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+# From Chaos to Clarity: 3DGS in the Dark
+
+Zhihao Li* Yufei Wang* Alex Kot Bihan Wen†
+
+Department of EEE, Nanyang Technology University, Singapore
+
+https://lizhihao6.github.io/Raw3DGS
+
+# Abstract
+
+Novel view synthesis from raw images provides superior high dynamic range (HDR) information compared to reconstructions from low dynamic range RGB images. However, the inherent noise in unprocessed raw images compromises the accuracy of 3D scene representation. Our study reveals that 3D Gaussian Splatting (3DGS) is particularly susceptible to this noise, leading to numerous elongated Gaussian shapes that overfit the noise, thereby significantly degrading reconstruction quality and reducing inference speed, especially in scenarios with limited views. To address these issues, we introduce a novel self-supervised learning framework designed to reconstruct HDR 3DGS from a limited number of noisy raw images. This framework enhances 3DGS by integrating a noise extractor and employing a noise-robust reconstruction loss that leverages a noise distribution prior. Experimental results show that our method outperforms LDR/HDR 3DGS and previous state-of-the-art (SOTA) self-supervised and supervised pre-trained models in both reconstruction quality and inference speed on the RawNeRF dataset across a broad range of training views. Code can be found in https://lizhihao6.github.io/Raw3DGS.
+
+# 1 Introduction
+
+Novel view synthesis (NVS) is fundamental to 3D vision, with extensive applications in virtual and augmented reality (VR/AR) [16; 17], autonomous driving [18; 34], and 3D asset creation [22; 4; 20]. Neural Radiance Fields (NeRF) [24] have revolutionized this field by rendering colors through the accumulation of RGB values along sampling rays, employing an implicit MultiLayer Perceptron (MLP) representation. Typically, this method uses low dynamic range (LDR) RGB images processed by image signal processing (ISP) modules, leading to a significant loss of crucial scene details, especially in high-contrast areas like highlights and shadows, which can degrade performance in high dynamic range (HDR) environments such as tunnels, sunsets, or dimly lit scenes. Moreover, reliance on ISP-processed RGB images restricts post-capture color and tone adjustments, presenting significant challenges for photographers and modelers during post-production.
+
+In contrast, raw images before ISP offer a higher dynamic range and preserve more scene information. Recent research has indicated that utilizing raw images can significantly enhance the performance of downstream computer vision tasks in complex lighting conditions [19] and offer greater flexibility in post-production adjustments [37; 5]. Building on this advantage, RawNeRF [23] first employed raw images as the optimization target in NeRF, achieving marked improvements over traditional RGB-based LDR NeRF approaches. However, RawNeRF's reliance on implicit 3D representation is computationally demanding, requiring up to 48 hours to train a single scene and about one minute to render a single view, which limits its practicality for real-time applications.
+
+![](images/b1f8a3d2a0184aa1562ad60483a0ea5ac3bdc74edfafde678169d47d70eedd64.jpg)  
+(a) Impact of noise on PSNR
+
+![](images/432142054b07ea3affa875df272c210fd1673978f37eb5905369d31b15326023.jpg)  
+(b) Impact of noise on FPS
+
+![](images/a4941d8d582df18717046b014b00e3ad4364548605d420951a171ea89961a603.jpg)  
+(c) Visualization of test views  
+Figure 1: Comparative analysis of 3DGS trained with clean raw images, denoted  $\mathcal{X}$ , versus noisy raw images, denoted  $\tilde{\mathcal{X}}$ , across various training view counts  $N$ . The clean raw images, captured in daylight, are selected from the RawNeRF dataset [23]. The noisy raw images are generated from these clean images using the noise model from PMN [6] with calibrated camera noise parameters. (a) Training with noisy raw images results in decreased PSNR in the test views, with a widening performance gap as the number of training views is reduced. (b) The rendering speed (FPS) shows a similar trend to PSNR. (c) Test view visualizations show that training with noisy images causes 3DGS to produce numerous thin, flat Gaussian shapes, leading to visual artifacts and reduced FPS, especially with fewer training views.
+
+Very recently, the advance on 3D Gaussian Splatting (3DGS) [13] offers an explicit 3D representation that employs a set of learnable 3D Gaussian points to depict color, shape, and opacity, thereby enabling real-time novel view rendering. However, utilizing raw images directly as the optimization target in 3DGS can introduce significant artifacts and adversely affect rendering speed, especially in scenarios with limited training views—a common challenge in real-world applications, as depicted in Fig. 1. These challenges arise primarily from the inherent noise in raw images, which is more pronounced than in RGB images due to the absence of noise reduction and smoothing processes typically performed by ISPs. As demonstrated in Fig. 2a, the presence of noise in raw images is an inevitable consequence of physical phenomena such as the photoelectric effect and hardware limitations like dark current leakage. Unlike NeRF, where the Multilayer Perceptron (MLP) acts as a low-frequency filter [38] to mitigate high-frequency noise, 3DGS tends to produce numerous thin, elongated Gaussians to represent noise, as seen in Fig. 1c.
+
+To address these challenges, we propose a novel self-supervised framework that jointly denoises and constructs HDR 3DGS representations within noisy raw images using a noise robust reconstruction loss. Specifically, our method utilizes a noise extractor to predict the underlying noise in raw images, constrained by a predefined noise model. Simultaneously, the 3DGS branch is optimized towards pseudo noise-free raw images. This dual-branch approach allows the noise extractor to accurately predict and separate noise from the raw images. Compared to using the standard RawNeRF loss function with conventional 3DGS, our approach significantly reduces noise artifacts in rendered views and enhances the rendering speed. Our contributions can be summarized as follows:
+
+- We discover that noisy raw images significantly degrade the rendering quality and speed of 3DGS, especially in scenarios with limited training views. We provide a detailed analysis of how noise impacts the optimization of 3DGS, and we model its relationship with the number of training views and the noise distribution.  
+- We propose a novel self-supervised framework that incorporates a noise robust reconstruction loss. This framework leverages a physical-based noise model to jointly denoise and enhance the HDR representation of 3DGS within noisy raw images.  
+- Our method substantially outperforms conventional 3DGS methods that employ RawNeRF loss, along with both self-supervised and pre-trained supervised denoisers, in terms of rendering quality and processing speed across various training view counts.
+
+# 2 Related work
+
+# 2.1 Neural Radiance Fields and 3D Gaussian Splitting
+
+Recently, Neural Radiance Fields (NeRF) [24] have spearheaded a significant advancement in novel-view synthesis by reconstructing 3D scenes. NeRF models achieve scene representation by optimizing coordinate-based multi-layer perceptrons (MLPs) to estimate color and density values through differentiable volume rendering. Typically, NeRF utilizes tone-mapped low dynamic range (LDR) images as inputs. These images are processed by image signal processing (ISP), which, while reducing noise and smoothing details, also maps high dynamic range (HDR) to LDR, often clipping highlights and shadows. To counter these limitations, RawNeRF [23] modified NeRF with a scaling loss to directly train on linear raw images, thus preserving the scene's full dynamic range. However, like most previous NeRF-based methods, RawNeRF is computationally demanding, requiring up to 48 hours to train a single scene and about one minute to render a single view, which constrains its usability in real-time applications. To accelerate rendering, various methods have been explored, including space discretization [7; 8; 10; 26; 29; 33; 35], codebooks [28], and hash tables [25]. Yet, these approaches still require a substantial number of queries to render a single pixel, rendering them unsuitable for real-time applications. More recently, 3D Gaussian Splatting (3DGS) [13] introduced a novel approach by explicitly representing scenes with learnable 3D Gaussians. Utilizing differentiable splatting and tile-based rasterization, 3DGS enables real-time novel view rendering. However, the initial 3DGS models did not achieve rendering quality comparable to NeRF. To address this, Scaffold-GS [13] introduces an anchor-level feature to capture correlations between adjacent Gaussian points, which significantly enhances the rendering quality of 3DGS, achieving comparable results to NeRF while maintaining real-time rendering speeds.
+
+# 2.2 Raw image and video denoising methods
+
+Raw image denoising is critical in both photography [9] and scientific imaging [15; 12]. Traditional ISPs typically employ self-similarity-based methods such as BM3D [3] to reduce noise. Recent advancements such as SID [2] and SIDD [1] demonstrate that data-driven, deep-learning approaches can surpass these traditional techniques. Innovatively, noise model-based methods like the ELD [32] have extended denoising capabilities into extreme low-light scenarios. It challenges conventional Gaussian noise models by introducing a row-based noise model that uses the Tukey Lambda distribution, which more accurately captures long-tail noise patterns in low-light conditions. The PMN [6] further investigates Fixed Pattern Noise in dark frames, enhancing alignment with the Poisson-Gaussian (P-G) distribution. These methods, however, often rely on large datasets of paired or clean images for training, as addressed by Neighbor2Neighbor [11], which trains denoising models solely on noisy images without clean counterparts. Building on the Neighbor2Neighbor approach, LGBPN [31] introduces more diverse masks to denoise spatially relevant noise. Nevertheless, Neighbor2Neighbor and LGBPN lack integration of a precise noise model or a physical prior of the 3D world, aspects our framework incorporates by merging a noise model with 3DGS to effectively separate noise from noisy images, eliminating the need for clean images. Complementing image-based methods, Yue et al. [36] expanded the scope to raw video. They introduced a dataset specifically curated for raw video denoising, along with multi-frame-based networks that leverage temporal information across frames to enhance the denoising process.
+
+# 3 Preliminaries
+
+In this section, we describe the noise model that we have adopted to regularize the extracted noise from raw images, provide an overview of the background of 3D Gaussian Splatting (3DGS)[13], and review the RawNeRF scaling loss[23] for handling the wide dynamic range in raw images.
+
+Noise model. In the camera imaging process, depicted in Fig. 2a, the raw image  $\tilde{\mathbf{x}}$  derived from the ideal clean raw image  $\mathbf{x}$  inevitably contains noise  $\mathbf{n}$ . This relationship is expressed as:
+
+$$
+\tilde {\mathbf {x}} = \mathbf {x} + \mathbf {n}. \tag {1}
+$$
+
+Drawing on existing noise models [32; 6], we decompose  $\mathbf{n}$  into various noise components as:
+
+$$
+\mathbf {n} = \mathbf {n} _ {\text {s h o t}} + \mathbf {n} _ {\text {r e a d}} + \mathbf {n} _ {f p}, \tag {2}
+$$
+
+where  $\mathbf{n}_{\text{shot}}$ ,  $\mathbf{n}_{\text{read}}$ , and  $\mathbf{n}_{fp}$  denote shot noise, read noise, and fixed pattern noise, respectively. A comprehensive description of each noise component is provided in the Supplemental Material. Given that  $\mathbf{n}_{\text{shot}}$  can be approximated from the Poisson distribution  $\mathcal{P}(\frac{\mathbf{x}}{k}) \cdot k - \mathbf{x}$  to a Gaussian distribution  $\mathcal{N}(0, \mathbf{x} \cdot k)$ , where  $k$  represents the camera system gain, it can be seamlessly combined with  $\mathbf{n}_{\text{read}}$  which also follows a Gaussian distribution  $\mathcal{N}(0, \sigma_{\text{read}}^2)$ . This integration yields a heteroscedastic Gaussian noise model:
+
+$$
+\mathbf {n} _ {h g} \sim \mathcal {N} \left(0, \sigma_ {h g} ^ {2}\right), \quad \sigma_ {h g} ^ {2} = \sigma_ {\text {r e a d}} ^ {2} + \mathbf {x} \cdot k. \tag {3}
+$$
+
+Thus, the overall noise model simplifies to:
+
+$$
+\mathbf {n} = \mathbf {n} _ {h g} + \mathbf {n} _ {f p}. \tag {4}
+$$
+
+Utilizing Eq. (4) as the prior for noise modeling allows for the potential separation of the clean image  $\mathbf{x}$  from noise corruption, enabling more precise 3DGS reconstructions.
+
+3D Gaussian Splatting. 3D Gaussian Splatting (3DGS) [13] models a 3D scene using a collection of 3D Gaussians, i.e.,  $\mathcal{G} = \{\mathbf{g}_1,\dots ,\mathbf{g}_M\}$ , which are rendered from various viewpoints via a differentiable splatting and tile-based rasterization process. Each Gaussian  $\mathbf{g}_i$  is initialized from Structure-from-Motion (SfM) and characterized by a 3D covariance matrix  $\pmb{\Sigma}_{i} = \mathbb{R}^{3\times 3}$ , position vectors  $\pmb{\mu}_i\in \mathbb{R}^3$ , a color vector  $\mathbf{c}_i\in \mathbb{R}^3$ , and opacity  $o_i$ . To maintain the positive semi-definiteness of  $\pmb{\Sigma}_{i}$ , it is parameterized as  $\pmb{\Sigma}_{i} = \mathbf{R}_{i}\mathbf{S}_{i}\mathbf{S}_{i}^{T}\mathbf{R}_{i}^{T}$ , where  $\mathbf{R}_i\in \mathbb{R}^{3\times 3}$  and diagonal matrix  $\mathbf{S}_i\in \mathbb{R}^{3\times 3}$  represent rotation and scaling matrices, respectively. To render a pixel  $\hat{\mathbf{x}} (\mathbf{p})$  at a pixel position  $\mathbf{p}$  in the reconstructed image  $\hat{\mathbf{x}}$  from a selected viewpoint within the camera projection  $\mathcal{I}$ , 3DGS utilizes a differentiable splatting operation to splat 3D Gaussians onto the 2D surface as follows:
+
+$$
+\hat {\mathbf {x}} (\mathbf {p}) = \sum_ {i = 1} ^ {M} \mathbf {c} _ {i} \alpha_ {i} (\mathbf {p}) \prod_ {j = 1} ^ {i - 1} (1 - \alpha_ {j} (\mathbf {p})), \tag {5}
+$$
+
+$$
+\alpha_ {i} (\mathbf {p}) = o _ {i} \cdot \exp \left(- \frac {1}{2} (\mathbf {p} - \mathcal {J} (\boldsymbol {\mu} _ {i})) ^ {T} \mathcal {J} (\boldsymbol {\Sigma} _ {i} ^ {- 1}) (\mathbf {p} - \mathcal {J} (\boldsymbol {\mu} _ {i}))\right).
+$$
+
+All attributes, i.e.,  $\mathbf{g}_i = \{\pmb {\mu}_i,\mathbf{R}_i,\mathbf{S}_i,\mathbf{c}_i,o_i\}$ , are learnable and optimized through the loss function:
+
+$$
+\mathcal {L} _ {\mathrm {3 D G S}} (\hat {\mathbf {x}}, \mathbf {x}) = (1 - \lambda) \mathcal {L} _ {1} (\hat {\mathbf {x}}, \mathbf {x}) + \lambda L _ {\mathrm {D - S S I M}} (\hat {\mathbf {x}}, \mathbf {x}). \tag {6}
+$$
+
+RawNeRF scaling loss. As discussed in RawNeRF [23], the dynamic range of colors in a raw image is vast, which complicates the application of standard  $\mathcal{L}_1$  or  $\mathcal{L}_2$  loss. These losses tend to overemphasize errors in brighter areas, leading to tone-mapped images with poorly rendered, low-contrast dark regions. To address this, RawNeRF proposes a scaling loss as:
+
+$$
+\mathcal {L} _ {\mathrm {R a w N e R F}} (\hat {\mathbf {x}}, \mathbf {x}) = \left(\frac {\hat {\mathbf {x}} - \mathbf {x}}{\operatorname {s g} (\hat {\mathbf {x}}) + \epsilon}\right) ^ {2}, \tag {7}
+$$
+
+where  $\mathrm{sg}(\cdot)$  is the stop gradient function, and  $\epsilon$  is a small constant to prevent division by zero.
