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+[
+    {
+        "type": "text",
+        "text": "ViSER: Video-Specific Surface Embeddings for Articulated 3D Shape Reconstruction ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Gengshan Yang1 Deqing Sun2 Varun Jampani2 Daniel Vlasic2 Forrester Cole2 Ce Liu4∗ Deva Ramanan1,3 ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1Carnegie Mellon University ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2Google Research 3Argo AI 4Microsoft Azure AI ",
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+    },
+    {
+        "type": "text",
+        "text": "ViSER-webpage",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We introduce ViSER, a method for recovering articulated 3D shapes and dense 3D trajectories from monocular videos. Previous work on high-quality reconstruction of dynamic 3D shapes typically relies on multiple synchronized cameras, strong category-specific priors, or 2D keypoint supervision. We show that none of these are required if one can reliably estimate long-range correspondences in a video, making use of only 2D object masks and two-frame optical flow as inputs. ViSER infers correspondences by matching 2D pixels to a canonical, deformable 3D mesh via video-specific surface embeddings that capture the view-independent appearance features of each surface point. These embeddings behave as a continuous set of keypoint descriptors defined over the mesh surface, which can be used to establish dense long-range correspondences across pixels. The surface embeddings are implemented as coordinate-based MLPs that are fit to each video via self-supervised losses. Experimental results show that ViSER compares favorably against prior work on challenging videos of humans with loose clothing and unusual poses as well as animal videos from DAVIS and YTVOS. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Reconstructing the world from a sequence of monocular frames is a long-standing task in computer vision. While there has been tremendous progress in reconstructing rigid scenes (via SfM and SLAM [7, 39, 43], or recent techniques based on neural rendering [28]), reconstructing dynamic scenes with articulated objects remains elusive. For example, given a monocular video, it is still challenging to reconstruct an everyday scene of a moving person with loose clothing. In this work, we tackle the problem of estimating the deforming mesh of articulated objects given a segmented monocular video of that object. Our method avoids the use of any mesh templates or category-specific priors and generalizes to unknown deformable articulated objects in the wild. ",
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+    {
+        "type": "text",
+        "text": "Nonrigid shape recovery is highly under-constrained due to fundamental ambiguities between shape, appearance, and time-varying deformation. Current approaches for addressing these challenges fall into two camps: better data “likelihoods” or better “priors”. The first camp extracts richer sensor data, via multi-camera studio setups [15] or depth sensors [30], but requires substantial efforts to work in the wild. The second camp makes use of category-level priors over object shapes [18, 20] and is particularly effective for human reconstruction. However, building such models requires considerable offline efforts in the form of registered 3D scans [26] or manual keypoint annotations [12], both of which are difficult to scale to arbitrary object categories. ",
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+    {
+        "type": "image",
+        "img_path": "images/c5024c4b9aaa8a762cdc7d035eea0072af4cfb73583565fe4600659f978d57ee.jpg",
+        "image_caption": [
+            "Figure 1: Given a long video (or multiple short videos), ViSER jointly learns articulated 3D shapes (represented as a mesh with vertices $\\bar { \\bf V }$ and faces $\\mathbf { F }$ ) and joint pixel-surface embeddings (including a surface embedding $\\mathbf { F _ { S } }$ and a pixel embedding $\\mathbf { F _ { I } }$ ) that establishes dense long-range pixel correspondences over time. As a result, ViSER produces accurate shapes, long term trajectories and meaningful part segmentation. "
+        ],
+        "image_footnote": [],
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+    {
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+        "text": "In this work, we use a practical but less explored variant of the data-likelihood camp: we use multiple frames of a video rather than multiple cameras or depth sensors. This considerably complicates analysis for dynamic, non-rigid scenes. Nonrigid structure-from-motion (NRSfM) [4, 38] attempts to constrain the problem by relying on motion correspondences such as 2D point tracks. While 2D correspondences over short time scales (i.e., optical flow) are relatively robust to extract, correspondences over long time scales are notoriously difficult to estimate because of appearance variations arising from viewpoint changes, occlusion and fast motion. In practice, this limits the applicability of NRSfM methods to controlled lab sequences. ",
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+    {
+        "type": "text",
+        "text": "We propose ViSER (Video-Specific Surface Embeddings for Reconstruction), which establishes long-range correspondence and reconstructs articulated 3D shapes from a monocular video. Fig. 1 shows a sample outdoor video and the corresponding ViSER results. The key insight behind ViSER is to force long-range video pixel correspondences to be consistent with an underlying canonical 3D mesh through the use of video-specific embeddings that capture the pixel appearance of each surface point. These embeddings behave as a continuous set of keypoint descriptors defined over the surface mesh, learned with coordinate-based MLPs that are fit to each video via self-supervised losses. ViSER simultaneously optimizes the image CNN, surface MLP, and 3D shape so as to fit the observed video frames. It reconstructs state-of-the-art articulated 3D shape and 3D trajectories without using category-specific priors, making it easily scalable to diverse videos including humans with challenging clothing and poses as well as animals. Lastly, we demonstrate that ViSER recovers meaningful part segmentation and blend skinning weights from videos, which typically require considerable manual effort from 3D artists. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Low-level correspondence. Optical flow is a well-studied representation for short-term correspondence between adjacent frames of a video. After decades of research, recent CNN models [40, 42, 48] for optical flow have achieved an impressive level of accuracy as evidenced by the Sintel and KITTI benchmarks [5, 9]. However, it is challenging to concatenate optical flow for reliable long-range correspondence due to occlusions and strong appearance changes [33, 37, 41]. ViSER does not concatenate optical flow but use it as a constraint to establishes long-range correspondence. ",
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+    {
+        "type": "text",
+        "text": "The layered approach [6, 14, 45] segments a video into different moving objects with coherent motion, thereby establishing long-range correspondence for every frames through the shared layers. Early layered methods assume parameter motion for each layer and can only handle limited scenes. Unwrap Mosiacs [32] uses a dense 2D-to-2D mapping from a texture map to every input frame, and editing operations on the texture map naturally transfers to each individual frame. However, the 2D representation cannot flexibly model complex 3D phenomena, such as occlusions. ",
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+    {
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+        "img_path": "images/85a0b0d499cfbf36b2e52416eddbbd27a71c5a591aa10f920a05d7b53bc2d6ab.jpg",
+        "image_caption": [
+            "Figure 2: We learn a joint pixel-surface embedding space for dense correspondence between pixels in video frames $I _ { t }$ and points on a canonical 3D surface $( \\bar { \\mathbf { V } } , \\mathbf { \\check { F } } )$ . Such embedding space is optimized through “top-down” differentiable rendering $\\mathcal { R } ( \\cdot )$ and “bottom-up” correspondence matching $\\hat { \\bf S } [ x , y ]$ (Sec 3.2). We introduce a 3D matching loss to optimize the embeddings, where the matched surface locations are encouraged to be close to the rendered surface locations. The embedding further enables articulated shape optimization through a 2D-3D-2D cycle reprojection: pixel $[ x , y ] $ matched surface ${ \\hat { \\mathbf { S } } } [ x , y ] \\to$ re-projected pixel $\\pi ( \\hat { \\mathbf { S } } [ x , y ] )$ (Sec. 3.3). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Dense pose and surface mappings. DensePose [12] directly maps pixels to the 3D surface of a human body model. It requires large amounts of training data with annotated image-to-surface correspondence and is hard to generalize to other categories. Articulation-aware Canonical Surface Mapping (A-CSM) [20] uses geometric cycle consistency for learning to map pixels to corresponding points on a template shape without using keypoint annotations. However, it requires a pre-defined template shape for each category. Continuous Surface Embeddings (CSE) [29] establishes dense correspondences between image pixels and 3D object geometry by predicting an embedding vector of the corresponding vertex in the object mesh for each pixel in a 2D image. While applicable to multiple categories, CSE requires annotations and only applies to categories in the training set. ViSER requires neither a template shape nor annotations to work on categories in the wild. ",
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+    {
+        "type": "text",
+        "text": "Nonrigid shape reconstruction. One way to accurately reconstruct articulated shapes is to rely on rich sensor data, e.g., multi-view [15] or depth sensors [30], which requires substantial efforts to setup and reconstruct objects in the wild. For monocular videos/images, one popular approach is to adopt strong 3D shape and pose priors [18, 26, 35, 36, 53, 54] but it works well only on limited categories, whose 3D data are easy to collect. To deal with more nonrigid object categories, a recent trend is to learn a category-level 3D shape model from a collection of images or videos with 2D annotations, such as keypoints and object silhouettes [10, 16, 20, 22, 23, 44, 46, 50]. Although they are able to reconstruct more object categories, such as birds and quadruped animals, the reconstruction usually lacks details, and the level of deformation recovered tends to be low. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Category-agnostic methods, such as nonrigid structure from motion (NRSfM) methods [4, 11, 19, 38] reconstruct nonrigid 3D shapes from a set of 2D point trajectories. However, due to the difficulty in obtaining accurate long-range correspondences [37, 41] they do not work well for videos in the wild. A recent work, LASR [49], uses two-frame optical flow to reconstruct articulate shapes from a monocular video with differentiable rendering. Despite the promising results, LASR does not reason about long-range correspondences and can only reliably reconstruct what is visible in a short video. ViSER establishes reliable long-range correspondence that are robust to moderate shape variations and appearance changes. Thus, ViSER can obtain much higher-quality reconstruction by using either a long video or several videos of a category. ",
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+    {
+        "type": "text",
+        "text": "3 Approach ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 provides an overview of our approach, which follows a typical framework of differentiable rendering [16, 25]. Borrowing the notation from LASR [49], we formalize our task as follows. Given a set of video observations including RGB pixel color, segmentation masks, and optical flow estimates $\\{ I _ { t } , S _ { t } , u _ { t } \\} _ { t = \\{ 0 , \\ldots , T \\} }$ , our goal is to recover a set of shape and motion parameters $\\{ \\mathbf { S } , \\mathbf { D } _ { t } \\}$ that produce reconstructions $\\{ \\hat { I } _ { t } , \\hat { S } _ { t } , \\hat { u } _ { t } \\} _ { t = \\{ 0 , \\dots , T \\} }$ that match the video observations. We refer to supplementary material for a complete list of notations defined in the paper. ",
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+    {
+        "type": "text",
+        "text": "3.1 Preliminaries ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We represent an object’s shape as a triangular mesh $\\mathbf { S } = \\{ \\bar { \\mathbf { V } } , \\mathbf { F } \\}$ with canonical vertices $\\bar { \\mathbf { V } } \\in \\mathbb { R } ^ { 3 \\times N }$ and a fixed topology (edge connectivity) $\\mathbf { \\bar { F } } \\in \\mathbb { R } ^ { 3 \\times M }$ . To render an object, we displace mesh vertices with motion parameters $\\mathbf { D } _ { t }$ , apply a perspective projection with camera intrinsics $\\mathbf { K } _ { t }$ , and rasterize. ",
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+    {
+        "type": "text",
+        "text": "We model vertex motion with root body transformations $\\mathbf { G } _ { 0 }$ and object articulations $\\{ \\mathbf { G _ { 1 } } , \\cdots , \\mathbf { G _ { B } } \\}$ using linear blend skinning (LBS) [20, 21]. LBS constrains vertex motion by linearly blending $B$ rigid “bone” transformations with a skinning weight matrix $\\mathbf { W } \\in \\mathbb { R } ^ { B \\times N }$ , transforming the canonical shape into frame $t$ as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b232677f352fdb21207c4a63d3b9c24be9e44c2dd26fada5f600774a4532283d.jpg",
+        "text": "$$\n\\mathbf { V } _ { i , t } = \\mathbf { G _ { 0 , t } } \\left( \\sum _ { b } \\mathbf { W } _ { b , i } \\mathbf { G } _ { b , t } \\right) \\bar { \\mathbf { V } } _ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $i$ is the vertex index, $b$ is the bone index. Similar to LASR, the root body and bone transformations are represented as the outputs of a pose CNN given an input image, $( \\mathbf { G _ { 0 } } , \\cdot \\cdot \\cdot , \\mathbf { G _ { B } } ) = \\psi _ { p } ( I _ { t } )$ . ",
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+    {
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+        "text": "We define a set of surface properties for rendering, including vertex 3D coordinates, textures and features, and rasterize them in a differentiable manner [25]. We denote the differentiable rendering function that renders the property $\\mathbf { C }$ defined on a canonical surface to an image as $\\mathcal { R } ( \\mathbf { C } ; \\mathbf { V } , \\mathbf { W } , \\mathbf { G } )$ , which executes the blending skinning function in Eq. (1) and softly blends the surface property based on their depth and barycentric coordinates [25]. For simplicity, we omit the shape, skinning, and motion parameters parameters and write the differentiable rendering function as $\\mathcal { R } ( \\mathbf { C } )$ . To render optical flow, we rasterize and project vertex coordinates in two consecutive frames and compute their 2D displacements [49]. Such renderings are compared against video observations to compute gradients for updating model parameters. ",
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+        "type": "text",
+        "text": "3.2 Video-specific Surface Embedding ",
+        "text_level": 1,
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+        "text": "Pixel-surface embeddings. We learn pixel and surface embeddings that map corresponding pixels in different frames to the same point on a canonical 3D surface. Intuitively, consider a particular region on the canonical surface mesh that is the “nose” of an articulated human. The surface embedding captures a descriptor for the nose, which can then be matched to pixel-level descriptors at each frame. ",
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+    {
+        "type": "text",
+        "text": "Given an input image $I _ { t }$ , the pixel-wise descriptor embedding is computed by a U-Net [34] encoder: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/607622e98146587ef59da1a23392816325397cde16e72d709e8e17df671a5134.jpg",
+        "text": "$$\n\\mathbf { F _ { I } } [ x , y , t ] = \\psi _ { e } ( I _ { t } ) [ x , y ] \\in \\mathbb { R } ^ { 1 6 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $[ x , y , t ]$ are pixel locations at frame $t$ . The surface embedding is computed by a positionencoded MLP: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/92a30e32ed6abae077e5eb1d513735d8edd37839e95433d02151015812e52707.jpg",
+        "text": "$$\n\\mathbf { F _ { S } } ( X , Y , Z ) = \\phi _ { e } ( X , Y , Z ) \\in \\mathbb { R } ^ { 1 6 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\phi _ { e } ( \\cdot )$ is an MLP defined over 3D points $( X , Y , Z )$ in the canonical space, augmented with Fourier positional encoding [28]. The two embeddings are optimized on test videos such that pixels representing the same surface location in different frames are mapped to the same canonical surface point [20]. ",
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+        "text": "Correspondence via soft-argmax regression. Given the pixel and surface embeddings, we construct a per-frame cost volume $D ( \\mathbf { F _ { I } } , \\mathbf { F _ { S } } )$ of size $H \\times W \\times N _ { s }$ over pixels and surface points (we randomly sample $N _ { s } = 2 0 0$ surface points at each step) by considering their cosine feature distances, ",
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+    {
+        "type": "equation",
+        "img_path": "images/ba69bc5c06566b57dcfc5c3471dc82f804654ed2f16feb096b2a636914183f49.jpg",
+        "text": "$$\nD ( \\mathbf { F } _ { \\mathbf { I } } , \\mathbf { F } _ { \\mathbf { S } } ) [ x , y , i ] = 1 - \\cos \\big ( \\mathbf { F } _ { \\mathbf { I } } [ x , y ] , \\mathbf { F } _ { \\mathbf { S } } ( X _ { i } , Y _ { i } , Z _ { i } ) \\big ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Normalizing the cost volume over the surface point dimension yields a softmax “heatmap” over surface points that potentially match to pixel $( x , y )$ , as shown in Fig. 3 (Left): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/df4f5d6f3b48daca909c5978cc0934cd194922176d10a28573f319720358cda5.jpg",
+        "text": "$$\n\\sigma _ { \\left( x , y \\right) } [ i ] = \\frac { e ^ { - D [ x , y , i ] / \\tau } } { \\sum _ { j } e ^ { - D [ x , y , j ] / \\tau } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\tau$ is a temperature scaling parameter that is jointly optimized with the feature embeddings. To output a single surface point for pixel $( x , y )$ , we can compute a “soft” argmax [17, 48] by taking the expectation of the softmax distribution over the 3D locations of the points samples, ",
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+    {
+        "type": "equation",
+        "img_path": "images/612885cb8748fe6c95f598d71fbe572ccfbd45cdb59129a756a077964572bedd.jpg",
+        "text": "$$\n\\hat { \\mathbf { S } } [ x , y ] = \\sum _ { i } \\sigma _ { ( x , y ) } [ i ] ( X _ { i } , Y _ { i } , Z _ { i } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $( X _ { i } , Y _ { i } , Z _ { i } )$ is the i-th sampled surface point and $\\sigma _ { ( } x , y ) [ i ]$ is the matching probability of pixel $( x , y )$ over the sampled points $i \\in \\{ 1 , 2 , \\ldots , N _ { s } \\}$ . ",
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+    {
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+        "img_path": "images/2ebf45b5b43939195594f2e62ce87906cefbbdcdadb94eac45fe28f932ffe5f0.jpg",
+        "image_caption": [
+            "Figure 3: Pixel-surface embeddings establish a continuous mapping between pixels and points on a canonical surface. Left: Given a query pixel at $( \\mathbf { x } , \\mathbf { y } )$ , we match it to a set of canonical surface points, where the matching distribution is used to regress a continuous mapping to the canonical surface. Right: Given a query surface point (X,Y,Z), a matching distribution over pixels can be computed. Warm color indicates high matching probability. "
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+        "text": "We can also normalize the $H \\times W \\times N _ { s }$ cost volume over spatial positions to capture a distribution of pixel locations that match to each surface point $( X _ { i } , Y _ { i } , Z _ { i } )$ : ",
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+        "text": "$$\n\\sigma _ { ( X _ { i } , Y _ { i } , Z _ { i } ) } [ x , y ] = \\frac { e ^ { - D [ x , y , i ] / \\tau } } { \\sum _ { [ x , y ] } e ^ { - D [ x , y , j ] / \\tau } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and compute a similar soft argmax mapping of surface points to pixels, as shown in Fig. 3 (Right). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Relation to keypoints. The output of classic keypoint detectors are often represented as $K$ -channel heatmaps over the pixel grid, where $K$ is the number of keypoints. To define dense keypoints, one may increase the number of channels, which is computationally heavy. Similar to CSE [29], we represent dense keypoints as low-dimensional pixel-surface embeddings, which establishes a mapping between pixels and a canonical 3D surface, but far more efficiently. DensePose [12] and CSM [20] use an alternative pixel-to-surface mapping that regresses a surface coordinate at every pixel. In contrast, our pixel-surface embedding captures multimodal uncertainties over keypoints; for example, $\\sigma _ { ( x , y ) } [ i ]$ can capture the fact that a particular pixel matches well to both the left and right ankle, as visualized in Fig. 3, while a regressor may “regress” to the mean of the two surface coordinates. ",
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+    {
+        "type": "text",
+        "text": "3.3 Learning Embeddings and Articulated Shapes ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Next we will introduce the loss functions that enable learning both embeddings and articulated shapes from monocular videos without a pre-defined shape template or annotated correspondence. To learn non-degenerate embeddings and overcome the local optima issue in differertiable renderers, we carefully construct a 3D matching loss and a 2D cycle loss. ",
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+    {
+        "type": "text",
+        "text": "3D match loss. Arguably, the simplest loss to learn embeddings is to minimize the difference between the rendered surface features and the observed pixel features: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4480714044003a0408da164975249413dcc289a992af832b31b01be88e9855ba.jpg",
+        "text": "$$\nL _ { \\mathrm { f e a t u r e - c o n s i s t e n c y } } = \\sum _ { x , y } { \\Big ( } 1 - \\cos ( \\mathcal { R } ( \\mathbf { F } _ { \\mathbf { S } } ) [ x , y ] , \\mathbf { F } _ { \\mathbf { I } } [ x , y ] ) { \\Big ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\cos ( \\cdot )$ denotes the inner product between two normalized vectors, and $\\mathcal { R } ( \\mathbf { F _ { S } } )$ is the differentiably rendered surface descriptors. However, the feature consistency loss admits a trivial solution, where all pixel and surface features are the same constant (yielding zero error). To address this, we introduce a 3D matching loss that ensures pixel embeddings only match to surface embeddings rendered at the pixel location: ",
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+    {
+        "type": "equation",
+        "img_path": "images/8c233702031121c18da60c5f0a27139a7c4b1ffe8577986ded6111c763c65eb1.jpg",
+        "text": "$$\nL _ { \\mathrm { m a t c h i n g } } = \\sum _ { x , y } \\left. \\mathcal { R } ( \\bar { \\mathbf { V } } ) [ x , y ] - \\hat { \\mathbf { S } } [ x , y ] \\right. _ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { R } ( \\bar { \\bf V } )$ is the rendered 3D surface location and $\\hat { \\mathbf { S } } [ x , y ]$ is the estimated pixel-to-surface mapping from Eq. (6), computed through sampling and computing the softmax distributions $\\sigma [ i ]$ over surface points [17]. To minimize the loss, the embeddings of surface points that do not project to $( x , y )$ will be pulled away from the pixel embedding of $( x , y )$ in a contrastive way [13]. ",
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+    {
+        "type": "text",
+        "text": "2D cycle loss. The match loss aims to learn pixel-surface embeddings that are consistent over video frames and discriminative over difference surface locations. However for articulation optimization, the match loss suffers from bad local optima issue similar to other losses based on differentiable rendering [25]. For instance, when the rendering of a body part is outside the ground-truth object silhouette, a gradient update of articulation parameters would likely not incur a lower loss. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To guide articulated 3D shape learning using the learned pixel-surface embeddings, we further define a cycle-based re-projection loss, inspired by prior approaches in 3D model fitting with keypoints [3] and canonical surface mappings [20]. Given an input image, we establish a 2D-3D mapping by extracting a pixel embedding and matching it to surface embedding. Then, we compute the expected surface coordinate $\\hat { \\mathbf { S } } [ x , y ]$ at every pixel using Eq. (6), and ensure the differentiably rendered canonical surface coordinate lands back on the original pixel coordinate $( x , y )$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/ca73a351dacef92a996329c65ac61a83730e90cf00cae1bd2b273d52d2a4c18e.jpg",
+        "text": "$$\nL _ { \\mathrm { r e p r o j } } = \\sum _ { x , y } \\left\\| \\mathcal { R } ( \\hat { \\mathbf { S } } [ x , y ] ) - ( x , y ) \\right\\| _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Reconstruction loss. Finally, we make use of reconstruction losses to ensure that generated images, masks, and flows match their estimated counterparts: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a0cd5e990fe6f8fc8a830fe02ff82113fd1fba2897ef195d213f4b8ec4e3645.jpg",
+        "text": "$$\nL _ { \\mathrm { r e c o n } } = \\beta _ { 1 } \\lvert | \\hat { S } _ { t } ^ { i } - S _ { t } \\rvert | _ { 2 } ^ { 2 } + \\beta _ { 2 } \\lvert | \\hat { I } _ { t } ^ { i } - I _ { t } \\rvert | _ { 2 } ^ { 2 } + \\beta _ { 3 } \\sigma _ { t } \\lvert | \\hat { u } _ { t } ^ { i } - u _ { t } \\rvert | _ { 2 } + \\beta _ { 4 } \\mathrm { p d i s t } ( \\hat { I } _ { t } , I _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            261,
+            735,
+            279
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ \\beta _ { 1 } , \\cdot \\cdot \\cdot , \\beta _ { 4 } \\}$ are weights empirically chosen, $\\sigma _ { t }$ is the normalized confidence map for flow measurement, and pdist $( \\cdot , \\cdot )$ is the perceptual distance [51] measured by an ImageNet-pretrained AlexNet. The reconstruction losses ensure the match between rendered and observed optical flow, texture and silhouette images. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Regularization. To avoid degenerate shapes, we use mesh Laplacian regularization [16, 49] to enforce the recovered shape to be smooth, and as-rigid-as-possible (ARAP) regularization to enforce the deformation to be locally rigid [44]. Different from prior work that only preserves the length of edges after articulation, we encourage both the area and length of faces to be the same after articulation. The area preserving term is defined as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f576b8c25a54b6dcff2628cc938c1e318fdf6b42b34da047a4a6035f04e576e.jpg",
+        "text": "$$\nL _ { \\mathrm { A R A P - a r e a } } = \\sum _ { i = 1 } ^ { | E | } \\sum _ { j \\in N _ { i } } \\mid \\left| \\mathbf { E _ { i } ^ { t } } \\times \\mathbf { E _ { j } ^ { t } } \\right| - \\left| \\mathbf { E _ { i } ^ { t + 1 } } \\times \\mathbf { E _ { j } ^ { t + 1 } } \\right| \\mid ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "where $| E |$ is the number of edges and $N _ { i }$ the indices of neighbouring edges. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.4 Representing Surface Properties with MLPs ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "By extending surface embedding MLPs with additional dimensions, we can model other surface properties including textures and even surface-based geometric deformations. Compared to explicitly defined textures, such continuous implicit representations have the capacity to encode arbitrary amount of details and are empirically easier to optimize. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Surface appearance. The appearance of the object is represented by a coordinate-MLP queried at points on the canonical mesh surface. To handle view-dependent appearance (such as shadow and lighting), we further concatenate the Fourier features of the $( X , Y , Z )$ coordinates with a frame appearance code, as the input to the texture MLP, ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcf4adbc2333e8fc98d5a11510f2e48379987021d4c360ac52fbe3cf382a5091.jpg",
+        "text": "$$\n\\mathbf { C _ { i , t } } = \\phi _ { t e x } ( \\mathcal { F } ( \\bar { \\mathbf { V } } _ { i } ) , \\omega _ { t } ) \\in \\mathbb { R } ^ { 3 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            648,
+            594,
+            666
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\bar { \\bf V } _ { i }$ is the i-th canonical mesh vertex, which is passed through a Fourier encoder $\\mathcal F ( \\cdot )$ as used in NeRF [28], and concatenated with $\\omega _ { t }$ , a 64-dimensional frame appearance code associated each image frame $t$ , predicted from a ResNet-18, as $\\omega _ { t } = \\psi _ { t e x } ( I _ { t } )$ . ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Instance shape deformation fields. To deal with videos of multiple instances of the same category, as experiments in Sec. 4.3, we model shape variations across instances by a continuous surface deformation field defined on the canonical surface. Similar to the surface texture, we represent the surface deformation field by a shape MLP, ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/66bc87475a1b47a318540b9821a279307a2fad3a1c36d27613d1a3ec65bcc76f.jpg",
+        "text": "$$\n\\mathbf { V _ { i , k } } = \\bar { \\mathbf { V _ { i } } } + \\phi _ { s h a p e } ( \\mathcal { F } ( \\bar { \\mathbf { V } } _ { i } ) , \\alpha _ { k } ) \\in \\mathbb { R } ^ { 3 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            781,
+            620,
+            799
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { V _ { k } }$ is the rest shape of instance $k$ and $\\alpha _ { k }$ is a video-specific 64-dimensional shape code that is randomly initialized and optimized together with the shape MLP. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate ViSER in three different scenarios where objects are highly articulating, making it challenging to reconstruct and estimate long-range correspondences. First, we consider long human videos with loose clothing and unusual poses. Next, we evaluate on videos of articulated animals for which accurate shape templates are missing. Finally, we analyze a multi-video variant of ViSER that learns a single model from multiple videos of the same category. All scenarios require jointly establishing long-range correspondences and reconstructing articulated 3D shapes at the same time. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/71404e8e10df833bf3b4fcbe5ee20874dc4e228d520c4c614a599c4cf65ce27c.jpg",
+        "table_caption": [
+            "Table 1: 2D Keypoint transfer accuracy on athletic videos. Methods with ∗ use keypoint annotations to train. Best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>break-1</td><td>break-2</td><td>dance</td><td>parkour</td><td>ballet-1</td><td>ballet-2</td><td>ballet-3</td><td>Ave.</td></tr><tr><td>*DensePose CSE [29]</td><td>56.0</td><td>13.2</td><td>77.2</td><td>85.9</td><td>45.6</td><td>49.0</td><td>64.5</td><td>55.9</td></tr><tr><td>*VIBE+SMPLify [18]</td><td>37.1</td><td>8.2</td><td>70.4</td><td>83.8</td><td>55.4</td><td>53.0</td><td>78.8</td><td>55.2</td></tr><tr><td>LASR [49]</td><td>29.1</td><td>18.1</td><td>56.6</td><td>49.8</td><td>44.5</td><td>47.4</td><td>48.6</td><td>42.0</td></tr><tr><td>ViSER (Ours)</td><td>70.5</td><td>22.5</td><td>80.7</td><td>62.9</td><td>52.7</td><td>56.1</td><td>59.9</td><td>57.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/766ee373c34b5ffd15ea33c7f0788bf95f6c1ce71c385b46be8c8a7ccb82fc17.jpg",
+        "table_caption": [
+            "Table 2: 2D Keypoint transfer accuracy on multiple elephant videos. Best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>inner</td><td>across</td></tr><tr><td>CSE [29]</td><td>55.7</td><td>52.2</td></tr><tr><td>Flow-VCN[48]</td><td>51.1</td><td>41.2</td></tr><tr><td>LASR [49]</td><td>57.8</td><td>-</td></tr><tr><td>ViSER (Ours)</td><td>80.4</td><td>68.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/d8c4d0b4da545631712caf9b91bd19168a613e8044cf52c7d31703de68b62f25.jpg",
+        "table_caption": [
+            "Table 3: 2D Keypoint transfer accuracy on BADJA dataset. Best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>camel</td><td>dog</td><td>cows</td><td>horse</td><td>bear</td><td>Ave.</td></tr><tr><td>CSE [29]</td><td>48.8</td><td>38.6</td><td>63.8</td><td>60.2</td><td>76.6</td><td>57.6</td></tr><tr><td>Flow-VCN [48]</td><td>47.9</td><td>25.7</td><td>60.7</td><td>14.4</td><td>63.8</td><td>42.5</td></tr><tr><td>N-NRSfM[38]</td><td>67.8</td><td>17.9</td><td>70.0</td><td>8.7</td><td>60.2</td><td>44.9</td></tr><tr><td>LASR [49]</td><td>81.9</td><td>65.8</td><td>83.7</td><td>49.3</td><td>85.1</td><td>73.2</td></tr><tr><td>ViSER (Ours)</td><td>80.1</td><td>73.8</td><td>82.9</td><td>76.3</td><td>87.3</td><td>80.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Optimization details We use the AdamW [27] optimizer with a batch of 4 consecutive image pairs. We reconstruct a long video sequence in an incremental manner similar to classic SfM. First, we use an initial set of around 20 consecutive frames to initialize the shape and pixel surface embeddings. The initial set is selected such that the viewpoint coverage is large enough. Then we gradually add in new frames. When a new frame is added, we first apply the 2D cycle loss $L _ { r e p r o j }$ to optimize its articulations, and then jointly optimize all frames with all losses. Empirically, simultaneously optimizing all the video frames produces unstable results of root body poses (or equivalently camera poses). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Athletic Video Reconstruction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            564,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Dataset. To evaluate ViSER on long-videos, we construct an athletic video dataset that is challenging due to loose clothing and unusual body poses. It consists of four videos from DAVIS [31] and three ballet videos. All videos are segmented and manually annotated with keypoints following the MSCOCO format [24]. We only use keypoint annotations for evaluation purposes. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Metrics. Due to the lack of ground-truth 3D data for challenging athletic human videos, we use 2D keypoint transfer as a proxy metric [2, 52]. Given any two frames from a video, the goal is to transfer an annotated 2D keypoint from one frame to another. The accuracy is measured by percentage of correctly transferred keypoints over all T(T-1) pairs of frames in a T-frame video. A transferred keypoint is marked as correct when its distance to the ground-truth annotation is lower than $d _ { t h } = 0 . 2 \\sqrt { | \\boldsymbol { S } | }$ , where $| S |$ is the area of the ground-truth silhouette [2]. In general, a more accurate reconstruction leads to a higher transfer accuracy. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Baselines. To compare with template-based approaches for video human reconstruction, we use VIBE with SMPLify temporal smoothing [18]. To compare with template-free methods, we use LASR [49], which also reconstructs articulated shapes using the same input setting as ours. To transfer keypoints from a reference frame to a target frame, we back-project the annotated keypoint in the reference frame to the canonical surface, and then project the intersected 3D point to the target frame. We also compare against Densepose CSE [29], which produces dense pixel-to-surface correspondences for a given category, but does not produce 3D reconstructions. To transfer keypoints for Densepose CSE, we compute pixelwise surface mappings for both frames and find the best matching w.r.t.geodesic distance on the surface. We further qualitatively compare against a state-ofthe-art human reconstruction method, PiFUHD [36] in Fig. 4, which only produces reconstruction, but not correspondence. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/fbe091753c2f0495ced8cb953c6fbf2f59e8ee2d1da65f5d598e70f3690e086d.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative comparisons for athletic video articulated shape reconstruction. Compared to methods that uses shape and pose priors (VIBE $^ +$ SMPLify and PiFUHD), our method achieves comparable performance for common appearance and poses, and does much better on unusual poses such as break-dancers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/166855c4415d8702b655a7dc42e4809baf2c3b46ebfba6bba2cf6a0b7a6990f9.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative comparisons for elephant shape reconstruction from multiple videos. Notice that ViSER is able to take advantage of multiple videos to improve the category-level shape reconstruction but also reconstruct instance-specific details (as shown in red circles). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
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+    {
+        "type": "image",
+        "img_path": "images/16aec4a0675c4cd1f54df8a58f8265aa7b4db992bddc92c4e4686871ea10a196.jpg",
+        "image_caption": [
+            "Figure 6: Part segmentation results. Colors are determined by hard-assigning vertices to the closest rigid bones. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/8d21ac3a175e69d94bbae955b587e18f58587f3cb36c3a814a1ad4d8dd719e12.jpg",
+        "image_caption": [
+            "Figure 7: Comparison between single video ViSER and multi-video ViSER in terms of reconstructing YTVOS elephants. We find using multiple videos helps reconstructing the body parts that may be occluded in a single video. While single-video ViSER reconstructs a flattened shape and misses the hidden rear leg of the elephant, multiple video ViSER reconstructs a more plausible shape and recovers both the two rear limbs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Results. Fig. 4 shows visual reconstruction results on sample videos and for different techniques. ViSER estimates reconstructions that are more faithful to the input than the baselines, especially when the humans have unusual poses like in the first two rows. The accurate long-range correspondence enables ViSER to reconstruct finer details than LASR that does not explicitly try to estimate longrange correspondences. We summarize quantitative comparisons in Tab. 1. There is a moderate performance gap between ViSER and template-based methods when the input fits the latter, such as parkour with tight clothing and usual pose. Note that the supervised Densepose CSE and OpenPose methods fail on breakdance videos due to the novel pose, and also do not work well on ballet dancers due to loose clothing. As a result, template-based approaches that rely on accurate pose recognition, such as VIBE [18] fails. In contrast. our method does not suffer from such poor out-of-distribution generalization. By establishing long-range correspondences, ViSER achieves higher keypoint transfer accuracy and better 3D reconstruction than LASR. ",
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+    {
+        "type": "text",
+        "text": "4.2 Reconstructing Animals from a Video ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use BADJA [2] to evaluate ViSER on animal videos including camel, cow, dog, bear and horse. Similar to the athletic human video dataset, we compare against template-free methods such as LASR and neural-dense-NRSfM (N-NRSfM) [38]. Similar to LASR and our setup, N-NRSfM learns a video-specific model for object shape, deformation and camera parameters from multi-frame optical flow estimations [8]. We further report performance comparison with dense correspondence methods such as CSE and an optical flow method, VCN. We use the CSE model trained on corresponding animal categories (except that we use the horse model for camel), and the “robust” model of VCN [1], which is the input to our method. As shown in Tab. 3, ViSER achieves better or similar accuracy on all five animal videos compared to LASR and N-NRSfM. While the input optical flow is not robust at estimating long-range correspondences, our method integrates local optical flow to a dense long-range correspondences via a canonical shape, and achieves much better keypoint transfer accuracy. Note that CSE performs well for categories it has been trained on, such as cow, horse and bear, but performs poorly on novel animal categories, such as camel and a novel breed of dog. ",
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+    {
+        "type": "text",
+        "text": "4.3 Multi-video Shape and Correspondence ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We curate a set of seven videos of different elephants from YTVOS [47] for multi-video shape and correspondence recovery. The annotations will be released for further research. We treat multiple videos as a single long video with strong appearance changes and shape variations. In the multi-video setup, We evaluate keypoint transfer accuracy on both the same instance (with video frames) and over different instances (across video frames), as denoted by “inner” and “across”. Quantitative results in Tab. 2 shows that ViSER is more accurate than the baseline methods in both cross-video keypoint transfer and inner-video keypoint transfer by a large margin, without using any keypoint annotations or pre-defined shape templates. Fig. 5 show visual result comparisons. While LASR recovers the visible surfaces in a video, it cannot infer the invisible parts. In contrast, our method is able to take advantage of multiple videos from the same category and produce a much better shape reconstruction. Note that LASR cannot handle multiple videos as it requires optical flow computed between every adjacent frame pairs. ViSER, on the other hand, also uses correspondences via estimated 3D shape, thereby allowing the use of multiple videos even when the optical flow is missing across videos. ",
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+    {
+        "type": "table",
+        "img_path": "images/fe4aea42ac94db76a757c6f5bf941663934cb07e95c778a9a7781c3c8211d1c4.jpg",
+        "table_caption": [
+            "Table 4: Ablation study on keypoint transfer. Best results are in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>break-1</td><td>elephants-inner</td><td>elephants-cross</td></tr><tr><td>Full</td><td>70.5</td><td>80.4</td><td>68.9</td></tr><tr><td>w/o matching loss Lmatching, Eq. (9)</td><td>36.2</td><td>51.3</td><td>42.6</td></tr><tr><td>w/o reprojection loss Lreproj, Eq. (10)</td><td>38.3</td><td>80.1</td><td>62.5</td></tr><tr><td>CSM regression [20]</td><td>47.1</td><td>77.4</td><td>63.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Benefit of Using Multiple Videos. To examine the benefits of using multiple videos, we further compare multi-video ViSER with single-video ViSER, as shown in Fig. 7. We find using multiple videos helps reconstructing the body parts that may be occluded in a single video. ",
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+    {
+        "type": "text",
+        "text": "4.4 Part Discovery and Ablations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Part discovery. ViSER can discover detailed 3D part segmentation without any manual annotation, as shown in Fig. 6. After training either on a collection of videos or a long video, ViSER can segment the 3D shape into meaningful parts, such as the trunk of the elephants and the feet of the dancer. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ablation study. We perform an ablation study on break-1 and elephants, as shown in Tab. 4. Without the contrastive matching loss, the pixel-surface embedding converges to a trivial solution with a significant decrease of accuracy. Removing the re-projection loss leads to much lower keypoint transfer (KPT) accuracy on the breakdance-1 sequence and cross-video KPT accuracy on the elephant videos. Likely the surface reprojection loss plays an important role in learning correct articulation that follows the bottom-up dense keypoint predictions. This may effectively avoid the local minimum issue for the differentiable rendering optimization. Finally, replacing the pixel-surface embedding with direct CSM regression [20] does not reason about distribution of possible matches and results in worse performance. ",
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+    {
+        "type": "text",
+        "text": "Limitations. We find ViSER to be sensitive to the random initialization of network parameters. We run optimization with different random seeds for initializing the network parameters and find some perform considerably worse than the others, due to the convergence to bad local optima. Although in practice, one could spot the convergence to a bad local optimum by visualizing the articulated shapes and re-run the optimization with a different random seed, an automatic method for selecting the best model parameters over different trials is desired. We leave how to make the optimization of ViSER robust for future research. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ViSER also relies on optical flow to kick-start with a reasonable initial shape and pose for learning pixel-surface embeddings. Although recent optical flow models generalize well in many scenarios, they may fail when a video is of low resolution or contains significant motion blur. In such challenging cases, using category shape and pose priors to initialize ViSER would be a promising direction. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have introduced ViSER, a method to reconstruct articulate shapes, dense trajectories, and object parts from monocular videos. ViSER establishes long-range correspondence by matching 2D pixels to a canonical 3D mesh via learned video-specific surface embeddings. Experimental results show that ViSER, without a template shape or keypoint annotations, compares favorably against prior work on challenging human and animal videos. ViSER shows that it could be fruitful to reconstruct articulate shapes for categories in the wild, and we hope to see more work in this direction. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Broader impact. ViSER has many potential applications, e.g., in robotics, AR/VR, and film industry, but may be used for malicious purposes, e.g., producing fake videos or extracting bio-metric information without prior consent. ViSER is only suitable to offline applications as it takes about several hours to process a 80-frame video on one NVIDIA P100 GPU. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            838,
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+    },
+    {
+        "type": "text",
+        "text": "This work was supported by Google Cloud Platform (GCP) awards received from Google and the CMU Argo AI Center for Autonomous Vehicle Research. We thank William T. Freeman and many others from CMU and Google for valuable feedback. ",
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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parse/train/B1exrnCcF7/B1exrnCcF7.md

@@ -0,0 +1,527 @@
+# DISJOINT MAPPING NETWORK FOR CROSS-MODAL MATCHING OF VOICES AND FACES
+
+Yandong Wen†, Mahmoud Al Ismail†, Weiyang $\mathbf { L i u ^ { \ S } }$ , Bhiksha Raj†, Rita Singh† †Carnegie Mellon University §Georgia Institute of Technology yandongw@andrew.cmu.edu, mahmoudi@andrew.cmu.edu, wyliu@gatech.edu
+
+# ABSTRACT
+
+We propose a novel framework, called Disjoint Mapping Network (DIMNet), for cross-modal biometric matching, in particular of voices and faces. Different from the existing methods, DIMNet does not explicitly learn the joint relationship between the modalities. Instead, DIMNet learns a shared representation for different modalities by mapping them individually to their common covariates. These shared representations can then be used to find the correspondences between the modalities. We show empirically that DIMNet is able to achieve better performance than the current state-of-the-art methods, with the additional benefits of being conceptually simpler and less data-intensive. The code is made available at https://github.com/ydwen/DIMNet.
