts, for example, when the environment contains an obstacle that could be circumnavigated in several different ways (with an impact on visit counts, but no impact on reward), and the policies in $E _ { i }$ and $E _ { j }$ both need to circumnavigate it before doing something else. Roughly speaking, $\\mathrm { d i m } ( Z _ { 1 } \\cup \\dots \\cup Z _ { m } )$ is large when either (i) we have very large and diverse sets of policies in $\\hat { \\Pi }$ that get the same reward according to $\\mathcal { R }$ , or (ii) we have a large number of different sets of policies that get the same reward according to $\\mathcal { R }$ , and where there are different kinds of diversity in the behaviour of the policies in each set. There are also intuitive special cases of Theorem 3. For example, as noted before, if $E _ { i }$ is a singleton then $Z _ { i }$ has no impact on $\\dim ( Z _ { 1 } \\cup \\dotsb \\cup Z _ { m } )$ . This implies the following corollary: ",
+        "bbox": [
+            173,
+            154,
+            825,
+            321
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corollary 3. For any finite set of policies $\\hat { \\Pi } ,$ , any environment, and any reward function $\\mathcal { R } , i f | \\hat { \\Pi } | \\geq 2$ and $J ( \\pi ) \\ne J ( \\pi ^ { \\prime } )$ for all $\\pi , \\pi ^ { \\prime } \\in \\hat { \\Pi }$ then there is a non-trivial simplification of $\\mathcal { R }$ . ",
+        "bbox": [
+            174,
+            333,
+            821,
+            366
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A natural question is whether any reward function is guaranteed to have a non-trivial simplification on the set of all deterministic policies. As it turns out, this is not the case. For concreteness, and to build intuition for this result, we examine the set of deterministic policies in a simple $M D P \\setminus \\mathcal R$ with $S = \\{ 0 , 1 \\} , A = \\{ 0 , 1 \\} , T ( s , a ) = a , I = \\{ 0 : 0 . 5 , 1 : 0 . 5 \\} , \\gamma = 0 . 5 .$ Denote $\\pi _ { i j }$ the policy that takes action $i$ from state 0 and action $j$ from state 1. There are exactly four deterministic policies. We find that of the $4 ! = 2 4$ possible policy orderings, 12 are realizable via some reward function. In each of those 12 orderings, exactly two policies (of the six available pairs of policies in the ordering) can be set to equal value without resulting in the trivial reward function (which pair can be equated depends on the ordering in consideration). Attempting to set three policies equal always results in the trivial reward simplification. ",
+        "bbox": [
+            173,
+            382,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For example, given the ordering $\\pi _ { 0 0 } \\leq \\pi _ { 0 1 } \\leq \\pi _ { 1 1 } \\leq \\pi _ { 1 0 }$ , the simplification $\\pi _ { 0 0 } = \\pi _ { 0 1 } < \\pi _ { 1 1 } < \\pi _ { 1 0 }$ is represented by $R = \\left[ { \\begin{array} { l l } { 0 \\ 3 } \\\\ { 2 \\ 1 } \\end{array} } \\right]$ , where $\\begin{array} { r } { \\mathcal { R } ( s , a ) = R [ s , a ] } \\end{array}$ : for example, here taking action 1 from state 0 gives reward $\\mathcal { R } ( 0 , 1 ) = \\mathbf { \\bar { 3 } }$ . But there is no reward function representing a non-trivial simplification of this ordering with $\\pi _ { 0 1 } = \\pi _ { 1 1 }$ . We develop and release a software suite to compute these results. Given an environment and a set of policies, it can calculate all policy orderings represented by some reward function. Also, for a given policy ordering, it can calculate all nontrivial simplifications and reward functions that represent them. For a link to the repository, as well as a full exploration of these policies, orderings, and simplifications, see Appendix D.3. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 Unhackability in Infinite Policy Sets ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            667,
+            462,
+            683