+
+# 4 Methodology
+
+# 4.1 Motivation
+
+Consider a set of noisy raw images, denoted as  $\tilde{\mathcal{X}} = \{\tilde{\mathbf{x}}_1,\dots ,\tilde{\mathbf{x}}_N\}$ , which are captured from different viewpoints using the same camera settings. For any real-world point, e.g.,  $\mathbf{r}$ , that is captured by all cameras, its intensity is recorded at a specific pixel coordinate  $\mathbf{p}_i$  in each raw image  $\tilde{\mathbf{x}}_i$ , with the corresponding pixel value  $\tilde{\mathbf{x}}_i(\mathbf{p}_i)$ . As discussed in Sec. 3, the noisy pixel value is the sum of the noise-free value and the noise at that location, i.e.,  $\tilde{\mathbf{x}}_i(\mathbf{p}_i) = \mathbf{x}(\mathbf{p}) + \mathbf{n}_i(\mathbf{p}_i)$ . Given the same camera settings and real-world point  $\mathbf{r}$ , the clean value  $\mathbf{x}(\mathbf{p})$  remains constant across images.
+
+To further explore the noise impact on 3D Gaussian Splatting (3DGS) [13] optimization, we simplify our analysis by utilizing an  $\mathcal{L}_2$  loss function. The optimal target for the 3DGS output  $\hat{\mathbf{x}} (\mathbf{p})$ , which corresponds to different positions  $\mathbf{p}_i$  in each image  $\tilde{\mathbf{x}}_i$  after the splatting process with camera projection  $\mathcal{I}_i$ , is the following:
+
+$$
+\hat {\mathbf {x}} (\mathbf {p}) = \arg \min  _ {\tilde {\mathbf {x}} (\mathbf {p})} \sum_ {i = 1} ^ {N} \mathcal {L} _ {2} \left(\hat {\mathbf {x}} (\mathbf {p}), \tilde {\mathbf {x}} _ {i} \left(\mathbf {p} _ {i}\right)\right) = \mathbf {x} (\mathbf {p}) + \frac {1}{N} \cdot \sum_ {i = 1} ^ {N} \mathbf {n} _ {i} \left(\mathbf {p} _ {i}\right). \tag {8}
+$$
+
+![](images/824f074e53f506108462f75e041c65441dbd69c58d02d10dcc25f179137b0bd2.jpg)  
+(a) Noise is inevitable in the imaging process
+
+![](images/3b5670fb440ea91c72ff717d85816b8a74771f01cab14e7cefcae46a6bea2ce2.jpg)  
+(b) Impact of noise on the 3DGS optimal target
+
+![](images/4d5d8aebddfed417b10e90cd811ab00cd1189748c2b205ccc6b7719b0fa92822.jpg)  
+Figure 2: An illustration of how prevalent noise in raw images impacts the 3DGS optimization. (a) The imaging process inherently introduces additive noise at various stages due to physical principles and hardware limitations, represented as  $\tilde{\mathbf{x}} = \mathbf{x} + \mathbf{n}$ , where  $\tilde{\mathbf{x}}$  and  $\mathbf{x}$  denote the noisy and clean images, respectively. (b) For a real-world point  $\mathbf{r}$ , a collection of raw images  $\tilde{\mathcal{X}} = \{\tilde{\mathbf{x}}_1,\tilde{\mathbf{x}}_2\}$  records its intensity at pixel coordinates  $\{\mathbf{p}_1,\mathbf{p}_2\}$ , influenced by noise. The optimal target of 3DGS for this point, denoted as  $\hat{\mathbf{x}} (\mathbf{p}) = \mathbb{E}_{\tilde{\mathbf{x}} (\mathbf{p})\sim \tilde{\mathcal{X}}}$ , has a discrepancy from the clean pixel intensity  $\mathbf{x}(\mathbf{p})$ . The variance of this discrepancy is detailed in Eq. (9).  
+Figure 3: Visualization of the 3DGS test view changes across optimization iterations. Initially, the 3DGS model fits the clean signal (at 1,000 and 3,000 iterations). However, as the iterations progress (from 10,000 to 30,000), the model starts to overfit the noise.
+
+Considering a real-world flat surface around point  $\mathbf{r}$  where intensities remain consistent, independent noise will cause variations in the optimal target across the surface as:
+
+$$
+\operatorname {V a r} (\hat {\mathbf {x}} (\mathbf {p})) = \frac {1}{N ^ {2}} \cdot \operatorname {V a r} \left(\mathbf {n} _ {i} \left(\mathbf {p} _ {i}\right)\right) = \frac {1}{N ^ {2}} \cdot \boldsymbol {\sigma} _ {h g} ^ {2}. \tag {9}
+$$
+
+With the same camera settings, it is evident that the variance of the optimal target  $\hat{\mathbf{x}}(\mathbf{p})$  is inversely proportional to the number of training views  $N$ . Since 3D Gaussians are splatted according to Eq. (5), the splatting process tends to produce numerous thin, flat Gaussian shapes to compensate for this variance. With fewer viewpoints, which is a common setting in real-world applications, these elongated Gaussians dominate, as illustrated in Fig. 1. This dominance not only degrades the reconstruction quality but also affects rendering speed, as the performance is directly linked to the number of Gaussian points  $M$ .
+
+Although 3DGS decomposes the variance on a flat surface into numerous elongated Gaussians, we observed that these Gaussian points tend to collapse and are subsequently divided during the later stages of the optimization process. In this process, the low-frequency signals are the first to be approximated before noise, as depicted in Fig. 3, which aligns with Deep Image Prior (DIP) [30], in which the low-frequency image components are typically reconstructed before noise. Based on such an observation, we propose incorporating a noise model as a prior in Sec. 4.3 to relax the constraints in the 3DGS optimization framework.
+
+# 4.2 Lens distortion correction
+
+Unlike NeRF [24], which employs ray-tracing directly from image data, 3DGS maps 3D Gaussians onto a 2D image plane through a splatting operation. To enhance processing efficiency, 3DGS adopts the straightforward pinhole camera model. However, this model does not account for the nonlinear distortions—primarily radial—that are typical with real-world camera lenses. These distortions are more significant in raw images, where the typical lens corrections provided by ISP are absent.
+
+To effectively correct for both radial and tangential lens distortions, we employ a distortion mapping function  $\mathcal{D}(\cdot)$ . This function transforms the undistorted image coordinates  $(x,y)$  into their
+
+![](images/eed0804a61456e970ceb59a4582f336b96564aa09fc1f885dd7b3e88fa8e7a0b.jpg)  
+Figure 4: Illustration of the noise-robust reconstruction loss,  $\mathcal{L}_{\mathrm{nr}}$  , which comprises three components: the reconstruction loss  $\mathcal{L}_{\mathrm{RawNeRF}}$  , the negative likelihood loss (NLL), and the covariance loss  $\mathcal{L}_{\mathrm{cov}}$  A noisy raw image,  $\tilde{\mathbf{x}}$  , is first input to the noise extractor  $F_{n}(\cdot ;\Omega)$  to estimate the noise,  $\hat{\mathbf{n}}$  . The estimated noise  $\hat{\mathbf{n}}$  is then used to calculate the NLL loss relative to the noise distribution. After that, the normalized noise,  $\hat{\mathbf{z}}$  , undergoes a covariance loss,  $\mathcal{L}_{\mathrm{cov}}$  , to minimize spatial dependencies among noise components. Finally, the reconstruction loss,  $\mathcal{L}_{\mathrm{RawNeRF}}$  , is computed between the rendered distorted image  $\mathcal{D}(\hat{\mathbf{x}})$  and the pseudo clean image  $\tilde{\mathbf{x}} -\hat{\mathbf{n}}$
+
+corresponding distorted coordinates  $(x_d, y_d)$ , based on the distortion model used in COLMAP [27]. The distortion map is derived through an iterative Newton-Raphson method. This computation is performed once prior to the training phase, thereby not affecting the training duration. Detailed explanations of the distortion mapping process are provided in the Supplemental Material.
+
+# 4.3 Noise robust reconstruction loss
+
+To mitigate overfitting to noise in 3DGS, we have revised the reconstruction loss function, now termed the noise-robust reconstruction loss function  $\mathcal{L}_{\mathrm{nrr}}$ , as depicted in Fig. 4. This function is defined as follows:
+
+$$
+\mathcal {L} _ {\mathrm {n r r}} (\mathcal {D} (\hat {\mathbf {x}}), \tilde {\mathbf {x}}) = \mathcal {L} _ {\mathrm {R a w N e R F}} (\mathcal {D} (\hat {\mathbf {x}}), \tilde {\mathbf {x}} - \hat {\mathbf {n}}) + \lambda_ {\mathrm {n d}} \cdot \mathcal {L} _ {\mathrm {n d}} (\hat {\mathbf {n}}, \mathbf {n}), \tag {10}
+$$
+
+where  $\mathcal{D}(\cdot)$  denotes the lens distortion mapping as described in Sec. 4.2. The function  $\mathcal{L}_{\mathrm{nd}}(\hat{\mathbf{n}},\mathbf{n})$  measures the noise divergence between the estimated noise  $\hat{\mathbf{n}}\sim q_{\hat{\mathbf{n}}}$  and the physical noise model  $\mathbf{n}\sim p_{\mathbf{n}}$ . The term  $\lambda_{\mathrm{nd}}$  is a Lagrangian relaxation parameter that balances fidelity and noise reduction. To estimate the noise, we introduce a noise extractor  $F_{n}(\cdot ;\Omega)$ , a neural network parameterized by  $\Omega$ , which predicts the noise  $\hat{\mathbf{n}}$  from the input raw image  $\tilde{\mathbf{x}}$  as:
+
+$$
+\hat {\mathbf {n}} = F _ {n} \left(\bar {\mathbf {x}} - \mathbf {n} _ {f p}; \Omega\right) + \mathbf {n} _ {f p}. \tag {11}
+$$
+
+Here, we subtract the input by the fixed pattern noise  $\mathbf{n}_{fp}$  before feeding it into the noise extractor  $F_{n}(\cdot ;\Omega)$ , due to the consistent nature of fixed pattern noise across different captures.
+
+For the noise divergence loss  $\mathcal{L}_{\mathrm{nd}}(\hat{\mathbf{n}},\mathbf{n})$ , we first use the NLL Loss to measure the divergence between the estimated noise and the physical noise in corresponding pixel coordinates. The NLL loss is defined as:
+
+$$
+\operatorname {N L L} (\hat {\mathbf {n}}, \mathbf {n}) = \mathbb {E} _ {q _ {\hat {\mathbf {n}}}} [ - \log p _ {\mathbf {n}} (\hat {\mathbf {n}}) ], \quad p _ {\mathbf {n}} = \mathcal {N} (0, \sigma_ {h g} ^ {2}) + \mathbf {n} _ {f p}, \tag {12}
+$$
+
+where  $\sigma_{hg}^2 = \sigma_{read}^2 +\hat{\mathbf{x}}\cdot k$  is calculated according to Eq. (3).
+
+The NLL loss models the distribution of predicted noise across each pixel, but it does not address the inter-pixel correlations, which ideally should be minimal. To tackle this, we then standardize the predicted noise  $\hat{\mathbf{n}}$  to a standard Gaussian distribution  $\mathcal{N}(0,1)$  and introduce a covariance loss  $\mathcal{L}_{\mathrm{cov}}$  to minimize spatial dependencies:
+
+$$
+\mathcal {L} _ {\mathrm {c o v}} (\hat {\mathbf {n}}, \mathbf {n}) = \mathbb {E} _ {q _ {\hat {Z} | \hat {X}}} [ I - \hat {\mathbf {z}} ^ {T} \hat {\mathbf {z}} ], \quad \hat {\mathbf {z}} = (\hat {\mathbf {n}} - \mathbf {n} _ {f p}) / \sigma_ {h g}, \tag {13}
+$$
+
+where  $I$  is the identity matrix, ensuring the decorrelation of noise across pixels.
+
+Consequently, the comprehensive noise divergence loss  $\mathcal{L}_{\mathrm{nd}}(\hat{\mathbf{n}},\mathbf{n})$  is formulated as:
+
+$$
+\mathcal {L} _ {\mathrm {n d}} (\hat {\mathbf {n}}, \mathbf {n}) = \operatorname {N L L} (\hat {\mathbf {n}}, \mathbf {n}) + \lambda_ {\mathrm {c o v}} / \lambda_ {\mathrm {n d}} \cdot \mathcal {L} _ {\mathrm {c o v}} (\hat {\mathbf {n}}, \mathbf {n}). \tag {14}
+$$
+
+Finally, the complete noise-robust reconstruction loss can be formulated as:
+
+$$
+\mathcal {L} _ {\mathrm {n r r}} (\mathcal {D} (\hat {\mathbf {x}}), \tilde {\mathbf {x}}) = \mathcal {L} _ {\mathrm {R a w N e R F}} (\mathcal {D} (\hat {\mathbf {x}}), \tilde {\mathbf {x}} - \hat {\mathbf {n}}) + \lambda_ {\mathrm {n d}} \cdot \operatorname {N L L} (\hat {\mathbf {n}}, \mathbf {n}) + \lambda_ {\mathrm {c o v}} \cdot \mathcal {L} _ {\mathrm {c o v}} (\hat {\mathbf {n}}, \mathbf {n}). \tag {15}
+$$
+
+![](images/2135636621b0f5deddbc1e07c0395ff7c9eb40f2dce232bde1ea6bd930948e0b.jpg)
+
+![](images/88f0567f1e9dee4183979cddf731b2f96cea65b685600dbb37d468b5cfe9331a.jpg)  
+(b) SSIM over the number of views in training.
+
+![](images/bb2c879f380c1d9283ac460fe3671888a1f41f3ef150cc61da90dbb2478e8e80.jpg)  
+(a) PSNR over the number of views in training.  
+(c) LPIPS over the number of views in training.  
+Figure 5: Comparative evaluation of various baselines and our method on rendering quality and speed in limited views training settings. The two-stage denoiser + 3DGS methods are represented by dotted lines, while training on RGB images is indicated by square markers. All metrics are evaluated on test views within the RGB domain.
+
+![](images/934d7c003371373c35ec50bdc8ba54ef0be34b233f8daac4fa7cab8f4d30af0b.jpg)  
+(d) FPS over the number of views in training.
+
+# 5 Experiments
+
+In this section, we introduce the experimental setup, followed by comparing our methods with existing baselines. Finally, we conduct ablation studies to facilitate a thorough discussion of our approach.
+
+# 5.1 Implementation details
+
+Our framework is built upon Scaffold-GS [21]. Considering the challenges associated with training  $F_{n}(\cdot ,\Omega)$  from scratch within only 30,000 iterations, we opted to pretrain it on the SID dataset [2], which features paired images of noise and noise-free scenes. Notably, the SID dataset is captured using a Sony DSLR camera, distinctly different from the iPhone X used in the RawNeRF dataset [23] for evaluation (e.g., bit depth differences: 14 vs 12), thereby mitigating the risk of data leakage.