+
+# 1 INTRODUCTION
+
+A person’s face is predictive of their voice. Biologically, the genetic, physical and environmental influences that affect the face also affect the voice. Humans have been shown to be able to associate voices of unknown individuals to pictures of their faces (Kamachi et al., 2003). Humans also show improved ability to memorize and recall voices when previously exposed to pictures of the speaker’s face, but not imposter faces (McAllister et al., 1993; Schweinberger et al., 2007; 2011). Cognitively, studies indicate that neuro-cognitive pathways for voices and faces share common structure (Ellis, 1989), possibly following parallel pathways within a common recognition framework (Belin et al., 2004; 2011). The above studies lend credence to the hypothesis that it may be possible to find associations between voices and faces algorithmically as well. With this in perspective, this paper focuses on the task of devising computational mechanisms for cross-modal matching of voice recordings and images of the speakers’ faces.
+
+The specific problem we look at is the one wherein we have an existing database of samples of people’s voices and images of their faces, and we aim to automatically and accurately determine which voices match to which faces. This problem has seen significant research interest, in particular since the recent introduction of the VoxCeleb corpus (Nagrani et al., 2017), which comprises collections of video and audio recordings of a large number of celebrities. The existing approaches (Nagrani et al., 2018b;a; Kim et al., 2018) have generally attempted to directly relate subjects’ voice recordings and their face images, in order to find the correspondences between the two. Nagrani et al. (2018b) formulates the mapping as a binary selection task: given a voice recording, one must successfully select the speaker’s face from a pair of face images (or the reverse – given a face image, one must correctly select the subject’s voice from a pair of voice recordings). They model the mapping as a neural network that is trained through joint presentation of voices and faces to determine if they belong to the same person. In Kim et al. (2018); Nagrani et al. (2018a), the authors attempt to learn common embeddings (i.e., vector representations) for voices and faces that can be compared to one another to identify associations. The networks that compute the embeddings are also trained through joint presentation of voices and faces, to maximize the similarity of embeddings derived from them if they belong to the same speaker. In all cases, the voice and face are implicitly assumed to directly inform about one another.
+
+In reality, though, it is unclear how much these models capture the direct influence of the voice and face on one another, and how much is explained through implicit capture of higher-level variables such as gender, age, ethnicity etc., which individually predict the two. These higher-level variables, which we will refer to as covariates1 can, in fact, explain much of our ability to match voices to faces (and vice versa) under the previously mentioned “select-from-a-pair” test (where a voice must be used to distinguish the speaker’s face from a randomly-chosen imposter). For instance, simply matching the gender of the voice and the face can result in an apparent accuracy of match of up to $7 5 \%$ in a gender-balanced testing setting. Even in a seemingly less constrained “verification” test, where one must only verify if a given voice matches a given face, matching them based on gender alone can result in an equal error rate of $33 \%$ (Appendix B). Even matching the voice and the face by age (e.g.matching older-looking faces to older-sounding voices) could result in match accuracy that’s significantly better than random.
+
+![](images/f2fea1fb385e890e39632022f9764fe9659b830a573b1ccf71a9c33ee2cb6524.jpg)  
+Figure 1: Overview of the proposed DIMNet and its comparison to the existing approaches. (a) Seeing faces and hearing voices from Nagrani et al. (2018b). (b) Learnable PINs from Nagrani et al. (2018a). (c) Learning face-voice association from Kim et al. (2018). (d) Our proposed DIMNets. DIMNets present a joint voice-face embedding framework via multi-task classification and require no pair construction (i.e., both voices and faces can be input sequentially without forming pairs).
+
+Previous studies (Nagrani et al., 2018b; Kim et al., 2018) attempt to disambiguate the effect of multiple covariates through stratified tests that separate the data by covariate value. The results show that at least some of the learned associations are explained by the covariate, indicating that their learning approaches do utilize the covariate information, albeit only implicitly.
+
+In this paper, we propose a novel framework to learn mappings between voices and faces that do not consider any direct dependence between the two, but instead explicitly exploit their individual dependence on the covariates. We define covariate as the identity-sensitive factors that can simultaneously affect voice and face, e.g. nationality, gender, identity (ID), etc. We do not require the value these factors take to be the same between the training and test set, since what we are learning is the nature of the covariation with the variable in general, not merely the covariation with the specific values the variable takes in the training set. In contrast to existing methods where supervision is provided through the correspondence of voices and faces, our learning framework, Disjoint Mapping Network (DIMNet), obtains supervision from common covariates, applied separately to voices and faces, to learn common embeddings for the two. The comparison between the existing approaches and DIMNets are illustrated in Fig. 1.
+
+DIMNet comprises individual feature learning modules which learn identically-dimensioned features for data from each modality, and a unified input-modality-agnostic classifier that attempts to predict covariates from the learned feature. Data from each modality are presented separately during learning; however the unified classifier forces the feature representations learned from the individual modalities to be comparable. Once trained, the classifier can be removed and the learned feature representations are used to compare data across modalities.
+
+The proposed approach greatly simplifies the learning process and, by considering the modalities individually rather than as coupled pairs, makes much more effective use of the data. Moreover, if multiple covariates are known, they can be simultaneously used for the training through multi-task learning in our framework (see Fig. 2).
+
+Compared to current methods (Nagrani et al., 2018b;a; Kim et al., 2018), DIMNets achieve consistently better performance, indicating that direct supervision through covariates is more effective in these settings. We find that of all the covariates, ID provides the strongest supervision. The results obtained from supervision through other covariates also match what may be expected.
+
+![](images/a34872da4d5a6db32a554692a5476cc9a920252354a0d5df3a85d1f787976797.jpg)  
+Figure 2: Our DIMNet framework. The input training data can be either voice or face, and there is no need for voices and faces to form pairs. Modality switch is to control which embedding network (voice or face) to process the data. While the embeddings are obtained, a multi-task classification network is applied to supervise the learning.
+
+Our contributions are summarized as follows:
+
+• We propose DIMNets, a framework that formulates the problem of cross-modal matching of voices and faces as learning common embeddings for the two through individual supervision from one or more covariates, in contrast to current approaches that attempt to map voices to faces directly. An overview of our framework is given in Fig. 2.   
+• In this framework, we can make full use of multiple kinds of label information (provided by covariates) with a multi-task objective function.   
+• We achieve the state-of-the-art results on multiple tasks. We are also able to isolate and analyze the effect of the individual covariate on the performance.
+
+Moreover, we note that the proposed framework is applicable in any setting where matching of different types of data which have common covariates is required.
+
+# 2 THE PROPOSED FRAMEWORK
+
+Our goal is to learn common vector representations for both voices and faces, that permit them to be compared to one another. In the following sections we first describe how we learn them from their relationship to common covariates. Subsequently, we describe how we will use them for comparison of voices to faces.
+
+# 2.1 LEVERAGING COVARIATES TO LEARN EMBEDDINGS
+
+The relationship between voices and faces is largely predicted by covariates – factors that individually relate to both the voice and the face. To cite a trivial example, a person’s gender relates their voice to their face: male subjects will have male voices and faces, while female subjects will have female voices and faces. More generally, many covariates may be found that relate to both voice and face (Lippert et al., 2017).
+
+Our model attempts to find common representations for both face images and voice recordings by leveraging their relationship to these covariates (rather than to each other). We will do so by attempting to predict covariates from voice and face data in a common embedding space, such that the derived embeddings from the two types of data can be compared to one another.
+
+Let $\nu$ represent a set of voice recordings, and $\mathcal { F }$ represent a set of face images. Let $\mathcal { C }$ be the set of covariates we consider. For the purpose of this paper, we assume that all covariates are discrete valued (although this is not necessary). Every voice recording in $\nu$ and every face in $\mathcal { F }$ can be related to each of the covariates in $\mathcal { C }$ . For every covariate $C \in { \mathcal { C } }$ we represent the value of that covariate for any voice recording $v$ as $C ( v )$ , and similarly the value of the covariate for any face $f$ as $C ( f )$ . For example, $C$ could be ID, gender, or nationality. When $C$ is ID, $C ( v )$ and $C ( f )$ are the $\mathrm { I D }$ of voice $v$ and face $f$ , respectively.
+
+Let $F _ { v } ( v ; \theta _ { v } ) : v \mapsto \mathbb R ^ { d }$ be a voice embedding function with parameters $\theta _ { v }$ that maps any voice recording $v$ into a $d$ -dimensional vector. Similarly, let $F _ { f } ( f ; \theta _ { f } )$ be a face embedding function that maps any face $f$ into a $d$ -dimensional vector. We aim to learn $\theta _ { v }$ and $\theta _ { f }$ such that the embeddings of the voice and face for any person are comparable.
+
+For each covariate $C \in { \mathcal { C } }$ we define a classifier $H _ { C } ( x ; \phi _ { C } )$ with parameter $\phi _ { C }$ , which assigns any input $x \in \mathbb { R } ^ { d }$ to one of the values taken by $C$ . The classifier $H _ { C } ( \cdot )$ is agnostic to which modality its input $x$ was derived from; thus, given an input voice $v$ , it operates on features $F _ { v } ( v ; \theta _ { v } )$ derived from the voice, whereas given a face $f$ , it operates on $F _ { f } ( f ; \theta _ { f } )$ .
+
+For each $v$ (or $f$ ) and each covariate $C$ , we define a loss $L ( H _ { C } ( F _ { v } ( v ; \theta _ { v } ) ; \phi _ { C } ) , C ( v ) )$ between the covariate predicted by $H _ { C } ( . )$ and the true value of the covariate for $v$ , $C ( v )$ . We can now define a total loss $\mathcal { L }$ over the set of all voices $\nu$ and the set of all faces $\mathcal { F }$ , over all covariates as
+
+$$
+\begin{array} { r l } { \mathcal { L } ( \theta _ { v } , \theta _ { f } , \{ \phi _ { C } \} ) } & { = \displaystyle \sum _ { C \in \mathcal { C } } \lambda _ { C } \bigg ( \sum _ { v \in \mathcal { V } } L ( H _ { C } ( F _ { v } ( v ; \theta _ { v } ) ; \phi _ { C } ) , C ( v ) ) } \\ & { \quad \quad \quad + \displaystyle \sum _ { f \in \mathcal { F } } L ( H _ { C } ( F _ { f } ( f ; \theta _ { f } ) ; \phi _ { C } ) , C ( f ) ) \bigg ) } \end{array}
+$$
+
+In order to learn the parameters of the embedding functions, $\theta _ { f }$ and $\theta _ { v }$ , we perform the following optimization.
+
+$$
+\theta _ { v } ^ { * } , \theta _ { f } ^ { * } = \arg \operatorname* { m i n } _ { \theta _ { v } , \theta _ { f } } \operatorname* { m i n } _ { \left\{ \phi _ { C } \right\} } \mathcal { L } ( \theta _ { v } , \theta _ { f } , \left\{ \phi _ { C } \right\} )
+$$
+
+# 2.2 DISJOINT MAPPING NETWORKS
+
+In DIMNet, we instantiate $F _ { v } ( v ; \theta _ { v } )$ , $F _ { f } ( f ; \theta _ { f } )$ and $H _ { C } ( x ; \phi _ { C } )$ as neural networks. Fig. 2 shows the network architecture we use to train our embeddings. It comprises three components. The first, labelled Voice Network in the figure, represents $F _ { v } ( v ; \theta _ { v } )$ and is a neural network that extracts $d$ - dimensional embeddings of the voice recordings. The second, labelled Face Network in the figure, represents $F _ { f } ( f ; \theta _ { f } )$ and is a network that extracts $d$ -dimensional embeddings of face recordings. The third component, labelled Classification Networks in the figure, is a bank of one or more classification networks, one per covariate considered. Each of the classification networks operates on the $d$ -dimensional features output by the embedding networks to classify one covariate, e.g.gender.
+
+The training data comprise voice recordings and face images. Voice recordings are sent to the voiceembedding network, while face images are sent to the face-embedding network. This switching operation is illustrated by the switch at the input in Fig.2. In either case, the output of the embedding network is sent to the covariate classifiers.
+
+As can be seen, at any time the system either operates on a voice, or on a face, i.e.the operations on voices and faces are disjoint. During the learning phase too, the updates of the two networks are disjoint – loss gradients computed when the input is voice only update the voice network, while loss gradients derived from face inputs update the face network, while both contribute to updates of the classification networks.
+
+In our implementation, specifically, $F _ { v } ( \cdot )$ is a convolutional neural network that operates on MelSpectrographic representations of the speech signal. The output of the final layer is pooled over time to obtain a final $d$ -dimensional representation. $F _ { f } ( \cdot )$ is also a convolutional network with a pooled output at the final layer that produces a $d$ -dimensional representation of input images. The classifiers $H _ { C } ( \cdot )$ are all simple multi-class logistic-regression classifiers comprising a single softmax layer.
+
+Finally, in keeping with the standard paradigms for training neural network systems, we use the cross-entropy loss to optimize the networks. Also, instead of the optimization in Eq. 2, the actual optimization performed is the one below. The difference is inconsequential.
+
+$$
+\theta _ { v } ^ { * } , \theta _ { f } ^ { * } , \{ \phi _ { C } ^ { * } \} = \operatorname * { a r g m i n } _ { \theta _ { v } , \theta _ { f } , \{ \phi _ { C } \} } \mathcal { L } ( \theta _ { v } , \theta _ { f } , \{ \phi _ { C } \} )
+$$
+
+# 2.3 TRAINING THE DIMNET
+
+All parameters of the network are trained through backpropagation, using stochastic gradient descent. During training, we construct the minibatches with a mixture of speech segments and face images, as the network learns more robust cross-modal features with mixed inputs. Taking voice as an example, we compute the voice embeddings using $F _ { v } ( v ; \theta _ { v } )$ , and obtain the losses using classifiers $H _ { C } { \bar { ( } } \cdot { \bar { ) } }$ for all the covariates. We back-propagate the loss gradient to update the voice network as well as the covariate classifiers. The same procedure is also applied to face data: the backpropagated loss gradients are used to update the face network and the covariate classifiers. Thus, the embedding functions are learned using the data from their modalities individually, while the classifiers are learned using data from all modalities.
+
+# 2.4 USING THE EMBEDDINGS
+
+Once trained, the embedding networks $F _ { v } ( v ; \theta _ { v } )$ and $F _ { f } ( f ; \theta _ { f } )$ can be used to extract embeddings from any voice recording or face image. Given a voice recording $v$ and a face image $f$ , we can now compute a similarity between the two through the cosine similarity $\begin{array} { r } { S ( v , f ) = \frac { \boldsymbol { F } _ { v } ^ { \top } \boldsymbol { F } _ { f } } { \Vert \boldsymbol { F } _ { v } \Vert _ { 2 } \Vert \boldsymbol { F } _ { f } \Vert _ { 2 } } } \end{array}$ . We can employ this similarity to evaluate the match of any face image to any voice recording. This enables us, for instance, to attempt to rank a collection of faces $f _ { 1 } , \cdots , f _ { K }$ in order of estimated match to a given voice recording $v$ , according to $S ( v , f _ { i } )$ , or conversely, to rank a collection of voices $v _ { 1 } , \cdots , v _ { K }$ according to their match to a face $f$ , on order of decreasing $S ( v _ { i } , f )$ .
+
+# 3 EXPERIMENTS
+
+We ran experiments on matching voices to faces, to evaluate the embeddings derived by DIMNets.   
+The details of the experiments are given below and Appendix A.
+
+Datasets. Our experiments were conducted on the Voxceleb (Nagrani et al., 2017) and VGGFace (Parkhi et al., 2015) datasets, which are specified in appendix A.1. We use the intersection of the two datasets, i.e.subjects who figure in both corpora, for our final corpus, which thus includes 1,225 IDs with 667 males and 558 females from 36 nationalities. The data are split into train/validation/test sets, following the settings in Nagrani et al. (2018b). Details can be found in Appendix A.1. We use ID, gender and nationality as our covariates, all of which are provided by the datasets. Separated data preprocessing pipelines are employed to audio segments and face images (see Appendix A.2).
+
+Training. The detailed network configurations are elaborated in appendix A.3. Note that the classification networks are single-layer softmax units with as many outputs as the number of unique values the class can take (2 for gender, 32 for nationalities, and 924 for IDs in our case). The networks are trained to minimize the cross entropy loss, following the typical settings of stochastic gradient descent (SGD) in appendix A.3
+
+Testing. We use the following protocols for evaluation:
+
+• 1:2 Matching. Here, we are given a probe input from one modality (voice or face), and a gallery of two inputs from the other modality (face or voice), including one that belongs to the same subject as the probe, and another of an “imposter” that does not match the probe. The task is to identify which entry in the gallery matches the probe. We report performance in terms of matching accuracy – namely what fraction of the time we correctly identify the right instance in the gallery.
+
+To minimize the influence of random selection, we construct as many testing instances as possible through exhaustive enumeration all positive matched pairs (of voice and face). To each pair, we include a randomly drawn imposter in the gallery. We thus have a total of 4,678,897 trials in the validation set, and 6,780,750 trials in the test set.
+
+• 1:N Matching. This is the same as the 1:2 matching, except that the gallery now includes $N - 1$ imposters. Thus, we must now identify which of the $N$ entries in the gallery matches the probe. Here too results are reported in terms of matching accuracy. We use the same validation and test sets as the 1:2 case, by augmenting each trial with $N - 2$ additional imposters. So the number of trials in validation and test sets is the same as earlier.
+
+• Verification. We are given two inputs, one a face, and another a voice. The task is to determine if they are matched, i.e.both belong to the same subject. In this problem setting the similarity between the two is compared to a threshold to decide a match. The threshold can be adjusted to trade off false rejections $( F _ { R } )$ , i.e.wrongly rejecting true matches, with false alarms $( F _ { A } )$ , i.e.wrongly accepting mismatches. We report results in terms of equal error rate, i.e.when $F _ { R } =$ $F _ { A }$ . We construct our validation and test sets from those used for the 1:2 matching tests, by separating each trial into two, one comprising a matched pair, and the other a mismatched pair. Thus, our validation and test sets are exactly twice as large as those for the 1:2 test.
+
+• Retrieval. The gallery comprises a large number of instances, one or more of which might match the probe. The task is to order the gallery such that the entries in the gallery that match the probe lie at the top of the ordering. Here, we report performance in terms of Mean Average Precision (MAP) (Manning et al., 2008). Here we use the entire collection of 58,420 test faces as the gallery for each of our 21,799 test voices, when retrieving faces from voices. For the reverse (retrieving voices from faces), the numbers are reversed.
+
+Each result is obtained by averaging the performances of 5 models, which are individually trained.
+
+Covariates in Training and Testing. We use the three covariates provided in the dataset, namely identity (I), gender (G), and nationality (N) for our experiments. The treatment of covariates differs for training and test.
+
+• Training. For training, supervision may be provided by any set of (one two or three) covariates. We consider all combinations of covariates, I, G, N, (I,G), (I,N), (G,N) and (I,G,N). Increasing the number of covariates effectively increases the supervision provided to training. All chosen covariates were assigned a weight of 1.0. • Testing. As explained in Appendix B, simply recognizing a covariate such as gender can result in seemingly significant matching performance. For instance, just recognizing the subjects’ gender from their voice and images can result in a $33 \%$ EER for verification, and $2 5 \%$ error in matching for the $1 : 2$ tests. In order to isolate the effect of covariates on performance hence we also stratify our test data by them. Thus we construct 4 testing groups based on the covariates, including the unstratified (U) group, stratified by gender (G), stratified by nationality $( \mathrm { N } )$ , and stratified by gender and nationality (G, N). In each group the test set itself is separated into multiple strata, such that for all instances within any stratum the covariate values are the same.
+
+# 3.1 CROSS-MODAL MATCHING
+
+In this section we report results on the 1:2 and $1 { : } N$ matching tests. In order to ensure that the embedding networks do indeed leverage on accurate modelling of covariates, we first evaluate the classification accuracy of the classification networks for the covariates themselves. Table 1 shows the results.
+
+<table><tr><td rowspan="2">method</td><td colspan="2">gender classification</td><td colspan="2">nationalityclassification</td></tr><tr><td>voice</td><td>face</td><td>voice</td><td>face</td></tr><tr><td>DIMNet-I</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>DIMNet-G</td><td>97.48</td><td>99.22</td><td>-</td><td>-</td></tr><tr><td>DIMNet-N</td><td>=</td><td>=</td><td>74.86</td><td>60.13</td></tr><tr><td>DIMNet-IG</td><td>97.70</td><td>99.42</td><td>-</td><td>-</td></tr><tr><td>DIMNet-IN</td><td>=</td><td>=</td><td>74.17</td><td>60.27</td></tr><tr><td>DIMNet-GN</td><td>97.59</td><td>99.06</td><td>74.62</td><td>60.50</td></tr><tr><td>DIMNet-IGN</td><td>97.69</td><td>99.15</td><td>74.37</td><td>59.88</td></tr></table>
+
+Table 1: Acc. $( \%$ ) of covariate prediction.
+
+The rows of the table show the covariates used to supervise the learning. Thus, for instance, the row labelled “DIMNet-I” shows results obtained when the networks have been trained using ID alone as covariate, the row labelled “DIMNet-G” shows results when supervision is provided by gender, “DIMNetIG” has been trained using ID and gender, etc.
+
+The columns of the table show the specific covariate being evaluated. Since the identities of subjects in the training and test set do not overlap, we are unable to evaluate the accuracy of ID classification. Note that we can only test the accuracy of the classification network for a covariate if it has been used in the training. Thus, classification accuracy for gender can be evaluated for DIMNet-G, DIMNetGN and DIMNet-IGN, while that for nationality can be evaluated for DIMNet-N, DIMNet-GN and DIMNet-IGN.
+
+The results in Table 1 show that gender is learned very well, and in all cases gender recognition accuracy is quite high. Nationality, on the other hand, is not a well-learned classifier, presumably because the distribution of nationalities in the data set is highly skewed (Nagrani et al., 2018b), with nearly $65 \%$ of all subjects belonging to the USA. It is to be expected therefore that nationality as a covariate will not provide sufficient supervision to learn good embeddings.
+
+1:2 matching. Table 2 shows the results for the 1:2 matching tests. In the table, the row labelled “SVHF-Net” gives results obtained with the model of Nagrani et al. (2018b).
+
+The columns are segregated into two groups, one labelled “voice face” and the other labelled “face voice”. In the former, the probe is a voice recording, while the gallery comprises faces. In the later the modalities are reversed. Within each group the columns represent the stratification of the test set. “U” represents test sets that are not stratified, and include the various covariates in the same proportion that they occur in the overall test set. The columns labelled “G” and “N” have been stratified by gender and nationality, respectively, while the column $\mathbf { \ddot { G } }$ , N” represents data that have been stratified by both gender and nationality. In the stratified tests, we have ensured that all data within a test instance have the same value for the chosen covariate. Thus, for instance, in a test instance for voice face in the “G” column, the voice and both faces belong to the same gender. This does not reduce the overall number of test instances, since it only requires ensuring that the gender of the imposter matches that of the probe instance.
+
+<table><tr><td rowspan="2">method</td><td colspan="4">voice→face (ACC%)</td><td colspan="4">face→voice (ACC %)</td></tr><tr><td>U</td><td>G</td><td>N</td><td>G,N</td><td>U</td><td>G</td><td>N</td><td>G,N</td></tr><tr><td>SVHF-Net</td><td>81.00</td><td>63.90</td><td>-</td><td>-</td><td>79.50</td><td>63.40</td><td>-</td><td>=</td></tr><tr><td>DIMNet-I</td><td>83.45±0.42</td><td>70.91±0.56</td><td>81.97±0.51</td><td>69.89±0.78</td><td>83.52±0.45</td><td>71.78±0.55</td><td>82.41±0.48</td><td>70.90±0.81</td></tr><tr><td>DIMNet-G</td><td>72.90±0.55</td><td>50.32±0.70</td><td>71.92±0.51</td><td>50.21±0.65</td><td>72.47±0.54</td><td>50.48±0.71</td><td>72.15±0.54</td><td>50.61±0.68</td></tr><tr><td>DIMNet-N</td><td>57.53±0.45</td><td>55.33±0.67</td><td>53.04±0.43</td><td>51.96±0.59</td><td>56.20±0.43</td><td>54.34±0.61</td><td>53.90±0.44</td><td>51.97±0.57</td></tr><tr><td>DIMNet-IG</td><td>84.12±0.44</td><td>71.32±0.60</td><td>82.65±0.57</td><td>70.39±0.80</td><td>84.03±0.39</td><td>71.65±0.60</td><td>82.96±0.49</td><td>70.78±0.47</td></tr><tr><td>DIMNet-IN</td><td>82.95±0.40</td><td>70.04±0.67</td><td>81.04±0.55</td><td>68.59±0.76</td><td>82.86±0.35</td><td>70.91±0.59</td><td>81.91±0.52</td><td>70.22±0.77</td></tr><tr><td>DIMNet-GN</td><td>75.92±0.42</td><td>56.66±0.55</td><td>72.94±0.48</td><td>53.48±0.73</td><td>73.78±0.69</td><td>54.90±0.54</td><td>72.63±0.48</td><td>53.45±0.85</td></tr><tr><td>DIMNet-IGN</td><td>83.73±0.53</td><td>70.76±0.34</td><td>81.75±0.48</td><td>69.17±0.71</td><td>83.63±0.66</td><td>71.42±0.49</td><td>82.50±0.43</td><td>70.46±0.62</td></tr></table>
+
+Table 2: Performance comparison of 1:2 matching for models trained using different sets of covariates.
+
+We make several observations. First, DIMNet-I performs better than SVHF-Net, improving the accuracies by $2 . 4 5 \% - 4 . 0 2 \%$ for the U group, and $7 . 0 1 \% 8 - 8 . 3 8 \%$ for the G group. It shows that mapping voices and faces to their common covariates is an effective strategy to learn representations for cross-modal matching.
+
+Second, DIMNet-I produces significantly better embeddings that DIMNet-G and DIMNet-N, highlighting the rather unsurprising fact that ID provides the more useful information than the other two covariates. In particular, DIMNet-G respectively achieves $7 2 . 9 0 \%$ and $7 2 . 4 7 \%$ for voice to face and face to voice matching using only gender as a covariate. This verifies our hypothesis that we can achieve almost $7 5 \%$ matching accuracy by only using the gender. These numbers also agree with the performance expected from the numbers in Table 1 and the analysis in Appendix B. As expected, nationality as a covariate does not provide as good supervision as gender. DIMNet-IG is marginally better than DIMNet-I, indicating that gender supervision provides additional support over ID alone.
+
+Third, we note that while DIMNet-I is able to achieve good performance on the dataset stratified by gender, DIMNet-G only achieves random performance. The performance achieved by DIMNet-G on the U dataset is hence completely explained by gender matching. Once again, the numbers match our expectations (Appendix B).
+
+1:N matching. We also experiment for $N > 2$ . Unlike SVHF-Net (Nagrani et al., 2018b) that needs to train different models for different $N$ in this setting, we use the same model for different $N$ . The results in Fig. 3 shows accuracy as a function of $N$ for various models.
+
+All the results in Fig. 3 are consistent with Table 2. As expected, the performance of all methods degrades with increasing $N$ . In general, DIMNets that use ID as supervision outperform SVHF-Net by a considerable margin, showing that DIMNets are able to make best use of the ID information. We obtain the best results when both ID and gender are used as supervision covariates. However, The results obtained using only gender information as covariate is much worse, which is also consistent with our analysis in Appendix B.
+
+![](images/b79c521c670a5cd138cb6b223592e12b6368c3df40abf1623769d7cd302d4f1f.jpg)  
+Figure 3: Performance of $1 { : } N$ matching
+
+# 3.2 CROSS-MODAL VERIFICATION
+
+For verification, we need to determine whether an audio segment and a face image are from the same ID or not. We report the equal error rate (EER) for verification in Table 3.
+
+In general, DIMNets that use ID as a covariate achieve an EER of about $2 5 \%$ , which is considerably lower than the $33 \%$ expected if the verification were based on gender matching alone. The results in Table 3 show that using both gender and ID information as covariates can further improve the performance over using ID alone, well validating the superiority of our multi-task learning framework. Using proper combination of covariates is crucial to the performance. ID is arguably the most effective covariate supervision. More interestingly, nationality is seen to be an ineffective covariate, while gender alone as a covariate produces results that well matches our expectation.
+
+Table 3: Verification results.   
+
+<table><tr><td rowspan="2">method</td><td colspan="4">verification (EER %)</td></tr><tr><td>U</td><td>G</td><td>N</td><td>G,N</td></tr><tr><td>DIMNet-I</td><td>24.95±0.20</td><td>34.95±0.45</td><td>25.92±0.68</td><td>35.74±0.87</td></tr><tr><td>DIMNet-G</td><td>34.86±0.11</td><td>49.69±0.24</td><td>35.13±0.36</td><td>49.67±0.51</td></tr><tr><td>DIMNet-N</td><td>45.89±0.39</td><td>46.97±0.55</td><td>47.89±0.82</td><td>48.87±1.14</td></tr><tr><td>DIMNet-IG</td><td>24.56±0.23</td><td>34.84±0.41</td><td>25.54±0.65</td><td>35.73±0.79</td></tr><tr><td>DIMNet-IN</td><td>25.54±0.18</td><td>36.22±0.40</td><td>27.25±0.72</td><td>37.39±0.79</td></tr><tr><td>DIMNet-GN</td><td>33.28±0.52</td><td>46.65±0.16</td><td>34.77±0.26</td><td>48.08±0.52</td></tr><tr><td>DIMNet-IGN</td><td>25.00±0.19</td><td>35.76±0.36</td><td>26.80±0.69</td><td>37.30±0.74</td></tr></table>
+
+![](images/17bc51c0aed33d37f56fca7771459bb2110f6a95cd06e7d31347c528ba3a5201.jpg)  
+Figure 4: Visualization of voice and face embeddings using multi-dimensional scaling Wickelmaier (2003) . The left panel shows subjects from the training set, while the right panel is from the test set.
+
+# 3.3 CROSS-MODAL RETRIEVAL
+
+We also perform retrieval experiments using voice or face as query. Table 4 lists the mean average precision (mAP) of the retrieval for various models.
+
+The columns in the table represent the covariate being retrieved. Thus, for example, in the “ID” column, the objective is to retrieve
+
+Table 4: Retrieval performance (mAP).   
+
+<table><tr><td rowspan="2">method</td><td colspan="3">voice →face(mAP%)</td><td colspan="3">face→voice (mAP %)</td></tr><tr><td>ID</td><td>gender</td><td>nationality</td><td>ID</td><td>gender</td><td>nationality</td></tr><tr><td>Random</td><td>0.58</td><td>52.61</td><td>40.70</td><td>0.55</td><td>52.60</td><td>40.69</td></tr><tr><td>DIMNet-I</td><td>4.25±0.11</td><td>89.57±0.35</td><td>43.26±0.16</td><td>4.17±0.10</td><td>88.50±0.37</td><td>43.68±0.20</td></tr><tr><td>DIMNet-G</td><td>1.07±0.07</td><td>97.84±0.58</td><td>41.56±0.23</td><td>1.15±0.12</td><td>97.15±0.62</td><td>41.97±0.20</td></tr><tr><td>DIMNet-N</td><td>1.24±0.13</td><td>56.99±0.32</td><td>45.69±0.65</td><td>1.03±0.10</td><td>56.90±0.34</td><td>49.30±0.57</td></tr><tr><td>DIMNet-IG</td><td>4.42±0.12</td><td>93.10±0.45</td><td>43.22±0.14</td><td>4.23±0.09</td><td>92.16±0.42</td><td>43.86±0.17</td></tr><tr><td>DIMNet-IN</td><td>3.94±0.11</td><td>89.72±0.39</td><td>43.95±0.69</td><td>3.99±0.14</td><td>88.39±0.39</td><td>45.93±0.66</td></tr><tr><td>DIMNet-GN</td><td>1.89±0.11</td><td>95.89±0.44</td><td>45.20±0.63</td><td>1.64±0.11</td><td>93.95±0.43</td><td>48.39±0.54</td></tr><tr><td>DIMNet-IGN</td><td>4.07±0.09</td><td>92.30±0.57</td><td>44.10±0.62</td><td>4.05±0.09</td><td>91.31±0.61</td><td>45.82±0.59</td></tr></table>
+
+gallery items with the same ID as the query, whereas in the “gender” column the objective is to retrieve the same gender.
+
+We note that ID-based DIMNets produce the best features for retrieval, with the best performance obtained with DIMNet-IG. Also, as may be expected, the covariates used in training result in the best retrieval of that covariate. Thus, DIMNet-G achieves an mAP of nearly $98 \%$ on gender, though on retrieval of ID it is very poor. As in other experiments, nationality remains a poor covariate in general. Compared to gender (2 classes) and nationality (unbalanced 28 classes), retrieving ID is a challenging problem given the large amount of identities (182 classes). The significant and consistent improvements over chance-level results show that the DIMNet models do learn some useful associations between voices and faces.
+
+3.4 COMPARISONS TO THE CURRENT STATE-OF-THE-ART  
+Table 5: AUCs $( \% )$ of DIMNets under different testing groups.   
+
+<table><tr><td></td><td colspan="5">Seen-Heard</td><td colspan="5">Unseen-Unheard</td></tr><tr><td></td><td>U</td><td>G</td><td>N</td><td>A</td><td>G,N,A</td><td>U</td><td>G</td><td>N</td><td>A</td><td>G,N,A</td></tr><tr><td>Nagrani et al. (2018a)</td><td>87.0</td><td>74.2</td><td>85.9</td><td>86.6</td><td>74.0</td><td>78.5</td><td>61.1</td><td>77.2</td><td>74.9</td><td>58.8</td></tr><tr><td>DIMNet-I</td><td></td><td></td><td></td><td>95.1±0.23 90.8±0.25 93.4±0.15 95.2±0.11 88.9±0.21</td><td></td><td>82.5±0.12 71.0±0.33 81.1±0.10 77.7±0.14 62.8±0.36</td><td></td><td></td><td></td><td></td></tr><tr><td>DIMNet-IG</td><td></td><td></td><td>94.7±0.23 89.8±0.22 93.2±0.13 94.8±0.12 87.8±0.18</td><td></td><td></td><td>83.2±0.11 71.2±0.37 81.9±0.18 78.0±0.13 62.8±0.39</td><td></td><td></td><td></td><td></td></tr></table>
+
+We compare DIMNet with the state of the art (Nagrani et al., 2018a). The results are reported in Table 5. Note that it is fair comparison because the DIMNet models in this section are trained with and evaluated on the same released datasets in Nagrani et al. (2018a). Detailed statistics and splits of the dataset can be found in Appendix A.1. There are two evaluation protocols, including SeenHeard and Unseen-Unheard scenarios. The identities of the training and testing set have overlaps in Seen-Heard scenario (closed-set), while they are fully disjoint in Unseen-Unheard scenario (openset). For each scenario, there are 5 testing groups based on the covariates, including the unstratified group (U), group, stratified by gender (G), stratified by nationality (N), stratified by age (A), and stratified by (G, N, A). We compute the area under the curve (AUC) for different testing groups.
+
+It is clear that DIMNets produce better embeddings than Nagrani et al. (2018a) for pair-wise verification on both seen-heard and unseen-unheard scenarios. Specifically, DIMNets achieve $8 \% - 1 5 \%$ absolute and $3 \%$ - $10 \%$ absolute improvements on seen-heard and unseen-unheard test set, respectively.
+
+Compared to DIMNet-IG, DIMNet-I performs better on the seen-heard test set while DIMNet-IG is better on the unseen-unheard test set. It implies that introducing useful covariates improves the generalization capability of DIMNet.
+
+# 4 DISCUSSIONS AND CONCLUDING REMARKS
+
+We have proposed that it is possible to learn common embeddings for multi-modal inputs, particularly voices and faces, by mapping them individually to common covariates. In particular, the proposed DIMNet architecture is able to extract embeddings for both modalities that achieves consistently better performance than the methods that directly map faces to voices.
+
+The approach also provides us the ability to tease out the influence of each of the covariates of voice and face data, in determining their relation. The results show that the strongest covariate, not unexpectedly, is ID. The results also indicate that prior results by other researchers who have attempted to directly match voices to faces may perhaps not be learning any direct relation between the two, but implicitly learning about the common covariates, such as ID, gender, etc.