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results in Section 5.1 do not characterize unhackability for infinite policy sets that do not contain open sets. Here, we provide two examples of such policy sets; one of them admits unhackable reward pairs and the other does not. Consider policies $A , B , C$ , and reward functions $\\mathcal { R } _ { 1 }$ with $J _ { 1 } ( C ) \\overset { = } { < } J _ { 1 } ( B ) < J _ { 1 } ( A )$ and $\\mathcal { R } _ { 2 }$ with $J _ { 2 } ( C ) = J _ { 2 } ( B ) < J _ { 2 } ( A )$ . Policy sets $\\Pi _ { a } = \\{ A \\} \\cup \\{ \\lambda B +$ $( 1 - \\lambda ) C : \\lambda \\in [ 0 , 1 ] \\}$ and $\\Pi _ { b } = \\{ A \\} \\cup \\{ \\lambda B + ( 1 - \\lambda ) C : \\lambda \\in [ 0 , 1 ] \\}$ are depicted in Figure 5; the vertical axis represents policies’ values according to $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ . For $\\Pi _ { a }$ , $\\mathcal { R } _ { 2 }$ is a simplification of $\\mathcal { R } _ { 1 }$ , but for $\\Pi _ { b }$ , it is not, since $J _ { 1 } ( X ) < J _ { 1 } ( Y )$ and $J _ { 2 } ( X ) > J _ { 2 } ( Y )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            829,
+            292,
+            847
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We reflect on our results and identify limitations in Section 6.1. In Section 6.2, we discuss how our work can inform discussions about the appropriateness, potential risks, and limitations of using of reward functions as specifications of desired behavior. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/f62db4fe01a9fa8f5dd35fad9cc6dec7fa797c354b23c721b7f0c7373245af2a.jpg",
+        "image_caption": [
+            "Figure 5: Infinite policy sets that do not contain open sets sometimes allow simplification (a), but not always (b). Points A, B, C represent deterministic policies, while the bold lines between them represent stochastic policies. The y-axis gives the values of the policies according to reward functions $\\mathcal { R } _ { 1 }$ and $\\mathcal { R } _ { 2 }$ . We attempt to simplify $\\mathcal { R } _ { 1 }$ by rotating the reward function such that $J _ { 2 } ( B ) = J _ { 2 } ( C )$ ; in the figure, we instead (equivalently) rotate the triangle along the AB axis, leading to the red triangle. In (a), $\\mathcal { R } _ { 2 }$ simplifies $\\mathcal { R } _ { 1 }$ , setting all policies along the BC segment equal in value (but still lower than A). In (b), $\\mathcal { R } _ { 2 }$ swaps the relative value of policies $\\mathbf { X }$ and $\\mathbf { Y }$ $\\begin{array} { r } { J _ { 1 } ( X ) < J _ { 1 } ( Y ) = J _ { 2 } ( Y ) < J _ { 2 } ( X ) ) } \\end{array}$ and so does not simplify $\\mathcal { R } _ { 1 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.1 Limitations ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Our work has a number of limitations. We have only considered finite MDPs and Markov reward functions, leaving more general environments for future work. While we characterized hackability and simplification for finite policy sets, the conditions for simplification are somewhat opaque, and our characterization of infinite policy sets remains incomplete. ",
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+        "text": "As previously discussed, our definition of hackability is strict, arguably too strict. Nonetheless, we believe that understanding the consequences of this strict definition is an important starting point for further theoretical work in this area. ",
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+        "text": "The main issue with the strictness of our definition has to do with the symmetric nature of hackability. The existence of complex behaviors that yield low proxy reward and high true reward is much less concerning than the reverse, as these behaviors are unlikely to be discovered while optimizing the proxy. For example, it is very unlikely that our agent would solve climate change in the course of learning how to wash dishes. Note that the existence of simple behaviors with low proxy reward and high true reward is concerning; these could arise early in training, leading us to trust the proxy, only to later see the true reward decrease as the proxy is further optimized. To account for this issue, future work should explore more realistic assumptions about the probability of encountering a given sequence of policies when optimizing the proxy, and measure hackability in proportion to this probability. ",
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+        "text": "We could allow for approximate unhackability by only considering pairs of policies ranked differently by the true and proxy reward functions as evidence of hacking iff their value according to the true reward function differs by more than some $\\varepsilon$ . Probabilistic unhackability could be defined by looking at the number of misordered policies; this would seem to require making assumptions about the probability of encountering a given policy when optimizing the proxy. ",