+
+Noise extractor  $F_{n}(\cdot ,\Omega)$  is optimized using the Adam optimizer with an initial learning rate of  $1\times 10^{-4}$ , reduced to  $1\times 10^{-5}$  after 25,000 iterations. The hyperparameters  $\lambda_{nd}$  and  $\lambda_{\mathrm{cov}}$ , as defined in Eq. (15) and Eq. (14), are set to 5 and 20 for full-views settings, and 3 and 20 for limited views settings, respectively. All 3DGS models are trained on a single NVIDIA RTX 3090 GPU, following the default parameter settings of Scaffold-GS.
+
+# 5.2 Comparison with baselines
+
+Baselines. We first compare our method with LDR Scaffold-GS [21], which uses RGB images processed by an ISP as the optimal target, to demonstrate the benefits of training on raw images. The ISP settings are the same as those used in RawNeRF. We also benchmark our method against HDR Scaffold-GS optimized with the  $\mathcal{L}_{\mathrm{RawNeRF}}$  loss using raw images as a fair baseline. Additionally, we evaluate our approach against two-stage denoiser+3DGS pipelines optimized using the  $\mathcal{L}_{\mathrm{RawNeRF}}$  loss, where 3DGS is optimized on denoised raw images produced by the denoiser. These comparisons
+
+![](images/3633cc0b3afccf26009e2eaffee67ff95ee5b6a261936cf2c9ee2fa5e1881496.jpg)  
+(a) BM3D
+
+![](images/8a67399e009018e6b1d2ce19f2ed602a1f6f28135474dd25049798dabcbcd7f7.jpg)  
+(b) ELD
+
+![](images/9f4c8cbb273e093126de4d407eb24d0a575f7805f773a711273a7bcbab2bbd1a.jpg)  
+(c) PMN
+
+![](images/8793b87ed47f692ba12728bfb5743c8cf6f7e1aa47c59f7c753f036d4fdc04d6.jpg)  
+(d) LGBPN
+
+![](images/a7fa1919a186721600dfb16d0423e28cff5fa2a3eb928bc8f2a21309a8e21540.jpg)  
+(e) Ne2Ne
+
+![](images/94fbb3f892923c507aed4b5d87695bb3718e02ca5deccf9beb9a631be8852f01.jpg)  
+22.13/0.527/0.560  
+(f) RViDeNet  
+22.96/0.545/0.548
+
+![](images/49b6b6abd0ea362ae0adfa9e25b1627be7a2cfa38ba9969198d9979a0221bd33.jpg)  
+21.73/0.544/0.521  
+(g) LDR Scaffold-GS  
+17.20/0.366/0.701
+
+![](images/96362b3e6753a66f71016eb9ede809868ce25a916094c55ca6e19e2b2e245346.jpg)  
+19.90/0.523/0.586  
+(h) HDR Scaffold-GS  
+20.18/0.457/0.605
+
+![](images/3af37ef26a24a96eec5b6de8de8b846f87fafcf0ff006bfeda550d4f32dc4acf.jpg)  
+22.38/0.520/0.593  
+(i) Ours  
+23.63/0.551/0.521
+
+![](images/7249bba4a22cda9870caae18331196e597171b7a074227d166e244db10e85da1.jpg)  
+22.96/0.547/0.535  
+(i) Ground truth  
+PSNR↑/SSIM↑/LPIPS↓
+
+![](images/22343b2c2219319e7d9d3f7ea221c9c95e6f5c4775b3e14812d8303511363c93.jpg)  
+Figure 6: Visual comparison using ours and competing methods under the 12-view training settings.  
+(a) BM3D
+
+![](images/f836271067479771a319c70d7247808928e4de868f516c6b7151d295d0a3a086.jpg)  
+(b) ELD
+
+![](images/339648d4fe98de1049350ddc6c7d9d9082b03f0ea13b6430d974d4b3982f9e95.jpg)  
+(c) PMN
+
+![](images/055f404504ef90b5468e6e7ca6dafb15d976d184a7c619df016781769c900086.jpg)  
+(d) LGBPN
+
+![](images/9413a0b955a4c7436adf2cab133d2708f6f9566dd48c93dbcd6ead86936548a6.jpg)  
+(e) Ne2Ne
+
+![](images/49c0b913f59daa4037a6b9d49068b167062c0bedd595f0ea8c09acf812453fef.jpg)  
+23.96/0.560/0.487  
+(f) RViDeNet  
+24.12/0.561/0.500  
+Figure 7: Visual comparison across various methods in full views (100-view) training settings.
+
+![](images/4dd1a9eadb9a797bd3a64fe71392155eb8199d27e1264822dc304fbb62d2bae4.jpg)  
+21.15/0.551/0.498  
+(g) LDR Scaffold-GS  
+19.24/0.527/0.557
+
+![](images/9030abbffb8876c6b463c6b5a4c387cdd47a0307008e9a757568883f3cfd6cbf.jpg)  
+19.64/0.532/0.549  
+(h) HDR Scaffold-GS  
+24.01/0.554/0.466
+
+![](images/7a632a707fdcb0541dd44cc91843e55ba39db8459411b5e4208f3b36a4b473b6.jpg)  
+20.46/0.518/0.603  
+(i) Ours  
+24.47/0.562/0.457
+
+![](images/a44c22c69c5c222c1a1fafc1c22e0de7a2e5b2596814cf311f5f8478af7c77dd.jpg)  
+22.58/0.566/0.496  
+(j) Ground truth  
+PSNR↑/SSIM↑/LPIPS↓
+
+include traditional denoiser BM3D [3], state-of-the-art pretrained image denoisers such as ELD [32] and PMN [6], the pretrained video denoiser RViDeNet [36], and advanced self-supervised methods like Neighbor2Neighbor [11] and LGBPN [31] for thorough evaluations. For the pretrained supervised denoisers, we utilized the checkpoints provided by the authors. For the self-supervised methods, we initially pretrained the denoisers on the SID dataset [2] and subsequently fine-tuned them on the training views. All compared methods apply the lens distortion function  $\mathcal{D}(\cdot)$  to the rendered images to align with distorted raw images, ensuring a fair comparison.
+
+Datasets and metrics. We utilize the RawNeRF dataset<sup>3</sup>, which comprises raw images captured in dark scenes using an iPhone X with various ISO settings. For the full-views training setting, we adhere to the same train-test view splits as in RawNeRF. For limited views settings, we randomly select subsets (4, 8, 12, 16, 20 views) from the training views, while maintaining a consistent test view set across all experiments to ensure fair comparison. Following RawNeRF, we assess the rendering quality of various methods using PSNR, SSIM, and LPIPS metrics in the RGB domain. Additionally, we report the frames per second (FPS) to evaluate the real-time rendering capabilities of each method.
+
+Limited views training results. In Fig. 5, we present the quantitative results of our approach compared to other baselines in limited views training settings. At first glance, it is evident that our method (indicated by red lines) significantly surpasses both the LDR Scaffold-GS, HDR Scaffold-GS, and two-stage methods across all view numbers. Compared to LDR Scaffold-GS (green lines), which utilizes RGB images processed by an ISP, our method achieves approximately a 4 dB improvement in PSNR, demonstrating the benefits of training with raw images. Additionally, when both are trained with raw images, our approach outperforms HDR Scaffold-GS (pink lines), optimized by  $\mathcal{L}_{\mathrm{RawNeRF}}$  by approximately 3 dB in PSNR across different training views, and about 4 times faster in FPS with limited 4-view settings, highlighting the efficiency of our noise robust reconstruction loss  $\mathcal{L}_{\mathrm{nrr}}$ . Furthermore, our method also exceeds the two-stage denoiser + 3DGS pipelines by 1 dB, which may lose high-frequency details during the denoising process. Fig. 6 provides a visualization comparison. Notably, with 12 views, both LDR- and HDR- Scaffold-GS exhibit significant elongated Gaussian artifacts, while our method produces more accurate and detailed reconstructions.
+
+Table 1: Quantitative results of our approach and other baselines in full views training settings.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Loss</td><td rowspan="2">Raw PSNR↑</td><td colspan="3">Affine-aligned RGB</td><td rowspan="2">FPS↑</td></tr><tr><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td></tr><tr><td>BM3D [3]</td><td></td><td>58.46</td><td>22.70</td><td>0.526</td><td>0.521</td><td>76</td></tr><tr><td>ELD [32]</td><td></td><td>54.70</td><td>19.82</td><td>0.511</td><td>0.544</td><td>80</td></tr><tr><td>PMN [6]</td><td rowspan="2">\( \mathcal{L}_{\text{RawNeRF}} \)</td><td>53.69</td><td>19.00</td><td>0.498</td><td>0.584</td><td>94</td></tr><tr><td>Neighbor2Neighbor [11]</td><td>58.40</td><td>22.22</td><td>0.531</td><td>0.518</td><td>78</td></tr><tr><td>LGBPN [31]</td><td></td><td>53.62</td><td>19.09</td><td>0.481</td><td>0.624</td><td>84</td></tr><tr><td>RViDeNet [36]</td><td></td><td>57.59</td><td>22.05</td><td>0.518</td><td>0.548</td><td>74</td></tr><tr><td>LDR Scaffold-GS [21]</td><td>\( \mathcal{L}_{\text{3DGS}} \)</td><td>-</td><td>17.34</td><td>0.486</td><td>0.622</td><td>56</td></tr><tr><td>HDR Scaffold-GS [21]</td><td>\( \mathcal{L}_{\text{RawNeRF}} \)</td><td>58.08</td><td>22.69</td><td>0.521</td><td>0.513</td><td>73</td></tr><tr><td>Ours</td><td>\( \mathcal{L}_{\text{nrr}} \)</td><td>59.49</td><td>23.53</td><td>0.535</td><td>0.499</td><td>80</td></tr></table>
+
+Table 2: Ablation study on the lens distortion  $\mathcal{D}(\cdot)$ . All models are trained using full views.  
+
+<table><tr><td>D(·)</td><td>Raw PSNR</td><td>RGB PSNR</td><td>RGB SSIM</td><td>RGB LPIPS</td></tr><tr><td>w/o</td><td>59.33</td><td>23.42</td><td>0.531</td><td>0.509</td></tr><tr><td>w/</td><td>59.49</td><td>23.53</td><td>0.535</td><td>0.499</td></tr></table>
+
+Table 3: Ablation study on the training time.  
+
+<table><tr><td>Method</td><td>Ne2Ne [11] 
+Scaffold-GS [21]</td><td>LGBPN [31] 
+Ours</td></tr><tr><td rowspan="2">Time cost</td><td>2.5h</td><td>5.3h</td></tr><tr><td>1.6h</td><td>3.1h</td></tr></table>
+
+Full-views training results. We also benchmarked our method against other baselines in full-views training settings. As shown in Table. 1 and Fig. 7, our method consistently outperforms all other baselines across PSNR, SSIM, and LPIPS, even with a large number of training views. Besides, visualizations demonstrate that our method recovers images with better detail and color accuracy.
+
+Table 4: Ablation study on the selection of  ${\lambda }_{\mathrm{{nd}}}$  and  ${\lambda }_{\text{cov }}$  . All models are trained using full views.  
+
+<table><tr><td>λnd</td><td>λcov</td><td>Raw PSNR</td><td>RGB PSNR</td><td>RGB SSIM</td><td>RGB LPIPS</td></tr><tr><td>0.5</td><td>20</td><td>56.36</td><td>21.95</td><td>0.508</td><td>0.591</td></tr><tr><td>10</td><td>20</td><td>56.84</td><td>22.18</td><td>0.509</td><td>0.539</td></tr><tr><td>5</td><td>0</td><td>55.38</td><td>22.41</td><td>0.515</td><td>0.523</td></tr><tr><td>5</td><td>2</td><td>57.10</td><td>22.52</td><td>0.516</td><td>0.520</td></tr><tr><td>0.5</td><td>50</td><td>56.72</td><td>21.98</td><td>0.514</td><td>0.576</td></tr><tr><td>5</td><td>20</td><td>59.49</td><td>23.53</td><td>0.535</td><td>0.499</td></tr></table>
+
+# 5.3 Ablation study and discussions
+
+Impact of lens distortion function  $\mathcal{D}(\cdot)$ . As detailed in Table 2, the absence of lens distortion correction results in a marginal decrease in reconstruction quality. In RawNeRF datasets, this effect is negligible due to the minimal lens distortion inherent in the iPhone X. However, for cameras with more pronounced distortion characteristics, the impact could be significantly more substantial.
+
+Impact of hyperparameters. The hyperparameters  $\lambda_{\mathrm{nd}}$  and  $\lambda_{\mathrm{cov}}$  within the  $\mathcal{L}_{\mathrm{nr}}$  play a pivotal role in striking the right balance between denoising efficacy and gradient accumulation. As delineated in Table 4, an undersized  $\lambda_{\mathrm{nd}}$  leads to inadequate constraints on the noise model, culminating in imprecise noise extraction. On the flip side, an over-amplified  $\lambda_{\mathrm{nd}}$  engenders excessively strong gradients emanating from the noise regularization loss, which can detrimentally impact the optimization process of 3DGS. The parameter  $\lambda_{\mathrm{cov}}$ , when assigned values that are either too low or too high, can incite similar challenges.
+
+Limitations. Our method has several limitations. The noise extractor and covariance loss computation require more training time compared to standard Scaffold-GS, as shown in Table 3. Additionally, some high-frequency signal distributions are similar to noise, making it difficult to differentiate them using the distribution divergence loss. This can result in over-smoothing in full-views training settings, as discussed in the Supplemental Material.
+
+# 6 Conclusion
+
+In this work, we identify the unavoidable noise in raw images as a significant detriment to the rendering quality and speed of HDR 3DGS, particularly in scenarios with limited training views. We provide a comprehensive analysis of how noise affects the optimization of 3DGS, modeling its relationship with both the number of training views and the noise distribution. To tackle these challenges, we
+
+propose a novel self-supervised framework that incorporates a noise-robust reconstruction loss. This framework utilizes a physically-based noise model to simultaneously denoise and enhance the HDR representation within noisy raw images. Our approach markedly outperforms LDR/HDR 3DGS that employs 3DGS/RawNeRF loss, as well as both self-supervised and supervised pre-trained two-stage methods, in terms of rendering quality and speed across various training view counts.
+
+# Acknowledgements
+
+This work was partially supported by the National Research Foundation Singapore Competitive Research Program (award number CRP29-2022-0003).
+
+# References
+
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+
+# From Chaos to Clarity: 3DGS in the Dark — Supplemental Material
+
+In this supplement, we begin by discussing the broader impacts of our method. Next, we outline the distribution and calibration of the camera noise model. Following that, we elaborate on the lens distortion function  $\mathcal{D}(\cdot)$ . Finally, we address the failure cases of our method as mentioned in the limitations section.
+
+# A Broader Impacts
+
+The broader social impacts of our work can be summarized as follows:
+
+1. Enhanced Applications: By significantly improving 3DGS performance in low-light conditions, our work has the potential to broaden the scope of its applications across various fields such as medical imaging, autonomous driving, and surveillance systems. These advancements could lead to better outcomes in scenarios where lighting is a critical factor.  
+2. Open Access and Collaboration: In the spirit of fostering innovation and collaboration within the research community, we will release our code as open-source. This will allow other researchers and developers to build upon our work, potentially leading to further advancements and new applications.