+
+Our experiments also show that although we have achieved possibly the best reported performance on this task, thus far, the performance is not anywhere close to prime-time. In the $1 : N$ matching task, performance degrades rapidly with increasing $N$ , indicating a rather poor degree of true match.
+
+To better understand the problem, we have visualized the learned embeddings from DIMNet-I in Fig. 4 to provide more insights. The visualization method we used is multi-dimensional scaling (MDS) (Wickelmaier, 2003), rather than the currently more popular t-SNE (van der Maaten & Hinton, 2008). This is because MDS tends to preserve distances and global structure, while t-SNE attempts to retain statistical properties and highlights clusters, but does not preserve distances.
+
+From Fig. 4, we immediately notice that the voice and face data for a subject are only weakly proximate. While voice and face embeddings for a speaker are generally relatively close to each other, they are often closer to other subjects. Interestingly, the genders separate (even though gender has not been used as a covariate for this particular network), showing that at least some of the natural structure of the data is learned. Fig. 4 shows embeddings obtained from both training and test data. We can observe similar behaviors in both, showing that the the general characteristics observed are not just the outcome of overfitting to training data. The visualization in Fig. 4 also shows that there is still significant room for improvement. For example, it may be possible to force compactness of the distributions of voice and face embeddings through modified loss functions such as the center loss (Wen et al., 2016) or angular softmax loss (Liu et al., 2016; 2017a;b), or through an appropriately designed loss function that is specific to this task.
+
+# REFERENCES
+
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+Pascal Belin, Shirley Fecteau, and Catherine Bedard. Thinking the voice: neural correlates of voice perception. Trends in cognitive sciences, 8(3):129–135, 2004. 1   
+Joon Son Chung and Andrew Zisserman. Out of time: automated lip sync in the wild. In Asian Conference on Computer Vision, pp. 251–263. Springer, 2016. 11   
+Andrew W Ellis. Neuro-cognitive processing of faces and voices. In Handbook of research on face processing, pp. 207–215. Elsevier, 1989. 1   
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015. 11   
+Miyuki Kamachi, Harold Hill, Karen Lander, and Eric Vatikiotis-Bateson. Putting the face to the voice’: Matching identity across modality. Current Biology, 13(19):1709–1714, 2003. 1   
+Changil Kim, Hijung Valentina Shin, Tae-Hyun Oh, Kaspar Alexandre, Mohamed Elgharib, and Wojciech Matusik. On learning associations of faces and voices. arXiv preprint arXiv:1805.05553, 2018. 1, 2
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012. 11
+
+Christoph Lippert, Riccardo Sabatini, M Cyrus Maher, Eun Yong Kang, Seunghak Lee, Okan Arikan, Alena Harley, Axel Bernal, Peter Garst, Victor Lavrenko, et al. Identification of individuals by trait prediction using whole-genome sequencing data. Proceedings of the National Academy of Sciences, 114(38):10166–10171, 2017. 3
+
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+Weiyang Liu, Yan-Ming Zhang, Xingguo Li, Zhiding Yu, Bo Dai, Tuo Zhao, and Le Song. Deep hyperspherical learning. In NIPS, 2017b. 9
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+Christopher D. Manning, Prabhakar Raghavan, and Hinrich Schutze. Evaluation in information ¨ retrieval. In Introduction to Information Retrieval, chapter 8, pp. 159–160. Cambridge University Press, 2008. 6
+
+Hunter A McAllister, Robert HI Dale, Norman J Bregman, Allyssa McCabe, and C Randy Cotton. When eyewitnesses are also earwitnesses: Effects on visual and voice identifications. Basic and Applied Social Psychology, 14(2):161–170, 1993. 1
+
+A. Nagrani, J. S. Chung, and A. Zisserman. Voxceleb: a large-scale speaker identification dataset. In INTERSPEECH, 2017. 1, 5
+
+Arsha Nagrani, Samuel Albanie, and Andrew Zisserman. Learnable pins: Cross-modal embeddings for person identity. arXiv preprint arXiv:1805.00833, 2018a. 1, 2, 8, 11
+
+Arsha Nagrani, Samuel Albanie, and Andrew Zisserman. Seeing voices and hearing faces: Crossmodal biometric matching. arXiv preprint arXiv:1804.00326, 2018b. 1, 2, 5, 6, 7, 11
+
+Omkar M Parkhi, Andrea Vedaldi, Andrew Zisserman, et al. Deep face recognition. In BMVC, 2015. 5
+
+Daniel Povey, Arnab Ghoshal, Gilles Boulianne, Lukas Burget, Ondrej Glembek, Nagendra Goel, Mirko Hannemann, Petr Motlicek, Yanmin Qian, Petr Schwarz, et al. The kaldi speech recognition toolkit. In IEEE 2011 workshop on automatic speech recognition and understanding. IEEE Signal Processing Society, 2011. 11
+
+Stefan R Schweinberger, David Robertson, and Jurgen M Kaufmann. Hearing facial identities. ¨ Quarterly Journal of Experimental Psychology, 60(10):1446–1456, 2007. 1
+
+Stefan R Schweinberger, Nadine Kloth, and David MC Robertson. Hearing facial identities: Brain correlates of face–voice integration in person identification. Cortex, 47(9):1026–1037, 2011. 1
+
+Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008. 9
+
+Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao. A discriminative feature learning approach for deep face recognition. In ECCV, 2016. 9
+
+Yandong Wen, Mahmoud Al Ismail, Bhiksha Raj, and Rita Singh. Optimal strategies for matching and retrieval problems by comparing covariates. arXiv preprint arXiv:1807.04834, 2018. 12
+
+Florian Wickelmaier. An introduction to mds. Sound Quality Research Unit, Aalborg University, Denmark, 46(5), 2003. 8, 9
+
+Kaipeng Zhang, Zhanpeng Zhang, Zhifeng Li, and Yu Qiao. Joint face detection and alignment using multitask cascaded convolutional networks. IEEE Signal Processing Letters, 23(10):1499–1503, 2016. 11
+
+# APPENDIX A EXPERIMENTAL DETAILS
+
+# A.1 DATASET
+
+The Voxceleb dataset consists of 153,516 audio segments from 1,251 speakers. Each audio segment is taken from an online video clip with an average duration of 8.2 seconds. For the face dataset, we used a manually filtered version of VGGFace. After face detection, there remain 759,643 images from 2,554 subjects. The data are split into train/validation/test sets, following the settings in Nagrani et al. (2018b). Details are shown in Table 6
+
+<table><tr><td># of samples</td><td>train</td><td>validation</td><td>test</td><td>total</td></tr><tr><td>speech segments</td><td>112,697</td><td>14,160</td><td>21,799</td><td>148,656</td></tr><tr><td>face images</td><td>313,593</td><td>36,716</td><td>58.420</td><td>408,729</td></tr><tr><td>IDs</td><td>924</td><td>112</td><td>189</td><td>1,225</td></tr><tr><td>genders</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>nationalities</td><td>32</td><td>11</td><td>18</td><td>36</td></tr><tr><td>testing instances</td><td>1</td><td>4,678,897</td><td>6,780,750</td><td>11,459,647</td></tr></table>
+
+Table 6: Statistics for the data appearing in VoxCeleb and VGGFace.
+
+The visual data used in Section 3.4 is densely extracted from the video in VoxCeleb dataset at 25/6 fps. It contains 100,000 segmented speaking face-tracks obtained by SyncNet (Chung & Zisserman, 2016), leading to 1,218,575 frames (images). For fair comparison, we follow the train/val/test split strategy from Nagrani et al. (2018a) in our experiments. The evaluations are performed based on the provided lists (Nagrani et al., 2018a), which specify the testing pairs of voices and faces.
+
+# A.2 PREPROCESSING
+
+We employ separated data preprocessing pipelines for audio segments and face images. For audio segments, we use an energy-based voice activity detector (Povey et al., 2011) to isolate speechbearing regions of the recordings. Subsequently, 64-dimensional log mel-spectrograms are generated, using an analysis window of $2 5 \mathrm { m s }$ , with hop of 10ms between frames. We perform mean and variance normalization of each mel-frequency bin.
+
+For training, we randomly crop out regions of varying lengths of 300 to 800 frames (so the size of the input spectrogram ranges from $3 0 0 \times 6 4$ to $8 0 0 \times 6 4$ for each mini-batch, around 3 to 8 seconds). For the face data, facial landmarks in all images are detected using MTCNN (Zhang et al., 2016). The cropped RGB face images of size $1 2 8 \times 1 2 8 \times 3$ are obtained by similarity transformation. Each pixel in the RGB images is normalized by subtracting 127.5 and then dividing by 127.5. We perform data augmentation by horizontally flipping the images with $5 0 \%$ probability in minibatches (effectively doubling the number of face images).
+
+# A.3 TRAINING
+
+The details of network architectures are shown in Table 7. For the voice network, we use 1D convolutional layers, where the convolution is performed along the axis that corresponds to time. The face network employs 2D convolutional layers. For both, the convolutional layers are followed by batch normalization (BN) (Ioffe & Szegedy, 2015) and rectified linear unit activations (ReLU) (Krizhevsky et al., 2012). The final face embedding is obtained by averaging the feature maps from the final layer, i.e.through average pooling. The final voice embedding is obtained by averaging the feature maps at the final convolutional layer along the time axis alone.
+
+We follow the typical settings of SGD for optimization. Minibatch size is 256. The momentum and weight decay values are 0.9 and 0.001 respectively. To learn the networks from scratch, the learning rate is initialized at 0.1 and divided by 10 after 16K iterations and again after 24K iterations. The training is completed at 28K iterations.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>layer</td><td rowspan=1 colspan=1>voice</td><td rowspan=1 colspan=1>face</td></tr><tr><td rowspan=6 colspan=1>embeddingnetwork</td><td rowspan=5 colspan=1>Conv</td><td rowspan=1 colspan=1>(3,256)/2,1[(3,256)/1,1][(3,256)/1,1]</td><td rowspan=1 colspan=1>(3× 3,64)/2,1[(3 × 3,64)/1,1][(3 × 3,64)/1,1]</td></tr><tr><td rowspan=1 colspan=1>(3,384)/2,1[(3,384)/1,1][(3,384)/1,1]</td><td rowspan=1 colspan=1>(3×3,128)/2,1[(3 × 3,128)/1,1][(3 × 3,128)/1,1]</td></tr><tr><td rowspan=1 colspan=1>(3,576)/2,1[(3,576)/1,1][(3,576)/1.1]</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>(3,864)/2,1[(3,864)/1,1][(3,864)/1,1]</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>(3,64)/2,1</td><td rowspan=1 colspan=1>(3× 3,64)/2,1</td></tr><tr><td rowspan=1 colspan=1>AvgPool</td><td rowspan=1 colspan=1>t×1</td><td rowspan=1 colspan=1>h×w×1</td></tr><tr><td rowspan=1 colspan=1>classificationnetwork</td><td rowspan=1 colspan=1>FC</td><td rowspan=1 colspan=2>64×924,64×2,64×32</td></tr></table>
+
+Table 7: The detailed CNNs architectures. The numbers within the parentheses represent the size and number of filters, while the subscripts represent the stride and padding. So, for example, $( 3 , 6 4 ) _ { / 2 , 1 }$ denotes a 1D convolutional layer with 64 filters of size 3, where the stride and padding are 2 and 1 respectively, while $( 3 \times 3 , 6 4 ) _ { / 2 , 1 }$ represents a 2-D convolutional layer of $6 4 3 \times 3$ filters, with stride 2 and padding 1 in both directions. Note that 924, 2, and 32 are the number of unique values taken by the ID, gender, and nationality covariates, respectively.
+
+# A.4 EXPERIMENTS ON THE EMBEDDING DIMENSION
+
+To investigate the affect of embedding dimension to the performance, we train DIMNet-I models with various embedding dimensions of 32, 64, 128, 256, and 512. Table 8 shows the results on 1:2 matching experiment (voice face).
+
+Table 8: The accuracies of DIMNet-I with different embedding dimensions on 1:2 matching experiments   
+
+<table><tr><td rowspan=1 colspan=1>embedding dimension</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>128</td><td rowspan=1 colspan=1>256</td><td rowspan=1 colspan=1>512</td></tr><tr><td rowspan=1 colspan=1>DIMNet-I</td><td rowspan=1 colspan=1>82.20</td><td rowspan=1 colspan=1>83.45</td><td rowspan=1 colspan=1>83.87</td><td rowspan=1 colspan=1>83.43</td><td rowspan=1 colspan=1>83.16</td></tr></table>
+
+It could be observed that the performance of cross-modal matching is very stable within a wide range of embedding dimension, showing that the accuracy is not sensitive to the embedding dimension.
+
+# APPENDIX B EXPECTED PERFORMANCE BASED ON GENDER MATCHING
+
+In this appendix we discuss the performance to be expected in the matching and verification tests, when the matching is done based purely on gender.
+
+We assume below that in any distribution of human-subject data, the division of subjects between male and female genders to be half and half.
+
+It is to be noted that gender is merely an illustrative example here; the analysis can be extended to other covariates. For a more detailed analysis of covariates with more values and unbalanced distributions, please refer to Wen et al. (2018).
+
+# B.1 ACCURACY OF 1:2 MATCHING BASED ON GENDER
+
+We show that the equal-error-rate for 1:2 matching can be as high as $2 5 \%$ , through gender matching alone.
+
+The problem is as follows: a probe input (voice or face), and a gallery consisting of two inputs (face or voice), one of which is from the same subject as the probe. We must identify which of the two is the true match.
+
+# B.1.1 PERFECT GENDER IDENTIFICATION
+
+Consider the situation where we are able to identify the gender of the subject of the data (face or voice) perfectly.
+
+There are two possibilities: (a) both probe instances are the same gender, and (b) they are different genders. Each of the two possibilities occurs with a probability of 0.5
+
+We employ the following simple strategy: If the two gallery instances are different genders, then we select the instance whose gender matches the probe. In this case, clearly, the probability of error is 0. If the two instances are the same gender, we select one of them randomly with a probability of 0.5. The probability of error here is 0.5.
+
+Thus, the overall probability of error is
+
+$$
+P r o b ( e r r o r ) = 0 . 5 \times 0 + 0 . 5 \times 0 . 5 = 0 . 2 5 .
+$$
+
+# B.1.2 IMPERFECT GENDER IDENTIFICATION
+
+Now let us consider the situation where gender identification itself is imperfect, and we have error rates $e _ { f }$ and $e _ { v }$ in identifying the gender of faces and voices, respectively. Assume the error rates are known. We will assume below that gallery entries are faces, and probe entries are voices. (The equations are trivially flipped to handle the converse case).
+
+Since we are aware that we sometimes make mistakes in identifying gender, we modify our strategy as follows: when the two gallery items are found to have different genders, we select the entry with the same gender as the probe $P$ of the time (so that if the gender classification was correct, we would have a match error rate of $\left( 1 - P \right)$ ). When both gallery items are found to be the same gender, we choose randomly.
+
+The actual error can now be computed as follows. The gallery items are both of the same gender in 0.5 of the trials, and of mismatched gender in the remaining 0.5 of the trials.
+
+When both gallery items have the same gender, regardless of the strategy chosen, the probability of error is 0.5 (by symmetry).
+
+When both gallery items are of mismatched gender, we have 8 combinations of correctness of gender-classification. Table 9 lists all eight, along with the probability of matching error (in the final column). Taking type 1 as an example, we have probability $( 1 - e _ { v } ) ( 1 - e _ { f } ) ^ { 2 }$ that the gender of both probe and galleries are correctly classified. In this case, our strategy gives us an error of $( 1 - P )$ . For type 2, the gender of probe and one of the gallery items is correctly classified, while the other gallery item is misclassified, we have an error of 0.5. If we go through all the cases, the total error $P r o b ( e r r o r )$ can be computed as
+
+Table 9: the possible error types with probabilities.   
+
+<table><tr><td rowspan=1 colspan=1>type</td><td rowspan=1 colspan=1>probe</td><td rowspan=1 colspan=1>gallery1</td><td rowspan=1 colspan=1>gallery2</td><td rowspan=1 colspan=1>Prob(error,type)</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td><td rowspan=8 colspan=1>(1-ev)(1-ef)²:(1-P)(1-ev)ef(1-ef)·0.5(1-ev)(i-ef)ef : 0.5(1-ev)(ef)²:Pe(1-ef)².Peuef(1-ef) ·0.5eu(i-ef)ef :0.5ev(ef)²:(1-P)</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>√</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>丁</td><td rowspan=1 colspan=1>√</td><td rowspan=2 colspan=1>××</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>×</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>√</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>√</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>√</td><td rowspan=1 colspan=1>×</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td><td rowspan=1 colspan=1>×</td></tr></table>
+
+$$
+\begin{array} { r } { P r o b ( e r r o r ) = 0 . 2 5 + 0 . 5 \sum _ { t y p e = 1 } ^ { 8 } P r o b ( e r r o r , t y p e ) } \\ { = 0 . 2 5 + 0 . 5 ( 2 e _ { f } e _ { v } - e _ { v } - e _ { f } + 1 } \\ { + P ( 2 e _ { f } + 2 e _ { v } - 4 e _ { f } e _ { v } - 1 ) ) } \end{array}
+$$
+
+Our objective is to minimize $P r o b ( e r r o r )$ , so we must choose $P$ to minimize the above term. I.e. we must solve
+
+$$
+\arg \operatorname* { m i n } _ { P } \ 2 e _ { f } e _ { v } - e _ { v } - e _ { f } + 1 + P ( 2 e _ { f } + 2 e _ { v } - 4 e _ { f } e _ { v } - 1 )
+$$
+
+Its easy to see that the solution for $P$ is 1.0 if its multiplicative factor is negative in the above equation, and 0 otherwise, i.e.
+
+$$
+P = \{ _ { 0 , } ^ { 1 , } \mathrm { i f } e _ { f } + e _ { v } < 2 e _ { f } e _ { v } + 0 . 5
+$$
+
+The corresponding match error rates are $P r o b ( e r r o r ) = 0 . 2 5 + 0 . 5 ( e _ { f } + e _ { v } - 2 e _ { v } e _ { f } )$ and $0 . 7 5 +$ $e _ { f } e _ { f } - 0 . { \bar { 5 } } ( e _ { v } + \bar { e } _ { f } )$ respectively.
+
+Although complicated looking, the solution is, in fact, quite intuitive. When gender classification is either better than random for both modalities $( i . e . e _ { f } , e _ { v } > 0 . 5 )$ or worse than random for both $( e _ { f } , e _ { v } < 0 . 5 )$ , the best strategy is to select the gallery item that matches the gender of the probe. If either of these is random (i.e.either $e _ { f }$ or $e _ { v }$ is 0.5) , the choice of $P$ does not matter, and the error is 0.5. If one of the two is correct more than half the time, and the other is wrong more than half the time $( e . g . e _ { f } < 0 . 5$ , $e _ { v } > 0 . 5 )$ , the optimal choice is to select the gallery item that is classified as mismatched in gender with the probe.
+
+# B.2 ACCURACY OF 1:N MATCHING BASED ON GENDER
+
+We now consider the best achievable performance on $1 { : } N$ matching, when the only information known is the gender of the voices and faces.
+
+# B.2.1 PERFECT GENDER IDENTIFICATION
+
+Consider the situation where the gender of the faces and voices in each test trial is perfectly known.
+
+We employ the following strategy: we randomly select one of the gallery instances that have the same gender as the probe instance. If there are $K$ imposter gallery instances of the same gender as the probe instance, the expected accuracy is $\frac { 1 } { K + 1 }$ . The probability of randomly having $K$ of $N - 1$ imposters of the same gender as the probe is given by
+
+$$
+P r o b ( K ; N - 1 ) = { \binom { N - 1 } { K } } 0 . 5 ^ { N - 1 }
+$$
+
+The overall accuracy is given by:
+
+$$
+\begin{array} { l } { { P r o b ( c o r r e c t ) = \displaystyle \sum _ { K = 0 } ^ { N - 1 } \frac { P r o b ( K , N - 1 ) } { K + 1 } } } \\ { { = 0 . 5 ^ { N - 1 } \displaystyle \sum _ { K = 0 } ^ { N - 1 } \left( { ^ { N - 1 } \atop K = 0 } \right) \displaystyle \frac { 1 } { K + 1 } } } \\ { { = \displaystyle \frac { 0 . 5 ^ { N - 1 } } { N } \displaystyle \sum _ { k = 1 } ^ { N } \left( { ^ { N } \atop K } \right) } } \\ { { = \displaystyle \frac { 0 . 5 ^ { N - 1 } ( 2 ^ { N } - 1 ) } { N } } } \\ { { = \displaystyle \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N } , } } \end{array}
+$$
+
+giving us the error
+
+$$
+P ( e r r o r ) = 1 - \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N } .
+$$
+
+# B.2.2 IMPERFECT GENDER IDENTIFICATION
+
+Consider now that the gender recognition is erroneous for voices with probability $e _ { v }$ and for faces with probability $e _ { f }$ . Note that regardless of the error in gender recognition, the probability of any noisy gallery entry having any gender remains 0.5.
+
+To account for the possible error in gender classification, we consider the following stochastic policy: with probability $P$ we select one of the gallery entries with the same gender assigned to probe (by the gender classifier), and with probability $1 - P$ we choose one of the entries with the opposite gender assigned to the probe.
+
+Let $\alpha$ represent the probability that the genders assigned to probe and the corresponding gallery entry by their respective classifiers are identical.
+
+$$
+\alpha = e _ { v } e _ { f } + ( 1 - e _ { v } ) ( 1 - e _ { f } ) .
+$$
+
+The equation above considers both possibilities: that both the probe and its matching gallery entry are correctly classified, and that both of them are misclassified. It follows that the probability and its matching gallery entries are assigned different genders is $1 - \alpha$ .
+
+Given that we have selected the correct gender for retrieval from the the gallery (i.e.that the gender we have selected is the same as that assigned to the gallery entry matching the probe by the face classifier), using the same analysis as in Section B.2.1, we obtain the following probability of being correct:
+
+$$
+P ( c o r r e c t | c o r r e c t g e n d e r ) = \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N }
+$$
+
+The probability of selecting the correct gender is given by
+
+$$
+P ( c o r r e c t g e n d e r ) = P \alpha + ( 1 - P ) ( 1 - \alpha )
+$$
+
+Since the probability of being correct when we choose the wrong gender is $_ 0$ , the overall probability of being correct is
+
+$$
+\begin{array} { r } { P ( c o r r e c t ) = ( P \alpha + ( 1 - P ) ( 1 - \alpha ) ) \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N } } \\ { = ( P ( 2 \alpha - 1 ) + 1 - \alpha ) \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N } } \end{array}
+$$
+
+Maximizing the probability requires us to solve
+
+$$
+\arg \operatorname* { m a x } _ { P } P ( 2 \alpha - 1 ) + 1 - \alpha , \ \mathrm { s } . t . \ 1 \geq P \geq 0
+$$
+
+which gives us the optimal $P$ as
+
+$$
+P = { \left\{ \begin{array} { l l } { 1 { \mathrm { ~ i f ~ } } \alpha > 0 . 5 } \\ { 0 { \mathrm { ~ o t h e r w i s e } } } \end{array} \right. }
+$$
+
+and the optimal error as
+
+$$
+\begin{array} { r } { P ( e r r o r ) = \left\{ \begin{array} { l l } { 1 - \alpha \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N } , \mathrm { ~ i f ~ } \alpha > 0 . 5 } \\ { 1 - ( 1 - \alpha ) \frac { ( 2 - 0 . 5 ^ { N - 1 } ) } { N } \mathrm { ~ o t h e r w i s e } . } \end{array} \right. } \end{array}
+$$
+
+# B.3 EER OF VERIFICATION BASED ON GENDER
+
+Here we show that the equal-error-rate for verification (determining if the the subjects in two recordings are the same) can be as high as $33 \%$ , through gender matching alone.
+
+The problem is as follows: we are given a pair of inputs, one (features extracted from) a face, and the other a voice. We must determine whether they are both from the same speaker.
+
+The test set include some number of “positives”, where both do belong to the same subject, and some “negatives”, where both do not. If a positive is falsely detected as a negative, we have an instance of false rejection. If a negative is wrongly detected as a positive, we have an instance of false acceptance.
+
+Let $F _ { R }$ represent the ‘false rejection rate”, i.e.the fraction of all positives that are wrongly rejected. Let $F _ { A }$ represent the “false acceptance rate”, i.e.the fraction of negatives that are wrongly accepted. Any classifier can generally be optimized to trade off $F _ { R }$ against $F _ { A }$ . The “Equal Error Rate” (EER) is achieved when $F _ { R } = F _ { A }$ .
+
+Among the “positive” test pairs, both voice and face in each pair have the same gender. We assume the “negative” test instances are drawn randomly, i.e., 0.5 of all negative pairs have the same gender, while the remaining 0.5 do not.
+
+# B.3.1 PERFECT GENDER IDENTIFICATION
+
+Consider the situation where we know the subject’s gender for both the voices and faces (or, alternately, are able to identify the gender from the voice or face perfectly).
+
+We employ the following strategy: if the gender of the voice and face are different, we declare it as a negative $100 \%$ of the time. If the two are from the same gender, we randomly call it a positive $P$ of the time, where $0 \leq P \leq 1 . 0$ .
+
+Using this strategy, the false acceptance rate is:
+
+$$
+F _ { A } = 0 . 5 \times 0 + 0 . 5 \times P = 0 . 5 P .
+$$
+
+Here we’re considering that using our strategy we never make a mistake on the $50 \%$ of negative pairs that have mismatched genders, but are wrong $P$ of the time on the negative pairs with matched genders.
+
+Among the positives, where all pairs are gender matched, our strategy of accepting only a fraction $P$ of them as positives will give us a false rejection rate $F _ { R } = 1 - P$ .
+
+The equal error rate is achieved when $F _ { R } = F _ { A }$ , i.e.
+
+$$
+0 . 5 P = 1 - P ,
+$$
+
+giving us $\textstyle P = { \frac { 2 } { 3 } }$ , i.e.the best EER is achieved when we accept gender-matched pairs two-thirds of the time.
+
+The EER itself is $\begin{array} { r } { 0 . 5 P = \frac { 1 } { 3 } } \end{array}$
+
+Thus, merely by being able to identify the gender of the subject accurately, we are able to verification EER of 0.33.
+
+# B.3.2 IMPERFECT GENDER IDENTIFICATION
+
+Now let us consider the situation where gender identification itself is imperfect, and we have error rates $e _ { f }$ and $e _ { v }$ in identifying the gender of the face and the voice, respectively. Assume these error rates are known.
+
+To account for this, we modify our strategy: when we find the genders of the voice and face to match, we accept the pair as positive $P$ of the time, but when they are mismatched we still accept them as positive $Q$ of the time.
+
+Let $\alpha$ represent the probability that we will correctly call the polarity of the gender match between the voice and the face. I.e. $\alpha$ is the probability that if the two have the same gender, we will correctly state that they have the same gender, or if they are of opposite gender, we will correctly state they are of opposite gender.
+
+$$
+\alpha = ( 1 - e _ { f } ) ( 1 - e _ { v } ) + e _ { f } e _ { v } .
+$$
+
+This combines two terms: that we call the genders of both the voice and face correctly, and that we call them both wrongly (which also results in finding the right polarity of the relationship). Its easy to see that $0 \leq \alpha \leq 1$ , and to verify that when gender identification is perfect, $\alpha = 1 . 0$ . The probability of calling the polarity of the gender relationship wrongly is $1 - \alpha$ .
+
+Among the positive test pairs, all pairs are gender matched. We will correctly call $\alpha$ of these as gender matched. Using our strategy, our error on these instances is $( 1 - P )$ . We will incorrectly call
+
+$1 - \alpha$ of these as gender mismatched, and the error on these instances is $( 1 - Q )$ . So the overall false rejection rate is given by
+
+$$
+F _ { R } = \alpha ( 1 - P ) + ( 1 - \alpha ) ( 1 - Q ) = 1 - \alpha P - ( 1 - \alpha ) Q
+$$
+
+Among the negative pairs, half are gender matched, and half are gender mismatched. Using the same logic as above, the error on the gender-matched negative pairs is $\alpha P + ( 1 - \alpha ) Q$ . Among the gender mismatched pairs the error is $\alpha Q + ( 1 - \alpha ) P$ . The overall false acceptance rate is given by
+
+$$
+F _ { A } = 0 . 5 ( \alpha P + ( 1 - \alpha ) Q ) + 0 . 5 ( \alpha Q + ( 1 - \alpha ) P ) = 0 . 5 ( P + Q ) .
+$$
+
+Equating $F _ { A }$ and $F _ { R }$ as the condition for EER, we obtain
+
+$$
+\begin{array} { l l } { { } } & { { 1 - \alpha P - ( 1 - \alpha ) Q = 0 . 5 ( P + Q ) } } \\ { { \implies } } & { { ( 3 - 2 \alpha ) Q + ( 1 + 2 \alpha ) P = 2 . } } \end{array}
+$$
+
+Since at EER, the EER equals $F _ { A }$ , and we would like to minimize it, we obtain the following solution to determine the optimal $P$ and $Q$ :
+
+$$
+\begin{array} { l } { \arg \underset { P , Q } { \operatorname* { m i n } } P + Q } \\ { \mathrm { s . } t . 1 \geq P , Q \geq 0 , ( 3 - 2 \alpha ) Q + ( 1 + 2 \alpha ) P = 2 . } \end{array}
+$$
+
+For $\alpha > 0 . 5$ it is easy to see that the solution to this is obtained at
+
+$$
+\begin{array} { l } { \displaystyle Q = 0 } \\ { \displaystyle P = \frac { 2 } { 1 + 2 \alpha } . } \end{array}
+$$
+
+For $\alpha < 0 . 5$ the optimal solution is at
+
+$$
+\begin{array} { l } { \displaystyle P = 0 } \\ { \displaystyle Q = \frac { 2 } { 3 - 2 \alpha } . } \end{array}
+$$
+
+That is, when the probe and gallery classifiers are likely to make the same error more than half the time, the optimal solution is to always reject pairs detected as having mismatched genders, and to accept matched-gender pairs $\frac { 2 } { 1 + 2 \alpha }$ of the time. The optimal EER is $\frac { \mathbf { \bar { \alpha } } _ { 1 } } { 1 + 2 \alpha }$ .
+
+When they are more likely to make different errors, the optimal solution is to always reject pairs detected as having matched genders, and to accept mismatched-gender pairs $\frac { 2 } { 3 - 2 \alpha }$ of the time. The optimal EER now is $\frac { 1 } { 3 - 2 \alpha }$ .
+
+Note that if an operating point other than EER were chosen to quantify performance $( e . g . F _ { A } = \beta F _ { R }$ for $\beta \neq 1$ , or for some fixed $F _ { A }$ or $F _ { R }$ ), the above analysis can be modified to accommodate it, provided a feasible solution exists.

parse/train/HklxbgBKvr/HklxbgBKvr.md

@@ -0,0 +1,397 @@
+# MODEL-BASED REINFORCEMENT LEARNING FOR BIOLOGICAL SEQUENCE DESIGN
+
+Christof Angermueller Google Research {christofa}@google.com
+
+David Dohan   
+Google Research   
+{ddohan}@google.com   
+David Belanger   
+Google Research   
+{dbelanger}@google.com
+
+Ramya Deshpande∗ Caltech {rdeshpan}@caltech.edu
+
+Kevin Murphy   
+Google Research   
+{kpmurphy}@google.com   
+Lucy Colwell   
+Google Research   
+University of Cambridge   
+{lcolwell}@google.com
+
+# ABSTRACT
+
+The ability to design biological structures such as DNA or proteins would have considerable medical and industrial impact. Doing so presents a challenging black-box optimization problem characterized by the large-batch, low round setting due to the need for labor-intensive wet lab evaluations. In response, we propose using reinforcement learning (RL) based on proximal-policy optimization (PPO) for biological sequence design. RL provides a flexible framework for optimization generative sequence models to achieve specific criteria, such as diversity among the high-quality sequences discovered. We propose a model-based variant of PPO, DyNA PPO, to improve sample efficiency, where the policy for a new round is trained offline using a simulator fit on functional measurements from prior rounds. To accommodate the growing number of observations across rounds, the simulator model is automatically selected at each round from a pool of diverse models of varying capacity. On the tasks of designing DNA transcription factor binding sites, designing antimicrobial proteins, and optimizing the energy of Ising models based on protein structure, we find that DyNA PPO performs significantly better than existing methods in settings in which modeling is feasible, while still not performing worse in situations in which a reliable model cannot be learned.
+
+# 1 INTRODUCTION
+
+Driven by real-world obstacles in health and disease requiring new drugs, treatments, and assays, the goal of biological sequence design is to identify new discrete sequences $x$ which optimize some oracle, typically an experimentally-measured functional property $f ( \bar { \boldsymbol { x } } )$ . This is a difficult black-box optimization problem over a combinatorially large search space in which function evaluation relies on slow and expensive wet-lab experiments. The setting induces unusual constraints in black-box optimization and reinforcement learning: large synchronous batches with few rounds total.
+
+The current gold standard for biomolecular design is directed evolution, which was recently recognized with a Nobel prize (Arnold, 1998) and is a form of randomized local search. Despite its impact, directed evolution is sample inefficient and relies on greedy hillclimbing to the optimal sequences. Recent work has demonstrated that machine-learning-guided optimization (Section 3) can find better sequences faster.
+
+Reinforcement learning (RL) provides a flexible framework for black-box optimization that can harness modern deep generative sequence models. This paper proposes a simple method for improving the sample efficiency of policy gradient methods such as PPO (Schulman et al., 2017) for black-box optimization by using surrogate models that are trained online to approximate $f ( x )$ . Our method updates the policy’s parameters using sequences $x$ generated by the current policy $\pi _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ , but evaluated using a learned surrogate $f ^ { \prime } ( x )$ , instead of the true, but unknown, oracle reward function $f ( x )$ . We learn the parameters of the reward model, $w$ , simultaneously with the parameters of the policy. This is similar to other model-based RL methods, but simpler, since in the context of sequence optimization, the state-transition model is deterministic and known. Initially the learned reward model, $f ^ { \prime } ( x )$ , is unreliable, so we rely entirely on $f ( x )$ to assess sequences and update the policy. This allows a graceful fallback to PPO when the model is not effective. Over time, the reward model becomes more reliable and can be used as a cheap surrogate, similar to Bayesian optimization methods (Shahriari et al., 2015). We show empirically that cross-validation is an effective heuristic for assessing the model quality, which is simpler than the inference required by Bayesian optimization.
+
+We rigorously evaluate our method on three in-silico sequence design tasks that draw on experimental data to construct functions $f ( x )$ characteristic of real-world design problems: optimizing binding affinity of DNA sequences of length 8 (search space size $4 ^ { 8 }$ ); optimizing anti-microbial peptide sequences (search space size $2 0 ^ { 5 0 }$ ), and optimizing binary sequences where $f ( x )$ is defined by the energy of an Ising model for protein structure (search space size $2 0 ^ { 5 0 }$ ). These do not rely on wet lab experiments, and thus allow for large-scale benchmarking across a range of methods. We show that our DyNA PPO method achieves higher cumulative reward for a given budget (measured in terms of number of calls to $f ( x ) )$ ) than existing methods, such as standard PPO, various forms of the cross-entropy method, Bayesian optimization, and evolutionary search.
+
+In summary, our contributions are as follows:
+
+• We provide a model-based RL algorthm, DyNA PPO, and demonstrate its effectiveness in performing sample efficient batched black-box function optimization.   
+• We address model bias by quantifying the reliability and automatically selecting models of appropriate complexity via cross-validation.   
+• We propose a visitation-based exploration bonus and show that it is more effective than entropy-regularization in identifying multiple local optima.   
+• We present a new optimization task for benchmarking methods for biological sequence design based on protein energy Ising models.
+
+# 2 METHODS
+
+Let $f ( x )$ be the function that we want to optimize and $x \in \ V ^ { T }$ a sequence of length $T$ over a vocabulary $V$ such as DNA nucleotides $( | V | = 4 )$ or amino acids $( | \bar { V } | = 2 0 )$ . We assume $N$ experimental rounds and that $B$ sequences can be measured per round. Let $D _ { n } = \{ ( x , f ( x ) ) \}$ be the data acquired in round $n$ with $| \bar { D } _ { n } | = B$ . For simplicity, we assume that the sequence length $T$ is constant, but our approach based on generating sequences autoregressively easily generalizes to variable-length sequences.
+
+# 2.1 MARKOV DECISION PROCESS
+
+We formulate the design of a single sequence $x$ as a Markov decision process $\mathcal { M } = ( S , A , p , r )$ with state space $S$ , action space $A$ , transition function $p$ , and reward function $r$ . The state space $S = \cup _ { t = 1 \dots T } V ^ { t }$ is the set of all possible sequence prefixes and $A$ corresponds to the vocabulary $V$ . A sequence is generated left to right. At time step $t$ , the state $s _ { t } = a _ { 0 } , . . . , a _ { t - 1 }$ corresponds to the $t$ last tokens and the action $a _ { t } \in A$ to the next token. The transition function $p ( s _ { t } + 1 | s _ { t } ) = s _ { t } a _ { t }$ is deterministic and corresponds to appending $a _ { t }$ to $s _ { t }$ . The reward $r ( s _ { t } , a _ { t } )$ is zero except at the last step $T$ , where it corresponds to the functional measurement $f ( s _ { T - 1 } )$ . For generating variable-length sequences, we extend the vocabulary by a special end-of-sequence token and terminate sequence generation when this token is selected.
+
+# Algorithm 1: DyNA PPO
+
+1: Input: Number of experiment rounds $\mathbf { N }$   
+2: Input: Number of model-based training rounds M   
+3: Input: Set of candidate models ${ \cal { S } } = \{ { \bar { f } } ^ { \prime } \}$   
+4: Input: Minimum model score $\tau$ for model-based training   
+5: Input: Policy $\pi \theta$ with initial parameters $\theta$   
+6: for $n = 1 , 2 , . . . \mathcal { N }$ do   
+7: Collect samples $\mathcal { D } _ { n } = \{ x , f ( x ) \}$ using policy $\pi _ { \theta }$   
+8: Train policy $\pi _ { \theta }$ on $\mathcal { D } _ { n }$   
+9: Fit candidate models $f ^ { \prime } \in { \mathcal { S } }$ on $\textstyle \bigcup _ { i = 1 } ^ { n } { \mathcal { D } } _ { i }$ and compute their score by cross-validation   
+10: Select the subset of models $S ^ { \prime } \subseteq { \bar { S } }$ with a score $\geq \tau$   
+11: if $\boldsymbol { S ^ { \prime } } \neq \boldsymbol { \emptyset }$ then   
+12: for $m = 1 , 2 , . . . \mathbf { M }$ do   
+13: Sample a batch of sequences $x$ from $\pi \theta$ and observe the reward $\begin{array} { r } { f ^ { \prime \prime } ( x ) = \frac { 1 } { | S ^ { \prime } | } \sum _ { f ^ { \prime } \in S ^ { \prime } } f ^ { \prime } ( x ) } \end{array}$   
+14: Update $\pi \theta$ on $\{ x , f ^ { \prime \prime } ( x ) \}$   
+15: end for   
+16: end if   
+17: end for
+
+# 2.2 POLICY OPTIMIZATION
+
+We train a policy $\pi _ { \boldsymbol { \theta } } ( a _ { t } | \boldsymbol { s } _ { t } )$ to optimize the expected sum of rewards :
+
+$$
+\mathbb { E } [ R ( s _ { 1 : t } ) | s _ { 0 } , \theta ] = \sum _ { s _ { t } } \sum _ { a _ { t } } \pi _ { \theta } ( a _ { t } | s _ { t } ) r ( s _ { t } , a _ { t } ) .