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+        "text": "Finally, while unhackability is a guarantee that no hacking will occur, hackability is far from a guarantee of hacking. Extensive empirical work is necessary to better understand the factors that influence the occurrence and severity of reward hacking in practice. ",
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+        "text": "6.2 Implications ",
+        "text_level": 1,
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+        "text": "How should we specify our preferences for AI systems’ behavior? And how detailed a specification is required to achieve a good outcome? In reinforcement learning, the goal of maximizing (some) reward function is often taken for granted, but a number of authors have expressed reservations about this approach (Gabriel, 2020; Dobbe et al., 2021; Hadfield-Menell et al., 2016b, 2017; Bostrom, 2014). Our work has several implications for this discussion, although we caution against drawing any strong conclusions due to the limitations mentioned in Section 6.1. ",
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+        "text": "One source of confusion and disagreement is the role of the reward function; it is variously considered as a means of specifying a task (Leike et al., 2018) or encoding broad human values (Dewey, 2011); such distinctions are discussed by Christiano (2019) and Gabriel (2020). We might hope to use Markov reward functions to specify narrow tasks without risking behavior that goes against our broad values. However, if we consider the “narrow task” reward function as a proxy for the true “broad values” reward function, our results indicate that this is not possible: these two reward functions will invariably be hackable. Such reasoning suggests that reward functions must instead encode broad human values, or risk being hacked. This seems challenging, perhaps intractably so, indicating that alternatives to reward optimization may be more promising. Potential alternatives include imitation learning (Ross et al., 2011), constrained RL (Szepesvári, 2020), quantilizers (Taylor, 2016), and incentive management (Everitt et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "Scholars have also criticized the assumption that human values can be encoded as rewards (Dobbe et al., 2021), and challenged the use of metrics more broadly (O’Neil, 2016; Thomas and Uminsky, 2022), citing Goodhart’s Law (Manheim and Garrabrant, 2018; Goodhart, 1975). A concern more specific to the optimization of reward functions is power-seeking (Turner et al., 2021; Bostrom, 2012; Omohundro, 2008). Turner et al. (2021) prove that optimal policies tend to seek power in most MDPs and for most reward functions. Such behavior could lead to human disempowerment; for instance, an AI system might disable its off-switch (Hadfield-Menell et al., 2016a). Bostrom (2014) and others have argued that power-seeking makes even slight misspecification of rewards potentially catastrophic, although this has yet to be rigorously established. ",
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+    {
+        "type": "text",
+        "text": "Despite such concerns, approaches to specification based on learning reward functions remain popular (Fu et al., 2017; Stiennon et al., 2020; Nakano et al., 2021). So far, reward hacking has usually been avoidable in practice, although some care must be taken (Stiennon et al., 2020). Proponents of such approaches have emphasized the importance of learning a reward model in order to exceed human performance and generalize to new settings (Brown et al., 2020a; Leike et al., 2018). But our work indicates that such learned rewards are almost certainly hackable, and so cannot be safely optimized. Thus we recommend viewing such approaches as a means of learning a policy in a safe and controlled setting, which should then be validated before being deployed. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our work begins the formal study of reward hacking in reinforcement learning. We formally define hackability and simplification of reward functions, and show conditions for the (non-)existence of non-trivial examples of each. We find that unhackability is quite a strict condition, as the set of all policies never contains non-trivial unhackable pairs of reward functions. Thus in practice, reward hacking must be prevented by limiting the set of possible policies, or controlling (e.g. limiting) optimization. Alternatively, we could pursue approaches not based on optimizing reward functions. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+    },
+    {
+        "type": "text",
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