+
+We have carefully considered the potential impacts of our work and do not foresee any serious negative consequences. Our contributions are intended to promote positive developments and collaboration in the field of 3DGS technology.
+
+# B Distribution of the camera noise model
+
+Building on the work of ELD [32] and PMN [6], the noise components in Eq.(4) adhere to specific distributions:
+
+$$
+\begin{array}{l} \mathbf {n} _ {s h o t} \sim \mathcal {P} \left(\frac {\mathbf {x}}{k}\right) \cdot k - \mathbf {x}, \\ \mathbf {n} _ {\text {r e a d}} \sim \mathcal {N} \left(0, \sigma_ {\text {r e a d}} ^ {2}\right), \tag {16} \\ \mathbf {n} _ {f p} = \operatorname {I S O} \cdot \mathbf {n} _ {f p _ {k}} + \mathbf {n} _ {f p _ {b}}, \\ \end{array}
+$$
+
+where  $k$  is the overall system gain associated with the ISO setting, and  $\mathcal{P}$  and  $\mathcal{N}$  represent Poisson and Gaussian distributions, respectively. The terms  $n_{fp_k}$  and  $n_{fp_b} \in \mathbb{R}^{H \times W}$  are pixel-wise dark frame noise components. Consistent with ELD[32] and PMN [6], the relationships among  $k$ ,  $\sigma_{\text{read}}$ , and ISO are expressed as:
+
+$$
+\begin{array}{l} k = a _ {k} \cdot \operatorname {I S O} + b _ {k}, \\ \log \left(\sigma_ {\text {r e a d}}\right) = a _ {\text {r e a d}} \cdot \log (k) + b _ {\text {r e a d}}, \tag {17} \\ \end{array}
+$$
+
+Parameters  $n_{fp_k}$ ,  $n_{fp_b}$ ,  $a_k$ ,  $b_k$ ,  $a_{read}$ , and  $b_{read}$  are calibrated for each camera using a series of flat-frame and dark-frame images captured at various ISO levels.
+
+# C Calibration of the camera noise model parameters
+
+The calibration process comprises three steps:
+
+1. Calibrating  $k_{i}$  for each ISO level:
+
+- Capture 25 flat frames under consistent lighting for each exposure time  $\mathrm{Exp}_j$  
+- Calculate the mean and variance for each color block, yielding 24 mean-variance pairs per exposure time.  
+- With three exposure times per ISO, gather 72 mean-variance pairs per ISO.  
+- Model  $\mathbf{n}_{\text{shot}}$  as  $\mathcal{N}(\mathbf{x}, \mathbf{x} \cdot k)$ , where  $\mathbf{x}$  is the mean and  $\mathbf{x} \cdot k$  the variance, and calibrate  $k$  from the mean-variance relationship. Exclude points with mean values beyond 1/4 saturation due to clipping effects.
+
+2. Calibrating  $\mathbf{n}_{fp_i}$  and  $\sigma_{read_i}$ :
+
+![](images/10447b04c58d36ac1386621efdb34f586a47af6ac9bf1dc09f033c7cab228780.jpg)  
+(a) HDR Scaffold-GS (PSNR: 24.59 dB)
+
+![](images/6fbb6ad1ff320f7a17b6c49ffc2c247139415a420eadf63c4729208a3a554c1f.jpg)  
+(b) Ours (PSNR: 24.85 dB)  
+Figure 8: Visualization of our failure cases on the RawNeRF dataset [23]. Despite achieving a higher PSNR than HDR Scaffold-GS by eliminating noise-induced artifacts, our approach resulted in overly smooth outcomes in areas with reflections.
+
+- Capture 100 dark frames at each ISO in a dark room.  
+- The mean of these dark frames gives  $\mathbf{n}_{fp_i}$ , representing fixed pattern noise.  
+- Subtract  $\mathbf{n}_{fp_i}$  from all dark frames and calculate variance across the total frame for  $\mathbf{n}_{read}$ .
+
+# 3. Fitting ISO-related parameters:
+
+- Repeat the above steps for different ISO levels to obtain a set of parameters  $\mathbf{n}_{fp_i}$ ,  $\sigma_{read_i}$ .  
+- Fit  $a_{k}, b_{k}, n_{fpk}, nfp_{b}, a_{read}, b_{read}$  based on these parameters and equations Eq.16.
+
+# D Lens distortion mapping process
+
+To align the distorted coordinates with their accurate positions, we compute a reverse mapping from  $(x_d,y_d)$  to  $(x,y)$  using a Newton-Raphson iterative method. This method minimizes the residuals between the distorted and true coordinates until the adjustments are beneath a predefined accuracy threshold. Crucially, this distortion mapping is computed once before the training process, thereby minimizing its impact on computational efficiency.
+
+using the following equations:
+
+$$
+x _ {d} = x \left(1 + k _ {1} r ^ {2} + k _ {2} r ^ {4} + k _ {3} r ^ {6} + k _ {4} r ^ {8}\right) + 2 p _ {1} x y + p _ {2} \left(r ^ {2} + 2 x ^ {2}\right), \tag {18}
+$$
+
+$$
+y _ {d} = y \left(1 + k _ {1} r ^ {2} + k _ {2} r ^ {4} + k _ {3} r ^ {6} + k _ {4} r ^ {8}\right) + p _ {1} \left(r ^ {2} + 2 y ^ {2}\right) + 2 p _ {2} x y,
+$$
+
+where  $r = \sqrt{x^2 + y^2}$  represents the radial distance from the center, and  $k_1, k_2, k_3, k_4, p_1, p_2$  are the coefficients for radial and tangential distortions, respectively.
+
+Since  $x$  and  $y$  are not integers, we use bilinear interpolation to obtain the values of  $x_{d}$  and  $y_{d}$ .
+
+# E Failure case
+
+Upon examination, we have observed that certain high-frequency signals exhibit a distribution pattern that closely resembles noise, making it difficult for the noise divergence loss to differentiate between them effectively. Consequently, this has resulted in overly smooth outputs in some areas rich in detail, particularly when training with full views.
+
+Differentiating high-frequency signals from noise is an intricate challenge. We acknowledge the complexity of this issue and are committed to further exploring potential solutions in our future research endeavors.
+
+Table 5: Quantitative Comparison on LLFF Datasets (3 Training Views)  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Loss</td><td colspan="3">1/4*</td><td colspan="3">1/8*</td></tr><tr><td>Raw PSNR↑</td><td>RGB PSNR↑</td><td>FPS↑</td><td>Raw PSNR↑</td><td>RGB PSNR↑</td><td>FPS↑</td></tr><tr><td>BM3D</td><td>LRawNeRF</td><td>42.902</td><td>18.032</td><td>163</td><td>22.596</td><td>17.242</td><td>272</td></tr><tr><td>PMN</td><td>LRawNeRF</td><td>42.585</td><td>18.064</td><td>201</td><td>23.208</td><td>18.572</td><td>389</td></tr><tr><td>Neighbor2Neighbor</td><td>LRawNeRF</td><td>42.774</td><td>17.977</td><td>152</td><td>23.008</td><td>18.297</td><td>372</td></tr><tr><td>FSGS</td><td>LRawNeRF</td><td>41.197</td><td>15.496</td><td>65</td><td>21.183</td><td>15.166</td><td>132</td></tr><tr><td>Ours</td><td>Lnrr</td><td>43.418</td><td>18.753</td><td>216</td><td>23.799</td><td>18.902</td><td>370</td></tr></table>
+
+*denotes the downsample ratio of the resolution, as same as [39].
+
+# F Extended evaluation on LLFF dataset
+
+Given the limitations of the RawNeRF dataset, we conducted additional quantitative comparisons using the LLFF dataset with simulated noisy raw images and provided qualitative comparisons for the training scenes in the RawNeRF dataset. Specifically, we used the inverse ISP proposed in [14] to convert LLFF RGB images into RAW images, added synthetic noise following the iPhone X noise model, and adhered to baseline methods' settings. We used FSGS [39] as the 3D representation baseline, modifying only the loss function to our proposed  $\mathcal{L}_{\mathrm{nl}}$ . The results are listed in Table 1.
+
+# G Visual comparison between NeRF-based and 3DGS-based methods
+
+![](images/f9946cbce90c29d21cd192f90fc4099091bf2973f2b8f4fd9ed44a38d9616c55.jpg)  
+(a) RawNeRF
+
+![](images/dbd9d6b4f30f401169e539a179769835ddea2d6d8e1942705b3826c3a5c47097.jpg)  
+(b) HDR Scaffold-GS  
+Figure 9: Visual comparison between NeRF-based and 3DGS-based methods in full views training settings (Zoom in for best view). Noise impact is unique to 3DGS. Compared to RawNeRF, HDR Scaffold-GS is more vulnerable to noise, generating many thin-flat 3D Gaussian points to overfit it. The MLP in RawNeRF, acting as a low-pass filter, does not have this issue. Additionally, the rendering speed of 3DGS is affected by the number of Gaussian points, degrading due to noise points. Our method avoids these noisy Gaussian points, boosting rendering speed and quality.
+
+![](images/5c4fa18544545eb684b73756776a04ec30cec60ca846799ebcd661929d65f8f4.jpg)  
+(c) Ours
+
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+
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+
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+# From Text to Trajectory: Exploring Complex Constraint Representation and Decomposition in Safe Reinforcement Learning
+
+Pusen Dong $^{1}$ , Tianchen Zhu $^{1}$ , Yue Qiu $^{1}$ , Haoyi Zhou $^{1,2}$ , Jianxin Li $^{1,2}$
+
+<sup>1</sup>Beihang University, Beijing, China
+
+$^{2}$ Zhongguancun Laboratory, Beijing, China
+
+{dongps, zhutc, qiuyue, zhouhy, lijx}@act.buaa.edu.cn
+
+# Abstract
+
+Safe reinforcement learning (RL) requires the agent to finish a given task while obeying specific constraints. Giving constraints in natural language form has great potential for practical scenarios due to its flexible transfer capability and accessibility. Previous safe RL methods with natural language constraints typically need to design cost functions manually for each constraint, which requires domain expertise and lacks flexibility. In this paper, we harness the dual role of text in this task, using it not only to provide constraint but also as a training signal. We introduce the Trajectory-level Textual Constraints Translator (TTCT) to replace the manually designed cost function. Our empirical results demonstrate that TTCT effectively comprehends textual constraint and trajectory, and the policies trained by TTCT can achieve a lower violation rate than the standard cost function. Extra studies are conducted to demonstrate that the TTCT has zero-shot transfer capability to adapt to constraint-shift environments.
+
+# 1 Introduction
+
+In recent years, reinforcement learning (RL) has achieved remarkable success in multiple domains, such as go game [37, 38] and robotic control [21, 45]. However, deploying RL in real-world scenarios still remains challenging. Many real-world decision-making applications, such as autonomous driving [13, 9] require agents to obey certain constraints while achieving the desired goals. To learn a safe constrained policy, some safe RL works [2, 33, 44, 49, 39, 48, 6] have proposed methods to maximize the reward while minimizing the constraint violations after training or during training.
+
+However, several limitations prevent the existing safe RL methods' widespread use in real-world applications. Firstly, these methods often require mathematical or logical definitions of cost functions, which require domain expertise (Limitation 1). Secondly, their cost function definitions are frequently specific to a particular context and cannot be easily generalized to new tasks with similar constraints (Limitation 2). Lastly, most current safe RL methods focus on constraints that are logically simple, typically involving only one single entity or one single state [28], which can't represent the real-world safety requirements and lack universality (Limitation 3).
+
+Using natural language to provide constraints [46, 29, 20] is a promising approach to overcome Limitation 1 and 2 because natural language allows for flexible, high-level expression of constraints that can easily adapt to different scenarios. Regarding Limitation 3, previous approaches primarily employ what we call the single state/entity textual constraint. The single-state/entity textual constraint focuses solely on constraints related to one specific state or entity, limiting the ability to model complex safety requirements in real-world scenarios. Many safety requirements involve interactions and dependencies among multiple states or entities over time. By only addressing a single state or
+
+Table 1: Comparison of trajectory-level constraints and previous single state/entity constraints. Universal Trajectory-level constraints can model any constraint requirements in real-world scenarios. But single state/entity constraint can only model the constraint requirements on individual state/entity.  
+
+<table><tr><td>Single state/entity constraint (previous)</td><td>Trajectory-level constraint (ours)</td></tr><tr><td>Don’t drive car. (single entity)</td><td>Don’t drive car after you drink wine. (multi entities)</td></tr><tr><td>Don’t touch lava. (single state)</td><td>Don’t touch lava more that three times. (multi states)</td></tr><tr><td>Don’t step in the gassed area. (single state)</td><td>Don’t stay in the gas area for more than 5 minutes, your gas mask will fail. (multi states)</td></tr></table>
+
+entity, these constraints fail to capture the dynamic relationships and temporal aspects that are crucial for ensuring safety in complex environments. So we suggest using a more generalizable constraint type trajectory-level textual constraint. The trajectory-level textual constraint is a more universal constraint with complex logic semantics involving multiple states/entities. It is a nonlinear combination of multiple entities or multiple environment states and can model any constraint requirements in real-world scenarios. The trajectory-level constraints in natural language form are the highly abstract expression of the agent's behavior guidelines, serving as a more natural and straightforward way to introduce constraints. Notably, the set of trajectory-level constraints encompasses single-state constraints, as any single-state constraint can be reformulated as an equivalent trajectory-level constraint. Examples of trajectory-level textual constraint and single state/entity textual constraint are presented in Table 1.
+
+Employing trajectory-level textual constraints across the entire trajectory poses two significant challenges. Firstly, determining whether an RL agent violates textual constraint over a trajectory is non-trivial, it needs to have the perception of the historical states and actions observed along the trajectory (Challenge 1). Secondly, the trajectory-level safety problem is susceptible to sparse cost [34], where cost is only imposed when the agent violates textual constraints at the final time step, making it challenging for the agent to learn which actions contribute to a gradual escalation of risk (Challenge 2). For instance, the agent needs to learn constraints like, "Don't touch lava after you touch grass." Without intermediate feedback, however, it struggles to understand how early actions, such as stepping on grass, contribute to eventual violations.
+
+To address Challenge 1, we propose a general approach to align the trajectory's factual logic with the text's semantic logic, eliminating the need for manual encoding or separate models for each type of constraint. Our method employs a sequence model [41, 15] for modeling agent historical interactions with the environment, and a pre-trained language model (LM) [7] to comprehend natural language constraints. We then maximize the embedding similarities between matching pairs (trajectory, text) and minimize the embedding similarities between non-matching pairs using a contrastive learning approach [25] similar to CLIP [30]. By calculating the similarity between the textual embeddings of the constraint and the trajectory, we can predict whether a constraint is violated in this trajectory. Our method uniquely leverages text as both a source of constraints and a unified supervisory signal for trajectory encoding. In this dual role, text not only provides constraints but also guides the training process, enabling the model to naturally handle diverse semantic constraints without requiring specific model adjustments for each type. This design allows for a more flexible and generalizable system, significantly simplifying the handling of complex, multi-dimensional constraints.