+$$
+
+We use proximal policy optimization (PPO) with KL trust-region constraint (Schulman et al., 2017), which we have found to be more stable and sample efficient than REINFORCE (Williams, 1992). We have also considered off-policy deep Q-learning (DQN) (Mnih et al., 2015), and categorical distributional deep Q-learning (CatDQN) (Bellemare et al., 2017), which are in principle more sampleefficient than on-policy learning using PPO since they can reuse samples multiple times. However, they performed worse than PPO in our experiments (Appendix C). We implement algorithms using the TF-Agents RL library (Guadarrama et al., 2018).
+
+We employ autoregressive models with one fully-connected layer as policy and value networks since they are faster to train and outperformed recurrent networks in our experiments. At time step $t$ , the network takes as input the $W$ last characters $a _ { t - W } , . . . , a _ { t - 1 }$ that are one-hot encoded, where the context window size $W$ is a hyper-parameter. To provide the network with information about the current position of the context window, it also receives the time step $t$ , which is embedded using a sinusoidal positional encoding (Vaswani et al., 2017), and concatenated with the one-hot characters. The policy network outputs a distribution $\pi _ { \boldsymbol { \theta } } ( a _ { t } | \boldsymbol { s } _ { t } )$ over next the token $a _ { t }$ . The value network $V ( s _ { t } )$ , which approximates the expected future reward for being in state $s _ { t }$ , is used as a baseline to reduce the variance of stochastic estimates of equation 1 (Schulman et al., 2017).
+
+# 2.3 MODEL-BASED POLICY OPTIMIZATION
+
+Model-based RL learns a model of the environment that is used as a simulator to provide additional pseudo-observations. While model-free RL has been successful in domains where interaction with the environment is cheap, such as those where the environment is defined by a software program, its high sample complexity may be unrealistic for biological sequence design. In model-based RL, the MDP $\mathcal { M } = ( S , A , p , r )$ is approximated by a model $\mathcal { M } ^ { \prime } \overset { \cdot } { = } ( S , A , p ^ { \prime } , \overset { \cdot } { r ^ { \prime } } )$ with the same state space $S$ and action space $A$ as $\mathcal { M }$ (Sutton $\&$ Barto, 2018, Ch. 8). Since the transition function $p$ is deterministic in our case, only the reward function $r ( s _ { t } , a _ { t } )$ needs to be approximated by $r ^ { \prime } ( s _ { t } , a _ { t } )$ . Since $r ( s _ { T } , a _ { T } )$ is non-zero at the last step $T$ and then corresponds to $f ( x )$ with ${ x = } = s _ { T - 1 }$ , the problem reduces to approximating $f ( x )$ . This can be done by supervised regression by fitting a regressor $f ^ { \prime } ( x )$ on the data $\cup _ { n ^ { \prime } < = n } D _ { n ^ { \prime } }$ collected so far. We then use the resulting model to collect additional observations $( x , f ^ { \prime } ( x ) )$ and update the policy in a simulation phase, instead of only using observations $( x , f ( x ) )$ from the the true environment, which are expensive to collect. We call our method DyNA PPO since it is similar to the DYNA architecture (Sutton (1991); Peng et al. (2018)) and since can be used for DNA sequence design.
+
+Model-based RL provides the promise of improved sample efficiency when the model is accurate, but it can reduce performance if insufficient data are available for training a trustworthy model. In this case, the policy is prone to exploit regions where the model is inaccurate (Janner et al., 2019). To reap the benefit of model-based RL when the model is accurate and avoid reduced performance when it is not, we (i) automatically select the model from a set of candidate models of varying complexity, (ii) only use the selected model if it is accurate, and iii) stop model-based training as soon the the model uncertainty increases by a certain threshold. After each round of experiment, we fit a set of candidate models on all available data to estimate $f ( x )$ via supervised regression. We quantify the accuracy of each candidate model by the $R ^ { 2 }$ score, which we estimate by five-fold cross-validation. See Appendix $\mathbf { G }$ for a discussion of different data splitting strategies to select models using crossvalidation. If the $R ^ { 2 }$ score of all candidate model is below a pre-specified threshold $\tau$ , we do not perform model-based training in that round. Otherwise, we build an ensemble model that includes all models with a score greater or equal than $\tau$ , and use the average prediction as reward for training the policy. We considered $\tau$ as a tunable hyper-parameter, were we found $\tau = 0 . 5$ to be optimal for all problems (see Figure 14. By ignoring the model if it is inaccurate, we aim to prevent the policy from exploiting deficiencies of the model (Janner et al., 2019).
+
+We perform up to $M$ model-based optimization rounds (see Algorithm 1) and stop as soon as the model uncertainty increased by a certain factor relative to the model uncertainty at the first round $( m = 1$ ). This is motivated by our observation that the model uncertainty is strongly correlated with the unknown model error, and prevents from training the policy with inaccurate model predictions (see Figure 12, 13) as soon as the model starts to explore regions on which the model was not trained on.
+
+For models, we consider nearest neighbor regression, Bayesian ridge regression, random forests, gradient boosting trees, Gaussian processes, and ensemble of deep neural networks. Within each model family, we additionally use cross-validation for tuning hyper-parameters, such as the number of trees, tree depth, kernels and kernel parameters, or the number of hidden layers and units (see Appendix A.7 for details). By testing and optimizing the hyper-parameters of different models automatically, the model capacity can dynamically increase as data becomes available.
+
+In Bayesian optimization, non-parametric models such as Gaussian processes are popular regressors, and they also automatically grow model capacity as more data arrives (Shahriari et al., 2015). However, with Bayesian optimization there is no opportunity to ignore the regressor entirely if it is unreliable. Furthermore, Bayesian optimization relies on performing (approximate) Bayesian inference, which in practice is sensitive to the choice of hyper-parameter (Snoek et al., 2012).
+
+Overall, our method combines the positive attributes of both generative and discriminative approaches to sequence design. Our experiments do not compare to prior work on model-based RL, since these methods primarily focus on estimating a dynamics model for state transitions.
+
+# 2.4 DIVERSITY-PROMOTING REWARD FUNCTION
+
+Learning policies to generate diverse sequences is important because of several reasons. In many applications, $f ( x )$ is an in-vitro (taking place outside a living organism) surrogate for an in-vivo taking place inside a living organism) functional measurement that is even more expensive to evaluate than $f ( x )$ . The in-vivo measurement may depend on properties that are correlated with $f ( x )$ and others that are not captured at all in-vitro, such as off-target effects or toxicity. To improve the chance that a sequence satisfying the ultimate in-vivo criteria is found, it is therefore desirable for the optimization procedure to discover a diverse set of candidate optima. Here, diversity is a downstream metric, for which training the policy $\pi _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ to maximize equation 1 will not necessarily yield good performance. For example, a high-quality policy can learn to always generate the same sequence $x$ with a high value of $f ( x )$ , and will therefore result in zero diversity. An additional reason that diversity matters is that it yields a good exploration strategy, even for scenarios where optimizing equation 1 is sufficient. Finally, use of strategies that reward high-diversity policies can reduce the policies’ tendency to generate exact duplicates.
+
+To increase sequence diversity, we employ a simple exploration reward bonus based on the density of proposed sequences, similar to existing exploration techniques based on state visitation frequency (Bellemare et al., 2016). Specifically, we define the final reward as $r _ { T } = f ( x ) - \lambda { \cdot } \mathrm { d e n s } _ { \epsilon } ( x )$ , where $\mathrm { d e n s } _ { \epsilon } ( x ) \in \mathbb { N } ^ { + }$ is the weighted number of sequences that have been proposed in previous rounds with a distance of less than $\epsilon$ away from $x$ , where the weight decays linearly with the distance. This reward penalizes proposing similar sequences multiple times, where the strength of the penalty is controlled by $\lambda$ . As a result, the policy learns not to generate related sequences and hence explores the search space more effectively. We used the edit distance as distance metric and tuned the distance radius $\epsilon$ , where setting $\epsilon > 0$ improved exploration on high-dimensional problems (see Figure 11). We also considered an alternative penalty based on the nearest neighbor distance of the proposed sequence to past sequences, which we found to be less effective (see Figure 9).
+
+# 3 RELATED WORK
+
+Recently, machine learning approaches have been shown to be effective in optimizing real-world DNA and protein sequences (Wang et al., 2019; Chhibbar & Joshi, 2019; de Jongh et al., 2019; Liu et al., 2019; Sample et al., 2019; Wu et al., 2019). Existing methods for biological sequence design fall into three broad categories: evolutionary search, optimization using discriminative models (e.g. Bayesian optimization), and optimization using generative models (e.g. the cross entropy method).
+
+Evolutionary approaches perform direct local search in the space of sequences. They include the aforementioned directed evolution and derivatives with application-specific mutation and recombination steps. Evolutionary approaches are appealing since they are simple and can easily incorporate human intuition into the design process, but generally suffer from low sample efficiency.
+
+Optimization methods based on discriminative models alternate between two steps: (i) using the data that have been collected so far to fit a regressor $f ^ { \prime } ( x )$ to approximate $f ( x )$ , and (ii) using $f ^ { \prime } ( x )$ to define an acquisition function that is optimized to select the next batch of sequences. Recently, such an approach was used to optimize the binding affinity of IgG antibodies (Liu et al., 2019), where a neural network ensemble was used for $f ^ { \prime } ( \bar { x } )$ . In general, optimizing the acquisition function is a non-trivial combinatorial optimization problem. Liu et al. (2019) employed activation maximization, where gradient-based optimization is performed on a continuous relaxation of the discrete search space. However, this requires $f ^ { \prime } ( x )$ to be differentiable and optimization of a continuous relaxation is vulnerable to leaving the data manifold (cf. deep dream (Mordvintsev et al., 2015)).
+
+Bayesian optimization defines an acquisition function such as the expected improvement (Mockus et al., 2014) based on the uncertainty of $f ^ { \prime } ( x )$ , which enables balancing exploration and exploitation (overview provided in Shahriari et al. (2015)). Gaussian processes (GPs) are commonly used for Bayesian black-box optimization since they provide calibrated uncertainty estimates. Unfortunately, GPs are hard to scale to large, high-dimensional datasets and are sensitive to the choice of hyperparameters. In response, recent work has performed continuous black-box optimization in the latent space of a deep generative model (Gomez-Bombarelli et al., 2018). However, this approach requires ´ a pre-trained model such as a variational autoencoder to obtain the latent embeddings. Our modelbased reinforcement learning approach is similar to these approaches in that we train a reinforcement learning policy to optimize a model $f ^ { \prime } ( x )$ . However, our policy is also trained directly on observations of $f ( x )$ and is able to resort to model-free training by automatically identifying if the model $f ^ { \prime } ( x )$ is too inaccurate to be used as surrogate of $f ( x )$ . Janner et al. (2019) investigated conditions in which an estimate of model generalization (their analysis uses validation accuracy) could justify model usage in such model-based policy optimization settings. Hashimoto et al. (2018) proposed using a cascade of classifiers, one per round, to guide sampling progressively better candidates.
+
+Optimization methods based on generative models seek to learn a distribution $p _ { \theta } ( x )$ parameterized by $\theta$ that maximizes the expected value of $f ( x ) \colon \mathbb { E } _ { x \sim P _ { \theta } ( x ) } [ f ( x ) ]$ . We note that this is the same form as variational optimization objectives, which allow the use of parameter-space evolutionary strategies (Staines & Barber, 2013; Wierstra et al., 2014; Salimans et al., 2017). Variants of the cross entropy method (De Boer et al., 2005; Brookes et al., 2019a) optimize $\theta$ , by alternating two steps: (i) sampling $x \sim p _ { \theta } ( x )$ and evaluating $\operatorname { f } ( \mathbf { x } )$ , and (ii) updating $\theta$ to maximize this expectation. Methods differ in how step (ii) is performed. For example, hillclimb-MLE (Neil et al., 2018) performs maximum-likelihood training on the top $\mathbf { k }$ sequences from step (i). Similarly, Feedback GAN (FBGAN) uses samples whose target function value $f ( x )$ exceeds a fixed threshold for training a generative adversarial network (Gupta & Zou, 2018). Design by Adaptive Sampling (DbAs) performs weighted MLE of variational autoencoders (Kingma $\&$ Welling, 2014), where a sample’s weight corresponds to the probability that $f ( x )$ is greater than a quantile cutoff under an noise model (Brookes & Listgarten, 2018). In Brookes et al. (2019b), $p _ { \theta } ( x )$ is further restricted to stay close to a prior distribution over sequences.
+
+An alternative approach for optimizing the above expectation is RL. While RL has been used for generating natural text (Bahdanau et al., 2016), small molecules (Zhou et al., 2019), and RNA sequences that fold into a particular structure (Runge et al., 2018), we are not aware of applications of RL to optimizing DNA and protein sequences.
+
+DyNA PPO is related to existing work on model-based RL for sample efficient control (Deisenroth & Rasmussen, 2011; Kurutach et al., 2018; Peng et al., 2018; Kaiser et al., 2019; Janner et al., 2019), with the key difference that the state transition function is known and the reward function is unknown in our work, whereas most existing model-based RL approaches seek to model the state-transition function and consider the reward function as known.
+
+Prior work on sequence generation incorporates non-differentiable rewards, like BLEU in machine translation, via weighted maximum likelihood (MLE). Norouzi et al. (2016) introduce reward augmented MLE, while Bahdanau et al. (2016) fine tune an MLE-pretrained model using actor-critic methods. Reinforcement learning has also been applied to solving combinatorial optimization problems (Bello et al., 2016; Bengio et al., 2018; Dai et al., 2017; Kool et al., 2018). In this setting sample complexity is less important because evaluating $f ( x )$ only involves a fast software program.
+
+Recent work has proposed generative models of protein structures (Sabban & Markovsky, 2019) or generative models of amino acids conditional on protein structure (Ingraham et al., 2019). Such methods are outside of the scope of this paper’s experiments, since they could only be used in experimental settings where protein structures, which are expensive to measure, are available.
+
+Finally, DNA and protein design differs from small molecule design (Griffiths & Hernandez-Lobato, ´ 2017; Kusner et al., 2017; Gomez-Bombarelli et al., 2018; Jin et al., 2018; Sanchez-Lengeling & ´ Aspuru-Guzik, 2018; Korovina et al., 2019) in the following points: (i) the number of sequences measured in parallel in the lab is typically higher (hundred or thousands vs. dozens) due to the maturity of DNA synthesis and sequencing technology, (ii) the search space is a set of sequences instead of molecular graphs, which require specialized network architectures for both discriminative and generative models, and (iii) molecules must be optimized subject to the constraint that there is a set of reactions to synthesize them, whereas practically all DNA or protein sequences are synthesizable.
+
+# 4 EXPERIMENTS
+
+In the next three sections, we compare DyNA PPO to existing methods on three in-silico optimization problems that we designed in collaboration with life scientists to faithfully simulate the behavior of real wet-lab experiments, which would be cost prohibitive for a comprehensive methodological evaluation. Along the way, we present ablation experiments to help to better understand the behavior of DyNA PPO.
+
+We compare the performance of model-free policy optimization (PPO) and model-based optimization (DyNA PPO) with the following methods that we discussed in Section 3. Further details for each method can be found in Appendix A:
+
+• RegEvolution: Local search based on regularized evolution (Real et al., 2019), which has performed well on other black-box optimization tasks and can be seen as an instance of directed evolution. DbAs: Cross-entropy optimization using variational autoencoders (Brookes & Listgarten, 2018). FBGAN: Cross entropy optimization using generative adversarial networks (Gupta & Zou, 2018). Bayesopt GP: Bayesian optimization using a Gaussian process regressor and activation maximization as acquisition function solver.   
+• Bayesopt ENN Bayesian optimization using an ensemble of neural network regressors and activation maximization as acquisition function solver.
+
+• Random: Guessing sequences uniformly at random.
+
+We quantify optimization performance by the cumulative maximum reward $f ( x )$ for sequences proposed up to a given round, and we use the area under the cumulative maximum reward curve to summarize one optimization trajectory as a single number. We quantify sequence diversity (Section 2.4) in terms of the mean pairwise hamming distance between the sequences proposed at each round. For problems with known optima, we also report the fraction of global optima found. We replicate experiments with 50 random seeds.
+
+![](images/a8c895b2dfeed479fa28cd4073c9bd3436d944ac8b8dcf2dd16e1ee7df5054e8.jpg)  
+4.1 OPTIMIZATION OF PROTEIN CONTACT ISING MODELS
+
+![](images/352b5c48a23f7ec5b53cc1e2d218a1319bff4e9239991679c92ef28c14103214.jpg)  
+Figure 1: Comparison of methods on optimizing the energy of protein contact Ising models. Left: the cumulative maximum reward depending on the number of rounds for one selected protein target (1A3N). Right: the mean cumulative maximum relative to Random for alternative protein targets. Since ${ \overline { { f ( x ) } } }$ can be wellapproximated by a model trained on few examples, model-based training (DyNA PPO) results in a clear improvement over model-free training (PPO).   
+Figure 2: Analysis of the performance of DyNA PPO on the Ising model. Left: Performance of DyNA PPO depending on the number of inner policy optimization rounds using the surrogate model. Using 0 rounds corresponds to PPO training. Since the surrogate model is sufficiently accurate, it is useful to perform many inner loop optimization rounds before querying $f ( x )$ again. Right: the $R ^ { 2 }$ of the surrogate model. Since it is always above the threshold for model-based training (0.5; dashed line), it is always used for training.
+
+We first consider synthetic black-box optimization problems based on the 3D structure of naturallyoccurring proteins. Ising models fit on sets of evolutionary-related protein sequences have been shown to be accurate predictors for proteins’ 3D structure (Shakhnovich & Gutin, 1993; Weigt et al., 2009; Marks et al., 2011; Sułkowska et al., 2012). We consider the inverse problem: given a protein, we seek to find the amino acid sequence that minimizes the energy of the Ising model parameterized by its structure. Optimizers are given a budget of 10 rounds with batch size 1000 and we consider sequences of length 50 (search space size $2 0 ^ { \overline { { 5 } } 0 }$ ). The functional form of the energy function is given in Appendix B.1.
+
+On the left of Figure 1 we consider the optimization trajectory for a representative protein and on the right we compare the best $f ( x )$ found for each method across a range of proteins. We find that DyNA PPO considerably outperforms the other methods. We expect that this is because this synthetic reward landscape can be well-described by a model fit using few examples, which also explains the good performance of Bayesian optimization. On the left of Figure 2 we vary the number of inner-loop policy optimization rounds with observations from the model-based environment, where using 0 rounds corresponds to performing standard PPO. Since the surrogate model is of sufficient accuracy already at the beginning (right plot), performing more inner policy optimization rounds increases performance and enables DyNA PPO to generate high-quality sequences using very few evaluations of $f ( x )$ .
+
+![](images/c5743d6fa6f7de86e53f31654dd259cca17a6e4a0bfa425e8c772c2420d37afe.jpg)  
+4.2 OPTIMIZATION OF TRANSCRIPTION FACTOR BINDING SITES   
+Figure 3: Comparison of methods on optimization transcription factor binding sites. Left: maximum cumulative reward $f ( x )$ as a function of samples. Right: fraction of local optima found. DyNA PPO optimizes $f ( x )$ faster and finds more optima than PPO and baseline methods. Results are shown for one representative transcription factor target (SIX6 REF R1).
+
+Table 1: Mean rank of methods across transcription factor binding targets. Mean rank of methods across all 41 hold-out transcription factor targets. Ranks were computed within each target using the average of metrics across optimization rounds, and then averaged across target. The higher the rank the better. 7 is the maximum rank. DyNA PPO outperforms the other methods on both optimization of $f ( x )$ and its ability to identify multiple well-separated local optima.   
+
+<table><tr><td></td><td>DyNA PPO</td><td>PPO</td><td>BO-GP</td><td>DbAs</td><td>RegEvol</td><td>FBGAN</td><td>Random</td><td></td></tr><tr><td>Cumulative maximum</td><td>6.4</td><td>5.8</td><td>5.0</td><td>3.7</td><td>3.7</td><td>2.2</td><td>1.3</td><td></td></tr><tr><td>Fraction optima found</td><td>6.8</td><td>5.6</td><td>5.4</td><td>3.3</td><td>3.3</td><td>2.5</td><td>1.0</td><td></td></tr><tr><td>Mean hamming distance</td><td>5.6</td><td>5.4</td><td>4.0</td><td>2.5</td><td>1.0</td><td>2.5</td><td>7.0</td><td></td></tr></table>
+
+Transcription factors are protein sequences that bind to DNA sequences and regulate their activity. Barrera et al. (2016) measured the binding affinity of numerous transcription factors against all possible length-8 DNA sequences $( V = 4 )$ ). The resulting dataset defines 158 different discrete optimization tasks, where the goal of each task is to find a DNA sequence of length eight that maximizes the affinity towards one of the transcription factors. It is well suited for in-silico benchmarking since (i) it is exhaustive and thereby does not require estimating missing $f ( x )$ and (ii) the distinct local optima of all tasks are known and can be used to quantify exploration (see Appendix B.2 for details). The optimization methods are given a budget of 10 rounds with a batch size of $B = 1 0 0$ sequences. The search space size is $4 ^ { 8 }$ . We use one task (CRX REF R1) for optimizing the hyper-parameters of all methods, and test performance on 41 heterogeneous hold-out tasks.
+
+Figure 3 plots the performance of methods on a single representative binding target (SIX REF R1) as a function of the total number of sequences measured so far. We find that DyNA PPO and PPO outperform all other methods in terms of both the cumulative maximum $f ( x )$ found as well as the fraction of local optima discovered. We also find that the diversity of proposed sequences quantified by the fraction of global optima found is high compared to other generative approaches. This shows that our method continues to explore the search space by proposing novel sequences instead of converging to a single sequence or a handful of sequences–a desired property as discussed in Section 2.4. Across all tasks DyNA PPO and PPO rank highest compared with other methods (Table 1).
+
+![](images/b0ca15567d665cd8a6018d3425169057d4da3643d7cf7fad42f7010dc0f580dd.jpg)  
+Figure 4: Analysis of the progression of model-based training on the transcription factor task. Left: the mean $R ^ { 2 }$ model score averaged across replicas as a function of the number of training samples. The horizontal dashed line indicates the minimum threshold (0.5) for model-based training. Right: the fraction of replicates that performed model-based training based on this threshold. Shows that models tend to be inaccurate in early rounds and are therefore not used for model-based training. This explains the relatively small improvement of DyNA PPO over PPO in Figure 3.
+
+![](images/7428acf6943c912de8c0f49fb2113247d2cceb5bae0de12e4bcc636023b01be5.jpg)  
+Figure 5: Comparison of the proposed exploration bonus vs. entropy regularization on the transcription factor task. Left: performance with exploration bonus as a function of the density penalty $\lambda$ (Section 2.4). Right: performance of entropy regularization as a function of the regularization strength. The top row shows that PPO finds about $80 \%$ of local optima with a relatively mild density penalty of $\lambda = 0 . 1$ , whereas only about $45 \%$ local optima are found when using entropy regularization. The bottom row shows that varying the density penalty enables to control the sequence diversity quantified by the mean pairwise hamming distance between sequences.
+
+In Figure 4 and 5, we analyze the effects two key design decisions of DyNA PPO: model-based training and promoting exploration. We find that automated model selection automatically increases the complexity of the model, but that the models are not always accurate enough to be used for modelbased training. This explains the relatively small improvement of DyNA PPO over PPO. We also find that the exploration bonus outlined in Section 2.4 is more effective than entropy regularization in finding multiple local optima and promoting sequence diversity.
+
+# 4.3 OPTIMIZATION OF ANTI-MICROBIAL PEPTIDES
+
+Next, we seek to design antimicrobial peptides (AMPs). AMPs are relatively short (8 - 75 amino acids) protein sequences $\left( \left| V \right| \right. = 2 0$ amino acids), which are promising candidates against multiresistant pathogens due to their wide range of antimicrobial activities. We use the dataset proposed by Witten & Witten (2019), which contains 6,760 unique AMP sequences and their antimicrobial activity towards multiple pathogens. We follow Witten & Witten (2019) for preprocessing the dataset and generating non-AMP sequences as negative training samples. Unlike the transcription factor binding site dataset, we do not have wet-lab experiments for every sequence in the search space. Therefore, we fit random forest classifiers to predict if a sequence is antimicrobial towards a certain pathogen in the dataset (see Section B.3), and use the predicted probability as the functional measurement $f ( x )$ to optimize. Given the high accuracy of the classifiers (cross-validated AUC 0.94 and 0.99), we expect that the reward landscape of $f ( x )$ is of realistic difficulty. We perform 8 rounds with a batch size 250 and restrict the sequence length to at most 50 characters (search space size $2 0 ^ { 5 0 }$ ).
+
+![](images/28e89c97ea9aa8a0c13a60490cf968a607cf9caa02a7694150330ac5028b4918.jpg)  
+Figure 6: Comparison of methods on the AMP design task. Left: Model-based training using DyNA PPO and model-free PPO clearly outperform the other methods in terms of the maximum cumulative reward. Right: The mean pairwise hamming distance between sequences proposed at each round, which is lower for DyNA PPO and PPO but does not converge to zero due to the density-based exploration bonus (Figure 11).
+
+Figure 6 compares methods on C. alibicani. We find that model-based optimization using DyNA PPO enables finding high reward sequences in early rounds, though model-free PPO slightly surpasses the performance of DyNA PPO later on. Both DyNA PPO and PPO considerably outperform the other methods in terms of the maximum $f ( x )$ found. The density based exploration bonus prevents PPO and DyNA PPO from generating non-unique sequences (Figure 11). Stopping modelbased training as soon as the model uncertainty increased by a certain factor prevents DyNA PPO from converging to a sub-optimal solution when performing many model-based optimization rounds (Figure 12,13).
+
+# 5 CONCLUSION
+
+We have shown that RL is an attractive alternative to existing methods for designing DNA and protein sequences. We have proposed DyNA PPO, a model-based extension of PPO (Schulman et al., 2017) with automatic model selection that improves sample efficiency, and incorporates a reward function that promotes exploration by penalizing identical sequences. By approximating an expensive wet-lab experiment with a surrogate model, we can perform many rounds of optimization in simulation. While this work has been focused on showing the benefit of DyNA PPO for biological sequence design, we believe that the large-batch, low-round optimization setting described here may well be of general interest, and that model-based RL may be applicable in other scientific and economic domain.
+
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+
+# A IMPLEMENTATION DETAILS
+
+# A.1 REGULARIZED EVOLUTION
+
+Regularized evolution is a variant of directed evolution that regularizes the search by keeping a fixed number of individuals alive as candidates for selection (analogous to death by aging). At each round, it generates a batch of child sequences by sampling two parent sequences per child from the population via tournament selection, i.e. selecting the fittest out of K randomly sampled individuals. It then performs crossover of the two parent sequences by copying the characters of one parent from left to right and randomly transitioning to transcribing from the other parent sequence with some crossover probability at each step. Child sequences are mutated by substituting characters independently by other characters with some substitution probability. For variable-length sequences, we also allowed insertion and deletion mutations. As hyper-parameters, we tune the tournament size, substitution-, insertion-, and deletion-probabilities.
+
+# A.2 MCMC AND SIMULATED ANNEALING
+
+MCMC and simulated annealing (Kirkpatrick et al., 1983) resemble evolution with no crossover, and selection only occurring between an individual and its parent. Beginning with a random population, each individual evolves as a single chain, with neighborhood structure defined by the mutation operator described in section A.1. We denote $x$ and $x ^ { \prime }$ as a parent and child sequence, respectively. A transition $x  x ^ { \prime }$ is always accepted if the reward increases $( f ( x ^ { \prime } ) > f ( \bar { x } ) )$ . Otherwise, the transition is accepted with some acceptance probability. For MCMC, the acceptance probability is $f ( x ^ { \prime } ) / f ( x )$ , while for simulated annealing it is $\exp ( ( { \dot { f } } ( x ^ { \prime } ) - f ( x ) ) / T )$ for some temperature $T$ . A high temperature increases the likelihood of accepting a move that decreases the reward. The next mutation on the chain begins from $x$ if the transition is rejected, and from $x ^ { \prime }$ otherwise. We treated the temperature $T$ as a tunable hyper-parameter in addition to the evolution hyper-parameters described in section A.1.
+
+# A.3 FEEDBACK GAN
+
+We follow the methodology suggested by Gupta & Zou (2018). Instead of using a constant threshold for selecting positive sequences as described in the original publication, we used a quantile cutoff, which does not depend on the absolute scale of $f ( x )$ and performed better in our experiments. As hyper-parameters, we tuned the quantile cutoff, learning rate, batch size, discriminator and generator training epochs, the gradient penalty weight, the Gumble softmax temperature, and the number of latent variables of the generator.
+
+# A.4 DBAS
+
+We follow the methodology suggested by Brookes & Listgarten (2018). As hyper-parameters, we optimized the quantile for selecting training samples, learning rate, batch size, training epochs, number of hidden units of the MLP generator and discriminator, and number of latent variables. The generative model is an variational autoencoder with a multi-layer perceptron decoder. We also considered DbAs with a LSTM as generative model, which performed slightly better than a VAE on the TfBind8 problem but worse on the PdbIsing and AMP problem (see Figure 8).
+
+# A.5 BAYESIAN OPTIMIZATION
+
+As regressors, we considered a Gaussian process (GP) with RBF kernel on one-hot features, and an ensemble of ten fully-connected neural networks with one fully connected layer and 128 hidden units. We used the regressor output to compute the expected improvement or posterior mean acquisition function, which we maximized by gradient ascent for a certain number of acquisition steps following Killoran et al. (2017). We took the resulting $B$ unique sequences with highest acquisition function value as sequences to measure in the next round. We tuned the length scale and variance of the RBF kernel, and the learning rate, batch size, and number of training epochs of the neural network ensemble. We further tuned the number of gradient ascent steps for activation maximization.
+
+# A.6 PPO AND DYNA PPO
+
+We used the PPO implementation of the TF-Agents RL library (Guadarrama et al., 2018). After each round, we trained trained the agent on the collected batch of sequences for a relatively high number of steps (about 72) since it resulted in a performance increase compared with performing only a single training step. We used the adaptive KL trust region penalty, which performed slightly better than importance ratio clipping in our experiments Schulman et al. (2017). We used a policy and value network with one fully connected layer and 128 hidden units. Both networks take the current position and the $W$ last generated characters as input, which we padded at the beginning of the sequence. We set the context window $W$ to the minimum of the total sequence length and 50. As hyper-parameters, we tuned the learning rate, number of training steps, adaptive KL target, and entropy regularization. For DyNA PPO, we also tuned the maximum number of model-based optimization rounds $M$ (see Section 2.3).
+
+# A.7 AUTOMATED MODEL SELECTION
+
+Automatic model selection optimizes the hyper-parameters of a set of candidate models by randomized search, and evaluates each hyper-parameter configuration by five-fold cross-validation using the $R ^ { 2 }$ score. To account for randomness in the $R ^ { 2 }$ score between models due to different crossvalidation splits, we used the same split for evaluating each of the models per round. We considered the following candidate models (implemented in Scikit-learn (Pedregosa et al., 2011)) and corresponding hyper-parameters:
+
+• KNeighborsRegressor: n neighbors   
+• BayesianRidge: alpha 1, alpha 2, lambda 1, lamdba 2   
+• RandomForestRegressor: max depth, max features, n estimators   
+• ExtraTreesRegressor: max depth, max features, n estimators   
+• GradientBoostingRegressor: learning rate, max depth, n estimators   
+• GaussianProcessRegressor: with RBF, RationalQuadratic, and Matern kernel
+
+We also considered an ensemble of 10 neural networks with two convolutional layers and one fully connected layer, and optimized the learning rate and number of training epochs.
+
+# B DATASET DETAILS
+
+# B.1 PROTEIN CONTACT ISING MODELS
+
+Given a protein from the Protein Data Bank (Berman et al., 2003), we compute the energy $E ( x )$ for sequence $x$ as $\begin{array} { r } { E ( x ) = \sum _ { i } \phi _ { i } ( x _ { i } ) + \sum _ { i j } C _ { i j } \phi ( x _ { i } , x _ { j } ) } \end{array}$ , where $x _ { i }$ refers to the character in the $i$ -th position of sequence $x$ . $C _ { i j }$ is an indicator for whether the $C \alpha$ atoms of the residues at positions $i$ and $j$ are separated by less than 6 Angstroms when the protein folds. $\phi ( x _ { i } , x _ { j } )$ is a widely-used ‘pair potential’ based on co-occurence probabilities derived from the structures of real-world proteins (Miyazawa & Jernigan, 1996). The same $2 0 \times 2 0$ table of pair potentials is used at all positions in the sequence, and thus the difference in energy functions across proteins is dictated only by their differing contact map structure. We set the local term $\phi _ { i } ( x _ { i } )$ to zero. In future work, it would be interesting to consider non-zero local terms.
+
+Our experiments consider a set of qualitatively-different proteins listed at the bottom-right of Figure 1. We identify the local optima using the same procedure as in Section B.2, except without accounting for reverse complements.
+
+# B.2 TRANSCRIPTION FACTOR BINDING SITE DATASET
+
+We used the dataset described by Barrera et al. (2016), and min-max normalized binding affinities between zero and one. To reduce computational costs, we only considered the first replicate (REF R1) of each wild type transcription factor in the dataset, which resulted in 41 optimization targets that we used for comparing optimizers as described in Section 4.2. We extracted local optima for each binding target as follows. First, we separated sequences into forward and reverse sequences by ordering sequences lexicographically and including each sequence in the set of forward sequences unless the set already contained its reverse complement. We then chose the 100 forward sequences with the highest binding affinity and clustered them using the hamming distance metric, where we determined the number of clusters by finding the number of PCA components required to explain $9 5 \%$ of variance. We then used the sequences with the highest reward per cluster and their reverse complement as local optima.
+
+# B.3 ANITMICROBIAL PEPTIDE DATASET
+
+We downloaded the dataset1 provided by Witten & Witten (2019), and followed the paper for preprocessing sequences and generating non-AMP sequences as negative training samples. We additionally excluded sequences containing cysteine and sequences shorter than 15 or longer than 50 amino acids. We fit one classifier to predict if a sequence is antimicrobial towards either E.coli, S.aureus, P.aeruginosa, or B.subtilis, which we used for hyper-parameter tuning, and a second classifier for C. alibicani, which we used for hold-out evaluation. We used C. alibicani as hold-out target since its antimicrobial activity was least correlated with the activity of other pathogenes in the dataset with more than 1000 AMP sequences. We used random forest classifiers since they were more accurate (cross-validated AUC 0.99 and 0.94) than alternative models such as k-nearest neighbors, Gaussian processes, or neural networks. Since sequences are variable-length, we padded them to the maximum sequence length of 50 and extended the vocabulary by an additional end of sequence token. Tokens after the fist end of sequence token were ignored when evaluating $f ( x )$ .
+
+![](images/f05395fabb70d742627f137243f2b5a8b60f46c3de926e2b67a4ce6292c031ca.jpg)  
+Figure 7: Comparison of PPO against alternative RL methods. Shown are the cumulative maximum reward and mean pairwise hamming distance for the transcription factoring binding, protein Ising, and AMP problem (Section 4).
+
+DyNA PPO is built on PPO, which we have found to outperform other policy-based and value-based RL methods in practice on our problems. In Figure 7 we contrast the performance of PPO (Schulman et al., 2017), REINFORCE (Williams, 1992), deep Q-learning (DQN) (Mnih et al., 2015), and categorical distributional deep Q-learning (CatDQN) (Bellemare et al., 2017) on all problems considered in Section 4. We find that PPO has better exploration properties than REINFORCE, which tends to converge too soon to a local optimum. The poor performance of DQN and CatDQN can be explained by the sparse reward (the reward is only non-zero at the terminal state), such that the Bellman error and training loss for updating the Q network are zero in most states. We also found the performance of DQN and CatDQN to be sensitive to the choice of the epsilon greedy rate and Boltzmann temperature for trading-off exploration and exploitation and increasing diversity.
+
+![](images/5faac59e0be4818c1ddf3058a7f7c96e32fcb97fd127c73e2939aadd775baa75.jpg)  
+Figure 8: Comparison of additional baselines. We consider the performance of optimizers based on MCMC (Section A.2). Such methods are known to be effective optimizers when evaluating the black-box function is inexpensive, and thus many iterations of sampling can be performed. The focus of our experiments is on resource-constrained black-box optimization. We find that their low sample efficiency makes them undesirable for biological sequence design. We also consider DbAS with a LSTM generative model instead of a VAE with multi-layer perceptron decoder to disentangle the choice of generative model in DbAS from the overall optimization strategy. DbAs VAE outperforms DbAs RNN on all problems except for TF bind.
+
+![](images/340227aabbbf9c2cbeaac43d335205bb9d9c3be21f4f1bec8349e66f7df20277.jpg)  
+Figure 9: Comparison of alternative approaches for promoting diversity. Left column: The proposed density based exploration bonus as described in Section 2.4, which adds a penalty to the reward of a sequence $\mathbf { X }$ that is proportional to the distance-weighted number of past sequence that are less than a specified distance away from $\mathbf { X }$ (here edit distance one). Middle column: An alternative approach where the exploration bonus of a sequence is proportional to the distance to the nearest neighboring past sequence. Right column: standard entropy regularization. Shown are the cumulative maximum reward and alternative metrics for quantifying diversity depending on the penalty strength $\boldsymbol { \lambda }$ in Section 2.4) of each exploration approach. Without exploration bonus (penalty $= 0 . 0$ ; red line), PPO does not find the optimum (cumulative maximum is below 1.0) and the hamming distance and uniqueness of sequences within a batch converge to zero. PPO finds the optimal solutions and continues to generate diverse sequences by increasing the strength of any of the three exploration approaches. The density based exploration bonus is most effective in recovering all optima (second row, left plot) and enables a more fine-grained control of diversity compared to the distance based approach. Results are shown for target CRX REF R1 of the transcription factor binding problem.
+
+![](images/39ec12d418b486751f8e80ffd430c1f960d6e0c27cff30d3521f62647f95aa98.jpg)  
+Figure 10: tSNE embedding of proposed sequences with and without exploration bonus colored by the optimization round when they were proposed. Without exploration bonus (penalty $= 0 . 0$ ), sequences cluster into few groups of low diversity at different rounds. With the exploration bonus, the diversity of proposed sequences remain high also in later rounds. Results are shown for target CRX REF R1 of the transcription factor binding problem.
+
+![](images/c0293c8a3e0795a5d864d6fb7ffd35ab2e272b4b0f3cdb80795b74ab158fdf05.jpg)  
+Figure 11: Analysis of density-based exploration bonus on the AMP problem. The top row shows the sensitivity to the distance radius $\epsilon$ and the bottom row to the regularization strength $\lambda$ (Section 2.4). Diversity correlates positively with the distance radius and regularization strength. $\epsilon = 2$ and $\lambda = 0 . 1$ provides the best trade-off between optimization performances (cumulative maximum reward) and diversity (mean pairwise hamming distance and uniqueness). Penalizing only exact duplicates ${ \epsilon = 0 }$ ) is less effective in maintaining a high hamming distance than taking neighboring sequences into account $( \epsilon > 0 )$ ).
+
+![](images/3f6ea9695fab459b1c00e58c1b2eef7225f62a6553639551e1ba07958dba51c2.jpg)  
+Figure 12: Analysis of model accuracy during model based training on the AMP problem. Shown are the mean absolute error (top row) and model uncertainty (bottom row; standard deviation) depending on the number of inner model-based optimization rounds $m$ $\mathbf { \widetilde { x } }$ -axis; see Algorithm 1) and outer optimization rounds $n$ (colors). Columns correspond to different thresholds for stopping model optimization (see Section 2.3). Without threshold (right column), the model error and model uncertainty increase rapidly after only a few inner optimization rounds. The model uncertainty is strongly correlated with the model error $\cdot R ^ { 2 } = 0 . 8 7 )$ , and can be hence used as proxy for the unknown model error. Stopping model-based optimization as soon the the model uncertainty increases by a factor of 0.5 (left column) upper-bounds the model error and prevents DyNA PPO from performing policy updates with inaccurate model predictions.
+
+![](images/777e298e95d756eb4d5fa66cb8e4c2ea6721ad134bc81384d86376c97929f265.jpg)  
+Figure 13: Optimization performance on the AMP problem depending on the uncertainty threshold for stopping model-based optimization and the maximum number model optimization rounds M. Without threshold (Inf; red line), DyNA PPO converges to a sub-optimal solution, in particular when the maximum number of model-based optimization rounds M is high. A threshold of 0.5 prevents a performance decrease due to inaccuracy of the model (see Figure 12).
+
+# G COMPARISON OF CROSS-VALIDATION SPLITTING STRATEGIES
+
+We used the $\mathbf { k }$ -fold cross-validation tools in scikit-learn for performing model selection. After publication of the paper, we discovered that the default behavior in sklearn.model selection.KFold is to not shuffle the input data but to slice them into chunks based on the input ordering.
+
+When we switched to using random cross-validation folds, we found that the predictive accuracy of models was considerably higher than when using folds based on the data order. This led to different models being selected, which led to a slight decrease in black-box optimization performance compared to when not shuffling the input data (Figure 15).
+
+Our data was sorted in the order in which it appeared in the optimization rounds. Hence, k-fold cross-validation without shuffling corresponds to splitting the data approximately by rounds, which favorably selects models that generalize across rounds. This is desired since samples in the same round tend to be correlated. It splits the data only approximately by rounds if the number of folds $\mathbf { k }$ is not equal to the number of rounds performed so far.
+
+![](images/b498a304ad78bf08a5cd315e90afc70bdb2d2d6151f86be1c5f2d18b281f88c5.jpg)  
+Figure 14: Sensitivity of DyNA PPO depending on the choice of the minimum cross-validation score $\tau$ for model-based optimization. Shown are the results for the transcription factor binding-, protein contact Ising-, and AMP problem. DyNA PPO reduces to PPO if $\tau$ is above the maximum cross-validation score of models that are considered during model selection, e.g. if $\tau = 1 . 0$ . If $\tau$ is too low, also inaccurate models are selected, which reduces the overall accuracy of the ensemble model and optimization performance. A cross-validation score between 0.4 and 0.5 is best for all problems.
+
+In response, we ran experiments using a true round-based split, which performed similarly to splitting the data approximately by rounds using sklearn.model selection.KFold with shuffle $\vDash$ False. Based on the similar performance of these two slitting strategies, we did not change the experiments in the paper.
+
+![](images/b0580457816357a86383122cefc690808297b4ca621563761433a16143346551.jpg)  
+Figure 15: Comparison of cross-validation splitting strategies. Shown are the mean optimization trajectories over 12 TfBind8 targets and 10 Protein Ising models when splitting the input data into 5 folds with shuffling the input data (Shuffle), without shuffling the input data (NoShuffle), and splitting the input data by optimization rounds (BatchShuffle). Splitting the input data approximately by rounds (NoShuffle) or exactly by rounds (BatchShuffle) results in a performance increase on Protein Ising problems compared with splitting the data randomly (Shuffle). NoShuffle performs as well as BatchShuffle on TfBind8, and better than BatchShuffle and Shuffle on all PdbIsing targets.

parse/train/KIfbqntFnOc/KIfbqntFnOc.md

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+# ROBUST ENSEMBLES OF NEURAL NETWORKS USING ITOˆ PROCESSES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Residual neural networks (ResNets) can be modeled as dynamical systems where the evolution of dynamical systems represents the inference in ResNets. We exploit this connection and the theory of stochastic dynamical systems to construct a novel ensemble of Ito processes as a new deep learning representation ˆ that is more robust than classical residual networks. An Ito process obtained by ˆ solving a suitably-formulated stochastic differential equation derived from a residual network has a probability density function that is not readily perturbed by small changes in the neural network’s inputs. Our robust stochastic Itoˆ ensemble of neural networks achieve an accuracy of $7 3 . 9 1 \%$ on the CIFAR-10 dataset against the PGD attack with $\epsilon \ : = \ : 2 . 0$ under the $L _ { 2 }$ norm, while the accuracy of Madry’s robustness toolbox on the same attack is $1 8 . 5 9 \%$ . Similarly, our stochastic Ito ensemble of neural networks achieves an accuracy of ˆ $7 9 . 6 6 \%$ on PGD attack with $\epsilon = 1 6 / 2 5 5$ under the $L _ { \infty }$ norm, while the accuracy of Madry’s robustness toolbox on the same attack is $1 8 . 1 3 \%$ . The Ito ensemble ˆ trained on ImageNet achieves an accuracy of $2 8 . 5 3 \%$ against PGD attacks under the $L _ { \infty }$ norm with $\epsilon = 1 6 / 2 5 5$ and accuracy of $6 5 . 7 4 \%$ under the $L _ { 2 }$ norm with $\epsilon = 3 . 0$ , respectively. This significantly improves state-of-the-art accuracy of $5 \%$ and $3 5 . 1 6 \%$ for Madry’s robustness tool against the same PGD attacks under the $L _ { \infty }$ and $L _ { 2 }$ norms, respectively. Further, our approach achieves these high robustness values without any explicit adversarial training or a significant loss of accuracy on benign inputs.
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) have emerged as a very effective learning representation achieving near human-level performance in many domains such as computer vision (Gkioxari et al., 2015), natural language processing (Majumder et al., 2017), and speech recognition (Hannun et al., 2014). Despite this success, the use of deep learning models in high-assurance systems with safety and security requirements such as autonomous vehicles (Bojarski et al., 2016) and medical diagnoses (De Fauw et al., 2018) faces a trust deficit. The lack of robustness of these models and their susceptibility to adversarial attacks (Kurakin et al., 2016; Szegedy et al., 2013) that can change the prediction of a deep neural network via small imperceptible perturbations make deep learning models less trustworthy. This limitation is further aggravated by deep neural networks generally exhibiting very high confidence on incorrect predictions (Guo et al., 2017a; Hendrycks & Gimpel, 2016). Consequently, this lack of robustness hinders their deployment in safety-critical applications. There is a pressing need for a principled approach to learning robust deep learning models that are resilient to adversarial attacks and can abstain from making decisions on inputs for which they are likely to make a wrong prediction.
+
+A number of approaches have been recently proposed to increase the robustness of deep learning models. Adversarial training (Tramer et al., 2017; Engstrom et al., 2020) uses adversarial samples \` in the training phase to make the models more robust. Another set of alternative approaches use the projection of inputs to data manifold (Lamb et al., 2018; Ilyas et al., 2017; Jang et al., 2020) or other preprocessing methods (Xie et al., 2019; Guo et al., 2017b). These approaches are robust to existing attack methods but their use of adversarial samples or predefined transformations (often achieved via another deep neural network such as autonecoders) makes these approaches susceptible to newer attack strategies. Certifiable-defense approaches (Wong et al., 2018; Dvijotham et al., 2018;
+
+Raghunathan et al., 2018; Dutta et al., 2018) have also been recently proposed to make deep learning models robust against worst-case input over a defined range of perturbations. These theoretical guarantees on worst-case inputs hold only for small perturbations; consequently, their use is limited in practice and their performance is typically inferior to approaches based on adversarial training, particularly for high-dimensional inputs.
+
+In this paper, we address this challenge of robust and trustworthy deep learning using a new representation that exploits the connection between dynamical systems and residual neural networks (ResNets), and uses the theory of stochastic dynamical systems. Dynamical systems can model ResNets where the inference in the network is represented by the evolution of the dynamical system (Chen et al., 2015; Chang et al., 2017; Sonoda & Murata, 2017; Chen et al., 2018; Lu et al., 2018). We construct a novel deep ensemble using a special class of stochastic dynamical systems, namely the Ito drift-diffusion process with suitably bounded diffusion term. It ˆ o process is the sum ˆ of the integral of a process over time and of another process over a Brownian motion. The drift over time models the typical inference in a ResNet and the diffusion Brownian motion models the added stochastic noise that makes the model robust to adversarial perturbations. We form an ensemble of these Ito processes by considering multiple such models and multiple inferences over the same model. If a majority of the ensemble agrees on a particular prediction, Ito ensemble ˆ makes that prediction; otherwise, it abstains from making a decision.
+
+<table><tr><td>Robustness Approach</td><td>Accuracy (%)</td><td rowspan="2">Benchmark</td><td rowspan="2">Norm</td><td colspan="2">Accuracy (%)</td></tr><tr><td>Ito Ensemble</td><td>84.60</td><td>Ito Ensemble</td><td>Madry toolbox</td></tr><tr><td>Engstrom et al. (2020)</td><td>53.49</td><td>CIFAR-10</td><td>L2</td><td>73.91</td><td>18.59</td></tr><tr><td>Balunovic &amp; Vechev (2020)</td><td>46.2</td><td>ImageNet</td><td>L2</td><td>69.51</td><td>43.04</td></tr><tr><td>Zhang et al. (2019)</td><td>40.5</td><td>CIFAR-10</td><td>L</td><td>79.66</td><td>18.13</td></tr><tr><td>Pang et al. (2019) (∈=0.01)</td><td>48.4</td><td>ImageNet</td><td>L8</td><td>28.53</td><td>5.00</td></tr></table>
+
+Table 1: (left) Our Ito ensemble approach outperforms SOTA defenses for the PGD attack on ˆ CIFAR-10 with $\epsilon = 8 / 2 5 5$ unless specified otherwise. (right) Ito ensemble outperforms Madry ˆ toolbox (Engstrom et al., 2020) under PGD attack with $L _ { 2 }$ norm, $\epsilon = 2 . 0$ , and $L _ { \infty }$ norm $\epsilon = 1 6 / 2 5 5$
+
+We highlight a few results demonstrating the robustness of Ito process ensembles in Table 1. Our It ˆ oˆ ensemble approach has higher accuracy compared to several state-of-the-art robustness approaches. The accuracy of our approach is $8 4 . 6 0 \%$ on CIFAR-10 against the PGD attack in $L _ { \infty }$ norm with $\epsilon = 8 / 2 5 5$ and the next best approach is Engstrom et al. (2020) (Madry toolbox) with an accuracy of $5 3 . 4 9 \%$ . On CIFAR-10 and ImageNet benchmarks, our Ito ensemble approach is significantly more ˆ robust than Engstrom et al. (2020) (Madry toolbox) against PGD attacks in both $L _ { 2 }$ and $L _ { \infty }$ norms for different values of attack strength $\epsilon$ . Thus, our Ito ensembles exhibit remarkable robustness ˆ against adversarial attacks without any explicit adversarial training.
+
+![](images/ee651e75a7ef35f0c3b36ce537787418b8143175e1fba6412c1b4db6a90edb24.jpg)  
+Figure 1: Examples of benign images on which our approach using Ito processes abstains from ˆ making a decision while the original ResNet model makes a decision despite high uncertainty. The first image is found by Ito process ensemble to be confusing between ˆ binoculars and cannon, the second between a radiator and a projector, the third between a trench-coat and bicycle, and the last one between stove and coffee-pot. This uncertainty in Ito ensemble resembles human judgement. ˆ
+
+Our approach using Ito ensembles can abstain from making decisions on a confusing input. In ˆ Section 4, we demonstrate that abstentions further improve the robustness of the Ito ensembles ˆ to adversarial examples compared to the state-of-the-art approaches. Further, we notice that Itoˆ ensembles abstain even on benign data inputs where manual inspection demonstrates high aleatoric or epistemic uncertainty as shown in Figure 1. Our experiments show that this new approach of using Ito ensembles achieves high robustness without significant loss in accuracy on benign inputs. ˆ
+
+# 2 RELATED WORK
+
+Residual neural networks are a common neural network architecture that learn only the residuals not learned by the previous layers. ResNets (He et al., 2016) are residual neural networks where residual learning is adopted for every few stacked neural network layers and such building blocks are used to design the complete residual neural network. The dynamics of ResNets and other similar neural networks can be described using ordinary and partial differential equations (Chen et al., 2015; Chang et al., 2017; Sonoda & Murata, 2017; Weinan, 2017; Chen et al., 2018; Lu et al., 2018). One timestep of the dynamics models each building block of the ResNets. Such a dynamical model enables memory efficiency in training and adaptive inference. In contrast, we use stochastic differential equations (Ito processes) and demonstrate their robustness to adversarial attacks. ˆ
+
+A number of adversarial attacks on deep neural networks have been proposed in literature and shown to be effective across different architectures. Attacks such as the fast gradient sign method (FGSM) (Szegedy et al., 2013), the projected gradient decent (PGD) (Madry et al., 2017) and other approaches (Nicolae et al., 2018) have demonstrated the fragility of deep neural networks to small perturbations in their inputs. The most effective state-of-art defenses use adversarial training (Tramer et al., 2017; Engstrom et al., 2020) or some projection or transformation of \` inputs (Lamb et al., 2018; Ilyas et al., 2017; Jang et al., 2020; Xie et al., 2019; Guo et al., 2017b). The use of adversarial examples or predefined transformations makes these approaches vulnerable to new attacks. In contrast, our approach using Ito process does not need adversarial examples. ˆ
+
+Our use of stochastic dynamical systems is inspired by their presence in biological systems (Kitano, 2004; Bressloff, 2014; Allen, 2010) where they impart robustness to external perturbations. As an example, (Arkin et al., 1998) study gene expression using Gillespie’s stochastic formulation of chemical kinetics and show that protein numbers can vary markedly from one cell to another with important consequences for biological robustness (Gonze et al., 2002).
+
+# 3 ROBUST LEARNING USING ITOˆ ENSEMBLES
+
+Residual networks (ResNets) can be modeled as dynamical systems where the evolution of the dynamical system represents the inference in ResNets (Chen et al., 2015; Chang et al., 2017; Sonoda & Murata, 2017; Chen et al., 2018; Lu et al., 2018). We connect this view to the theory of stochastic differential equations and construct an ensemble using a class of Ito processes with suitably bounded ˆ diffusion term. This Ito process ensemble exhibits remarkable robustness against adversarial attacks ˆ without any explicit adversarial training.
+
+# FROM RESNETS TO STOCHASTIC ITOˆ PROCESSES
+
+A building block of a residual neural network (He et al., 2016) with the residual mapping $\mathcal { F } ( \mathbf { x } ( i ) , \bar { \mathbf { W } } ( i ) )$ can be described using the following equation: $\mathbf { x } ( i + 1 ) = \mathcal { F } ( \mathbf { x } ( i ) , \mathbf { W } ( i ) ) + \bar { \mathbf { x } } ( i )$ . Here, ${ \bf x } ( i )$ is the input to the $i ^ { t h }$ residual network building block and $\mathbf { x } ( i + 1 )$ is the corresponding output that serves as an input to the next building block. The weights of the neural network layers in this ResNet building block are denoted by $\mathbf { W } ( i )$ .
+
+After taking suitable limits, the evolution of the ResNet can be described by the ResNet ordinary differential equation (ODE): $\begin{array} { r } { \frac { d \mathbf { x } ( t ) } { d t } = \mathcal { G } ( \mathbf { x } ( t ) , \mathbf { W } ( t ) ) } \end{array}$ . Here, $\begin{array} { r } { \mathcal { G } ( \mathbf { x } ( t ) , \mathbf { W } ( t ) ) = \operatorname* { l i m } _ { \delta t  0 } \frac { \mathcal { F } ( \mathbf { x } ( t ) , \mathbf { W } ( t ) ) } { \delta t } } \end{array}$ and ${ \bf x } ( 0 )$ is the input to the neural network. The ResNet ODE can be naturally generalized into an Ito process by using a Brownian motion term with diffusion coefficient ˆ $\Sigma ( t ) \dot { = } \overline { { ( \sigma _ { i j } ( t ) ) } }$ : $d { \bf x } ( t ) =$ $\mathcal { G } ( \bar { \mathbf { x } } ( t ) , \mathbf { W } ( t ) ) ~ d t + \bar { \Sigma } ( t ) ~ d B ( t )$ . There are two competing objectives here:
+
+• Very large values of the diffusion term $\Sigma ( t )$ can completely overshadow the drift term $\mathcal { G } ( \mathbf { x } ( t ) , \mathbf { W } ( t ) )$ leading to a poor accuracy even on benign inputs. When the diffusion term is very large, the paths of the Ito process can completely diverge from the solution of the ˆ original ResNet from which the Ito process was obtained. ˆ   
+Very small values of $\Sigma ( t )$ make the model closer to the original ResNet and equally non-robust. $\Sigma ( t ) = 0$ reproduces the original non-stochastic ResNet with no additional robustness. As we increase the diffusion term, the robustness of the neural network increases; this is experimentally demonstrated in Section 4.
+
+So, a natural question to ask is: How do we select the diffusion term $\Sigma ( t )$ such that the Ito process ˆ satisfies these two competing objectives of accuracy and robustness?
+
+At one hand, the generated Ito process must retain similar accuracy on benign models as the original ˆ ResNet, that is, its solutions are determined mainly by the term $\mathcal { G } ( \mathbf { \dot { x } } ( t ) , \mathbf { W } ( \bar { t } ) )$ and Brownian motion noise does not make it diverge significantly. On the other hand, the choice of added diffusion term $\Sigma ( t )$ must make the model robust enough to be resilient to adversarial perturbations on the inputs.
+
+ROBUSTNESS OF STOCHASTIC ITOˆ RESNET ENSEMBLES
+
+Given the Ito process ˆ ${ \bf x } ( t )$ satisfying the stochastic differential equation $d \mathbf { x } ( t ) = { \mathcal { G } } ( \mathbf { x } ( t ) , \mathbf { W } ( t ) ) d t + \Sigma ( t ) d { \dot { B } } ( t )$ , it is known (Oksendal, 1992) that the probability density $\boldsymbol { p } ( \mathbf { x } , t )$ can be mathematically characterized by the following equation:
+
+$$
+\frac { \mathrm { \partial } p ( \mathbf { x } , t ) } { \partial t } + \mathcal { G } ( \mathbf { x } , \mathbf { W } ) \nabla p ( \mathbf { x } , t ) = - p ( \mathbf { x } , t ) \sum _ { i } \frac { \partial \mathcal { G } } { \partial \mathbf { x } _ { i } } + \frac { 1 } { 2 } \sum _ { i } \sum _ { j } \frac { \partial ^ { 2 } } { \partial \mathbf { x } _ { i } \partial \mathbf { x } _ { j } } \left( ( \sum _ { k } \sigma _ { i k } ( t ) \sigma _ { j k } ( t ) ) p ( \mathbf { x } , t ) \right)
+$$
+
+map itself, that is For robust networks, the rate of change of the residual learning map is much smaller than the residual $\begin{array} { r } { \sum _ { i } \frac { \partial \mathcal { G } } { \partial \mathbf { x } _ { i } } < \eta _ { 1 } \mathcal { G } ( \mathbf { \bar { x } } , \mathbf { W } ) \frac { \nabla p ( \mathbf { x } , t ) } { p ( \mathbf { x } , t ) } } \end{array}$ for some small $\eta _ { 1 }  0$ . Hence, the probability density $\boldsymbol { p } ( \mathbf { x } , t )$ can be simplified to the following equation:
+
+$$
+\frac { \partial p ( { \bf x } , t ) } { \partial t } + ( 1 + \eta _ { 1 } ) \mathcal { G } ( { \bf x } , { \bf W } ) \nabla p ( { \bf x } , t ) = \frac { 1 } { 2 } \sum _ { i } \sum _ { j } \frac { \partial ^ { 2 } } { \partial { \bf x } _ { i } \partial { \bf x } _ { j } } \left( \left( \sum _ { k } \sigma _ { i k } ( t ) \sigma _ { j k } ( t ) \right) p ( { \bf x } , t ) \right)
+$$
+
+Using the fact that the double derivative of the probability density for robust neural networks is much smaller thfor some small an the re tself, that is 12 Pi Pj ∂2∂xi∂xj sidual map i, we choose $\begin{array} { r } { \frac { 1 } { 2 } \sum _ { i } \sum _ { j } \frac { \partial ^ { 2 } } { \partial \mathbf { x } _ { i } \partial \mathbf { x } _ { j } } \left( p ( \mathbf { x } , t ) \right) < \eta _ { 2 } \mathcal { G } ( \mathbf { x } , \mathbf { W } ) \frac { \nabla p ( \mathbf { x } , t ) } { p ( \mathbf { x } , t ) } } \end{array}$ $\eta _ { 2 }  0$ $\textstyle \sigma _ { i j } ( t ) \leq { \frac { \omega } { 1 + t } }$ $\omega$ probability density function of a ResNet with $n$ -dimensional inputs can be further simplified as
+
+$$
+\frac { \partial p ( { \bf x } , t ) } { \partial t } + ( 1 + \eta _ { 1 } - n \omega ^ { 2 } \eta _ { 2 } ) \mathcal { G } ( { \bf x } , { \bf W } ) \nabla p ( { \bf x } , t ) = 0
+$$
+
+As $\eta _ { 1 }  0$ and $\eta _ { 2 }  0$ for robust neural networks, our choice of $\textstyle \sigma _ { i j } ( t ) \leq { \frac { \omega } { 1 + t } }$ for a constant $\omega$ reduces the equation describing the probability density function to ∂p(x,t)∂t + G(x, W)∇p(x, t) = 0. Interpreting p(x, t) as a function that is constant along the trajectories of a ordinary differential equation i.e. dp(x,t)dt = 0, p(x, t) corresponds to the following differential equation: $\begin{array} { r } { \frac { d \mathbf { x } ( t ) } { d t } \ = \ \mathcal { G } ( \mathbf { x } ( t ) , \mathbf { W } ( t ) ) } \end{array}$ . Hence, under our choice of $\textstyle \sigma _ { i j } ( t ) \leq { \frac { \omega } { 1 + t } }$ for a constant $\omega$ , the solution to the stochastic differential equation agrees with the ResNet ODE for robust neural networks. Our implementation of the stochastic robust Ito ensemble of residual ˆ neural network is formed by discretizing the stochastic differential equation dx(t) = G(x(t), W(t)) dt + Σ(t) dB(t) with the constraint that σij (t) = ω1+t .
+
+# 4 RESULTS
+
+We train stochastic Ito ensembles of residual neural networks using both CIFAR-10 (Krizhevsky ˆ et al., 2014) and ImageNet (Deng et al., 2009) benchmarks. We evaluate the robustness of our stochastic Ito ensembles against two popular adversarial attacks: the fast gradient sign method ˆ (FGSM) (Szegedy et al., 2013) and the projected gradient descent (PGD) (Kurakin et al., 2016) under both $L _ { 2 }$ and $L _ { \infty }$ norms. We use the conformance in prediction of our Ito ensemble to exploit ˆ their robustness. If a majority of residual network models in our ensemble predict the same label for a given data item, the ensemble makes a prediction as this majority label. Otherwise, the stochastic Ito ensemble assigns no label and abstains from making any decision on the given input ˆ data. Our experiments indicate that this capability of the Ito ensemble to abstain from making ˆ decisions on non-conforming inputs not only helps defend the model against adversarial attacks, it also decreases incorrect predictions on benign data.
+
+# CIFAR-10 RESULTS
+
+Our experiments are performed on a 40-core 256GB RAM server with 4 NVIDIA V100 GPUs
+
+# Question 1: Does the Ito ensemble achieve competitive accuracy on benign inputs? ˆ
+
+We train the standard ResNet models on benign data and compare their accuracy with the accuracy of our stochastic Ito ensembles on benign data. We obtain the stochastic ResNet models in the ˆ stochastic Ito ensemble by starting with the weights of a standard ResNet model and training them ˆ for 40 epochs with a learning rate of 0.0001 using the Adam optimizer. Table 2 compares the accuracy of the standard model with Ito ensembles. ˆ
+
+<table><tr><td>Architecture</td><td colspan="2">Accuracy (%)</td><td colspan="2">Incorrect Prediction (%)</td><td rowspan="2">Correct + Abstention (%) Ito Ensemble (%)</td></tr><tr><td></td><td>Original</td><td>Ito Ensemble</td><td>Original</td><td>Itó Ensemble</td></tr><tr><td>ResNet-18</td><td>93.33</td><td>91.51</td><td>6.67</td><td>5.72</td><td>94.28</td></tr><tr><td>ResNet-34</td><td>92.92</td><td>91.33</td><td>7.08</td><td>5.80</td><td>94.20</td></tr><tr><td>ResNet-50</td><td>93.86</td><td>91.59</td><td>6.14</td><td>4.64</td><td>95.29</td></tr></table>
+
+Table 2: Our stochastic Ito ensembles and the standard ResNet neural network architectures haveˆ similar accuracy on CIFAR-10 test data. Because of its ability to abstain from assigning a label when majority of predictions do not conform, the fraction of data where the Ito ensemble predicts ˆ an incorrect label is lower that the fraction of data where the original ResNet model is incorrect.
+
+Our stochastic Ito ensembles ofˆ 20 models in Table 2 are trained using a diffusion term corresponding to $\omega ~ = ~ 0 . 2$ . 20 inferences are obtained from each stochastic model in our Itoˆ ensemble. The accuracy of the stochastic Ito ensembles on CIFAR-10 test data is comparable to ˆ that of the standard models on three ResNet architectures: ResNet-18, ResNet-34, and ResNet-50. Our stochastic Ito ensemble abstains when a majority of the ensemble models do not agree on a ˆ single prediction. This lowers the incorrect predictions of Ito ensemble compared to the original ˆ model. As shown in Figure 1, some of the correct predictions by original ResNet are on images with high aleatoric uncertainty on which Ito ensemble correctly abstains. ˆ
+
+# Question 2: Is the stochastic Ito ensemble robust against adversarial attacks? ˆ
+
+We train stochastic Ito ensembles of 20 ResNet-50 models on CIFAR-10 data with diffusion terms ˆ corresponding to $\omega = 0 . 2$ and $\omega = 0 . 4$ . 20 independent inferences are drawn from each stochastic model in the Ito ensemble. We evaluate the robustness of our stochastic It ˆ o ensemble against FGSM ˆ and PGD under both $L _ { 2 }$ and $L _ { \infty }$ norms. We compare the accuracy of predictions from our stochastic Ito ensembles with that of Madry’s robustness toolbox (Engstrom et al., 2020). ˆ
+
+<table><tr><td>E</td><td>Accuracy for PGD (%)</td><td>Correct + Abstention for PGD (%)</td><td>Accuracy for FGSM (%)</td><td>Correct + Abstention for FGSM (%)</td></tr><tr><td>0.2</td><td>91.34</td><td>94.83</td><td>91.41</td><td>94.93</td></tr><tr><td>0.5</td><td>90.37</td><td>94.53</td><td>90.78</td><td>94.87</td></tr><tr><td>1.0</td><td>86.39</td><td>91.41</td><td>89.22</td><td>93.82</td></tr><tr><td>2.0</td><td>73.91</td><td>79.80</td><td>83.31</td><td>89.41</td></tr></table>
+
+Table 3: The accuracy of our stochastic Ito ensemble with diffusion term corresponding to ˆ $\omega = 0 . 2$ on the PGD attack for different values of $\epsilon$ under the $L _ { 2 }$ norm.
+
+Robustness under the $L _ { 2 }$ norm. The accuracy of our stochastic Ito ensembles on the fast gradient ˆ sign method (FGSM) (Szegedy et al., 2013) and the projected gradient descent (PGD) (Kurakin et al., 2016) under the $L _ { 2 }$ norm is shown in Table 3. Our stochastic Ito ensembles use the ResNet-50 ˆ architecture with the diffusion term corresponding to $\omega = 0 . 2$ .
+
+Our stochastic Ito ensemble approach with a diffusion terms corresponding to ˆ $\omega = 0 . 2$ shows an accuracy of $7 3 . 9 1 \%$ against the PGD attack with $\epsilon = 2 . 0$ under the $L _ { 2 }$ norm. This compare favorably with the $1 8 . 5 9 \%$ accuracy of Madry’s robustness toolbox on the same PGD attack. The accuracy of our stochastic Ito ensemble approach improves to ˆ $7 9 . 0 7 \%$ when the diffusion term corresponds to $\omega = 0 . 4$ . Further, the sum of correct labels and abstentions from our Ito ensemble approach is ˆ $8 8 . 4 3 \%$ against the PGD attack with $\epsilon = 2 . 0$ under the $L _ { 2 }$ norm.
+
+![](images/73db714b5b8a9637033c38ef79b706340e1d993d9ff32a9950e1ff1232084cfa.jpg)  
+Figure 2: The accuracy of our stochastic Ito ensemble with a diffusion term corresponding to ˆ $\omega =$ 0.2 (left) and $\omega = 0 . 4$ (right) on CIFAR-10 compares favorably with Madry’s Robustness Toolbox using the $L _ { 2 }$ norm for different values of $\epsilon$ .
+
+Figure 2 shows the accuracy of our Ito ensemble approach and compares it with the accuracy of ˆ Madry’s robustness toolbox (Engstrom et al., 2020) on CIFAR-10 test data. Both our stochastic Ito ensembles with ˆ $\omega = 0 . 2$ and $\omega = 0 . 4$ have higher accuracy on benign data and their accuracy remains higher than Madry’s robustness toolbox for all values of $\epsilon$ under the $L _ { 2 }$ norm. The accuracy of the Ito ensemble degrades more gracefully as the value of ˆ $\epsilon$ increases under the $L _ { 2 }$ norm.
+
+Robustness under the $L _ { \infty }$ norm. We investigate the accuracy of our stochastic Ito ensemble ˆ approach under the $L _ { \infty }$ norm and compare it to the accuracy of Madry’s robustness toolbox. Table 4 shows the accuracy of our Ito ensemble with diffusion term corresponding to ˆ $\omega = 0 . 2$ .
+
+<table><tr><td>E</td><td>Accuracy for PGD (%)</td><td>Correct + Abstention for PGD (%)</td><td>Accuracy for FGSM (%)</td><td>Correct + Abstention for FGSM (%)</td></tr><tr><td>4 255</td><td>85.82</td><td>92.91</td><td>86.02</td><td>93.00</td></tr><tr><td>8 255</td><td>84.60</td><td>92.48</td><td>85.24</td><td>92.84</td></tr><tr><td>16 255</td><td>79.66</td><td>89.06</td><td>82.74</td><td>91.20</td></tr><tr><td>32 255</td><td>62.97</td><td>74.05</td><td>72.49</td><td>84.08</td></tr></table>
+
+Table 4: The accuracy of our stochastic Ito ensemble with diffusion term corresponding to ˆ $\omega = 0 . 2$ on the PGD attack for different values of $\epsilon$ under the $L _ { \infty }$ norm.
+
+We study the performance of our stochastic Ito ensemble on PGD and FGSM and attacks of varying ˆ magnitudes under the $L _ { \infty }$ norm, and determine that our stochastic Ito ensemble is robust against ˆ adversarial noise. Figure 3 shows the accuracy of Madry’s robustness toolbox and our Ito ensemble ˆ on adversarial images under the PGD attack. The accuracy of our stochastic Ito ensemble with ˆ diffusion term corresponding to $\omega = 0 . 4$ is higher than that of the model from Madry’s toolbox for both the original unperturbed images and PGD adversarial images with $\epsilon = 8 / 2 5 5$ and $\epsilon = 1 6 / 2 5 5$ .
+
+# Question 3: How does the robustness of the stochastic Ito ensemble approach change with the ˆ number of models in the ensemble and the number of inferences?
+
+The Ito ensemble has two sources of diversity - different stochastic models and multiple inferences ˆ on the same model. We investigate the accuracy of our Ito ensemble with different number of ˆ stochastic models and different number of inferences from each stochastic model. Figure 4 (left) illustrates the results of our investigations. A significant increase of $5 . 1 \%$ is observed for the sum of correct outcomes and abstentions by increasing the number of stochastic models from 3 to 20 and the number of sampled independent inferences for each stochastic model from 3 to 20. Increasing the number of stochastic models in the ensemble has more significant influence on the performance of the Ito ensemble than increasing the number of independent inferences from each stochastic model. ˆ
+
+![](images/8e6fb227444f94e631b31f923b7f994419a2350bbea61dae1d1b4df91709e6e0.jpg)  
+Figure 3: The accuracy of our stochastic Ito ensemble on CIFAR-10 compares favorably withˆ Madry’s Robustness Toolbox using the $L _ { \infty }$ norm. (left) $\omega = 0 . 2$ (right) $\omega = 0 . 4$ .
+
+![](images/069a158f9f8721be1248155639fd870d1b9c30fb5fa3bdb99692a524a67895e6.jpg)  
+Figure 4: (left) The impact of the number of models and inferences on the sum of correct outcomes and abstentions for our stochastic Ito ensemble with ˆ $\omega = 0 . 4$ . (right) Diffusion with different values of $\omega$ in stochastic Ito ensemble vs. PGD accuracy under the ˆ $L _ { \infty }$ norm.
+
+# Question 4: How can we control the diffusion term $\omega$ to trade-off robustness and accuracy?
+
+Figure 4 (right) shows the effect of varying diffusion terms with different $\omega$ on the accuracy of the stochastic Ito ensemble under the ˆ $L _ { \infty }$ norm for PGD attacks with $\epsilon = 8 / 2 5 5 , 1 6 / 2 5 5$ and 32/255. The accuracy of the stochastic Ito ensemble first increases as the value ofˆ $\omega$ increases and then starts decreasing for any given value of $\omega$ . This shows a tradeoff between the accuracy of the neural network on benign data and its ability to be robust to large adversarial perturbations.
+
+# IMAGENET RESULTS
+
+These experiments are performed on a 92-core 480GB RAM server with 8 NVIDIA V100 GPUs.
+
+# Question 1: Does the Ito ensemble achieve competitive accuracy on benign inputs? ˆ
+
+We study the accuracy of our Ito ensemble of 5 ResNet-50 models with 20 independent inferences ˆ per model for different values of $\omega$ and associated diffusion terms. As shown in Table 5, small values of $\omega$ do not significantly reduce the accuracy of the Ito ensemble on benign data. ˆ
+
+# Question 2: Is the Ito ensemble approach robust against adversarial attacks? ˆ
+
+Figure 5 shows the accuracy of our stochastic Ito ensemble approach on ImageNet against the PGD ˆ attack with various values of $\epsilon$ under $L _ { 2 }$ as well as $L _ { \infty }$ norms. The accuracy of our Ito ensemble ˆ compares favorably with the results from Madry’s robustness toolbox Engstrom et al. (2020).
+
+Table 5: Accuracy of Ito ensembles on benign data with diffusion corresponding to different ˆ $\omega$   
+
+<table><tr><td>Diffusion Term w</td><td>Original Accuracy</td><td>Itó Ensemble Accuracy</td><td>% Decrease in Accuracy</td><td>Correct + Abstentions (%)</td></tr><tr><td>0.1</td><td>76.13</td><td>76.04</td><td>0.09</td><td>77.87</td></tr><tr><td>0.2</td><td>76.13</td><td>73.59</td><td>2.54</td><td>78.29</td></tr><tr><td>0.3</td><td>76.13</td><td>67.79</td><td>8.34</td><td>76.40</td></tr><tr><td>0.4</td><td>76.13</td><td>61.66</td><td>14.47</td><td>76.42</td></tr></table>
+
+![](images/21162b88fb4ee53ccc2c070dd16b40f76e3af9b8a9bd3276f4cb2b4c88ffdb94.jpg)  
+Figure 5: The accuracy of our stochastic Ito ensemble with ˆ $\omega { = } 0 . 2$ on ImageNet compares favorably with Robustness Toolbox using the $L _ { 2 }$ norm (left) and the $L _ { \infty }$ norm (right) for different values of $\epsilon$
+
+# Question 3: How can we control the diffusion term $\omega$ to trade-off robustness and accuracy?
+
+Table 5 and Table 6 show that the diffusion parameter $\omega$ can be used to establish a desired trade-off between robustness and benign accuracy for the ImageNet data set.   
+
+<table><tr><td>Diffusion Term w</td><td>Ito Ensemble Benign Accuracy (%)</td><td>Ito Ensemble PGD Accuracy (%)</td><td>Correct + Abstentions (%)</td></tr><tr><td>0.1</td><td>76.04</td><td>26.23</td><td>27.89</td></tr><tr><td>0.2</td><td>73.59</td><td>53.42</td><td>61.32</td></tr><tr><td>0.3</td><td>67.79</td><td>60.11</td><td>70.01</td></tr><tr><td>0.4</td><td>61.66</td><td>58.57</td><td>73.55</td></tr></table>
+
+Table 6: Accuracy of our Ito ensemble approach on PGD attack ˆ $( \epsilon = 8 / 2 5 5 )$ with different diffusion parameters $\omega$ . Higher values of $\omega$ lead to more robust models.
+
+# 5 CONCLUSION
+
+We have shown that ensembles of neural networks corresponding to a class of Ito processes are ˆ more robust than classical residual networks. An Ito process obtained by solving a ˆ suitably-formulated stochastic differential equation derived from a residual network has a probability density function that is robust to adversarial input perturbations. Further, the achieved robustness does not require any explicit adversarial training; hence, it is likely to generalize to unforeseen attacks. We empirically evaluated the robustness of our Ito ensembles and demonstrated ˆ that they achieve higher accuracy under FGSM/PGD attacks over the $L _ { 2 } / L _ { \infty }$ norm compared to state-of-the-art methods. This robustness is attained without significantly sacrificing accuracy on benign data. Further, Ito ensemble abstains on benign inputs with high uncertainty reflecting ˆ uncertainty-aware learning. Our paper is a step towards the use of Ito processes and stochastic ˆ differential equation models to build robust ensembles in deep learning. This will aid the adoption of deep learning in safety-critical applications.