+
+In addition, to address the issue of cost sparsity (Challenge 2), we introduce a method for temporal credit assignment [40]. The proposed approach involves decomposing the one episodic cost of the textual constraint into multiple parts and allocating them to each state-action pair within the trajectory. This method offers denser cost signals regarding the relationship between the textual constraint and the agent's every action. It informs the agent which behaviors are risky and which are safer, thereby enhancing safety and aiding model performance.
+
+Our experiments demonstrate that the proposed method can effectively address Challenge 1 and Challenge 2. In both 3D navigation [19] and 2D grid exploration [46] tasks, agents trained using our method achieve significantly lower violation rates (up to  $4.0\mathrm{x}$ ) compared to agents trained with ground-truth cost functions while maintaining comparable rewards and more importantly, our method obtains the Pareto frontier [10]. In addition to this, our method has zero-shot adaptation capability to adapt to constraint-shift environments without fine-tuning.
+
+# 2 Related Work
+
+Safe RL. Safe RL aims to train policies that maximize reward while minimizing constraint violations [11, 13]. In prior work, there are usually two ways to learn safe policies: (1) consider cost as one of the optimization objectives to achieve safety [48, 6, 49, 39, 44, 33], and (2) achieve safety by leveraging external knowledge (e.g. expert demonstration) [35, 32, 1, 47]. These typical safe RL algorithms require either human-defined cost functions or human-specified cost constraints which are unavailable in the tasks that constraints are given by natural language.
+
+RL with Natural Language. Prior works have integrated natural language into RL to improve generalization or learning efficiency in various ways. For example, Hermann et al. [14] studied how to train an agent that can follow natural language instructions to reach a specific goal. Additionally, natural language has been used to constrain agents to behave safely. For instance, Prakash et al. [29] trained a constraint checker to predict whether natural language constraints are violated. Yang et al. [46] trained a constraint interpreter to predict which entities in the environment may be relevant to the constraint and used the interpreter to predict costs. Lou et al. [28] used pre-trained language models to predict the cost of specific states, avoiding the need for artificially designed cost functions. However, previous methods cannot uniformly handle textual constraints with one framework, which limits their applicability.
+
+Credit Assignment in RL. Credit assignment studies the problem of inferring the true reward from the designed reward. Prior works have studied improving sample efficiency of RL algorithms through credit assignment, for example by using information gain [17], as an intrinsic bonus reward to aid exploration. Goyal et al. [12] proposed the use of natural language instructions to perform reward shaping to improve the sample efficiency of RL algorithms. Liu et al. [26] learned to decompose the episodic return as the reward for policy optimization. However, to the best of our knowledge, our work is the first to apply credit assignment to safe RL.
+
+# 3 Preliminaries
+
+Problem formulation. Trajectory-level constraint problem can be formed as the Constrained Non-Markov Decision Process (CNMDP) [3, 43], and it can be defined by the tuple  $< S, A, T, R, \gamma, C, Y, \tau^{*}>$ . Here  $S$  represents the set of states,  $A$  represents the set of actions,  $T$  represents the state transition function,  $R$  represents the reward function, and  $\gamma \in (0,1)$  represents the discount factor. In addition,  $Y$  represents the set of trajectory-level textual constraints (e.g., "You have 10 HP, you will lose 3 HP every time you touch the lava, don't die"), which describes the constraint that the agent needs to obey across the entire trajectory.  $C$  represents the cost function determined by  $y \in Y$ .  $\tau^{*}$  represents the set of historical trajectories.
+
+RL with constraints. The objective for the agent is to maximize reward while obeying the specified textual constraint as much as possible [27]. Thus, in our task setting, the agent needs to learn a policy  $\pi: S \times Y \times \tau^{*} \to P(A)$  which maps from the state space  $S$ , textual constraints  $Y$  and historical trajectories  $\tau^{*}$  to the distributions over actions  $A$ . Given a  $y$ , we learn a policy  $\pi$  that maximizes the cumulative discounted reward  $J_{R}$  while keeping the cumulative discounted cost (average violation rate)  $J_{C}$  below a constraint violation budget  $B_{C}(y)$ :
+
+$$
+\max  _ {\pi} J _ {R} (\pi) = \mathbb {E} _ {\pi} \left[ \sum_ {t = 0} ^ {\infty} \gamma R (s _ {t}) \right] \quad \text {s . t .} \quad J _ {C} (\pi) = \mathbb {E} _ {\pi} \left[ \sum_ {t = 0} ^ {\infty} \gamma C (s _ {t}, a _ {t}, y, \tau_ {t}) \right] \leq B _ {C} (y). \tag {1}
+$$
+
+Here  $B_{C}(y)$  and  $C(s_{t},a_{t},y,\tau_{t})$  are two functions both depending on textual constraint  $y$ .  $\tau_{t}$  represents the historical trajectory at time step  $t$ .
+
+Episodic RL. Similar to the task with episodic rewards [4], in our task setting, a cost is only given at the end of each trajectory  $\tau$  when the agent violates the textual constraint  $y$ . In other words, before violating  $y$ , the cost  $C(s_{t},a_{t},y,\tau_{t}) = 0$  for all  $t < T$ . For simplicity, we omit the discount factor and assume that the trajectory length is at most  $T$  so that we can denote  $a_{T}$  as the final action that causes the agent to violate  $y$  without further confusion. Therefore, the constraint qualification of the objective in RL with constraints becomes  $J_{C}(\pi) = \mathbb{E}_{\pi}[C(s_{T},a_{T},y,\tau_{T})]\leq B_{C}(y)$ . Due to the sparsity of cost, a large amount of rollout trajectories are needed to help the agent distinguish the subtle effects of actions on textual constraint [24]. This situation will become more serious when trajectory-level constraints are complex and difficult to understand.
+
+![](images/fa2c1421ed15830a9efb1747da0d39d0a1f47c8869f5125eec7f6826e8aeaea6.jpg)  
+Figure 1: TTCT overview. TTCT consists of two training components: (1) the text-trajectory alignment component connects trajectory to text with multimodal architecture, and (2) the cost assignment component assigns a cost value to each state-action based on its impact on satisfying the constraint. When training RL policy, the text-trajectory alignment component is used to predict whether a trajectory violates a given constraint and the cost assignment component is used to assign non-violation cost.
+
+![](images/cf96bb7b33b63343bc9960ad0ae5db57233adc79fcc1f43ec35aec29bc076645.jpg)
+
+# 4 TTCT: Trajectory-level Textual Constraints Translator
+
+In this section, we introduce our proposed framework TTCT (Trajectory-level Textual Constraints Translator) as shown in Figure 1. TTCT has two key components: the text-trajectory alignment component and the cost assignment component. The text-trajectory alignment component is used to address the violations prediction problem. The cost assignment component is used to address the sparse cost problem.
+
+# 4.1 Text-Trajectory Alignment Component
+
+We propose a component to learn from offline data to predict whether a given trajectory violates textual constraints. The core idea of this component is to learn trajectory representations under textual supervision and connect trajectory representation to text representation. If the distance between the two representations in the embedding space is sufficiently close, we can consider that the given trajectory violates the constraint. Our approach does not require modeling entities of the environment like previous work, such as [46], which involves labeling hazardous items artificially in every observation. Instead, we model this task as a trajectory-text multimodal learning problem. Hence, our method can learn trajectory representations and text representations from the pairs (trajectory, trajectory-level textual constraint). We believe that learning from the supervision of natural language could not only enhance the representation power but also enable flexible zero-shot transfer [30].
+
+Formally, given a batch of  $N$  (trajectory  $\tau$ , trajectory-level textual constraint  $y$ ) pairs. For each pair, the trajectory corresponds to the text, indicating that the given trajectory violates the given textual constraint. The trajectory can be defined as  $\tau = (s_1, a_1, s_2, a_2, \dots, s_{T-1}, a_{T-1}, s_T, a_T)$ , where  $T$  is the step at which the textual constraint  $y$  is first violated by the trajectory. Here  $s_t$  is a  $d_s$ -dimensional observation vector. Each state in the trajectory is processed by a state encoder to obtain a representation  $v_t^s$ , also action is processed by an action encoder to obtain a representation  $v_t^a$ . Then, we concatenate  $v_t^s$  and  $v_t^a$  to obtain a vector representation  $v_t$  for each state-action pair. After that, we learn separate unimodal encoders  $g_T$  and  $g_C$  for the trajectory and textual constraint, respectively. The trajectory encoder  $g_T$  utilizes a causal transformer to extract the trajectory representation from the input state-action representation sequence  $\{v_t\}_{t=1}^T$ :
+
+$$
+H _ {1}, H _ {2}, H _ {3}, \dots , H _ {T - 1}, H _ {T} = g _ {T} \left(\left\{v _ {t} \right\} _ {t = 1} ^ {T}\right), \tag {2}
+$$
+
+where  $H_{t}$  is a  $d_{H}$ -dimensional vector. The final embedding  $H_{T}$  is used as the representation for the entire trajectory. Specifically, the causal Transformer processes the trajectory sequence by maintaining a left-to-right context while generating embeddings. This allows the model to capture temporal dependencies within the trajectory and obtain the embeddings for time steps before  $T$ . The textual constraint encoder  $g_{C}$  is used to extract features that are related to the constraints and it could be one of a wide variety of language models:
+
+$$
+L = g _ {C} (y), \tag {3}
+$$
+
+where  $L$  is a  $d_{L}$ -dimensional vector representation. Then we define symmetric similarities between the two modalities with cosine distance:
+
+$$
+\sin_ {T} (\tau , y) = \exp (\alpha) * \frac {H _ {T} \cdot L ^ {\top}}{\| H _ {T} \| \| L \|}, \quad \sin_ {T} (y, \tau) = \exp (\alpha) * \frac {L ^ {\top} \cdot H _ {T}}{\| L \| \| H _ {T} \|}, \tag {4}
+$$
+
+where  $\alpha$  is a learnable parameter and  $T$  means that use last embedding of trajectory to calculate similarity. And we use softmax to calculate trajectory  $\rightarrow$  text and text  $\rightarrow$  trajectory similarities scores:
+
+$$
+p _ {i} ^ {\tau \rightarrow y} (\tau) = \frac {\exp \left(\operatorname {s i m} _ {T} \left(\tau , y _ {i}\right)\right)}{\sum_ {j = 1} ^ {N} \exp \left(\operatorname {s i m} _ {T} \left(\tau , y _ {j}\right)\right)}, p _ {i} ^ {y \rightarrow \tau} (y) = \frac {\exp \left(\operatorname {s i m} _ {T} \left(y , \tau_ {i}\right)\right)}{\sum_ {j = 1} ^ {N} \exp \left(\operatorname {s i m} _ {T} \left(y , \tau_ {j}\right)\right)}. \tag {5}
+$$
+
+Let  $q^{\tau \rightarrow y}(\tau), q^{y \rightarrow \tau}(y)$  indicate the ground-truth similarity scores, where the negative pair (trajectory doesn't violate textual constraint) has a probability of 0 and the positive pair (trajectory violate textual constraint) has a probability of 1. In our task setting, a trajectory can correspond to multiple textual constraints, and vice versa. For example, two textual constraints such as "Do not touch lava" and "After stepping on water, do not touch lava" might both be violated by a single given trajectory. This many-to-many relationship between trajectories and textual constraints implies that the same textual constraints (or different textual constraints with the same semantics) can apply to multiple trajectories, while a single trajectory may comply with several textual constraints. So there may be more than one positive pair in  $q_i^{\tau \rightarrow y}(\tau)$  and  $q_i^{y \rightarrow \tau}(y)$ . Therefore, we use Kullback-Leibler (KL) divergence [23] as the multimodal contrastive (MC) loss to optimize our encoder similar to [42]:
+
+$$
+\mathcal {L} _ {M C} = \frac {1}{2} \mathbb {E} _ {(\tau , y) \sim \mathcal {D}} \left[ \mathrm {K L} \left(p ^ {\tau \rightarrow y} (\tau), \operatorname {s o f t m a x} \left(q ^ {\tau \rightarrow y} (\tau)\right)\right) + \mathrm {K L} \left(p ^ {y \rightarrow \tau} (y), \operatorname {s o f t m a x} \left(q ^ {y \rightarrow \tau} (y)\right)\right)\right], \tag {6}
+$$
+
+where  $\mathcal{D}$  is the training set.
+
+In addition to the multimodal contrastive (MC) loss, we also introduce a within-trajectory (WT) loss. Specifically, suppose we have a trajectory's representation sequence  $(H_{1}, H_{2}, H_{3}, \dots, H_{(T-1)}, H_{T})$  and its corresponding textual constraint embedding  $L$ , we can calculate cosine similarity  $\mathrm{sim}_t(\tau, y)$  between embedding  $H_{t}$  and  $L$  using Equation 4. Then we can calculate similarity scores within the trajectory:
+
+$$
+p _ {t} ^ {\tau} (y) = \frac {\exp \left(\operatorname {s i m} _ {t} (\tau , y)\right)}{\sum_ {k = 1} ^ {T} \exp \left(\operatorname {s i m} _ {k} (\tau , y)\right)}. \tag {7}
+$$
+
+Different from  $p_i^{y \to \tau}(y)$  in Equation 5, which measures similarity scores across  $N$  trajectories, Equation 7 is used to measure the similarity scores of different time steps within a trajectory. The reason for doing this is that the textual constraint is violated at time step  $T$ , while in the previous time steps the constraint is not violated. Therefore, the instinct is to maximize the similarity score between the final trajectory embedding  $H_T$  and the textual constraint embedding  $L$ , while minimizing the similarity score between all previous time step embeddings  $(H_1, H_2, H_3, \dots, H_{(T-1)})$  and the textual constraint embedding  $L$ . Based on this instinct, we introduce within-trajectory (WT) loss:
+
+$$
+\mathcal {L} _ {W T} = \mathbb {E} _ {(\tau , y) \sim \mathcal {D}} \left[ - \frac {1}{T} \left(\sum_ {t = 1} ^ {T - 1} \log \left(1 - p _ {t} ^ {\tau} (y)\right) + \log \left(p _ {T} ^ {\tau} (y)\right)\right) \right], \tag {8}
+$$
+
+where the first term is responsible for minimizing the similarity score for the embedding of time steps before  $T$  in the trajectory sequence, while the second term is responsible for maximizing the similarity score for the embedding of time step  $T$ .
+
+By combining these two losses  $\mathcal{L}_{WT},\mathcal{L}_{MC}$ , we can train a text encoder to minimize the distance between embeddings of semantically similar texts, while simultaneously training a trajectory encoder to minimize the distance between embeddings of semantically similar trajectories. Crucially, this approach enables us to align the text and trajectory embeddings that correspond to the same semantic constraint, fostering a cohesive representation of their shared meaning, and further determining whether the trajectory violates the constraint by calculating embedding similarity.
+
+# 4.2 Cost Assignment Component
+
+After solving the problem of predicting whether a given trajectory violates a textual constraint, there is still the issue of cost sparsity. Motivated by the works of temporal credit assignment [26], we propose
+
+a component to capture the relationship between the state-action pair and the textual constraint and assign a cost value to each state-action based on its impact on satisfying the constraint.