+
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parse/train/fgBxcKcz9ea/fgBxcKcz9ea.md

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+# Differential Privacy of Dirichlet Posterior Sampling
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 We study the inherent privacy of releasing a single sample from a Dirichlet posterior   
+2 distribution. As a complement to the previous study that provides general theories   
+3 on the differential privacy of posterior sampling from exponential families, this   
+4 study focuses specifically on the Dirichlet posterior sampling and its privacy   
+5 guarantees. With the notion of truncated concentrated differential privacy (tCDP),   
+6 we are able to derive a simple privacy guarantee of the Dirichlet posterior sampling,   
+7 which effectively allows us to analyze its utility in various settings. Specifically,   
+8 we provide accuracy guarantees of the Dirichlet posterior sampling in Multinomial  
+9 Dirichlet sampling and private normalized histogram publishing.
+
+# 10 1 Introduction
+
+11 The Bayesian framework provides a way to perform statistical analysis by combining prior beliefs   
+12 with real-life evidence. At a high level, the belief and the evidence are assumed to be described   
+13 by probabilistic models. As we receive new data, our belief is updated accordingly via the Bayes’   
+14 theorem, resulting in the so-called posterior belief. The posterior tells us how much we are uncertain   
+15 about the model’s parameters.   
+16 The Dirichlet distribution is usually chosen as the prior when performing Bayesian analysis on discrete   
+17 variables, as it is a conjugate prior to the categorical and multinomial distributions. Specifically,   
+18 Dirichlet distributions are often used in discrete mixture models, where a Dirichlet prior is put on   
+19 the mixture weights [LW92; MMR05]. Such models have applications in NLP [PB98], biophysical   
+20 systems [Hin15], accident analysis [de 06], and genetics [BHW00; PM01; CWS03]. In all of these   
+21 studies, samplings from Dirichlet posteriors arise when performing Markov chain Monte Carlo   
+22 methods for approximate Bayesian inference.   
+23 Dirichlet posterior sampling also appears in other learning tasks. For example, in Bayesian active   
+24 learning, it arises in Gibbs sampling, which is used to approximate the posterior of the classifier over   
+25 the labeled sample [NLYCC13]. In Thompson sampling for multi-armed bandits, one repeatedly   
+26 draws a sample from the Dirichlet posterior of each arm, and picks the arm whose sample maximizes   
+27 the reward [ZHGSY20; AAFK20; NIK20]. And in Bayesian reinforcement learning, state-transition   
+28 probabilities are sampled from the Dirichlet posterior over past observed states [Str00; ORR13].   
+29 Dirichlet posterior sampling can also be used for data synthesis. Suppose that we have a histogram   
+30 $( x _ { 1 } , \ldots , x _ { d } )$ of actual data. An approximate discrete distribution of this histogram can be obtained by   
+31 drawing a sample $\mathbf { Y }$ from Dirichlet $( x _ { 1 } + \alpha _ { 1 } , \ldots , x _ { d } + \alpha _ { d } )$ , where $\alpha _ { 1 } , \ldots , \alpha _ { d }$ are prior parameters.   
+32 Then synthetic data is produced by repeatedly drawing from Multinomial(Y). There are many   
+33 studies on data synthesis that followed this approach [AV08; MKAGV08; RWZ14; PG14; SJGLY17].   
+34 In the above examples, the data that we integrate into these tasks might contain sensitive information.   
+35 Thus it is important to ask: how much of the information is protected from the Dirichlet samplings?   
+36 The goal of this study is to find an answer to this question.   
+37 The mathematical framework of differential privacy (DP) [DMNS06] allows us to quantify how much   
+38 the privacy of the Dirichlet posterior sampling is affected by the prior parameters $\alpha _ { 1 } , \ldots , \alpha _ { d }$ . In the   
+39 definition of DP, the privacy of a randomized algorithm is measured by how much its distribution   
+40 changes upon perturbing a single data point of the input. Nonetheless, this notion might be too   
+41 strict for the Dirichlet distribution, as a small perturbation of a near-zero parameter can cause a large   
+42 distribution shift. Thus, it might be more appropriate to rely on one of several relaxed notions of   
+43 DP, such as approximate differential privacy, Rényi differential privacy, or concentrated differential   
+44 privacy. It is natural to wonder if the Dirichlet posterior sampling satisfies any of these definitions.
+
+# 45 1.1 Overview of Our results
+
+46 This study focuses on the privacy and utility of Dirichlet posterior sampling. In summary, we provide   
+47 a closed-form privacy guarantee of the Dirichlet posterior sampling, which in turn allows us to   
+48 effectively analyze its utility in various settings.
+
+$\ S 3$ Privacy. We study the role of the prior parameters in the privacy of the Dirichlet posterior sampling. Theorem 1 is our main result, where we provide a guaranteed upper bound for truncated concentrated differential privacy (tCDP) of the Dirichlet posterior sampling. In addition, we convert the tCDP guarantee into an approximate differential privacy guarantee in Corollary 2.
+
+$\ S 4$ Utility. Using the tCDP guarantee, we investigate the utility of Dirichlet posterior sampling applied in two specific applications:
+
+• In Section 4.1, we consider one-time sampling from a Multinomial-Dirichlet distribution. But instead of directly sampling from this distribution, we sample from another distribution with larger prior parameters. The accuracy is then measured by the KL-divergence between the original and the private distributions.   
+• In Section 4.2, we use the Dirichlet posterior sampling for a private release of a normalized histogram. In this case, the accuracy is measured by the mean-squared error between the sample and the original normalized histogram.
+
+62 In both tasks, we compute the sample size that guarantees the desired level of accuracy. In the case   
+63 of private histogram publishing, we also compare the Dirichlet posterior sampling to the Gaussian   
+64 mechanism.
+
+# 65 1.2 Related work
+
+66 There are several studies on the differential privacy of posterior sampling. Wang, Fienberg, and   
+67 Smola [WFS15] showed that any posterior sampling with the log-likelihood bounded by $B$ is $4 B$ -   
+68 differentially private. However, the likelihoods that we study are not bounded away from zero; they   
+69 have the form $\Pi _ { i } p _ { i } ^ { x _ { i } }$ which becomes small when one of the $p _ { i }$ ’s is close to zero. Dimitrakakis, Nelson,   
+70 Zhang, Mitrokotsa, and Rubinstein [DNZMR17] showed that if the condition on the log-likelihood is   
+71 relaxed to the Lipschitz continuity with high probability, then one can obtain the approximate DP.   
+72 Nonetheless, with the Dirichlet density, it is difficult to compute the probability of events in which   
+73 the Lipschitz condition is satisfied.   
+74 In the case that the sufficient statistics $\mathbf { x }$ has finite $\ell ^ { 1 }$ -sensitivity, Foulds, Geumlek, Welling and   
+75 Chaudhuri [FGWC16] suggested adding Laplace noises to $\mathbf { x }$ . Suppose that y is the output; they   
+76 showed that sampling from $p ( \boldsymbol { \theta } | \mathbf { y } )$ is differentially private and as asymptotically efficient as sampling   
+77 from $p ( \boldsymbol { \theta } | \mathbf { x } )$ . However, for a small sample size, the posterior over the noisy statistics might be too   
+78 far away from the actual posterior. Bernstein and Sheldon [BS18] thus proposed to approximate the   
+79 joint distribution $p ( \boldsymbol { \theta } , \mathbf { x } , \mathbf { y } )$ using Gibbs sampling, which is then integrated over $\mathbf { x }$ to obtain a more   
+80 accurate posterior over $\mathbf { y }$ .   
+81 Geumlek, Song, and Chaudhuri [GSC17] were the first to study the posterior sampling with the   
+82 RDP. Even though they provided a general framework to find $( \lambda , \epsilon )$ -RDP guarantees for exponential   
+83 families, explicit forms of $\epsilon$ and the upper bound of $\lambda$ were not given. In contrast, our tCDP guarantees   
+84 of the Dirichlet posterior sampling imply an explicit expression for $\epsilon$ , and also an upper bound for $\lambda$ .   
+85 The privacy of data synthesis via sampling from Multinomial $( \mathbf { Y } )$ , where $\mathbf { Y }$ is a discrete distri  
+86 bution drawn from the Dirichlet posterior, was first studied by Machanavajjhala, Kifer, Abowd,   
+87 Gehrke, and Vilhuber [MKAGV08]. They showed that the data synthesis is $( \varepsilon , \delta )$ -probabilistic DP,   
+88 which implies $( \varepsilon , \delta )$ -approximate DP. However, as their privacy analysis includes the sampling from   
+89 Multinomial $( \mathbf { Y } )$ , their privacy guarantee depends on the number of synthetic samples. In contrast,   
+90 we show that the one-time sampling from the Dirichlet posterior is approximate DP, which by the   
+91 post-processing property allows us to sample from Multinomial $( \mathbf { Y } )$ as many times as we want while   
+92 retaining the same privacy guarantee.   
+93 The Dirichlet mechanism was first introduced by Gohari, Wu, Hawkins, Hale, and Topcu [GWHHT21].   
+94 Originally, the Dirichlet mechanism takes a discrete distribution $\mathbf { p } : = ( p _ { 1 } , \ldots , p _ { d } )$ and draws one   
+95 sample $\mathbf { Y } \sim \mathrm { D i r i c h l e t } ( r p _ { 1 } , \dots , r p _ { d } )$ . Note the absence of the prior parameters, which makes $\mathbf { Y }$ an   
+96 unbiased estimator of $\mathbf { p }$ . But this comes with a cost, as the worst case of privacy violation occurs   
+97 when almost all of the parameters are close to zero. The authors avoided this issue by restricting   
+98 the input space to a subset of the unit simplex, with some of the $p _ { i }$ ’s bounded below by a fixed   
+99 positive constant. This results in complicated expressions for the privacy guarantees as they involve   
+100 a minimization problem over the restricted domain. In this study, we take a different approach by   
+101 adding prior parameters to the Dirichlet mechanism. As a result, we obtain a biased algorithm that   
+102 requires no assumption on the input space and has simpler forms of privacy guarantees.
+
+# 1.3 Notations
+
+104 We let $\mathbb { R } _ { \geq 0 } ^ { d }$ be the set of $d$ -tuples of non-negative real numbers and $\mathbb { R } _ { > 0 } ^ { d }$ be the set of $d$ -tuples of   
+105 positive real numbers. We assume that all vectors are $d$ -dimensional where $d \geq 2$ . The notations for   
+106 all vectors are always in bold. Specifically, $\mathbf { x } : = ( x _ { 1 } , \ldots , x _ { d } ) \in \mathbb { R } _ { \geq 0 } ^ { d }$ consists of sample statistics of   
+107 the data and $\pmb { \alpha } : = ( \alpha _ { 1 } , \ldots , \alpha _ { d } ) \in \mathbb { R } _ { > 0 } ^ { d }$ consists of the prior parameters. The vector $\mathbf { p } : = ( p _ { 1 } , \ldots , p _ { d } )$   
+108 always satisfies $\textstyle \sum _ { i } p _ { i } = 1$ . The number of observations is always $N$ . We also denote $x _ { 0 } : = \textstyle \sum _ { i } x _ { i }$   
+109 and $\alpha _ { 0 } : = \textstyle \sum _ { i } \alpha _ { i }$ . For any vectors $\mathbf { x } , \mathbf { x } ^ { \prime }$ and scalar $r > 0$ , we write $\mathbf { x } + \mathbf { x } ^ { \prime } : = ( x _ { 1 } + x _ { 1 } ^ { \prime } , \ldots , x _ { d } + x _ { d } ^ { \prime } )$   
+110 and $r \mathbf { x } : = ( r x _ { 1 } , \ldots , r x _ { d } )$ . For any positive reals $x$ and $x ^ { \prime }$ , the notation $x \propto x ^ { \prime }$ means $x = C x ^ { \prime }$ for   
+111 some constant $C > 0$ , $x \approx x ^ { \prime }$ means $c x ^ { \prime } \leq x \leq C x ^ { \prime }$ for some $c , C > 0$ , and $x \lesssim x ^ { \prime }$ means $x \leq C x ^ { \prime }$   
+112 for some $C > 0$ . Lastly, $\| \mathbf { x } \| _ { \infty } : = \operatorname* { m a x } _ { i } | x _ { i } |$ is the $\ell ^ { \infty }$ norm of $\mathbf { x }$ .
+
+# 13 2 Background
+
+# 2.1 Privacy models
+
+Definition 2.1 (Pure and Approximate DP [DMNS06]). A randomized mechanism $M : \mathcal { X } ^ { n }  \mathcal { Y }$ is $( \varepsilon , \delta )$ -differentially private $( \varepsilon , \delta )$ -DP) if for any datasets $x , x ^ { \prime }$ differing on a single entry, and all events $E \subset \mathcal { V }$ ,
+
+$$
+\mathbb { P } [ M ( x ) \in E ] \leq e ^ { \varepsilon } \mathbb { P } [ M ( x ^ { \prime } ) \in E ] + \delta .
+$$
+
+118 If $M$ is $( \varepsilon , 0 )$ -DP, then we say that it is $\varepsilon$ -differential privacy $\dot { \varepsilon }$ -DP).
+
+19 The term pure differential privacy (pure DP) refers to $\epsilon$ -differential privacy, while approximate   
+20 differential privacy (approximate DP) refers to $( \varepsilon , \delta )$ -DP when $\delta > 0$ .
+
+121 In contrast to pure and approximate DP, the next definitions of differential privacy are defined in terms of the Rényi divergence between 122 $M ( x )$ and $M ( x ^ { \prime } )$ :
+
+123 Definition 2.2 (Rényi Divergence [Rén61]). Let $P$ and $Q$ be probability distributions. $\mathrm { F o r } \lambda \in ( 1 , \infty )$   
+124 the Rényi divergence of order $\lambda$ between $P$ and $Q$ is defined as
+
+$$
+\mathrm { D } _ { \lambda } ( P \| Q ) : = { \frac { 1 } { \lambda - 1 } } \log \int P ( y ) ^ { \lambda } Q ( y ) ^ { 1 - \lambda } d y = { \frac { 1 } { \lambda - 1 } } \log \biggl ( \operatorname { \mathbb { E } } _ { y \sim P } \biggl [ { \frac { P ( y ) ^ { \lambda - 1 } } { Q ( y ) ^ { \lambda - 1 } } } \biggr ] \biggr . . \biggr )
+$$
+
+125 Definition 2.3 (tCDP and zCDP [BDRS18; BS16]). A randomized mechanism $M : \mathcal { X } ^ { n }  \mathcal { Y }$ is   
+126 $\omega$ -truncated $\rho$ -concentrated differentially private $( ( \rho , \omega )$ -tCDP) if for any datasets $x , x ^ { \prime }$ differing on a   
+127 single entry and for all $\lambda \in ( 1 , \omega )$ ,
+
+$$
+\begin{array} { r } { \mathrm { D } _ { \lambda } ( M ( x ) \| M ( x ^ { \prime } ) ) \leq \lambda \rho . } \end{array}
+$$
+
+128 If $M$ is $( \rho , \infty )$ -tCDP, then we say that it is $\rho$ -zero-concentrated differential privacy ( $\rho$ -zCDP).
+
+129 Note that both tCDP and zCDP have the composition and post-processing properties. Intuitively, $\rho$ con  
+130 trols the expectation and standard deviation of the privacy loss random variable: Z = log P [M(x)=Y ]P [M(x0)=Y ] ,   
+131 where $Y$ has density $M ( x )$ , and $\omega$ controls the number of standard deviations for which $Z$ concen  
+132 trates like a Gaussian. A smaller $\rho$ and larger $\omega$ correspond to a stronger privacy guarantee. It turns   
+133 out that tCDP implies approximate DP:   
+134 Lemma 1 (From tCDP to Approximate DP [BDRS18]). Let $\delta > 0$ . If M is a $( \rho , \omega )$ -tCDP mechanism,   
+135 then it also satisfies $( \varepsilon , \delta )$ - $D P$ with
+
+$$
+\varepsilon = \left\{ \begin{array} { l l } { \rho + 2 \sqrt { \rho \log ( 1 / \delta ) } \quad } & { i f \log ( 1 / \delta ) \leq ( \omega - 1 ) ^ { 2 } \rho } \\ { \rho \omega + \frac { \log ( 1 / \delta ) } { \omega - 1 } } & { i f \log ( 1 / \delta ) > ( \omega - 1 ) ^ { 2 } \rho } \end{array} \right. .
+$$
+
+# 136 2.2 Dirichlet distribution
+
+137 For $\alpha \in \mathbb { R } _ { > 0 } ^ { d }$ , the Dirichlet distribution Dirichlet $( \alpha )$ is a continuous distribution of $d$ -dimensional   
+138 probability vectors i.e. vectors whose coordinate sum is equal to 1. The density function of $\mathbf { Y } \sim$   
+139 Dirichlet $( \alpha )$ is given by:
+
+$$
+p ( \mathbf { y } ) = \frac { 1 } { B ( \pmb { \alpha } ) } \prod _ { i = 1 } ^ { d } y _ { i } ^ { \alpha _ { i } - 1 } ,
+$$
+
+140 where $B ( \alpha )$ is the beta function, which can be written in terms of the gamma function:
+
+$$
+B ( \pmb { \alpha } ) = \frac { \prod _ { i } \Gamma ( \alpha _ { i } ) } { \Gamma ( \sum _ { i } \alpha _ { i } ) } .
+$$
+
+# 141 2.3 Dirichlet posterior sampling
+
+142 We consider the prior Dirichlet $( \alpha )$ and the likelihood of the form $\begin{array} { r } { p ( \mathbf { x } | \mathbf { y } ) \propto \prod _ { i = 1 } ^ { d } y _ { i } ^ { x _ { i } } } \end{array}$ where   
+143 $\mathbf { x } \in \mathbb { R } _ { \geq 0 } ^ { d }$ consists of sample statistics of the dataset. The Dirichlet posterior sampling is a one-time   
+144 sampling:
+
+$$
+\mathbf { Y } \sim { \mathrm { D i r i c h l e t } } ( \mathbf { x } + \pmb { \alpha } ) .
+$$
+
+145 There is a modification of the sampling which introduces a concentration parameter $r > 0$ , and   
+146 instead we sample from Dirichlet $( r \mathbf { x } + \pmb { \alpha } )$ [GSC17; GWHHT21]. Smaller values of $r$ make the   
+147 sampling more private, and larger values of $r$ make $\mathbf { Y }$ a closer approximation of $\mathbf { x }$ . Even though the   
+148 case $r = 1$ is the main focus of this study, our main privacy results can be easily extended to other   
+149 values of $r$ as we will see at the end of Section 3.1.   
+150 Consider a special case where $\mathbf x = \mathbf p$ is an empirical distribution derived from the dataset, and we   
+151 want $\mathbf { Y }$ to be a private approximation of $\mathbf { p }$ ; the sampling $\mathbf { Y } \sim { \mathrm { D i r i c h l e t } } ( r \mathbf { p } + \alpha )$ is called the   
+152 Dirichlet mechanism [GWHHT21]. It is interesting to note that the Dirichlet mechanism is a form of   
+153 the exponential mechanism [MT07]: let $r > 0$ be the privacy parameter, Dirichlet $( \alpha )$ be the prior,   
+154 and the negative KL-divergence be the score function of the exponential mechanism. Then the output   
+155 $\mathbf { Y }$ of this mechanism is distributed according to the following density function:
+
+$$
+\begin{array} { r l r } & { } & { \frac { \exp \left( - r \mathrm { D } _ { \mathrm { K L } } ( \mathbf { p } , \mathbf { y } ) \right) \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } } { \int \exp \left( - r \mathrm { D } _ { \mathrm { K L } } ( \mathbf { p } , \mathbf { y } ) \right) \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } d \mathbf { y } } \propto \exp \left( r \sum _ { i , p _ { i } \neq 0 } p _ { i } \log ( y _ { i } / p _ { i } ) \right) \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } } \\ & { } & { \propto \displaystyle \prod _ { i , p _ { i } \neq 0 } y _ { i } ^ { r p _ { i } } \prod _ { i } y _ { i } ^ { \alpha _ { i } - 1 } = \prod _ { i } y _ { i } ^ { r p _ { i } + \alpha _ { i } - 1 } , } \end{array}
+$$
+
+156 which is exactly the density function of Dirichlet $( r \mathbf { p } + \alpha )$ .
+
+58 In most of this study, we take advantage of several nice properties of the log-gamma function and its derivatives. Specifically, $\begin{array} { r } { \psi ( x ) : = \frac { d } { d x } \log \Gamma ( x ) } \end{array}$ is concave and increasing, while its derivative $\psi ^ { \prime } ( x )$ is positive, convex, and decreasing. In addition, 0 $\psi ^ { \prime }$ can be approximated by the reciprocals:
+
+$$
+{ \frac { 1 } { x } } + { \frac { 1 } { 2 x ^ { 2 } } } < \psi ^ { \prime } ( x ) < { \frac { 1 } { x } } + { \frac { 1 } { x ^ { 2 } } } ,
+$$
+
+which implies that 161 $\textstyle \psi ^ { \prime } ( x ) \approx { \frac { 1 } { x ^ { 2 } } }$ as $x \to 0$ and $\begin{array} { r } { \psi ^ { \prime } ( x ) \approx \frac { 1 } { x } } \end{array}$ as $x \to \infty$
+
+# 3 Main privacy results
+
+# 3.1 Truncated concentrated differential privacy
+
+Theorem 1. Let $\alpha \in \mathbb { R } _ { > 0 } ^ { d }$ and $\alpha _ { m } : = \operatorname* { m i n } _ { i } \alpha _ { i }$ . Let $\gamma \in ( 0 , \alpha _ { m } )$ . Let $\Delta _ { 2 } , \Delta _ { \infty } > 0$ be constants that satisfy $\begin{array} { r } { \sum _ { i } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \leq \Delta _ { 2 } ^ { 2 } } \end{array}$ and $\mathrm { m a x } _ { i } \left| x _ { i } - x _ { i } ^ { \prime } \right| \leq \Delta _ { \infty }$ whenever x, $\mathbf { \Delta } , \mathbf { x } ^ { \prime } \in \mathbb { R } _ { \geq 0 } ^ { 2 }$ are sample statistics of any two datasets differing on a single entry. The one-time sampling from Dirichlet $\left( \mathbf { x } + \alpha \right)$ is $( \rho , \omega )$ -tCDP, where $\begin{array} { r } { \omega = \frac { \gamma } { \Delta _ { \infty } } + 1 } \end{array}$ and
+
+$$
+\rho = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) .
+$$
+
+168 Note that $( \rho , \infty )$ -tCDP is not obtainable, as the ratio between two Dirichlet densities blows up as   
+169 $\omega  \infty$ . We present here a short proof that skips some calculations (see Appendix 1 for a full proof).
+
+proof. Consider any 170 $\begin{array} { r } { \lambda \in \left( 1 , \frac { \gamma } { \Delta _ { \infty } } + 1 \right) } \end{array}$ . Let $\mathbf { u } : = \mathbf { x } + \pmb { \alpha }$ and $\mathbf { u } ^ { \prime } : = \mathbf { x } ^ { \prime } + \alpha ^ { \prime }$ . Let $P ( \mathbf { y } )$ be the density of Dirichlet171 $\mathbf { \Pi } ( \mathbf { u } )$ and $P ^ { \prime } ( \mathbf { y } )$ be the density of Dirichlet $\mathbf { \Pi } ^ { ( \mathbf { u } ^ { \prime } ) }$ . A quick calculation shows that:
+
+$$
+\mathbb { E } _ { { \mathbf { y } } \sim P ( { \mathbf { y } } ) } \left[ \frac { P ( { \mathbf { y } } ) ^ { \lambda - 1 } } { P ^ { \prime } ( { \mathbf { y } } ) ^ { \lambda - 1 } } \right] = \frac { B ( { \mathbf { u } } ^ { \prime } ) ^ { \lambda - 1 } } { B ( { \mathbf { u } } ) ^ { \lambda - 1 } } \cdot \frac { B ( { \mathbf { u } } + ( \lambda - 1 ) ( { \mathbf { u } } - { \mathbf { u } } ^ { \prime } ) ) } { B ( { \mathbf { u } } ) } .
+$$
+
+172 We take the logarithm on both sides and apply the second-order Taylor expansion to the following   
+173 $G ( u _ { i } , u _ { i } ^ { \prime } )$ and $H ( u _ { i } , u _ { i } ^ { \prime } )$ terms that appear on the right-hand side. As a result, there exist $\xi$ between   
+174 $u _ { i } + ( \lambda - 1 ) ( u _ { i } - u _ { i } ^ { \prime } )$ and $u _ { i }$ , and $\xi ^ { \prime }$ between $u _ { i }$ and $u _ { i } ^ { \prime }$ such that
+
+$$
+\begin{array} { r l r } {  { G ( u _ { i } , u _ { i } ^ { \prime } ) : = ( \lambda - 1 ) ( \log \Gamma ( u _ { i } ^ { \prime } ) - \log \Gamma ( u _ { i } ) ) } } \\ & { } & \\ & { } & { = - ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) \psi ( u _ { i } ) + \frac { 1 } { 2 } ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \xi ^ { \prime } ) } \\ & { } & { H ( u _ { i } , u _ { i } ^ { \prime } ) : = \log \Gamma ( u _ { i } + ( \lambda - 1 ) ( u _ { i } - u _ { i } ^ { \prime } ) ) - \log \Gamma ( u _ { i } ) } \\ & { } & { = ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) \psi ( u _ { i } ) + \frac { 1 } { 2 } ( \lambda - 1 ) ^ { 2 } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \xi ) , } \end{array}
+$$
+
+Note that 175 $\psi ^ { \prime }$ is increasing. If $x _ { i } > x _ { i } ^ { \prime }$ , then $\xi$ and $\xi ^ { \prime }$ are bounded below by $u _ { i } ^ { \prime } \geq \alpha _ { m }$ . On the other hand, if 176 $x _ { i } \leq x _ { i } ^ { \prime }$ , then $\xi$ and $\xi ^ { \prime }$ are bounded below by $u _ { i } - ( \lambda - 1 ) | u _ { i } - u _ { i } ^ { \prime } |$ . The condition 177 $\begin{array} { r } { \lambda < \frac { \gamma } { \Delta _ { \infty } } + 1 } \end{array}$ guarantees that $u _ { i } - ( \lambda - 1 ) | u _ { i } - u _ { i } ^ { \prime } | > \alpha _ { m } - \gamma$ . All cases considered, we have
+
+$$
+\begin{array} { l } { G ( u _ { i } , u _ { i } ^ { \prime } ) + H ( u _ { i } , u _ { i } ^ { \prime } ) \leq \displaystyle \frac { 1 } { 2 } \big ( ( \lambda - 1 ) + ( \lambda - 1 ) ^ { 2 } \big ) ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) } \\ { \displaystyle \qquad = \frac { 1 } { 2 } \lambda ( \lambda - 1 ) ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) . } \end{array}
+$$
+
+Denoting 178 $u _ { 0 } : = \textstyle \sum _ { i } u _ { i }$ and $u _ { 0 } ^ { \prime } : = \textstyle \sum _ { i } u _ { i } ^ { \prime }$ , the same argument shows that $G ( u _ { 0 } , u _ { 0 } ^ { \prime } ) + H ( u _ { 0 } , u _ { 0 } ^ { \prime } ) > 0$ . 179 Therefore,
+
+$$
+\begin{array} { r l } & { D _ { \boldsymbol { \lambda } } ( P ( \mathbf { y } ) \| P ^ { \prime } ( \mathbf { y } ) ) = \displaystyle \frac { 1 } { \boldsymbol { \lambda } - 1 } \Biggl ( \sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \prime } ) + H ( u _ { i } , u _ { i } ^ { \prime } ) ) - G ( u _ { 0 } , u _ { 0 } ^ { \prime } ) - H ( u _ { 0 } , u _ { 0 } ^ { \prime } ) \Biggr ) } \\ & { \quad \quad < \displaystyle \frac { 1 } { \boldsymbol { \lambda } - 1 } \sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \prime } ) + H ( u _ { i } , u _ { i } ^ { \prime } ) ) } \\ & { \quad \quad \le \displaystyle \frac { 1 } { 2 } \boldsymbol { \lambda } \sum _ { i } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) \le \frac { 1 } { 2 } \boldsymbol { \lambda } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) . } \end{array}
+$$
+
+![](images/9b723c7be81bab0fe812229760033b22531dcf9d6fe77c60ac3901a2063deaff.jpg)  
+Figure 1: Left: the actual values of $\begin{array} { r } { \rho = \frac { 1 } { 2 } \operatorname { D } _ { 2 } ( P \| P ^ { \prime } ) } \end{array}$ and the worst case $( \rho , 2 )$ -tCDP guarantees (2) at $\Delta _ { 2 } ^ { 2 } = \Delta _ { \infty } = 1$ . Here, $P$ and $P ^ { \prime }$ are Dirichlet posterior densities over $\mathbf { x } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 0 )$ , $\mathbf { x } ^ { \prime } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 1 )$ , and $\pmb { \alpha } = ( \alpha , \ldots , \alpha )$ . Right: comparison between $( \varepsilon , \delta )$ -DP guarantees of the Dirichlet posterior samplings (8) with different uniform priors: $\pmb { \alpha } = ( \alpha , \ldots , \alpha )$ .
+
+180 The guaranteed upper bound (2) is independent of the sample statistics. As a result, the bound applies   
+181 even in worst settings i.e., when $x _ { i } = 0$ and $x _ { i } ^ { \prime } = \Delta _ { \infty }$ , or vice versa, for some $i$ . As we can see in   
+182 Figure 1, the upper bound is a close approximation to the actual value of $\rho$ when $x _ { 6 } = 0$ and $x _ { 6 } ^ { \prime } = 1$ .   
+183 However, being a sample independent bound, the difference becomes substantial when all $x _ { i }$ ’s are   
+184 large. There is one way to get around this issue: if there is no privacy violation in assuming that   
+185 the sample statistics are always bounded below by some threshold $\tau$ , then we can incorporate the   
+186 threshold into the prior (thus $\psi ^ { \prime } ( \alpha _ { m } - \gamma )$ in (2) is replaced by $\psi ^ { \prime } ( \alpha _ { m } + \tau - \gamma ) )$ .   
+187 The parameter $\gamma$ allows us to adjust the moment bound $\omega$ as desired. Even though a higher $\omega$ usually   
+188 leads to a better privacy guarantee, there are two downsides to picking $\gamma$ close to $\alpha _ { m }$ in this case.   
+189 First, note that $\rho$ contains $\psi ^ { \prime } ( \alpha _ { m } - \gamma )$ ; as $\gamma \to \alpha _ { m }$ , the value of $\rho$ diverges to $\infty$ , leading to a weaker   
+190 privacy guarantee instead. Second, as the Taylor approximation (5) is accurate when $u _ { i }$ is close to   
+191 $u _ { i } + ( \lambda - 1 ) ( u _ { i } - u _ { i } ^ { \prime } )$ , having a large value of $\lambda$ would push the guaranteed upper bound away from   
+192 the actual privacy loss. Thus it is recommended to pick $\gamma$ so that $\gamma / \Delta _ { \infty } \geq 1$ and $\alpha _ { m } - \gamma \gg 0$ .   
+193 Alternatively, we can choose the value of $\gamma$ that minimizes $\varepsilon$ when converting from tCDP to $( \varepsilon , \delta )$ -DP   
+194 using Lemma 1—this method will be explored in the next subsection.
+
+Theorem 1 can be easily applied to sampling from Dirichlet $( r \mathbf { x } + \alpha )$ . Replacing $\mathbf { x }$ with $r \mathbf { x }$ , we have $\Delta _ { 2 }$ replaced by $r \Delta _ { 2 }$ and $\Delta _ { \infty }$ replaced by $r \Delta _ { \infty }$ . Consequently, the sampling is $\begin{array} { r } { \left( \rho , \frac { \gamma } { r \Delta _ { \infty } } + 1 \right) } \end{array}$ -tCDP, where $\rho = { \textstyle { \frac { 1 } { 2 } } r ^ { 2 } \Delta _ { 2 } ^ { 2 } } \psi ^ { \prime } ( \alpha _ { m } - \gamma )$ . In Appendix 4, we analyze the scaling of $r$ in conjunction with $\alpha _ { m }$ at a fixed privacy budget $\rho$ .
+
+# 3.2 Approximate differential privacy
+
+200 We now convert the tCDP guarantee to an approximate DP guarantee. Let $\delta \in ( 0 , 1 )$ . Using Lemma 1,   
+201 the Dirichlet posterior sampling with Dirichlet $( \alpha )$ as the prior is $( \varepsilon , \delta )$ -DP with
+
+$$
+\varepsilon = \left\{ \begin{array} { l l } { \rho ( \gamma ) + 2 \sqrt { \rho ( \gamma ) \log ( 1 / \delta ) } } & { \mathrm { i f ~ } \log ( 1 / \delta ) \leq \gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 } } \\ { \rho ( \gamma ) \Big ( \frac { \gamma } { \Delta _ { \infty } } + 1 \Big ) + \frac { \log ( 1 / \delta ) \Delta _ { \infty } } { \gamma } } & { \mathrm { i f ~ } \log ( 1 / \delta ) > \gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 } } \end{array} , \right.
+$$
+
+where 202 $\begin{array} { r } { \rho ( \gamma ) = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) } \end{array}$
+
+203 We try to minimize $\epsilon$ by adjusting the value of $\gamma$ . First, we consider the case $\log ( 1 / \delta ) \leq \gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 }$   
+204 Since $\rho ( \gamma )$ is a strictly increasing function of $\gamma$ , both $\rho ( \gamma ) + 2 \sqrt { \rho ( \gamma ) \log ( 1 / \delta ) }$ and $\gamma ^ { 2 } \rho ( \gamma ) / \Delta _ { \infty } ^ { 2 }$   
+205 are both strictly increasing function of $\gamma$ . Therefore, $\varepsilon$ is minimized at the minimum possible   
+206 value of $\gamma$ in this case, that is, at the unique $\gamma _ { M }$ that satisfies $\log ( 1 / \delta ) = \gamma _ { M } ^ { 2 } \rho ( \gamma _ { M } ) / \Delta _ { \infty } ^ { 2 } =$   
+207 $\begin{array} { r } { \frac { 1 } { 2 } \gamma _ { M } ^ { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma _ { M } ) / \Delta _ { \infty } ^ { 2 } } \end{array}$ .   
+208 Now we consider the second case, when $\gamma < \gamma _ { M }$ . As $\rho ( \gamma )$ is an increasing positive convex function   
+209 of $\gamma$ , the function
+
+$$
+f ( \gamma ) : = \frac 1 2 \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) \left( \frac { \gamma } { \Delta _ { \infty } } + 1 \right) + \frac { \log ( 1 / \delta ) \Delta _ { \infty } } { \gamma } ; \qquad \gamma \in ( 0 , \gamma _ { M } ] ,
+$$
+
+is also convex in $\gamma$ , and thus has a unique minimizer $\underline { { \gamma _ { m } } } \in ( 0 , \gamma _ { M } ]$ . Comparing to the first case, we have $f ( \gamma _ { m } ) \leq f ( \gamma _ { M } ) = \rho ( \gamma _ { M } ) + 2 \sqrt { \rho ( \gamma _ { M } ) \log ( 1 / \delta ) } .$ We then conclude that $\varepsilon = f ( \gamma _ { m } )$ .
+
+212 Theorem 2. Let $\alpha \in \mathbb { R } _ { > 0 } ^ { 2 }$ and denote $\alpha _ { m } = \operatorname* { m i n } _ { i } \alpha _ { i }$ . Let $\Delta _ { 2 } , \Delta _ { \infty } > 0$ be constants that satisfy   
+213 $\begin{array} { r } { \sum _ { i } ( x _ { i } - x _ { i } ^ { \prime } ) ^ { 2 } \leq \Delta _ { 2 } ^ { 2 } } \end{array}$ and $\operatorname* { m a x } _ { i } \left| x _ { i } - x _ { i } ^ { \prime } \right| \leq \Delta _ { \infty }$ whenever x, $\mathbf { x } ^ { \prime } \in \mathbb { R } _ { \geq 0 } ^ { d }$ are sample statistics of any   
+214 two datasets differing on a single entry. For any $\delta \in ( 0 , 1 )$ , let $\gamma _ { M }$ be the solution to the equation   
+215 $\begin{array} { r } { \log ( 1 / \delta ) = \frac { 1 } { 2 } \gamma ^ { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma ) / \Delta _ { \infty } ^ { 2 } } \end{array}$ . The one-time sampling from Dirichlet $\left( \mathbf { x } + \alpha \right)$ is $( \varepsilon , \delta ) – D P$   
+216 where
+
+$$
+\varepsilon = \operatorname* { m i n } _ { \gamma \in ( 0 , \gamma _ { M } ] } f ( \gamma ) .
+$$
+
+17 Figure 1 shows how $\delta$ decays as a function of $\varepsilon$ at three different values of $\alpha _ { m }$
+
+# 4 Utility
+
+Using the results from the previous section, we analyze the Dirichlet posterior sampling’s utility in two specific tasks.
+
+# 4.1 Multinomial-Dirichlet sampling
+
+Suppose that we are observing $N$ trials, each of which has $d$ possible outcomes. For each $i \in$ $\{ 1 , \ldots , d \}$ , let $x _ { i }$ be the number of times the $i$ -th outcome was observed. Then we have the multinomial likelihood $\begin{array} { r } { p ( \mathbf { x } | \mathbf { y } ) \propto \prod _ { i } y _ { i } ^ { x _ { i } } } \end{array}$ . From this, we sample from the Dirichlet posterior:
+
+$$
+\mathbf { Y } \sim { \mathrm { D i r i c h l e t } } ( \mathbf { x } + \pmb { \alpha } ) .
+$$
+
+225 Suppose that we want to sample from a true distribution $P _ { \mathbf { X } } \sim \mathrm { D i r i c h l e t } ( \mathbf { x } + { \pmb { \alpha } } )$ , but for privacy   
+226 reasons, we instead sample from $Q _ { \mathbf { x } } \sim \mathrm { D i r i c h l e t } ( \mathbf { x } + \pmb { \alpha } ^ { \prime } )$ where $\alpha _ { i } ^ { \prime } > \alpha _ { i }$ for all $i$ . The utility of the   
+227 privacy scheme is then measured by the KL-divergence between $P _ { \mathbf { x } }$ and $Q _ { \mathbf { x } }$ . Assuming that $\mathbf { x }$ is an   
+228 observation of Multinomial $\mathbf { \tau } ( \mathbf { p } )$ , the following Theorem tells us that, on average, the KL-divergence   
+229 is small when the sample size is large, and the $p _ { i }$ ’s are evenly distributed.
+
+Theorem 3. Let 30 $\mathbf { p } : = \left( p _ { 1 } , \ldots , p _ { d } \right)$ where $p _ { i } > 0$ for all $i$ and $\textstyle \sum _ { i } p _ { i } \ = \ 1$ . Define a random variable 31 $\mathbf { X } \sim$ Multinomial $\mathbf { \tau } ( \mathbf { p } )$ . Let $P _ { \mathbf { X } } \sim \mathrm { D i r i c h l e t } ( \mathbf { X } + { \boldsymbol { \alpha } } )$ and $Q _ { \mathbf { X } } \sim { \mathrm { D i r i c h l e t } } ( \mathbf { X } + \mathbf { \alpha } \mathbf { \alpha } ^ { \prime } )$ where 32 $\alpha _ { i } ^ { \prime } \geq \alpha _ { i } \geq 1$ for all $i$ . The following estimate holds:
+
+$$
+\mathbb { E } _ { \mathbf { X } } [ \mathrm { D } _ { \mathrm { K L } } ( P _ { \mathbf { X } } \| Q _ { \mathbf { X } } ) ] \leq \frac { 1 } { N + 1 } \sum _ { i } ( \alpha _ { i } ^ { \prime } - \alpha _ { i } ) ^ { 2 } \cdot \frac { 1 } { p _ { i } } .
+$$
+
+233 The proof is given in Appendix 2. Let us consider a simple privacy scheme where we fix $s > 0$ and let 234 $\alpha _ { i } ^ { \prime } = \alpha _ { i } + s$ for all $i$ . Thus (10) becomes:
+
+$$
+\mathbb { E } _ { \mathbf { X } } [ \mathrm { D } _ { \mathrm { K L } } ( P _ { \mathbf { X } } \| Q _ { \mathbf { X } } ) ] \leq \frac { G ( \mathbf { p } ) s ^ { 2 } } { N + 1 } ,
+$$
+
+where 235 $G ( \mathbf { p } ) : = \textstyle \sum _ { i } 1 / p _ { i }$ . Now we take into account the privacy parameters. Let $\rho = \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } - \gamma )$ and 236 $\rho ^ { \prime } = \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha _ { m } ^ { \prime } - \gamma )$ , where $\alpha _ { m } = \operatorname* { m i n } _ { i } \alpha _ { i }$ , $\alpha _ { m } ^ { \prime } = \operatorname* { m i n } _ { i } \alpha _ { i } ^ { \prime }$ , and $\gamma < \alpha _ { m }$ . Here, we approximate the values of 237 $\psi ^ { \prime } ( \alpha _ { m } - \gamma )$ and $\psi ^ { \prime } ( \alpha _ { m } ^ { \prime } - \gamma )$ under two regimes:
+
+High-privacy regime: 238 $\alpha _ { m } ^ { \prime } \mathrm { ~ - ~ } \gamma \mathrm { ~ > ~ } 1$ . We have $\psi ^ { \prime } ( \alpha _ { m } ^ { \prime } - \gamma ) \approx 1 / ( \alpha _ { m } ^ { \prime } - \gamma )$ , which implies 239 $\alpha _ { m } ^ { \prime } - \gamma \approx \Delta _ { 2 } ^ { 2 } / \rho ^ { \prime }$ . We also have $\alpha _ { m } - \gamma \approx \Delta _ { 2 } ^ { 2 } / \rho$ for $\alpha _ { m } - \gamma \geq 1$ and $\alpha _ { m } - \gamma > ( \alpha _ { m } - \gamma ) ^ { 2 } \approx \Delta _ { 2 } ^ { 2 } / \rho$ 240 for $\alpha - \gamma < 1$ . Thus we have the following bound for the right-hand side of (11):
+
+$$
+\frac { G ( { \bf p } ) s ^ { 2 } } { N + 1 } = \frac { G ( { \bf p } ) ( \alpha _ { m } ^ { \prime } - \alpha _ { m } ) ^ { 2 } } { N + 1 } \lesssim \frac { \Delta _ { 2 } ^ { 4 } G ( { \bf p } ) } { N + 1 } \biggl ( \frac { 1 } { \rho ^ { \prime } } - \frac { 1 } { \rho } \biggr ) ^ { 2 } < \frac { \Delta _ { 2 } ^ { 4 } G ( { \bf p } ) } { \rho ^ { \prime 2 } ( N + 1 ) } .
+$$
+
+Consequently, we have 241 $\mathrm { D } _ { \mathrm { K L } } ( P \| Q ) < \epsilon$ for $\begin{array} { r } { N = \Omega \left( \frac { \Delta _ { 2 } ^ { 4 } G ( \mathbf { p } ) } { \rho ^ { \prime 2 } \epsilon } \right) } \end{array}$
+
+242 Low-privacy regime: $1 > \alpha _ { m } ^ { \prime } - \gamma > 0$ . This is similar as above, except we have $\alpha _ { m } ^ { \prime } - \gamma \approx$   
+243 $\Delta _ { 2 } / \rho ^ { \prime 1 / 2 }$ and $\alpha _ { m } - \gamma \approx \Delta _ { 2 } / \rho ^ { 1 / 2 }$ . Similar computation as (12) shows that $\mathrm { D } _ { \mathrm { K L } } ( P \| Q ) < \epsilon$ when   
+244 $\begin{array} { r } { N = \Omega \left( \frac { \Delta _ { 2 } ^ { 2 } G ( \mathbf { p } ) } { \rho ^ { \prime } \epsilon } \right) } \end{array}$
+
+We observe that, in both regimes, the sample size scales faster with respect to $\epsilon$ with a higher value of $G ( \mathbf { p } )$ , which is associated with a higher number of outcomes $d$ , and more concentrated multinomial parameter $\mathbf { p }$ ; this agrees with the result of our simulation in Appendix 3. Moreover, for small $\rho ^ { \prime }$ the sample size scales as $1 / \rho ^ { \prime 2 }$ , while for large $\rho ^ { \prime }$ the sample size scales as $1 / \rho ^ { \prime }$ .
+
+# 4.2 Private normalized histograms
+
+250 Let $\mathbf { x } = ( x _ { 1 } , \ldots , x _ { d } )$ be a histogram of $N$ observations and $\mathbf { p } : = \mathbf { x } / N$ . We can privatize $\mathbf { p }$ by   
+251 sampling a probability vector: $\mathbf Y \sim \mathrm { D i r i c h l e t } ( { \mathbf x } + { \boldsymbol \alpha } )$ . Note that $\mathbf { Y }$ is a biased estimator of $\mathbf { p }$ .   
+252 Denoting $\alpha _ { 0 } : = \textstyle \sum _ { i } \alpha _ { i }$ , the bias of each component of $\mathbf { Y }$ is given by $\mathbb { E } [ \mathbf { Y } ] - p _ { i }$ . Hence,
+
+$$
+| \mathrm { B i a s } ( Y _ { i } ) | = \left| \frac { x _ { i } + \alpha _ { i } } { N + \alpha _ { 0 } } - p _ { i } \right| = \frac { | x _ { i } \alpha _ { 0 } - N \alpha _ { i } | } { N ( N + \alpha _ { 0 } ) } \leq \frac { N \alpha _ { 0 } } { N ( N + \alpha _ { 0 } ) } = \frac { \alpha _ { 0 } } { N + \alpha _ { 0 } } .
+$$
+
+Since 253 $Y _ { i } \sim \mathrm { B e t a } ( x _ { i } + \alpha _ { i } , N + \alpha _ { 0 } - x _ { i } - \alpha _ { i } )$ is $\frac { 1 } { 4 ( N + \alpha _ { 0 } + 1 ) }$ -sub-Gaussian [MA17], we have,
+
+$$
+\begin{array} { r l } & { \mathbb { P } [ | Y _ { i } - p _ { i } | > t + | \mathrm { B i a s } ( Y _ { i } ) | ] \le \mathbb { P } [ | Y _ { i } - \mathbb { E } [ Y _ { i } ] | + | \mathrm { B i a s } ( Y _ { i } ) | > t + | \mathrm { B i a s } ( Y _ { i } ) | ] } \\ & { \qquad = \mathbb { P } [ | Y _ { i } - \mathbb { E } [ Y _ { i } ] | > t ] } \\ & { \qquad \le 2 e ^ { - 2 t ^ { 2 } ( N + \alpha _ { 0 } + 1 ) } . } \end{array}
+$$
+
+With the union bound, we plug in 254 $\begin{array} { r } { t = \sqrt { \frac { \log ( 2 d / \beta ) } { 2 ( N + \alpha _ { 0 } + 1 ) } } } \end{array}$ , for any $\beta \in ( 0 , 1 )$ , to obtain the following 255 accuracy guarantee of the private normalized histogram:
+
+256 Theorem 4. Let $\mathbf Y \sim \mathrm { D i r i c h l e t } ( { \mathbf x } + { \boldsymbol \alpha } )$ , where $\mathbf { x } \in \mathbb { R } _ { \geq 0 } ^ { d }$ and $\alpha \in \mathbb { R } _ { > 0 } ^ { d }$ , and $\mathbf { p } : = \mathbf { x } / N$ . For any   
+257 $\beta \in ( 0 , 1 )$ , with probability at least $1 - \beta$ , the following inequality holds:
+
+$$
+\| \mathbf { Y } - \mathbf { p } \| _ { \infty } \leq \sqrt { \frac { \log ( 2 d / \beta ) } { 2 ( N + \alpha _ { 0 } + 1 ) } } + \frac { \alpha _ { 0 } } { N + \alpha _ { 0 } } .
+$$
+
+258 Given $\epsilon > 0$ , we use (13) to find a lower bound for $N$ that gives $\| \mathbf { Y } - \mathbf { p } \| _ { \infty } < \epsilon$ w.p. $1 - \beta$ when   
+259 $\mathbf { Y }$ is sampled with $\rho$ -tCDP. For simplicity, we consider a uniform prior: $\alpha _ { i } = \alpha > 0$ for all $i$ .   
+260 Thus, $\begin{array} { r } { \rho = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha - \gamma ) } \end{array}$ , where $\gamma$ might be chosen according to Corollary 2. We consider the two   
+261 following regimes:
+
+High-privacy regime: $\alpha - \gamma > 1$ . In this case, $\psi ^ { \prime } ( \alpha - \gamma ) \approx 1 / ( \alpha - \gamma )$ . From $\begin{array} { r } { \rho = \frac { 1 } { 2 } \Delta _ { 2 } ^ { 2 } \psi ^ { \prime } ( \alpha - \gamma ) } \end{array}$ we have $\alpha \approx \Delta _ { 2 } ^ { 2 } / 2 \rho + \gamma$ . Replacing $\alpha _ { 0 }$ by $d \alpha$ in (13) yields the sample size:
+
+$$
+N = \Omega \biggl ( \frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { d } { \epsilon } \biggl ( \frac { \Delta _ { 2 } ^ { 2 } } { 2 \rho } + \gamma \biggr ) \biggr ) ,
+$$
+
+264 for the desired accuracy.
+
+265 Low-privacy regime: $\alpha - \gamma < 1$ . This is the same as above, except now we have $\psi ^ { \prime } ( \alpha - \gamma ) \approx$   
+266 $1 / ( \alpha - \gamma ) ^ { 2 }$ , which implies $\alpha \approx \Delta _ { 2 } / ( 2 \rho ) ^ { 1 / 2 } + \gamma$ . The sample size that guarantees the desired   
+267 accuracy is:
+
+$$
+N = \Omega \biggl ( \frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { d } { \epsilon } \biggl ( \frac { \Delta _ { 2 } } { \sqrt { 2 \rho } } + \gamma \biggr ) \biggr ) .
+$$
+
+268 Let us compare this result to the Gaussian mechanism, which adds a noise $\mathbf { Z } \sim N ( 0 , \sigma ^ { 2 } I _ { d } )$ to the   
+269 normalized histogram $\mathbf { p }$ directly. Thus the $\ell _ { 2 }$ -sensitivity in this case is $\Delta _ { 2 } / N$ . We have that the   
+270 Gaussian mechanism is $\rho$ -zCDP where $\begin{array} { r } { \rho = \frac { \Delta ^ { 2 } } { 2 N ^ { 2 } \sigma ^ { 2 } } } \end{array}$ [BS16]. Using the same argument as above, with   
+271 probability at least $1 - \beta$ , the following inequality holds for all $i$ :
+
+$$
+\| \mathbf { Z } \| _ { \infty } \leq \sqrt { \frac { \log ( 2 d / \beta ) \Delta _ { 2 } ^ { 2 } } { N ^ { 2 } \rho } } .
+$$
+
+![](images/4d398e42217bd393cee6a912cf2c04e26395de3300d829c872c8f73d8e70fe67.jpg)  
+Figure 2: The $\ell ^ { \infty }$ -accuracy, as a function of $N$ , of Dirichlet posterior sampling $( \gamma = 1 )$ ) and Gaussian mechanisms for private normalized histograms $\Delta _ { 2 } ^ { 2 } = 2$ and $\Delta _ { \infty } = 1 $ ). For each $N , d$ and $\rho$ , we generated the inputs $\mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { 2 0 0 }$ , where $\mathbf { x } _ { k } \sim$ Multinomial $\left( \mathbf { q } _ { k } \right)$ and $\mathbf { q } _ { k } \sim \mathrm { D i r i c h l e t } ( 5 , \ldots , 5 )$ .
+
+Hence, the sample size of 272 $N = \Omega \Big ( \sqrt { \log ( 2 d / \beta ) \Delta _ { 2 } ^ { 2 } / \rho \epsilon ^ { 2 } } \Big )$ guarantees the desired accuracy. Compar273 ing this to (14), if we assume $\epsilon < 1$ , the AM-GM inequality tells us that
+
+$$
+\frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { d \Delta _ { 2 } ^ { 2 } } { \rho \epsilon } > \frac { \log ( 2 d / \beta ) } { \epsilon ^ { 2 } } + \frac { \Delta _ { 2 } ^ { 2 } } { \rho } \geq 2 \sqrt { \frac { \log ( 2 d / \beta ) \Delta _ { 2 } ^ { 2 } } { \rho \epsilon ^ { 2 } } } .
+$$
+
+274 The inequality (17) implies that the Gaussian mechanism requires less sample than the Dirichlet   
+275 mechanism in order to guarantee the same level of accuracy. The Gaussian mechanism is also better   
+276 in the low-privacy regime as the $\rho$ in (15) satisfies ${ \sqrt { \rho } } < \rho$ and $\Delta _ { 2 } \approx \Delta _ { 2 } ^ { 2 }$ , leading to the same   
+277 inequality (17). Nonetheless, the decay in (16) is linear in $d$ , while that in (13) has $\alpha _ { 0 } = d \alpha$ in   
+278 the denominators. This observation suggests that, when $\mathbf { x }$ is a sparse histogram i.e. when $N \leq d$ ,   
+279 the $\ell ^ { \infty }$ -accuracy of the Dirichlet mechanism is smaller than that of the Gaussian mechanism. This   
+280 conclusion is supported by our simulation in Figure 2. We see that the $\ell ^ { \infty }$ -accuracy of the Dirichlet   
+281 mechanism is smaller than that of the Gaussian mechanism for small $N$ when $d = 1 0 0 0$ . The code   
+282 for all experiments in this study can be found in the supplemental material.
+
+# 283 Potential negative societal impacts
+
+It is important to note that, when $\rho$ becomes unacceptably large (e.g., $\rho = 1 0 ^ { 4 }$ ), the sampling is far away from being private. Thus any organization that deploys the posterior sampling on sensitive data must not vacuously refer to this study and claim that its algorithm is private. It is the organization’s responsibility to fully publish the prior parameters, and educate its users/customers on differential privacy and how the privacy guarantees are calculated.
+
+It is desirable that differentially private algorithms are accurate for the task at hand, especially when the data is used for important decision-making. Thus, one needs to make sure that there is enough sample to achieve the desired level of accuracy. For a large differentially private system, privacy budgets need to be allocated to the parts that require accurate outputs.
+
+293 Lastly, one must be careful with the choice of prior parameters; if a uniform prior is used, smaller   
+294 groups will suffer a relatively larger statistical bias. As a result, private statistics of small populations   
+295 (such as ethnic or racial minorities) will be relatively less accurate. One way to get around this issue   
+296 is to (privately) impose larger prior parameters on larger populations.   
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+
+# 401 Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We gave a simple guaranteed upper bound of tCDP (2) for the Dirichlet posterior sampling and illustrated how it can be used to derive accuracy guarantees in Section 4.   
+(b) Did you describe the limitations of your work? [Yes] We discussed a limitation of the guaranteed upper bound of tCDP in the paragraph following Theorem 1. We also described a situation under which the Gaussian mechanism is preferable to the Dirichlet posterior sampling at the end of Section 4.2.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See the section on potential negative societal impacts at the end of the paper.
+
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes]   
+(b) Did you include complete proofs of all theoretical results? [No] The proofs of all theorems are given in the main paper, except that of Theorem 3 which is given in Appendix 2.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code and the instructions for our simulations are included in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We specified the details of our simulations in the figures’ captions.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We reported the error bars in Figure 2   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [N/A] Our experiments are not computationally intensive.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A]   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/fgBxcKcz9ea/fgBxcKcz9ea_content_list.json

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+    {
+        "type": "text",
+        "text": "Differential Privacy of Dirichlet Posterior Sampling ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        ],
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+    },
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+        "text": "1 We study the inherent privacy of releasing a single sample from a Dirichlet posterior   \n2 distribution. As a complement to the previous study that provides general theories   \n3 on the differential privacy of posterior sampling from exponential families, this   \n4 study focuses specifically on the Dirichlet posterior sampling and its privacy   \n5 guarantees. With the notion of truncated concentrated differential privacy (tCDP),   \n6 we are able to derive a simple privacy guarantee of the Dirichlet posterior sampling,   \n7 which effectively allows us to analyze its utility in various settings. Specifically,   \n8 we provide accuracy guarantees of the Dirichlet posterior sampling in Multinomial  \n9 Dirichlet sampling and private normalized histogram publishing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "10 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
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+        ],
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+    },
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+        "text": "11 The Bayesian framework provides a way to perform statistical analysis by combining prior beliefs   \n12 with real-life evidence. At a high level, the belief and the evidence are assumed to be described   \n13 by probabilistic models. As we receive new data, our belief is updated accordingly via the Bayes’   \n14 theorem, resulting in the so-called posterior belief. The posterior tells us how much we are uncertain   \n15 about the model’s parameters.   \n16 The Dirichlet distribution is usually chosen as the prior when performing Bayesian analysis on discrete   \n17 variables, as it is a conjugate prior to the categorical and multinomial distributions. Specifically,   \n18 Dirichlet distributions are often used in discrete mixture models, where a Dirichlet prior is put on   \n19 the mixture weights [LW92; MMR05]. Such models have applications in NLP [PB98], biophysical   \n20 systems [Hin15], accident analysis [de 06], and genetics [BHW00; PM01; CWS03]. In all of these   \n21 studies, samplings from Dirichlet posteriors arise when performing Markov chain Monte Carlo   \n22 methods for approximate Bayesian inference.   \n23 Dirichlet posterior sampling also appears in other learning tasks. For example, in Bayesian active   \n24 learning, it arises in Gibbs sampling, which is used to approximate the posterior of the classifier over   \n25 the labeled sample [NLYCC13]. In Thompson sampling for multi-armed bandits, one repeatedly   \n26 draws a sample from the Dirichlet posterior of each arm, and picks the arm whose sample maximizes   \n27 the reward [ZHGSY20; AAFK20; NIK20]. And in Bayesian reinforcement learning, state-transition   \n28 probabilities are sampled from the Dirichlet posterior over past observed states [Str00; ORR13].   \n29 Dirichlet posterior sampling can also be used for data synthesis. Suppose that we have a histogram   \n30 $( x _ { 1 } , \\ldots , x _ { d } )$ of actual data. An approximate discrete distribution of this histogram can be obtained by   \n31 drawing a sample $\\mathbf { Y }$ from Dirichlet $( x _ { 1 } + \\alpha _ { 1 } , \\ldots , x _ { d } + \\alpha _ { d } )$ , where $\\alpha _ { 1 } , \\ldots , \\alpha _ { d }$ are prior parameters.   \n32 Then synthetic data is produced by repeatedly drawing from Multinomial(Y). There are many   \n33 studies on data synthesis that followed this approach [AV08; MKAGV08; RWZ14; PG14; SJGLY17].   \n34 In the above examples, the data that we integrate into these tasks might contain sensitive information.   \n35 Thus it is important to ask: how much of the information is protected from the Dirichlet samplings?   \n36 The goal of this study is to find an answer to this question.   \n37 The mathematical framework of differential privacy (DP) [DMNS06] allows us to quantify how much   \n38 the privacy of the Dirichlet posterior sampling is affected by the prior parameters $\\alpha _ { 1 } , \\ldots , \\alpha _ { d }$ . In the   \n39 definition of DP, the privacy of a randomized algorithm is measured by how much its distribution   \n40 changes upon perturbing a single data point of the input. Nonetheless, this notion might be too   \n41 strict for the Dirichlet distribution, as a small perturbation of a near-zero parameter can cause a large   \n42 distribution shift. Thus, it might be more appropriate to rely on one of several relaxed notions of   \n43 DP, such as approximate differential privacy, Rényi differential privacy, or concentrated differential   \n44 privacy. It is natural to wonder if the Dirichlet posterior sampling satisfies any of these definitions. ",
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+    {
+        "type": "text",
+        "text": "45 1.1 Overview of Our results ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            281,
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+        ],
+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "46 This study focuses on the privacy and utility of Dirichlet posterior sampling. In summary, we provide   \n47 a closed-form privacy guarantee of the Dirichlet posterior sampling, which in turn allows us to   \n48 effectively analyze its utility in various settings. ",
+        "bbox": [
+            148,
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+    },
+    {
+        "type": "text",
+        "text": "$\\ S 3$ Privacy. We study the role of the prior parameters in the privacy of the Dirichlet posterior sampling. Theorem 1 is our main result, where we provide a guaranteed upper bound for truncated concentrated differential privacy (tCDP) of the Dirichlet posterior sampling. In addition, we convert the tCDP guarantee into an approximate differential privacy guarantee in Corollary 2. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "$\\ S 4$ Utility. Using the tCDP guarantee, we investigate the utility of Dirichlet posterior sampling applied in two specific applications: ",
+        "bbox": [
+            163,
+            422,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• In Section 4.1, we consider one-time sampling from a Multinomial-Dirichlet distribution. But instead of directly sampling from this distribution, we sample from another distribution with larger prior parameters. The accuracy is then measured by the KL-divergence between the original and the private distributions.   \n• In Section 4.2, we use the Dirichlet posterior sampling for a private release of a normalized histogram. In this case, the accuracy is measured by the mean-squared error between the sample and the original normalized histogram. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "62 In both tasks, we compute the sample size that guarantees the desired level of accuracy. In the case   \n63 of private histogram publishing, we also compare the Dirichlet posterior sampling to the Gaussian   \n64 mechanism. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "65 1.2 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+        "text": "66 There are several studies on the differential privacy of posterior sampling. Wang, Fienberg, and   \n67 Smola [WFS15] showed that any posterior sampling with the log-likelihood bounded by $B$ is $4 B$ -   \n68 differentially private. However, the likelihoods that we study are not bounded away from zero; they   \n69 have the form $\\Pi _ { i } p _ { i } ^ { x _ { i } }$ which becomes small when one of the $p _ { i }$ ’s is close to zero. Dimitrakakis, Nelson,   \n70 Zhang, Mitrokotsa, and Rubinstein [DNZMR17] showed that if the condition on the log-likelihood is   \n71 relaxed to the Lipschitz continuity with high probability, then one can obtain the approximate DP.   \n72 Nonetheless, with the Dirichlet density, it is difficult to compute the probability of events in which   \n73 the Lipschitz condition is satisfied.   \n74 In the case that the sufficient statistics $\\mathbf { x }$ has finite $\\ell ^ { 1 }$ -sensitivity, Foulds, Geumlek, Welling and   \n75 Chaudhuri [FGWC16] suggested adding Laplace noises to $\\mathbf { x }$ . Suppose that y is the output; they   \n76 showed that sampling from $p ( \\boldsymbol { \\theta } | \\mathbf { y } )$ is differentially private and as asymptotically efficient as sampling   \n77 from $p ( \\boldsymbol { \\theta } | \\mathbf { x } )$ . However, for a small sample size, the posterior over the noisy statistics might be too   \n78 far away from the actual posterior. Bernstein and Sheldon [BS18] thus proposed to approximate the   \n79 joint distribution $p ( \\boldsymbol { \\theta } , \\mathbf { x } , \\mathbf { y } )$ using Gibbs sampling, which is then integrated over $\\mathbf { x }$ to obtain a more   \n80 accurate posterior over $\\mathbf { y }$ .   \n81 Geumlek, Song, and Chaudhuri [GSC17] were the first to study the posterior sampling with the   \n82 RDP. Even though they provided a general framework to find $( \\lambda , \\epsilon )$ -RDP guarantees for exponential   \n83 families, explicit forms of $\\epsilon$ and the upper bound of $\\lambda$ were not given. In contrast, our tCDP guarantees   \n84 of the Dirichlet posterior sampling imply an explicit expression for $\\epsilon$ , and also an upper bound for $\\lambda$ .   \n85 The privacy of data synthesis via sampling from Multinomial $( \\mathbf { Y } )$ , where $\\mathbf { Y }$ is a discrete distri  \n86 bution drawn from the Dirichlet posterior, was first studied by Machanavajjhala, Kifer, Abowd,   \n87 Gehrke, and Vilhuber [MKAGV08]. They showed that the data synthesis is $( \\varepsilon , \\delta )$ -probabilistic DP,   \n88 which implies $( \\varepsilon , \\delta )$ -approximate DP. However, as their privacy analysis includes the sampling from   \n89 Multinomial $( \\mathbf { Y } )$ , their privacy guarantee depends on the number of synthetic samples. In contrast,   \n90 we show that the one-time sampling from the Dirichlet posterior is approximate DP, which by the   \n91 post-processing property allows us to sample from Multinomial $( \\mathbf { Y } )$ as many times as we want while   \n92 retaining the same privacy guarantee.   \n93 The Dirichlet mechanism was first introduced by Gohari, Wu, Hawkins, Hale, and Topcu [GWHHT21].   \n94 Originally, the Dirichlet mechanism takes a discrete distribution $\\mathbf { p } : = ( p _ { 1 } , \\ldots , p _ { d } )$ and draws one   \n95 sample $\\mathbf { Y } \\sim \\mathrm { D i r i c h l e t } ( r p _ { 1 } , \\dots , r p _ { d } )$ . Note the absence of the prior parameters, which makes $\\mathbf { Y }$ an   \n96 unbiased estimator of $\\mathbf { p }$ . But this comes with a cost, as the worst case of privacy violation occurs   \n97 when almost all of the parameters are close to zero. The authors avoided this issue by restricting   \n98 the input space to a subset of the unit simplex, with some of the $p _ { i }$ ’s bounded below by a fixed   \n99 positive constant. This results in complicated expressions for the privacy guarantees as they involve   \n100 a minimization problem over the restricted domain. In this study, we take a different approach by   \n101 adding prior parameters to the Dirichlet mechanism. As a result, we obtain a biased algorithm that   \n102 requires no assumption on the input space and has simpler forms of privacy guarantees. ",
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+    {
+        "type": "text",
+        "text": "1.3 Notations ",
+        "text_level": 1,
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+        "text": "104 We let $\\mathbb { R } _ { \\geq 0 } ^ { d }$ be the set of $d$ -tuples of non-negative real numbers and $\\mathbb { R } _ { > 0 } ^ { d }$ be the set of $d$ -tuples of   \n105 positive real numbers. We assume that all vectors are $d$ -dimensional where $d \\geq 2$ . The notations for   \n106 all vectors are always in bold. Specifically, $\\mathbf { x } : = ( x _ { 1 } , \\ldots , x _ { d } ) \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ consists of sample statistics of   \n107 the data and $\\pmb { \\alpha } : = ( \\alpha _ { 1 } , \\ldots , \\alpha _ { d } ) \\in \\mathbb { R } _ { > 0 } ^ { d }$ consists of the prior parameters. The vector $\\mathbf { p } : = ( p _ { 1 } , \\ldots , p _ { d } )$   \n108 always satisfies $\\textstyle \\sum _ { i } p _ { i } = 1$ . The number of observations is always $N$ . We also denote $x _ { 0 } : = \\textstyle \\sum _ { i } x _ { i }$   \n109 and $\\alpha _ { 0 } : = \\textstyle \\sum _ { i } \\alpha _ { i }$ . For any vectors $\\mathbf { x } , \\mathbf { x } ^ { \\prime }$ and scalar $r > 0$ , we write $\\mathbf { x } + \\mathbf { x } ^ { \\prime } : = ( x _ { 1 } + x _ { 1 } ^ { \\prime } , \\ldots , x _ { d } + x _ { d } ^ { \\prime } )$   \n110 and $r \\mathbf { x } : = ( r x _ { 1 } , \\ldots , r x _ { d } )$ . For any positive reals $x$ and $x ^ { \\prime }$ , the notation $x \\propto x ^ { \\prime }$ means $x = C x ^ { \\prime }$ for   \n111 some constant $C > 0$ , $x \\approx x ^ { \\prime }$ means $c x ^ { \\prime } \\leq x \\leq C x ^ { \\prime }$ for some $c , C > 0$ , and $x \\lesssim x ^ { \\prime }$ means $x \\leq C x ^ { \\prime }$   \n112 for some $C > 0$ . Lastly, $\\| \\mathbf { x } \\| _ { \\infty } : = \\operatorname* { m a x } _ { i } | x _ { i } |$ is the $\\ell ^ { \\infty }$ norm of $\\mathbf { x }$ . ",
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+        "type": "text",
+        "text": "13 2 Background ",
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+        "type": "text",
+        "text": "2.1 Privacy models ",
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+        "type": "text",
+        "text": "Definition 2.1 (Pure and Approximate DP [DMNS06]). A randomized mechanism $M : \\mathcal { X } ^ { n }  \\mathcal { Y }$ is $( \\varepsilon , \\delta )$ -differentially private $( \\varepsilon , \\delta )$ -DP) if for any datasets $x , x ^ { \\prime }$ differing on a single entry, and all events $E \\subset \\mathcal { V }$ , ",
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+        "img_path": "images/a03fcdf947f012874d645b12dc14503ac64d8f6d03d5b3a81a3e0270a6008422.jpg",
+        "text": "$$\n\\mathbb { P } [ M ( x ) \\in E ] \\leq e ^ { \\varepsilon } \\mathbb { P } [ M ( x ^ { \\prime } ) \\in E ] + \\delta .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "118 If $M$ is $( \\varepsilon , 0 )$ -DP, then we say that it is $\\varepsilon$ -differential privacy $\\dot { \\varepsilon }$ -DP). ",
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+        "text": "19 The term pure differential privacy (pure DP) refers to $\\epsilon$ -differential privacy, while approximate   \n20 differential privacy (approximate DP) refers to $( \\varepsilon , \\delta )$ -DP when $\\delta > 0$ . ",
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+        "type": "text",
+        "text": "121 In contrast to pure and approximate DP, the next definitions of differential privacy are defined in terms of the Rényi divergence between 122 $M ( x )$ and $M ( x ^ { \\prime } )$ : ",
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+        "type": "text",
+        "text": "123 Definition 2.2 (Rényi Divergence [Rén61]). Let $P$ and $Q$ be probability distributions. $\\mathrm { F o r } \\lambda \\in ( 1 , \\infty )$   \n124 the Rényi divergence of order $\\lambda$ between $P$ and $Q$ is defined as ",
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+        "img_path": "images/93eccc79cfb463d3011599e5ccc70bc2df5473e9e983b54fac6d7edb496536f4.jpg",
+        "text": "$$\n\\mathrm { D } _ { \\lambda } ( P \\| Q ) : = { \\frac { 1 } { \\lambda - 1 } } \\log \\int P ( y ) ^ { \\lambda } Q ( y ) ^ { 1 - \\lambda } d y = { \\frac { 1 } { \\lambda - 1 } } \\log \\biggl ( \\operatorname { \\mathbb { E } } _ { y \\sim P } \\biggl [ { \\frac { P ( y ) ^ { \\lambda - 1 } } { Q ( y ) ^ { \\lambda - 1 } } } \\biggr ] \\biggr . . \\biggr )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "125 Definition 2.3 (tCDP and zCDP [BDRS18; BS16]). A randomized mechanism $M : \\mathcal { X } ^ { n }  \\mathcal { Y }$ is   \n126 $\\omega$ -truncated $\\rho$ -concentrated differentially private $( ( \\rho , \\omega )$ -tCDP) if for any datasets $x , x ^ { \\prime }$ differing on a   \n127 single entry and for all $\\lambda \\in ( 1 , \\omega )$ , ",
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+        "img_path": "images/ada4dfd5ac3d4ef86be7aac385827fcfd1809248cc9c468a693ec0b99e679e3a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { D } _ { \\lambda } ( M ( x ) \\| M ( x ^ { \\prime } ) ) \\leq \\lambda \\rho . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "128 If $M$ is $( \\rho , \\infty )$ -tCDP, then we say that it is $\\rho$ -zero-concentrated differential privacy ( $\\rho$ -zCDP). ",
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+        "text": "129 Note that both tCDP and zCDP have the composition and post-processing properties. Intuitively, $\\rho$ con  \n130 trols the expectation and standard deviation of the privacy loss random variable: Z = log P [M(x)=Y ]P [M(x0)=Y ] ,   \n131 where $Y$ has density $M ( x )$ , and $\\omega$ controls the number of standard deviations for which $Z$ concen  \n132 trates like a Gaussian. A smaller $\\rho$ and larger $\\omega$ correspond to a stronger privacy guarantee. It turns   \n133 out that tCDP implies approximate DP:   \n134 Lemma 1 (From tCDP to Approximate DP [BDRS18]). Let $\\delta > 0$ . If M is a $( \\rho , \\omega )$ -tCDP mechanism,   \n135 then it also satisfies $( \\varepsilon , \\delta )$ - $D P$ with ",
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+        "text": "$$\n\\varepsilon = \\left\\{ \\begin{array} { l l } { \\rho + 2 \\sqrt { \\rho \\log ( 1 / \\delta ) } \\quad } & { i f \\log ( 1 / \\delta ) \\leq ( \\omega - 1 ) ^ { 2 } \\rho } \\\\ { \\rho \\omega + \\frac { \\log ( 1 / \\delta ) } { \\omega - 1 } } & { i f \\log ( 1 / \\delta ) > ( \\omega - 1 ) ^ { 2 } \\rho } \\end{array} \\right. .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "136 2.2 Dirichlet distribution ",
+        "text_level": 1,
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+        "text": "137 For $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { d }$ , the Dirichlet distribution Dirichlet $( \\alpha )$ is a continuous distribution of $d$ -dimensional   \n138 probability vectors i.e. vectors whose coordinate sum is equal to 1. The density function of $\\mathbf { Y } \\sim$   \n139 Dirichlet $( \\alpha )$ is given by: ",
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+        "img_path": "images/cb865a389ca943847c9b7ca4d3ac4e3e6cafefda1521cf6aeef1ab38819aa540.jpg",
+        "text": "$$\np ( \\mathbf { y } ) = \\frac { 1 } { B ( \\pmb { \\alpha } ) } \\prod _ { i = 1 } ^ { d } y _ { i } ^ { \\alpha _ { i } - 1 } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "140 where $B ( \\alpha )$ is the beta function, which can be written in terms of the gamma function: ",
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+        "img_path": "images/4de47ec9a7935d45799b02e33e5087d394647393a8890ed4359090d66b6b2981.jpg",
+        "text": "$$\nB ( \\pmb { \\alpha } ) = \\frac { \\prod _ { i } \\Gamma ( \\alpha _ { i } ) } { \\Gamma ( \\sum _ { i } \\alpha _ { i } ) } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "141 2.3 Dirichlet posterior sampling ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "142 We consider the prior Dirichlet $( \\alpha )$ and the likelihood of the form $\\begin{array} { r } { p ( \\mathbf { x } | \\mathbf { y } ) \\propto \\prod _ { i = 1 } ^ { d } y _ { i } ^ { x _ { i } } } \\end{array}$ where   \n143 $\\mathbf { x } \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ consists of sample statistics of the dataset. The Dirichlet posterior sampling is a one-time   \n144 sampling: ",
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+        "img_path": "images/81c53d2e40427e02c3b08c62fc781df0d80965bd295a8ecfc387de285a339621.jpg",
+        "text": "$$\n\\mathbf { Y } \\sim { \\mathrm { D i r i c h l e t } } ( \\mathbf { x } + \\pmb { \\alpha } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "145 There is a modification of the sampling which introduces a concentration parameter $r > 0$ , and   \n146 instead we sample from Dirichlet $( r \\mathbf { x } + \\pmb { \\alpha } )$ [GSC17; GWHHT21]. Smaller values of $r$ make the   \n147 sampling more private, and larger values of $r$ make $\\mathbf { Y }$ a closer approximation of $\\mathbf { x }$ . Even though the   \n148 case $r = 1$ is the main focus of this study, our main privacy results can be easily extended to other   \n149 values of $r$ as we will see at the end of Section 3.1.   \n150 Consider a special case where $\\mathbf x = \\mathbf p$ is an empirical distribution derived from the dataset, and we   \n151 want $\\mathbf { Y }$ to be a private approximation of $\\mathbf { p }$ ; the sampling $\\mathbf { Y } \\sim { \\mathrm { D i r i c h l e t } } ( r \\mathbf { p } + \\alpha )$ is called the   \n152 Dirichlet mechanism [GWHHT21]. It is interesting to note that the Dirichlet mechanism is a form of   \n153 the exponential mechanism [MT07]: let $r > 0$ be the privacy parameter, Dirichlet $( \\alpha )$ be the prior,   \n154 and the negative KL-divergence be the score function of the exponential mechanism. Then the output   \n155 $\\mathbf { Y }$ of this mechanism is distributed according to the following density function: ",
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+        "img_path": "images/32fa8491332f190dd97efb2a6ec88182fc644c266ee63558b2bdeccc8187550c.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\frac { \\exp \\left( - r \\mathrm { D } _ { \\mathrm { K L } } ( \\mathbf { p } , \\mathbf { y } ) \\right) \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } } { \\int \\exp \\left( - r \\mathrm { D } _ { \\mathrm { K L } } ( \\mathbf { p } , \\mathbf { y } ) \\right) \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } d \\mathbf { y } } \\propto \\exp \\left( r \\sum _ { i , p _ { i } \\neq 0 } p _ { i } \\log ( y _ { i } / p _ { i } ) \\right) \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } } \\\\ & { } & { \\propto \\displaystyle \\prod _ { i , p _ { i } \\neq 0 } y _ { i } ^ { r p _ { i } } \\prod _ { i } y _ { i } ^ { \\alpha _ { i } - 1 } = \\prod _ { i } y _ { i } ^ { r p _ { i } + \\alpha _ { i } - 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "156 which is exactly the density function of Dirichlet $( r \\mathbf { p } + \\alpha )$ . ",
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+        "text": "58 In most of this study, we take advantage of several nice properties of the log-gamma function and its derivatives. Specifically, $\\begin{array} { r } { \\psi ( x ) : = \\frac { d } { d x } \\log \\Gamma ( x ) } \\end{array}$ is concave and increasing, while its derivative $\\psi ^ { \\prime } ( x )$ is positive, convex, and decreasing. In addition, 0 $\\psi ^ { \\prime }$ can be approximated by the reciprocals: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/43598d4f5aab518fd3cbc576f85848eaad039f6a414b2615b48ed5e3a2340362.jpg",