+
+Specifically, suppose we have a (trajectory  $\tau$ , textual constraint  $y$ ) pair and its representation  $(\{H_t\}_{t=1}^T, L)$  obtained from text-trajectory alignment component. The textual constraint representation  $L$  is processed by an episodic-cost prediction layer  $F^e$  to obtain a predicted episodic cost  $\hat{C}(y) = \text{sigmoid}(F^e(L))$  for the entire trajectory. We expect the episodic cost can be considered as the sum of cost on all non-violation state-action pairs:  $\hat{C}(y) = \sum_{t=1}^{T-1} \hat{c}(s_t, a_t, y, \tau_t)$ . To evaluate the significance of each timestep's action relative to textual constraints, we employ an attention mechanism:
+
+$$
+e _ {t} = \operatorname {s i g m o i d} \left(\operatorname {s i m} _ {t} (\tau , y)\right). \tag {9}
+$$
+
+Here we regard the text representation as the query, each time step's representation in the trajectory as the key, and compute the attention score  $e_t$  based on the cosine similarity metric. After that, we use the sigmoid function to make sure the score falls within the range of 0 to 1. Each attention score  $e_t$  quantifies the degree of influence of the state-action pair  $(s_t, a_t)$  on violating the textual constraint. Then we obtain an influence-based representation  $H_t^* = e_t H_t$ . To predict the cost  $\hat{c}(s_t, a_t, y, \tau_t)$ , we incorporate a feed-forward layer called the cost assignment layer  $F^c$  and output the predicted non-violation single step cost as:
+
+$$
+\hat {c} \left(s _ {t}, a _ {t}, y, \tau_ {t}\right) = \operatorname {s i g m o i d} \left(F ^ {c} \left(\operatorname {C o n c a t} \left(H _ {t} ^ {*}, L\right)\right)\right). \tag {10}
+$$
+
+The loss function for measuring inconsistency of the episodic cost  $\hat{C}(y)$  and the sum of non-violation single step costs  $\hat{c}(s_t, a_t, y, h_t)$  can be formed as:
+
+$$
+\mathcal {L} _ {C A} = \mathbb {E} _ {(\tau , y) \sim \mathcal {D}} \left[ \left(\sum_ {t = 1} ^ {T - 1} \hat {c} \left(s _ {t}, a _ {t}, y, \tau_ {t}\right) - \hat {C} (y)\right) ^ {2} \right]. \tag {11}
+$$
+
+This mutual prediction loss function  $\mathcal{L}_{CA}$  is only used to update the episodic-cost prediction layer and cost assignment layer, and not to update the parameters of the trajectory encoder or text encoder during backpropagation. This helps ensure the validity of the predictions by preventing overfitting or interference from other parts of the model.
+
+The effectiveness of this component comes from two main sources. First, the text-trajectory alignment component projects semantically similar text representations to nearby points in the embedding space, allowing the episodic-cost prediction layer to assign similar values to embeddings with close distances. This aligns with the intuition that textual constraints with comparable violation difficulty should yield similar episodic costs. Second, the cost assignment layer leverages the representational power of the text-trajectory alignment component to capture complex relationships between state-action pairs and constraints, enabling accurate single-step cost predictions.
+
+In our experiment, we simultaneously train the text-trajectory alignment component and cost assignment component end-to-end with a uniform loss function  $\mathcal{L}_{TTCT}$ :
+
+$$
+\mathcal {L} _ {T T C T} = \mathcal {L} _ {M C} + \mathcal {L} _ {W T} + \mathcal {L} _ {C A}. \tag {12}
+$$
+
+By doing this, we can avoid the need for separate pre-training or fine-tuning steps, which can be time-consuming and require additional hyperparameter tuning. Also, this can enable the cost assignment component to gradually learn from the text-trajectory alignment component and make more accurate predictions over time.
+
+In the test phase, at time step  $t$  we encode trajectory  $\tau_{t}$  and textual constraint  $y$  with Equation 3 and Equation 2 to obtain the entire trajectory embedding  $H_{t}$  and text embedding  $L$ . Then we calculate distance score  $\mathrm{sim}(\tau_t,y)$  using Equation 4. The predicted cost function  $\hat{c}$  is given by:
+
+$$
+\hat {c} = \left\{ \begin{array}{l l} 1, & \text {i f} \operatorname {s i m} \left(\tau_ {t}, y\right) \geq \beta , \\ \operatorname {s i g m o i d} \left(F ^ {c} \left(\operatorname {C o n c a t} \left(H _ {t} ^ {*}, L\right)\right)\right), & \text {o t h e r w i s e} \end{array} , \right. \tag {13}
+$$
+
+where  $\beta$  is a hyperparameter that defines the threshold of the cost prediction.  $\hat{c} = 1$  indicates that the TTCT model predicts that the textual constraint is violated. Otherwise, if the textual constraint is not violated by the given trajectory, we assign a predicted cost using Equation 10.
+
+# 5 Policy Training
+
+Our Trajectory-level textual constraints Translator framework is a general method for integrating free-form natural language into safe RL algorithms. In this section, we introduce how to integrate our TTCT into safe RL algorithms so that the agents can maximize rewards while avoiding early termination of the environment due to violation of textual constraints. To enable perception of historical trajectory, the trajectory encoder and text encoder are not only used as frozen plugins  $g_{T}$  and  $g_{C}$  for cost prediction but also as trainable sequence models  $g_{T}^{*}$  and  $g_{C}^{*}$  for modeling historical trajectory. This allows the agent to take into account historical context when making decisions. To further improve the ability to capture relevant information from the environment, we use LoRA [18] to fine-tune both the  $g_{T}^{*}$  and  $g_{C}^{*}$  during policy training. The usage of  $g_{T}$ ,  $g_{C}$  and  $g_{T}^{*}$ ,  $g_{C}^{*}$  is illustrated in Appendix A.4 Figure 8.
+
+Formally, let's assume we have a policy  $\pi$  with parameter  $\phi$  to gather transitions from environments. We maintain a vector to record the history state-action pairs sequence, and at time step  $t$  we use  $g_{T}^{*}$  and  $g_{C}^{*}$  to encode  $\tau_{t-1}$  and textual constraint  $y$  so that we can get historical context representation  $H_{t-1}$  and textual constraint representation  $L$ . The policy selects an action  $a_{t} = \pi_{\theta}(o_{t}, H_{t-1}, L)$  to interact with environment to get a new observation  $o_{t+1}$ . And we update  $\tau_{t}$  with the new state-action pair  $(o_{t}, a_{t})$  to get  $\tau_{t}$ . With  $\tau_{t}$  and  $L$ ,  $\hat{c}_{t}$  can be predicted according to Equation 13. Then we store the transition into the buffer and keep interacting until the buffer is full. In the policy updating phase, after calculating the specific loss function for different safe RL algorithms, we update the policy  $\pi$  with gradient descent and update  $g_{T}^{*}$ ,  $g_{C}^{*}$  with LoRA. It is worth noting that  $g_{T}$  and  $g_{C}$  are not updated during the whole policy training phase, as they are only used for cost prediction. The pseudo-code and more details of the policy training can be found in Appendix A.4.
+
+# 6 Experiments
+
+Our experiments aim to answer the following questions: (1) Can our TTCT accurately recognize whether an agent violates the trajectory-level textual constraints? (2) Does the policy network, trained with predicted cost from TTCT, achieve fewer constraint violations than trained with the ground-truth cost function? (3) How much performance improvement can the cost assignment (CA) component achieve? (4) Does our TTCT have zero-shot capability to be directly applicable to constraint-shift environments without any fine-tuning? We adopt the following experiment setting to address these questions.
+
+![](images/f80988627c7f2a3ad7623484edb373ebf9dc7cfbffe46a477cd6449aaa057bbb.jpg)  
+(a) Hazard-World-Grid
+
+![](images/e8be2d94e57a955e4182a3b4e32effd0f54a943ddab6d884b48898bf30bfffff.jpg)  
+(b) SafetyGoal  
+Figure 2: (a) One layout in Hazard-World-Grid [46], where orange tiles are lava, blue tiles are water and green tiles are grass. Agents need to collect reward objects in the grid while avoiding violating our designed textual constraint for the entire episode. (b) Robot navigation task SafetyGoal that is built in Safety-Gymnasium [19], where there are multiple types of objects in the environment. Agents need to reach the goal while avoiding violating our designed textual constraint for the entire episode. (c) LavaWall [5], a task has the same goal but different hazard objects compared to Hazard-World-Grid.
+
+![](images/08897aac783439c321b8adb44b85d664bd75f9dbc91a586b06fdd38c497dd397.jpg)  
+(c) LavaWall
+
+# 6.1 Setup
+
+Task. We evaluate TTCT on two tasks (Figure 2 (a,b)): 2D grid exploration task Hazard-WorldGrid (Grid) [46] and 3D robot navigation task SafetyGoal (Goal) [19]. And we designed over 200 trajectory-level textual constraints which can be grouped into 4 categories, to constrain the agents. A detailed description of the categories of constraints will be given in Appendix A.1. Different from the default setting, in our task setting, when a trajectory-level textual constraint is violated, the
+
+environment is immediately terminated. This is a more difficult setup than the default. In this setup, the agents must collect as many rewards as possible while staying alive.
+
+Baselines. We consider the following baselines: PPO [36], PPO_Lagrangian(PPO_Lag) [33], CPP0_PID [39], FOCOPS [49]. PPO does not consider constraints and simply aims to maximize the average reward. We use PPO to compare the ability of our methods to obtain rewards. As for the last three algorithms, we design two training modes for them. One is trained with standard ground-truth cost, where the cost is given by the human-designed violation checking functions, and we call it ground-truth (GC) mode. The other is trained with the predicted cost by our proposed TTCT, which we refer to as cost prediction (CP) mode. More information about the baselines and training modes can be found in Appendix A.1 A.2.
+
+Metrics. We take average episodic reward (Avg. R) and average episodic cost (Avg. C) as the main comparison metrics. Average episodic cost can also be considered as the average probability of violating the constraints. The higher Avg. R, the better performance, and the lower Avg. C, the better performance.
+
+![](images/2dcd8d669180c730572d8e1231141ea003109d7deeade2758a16a4a1a66c5e4c.jpg)  
+(a) Hazard-World-Grid
+
+![](images/d2b379c097bb185f9c61669ed32079d3bec635a963836a8d5d86957aac7e1070.jpg)  
+Figure 3: Evaluation results of our proposed method TTCT. The blue bars are our proposed cost prediction (CP) mode performance and the orange bars are the ground-truth cost (GC) mode performance. The black dashed lines are PPO performance. (a) Results on Hazard-World-Grid task. (b) Results on SafetyGoal task.
+
+![](images/7f90dda19a394040dc07002f75dfaae2a92d7008e64ee4f494edfa3893d7efe9.jpg)  
+(b) SafetyGoal
+
+![](images/8084c70a6de6b98700959feda71eac9385ed9844786c20d02689a97ea6e8593e.jpg)
+
+# 6.2 Main Results and Analysis
+
+The evaluation results are shown in Figure 3 and the learning curves are shown in Figure 4. We can observe that in the Hazard-World-Grid task, compared with PP0, the policies trained with GC can reduce the probability of violating textual constraints to some extent, but not significantly. This is because the sparsity of the cost makes it difficult for an agent to learn the relevance of the behavior to the textual constraints, further making it difficult to find risk-avoiding paths of action. In the more difficult SafetyGoal task, it is even more challenging for GC-trained agents to learn how to avoid violations. In the CPP0_PID and F0COPS algorithms trained with GC mode, the probability of violations even rises gradually as the training progresses. In contrast, the agents trained with predicted cost can achieve lower violation probabilities than GC-trained agents across all algorithms and tasks and get rewards close to GC-trained agents.
+
+These results show that TTCT can give an accurate predicted episodic cost at the time step when the constraint is violated, it can also give timely cost feedback to non-violation actions through the cost assignment component so that the agents can find more risk-averse action paths. And these results answer questions (1) and (2). The discussion about the violation prediction capability of the text-trajectory component can be found in Appendix B.1. The interpretability and case study of the cost assignment component can be found in Appendix B.2.
+
+# 6.3 Ablation Study
+
+To study the influence of the cost assignment component. We conduct an ablation study by removing the cost assignment component from the full TTCT. The results of the ablation study experiment are shown in Figure 4. We can observe that even TTCT without cost assignment can achieve similar performance as GC mode. And in most of the results if we remove the cost assignment component, the performance drops. This shows that our text trajectory alignment component can accurately predict the ground truth cost, and the use of the cost assignment component can further help us learn a safer agent. These results answer questions (3).
+
+![](images/41603a8bde61d4188ee8dff450864e2a77567f31745deb3bf8008be2288c7168.jpg)  
+Figure 4: Learning curve of our proposed method TTCT. Each column is an algorithm. The six figures on the left show the results of experiments on the Hazard-World-Grid task and the six figures on the right show the results of experiments on the SafetyGoal task. The solid line is the mean value, and the light shade represents the area within one standard deviation.
+
+![](images/5f19256b723a5923ef2ceaccddeba0d01f0fb41b4dc493b06b68a19070747918.jpg)  
+(a) Hazard-World-Grid
+
+![](images/fe6a0db69ea8bd838f2f5fcd4f6f962d90660190a4ebb3efd7491087d19a2ab2.jpg)  
+Figure 5: Ablation study of removing the cost assignment (CA) component. The blue bars are cost prediction (CP) mode performance with full TTCT and the orange bars is the cost prediction (CP) mode performance without CA component. The black dashed lines are PPO performance. (a) Ablation results on Hazard-World-Grid task. (b) Ablation results in SafetyGoal task.
+
+![](images/b5aa611e5eb4643f36a5212915c066aedb529fbef01e5cc79f64aa4949a4dfac.jpg)  
+(b) SafetyGoal
+
+![](images/00f5ef5743253eda00f4420acb09d9de014c0e61acfc3c437ea5db0edcf5e7a4.jpg)
+
+# 6.4 Further Results
+
+Pareto frontier. Multi-objective optimization typically involves finding the best trade-offs between multiple objectives. In this context, it is important to evaluate the performance of different methods based on their Pareto frontier [10], which represents the set of optimal trade-offs between the reward and cost objectives. We plot the Pareto frontier of policies trained with GC and policies trained with CP on a two-dimensional graph, with the vertical axis representing the reward objective and the horizontal axis representing the cost objective as presented in Figure 6. The solution that has the Pareto frontier closer to the origin is generally considered more effective than those that have the Pareto frontier farther from the origin. We can observe from the figure that the policies trained with predicted cost by our TTCT have a Pareto frontier closer to the origin. This proves the effectiveness of our method and further answers Questions (1) and (2).
+
+![](images/fb19c5f995f56263e2a50302773673c97f1f7c201cf5b286beb8517df6c99893.jpg)  
+(a) PPO_Lag
+
+![](images/d4c6744cd45fe4e8b552968db7d3568b1874344060b5d573b43051cc5a291f40.jpg)  
+(b) CPP0_PID  
+Figure 6: Results of Pareto frontiers. We compare the performance of 200 policies trained using cost prediction (CP) and 200 policies trained with ground-truth cost (GC). The  $\star$  symbol represents the policy on the Pareto frontier. And we connect the Pareto-optimal policies with a curve.
+
+![](images/1ec9be577fe0c4edd978c126d607de93e525449dd38d345bd3bf034bebcab9de.jpg)  
+(c) FOCOPS
+
+Zero-shot transfer capability. To explore whether our method has zero-shot transfer capability, we use the TTCT trained under the Hazard-World-Grid environment to apply directly to a new environment called LavaWall (Figure 2 (c)) [5], without fine-tuning. The results are shown in Figure 7. We can observe that the policy trained with cost prediction (CP) from TTCT trained under the Hazard-World-Grid environment can still achieve a low violation rate comparable to the GC-trained policy. This answers Question (4).
+
+![](images/2b0f0945d9d48b0101de66b56cc89a1201e702e69ae84f40c2275ab1e3193985.jpg)  
+(a) Zero-shot reward
+
+![](images/c2646f71eb6b8cc41f4f47daeb22dce104943c026e20da7900916bbf22c533c0.jpg)  
+(b) Zero-shot cost  
+Figure 7: Zero-shot adaptation capability of TTCT on LavaWall task. The left figure shows the average reward and the right figure shows the average cost.
+
+# 7 Conclusion and Future Work
+
+In this paper, we study the problem of safe RL with trajectory-level natural language constraints and propose a method of trajectory-level textual constraints translator (TTCT) to translate constraints into a cost function. By combining the text-trajectory alignment (CA) component and the cost assignment (CA) component, our method can elegantly solve the problems of predicting constraint violations and cost sparsity. We demonstrated that our TTCT method achieves a lower violation probability compared to the standard cost function. Thanks to its powerful multimodal representation capabilities, our method also has zero-shot transfer capability to help the agent safely explore the constraint-shift environment. This work opens up new possibilities for training agents in safe RL tasks with total free-form and complex textual constraints.
+
+Our work still has room for improvement. The violation rate of our method is not absolute zero. In future work, we plan to investigate the application of TTCT in more complex environments and explore the integration of other techniques such as meta-learning [16] to further improve the performance and generalization capabilities of our method.
+
+# Acknowledgments
+
+This work was supported by the grants from the Natural Science Foundation of China (62225202, 62202029), and Young Elite Scientists Sponsorship Program by CAST (No. 2023QNRC001). Thanks for the computing infrastructure provided by Beijing Advanced Innovation Center for Big Data and Brain Computing. This work was also sponsored by CAAI-Huawei MindSpore Open Fund. Jianxin Li is the corresponding author.
+
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+
+# A Dataset and Training Details
+
+# A.1 Dataset
+
+In our task setting, humans need to provide high-level textual instruction to the agent for the entire trajectory, and then TTCT can predict the cost based on the real-time state of the agent's exploration so that the agent can learn a safe policy with the predicted cost. Thus our dataset is comprised of two parts: the trajectory-level textual constraints and the environments.
+
+Trajectory-level textual constraints: To generate textual constraints, we first explore the environment using a random policy, collecting a large amount of offline trajectory data. Then we design a descriptor that automatically analyzes trajectories and gives natural language descriptions based on predefined templates. To validate whether our method can understand different difficulty levels of textual constraints, we design four types of trajectory-level textual constraints. The four types of textual constraints are:
+
+1. Quantitative textual constraint describes a quantitative relationship in which an entity in the environment cannot be touched beyond a specific number of times, which can be interpreted as the entity's tolerance threshold, and when the threshold is exceeded, the entity may experience irrecoverable damage.  
+2. Sequential textual constraint describes a sequence-based relationship, where the occurrence of two or more distinct actions independently may not pose a risk, but when they occur in sequence, it does. For instance, it's safe to drink or drive when they happen independently, but when they occur in sequence (i.e., drinking first and then driving), it becomes dangerous.  
+3. Relational textual constraint describes constraints on the relationships between an agent and entities in its environment, such as maintaining a certain distance, always being in front of that entity, or not staying too far from it.  
+4. Mathematical textual constraints often do not provide explicit instructions to the agent regarding what actions to avoid, but rather present logical descriptions that demand the model's ability to reason mathematically. This type of constraint thereby presents a higher cognitive burden for our TTCT to comprehend.
+
+Examples of four types of trajectory-level textual constraints are included in Table 2. Then we randomly split the (trajectory, textual constraint) pairs into  $80\%$  training and  $20\%$  test sets. And we use the training set to train our TTCT end-to-end.
+
+**Environments:** We use two environments Hazard-World-Grid and SafetyGoal as main benchmarks and a environment LavaWall to evaluate the zero-shot transfer capability:
+
+1. Hazard-World-Grid. The environment is a  $12 \times 12$  grid, with the gray walls surrounding the perimeter. The agent can only explore within the grid. Inside the grid, some items provide rewards: blue keys, red balls, and yellow boxes. Collecting all of these items will be considered as completing the task. Additionally, there are hazardous materials in the grid, where orange tiles are lava, cyan tiles are water, and green tiles are grass. During exploration, the agent can only see a range of  $7 \times 7$  pixels ahead, resulting in an observation space with size  $7 \times 7 \times 3$ .  
+2. **SafetyGoal.** Robot navigation task, the environment has a navigation target, which is completed when the robot reaches the target. The environment also contains vases (cyan squares) and Hazards (Mediumslateblue circles). The vases can be moved by the robot and hazards are fixed.  
+3. LavaWall. The LavaWall environment shares the same task goal as Hazard-World-Grid but with different hazardous materials. Whenever the environment is initialized, a random lava wall with only one entrance is generated in the grid, and the agent must learn to obtain the reward on the other side while avoiding stepping on lava.
+
+For each episode, we place the agent at a randomized start location, fill the environment with objects, and randomly select a textual constraint from the constraint pool. We manually design the violation checking function for every constraint to determine whether the environment violates the textual constraint, and when the textual constraint is violated, the environment will provide a cost with a
+
+Table 2: Examples from the different trajectory-level textual constraint categories.  
+
+<table><tr><td>Constraint type</td><td>Examples</td></tr><tr><td rowspan="6">Quantitative</td><td>Do not cross lava more than five times.</td></tr><tr><td>Never reach grass more than five times.</td></tr><tr><td>Please touch water less than three times.</td></tr><tr><td>The vase is very fragile, please don’t touch it!</td></tr><tr><td>If you touch the vase more than three times, the vase will break.</td></tr><tr><td>The lava is very dangerous. Please don’t touch it.</td></tr><tr><td rowspan="3">Sequential</td><td>After stepping through the water, your shoes can’t touch the lava.</td></tr><tr><td>Once you’ve walked past the grass, make sure you don’t set foot on water.</td></tr><tr><td>After walking past lava, don’t go back to the water.</td></tr><tr><td rowspan="3">Mathematical</td><td>You only have 20 HP, grass and lava are very dangerous, you will lose 3 and 2 HP respectively if you step on them, please don’t die!</td></tr><tr><td>You only have 25 HP, stepping on grass and lava will lose 3 and 2 HP respectively, but water will help you regain 1 HP, please don’t die!</td></tr><tr><td>You only have 10 HP, and stepping on dangerous water and lava will cause you to lose 3 and 2 HP respectively, your fate hangs by a thread. don’t let it snap!</td></tr><tr><td rowspan="3">Relational</td><td>Please keep a distance of 0.2 from dangerous chemicals or you will die!</td></tr><tr><td>Maintain a safe distance of 0.25 from the hazard.</td></tr><tr><td>It’s important to keep a distance of 0.3 from the hazard.</td></tr></table>
+
+value of 1.0 to the agent and terminate the environment (the manually designed function will not be used in the cost prediction (CP) mode, all the costs will be predicted by our TTCT in CP mode).
+
+# A.2 Baselines
+
+(1) PPO [36]: This algorithm does not consider constraints, and simply considers maximizing the average reward, which we use to compare the ability of our methods to obtain rewards.  
+(2) PPO_Lagrangian(PPO_Lag) [33]: This algorithm transforms a constrained optimization problem into an unconstrained optimization problem via Lagrange multipliers  
+(3) CPPO_PID [39]: This algorithm PID to control Lagrange multiplier, which solves the cycle fluctuation problem of PPO-Lagrange.  
+(4) FOCOPS [49]: This algorithm finds the optimal update policy by solving a constrained optimization problem in the nonparameterized policy space, then projects the update policy back into the parametric policy space.
+
+# A.3 Training Detail of TTCT
+
+We provide training pseudocode of TTCT in Figure 9. The text encoder we use is Bert (https://huggingface.co/google-bert/bert-base-uncased/tree/main) [7]. And the hyperparameters we use are shown in Table 3. We use the same hyperparameters across different tasks' datasets. We conduct the training on the machine with A100 GPU for about 1 day.
+
+# A.4 Policy Training Detail
+
+We provide pseudocode of training policy with predicted from TTCT in Algorithm 1. To enable the agent to comply with diverse types of constraints, we launch async vectorized environments with different types of constrained environments during roll-out collection. The agent interacts with these environments and collects transitions under different constraint restrictions to update its policy. During policy updates, we fine-tune the trajectory encoder  $g_{H}^{*}$  and text encoder  $g_{L}^{*}$  using LoRA [18] at every epoch's first iteration, while not updating encoder parameters in other iterations, which saves training time.
+
+Table 3: Hyperparameters used in TTCT  
+
+<table><tr><td colspan="2">Hyperparameters</td></tr><tr><td>Batch size</td><td>194</td></tr><tr><td>Epochs</td><td>32</td></tr><tr><td>Learning rate</td><td>10-6</td></tr><tr><td>Embedding dimension of trajectory dH</td><td>512</td></tr><tr><td>Embedding dimension of textdT</td><td>512</td></tr><tr><td>Trajectory length</td><td>200</td></tr><tr><td>Transformer width</td><td>512</td></tr><tr><td>Transformer number of heads</td><td>8</td></tr><tr><td>Transformer number of layers</td><td>12</td></tr><tr><td>Optimizer</td><td>Adam [22]</td></tr></table>
+
+# Algorithm 1 Safe RL algorithm with TTCT
+
+1: Initialize value function network  $\phi$ , cost value function network  $\phi_c$ , policy network  $\theta$ , trajectory encoder  $g_T^*$ , textual constraint encoder  $g_C^*$ , frozen trajectory encoder  $g_T$  for cost prediction, frozen textual constraint encoder  $g_C$  for cost prediction, cost alignment network  $F^c$  
+2: for each epoch do  
+3: for each episode do  
+4: Sample a textual constraint  $y$  for this episode  
+5: Get text representation  $L$  of  $y$  with  $g_{C}^{*}$  
+6: Reset environment to get observation  $o_1$  
+7: Initialize sequence  $\tau_0 = \{o_{pad},a_{pad}\}$  and encoded sequence representation  $H_0 = g_T^* (\tau_1)$  
+8:  $\mathbf{for}t = 1,T$  do  
+9: Select a action  $a_{t} = \pi_{\theta}(o_{t},H_{t - 1},L)$  
+10: Execute action  $a_{t}$  in emulator and observe reward  $r_{t}$  and next observation  $o_{t+1}$ .  
+11: Append  $(o_t, a_t)$  into  $\tau_{t-1}$  to get  $\tau_t$  and update sequence representation  $H_t = g_T^*(\tau_t)$ .  
+12: Predict cost  $\hat{c}_t$  according to Equation 13  
+13: Store transition  $(o_{t},a_{t},r_{t},\hat{c}_{t},\tau_{t - 1},y)$  in buffer  
+14: end for  
+15: end for  
+16: Sample batch of  $N$  transitions from buffer.  
+17: Encode  $\{\tau_j\}_{i = 1}^N$  and  $\{y_{j}\}_{i = 1}^{N}$  with  $g_{T},g_{C}$  
+18: Calculate specific loss function  
+19: Update value function  $\phi$ , cost value function  $\phi_c$ , policy network  $\theta$  
+20: Update trajectory encoder  $g_{T}^{*}$  and textual constraint encoder  $g_{C}^{*}$  with LoRA  
+21: end for
+
+![](images/3128cc57f135c3218aa7a9f68bf4b24476269d685235df4c3bbb4ac2a582d207.jpg)  
+Figure 8: A diagram illustrating the usage of  $g_{T}$ ,  $g_{C}$  and  $g_{T}^{*}$ ,  $g_{C}^{*}$  when training policy.
+
+```python
+stateEncoder - CNN or DNN   
+#actionEncoder - DNN   
+#trajectoryEncoder - Causal Transformer   
+#textEncoder - Any Text Transformer   
+#episodic_cost_predicted - DNN   
+#single_step_cost_predicted - DNN   
+#labelsMC - [n, n] Labels for multimodal   
+#W_tra[d_tra, d_e] - Learned proj of trajectory to embed   
+#W_t[d_t, d_e] - Learned proj of text to embed   
+#S[n, l, d_s] - minibatch of n state sequences   
+#A[n, l, d_a] - minibatch of n action sequences   
+#T[n, w] - minibatch of n text constrains   
+#extract feature representations of different inputs   
+S_f = stateEncoder(S)   
+A_f = actionEncoder(A)   
+Tra_f = trajectory Encoder(Concat([S_f, A_f], dim=-1)) # [n, L, d_tra]   
+T_f = textEncoder(T) # [n, 1, d_t]   
+# normalization   
+T_e = 12_normalize(np.dot(T_f, W_t), axis=-1) # [n, 1, d_e]   
+Tra_e = 12_normalize(np.dot(Tra_f, W_tra), axis=-1) # [n, L, d_e]   
+#cost assignment   
+EpCost = episodic_cost Predictor(T_edetach()) # [n]   
+logits_WT = torch/matmul(Tra_e * np.exp(t), T_e) # [n, L, 1]   
+atten_score = sigmoid(logits_WT) # [n, L, 1]   
+Tra_atten = atten_score * Tra_e   
+cost = single_step_cost Predictor(Concat([Tra_atten[., :l-1, :, ], T_e.repeata(1, l-1, 1).detach()], dim=-1)) # [n, L]   
+sum_cost = sum(cost, dim=1) # [n]   
+loss_CA = (mse_loss(EpCost, sum_cost) + mse_loss(sum_cost, EpCost))/2   
+#within-trajectory (WT)   
+labels_WT = torch.tensord([l-1] * n) # only the last time step is positive   
+loss_WT = cross_entropy_loss(logits_WT, labels_WT)   
+#multimodal contrastive (MC)   
+logits_VC = np.dot(Tra_e[., -1, :] * np.exp(t), T_e) # [n, n]   
+loss_tra = KL_divergence_loss(logits_VC, labels_VC, axis=0)   
+loss_t = KL_divergence_loss(logits_VC, labels_VC, axis=1)   
+loss_VC = (loss_tra + loss_t)/2   
+#total loss   
+loss = loss_VC + loss_WT + loss_CA
+```
+
+Figure 9: Pseudocode for the code of training our TTCT.
+
+# B Additional Experiments
+
+# B.1 Violations Prediction Capability of Text-Trajectory Alignment Component
+
+![](images/c3523fe8df76e9e87889c94d75fbba810eb77fa0218cf3437bb844e1d679bd28.jpg)  
+(a) Cosine similarity (after scale)
+
+![](images/8a998205789292dfc022e857faf01e7c74cf19bee75c0b9b67c5eab123c77bcc.jpg)  
+(b) Ground-truth similarity  
+Figure 10: Heatmap of cosine similarity between trajectory and text embeddings.