+        "text": "$$\n{ \\frac { 1 } { x } } + { \\frac { 1 } { 2 x ^ { 2 } } } < \\psi ^ { \\prime } ( x ) < { \\frac { 1 } { x } } + { \\frac { 1 } { x ^ { 2 } } } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which implies that 161 $\\textstyle \\psi ^ { \\prime } ( x ) \\approx { \\frac { 1 } { x ^ { 2 } } }$ as $x \\to 0$ and $\\begin{array} { r } { \\psi ^ { \\prime } ( x ) \\approx \\frac { 1 } { x } } \\end{array}$ as $x \\to \\infty$ ",
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+    {
+        "type": "text",
+        "text": "3 Main privacy results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 Truncated concentrated differential privacy ",
+        "text_level": 1,
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+        "text": "Theorem 1. Let $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { d }$ and $\\alpha _ { m } : = \\operatorname* { m i n } _ { i } \\alpha _ { i }$ . Let $\\gamma \\in ( 0 , \\alpha _ { m } )$ . Let $\\Delta _ { 2 } , \\Delta _ { \\infty } > 0$ be constants that satisfy $\\begin{array} { r } { \\sum _ { i } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\leq \\Delta _ { 2 } ^ { 2 } } \\end{array}$ and $\\mathrm { m a x } _ { i } \\left| x _ { i } - x _ { i } ^ { \\prime } \\right| \\leq \\Delta _ { \\infty }$ whenever x, $\\mathbf { \\Delta } , \\mathbf { x } ^ { \\prime } \\in \\mathbb { R } _ { \\geq 0 } ^ { 2 }$ are sample statistics of any two datasets differing on a single entry. The one-time sampling from Dirichlet $\\left( \\mathbf { x } + \\alpha \\right)$ is $( \\rho , \\omega )$ -tCDP, where $\\begin{array} { r } { \\omega = \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 } \\end{array}$ and ",
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+    {
+        "type": "equation",
+        "img_path": "images/9c071f26e5c9b105ed606cd08d89805f7dc5069443a3294706e91b05f41c9b8e.jpg",
+        "text": "$$\n\\rho = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            351,
+            575,
+            381
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "168 Note that $( \\rho , \\infty )$ -tCDP is not obtainable, as the ratio between two Dirichlet densities blows up as   \n169 $\\omega  \\infty$ . We present here a short proof that skips some calculations (see Appendix 1 for a full proof). ",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "proof. Consider any 170 $\\begin{array} { r } { \\lambda \\in \\left( 1 , \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 \\right) } \\end{array}$ . Let $\\mathbf { u } : = \\mathbf { x } + \\pmb { \\alpha }$ and $\\mathbf { u } ^ { \\prime } : = \\mathbf { x } ^ { \\prime } + \\alpha ^ { \\prime }$ . Let $P ( \\mathbf { y } )$ be the density of Dirichlet171 $\\mathbf { \\Pi } ( \\mathbf { u } )$ and $P ^ { \\prime } ( \\mathbf { y } )$ be the density of Dirichlet $\\mathbf { \\Pi } ^ { ( \\mathbf { u } ^ { \\prime } ) }$ . A quick calculation shows that: ",
+        "bbox": [
+            142,
+            433,
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+            472
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d51b1105be5d627508b18a044d72525a2b62b72c1b7e105ea02b3abafae54e6.jpg",
+        "text": "$$\n\\mathbb { E } _ { { \\mathbf { y } } \\sim P ( { \\mathbf { y } } ) } \\left[ \\frac { P ( { \\mathbf { y } } ) ^ { \\lambda - 1 } } { P ^ { \\prime } ( { \\mathbf { y } } ) ^ { \\lambda - 1 } } \\right] = \\frac { B ( { \\mathbf { u } } ^ { \\prime } ) ^ { \\lambda - 1 } } { B ( { \\mathbf { u } } ) ^ { \\lambda - 1 } } \\cdot \\frac { B ( { \\mathbf { u } } + ( \\lambda - 1 ) ( { \\mathbf { u } } - { \\mathbf { u } } ^ { \\prime } ) ) } { B ( { \\mathbf { u } } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "172 We take the logarithm on both sides and apply the second-order Taylor expansion to the following   \n173 $G ( u _ { i } , u _ { i } ^ { \\prime } )$ and $H ( u _ { i } , u _ { i } ^ { \\prime } )$ terms that appear on the right-hand side. As a result, there exist $\\xi$ between   \n174 $u _ { i } + ( \\lambda - 1 ) ( u _ { i } - u _ { i } ^ { \\prime } )$ and $u _ { i }$ , and $\\xi ^ { \\prime }$ between $u _ { i }$ and $u _ { i } ^ { \\prime }$ such that ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eda561c84699fa451ed7b193c850b548da587691345bf8f7becc94c8820d1cfd.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { G ( u _ { i } , u _ { i } ^ { \\prime } ) : = ( \\lambda - 1 ) ( \\log \\Gamma ( u _ { i } ^ { \\prime } ) - \\log \\Gamma ( u _ { i } ) ) } } \\\\ & { } & \\\\ & { } & { = - ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) \\psi ( u _ { i } ) + \\frac { 1 } { 2 } ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\xi ^ { \\prime } ) } \\\\ & { } & { H ( u _ { i } , u _ { i } ^ { \\prime } ) : = \\log \\Gamma ( u _ { i } + ( \\lambda - 1 ) ( u _ { i } - u _ { i } ^ { \\prime } ) ) - \\log \\Gamma ( u _ { i } ) } \\\\ & { } & { = ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) \\psi ( u _ { i } ) + \\frac { 1 } { 2 } ( \\lambda - 1 ) ^ { 2 } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\xi ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            556,
+            723,
+            655
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that 175 $\\psi ^ { \\prime }$ is increasing. If $x _ { i } > x _ { i } ^ { \\prime }$ , then $\\xi$ and $\\xi ^ { \\prime }$ are bounded below by $u _ { i } ^ { \\prime } \\geq \\alpha _ { m }$ . On the other hand, if 176 $x _ { i } \\leq x _ { i } ^ { \\prime }$ , then $\\xi$ and $\\xi ^ { \\prime }$ are bounded below by $u _ { i } - ( \\lambda - 1 ) | u _ { i } - u _ { i } ^ { \\prime } |$ . The condition 177 $\\begin{array} { r } { \\lambda < \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 } \\end{array}$ guarantees that $u _ { i } - ( \\lambda - 1 ) | u _ { i } - u _ { i } ^ { \\prime } | > \\alpha _ { m } - \\gamma$ . All cases considered, we have ",
+        "bbox": [
+            145,
+            656,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/924a0a8d6141a741238a6777371a02cd2f841049f9963563d324459104d00d79.jpg",
+        "text": "$$\n\\begin{array} { l } { G ( u _ { i } , u _ { i } ^ { \\prime } ) + H ( u _ { i } , u _ { i } ^ { \\prime } ) \\leq \\displaystyle \\frac { 1 } { 2 } \\big ( ( \\lambda - 1 ) + ( \\lambda - 1 ) ^ { 2 } \\big ) ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\lambda ( \\lambda - 1 ) ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Denoting 178 $u _ { 0 } : = \\textstyle \\sum _ { i } u _ { i }$ and $u _ { 0 } ^ { \\prime } : = \\textstyle \\sum _ { i } u _ { i } ^ { \\prime }$ , the same argument shows that $G ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) + H ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) > 0$ . 179 Therefore, ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdd25e6e1558eba61028d51c44db09f492a18b03b4ee8237fc3befaaf821011a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { D _ { \\boldsymbol { \\lambda } } ( P ( \\mathbf { y } ) \\| P ^ { \\prime } ( \\mathbf { y } ) ) = \\displaystyle \\frac { 1 } { \\boldsymbol { \\lambda } - 1 } \\Biggl ( \\sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \\prime } ) + H ( u _ { i } , u _ { i } ^ { \\prime } ) ) - G ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) - H ( u _ { 0 } , u _ { 0 } ^ { \\prime } ) \\Biggr ) } \\\\ & { \\quad \\quad < \\displaystyle \\frac { 1 } { \\boldsymbol { \\lambda } - 1 } \\sum _ { i } ( G ( u _ { i } , u _ { i } ^ { \\prime } ) + H ( u _ { i } , u _ { i } ^ { \\prime } ) ) } \\\\ & { \\quad \\quad \\le \\displaystyle \\frac { 1 } { 2 } \\boldsymbol { \\lambda } \\sum _ { i } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) \\le \\frac { 1 } { 2 } \\boldsymbol { \\lambda } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            797,
+            784,
+            916
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9b723c7be81bab0fe812229760033b22531dcf9d6fe77c60ac3901a2063deaff.jpg",
+        "image_caption": [
+            "Figure 1: Left: the actual values of $\\begin{array} { r } { \\rho = \\frac { 1 } { 2 } \\operatorname { D } _ { 2 } ( P \\| P ^ { \\prime } ) } \\end{array}$ and the worst case $( \\rho , 2 )$ -tCDP guarantees (2) at $\\Delta _ { 2 } ^ { 2 } = \\Delta _ { \\infty } = 1$ . Here, $P$ and $P ^ { \\prime }$ are Dirichlet posterior densities over $\\mathbf { x } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 0 )$ , $\\mathbf { x } ^ { \\prime } = ( 1 1 , 8 , 6 5 , 2 5 , 3 8 , 1 )$ , and $\\pmb { \\alpha } = ( \\alpha , \\ldots , \\alpha )$ . Right: comparison between $( \\varepsilon , \\delta )$ -DP guarantees of the Dirichlet posterior samplings (8) with different uniform priors: $\\pmb { \\alpha } = ( \\alpha , \\ldots , \\alpha )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            102,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "180 The guaranteed upper bound (2) is independent of the sample statistics. As a result, the bound applies   \n181 even in worst settings i.e., when $x _ { i } = 0$ and $x _ { i } ^ { \\prime } = \\Delta _ { \\infty }$ , or vice versa, for some $i$ . As we can see in   \n182 Figure 1, the upper bound is a close approximation to the actual value of $\\rho$ when $x _ { 6 } = 0$ and $x _ { 6 } ^ { \\prime } = 1$ .   \n183 However, being a sample independent bound, the difference becomes substantial when all $x _ { i }$ ’s are   \n184 large. There is one way to get around this issue: if there is no privacy violation in assuming that   \n185 the sample statistics are always bounded below by some threshold $\\tau$ , then we can incorporate the   \n186 threshold into the prior (thus $\\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ in (2) is replaced by $\\psi ^ { \\prime } ( \\alpha _ { m } + \\tau - \\gamma ) )$ .   \n187 The parameter $\\gamma$ allows us to adjust the moment bound $\\omega$ as desired. Even though a higher $\\omega$ usually   \n188 leads to a better privacy guarantee, there are two downsides to picking $\\gamma$ close to $\\alpha _ { m }$ in this case.   \n189 First, note that $\\rho$ contains $\\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ ; as $\\gamma \\to \\alpha _ { m }$ , the value of $\\rho$ diverges to $\\infty$ , leading to a weaker   \n190 privacy guarantee instead. Second, as the Taylor approximation (5) is accurate when $u _ { i }$ is close to   \n191 $u _ { i } + ( \\lambda - 1 ) ( u _ { i } - u _ { i } ^ { \\prime } )$ , having a large value of $\\lambda$ would push the guaranteed upper bound away from   \n192 the actual privacy loss. Thus it is recommended to pick $\\gamma$ so that $\\gamma / \\Delta _ { \\infty } \\geq 1$ and $\\alpha _ { m } - \\gamma \\gg 0$ .   \n193 Alternatively, we can choose the value of $\\gamma$ that minimizes $\\varepsilon$ when converting from tCDP to $( \\varepsilon , \\delta )$ -DP   \n194 using Lemma 1—this method will be explored in the next subsection. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 can be easily applied to sampling from Dirichlet $( r \\mathbf { x } + \\alpha )$ . Replacing $\\mathbf { x }$ with $r \\mathbf { x }$ , we have $\\Delta _ { 2 }$ replaced by $r \\Delta _ { 2 }$ and $\\Delta _ { \\infty }$ replaced by $r \\Delta _ { \\infty }$ . Consequently, the sampling is $\\begin{array} { r } { \\left( \\rho , \\frac { \\gamma } { r \\Delta _ { \\infty } } + 1 \\right) } \\end{array}$ -tCDP, where $\\rho = { \\textstyle { \\frac { 1 } { 2 } } r ^ { 2 } \\Delta _ { 2 } ^ { 2 } } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ . In Appendix 4, we analyze the scaling of $r$ in conjunction with $\\alpha _ { m }$ at a fixed privacy budget $\\rho$ . ",
+        "bbox": [
+            163,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Approximate differential privacy ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            699,
+            442,
+            715
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "200 We now convert the tCDP guarantee to an approximate DP guarantee. Let $\\delta \\in ( 0 , 1 )$ . Using Lemma 1,   \n201 the Dirichlet posterior sampling with Dirichlet $( \\alpha )$ as the prior is $( \\varepsilon , \\delta )$ -DP with ",
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+    {
+        "type": "equation",
+        "img_path": "images/6298ff4207b3ce4534b84df6d385b7bcce1b93402f660e9b6ae174c7d896ac30.jpg",
+        "text": "$$\n\\varepsilon = \\left\\{ \\begin{array} { l l } { \\rho ( \\gamma ) + 2 \\sqrt { \\rho ( \\gamma ) \\log ( 1 / \\delta ) } } & { \\mathrm { i f ~ } \\log ( 1 / \\delta ) \\leq \\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 } } \\\\ { \\rho ( \\gamma ) \\Big ( \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 \\Big ) + \\frac { \\log ( 1 / \\delta ) \\Delta _ { \\infty } } { \\gamma } } & { \\mathrm { i f ~ } \\log ( 1 / \\delta ) > \\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 } } \\end{array} , \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where 202 $\\begin{array} { r } { \\rho ( \\gamma ) = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) } \\end{array}$ ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "203 We try to minimize $\\epsilon$ by adjusting the value of $\\gamma$ . First, we consider the case $\\log ( 1 / \\delta ) \\leq \\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 }$   \n204 Since $\\rho ( \\gamma )$ is a strictly increasing function of $\\gamma$ , both $\\rho ( \\gamma ) + 2 \\sqrt { \\rho ( \\gamma ) \\log ( 1 / \\delta ) }$ and $\\gamma ^ { 2 } \\rho ( \\gamma ) / \\Delta _ { \\infty } ^ { 2 }$   \n205 are both strictly increasing function of $\\gamma$ . Therefore, $\\varepsilon$ is minimized at the minimum possible   \n206 value of $\\gamma$ in this case, that is, at the unique $\\gamma _ { M }$ that satisfies $\\log ( 1 / \\delta ) = \\gamma _ { M } ^ { 2 } \\rho ( \\gamma _ { M } ) / \\Delta _ { \\infty } ^ { 2 } =$   \n207 $\\begin{array} { r } { \\frac { 1 } { 2 } \\gamma _ { M } ^ { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma _ { M } ) / \\Delta _ { \\infty } ^ { 2 } } \\end{array}$ .   \n208 Now we consider the second case, when $\\gamma < \\gamma _ { M }$ . As $\\rho ( \\gamma )$ is an increasing positive convex function   \n209 of $\\gamma$ , the function ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            138,
+            90,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dcd8f2f31dac6a5156ad364cfe9c4a90ce60b51aeb6769e198459e174b0e14bb.jpg",
+        "text": "$$\nf ( \\gamma ) : = \\frac 1 2 \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) \\left( \\frac { \\gamma } { \\Delta _ { \\infty } } + 1 \\right) + \\frac { \\log ( 1 / \\delta ) \\Delta _ { \\infty } } { \\gamma } ; \\qquad \\gamma \\in ( 0 , \\gamma _ { M } ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            121,
+            743,
+            156
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "is also convex in $\\gamma$ , and thus has a unique minimizer $\\underline { { \\gamma _ { m } } } \\in ( 0 , \\gamma _ { M } ]$ . Comparing to the first case, we have $f ( \\gamma _ { m } ) \\leq f ( \\gamma _ { M } ) = \\rho ( \\gamma _ { M } ) + 2 \\sqrt { \\rho ( \\gamma _ { M } ) \\log ( 1 / \\delta ) } .$ We then conclude that $\\varepsilon = f ( \\gamma _ { m } )$ . ",
+        "bbox": [
+            161,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "212 Theorem 2. Let $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { 2 }$ and denote $\\alpha _ { m } = \\operatorname* { m i n } _ { i } \\alpha _ { i }$ . Let $\\Delta _ { 2 } , \\Delta _ { \\infty } > 0$ be constants that satisfy   \n213 $\\begin{array} { r } { \\sum _ { i } ( x _ { i } - x _ { i } ^ { \\prime } ) ^ { 2 } \\leq \\Delta _ { 2 } ^ { 2 } } \\end{array}$ and $\\operatorname* { m a x } _ { i } \\left| x _ { i } - x _ { i } ^ { \\prime } \\right| \\leq \\Delta _ { \\infty }$ whenever x, $\\mathbf { x } ^ { \\prime } \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ are sample statistics of any   \n214 two datasets differing on a single entry. For any $\\delta \\in ( 0 , 1 )$ , let $\\gamma _ { M }$ be the solution to the equation   \n215 $\\begin{array} { r } { \\log ( 1 / \\delta ) = \\frac { 1 } { 2 } \\gamma ^ { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma ) / \\Delta _ { \\infty } ^ { 2 } } \\end{array}$ . The one-time sampling from Dirichlet $\\left( \\mathbf { x } + \\alpha \\right)$ is $( \\varepsilon , \\delta ) – D P$   \n216 where ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5347fdf65d8f5bd7b8451bda92c6cba6c425f16d7d5ff29dc3ed4b319a7a3d60.jpg",
+        "text": "$$\n\\varepsilon = \\operatorname* { m i n } _ { \\gamma \\in ( 0 , \\gamma _ { M } ] } f ( \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "17 Figure 1 shows how $\\delta$ decays as a function of $\\varepsilon$ at three different values of $\\alpha _ { m }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Utility ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            328,
+            258,
+            345
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Using the results from the previous section, we analyze the Dirichlet posterior sampling’s utility in two specific tasks. ",
+        "bbox": [
+            176,
+            359,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Multinomial-Dirichlet sampling ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            404,
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+            420
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Suppose that we are observing $N$ trials, each of which has $d$ possible outcomes. For each $i \\in$ $\\{ 1 , \\ldots , d \\}$ , let $x _ { i }$ be the number of times the $i$ -th outcome was observed. Then we have the multinomial likelihood $\\begin{array} { r } { p ( \\mathbf { x } | \\mathbf { y } ) \\propto \\prod _ { i } y _ { i } ^ { x _ { i } } } \\end{array}$ . From this, we sample from the Dirichlet posterior: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e9407c845bb1087a07c2694a9563c1955bd334025bf84648bde6447d9bc518b.jpg",
+        "text": "$$\n\\mathbf { Y } \\sim { \\mathrm { D i r i c h l e t } } ( \\mathbf { x } + \\pmb { \\alpha } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "225 Suppose that we want to sample from a true distribution $P _ { \\mathbf { X } } \\sim \\mathrm { D i r i c h l e t } ( \\mathbf { x } + { \\pmb { \\alpha } } )$ , but for privacy   \n226 reasons, we instead sample from $Q _ { \\mathbf { x } } \\sim \\mathrm { D i r i c h l e t } ( \\mathbf { x } + \\pmb { \\alpha } ^ { \\prime } )$ where $\\alpha _ { i } ^ { \\prime } > \\alpha _ { i }$ for all $i$ . The utility of the   \n227 privacy scheme is then measured by the KL-divergence between $P _ { \\mathbf { x } }$ and $Q _ { \\mathbf { x } }$ . Assuming that $\\mathbf { x }$ is an   \n228 observation of Multinomial $\\mathbf { \\tau } ( \\mathbf { p } )$ , the following Theorem tells us that, on average, the KL-divergence   \n229 is small when the sample size is large, and the $p _ { i }$ ’s are evenly distributed. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3. Let 30 $\\mathbf { p } : = \\left( p _ { 1 } , \\ldots , p _ { d } \\right)$ where $p _ { i } > 0$ for all $i$ and $\\textstyle \\sum _ { i } p _ { i } \\ = \\ 1$ . Define a random variable 31 $\\mathbf { X } \\sim$ Multinomial $\\mathbf { \\tau } ( \\mathbf { p } )$ . Let $P _ { \\mathbf { X } } \\sim \\mathrm { D i r i c h l e t } ( \\mathbf { X } + { \\boldsymbol { \\alpha } } )$ and $Q _ { \\mathbf { X } } \\sim { \\mathrm { D i r i c h l e t } } ( \\mathbf { X } + \\mathbf { \\alpha } \\mathbf { \\alpha } ^ { \\prime } )$ where 32 $\\alpha _ { i } ^ { \\prime } \\geq \\alpha _ { i } \\geq 1$ for all $i$ . The following estimate holds: ",
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+    {
+        "type": "equation",
+        "img_path": "images/dcc77176e1fb57707beeb51e095d0ecb850dd9043b7f7c0e2b40ea53a17d8b2f.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbf { X } } [ \\mathrm { D } _ { \\mathrm { K L } } ( P _ { \\mathbf { X } } \\| Q _ { \\mathbf { X } } ) ] \\leq \\frac { 1 } { N + 1 } \\sum _ { i } ( \\alpha _ { i } ^ { \\prime } - \\alpha _ { i } ) ^ { 2 } \\cdot \\frac { 1 } { p _ { i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "233 The proof is given in Appendix 2. Let us consider a simple privacy scheme where we fix $s > 0$ and let 234 $\\alpha _ { i } ^ { \\prime } = \\alpha _ { i } + s$ for all $i$ . Thus (10) becomes: ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c990b3fed25d5be5f403f085524958192030cc19922bc62f624f795a50d9cc4e.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\mathbf { X } } [ \\mathrm { D } _ { \\mathrm { K L } } ( P _ { \\mathbf { X } } \\| Q _ { \\mathbf { X } } ) ] \\leq \\frac { G ( \\mathbf { p } ) s ^ { 2 } } { N + 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            712,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where 235 $G ( \\mathbf { p } ) : = \\textstyle \\sum _ { i } 1 / p _ { i }$ . Now we take into account the privacy parameters. Let $\\rho = \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ and 236 $\\rho ^ { \\prime } = \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha _ { m } ^ { \\prime } - \\gamma )$ , where $\\alpha _ { m } = \\operatorname* { m i n } _ { i } \\alpha _ { i }$ , $\\alpha _ { m } ^ { \\prime } = \\operatorname* { m i n } _ { i } \\alpha _ { i } ^ { \\prime }$ , and $\\gamma < \\alpha _ { m }$ . Here, we approximate the values of 237 $\\psi ^ { \\prime } ( \\alpha _ { m } - \\gamma )$ and $\\psi ^ { \\prime } ( \\alpha _ { m } ^ { \\prime } - \\gamma )$ under two regimes: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "High-privacy regime: 238 $\\alpha _ { m } ^ { \\prime } \\mathrm { ~ - ~ } \\gamma \\mathrm { ~ > ~ } 1$ . We have $\\psi ^ { \\prime } ( \\alpha _ { m } ^ { \\prime } - \\gamma ) \\approx 1 / ( \\alpha _ { m } ^ { \\prime } - \\gamma )$ , which implies 239 $\\alpha _ { m } ^ { \\prime } - \\gamma \\approx \\Delta _ { 2 } ^ { 2 } / \\rho ^ { \\prime }$ . We also have $\\alpha _ { m } - \\gamma \\approx \\Delta _ { 2 } ^ { 2 } / \\rho$ for $\\alpha _ { m } - \\gamma \\geq 1$ and $\\alpha _ { m } - \\gamma > ( \\alpha _ { m } - \\gamma ) ^ { 2 } \\approx \\Delta _ { 2 } ^ { 2 } / \\rho$ 240 for $\\alpha - \\gamma < 1$ . Thus we have the following bound for the right-hand side of (11): ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43f3c38a4e99d254010fca3b14c8d659b5a9d81394f89e4fcf805698b459a86b.jpg",
+        "text": "$$\n\\frac { G ( { \\bf p } ) s ^ { 2 } } { N + 1 } = \\frac { G ( { \\bf p } ) ( \\alpha _ { m } ^ { \\prime } - \\alpha _ { m } ) ^ { 2 } } { N + 1 } \\lesssim \\frac { \\Delta _ { 2 } ^ { 4 } G ( { \\bf p } ) } { N + 1 } \\biggl ( \\frac { 1 } { \\rho ^ { \\prime } } - \\frac { 1 } { \\rho } \\biggr ) ^ { 2 } < \\frac { \\Delta _ { 2 } ^ { 4 } G ( { \\bf p } ) } { \\rho ^ { \\prime 2 } ( N + 1 ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Consequently, we have 241 $\\mathrm { D } _ { \\mathrm { K L } } ( P \\| Q ) < \\epsilon$ for $\\begin{array} { r } { N = \\Omega \\left( \\frac { \\Delta _ { 2 } ^ { 4 } G ( \\mathbf { p } ) } { \\rho ^ { \\prime 2 } \\epsilon } \\right) } \\end{array}$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "242 Low-privacy regime: $1 > \\alpha _ { m } ^ { \\prime } - \\gamma > 0$ . This is similar as above, except we have $\\alpha _ { m } ^ { \\prime } - \\gamma \\approx$   \n243 $\\Delta _ { 2 } / \\rho ^ { \\prime 1 / 2 }$ and $\\alpha _ { m } - \\gamma \\approx \\Delta _ { 2 } / \\rho ^ { 1 / 2 }$ . Similar computation as (12) shows that $\\mathrm { D } _ { \\mathrm { K L } } ( P \\| Q ) < \\epsilon$ when   \n244 $\\begin{array} { r } { N = \\Omega \\left( \\frac { \\Delta _ { 2 } ^ { 2 } G ( \\mathbf { p } ) } { \\rho ^ { \\prime } \\epsilon } \\right) } \\end{array}$ ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We observe that, in both regimes, the sample size scales faster with respect to $\\epsilon$ with a higher value of $G ( \\mathbf { p } )$ , which is associated with a higher number of outcomes $d$ , and more concentrated multinomial parameter $\\mathbf { p }$ ; this agrees with the result of our simulation in Appendix 3. Moreover, for small $\\rho ^ { \\prime }$ the sample size scales as $1 / \\rho ^ { \\prime 2 }$ , while for large $\\rho ^ { \\prime }$ the sample size scales as $1 / \\rho ^ { \\prime }$ . ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Private normalized histograms ",
+        "text_level": 1,
+        "bbox": [
+            163,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "250 Let $\\mathbf { x } = ( x _ { 1 } , \\ldots , x _ { d } )$ be a histogram of $N$ observations and $\\mathbf { p } : = \\mathbf { x } / N$ . We can privatize $\\mathbf { p }$ by   \n251 sampling a probability vector: $\\mathbf Y \\sim \\mathrm { D i r i c h l e t } ( { \\mathbf x } + { \\boldsymbol \\alpha } )$ . Note that $\\mathbf { Y }$ is a biased estimator of $\\mathbf { p }$ .   \n252 Denoting $\\alpha _ { 0 } : = \\textstyle \\sum _ { i } \\alpha _ { i }$ , the bias of each component of $\\mathbf { Y }$ is given by $\\mathbb { E } [ \\mathbf { Y } ] - p _ { i }$ . Hence, ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/acfa7af0b43f43c6c250eb27ee7cd6953e76746f9898d49a1224abbc50b2e3f6.jpg",
+        "text": "$$\n| \\mathrm { B i a s } ( Y _ { i } ) | = \\left| \\frac { x _ { i } + \\alpha _ { i } } { N + \\alpha _ { 0 } } - p _ { i } \\right| = \\frac { | x _ { i } \\alpha _ { 0 } - N \\alpha _ { i } | } { N ( N + \\alpha _ { 0 } ) } \\leq \\frac { N \\alpha _ { 0 } } { N ( N + \\alpha _ { 0 } ) } = \\frac { \\alpha _ { 0 } } { N + \\alpha _ { 0 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since 253 $Y _ { i } \\sim \\mathrm { B e t a } ( x _ { i } + \\alpha _ { i } , N + \\alpha _ { 0 } - x _ { i } - \\alpha _ { i } )$ is $\\frac { 1 } { 4 ( N + \\alpha _ { 0 } + 1 ) }$ -sub-Gaussian [MA17], we have, ",
+        "bbox": [
+            142,
+            338,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/abf09d4f3a689aaa55b0df6994813c7534a229d387d704d8a3da2faeda62cfbe.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathbb { P } [ | Y _ { i } - p _ { i } | > t + | \\mathrm { B i a s } ( Y _ { i } ) | ] \\le \\mathbb { P } [ | Y _ { i } - \\mathbb { E } [ Y _ { i } ] | + | \\mathrm { B i a s } ( Y _ { i } ) | > t + | \\mathrm { B i a s } ( Y _ { i } ) | ] } \\\\ & { \\qquad = \\mathbb { P } [ | Y _ { i } - \\mathbb { E } [ Y _ { i } ] | > t ] } \\\\ & { \\qquad \\le 2 e ^ { - 2 t ^ { 2 } ( N + \\alpha _ { 0 } + 1 ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "With the union bound, we plug in 254 $\\begin{array} { r } { t = \\sqrt { \\frac { \\log ( 2 d / \\beta ) } { 2 ( N + \\alpha _ { 0 } + 1 ) } } } \\end{array}$ , for any $\\beta \\in ( 0 , 1 )$ , to obtain the following 255 accuracy guarantee of the private normalized histogram: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "256 Theorem 4. Let $\\mathbf Y \\sim \\mathrm { D i r i c h l e t } ( { \\mathbf x } + { \\boldsymbol \\alpha } )$ , where $\\mathbf { x } \\in \\mathbb { R } _ { \\geq 0 } ^ { d }$ and $\\alpha \\in \\mathbb { R } _ { > 0 } ^ { d }$ , and $\\mathbf { p } : = \\mathbf { x } / N$ . For any   \n257 $\\beta \\in ( 0 , 1 )$ , with probability at least $1 - \\beta$ , the following inequality holds: ",
+        "bbox": [
+            145,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0d54e562ae47567af0e5d3fafa75d73322673a6540b4c7b9479beacf8ae913f.jpg",
+        "text": "$$\n\\| \\mathbf { Y } - \\mathbf { p } \\| _ { \\infty } \\leq \\sqrt { \\frac { \\log ( 2 d / \\beta ) } { 2 ( N + \\alpha _ { 0 } + 1 ) } } + \\frac { \\alpha _ { 0 } } { N + \\alpha _ { 0 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "258 Given $\\epsilon > 0$ , we use (13) to find a lower bound for $N$ that gives $\\| \\mathbf { Y } - \\mathbf { p } \\| _ { \\infty } < \\epsilon$ w.p. $1 - \\beta$ when   \n259 $\\mathbf { Y }$ is sampled with $\\rho$ -tCDP. For simplicity, we consider a uniform prior: $\\alpha _ { i } = \\alpha > 0$ for all $i$ .   \n260 Thus, $\\begin{array} { r } { \\rho = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha - \\gamma ) } \\end{array}$ , where $\\gamma$ might be chosen according to Corollary 2. We consider the two   \n261 following regimes: ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "High-privacy regime: $\\alpha - \\gamma > 1$ . In this case, $\\psi ^ { \\prime } ( \\alpha - \\gamma ) \\approx 1 / ( \\alpha - \\gamma )$ . From $\\begin{array} { r } { \\rho = \\frac { 1 } { 2 } \\Delta _ { 2 } ^ { 2 } \\psi ^ { \\prime } ( \\alpha - \\gamma ) } \\end{array}$ we have $\\alpha \\approx \\Delta _ { 2 } ^ { 2 } / 2 \\rho + \\gamma$ . Replacing $\\alpha _ { 0 }$ by $d \\alpha$ in (13) yields the sample size: ",
+        "bbox": [
+            163,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b2e0f2a9c4ecb55ff40d739b01c28e86d4c732b58831a487349c9317886cd59.jpg",
+        "text": "$$\nN = \\Omega \\biggl ( \\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { d } { \\epsilon } \\biggl ( \\frac { \\Delta _ { 2 } ^ { 2 } } { 2 \\rho } + \\gamma \\biggr ) \\biggr ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "264 for the desired accuracy. ",
+        "bbox": [
+            142,
+            694,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "265 Low-privacy regime: $\\alpha - \\gamma < 1$ . This is the same as above, except now we have $\\psi ^ { \\prime } ( \\alpha - \\gamma ) \\approx$   \n266 $1 / ( \\alpha - \\gamma ) ^ { 2 }$ , which implies $\\alpha \\approx \\Delta _ { 2 } / ( 2 \\rho ) ^ { 1 / 2 } + \\gamma$ . The sample size that guarantees the desired   \n267 accuracy is: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7142b5d9c6b72b545f886db6868236a551d39d6dbd61e35e32d88d5fc2393253.jpg",
+        "text": "$$\nN = \\Omega \\biggl ( \\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { d } { \\epsilon } \\biggl ( \\frac { \\Delta _ { 2 } } { \\sqrt { 2 \\rho } } + \\gamma \\biggr ) \\biggr ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "268 Let us compare this result to the Gaussian mechanism, which adds a noise $\\mathbf { Z } \\sim N ( 0 , \\sigma ^ { 2 } I _ { d } )$ to the   \n269 normalized histogram $\\mathbf { p }$ directly. Thus the $\\ell _ { 2 }$ -sensitivity in this case is $\\Delta _ { 2 } / N$ . We have that the   \n270 Gaussian mechanism is $\\rho$ -zCDP where $\\begin{array} { r } { \\rho = \\frac { \\Delta ^ { 2 } } { 2 N ^ { 2 } \\sigma ^ { 2 } } } \\end{array}$ [BS16]. Using the same argument as above, with   \n271 probability at least $1 - \\beta$ , the following inequality holds for all $i$ : ",
+        "bbox": [
+            138,
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02c4415ff1cb154a29b826fadec2a5e08c020ede9ac6400f3cb8e56b453a6080.jpg",
+        "text": "$$\n\\| \\mathbf { Z } \\| _ { \\infty } \\leq \\sqrt { \\frac { \\log ( 2 d / \\beta ) \\Delta _ { 2 } ^ { 2 } } { N ^ { 2 } \\rho } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/4d398e42217bd393cee6a912cf2c04e26395de3300d829c872c8f73d8e70fe67.jpg",
+        "image_caption": [
+            "Figure 2: The $\\ell ^ { \\infty }$ -accuracy, as a function of $N$ , of Dirichlet posterior sampling $( \\gamma = 1 )$ ) and Gaussian mechanisms for private normalized histograms $\\Delta _ { 2 } ^ { 2 } = 2$ and $\\Delta _ { \\infty } = 1 $ ). For each $N , d$ and $\\rho$ , we generated the inputs $\\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { 2 0 0 }$ , where $\\mathbf { x } _ { k } \\sim$ Multinomial $\\left( \\mathbf { q } _ { k } \\right)$ and $\\mathbf { q } _ { k } \\sim \\mathrm { D i r i c h l e t } ( 5 , \\ldots , 5 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            97,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Hence, the sample size of 272 $N = \\Omega \\Big ( \\sqrt { \\log ( 2 d / \\beta ) \\Delta _ { 2 } ^ { 2 } / \\rho \\epsilon ^ { 2 } } \\Big )$ guarantees the desired accuracy. Compar273 ing this to (14), if we assume $\\epsilon < 1$ , the AM-GM inequality tells us that ",
+        "bbox": [
+            138,
+            426,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71774da6e8c49643e0365d0aff1d38983a994504e1f2b72a5cccd376a2a41f0c.jpg",
+        "text": "$$\n\\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { d \\Delta _ { 2 } ^ { 2 } } { \\rho \\epsilon } > \\frac { \\log ( 2 d / \\beta ) } { \\epsilon ^ { 2 } } + \\frac { \\Delta _ { 2 } ^ { 2 } } { \\rho } \\geq 2 \\sqrt { \\frac { \\log ( 2 d / \\beta ) \\Delta _ { 2 } ^ { 2 } } { \\rho \\epsilon ^ { 2 } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            468,
+            705,
+            511
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "274 The inequality (17) implies that the Gaussian mechanism requires less sample than the Dirichlet   \n275 mechanism in order to guarantee the same level of accuracy. The Gaussian mechanism is also better   \n276 in the low-privacy regime as the $\\rho$ in (15) satisfies ${ \\sqrt { \\rho } } < \\rho$ and $\\Delta _ { 2 } \\approx \\Delta _ { 2 } ^ { 2 }$ , leading to the same   \n277 inequality (17). Nonetheless, the decay in (16) is linear in $d$ , while that in (13) has $\\alpha _ { 0 } = d \\alpha$ in   \n278 the denominators. This observation suggests that, when $\\mathbf { x }$ is a sparse histogram i.e. when $N \\leq d$ ,   \n279 the $\\ell ^ { \\infty }$ -accuracy of the Dirichlet mechanism is smaller than that of the Gaussian mechanism. This   \n280 conclusion is supported by our simulation in Figure 2. We see that the $\\ell ^ { \\infty }$ -accuracy of the Dirichlet   \n281 mechanism is smaller than that of the Gaussian mechanism for small $N$ when $d = 1 0 0 0$ . The code   \n282 for all experiments in this study can be found in the supplemental material. ",
+        "bbox": [
+            138,
+            515,
+            826,
+            651
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "283 Potential negative societal impacts ",
+        "text_level": 1,
+        "bbox": [
+            140,
+            670,
+            464,
+            688
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is important to note that, when $\\rho$ becomes unacceptably large (e.g., $\\rho = 1 0 ^ { 4 }$ ), the sampling is far away from being private. Thus any organization that deploys the posterior sampling on sensitive data must not vacuously refer to this study and claim that its algorithm is private. It is the organization’s responsibility to fully publish the prior parameters, and educate its users/customers on differential privacy and how the privacy guarantees are calculated. ",
+        "bbox": [
+            166,
+            700,
+            825,
+            777
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It is desirable that differentially private algorithms are accurate for the task at hand, especially when the data is used for important decision-making. Thus, one needs to make sure that there is enough sample to achieve the desired level of accuracy. For a large differentially private system, privacy budgets need to be allocated to the parts that require accurate outputs. ",
+        "bbox": [
+            171,
+            784,
+            823,
+            844
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "293 Lastly, one must be careful with the choice of prior parameters; if a uniform prior is used, smaller   \n294 groups will suffer a relatively larger statistical bias. As a result, private statistics of small populations   \n295 (such as ethnic or racial minorities) will be relatively less accurate. One way to get around this issue   \n296 is to (privately) impose larger prior parameters on larger populations.   \n297 References   \n298 [AAFK20] I. Aykin, B. Akgun, M. Feng, and M. Krunz. “MAMBA: A Multi-armed Bandit Framework for   \n299 Beam Tracking in Millimeter-wave Systems”. In: 39th IEEE Conference on Computer Com  \n300 munications, INFOCOM 2020, Toronto, ON, Canada, July 6-9, 2020. IEEE, 2020, pp. 1469–   \n301 1478.   \n302 [AV08] J. M. Abowd and L. Vilhuber. “How Protective Are Synthetic Data?” In: Privacy in Statistical   \n303 Databases, UNESCO Chair in Data Privacy International Conference, PSD 2008, Istan  \n304 bul, Turkey, September 24-26, 2008. 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