+
+To further study the ability of our text-trajectory alignment component to predict violations, we conduct an experiment given a batch of trajectory-text pairs and we use the text-trajectory alignment component to encode the trajectory and textual constraint, and then calculate the cosine distance with Equation 5 between every two embeddings across two modal. We plot a sample of heatmap of calculated cosine similarity and ground-truth as presented in 10. Further, We plot the receiver operating characteristic (ROC) [8] curve to evaluate the performance of the text-trajectory alignment component as presented in Figure 11. AUC (Area Under the Curve) [8] values indicate the area under the ROC curve. The AUC value of our violations prediction result is 0.98. Then We set threshold  $\beta$  equal to the best cutoff value of the ROC curve. We determine whether the trajectory violates a given textual constraint by:
+
+$$
+\left\{ \begin{array}{l l} y e s, & \text {i f} \operatorname {s i m} (\tau , y) \geq \beta , \\ n o, & \text {o t h e r w i s e} \end{array} \right., \tag {14}
+$$
+
+The results are shown in Table 4. These results indicate that our text-trajectory alignment component can accurately predict whether a given trajectory violates a textual constraint.
+
+Table 4: Violations prediction results of text-trajectory alignment component.  
+
+<table><tr><td>Accuracy</td><td>Recall</td><td>Precision</td><td>F1-score</td></tr><tr><td>0.98</td><td>0.9824</td><td>0.8079</td><td>0.8866</td></tr></table>
+
+# B.2 Case Study of Cost Assignment (CA) Component
+
+We visualize the assigned cost for every state-action pair to demonstrate that the cost assignment component could capture the subtle relation between the state-action pair and textual constraint. Our intuition is that we should assign larger costs to state-action pairs that lead to violations of trajectory-level textual constraints, and smaller or negative values to pairs that do not contribute to constraint violations. Using the Hazard-World-Grid environment as an example, we choose three different types of constraints to show our results in Figure 12. The first row shows the textual constraint, the second row shows the trajectory of the agent in the environment, and to make it easier to visualize, we simplify the observation  $s_t$  by representing it as a square, denoting the entity stepped on by the agent at time step  $t$ . The square emphasized by the red line indicates the final entity that makes the agent violate textual constraint at time step  $T$ . The third row shows the predicted cost of the agent at every time step  $t$  and deeper colors indicate larger cost values.
+
+The (a) constraint is mathematical textual constraint: "You only have 20 HP. Lava and grass are dangerous, they will make you lose 3 and 2 HP, respectively. However, water can regenerate 1 HP. Please don't die." This constraint describes the two dangerous entities lava and grass, and the beneficial entity water. From the third-row heat map, we can observe that our cost assignment
+
+![](images/2158e80b24528ded83a5d9d19cbf48c52a8462ee9aef9b600a2454f61cfd1a31.jpg)  
+Figure 11: ROC curve of text-trajectory alignment component. The x-axis represents the false positive rate, and the y-axis represents the true positive rate. The closer the AUC value is to 1, the better the performance of the model; conversely, the closer the AUC value is to 0, the worse the performance of the model.
+
+![](images/a0b5dbc7650d212b2e9733ee40948a5a3796f7b88b49024fd051087f1558a247.jpg)  
+Figure 12: Case study of cost assignment component on three types of textual constraints. The first row of every case shows the textual constraint, the second row shows the trajectory of the agent in the environment and each square represents the object stepped on by the agent at that time step, the third row shows the assigned cost of the agent at each time step, and the fourth row shows the time steps. The red line indicates the final observation where the agent violates the textual constraint.
+
+component assigns a high cost to the action that steps on lava or grass, with the cost increasing as the agent approaches the constraint-violating situation. Not only that, the CA component also recognizes the different levels of danger posed by lava and grass. Since stepping on lava will deduct 3 HP while stepping on grass will deduct 2 HP, the CA component assigns a larger cost value at time step 13 - 14 compared to the cost value at time step 3 - 5.
+
+The (b) constraint is quantitative textual constraint: "Lava is dangerous. Don't touch them more than eight times!". When stepping on the floor, the CA component considers these actions to be safe and assigns a cost of nearly 0. However, when the agent steps onto lava, it assigns a higher cost, especially when the agent steps on lava for the eighth time. At this point, our CA component concludes that the situation has become extremely dangerous, and one more step on lava will violate the constraint, thus giving the highest cost compared to the before time steps.
+
+The (c) constraint is sequential textual constraint: "After you touch lava, don't step on grass!". Our CA component captures two relevant entities: lava and grass, and understands the sequential relationship between entities. When the agent first steps onto the grass, the text-trajectory alignment component determines that this action does not violate the textual constraint. However, after the
+
+agent steps onto lava and then steps onto the grass for the second time, the text-trajectory alignment component detects the cosine similarity  $\sin (\tau ,y)$  greater than the threshold  $\beta$ , thereby violating the textual constraint. And CA component captures that the key trigger condition for violating constraint is stepping onto the lava, therefore assigning a relatively larger cost to such actions at time step  $7 - 9$ . The cost assignment component also assigns relatively small costs to some safe actions that are stepping on the floor, such as steps  $13 - 15$ . This is because it detects that the agent, although hasn't stepped on grass yet, is trending towards approaching grass, which is a hazardous trend, thus providing a series of gradually increasing and small costs. This demonstrates that our component not only monitors key actions that lead to constraint violations but also monitors the hazardous trend and nudging the agent to choose relatively safer paths.
+
+# B.3 Results for Different Types of Constraints
+
+To evaluate the agent's understanding of different types of trajectory-level textual constraints, we conducted an additional experiment using the CPP0_PID algorithm. During training, we separately tracked the average episodic reward and the average episodic cost for three types of textual constraints as presented in Figure 13. From the learning curves, we can observe that for every type of constraint, our CP mode can achieve the lowest violation rate compared to CP without the CA component mode and ground-truth cost (GC) mode.
+
+![](images/7c7b1ae6c23aa1433634255e27a57afb100e372af08f214c8d95b35e816519bb.jpg)  
+(a) Quantitative constraints
+
+![](images/6d1d4c0f329bf1a78df3b79204e75cc56485a42be1dd7dddfca642365015200b.jpg)
+
+![](images/916d3d3c621d63d909e77ac9fb987cf0c828c522870d8bc0f5029346a2a28272.jpg)  
+(b) Sequential constraints
+
+![](images/0cdad850be7ad546491a7c5d7cdf9078a1b3e9614a420eb56b665fc248cecfd9.jpg)
+
+![](images/d1f283061cec56feec41397290a0724100d1bef73f3343157bba8c2f325c079a.jpg)  
+(c) Mathematical constraints
+
+![](images/233c2ff0b7644c45c5131db503128aee27394c7516e19e8daad16ef28551c8ce.jpg)  
+Figure 13: Learning curves for different types of textual constraints. The left figure shows the average reward and the right figure shows the average cost. The result shows for each type of textual constraint, the policies trained by predicted cost from the TTCT can achieve lower constraint violations.
+
+# B.4 Inference time
+
+We perform the trajectory length sensitivity analysis on Hazard-World-Grid. Since our framework is mainly used for reinforcement learning policy training where data is typically provided as input in batches, we counted the inference time for different trajectory lengths with 64 as the batch size, using the hardware device V100-32G. Figure 14 shows that the average inference time per trajectory is 10ms for trajectories of length 100.
+
+![](images/0d5485c5499eaebf023a7cae29a4a87d9e2e76e22c376c9e49bee521b4a938fc.jpg)  
+Figure 14: Inference time of different trajectory lengths for Hazard-World-Grid on the V100-32G hardware device. Batch size is 64.
+
+# B.5 Different Text Encoder
+
+![](images/d94035564043c2b763f93ade0ea39bb3736d7295c180413f30ace6477a6df85c.jpg)  
+(a) Average episodic reward
+
+![](images/907d21d7ca693a1927e028b7bba85d408c85d8ca579827513abf4e4b12cfbffc.jpg)  
+(b) Average episodic cost  
+Figure 15: Empirical analyses on Hazard-World-Grid with varying text encoders. We choose three different models, transformer-25M [41], gpt2-137M [31], and bert-base-uncased-110M [7].
+
+# B.6 Additional Results
+
+We present the learning curve of our main experiment in Figure 4. We also present the main results and ablation study in Table 5 and 6.
+
+Table 5: Evaluation results of our proposed method TTCT.  $\uparrow$  means the higher the reward, the better the performance.  $\downarrow$  means the lower the cost, the better the performance. Each value is reported as mean  $\pm$  standard deviation and we shade the safest agents with the lowest averaged cost violation values for every algorithm.  
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Stats</td><td colspan="2">PP0_Lag</td><td colspan="2">CPPO_PID</td><td colspan="2">FOCOPS</td><td rowspan="2">PP0</td></tr><tr><td>CP(our)</td><td>GC</td><td>CP(our)</td><td>GC</td><td>CP(our)</td><td>GC</td></tr><tr><td rowspan="2">Grid</td><td>Avg.R ↑</td><td>2.70±0.11</td><td>2.71±0.11</td><td>2.01±0.14</td><td>2.00±0.26</td><td>1.52±0.12</td><td>1.53±0.28</td><td>2.55±0.23</td></tr><tr><td>Avg.C ↓</td><td>0.28±0.08</td><td>0.61±0.07</td><td>0.28±0.03</td><td>0.40±0.07</td><td>0.48±0.20</td><td>0.58±0.12</td><td>0.88±0.05</td></tr><tr><td rowspan="2">Goal</td><td>Avg.R ↑</td><td>0.23±0.12</td><td>0.22±0.29</td><td>0.36±0.20</td><td>0.35±0.77</td><td>0.02±0.36</td><td>-0.63±0.68</td><td>0.23±0.55</td></tr><tr><td>Avg.C ↓</td><td>0.19±0.02</td><td>0.26±0.03</td><td>0.11±0.03</td><td>0.35±0.20</td><td>0.10±0.03</td><td>0.41±0.07</td><td>0.32±0.19</td></tr></table>
+
+Table 6: Ablation study of removing the cost assignment (CA) component.  $\uparrow$  means the higher the reward, the better the performance.  $\downarrow$  means the lower the cost, the better the performance. Each value is reported as mean  $\pm$  standard deviation and we shade the safest agents with the lowest averaged cost violation values for every algorithm.  
+
+<table><tr><td rowspan="2">Task</td><td rowspan="2">Stats</td><td colspan="2">PPO_Lag</td><td colspan="2">CPPO_PID</td><td colspan="2">FOCOPS</td><td rowspan="2">PPO</td></tr><tr><td>CP(Full)</td><td>CP w/o CA</td><td>CP(Full)</td><td>CP w/o CA</td><td>CP(Full)</td><td>CP w/o CA</td></tr><tr><td rowspan="2">Grid</td><td>Avg.R ↑</td><td>2.70±0.11</td><td>2.57±0.11</td><td>2.01±0.14</td><td>2.00±0.16</td><td>1.52±0.12</td><td>0.79±0.06</td><td>2.55±0.23</td></tr><tr><td>Avg.C ↓</td><td>0.28±0.08</td><td>0.56±0.08</td><td>0.28±0.03</td><td>0.52±0.14</td><td>0.48±0.20</td><td>0.43±0.04</td><td>0.88±0.05</td></tr><tr><td rowspan="2">Goal</td><td>Avg.R ↑</td><td>0.23±0.12</td><td>-0.62±0.10</td><td>0.36±0.20</td><td>-0.08±0.97</td><td>0.02±0.36</td><td>-0.63±0.21</td><td>0.23±0.55</td></tr><tr><td>Avg.C ↓</td><td>0.19±0.02</td><td>0.35±0.10</td><td>0.11±0.03</td><td>0.29±0.04</td><td>0.10±0.03</td><td>0.23±0.09</td><td>0.32±0.19</td></tr></table>
+
+# C Broader Impacts and Limitation
+
+Our method can help train agents in reinforcement learning tasks with total free-form natural language constraints, which can be useful in various real-world applications such as autonomous driving, robotics, and game playing. There are still limitations to our work. Our method may not be able to completely eliminate constraint violations. Our method has the contextual bottleneck as the length of the trajectory increases. We performed a trajectory length sensitivity analysis on Hazard-World-Grid. As shown in Figure 16, initially increasing the trajectory length improves performance because longer trajectories may provide more dependencies. However, beyond a certain point, further increases in trajectory length result in a slight drop in AUC. This decline is because the trajectory encoder has difficulty capturing global information. We consider this bottleneck to be related to the transformer's encoding capability.
+
+![](images/1f855c7bd22654f9541808294c1a06171b92793babad3a2b592601bca1eb06ae.jpg)  
+Figure 16: Evaluation results of different trajectory lengths for Hazard-World-Grid. Sets of trajectories with varying lengths shared the same set of textual constraints.
+
+# NeurIPS Paper Checklist
+
+# 1. Claims
+
+Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
+
+Answer: [Yes]
+
+Justification: We show our approach can predict cost with trajectory-level natural language. The predicted cost by our approach can help train a safer policy than the ground-truth cost.
+
+Guidelines:
+
+- The answer NA means that the abstract and introduction do not include the claims made in the paper.  
+- The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.  
+- The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.  
+- It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
+
+# 2. Limitations
+
+Question: Does the paper discuss the limitations of the work performed by the authors?
+
+Answer: [Yes]
+
+Justification: See Section 7
+
+Guidelines:
+
+- The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.  
+- The authors are encouraged to create a separate "Limitations" section in their paper.  
+- The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.  
+- The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.  
+- The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.  
+- The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.  
+- If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.  
+- While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
+
+# 3. Theory Assumptions and Proofs
+
+Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
+
+Answer: [NA]
+
+Justification: This paper does not include theoretical results.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include theoretical results.  
+- All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.  
+- All assumptions should be clearly stated or referenced in the statement of any theorems.  
+- The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.  
+- Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.  
+- Theorems and Lemmas that the proof relies upon should be properly referenced.
+
+# 4. Experimental Result Reproducibility
+
+Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
+
+Answer: [Yes]
+
+Justification: We show the used parameters and the pseudocode of TTCT in Appendix A.3.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.  
+- If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.  
+- Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general, releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.  
+- While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example  
+(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.  
+(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.  
+(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).  
+(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
+
+# 5. Open access to data and code
+
+Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
+
+# Answer: [Yes]
+
+Justification: Code to reproduce our results is available in the anonymous repository https://github.com/formember/TTCT.
+
+# Guidelines:
+
+- The answer NA means that paper does not include experiments requiring code.  
+- Please see the NeurIPS code and data submission guidelines (https://nips.cc/public/guides/CodeSubmissionPolicy) for more details.  
+- While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).  
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+- The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.  
+- At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).  
+- Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
+
+# 6. Experimental Setting/Details
+
+Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
+
+# Answer: [Yes]
+
+Justification: See Section 6 and Appendix A.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.  
+- The full details can be provided either with the code, in appendix, or as supplemental material.
+
+# 7. Experiment Statistical Significance
+
+Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
+
+# Answer: [Yes]
+
+Justification: See Section 6.
+
+# Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.  
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+- The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)  
+- The assumptions made should be given (e.g., Normally distributed errors).
+
+- It should be clear whether the error bar is the standard deviation or the standard error of the mean.  
+- It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a  $96\%$  CI, if the hypothesis of Normality of errors is not verified.  
+- For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).  
+- If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
+
+# 8. Experiments Compute Resources
+
+Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
+
+Answer: [Yes]
+
+Justification: See the Appendix A.3.
+
+Guidelines:
+
+- The answer NA means that the paper does not include experiments.  
+- The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.  
+- The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.  
+- The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
+
+# 9. Code Of Ethics
+
+Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?
+
+Answer: [Yes]
+
+Justification: All the links, data, code are provided in the anonymous form.
+
+Guidelines:
+
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